Energy-Based Control of Electromechanical Systems: A Novel Passivity-Based Approach [1st ed.] 9783030587857, 9783030587864

Table of contents :
Front Matter ....Pages i-xx
Introduction (Victor Manuel Hernández-Guzmán, Ramón Silva-Ortigoza, Jorge Alberto Orrante-Sakanassi)....Pages 1-6
Mathematical Preliminaries (Victor Manuel Hernández-Guzmán, Ramón Silva-Ortigoza, Jorge Alberto Orrante-Sakanassi)....Pages 7-47
Permanent Magnet Brushed DC-Motor (Victor Manuel Hernández-Guzmán, Ramón Silva-Ortigoza, Jorge Alberto Orrante-Sakanassi)....Pages 49-96
Permanent Magnet Synchronous Motor (Victor Manuel Hernández-Guzmán, Ramón Silva-Ortigoza, Jorge Alberto Orrante-Sakanassi)....Pages 97-203
Induction Motor (Victor Manuel Hernández-Guzmán, Ramón Silva-Ortigoza, Jorge Alberto Orrante-Sakanassi)....Pages 205-276
Switched Reluctance Motor (Victor Manuel Hernández-Guzmán, Ramón Silva-Ortigoza, Jorge Alberto Orrante-Sakanassi)....Pages 277-341
Synchronous Reluctance Motor (Victor Manuel Hernández-Guzmán, Ramón Silva-Ortigoza, Jorge Alberto Orrante-Sakanassi)....Pages 343-359
Bipolar Permanent Magnet Stepper Motor (Victor Manuel Hernández-Guzmán, Ramón Silva-Ortigoza, Jorge Alberto Orrante-Sakanassi)....Pages 361-391
Brushless DC-Motor (Victor Manuel Hernández-Guzmán, Ramón Silva-Ortigoza, Jorge Alberto Orrante-Sakanassi)....Pages 393-434
Magnetic Levitation Systems and Microelectromechanical Systems (Victor Manuel Hernández-Guzmán, Ramón Silva-Ortigoza, Jorge Alberto Orrante-Sakanassi)....Pages 435-467
Trajectory Tracking for Robot Manipulators Equipped with PM Synchronous Motors (Victor Manuel Hernández-Guzmán, Ramón Silva-Ortigoza, Jorge Alberto Orrante-Sakanassi)....Pages 469-487
PID Control of Robot Manipulators Equipped with SRMs (Victor Manuel Hernández-Guzmán, Ramón Silva-Ortigoza, Jorge Alberto Orrante-Sakanassi)....Pages 489-510
Back Matter ....Pages 511-619

Citation preview

Advances in Industrial Control

Victor Manuel Hernández-Guzmán Ramón Silva-Ortigoza Jorge Alberto Orrante-Sakanassi

Energy-Based Control of Electromechanical Systems A Novel Passivity-Based Approach

Advances in Industrial Control Series Editors Michael J. Grimble, Industrial Control Centre, University of Strathclyde, Glasgow, UK Antonella Ferrara, Department of Electrical, Computer and Biomedical Engineering, University of Pavia, Pavia, Italy Editorial Board Graham Goodwin, School of Electrical Engineering and Computing, University of Newcastle, Callaghan, NSW, Australia Thomas J. Harris, Department of Chemical Engineering, Queen’s University, Kingston, ON, Canada Tong Heng Lee, Department of Electrical and Computer Engineering, National University of Singapore, Singapore, Singapore Om P. Malik, Schulich School of Engineering, University of Calgary, Calgary, AB, Canada Kim-Fung Man, City University Hong Kong, Kowloon, Hong Kong Gustaf Olsson, Department of Industrial Electrical Engineering and Automation, Lund Institute of Technology, Lund, Sweden Asok Ray, Department of Mechanical Engineering, Pennsylvania State University, University Park, PA, USA Sebastian Engell, Lehrstuhl für Systemdynamik und Prozessführung, Technische Universität Dortmund, Dortmund, Germany Ikuo Yamamoto, Graduate School of Engineering, University of Nagasaki, Nagasaki, Japan

Advances in Industrial Control is a series of monographs and contributed titles focusing on the applications of advanced and novel control methods within applied settings. This series has worldwide distribution to engineers, researchers and libraries. The series promotes the exchange of information between academia and industry, to which end the books all demonstrate some theoretical aspect of an advanced or new control method and show how it can be applied either in a pilot plant or in some real industrial situation. The books are distinguished by the combination of the type of theory used and the type of application exemplified. Note that “industrial” here has a very broad interpretation; it applies not merely to the processes employed in industrial plants but to systems such as avionics and automotive brakes and drivetrain. This series complements the theoretical and more mathematical approach of Communications and Control Engineering. Indexed by SCOPUS and Engineering Index. Proposals for this series, composed of a proposal form downloaded from this page, a draft Contents, at least two sample chapters and an author cv (with a synopsis of the whole project, if possible) can be submitted to either of the: Series Editors Professor Michael J. Grimble Department of Electronic and Electrical Engineering, Royal College Building, 204 George Street, Glasgow G1 1XW, United Kingdom e-mail: [email protected] Professor Antonella Ferrara Department of Electrical, Computer and Biomedical Engineering, University of Pavia, Via Ferrata 1, 27100 Pavia, Italy e-mail: [email protected] or the In-house Editor Mr. Oliver Jackson Springer London, 4 Crinan Street, London, N1 9XW, United Kingdom e-mail: [email protected]

Proposals are peer-reviewed. Publishing Ethics Researchers should conduct their research from research proposal to publication in line with best practices and codes of conduct of relevant professional bodies and/or national and international regulatory bodies. For more details on individual ethics matters please see: https://www.springer.com/gp/authors-editors/journal-author/journal-author-helpdesk/ publishing-ethics/14214

More information about this series at http://www.springer.com/series/1412

Victor Manuel Hernández-Guzmán Ramón Silva-Ortigoza Jorge Alberto Orrante-Sakanassi

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Energy-Based Control of Electromechanical Systems A Novel Passivity-Based Approach

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Victor Manuel Hernández-Guzmán School of Engineering Autonomous University of Queretaro Querétaro, Mexico

Ramón Silva-Ortigoza CIDETEC Instituto Politécnico Nacional Mexico City, Mexico

Jorge Alberto Orrante-Sakanassi Graduate Studies and Research Instituto Tecnológico de Matamoros Matamoros, Mexico

ISSN 1430-9491 ISSN 2193-1577 (electronic) Advances in Industrial Control ISBN 978-3-030-58785-7 ISBN 978-3-030-58786-4 (eBook) https://doi.org/10.1007/978-3-030-58786-4 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Series Editor’s Foreword

A new book on control of electromechanical systems may seem pleonastic in the literature on industrial control systems. Actually, this is an original book both from the point of view of the methodological tools introduced, and of the results that the use of such tools allows to obtain in terms of industrial applications. Electromechanical systems are of fundamental importance in industrial automation and process control. The actuators of robotic systems, of goods-handling systems, of numerical control machines, of servomotors that open and close the valves and shift the moving surfaces of relevant industrial plants are, in the majority of the cases, electromechanical systems. The dynamic models of electromechanical systems are typically formulated by referring to the balance between the potential energy and the kinetic energy associated with the systems themselves. For this reason, the use of a control approach based on passivity concepts appears to be absolutely natural: the most direct and obvious way of controlling that class of systems. Passivity is a classical concept in control theory. It is strictly related to the concept of energy conservation. As a matter of fact, given a dynamical system and a certain time interval, the energy stored in the system is equal to the energy transferred to the system minus the energy dissipated. Then, in a physical system, since the dissipated energy is non-negative, the energy transferred to the system is always larger than or equal to the energy stored. In other words, a passive system is a system not able to produce energy. Passivity in linear time-invariant systems implies stability, as was highlighted in the early sixties with the well-known Kalman–Yakubovich–Popov Lemma. The extension of passivity concepts to nonlinear systems and its implications for Lyapunov stability was worked out through a series of fundamental papers published in the nineteen-seventies and nineties. Passivity can be enforced in a system, under certain conditions, by means of an appropriate control synthesis, and this is often done with the main purpose of conferring the prescribed stability properties to the controlled system. The application of the classical theory of passivity to electromechanical systems can be found in many works that have appeared in the scientific literature in recent decades. Yet,

what distinguishes this book from the previously published works is the use of the classical concept of passivity revisited in a new and original way. Different types of motors are considered in the book: permanent magnet brushed DC-motors and synchronous motors, induction motors, switched and synchronous reluctance motors, bipolar permanent magnet stepper motors and brushless DC-motors. They are all motors widely used in industrial plants. They often constitute the set of actuators upon which most of the process control and industrial automation systems rely. For every motor the authors describe the dynamic model formulation, starting from the working principle, and also introduce the standard control schemes. Then, they discuss and illustrate multi-loop control schemes based on their novel approach to passivity. The advantage of these schemes is that, by virtue of their simplicity, they are easily implementable and understandable, even to non-experts in control theory. The book also includes a part dedicated to magnetic levitation systems and micro-electromechanical systems, as well as a final part where robotic systems driven by permanent magnet motors and by synchronous reluctance motors are considered. Given the importance that robotic applications are acquiring on the international industrial scene, this part may be of great interest even to practitioners wishing to improve the performance of robotic automation systems by adopting simple control schemes, which do not require complex computations and are actually compatible with the typical sampling times of the industrial world. This book is very rich in content. The theoretical part is detailed and treated with precision. It is however, pleasant to read and understandable to an audience even of non-engineers, precisely because it focuses on very simple physical concepts, on which it is possible to have a natural intuition, even if the theoretical bases are lacking. The part in which electromechanical drives are described can also be useful to students who do not deal with control, but only with the modeling and design of electric machines. The robotic part can also be appreciated by readers who are interested in robot control, without necessarily having an expertise on electromechanical drives. For its richness of theoretical details, its methodological rigor and its well-defined structure, I think that this book will also be beneficial to researchers and doctoral students. For all the reasons mentioned, I am particularly happy to welcome this new monograph in the series on Advances in Industrial Control, certain that readers will also be able to appreciate it and get ideas for their application activities in the field of industrial control. Antonella Ferrara University of Pavia Pavia, Italy

To Judith, my parents and my brothers. Victor Manuel Hernández-Guzmán. To my wonderful children—Rhomy, Robert, Joserhamón, and Alessa—and to my mother. Ramón Silva-Ortigoza. To God, Virgin Mary, my parents and my brother. Jorge Alberto Orrante-Sakanassi.

Preface

Electromechanical systems were introduced when electricity was employed for the first time to generate force and torque. At the beginning, electromechanical systems were controlled in open-loop. Once Automatic Control became a mature discipline, it was recognized that closed-loop control of electromecanical systems is instrumental to improve performance. Since then, much research work has been devoted to closed-loop control of electromechanical systems and many control techniques have been applied. However, since many modern control techniques employ complex mathematical tools, most works on closed-loop control of electromechanical systems have resulted in complex mathematical algorithms which are difficult to understand and, hence, they have not been welcome by practitioners. Moreover, many formally supported controllers result in control laws that require lots of on-line computations which, besides requiring powerful and, hence, expensive hardware, deteriorate performance because they amplify noise, increase numerical errors and produce actuator saturation. The above situation has motivated the application, by several authors in the past, of passivity-based ideas for closed-loop control of electromechanical systems. This approach takes advantage of the natural structure of the plant to be controlled and, hence, it has been demonstrated that results in simpler control laws. Thus, the designed controllers require a fewer number of on-line computations, improve performance because noise amplification and numerical errors are reduced and actuator saturation is avoided. Moreover, since passivity-based control is supported by energy ideas, a fundamental concept in engineering, this approach can be better understood by practitioners. However, despite these advancements, passivity-based controllers are not as simple as controllers that are employed in industrial practice for electromechanical systems. This is the case of field oriented control (FOC) of alternating current (AC) motors. It is important to stress at this point that FOC of AC-motors is a control scheme which is not provided, until now, with a formal global asymptotic stability proof although presenting such a result has been the aim of several authors

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in the past. The reason for such a search is to explain why FOC of AC-motors works well in practice and to provide tuning guide lines. It must be recognized that presenting a solution for the above described control problem has not attracted attention of the control community in the recent years. This, however, is neither because the problem has lost relevance or nothing is remaining to solve. We believe that the mere reason for such a lack of interest is that the leading researchers have moved to other subjects. As a matter of fact, one of the leading researchers that tried in the past to present a global asymptotic stability proof for FOC of AC-motors has recently presented in [205] a work on such a subject. He states there the importance of presenting a global asymptotic stability proof for FOC of AC-motors providing tuning guide lines. The present book is devoted to introduce recent advancements in the design of controllers for electromechanical systems. Most of our proposals consist of multi-loop control schemes possessing an internal proportional-integral (PI) electric current loop and an external PI velocity loop or proportional-integral-derivative (PID) position loop. Aside from these simple controllers, some additional simple terms are included to ensure global asymptotic stability. As we demonstrate along the book, these proposals are simpler control laws than the passivity-based controllers that have been proposed in the past. The theoretical key to achieve these results is a novel passivity-based approach that we have developed during the last 12 years. This approach exploits the fact that the electrical and the mechanical subsystems exchange energy naturally during their normal operation. From the stability proof point of view, this allows the natural cancellation of several high-order terms. This allows to obtain simpler control laws because these terms must be computed on-line to be exactly cancelled when employing other control approaches, including the passivity-based approaches presented in the past. It is the authors belief that, because of simpler control laws and a simpler rationale behind their design, our proposals can be welcome by practitioners. At this point, it is important to stress that simple Lyapunov stability analysis is the main mathematical tool that is employed to present the complete stability proofs. In this respect, in order to render attractive the book for both theorists and practitioners, we include the complete general mathematical modeling of all the electromechanical systems that we control. We also include, at the end of each chapter, how to obtain the mathematical model of practical AC-motors: (i) we dismantled the motor to analyze how the phase windings are distributed on the stator and how the permanent magnet (PM) poles are distributed on the rotor, (ii) based on the previous analysis, we compute the magnetic flux at the air gap using Ampère’s Law, (iii) these results were employed to derive the motor mathematical model. Finally, let us say that most of the complete formal stability proofs are presented in appendices, for the interested theorist readers. Simple sketches of the proofs are presented in the corresponding chapters to allow practitioners to understand the main results using energy interpretations.

Preface

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We apply our approach to PM bushed-DC motors, PM synchronous motors, induction motors, switched reluctance motors, synchronous reluctance motors, PM stepper motors, brushless DC-motors, magnetic levitation systems, microelectromechanical systems, and rigid robot manipulators equipped with PM synchronous motors and switched reluctance motors. Querétaro, Mexico México City, Mexico Matamoros, Mexico

Victor Manuel Hernández-Guzmán Ramón Silva-Ortigoza Jorge Alberto Orrante-Sakanassi

Acknowledgments

The first author acknowledges the work of his coauthors. Their collaboration has been instrumental to accomplish the writing of this book. Thanks to both of them for the collaborative work that we have performed along the years, with Ramón since we were Ph.D. students, and with Sakanassi since he pursued postdoctoral studies. Both of them have been very enthusiastic when collaborating with the first author. Thanks to Universidad Autónoma de Querétaro, Facultad de Ingeniería, for economical support since 1995, and to the Mexican Researchers National System (CONACYT-SNI) for its economical support since 2005. The ideas that have resulted in this book arose during the early years when the first author began his career as a researcher. Permanent magnet brushed DC-motors were studied first during the Ph.D. studies where he has Dr. Hebertt Sira-Ramírez as advisor. Very special thanks to him. After realizing that energy exchange between the electrical and the mechanical subsystems in these motors allows the natural cancellation of some cross terms, it was also natural to wonder whether these cancellations also exist in other classes of electric motors. After all, energy exchange between the electrical and the mechanical subsystems are the fundamental Physics phenomenon behind any electric motor operation. The present book is the answer for such a question. Since those early years, a source of motivational support has been Dr. Victor Santibanez, a researcher at Instituto Tecnológico de La Laguna, in Torreón, Coah., México. Thanks and a special acknowledgment to him. Also thanks to my former Ph.D. students, Fortino Mendoza-Mondragón, Moises Martínez-Hernández, Mayra Antonio-Cruz, José Rafael García-Sánchez, and Celso Márquez-Sánchez, for their collaboration in diverse research subjects. Special thanks to my wife Judith for her continuous moral support and understanding when I spend so much time in researching and writing. Also thanks to my parents and my brothers for their fundamental teachings about life. The second author acknowledges and thanks the first author for his invitation to participate in the creation of this book and for other ambitious academic and research projects. Special thanks to Dr. Gilberto Silva-Ortigoza and Dr. Hebertt

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Sira-Ramírez, researchers at Benemérita Universidad Autónoma de Puebla and CINVESTAV-IPN. The former has been his mentor throughout his entire professional formation and the latter was his mentor during his graduate years. He also acknowledges the important academic and research collaboration of Dr. Magdalena Marciano-Melchor (CIDETEC-IPN), Dr. Mariana Marcelino-Aranda (UPIICSAIPN), and Dr. Hind Taud (CIDETEC-IPN). He would like to thank all of his former bachelor, master, and doctoral students for their collaboration in research and projects development. He especially thanks his former doctoral students Dr. José Rafael García-Sánchez and Dr. Eduardo Hernández-Márquez, for their continuous support, generosity and their willingness to undertake new theoretical and practical problems. The second author is grateful to CIDETEC of Instituto Politécnico Nacional (IPN), the Research Center where he has been based since 2006, SIP-IPN, programs EDI and SIBE from IPN, and to the SNI from CONACYT-México for financial support. A special loving mention is deserved by my children—Alessa, Robert, Rhomy, and Joserhamón—and my mother. They are the inspiration I need to improve and reinvent myself every day. The third author specially thanks to God for all the blessings received throughout his life and the Virgin Mary for always interceding for his well-being and helping him to have his feet on the ground. Also special thanks to my four greatest mentors in my life, José Aranda González, Mario Rodríguez Franco (may he rest in peace), Víctor Santibáñez Dávila and Victor Hernández Guzmán (first author), who inspired my professional life. Finally, the greatest of thanks to my parents and my brother Jesús Yusen for their moral support and for all the unforgettable moments we have had. Querétaro, Mexico México City, Mexico Matamoros, Mexico

Victor Manuel Hernández-Guzmán Ramón Silva-Ortigoza Jorge Alberto Orrante-Sakanassi

Contents

1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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2

Mathematical Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Control of Linear Systems . . . . . . . . . . . . . . . . . . . . . . 2.2 Mathematical Tools for the Study of Nonlinear Control Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Passivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 A Novel Passivity-Based Approach for Control of Electromechanical Systems . . . . . . . . . . . . . . . . . . . 2.5 The Electromechanical Systems that Are Studied in This Book . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Permanent Magnet Brushed DC-Motor . . . . . . . . . . . . . . . . 3.1 Motor Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.1 A Simple Methodology . . . . . . . . . . . . . . . . . . . 3.1.2 A General Methodology . . . . . . . . . . . . . . . . . . 3.2 Standard Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 Case 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.2 Case 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.3 Case 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.4 Case 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.5 An Improved PI Velocity Controller . . . . . . . . . 3.2.6 An Improved PID Position Controller . . . . . . . . 3.3 The Standard Control Scheme Revisited . . . . . . . . . . . . . 3.4 Open-Loop Energy Exchange . . . . . . . . . . . . . . . . . . . . 3.4.1 The Velocity Model . . . . . . . . . . . . . . . . . . . . . 3.4.2 The Position Model . . . . . . . . . . . . . . . . . . . . . 3.5 Velocity Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6 Position Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.7 Velocity Control Using a DC/DC Buck Power Converter As Power Amplifier . . . . . . . . . . . . . . . . . . . . . . . . . . .

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3.7.1 3.7.2 3.7.3 3.7.4 4

Dynamic Model . . . . . . . . . . . . . . . . . . . . . . . . . . . Open-Loop Energy Exchange . . . . . . . . . . . . . . . . . Control of the DC to DC Buck Converter DC-Motor System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Experimental Results . . . . . . . . . . . . . . . . . . . . . . .

Permanent Magnet Synchronous Motor . . . . . . . . . . . . . . . . . 4.1 Motor Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.1 The Working Principle . . . . . . . . . . . . . . . . . . . . 4.1.2 Three-Phase Dynamic Model . . . . . . . . . . . . . . . 4.1.3 Park’s Transformation or dq Transformation . . . . 4.1.4 The dq Dynamic Model . . . . . . . . . . . . . . . . . . . 4.1.5 dq Decomposition of the Stator Magnetic Flux . . 4.1.5.1 The Open-Loop Working Principle . . . 4.1.5.2 Closed-Loop Operation . . . . . . . . . . . . 4.1.5.3 Field Weakening . . . . . . . . . . . . . . . . . 4.1.5.4 The Home Position . . . . . . . . . . . . . . . 4.1.6 Standard Field-Oriented Control . . . . . . . . . . . . . 4.2 Open-Loop Energy Exchange . . . . . . . . . . . . . . . . . . . . . 4.2.1 The Velocity Model . . . . . . . . . . . . . . . . . . . . . . 4.2.2 The Position Model . . . . . . . . . . . . . . . . . . . . . . 4.3 Velocity Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.1 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . 4.4 Position Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.1 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . 4.5 Velocity Ripple Minimization . . . . . . . . . . . . . . . . . . . . . 4.5.1 Mutual Torque . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.1.1 Nonsinusoidal Distribution of Stator Windings . . . . . . . . . . . . . . . . . . . . . . 4.5.1.2 Errors in Stator Electric Current Measurements . . . . . . . . . . . . . . . . . . . 4.5.2 Reluctance Torque . . . . . . . . . . . . . . . . . . . . . . . 4.5.3 Cogging Torque . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.4 Torque Ripple . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.5 The Problem to Solve . . . . . . . . . . . . . . . . . . . . . 4.5.6 Stability Analysis . . . . . . . . . . . . . . . . . . . . . . . . 4.5.7 Experimental Results . . . . . . . . . . . . . . . . . . . . . 4.6 A Practical PM Synchronous Motor . . . . . . . . . . . . . . . . . 4.6.1 Magnetic Field at the Air Gap . . . . . . . . . . . . . . . 4.6.1.1 Magnetic Field Produced by the Stator Windings . . . . . . . . . . . . . . . . . . . . . . 4.6.1.2 Magnetic Field Produced by Rotor . . . .

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Induction Motor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Motor Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.1 The Working Principle . . . . . . . . . . . . . . . . . . . . 5.1.2 Three-Phase Dynamical Model . . . . . . . . . . . . . . 5.1.3 ab Transformation . . . . . . . . . . . . . . . . . . . . . . . 5.1.4 The ab Dynamical Model . . . . . . . . . . . . . . . . . . 5.1.5 Park’s Transformation or dq Transformation . . . . 5.1.6 The dq Dynamical Model . . . . . . . . . . . . . . . . . . 5.1.7 dq Decomposition of the Magnetic Flux . . . . . . . 5.1.7.1 The Open-Loop Working Principle . . . 5.1.7.2 Closed-Loop Operation . . . . . . . . . . . . 5.1.7.3 Field Weakening . . . . . . . . . . . . . . . . . 5.1.8 Standard Indirect Field-Oriented Control . . . . . . . 5.2 Open-Loop Energy Exchange . . . . . . . . . . . . . . . . . . . . . 5.2.1 The Velocity Model . . . . . . . . . . . . . . . . . . . . . . 5.2.2 The Position Model . . . . . . . . . . . . . . . . . . . . . . 5.3 Velocity Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.1 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . 5.4 Position Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.1 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . 5.5 A Practical Induction Motor . . . . . . . . . . . . . . . . . . . . . . 5.5.1 Magnetic Field at the Air Gap . . . . . . . . . . . . . . . 5.5.1.1 Magnetic Field Produced by the Stator Windings . . . . . . . . . . . . . . . . . . . . . . 5.5.1.2 Magnetic Field Produced by Rotor . . . . 5.5.2 The Magnetic Flux Linkages . . . . . . . . . . . . . . . . 5.5.3 The Motor Dynamic Model . . . . . . . . . . . . . . . .

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Switched Reluctance Motor . . . . . . . . . . . . . . . . . 6.1 Motor Modeling . . . . . . . . . . . . . . . . . . . . . 6.1.1 The Working Principle . . . . . . . . . . 6.1.2 Magnetic Circuits . . . . . . . . . . . . . . 6.1.3 SRM Unsaturated Dynamical Model 6.1.4 SRM Saturated Dynamical Model . .

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4.6.2 Magnetic Flux Linkages . . . . . . . . . . . . . . . . . . . 4.6.3 The Motor dq Dynamical Model . . . . . . . . . . . . . Another Practical PM Synchronous Motor . . . . . . . . . . . . 4.7.1 Magnetic Field at the Air Gap . . . . . . . . . . . . . . . 4.7.1.1 Magnetic Field Produced by the Stator Windings . . . . . . . . . . . . . . . . . . . . . . 4.7.1.2 Magnetic Field Produced by Rotor . . . . 4.7.2 Magnetic Flux Linkages . . . . . . . . . . . . . . . . . . . 4.7.3 The Motor dq Dynamical Model . . . . . . . . . . . . .

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6.2

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6.1.5 The Torque Sharing Approach . . . . . . . . . . . . . . . 6.1.6 Standard Control . . . . . . . . . . . . . . . . . . . . . . . . . Open-Loop Energy Exchange . . . . . . . . . . . . . . . . . . . . . . 6.2.1 The Unsaturated Velocity Model . . . . . . . . . . . . . . 6.2.2 The Saturated and Unsaturated Position Models . . . 6.2.2.1 The Unsaturated Model . . . . . . . . . . . . . 6.2.2.2 The Saturated Model . . . . . . . . . . . . . . . Velocity Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.1 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . Position Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.1 Position Control Without Velocity Measurements . . 6.4.1.1 Closed-Loop Dynamics . . . . . . . . . . . . . 6.4.1.2 A Positive Definite Decrescent Function . 6.4.1.3 Stability Analysis . . . . . . . . . . . . . . . . . 6.4.1.4 Proof of Proposition 6.11 . . . . . . . . . . . 6.4.2 Position Control Taking into Account Magnetic Saturation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.2.1 Simulation Results . . . . . . . . . . . . . . . . A Practical Switched Reluctance Motor . . . . . . . . . . . . . . . 6.5.1 Magnetic Field at the Air Gap . . . . . . . . . . . . . . . . 6.5.2 Magnetic Flux Linkages . . . . . . . . . . . . . . . . . . . . 6.5.3 SRM Dynamical Model . . . . . . . . . . . . . . . . . . . .

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316 325 328 328 337 338

7

Synchronous Reluctance Motor . . . . . . . . . . . . . . . . . . . . 7.1 Motor Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1.1 The Working Principle . . . . . . . . . . . . . . . . 7.1.2 dq Dynamical Model . . . . . . . . . . . . . . . . . 7.1.3 Standard Field-Oriented Control of a SYRM 7.2 Open-Loop Energy Exchange . . . . . . . . . . . . . . . . . 7.2.1 The Velocity Model . . . . . . . . . . . . . . . . . . 7.2.2 The Position Model . . . . . . . . . . . . . . . . . . 7.3 Velocity Control . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4 Position Control . . . . . . . . . . . . . . . . . . . . . . . . . . .

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343 344 344 345 346 347 347 349 349 354

8

Bipolar Permanent Magnet Stepper Motor . 8.1 Motor Modeling . . . . . . . . . . . . . . . . . 8.1.1 The Working Principle . . . . . . 8.1.2 Dynamical Model . . . . . . . . . . 8.1.3 Standard Control . . . . . . . . . . 8.2 Open-Loop Energy Exchange . . . . . . . 8.2.1 The Velocity Model . . . . . . . . 8.2.2 The Position Model . . . . . . . .

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370 376 378 382 385

Brushless DC-Motor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1 Motor Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1.1 Dynamic Model . . . . . . . . . . . . . . . . . . . . . . . . . 9.1.2 Standard Control . . . . . . . . . . . . . . . . . . . . . . . . 9.2 Open-Loop Energy Exchange . . . . . . . . . . . . . . . . . . . . . 9.2.1 The Velocity Model . . . . . . . . . . . . . . . . . . . . . . 9.2.2 The Position Model . . . . . . . . . . . . . . . . . . . . . . 9.3 Velocity Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3.1 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . 9.4 Position Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4.1 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . 9.5 A Practical BLDC Motor . . . . . . . . . . . . . . . . . . . . . . . . 9.5.1 Magnetic Field at the Air Gap . . . . . . . . . . . . . . . 9.5.1.1 Magnetic Field Produced by the Stator Windings . . . . . . . . . . . . . . . . . . . . . . 9.5.1.2 Magnetic Field Produced by Rotor . . . . 9.5.2 Magnetic Flux Linkages . . . . . . . . . . . . . . . . . . . 9.5.3 Mathematical Model . . . . . . . . . . . . . . . . . . . . . .

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xix

Velocity Control . . . . . . . . . . . 8.3.1 Simulation Results . . . Position Control . . . . . . . . . . . 8.4.1 Simulation Results . . . A Practical PM Stepper Motor .

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10 Magnetic Levitation Systems and Microelectromechanical Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.1 Magnetic Levitation Systems . . . . . . . . . . . . . . . . . . . 10.1.1 Mathematical Model . . . . . . . . . . . . . . . . . . . 10.1.2 Open-Loop Energy Exchange . . . . . . . . . . . . 10.1.3 Position Control . . . . . . . . . . . . . . . . . . . . . . 10.1.4 Simulation Results . . . . . . . . . . . . . . . . . . . . 10.2 Microelectromechanical Systems . . . . . . . . . . . . . . . . 10.2.1 Mathematical Model . . . . . . . . . . . . . . . . . . . 10.2.2 Open-Loop Energy Exchange . . . . . . . . . . . . 10.2.3 Position Control . . . . . . . . . . . . . . . . . . . . . . 10.2.4 Simulation Study . . . . . . . . . . . . . . . . . . . . .

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11 Trajectory Tracking for Robot Manipulators Equipped with PM Synchronous Motors . . . . . . . . . . . . . . . . . . . . . . . . . . . . 469 11.1 Dynamical Model of Robot Manipulators Equipped with PM Synchronous Motors . . . . . . . . . . . . . . . . . . . . . . . . . 471 11.2 Main Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 473

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11.3 Simulation and Experimental Results . . . . . . . . . . . . . . . . . . . . 480 11.3.1 Simulation Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . 480 11.3.2 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . 485 12 PID Control of Robot Manipulators Equipped with SRMs . 12.1 Dynamical Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.2 The Torque Sharing Approach . . . . . . . . . . . . . . . . . . . 12.3 Main Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.3.1 The Rationale Behind Controller in Proposition 12.1 . . . . . . . . . . . . . . . . . . . . . 12.3.2 Sketch of Proof of Proposition 12.1 . . . . . . . . . 12.3.3 Closed-Loop Dynamics . . . . . . . . . . . . . . . . . . 12.3.4 A Positive Definite and Decrescent Function . . 12.3.5 Time Derivative of VðyÞ . . . . . . . . . . . . . . . . . 12.3.6 Proof of Proposition 12.1 . . . . . . . . . . . . . . . . 12.4 Simulation Study . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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495 496 496 498 500 505 506

Appendix A: Energy Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 511 Appendix B: Proofs of Propositions for Brushed DC-Motors . . . . . . . . . 533 Appendix C: Proofs of Propositions for PM Synchronous Motors . . . . . 545 Appendix D: Proofs of Propositions for Induction Motors . . . . . . . . . . . 561 Appendix E: Proofs of Propositions for Switched Reluctance Motors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 577 Appendix F: Proofs for BLDC Motors . . . . . . . . . . . . . . . . . . . . . . . . . . . 589 Appendix G: Derivation of Some Expressions for the Proofs in Chap. 10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 599 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 603 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 617

Chapter 1

Introduction

Electric motors are fundamental devices for force and torque generation in industrial facilities. In fact, electric motors were introduced in the nineteenth century, by the incipient companies that tried to expand the use of electric energy, to provide industry with a device capable to generate force and torque. These companies distributed the electric energy in the form of what we know now as direct current (DC). This situation motivated T.A. Edison to introduce the brushed DC-motor. The main advantage of brushed DC-motor is that it can be easily controlled. In fact, in order to put to work a brushed DC-motor it suffices to connect its terminals. Alternating current (AC) was introduced later by N. Tesla as a means to improve the distribution of electric energy. Tesla recognized the need for a new motor working on the basis of AC, and thus, he introduced the induction motor. One important advantage of induction motor is that it does not employ brushes nor commutators to operate. However, it was soon realized that induction motor was difficult to control compared to brushed DC-motor. For instance, in order to vary the motor velocity it is required to vary the frequency of the AC flowing through the motor phase windings. Hence, complex hardware was required to operate induction motors compared to hardware required to operate brushed DC-motors. Moreover, performance was not as good as that achieved by brushed DC-motors: this control strategy, also known as open-loop, was found to be stable in practice only if the motor is lightly loaded.1 At that point of time, electric motors were controlled empirically, i.e., without taking into account their mathematical models. When Automatic Control became a mature discipline, it was recognized the necessity to take into account the motor model to improve performance. It was then found that brushed DC-motor has a linear and singe-input single-output model which facilitates the control design task. On the other hand, the induction motor model is nonlinear and it has several inputs. Thus, controller design for induction motors was soon recognized to be a difficult task. Several classes AC-motors were introduced later in the twentieth century which, 1 It

is interesting to remark that this was mathematically demonstrated recently in [212]. Moreover, global exponential stability was demonstrated later in [270]. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 V. M. Hernández-Guzmán et al., Energy-Based Control of Electromechanical Systems, Advances in Industrial Control, https://doi.org/10.1007/978-3-030-58786-4_1

1

2

1 Introduction

contrary to brushed DC-motors, do not require brushes nor commutators to work, an attractive feature because of the reductions in maintenance periods and costs. However, all of these AC-motors have a similar disadvantage than induction motors: they are difficult to control because they have nonlinear models and multiple inputs. The ease to operate brushed DC-motors motivated their employment in most of the applications of electric motors in the twentieth century. In 1972, a new control methodology for induction motors was empirically introduced [22]. It is known now as vectorial control of AC-motors or field-oriented control (FOC) of AC-motors. However, because of the hardware limitations at that moment, the practical applications of FOC appeared after several years when microprocessor and digital-signal-processor (DSP) technologies became popular [20, 264]. The use of what we know now as standard FOC (SFOC) importantly improved performance of induction motors in practical applications and SFOC has become now the standard in industrial applications. This success motivated the development of SFOC ideas to control other AC-motors. However, despite the success of SFOC in practice, any formal study does not exist ensuring its global asymptotic stability. This has motivated several formal works trying to (a) propose new control design methodologies which be provided with a formal stability proof and improving the performance achievable with SFOC and (b) find a global asymptotic stability proof for SFOC which formally explains its practical success providing control tuning guidelines. Following the line of (a), the book [60] has summarized the work of a group of researchers who proposed a series of controllers for different AC-motors which exploit the backstepping ideas first introduced in [152]. The main drawback of such proposals is the large amount of online computations that are required by the corresponding control laws. It is important to stress that this is not just a matter of finding fast enough hardware to perform the required computations, but it is a matter that pertains to performance deterioration due to numerical errors and noise amplification introduced by complex computations [204]. Furthermore, as it is stated in [222], the drives community does not like complex controllers. On the other hand, following the line of (b), the book [204] has summarized the work of a group of researchers who proposed a series of controllers for different AC-motors which rely on standard passivity-based control, an approach that has its roots in the work published in [209]. It is shown that using this approach, several basic concepts of FOC for AC-motors can be explained. In particular, in [210] is formally proven global asymptotic stability of standard indirect field-oriented control (SIFOC) for current-fed induction motors. This means that the stator electrical dynamics is neglected, i.e., the stator electric current is assumed to be the control signal for motor instead of the applied voltages. It is important to stress that SIFOC is the workhorse for induction motor control in industrial applications at present. However, when taking into account the stator electrical dynamics, i.e., when the motor control signals are voltages applied at the stator terminals, a number of important

1 Introduction

3

differences appear between the solutions in [204] and SIFOC of induction motors.2 In particular, a number of additional terms which require the exact knowledge of several motor parameters have to be computed on line. The proposed solution relies on torque observers instead of PI velocity controllers. Moreover, the PID position control problem is not studied. This is important to stress since PID control defines the application of SIFOC in position control problems. It is interesting to remark that instead of PID position regulation, position trajectory tracking control of robot manipulators equipped with induction motors is solved in [204]. Notice that trajectory tracking requires more complex controllers than simple PID regulators. Furthermore, recall that the motor controllers proposed in [204] are more complex than SIFOC. Thus, it is the authors belief that solving the trajectory tracking control problem instead of the simpler PID position regulation problem can be explained by recalling that it is easier to justify a complex controller if the task is also complex. After [60, 204] were published, the interest in these problems diminished and the authors that led those works moved to other research subjects. Moreover, current research on AC electric machines is focused on sensorless control. See [18, 54, 86, 117, 118, 156, 183, 193, 207, 208, 245, 268, 281] and references therein for instance. Despite this, as we have shown in the above discussion, it is not because the problem has been solved or it has no relevance. As a matter of fact, in the recent paper [205], the leading author of [204] recognizes the importance of presenting a global asymptotic stability proof for field-oriented control. Moreover, efforts in that paper focus in proving global asymptotic stability when internal PI electric current controllers are employed. However, this is performed when controlling the mechanical subsystem in open-loop and the use of either a PI velocity controller or a PID position controller still remains without a formal solution. Motivated by the control problems that remain open, which we describe below, in the present book, we introduce a novel passivity-based control approach for electromechanical systems. The main advantages of this new approach with respect to previous passivity-based methodologies in the literature are the following: 1. The energy exchange that naturally appears between the electrical and the mechanical subsystems is exploited. This feature is important to stress because it allows the derivation of simpler control laws and it has not been exploited in previous works in the literature. Moreover, previous passivity-based controllers rely on exact cross term cancellations that require the exact knowledge of several motor parameters. 2. The previous passivity-based approaches in the literature rely on deriving isolated closed-loop error equations for the electrical dynamics in order to take advantage of the system passivity properties. Instead of this, we dominate the cross terms existing between the electrical and the mechanical dynamics.

2 We

consider important to stress that it is stated in [204] that their control approach is also valid for many AC-motors; however, an explicit controller is only presented for induction motors. As we show in the present book much simpler controllers are designed for other AC-motors when using our approach than following the steps suggested by the approach in [204].

4

1 Introduction

3. In order to achieve the previous item, the previous passivity-based approaches in the literature require to compute online and to cancel the complete expression for the time derivative of the desired electric current. This commonly requires an important number of additional online computations rendering the controller more complex. The novel passivity-based approach that we propose does not require to perform these computations nor cancellations because we dominate these terms. 4. A nested-loop passivity-based control approach is exploited in [204]. This means that the electric current error is first proven to converge exponentially to zero and this allows to use this variable as a vanishing perturbation for the mechanical subsystem. This, however, requires the online computations referred in the previous item. Instead of that we use an approach which is similar to what was called in [204], pp. 243, passivity-based control with total energy shaping. Although the latter approach has been disregarded in [204] arguing that it results in more complex controllers; we prove the opposite in the present book. This is another important contribution of our proposal. 5. The previous passivity-based approaches in the literature replace velocity measurements with high-pass position filtering explaining that this reduces the effects of noise that is present in velocity measurements, which is true. However, it is also true that the time derivative of the desired electric current would become even more complex if velocity measurements were allowed. Hence, an even more complex controller would result. On the other hand, it is important to stress that allowing velocity measurements is instrumental to successfully design PID position controllers in nonlinear systems.3 This is concluded from results in PID robot control where the few control schemes that have been proposed without requiring velocity measurements impose severe constraints to the controller gains that can be employed and, hence, the achieved performances are far from satisfactory (try to perform simulations with the controllers in [98, 172, 203, 252, 253], for instance). Furthermore, some of these works rely on the presence of significant viscous friction that the mechanical subsystem must naturally possess. This might be the reason why PID position regulation is not reported together with SIFOC in AC-motors as pointed out above. It is shown in the subsequent chapters that the novel passivity-based approach that we propose allows velocity measurements without increasing the complexity of the resulting controller. As a matter of fact, simple PID position controllers are presented for several AC-motors. 6. Aside from three simple nonlinear terms, the main controllers that we propose are identical to SFOC for voltage-fed AC-motors, i.e., when the complete motor dynamics is taken into account. Hence, we propose the most similar control scheme to SFOC but provided with a global asymptotic stability proof. In this book, we are interested in proposing control strategies that are simple to implement. In particular, those control strategies that are very similar to SFOC 3 In [204], Ch. 8, is designed a PID position controller using velocity low-pass filtering for a magnetic levitation system. However, this is possible because the mechanical subsystem is linear and, hence, linear concepts such as Hurwitz matrices can be used to ensure stability.

1 Introduction

5

or other control schemes that are recognized to be the standard control techniques for each one of the electromechanical systems that we consider in the book. Our contribution consists in providing formal stability proofs for the control schemes that we propose such that, under mild assumptions, our proposals explain to some extent why SFOC and other standard controllers work well in practice. We are also interested in presenting the details behind the mathematical models that are employed to perform the control design task. It is the authors experience that designing controllers for electromechanical systems without a clear understanding of the principles that determine how the plant works, and in particular how the electromagnetic subsystem works, is something like being blind. Thus, we begin each chapter by explaining how the electromechanical system is modeled and the rationale behind the coordinate changes that are employed. After that, we describe the standard control scheme that is traditionally employed for the electromechanical system under study, and finally, we present the controller that we propose with the corresponding stability proof. Aside from theorists, this book also tries to attract attention of practitioners. Hence, the complete formal stability proofs are sent to appendices for the interested readers, and simple sketches of the proofs are presented in the body of the chapters. Reason for this is to explain, using simple energy-based arguments, how the main ideas of the novel passivity-based approach that we introduce are exploited. On the other hand, it is usual in the control community concerned with the study of electromechanical systems to use the dynamical models that are at disposal but they have never seen the inside of an AC-motor, for instance. Although this is not necessary to design controllers, performing this gives a lot of insight on the main assumptions employed when modeling an AC-motor and the main ideas behind the coordinate changes that are usual when using SFOC. In order to reduce this gap between theory and practice, we present the study of the stator windings of several classes of practical AC-motors at the end of most chapters. First, we have dismantled the motor and we describe the physical distribution of the stator phase windings and the rotor permanent magnets if any. Then, we employ a procedure described in [55], based on Ampère’s Law, to mathematically model the magnetic field distribution produced by the stator phase windings. We also describe mathematically the magnetic field distribution produced by the rotor permanent magnets, if any. This allows to compute the stator three-phase windings flux linkages, and thus, (i) the three-phase electrical subsystem dynamical model is derived and (ii) the electromagnetic torque generated by the three phases of the motor is also computed. After this, the application of Newton’s Second Law results in the mechanical subsystem dynamical model. Then, as an application example, the formulas derived at the beginning of each chapter are employed to obtain the dq dynamical model of the motor. Finally, we explain how these derived models are correctly represented by the general dq dynamical model derived in the first part of the chapter for a general motor of the class under study.

6

1 Introduction

In the following chapter, i.e., Chap. 2, we review the mathematical tools required by the formal part of the studies presented in this book. The main ideas of our novel control approach are illustrated using the permanent magnet brushed DC-motor as an example. Finally, we also present there a description of the book content and how the subsequent chapters relate among them.

Chapter 2

Mathematical Preliminaries

2.1 Control of Linear Systems From the classical control point of view physical systems are modeled using the following ordinary linear n−order differential equation: y (n) + an−1 y (n−1) + · · · + a2 y¨ + a1 y˙ + a0 y = bm u (m) + bm−1 u (m−1) + · · · + b2 u¨ + b1 u˙ + b0 u,

(2.1)

where n ≥ m, ai , b j , i = 0, . . . , n − 1, j = 0, . . . , m, are real constant scalars, y(t) is the variable representing the system response or output and u(t) is the system excitation or input. Applying the Laplace transform to the previous expression and assuming that all of the initial conditions are zero, we obtain the following transfer function: G(s) =

bm s m + bm−1 s m−1 + · · · + b2 s 2 + b1 s + b0 Y (s) = , U (s) s n + an−1 s n−1 + · · · + a2 s 2 + a1 s + a0

(2.2)

where Y (s) = L{y(t)} and U (s) = L{u(t)} are the Laplace transforms of y(t) and u(t), respectively. The n roots of polynomial at the denominator are known as the system poles and the m roots of polynomial at the numerator are known as the system L(s) , where L(s) and zeros. Given a known function of time u(t) such that U (s) = M(s) M(s) are polynomials, the solution Y (s) is found using fraction expansion, i.e., bm s m + bm−1 s m−1 + · · · + b2 s 2 + b1 s + b0 U (s), s n + an−1 s n−1 + · · · + a2 s 2 + a1 s + a0 bm s m + bm−1 s m−1 + · · · + b2 s 2 + b1 s + b0 L(s) , = s n + an−1 s n−1 + · · · + a2 s 2 + a1 s + a0 M(s)

Y (s) =

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 V. M. Hernández-Guzmán et al., Energy-Based Control of Electromechanical Systems, Advances in Industrial Control, https://doi.org/10.1007/978-3-030-58786-4_2

7

8

2 Mathematical Preliminaries

=

l 

fi s + di i=1   q  g1i g2i g(k−1)i gki , + + + ··· + + (s + h i )k (s + h i )k−1 (s + h i )2 s + hi i=1 r 

βi s + γi 2 2 + 2ζ ω s i p ni p s + ωni p i=1 v   δ1i s + α1i δ2i s + α2i + + 2 + 2 k 2 k−1 2 (s + 2ζi ωni s + ωni ) (s + 2ζi ωni s + ωni ) i=1 +

δ(k−1)i s + α(k−1)i δki s + αki ··· + 2 + 2 2 2 2 (s + 2ζi ωni s + ωni ) s + 2ζi ωni s + ωni +



Q(s) , M(s)

where it is taken into account that the plant poles may be real, single or repeated k times, or complex conjugate, single or repeated k times. The symbols f i , g ji , βi , γi , δ ji , α ji , j = 1, . . . , k, represent real constants to be computed. It is assumed that G(s) has (i) l different real single poles located at s = −di , (ii) q different real repeated poles located at s = −h i , iii) r different single pairs  of complex conjugate

poles located at s = −ζi p ωni p ± jωdi p , where ωdi p = ωni p 1 − ζi2p , and iv) v different repeated  pairs of complex conjugate poles located at s = −ζi ωni ± jωdi , where

Q(s) , where Q(s) is another polyωdi = ωni 1 − ζi2 . Moreover, the latter fraction M(s) nomial to be computed, represents the effect of all of the input poles. Thus, we can apply the inverse Laplace transform to find the general response of a linear system:

y(t) = L−1 {Y (s)} =

l 

f i e−di t

i=1

+ +

q   i=1 r  i=1

+

v 

 g1i t k−1 e−h i t + g2i t k−2 e−h i t + · · · + g(k−1)i te−h i t + gki e−h i t ,

 ⎛

λi 1−

ζi2p

e−ζi p ωni p t sin(ωdi p t + φi p )

⎝  ρ1i t k−1 e−ζi ωni t sin(ωdi t + φi p ) 1 − ζi2 i=1 ρ2i + t k−2 e−ζi ωni t sin(ωdi t + φi )+ 2 1 − ζi

2.1 Control of Linear Systems

ρ(k−1)i −ζi ωni t ··· +  te sin(ωdi t + φi ) 1 − ζi2 ⎞ ρki e−ζi ωni t sin(ωdi t + φi )⎠ + 2 1 − ζi

Q(s) −1 + p(t), where p(t) = L . M(s)

9

(2.3)

The symbols λi , ρ ji , j =1,√. . . , k,  represent real constants and φi p = √ 1−ζi2p 1−ζi2 , φi = arctan . arctan ζi p ζi In the previous expression, p(t) is known as the forced response, y f (t), and all of the remaining terms are known as the natural response, yn (t). Since p(t) =  −1 Q(s) , the forced response is “similar” to the input u(t). Because of this, L M(s) in a control system the input u(t) (through p(t)) is employed to specify the behavior that is expected for y(t), i.e., a closed-loop control system is designed such that limt→∞ y(t) = y f (t) = p(t). Hence, it is desirable that p(t) = u(t) and limt→∞ yn (t) = 0. A transfer function is said to be stable if and only if this limit is accomplished for all initial conditions. According to the general solution given in (2.3), we have the following conclusions on the stability of an arbitrary linear system. 1. The differential equation in (2.1) or, equivalently, the transfer function in (2.2) is stable if and only if all of the poles of G(s) have negative real parts, i.e., if and only if −di < 0, −h i < 0, −ζi p ωni p < 0, and −ζi ωni < 0. 2. The differential equation in (2.1) or, equivalently, the transfer function in (2.2) is unstable if at least one of the poles of G(s) has a positive real part, i.e., if −di > 0 or −h i > 0 or −ζi p ωni p > 0 or −ζi ωni > 0. 3. The differential equation in (2.1) or, equivalently, the transfer function in (2.2) is unstable if G(s) has at least one pole with zero real part which is repeated at least two times, i.e., if −h i = 0 or −ζi ωni = 0 and k ≥ 2. 4. The differential equation in (2.1) or, equivalently, the transfer function in (2.2) is marginally stable if all of the poles of G(s) have negative real parts excepting some single poles which have zero real parts, i.e., if −di < 0, −h i < 0, −ζi p ωni p < 0, −ζi ωni < 0 and k = 1 for the poles such that −h i = 0 or −ζi ωni = 0. From the expression in (2.3) it is clear that the poles of a transfer function not only determine the system stability but also the system transient response. As a matter of fact, in Fig. 2.1 we present several examples of typical transient responses due to real poles and complex conjugate poles when u(t) = 1. The relative location of these poles on the plane s is depicted in Fig. 2.2. We conclude that a faster response is accomplished as these poles are located farther from the origin on the left-hand halfcomplex plane. A more oscillatory response is obtained as the complex conjugate

10

2 Mathematical Preliminaries

Fig. 2.1 Transient responses when an unit step input is applied 5 4 3 2 1

Im(s)

Fig. 2.2 Location on the plane s of poles corresponding to the transient responses in Fig. 2.1. Figure 2.1a: dash-dot x, continuous o, dashed +. Figure 2.1b: dash-dot triangle-right, continuous square, dashed triangle-left. Figure 2.1c: dash-dot triangle-down, dashed triangle-up

0 -1 -2 -3 -4 -5 -5

-4

-3

-2

-1

0

1

Re(s)

poles approach the imaginary axis. Real poles do not produce any oscillation. This is depicted in Fig. 2.3.

2.1 Control of Linear Systems

11

Fig. 2.3 Regions of the plane s. a Arrows of the straight lines indicate where to locate poles to achieve a faster response. A slower response is obtained assigning the poles in the opposite direction of the arrows. b Arrows of the circular lines indicate where to locate the poles to obtain a more oscillatory response. A less oscillatory response is obtained assigning the poles in the opposite direction of the arrows

Also notice that in the case where u(t) = A = constant, that is U (s) = steady-state response can be computed using the final value theorem, i.e., lim y(t) = lim sY (s) = lim sG(s)U (s) = lim sG(s)

t→∞

s→0

s→0

s→0

A , s

the

A = G(0)A. s

This means that the system steady-state response depends on the locations of both poles and zeros of G(s). Thus, the problem of designing a controller in classical control is to choose a new differential equation with transfer function G c (s), known as the controller, such that when it is feedback connected to the plant transfer function G p (s), i.e., as shown in Fig. 2.4, the closed-loop transfer function: G c (s)G p (s) Y (s) = G cl (s) = , Yd (s) 1 + G c (s)G p (s) where Yd (s) = requirements.

A s

represents the reference or the desired output, satisfies the following

• G cl (s) is stable. • All poles and zeros of G cl (s) are suitably assigned on the left-hand half-plane such that the transient response satisfies the desired specifications. • All poles and zeros of G cl (s) are suitably assigned on the left-hand half-plane such that the steady-state response satisfies limt→∞ y(t) = G cl (0)A = A = yd (t) = p(t) = y f (t), where yd (t) = L−1 {Yd (s)}. Two powerful tools are employed in classical control to achieve simultaneously these requirements: the root locus method (see Sect. 3.2) and the frequency response

12

2 Mathematical Preliminaries

Fig. 2.4 Feedback connection of the plant G p (s) and the controller G c (s)

method (see [106], Ch. 6). Both of these successful approaches exist, thanks to the linear property of control systems considered in classical control: (1) linear systems have a known general solution (see (2.3)), (2) poles and zeros are well defined for linear systems, (3) linear systems, by definition, satisfy the superposition principle which renders possible the existence of the frequency response approach, (4) the frequency response approach allows to see control systems as filters, which is important to analyze disturbance rejection as in Sect. 3.2.5 and transient response to a desired reference. Hence, many control problems can be understood and solved if the control system is linear. Unfortunately, most control systems are not linear, i.e., they are known as nonlinear. Moreover, most of the classes of electric machines that are of interest in this book are nonlinear. In such a case all of the advantages listed in the previous paragraph vanish: (a) a general analytical solution method for nonlinear systems does not exist and, thus, it is not possible in general to design a controller to satisfy the transient response specifications, (b) the superposition principle cannot be applied nor Laplace transform and, thus, poles and zeros do not exist either, (c) the root locus and frequency response methods do not exist for nonlinear systems. Thus, new analysis and design methodologies must be introduced to control nonlinear systems. It is important to say that several successful approaches exist at present for this. In the following section, we present some mathematical tools that are fundamental for analysis and design of nonlinear control systems.

2.2 Mathematical Tools for the Study of Nonlinear Control Systems Any arbitrary ordinary linear n−order differential equation of the form (2.1) can be rewritten in the general linear state space representation: x˙ = Ax + Bu, y = C x + Du,

(2.4)

where x ∈ Rn is the system state and A ∈ Rn×n . Moreover, this notation can be generalized to the case where l outputs and p inputs exist, i.e., y ∈ Rl and u ∈ R p , by defining C ∈ Rl×n , B ∈ Rn× p , and D ∈ Rl× p . It is said that any system that cannot be written in the form given in (2.4) is nonlinear and the following general nonlinear state space representation is employed:

2.2 Mathematical Tools for the Study of Nonlinear Control Systems

13

x˙ = f (x, u),

(2.5)

y = h(x, u),

(2.6)

where x ∈ Rn , whereas f : Rn × R p → Rn and h : Rn × R p → Rl are arbitrary nonlinear functions and f is assumed to be locally Lipschitz [132], Ch. 3, to ensure local existence and uniqueness of the solution x(t). Example 2.1 The dq dynamical model of a salient rotor permanent magnet (PM) synchronous motor is given in (4.29), (4.30), (4.31), and it is rewritten here for the ease of reference: L q I˙q = −R Iq − n p L d Id ω − Φ M ω + Vq , L d I˙d = −R Id + n p L q Iq ω + Vd , J ω˙ + bω = n p (L d − L q )Id Iq + Φ M Iq − τ L . Defining     x = Iq , Id , ω , u = Vq , Vd , τ L ,

y = ω,

i.e., n = 3, l = 1, and p = 3, we can write this model in the form (2.5), (2.6), as x˙ = f (x, u), y = h(x, u), ⎡ ⎤ (−R Iq − n p L d Id ω − Φ M ω + Vq )/L q ⎦ , h(x, u) = ω. (−R Id + n p L q Iq ω + Vd )/L d f (x, u) = ⎣ (−bω + n p (L d − L q )Id Iq + Φ M Iq − τ L )/J Notice that this model is nonlinear, i.e., it cannot be written in the form (2.4), because of the products Id ω, Iq ω, and Id Iq , although h(x, u) is linear. Example 2.2 The unsaturated dynamic model of a switched reluctance motor (SRM) is given in (6.9), (6.10), and it is rewritten here for the ease of reference: D(q) I˙ + K (q)I ω + R I = U, 3  1 K j (q)I j2 − τ L , q˙ = ω, J ω˙ + bω = 2 j=1 U = [U1 , U2 , U3 ] , I = [I1 , I2 , I3 ] , D(q) = diag{L 1 (q), L 2 (q), L 3 (q)},   2π , j = 1, 2, 3, L j (q) = l0 − l1 cos Nr q − ( j − 1) 3 K (q) = diag{K 1 (q), K 2 (q), K 3 (q)},   ∂ L j (q) 2π , j = 1, 2, 3. K j (q) = = l1 Nr sin Nr q − ( j − 1) ∂q 3

14

2 Mathematical Preliminaries

Defining     x = I1 , I2 , I3 , ω, q , u = U1 , U2 , U3 , τ L ,

y = q,

i.e., n = 5, l = 1, and p = 4, we can write this model in the form (2.5), (2.6), as x˙ = f (x, u), y = h(x, u), ⎤ ⎡ (−K 1 (q)I1 ω − R I1 + U1 )/L 1 (q) ⎢ (−K 2 (q)I2 ω − R I2 + U2 )/L 2 (q) ⎥ ⎥ ⎢ ω − R I + U3 )/L 3 (q) ⎥ f (x, u) = ⎢ ⎥ , h(x, u) = q. ⎢ (−K 3 (q)I 33 1 3 ⎣ (−bω + j=1 K j (q)I j2 − τ L )/J ⎦ 2 ω Notice that this model is nonlinear, i.e., it cannot be written in the form (2.4), because of the products K j (q)I j ω, K j (q)I j2 , the trigonometric functions in K j (q) and L j (q), and the divisions by L j (q), j = 1, 2, 3, although h(x, u) is linear. In paragraph after (2.5), (2.6), it is stated that f : Rn × R p → Rn is assumed to be locally Lipschitz to ensure local existence and uniqueness of the solution x(t). This means that given an initial condition x(t0 ), at t = t0 , the solution x(t) is ensured to exist only in the interval t ∈ [t0 , t0 + δ] for some small δ > 0. The reason for this constrained interval of existence is because, after this interval of time, x(t) may reach some region where f is not locally Lipschitz and, thus, the solution x(t) is not ensured to exist and/or to be unique. Thus, the local Lipschitz property of f to ensure existence and uniqueness of a solution may appear to be a limited result. However, the existence and uniqueness of a solution may be extended to the open infinite interval of time t ∈ [t0 , ∞) if x(t) is ensured to remain in a region D ⊂ Rn where f is locally Lipschitz for all t0 ≤ t < ∞. From the Mathematics point of view, this is the basic reason for the study of system stability. From the Engineering point of view, we must ensure closed-loop system stability to avoid the whole state or some part of the state to reach some dangerous or undesired region, i.e., to avoid the system state to remain far from its desired values. Moreover, since we must ensure that the state converges to its desired values the concept of asymptotic stability is introduced. These ideas are rendered precise in the following. Since Lyapunov stability is a powerful tool for the study of nonlinear systems, in the following we present some basic ideas of this approach. Definition 2.3 ((Vector norms) [132], pp. 647, [241], pp. 78) The norm x of a vector x is a real-valued function with the properties: • x ≥ 0 for all x ∈ Rn , with x = 0 if and only if x = 0. • x + y ≤ x + y , for all x, y ∈ Rn (triangle inequality). • αx = |α| x , for all α ∈ R and x ∈ Rn . The symbol | · | stands for the absolute value of a scalar.

2.2 Mathematical Tools for the Study of Nonlinear Control Systems

15

According to the above definition, a vector norm is useful to formalize the concept of distance. In this respect, there are several different norms and we define some of them at the end of this section. In all of the formal results that we present in the remaining of this chapter we refer to any norm satisfying the properties in Definition 2.3 if nothing else is specified. Definition 2.4 ((Equilibrium point) [132], pp. 112, [241], pp. 184) The following system: x˙ = f (x), x ∈ Rn ,

(2.7)

where f : D → Rn is a locally Lipschitz map from a domain D ⊂ Rn into Rn , is said to be autonomous if f (x) does not depend explicitly on time or some other variables different from x. We define an equilibrium point x = x¯ as that point where f (x) ¯ = 0. Notice that the system in (2.7) has not input. Hence, the above definition means that if the system (represented by the state x) is taken to the equilibrium point, i.e., x(t0 ) = x¯ at some t = t0 , and it is left there without perturbing it (without applying any external input) then the system will remain there for any future time because x˙¯ = f (x) ¯ = 0 means that the state remains constant. Using a suitable change of coordinates such as z = x − x, ¯ any equilibrium point can always be translated to the origin. Hence, without any loss of generality, we can always assume that the equilibrium point is the origin, i.e., that x¯ = 0. Now, we define stability in the sense of Lyapunov. Definition 2.5 ((Stability in the sense of Lyapunov) [132], pp. 112) The equilibrium point x = 0 of (2.7) is • stable if, for each ε > 0, there is a δ = δ(ε) > 0 such that

x(0) < δ ⇒ x(t) < ε, ∀t ≥ 0. • unstable if it is not stable. • asymptotically stable if it is stable and δ can be chosen such that

x(0) < δ ⇒ lim x(t) = 0. t→∞

According to this definition, 0 is the desired value for the state x. In this respect, we must think that the state x is in fact the system error which we desire to become zero. Thus, the control objective is to accomplish x(t) → 0 as t → ∞, i.e., asymptotic stability is the main objective. On the other hand, stability implies that x(t) remains close to its desired value 0. Moreover, according to the definition of stability, we succeed to keep x(t) closer to 0 just by choosing an initial value for x which is closer to 0. This idea implies that the system state can be ensured to remain arbitrarily close to its desired value, i.e., the equilibrium point x = 0 is not repulsive.

16

2 Mathematical Preliminaries

Fig. 2.5 Level surfaces of V (x) on D, when n = 2

The problem with Definition 2.5 is that (2.7) is required to be solved to observe the evolution of x(t). Recall that one fundamental problem in the study of the nonlinear system (2.7) it is that a general analytical solution method does not exist. Thus, an alternative criterion must be formulated to check the conditions in Definition 2.5. This is the problem that solves the following result. Theorem 2.6 ((Lyapunov Theorem) [132], pp. 114) Let x = 0 be an equilibrium point for (2.7) and D ⊂ Rn be a domain containing x = 0. Let V : D → R be a continuously differentiable function such that V (0) = 0 and V (x) > 0 in D − {0}, V˙ (x) ≤ 0 in D.

(2.8) (2.9)

Then, x = 0 is stable. Moreover, if V˙ (x) < 0 in D − {0},

(2.10)

then x = 0 is asymptotically stable. In the case n = 2, the continuous differentiability of V (x) and the conditions in (2.8) imply that the level surfaces of V (x) constitute closed trajectories which are concentric to the equilibrium under study, i.e., the origin (see Fig. 2.5), at least in a domain Ω ⊂ D. Notice that more internal level surfaces correspond to smaller values of V (x). Hence, if (2.10) is true (V (x) decreases as time grows), the solution x(t) always moves toward more internal level surfaces until x = 0 is reached. This is what asymptotic stability implies. On the other hand, if (2.9) is true, then the solution x(t) might move toward more internal level surfaces, if V˙ < 0, but it is also possible for the solution x(t) to remain on a level surface where V > 0 is constant (i.e., V˙ = 0). This implies that x(t) → 0 is not ensured but it is ensured that the origin is stable. The function V (x) is arbitrary as long as it is useful to apply the above theorem. A function V (x) satisfying (2.8) is called a Lyapunov function candidate. If some of the conditions in (2.9) or (2.10) are satisfied, then the function V (x) is said to

2.2 Mathematical Tools for the Study of Nonlinear Control Systems

17

Fig. 2.6 Simple pendulum

be a Lyapunov function. The function V (x) is very general and it must be proposed since a general method to find it does not exist. However, some times V (x) can be proposed as the system energy. Example 2.7 Consider the simple pendulum model (see Fig. 2.6): 

 x2 x˙ = f (x) = , x = [x1 , x2 ] ∈ R2 , − mlb 2 x2 − mgl sin(x1 ) ml 2

(2.11)

˙ the constants m, g, l, are all positive, whereas b > 0 if friction where x1 = θ, x2 = θ, is present and b = 0 if no friction is present. The equilibrium points x¯ = [x¯1 , x¯2 ] are those points satisfying  f (x) ¯ =

   x¯2 0 = , mgl 0 − mlb 2 x¯2 − ml 2 sin(x¯1 )

i.e., x¯2 = 0, sin(x¯1 ) = 0, or x¯1 = ±nπ,

(2.12)

where n = 0, 1, 2, 3, . . .. Let us analyze the stability of the equilibrium point x¯1 = 0, x¯2 = 0. For this, consider the following scalar function: V (x) = V (x1 , x2 ) =

1 2 2 ml x2 + mgl(1 − cos(x1 )). 2

(2.13)

We stress that this function V (x) is the sum of the pendulum kinetic energy K (x2 ) = 1 ml 2 x22 and the pendulum potential energy E p (x1 ) = mgl(1 − cos(x1 )). Moreover, 2 notice that V (0, 0) = 0 and V (x1 , x2 ) > 0 for all (x1 , x2 ) ∈ D − {(0, 0)} where D = {(x1 , x2 ) ∈ R2 |x2 ∈ R, −2π < x1 < 2π}. Hence, we conclude that V (x) defined in (2.13) qualifies as a Lyapunov function candidate. Now, let us compute V˙ , which is also known as the time derivative of V along the trajectories of (2.11):

18

2 Mathematical Preliminaries

V˙ = ml 2 x2 x˙2 + mgl x˙1 (−(−1) sin(x1 )),   b mgl = ml 2 x2 − 2 x2 − sin(x ) + mglx2 sin(x1 ), 1 ml ml 2 = −bx22 , where x˙1 and x˙2 have been replaced by the expressions in (2.11). Notice that V˙ = 0 if x2 = 0 (for b > 0) but x1 may take any value in D and V˙ = 0 still stands. Thus, we cannot conclude that V˙ (x) < 0 in D − {(0, 0)}. What we can conclude is that V˙ ≤ 0 in D. Notice that this result stands for b ≥ 0. According to Theorem 2.6 this implies that the equilibrium point (x1 , x2 ) = (0, 0) is stable if b ≥ 0. Remark 2.8 Notice that the pendulum total energy function V (x), defined in (2.13), has a local minimum at the equilibrium point under study (x1 , x2 ) = (0, 0). In the case where b = 0, it is well known from Physics that the total energy remains constant as pendulum moves, i.e., V˙ = 0. This implies that, given a different from zero initial condition which is close enough to (x1 , x2 ) = (0, 0), the pendulum oscillates forever and this oscillatory movement does not increase nor diminish its amplitude. Moreover, the pendulum oscillates with a smaller amplitude if the initial condition is chosen to be closer to (x1 , x2 ) = (0, 0). This behavior is what the concept of stability, introduced in Definition 2.5, describes. When b > 0 friction exists and Physics predicts that the total energy continuously decreases as long as the pendulum is in movement. Furthermore, the pendulum moves as long as its energy is different from zero. Since the pendulum energy is zero (minimal) when (x1 , x2 ) = (0, 0), then the pendulum evolution must converge to the equilibrium point (x1 , x2 ) = (0, 0) as t → ∞. This behavior is described by the concept of asymptotic stability introduced in Theorem 2.6. Thus, asymptotic stability should be concluded which is not the case when applying Theorem 2.6 in Example 2.7. This is an example of the fact that the conditions given in Theorem 2.6 are sufficient but not necessary. This means that (i) if one of the conditions in Theorem 2.6 are satisfied, then either stability or asymptotic stability are ensured, but (ii) if one or any of the conditions in Theorem 2.6 are not satisfied we can not conclude instability or that asymptotic stability is not possible. In such a case we must look for another function V (x) and, again, apply Theorem 2.6 trying to arrive to some conclusion on the system stability. These ideas are applied in the next example. Let us first present some results that are useful in Lyapunov stability analysis. The scalar function V , defined in Theorem 2.6, when satisfying (2.8) is called a positive definite function. If (2.8) is modified as V (0) = 0 and V (x) ≥ 0 for x = 0, V (x) is said to be a positive semidefinite function. If (2.8) is modified as V (0) = 0 and V (x) < 0 for x = 0, V (x) is said to be a negative definite function. If (2.8) is modified as V (0) = 0 and V (x) ≤ 0 for x = 0, V (x) is said to be a negative semidefinite function. Using the above nomenclature the results in Theorem 2.6 can be restated as • If V (x) is positive definite and V˙ is negative semidefinite, then the origin is stable.

2.2 Mathematical Tools for the Study of Nonlinear Control Systems

19

• If V (x) is positive definite and V˙ is negative definite,1 then the origin is asymptotically stable. A special class of scalar functions are quadratic forms, which can be written as V (x) = x  Ax,

(2.14)

where x ∈ Rn and the matrix A ∈ Rn×n is real and symmetric. The definiteness of matrix A is defined depending on the definiteness of the quadratic form in (2.14) as it is shown below. Definition 2.9 ([28], pp. 264) The matrix A is • positive definite if the quadratic form in (2.14) is positive definite. • positive semidefinite if the quadratic form in (2.14) is positive semidefinite. We have the following results: Theorem 2.10 ([109], pp. 170) The n eigenvalues of the symmetric n × n matrix A are real. Theorem 2.11 ([109], pp. 402, [28], pp. 265, [277], pp. 582) Let ai j represent the entry of a n × n symmetric matrix A located at row i and column j. Define the following n determinants as the leading principal minors of the symmetric matrix A: ⎛ ⎞   a11 a12 a13 a11 a12 , Δ3 = det ⎝ a21 a22 a23 ⎠ , . . . , Δn = det(A). Δ1 = a11 , Δ2 = det a21 a22 a31 a32 a33 The matrix A is • positive definite if and only if (a) all of its eigenvalues are strictly positive or, equivalently, (b) all of its leading principal minors are strictly positive. • positive semidefinite if and only if (a) all of its eigenvalues are positive or zero. Equivalently, (b) all of its leading principal minors are positive or zero. The following result is useful when trying to find the upper and lower bounds of a quadratic form. Theorem 2.12 ((Rayleigh–Ritz), [109], pp. 176) Let A be a n × n symmetric matrix, then λmin (A) x 2 ≤ x  Ax ≤ λmax (A) x 2 , ∀x ∈ Rn ,

(2.15)

where λmin (A) and λmax (A) stand, respectively, for the smallest and the largest eigenvalues of matrix A and x is the Euclidean norm of vector x. 1 In

this case, V (x) is said to be a strict Lyapunov function.

20

2 Mathematical Preliminaries

The following theorem is useful when trying to find the conditions to ensure that the eigenvalues of a matrix can be enlarged arbitrarily. Theorem 2.13 ((Geršgorin), [109], pp. 344) Let Ai j represent the entry of the n × n matrix A located at the ith row and the jth column. Define Ri (A) :=

j=n 

|Ai j |, 1 ≤ i ≤ n.

j=1, j =i

Then all eigenvalues of matrix A are located in the union of the n discs:   n ∪i=1 z ∈ C : |z − Aii | ≤ Ri (A) . Furthermore, if an union of k of these n discs forms a connected region that is disjoint from all the remaining n − k discs, then there are precisely k eigenvalues of A in this region. Another useful tool to study Lyapunov stability is the mean value Theorem. Theorem 2.14 ((Mean Value Theorem), [132], pp. 651) Assume that f : Rn → R is continuously differentiable at each point x of an open set S ⊂ Rn . Let x and y be two points of S such that the segment of line joining x and y belongs to S. Then there exists a point z belonging to this segment of line such that  ∂ f  f (y) − f (x) = (y − x). ∂x x=z Now, let us continue analyzing the simple pendulum equilibrium points. Example 2.15 Consider again the simple pendulum model given in (2.11) and assume that b > 0. Let us study the stability of the equilibrium point x¯1 = 0, x¯2 = 0. We now propose the following scalar function: V (x) = V (x1 , x2 ) =

1 2 2 1 ml x2 + αml 2 x1 x2 + mgl(1 − cos(x1 )) + αbx12 . (2.16) 2 2

Notice that we can write    1 x1 αb αml 2 + mgl(1 − cos(x1 )). V (x) = [x1 , x2 ] 2 2 αml ml x 2 2

(2.17)

According to Theorem 2.11, the following conditions: α > 0, b > αml 2 , ensure positive definiteness of the matrix:

(2.18)

2.2 Mathematical Tools for the Study of Nonlinear Control Systems

 A=

 αb αml 2 . αml 2 ml 2

21

(2.19)

Notice that both conditions in (2.18) are always satisfied using a small enough α > 0. Thus, since mgl(1 − cos(x1 )) ≥ 0, for all x1 ∈ R, we conclude that V (x1 , x2 ) = 0 for (x1 , x2 ) = (0, 0) and V (x1 , x2 ) > 0 in R2 − {(0, 0)}. Now, let us compute V˙ : V˙ = ml 2 x2 x˙2 + αml 2 x˙1 x2 + αml 2 x1 x˙2 + mgl x˙1 (−(−1) sin(x1 )) + αbx1 x˙1 ,   b mgl 2 = ml x2 − 2 x2 − sin(x1 ) + αml 2 x22 ml ml 2 +αx1 (−bx2 − mgl sin(x1 )) + mglx2 sin(x1 ) + αbx1 x2 , = −(b − αml 2 )x22 − αmglx1 sin(x1 ), where x˙1 and x˙2 have been replaced by the expressions in (2.11). Notice that x1 sin(x1 ) = 0 if x1 = 0 and x1 sin(x1 ) > 0 if 0 < |x1 | < π. Thus, we conclude that V˙ (x) < 0 in D − {(0, 0)} where D = {(x1 , x2 ) ∈ R2 |x2 ∈ R, −π < x1 < π}, if the conditions in (2.18) are satisfied. Since α > 0, involved in (2.18), is required just to exist, then the conditions (2.18) are always satisfied. Hence, according to Theorem 2.6 the equilibrium point (x1 , x2 ) = (0, 0) is asymptotically stable. This result is consistent with the Physics-based reasoning presented after Remark 2.8. It is important to stress that the above result does not ensure that the solution x(t) will converge to (x1 , x2 ) = (0, 0) when beginning from any initial condition belonging to D, defined above. We can resort, again, to Physics to realize that if the initial position satisfies −π < x1 (0) < π and the initial velocity x2 (0) is large enough, the pendulum will rotate several times before stopping at some x2 = 0 and x1 = 0. Thus, Theorem 2.6 only ensures that there exists a domain Ω ⊂ D, called the domain of attraction, such that any solution beginning from any initial condition belonging to Ω will converge to (x1 , x2 ) = (0, 0) as t → ∞. However, finding Ω is an elaborate task which is desirable to avoid. This explains the interest for the search of global asymptotic stability which ensures convergence to the desired equilibrium point no matter the initial condition. The conditions ensuring global asymptotic stability are stated in the following theorem. Theorem 2.16 ((Global asymptotic stability) [132], pp. 124) Let x = 0 be an equilibrium point for (2.7). Let V : Rn → R be a continuously differentiable function such that V (0) = 0 and V (x) > 0, ∀x = 0,

x → ∞ ⇒ V (x) → ∞, ˙ V (x) < 0, ∀x = 0,

(2.20)

then x = 0 is globally asymptotically stable. The fundamental difference between Theorems 2.6 and 2.16 is the condition in (2.20). A scalar function V (x) satisfying (2.20) is said to be radially unbounded.

22

2 Mathematical Preliminaries

Radial unboundedness means that V (x) grows as x is farther from the origin, i.e., from the equilibrium point under study. The basic idea is that, in the n = 2 case for instance, the level surfaces of a radially unbounded function are concentric closed trajectories laying on the whole plane R2 with the equilibrium x = 0 located at the most internal level surface (constituted by a single point). Hence, if V˙ < 0 (V (x) decreases as time grows) the solution x(t) always moves toward the most internal level surface, i.e., toward the equilibrium point located at the origin, no matter how far is the initial condition from the origin. If V (x) is not radially unbounded, the level surfaces may be open trajectories and, thus, the solution may move far away from the origin despite V˙ < 0. Remark 2.17 Notice that the simple pendulum is a nonlinear system because of the nonlinear function sin(x1 ). This function is responsible for the simple pendulum to have an infinite number of equilibrium points (see (2.12)). According to Physics, if b > 0, the equilibrium points where n is even are asymptotically stable and the equilibrium points where n is odd are unstable. This is another important property of nonlinear systems: several isolated equilibrium points may exist having different stability properties. Thus, in nonlinear systems we do not talk about stable or unstable systems but we talk about stable or unstable equilibrium points. On the other hand, global asymptotic stability means that convergence to the desired equilibrium point is achieved from any initial condition. Notice that, by definition, any solution beginning from an initial condition which corresponds to an equilibrium point will remain forever at that equilibrium point. Thus, global asymptotic stability implies that only one equilibrium point exists: the desired one. This is another interesting feature of global asymptotic stability. In Example 2.15, we have shown that despite stability has been concluded when using Theorem 2.6, the more exigent asymptotic stability property may be still concluded using Theorem 2.6 if a new Lyapunov function candidate V (x) is employed. In the following, we present another manner to solve the same problem. The new tool is known as the LaSalle Invariance Principle and it can only be applied to autonomous systems. This result is commonly used through the two following corollaries. The first one establishes conditions for (local) asymptotic stability and the second one establishes conditions for global asymptotic stability. Corollary 2.18 ((The (local) LaSalle’s Invariance Principle), [132], pp. 128) Let x = 0 be an equilibrium point for (2.7). Let V : D → R be a continuously differentiable, positive definite function on a domain D containing the origin x = 0, such that V˙ (x) ≤ 0 in D. Let S = {x ∈ D|V˙ (x) = 0} and suppose that no solution can stay identically in S, other than the trivial solution x(t) ≡ 0. Then, the origin is asymptotically stable. Corollary 2.19 ((The (global) LaSalle’s Invariance Principle), [132], pp. 129) Let x = 0 be an equilibrium point for (2.7). Let V : Rn → R be a continuously differentiable, radially unbounded, positive definite function such that V˙ (x) ≤ 0 for all

2.2 Mathematical Tools for the Study of Nonlinear Control Systems

23

x ∈ Rn . Let S = {x ∈ Rn |V˙ (x) = 0} and suppose that no solution can stay identically in S, other than the trivial solution x(t) ≡ 0. Then, the origin is globally asymptotically stable. In the following example we render clear these ideas. Example 2.20 Consider again the simple pendulum model given in (2.11) and assume that b > 0. Let us study the stability of the equilibrium point x¯1 = 0, x¯2 = 0. To this aim consider again the Lyapunov function candidate in (2.13), i.e., V (x) = V (x1 , x2 ) =

1 2 2 ml x2 + mgl(1 − cos(x1 )). 2

Recall that V˙ = −bx22 , was found in Example 2.7 and the set D = {(x1 , x2 ) ∈ R2 |x2 ∈ R, −2π < x1 < 2π} was defined. Hence, stability was concluded. From this result, we can apply Corollary 2.18 to define the set S = {x ∈ D|V˙ (x) = 0} = {(x1 , x2 ) ∈ D|x2 = 0, −2π < x1 < 2π}. Now, in order to check that no solution can stay identically in S, other than the trivial solution x(t) ≡ 0, we evaluate the dynamics (2.11) in the set S defined above. This is achieved by replacing x2 = 0 and x˙2 = 0 because x2 = 0 is constant in S, i.e., 

x˙1 x˙2





x˙ = 1 0





 0 = . − mlb 2 (0) − mgl sin(x1 ) ml 2

This means that x1 is constant, because x˙1 = 0, taking only one of the values x1 = ±nπ, n = 0, 1, 2, . . .. In order to prove asymptotic stability, Corollary 2.18 requires that only one of these points be contained in S. Thus, we need to go back to redefine the sets D and S as D = {(x1 , x2 ) ∈ R2 |x2 ∈ R, −π < x1 < π} and S = {(x1 , x2 ) ∈ D|x2 = 0, −π < x1 < π}. This allows us to conclude that the origin is asymptotically stable. As stated in the paragraph before Example 2.7, some times the scalar function V (x) can be chosen to be the system energy or to be interpreted as the system energy. This idea is formalized through the concept of passivity which will be reviewed in Sect. 2.3. To this point, we have only considered autonomous systems. However, nonautonomous systems2 are also important for our purposes in this book. In the following, we present some basic tools for Lyapunov stability analysis of nonautonomous nonlinear systems. Consider the nonautonomous system: means that the system is written in the form x˙ = f (t, x), where f (t, x) depends not only on the state x but also on time t (or any other variable different from x).

2 This

24

2 Mathematical Preliminaries

x˙ = f (t, x),

(2.21)

where f : [0, ∞) × D → Rn is piecewise continuous in t and locally Lipschitz in x on [0, ∞) × D and D ⊂ Rn is a domain that contains the origin x = 0. The origin is an equilibrium point for (2.21) at t = 0 if f (t, 0) = 0, ∀t ≥ 0. In general, in nonautonomous systems the stability properties change with time. Thus, in the following definitions the time when the stability properties are tested, t0 , is important. Definition 2.21 ([132], pp. 149) The equilibrium point x = 0 of (2.21) is • stable if, for each ε > 0, there is a δ = δ(ε, t0 ) > 0 such that

x(t0 ) < δ ⇒ x(t) < ε, ∀t ≥ t0 ≥ 0.

(2.22)

• uniformly stable if, for each ε > 0, there is δ = δ(ε) > 0, independent of t0 , such that (2.22) is satisfied. • unstable if it is not stable. • asymptotically stable if it is stable and there is a positive constant c = c(t0 ) such that x(t) → 0 as t → ∞ for all x(t0 ) < c. • uniformly asymptotically stable if it is uniformly stable and there is a positive constant c, independent of t0 , such that for all x(t0 ) < c, x(t) → 0 as t → ∞, uniformly in t0 . That is, for each η > 0, there is T = T (η) > 0 such that

x(t) < η, ∀t ≥ t0 + T (η), ∀ x(t0 ) < c. • globally uniformly asymptotically stable if it is uniformly stable, δ(ε) can be chosen to satisfy limε→∞ δ(ε) = ∞, and, for each pair of positive numbers η and c there is T (η, c) > 0 such that

x(t) < η, ∀t ≥ t0 + T (η, c), ∀ x(t0 ) < c. Stability and asymptotic stability of the nonautonomous system in (2.21) are proven if the same conditions given in Theorem 2.6 are satisfied, but in this case the Lyapunov function may also be a function of time, i.e., V = V (t, x). However, stability and asymptotic stability of nonautonomous systems, as defined above, are limited results because they change with time. The desirable stability properties that we are interested in for nonautonomous systems are uniform stability and uniform asymptotic stability. This is because these stability properties are independent of time. Thus, the uniform stability properties are further characterized in the following. Definition 2.22 ((Class K functions) [241], pp. 188) A continuous function α : R+ → R+ is said to belong to class K if it is strictly increasing and α(0) = 0. It is said to belong to class K∞ if it is of class K and in addition α(r ) → ∞ as r → ∞.

2.2 Mathematical Tools for the Study of Nonlinear Control Systems

25

An example of a class K function is α(r ) = arctan(r ) and an example of a class K∞ function is α(r ) = r c for any positive real number c, [132] pp. 144. Definition 2.23 ((Class KL functions) [132], pp. 144) A continuous function β : [0, a) × [0, ∞) → [0, ∞) is said to belong to class KL if, for each fixed s, the mapping β(r, s) belongs to class K with respect to r and, for each fixed r , the mapping β(r, s) is decreasing with respect to s and β(r, s) → 0 as s → ∞. An example of a class KL function is β(r, s) = r c e−s for any positive real number c, [132] pp. 145. Lemma 2.24 The equilibrium point x = 0 of (2.21) is • uniformly stable if and only if there exist a class K function α and a positive constant c independent of t0 , such that

x(t) ≤ α( x(t0 ) ), ∀t ≥ t0 ≥ 0, ∀ x(t0 ) < c. • uniformly asymptotically stable if and only if there exist a class KL function β and a positive constant c, independent of t0 , such that

x(t) ≤ β( x(t0 ) , t − t0 ), ∀t ≥ t0 ≥ 0, ∀ x(t0 ) < c.

(2.23)

• globally uniformly asymptotically stable if and only if the inequality (2.23) is satisfied for any initial state x(t0 ). The problem with these characterizations is that checking them would require to solve the nonlinear system in (2.21) which, in general, is not possible. The following results allow to check the characterizations in Lemma 2.24 without solving (2.21). Theorem 2.25 ((Uniform stability), [132], pp. 151) Let x = 0 be an equilibrium point for (2.21) and D ⊂ Rn be a domain containing x = 0. Let V : [0, ∞] × D → R be a continuously differentiable function such that W1 (x) ≤ V (t, x) ≤ W2 (x), ∂V ∂V + f (t, x) ≤ 0, ∂t ∂x ∀t ≥ 0 and ∀x ∈ D, where W1 (x) and W2 (x) are continuous positive definite functions on D. Then, x = 0 is uniformly stable. Since V (x) satisfies V (t, x) ≤ W2 (x) with W2 (x) a positive definite function, it is said that V (x) is a decrescent function. Theorem 2.26 ((Uniform asymptotic stability), [132], pp. 152) Let x = 0 be an equilibrium point for (2.21) and D ⊂ Rn be a domain containing x = 0. Let V : [0, ∞] × D → R be a continuously differentiable function such that

26

2 Mathematical Preliminaries

W1 (x) ≤ V (t, x) ≤ W2 (x), ∂V ∂V + f (t, x) ≤ −W3 (x), ∂t ∂x ∀t ≥ 0 and ∀x ∈ D, where W1 (x), W2 (x), and W3 (x) are continuous positive definite functions on D. Then, x = 0 is uniformly asymptotically stable. Moreover, if D = Rn and W1 (x) is radially unbounded, then x = 0 is globally uniformly asymptotically stable. For particular classes of functions W1 (x), W2 (x), and W3 (x), uniform asymptotic stability becomes exponential stability as formalized in the following. Definition 2.27 The equilibrium point x = 0 of (2.21) is exponentially stable if there exist positive constant scalars c, k, and λ such that

x(t) ≤ k x(t0 ) e−λ(t−t0 ) , ∀ x(t0 ) < c. Theorem 2.28 ((Exponential stability), [132], pp. 154) Let x = 0 be an equilibrium point for (2.21) and D ⊂ Rn be a domain containing x = 0. Let V : [0, ∞] × D → R be a continuously differentiable function such that k1 x a ≤ V (t, x) ≤ k2 x a , ∂V ∂V + f (t, x) ≤ −k3 x a , ∂t ∂x ∀t ≥ 0 and ∀x ∈ D, where k1 , k2 , k3 and a are positive constants. Then, x = 0 is exponentially stable. If the assumptions hold globally, then x = 0 is globally exponentially stable. It is clear that the main advantage of exponential stability is that it guarantees a convergence rate to the equilibrium point. On the other hand, both uniform asymptotic stability and exponential stability have some robustness properties with respect to bounded perturbations. This is formalized in the following result. Theorem 2.29 (Theorem 4.18 in [132], pp. 172) Let D ⊂ Rn be a domain that contains the origin and V : [0, ∞) × D → R be a continuously differentiable function such that α1 ( x ) ≤ V (t, x) ≤ α2 ( x ), ∂V ∂V + f (t, x) ≤ −W3 (x), ∀ x ≥ μ > 0, ∂t ∂x

(2.24) (2.25)

∀t ≥ 0 and ∀x ∈ D, where α1 and α2 are class K functions and W3 (x) is a continuous positive definite function. Take r > 0 such that Br ⊂ D and suppose that μ < α2−1 (α1 (r )).

(2.26)

2.2 Mathematical Tools for the Study of Nonlinear Control Systems

27

Then, there exists a class KL function β and for every initial state x(t0 ), satisfying

x(t0 ) ≤ α2−1 (α1 (r )), there is T ≥ 0 (dependent on x(t0 ) and μ) such that the solution of x˙ = f (t, x) satisfies

x(t) ≤ β( x(t0 ) , t − t0 ), ∀t0 ≤ t < t0 + T,

x(t) ≤ α1−1 (α2 (μ)), ∀t ≥ t0 + T.

(2.27) (2.28)

Moreover, if D = Rn and α1 belongs to class K∞ , then (2.27) and (2.28) hold for any initial state x(t0 ), with no restriction on how large μ is. Notice that Theorem 2.29 considers the possibility that any equilibrium point does not exist. We stress that this is a common situation when a time varying disturbance is present. Thus, instead of ensuring convergence to an equilibrium point Theorem 2.29 establishes that the system state x converges, in finite time, into a ball whose radius is given in (2.28). This radius is known as the ultimate bound. Finally, some times asymptotic stability cannot be proven directly using the Lyapunov stability analysis theorems but it is still possible to prove convergence of some part of the state to its equilibrium values. Barbalat’s Lemma is very useful for this. Lemma 2.30 (Barbalat’s Lemma) [249], pp. 123) If the differentiable function f (t) has a finite limit as t → ∞, and if f˙(t) is uniformly continuous, then f˙(t) → 0 as t → ∞. A very simple sufficient condition for a differentiable function to be uniformly continuous is that its derivative be bounded [249], pp. 123. A different approach to study stability in nonlinear systems if known as inputoutput stability. In the following, we present some basic results to apply these ideas. Given a function u : [0, ∞) → Rm , we say that u ∈ Lmp , for 1 ≤ p < ∞, if [132, 242]: 

u L p =

∞

1/ p

u(t) dt p

< ∞,

(2.29)

0

where u(t) is the Euclidean norm of u(t). On the other hand, u ∈ Lm ∞ if [132, 242]:

u L∞ = sup u(t) < ∞.

(2.30)

t≥0

Now suppose that a system is represented by the following input-output relation: y = H u,

(2.31)

where H is some mapping or operator that specifies y (the output) in terms of u (the input). Input-output stability is a property of H which can be understood by considering u as an input “disturbance” and y as the output error. It is expected that

28

2 Mathematical Preliminaries

an input disturbance deviates the output error from zero. Furthermore, in an inputoutput stable system it is expected that deviation of y from zero be smaller as the disturbance be smaller. In order to measure these deviations and disturbances the

u L p norm defined in (2.29) is very useful for disturbances u which disappear with time. However, input-output stability also formalizes this idea even for the cases when the disturbance u and the output error y do not disappear with time. For this aim, it is assumed that u belongs to an extended space Lm e and y belongs to an q is defined by extended space Le , where Lm e m Lm e = {u|u τ ∈ L , ∀τ ∈ [0, ∞)}

and u τ is a truncation of u defined by uτ =

u(t), 0 ≤ t ≤ τ 0, t > τ

q

The extended space Le and the truncation yτ of y are defined correspondingly. Thus, the input-output stability of H can be established as a property of the system which ensures that the L p −norm of the truncated output error yτ is less or equal to the L p −norm of the truncated input u τ in some sense. This idea is rendered precise in the following definition. q

Definition 2.31 ([132], pp. 197) A mapping H : Lm e → Le is L stable if there exist a class K function α, defined on [0, ∞), and a nonnegative constant β such that

(H u)τ L ≤ α( u τ L ) + β, for all u ∈ Lm e and τ ∈ [0, ∞). It is finite-gain L stable if there exist nonnegative constants γ and β such that

(H u)τ L ≤ γ u τ L + β, for all u ∈ Lm e and τ ∈ [0, ∞). Once the above definitions have been included, we are ready to present the following Corollary of Barbalat’s Lemma, which is also very useful under certain conditions. Corollary 2.32 ((Barbalat’s Lemma) [242], pp. 19) If g, g˙ ∈ L∞ , and g ∈ L p , for some p ∈ [1, ∞), then g(t) → 0 as t → ∞. Some times, control systems are subject to periodic disturbances. In such a case Fourier analysis is useful. Among the important results in Fourier analysis is Parseval’s Theorem. Theorem 2.33 ((Parseval), [110], pp. 16) Assume that a function f (t), which is periodic with T as period, can be represented by the following Fourier series:

2.2 Mathematical Tools for the Study of Nonlinear Control Systems

f (t) =

29

∞  1 a0 + (an cos(ω0 t) + bn sin(ω0 t)), 2 n=1

where ω0 = 2π/T . Then 1 T



T /2 −T /2

∞

f 2 (t)dt =

1 2 1 2 a + (a + bn2 ). 4 0 2 n=1 n

The following class of saturation functions is important for the purposes of this book. Definition 2.34 Given positive constants L and M, with L < M, a function σ : R → R : ς → σ(ς) is said to be a strictly increasing linear saturation for (L , M) if it is locally Lipschitz, strictly increasing, and satisfies [286]: σ(ς) = ς, when |ς| ≤ L , |σ(ς)| < M, ∀ς ∈ R n Given a x = [x1 , . . . , xn ] ∈ Rn , the 1−norm of x is defined as x 1 = i=1 |xi |, where the symbol | · | stands for the absolute value of an scalar and the Euclidean  n 2 norm is defined as x 2 = i=1 x i . A property that relates these norms is x 2 ≤ √

x 1 ≤ n x 2 [132].  The spectral norm of an n × n matrix A(x) is defined as A = λmax (A A) where λmax (A A) stands for the largest eigenvalue of the symmetric matrix A A. If A(x) is an n × n symmetric matrix then symbol λmin (A) represents the smallest eigenvalue of A for all x ∈ Rn and A = maxi |λi (A)|, where λi (A) stands for eigenvalues of A(x). In the remaining of this book, we will employ the symbols x to represent the Euclidean norm of the vector x and A to represent the spectral norm of the matrix A, if anything else is not specified. Given a bounded matrix A, a bounded vector x or a bounded scalar y, the symbols A M , x M , and |y| M stand for the supreme values over the norm and the absolute value, respectively. Some important properties of the Euclidean and the spectral norms are ± y  D(x)w ≤ D y w ,

D(x)y ≤ D y ,

(2.32)

|y  w| ≤ y w ,

D(x)G(x) ≤ D(x) G(x) , ±y  ABw ≤ B A y w , for all x, w, y ∈ Rn , n × n matrices D(x), G(x), and n × n symmetric matrices B and A.

30

2 Mathematical Preliminaries

Fig. 2.7 Ampère’s Law

Finally, let us state Ampère’s Law, which is a fundamental result in electromagnetism that will be used in subsequent chapters to compute the distribution along the stator of the magnetic field produced by the stator phase windings in practical AC-motors. Electric currents are the sources of magnetic field and Amperère’s Law establishes a precise relation between them in the following terms. Ampère’s Law [274] Suppose that the electric current Ienc is enclosed by an arbitrary closed trajectory C in the space, as depicted in Fig. 2.7. The magnetic field H along C relates to Ienc through H · dl = Ienc ,

(2.33)

C

where dl is the differential of C, the symbol “·” stands for the standard scalar product of vectors, and the integral on the left side represents the line integral of H along the closed trajectory C. The symbol Ienc represents the net electric current enclosed by C. This means that Ienc is given as the algebraic sum of all electric currents that are enclosed by C and, hence, each enclosed current must be accompanied by a “+” sign or a “−” sign. This is determined using the right-hand rule. Right-Hand Rule Hold the right hand such that the curl of the fingers indicates the orientation of the closed trajectory C. Each electric current enclosed by the trajectory C must be affected by a “+” if it has the sense indicated by the thumb. Otherwise employ a “−”. See Fig. 2.8.

2.3 Passivity

31

Fig. 2.8 Right-hand rule Fig. 2.9 Passive circuit elements

2.3 Passivity Passivity is a concept that is closely related to electric circuits and mechanical systems and, thus, passivity is well suited for the study of electromechanical systems. In electric circuits, for instance, the basic phenomena are described by inductances, capacitances, and resistances. These circuit elements have the property to either store energy (inductors and capacitors) or dissipate energy (resistors). Moreover, energy stored by inductors and capacitors is never greater than energy that they receive and energy dissipated by resistors is never greater than energy that they receive. This means that inductances, capacitances, and resistances do not generate energy and, because of that, they are called passive circuit elements. On the other hand, energy is supplied to each circuit element with a rate given by the electrical power w = vi which is defined as the product of voltage v applied at the circuit element terminals and electric current i flowing through the circuit element (see Fig. 2.9). Furthermore, since power is the time derivative of energy E in electric circuits, then it is natural to define passivity as a phenomenon described by  E=

t

vi dr ≥ β,

(2.34)

0

for all t ≥ 0 and some finite β ≤ 0. According to the standard convention in electric circuits, if power is positive w = iv > 0, then energy is supplied from the source to the circuit element. If power is negative w = iv < 0, then energy is given back from the circuit element to the source. In passive circuit elements both cases are possible:

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2 Mathematical Preliminaries

a negative power may appear when terminals of a capacitor or an inductance are put in short-circuit when electric charge in the capacitor is not zero or when the magnetic flux linkage in an inductor is not zero, for instance. In such a case, the integral of a negative power must be energy that is stored in the circuit element at t = 0 which cannot be infinite. Thus, β ≤ 0 in !(2.34) represents the stored energy at t = 0 which t is finite. Notice that the integral 0 vi dr is not upper bounded in (2.34) because energy supplied by the source only depends on the source energy capacity. On the contrary, negative values of this integral are lower bounded by energy that is initially stored in the circuit element. !t Suppose that a large value of energy 0 1 vi dr > 0 is supplied. Hence, one might argue that since there is not an upper bound on this integral, ! t neither does exist a lower bound on the extracted energy from the system, i.e., on t1 vi dr < 0. However, notice !t !t !t that in such a case 0 vi dr = 0 1 vi dr + t1 vi dr = 0 ≥ β, i.e., (2.34) still stands. According to the above discussion, we have the following general result. Definition 2.35 ([29]) A system with input u ∈ Rm and output y ∈ Rm , is passive if there is a constant β ≤ 0 such that 

t

y  (r )u(r )dr ≥ β,

(2.35)

0

for all functions u and all t ≥ 0. We also have the following result taken from [29]. Theorem 2.36 Assume that there is a continuous function V ≥ 0 such that 

t

V (t) − V (0) ≤

y(r ) u(r )dr,

(2.36)

0

for all functions u, for all t ≥ 0 and all V (0). Then the system with input u and output y is passive. We can show that (2.36) is equivalent to (2.35) by noticing that because of V ≥ 0, we can write (2.36) as  t

−V (0) ≤

y(r ) u(r )dr,

0

and setting β = −V (0) ≤ 0. It is common in the literature to designate V as the (energy) storage function and the product w = y  u as the supply rate. When the supply rate is a more elaborate function of y and u, it is employed the nomenclature w = w(y, u). Then, if a nonlinear system satisfies (2.36) with some w = w(y, u) it is said to be a dissipative system [29, 204, 267]. Differentiating (2.36) it is clear that V˙ ≤ y  u,

2.3 Passivity

33

Fig. 2.10 A passive network is composed as the connection of passive circuit elements

which is another manner to define passivity (see Definition 2.42). Notice that using u = 0 we have V˙ ≤ 0 and, hence, a passive system is stable in the sense of Lyapunov if the storage function V is a positive definite function of the system state x. On the other hand, it is important to say that (2.34) also stands for any electric network, as that shown in Fig. 2.10, when it is only composed by passive inductors, capacitors, and resistors. Such a network is called a passive network. The impedance (s) , where V (s), I (s) are the Laplace transforms of v, i, of this network Z (s) = VI (s) and is a transfer function having the property to be positive real (PR), which means that Theorem 2.37 ([29]) A rational (transfer) function G(s) is PR if and only if 1. G(s) has no poles in Re(s) > 0. 2. Re(G( jω)) ≥ 0 for all ω ∈ [−∞, +∞] such that jω is not a pole in G(s). 3. If s = jω0 is a pole in G(s), then it is a simple pole, and if ω0 is finite, then the residual: lim (s − jω0 )G(s), s→ jω0

is real and positive. If ω0 is infinite, then R∞  limω→∞

G( jω) jω

is real and positive.

This result basically establishes the case of an electric network only composed by inductances, capacitances, and resistances that is marginally stable (in the classical sense). Under these conditions the sinusoidal steady-state voltage and current at terminals of such a network only present a phase shift φ in the closed range [−90◦ , +90◦ ]. We stress that this latter property implies (2.34). For instance, in Fig. 2.11, we show the case when the phase shift satisfies |φ| < 90◦ . Notice that power w = vi takes alternate positive and negative values but ! t it has a mean value !t that is positive. Thus, 0 vi dr satisfies (2.34) and the integral 0 vi dr grows without a limit as t → ∞. In Fig. 2.12, we show the case when the phase shift!satisfies |φ| = 90◦ . Notice t that the mean value of power w = vi is zero and, hence, 0 vi dr satisfies (2.34) and remains bounded. ◦ In Fig. 2.13, we show the case when the phase shift satisfies ! t |φ| > 90 . Notice that the mean value of power w = vi is negative and, hence, 0 vi dr does not satisfy (2.34) because it is not bounded from below. PR systems were first studied in the electric networks community. See [29–31, 38, 44–46, 72, 218]. We also have the following.

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2 Mathematical Preliminaries

Fig. 2.11 Electric current lags |φ| < 90◦ the ! t applied voltage. Top: continuous, voltage v; dashed, current i. Middle: power w = vi. Bottom: 0 vi dr

Fig. 2.12 Electric current lags |φ| = 90◦ the ! t applied voltage. Top: continuous, voltage v; dashed, current i. Middle: power w = vi. Bottom: 0 vi dr

2.3 Passivity

35

Fig. 2.13 Electric current lags |φ| > 90◦ the ! t applied voltage. Top: continuous, voltage v; dashed, current i. Middle: power w = vi. Bottom: 0 vi dr

Theorem 2.38 ([29]) A rational (transfer) function G(s) is strictly positive real (SPR) if 1. G(s) has no poles in Re(s) ≥ 0. 2. Re(G( jω)) > 0 for all ω ∈ (−∞, +∞). 3. a) limω2 →∞ ω 2 Re(G( jω)) > 0 when r = 1, b) limω2 →∞ Re(G( jω)) > 0, lim|ω|→∞ G(jωjω) > 0 when r = −1, where r = n − m is known as the relative degree of the system with n and m the number of poles and number of zeros, respectively, of G(s). This result basically establishes the case when the electric network only composed by inductances, capacitances, and resistances is stable (in the classical sense) and that the sinusoidal steady-state voltage and current at terminals of such a network only present a phase shift φ in the open range (−90◦ , +90◦ ). The connection between PR systems and passivity is formally established in the so-called Kalman-Yakubovich-Popov Lemma or KYP Lemma for short. The following is one among lots of versions of KYP Lemma. Lemma 2.39 ((KYP Lemma) [29, 204]) Consider a controllable and observable linear-time-invariant system defined as x˙ = Ax + Bu, x(0) = x0 , y = C x,

36

2 Mathematical Preliminaries

where x ∈ Rn and u, y ∈ R, with the corresponding transfer function G(s) = C(s I − A)−1 B. The following two statements are equivalent: (a) The transfer function G(s) is PR. (b) There exists a symmetric positive definite matrix P and a symmetric positive semidefinite matrix Q such that A P + P A = −Q, 

PB = C .

(2.37) (2.38)

The following two statements are equivalent: (c) The transfer function G(s) is SPR. (d) There exist symmetric positive definite matrices P and Q such that (2.37) and (2.38) hold. Result in (a) and (b) means that system defined by transfer function UY (s) = G(s) is (s) passive. As a matter of fact, consider the continuously differentiable positive definite storage function V (x) = 21 x  P x and take its time derivative: 1 1 ˙ V˙ = x˙  P x + x  P x, 2 2 1 1 = (Ax + Bu) P x + x  P(Ax + Bu), 2 2 1   1   1 1 = x A P x + u B P x + x  P Ax + x  P Bu, 2 2 2 2 1 = x  (A P + P A)x + u  B  P x, 2 1 = − x  Qx + u  C x, 2 1 = − x  Qx + u  y, 2  t u  (r )y(r )dr, ⇒ V (x(t)) − V (x(0)) ≤

(2.39)

0

= because − 21 x  Qx ≤ 0 for all x ∈ Rn . Hence, according to (2.36), system UY (s) (s) 1  ˙ G(s) is passive. Moreover, using u = 0 in (2.39) we obtain V = − 2 x Qx ≤ 0 for all x ∈ Rn which means that system is also stable (in the sense of Lyapunov), i.e., it is marginally stable in the classical sense. Proceeding similarly, we realize that (c) and (d) mean that system defined by = G(s) is passive and it is asymptotically stable (in the sense transfer function UY (s) (s) of Lyapunov), i.e., stable in the classical sense, because − 21 x  Qx < 0 for all x ∈ Rn excepting x = 0 in this case.

2.3 Passivity

37

The KYP Lemma stated above for linear systems can be extended as follows for nonlinear systems having a state space representations that is affine in the input. Consider the following nonlinear system: x˙ = f (x) + g(x)u, y = h(x),

(2.40) (2.41)

with x(0) = x0 , x ∈ Rn , u, y ∈ Rm , and f : Rn → Rn , g : Rn → Rn×m , h : Rn → Rm , are smooth functions of x. Moreover f (0) = 0 and h(0) = 0. Lemma 2.40 ((KYP Lemma for nonlinear systems) [29]) Consider the nonlinear system (2.40), (2.41). The following statements are equivalent: (i) There exists a continuously differentiable storage function V (x) ≥ 0, V (x) = 0 when x = 0, and a function L(x) ≥ 0 such that for all t ≥ 0: 

t

V (x(t)) − V (x(0)) =





y (r )u(r )dr −

0

t

L(x(r ))dr.

(2.42)

0

(ii) There exists a continuously differentiable nonnegative function V : Rn → R with V (x) = 0 when x = 0, such that L f V (x) = −L(x), L g V (x) = h  (x), where L f V (x) =

∂V (x) ∂x

f (x) and L g V (x) =

(2.43)

∂V (x) g(x). ∂x

Expressions in (2.43) represent conditions for the nonlinear system in (2.40), (2.41), to be passive. As a matter of fact, the time derivative of V (x) is given as ∂V (x) x, ˙ V˙ = ∂x ∂V (x) ∂V (x) = f (x) + g(x)u, ∂x ∂x = L f V (x) + L g V (x)u, = −L(x) + h  u, = −L(x) + y  u, ⇒



V (x(t)) − V (x(0)) =

t

y  (r )u(r )dr −

0



(2.44) t

L(x(r ))dr.

0

!t Thus, (2.42) is retrieved. Moreover, since − 0 L(x(r ))dr ≤ 0 then V (x(t)) − !t V (x(0)) ≤ 0 y  (r )u(r )dr which, according to (2.36), proves that the nonlinear system in (2.40), (2.41), is passive. On the other hand, using u = 0 in (2.44) we have that V˙ = −L(x).

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2 Mathematical Preliminaries

The following observations are in order: • If V (x) is a continuously differentiable positive definite function and L(x) is a continuous positive semidefinite function, then the origin of system in (2.40), (2.41) is stable in the sense of Lyapunov. • If V (x) is a continuously differentiable positive definite function and L(x) is a continuous positive definite function, then the origin of system in (2.40), (2.41) is asymptotically stable in the sense of Lyapunov. See [267], Sect. 3.2, for further connections between passivity and stability. Moreover, assuming that L(x) = 0 in (2.44) we have that V˙ = y  u. Then, choose the control law u = −ρ(y) for some ρ(y) such that y  ρ(y) > 0 for all y = 0, to obtain V˙ = −y  ρ(y) < 0, ∀y = 0. Although this only ensures stability, it also ensures that a set where y ≡ 0 will be reached. Hence, asymptotic stability can be enforced if the closed-loop system (2.40), (2.41), u = −ρ(y), possesses the following property. Definition 2.41 ([29, 204, 267]) The nonlinear system x˙ = f (x), x ∈ Rn , is locally zero-state detectable (ZSD) from the output y if there is a neighborhood N of x = 0 such that for all x ∈ N : y(t) ≡ 0

⇒ lim x(t) = 0. t→∞

Global zero-state detectability is defined analogously by setting N = Rn . Thus, passivity together with zero-state detectability is very useful to design output-feedback controllers. See [204] for instance. We conclude that depending on properties of L(x) we have different stability properties. Thus, it is natural to define different passivity properties. This is stated as follows: Definition 2.42 ((Passivity) [132], pp. 236) The system: x˙ = f (x, u), y = h(x, u),

(2.45) (2.46)

where f : Rn × Rm → Rn is locally Lipschitz, h : Rn × Rm → Rm is continuous, f (0, 0) = 0 and h(0, 0) = 0 is said to be passive if there exists a continuously differentiable positive semidefinite function V (x), called the storage function, such that ∂V f (x, u), ∀(x, u) ∈ Rn × Rm . u  y ≥ V˙ = ∂x

(2.47)

2.3 Passivity

39

Moreover, it is said to be • lossless if V˙ = u  y. • input strictly passive if V˙ ≤ u  y − u  ϕ(u) and u  ϕ(u) > 0 for all u = 0. • output strictly passive if V˙ ≤ u  y − y  ρ(y) and y  ρ(y) > 0 for all y = 0. • very strictly passive if V˙ ≤ u  y − u  ϕ(u) − y  ρ(y) and u  ϕ(u) > 0 for all u = 0, y  ρ(y) > 0 for all y = 0. • strictly passive if V˙ ≤ u  y − ψ(x) for some positive definite function ψ(x). In all cases, the inequality should hold for all (x, u) ∈ Rn × Rm . Finally, the following result defines a connection between L2 −stability and passivity. Lemma 2.43 ([132, 267], pp. 242) If the system (2.45), (2.46) is output strictly passive with u  y ≥ V˙ + δ y  y, for some δ > 0, then it is finite-gain L2 −stable and its L2 gain is less than or equal to 1/δ. As we have explained in the above discussion, passivity does not require a positive definite storage function but just a positive semidefinite storage function. Thus, if the storage function V (x) is restricted to be positive definite, then a direct relationship exists between passivity and global asymptotic stability. This result is exploited along the present book to construct our main contributions as it is explained in the next section.

2.4 A Novel Passivity-Based Approach for Control of Electromechanical Systems In this section, we describe the main features of the novel passivity-based approach that is introduced in this book for control of electromechanical systems. This is done through a simple example: proportional-integral-derivative (PID) position control of a PM brushed DC-motor actuating on a nonlinear position dependent load. The mathematical model of this electromechanical system is presented in Proposition 3.5 and it is given as L

di = −Ri − ke q˙ + v, dt J q¨ = km i − bq˙ − g(q),

(2.48) (2.49)

where it is assumed that g(q), the nonlinear load, possesses the following properties: |g(x) − g(y)| ≤ kg |x − y|, ∀x, y ∈ R, (Lipschitz condition)    dg(q)  ,  kg > max  q∈R dq  dU (q) g(q) = , ∀q ∈ R, dq

(2.50)

40

2 Mathematical Preliminaries

with U (q) a lower bounded twice continuously differentiable scalar function and kg > 0 is a finite constant. Let us first show that the mechanical subsystem in (2.49) is passive. To this aim, consider the following positive semidefinite storage function, which represents the addition of kinetic energy and potential energy: ˙ = V1 (q, q)

1 2 J q˙ + U (q) − U0 , 2

˙ = 0 at q˙ = 0 and q = q0 , where q0 is with U0 = minq∈R U (q). Notice that V1 (q, q) one minimum of U (q). Then ˙ m i − bq˙ − g(q)] + g(q)q˙ = km i q˙ − bq˙ 2 . V˙1 = q[k This means that (2.49) is output strictly passive if we define the input as the applied ˙ See Definition 2.42. torque u 1 = km i and output as the velocity y1 = q. On the other hand, consider the magnetic energy stored in the armature inductance: V2 (i) =

1 2 Li , 2

and compute its time derivative along the trajectories of the electrical subsystem (2.48), i.e., V˙2 = −Ri 2 − ke i q˙ + vi. If we define the input as the applied voltage v and the output as electric current, then the only obstacle to conclude that the electrical subsystem (2.48) is output strictly ˙ However, if we consider (2.48), (2.49) as passive is the presence of the term −ke i q. ˙ i) = V1 (q, q) ˙ + V2 (i), with V1 and a single system with storage function V3 (q, q, V2 defined as above, then V˙3 = −bq˙ 2 − Ri 2 + vi ≤ −Ri 2 + vi,

(2.51)

where terms km i q˙ − ke i q˙ naturally cancel because km = ke as demonstrated in Chap. 3. The above expression proves that system (2.48), (2.49) is output strictly passive if we define the input as the applied voltage u 3 = v and the output as electric current y3 = i. Remark 2.44 The cancellation of terms km i q˙ − ke i q˙ represents the energy exchange that exists between the mechanical and electrical subsystems of a brushed DC-motor and it is exploited along the whole book. These terms are simple in a brushed DCmotor. However, in AC-motors terms that are involved in this cancellation are complex and nonlinear and, hence, its natural cancellation results in an important simplification of control law. Other approaches [204] are forced to exactly cancel such

2.4 A Novel Passivity-Based Approach for Control of Electromechanical Systems

41

terms via feedback, i.e., rendering more complex the control law. Another cancellation is that of terms −qg(q) ˙ + g(q)q˙ which represents the exchange of kinetic and potential energies in the mechanical subsystem. Notice that cancellation of cross terms is common in passivity-based approaches and this is the reason why, despite its complexity, the system under study admits the simple representation V˙ ≤ uy that defines a passive system. Closed-loop control design begins by defining a desired electric current i ∗ which, when reached, ensures that a desired torque km i ∗ is applied to the mechanical part of the system. This task can be better understood by adding and subtracting km i ∗ in (2.49) to find J q¨ = km i˜ + km i ∗ − bq˙ − g(q),

(2.52)

where i˜ = i − i ∗ . Let us propose 1 [−k p h(q) ˜ − kd q˙ − ki σ(z)], km  t h(q(r ˜ ))dr, z=

i∗ =

(2.53)

0

where q˜ = q − q ∗ with q ∗ the desired position, and h(·), σ(·), are some saturation functions to be defined. Replacing this in (2.52) we find ˜ − kd q˙ − ki s(z) − bq˙ − g(q) + g(q ∗ ), J q¨ = km i˜ − k p h(q) 1 s(z) = σ(z) + g(q ∗ ). ki

(2.54)

This means that the desired torque km i ∗ = −k p h(q) ˜ − kd q˙ − ki σ(z) is defined as a nonlinear PID position controller which appears applied to the mechanical subsystem (2.52) if i˜ → 0. Exploiting the previous experience with the storage function ˙ = 21 J q˙ 2 + U (q) − U0 , we may propose a storage function containing the V1 (q, q) terms: V4 (q, ˜ q, ˙ z) =

1 2 J q˙ + U (q) − U (q ∗ ) + β1 J h(q) ˜ q˙ + β2 J s(z)q˙ + V5 , (2.55) 2

where V5 represents some additional terms which are unknown up to now. ˜ q˙ and β2 J s(z)q, ˙ β1 > 0, β2 > 0, The differentiation of terms 21 J q˙ 2 , β1 J h(q) 2 2 2 ˜ will introduce terms −kd q˙ , −β1 k p h (q), ˜ −β2 ki s (z), and km i q, ˙ in V˙4 . This represents a suitable exploitation of the open-loop output strictly passivity property of the mechanical subsystem (2.49) discussed above. On the other hand, let us propose the applied voltage as a classical proportionalintegral (PI) electric current controller:

42

2 Mathematical Preliminaries

v = −α p i˜ − αi



t

˜ )dr. i(r

(2.56)

0 ∗

Replace this in (2.48) and add and subtract Ri ∗ , L didt to find L

d i˜ = −(α p + R)i˜ − αi dt



t 0

∗

˜ )dr − Ri ∗ − ke q˙ − L di . i(r dt

Finally, replacing i ∗ form (2.53) yields L

Rk p Rkd Rki di ∗ d i˜ = −(α p + R)i˜ − αi z i + , h(q) ˜ + q˙ + s(z) − ke q˙ − L dt km km km dt  t  ˜ )dr + R g(q ∗ ) . z i = −αi i(r km αi 0

˜ where V7 (i) ˜ = 1 L i˜2 + Defining the storage function V6 = V4 (q, ˜ q, ˙ z) + V7 (i), 2 we find V˙6 = V˙4 − (α p + R)i˜2 +

Rk p Rkd ˜ Rki di ∗ . h(q) ˜ i˜ + q˙ i + s(z)i˜ − ke q˙ i˜ − i˜ L km km km dt

αi 2 z 2 i

(2.57)

The presence of term −(α p + R)i˜2 in this expression is a direct consequence of the output strictly passivity property of the brushed DC-motor that is shown in (2.51). ˜ in (2.57), and km i˜q, ˙ in V˙4 , cancel, which is a direct conNotice that terms −ke q˙ i, sequence of the open-loop property described in Remark 2.44. Also notice that the ˜ −β2 ki s 2 (z), in V˙4 , described above, together presence of terms −kd q˙ 2 , −β1 k p h 2 (q), Rk ˜ Rkd q˙ i, ˜ ˜ i, with −(α p + R)i˜2 in (2.57) enables us to dominate the cross terms kmp h(q) km Rki ˜ in (2.57) by using a large α p > 0. Finally, notice that differentiating (2.53) s(z)i, km we have that − i˜ L

Lk p dh(q) Lkd ˜ Lki dσ(z) di ∗ = i˜ q˙ + i˜ h(q). ˜ q¨ + i˜ dt km d q˜ km km dz

(2.58)

If we define h(·) and σ(·) as the saturation functions introduced in Definition 2.34 with the additional requirement: 0
0 is employed. In this respect, let us say that it is shown in [130] that ˜ ∀q˜ ∈ R, | − g(q) + g(q ∗ )| ≤ c1 |h(q)|,

2.4 A Novel Passivity-Based Approach for Control of Electromechanical Systems

43

for some c1 > 0. Function V5 , introduced in (2.55) is to be designed to ensure positive definiteness and radial unboundedness of the storage function V6 . Another conse˜ −β2 ki s 2 (z) present in V˙4 , and −(α p + R)i˜2 , quence of terms −kd q˙ 2 , −β1 k p h 2 (q), present in (2.57), is that the LaSalle invariance principle (see Corollary 2.19), instead of zero-state detectability, can be invoked to ensure global asymptotic stability when the closed-loop system is autonomous. In this section, we have just tried to highlight the rationale behind our novel passivity-based approach and the particular details have been avoided. We refer to Chap. 3 for the complete procedure when this novel approach is applied to PM brushed DC-motors. In brushed DC-motors, it is not necessary that the definition of i ∗ , given in (2.53), includes the nonlinear functions h(·) and σ(z). However, this trick is very useful for AC-motors where the torque error is given by a nonlinear function of the state. This allows to dominate several third-order terms by second-order terms. This is because such third-order terms are given as the product of three functions being either h(·) or σ(z) one of them. Control law is given by (2.56), (2.53), and represents a double loop controller. The internal loop is driven by a classical PI electric current controller. The external loop is driven by a nonlinear PID position controller for AC-motors which, as described in the previous paragraph, can be designed as a classical PID controller for brushed DC-motors. Thus, our approach renders possible the presentation of a formal stability proof for a controller that has several similarities with the standard control scheme that is commonly employed in industrial practice. The aim of our work is to achieve this goal for several nonlinear electromechanical systems. In this respect, it is important to stress that this has also been the goal of several previous works in the literature [204, 210] but the controllers that they propose have still significant differences with industrial practice. This justifies the merit of our proposal. It is also interesting to highlight the fact that, according to the above arguments, the proportional constant α p is proposed to be large in order to dominate several cross terms. This idea is consistent with current practice, where the PI electric current gains are chosen large in order to render the convergence i˜ → 0 fast enough to produce satisfactory torque tracking. Remark 2.45 In standard passivity-based control approaches [204], a voltage input is designed as di ∗ . (2.59) v = −α p i˜ + Ri ∗ + ke q˙ + L dt This allows to write (2.48) as L

d i˜ ˜ = −(α p + R)i, dt

ensuring that i˜ → 0 exponentially with a rate that can be arbitrarily assigned using some α p > 0. Then, the designer can focus attention in stabilizing the closed-loop mechanical dynamics in (2.54) by considering the torque error km i˜ as a vanishing

44

2 Mathematical Preliminaries

disturbance. However, according to (2.59), proceeding like this has several drawbacks: • The back electromotive force term ke q˙ must be exactly cancelled. This is equivalent to use feedback to render passive the electric dynamics, idea that was introduced in [35] for nonlinear systems. In induction motors, the back electromotive force term depends on the unmeasurable rotor flux and, hence, such a term cannot be exactly cancelled via feedback. Because of that, in [204], Proposition 10.6, it feedback’s the desired value of the back electromotive force term. This allows to construct a term depending on the rotor flux error which converges exponentially to zero with a rate (that cannot be modified) depending on the natural time constant of the rotor flux dynamics. Although in [204] is stated that the design procedure introduced there can be applied to several classes of electric motors, an explicit controller is only presented for induction motors. Thus, reviewing that procedure we conclude that in the case of PM brushed DC-motors feedback cancellation of the back electromotive force term is unavoidable to force i˜ → 0 exponentially. Also see Remark 4.11 in the present book for PM synchronous motors. Notice that, as explained above, in the novel approach that we propose this term naturally cancels without online computing it. We achieve this by taking advantage from the passivity property that arises when working together the mechanical and the electrical subsystems. See Remark 2.44. Although this term is simple in brushed DC-motors, it is complex in most AC-motors which renders more complex the control law. Moreover, this requires the exact knowledge of some motor parameters. • The time derivative of the desired current must be online computed which implies further additional computations. Although this is relaxed in [204] because i ∗ designed in that work does not involve velocity measurements (i.e., computing di ∗ does not require to know acceleration), this, however, has represented for dt them an obstacle to design simple PID position regulators. This is because outputfeedback PID control designs impose restrictive conditions on the controller gains which results in important performance deterioration. See [172, 203, 252, 253], ∗ for instance. Finally, let us say that computing L didt has the additional requirement to exactly know the armature inductance L. • Contrary to the passivity-based approach that we propose, an integral action on the electric current error is not included and this forces term Ri ∗ in (2.59). The problem with this term is that armature resistance R is not exactly known because its value changes under normal operation conditions. Also contrary to our proposal, the integral action on the electric current error is not included in [204] because the resulting cross terms, see (2.57), complicate the formal procedure to show that i˜ → 0 exponentially. Although it is shown in [204] that such integral term can be included for the case of induction motors, term Ri ∗ is still employed and, hence, the integral term has noting to compensate. Remark 2.46 The approach employed in [204], which is described in Remark 2.45, is know as nested-loop passivity-based control. This approach was preferred in [204]

2.4 A Novel Passivity-Based Approach for Control of Electromechanical Systems

45

over an approach similar to that proposed in the present section, that they called passivity-based control with total energy shaping , by arguing that the approach in Remark 2.45 results in simpler control laws. However, as we have just explained, our approach does result in simpler control laws than those in [204] and we show this in the present monograph. Remark 2.47 In passivity-based approaches that are common in the literature, it is usual to separate the control law into two parts. One part is intended to shape the potential energy such that this has a unique minimum at the desired equilibrium point. The second part is intended to introduce suitable damping and, thus, ensuring global asymptotic stability. Such a control law separation is well suited when designing proportional-derivative (PD)-like controllers, i.e., when controllers provided with integral actions are not employed. Moreover, in order to render robust the closedloop system with respect to constant external disturbances, disturbance observers are preferred in [204] instead of PI or PID controllers. The passivity-based approach described in the present section does employ PID position controllers (or PI velocity controllers). This is the reason why we do not focus our attention on separating the control law in energy shaping and damping injection.

2.5 The Electromechanical Systems that Are Studied in This Book As stated above, the aim of this book is to present, for several nonlinear electromechanical systems, formal stability proofs for controllers that have several similarities with the standard control schemes that are commonly employed in industrial practice. The common thread among these results is the novel passivity-based approach described in the previous section. These results are organized along the subsequent chapters as described in the following. Most standard industrial control schemes for nonlinear electromechanical systems are based on obtaining a description of the plant that allows to control it as it was a PM brushed DC-motor. Thus, in Chap. 3 we study PM brushed DC-motor: how it works and how it is modeled and controlled. In Chap. 4, PM synchronous motors are studied. It is important to stress that all of the controllers that we propose in this chapter are valid for both salient and round rotor motors. Hence, contrary to some proposals in the literature, considering the salient rotor case does not result in a complex controller (see [65], for instance). The induction motor is studied in Chap. 5 where our main proposals only differ from SIFOC in a single nonlinear term in the part of the controller devoted to the electrical dynamics. Thus, our controller is very similar to SIFOC but provided with a global convergence proof. Chapter 6 is devoted to switched reluctance motor (SRM). Proposing controllers for this class of motors provided with formal stability proofs has demonstrated to be a difficult task. As a matter of fact, aside from the authors of this book, only one

46

2 Mathematical Preliminaries

group of researchers has been working in the subject in recent years [57, 69, 168– 171]. Their proposals are based on the passivity-based approach exploited in [204]. However, one important drawback of this approach has been pointed out in [92]. This refers to the fact that a singularity appears when the desired torque crosses by zero, a situation that is common when a change in the motor sense of rotation is commanded. In this respect, in Chap. 10, in the present book, is shown that magnetic levitation systems represent the one phase and one pole case of SRMs. It is interesting to recall that in [204], Ch. 8, Remark 8.5, it is pointed out that when the passivity-based approach introduced in [204] is applied to a magnetic levitation system, a singularity exists when the desired force to be exerted by the electromagnet crosses by zero. Our proposals avoid such a singularity and, in addition, solve the position regulation problem when taking into account the magnetic saturation that is common to appear in normal operation conditions of SRM. The synchronous reluctance motor (SYRM) is studied in Chap. 7. We include this class of AC-motors because, although it is a somewhat unknown motor, it works on the basis of a principle that is a combination of the working principles of synchronous motors and reluctance motors. We solve both velocity and position regulation problems. Chapter 8 is devoted to study PM stepper motor. In particular, we study bipolar PM stepper motors. Our proposals rely on an industrial strategy know as commutation which does not require a dq transformation and the controller is designed on the basis of the actual motor coordinates. Designing a controller with these features is the reason to include the study of PM stepper motors. Brushless DC (BLDC) motor is studied in Chap. 9. This class of motors is controlled in practice on the basis of intuitive ideas, and formally designed controllers are sparse in the literature. The motivation to include this chapter has been the recent formal work [80] where a passivity-based approach is employed. Thus, we want to show that our novel passivity-based approach can also improve the existing results in the literature on BLDC motors. In Chap. 10, we apply our novel passivity-based approach to magnetic levitation systems and microelectromechanical systems. The aim is to show that our novel passivity-based approach is not only intended for AC-motors but also, in general, for electromechanical systems. In Chap. 11 we present an application of our approach to the case of trajectory tracking control in robot manipulators when taking into account the electrical dynamics of the actuators, in particular PM synchronous motors. Our novel approach is applied to study the performance of a robot controller previously proposed in the literature under the assumption that the motors electrical dynamics can be neglected (see [159, 240], for instance). However, we do consider the electrical dynamics of the motors in our study and this allows us to explain some performance deterioration that has been reported in the previous literature. We found that the traditional explanation for such a performance deterioration is incorrect: the nonzero tracking error is usually attributed to bad high velocity computation via a backward difference algorithm and wrong friction compensation (see [230], for instance). We found, instead, that this problem is produced by the fact that a different from zero tracking error exists between the generated torque and the desired torque. Moreover, such a difference in torques is impossible to eliminate if simple internal PI electric current

2.5 The Electromechanical Systems that Are Studied in This Book

47

controllers are employed when trying to solve the position trajectory tracking task. This is an interesting result that has not been presented in the literature and justifies our efforts to study control of mechanical systems when taking into account the electrical dynamics of the actuators. It is important to stress this fact since such studies are commonly underestimated by the control community. Moreover, this class of results underline the advantages of our novel passivity-based approach. Finally, in Chap. 12, we present another application to robot control of the approach that we propose: PID position regulation in robot manipulators actuated by switched reluctance motors. We assume that all of the switched reluctance motors have saturated magnetic fluxes and hysteresis electric current controllers are employed as usual in current practice [64]. Thus, contrary to previous works in the literature, we succeed to control simultaneously the position of n switched reluctance motors, under saturated flux conditions, when actuating a complex, highly coupled, nonlinear mechanical load. Solving this class of control problems is another advantage of our novel passivity-based approach.

Chapter 3

Permanent Magnet Brushed DC-Motor

Electric motors are mainly composed of two parts: the stator and the rotor. The former is the part of motor that remains at rest, whereas motor works and the latter is the part of motor that rotates when motor works. In brushed DC-motors the stator contains a winding (also know as the “field”) that produces a magnetic field which has to react to the magnetic field that produces another winding that is located on the rotor (also know as the “armature”). The manner these windings are fed defines several classes of brushed DC-motors: the series motor has series connected to the field and the armature windings and the shunt motor has parallel connected to the field and the armature windings. The permanent magnet (PM) brushed DC-motor only has one winding: that is of the armature on rotor, whereas the field winding (on the stator) is replaced by a permanent magnet. This is the class of brushed DC-motor that we study in this chapter. PM brushed DC-motors can be found in a wide range of applications. The largest market segment for this type of motor is found in the low-power range. Moreover, new developments in the area of high-energy permanent magnet material offer the opportunity of miniaturization and promise a cost-effective design [88]. Thus, study of PM brushed DC-motors in this chapter is not only motivated by its simplicity in a first approach to a novel control strategy but also by its practical applications.

3.1 Motor Modeling 3.1.1 A Simple Methodology In Fig. 3.1, a square loop is shown, with area A = l 2 , which is placed inside a magnetic ¯ An external voltage v forces an electric current i to flow through the loop. field B. Symbols L and R represent the loop inductance and resistance. Force produced by the magnetic field on the left side of the loop is given as [10], Chap. 24B: © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 V. M. Hernández-Guzmán et al., Energy-Based Control of Electromechanical Systems, Advances in Industrial Control, https://doi.org/10.1007/978-3-030-58786-4_3

49

50

3 Permanent Magnet Brushed DC-Motor

Fig. 3.1 PM brushed DC-motor

¯ F¯ = il iˆ × B.

(3.1)

A bar is used over variables to indicate that they represent vectors. iˆ is an unit vector having the same direction as the electric current. Symbol “×” stands for the standard vector product. Hence, F¯ is applied as shown in Fig. 3.1b. This means that a torque: τ¯ =

l l ¯ rˆ × F¯ + (−ˆr ) × (− F) 2 2

(3.2)

appears around the loop axis (notice that a force − F¯ is applied on the right side of the loop and rˆ is an unit vector applied from the center to the left side of the loop). The reader can verify that, according to (3.1), no torque is produced by electric current in the front and the rear sides of the loop. It is clear that the loop rotates with an angular velocity ω = −γ, ˙ i.e., in the opposite direction defined by γ in Fig. 3.3b (γ is the angle between the magnetic field B¯ and a line orthogonal to the loop area). According to Faraday’s Law [274], Chap. 17, this rotation induces a voltage vb at the loop terminals which is given as

3.1 Motor Modeling

51

vb =

dλ , dt

where λ is the flux linkage, i.e., λ = B A cos(γ).

(3.3)

Using these expressions it is possible to write vb = −B Aγ˙ sin(γ) = B Aω sin(γ). Notice that, according to Lenz’s Law [274], Chap. 17, polarity of the induced voltage vb (see Fig. 3.1a) is such that it opposes to the electric current flow. Now, assume that several loops with a common axis are placed at different angles γ. Also assume that this set of loops rotates together (because of the applied torque τ¯ ) and that a mechanical switch automatically connects only the loop for which γ = 90◦ , i.e., electric current i only flows through this loop. Then, the induced voltage is given as vb = B Aω

(3.4)

and, according to (3.1), (3.2), the magnitude of torque on the loop is given as τ = il 2 B = i AB,

(3.5)

which is applied in the same direction as velocity ω. The working principle of a permanent magnet (PM) brushed DC-motor is what we have just described. In a more elaborate description of a PM brushed DC-motor it is common to write (3.4), (3.5) as vb = ke ω,

(3.6)

τ = km i,

(3.7)

where ke = AB > 0 and km = AB > 0 are known, respectively, as the motor back electromotive force constant and the motor torque constant. These constants are introduced in order to take into account several issues such as the facts that all of the loops are wound on a piece of iron and that what we have considered here as a single loop in practice is composed by several turns of wire. According to the Energy Conservation Law, Pe = Pm where Pe is the electrical power which converts into the mechanical power Pm . Hence, using (3.6) and (3.7): Pe = vb i = ke ω which means that

1 τ = τ ω = Pm , km

52

3 Permanent Magnet Brushed DC-Motor

ke = km ,

(3.8)

in any PM brushed DC-motor. This can be verified in commercial motor datasheets by consulting the values given for these parameters in the International System of Units (SI). Assuming that all of the loops are wound on a piece of iron and J represents the total inertia of the rotor, the Newton’s Second Law can be used to find J ω˙ = τ − bω − τ L ,

(3.9)

where b is a positive constant scalar representing the viscous friction coefficient, τ is the generated torque, and τ L is the load torque which represents torque exerted by an external agent (another body, for instance) which opposes to rotor movement. Note that the generated torque τ appears affected by a “+” whereas both, torque due to friction bω and the load torque τ L , appear affected by a “−”. This is because τ is applied in the same direction as the rotor velocity ω whereas friction and load torque oppose to the rotor movement. On the other hand, using the Kirchhoff’s Voltage Law in the circuit shown in Fig. 3.1a it is obtained the following: L

di + Ri + vb = v. dt

(3.10)

Since τ and vb are given by (3.7) and (3.6), respectively, the final form of the dynamical model of a PM brushed DC-motor is, according to (3.9) and (3.10), the following: L

di = −Ri − ke ω + v, dt J ω˙ = km i − bω − τ L .

(3.11) (3.12)

Remark 3.1 Another manner to explain torque generation is thinking that torque is produced by the interaction between two magnetic fields: one magnetic field is due to the permanent magnets fixed to stator and the other magnetic field is produced by the electric current flowing through loops of wire wound around rotor. According to the right-hand rule and the disposition of pieces shown in Fig. 3.1, the magnetic field produced by the electric current flowing around the rotor has an equivalent north pole (N) located at the top of the loop in Fig. 3.1b (the S pole is located at the bottom of this loop). Further, according to (3.1), (3.2), and (3.5), a maximum torque is produced under conditions in Fig. 3.1. This means that a maximum torque is produced by the interaction between two magnetic fields when the directions of their fluxes are orthogonal (also see [55], Chap. 5, and [87], Chap. 33). Notice that this torque tries to align the magnetic fluxes of both magnetic fields, i.e., such that flux due to electric current through the rotor winding (from its S pole to its N pole) be in the same direction than flux due to the permanent magnet at stator in Fig. 3.1b

3.1 Motor Modeling

53

(from pole N to pole S). A zero torque is produced when poles N (S) of stator and S (N) of rotor are in front of each other. Remark 3.2 According to (3.7), the generated torque in a PM brushed DC-motor is given as the product of a constant km and electric current i through loops wound around the rotor. Moreover, according to the above exposition, the motor torque constant km summarizes the contribution to torque generation of (the constant magnetic field due to) the permanent magnets fixed to stator. Thus, a desired torque τ ∗ can be ensured to be generated just by forcing the electric current to reach the desired value i ∗ = τ ∗ /km .

3.1.2 A General Methodology The dynamic model in (3.11), (3.12) can also be obtained using a more general methodology which will result useful when modeling more complex electric motors. As it is shown in Fig. 3.1 a PM brushed DC-motor is basically constituted by a series RL circuit placed in a magnetic field produced by permanent magnets. Thus, use of the Kirchhoff’s Voltage Law and Faraday’s Law allows to write ψ˙ + Ri = v, ψ = Li + λ, with λ given in (3.3) and ψ is the flux linkage due to both the electric current through the armature circuits and the stator permanent magnets. Hence, last expression is equivalent to L

di − γ˙ B A sin(γ) + Ri = v. dt

Using ω = −γ, ˙ γ = 90◦ and defining ke = B A, as before, we retrieve (3.11). On the other hand, according to D’Alembert’s Principle [55], pp. 399, the generated torque (applied on the rotor in the sense of rotor velocity ω) is given as the co-energy’s derivative with respect to rotor position q = −γ, i.e., q˙ = ω:   ∂ 1 2 Li + iλ , ∂q 2 = i B A sin(γ),

τ =

since both L and i are independent of the rotor position. Using again γ = 90◦ we retrieve (3.5) or, equivalently, (3.7). Finally, according to the Newton’s Second Law we retrieve (3.12).

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3 Permanent Magnet Brushed DC-Motor

3.2 Standard Control It is clear that the PM brushed DC-motor model given in (3.11), (3.12) is linear. Hence, we can apply Laplace transform to write L(s I (s) − i(0)) = −R I (s) − ke ω(s) + V (s), J (sω(s) − ω(0)) = km I (s) − bω(s) − τ L (s), where I (s), ω(s), V (s), τ L (s), are the Laplace transforms of i, ω, v, τ L , whereas i(0), ω(0), represent the initial conditions of the functions of time i and ω. From the above expressions we obtain L 1 i(0), (V (s) − ke ω(s)) + Ls + R Ls + R 1 J ω(0), ω(s) = (km I (s) − τ L (s)) + Js + b Js + b I (s) =

and finally 1 (V (s) − ke ω(s)) , Ls + R 1 ω(s) = (km I (s) − τ L (s)) , Js + b I (s) =

(3.13) (3.14)

when assuming all of the initial conditions to be zero. In Fig. 3.2a, we present the corresponding block diagram. Notice that the electrical subsystem, given in (3.13), is placed between the applied voltage V (s) and the generated torque, τ (s) = km I (s). Moreover, under certain conditions the electrical subsystem may have an deleterious effect on the generated torque. Since the generated torque is the input signal for the mechanical subsystem in (3.14), which determines the evolution in time of velocity, it is desirable to design a methodology capable to reduce the effects of the electrical subsystem on the generated torque. This motivates the proportional-integral (PI) electric current loop shown in Fig. 3.2b. The task to be solved by this electric current controller is to force the actual electric current i flowing through the armature circuit to reach the commanded electric current i ∗ . This must be accomplished as fast as desired no matter the parameters of the electrical subsystem in (3.13). Thus, the PI electric current controller gains must be selected suitably. If i → i ∗ is accomplished fast, then we have that τ = km i → km i ∗ = τ ∗ , where ∗ τ is the commanded torque. Hence, it is desirable to design τ ∗ (s) as either a classical PI velocity controller or a classical proportional-integral-derivative (PID) position controller, i.e., to obtain the closed-loop block diagrams in Fig. 3.2c and d. Thus, the parameter or gains of the PI velocity controller and the PID position controller must be selected suitably to control the mechanical subsystem. First consider the case of a PI velocity controller.

3.2 Standard Control

55

Fig. 3.2 Block diagrams for velocity and position control of a PM brushed DC-motor

Using some block algebra in Fig. 3.2c, we obtain the block diagrams in Fig. 3.3a and b. The root locus diagram for the internal loop in the block diagram in Fig. 3.3b is depicted in Fig. 3.4a. From this result, we can construct the root locus diagram corresponding to the external loop in the block diagram in Fig. 3.3b. This root locus diagram is shown in Fig. 3.4b. Notice that the resulting closed-loop poles, at −s6 , −s7 , and the closed-loop zero, at − kkpiii , can be located far to the left such that their effects on the transient response can be neglected. Notice that this can be achieved by using large values for both k pi and kii , i.e., the PI electric current controller gains. Hence, we can assume that the transient response is determined only by the closed-loop poles at −s8 , −s9 and the closed-loop zero at − kkpi . Notice that both poles at −s8 and

56

3 Permanent Magnet Brushed DC-Motor

Fig. 3.3 Block diagrams for velocity control of a PM brushed DC-motor

−s9 as well as the zero at − kkpi can be suitably placed using the PI velocity controller gains k p and ki . Thus, let us analyze the time response obtained by suitably locating the openloop poles at s = 0 and s = −s3 as well as the zero at s = − kkpi , i.e., by considering that τ = τ ∗ as in the block diagram in Fig. 3.5. This analysis is verified through simulations using the MATLAB/Simulink diagram in Fig. 3.6 and the following motor parameters: km = ke = 0.573 (Nm/A), R = 2.3 (Ohm), J = 12.43 × 10−4 (kg m2 ), L = 9.6 × 10−3 (H). These parameters correspond to the PM brushed DCmotor model MT-4060 ALYBE from Baldor Electric Company [15]. We have also used b = 0.05 (Nm/(rad/s)), ω ∗ = 2000(2π/60) (rad/s), τ L = t p = 15 (Nm), k pi = 10, kii = 10000.

3.2.1 Case 1 Suppose that the zero at s = − kkpi is located between the open-loop poles at s = 0 and s = −s3 , as it is shown in Fig. 3.7a. The closed-loop transfer function (when k p (s+ki /k p ) τ L (s) = 0) is given as ωω(s) ∗ (s) = G ∗ (s) = J (s+s )(s+s ) . Notice that the slow closed10 11 loop pole at −s10 is approximately cancelled by zero at − kkpi . Hence, the traditional ideas in classical control assume that the effects of this pole and zero on the transient response are not appreciated and, thus, the transient response to a desired reference is determined only by the fast closed-loop pole at s = −s11 . This is a good feature of this approach. On the other hand, the response to a torque disturbance is given = G τ (s) = J (s+s10−s)(s+s11 ) . Notice that the slow closedby the transfer function τω(s) L (s)

3.2 Standard Control

57

Fig. 3.4 Root locus diagrams

Fig. 3.5 Block diagram for velocity control of a PM brushed DC-motor

loop zero at −s10 does not cancel this time and, hence, it has an important effect on the transient response, i.e., the effects of the torque disturbance are only slowly compensated. This is an important drawback of this approach. In order to verify these ideas, we present some numerical results in Figs. 3.7 and 3.8a using the parameters of the PM brushed DC-motor model MT-4060 ALYBE. The root locus diagram corresponding to the external loop in the block diagram of Fig. 3.3b is presented in Fig. 3.7b. We can see at the left the two poles and zero corresponding to the fast electrical dynamics and the PI electric current controller. At the right we observe the two poles and zero of the mechanical dynamics and the

58

3 Permanent Magnet Brushed DC-Motor

Fig. 3.6 Simulink diagram used to obtain the responses in Fig. 3.8

Fig. 3.7 Root locus diagram for Case 1

3.2 Standard Control

59

Fig. 3.8 Time responses with a classical PI velocity controller. Dashed: desired velocity response. Continuous: actual velocity response

PI velocity controller. A zoom-in on the latter poles and zero is presented in Fig. 3.7c. The lines conforming the root locus are interrupted at the points corresponding to the actual closed-loop poles determined by the controller gains that we employ, i.e., k p = 0.13 and ki = 0.5229. Notice that it is observed the same configuration as in root locus of Fig. 3.7a. The corresponding time response is presented in Fig. 3.8a using a continuous a with line. The dashed line represents the time response of the transfer function s+a a = 139.1, which represents the closed-loop pole at the left of Fig. 3.7c. Although both responses are very similar at the beginning, the actual velocity becomes much slower as time grows. This is because the effect of the slow closed-loop pole, which was assumed above to be negligible because of the approximate cancellation with the zero of the closed-loop transfer function is in fact not negligible. It is important to underline this kind of incorrect assumptions in classical control. We refer the reader to [8] to consult the first formal work stressing these problems. On the other hand, it is also observed the slow rejection of a step torque disturbance appearing at t = 1.5 (s). This slow response is due, again, to the slow closed-loop pole located between

60

3 Permanent Magnet Brushed DC-Motor

Fig. 3.9 Root locus diagrams for Case 2

the zero and the pole, at s = 0, which appear at the right of Fig. 3.7c. This was also pointed out in the above discussion.

3.2.2 Case 2 Suppose that the zero at s = − kkpi is located at the left of the open-loop poles at s = −s3 , as it is shown in Fig. 3.9a. If −s8 and −s9 are chosen as the closed-loop poles, k p (s+ki /k p ) ω(s) −s i.e., ωω(s) ∗ (s) = G ∗ (s) = J (s+s )(s+s ) and τ (s) = G τ (s) = J (s+s )(s+s ) , the closed-loop 8 9 L 8 9 transient response to both, a desired velocity reference and an external torque disturbance, may be chosen to be fast. This is a good feature of this approach. However, the zero at − kkpi has an important effect modifying the transient response expected when locating the closed-loop poles at −s8 and −s9 . This is a drawback of this approach. In order to verify these ideas, we present some numerical results in Figs. 3.9b and 3.8b using the parameters of the PM brushed DC-motor model MT-4060 ALYBE. The root locus diagram corresponding to the external loop in the block diagram of Fig. 3.3b is presented in Fig. 3.9b. We can see at the left the two poles and zero corresponding to the fast electrical dynamics and the PI electric current controller. At the right we observe the two poles and zero of the mechanical dynamics and the PI velocity controller. The lines conforming the root locus are interrupted at the

3.2 Standard Control

61

Fig. 3.10 Root locus diagram for Case 3

points representing the actual closed-loop poles determined by the controller gains that we employ, i.e., k p = 0.3 and ki = 36.2027. Notice that it is observed the same configuration as in root locus of Fig. 3.9a when −s8 and −s9 are the closed-loop poles. The corresponding time response is presented in Fig. 3.8b using a continuous line. The dashed line represents the time response of the transfer function 2.883×104 , which is determined by the complex conjugate closed(s+139.5+96.8 j)(s+139.5−96.8 j) loop poles at the right Fig. 3.9b. Notice that rejection of the torque disturbance appearing at t = 0.1 (s) is much faster than in Case 1 and the transient response to a desired velocity reference is also fast. However, a clear difference exists between the expected response (dashed line) and the actual response (continuous line). As explained above, this difference is due to the effects of zero at −120.6 in Fig. 3.9b.

3.2.3 Case 3 Consider Fig. 3.9a. If −s12 and −s13 are chosen as the closed-loop poles, i.e., k p (s+ki /k p ) J (s+s12 )(s+s13 )

ω(s) τ L (s)

−s , J (s+s12 )(s+s13 )

ω(s) ω ∗ (s)

=

G ∗ (s) = and = G τ (s) = the closed-loop transient response to both, a desired velocity reference and an external torque disturbance, may be chosen to be fast again. This is a good feature of this approach again. However, it is demonstrated in [79] that since the two closed-loop poles are located at the left of the zero at − kkpi the transient response to a desired velocity reference will present overshoot. Notice that this is not a desired feature since both closed-loop poles have been chosen to be real, i.e., at −s12 and −s13 . This is a drawback of this approach. In order to verify these ideas, we present some numerical results in Figs. 3.10 and 3.8c using the parameters of the PM brushed DC-motor model MT-4060 ALYBE. The controller gains that we have employed are k p = 0.55 and ki = 66.3. Notice that the root locus diagram in Fig. 3.10 has the same configuration as in root locus of Fig. 3.9a when −s12 and −s13 are the closed-loop poles.

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3 Permanent Magnet Brushed DC-Motor

The corresponding time response is presented in Fig. 3.8c using a continuous a , with line. The dashed line represents the time response of the transfer function s+a a = 276.1 which represent the real closed-loop pole at the left of Fig. 3.10. Notice that rejection of the torque disturbance appearing at t = 0.1 (s) is much faster than in Case 1 and slightly faster than in Case 2. This is because the slowest pole is faster for Case 3 than for Case 1 and slightly faster than for Case 2. On the other hand, although the transient response to a desired velocity reference is also fast, a clear difference exists between the expected response (dashed line) and the actual response (continuous line). Furthermore, the actual response presents an overshoot which is not expected since the two slowest closed-loop poles are real. As we have said above, this phenomenon is explained in [79] because all of the closed-loop poles are located at the left of the zero at −120.6 in Fig. 3.10.

3.2.4 Case 4 s14 If it is chosen kkpi = s3 , as in Fig. 3.11a, then ωω(s) and τω(s) = ∗ (s) = G ∗ (s) = s+s 14 L (s) −s G τ (s) = J (s+s14 )(s+b/J ) . In this case the closed-loop transient response to a desired velocity reference may be chosen to be first order and fast. This is a good feature of this approach. However, the transient response to a torque disturbance is dominated by the plant’s slow pole at −b/J , i.e., the effects of an external torque disturbance can only be compensated slowly. This is a drawback of this approach. In order to verify these ideas, we present some numerical results in Figs. 3.11b and 3.8d using the parameters of the PM brushed DC-motor model MT-4060 ALYBE. The controller gains that we have employed are k p = 0.175 and ki = 6.84. Notice that the root locus diagram in Fig. 3.11b has the same configuration as in root locus of Fig. 3.11a. The corresponding time response is presented in Fig. 3.8d using a continuous a , with line. The dashed line represents the time response of the transfer function s+a a = 138.5 which represent the real closed-loop pole at the left of Fig. 3.11b. Notice that the actual transient response to a desired velocity reference (continuous line) is fast and identical to the expected response (dashed line). However, rejection of the torque disturbance appearing at t = 0.1 (s) is slower than in Cases 2 and 3. As explained above, this is because the slowest pole, at −b/J , of the transfer function ω(s) = G τ (s) = J (s+s14−s depends exclusively on the plant and it cannot be τ L (s) )(s+b/J ) modified. Thus, we conclude that it is not possible to design an exact tuning rule for a classical PI velocity controller such that the performances achieved, when both a desired velocity reference and a torque disturbance are applied, they can be rendered arbitrarily as good as desired. We stress that, according to the above discussion, this problem cannot be attributed to an inadequate design but it is a fundamental limitation of classical PI velocity control.

3.2 Standard Control

63

Fig. 3.11 Root locus diagrams for Case 4

3.2.5 An Improved PI Velocity Controller In this section, we design an improved PI velocity controller to solve the problems of classical PI velocity control that have been described above. Consider the mechanical subsystem, (3.12) or (3.14), alone J ω˙ = τ − bω − τ L ,

(3.15)

where τ = km i, i.e., when assuming that the PI electric current controller has been designed to render fast the electrical dynamics, as explained above. Consider the following controller: ∗

∗



t

τ = τ = k p (ω − ω) + ki

(ω ∗ − ω(r ))dr + bω ∗ + k1 (ω(0) − ω ∗ ), (3.16)

0

where ω(0) is the initial velocity and ω ∗ is the constant desired velocity. Assume that the torque disturbance is constant. i.e., τ L = t p =constant.

64

3 Permanent Magnet Brushed DC-Motor

Replacing (3.16) in (3.15) and rearranging b + kp ki (ω − ω ∗ ) + ω˙ + J J



t

(ω(r ) − ω ∗ )dr =

0

k1 1 (ω(0) − ω ∗ ) − τ L . J J

Notice that this can be written as ω˙ +

b + k p J

(ω − ω ∗ ) +



ki J +

t

(ω(r ) − ω ∗ )dr

0

1 k1 [ω − ω ∗ − (ω(0) − ω ∗ )] = − τ L , J J

if we define k p = k p + k1 . On the other hand, using ω − ω ∗ − (ω(0) − ω ∗ ) =



t

(ω(r ˙ ) − ω˙ ∗ )dr,

0

defining ki = ki k1 and recalling that ω˙ ∗ = 0, because ω ∗ is a constant, we have that ω˙ +

b + k p J

k1 (ω − ω ) + J ∗



t

0

1 [ω(r ˙ ) + ki (ω(r ) − ω ∗ )]dr = − τ L . (3.17) J

Thus, defining ki =

b + k p 

ζ= 0

, J t  ω(r ˙ ) + ki (ω(r ) − ω ∗ ) dr,

we can write (3.17) as 1 k1 ζ˙ + ζ = − τ L . J J Applying Laplace transform with zero initial conditions: ζ(s) =

− 1J s+

k1 J

τ L (s),

or sζ(s) =

− 1J s s+

k1 J

τ L (s).

(3.18)

3.2 Standard Control

65

Notice that this is a high-pass filter with zero gain at zero frequency, which means that, applying the final value theorem: ˙ = lim s [sζ(s)] = 0, lim ζ(t)

t→∞

s→0

t

if τ L = t p =constant, i.e., τ L (s) = sp . Moreover, this filter is stable if kJ1 > 0 and ˙ approaches to zero faster as k1 > 0 is larger. Notice that limt→∞ ζ(t) ˙ = 0 also ζ(t) J implies that d dt



t 0

  ω(r ˙ ) + ki (ω(r ) − ω ∗ ) dr = ω˙ + ki (ω − ω ∗ ) → 0.

This means that effect of the constant disturbance disappears as time grows and the remaining is the differential equation ω˙ + ki ω = ki ω ∗ , which is stable if ki > 0, possesses the time constant T = k1 and has unitary gain in steady state. i Thus, we arrive to the following conclusions: ˙ = 0 for all t ≥ 0), the velocity response • If no perturbation exists (T p = 0 and ζ(t) is as that of a first order system with time constant T = k1 . The velocity ω reaches i the constant desired velocity ω ∗ in steady state. • If a constant disturbance appears (T p = td = 0), the deviation produced by such a disturbance vanishes as time grows. The disturbance rejection is accomplished arbitrarily fast by choosing an arbitrarily large k1 . Since initial conditions are always assumed to be zero in classical control, then ω(0) = 0 can be assumed and controller in (3.16) becomes ∗

∗



t

τ = τ = k p (ω − ω) + ki

(ω ∗ − ω(r ))dr + (b − k1 )ω ∗ ,

(3.19)

0

which constitutes a classical PI velocity controller with a constant feedforward term. Finally, the tuning rule is summarized as k p = k p + k1 , ki = ki k1 , b + k p , ki = J

(3.20)

where J • k p > 0 is chosen to select the desired time constant T = k1 = b+k  with respect a p i desired velocity reference. • k1 > 0 is chosen large for a fast rejection of the disturbance effects. This can be done by trial and error or it can be computed by recalling that kJ1 is the time constant

66

3 Permanent Magnet Brushed DC-Motor

Fig. 3.12 Time response with the improved PI velocity controller in (3.19). Dashed: desired velocity response. Continuous: actual velocity response

Fig. 3.13 Simulink diagram used to obtain the response in Fig. 3.12

of the filter in (3.18), which is responsible for elimination of the deviation due to disturbance. The above controller design was introduced by the first time in [106] where it is also explained that it is equivalent to design of a two-degrees-of-freedom velocity controller. In order to verify these ideas, we present some simulation results in Fig. 3.12 employing the simulation diagram in Fig. 3.13 and using the parameters of the PM brushed DC-motor model MT-4060 ALYBE. The controller gains that we have employed are k p = 0.9923, ki = 120.5089, k p = 0.1222, k1 = 0.8701. We have also used b = 0.05 (Nm/(rad/s)), ω ∗ = 2000(2π/60) (rad/s), τ L = t p = 15 (Nm), k pi = 10, kii = 10000. The corresponding time response is presented in Fig. 3.12 using a continuous line. The dashed line represents the time response of the transa , with a = ki = 138.5. Notice that the actual transient response to fer function s+a a desired velocity reference (continuous line) is fast and identical to the expected response (dashed line). On the other hand, rejection of the torque disturbance appearing at t = 0.1 (s) is also fast, i.e., faster than in Cases 1, 2, 3, and 4, presented above for a classical PI velocity controller. Thus, these results verify the above predictions.

3.2 Standard Control

67

3.2.6 An Improved PID Position Controller The classical PID position controller has similar problems as those described above for the classical PI velocity controller. These problems can be solved, again, by an improved PID position controller designed following the same ideas as the improved PI velocity controller presented in Sect. 3.2.5. In the following, we just summarize the design procedure for this improved PID position controller and we refer the reader to [106] for a complete presentation. Consider the mechanical subsystem, (3.12) or (3.14), alone J θ¨ = τ − bθ˙ − τ L ,

(3.21)

where θ˙ = ω and τ = km i, i.e., when assuming that the PI electric current controller has been designed to render fast the electrical dynamics, as explained at the beginning of Sect. 3.2. Suppose that the torque disturbance is constant, i.e., τ L = t p =constant. Consider the following improved PID position controller: τ ∗ = τ = K P (θ∗ − θ) − K D θ˙ + K I b + κd , K P = κ p + k1 J



t

(θ∗ − θ(r ))dr − k1

0

K D = κd + k 1 ,

KI =

b + kd ∗ θ , (3.22) J

k1 κ p , J

where θ∗ is the constant desired position, which constitutes a simple PID controller with a constant feedforward term. Notice that the derivative term does not contain the time derivative of the desired position. The closed-loop system (3.21), (3.22) has the following properties: • If no disturbance is present, i.e., τ L = 0, the response in position is as the response of a second-order system with rise time and overshoot that can be arbitrarily chosen by suitably selecting κ p and κd . The position θ reaches the constant desired position θ∗ in steady state. • If a constant disturbance appears, i.e., τ L = td = 0, the deviation that it produces vanishes as time grows. Moreover, if an arbitrarily large k1 is chosen (such that k1 /J > 0 is larger), then the deviation due to the disturbance vanishes arbitrarily faster. Finally, the tuning rule for controller in (3.22) is summarized as follows: 1. κ p > 0 and κd > 0 are chosen to fix the desired rise time and overshoot by suitably κ d s + Jp , i.e., assigning the roots of the characteristic polynomial s 2 + b+κ J b + κd = 2ζωn , J

κp = ωn2 . J

2. k1 > 0 is chosen larger for a faster disturbance rejection. This can be done by trial and error or it can be computed recalling that kJ1 is the time constant of the filter:

68

3 Permanent Magnet Brushed DC-Motor 2 1.8 1.6 1.4

theta [rad]

1.2 1 0.8 0.6 0.4 0.2 0 0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

t [s]

Fig. 3.14 Time response with the improved PID position controller in (3.22). Dashed: desired position response. Continuous: actual position response

Fig. 3.15 Simulink diagram used to obtain the response in Fig. 3.14

sξ(s) =

− 1J s

τ L (s), s + kJ1   t ¨ ) + b + κd θ(r ˙ ) + κ p (θ(r ) − θ∗ ) dr, ξ= θ(r J J 0

which is responsible for the disturbance rejection. Finally, let us point out that it is also demonstrated in [106] that the controller in (3.22) is equivalent to a two-degrees-of-freedom controller. In order to give some insight on the achievable performance we present some simulation results in Fig. 3.14 employing the simulation diagram in Fig. 3.15 and using the parameters of the PM brushed DC-motor model MT-4060 ALYBE. The controller gains that we have employed are computed from the basic data: tr = 0.02 (s), M p = 15%, k1 = TJ , with T = 0.001 (s), and using [106]:

3.2 Standard Control





ζ=

69

   M ln2 100p 1 − ζ2 1 , ωd = π − arctan , M tr ζ ln2 100p + π 2

ωd ωn =  , κd = 2J ζωn − b, κ p = J ωn2 , 1 − ζ2 κp b + κd K P = κ p + k1 , K D = κd + k 1 , K I = k 1 . J J We have also used b = 0.05 (Nm/(rad/s)), θ∗ = π2 (rad), τ L = t p = 15 (Nm), k pi = 10, kii = 10000. The corresponding time response is presented in Fig. 3.14 using a continuous line. The dashed line represents the time response of the transfer function ωn2 . Notice that the actual transient response to a desired position reference s 2 +2ζωn s+ωn2 (continuous line) is fast and identical to the expected response (dashed line). On the other hand, rejection of the step torque disturbance appearing at t = 0.1 (s) is also fast. Thus, these results verify the above predictions.

3.3 The Standard Control Scheme Revisited The standard control scheme for a PM brushed DC-motor is presented in Fig. 3.16a. This control scheme has been very successful in industrial applications [217] because of its simplicity and good performance. The rationale behind this controller is the following. The classical PI electric current controller: v = α p (i ∗ − i) + αi



t

(i ∗ (s) − i(s))ds,

(3.23)

0

Fig. 3.16 Standard velocity control of a PM brushed DC-motor. The electrical subsystem refers to (3.11), whereas the mechanical subsystem refers to (3.12)

70

3 Permanent Magnet Brushed DC-Motor

where α p and αi are, respectively, the proportional and integral gains, is used to force i to reach its desired value i ∗ very fast. In order to ensure this, the controller gains α p and αi are selected using classical control arguments such as pole placement [155], Chap. 4. Once it is ensured that i → i ∗ very fast, then τ = τ ∗ can be assumed because of i ∗ = τ ∗ /km and i = τ /km . This implies that the block diagram in Fig. 3.16a can be simplified to the block diagram in Fig. 3.16b. Thus, the proportional and integral gains k p and ki of the classical PI velocity controller (where ω ∗ represents the desired velocity): ∗

∗



t

τ = k p (ω − ω) + ki

(ω ∗ (s) − ω(s))ds

(3.24)

0

can be chosen using, again, classical control arguments such as pole placement by taking into account only the mechanical dynamics, i.e., using the simplified block diagram in Fig. 3.16b. Alternatively, since the complete motor dynamic model is linear, design can be performed by taking into account the complete block diagram in Fig. 3.16a, together with (3.11), (3.12), using classical control techniques such as root locus [155], Chap. 4. Also see the previous sections in the present chapter of this book. It is convenient to stress that integral actions are included in controllers (3.23) and (3.24) to formally ensure that i = i ∗ and ω = ω ∗ in steady state when ω ∗ is a constant and despite the presence of some unknown but constant external torque disturbance τL . A similar procedure is also valid for motor position control if the PI velocity controller in (3.24) is replaced by the classical PID position controller (where q ∗ represents the desired position): τ ∗ = k p (q ∗ − q) + kd

d(q ∗ − q) + ki dt



t

(q ∗ (s) − q(s))ds,

(3.25)

0

and the additional state equation is considered: q˙ = ω. Again, the integral action in controller (3.25) is included in order to ensure that q = q ∗ in steady state when q ∗ is constant and despite the presence of some unknown but constant external torque disturbance τ L . An alternative design method to classical control techniques is presented in the next sections of this chapter to find out some criteria for selection of the controller gains in (3.23), (3.24), and (3.25). This new methodology proves to be useful for our purposes since, as we show in latter chapters in this book, it can be extended to the design of field oriented-based controllers for AC-motors which, because of the nonlinear character of AC-motor models, cannot be designed using classical control tools.

3.3 The Standard Control Scheme Revisited

71

In the remaining of this book, we present some PI velocity controllers and some PID position controllers for several classes of electric machines. The reason to present these PI- and PID-based controllers is the quest for global stability proofs for standard field-oriented control of electric machines, a problem which remains unsolved until now. However, we have shown above that the improved PI and PID control schemes described in Sects. 3.2.5 and 3.2.6 improve the transient performance. Hence, it is also interesting to present global stability proofs for such control schemes when applied to different classes of electric machines. Thus, it is important to stress that this improved PI velocity controller and this improved PID position controller only differ from classical PI and PID controllers by an additional constant term. In fact, it is very easy to extend the energy-based controller design methodologies presented in the next sections and in the remaining chapters of this book to the case when the controllers in Sects. 3.2.5 and 3.2.6 are employed. This can be accomplished by handling the constant term as it was a constant torque disturbance. This allows to fit these control schemes to the stability proofs that are presented in subsequent chapters.

3.4 Open-Loop Energy Exchange 3.4.1 The Velocity Model According to Sect. 3.1.1, the dynamic model of a PM brushed DC-motor is given by (3.11), (3.12), which is rewritten here for the ease of reference: L

di = −Ri − ke ω + v, dt J ω˙ = km i − bω − τ L ,

(3.26) (3.27)

where i, v, ω ∈ R represent electric current through the armature circuits, voltage applied at the motor terminals and rotor angular velocity, respectively. All of the positive scalars L , R, J, b, ke , km are assumed to be constant standing for armature inductance and resistance, rotor inertia, viscous friction coefficient, the motor back electromotive force constant and the motor torque constant. Finally, τ L is the load torque. Recall that the following interesting property of model (3.26), (3.27): ke = km ,

(3.28)

which is related to energy conservation has been proven to be true in (3.8). The following scalar function represents the total energy stored in the motor: V (i, ω) =

1 2 1 2 Li + J ω , 2 2

(3.29)

72

3 Permanent Magnet Brushed DC-Motor

where the first term represents the magnetic energy stored in the electrical subsystem, whereas the second term stands for the kinetic energy stored in the mechanical subsystem. The time derivative of V is given as di V˙ = i L + ω J ω˙ dt which, according to (3.26), (3.27), can be written as V˙ = i(−Ri − ke ω + v) + ω(−bω + km i − τ L ). By virtue of (3.28) two cross terms cancel, which is a result from the natural energy exchange between the motor electrical and mechanical subsystems. Hence, V˙ = −Ri 2 − bω 2 + vi − τ L ω.

(3.30)

Defining the input u = [v, −τ L ] and te output y = [i, ω] , we can write (3.30) as V˙ = −y Qy + y u,

 Q=

 R0 . 0 b

(3.31)

Since Q is a positive definite matrix, (3.31) shows that model (3.26), (3.27) is output strictly passive (see Definition 2.42) for the output y and input u defined above. In this chapter, it will be shown that this is the basic theoretical principle behind the common and successful industrial velocity control scheme depicted in Fig. 3.16.

3.4.2 The Position Model If the rotor position is designated by q then ω = q. ˙

(3.32)

Assume that the load torque is a nonlinear function of position, i.e., τ L = G(q), which is given as the gradient of a positive semidefinite function P(q): G(q) =

d P(q) . dq

(3.33)

Using these ideas and (3.32), the dynamical model (3.26), (3.27) can be written as

3.4 Open-Loop Energy Exchange

L

73

di = −Ri − ke q˙ + v, dt J q¨ = km i − bq˙ − G(q).

(3.34) (3.35)

The following scalar function represents the total energy stored in motor and load: V (i, q, q) ˙ =

1 2 1 2 Li + J q˙ + P(q), 2 2

(3.36)

where the new third term stands for the potential energy stored in the mechanical subsystem (because of (3.33), i.e., torque is given as gradient of the potential energy, see [10], Chap. 11). Notice that V is a positive semidefinite function since P(q) also has this property. Proceeding as in the previous section it is not difficult to find that the cancellation of cross terms due to (3.28) appears again when computing V˙ . Notice, however, that a new cancellation appears between the cross terms ±G(q)q, ˙ which represents the kinetic and potential energy exchange in the mechanical subsystem. Thus, we find that: V˙ = −Ri 2 − bq˙ 2 + vi,

(3.37)

≤ −Ri + vi, because b > 0, ˙ V ≤ −Ry 2 + yu, 2

where the input u = v and the output y = i have been defined. This means that model (3.34), (3.35) is output strictly passive (see Definition 2.42) for the output y and input u defined above. It will be shown later in this chapter that this is the basic theoretical principle behind the common and successful industrial control scheme depicted in Fig. 3.16, when a linear PID position controller is used.

3.5 Velocity Control In the case of velocity control of PM brushed DC-motors, the electrical subsystem and the mechanical subsystem are linear. Hence, design of the standard velocity control scheme for a PM brushed DC-motor that has been presented in Sect. 3.3 (also see Fig. 3.17 for the ease of reference) can be performed using classical control tools as those in Sect. 3.2. The main result in this section is to present an alternative manner to design such a control scheme by ensuring global asymptotic stability. The stability proof is based on a Lyapunov stability analysis that exploits the concept of energy in the electromechanical system. This approach is important for the purposes of this book because, as it is shown in the subsequent chapters, it is the basis to design velocity controllers for AC-motors which have complex nonlinear models. Proposition 3.3 Consider the PM brushed DC-motor dynamical model (3.26), (3.27), in closed-loop with the following controller:

74

3 Permanent Magnet Brushed DC-Motor

Fig. 3.17 Standard velocity control of a PM brushed DC-motor. The electrical subsystem refers to (3.26) whereas the mechanical subsystem refers to (3.27)

v = α p (i ∗ − i) + αi



t

(i ∗ (s) − i(s))ds,

(3.38)

0

τ ∗ = k p (ω ∗ − ω) + ki i∗ =



τ∗ , kme

t

(ω ∗ (s) − ω(s))ds,

0

where kme is a positive real constant standing for the estimate of km . Assume that the load toque τ L is unknown but constant and ω ∗ is a real constant standing for the desired velocity. There always exist positive controller gains α p , αi , k p , ki , such that the closed-loop dynamics has an unique equilibrium point which is globally asymptotically stable. At this equilibrium point ω = ω ∗ and i = i ∗ . The complete proof of Proposition 3.3 is presented in Appendix B.1. In the following, we present just a sketch of such a proof. The main idea is to highlight the rationale behind the proof and ideas that are employed subsequently in the proofs of controllers for AC-motors. Sketch of the proof of Proposition 3.3. There always exists a positive constant ε such that kme = εkm . Hence, it is always possible to define two constants k p and ki such that k p = εk p and ki = εki . Thus, defining i˜ = i − i ∗ , ω˜ = ω − ω ∗ , and 

bω ∗ + τ L , k 0 i ∗   t bω + τ L 1 ∗ ˜ R + ke ω , zi = i(s)ds + αi km 0 z=

t

ω(s)ds ˜ +

the closed-loop dynamics can be written as follows (recall that τ L is constant): L

d i˜ dt J ω˙˜ z˙ z˙ i

= −(R + α p )i˜ − ke ω˜ + V,

(3.39)

= −(b + k p )ω˜ + km i˜ − TL , = ω, ˜ ˜ = i,

(3.40) (3.41) (3.42)

3.5 Velocity Control

75

TL = ki z, V =

Rk p km

ω˜ +

Rki L  ˙ −k p ω˜ − ki ω˜ . z − αi z i − km km

˜ z i , ω, The state of this closed-loop dynamics is x = [i, ˜ z] . This closed-loop ∗

dynamics is autonomous and x = [0, 0, 0, 0] is its only equilibrium point. Also notice that (3.39) and (3.40) are almost identical to the open-loop dynamics in (3.26) ˜ ω, ˜ TL , V, respectively. An important and (3.27), if we replace i, ω, τ L , v by i, difference is that electric resistance and viscous friction have been enlarged in the closed-loop dynamics, i.e., we have R + α p and b + k p in (3.39) and (3.40) instead of R and b in (3.26) and (3.27). Another important difference is the presence of two additional equations in (3.41), (3.42), which represent the integral parts of both, the PI velocity controller and the PI electric current controller, respectively. These observations motivate the use of the following “energy” storage function for the closed-loop dynamics: 1 1 ˜ z i , ω, ˜ z) = L i˜2 + αi z i2 + Vω (ω, ˜ z), V (i, 2 2 1 1 Vω (ω, ˜ z) = J ω˜ 2 + [ki + γ(b + k p )]z 2 + γ J z ω, ˜ 2 2

(3.43) (3.44)

˜ z i , ω, where γ is a positive constant. Reason for the first two terms in V (i, ˜ z) is to take into account “energy” in the electrical subsystem which is composed by the “magnetic energy” and “energy” stored in the integral part of the PI electric current ˜ z) is intended to take into account “energy” stored in the controller. Function Vω (ω, mechanical subsystem; it is composed by the “kinetic energy” and “energy” stored in the integral term of the PI velocity controller. The third cross term is included in order to obtain a V˙ having a negative quadratic term in z. In this respect, first notice that thanks to the above-cited similarities between (3.39), (3.40), and (3.26), (3.27), the cross term cancellations due to the natural energy exchange between the electrical and the mechanical subsystems described in Sect. 3.4.1, are also present in the closed-loop system. Second, this observation is useful to find that d dt



1 ˜2 1 2 L i + J ω˜ 2 2



= −(R + α p )i˜2 + i˜ V − (b + k p )ω˜ 2 − ωT ˜ L.

Third, since V depends on z, it is necessary a negative quadratic term in z to dominate Rk  the cross term kmi i˜ z (the quadratic term −(R + α p )i˜2 already exists). On the other   ˜ cancel with d 1 ki z 2 + 1 αi z i2 . hand, terms −ωT ˜ L and −αi i˜ z i , arising from iV, dt

2

2

˜ z) because its time derivative cancels Finally, term 21 γ(b + k p )z 2 is included in Vω (ω, d ˜ Thus, time derivative of V results some undesired cross terms arising from dt (γ J z ω). in

76

3 Permanent Magnet Brushed DC-Motor

V˙ = −(R + α p )i˜2 − (b + k p )ω˜ 2 − γki z 2 +

Rk p ˜ km

i ω˜ +

Rki km

z i˜ +

(3.45)

Lki

L  ˙˜ ˜ k ω˜ i + ω˜ i˜ + γ J ω˜ 2 + γkm z i. km p km

As explained in Sect. 2.4, terms in the first row appear thanks to the open-loop system passivity properties that are inherited to the closed-loop system. Stability is ensured by choosing the controller gains, present in the first row above, such that terms in the second row be dominated. Then, global asymptotic stability is proven invoking the LaSalle invariance principle. Conditions for this stability result are summarized by ki + γ(b + k p ) − J γ 2 > 0, γ > 0, αi > 0, b + k p > γ J, ki > 0, det(Q) > 0, with Q a matrix defined in (B.17). These are the conditions for positive definiteness ˜ z i , ω, ˜ z) and negative semidefiniteness of its time derivative. of fuction V (i, Remark 3.4 The closed-loop mechanical subsystem dynamics given in (3.40) can be written as J ω˙˜ = −bω˜ + τe − k p ω˜ − ki z, z˙ = ω. ˜

(3.46)

where τe = km i˜ (i.e., the difference between desired torque and generated torque) is given as a function of error of the electrical dynamics. The time derivative of func˜ z), defined in (3.44), along the trajectories of the mechanical subsystem tion Vω (ω, dynamics (3.46) is given as ˜ e + γzτe , V˙ω = −(b + k p )ω˜ 2 + γ J ω˜ 2 − γki z 2 + ωτ which can be written as   ω˜ + (ω˜ + γz)τe , ˜ z]Q  V˙ω = −[ω, z

Q =



 Q 11 0 , 0 Q 22

(3.47)

with Q 11 = b + k p − γ J , Q 22 = γki . We stress that matrix Q  is positive definite if and only if b + k p > γ J, γ > 0, ki > 0,

(3.48)

and hence λmin (Q  ) > 0. On the other hand, the time derivative of the first component ˜ z i ) = 1 L i˜2 + 1 αi z i2 , along the trajectories of the of V , defined in (3.43), i.e., Ve (i, 2 2 closed-loop electrical subsystem dynamics (3.39) which contains the negative term −(R + α p )i˜2 . Thus, when computing V˙ = V˙ω + V˙e to obtain (3.45), we realize that the following features are instrumental for the stability result in Proposition 3.3:

3.5 Velocity Control

77

• The scalar function Vω is a strict Lyapunov function when the desired torque is the input of the closed-loop mechanical subsystem dynamics, given in (3.46), i.e., when τe = 0. • Coefficient of the negative term −(R + α p )i˜2 in V˙ can be enlarged arbitrarily. This is important to dominate cross terms in V˙ , given in (3.45), involving variable ˜ i.e., when i˜ = 0. i, ˜ e. • Cancellation of the cross terms −ke i˜ω˜ and ωτ Notice that all of these features are possible thanks to the passivity properties described in Sect. 3.4.1 for the open-loop model. Furthermore, the above proof contains the main ideas of procedure described in Sect. 2.4. In the subsequent chapters, it is shown that these features are also instrumental for the stability results introduced for some velocity controllers designed for several AC-motors. An important difference is that, in those cases, τe is given as a nonlinear function of the electrical dynamics error. Moreover, this function is different for each different motor that is studied. Thus, the main difference in the stability proofs that are introduced in the subsequent chapters is the manner that nonlinear function defining τe is handled.

3.6 Position Control Design of the standard position control scheme for a PM brushed DC-motor that has been presented in Sect. 3.3 can be performed proceeding as in Sect. 3.2, for instance, or using other classical linear techniques such as root locus, frequency response, and pole assignment (see [155], for instance). Recall that this control scheme is identical to that shown in Fig. 3.17 with a PID position controller instead of a PI velocity controller. If it is assumed that the PI electric current loop is stable and “fast enough” then it can be also assumed that the generated torque reaches very fast the desired torque and, hence, either electrical current or torque can be considered to be the control input instead of the applied voltage. This introduces the concept of current-fed or torque-fed motors. As a result, it is common in the control literature to neglect the dynamics of the electrical subsystem and control design focuses in the mechanical subsystem dynamics alone with torque as the input (see [71, 129, 130, 257, 261], for instance). This approach is also preferred because taking into account the electrical dynamics further complicates the control design task in robotics [7]. However, in [258] was shown that neglecting the electrical subsystem dynamics may result in performance deterioration or even instability. This motivated the interest to control mechanical systems when taking into account the electrical dynamics of PM brushed DC-motors used as actuators [5–7, 33, 74, 176]. These works proposed rather complex control schemes instead of considering the simple standard control scheme presented in Sect. 3.3. In [100] is demonstrated that the combination of two controllers that were previously proposed in the literature for rigid robots under the assumption that the electrical dynamics of the actuators can be neglected, still ensures

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3 Permanent Magnet Brushed DC-Motor

semiglobal asymptotic stability when the electric dynamics of the PM brushed DCmotors used as actuators is taken into account. Finally, in [99] was presented a global asymptotic stability proof for PD control with desired gravity compensation of rigid robots actuated by PM brushed DC-motors. Voltage applied at motor terminals is given by proportional electric current controllers and the motor stator resistance is required to be exactly known. Contrary to several works in the literature, the result in [99] does not rely on friction that is naturally present in the mechanical subsystem nor in time scale separation arguments. The problem of designing the simple standard control scheme presented in Sect. 3.3 when the mechanical subsystem is nonlinear remains unsolved. In the following, proposition we present a solution to this problem ensuring global asymptotic stability. Proposition 3.5 Consider the PM brushed DC-motor dynamical model (3.26), ˙ i.e., (3.27), with τ L = g(q) standing for a position-dependent load and ω = q, L

di = −Ri − ke q˙ + v, dt J q¨ = km i − bq˙ − g(q),

(3.49) (3.50)

where J represents the addition of rotor and load inertias. Suppose that model (3.49), (3.50) is connected in closed-loop with the following controller: v = α p (i ∗ − i) + αi



t

(i ∗ (s) − i(s))ds, 0  t ∗ ∗ (q ∗ (s) − q(s))ds, τ = k p (q − q) − kd q˙ + ki τ∗ i = , kme

(3.51)

0

∗

where kme is a positive real constant standing for the estimate of km and q ∗ represents the constant desired position. Assume that g(q) possesses the following properties: |g(x) − g(y)| ≤ kg |x − y|, ∀x, y ∈ R, (Lipschitz condition)    dg(q)  , kg > max  q∈R dq  dU (q) g(q) = , ∀q ∈ R, dq

(3.52) (3.53) (3.54)

with U (q) a lower bounded twice continuously differentiable scalar function and kg > 0 is a finite constant. There always exist positive controller gains α p , αi , k p , kd , ki such that the closedloop dynamics has an unique equilibrium point which is globally asymptotically stable. At this equilibrium point q = q ∗ and i = i ∗ .

3.6 Position Control

79

The complete proof of Proposition 3.5 is presented in Appendix B.2. In the following, we just present a sketch of such a proof. The main idea is to highlight the rationale behind the proof and ideas that are employed subsequently in the proofs of controllers for AC-motors. Sketch of the proof of Proposition 3.5. Defining kme = εkm , for some positive ε, k p = εk p , kd = εkd , ki = εki , and q˜ = q − q ∗ , the following is found: i∗ =

1 km

   t −k p q˜ − kd q˙ − ki q(s)ds ˜ . 0

Using this and defining i˜ = i − i ∗ , k p = k p − ki , ki = α1 ki , and 

t

1 (αq(s) ˜ + q(s))ds ˙ + q(0) ˜ +  g(q ∗ ), ki 0    t R 1 ˜ g(q ∗ ) , zi = i(s)ds + αi km 0 z=

the following closed-loop dynamics is obtained: L

d i˜ = −(R + α p )i˜ − ke q˙ + V, dt J q¨ = −(b + kd )q˙ + km i˜ − G, ˜ z˙ = αq˜ + q, ˙ z˙ i = i, G= V=

k p q˜ + ki z + g(q) Rk p Rkd km

q˜ +

km

q˙ +

(3.55) (3.56) (3.57)

∗

− g(q ),  Rki L   −k p q˙ − kd q¨ − ki (αq˜ + q) z − αi z i − ˙ . km km

˜ z i ] . This closed-loop The state of this closed-loop dynamics is x = [q, ˜ q, ˙ z, i, ∗

dynamics is autonomous and x = [0, 0, 0, 0, 0] is the only equilibrium point. Notice that (3.55) and (3.56) are almost identical to the open-loop model in (3.49) ˜ q, and (3.50), if we replace i, q, ˙ g(q), v by i, ˙ G, V, respectively. An important difference is that electric resistance and viscous friction have been enlarged in the closed-loop dynamics, i.e., we have R + α p and b + kd in (3.55) and (3.56) instead of R and b in (3.49) and (3.50). Another important difference is the presence of two additional equations in (3.57) which represent the integral parts of both, the PID position controller and the PI electric current controller, respectively. These observations motivate use of the following “energy” storage function for the closed-loop dynamics:

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3 Permanent Magnet Brushed DC-Motor

1 ˜2 1 L i + αi z i2 + Vq (q, ˜ q, ˙ z), (3.58) 2 2 ˜ q, ˙ z) = V1 (q, ˜ q) ˙ + V2 (z, q) ˙ + V3 (q, ˜ z), Vq (q, 1 α 1 ˜ 2 − α2 J q˜ 2 + (b + kd )q˜ 2 + k p q˜ 2 ˜ q) ˙ = J (q˙ + 2αq) V1 (q, 4 2 2 ∗ +U (q) − U (q ∗ ) − qg(q ˜ ), 1 αβ  2 1  2 ˙ = J (q˙ + 2αβz)2 − α2 β 2 J z 2 + V2 (z, q) k z + ki z , 4 2 d 2 β  ˜ = (k p − αkd )(z − q) ˜ 2, V3 (z, q) 2

˜ zi ) = V (q, ˜ q, ˙ z, i,

where α and β are two positive constants. Reason for the first and second terms in (3.58) is to take into account “energy” in the electrical subsystem, which includes the “magnetic energy” and “energy” due to the integral part of the PI electric current ˜ q, ˙ z) represents “energy” in the mechanical subsystem. controller. Function Vq (q, ˜ q) ˙ This includes kinetic energy 21 J q˙ 2 whose contribution is separated in both V1 (q, ˙ the “potential energy”: and V2 (z, q), P(q) ˜ =

1  2 ∗ k q˜ + U (q) − U (q ∗ ) − qg(q ˜ ), 2 p

defined as in robot control [130], and “energy” due to the integral part of the PID position controller 21 ki z 2 . Cross terms between q˜ and q˙ as well as between z and q˙ ˜ q) ˙ and V2 (z, q) ˙ and they are intended to obtain a function V˙ appear in both V1 (q, possessing negative quadratic terms in both q˜ and z. In this respect, first notice that thanks to the above-cited similarities between (3.55), (3.56), and (3.49), (3.50), it is easy to verify that the cross term cancellations due to the natural energy exchange between the electrical and the mechanical subsystems, described in Sect. 3.4.2, are also present in the closed-loop system. Second, this observation is useful to find that d dt



 1 ˜2 1 2 L i + J q˙ + P(q) ˜ = −(R + α p )i˜2 − (b + kd )q˙ 2 + i˜ V − ki qz. ˙ 2 2

Third, since V depends on both q˜ and z, two negative quadratic terms in q˜ and z are necessary to dominate those cross terms arising from i˜ V (the quadratic term −(R + α p )i˜2 already exists). ˜ cancel with hand, terms −ki qz ˙ and −αi i˜ z i , arising from iV, On the other  d 1  2 1 2 . V k z + α z ( q, ˜ z) is a function which is required to be semidefinite pos3 dt 2 i 2 i i itive and it is included in order to avoid that the restrictive condition k p = αkd be ˜ q) ˙ and V2 (z, q) ˙ imposed on the controller gains. The remaining terms in both V1 (q, are useful to cancel some undesired cross terms in V˙ . Hence, time derivative of V along the trajectories of the closed-loop dynamics (3.55)–(3.57) is given as

3.6 Position Control

81

V˙ = −(R + α p )i˜2 − (b + kd )q˙ 2 − αk p q˜ 2 − αβki z 2 ∗ ˜ ) − g(q)) − βαk p q˜ 2 + βα2 kd q˜ 2 + +αJ q˙ 2 + αkm q˜ i˜ + αq(g(q

(3.59) Rk p km

q˜ i˜

+α2 β J q˙ q˜ + αβ J q˙ 2 − αβbz q˙ + αβkm z i˜ + αβz(g(q ∗ ) − g(q)) Rki ˜ Lk p ˜ Lkd ˜ Lk  ˜ Lk  Rk  zi + q˙ i + i q¨ + α i q˜ i˜ + i q˙ i. + d q˙ i˜ + km km km km km km Stability is ensured by choosing the controller gains appearing in the first row above, such that terms in the remaining rows be dominated. Then, global asymptotic stability is proven invoking the LaSalle invariance principle. Conditions for this stability result are summarized by k p > kg , α > 0,

1 (b + kd ) > αJ, αi > 0, 2

αβ  1 k + k  > α2 β 2 J, β > 0, k p > αkd , 2 d 2 i ⎛ ⎞ Q 11 Q 12 Q 13 Q 11 > 0, Q 11 Q 22 − Q 12 Q 21 > 0, det ⎝ Q 21 Q 22 Q 23 ⎠ > 0, det(Q) > 0, Q 31 Q 32 Q 33 where Q i j , for i, j = 1, 2, 3 are entries of matrix Q defined in (B.32). These conditions ensure that function V is positive definite and radially unbounded and V˙ is positive semidefinite. Remark 3.6 The closed-loop mechanical subsystem dynamics given in (3.56) can be written as J q¨ = −(b + kd )q˙ + τe − k p q˜ − ki z − g(q) + g(q ∗ ),

(3.60)

z˙ = αq˜ + q. ˙ where τe = km i˜ is the difference between desired torque and generated torque, as ˜ q, ˙ z), defined in (3.58), along the trabefore. The time derivative of function Vq (q, jectories of the mechanical subsystem dynamics (3.60) is given as ∗ ˙ e + αJ q˙ 2 + αqτ ˜ e − αk p q˜ 2 + αq(g(q ˜ ) − g(q)) V˙q = −(b + kd )q˙ 2 + qτ

+α2 β J q˙ q˜ + αβ J q˙ 2 − αβbz q˙ + αβzτe − αβki z 2 + αβz(g(q ∗ ) − g(q)) −βαk p q˜ 2 + βα2 kd q˜ 2 . This can be written as

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3 Permanent Magnet Brushed DC-Motor

⎡

⎤ |q| ˙ ˜ ⎦ + (q˙ + αq˜ + αβz)τe , ˙ |q|, ˜ |z|]Q  ⎣ |q| V˙q ≤ −[|q|, |z|

⎤ Q 11 Q 12 Q 13 Q  = ⎣ Q 21 Q 22 Q 23 ⎦ . Q 31 Q 32 Q 33 (3.61) ⎡

Entries of matrix Q  are identical to entries in the first three rows and columns of matrix Q defined in (B.32). Matrix Q  is positive definite if and only if ⎛

Q 11 > 0,

Q 11 Q 22 − Q 12 Q 21

⎞ Q 11 Q 12 Q 13 > 0, det ⎝ Q 21 Q 22 Q 23 ⎠ > 0, Q 31 Q 32 Q 33

(3.62)

which ensures λmin (Q  ) > 0. On the other hand, the time derivative of the first com˜ z i ) = 1 L i˜2 + 1 αi z i2 , along the trajectories ponent of V , defined in (3.58), i.e., Ve (i, 2 2 of the closed-loop electrical subsystem dynamics (3.55) contains the negative term −(R + α p )i˜2 . Thus, when computing V˙ = V˙q + V˙e to obtain (3.59), we realize that the following features are instrumental for the stability result in Proposition 3.5: • The scalar function Vq is a strict Lyapunov function when the desired torque is the input of the closed-loop mechanical subsystem dynamics, given in (3.60), i.e., when τe = 0. • Coefficient of the negative term −(R + α p )i˜2 in V˙ can be enlarged arbitrarily. This is important to dominate cross terms depending on i˜ and appearing in V˙ , given in (3.59), i.e., when i˜ = 0. • Cancellation of the cross terms −ke i˜q˙ and qτ ˙ e. Notice that all of these features are possible thanks to the passivity properties described in Sect. 3.4.2. Furthermore, the above proof contains the main ideas of procedure described in Sect. 2.4. In the subsequent chapters it is shown that these features are also instrumental for the stability results introduced for some position controllers designed for several AC-motors. As explained in the previous section, an important difference is that in AC-motors τe is given as a nonlinear function of the electrical dynamics error and this function is different for each different motor that is studied. Thus, the main difference in the stability proofs that are introduced in the subsequent chapters is the manner that nonlinear function defining τe is handled. In this respect, we stress that we do not present a general procedure valid for all AC-motors but, instead, we propose a particular procedure for each class of AC-motors which exploits the particular nonlinearities that are present in the model of each AC-motor. Remark 3.7 A control problem as that solved in Proposition 3.5 tends to be underestimated in the control systems community. The argument used for this is that “such a control problem is easy” because the model of a PM brushed DC-motor is linear and, hence, well-known approaches as those described in [155] (root locus, frequency response, pole assignment, etc.) can be employed. Furthermore, in [204], Chap. 9, is explained that control of PM brushed DC-motors is not of interest because of their

3.6 Position Control

83

well-known disadvantages; they use brushes and commutators and they are expensive when compared to AC-motors. Despite this criticism, we consider that it is important to stress the following: • The “easy control problem” argument cited above is employed even when solving the much more complex and nonlinear problem of controlling n−degrees of freedom rigid robots actuated by PM brushed DC-motors. Moreover, even the simple closed-loop system model resulting in Proposition 3.5 is in fact nonlinear and, thus, well-known approaches as those described in [155] cannot be applied. • It is explained in [204], Chap. 9, that an Euler-Lagrange-based model for motors with commutators is rather complicated to derive and, because of this, models for these motors are not derived using variational principles. After stating this, the disadvantages of the PM brushed DC-motor, listed above, are used just to justify why this class of motor is not studied in that book. Once these observations have been pointed out, let us remark that Proposition 3.5 is the first formal result on the standard control of PM brushed DC-motors when (a) actuating a nonlinear mechanical plant and (b) the electrical dynamics of this actuator is taken into account in the solution of the problem. This means that no previous work in the literature has solved such a problem. In this respect, notice that the result in Proposition 3.5 does not rely on time scale separation approximations, i.e., it is not required to assume that the closed-loop electrical dynamics is “fast enough”. The reader will realize that the solution to control n−degrees of freedom rigid robots when actuated by PM brushed DC-motors can also be straightforward solved by using Proposition 3.5 to adapt the results in [93] to PM brushed DC-motors. Hence, this problem is left as an exercise. According to the above arguments, we consider important to say that one advantage of our novel passivity-based approach is that it is capable to handle the PM brushed DC-motor control problem, a problem that becomes complicated when handled by the passivity-based approach in [204]. Finally, let us quote [42] where it is stated “although PM brushed DC-motor is well researched, successful application of a novel control technique motivates its extensions to the development of similar controllers for more complex electromechanical systems, such as permanent magnet stepper motor, brushless DC-motor, switched reluctance motor, and AC induction servo motors”. This is what we present in the remaining of the book. See [74] for the design of an advanced control strategy in a complex mechanism which is actuated by PM brushed DC-motors.

3.7 Velocity Control Using a DC/DC Buck Power Converter As Power Amplifier Control of brushed DC-motors has been traditionally based on use of pulse width modulation (PWM) techniques at the power amplifier stage. However, because of the underlying hard switching of PWM, this results in large forces that appear acting

84

3 Permanent Magnet Brushed DC-Motor

on the motor mechanics and large currents that detrimentally stress the electronic components of the motor and the power supply [11]. Motivated by these problems, it is proposed in [24] to replace the PWM techniques by the combination of DC to DC power converters with DC-motors. This configuration improves performance, because DC to DC power converters deliver smooth DC output voltages and currents with a very small ripple. A DC to DC Buck converter driven DC-motor system has been proposed in [163, 164]. Motivated by the work in [163, 164] several control schemes have been proposed for the DC to DC Buck converter driven DC-motor system [4, 11, 66, 161, 248, 255]. However, most of these controllers are complex in the sense that they require the exact knowledge of all (or, at least, many) of the system parameters (i.e., inductances, resistances, capacitances, inertia, viscous friction coefficient, torque and back electromotive force constants, etc.) and they are based on differential Flatness or Backstepping approaches which require lots of computations. Trying to avoid these problems, other proposals in the literature [81, 231, 232, 282] are based in rather elaborate ideas if we take into account that the DC to DC Buck converter driven DC-motor system is linear. On the other hand, it has been pointed out in [174] that an important reason why brushed DC-motors are extensively used in many industrial applications is the relative ease in devising the appropriate feedback control schemes, specially those of the PI and PID types [122]. This has motivated works in [4, 255] where PI controllers have been proposed for the DC to DC Buck converter driven DC-motor system. However, those solutions are far from complete since either the presence of a load torque at the motor shaft is not considered or the load torque is considered to be known. In this section, a controller for the DC to DC Buck converter driven DC-motor system is presented. This controller is simple, i.e., a reduced number of computations are required when compared to Flatness-based and Backstepping-based controls schemes presented in [4, 11, 66, 161, 163, 164, 248, 255]. It possesses integral actions on the velocity error, the motor armature electric current error and voltage error at the converter output capacitor. Moreover, a sliding modes controller ensures convergence of electric current through the converter inductor to its desired value. Hence, robustness with respect to load torque and parameter uncertainties is expected, which is verified through experiments at the end of the present section.

3.7.1 Dynamic Model A DC to DC Buck converter driven DC-motor system is presented in Fig. 3.18a, whereas in Fig. 3.18b it is shown how to model the transistor and the diode by means of ideal switches. Using Kirchhoff’s Laws it is found the following dynamical model:

3.7 Velocity Control Using a DC/DC Buck Power Converter As Power Amplifier

85

Fig. 3.18 Electric diagram of the DC to DC Buck converter driven DC-motor system

di dt dυ C dt di a La dt dω J dt L

= −υ + Eu, = i − ia −

υ , R

(3.63) (3.64)

= υ − Ra i a − ke ω,

(3.65)

= km i a − Bω − TL .

(3.66)

Variables i and υ represent the electric current through the converter inductance L and voltage at the converter output (i.e., the electric voltage applied at the motor armature terminals). Electric current through the DC-motor armature circuit and motor velocity are given, respectively, by i a and ω. The switch position is represented by u, which is considered to be the control input only taking two discrete values: 0 or 1. Constants C, R, E, L a , Ra , J , B, km and ke are positive standing for the converter capacitance, a resistance fixed at the converter output, power supply voltage, the armature inductance, the armature resistance, motor inertia, the viscous friction coefficient, motor torque constant, and motor back electromotive force constant. Finally, TL is a constant representing the unknown load torque.

86

3 Permanent Magnet Brushed DC-Motor

3.7.2 Open-Loop Energy Exchange The following scalar function represents the total energy stored in the DC to DC Buck converter driven DC-motor system: V (i, v, i a , ω) =

1 2 1 2 1 1 Li + Cv + L a i a2 + J ω 2 . 2 2 2 2

(3.67)

First term represents the magnetic energy stored in the Buck converter inductor. Second term stands for electric energy stored in the Buck converter output capacitor. Third term quantifies the magnetic energy in the motor electrical subsystem. Fourth term is kinetic energy in the motor mechanical subsystem. The time derivative of V is given as dv di a di + ia L a + ω J ω, ˙ V˙ = i L + vC dt dt dt which, according to (3.63)–(3.66), can be written as υ V˙ = i(−υ + Eu) + v i − i a − + i a (υ − Ra i a − ke ω) R +ω(km i a − Bω − TL ).

(3.68)

Notice that several cross terms cancel in this expression. Since V represents energy stored in the system, these cross term cancellations represent energy exchange among the system components and they illustrate how energy passes from the DC power supply to the motor mechanical subsystem. Hence, it can be written 1 V˙ = i Eu − v 2 − Ra i a2 − Bω 2 − τ L ω. R

(3.69)

Defining the input u ∗ = [Eu, −τ L ] and the output y = [i, ω] , we can write (3.69) as V˙ ≤ y u ∗ ,

(3.70)

since R, Ra and B are positive constants. This shows that model (3.63)–(3.66), is passive (see Definition 2.42) for the output y and the input u ∗ defined above.

3.7 Velocity Control Using a DC/DC Buck Power Converter As Power Amplifier

87

3.7.3 Control of the DC to DC Buck Converter DC-Motor System The following proposition summarizes the result introduced in this section. Proposition 3.8 Consider the DC to DC Buck converter driven DC-motor system (3.63)–(3.66) in closed-loop with the following controller:  1 +1, s ≥ 0 ∗ u = [1 − sign(s)], s = i − i , sign(s) = , (3.71) −1, s < 0 2  t υ + k p1 e + ki1 e(τ )dτ , (3.72) i∗ = R 0  t  t υ = −ra ea + Ra i a − γ ea (τ )dτ + f k p2 ω, ˜ i a = ki2 ω(τ ˜ )dτ , (3.73) 0

0

e = υ − υ, ea = i a − i a , ω˜ = ωd − ω,

(3.74)

where ωd is the time varying rest-to-rest desired velocity, i.e., it is given as ωd (t) = ωd∗ (t) + ω d with ω d a positive constant representing the final desired velocity and ωd∗ (t) is a function of time which has to be selected such that ω˙ d∗ (t) is bounded for all t ≥ 0, ωd (0) equals the initial desired velocity and ωd (t) = ω d , ∀t ≥ t f , where t f > 0 is a finite constant. There always exist positive constants k p1 , ki1 , k p2 , ki2 , f , ra and γ such that the origin of the closed-loop system is asymptotically stable as long as 0 0, en = −e, ω˜ n = −ω, 

t

1 (Bω d + TL ), km ki1 0  t 1 ω(τ ˜ )dτ − (Bω d + TL ), ξ= ki 0  t 1 ke ω d , ρ(τ )dτ + z1 = 2γ 0 ζ=

e(τ )dτ −

3.7 Velocity Control Using a DC/DC Buck Power Converter As Power Amplifier

 z2 =

t

σ(τ )dτ +

0

89

1 ke ω d , 2γ

and it is found that the closed-loop dynamics on the sliding surface s = 0 is given as (recall that ω d and TL are constants):  C e˙n = I − ea − L a ρ˙ = L a σ˙ = J ω˙˜ n = ζ˙ = I=

 1 + k p1 en , R

1 1 en − r ρ − ke ω˜ n − γz 1 − ke ωd∗ , 2 2 1 1 en − r σ − f 1 ω˜ n − γz 2 − ke ωd∗ , 2 2 km ρ + km σ − B ω˜ n − T L , −en , ξ˙ = −ω˜ n z˙ 1 = ρ, z˙ 2 = σ, ki dυ ki1 ζ − ξ − C , T L = −ki ξ + J ω˙ d + Bωd∗ . km dt

(3.76) (3.77) (3.78) (3.79) (3.80)

The state of this dynamics is ys = [ω, ˜ ξ, σ, ρ, z 2 , z 1 , e, ζ] ∈ R8 . Notice that (3.76)– (3.80) are almost identical to dynamics in (3.64)–(3.66) if v, i a , ω, i, τ L are replaced by en , ea , ω˜ n , I, T L , respectively. In this respect, it is stressed that (3.65) is replaced by (3.77) and (3.78). This is because the closed-loop motor electrical dynamics is divided into two parts: ρ is intended to represent dynamics of the armature electric current error, whereas σ is to be handled as a low-pass filtering of the velocity error. Adding expressions in (3.77) and (3.78) and using definition ea = ρ + σ it is possible to write: 1 1 L a e˙a = en − r ea − ke ω˜ n − γz 1 − ke ωd∗ − f 1 ω˜ n − γz 2 − ke ωd∗ , 2 2 where similarities with (3.65) are more evident (terms −γz 1 − 21 ke ωd∗ − f 1 ω˜ n − γz 2 − 21 ke ωd∗ have not any analogy with (3.65)). Four additional differential equations are present in (3.80) which represent the integrals of the state variables en , ω˜ n , ρ, σ. Finally, an additional difference exists between dynamics in (3.64)–(3.66) and dynamics in (3.76)–(3.80): coefficients R1 and Ra , appearing in (3.64)–(3.66), have been replaced by R1 + k p1 and r in (3.76)–(3.80). This difference is important since the latter parameters are larger, i.e., recalling (3.69) it is not difficult to realize that large values of these parameters are useful to ensure the desired closed-loop stability properties. In this respect, although the viscous friction coefficient B seems not to be larger in (3.79), it will be clear later that additional friction can be provided through σ, the low-pass filtered velocity error. Using ea = ρ + σ again, similarities between (3.79) and (3.66) are evident. With these ideas in mind, the following Lyapunov function candidate is proposed: ˜ ξ) + W2 (ρ, z 1 ) + W3 (e, ζ) + W4 (σ, z 2 ), W (ys ) = W1 (ω,

(3.81)

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3 Permanent Magnet Brushed DC-Motor

1 2 1 J ω˜ + αJ ωξ ˜ + ki ξ 2 , 2 2 1 1 2 W2 (ρ, z 1 ) = L a ρ + pL a z 1 ρ + γz 12 , 2 2 1 1 W3 (e, ζ) = Ce2 + δCeζ + ki1 ζ 2 , 2 2 1 km 1 km W4 (σ, z 2 ) = L a σ 2 + β L a z 2 σ + γ z 22 . 2 f1 2 f1 W1 (ω, ˜ ξ) =

Functions W1 , W2 and W3 contain the kinetic “energy” in the mechanical subsystem, the magnetic “energy” in the motor electrical subsystem and the electric “energy” in the converter output capacitor, respectively. These terms are suggested by the passivity property remarked in Sect. 3.7.2 and the similarities existing between the open-loop dynamics and the closed-loop dynamics pointed out above. Furthermore, W1 , W2 , W3 also contain what can be interpreted as the “energies” due to integrals of the velocity error, the armature electric current error and the capacitor voltage error. The cross terms are introduced in order to force W˙ to contain negative quadratic terms in the integrals of the above-cited errors. Reason for this is that closed-loop stability has to be shown despite the presence of the time dependent function ωd∗ which is to be handled as a bounded disturbance. Thus, a strict Lyapunov function is required for this. Finally, although W4 is built according to the above description, it is important to say that W4 is so contrived that velocity filtering σ can be handled in a similar manner as high-pass position filtering is handled in [128]. The time derivative of W along the trajectories of the closed-loop dynamics on the sliding surface s = 0, i.e., (3.76)–(3.80), can be bounded as W˙ ≤ −y Qy + y |x|,

(3.82)

where y = [|ω|, ˜ |ξ|, |σ|, |ρ|, |z 2 |, |z 1 |, |e|, |ζ|] , x is a bounded scalar function of ∗ ωd and ω˙ d which is zero when both ωd∗ = 0 and ω˙ d = 0, and Q is a 8 × 8 symmetric matrix whose entries are defined in (B.48). Using (3.82), Theorem 2.29 can be invoked to prove that ys ∈ R8 is bounded and converges to zero as t → ∞ because |x(t)| = 0, ∀t ≥ t f , i.e., ωd∗ = 0 and ω˙ d = 0, ∀t ≥ t f . Conditions ensuring this result are α > 0, k i = ki − α2 J > 0, k i1

p > 0, γ = γ − p 2 L a > 0, δ > 0, p = ki1 − δ 2 C > 0, β > 0, f 1 = km , β

and a suitable choice for α > 0, ki > 0, r > 0, β > 0, r > 0, p > 0, γ > 0, k p1 > 0, ki1 > 0 and δ > 0, such that the eight leading principal minors of matrix Q, defined in (B.48), are positive. These conditions ensure that Lyapunov function W is positive definite and radially unbounded and matrix Q is positive definite.

3.7 Velocity Control Using a DC/DC Buck Power Converter As Power Amplifier

91

Remark 3.9 The stability result in Proposition 3.8 is possible thanks to the fact that all diagonal entries of matrix Q, defined in (B.48), can suitably be selected to dominate the out-of-diagonal entries. Instrumental for this is the plant passivity property described in (3.69) because it indicates that coefficients in − R1 v 2 − Ra i a2 can be enlarged to dominate some undesired cross terms. In this respect, notice that i = i ∗ is to be used as the input of (3.64). Hence, the negative coefficient − R1 can be arbitrarily enlarged by defining i ∗ as possessing term k p1 e as in (3.72). Notice that this also allows to introduce an integral action driven by the capacitor voltage error. Similarly, the capacitor voltage v is to be used as the input for the motor electrical dynamics. Hence, the negative coefficient −Ra can be arbitrarily enlarged by defining υ as possessing term −ra ea as defined in (3.73). This also allows to include an integral action on the armature electric current error ea . On the other hand, notice that (3.69) also implies that the negative coefficient of term −Bω 2 can arbitrarily be enlarged to introduce damping. This would require to define the input to the mechanical dynamics, i.e., the desired armature electric ˜ However, because of the filtering current i a , as possessing a term such as −k p2 ω. property of the linear first order armature electric current dynamics, acceleration measurements would be required to force term −k p2 ω˜ to appear at the mechanical t dynamics. To avoid this drawback, i a = ki2 0 ω(τ ˜ )dτ is defined in (3.73) and term f k p2 ω˜ is included in definition of υ, the input of the motor electrical dynamics. Although this results in low-pass velocity error filtering, when it is seen from the motor mechanical dynamics, introduction of suitable damping is still possible by proceeding as in [128]. There, damping is obtained from high-pass position filtering which is equivalent to low-pass velocity filtering. Important for this approach is to recall that DC-motor classical PI velocity control is analogue to DC-motor classical PD position control. It is important to stress that despite the definitions k p2 = k p /km and ki2 = ki /km , it is not necessary to know exactly km : both k p and ki have to satisfy the stability conditions listed above but this does not require an exact value for them which implies that both expressions k p2 = k p /km and ki2 = ki /km can be rendered true just by using large enough values for k p2 , ki2 and the other controller gains. Moreover, according to the first expression in (3.73), the armature resistance Ra has to be exactly known. However, this requirement can be relaxed if the controller gain ra is chosen large enough such that ra  Ra , i.e., that r ≈ ra . Similarly, the exact knowledge condition on the Buck converter output resistance R, imposed in (3.72), can be relaxed if a large controller gain k p1 is selected. Furthermore, as explained after (3.75), this control strategy is expected to be robust and, hence, uncertainties in both Ra and R are expected to be compensated by the PI controllers acting on the converter output voltage error and the motor armature electric current. Some experiments are presented in Sect. 3.7.4 which confirm these observations. Finally, condition (3.75) can be ensured to be satisfied by designing a suitable desired velocity profile ωd as explained in the following. Using the differential Flatness property of the DC to DC Buck converter driven DC-motor system we have that [161]

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     B L a + J Ra B Ra + k e k m J La ω¨ + ω˙ + ω, km km km     J LaC B L a C + J Ra C + J L a G ω (3) + ω¨ i= km km     BG Ra + ke km G + B B L a G + Ra J G + J + B Ra C + k e k m C ω˙ + ω, + km km 

v=

where G = R1 . Assuming a perfect tracking, i.e., ω = ωd and i = i ∗ , the previous ∗ expressions can be used to evaluate numerically υ + L didt for a given ωd and verify whether (3.75) is satisfied or not. Note that this task is to be performed offline.

3.7.4 Experimental Results A block diagram of the experimental setup is shown in Fig. 3.20. The DC to DC Buck power converter employs the NTE2984 N-channel MOSFET and the MUR840 diode as switching devices. A brushed DC-motor model GNM5440E from Engel is used, which has been equipped with an E6B2-CWZ6C incremental encoder from Omron. Velocity is computed through numerical differentiation of the rotor position. Two Tektronix A622 AC/DC Current Probes measure i and i a and a Tektronix P5200 Differential Voltage Probe is used for measurement of υ. On the other hand, controller is implemented in the Control Block (also shown in Fig. 3.20). This is achieved by building a block diagram in Matlab-Simulink (shown in Fig. 3.21) which is executed using the DS1104 card from dSPACE. The block diagram shown in Fig. 3.21 has the following three components. (i) Input signals: this component acquires all the system measurements, i.e., i, υ, i a and ω. (ii) Desired velocity: the desired velocity trajectory ωd (t) is programmed in this block. (iii) Controller: the control law is implemented in this block. Output of this block is u, i.e., the inverted value of u. This is done because the opto-isolator NTE3087 delivers at the output the inverted value of signal at its input, hence, retrieving u. The NTE3087 provides electrical isolation between the DS1104 card and the DC to DC Buck power converter system. The numerical parameters of the brushed DC-motor model GNM5440E are the following: L a = 2.22 × 10−3 H, Ra = 0.965 , km = 120.1 × 10−3 Nm/A, b = 129.6 × 10−3 Nm/(rad/s), J = 118.2 × 10−3 kgm2 , ke = 120.1 × 10−3 V/(rad/s). The DC to DC Buck converter has the following numerical values: R = 28.5 /50 W, C = 114.4 µF, L = 68.6 mH, E = 52 V. The desired velocity was proposed to be given as

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93

Fig. 3.20 Block diagram of the DC to DC Buck converter driven DC-motor system used to perform the experiments

Fig. 3.21 Control Block implemented using Matlab-Simulink

      ωd (t) = ωd (ti ) + ωd t f − ωd (ti ) ϕ t, ti , t f ,

(3.83)

where ϕ(t, ti , t f ) is the following function which interpolates between 0 and 1 using a sixth degree polynomial:

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Fig. 3.22 Experimental results under nominal conditions

⎧ ⎨ 0,   for t ≤ ti ϕ t, ti , t f = p 3 (t) 20 − 45 p(t) + 36 p 2 (t) − 10 p 3 (t) , for t ∈ (ti , t f ) , ⎩ 1, for t ≥ t f t − ti p(t) = , (3.84) t f − ti 



with ti = 0 s, t f = 1.46 s, ωd (ti ) = 0, ωd (t f ) = 20. Using the nomenclature introω d = ωd (t f ) and ωd∗ (t) = ωd (ti ) + duced    3.8  this yields   in Proposition ωd t f − ωd (ti ) ϕ t, ti , t f − ωd t f . The controller parameters used in all of the experiments are k p1 = 29, ki1 = 2, k p2 = 0.8326, ki2 = 9.1590, ra = 0.5, r = Ra + ra = 1.465, γ = 50, f = 1, β = 0.15, p = 0.5, α = 0.78, f 1 = βp km = 0.4003, δ = 0.0005. It is important to say that these control parameters were chosen such that all of the stability conditions, summarized at the end of proof of Proposition 3.8, are satisfied. The experimental results obtained when the DC to DC Buck converter driven DC-motor system has the nominal parameters listed above are presented in Fig. 3.22. Notice the nice velocity response: the desired velocity ωd (t) and the actual motor velocity ω overlap all the time. This is achieved despite the noisy shapes of i and i ∗

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95

Fig. 3.23 Experimental results when changes on both R and E appear

and the slightly oscillatory shape of i¯a at the beginning of the experiment. Also note that v and v¯ overlap all the time. Also, condition (3.75) is satisfied all the time since E = 52 V. An experiment intended to test robustness of the controller is presented in Fig. 3.23. Resistance R, at the output capacitor of the Buck converter, changes to 7.56  for t > 3 s and the DC power supply voltage E changes to 30 V for t > 5 s. Notice, again, that the desired velocity ωd (t) and the actual motor velocity ω overlap all the time despite the above-cited changes in the numerical values of the system parameters. We observe that these parameter changes only have some effect on values of the variables i and i ∗ . Condition (3.75) is again satisfied all the time since E ≥ 30 V.

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Fig. 3.24 Experimental results when an external load torque is applied

Finally, a test is presented in Fig. 3.24 where the external torque disturbance due to a brake system is applied for t > 2.8 s. Again, the desired velocity ωd (t) and the actual motor velocity ω overlap all the time. The only effects produced by the external load torque are small changes in i a , i¯a , i and i ∗ . Condition (3.75) is satisfied all the time since E = 52 V.

Chapter 4

Permanent Magnet Synchronous Motor

It is widely recognized at present that use of permanent magnet (PM) synchronous motors1 presents a number of advantages with respect to use of PM brushed DCmotors [26, 89, 111, 133]. For instance, the normal operation of a PM synchronous motor does not require mechanical brushes nor commutators. This means that expensive maintenance procedures are not required and, hence, operation cost is importantly reduced. Moreover, there is a number of other advantages that motivate the use of PM synchronous motors in industrial applications: motor heat dissipation is more efficient since the motor windings are located on the stator, torque production is high, friction is reduced, size is small, and reliability is high [26, 111, 187, 222]. However, it is also known that control of PM synchronous motors is more complicated because of the nonlinear and multivariable nature of the mathematical model. As explained in Chap. 1, we are interested in proposing control strategies that are simple to implement. In particular, those control strategies that are very similar to field-oriented control (FOC). Our contribution consists in providing formal stability proofs and, hence, under mild assumptions these proposals explain, to some extent, why FOC works well in practice. Motivated by these challenges, in this chapter, we present several field-oriented-based controllers for both velocity and position regulation in PM synchronous motors. This chapter is organized as follows. The dq model of a PM synchronous motor is derived in Sect. 4.1. As a result of this analysis, the field-oriented control approach is straightforward explained for PM synchronous motors in this section. Paving the way for the main results in this chapter, in Sect. 4.2, it is shown how energy naturally exchanges between the electrical and the mechanical subsystems of the motor. The velocity control problem is solved in Sect. 4.3 and the position control problem is 1 Acronym PMSM is used in the literature to designate both permanent magnet synchronous motors

and permanent magnet stepper motors. In order to avoid any confusion, in the present book PM synchronous motor and PM stepper motor are used to designate each one of these classes of motors. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 V. M. Hernández-Guzmán et al., Energy-Based Control of Electromechanical Systems, Advances in Industrial Control, https://doi.org/10.1007/978-3-030-58786-4_4

97

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Fig. 4.1 A one pole pair PM synchronous motor

solved in Sect. 4.4. The velocity ripple minimization control problem is described and solved in Sect. 4.5. Finally, two practical PM synchronous motors are modeled in Sects. 4.6 and 4.7.

4.1 Motor Modeling 4.1.1 The Working Principle A PM synchronous motor is a three-phase AC-motor provided with phase windings on the stator and a permanent magnet at rotor. A one pole pair (n p = 1) PM synchronous motor is shown in Fig. 4.1. Its working principle is described as following: The three phases are symmetrical, i.e., they have the same structure, but they radians) the are wound on the stator at an orientation which differs by 120◦ ( 2π 3 orientation of the other phase windings. If electric current through each phase is given as I1 = Im sin(ω S t),

(4.1)

2π ), 3 2π I3 = Im sin(ω S t + ), 3

I2 = Im sin(ω S t −

then a constant magnitude flux which rotates counter-clockwise, with angular velocity ω S , around the internal surface of stator is obtained (see [55], Chaps. 4, 7, for an analytical manner to verify this fact). This rotating magnetic flux, due to the stator electric currents, exerts an attractive force on the permanent magnet at rotor producing its movement. Word synchronous is used to designate this class of motor to stress the fact that, in steady state, the angular velocity of rotor ω R equals the angular velocity of the magnetic flux due to the stator windings, i.e., ω R = ω S . Under these

4.1 Motor Modeling

99

conditions the generated torque is proportional to the sine function of the difference between the angular position of the magnetic flux due to the stator windings and the angular position of the magnetic flux of the permanent magnet at rotor (at this point it is important to recall Remark 3.1 where it is claimed that the maximal torque, due to interaction between two magnetic fields, is produced when axes of the respective fluxes are orthogonal, i.e., when their angular positions differ by 90◦ ). It is clear that this difference of angular positions is determined by load applied to rotor [55], Chap. 5, since the generated torque equals the load torque (friction effects included) in steady state. What we have just described is also known as open-loop control of a PM synchronous motor.

4.1.2 Three-Phase Dynamic Model The following assumptions are commonly considered when modeling PM synchronous motors [204], Chap. 9, [60], Appendix A, [55], Chap. 7: A1. A2. A3. A4. A5.

Core has infinite permeability and is fully laminated. Saturation, iron losses, end winding and slot effects are neglected. Only linear magnetic materials are considered. All parameters are constant. Rotor is composed by a permanent magnet whose magnetic flux is sinusoidally distributed along the internal side of stator. A6 . Three phases are wound on stator. These windings are symmetrical and star connected. Each one of these windings is sinusoidally distributed on the internal side of stator. Remark 4.1 The last part of Assumption A6 means that turn densities of each phase, Nφi , i = 1, 2, 3, are assumed to be given as (see Fig. 4.2): N 2 N Nφ2 (θ S ) = 2 N Nφ3 (θ S ) = 2 Nφ1 (θ S ) =

| sin(θ S )|,     sin(θ S − 2π ) ,  3      sin(θ S + 2π ) ,  3 

where N is the total number of turns in each phase and θ S is angle defined in Fig. 4.1 as the angular position on the stator. Thanks to this, it can be assumed that a magnetic field is produced by each phase which is sinusoidally distributed around the internal side of stator (see Fig. 4.3). In practice, however, this assumption is approximated by using two layer windings as shown in Fig. 4.1 which produces a staircase magnetic field which is approximately sinusoidal [55], Chap. 7 (see Fig. 4.3).

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Fig. 4.2 Sinusoidally distributed phase windings 1 and 2 Fig. 4.3 A staircase magnetic field distribution as an approximation to sinusoidally distributed magnetic flux

Remark 4.2 PM synchronous motors having an uniform air gap, as in Fig. 4.4, are said to be non-salient rotor motors or round rotor motors. However, a sinusoidally distributed magnetic flux (Assumption A5) can also be obtained using a permanent magnet with an uniform magnetic flux by shaping the magnet faces of the rotor appropriately as shown in Fig. 4.5. Although air gap is not constant, construction is less expensive in this case [55]. Motors having a rotor as in Fig. 4.5 are said to be salient rotor motors. Remark 4.3 Notice that the returning paths for all the magnetic fluxes produced by both, the stator phase windings and the rotor, is the external side of the stator. This explains importance of the linear magnetic materials and the no saturation Assumptions in A2 and A3. On the other hand, the infinite permeability core Assumption (A1) is useful to consider that reluctance of the magnetic circuits is due only to the air gap, i.e., simplifying computation of magnetic fluxes. Since the stator windings are star connected (Assumption A6), stator must be fed as shown in Fig. 4.6. Notice that because of the star connection of the stator phase windings and the source voltages we have that I1 (t) + I2 (t) + I3 (t) = 0.

(4.2)

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101

Fig. 4.4 A round rotor motor. Rotor is perfectly cylindric and, hence, the air gap is constant. If the permanent magnet is well designed, this also allows to produce a sinusoidally distributed magnetic flux along the air gap

Fig. 4.5 A salient rotor motor. Rotor has not a perfectly cylindric shape and, hence, the air gap is not constant. This is so contrived to achieve a sinusoidally distributed magnetic flux along the air gap even when magnetic flux produced by rotor is not designed to distribute sinusoidally. The main reason to proceed like this is that it is more expensive to built a permanent magnet with a sinusoidally distributed magnetic flux

Then, the phase electric currents are said to be balanced.2 Let v1 Nˆ , v2 Nˆ , v3 Nˆ denote the phase to motor neutral voltages. In [55], pp. 423, it is shown that in the case when the source voltages are balanced, i.e., when3 : is the case when I1 (t), I2 (t), I3 (t) are given as in (4.1) or in terms of cosine functions. is the case when V1 (t), V2 (t), V3 (t) are given as in (4.1), i.e., V1 = Vm sin(ω S t), V2 = 2π Vm sin(ω S t − 2π 3 ), V3 = Vm sin(ω S t + 3 ), or in terms of cosine functions.

2 This 3 This

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Fig. 4.6 Feeding a PM synchronous motor using a three phase generator

V1 (t) + V2 (t) + V3 (t) = 0,

(4.3)

then V1 = v1 Nˆ , V2 = v2 Nˆ , V3 = v3 Nˆ , v Nˆ N = v Nˆ − v N = 0. Hence, applying Kirchhoff’s Voltage Law, Faraday’s Law, and the Ohm’s Law to each phase winding it follows that Λ˙ + R I = V,

(4.4)

where V = [V1 , V2 , V3 ] ,

I = [I1 , I2 , I3 ] ,

(4.5)

R is a positive constant scalar representing resistance of each phase winding, Λ ∈ R3 stands for the stator phase winding flux linkages: Λ = L I + Γ,

(4.6)

where the contribution of the rotor permanent magnet is given as      2π 2π  , sin θ + , Γ = K E sin(θ), sin θ − 3 3

(4.7)

being K E > 0 a constant and θ represents the rotor position as defined in Fig. 4.1. The inductance matrix is given as:

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103

L = L 1 + L 2 + L 3,

(4.8)

L 1 = diag{L s + L A , L s + L A , L s + L A } ∈ R , ⎡ ⎤ 0 − 21 L A − 21 L A L 2 = ⎣ − 21 L A 0 − 21 L A ⎦ , − 1 L A − 21 L A 0 ⎡ 2 ⎤ −L B cos(2θ) −L B cos(2θ − 2π ) −L B cos(2θ + 2π ) 3 3 L 3 = ⎣ −L B cos(2θ − 2π ) −L B cos(2θ + 2π ) −L B cos(2θ) ⎦ . 3 3 2π ) −L B cos(2θ + 3 ) −L B cos(2θ) −L B cos(2θ − 2π 3 3×3

Remark 4.4 The simple expressions in (4.7) and (4.8) are obtained thanks to Assumptions A5 and A6, i.e., the magnetic fluxes due to rotor and the phase windings are sinusoidally distributed on the internal side of stator. Notice that defining the rotor position θ as in Fig. 4.1 (i.e., as the angular position of the line which divides the rotor permanent magnet in two poles), flux linkage due to rotor flux through phase 1 becomes zero when θ = 0 (because axis of the phase winding 1 is orthogonal to the rotor flux direction) which is correctly predicted by the first component in (4.7). See [55], Chap. 7, for an analytical manner to find (4.7). On the other hand, the inductance matrix L is explained as follows. L 1 stands for the self inductance component of L. Self inductance of each phase winding is composed by two parts: the leakage inductance L s and the magnetizing inductance L A . The leakage inductance L s is introduced to take into account the fact that a fraction of flux produced by electric current Ii , flowing through phase i, does not produce any effect on the other phase windings. This is called flux leakage and is due to imperfect magnetic coupling among phase windings. However, the leakage inductance must be taken into account as a part of the self inductance of each phase winding. Notice that, being a self inductance matrix, L 1 is given as a diagonal matrix. L 2 stands for the phase winding mutual inductance. The mutual inductance of two perfectly aligned windings equals the magnetizing inductance L A . Taking advantage from Assumption A6, i.e., the sinusoidal distribution of the phase windings on the internal side of stator, it is analytically shown in [55], Chap. 7, that mutual inductance between the phase windings in Fig. 4.1 is − A2 . Sign “−” indicates that a positive electric current through phase i (the positive direction of electric currents are defined in Fig. 4.1) produces a magnetic flux having a component in the opposite direction of flux produced by a positive electric current through the other phase windings. This explains the form of L 2 . L 3 is a matrix used to take into account the self inductance and mutual inductance variations due to a nonuniform air gap in a salient rotor motor. According to Fig. 4.5, whenever rotor completes a turn the air gap in phase 1 reaches two minimal values and two maximal values. Further, self inductance of phase 1 is maximal when the air gap is minimal, i.e., when θ = π/2[rad] and θ = 3π/2[rad], which is correctly predicted by the first diagonal element of L 3 which becomes −L B cos(2θ) = +L B in both cases. Notice, however, that L A > L B and, hence, self inductance variations due to a nonuniform air gap are always smaller than the magnetizing inductance. On the other hand, according to L 2 and L 3 , mutual induc-

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Fig. 4.7 Effect of the (salient) rotor orientation on the magnetic flux produced by the phase windings on the stator

tance between phase 1 and phase 2 is given by − 21 L A − L B cos(2θ − 2π ). When 3 1 θ = π3 it is obtained − 21 L A − L B cos(2θ − 2π ) = − L − L , which means that A B 3 2 mutual inductance between phases 1 and 2 are increased. Notice that, according to Fig. 4.7(a) this is because rotor material deviates flux produced by phase 2 in a further opposite direction to flux produced by a positive current through phase 1 (see description of matrix L 2 in the previous paragraph). On the other hand, when ) = − 21 L A + L B , which means that θ = − π6 it is obtained − 21 L A − L B cos(2θ − 2π 3 mutual inductance between phases 1 and 2 are decreased. According to Fig. 4.7(b), this is because rotor material deviates flux produced by phase 2 in a direction which reduces the opposition to flux produced by a positive current through phase 1 (see description of matrix L 2 in the previous paragraph). The other entries of matrix L 3 can be explained in a similar manner. Notice that it suffices to assume B = 0 to retrieve the case of a round rotor motor from (4.8). Finally, replacing (4.6) in (4.4) yields V = RI +

d (L I ) + Γ˙ dt

(4.9)

Expression in (4.9) represents the three-phase model of the electrical subsystem of a PM synchronous motor. This is a set of three ordinary nonlinear differential equations. The main problem with (4.9) is its structure: it is not clear how to design a controller from (4.9). It is shown in the following that this problem can be simplified by using a suitable coordinate transformation.

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Fig. 4.8 Projection of the three phases (1,2,3) in a PM synchronous motor on two stator phases (d,q) which rotate as if they were fixed to rotor

4.1.3 Park’s Transformation or dq Transformation In order to simplify the dynamical model in (4.9), it is common to define a coordinate transformation which converts this three-phase dynamical model in a two phase dynamical model. Two fictitious phase windings are defined which, although placed on stator, are assumed to rotate at the same angular velocity than rotor. These two phase windings are designated as “d” and “q” where “d” means “direct”, i.e., axis of this phase winding is aligned to the magnetic flux due to rotor, whereas “q” means “quadrature”, i.e., axis of this phase winding is orthogonal to axis of phase winding “d”. This situation is shown in Fig. 4.8. The disposition of components in Fig. 4.8 is intended to indicate that loops of phase 1 are wound on the stator horizontally. Hence, the positive direction of axis of phase 1 (also defined in Fig. 4.8 and determined by the positive sense of current shown in Fig. 4.1 and the right-hand rule) and the positive direction of rotor flux (from pole S to pole N) define a 90◦ angle when θ = 0. This means that “d” axis must be horizontal and pointing to the right when θ = 0 (the magnetic flux due to electric current through phase “d” is aligned to rotor flux). Notice that this description is in complete agreement with structure of Γ , the phase winding flux linkages due to rotor flux, defined in (4.7) which was explained in Remark 4.4. Also notice that the disposition of components in Fig. 4.8 agrees too with structure of matrix L, defined in (4.8), which was also explained in Remark 4.4. Phases “d” and “q” are introduced as an attempt to render the PM synchronous motor similar to a PM brushed DC-motor where the generated torque is given by

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4 Permanent Magnet Synchronous Motor

the product τ = km i. Recall that km is a constant that summarizes the contribution to torque generation of the permanent magnets fixed to stator (see Remark 3.2). On the other hand, i is the electric current flowing through the rotor windings. Furthermore, in Remark 3.1, it was claimed that a maximal torque is produced by the interaction between two magnetic fields when their axes are orthogonal. Moreover, this is exactly what happens in a PM brushed DC-motor where one magnetic field is due to permanent magnets on stator and the other magnetic field is produced by electric current flowing through the rotor windings. This situation is reproduced in a PM synchronous motor because phase “q” (on the stator) is always orthogonal to the rotor flux (due to a permanent magnet), i.e., orthogonal to phase “d”. Notice that under these conditions rotor flux is always constant from the phase “q” point of view since they are always orthogonal. Furthermore, because of this, a maximal torque is produced by the interaction between rotor flux and the magnetic field due to Iq , the electric current through phase “q” (on the stator). Hence, the generated torque can be obtained as the product τ = Φ M Iq , where Φ M is a constant that summarizes the contribution to torque generation of the permanent magnet fixed to rotor. Notice that Φ M plays a similar function in a PM synchronous motor as km in a PM brushed DC-motor and, because of this, Φ M is called the motor torque constant or the motor back electromotive force constant (recall (3.8)). According to the above discussion, it is concluded that the positive direction of phase “q” has to be defined as in Fig. 4.8. This ensures the generated torque to force rotor to move counter-clockwise, i.e., in the positive definition of θ, because the generated torque always tries to align the magnetic fluxes of phase “q” and rotor (see Remark 3.1). Finally, it is important to stress the following. Since rotor flux ensures suitable torque generation and the magnetic flux due to phase “d” is aligned to rotor flux, a zero electric current through phase “d”, i.e., Id = 0, does not avoid motor to work correctly. Voltages and electric currents in phases “d” and “q” are computed by projecting voltages and electric currents in phases 1, 2, and 3 on the positive directions defined for phases “d” and “q” in Fig. 4.8, i.e., Vq = V1 cos(θ) − V2 cos(θ + 60) − V3 cos(60 − θ), = V1 cos(θ) − V2 cos(θ + 60) − V3 cos(θ − 60), = V1 cos(θ) + V2 cos(θ + 60 − 180) + V3 cos(θ − 60 + 180), = V1 cos(θ) + V2 cos(θ − 120) + V3 cos(θ + 120),     2π 2π + V3 cos θ + , (4.10) Vq = V1 cos(θ) + V2 cos θ − 3 3 Vd = V1 cos(90 − θ) − V2 cos(120 − 90 − θ) + V3 cos(θ + 120 − 90), = V1 cos(θ − 90) − V2 cos(θ − 120 + 90) + V3 cos(θ + 120 − 90), = V1 sin(θ) + V2 sin(θ − 120) + V3 sin(θ + 120),     2π 2π  + V3 sin θ + , (4.11) Vd = V1 sin(θ) + V2 sin θ − 3 3

4.1 Motor Modeling

107

and, hence,     2π 2π + I3 cos θ + , Iq = I1 cos(θ) + I2 cos θ − 3 3     2π 2π + I3 sin θ + . Id = I1 sin(θ) + I2 sin θ − 3 3

(4.12) (4.13)

The direct Park’s or dq transformation is obtained by introducing  Vq =

2  V , Vd = 3 q



2  V , 3 d

 Iq =



2  I , 3 q

Id =

2  I , 3 d

to write (4.10), (4.11), (4.12), (4.13) as VN = T V,





 ⎡ cos(θ) cos θ − 2π cos θ + 2π ⎤ 3 3



2⎣ ⎦(4.14) sin θ + 2π sin(θ) sin θ − 2π I N = T I, T = 3 3 3 1 1 √ √ √1 2

2

2

To obtain this, (4.5) and VN = [Vq , Vd , V0 ] ,

I N = [Iq , Id , I0 ] ,

(4.15)

have been used, where V0 and I0 are the zero phase voltage and electric current, respectively. It is important to say that in the electric machines community there are several slightly different manners to define the transformation matrix T . Some of these definitions are introduced for power conservation in the new phases or in order to render orthogonal matrix  T . The latter is the reason for definition in (4.14), i.e., for

introduction of factor 23 . This means that T −1 = T  for matrix defined in (4.14). From (4.14) we obtain  I0 =

1 [I1 + I2 + I3 ], V0 = 3



1 [V1 + V2 + V3 ]. 3

Using this, the fact that the three phase windings are star connected and assuming that the source voltages are balanced, i.e., (4.2), (4.3), it is found that I0 = 0, V0 = 0

(4.16)

It is not difficult to show that the inverse Park’s transformation, i.e., the inverse dq transformation, is given as

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4 Permanent Magnet Synchronous Motor

V = T  VN , I = T  I N , ⎤ ⎡ ⎤  ⎡ Iq cos(θ) + Id sin(θ) + √12 I0 I1

⎥ 2π 2π 1 ⎣ I2 ⎦ = 2 ⎢ ⎣ Iq cos θ − 3 + Id sin θ − 3 + √2 I0 ⎦ , 3 I3 + Id sin θ + 2π + √12 I0 Iq cos θ + 2π 3 3 ⎡ ⎤ ⎡ ⎤  Vq cos(θ) + Vd sin(θ) + √12 V0 V1

⎥ 2π 2π 1 ⎣ V2 ⎦ = 2 ⎢ ⎣ Vq cos θ − 3 + Vd sin θ − 3 + √2 V0 ⎦ , 3 2π 2π 1 V3 Vq cos θ + 3 + Vd sin θ + 3 + √2 V0

(4.17)

which under conditions in (4.16) becomes ⎤  ⎡ ⎤ + Id sin(θ) I1

Iq cos(θ) 2 ⎣ I2 ⎦ = ⎣ Iq cos θ − 2π + Id sin θ − 2π ⎦ , 3 3 3 I3 Iq cos θ + 2π + Id sin θ + 2π 3 3 ⎤ ⎡ ⎤  ⎡ V q cos(θ) + Vd sin(θ) V1

2 ⎣ Vq cos θ − 2π + Vd sin θ − 2π ⎦ . ⎣ V2 ⎦ = 3 3 3 2π V3 sin θ + Vq cos θ + 2π + V d 3 3 ⎡

(4.18)

The expression for applied voltages in (4.18) is especially true if V0 = 0 is selected as a part of the control strategy. It is important to say at this point that Park’s transformation, or dq transformation, is very useful to design a controller in terms of the fictitious voltages Vq , Vd , V0 because this problem simplifies using these new coordinates. In the next section, it is shown that the simplest and most effective solution is obtained by choosing V0 = 0. This is due to the fact that the zero phase has not any effect on neither the mechanical subsystem nor the other phases, i.e., phases “d” and “q”. In this order of ideas, it is convenient to say that expressions in (4.14) are useful when computing the fictitious electric currents, Iq and Id , from the real three-phase electric currents, I1 , I2 and I3 . As it will become clear later, this is instrumental to construct inner electric current loops as in the case of a PM brushed DC-motor (see (3.23) and Fig. 3.16(a)). On the other hand, expressions in (4.18) are useful when computing the real three-phase voltages to be applied to motor, V1 , V2 , and V3 , from the fictitious voltages Vq and Vd , which are given directly by the controller.

4.1.4 The dq Dynamic Model Generally speaking, a PM synchronous motor may have several pole pairs as a means to increase the generated torque (a pole pair is composed by two poles: one N pole and one S pole). In order to achieve this, the stator windings and the rotor permanent magnets have to be modified as in Fig. 4.9 where a two pole pairs (n p = 2, i.e., four poles) PM synchronous motor is shown. Note that in this case each phase is composed by two windings a, −a and a  , −a  for phase 1, b, −b, and b , −b for phase 2 and c, −c and c , −c for phase 3. Also note that, although placed at different points

4.1 Motor Modeling

109

Fig. 4.9 A two pole pairs PM synchronous motor

of stator, these two windings at each phase receive the same magnetic effect from rotor, because the rotor permanent magnets have two N and S poles. This and the fact that these two windings are series connected, implies that the generated torque doubles torque generated by an identical motor having only one pole pair. The reader is referred to Sect. 4.6 to see another manner to built a PM synchronous motor with several pole pairs. Note, however, that because of the geometry shown in Fig. 4.9, all the above discussion is still valid for a motor with n p pole pairs if it is defined θ = n p q, where q is the (mechanical) rotor position (given in mechanical radians) and θ is also known as the electrical rotor position, given in electrical radians. This observation is important because the dq dynamic model is stated for a general PM synchronous motor having n p pole pairs. It was claimed earlier that dq transformation is introduced in order to obtain a simpler expression for (4.9) in the new coordinates. This is performed by using (4.14) to replace V and I in (4.9), using θ = n p q and mimicking [60], Appendix A, to obtain d (L T  I N ) + Γ˙ , dt d = RT  I N + (L T  )I N + L T  I˙N + Γ˙ . dt

T  VN = RT  I N +

Multiplying by T by the left and recalling that R is a constant scalar:

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4 Permanent Magnet Synchronous Motor

d (L T  )I N + T L T  I˙N + T Γ˙ , dt d VN = R I N + T (T  L N )I N + L N I˙N + T Γ˙ , dt     ∂  ∂ ˙ L N I N q˙ + L N I N + T VN = R I N + T T Γ q. ˙ ∂q ∂q VN = R I N + T

(4.19)

By direct computation, which includes an exhaustive use of trigonometric identities, the reader can verify that the matrix L N = T L T  = diag{L q , L d , L 0 } is constant, for positive scalars L q , L d , L 0 which are defined below, and  T

⎡ ⎤  0 Ld 0 ∂  L N = n p ⎣ −L q 0 0 ⎦ , T ∂q 0 0 0 ⎡ ⎤ 3   n K p E ∂ ⎢ 2 ⎥ Γ =⎣ T ⎦. 0 ∂q 0

(4.20)

(4.21)

Thus, replacing these expressions in (4.19), we obtain the following PM synchronous motor dynamic model: L q I˙q = −R Iq − n p L d Id q˙ − Φ M q˙ + Vq , L d I˙d = −R Id + n p L q Iq q˙ + Vd , L 0 I˙0 = −R I0 + V0 ,

(4.22) (4.23) (4.24)

where 3 3 L q = L s + (L A − L B ), L d = L s + (L A + L B ), 2 2  3 npKE. L 0 = L s , ΦM = 2

(4.25)

We stress that we have assumed (see Remark 4.4) that rotor geometry is that depicted in Fig. 4.5, i.e., that rotor saliency coincides with rotor poles. This means that rotor saliency coincides with the fictitious phase winding d which implies that air gap for phase d is smaller than for phase q. Moreover, since inductance is larger for a smaller air gap, this explains why L d > L q in (4.25). However, following ideas in [21], other rotor structures such as those with interior, or buried, permanent magnets render possible the case L q > L d . On the other hand, according to D’Alembert’s principle, the generated torque (applied on the rotor) is given as the co-energy’s derivative with respect to rotor position. Thus, mimicking again [60], Appendix A, we have

4.1 Motor Modeling

τ = = = = = =

111



 1  I LI + Γ I , 2   ∂Γ  1  ∂L I, I I+ 2 ∂q ∂q     ∂Γ   1   ∂L (T I N ) (T  I N ) + T IN , 2 ∂q ∂q     1  ∂ ∂Γ     I T (T L N T ) T I N + T IN , 2 N ∂q ∂q       ∂ 1  ∂  ∂Γ   LN T + T LN IN T T T T  IN + T IN , 2 ∂q ∂q ∂q     ∂  ∂Γ    L N IN + T IN T T IN . (4.26) ∂q ∂q ∂ ∂q

Using (4.20) and (4.21) in (4.26), we obtain τ = n p (L d − L q )Id Iq + Φ M Iq .

(4.27)

It was claimed in the paragraphs before (4.10) that this torque is applied on rotor in the positive sense of θ = n p q, i.e., in the positive sense of q. Thus, using Newton’s Second Law, it is found that the dynamic model of the mechanical subsystem is J ω˙ + bω = n p (L d − L q )Id Iq + Φ M Iq − τ L ,

(4.28)

where J and b are positive constant scalars standing for rotor inertia and viscous friction coefficient, respectively, τ L is load torque applied at rotor and ω = q˙ is the rotor mechanical velocity. Finally, notice that the zero phase voltage V0 and electric current I0 have no effect on any of the other phases nor the motor mechanical subsystem, i.e. (4.22), (4.23), (4.28). Also notice that (4.24) is linear and stable. Hence, a good design criterion is to choose V0 = 0, which was suggested at the end of Sect. 4.1.3 and justifies use of (4.18). Thus, the dq dynamical model of a salient rotor PM synchronous motor is given by L q I˙q = −R Iq − n p L d Id q˙ − Φ M q˙ + Vq , L d I˙d = −R Id + n p L q Iq q˙ + Vd ,

(4.29) (4.30)

J ω˙ + bω = n p (L d − L q )Id Iq + Φ M Iq − τ L .

(4.31)

Assuming L B = 0 (see Remark 4.4), it is obtained from (4.25) that L d = L q = L s + 23 L A = L R . Thus, the dq dynamic model of a non-salient rotor (or round rotor) PM synchronous motor is given as

112

4 Permanent Magnet Synchronous Motor

L R I˙q = −R Iq − n p L R Id q˙ − Φ M q˙ + Vq , L R I˙d = −R Id + n p L R Iq q˙ + Vd , J ω˙ + bω = Φ M Iq − τ L .

(4.32) (4.33) (4.34)

It is interesting to realize that in a round rotor motor the generated torque is given as τ = Φ M Iq , which was assumed previously in Sect. 4.1.3. In a salient rotor motor, the generated torque is given by (4.27) where the term n p (L d − L q )Id Iq is added and results from the air gap variations.

4.1.5 dq Decomposition of the Stator Magnetic Flux In the following, we use the dq decomposition of the magnetic flux produced by stator to explain some issues on the PM synchronous motor operation.

4.1.5.1

The Open-Loop Working Principle

Open-loop operation of a PM synchronous motor has been described in Sect. 4.1.1 and the corresponding situation is represented in Fig. 4.10 for n p = 1. According to the dq coordinate transformation introduced in the previous sections, the original three-phase windings are projected or decomposed in two fictitious phase windings called d and q. Electric current Id through the fictitious winding d produces a magnetic flux λd and electric current Iq through the fictitious winding q produces a magnetic flux λq . Hence, the total magnetic flux λ due to the three phase windings at stator has λd and λq as its orthogonal components. Moreover, in the previous sections it was stated that phase winding d is parallel to magnetic flux produced by the permanent magnet at rotor ψ. This means that λd is parallel to ψ and, thus, λq is orthogonal to ψ (see Fig. 4.10). This is possible in steady state since rotor, i.e., ψ, has the same angular velocity as the stator magnetic flux λ, ω R = ω S . Recall that this is the reason for the term synchronous motor. According to Sect. 4.1.1 and Remark 3.1, we have the following facts. The rotating magnetic flux λ exerts an attractive force on the magnetic flux ψ producing rotor movement. Torque on rotor is proportional to sin(γ), where γ is the difference between the angular position of λ and the angular position of ψ. The maximal torque is produced when γ =90◦ . The difference of angular positions γ is determined by load applied to rotor since the generated torque equals the load torque (friction effects included) in steady state. Notice that in the case when γ =90◦ we have that λq = λ and λd = 0. Since λd = L d Id = 0, this means that Id = 0, i.e., any electric current is not required to flow through the fictitious winding d when the maximal torque is generated. This is explained recalling that torque is produced by the interaction between the magnetic

4.1 Motor Modeling

113

fluxes λ and ψ and λ = λq = L q Iq is produced only by electric current through the fictitious winding q whereas ψ is produced only by the permanent magnet at rotor. When 0 ≤ γ 0 forcing ψ and λd to align. If θ = 0 is assigned at this configuration use of (4.18) ensures that λ and ψ will remain orthogonal once Id = 0.

4.1.6 Standard Field-Oriented Control Two block diagrams are shown in Fig. 4.11 which represent standard field-oriented control of a PM synchronous motor. Block diagram in Fig. 4.11(a) presents how the dq transformation in (4.14) has to be used to compute the fictitious electric currents Id , Iq , from the real three-phase electric currents I1 , I2 , I3 , and how the inverse dq transformation in (4.18) has to be used to compute the actual three-phase voltages V1 , V2 , V3 , to be applied to motor from the fictitious voltages Vd , Vq . On the other

4.1 Motor Modeling

115

Fig. 4.11 Standard field-oriented control of a PM synchronous motor

hand, Fig. 4.11(b) presents an equivalent block diagram which is more suitable for the exposition that follows. This control scheme has been very successful in high performance industrial applications because of simplicity and good performance. The rationale behind this control scheme is the following. Two classical PI electric current controllers: Vd = αd (Id∗ − Id ) + αdi Vq = αq (Iq∗ − Iq ) + αqi

 

t

0 t 0

(Id∗ (s) − Id (s))ds,

(4.35)

(Iq∗ (s) − Iq (s))ds,

(4.36)

are used to force Id and Iq to reach their desired values Id∗ and Iq∗ , respectively, very fast. The constant scalars αd , αdi stand for the proportional and integral gains for phase “d”, whereas αq , αqi are defined correspondingly for phase “q”. The desired electric current for phase “d” is chosen to be zero, i.e., Id∗ = 0, because this simplifies dynamics in (4.32), (4.33), (4.34) to4 L R I˙q = −R Iq − Φ M q˙ + Vq , J ω˙ + bω = Φ M Iq − τ L , and dynamics in (4.29), (4.30), (4.31) to: that a zero desired value for electric current in phase “d”, i.e., Id∗ = 0, is a suitable (and even a desired) selection since Id = 0 does not avoid a PM synchronous motor to work correctly (see the previous section and Sect. 4.1.3).

4 Notice

116

4 Permanent Magnet Synchronous Motor

Fig. 4.12 Standard field-oriented control of a PM synchronous motor presented in Fig. 4.11 reduces to control of a simple linear mechanical system

L q I˙q = −R Iq − Φ M q˙ + Vq , J ω˙ + bω = Φ M Iq − τ L , once Id = Id∗ = 0. Notice that these two dynamics are identical to the PM brushed DC motor dynamics given in (3.11), (3.12), i.e., L

di = −Ri − ke ω + v, dt J ω˙ = km i − bω − τ L .

Hence, mimicking control of a PM brushed DC-motor, the desired electric current for phase “q” is defined as Iq∗ = τ ∗ /Φ M , where τ ∗ is the desired torque. Assuming that use of (4.36) achieves Iq = Iq∗ very fast, then τ = τ ∗ can also be assumed because of Iq∗ = τ ∗ /Φ M and Iq = τ /Φ M .5 This implies that block diagram in Fig. 4.11(b) can be simplified to block diagram in Fig. 4.12. Thus, the proportional and integral gains k p and ki of the classical PI velocity controller (where ω ∗ represents the desired velocity): ∗

∗



t

τ = k p (ω − ω) + ki

(ω ∗ (s) − ω(s))ds,

(4.37)

0

can be chosen by taking into account only the mechanical dynamics, i.e., using the simplified block diagram in Fig. 4.12. A similar procedure is also valid for motor position control if the PI velocity controller in (4.37) is replaced by the classical PID position controller (where q ∗ represents the desired position): τ ∗ = k p (q ∗ − q) + kd

d(q ∗ − q) + ki dt



t

(q ∗ (s) − q(s))ds,

(4.38)

0

and the additional state equation is considered: q˙ = ω.

5 Notice

that this is just an approximation for a salient rotor motor where the generated torque is given by (4.27).

4.1 Motor Modeling

117

The integral action in controllers (4.37) and (4.38) is included in order to ensure that ω = ω ∗ and q = q ∗ are achieved in steady state when both q ∗ and ω ∗ are constant and despite the presence of some unknown but constant external torque disturbance τ L . Although dynamics in (4.29)–(4.31) and (4.32)–(4.34) are nonlinear, controller gains in (4.35), (4.36), (4.37), (4.38) are traditionally selected using approximate classical control arguments. An alternative design method is presented in the present chapter to find out some criteria for selection of the controller gains in (4.35), (4.36), (4.37), (4.38). This new methodology takes into account the nonlinear character of the closed loop dynamics and, contrary to the classical control techniques when applied to PM synchronous motors, does not rely on linear approximations. Moreover, this new methodology gives a complete explanation for the case of a salient rotor motor where the generated torque is given by (4.27). This feature must not be underestimated since term n p (L d − L q )Id Iq in the generated torque has traditionally complicated the control design task. See works in [60, 65, 222], where term n p (L d − L q )Id Iq has forced to design control schemes more complex that is shown in Fig. 4.11. Finally, recall that the star connection assumption on the stator phase windings as well as the balanced assumption on the source voltages V1 , V2 , V3 , i.e., condition stated in (4.3), are fundamental for the dynamic modeling presented in Sect. 4.1.2 and, hence, in Sect. 4.1.4. Notice that the star connection assumption on the stator phase windings is satisfied by construction of motor. On the other hand, it is not difficult to verify that (4.3) is always satisfied since the source voltages V1 , V2 , V3 are computed using (4.18) from the fictitious phase voltages Vd , Vq , which are directly given by controllers in (4.35), (4.36).

4.2 Open-Loop Energy Exchange 4.2.1 The Velocity Model According to Sect. 4.1.4, the dynamic model of a salient rotor PM synchronous motor is given by (4.29), (4.30), (4.31), which is rewritten here for the ease of reference: L q I˙q = −R Iq − n p L d Id ω − Φ M ω + Vq , L d I˙d = −R Id + n p L q Iq ω + Vd ,

(4.39) (4.40)

J ω˙ + bω = n p (L d − L q )Id Iq + Φ M Iq − τ L .

(4.41)

Variables Id , Iq , ω ∈ R represent the electric currents through the d and q phase windings as well as the motor angular velocity, respectively, with ω = q, ˙ being q the rotor angular position. Symbols Vd and Vq stand for voltages applied at phases d and q, respectively. The positive constants n p , L d , L q , R, Φ M , J, b represent the number of pole pairs, the stator inductances, the stator winding resistance, the torque constant, the rotor moment of inertia, and the viscous friction coefficient. Finally, τ L is the load torque.

118

4 Permanent Magnet Synchronous Motor

Notice that the presence of the torque constant Φ M in both expressions (4.39) and (4.41) is equivalent to property indicated in (3.28) for a PM brushed DC-motor, i.e., that torque constant equals the back electromotive force constant ke = km . Also note the presence of factors n p L d Id and n p L q Iq in (4.39), (4.40), and (4.41). It is shown in what follows that all of these terms appear because of the same reason: they are instrumental for the natural energy exchange between the motor electrical and mechanical subsystems. The following scalar function represents the total energy stored in the motor: V (Id , Iq , ω) =

1 1 1 L d Id2 + L q Iq2 + J ω 2 , 2 2 2

(4.42)

where the first and second terms represent the magnetic energy stored in the electrical subsystem whereas the third term stands for the kinetic energy stored in the mechanical subsystem. The time derivative of V is given as ˙ V˙ = Id L d I˙d + Iq L q I˙q + ω J ω, which, according to (4.39), (4.40), (4.41), can be written as: V˙ = Id (−R Id + n p L q Iq ω + Vd ) + Iq (−R Iq − n p L d Id ω − Φ M ω + Vq ) +ω(−bω + n p (L d − L q )Id Iq + Φ M Iq − τ L ). Notice that several cross terms cancel in this expression (these terms are referred in the paragraph before (4.42)). Since V represents energy stored in motor, we conclude that these cross term cancellations (appearing in V˙ ) represent the energy exchange between the motor electrical and mechanical subsystems. Hence, this yields V˙ = −R Id2 − R Iq2 − bω 2 + Id Vd + Iq Vq − τ L ω.

(4.43)

Defining the input u = [Vd , Vq , −τ L ] and the output y = [Id , Iq , ω] , we can write (4.43) as ⎡

V˙ = −y  Qy + y  u,

⎤ R 0 0 Q = ⎣ 0 R 0⎦. 0 0 b

(4.44)

Since Q is a positive definite matrix, (4.44) shows that model (4.39), (4.40), (4.41), is output strictly passive (see Definition 2.42) for the output y and input u defined above. In this chapter, it will be shown that this is the basic theoretical principle behind the common and successful industrial velocity control scheme depicted in Fig. 4.11.

4.2 Open-Loop Energy Exchange

119

4.2.2 The Position Model If the rotor position is designated by q then ω = q. ˙ Assume that the load torque is given as a nonlinear function of position, i.e., τ L = G(q), which is given as the . Using these ideas gradient of a positive semidefinite function P(q), G(q) = d P(q) dq and the dynamic model of a salient rotor PM synchronous motor given by (4.29), (4.30), (4.31), in Sect. 4.1.4, we can write L q I˙q = −R Iq − n p L d Id q˙ − Φ M q˙ + Vq , L d I˙d = −R Id + n p L q Iq q˙ + Vd ,

(4.45) (4.46)

J q¨ + bq˙ = n p (L d − L q )Id Iq + Φ M Iq − G(q).

(4.47)

The following scalar function represents the total energy stored in motor and load: ˙ = V (Iq , Id , q, q)

1 1 1 L q Iq2 + L d Id2 + J q˙ 2 + P(q), 2 2 2

(4.48)

where the last (new) term represents the potential energy stored in the load. Notice that V , given in (4.48), is a positive semidefinite function because P(q) is assumed to have this property. Proceeding as in the previous section it is found that the cross term cancellations referred before (4.43) appear again when computing V˙ . Notice, however, that a new cancellation exists between terms ±G(q)q, ˙ which represents exchange between kinetic and potential energies in the mechanical subsystem. Thus, we find that V˙ = −R Id2 − R Iq2 − bq˙ 2 + Id Vd + Iq Vq .

(4.49)

Recall that b > 0. Hence, defining the input u = [Vd , Vq ] and the output y = [Id , Iq ] , we can write (4.49) as V˙ ≤ −Ry  y + y  u.

(4.50)

Since R > 0, (4.50) shows that model (4.45), (4.46), (4.47), is output strictly passive (see Definition 2.42) for the output y and input u defined above. In this chapter, it will be shown that this is the basic theoretical principle behind the common and successful industrial control scheme depicted in Fig. 4.11, when a linear PID position controller is used.

4.3 Velocity Control Standard field-oriented control of PM synchronous motors has been described in Sect. 4.1.6. This successful control scheme is depicted again in Fig. 4.13 for the ease of reference. This control strategy was first proposed for induction motors

120

4 Permanent Magnet Synchronous Motor

[22] and it was extended later to other AC-motors. However, this was performed without presenting any stability proof explaining why this control approach works well. Since this task has been found to be difficult, researchers have focused on proposing new control schemes, provided with stability proofs, trying to achieve better performances than standard field-oriented control [26, 60, 65, 89, 111, 133, 158, 187, 202, 222, 254, 273, 289, 289]. Some of these control schemes are based on backstepping resulting in very complex control laws. Some others take advantage from the dq transformation but they employ controllers that are much more complex than standard field-oriented control. Also exist the group of controllers that employs PI electric current controllers, without any stability proof, but the controller intended for the mechanical part is not a simple PI velocity controller. The stability analysis is performed by assuming that torque is the control input. Any formal proof showing that standard field-oriented control is globally asymptotically stable does not exist at present. Motivated by this situation, in [103] was presented a controller provided with proportional electric current controllers plus two additional nonlinear terms. The mechanical controller is a linear PI velocity controller employing position filtering instead of velocity measurements. Global convergence to the constant desired velocity and boundedness of the state are proven. However, this approach relies on friction that is naturally present in the mechanical subsystem. This latter drawback is eliminated in [188] where standard field-oriented control is proven to ensure global convergence to the constant desired velocity and boundedness of the state, when some additional adaptive terms are included. Finally, let us say that, in the recent literature, works on the subject are sparse and most papers are devoted to sensorless velocity control. However, as we explain above, this does not mean that the problem is already solved or that it has no relevance. As a matter of fact, in the recent paper [205] the leading author of [204] recognizes the importance of presenting a global asymptotic stability proof for field-oriented control and efforts in that paper focus in proving global asymptotic stability when internal PI electric current controllers are employed. However, this is performed when controlling the mechanical subsystem in open loop and the use of a PI velocity controller still remains without a formal solution. In the following proposition, a control scheme is introduced which will be shown to be very similar to standard field-oriented control. Contrary to standard field-oriented control, the proposed control scheme is provided with a formal global asymptotic stability proof, i.e., velocity and the whole state converge to their desired values from any initial condition. Moreover, this result relies on energy ideas. Proposition 4.5 Consider the dq model for salient rotor (i.e., L d = L q ) PM synchronous motors given in (4.39), (4.40), (4.41), in closed loop with the controller:

4.3 Velocity Control

121

Fig. 4.13 Standard field-oriented control of a PM synchronous motor

 Vd = −αd Id − αdi

t

Id (s)ds − K d ω˜ 2 Id ,

(4.51)

0

 t Vq = −αq I˜q − αqi I˜q (s)ds − K q ω˜ 2 I˜q − K f |Id | I˜q , 0   t  1 ω(s)ds ˜ , Iq∗ =  −k p ω˜ − ki σ ΦM 0

(4.52) (4.53)

where I˜q = Iq − Iq∗ and ω˜ = ω − ω ∗ with ω ∗ a real constant which stands for  > 0 represents the estimate of Φ M , and σ(·) is a strictly the desired velocity, Φ M increasing linear saturation function (see Definition 2.34) for some (L , M) such that ∗ | which we additionally require to be continuously differentiable M > L > | τL +bω ki such that 0
0, αdi > 0, αqi > 0. Moreover, this closed-loop dynamics is autonomous. The expressions in (4.56)–(4.59) are almost identical to the open-loop model ˜ TL , Vd , Vq , given in (4.39)–(4.41), if we replace Iq , Id , ω, τ L , Vd , Vq , by I˜q , Id , ω, respectively. An important difference is that electric resistance and viscous friction have been enlarged, i.e., we have R + αq , R + αd , and b + k p in (4.56)–(4.59) instead of R and b in (4.39)–(4.41). Another important difference is the presence of three additional equations in (4.59) which represent the integral parts of both, the PI velocity controller and the PI electric current controllers, respectively. These observations motivate the use of the following “energy” storage function for the closed-loop dynamics:

4.3 Velocity Control

123

1 ˜2 1 1 1 τ L + bω ∗ L q Iq + L d Id2 + αdi z d2 + αqi z q2 + Vω (ω, ˜ x+ ), 2 2 2 2 ki

V (ξ) =

τ L + bω ∗ 1 Vω (ω, ˜ x+ ) = J ω˜ 2 + [ki + β(b + k p )]  ki 2



(4.60) x −

τ L +bω ∗ ki

z(r )dr + β J z(x)ω. ˜

The first terms in V (ξ) are included to take into account the “magnetic energy” stored in the electrical subsystem and “energy” due to integrals of electric currents. The first terms in Vω stand for the “kinetic energy” stored in the mechanical subsystem and “energy” stored in the integral of velocity. Since the integral term of the PI velocity controller is nonlinear, a nonlinear integral of velocity is used as a kind of nonlinear “energy” function. Finally, third term in Vω is a cross term intended to provide V˙ω with a negative quadratic term on the nonlinear function z(x). In this respect, notice that d dt



1 ˜2 1 1 L q Iq + L d Id2 + J ω˜ 2 2 2 2



= −(R + αq ) I˜q2 + I˜q Vq

(4.61)

− (R + αd )Id2 + Id Vd − (b + k p )ω˜ 2 − ωT ˜ L. Now, since both Vq and Vd depend on z(x), a negative quadratic term in z(x) is Rk  n L ω∗ k  required to dominate cross terms Φ Mi z(x) I˜q and − p ΦqM i z(x)Id (the quadratic terms −(R + αq ) I˜q2 and −(R + αd )Id2 already exist). On the other hand, a third-order term ˜ arises from product Id Vd . This cross term can be dominated depending on Id ωz(x) by quadratic negative terms in both Id and ω˜ if |z(x)| is bounded by a finite number. This is reason to saturate the integral action in (4.53). Although a term depending on by −K d ω˜ 2 Id included in (4.51). n p L q ω˜ 2 Id also appears, it is to be compensated  1 1 2  x 2 Terms 2 αdi z d + 2 αqi z q and ki − τL +bω∗ z(r )dr are included in V (ξ) because ki

their time derivatives cancel with cross terms −ki z(x)ω, ˜ arising from −ωT ˜ L, ˜ ˜ and −αdi z d Id , −αqi z q Iq arising from Id Vd , Iq Vq , respectively. Finally, β(b + x k p ) − τL +bω∗ z(r )dr is included to cancel an undesired cross term arising from time ki

derivative of the cross term β J z(x)ω. ˜ ∗ ˜ x + τL +bω ) can always be rendered positive definite and radially unbounded Vω (ω,  ki by suitable selection of controller gains. Taking advantage from term cancellations described above, it is found that time derivative of V along the trajectories of the closed-loop dynamics (4.56)–(4.59) can be upper bounded as V˙ ≤ −ζ  Qζ + +

n p L d k p ΦM

L q k p n p |L d − L q | J ΦM

|Id | I˜q2 − K f |Id | I˜q2

|Id |ω˜ 2 − K d1 ω˜ 2 Id2 − k p2 ω˜ 2

(4.62)

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4 Permanent Magnet Synchronous Motor

+

L q k p n p |L d − L q |k p 2 J ΦM

|Id | | I˜q | |ω| ˜ − K q ω˜ 2 I˜q2 − K d2 ω˜ 2 Id2 − αd2 Id2 − αq2 I˜q2 ,

where ζ = [|ω|, ˜ |z|, |Id |, | I˜q |] , k p = k p1 + k p2 , with k p1 > 0, k p2 > 0, K d = K d1 + K d2 , with K d1 > 0, K d2 > 0, αd = αd1 + αd2 , and αq = αq1 + αq2 , with αd1 , αd2 , αq1 , αq2 positive constants and entries of matrix Q are given in (C.16). Notice that V˙ above is rendered nonpositive if matrix Q is positive definite6 and the controller gains K f , K d1 , k p2 , K q , K d2 , αd2 , αq2 , are chosen such that the positive terms are dominated by the negative terms. This explains the nonlinear terms in (4.51) and (4.52). Hence, it is possible to write V˙ ≤ −ζ  Qζ ≤ 0, ∀ξ ∈ R6 , which means that the origin is stable. Global asymptotic stability of ξ = 0 is proven invoking the LaSalle invariance principle. Conditions for this stability result are summarized by (A.7), (C.13), (C.17), (C.18), (C.19), (C.20). These conditions ensure that V is positive definite and radially unbounded and V˙ is negative semidefinite. Remark 4.6 Instrumental to obtain (4.61) are some cancellations of cross terms which involve −n p L d ω˜ Id I˜q − Φ M ω˜ I˜q + n p L q ω˜ I˜q Id + n p L q ω˜ Id Iq∗ belonging to   d 1 ˜q2 + 1 L d Id2 , the time derivative of the“magnetic energy” stored in the L I q dt 2 2 electrical subsystem, and n p (L d − L q )Id I˜q ω˜ + Φ M I˜q ω˜ − n p L q Id Iq∗ ω˜ belonging to

d 1 J ω˜ 2 , the time derivative of the “kinetic energy” stored in the mechanical subdt 2 system. These cancellations are due to the natural energy exchange between the electrical and mechanical subsystems. They are obvious consequences of the almost identical structure of the closed-loop dynamics in (4.56)–(4.59), and the open-loop dynamics in (4.39)–(4.41). Remark 4.7 Notice that controller in Proposition 4.5 does not require the exact knowledge of any motor parameter. Only rough estimates for the motor torque con stant Φ M and load torque are required. In this respect, we stress that an estimate of load torque to be compensated is always taken into account in practice when selecting a motor. Moreover, only approximate values of the motor parameters are required to verify the stability conditions (A.7), (C.13), (C.17), (C.18), (C.19), (C.20). Remark 4.8 The closed-loop mechanical subsystem dynamics given in (4.58) can be written as J ω˙˜ = −bω˜ + τe − k p ω˜ − ki z, if we define

6 Which

is always possible by suitable selection of controller gains.

(4.63)

4.3 Velocity Control

125

τe = n p (L d − L q )Id I˜q + n p (L d − L q )Id Iq∗ + Φ M I˜q ,

(4.64)

= n p (L d − L q )Id Iq + Φ M I˜q , which includes the difference between actual and desired torques, as well as torque due to rotor saliency which becomes zero if Id = 0. Hence    ω˜ Q 11 0 ˙ + (ω˜ + βz(x))τe , ˜ z(x)] Vω ≤ −[ω, 0 Q 22 z(x) where function Vω is defined in (4.60) and Q 11 and Q 22 are entries of matrix Q defined in (C.16). On the other hand, the time derivative of the first component of V in (4.60), i.e., Ve =

1 ˜2 1 1 1 L q Iq + L d Id2 + αdi z d2 + αqi z q2 , 2 2 2 2

along the trajectories of the closed-loop electrical subsystem dynamics (4.56), (4.57), contains the quadratic negative terms −(R + α p ) I˜q2 and −(R + αd )Id2 . Thus, when computing V˙ = V˙ω + V˙e , we realize that instrumental for the stability result in Proposition 4.5 are the following features, which are very similar to those in Remark 3.4: • The scalar function Vω is a strict Lyapunov function when the desired torque is the input of the closed-loop mechanical subsystem dynamics, given in (4.63), and Id = 0, i.e., when τe = 0. • Coefficients of the negative terms −(R + αq ) I˜q2 , −(R + αd )Id2 , appearing in V˙ can be enlarged arbitrarily. This is important to dominate cross terms in V˙ depending on I˜q , Id , when both I˜q = 0 and Id = 0. • The cancellation of several cross terms belonging to V˙ω and V˙e which is explained in Remark 4.6. Notice that all of these features are inherited from the passivity properties of the openloop motor model described in Sect. 4.2.1. Furthermore, the above proof contains the main ideas of procedure described in Sect. 2.4. As pointed out in Remark 3.4, τe is given as a nonlinear function of the electrical dynamics error for AC-motors (see (4.64)). In the case of salient rotor PM synchronous motors we have resorted to introduction of term −K d ω˜ 2 Id , in (4.51), and terms −K q ω˜ 2 I˜q − K f |Id | I˜q , in (4.52), which are important to dominate some third-order terms appearing in V˙ . See (4.62). These nonlinear terms arise from the bilinear nature of the motor model, a feature that has traditionally complicated controller design for AC-motors. Thanks to introduction of the above terms, controller in Proposition 4.5 is very simple, i.e., only differs from standard field-oriented control for PM synchronous motors by the simple terms −K d ω˜ 2 Id , −K q ω˜ 2 I˜q − K f |Id | I˜q , which are zero at the desired equilibrium point, and saturation of the integral part of the velocity controller.

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4 Permanent Magnet Synchronous Motor

Fig. 4.14 Block diagram of controller in Proposition 4.5. yq = −K q ω˜ 2 I˜q − K f |Id | I˜q , y = −K d ω˜ 2 Id and PI velocity controller has a saturated integral part

Remark 4.9 Notice that Proposition 4.5 is stated for the more general and more complex salient rotor motor case. However, these results reduce to the simpler round rotor motor case just by assuming that L q = L d = L R . This means that all terms including the factor L d − L q vanish. Hence controller in Proposition 4.5 becomes simpler since K f = K q = 0 can be used. Remark 4.10 Standard field-oriented control of a PM synchronous motor was presented in Sect. 4.1.6, i.e., in (4.35), (4.36), (4.37), which are repeated here for the ease of reference:  t ∗ Vd = αd (Id − Id ) + αdi (Id∗ (s) − Id (s))ds, (4.65) 0  t (Iq∗ (s) − Iq (s))ds, Vq = αq (Iq∗ − Iq ) + αqi 0  t ∗ ∗ (ω ∗ (s) − ω(s))ds, τ = k p (ω − ω) + ki 0

where Iq∗ = τ ∗ /Φ M and Id∗ = 0. This control strategy is depicted in Fig. 4.13. Excepting the addition of terms in −K d ω˜ 2 Id , and −K q ω˜ 2 I˜q − K f |Id | I˜q , in (4.51), (4.52), and use of a saturated integral part of the PI velocity controller; the controller in Proposition 4.5 (see Fig. 4.14) is exactly the control scheme in (4.65), i.e., in Fig. 4.13. Moreover, Proposition 4.5 establishes global asymptotic stability of the desired equilibrium point (where ω˜ = 0) in the case of a salient rotor where the generated torque n p (L d − L q )Iq Id + Φ M Iq includes a nonlinear term not considered when assuming, as in (4.65), that the generated torque is given as Φ M Iq . Also notice that this stability result stands even when the torque constant is not exactly known,  = Φ M . Hence, Proposition 4.5 gives a formal justification for these i.e., when Φ M features that are always present in practical applications. Recall that controller in Proposition 4.5 is also valid for the round rotor motor case. Remark 4.11 The passivity-based approach introduced in [204] when applied to control PM synchronous motors is presented in Chap. 9, Example 9.38, of that book. There, it is shown that the electrical dynamics of the motor is given as De (qm )q¨s + W1 (qm )q˙m q˙s + W2 (qm )q˙m + Re q˙s = u,

(4.66)

4.3 Velocity Control

127

where u = V = [V1 , V2 , V3 ] , defined in (4.5). According to (9.16), (9.22), in [204], applying the following three-phase voltages u (control law) at the motor terminals: u = v + W2 (qm )q˙m − K 1 (qm , q˙m )q˙s ,

(4.67)

v = De (qm )q¨sd + Ce (qm q˙m )q˙sd + Res (qm , q˙m )q˙sd ,

(4.68)

results in the following closed-loop electrical dynamics: De (qm )q¨˜s + Ce (qm q˙m )q˙˜s + Res (qm , q˙m )q˙˜s = 0,

(4.69)

where 1 K 1 (qm , q˙m ) = − W1 (qm )q˙m + k I3 , k > 0, (4.70) 2 1 1 Ce (qm q˙m ) = W1 (qm )q˙m , Res (qm , q˙m ) = Re + W1 (qm )q˙m + K 1 (qm , q˙m ), 2 2 d De (qm ) dμ(qm ) W1 (qm ) = , W2 (qm ) = , Re = diag{rs , rs , rs }. (4.71) dqm dqm We have that De (qm ) = L, with L the inductance matrix defined in (4.8) in the present chapter, μ(qm ) = Γ , with Γ defined in (4.7) in the present chapter, k is an arbitrary positive constant, and I3 is the 3 × 3 identity matrix. The symbols qm , q˙s , q˙sd , rs stand for the motor position, the electric current through the three-phase windings at stator, its desired value, and the stator phase windings resistance, respectively. Notice that q˙s = I = [I1 , I2 , I3 ] , defined in (4.5), and qm = q, defined in θ = n p q. Finally, q˙˜s = q˙s − q˙sd . According to (4.14), the three-phase voltages u = V = [V1 , V2 , V3 ] relate to the voltages applied at the fictitious dq phases, VN = [Vq , Vd , V0 ] defined in (4.15), through



 ⎡ cos(θ) cos θ − 2π cos θ + 2π ⎤ 3 3



2⎣ ⎦. sin θ + 2π sin(θ) sin θ − 2π VN = T V, T = 3 3 3 1 1 √ √ √1 2

2

2

Thus, the control law (4.67), (4.68) can be written in the dq coordinates as VN = T u = T W2 (qm )q˙m − T K 1 (qm , q˙m )q˙s +T De (qm )q¨sd + T Ce (qm q˙m )q˙sd + T Res (qm , q˙m )q˙sd , Moreover, following Proposition 9.29 and Example 9.21 in [204], it is shown that the desired electric current and its time derivative is given as q˙sd = T  I N d , q¨sd =

dT  I N d + T  I˙N d , dt

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4 Permanent Magnet Synchronous Motor

where I N d = [Iq∗ , Id∗ , I0∗ ] = [Iq∗ , 0, 0] , according to (4.15), and Iq∗ = in Example 9.38 of [204] with τ ∗ the desired torque. Hence, 

τ∗ ΦM

is defined

dT  I N d + T  I˙N d VN = T W2 (qm )q˙m − T K 1 (qm , q˙m )q˙s + T De (qm ) dt



+T Ce (qm q˙m )T  I N d + T Res (qm , q˙m )T  I N d , which, after some straightforward although long computations, is shown to be equivalent to ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ 0 L q I˙q∗ Φ M q˙m Vq I˜q ⎣ Vd ⎦ = ⎣ 0 ⎦ − T K 1 (qm , q˙m )T  ⎣ Id ⎦ − ⎣ n p L q Iq∗ q˙m ⎦ + ⎣ 0 ⎦ V0 0 0 0 I0 ⎡ ∗⎤ ⎡ ∗⎤ r s Iq I d De (qm )  ⎣ q ⎦ +T T (4.72) 0 q˙m + ⎣ 0 ⎦ , dqm 0 0 where T De (qm ) = T L = L N T has been employed. We observe the following differences with respect to control scheme in Proposition 4.5 in the present book: • The back-electromotive force is required to be cancelled through the term [Φ M q˙m , 0, 0] in (4.72). We stress that instead of using exact feedback cancellation, our approach takes advantage from the natural energy exchange existing between the electrical and the mechanical subsystems as explained in Remark 4.6. Aside from requiring less computational effort, this also avoids numerical errors and the necessity to exactly know the motor back-electromotive force constant. • The term W1 (qm )q˙m q˙s that appears in (4.66) is separated in two terms in (4.69). The first part is grouped in Ce (qm q˙m )q˙˜s which is cancelled by energy exchange arguments in the stability analysis. However, it is important to remark that such cancellation would not be possible if this term separation was not performed. Furthermore, this cancellation of cross terms is not natural since it requires the term Ce (qm q˙m )q˙sd , appearing in (4.68) to be included. The second part is grouped in Res (qm , q˙m )q˙˜s , where Res (qm , q˙m ) is forced to be a positive definite matrix by ensuring that K 1 (qm , q˙m ) dominates 21 W1 (qm )q˙m . This requires to introduce the first (nonlinear) term in the definition of K 1 (qm , q˙m ) given in (4.70). The main reason why the term W1 (qm )q˙m q˙s appears in the motor model (4.66) is because the inductance matrix De (qm ) is position dependent (see (4.71)). This is a consequence of the fact that the Euler-Lagrange approach to model the motor is applied on the basis of a three-phase formulation of the Lagrangian in [204]. Thus, these additional computations required to handle the term W1 (qm )q˙m q˙s 7 would be avoided if a dq transformation was first performed. 7 See

    ∗ , 0, 0 e (qm )  I the terms −T K 1 (qm , q˙m )T  I˜q , Id , I0 and T d Ddq T q˙m in (4.72). q m

4.3 Velocity Control

129

• The time derivative of the desired electric current through the stator must be fed back using the term De (qm )q¨sd included in (4.68). This results in the terms     − 0, n p L q Iq∗ q˙m , 0 + L q I˙q∗ , 0, 0 in (4.72). This step is a fundamental feature in the passivity-based approach introduced in [204] and may require that several additional computations be performed online. This is because the desired electric current is given in terms of the desired torque which represents the controller employed to control the mechanical subsystem. Moreover, the mechanical part of the controller is forced not to depend on velocity measurements since this would require acceleration measurements (or an important number of additional computations) when obtaining the time derivative of the desired electric current. Notice that our novel passivity-based approach does not require to feedback the time derivative of the desired electric current but we simply dominate those terms. Moreover, we allow velocity measurements.   • Including the term rs Iq∗ , 0, 0 in (4.72) may result in performance deterioration since the stator resistance rs changes during the normal operation of the motor and the product rs Iq∗ depends on friction, motor velocity, and load torque. Notice that controller in Proposition 4.5 does not require the exact knowledge of this resistance because this term is compensated using the integral part of the PI electric current controller for phase q. In this respect, although in Proposition 10.12 of [204] it is proven that their approach also works when including the integral part of the electric current controller (for the induction motor case), in that stability proof this integral term has nothing to compensate since the term rs Iq∗ still appears in the control law. On the other hand, in the recent paper [205] is shown that the use of a PI electric current controller for PM synchronous motors ensures global asymptotic stability. However, it is required to exactly know the viscous friction coefficient, the torque constant, the dq inductances, and the motor inertia. Moreover, a load torque estimator must be constructed. All of these requirements are due to the fact that the mechanical subsystem is open-loop controlled, i.e. any external loop intended to control velocity or position is not included. • Taking advantage from the Euler-Lagrange structure of (4.69) and the positive definiteness of matrix Res (qm , q˙m ) it is demonstrated in [204] that q˙˜s → 0 exponentially. This step is instrumental to ensure asymptotic stability when the mechanical subsystem dynamics be taken into account. However, notice that this result relies on obtaining an isolated error equation for the closed-loop dynamics of the electrical subsystem, i.e., (4.69). This requirement becomes a drawback of such an approach because of all the previous steps that we have listed above and that are instrumental to achieve this. In our novel passivity-based approach, we do not need to prove exponential convergence to zero of electric current trough the stator. We prove global asymptotic stability of both the electrical subsystem and the mechanical subsystem by

130

4 Permanent Magnet Synchronous Motor

dominating the cross terms existing between these subsystems. This feature is instrumental to obtain the simpler control law in Proposition 4.5. • Notice that the terms: ⎡ ⎡ ∗⎤ ⎡ ∗⎤ ⎤ ⎤ ⎡ r s Iq Iq 0 Φ M q˙m (q ) d D e m ⎣ ⎣ 0 ⎦ − ⎣ n p L q Iq∗ q˙m ⎦ + T ⎣ ⎦ T 0 q˙m + 0 ⎦, dq m 0 0 0 0 appearing in (4.72) grow as both motor velocity and electric current at the stator also grow. This might result in important performance deterioration specially at high velocities and high load torques because this requires larger voltages as motor velocity and load torque increase. Recall that, in practical situations, voltage that can be applied at the motor terminals always has a limited range of variation. In this respect, we stress that the nonlinear terms −K d ω˜ 2 Id , −K q ω˜ 2 I˜q , and −K f |Id | I˜q , appearing in (4.51) and (4.52), in Proposition 4.5, vanish as the the desired currents or the desired velocity are reached. This means that performance deterioration is expected to be smaller at the desired equilibrium point.

4.3.1 Simulation Results In what follows, we present a numerical example to show the performance of controller in Proposition 4.5. To this aim, we employ the numerical parameters of the PM synchronous motor model Estun EMJ-04APB22 which were experimentally identified in [189]. These parameters are n p = 4, R = 2.7[Ohm], L d = L q = 8.5[mH], Φ M = 0.301[N m/A], J = 31.69 × 10−6 [kg m2 ], b = 52.79 × 10−6 [N m s/rad]. The controller parameters that were employed are k p = 0.01, ki = 0.5, αq = 60,  = 0.3612. αqi = 6000, αd = 60, αdi = 6000, K d = 20, K q = 10, K f = 10, Φ M We employ the following linear saturation function: x+L ⎧ ⎨ −L + (M − L) tanh M−L , if x < −L if |x| ≤ L σ(x) = x, x−L ⎩ L + (M − L) tanh M−L , if x > L

(4.73)

where L = 12.26, M = 13.49, and tanh(·) stands for the tangent hyperbolic function. The above controller gains were verified to satisfy all of the stability conditions ensuring the result in Proposition 4.5 when β = 40 is used. A step desired velocity was commanded at t = 0 with ω ∗ = 157[rad/s], and τ L = 0.25[N m] is applied as a step torque disturbance which appears for t ≥ 0.15[s]. These values of variables were employed in the experiments reported in [189]. The corresponding simulation results are presented in Figs. 4.15 and 4.16. We observe that the velocity step response is a little distorted if compared with the step response of a standard linear second-order system. This distortion is attributed to the saturated integral action on the velocity error that is employed. Despite this feature, the

4.3 Velocity Control

131

Fig. 4.15 Simulation results when controller in Proposition 4.5 is employed

transient response, described by a rise time and an overshoot of about 10[ms] and 11%, respectively, is acceptable if compared with the experimental transient response reported in [189] where a first-order-shaped response is obtained with a time constant of 50[ms]. Moreover, the 15% maximal velocity deviation that is observed in Fig. 4.15 when the torque disturbance is applied, has to be contrasted with the 50% maximal velocity deviation reported in [189] for a classical PI velocity controller. Our intention is not to demonstrate that the performance achieved in simulation with the controller in Proposition 4.5 is superior than performance observed experimentally in [189], since that would be an unfair comparison. We cite results in [189] just to give some insight on the experimentally achievable transient response. In Fig. 4.16 we observe that the d electric current and voltage remain small as usual in standard velocity control of PM synchronous motors. The q electric current never exceeds 5[A] and the q voltage remains around 50[V] in steady state. We stress that the rated phase to phase voltage is 200[Vrms] for this motor. In this respect, a saturation function with limits at ±200[V] has been employed at the q motor terminals to maintain Vq within this range. If this saturation is not employed, Vq reaches a very large initial peak which, however, is applied during a very short time interval after t = 0. The reason to include this saturation is twofold. 1) Such large voltage peaks are not possible in practice because the power supplies have a limited range of voltages and, thus, it is of interest to know what happens if such a large voltages are not applied. 2) It is also of interest to verify that performance

132

4 Permanent Magnet Synchronous Motor

Fig. 4.16 Simulation results when controller in Proposition 4.5 is employed (cont.)

4.3 Velocity Control

133

does not deteriorate if such a large voltage peak is not applied. In this respect, we have observed that the same performance is achieved whether the saturation on Vq is employed or not. We believe that this is because such a large voltage peak is applied only during a very short initial time interval. Finally, the corresponding three-phase voltages V1 , V2 , V3 , are also shown in Fig. 4.15. These, three-phase voltages have been computed using the inverse dq transformation presented in (4.18).

4.4 Position Control Some works have been proposed in the literature to control position in mechanical systems that are actuated by PM synchronous motors [26, 60, 89, 133, 187]. However, those control schemes are complex and very different from standard fieldoriented control, a simple and successful approach in industrial applications. Despite this fact, this control strategy is not provided with a formal global asymptotic stability proof explaining why it works well in practice presenting clear tuning guidelines. Motivated by the above-described situation, in [98] is presented a PID position regulator for rigid robots actuated by PM synchronous motors. The electrical dynamics of these actuators is taken into account in the stability analysis and global convergence to the desired position as well as state boundedness is ensured. A proportional electric current controller as well as some additional adaptive terms are employed. The same problem is solved in [102] where an output feedback saturated PD controller is employed for the mechanical subsystem and a proportional electric current controller is designed for the electrical subsystem. These are the only differences with respect to standard field-oriented control. Global asymptotic stability is proven. The design of the above two controllers relies on the the viscous friction that the mechanical subsystem naturally has. This drawback is avoided in [90] where a PID position regulator and proportional electric current loops plus some adaptive terms are employed. Global convergence to the desired position and state boundedness are ensured when the mechanical subsystem is a simple pendulum. Finally, in [41] is presented a PD with feedforward trajectory tracking controller for n−degrees of freedom rigid robots actuated by PM synchronous motors. Proportional electric current controller plus some adaptive terms are designed for the electrical subsystem. Semiglobal trajectory tracking and state boundedness are proven. Thus, the problem of finding a global asymptotic stability proof for standard field-oriented (position) control in mechanical systems actuated by PM synchronous motors still remains without solution. In this section, the following PM synchronous motor model is considered: L q I˙q = −R Iq − n p L d Id q˙ − Φ M q˙ + Vq , L d I˙d = −R Id + n p L q Iq q˙ + Vd , J q¨ + bq˙ = n p (L d − L q )Id Iq + Φ M Iq − g(q),

(4.74) (4.75) (4.76)

134

4 Permanent Magnet Synchronous Motor

where g(q) represents a position-dependent mechanical load which is assumed to possess the following properties: |g(x) − g(y)| ≤ kg |x − y|, ∀x, y ∈ R, (Lipschitz condition)    dg(q)  , kg > max  q∈R dq  dU (q) g(q) = , |g(q)| ≤ k  , ∀q ∈ R, dq

(4.77) (4.78) (4.79)

with U (q) the potential energy and kg , k  , some positive constants. Controller in Proposition 4.5 can be extended to the case of position control as shown in the following proposition Proposition 4.12 Consider the PM syncrhonous motor model (4.74), (4.75), (4.76), when L d = L q , together with the following controller: 

t

Vd = −αdp Id − αdi Vq = −αq p I˜q − αqi



Id dt − K d q˙ 2 Id ,

(4.80)

I˜q dt − K q q˙ 2 I˜q − K f |Id | I˜q ,

(4.81)

0 t

0

1 ˜ − kd q˙ − ki sat (z) , −k p h(q)  ΦM      t  βk p αβkd z= α 1+ h(q) ˜ + 1+ q˙ dt, ki ki 0

Iq∗ =

(4.82)

 where q˜ = q − q ∗ , with q ∗ a real constant standing for the desired position, Φ M is a positive constant representing an estimate of Φ M , I˜q = Iq − Iq∗ , whereas ˜ = σ(q), ˜ K d , K q , K f , k p , kd , ki , αdp , αdi , αq p , αqi , are constant scalars, and h(q) sat (z) = σ(z), where σ(·) is a strictly increasing linear saturation function for some (L , M) (see Definition 2.34). Furthermore, it is also required that function σ(·) be continuously differentiable such that

0
0 is an estimate of with q ∗ a real constant representing the desired position, Φ M Φ M , and Γi , i = 1, 2, 3, 4, are arbitrary positive constants. There always exist positive controller gains αd , αq , αqi , αdi , k p , kd , ki , α, β, such that the origin is a stable equilibrium point of the closed-loop dynamics and ˜ = 0 is ensured from any initial condition. limt→∞ q(t)

The complete proof of Proposition 4.18 is presented in Appendix C.3. In what follows, just a sketch of the proof is presented which is intended to highlight the rationale behind the proof and how energy ideas are employed.  Sketch of proof of Proposition 4.18. Since both Φ M and Φ M are positive, there  always exists a positive constant ε such that Φ M = εΦ M . Hence, defining k p = k p /ε, kd = kd /ε, ki = ki /ε, ki = ki /α, k p = k p − ki and  z=

t

(αq(s) ˜ + q(s))ds ˙ + q(0) ˜ +

0

1 g(q ∗ ), ki

it is found that Iq∗ =

1  −k p q˜ − kd q˙ − ki z + g(q ∗ ) . ΦM

γi , for i = 1, . . . , 4, where On the other hand, defining γ˜ i = γi −  ˜ + γ1 = n p L d , γ2 = n p (L d − L q ), γ3 = αβ(q(0) γ4 =

L q kd n p (L d − L q ). J ΦM

and replacing in (4.95) the following is found:

1 g(q ∗ ))n p (L d − L q ), ki

4.4 Position Control

141

γ˙˜ 1 = −Γ1 Id q˙ Iq∗ , γ˙˜ 2 = −Γ2 G 1 Id , γ˙˜ 3 = −Γ3 Id Iq , γ˙˜ 4 = −Γ4 Id Iq I˜q . (4.96) Finally, defining 

t

z1 = 0

 Id (s)ds, z 2 =

t

I˜q (s)ds +

0

R g(q ∗ ), αqi Φ M

the closed-loop dynamics is found to be given as L d I˙d = −(R + αd )Id + n p L q q˙ I˜q + Vd , L q I˙˜q = −(R + αq ) I˜q − n p L d q˙ Id − Φ M q˙ + Vq , J q¨ = −(b + kd )q˙ + n p (L d − L q )Id I˜q + Φ M I˜q − G, z˙ = αq˜ + q, ˙ z˙ 1 = Id , z˙ 2 = I˜q ,

(4.97) (4.98) (4.99) (4.100)

Vd = n p L q q˙ Iq∗ − αdi z 1 + h, L q  R  (k p q˜ + kd q˙ + ki z) − αqi z 2 + (k q˙ + kd q¨ + ki (αq˜ + q)), ˙ Vq = ΦM ΦM p G = −n p (L d − L q )Id Iq∗ + k p q˜ + ki z − g(q ∗ ) + g(q). The state of the closed-loop dynamics (4.97)–(4.100) is ξ = [q, ˜ q, ˙ z, Id , z 1 , I˜q , z 2 , γ˜ 1 , . . . , γ˜ 4 ] ∈ R11 , and ξ = 0 is always an equilibrium point. Expressions in (4.97)–(4.100) are almost identical to the open-loop model in ˙ G(q), Vd , Vq , by I˜q , Id , q, ˜ q, ˙ G, Vd , Vq , (4.45)–(4.47), if we replace Iq , Id , q, q, respectively. An important difference is that electric resistance and viscous friction have been enlarged, i.e., we have R + αq , R + αd , and b + kd in (4.97)–(4.100) instead of R and b in (4.45)–(4.47). Another important difference is the presence of three additional equations in (4.100) which represent the integral parts of both, the PID position controller and the PI electric current controllers, respectively. These observations motivate use of the following “energy” storage function for the closedloop dynamics:  1 1 1 1 ˜2 1 L q Iq + L d Id2 + αdi z 12 + αqi z 22 + Vq (q, ˜ q, ˙ z) + γ˜ i2 , 2 2 2 2 2Γ i i=1 4

V (ξ) =

(4.101) Vq (q, ˜ q, ˙ z) = V1 (q, ˜ q) ˙ + V2 (z, q) ˙ + V3 (q, ˜ z), 1 α 1 ˜ 2 − α2 J q˜ 2 + (b + kd )q˜ 2 + k p q˜ 2 ˜ q) ˙ = J (q˙ + 2αq) V1 (q, 4 2 2 ∗ +U (q) − U (q ∗ ) − qg(q ˜ ), 1 αβ  2 1  2 k z + ki z , ˙ = J (q˙ + 2αβz)2 − α2 β 2 J z 2 + V2 (z, q) 4 2 d 2 β   2 ˜ = (k p − αkd )(z − q) ˜ . V3 (z, q) 2

142

4 Permanent Magnet Synchronous Motor

Function Vq (q, ˜ q, ˙ z), which is intended for the mechanical part of the closed-loop dynamics, was introduced in Chap. 3 for the stability analysis of the standard position control scheme in PM brushed DC motors where its composition was explained. The reader is encouraged to see that part of Chap. 3 to understand how the concept of ˜ q, ˙ z). Moreover, first terms in V (ξ) energy is employed to construct function Vq (q, are included because of similar reasons as in Chap. 3, to take into account “energy” stored in the electrical subsystem. Finally, the last terms in V (ξ) are included to take into account the adaptive part of the controller, i.e., (4.96). Vq can be shown to be positive definite and radially unbounded by suitable selection of controller gains. Thus, V given in (4.101) qualifies as a Lyapunov function candidate because it is positive definite and radially unbounded provided that controller gains are suitably selected. After several straightforward cancellations (see Remark 4.19), the time derivative of V along the trajectories of the closed-loop dynamics (4.97)–(4.100) is found to be upper bounded as ˙ |q|, ˜ |z|, |Id |, | I˜q |] , V˙ ≤ −ζ  Mζ, ζ = [|q|,

(4.102)

where matrix M can always be rendered positive definite by suitable selection of controller gains, i.e., λmin (M) > 0. Moreover, V˙ ≤ −ζ  Mζ ≤ −λmin (M)(q˙ 2 + q˜ 2 + z 2 + Id2 + I˜q2 ) ≤ −λmin (M)q˜ 2 ≤ 0, ∀ξ ∈ R11 , which means that the origin is stable, i.e., that the whole state is bounded. Further˜ = more, this expression allows to invoke Corollary 2.32 to conclude that lim t→∞ q(t) 0 from any initial condition. This completes the proof of Proposition 4.18. Conditions for this result are summarized by (C.50), (C.51), (C.53), and the first three conditions in (B.33). These conditions ensure that V is a positive definite radially unbounded function, and V˙ is negative semidefinite. Remark 4.19 The cancellation of cross terms referred before (4.102)  involves d ∗ ˜ ˜ ˜ −n p L d q˙ Id Iq − Φ M q˙ Iq + n p L q q˙ Iq Id + n p L q q˙ Id Iq belonging to dt 21 L q I˜q2

+ 21 L d Id2 , the time derivative of the“magnetic energy” stored in the electrical sub

system, and n p (L d − L q )Id I˜q q˙ + Φ M I˜q q˙ − n p L q Id Iq∗ q˙ belonging to dtd 21 J q˙ 2 , the time derivative of the kinetic energy stored in the mechanical subsystem. These cancellations are due to the natural energy exchange between the electrical and mechanical subsystems. There are obvious consequences of the almost identical structure of the closed-loop dynamics in (4.97)–(4.100), and the open-loop dynamics in (4.45)– (4.47). Other important cross term cancellations are those related to the adaptive part of the controller defining function h. This function is introduced in order to cancel several third-order terms arising from the nonlinear part of the generated torque n p (L d − L q )Id Iq . In this respect, the adaptation laws in (4.95) are introduced to accomplish these cancellations without requiring the exact knowledge of any motor

4.4 Position Control

143

parameter. It is important to underline that only approximate values of the motor parameters are required to verify the stability conditions (C.50), (C.51), (C.53),  is required (B.33). Moreover, only a rough estimate for the motor torque constant Φ M to implement the controller (4.91)–(4.95), which is consistent with current practice. Remark 4.20 The closed-loop mechanical subsystem dynamics given in (4.99) can be written as J q¨ = −(b + kd )q˙ + τe − k p q˜ − ki z + g(q ∗ ) − g(q), z˙ = αq˜ + q, ˙

(4.103)

τe = n p (L d − L q )Id I˜q + n p (L d − L q )Id Iq∗ + Φ M I˜q ,

(4.104)

if we define = n p (L d − L q )Id Iq + Φ M I˜q . Notice that τe includes the difference between actual and desired torques, as well as torque due to rotor saliency which becomes zero if Id = 0. The time derivative ˜ q, ˙ z), defined in (4.101), along the trajectories of the mechanical of function Vq (q, subsystem dynamics (4.103), is given as ∗ ˙ e + αJ q˙ 2 + αqτ ˜ e − αk p q˜ 2 + αq(g(q ˜ ) − g(q)) V˙q = −(b + kd )q˙ 2 + qτ

+α2 β J q˙ q˜ + αβ J q˙ 2 − αβbz q˙ + αβzτe − αβki z 2 + αβz(g(q ∗ ) − g(q)) −βαk p q˜ 2 + βα2 kd q˜ 2 . This can be written as ⎡

⎤ |q| ˙ ˜ ⎦ + (q˙ + αq˜ + αβz)τe , V˙q ≤ −[|q|, ˙ |q|, ˜ |z|]Q  ⎣ |q| |z|

⎡

⎤ Q 11 Q 12 Q 13 Q  = ⎣ Q 21 Q 22 Q 23 ⎦ . Q 31 Q 32 Q 33 (4.105)

Entries of matrix Q  are identical to entries in the first three rows and columns of matrix Q defined in (B.32). Matrix Q  is positive definite if and only if ⎞ Q 11 Q 12 Q 13 > 0, det ⎝ Q 21 Q 22 Q 23 ⎠ > 0, (4.106) Q 31 Q 32 Q 33 ⎛

Q 11 > 0,

Q 11 Q 22 − Q 12 Q 21

which ensures λmin (Q  ) > 0. On the other hand, the time derivative of the first component of V , defined in (4.101), i.e.,  1 1 ˜2 1 1 1 L q Iq + L d Id2 + αdi z 12 + αqi z 22 + γ˜ i2 , 2 2 2 2 2Γ i i=1 4

Ve =

144

4 Permanent Magnet Synchronous Motor

along the trajectories of the closed-loop electrical subsystem dynamics (4.97), (4.98), and the adaptive dynamics in (4.96), contains the negative terms −(R + αq ) I˜q2 and −(R + αd )Id2 . Thus, when computing V˙ = V˙q + V˙e to obtain (4.102), we realize that instrumental for the stability result in Proposition 4.18 are the following features, which are similar to those in Remark 3.6: • The scalar function Vq is a strict Lyapunov function when the desired torque is the input of the closed-loop mechanical subsystem dynamics, given in (4.103), i.e., when I˜q = 0, Id = 0 and τe = 0. See (4.104) and (4.105). • Coefficients of the negative terms −(R + αq ) I˜q2 and −(R + αd )Id2 in V˙ can be enlarged arbitrarily. This is important to dominate cross terms in V˙ , given in (4.102), when both I˜q = 0 and Id = 0. • Cancellation of cross terms referred in Remark 4.19. Notice that all of these features are possible thanks to the passivity properties inherited by the open-loop system to the closed-loop system which are described in Sect. 4.2.2. Furthermore, the above proof contains the main ideas of procedure described in Sect. 2.4. As pointed out in Remark 3.6, τe is given as a nonlinear function of the electrical dynamics error for PM synchronous motors (see (4.104)). As it has been shown above, we have resorted to adaptive control to cope with the nonlinear function defining τe . This has been instrumental to ensure global stability and convergence results without requiring the exact knowledge of any motor parameter. We recall that the round rotor case is obtained when L q = L d = L R . This means that γi = 0, i = 2, 3, 4. Hence, we have the following simplification. Proposition 4.21 Assume that controller presented in Proposition 4.18 has the only following change: h = − γ1 Iq∗ q, ˙

(4.107)

with  γ1 defined as in Proposition 4.18. Consider the closed-loop connection of this controller with the PM synchronous motor dq model given in (4.74)–(4.76) when L d = L q = L R . Let Γ1 be an arbitrary positive constant. There always exist positive controller gains αd , αdi , αq , αqi , k p , ki , α, β, ensuring boundedness of the whole ˜ = 0 from any initial condition. state and limt→∞ q(t) Proof of Proposition 4.21 is exactly the same as the proof of Proposition 4.18 and the stability conditions ensuring this result are the same, i.e., (C.50), (C.51), (C.53), and the first three conditions in (B.33). It is only required to replace L q = L d = L R in definition of matrix M. Remark 4.22 Standard field-oriented control of a PM synchronous motor was presented in Sect. 4.1.6, i.e., in (4.35), (4.36), (4.38), which are repeated here for the ease of reference:

4.4 Position Control

145

Fig. 4.18 Standard field-oriented control of a PM synchronous motor

Vd = αd (Id∗ − Id ) + αdi Vq = αq (Iq∗ − Iq ) + αqi

 

t 0

(Id∗ (s) − Id (s))ds,

(4.108)

t

(Iq∗ (s) − Iq (s))ds,  t d(q − q) + ki (q ∗ (s) − q(s))ds, τ ∗ = k p (q ∗ − q) + kd dt 0 0 ∗

where Iq∗ = τ ∗ /Φ M and Id∗ = 0. This control strategy is depicted in Fig. 4.18, again for the ease of reference. Excepting addition of terms in h at the output of the PI controller for the d phase electric current (see (4.91)–(4.95) and Fig. 4.19), controller in Proposition 4.18 is exactly the control scheme in (4.108), i.e., standard field oriented in Fig. 4.18. Moreover, Proposition 4.18 establishes stability and global convergence of the position error to zero in the case of a salient rotor where the generated torque n p (L d − L q )Iq Id + Φ M Iq includes a nonlinear term not considered when assuming, as in (4.108), that the generated torque is given as Φ M Iq . Also notice that this stability result stands even when the torque constant is not exactly know, i.e., when  = Φ M . Hence, Proposition 4.18 gives a formal justification for these features ΦM that are always present in practical applications. Proposition 4.21 extends this result to the round rotor motor where differences between the proposed formally justified controller and standard field-oriented control of PM synchronous motors in (4.108) are represented by the simple term in (4.107) and  γ˙ 1 = Γ1 Id q˙ Iq∗ . Finally, let us stress that contrary to controller in Proposition 4.12, the controller gains in Propositions 4.18 and 4.21 do not result to be so large. This is because instead

146

4 Permanent Magnet Synchronous Motor

Fig. 4.19 Block diagram of controller in Proposition 4.18

of dominating many terms, adaptive control is employed to cancel such terms without requiring to exactly know any motor parameter. Result in Proposition 4.21 can be easily extended to velocity control. Thus, in the following proposition such a result is formally stated. Proposition 4.23 Consider the PM synchronous motor dq model given in (4.39)– (4.41), with L d = L q , together with the controller: 

t

Vd = −αd Id − αdi 

Id (s)ds + h,

(4.109)

0 t

I˜q (s)ds, I˜q = Iq − Iq∗ , Vq = −αq I˜q − αqi 0    t 1 ω(s)ds ˜ , Iq∗ =  −k p ω˜ − ki ΦM 0  t  t h = − γ1 I˜q2 −  γ2 Iq∗ I˜q −  γ3 Iq∗ ω˜ −  γ4 I˜q ω(s)ds ˜ − γ5 Iq∗ ω(s)ds ˜ − γ6 I˜q −  γ7 Iq∗ , 0

0

 γ˙ 1 = Γ1 Id I˜q2 ,  γ˙ 2 = Γ2 Iq∗ Id I˜q ,  γ˙ 3 = Γ3 Iq∗ ω˜ Id ,  t  t  γ˙ 4 = Γ4 Id I˜q ω(s)ds, ˜  γ˙ 5 = Γ5 Iq∗ Id ω(s)ds, ˜  γ˙ 6 = Γ6 I˜q Id ,  γ˙ 7 = Γ7 Iq∗ Id , 0

o

 where ω˜ = ω − ω ∗ , with ω ∗ and Φ M > 0 real constants which represent the desired velocity and the estimate of Φ M , respectively, whereas Γi , i = 1, . . . , 7, are arbitrary positive constants. There always exist positive controller gains αd , αdi , αq , αqi , k p , ki ensuring that the whole state is bounded and limt→∞ ω(t) ˜ = 0 from any initial condition.

Proof of Proposition 4.23 follows the same steps as in the proof of Proposition 4.21. The closed-loop system state is given by ξ = [ω, ˜ z, I˜q , Id , z 1 , z 2 , γ˜ 1 , . . . , γ˜ 7 ] ∈ R13 where  t 1 ω(s)ds ˜ +  (τ L + bω ∗ ), z= ki 0  t 1 n p L q ω∗ z1 = Id (s)ds − (τ L + bω ∗ ), α Φ di M 0    t R 1 (τ L + bω ∗ ) + Φ M ω ∗ , I˜q (s)ds + z2 = αqi Φ M 0

4.4 Position Control

147

and γ˜ i = γi −  γi , for i = 1, . . . , 7, are defined such that γ1 = γ2 =

L q k p n p (L d − L q ) J ΦM

, γ3 = n p L d ,

γ4 = γ5 = βn p (L d − L q ), γ6 =

γ4 γ5 ∗ ∗  (τ L + bω ), γ7 =  (τ L + bω ). ki ki

Finally, the following Lyapunov function is employed: V (ω, ˜ z, I˜q , Id , z 1 , z 2 , γ˜ 1 , . . . , γ˜ 7 ) =  1 1 ˜2 1 1 1 L q Iq + L d Id2 + αdi z 12 + αqi z 22 + Vω (ω, ˜ z) + γ˜ i2 , 2 2 2 2 2Γ i i=1 7

= Vω (ω, ˜ z) =

1 2 1  ˜ J ω˜ + [ki + γ(b + k p )]z 2 + γ J z ω, 2 2

where Vω (ω, ˜ z) is the same function introduced in (3.44). The complete proof of Proposition 4.23 was introduced for the first time in [188]. Again, controller in Proposition 4.23 is stated for the more general and more complex salient rotor motor case but it is also valid for the simpler round rotor motor case just by assuming that L d = Lq = L R . This is formalized in the following result. Proposition 4.24 Assume that controller presented in Proposition 4.23 has the only following change: h = − γ3 Iq∗ ω, ˜ with  γ3 defined as in Proposition 4.23. Consider the closed-loop connection of this controller with the PM synchronous motor dq model given in (4.39)–(4.41) when L d = L q = L R . Let Γ3 be an arbitrary positive constant. There always exist positive controller gains αd , αdi , αq , αqi , k p , ki , ensuring boundedness of the whole state ˜ = 0 from any initial condition. and limt→∞ ω(t) Proof of this proposition is exactly the same as the proof of Proposition 4.23. We only have to consider that L q = L d = L R . The complete proof of Proposition 4.24 was introduced for the first time in [188].

4.4.1 Simulation Results Simulation results for controllers in Propositions 4.23 and 4.24 have been presented in [188]. From those results, it is straightforward to perform simulations for controllers in Propositions 4.18 and 4.21. Thus, in this section, we present a numerical example to give some insight on performance that is achievable with controller in Proposition

148

4 Permanent Magnet Synchronous Motor

Fig. 4.20 Simulation results when controller in Proposition 4.12 is employed

4.12. We use the numerical parameters of the PM synchronous motor described in Sect. 4.3.1. We assume that a simple pendulum is fixed at the motor shaft in such a way that (see (4.76)) g(q) = mlg sin(q) where m = 0.2[kg], g = 9.81[m/s2 ], and l = 0.14[m]. Moreover, the mechanism inertia is given as J = 31.69 × 10−6 [kg m2 ]+ml 2 + 13 m(2l)2 , where 31.69 × 10−6 [kg m2 ] is inertia of the motor rotor. The controller gains were chosen as k p = 1, kd = 0.3, ki = 50, αq p = 750, αqi = 6000, αdp = 750, αdi = 6000, K d = 20, K q = 10, K f = 10, α = 2, and β = 2. We used again the saturation function presented in (4.73) with L = 0.79 and M = 0.87. These parameters were verified to satisfy all of the stability conditions for Proposition 4.12. The desired position was chosen as q ∗ = π2 [rad] and all the initial conditions were set to zero. We can see in Fig. 4.20 that the motor position reaches the desired position in steady state. Moreover, the transient response disappears in about 1.5[s], which represents a fast motor response. Notice that the position response does not exhibit any overshoot and it is very damped. Moreover, we have performed several additional simulations using larger values for ki and the position response remains without changing its damped shape. We have finally found that when the integral term in (4.82) is modified to   t  βk p h(q)dt, ˜ α 1+ z= ki 0

4.4 Position Control

Fig. 4.21 Simulation results when controller in Proposition 4.12 is employed (cont.)

149

150

4 Permanent Magnet Synchronous Motor

oscillations immediately appear when ki is increased. Thus, we conclude that the   αβkd term 1 + ki q˙ appearing in (4.82) has important damping properties. On the other hand, we can see in Fig. 4.21 that the electric current Iq has a peak of about 7[A], which only appears during a short time interval, and a steady-state value of less than 2[A]. These values are acceptable since the motor has a rated current of 2.7[A]. Although Vq remains within the ±20[V] range, there is a very large peak around t = 0. Since such large voltage values are not possible in practice (this motor has a rated voltage of 200[V]), we have saturated Vq to remain within the limits ±200[V]. We can still see in Fig. 4.21 a large voltage peak after t = 0 which reaches 200[V] and disappears very fast. In this figure, we have constrained the plot of Vq to the range [−10, 30][V] in order to give more detail of the evolution of this signal. We can also observe that both Vd and Id remain small as usual in PM synchronous motor operation. Finally, in Fig. 4.20 we also present the three-phase voltages V1 , V2 , V3 that are actually applied at the motor terminals. These three-phase voltages have been computed using the inverse dq transformation presented in (4.18).

4.5 Velocity Ripple Minimization The main drawback of PM synchronous motors is parasitic torque ripple which can be appreciated as a velocity ripple. These velocity variations severely degrade the servo performance [228, 280] and they are specially important at low velocities since the rotor and load inertias naturally filter these variations at high velocities [154, 197, 227, 228]. Torque produced in a PM synchronous motor can be classified as follows [223, 227, 228]: • Mutual torque, which is due to interaction between rotor flux and stator electric currents. • Reluctance torque, due to rotor saliency. • Cogging torque, due to stator slots.

4.5.1 Mutual Torque Torque ripple is produced in mutual torque due to two reasons: (i) high harmonics appearing because of a nonsinusoidal distribution of the stator windings or rotor magnet and (ii) high harmonics due to errors in stator electric current measurements.

4.5 Velocity Ripple Minimization

4.5.1.1

151

Nonsinusoidal Distribution of Stator Windings

When stator windings are ideally sinusoidally distributed, flux linkages between the permanent magnet and the stator currents are perfectly sinusoidal (as in (4.7)) producing a fixed torque constant and a fixed counter electromotive constant in the dq reference frame. However, as explained in Sect. 4.1.2, this assumption is approximated in practice by using two layer windings, as shown in Fig. 4.1, which produces a staircase flux linkage which is approximately sinusoidal (see Fig. 4.3). It is well known in the literature [154, 227, 228] that this produces flux linkage harmonics (with the motor electrical velocity θ˙ as the fundamental frequency) of the order 5, 7, 11, 13, etc., in the original three-phase coordinates, which appear as sixth and multiples of sixth-order harmonics in torque produced in the dq coordinate frame. In the following, we present an explanation for this. It is shown in [55], Chap. 4, Sect. 4.2, that a flux linkage having a staircase waveform as in Fig. 4.3 can be expanded as a Fourier series, only composed by sine functions of the electrical position θ, which only contains 1st, 3rd, 5th, 7th, 9th, 11th, 13th, etc., frequency components (i.e., this Fourier series replaces each component in (4.7) in such a case). Moreover, as stressed in [154], third harmonics and its multiples are internally cancelled in star-connected three-phase circuits. Hence, only 5th, 7th, 11th, 13th, etc., harmonics are present in the original three-phase frame as well as the fundamental frequency component. When each component of Γ is replaced by such a Fourier series in (4.26), then is written as a Fourier series only in terms of cosine functions each component of ∂Γ ∂q of θ having 1st, 5th, 7th, 11th, 13th, etc., frequency components. Using I = T  I N , with matrix T defined in (4.14) and I N = [Iq , Id , I0 ] , as well as some trigonometric identities it is possible to write 

∂Γ ∂q



I = (Φ M + Φ M6 cos(6θ) + Φ M12 cos(12θ) + · · · )Iq +(Φ6 sin(6θ) + Φ12 sin(12θ) + · · · )Id

which has to replace the simple term Φ M Iq in (4.27). However, since Id is regulated to zero in standard field-oriented control, then terms (Φ6 sin(6θ) + Φ12 sin(12θ) + · · · )Id are commonly negligible [154]. Furthermore, according to (4.26), term 1  ∂L I ∂q I also contributes with high harmonics due to non-perfect sinusoidal sta2 tor windings. However, in motors employing high-coercive PM material, the torque pulsation is mainly concerned with the flux harmonics in the PM and the effects of the inductance harmonics in the stator windings, i.e., torque harmonics in 21 I  ∂∂qL I , can be neglected [154, 197].

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4 Permanent Magnet Synchronous Motor

4.5.1.2

Errors in Stator Electric Current Measurements

There are two sources of error in measurements of electric current which are responsible of torque ripple. Both of them are studied in what follows. • a) Offset error. This refers to constant errors in electric current measurements which are due to an unbalanced DC supply voltage in electric current sensors and inherent offsets in the analog electronic devices [227, 228]. The offset errors appear in electric current measurements in the three-phase original frame. Let I1 , I2 , I3 be the actual electric currents in the original three-phase windings. The measured electric currents (obtained at the output of the A/D converters used for this purpose) are given as [227, 228] I1−AD = I1 + ΔI1 , I2−AD = I2 + ΔI2 , I3−AD = I3 + ΔI3 , where ΔI1 , ΔI2 , ΔI3 are offset errors in the original three-phase windings. Using the dq transformation defined in (4.14), the corresponding variables in the dq frame can be obtained as Iq−AD = Iq + ΔIq , Id−AD = Id + ΔId , I0−AD = I0 + ΔI0 , where ⎡

⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ Iq I1 ΔIq ΔI1 ⎣ Id ⎦ = T ⎣ I2 ⎦ , ⎣ ΔId ⎦ = T ⎣ ΔI2 ⎦ . I0 I3 ΔI0 ΔI3

(4.110)

In particular

⎤ ⎤  ⎡ ΔI1 cos(θ) ΔI2 cos θ − 2π ΔI3 cos θ + 2π ΔIq 3 3

⎦. ⎣ ΔId ⎦ = 2 ⎣ ΔI1 sin(θ) ΔI2 sin θ − 2π ΔI3 sin θ + 2π 3 3 3 1 1 1 √ √ √ ΔI1 ΔI2 ΔI3 ΔI0 2 2 2 ⎡

Adding and subtracting ΔI1 cos θ − 2π and ΔI1 cos θ + 2π , using the fact 3 3

2π 2π that ΔI1 cos(θ) + ΔI1 cos θ − 3 + ΔI1 cos θ + 3 = 0, and taking advantage from some trigonometric identities, the electric current offset in the q phase winding can be written as

4.5 Velocity Ripple Minimization

153

% $  $ √ 2 3 1 sin(θ) ΔIq = ΔI2 − cos(θ) + 3 2 2 % $ % √ 3 1 +ΔI3 − cos(θ) − sin(θ) + ΔI1 cos(θ) . 2 2 Defining √ 3/2 −1/2 , cos(β ) = , sin(β1 ) =   1 1 2  √3 2 1 2  √3 2 + 2 + 2 2 2 √ −1/2 − 3/2 sin(β2 ) =  , cos(β2 ) =  , 1 2  √3 2 1 2  √3 2 + 2 + 2 2 2 and using some trigonometric identities yields  ΔIq =

⎛

' ( 2 $ √ %2 ( 1 2⎜ 3 ) + sin(θ + β1 ) ⎝ΔI2 3 2 2

⎞ ' ( 2 $ √ %2 ( 1 3 ⎟ + sin(θ + β2 ) + ΔI1 cos(θ)⎠ . +ΔI3 ) 2 2 This means that the generated torque is given as Φ M Iq = Φ M (Iq−AD − ΔIq ) = T0 − T1 ,

(4.111)

T0 = Φ M Iq−AD , ⎛ ' ( 2 $ √ %2  ( 1 2⎜ 3 ) + sin(θ + β1 ) T1 = Φ M ⎝ΔI2 3 2 2 ⎞ ' ( 2 $ √ %2 ( 1 3 ⎟ +ΔI3 ) + sin(θ + β2 ) + ΔI1 cos(θ)⎠ , 2 2 where T0 is the nominal generated torque and T1 is torque ripple. Thus, offset errors in electric current measurements produce torque ripple at the fundamental frequency. • b) Scaling error. The output of the current sensor must be scaled to match the input of the A/D converter and the controller re-scales the value of the A/D converter

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4 Permanent Magnet Synchronous Motor

output to obtain the actual value of the electric current. Hence, scaling errors are unavoidable [227, 228]. Assume that the actual values of electric currents in the original three-phase windings are given as I1 = I cos(θ),

  2π , I2 = I cos θ − 3

  2π I3 = I cos θ + , 3

whereas the measured values are given as   I I 2π , cos(θ), I2−AD = cos θ − K1 K2 3   I 2π , = cos θ + K3 3

I1−AD = I3−AD

where K 1 , K 2 , K 3 are constants close to unity which represent scaling errors in phases 1, 2 and 3, respectively [227, 228]. Proceeding similarly to the previous case, the measurement errors are found to be given as ⎡

⎤

⎡

⎤

⎡

 I

ΔI1−AD − I1 ΔI1 ⎢  ⎣ ΔI2 ⎦ = ⎣ ΔI2−AD − I2 ⎦ = ⎢ ⎢ I K12 ⎣  ΔI3 ΔI3−AD − I3 I K13

⎤  − 1 cos(θ) 

⎥ ⎥ − 1 cos θ − 2π ⎥. 3 

⎦ 2π − 1 cos θ + 3 1 K1

Using again (4.110), matrix T defined in (4.14) and the trigonometric identity cos(A) cos(B) = 21 [cos(A − B) + cos(A + B)], it is straightforward to find that ΔIq contains cosine functions of 2θ. Thus, using a similar expression to (4.111), it is concluded that a torque ripple exists at twice the fundamental frequency.

4.5.2 Reluctance Torque Reluctance torque is nonzero only in the salient rotor case. If stator windings are perfectly sinusoidally distributed, the reluctance torque only has a DC component. If stator windings are not perfectly sinusoidally distributed, then reluctance torque has harmonics which are multiples of 6θ [223].

4.5.3 Cogging Torque The cogging torque spectrum depends only on the geometry and number of stator slots. Thus, cogging torque and its minimization is usually addressed in machine design with the aid of various numerical tools [223, 251].

4.5 Velocity Ripple Minimization

155

4.5.4 Torque Ripple According to the above exposition and (4.26) it is concluded that the generated torque, including torque ripple, can be written as   1  ∂L ∂Γ  I I I+ 2 ∂q ∂q = n p (L d − L q )Id Iq + (Φ M + Φ M6 cos(6θ) + Φ M12 cos(12θ) + · · · )Iq

τ =

+TM1 cos(θ + α1 ) + TM2 cos(2θ + α2 )

(4.112)

for some constants α1 , α2 . This means that torque ripple is periodic in the rotor electrical position θ and it consists of the 1st, 2nd, 6th and multiples of 6th harmonic components. Hence, a Fourier series can be used to express torque ripple as τ L = a¯ 0 +

∞  (a¯ k cos(kθ) + b¯k sin(kθ)),

(4.113)

k=1

where a¯ k , b¯k , k = 1, . . . , ∞ are unknown constants and the constant term a¯ 0 can be used to take into account a constant but unknown load torque. Thus, in the standard dq model of a PM synchronous motor given in (4.39), (4.40), (4.41) it must be assumed that load torque τ L is given as in (4.113), i.e., it includes torque ripple and the effect of a constant but unknown load torque. Recall that ω is the motor velocity and θ˙ = n p ω, θ = n p q.

4.5.5 The Problem to Solve Torque and velocity ripple minimization can be performed by motor design improvement [123, 251] or using active control of the stator electric currents. The present section is devoted to minimize velocity ripple using feedback to control the stator electric currents. Several works intended to minimize torque and velocity ripple have been reported by both, practitioners and theorists [73, 78, 84, 116, 186, 223, 227, 228, 256, 280]. Most of them use standard field-oriented control as the basic structure and they simply add, to the desired torque, some terms computed by the velocity ripple (or torque ripple) minimization algorithm that they design. These facts reflect the practical success of standard field-oriented control. However, in all of above-cited works is assumed that i) standard field-oriented control is globally stable, even when the torque/velocity ripple minimization algorithm is added, and ii) the mechanical subsystem dynamics is not taken into account when designing the torque/velocity ripple minimization algorithm. In this respect, it is important to stress that a formal global asymptotic stability proof does not exist until now even for the simple standard

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4 Permanent Magnet Synchronous Motor

field oriented control of PM synchronous motors. Hence, assuming that standard field oriented control is globally stable when a torque/velocity ripple is added is even more questionable. Furthermore, since the mechanical subsystem dynamics has a feedback effect on the electrical subsystem dynamics, any complete stability analysis must take into account the mechanical subsystem dynamics. In this section, a controller is introduced for velocity ripple minimization in PM synchronous motors. This controller uses standard field-oriented control, as the basic control scheme, plus a velocity ripple minimization algorithm. A formal stability proof is presented for this control scheme by taking into account, and simultaneously, a) standard field-oriented control, b) the velocity ripple minimization algorithm, and c) the electrical and the mechanical subsystem dynamics. Thus, we present a global stability proof for standard field-oriented control when designed to minimize velocity ripple.

4.5.6 Stability Analysis Consider the following controller for salient rotor PM synchronous motors, i.e., when L d = L q : 

t

Vd = −Rd Id − rdi

Id (s)ds + h,  t ∗ ˜ I˜q (s)ds + g, Vq = Rs Iq − Rq Iq − rqi 0 % $ N  1 ∗  ∗ Iq = ( γk cos(kθ) +  γk sin(kθ)) , −k p ω˜ + k0 z 0 + ΦM k=1

(4.114)

0

(4.115) (4.116)

where Lq k p I˜q , J ΦM   Lq k p  I˜q , k = 1, . . . , N ,  γ˙ k = Γk cos(kθ) −ω˜ − J ΦM   Lq k p ∗ ∗ I˜q , k = 1, . . . , N ,  γ˙ k = Γk sin(kθ) −ω˜ − J ΦM Lq n pk p h = −n p L d ω˜ Iq∗ − n p L q ω ∗ Iq∗ − (L d − L q )(Iq∗ I˜q + I˜q2 ), J ΦM Lq g = × ΦM ⎞ ⎛ N   ∗  ∗ ˙ γ sin(kθ) +  ˙ γ cos(kθ)]⎠ , ⎝k0 z˙ 0 + [ γ˙ k cos(kθ) − k θ γ˙ k sin(kθ) + k θ k k z˙ 0 = −ω˜ −

(4.117) (4.118) (4.119) (4.120)

k=1

(4.121)

4.5 Velocity Ripple Minimization

157

with ω˜ = ω − ω ∗ , where ω ∗ is a real constant representing the desired velocity, I˜q = Iq − Iq∗ is the electric current error for phase q, where Iq∗ is the corresponding desired current, and N is a positive integer to be chosen to improve torque ripple compensation. Finally, constant scalars k p , k0 , Γk , Γk∗ , k = 1, . . . , N , Rd , rdi , Rq , rqi , are the controller parameters which, in order to ensure stability, have to satisfy some conditions to be stated later. Replacing (4.115) in (4.39), (4.114) in (4.40), and (4.113) in (4.41), and defining r q = R s + Rq ,  t ΦM ∗ zq = ω , I˜q (s)ds + rqi 0 1 z˜ 0 = z 0 − (a¯ 0 + bω ∗ ), k0 γk , γ˜ k∗ = b¯k −  γk∗ , k = 1, . . . , N , γ˜ k = a¯ k −  ∞  (a¯ k cos(kθ) + b¯k sin(kθ)), ϕ=

(4.122) (4.123) (4.124) (4.125) (4.126)

k=N +1

r d = Rs + Rd ,  t zd = Id (s)ds,

(4.127) (4.128)

0

it is found that the closed-loop dynamics is given as L q I˙˜q = −rq I˜q − n p L d ω˜ Id − Φ M ω˜ + Vq , L d I˙d = −rd Id + n p L q ω˜ I˜q + Vd , J ω˙˜ = −(b + k p )ω˜ + n p (L d − L q )Id I˜q + Φ M I˜q − T L , Lq k p ˜ z˙ q = I˜q , z˙ d = Id , z˙˜ 0 = −ω˜ − Iq , J ΦM   Lq k p ˜ γ˙˜ k = −Γk cos(kθ) −ω˜ − Iq , k = 1, . . . , N , J ΦM   ˙γ˜ ∗ = −Γ ∗ sin(kθ) −ω˜ − L q k p I˜q , k = 1, . . . , N , k k J ΦM T L = −n p (L d − L q )Id Iq∗ − k0 z˜ 0 + ϕ −

(4.130) (4.131) (4.132) (4.133) (4.134)

N  [γ˜ k cos(kθ) + γ˜ k∗ sin(kθ)],

k=1 ∗ n p L q ω˜ Iq + n p L q ω ∗ Iq∗ ,

Vd = n p L q ω I˜q + h − rdi z d + Lq k p ˙ ω˜ − rqi z q . Vq = −n p L d ω ∗ Id + ΦM ∗

(4.129)

(4.135)

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4 Permanent Magnet Synchronous Motor

  +N Notice that terms k0 z 0 + k=1 ( γk cos(kθ) +  γk∗ sin(kθ)) in (4.116) are intended to + ¯ k cos(kθ) + b¯k sin(kθ)) in (4.113). compensate for torque ripple τ L = a¯ 0 + ∞ k=1 (a However, since terms in (4.116) are finite in number whereas terms in (4.113) are infinite in number, only the first N harmonics of torque ripple are expected to be compensated. Thus, ϕ defined in (4.126) represents the highest harmonics of torque ripple which will not be compensated. It is stressed that ϕ affects the mechanical subsystem dynamics (4.131) and, hence, the q electrical subsystem dynamics (4.129). The state of the closed-loop dynamics is [ω, ˜ z˜ 0 , γ˜ 1 , . . . , γ˜ N , γ˜ 1∗ , . . . , γ˜ N∗ , I˜q , Id ,  6+2N . Suppose that ω˜ = 0 with ω ∗ = 0, i.e., θ changes at a constant zq , zd ] ∈ R rate. Hence, due to+ the fact that functions sin(·) and cos(·) are linearly independent, N [γ˜ k cos(kθ) + γ˜ k∗ sin(kθ)] = 0 is true for constant values of expression k0 z˜ 0 + k=1  ∗ z˜ 0 , γ˜ k , and γ˜ k , k = 1, . . . , N , if and only if z˜ 0 = 0, γ˜ k = 0, γ˜ k∗ = 0, for k = 1, . . . , N . Hence, it is not difficult to verify that the only equilibrium point is the origin in the case when ϕ = 0 if constants k p , k0 , Rd , rdi , Rq , rqi , Γk , Γk∗ , k = 1, . . . , N , are different from zero. However, the origin is not an equilibrium point when ϕ = 0. Thus, it is reasonable to handle ϕ as an external disturbance. According to these ideas, it is possible to formulate the following proposition. Proposition 4.25 There always exist positive constant scalars k p , k0 , Rd , rdi , Rq , rqi , Γk , Γk∗ , k = 1, . . . , N , such that the closed-loop dynamics (4.129)–(4.135) is finite gain L2 −stable when ϕ and  y = −ω˜ −

Lq k p J ΦM



I˜q

(4.136)

are considered the system input and output, respectively. This stability result stands when starting from any initial condition in R6+2N . Proposition 4.25 was introduced for the first time in [91] where a complete proof was presented. In the following, a sketch of such a proof is presented to highlight the main ideas behind the proof and to give an insight on how energy ideas are exploited. Sketch of proof of Proposition 4.25. Notice that expressions in (4.129)–(4.135) are almost identical to the open-loop model given in (4.39)–(4.41), if we replace Iq , Id , ω, τ L , Vd , Vq , by I˜q , Id , ω, ˜ TL , Vd , Vq , respectively. An important difference is that electric resistance and viscous friction have been enlarged, i.e., we have rq , rd , which can be assigned arbitrarily, and b + k p in (4.129)–(4.131) instead of R and b in (4.39)–(4.41). Another important difference is the presence of three new equations in (4.132) which represent the integral parts of both, the PI velocity controller and the PI electric current controllers. Moreover, 2N additional equations are included in (4.133) and (4.134) which are introduced to compensate torque variations. These observations motivate use of the following “energy” storage function for the closedloop dynamics:

4.5 Velocity Ripple Minimization

159

V (ω, ˜ z˜ 0 , γ˜ 1 , . . . , γ˜ N , γ˜ 1∗ , . . . , γ˜ N∗ , I˜q , Id , z q , z d ) = +

1 2 1 2 J ω˜ + k0 z˜ 0 2 2

(4.137)

N N   1 1 ∗2 1 ˜2 1 1 1  2 ( γ ˜ ) + ˜ k + L q Iq + L d Id2 + rqi z q2 + rdi z d2 . k  ∗γ 2Γ 2Γ 2 2 2 2 k k k=1 k=1

Notice that this scalar function has a quadratic term for each one of the closedloop system states. Hence, (4.137) is positive definite and radially unbounded if constants k0 , rqi , rdi , Γk , Γk∗ , k = 1, . . . , N , are positive (recall that J , L q and L d are always positive). Since V defined above contains the “kinetic energy” 21 J ω˜ 2 stored in the mechanical subsystem and the “magnetic energy” 21 L q I˜q2 + 21 L d Id2 stored in the electrical subsystem, cancellation of cross terms referred in remark 4.6 are present again. Recall, that these cancellations are due to the natural energy exchange between the electrical and mechanical subsystems. They are obvious consequences of the almost identical structure of the closed-loop dynamics in (4.129)– (4.135), and the open-loop dynamics in (4.39)–(4.41). Also, the adaptive terms in +N + N 1 1  2 ∗2 ( γ ˜ ) + γ  ∗ k=1 2Γk k=1 2Γk ˜ k are included to cancel, together with (4.133) and k +N (4.134), terms k=1 [γ˜ k cos(kθ) + γ˜ k∗ sin(kθ)] appearing in the definition of T L . Using (4.120), ± xw ≤ |x| |w|, ∀x, w ∈ R, and taking advantage of several straightforward cancellations, which include the cancellations described in the previous paragraph, we find that the time derivative of V along the trajectories of the closed-loop dynamics (4.129)–(4.135) can be upper bounded as     Lq k p ˜ V˙ ≤ −ζ  Pζ + ϕ −ω˜ − Iq − rq2 I˜q2 , J ΦM

(4.138)

where ζ = [|ω|, ˜ | I˜q |, |Id |] , and Lq k p , J

P11 = b + k p ,

P22 = rq1 −

P12 = P21 = −

L q (b + k p )k p , 2J Φ M

P33 = rd ,

P23 = P32 = −

P13 = P31 = 0, n p (L d + L q )|ω ∗ | , 2

rq = rq1 + rq2 , with rq1 > 0 and rq2 > 0. Matrix P is positive definite if and only if P11 > 0, δ1 = P11 P22 − P12 P21 > 0,

P33 δ1 − P23 P11 P32 > 0. (4.139)

Notice that all of these conditions can always be rendered true by choosing large enough positive values for k p , rq1 and rd , i.e., λmin (P) > 0 can always be rendered true. Hence, we can write V˙ ≤ −λmin (P)ω˜ 2 − rq2 I˜q2 + ϕy which, choosing

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4 Permanent Magnet Synchronous Motor

 rq2 > λmin (P)

Lq k p J ΦM

2 ,

λmin (P) = 2δ,

(4.140) (4.141)

for some positive constant δ, can be written as ,   L q k p 2 ˜2 2 ˙ V ≤ −2δ ω˜ + Iq + ϕy. J ΦM Recalling the facts that |c − d| ≤ |c| + |d| and x1 ≤ written

√ 2x2 , ∀x ∈ R2 , it can be

V˙ ≤ −δ y 2 + ϕy which, according to Lemma 2.43, implies that the closed-loop dynamics (4.129)– (4.135) is finite gain L2 −stable with a finite gain less than or equal to 1δ = 2/λmin (P) > 0. This completes the proof of Proposition 4.25. We stress that conditions for this result are summarized by k0 > 0, rqi > 0, rdi > 0, Γk > 0, Γk∗ > 0, k = 1, . . . , N , and (4.139), (4.140). Remark 4.26 As it is clear from Definition 2.31, the finite gain L2 −stability of the closed-loop system characterizes the disturbance attenuation capabilities of the control scheme. In this respect, according to Definition 2.31, finite gain L2 −stability means that the output truncation L2 −norm: .

s

ys L2 =

y 2 (t)dt, s ∈ [0, ∞)

0

is smaller as the input truncation L2 −norm: .

s

ϕs L2 =

ϕ2 (t)dt, s ∈ [0, ∞)

0

is also smaller. Since (4.113) is assumed to be a convergent Fourier series, according to (4.126) and Parseval’s Theorem 2.33, the norm ϕs L2 exists for a finite s and decreases as N is chosen larger. Thus, it is recommended to select a larger N to further decrease the norm ys L2 . Remark 4.27 It is important to stress that coefficients of terms −rq I˜q and −rd Id appearing in (4.129) and (4.130), respectively, can be enlarged arbitrarily. This fact is instrumental for results in Proposition 4.25 because it allows to dominate cross terms in (4.138), where ρ or Id appear, such that conditions in (4.139) be satisfied. This task is also simplified by the inclusion of functions h and g in (4.114) and (4.115), respectively, since this allows to cancel several undesired terms. In this respect,

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161

Fig. 4.22 Standard field-oriented control of a PM synchronous motor

contrary to control schemes previously presented in this chapter, approach in the present section does employ feedback to complete the closed-loop error equations. In fact, function g defined in (4.121) is part of I˙q∗ . As a consequence, controller expressions are more complex requiring a higher number of computations. This increased number of computations is justified by the fact that the control problem to be solved is also more complex. Remark 4.28 Standard field-oriented control of a PM synchronous motor was presented in Sect. 4.1.6, i.e., in (4.35)–(4.37), which are repeated here for the ease of reference:  t (Id∗ (s) − Id (s))ds, (4.142) Vd = αd (Id∗ − Id ) + αdi 0  t (Iq∗ (s) − Iq (s))ds, Vq = αq (Iq∗ − Iq ) + αqi 0  t (ω ∗ (s) − ω(s))ds, τ ∗ = k p (ω ∗ − ω) + ki 0

where Iq∗ = τ ∗ /Φ M and Id∗ = 0. This control strategy is depicted in Fig. 4.22, again for the ease of reference. Since controller (4.114)–(4.121) is designed to control velocity with velocity ripple minimization, several terms intended to compensate torque ripple have to be added to the PI terms (see Fig. 4.23). This is the reason to define Iq∗ according to (4.116): terms −k p ω˜ + k0 z 0 represent the (modified) PI control terms, whereas +N γk cos(kθ) +  γk∗ sin(kθ)) represents the torque ripple compensation terms. k=1 ( Moreover, note that Iq∗ = τ ∗ /Φ M still stands if we define the desired torque as

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4 Permanent Magnet Synchronous Motor

Fig. 4.23 Block diagram of controller proposed in (4.114)–(4.121)

+N τ ∗ = −k p ω˜ + k0 z 0 + k=1 ( γk cos(kθ) +  γk∗ sin(kθ)). Also note that the first two expressions in (4.142) are retrieved from (4.114), (4.115) if Id∗ = 0, Rq = αq and Rd = αd are large enough such that rq ≈ Rq and rd ≈ Rd , i.e., R is negligible, and both h and g are not considered, which is a valid argument if all of Rd , rdi , Rq , rqi are chosen large. Thus, the main goal of this section is to present a formal stability proof which, under mild assumptions, explains why the traditional control scheme for PM synchronous motors described by (4.142), and depicted in Fig. 4.22, works well when modified as in Fig. 4.23 to minimize velocity ripple. Another important contribution of this result is to formally show that the design formula Iq∗ = τ ∗ /Φ M is still valid for the salient rotor case. We stress that Iq∗ = τ ∗ /Φ M has been used in practice for the salient rotor case based only on the intuitive argument that term n p (L d − L q )Id Iq is negligible compared to Φ M Iq in the generated torque expression τ = n p (L d − L q )Id Iq + Φ M Iq . Finally, it is stressed that although result in Proposition 4.25 is stated for the more general and more complex salient rotor motor case, it simplifies very easily to the round rotor motor case: just consider that L q = L d = L R in the proof of Proposition 4.25.

4.5.7 Experimental Results In order to test controller (4.114)–(4.121), some experiments have been performed using the TMDSHVMTRPFCKIT Texas Instruments Motor Control and Power Factor Correction Developer’s Kit. The Delfino F28335 floating point control card was selected together with a EMJ-04APB22 Estun PM synchronous motor. It is important to remark that the Power Factor Correction option was not enabled during experiments. Numerical parameters of the EMJ-04APB22 Estun PM synchronous motor are the following Rs = 4.7 Ohm, n p = 4, Φ M = 0.06 N-m/A, L d = 0.0065 H, L q = 0.0065 H, J = 0.00025034 Kg m2 and the viscous friction coefficient is assumed to be zero b = 0. The controller gains used in all the reported experimental tests are k p = 0.0477, k0 = 2.38, Rq = Rd = 19.8, rqi = rdi = 250, Γk = Γk∗ = 2, k = 1, 2, 6. All of these numerical values satisfy all the stability conditions sum-

4.5 Velocity Ripple Minimization

163

Fig. 4.24 Use of SFOC with ωd = 0.6283 rad/s. Above: velocity error response. Below: Velocity error spectrum

marized at the end of the proof of Proposition 4.25. For simplicity, acronym SFOC is used to designate standard field-oriented control as shown in Fig. 4.22. Acronym VRMC (velocity ripple minimization controller) is used to specify that the complete controller given in (4.114)–(4.121) is employed (see Fig. 4.23). The initial values γk∗ (0) = 0, k = 1, 2, 6. The same controller gains and were chosen to be  γk (0) =  initial conditions were used in all experiments reported in what follows. The motor velocity error ω˜ and the corresponding spectrum are presented in Fig. 4.24 when using SFOC. The desired velocity is ωd = 0.6283 rad/s (6 rev/min) which corresponds to 0.002 times the motor nominal velocity (3000 rev/min). Recall that the velocity ripple effects are more important at low velocities [154, 197, 227, 228]. Notice that 1st, 2nd, and 6th harmonics are the main frequency components of velocity ripple (recall that these harmonics have θ˙ as the fundamental frequency). Hence, only 1st, 2nd, and 6th harmonics have to be compensated. The velocity error and the corresponding spectrum are presented in Fig. 4.25 when using VRMC under the same conditions as in Fig. 4.24. Notice that an important reduction in the peakto-peak value of the velocity ripple has been accomplished (from approximately 1 rad/s in Fig. 4.24 to about 0.2 rad/s in Fig. 4.25) as well as in magnitude of 1st, 2nd, and 6th harmonics. An experiment is presented in Fig. 4.26, when ωd = 0.6283 rad/s, intended to evaluate how fast controller in (4.114)–(4.121) minimizes the velocity ripple. These results start from a steady state reached using SFOC. Then, VRMC is connected and the exact time when this occurs is indicated. Notice that a steady-state compensation is achieved very fast. In Fig. 4.26, the peak-to-peak value of the velocity ripple is maintained within the (approximately) 0.2 rad/s bound cited above in about 0.4 s.

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4 Permanent Magnet Synchronous Motor

Fig. 4.25 Use of VRMC with ωd = 0.6283 rad/s. Above: velocity error response. Below: Velocity error spectrum

Fig. 4.26 Transient response when VRMC is connected: ωd = 0.6283 rad/s

According to the shape of signals in this figure it is concluded that the steady-state compensation is achieved in acceptable short periods of time.

4.6 A Practical PM Synchronous Motor

165

Fig. 4.27 A view of the stator windings

4.6 A Practical PM Synchronous Motor In Fig. 4.27 is presented a picture of the stator windings of this motor. In Fig. 4.28 is presented a drawing showing the relative positions of the stator phase windings and the rotor permanent magnet poles. Terminals labeled as 1, 2, 3, represent terminals of the stator phases 1, 2, 3, whereas the three terminals labeled as N are connected together and represent the stator windings neutral. The neutral is isolated, i.e., only the terminals 1, 2, 3 are accessible. Thus, the three stator phases are star connected. ◦ = 40◦ = 2π [rad] angle There are nine stator windings. This means that a 360 9 9 exists between adjacent windings. Each one of these windings is wound on the stator around a strip-shaped core which is parallel to the rotor axis (see Fig. 4.29). Each phase is composed by three consecutive windings which are series connected. However, the second winding of each phase is wound in the opposite sense with respect to the first and the third winding of each phase. Reason for this feature is simple and can be easily understood as follows by observing the windings belonging to phase 1 in Fig. 4.28. Notice that the magnetic flux linkages produced by rotor in the first and third windings are produced by N poles, whereas the second winding receives the effect of a S pole. Since the second winding is inversely wound, the final effect is that flux linkages in the three windings of phase 1 add producing a maximal value for this variable at this rotor position. It is not difficult to see that this is also true for phases 2 and 3 when rotor is at a suitable position. The arrows flowing through each phase in Fig. 4.28 represent the sense of a electric current when assumed to be positive.

166

4 Permanent Magnet Synchronous Motor

Fig. 4.28 Relative positions of the stator phase windings and the rotor permanent magnets

Fig. 4.29 Another view of the stator windings

4.6 A Practical PM Synchronous Motor

167

Fig. 4.30 Two views of the rotor permanent magnets

This is a four pole pairs (n p = 4) PM synchronous motor, i.e., a total of four N and ◦ = four S poles are alternatively placed on the rotor surface. This means that a 360 8 45◦ = π4 [rad] angle exists between adjacent poles. Each pole is built as a magnetic strip laying on the rotor parallel to the rotor axis and a cylindrical ferromagnetic sheet is placed around the complete ensemble. See Fig. 4.30. Hence, this PM synchronous motor is considered as a round rotor motor by the manufacturer. In the following, we will obtain the mathematical model of this motor using a procedure described in [55].

4.6.1 Magnetic Field at the Air Gap 4.6.1.1

Magnetic Field Produced by the Stator Windings

In order to compute the magnetic field produced at the air gap by the stator phase winding 1, consider Fig. 4.31. There, the symbol  means that that electric current through phase 1 is coming out of the page at γ = 160◦ and γ = 240◦ , whereas the

168

4 Permanent Magnet Synchronous Motor

Fig. 4.31 Distribution of the stator phase winding 1

symbol ⊗ means that electric current is going into the page at γ = 120◦ and γ = 200◦ , to be consistent with the sense of a positive electric current flowing through phase 1, defined in Fig. 4.28. Let us apply Ampère’s Law (2.33) to the oriented closed trajectory 1–2–3–4–1 shown in Fig. 4.31: / H1 · dl = i 1enclosed ,  2  = H1 · dl + 1

3



4

H1 · dl +

2

 H1 · dl +

3

= g H1 (0) − g H1 (γ),

1

H1 · dl,

4

(4.143)

where 

2

 H1 · dl =

1



1 3

4

3



4



2



2

H1 (0)dl =

1

4

H1 · dl = 3

1

 H1 (0)ˆr · (dl rˆ ) =

H1 (0)dr = g H1 (0),

1

H1 · dl = 0,

2



2

H1 · dl = 0.



4

H1 (γ)ˆr · (−dl r) ˆ =− 3



4

H1 (γ)dl = 3

H1 (γ)dr = −g H1 (γ),

4.6 A Practical PM Synchronous Motor

169

In the above computations, the following assumptions have been taken into account [55]: • The phase loops are located at the surface of the stator. • Wires forming the loops have no width. • The segments of the closed trajectory 1–2–3–4–1 laying in the stator and the rotor are just below the surface of the stator and rotor. • The space between stator and rotor, known as the air gap, is constant and its width is represented by g. • The magnetic field H1 is radially oriented at the air gap. • H1 = 0 inside a ferromagnetic material with high relative magnetic permeability, i.e., the stator and rotor. • dl = dr in the segment 1–2, whereas dl = −dr in the segment 3–4. On the other hand, i 1enclosed depends on γ and is given as (see Fig. 4.31):

i 1enclosed

⎧ 0, ⎪ ⎪ ⎪ ⎪ ⎨ −N I1 , = N I1 , ⎪ ⎪ −N I1 , ⎪ ⎪ ⎩ 0,

0 ≤ γ < 120◦ 120 ≤ γ < 160◦ 160 ≤ γ < 200◦ , 200 ≤ γ < 240◦ 240 ≤ γ < 360◦

(4.144)

where N stands for the number of turns composing each winding and I1 is electric current flowing through phase 1 which is assumed to be positive. It is important to stress that in order to be consistent with Ampère’s Law the right-hand rule has been taken into account. See Fig. 2.8. Thus, from (4.143) we have 1 H1 (γ) = H1 (0) − i 1enclosed . g Since H1 (γ) is defined at the air gap, where of course only air is present, then the relation B1 (γ) = μ0 H1 (γ) can be employed to write   1 B1 (γ) = μ0 H1 (0) − i 1enclosed rˆ . g

(4.145)

In the following, H1 (0) is computed applying Gauss’ Law for the magnetic field, i.e., / B1 · ds = 0, to a Gaussian surface represented by a cylinder enclosing the rotor whose cylindrical surface (with radius r ) lies within the air gap. Hence,

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4 Permanent Magnet Synchronous Motor



  B1 · ds + B1 · ds + B1 · ds, S1 S2 S3   2π  1 H1 (0) − i 1enclosed dγ = 0, = l1r μ0 g 0

/

B1 · ds =

(4.146)

where (4.145) has been employed and  B1 · ds = 0,  S1 B1 · ds = 0, 

S2



l1

B1 · ds = S3

0



2π

B1 (γ)ˆr · (r dγdzrˆ ).

0

In the above computations, the following considerations have been taken into account [55]: • S1 and S2 are the disk-shaped surfaces at each end of the cylinder. At these surfaces B1 = 0. • S3 is the cylinder-shaped part of the Gaussian surface where ds = r dγdzrˆ , in cylindrical coordinates, with r the radius of S3 and l1 the length of rotor (laying on the z axis), i.e., the length of S3. From (4.146) and (4.144) we find  2π 1 i 1enclosed dγ, 2πg 0   1 2π 2π 2π −N I1 × + N I1 × − N I1 × , = 2πg 9 9 9 N H1 (0) = − I1 . 9g

H1 (0) =

Finally, replacing this in (4.145), we find the expression for the magnetic field at the air gap produced by electric current through phase 1:   N rR 1 rˆ , B1 (γ) = μ0 − I1 − i 1enclosed 9g g r

(4.147)

where the factor rrR , with r R the rotor radius, is suggested in [55] to be introduced in order to be consistent with magnetic flux conservation. Notice, however, that rrR ≈ 1 at the air gap and, hence, results obtained in the above procedure remain without change. Proceeding similarly for phases 2 and 3 we find (see Figs. 4.32 and 4.33):

4.6 A Practical PM Synchronous Motor Fig. 4.32 Distribution of the stator phase winding 2

Fig. 4.33 Distribution of the stator phase winding 3

171

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4 Permanent Magnet Synchronous Motor

  N rR 1 rˆ , B2 (γ) = μ0 − I2 − i 2enclosed 9g g r ⎧ 0, 0 ≤ γ < 240◦ ⎪ ⎪ ⎨ −N I2 , 240 ≤ γ < 280◦ i 2enclosed = , N I2 , 280 ≤ γ < 320◦ ⎪ ⎪ ⎩ −N I2 , 320 ≤ γ < 360◦

(4.148)

(4.149)

and   N rR 1 B3 (γ) = μ0 − I3 − i 3enclosed rˆ , 9g g r ⎧ ◦ ⎪ ⎪ −N I3 , 0 ≤ γ < 40 ◦ ⎨ N I3 , 40 ≤ γ < 80 i 3enclosed = . −N I , 80 ≤ γ < 120◦ ⎪ 3 ⎪ ⎩ 0, 120 ≤ γ < 360◦

(4.150)

(4.151)

Notice that (4.147), (4.148), (4.150) can be rewritten as B1 (γ) = b1 (γ)I1 rˆ , B2 (γ) = b2 (γ)I2 rˆ ,

(4.152) (4.153)

B3 (γ) = b3 (γ)I3 rˆ ,

(4.154)

where the scalar functions b1 (γ), b2 (γ), and b3 (γ), obviously defined from (4.147), (4.148), (4.150), are plotted in Fig. 4.34. According to these plots, if Ii > 0, i = 1, 2, 3, the magnetic fields B1 (γ), B2 (γ), and B3 (γ) are positive, i.e., radially directed toward the stator, at the first and third windings of each phase, but they are negative, i.e., radially directed toward the rotor axis, at the second winding of each phase. This is because polarity of the second winding of each phase is reversed with respect to the first and third windings, as explained at the beginning of this section. In Fig. 4.35 is depicted how B1 (γ), defined in (4.152), distributes along the air gap. The total magnetic field produced by the three stator phases at the air gap is finally given as B S (γ) = B1 (γ) + B2 (γ) + B3 (γ).

(4.155)

It is well known that in the case when the phase

electric currents are sinusoidal , I3 = cos ω S t + 2π , functions of time such as I1 = cos(ω S t), I2 = cos ω S t − 2π 3 3 the total magnetic field contributed by the three phases, i.e., B S (γ) defined in (4.155), rotates along the air gap as a function of time. This is verified in Fig. 4.36, where it is observed how the waveform depicted for b1 (γ) in Fig. 4.34, moves from 120◦ ≤ γ ≤ 240◦ to 240◦ ≤ γ ≤ 360◦ when passing from ω S t = 0 to ω S t = 2π . 3 In Fig. 4.37, it is observed how the waveform depicted for b2 (γ) in Fig. 4.34, to moves from 240◦ ≤ γ ≤ 360◦ to 0◦ ≤ γ ≤ 120◦ when passing from ω S t = 2π 3 ω S t = 4π . 3

4.6 A Practical PM Synchronous Motor

Fig. 4.34 Plots of b1 (γ), b2 (γ), and b3 (γ) Fig. 4.35 Distribution of the magnetic flux produced by the stator phase winding 1

173

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4 Permanent Magnet Synchronous Motor

Fig. 4.36 Total magnetic field B S (γ), defined in (4.155) when I1 = cos(ω S t), I2 = cos ω S t −

2π I3 = cos ω S t + 2π 3 , and passing from ω S t = 0 to ω S t = 3

2π 3

,

Finally, in Fig. 4.38, it is observed how the waveform depicted for b3 (γ) in Fig. 4.34, moves from 0◦ ≤ γ ≤ 120◦ to 120◦ ≤ γ ≤ 240◦ when passing from to ω S t = 2π. ω S t = 4π 3 Notice, however, that we cannot see how these waveforms continuously move but we just observe that these waveforms appear in another segment of the stator when an [rad] has elapsed in the variable ω S t. This phenomenon, introduced by increment of 2π 3 a nonsinusoidal distribution of the phase windings on stator, is responsible for torque ripple. A controller designed to cope with such a problem is studied in Sect. 4.5.

4.6.1.2

Magnetic Field Produced by Rotor

Consider again Fig. 4.31. Suppose that rotor moves counter-clockwise. Let Q represent the rotor position defined as the counter-clockwise angle between the stator position where γ = 0 and the center of the S pole at rotor9 which is located at γ = 0 in Fig. 4.31. Since rotor has n p = 4 pole pairs, then the magnetic field produced by the permanent magnets at rotor distributes along the air gap according to B R (γ − Q) = −Bm cos(4(γ − Q)) 9 Which

rR rˆ , r

(4.156)

has moved a counter-clockwise angle Q with respect to situation shown in Fig. 4.31.

4.6 A Practical PM Synchronous Motor

175

Fig. 4.37 Total magnetic field B S (γ), defined in (4.155) when I1 = cos(ω S t), I2 = cos ω S t −

2π 4π I3 = cos ω S t + 2π 3 , and passing from ω S t = 3 to ω S t = 3

2π 3

,

Fig. 4.38 Total magnetic field B S (γ), defined in (4.155) when I1 = cos(ω S t), I2 = cos ω S t −

4π I3 = cos ω S t + 2π 3 , and passing from ω S t = 3 to ω S t = 2π

2π 3

,

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4 Permanent Magnet Synchronous Motor

Fig. 4.39 Distribution along the air gap of the magnetic flux produced by rotor

if assumed that this distribution is sinusoidal. The sign “−” arises from the fact that a S pole is used as reference to define the rotor position Q. Moreover, if γ = Q then B R (γ − Q) = −Bm cos(4(γ − Q)) rrR rˆ = −Bm rrR rˆ indicates that the magnetic field points toward the axis of motor at the stator position γ = Q. This can be verified in Fig. 4.39. The following cases can also be verified in Fig. 4.39. If γ − Q = 45◦ then B R (γ − Q) = −Bm cos(180◦ ) rrR rˆ = Bm rrR rˆ . Notice that a N pole is located on rotor at this value of γ at stator. Moreover, if γ − Q = 22.5◦ then B R (γ − Q) = −Bm cos(90◦ ) rrR rˆ = 0. Notice that the border between a S pole and a N pole is located on rotor at this value of γ at stator. Thus, the total magnetic field produced by both the stator phase windings and permanent magnet at rotor distributes along the air gap according to B(γ, Q) = B S (γ) + kB R (γ − Q),

(4.157)

where B S (γ) is defined in (4.155), B R (γ − Q) is defined in (4.156) and 0 ≤ k ≤ 1 is the coupling factor which is included to account for leakage.

4.6 A Practical PM Synchronous Motor

177

4.6.2 Magnetic Flux Linkages Let Λ1 represent the magnetic flux linkage of phase 1 at stator. This flux linkage represents the magnetic flux through the stator windings located between γ = 120◦ and γ = 240◦ in Fig. 4.31. Define S A as the cylindrical surface subtended between these stator angular positions and the total length of stator. Thus,  Λ1 = N =N

B(γ, Q) · ds, SA  l1



0

=N



240◦

120◦ l1  240◦ 120◦

0



+N

l1



B(γ, Q) · ds, [B1 (γ) − k Bm cos(4(γ − Q))

240◦

120◦

0

(4.158)



l1

B2 (γ) · ds + N 0



240◦ 120◦

rR rˆ ] · ds r

B3 (γ) · ds.

Notice that r S is radius of stator and 

l1



240◦ 120◦

0



l1

B2 (γ) · ds = 0



120◦



l1

+ 

0

+ 

l1 0



240◦

120◦

160◦



l1

B3 (γ) · ds = 0



200◦

l1 

+ 

160◦ 120◦

 0

+ 0

200◦

160◦ l1  240◦

0



B2 (γ) · (r S dγdzrˆ ) B2 (γ) · (−r S dγdzrˆ ) B2 (γ) · (r S dγdzrˆ ) = −l1 μ0

N 2π I2 r R , 9g 9

B3 (γ) · (r S dγdzrˆ )

200◦

160◦ l1  240◦ 200◦

B3 (γ) · (−r S dγdzrˆ ) B3 (γ) · (r S dγdzrˆ ) = −l1 μ0

N 2π I3 r R , 9g 9

N N since B2 (γ) = −μ0 9g I2 rrR rˆ and B3 (γ) = −μ0 9g I3 rrR rˆ for 120◦ ≤ γ ≤ 240◦ , but ◦ recall that the phase 1 winding located at 160 ≤ γ ≤ 200◦ is wound in the opposite sense. Because of the same reason, we have that

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4 Permanent Magnet Synchronous Motor



l1



0

240◦ 120◦

 B1 (γ) · ds = 0

l1



120◦



+ 

160◦

0

+

B1 (γ) · (r S dγdzrˆ )

l1 

200◦

l1 

160◦ 240◦

0

= μ0 r R l1

200◦

B1 (γ) · (−r S dγdzrˆ ) B1 (γ) · (r S dγdzrˆ ),

26N 2π I1 , 9g 9

where (4.147) and (4.144) have been employed. On the other hand, 

l1

−N 0





240◦

120◦ 240◦

k Bm cos(4(γ − Q))

rR rˆ · (r S dγdzrˆ ) r

rR rˆ · (r S dγ rˆ ), r 120◦  240◦ rR k Bm cos(4(γ − Q)) rˆ · (r S dγ rˆ ) = −Nl1 ◦ r 200  200◦ rR + k Bm cos(4(γ − Q)) rˆ · (−r S dγ rˆ ) ◦ r 160   160◦ rR k Bm cos(4(γ − Q)) rˆ · (r S dγ rˆ ) , + r 120◦  240◦ 1 = − k N Bm l1r R cos(4(γ − Q))4dγ 4 200◦  200◦ − cos(4(γ − Q))4dγ 160◦   160◦ cos(4(γ − Q))4dγ , +

= −Nl1

k Bm cos(4(γ − Q))

120◦

1 1 ◦ 200◦ = − k N Bm l1r R [sin(4(γ − Q))]240 200◦ − [sin(4(γ − Q))]160◦ 4 ◦2 + [sin(4(γ − Q))]160 120◦ , 5.6713 k N Bm l1r R cos(4Q), = 4 5.6713 k N Bm l1r R cos(Θ), = 4 where Θ = 4Q, the trigonometric identity sin(A ± B) = sin A cos B ± cos A sin B has been employed several times, and we have used the surface differential (−r S dγ rˆ ) in the integral within the limits 160◦ and 200◦ because the winding in phase 1 laying

4.6 A Practical PM Synchronous Motor

179

between these angular values on the stator is wound in the opposite direction with respect to the other two windings of phase 1. Hence, (4.158) can be written as   5.6713 1 1 I2 − I3 + k N Bm l1r R cos(Θ), Λ1 = L A I1 − 26 26 4 26 2π LA = μ0 N 2 l1r R . 9 9g Proceeding similarly for phases 2 and 3 we find     1 2π 1 5.6713 I3 + k N Bm l1r R cos Θ − Λ2 = L A − I1 + I2 − , 26 26 4 3     1 5.6713 1 2π I2 + I3 + k N Bm l1r R cos Θ + , Λ3 = L A − I1 − 26 26 4 3 where Λ2 and Λ3 are the flux linkages of phases 2 and 3, respectively. According to definition of Q above and the fact that Θ = 4Q it is clear that Θ is also defined as a counter-clockwise rotor position aligned with a rotor S pole. Notice that the rotor position θ introduced in Fig. 4.1 is also a counter clockwise position but aligned with the border between a S and a N pole of rotor.10 Hence, we conclude form Fig. 4.1 that Θ = θ − π2 , and thus π cos(Θ) = cos(θ − ) = sin(θ),   2    π 2π 2π = cos θ − − = sin θ − cos Θ − 3 2 3      π 2π 2π = cos θ − + = sin θ + cos Θ + 3 2 3

 2π , 3  2π . 3

The leakage inductance L s , introduced in (4.8), is considered to take into account the fact that a fraction of flux produced by the electric current Ii , flowing through phase i, does not produce any effect on the other phase windings. This effect is called flux leakage and is due to unperfect magnetic coupling among phase windings. k N Bm l1r R and conIf we include the leakage inductance L s , define K E = 5.6713 4 sider that the motor under study has no saliency, i.e., that L B = 0 and, hence, L 3 = 0 in (4.8), then we can finally write

10 Recall that according to Sect. 4.1.4, the electrical rotor position θ and the mechanical rotor position

q relate through θ = n p q, with n p = 4 for this motor.

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4 Permanent Magnet Synchronous Motor

  1 1 I2 − I3 + K E sin(θ), Λ1 = L s I1 + L A I1 − 26 26    1 1 I3 + K E sin θ − Λ2 = L s I2 + L A − I1 + I2 − 26 26    1 1 I2 + I3 + K E sin θ + Λ3 = L s I3 + L A − I1 − 26 26

 2π , 3  2π . 3

Notice that if Λ = [Λ1 , Λ2 , Λ3 ] , I = [I1 , I2 , I3 ] and      2π 2π  , sin θ + , Γ = K E sin(θ), sin θ − 3 3 are defined, then we can write Λ = L I + Γ, where ⎤ 1 1 L A − 26 LA L s + L A − 26 1 1 L = ⎣ − 26 L A L s + L A − 26 LA ⎦. 1 1 − 26 L A − 26 L A L s + L A ⎡

Since

1 26

 1, we can approximate ⎡

⎤ Ls + L A 0 0 ⎦. 0 0 Ls + L A L=⎣ 0 0 Ls + L A

(4.159)

Then, (4.9) is retrieved, i.e., V = RI +

d (L I ) + Γ˙ , dt

(4.160)

which represents the three phase motor model.

4.6.3 The Motor dq Dynamical Model We can proceed as in Sect. 4.1.4 to obtain (4.19) from (4.160), i.e.,  VN = R I N + T

   ∂  ∂ ˙ L N I N q˙ + L N I N + T T Γ q, ˙ ∂q ∂q

(4.161)

4.6 A Practical PM Synchronous Motor

181

where matrix T is defined in (4.14) and matrix L N = T L T  = diag{L  q , Ld , L 0 } = ∂ T  L N and diag{L s + L A , L s + L A , L s + L A } is constant (see (4.159)), and T ∂q   ∂ Γ are given in (4.20), (4.21), respectively. Thus, the following dq dynamic T ∂q model is obtained from (4.161)

L q I˙q = −R Iq − n p L d Id q˙ − Φ M q˙ + Vq , L d I˙d = −R Id + n p L q Iq q˙ + Vd , L 0 I˙0 = −R I0 + V0 . On the other hand, recall that the generated  is given in (4.26)  and it is easy  torque ∂ ∂ T  L N and T ∂q Γ given in (4.20), to show that from this expression and T ∂q (4.21), we obtain τ = n p (L d − L q )Id Iq + Φ M Iq . Finally, since L q = L d = L s + L A = L R , neglecting the differential equation for I0 ,11 and inserting the generated torque in the motor mechanical dynamics, we obtain L R I˙q = −R Iq − n p L R Id q˙ − Φ M q˙ + Vq , L R I˙d = −R Id + n p L R Iq q˙ + Vd , J ω˙ + bω = Φ M Iq − τ L , which is identical to the dynamic model for a round rotor PM synchronous motor shown in (4.32), (4.33), (4.34).

4.7 Another Practical PM Synchronous Motor In Fig. 4.40 are presented two pictures of the stator windings of this motor. In Fig. 4.41 is presented a drawing showing the relative positions of the stator phase windings. The stator phases are labeled by the manufacturer as R, W, and B. In Fig. 4.41, we label these phases as BR, BW, and BB in order to stress that these terminals of the windings are provided with a blue color, whereas the other end of each winding, which connects to either N1 or N2 neutral terminals, is provided with a black color. It is observed in Fig. 4.41 that there are six phase windings. These phase windings are arranged in two star connections which are parallelly connected as shown in Fig. 4.42. This means that both terminals labeled as BR connect together to terminal labeled as R, both terminals labeled as BW connect together to terminal labeled as W and both terminals labeled as BB connect together to terminal labeled as B. The two 11 This

is possible because this differential equation is stable and I0 does not affect any of the other differential equations of the model.

182

4 Permanent Magnet Synchronous Motor

Fig. 4.40 Two views of the stator windings

neutral points, labeled as N1 and N2, are isolated, i.e., no connection exists between them. This means that the user only has access to terminals R, W, and B. In Fig. 4.43 is presented a more detailed drawing showing the relative positions of the stator phase windings and the rotor permanent magnet poles. For the ease of presentation, we show the stator windings arranged in two concentric layers. Each layer contains the stator windings composing one star: the internal layer constitutes the star connection whose neutral is designated as N1, the external layer constitutes the star connection whose neutral is designated as N2. Each star, i.e., each layer, has ◦ = 40◦ = 2π [rad] angle exists between nine stator windings. This means that a 360 9 9 adjacent windings. The stator is composed by 18 skewed slots (see Fig. 4.40), i.e., ◦ = 20◦ = 2π [rad] between adjacent slots. Each one of with a separation angle of 360 18 18 the nine stator windings composing each layer is wound leaving one free slot which is used to wound the other star phase windings. Each stator phase is composed by three adjacent windings. The end of one winding and the start of the adjacent winding, which are series connected since they belong to the same phase, are wound in the same slot. This, however, requires that the second winding of each phase be wound in the opposite sense than the first and the third windings of the same stator phase. This is also indicated in Fig. 4.43 by reversing the sense of the electric current flowing

4.7 Another Practical PM Synchronous Motor

Fig. 4.41 Relative positions of the stator phase windings

Fig. 4.42 Two star connection of the stator phase windings

183

184

4 Permanent Magnet Synchronous Motor

Fig. 4.43 A detailed view of the relative positions of the stator phase windings

through the second winding of each phase. One reason to wind the second winding in the opposite sense can be explained as follows. Observe the windings in the internal layer which belong to phase BB in Fig. 4.43, i.e., from 120◦ to 240◦ on stator. Notice that the magnetic flux linkages produced by rotor in the first and third windings are produced by N poles, whereas the second winding receives the effect of a S pole. Since the second winding is wound in the opposite sense, the final effect is that flux linkages in the three windings of phase BB add producing a maximal value for this variable at this rotor position. It is not difficult to see that this is also true for all of the motor phases when rotor is at a suitable position. The arrows flowing through each phase in Fig. 4.43 represent the sense of a electric current when assumed to be positive. This is a four pole pairs (n p = 4) PM synchronous motor (see Fig. 4.44), i.e., a total of four N and four S poles are alternatively placed on the rotor surface. This ◦ = 45◦ = π4 [rad] angle exists between adjacent poles. Each pole is means that a 360 8 built as a magnetic strip laying on the rotor parallel to the rotor axis. The skewed design of slots produces a smooth, uniform, sinusoidal torque constant with low harmonic content. This ensures linear, ripple-free torque response and minimum detent torque resulting in smoother motion, faster settling times, and reduced motor hunting, according to the manufacturer. This PM synchronous motor is considered a round rotor motor by the manufacturer.

4.7 Another Practical PM Synchronous Motor Fig. 4.44 Two views of the rotor permanent magnets

185

186

4 Permanent Magnet Synchronous Motor

In the following, we will obtain the mathematical model of this motor using a procedure described in [55]. For this purpose, we rename phases in the internal layer BB, BW, BR as 1a, 2a, 3a, respectively, and phases in the external layer BB, BW, BR as 1b, 2b, 3b, respectively. Electric currents flowing through these phases and voltages applied to these phases are obviously designated as I1a , I2a , I3a , I1b , I2b , I3b and V1a , V2a , V3a , V1b , V2b , V3b .

4.7.1 Magnetic Field at the Air Gap 4.7.1.1

Magnetic Field Produced by the Stator Windings

We stress that, aside from their physical construction, windings in the internal layer of Fig. 4.43 are identically distributed as phase windings of motor studied in Sect. 4.6. Thus, we refer to that part of this book for a description on how to conclude that the expressions for the magnetic fields at the air gap produced by electric currents through phases 1a, 2a, and 3a, are given as in (4.147), (4.144), (4.148), (4.149), (4.150), (4.151), which are rewritten here for the ease of reference:   N rR 1 rˆ , B1a (γ) = μ0 − I1a − i 1aenclosed 9g g r ⎧ 0, 0 ≤ γ < 120◦ ⎪ ⎪ ⎪ ⎪ ⎨ −N I1a , 120 ≤ γ < 160◦ i 1aenclosed = N I1a , 160 ≤ γ < 200◦ , ⎪ ⎪ ⎪ −N I1a , 200 ≤ γ < 240◦ ⎪ ⎩ 0, 240 ≤ γ < 360◦   N rR 1 rˆ , B2a (γ) = μ0 − I2a − i 2aenclosed 9g g r ⎧ 0, 0 ≤ γ < 240◦ ⎪ ⎪ ⎨ −N I2a , 240 ≤ γ < 280◦ i 2aenclosed = , ⎪ N I2a , 280 ≤ γ < 320◦ ⎪ ⎩ −N I2a , 320 ≤ γ < 360◦   N rR 1 B3a (γ) = μ0 − I3a − i 3aenclosed rˆ , 9g g r ⎧ −N I3a , 0 ≤ γ < 40◦ ⎪ ⎪ ⎨ N I3a , 40 ≤ γ < 80◦ i 3aenclosed = . −N I , 80 ≤ γ < 120◦ ⎪ 3a ⎪ ⎩ ◦ 0, 120 ≤ γ < 360

(4.162)

(4.163)

(4.164)

(4.165)

(4.166)

(4.167)

4.7 Another Practical PM Synchronous Motor

187

Fig. 4.45 Computing the magnetic field produced at the air gap by the stator phase winding 1b

In order to compute the magnetic field produced at the air gap by the stator phase winding 1b, consider Fig. 4.45. There, the symbol  means that that electric current through phase 1b is coming out of the page at γ = 340◦ and γ = 60◦ , whereas the symbol ⊗ means that electric current is going into the page at γ = 300◦ and γ = 20◦ , to be consistent with the sense of a positive electric current flowing through phase 1b, defined in Fig. 4.43. Let us apply Ampère’s Law (2.33) to the oriented closed trajectory 1–2–3–4–1 shown in Fig. 4.45: / H1b · dl = i 1benclosed ,  2  = H1b · dl + 1

2

3

 H1b · dl +

= g H1b (0) − g H1b (γ), where

3

4



1

H1b · dl +

H1b · dl,

4

(4.168)

188

 2 1

 3 2

 4 3

 1 4

4 Permanent Magnet Synchronous Motor

H1b · dl =

 2 1

H1b (0)ˆr · (dl rˆ ) =

 2 1

H1b (0)dl =

 2 1

H1b (0)dr = g H1b (0),

H1b · dl = 0, H1b · dl =

 4 3

H1b (γ)ˆr · (−dl rˆ ) = −

 4 3

H1b (γ)dl =

 4 3

H1b (γ)dr = −g H1b (γ),

H1b · dl = 0.

In the above computations, the following assumptions have been taken into account [55]: • The phase loops are located at the surface of the stator. • Wires forming the loops have no width. • The segments of the closed trajectory 1–2–3–4–1 laying in the stator and the rotor are just below the surface of the stator and rotor. • The space between stator and rotor, known as the air gap, is constant and its width is represented by g. • The magnetic field H1b is radially oriented at the air gap. • H1b = 0 inside a ferromagnetic material with high relative magnetic permeability, i.e., the stator and rotor. • dl = dr in the segment 1–2, whereas dl = −dr in the segment 3–4. On the other hand, i 1benclosed depends on γ and is given as (see Fig. 4.45)

i 1benclosed

⎧ 0 ≤ γ < 20◦ ⎪ ⎪ 0, ⎪ ⎪ ⎨ −2N I1b , 20 ≤ γ < 60◦ = −N I1b , 60 ≤ γ < 300◦ , ⎪ ⎪ −2N I1b , 300 ≤ γ < 340◦ ⎪ ⎪ ⎩ 0, 340 ≤ γ < 360◦

(4.169)

where N stands for the number of turns composing each winding and I1b is electric current flowing through phase 1b which is assumed to be positive. It is important to stress that in order to be consistent with Ampère’s Law the right-hand rule has been taken into account. See Fig. 2.8. Thus, from (4.168) we have 1 H1b (γ) = H1b (0) − i 1benclosed . g Since H1b (γ) is defined at the air gap, where of course only air is present, then the relation B1b (γ) = μ0 H1b (γ) can be employed to write   1 B1b (γ) = μ0 H1b (0) − i 1benclosed rˆ . g

(4.170)

4.7 Another Practical PM Synchronous Motor

189

In the following, H1b (0) is computed applying Gauss’ Law for the magnetic field, i.e., / B1b · ds = 0 to a Gaussian surface represented by a cylinder, enclosing the rotor, whose cylindrical surface (with radius r ) lies within the air gap. Hence, 

  B1b · ds + B1b · ds + B1b · ds, S1 S2 S3   2π  1 H1b (0) − i 1benclosed dγ = 0, = l1r μ0 g 0

/

B1b · ds =

(4.171)

where (4.170) has been employed and  B1b · ds = 0,  S1 B1b · ds = 0, 

S2



l1

B1b · ds = S3

0



2π

B1b (γ)ˆr · (r dγdzrˆ ).

0

In the above computations, the following considerations have been taken into account [55]: • S1 and S2 are the disk-shaped surfaces at each end of the cylinder. At these surfaces B1b = 0. • S3 is the cylinder-shaped part of the Gaussian surface where ds = r dγdzrˆ , in cylindrical coordinates, with r the radius of S3 and l1 the length of rotor (laying on the z axis), i.e., the length of S3. From (4.171) and (4.169) we find  2π 1 i 1benclosed dγ, 2πg 0   1 2π 6π 2π −2N I1b × − N I1b × (2π − ) − 2N I1b × , = 2πg 9 9 9 10N I1b . H1b (0) = − 9g H1b (0) =

Finally, replacing this in (4.170), we find the expression for the magnetic field at the air gap produced by electric current through phase 1b:

190

4 Permanent Magnet Synchronous Motor

Fig. 4.46 Computing the magnetic field produced at the air gap by the stator phase winding 2b

  10N rR 1 I1b − i 1benclosed rˆ , B1b (γ) = μ0 − 9g g r

(4.172)

where the factor rrR , with r R the rotor radius, is suggested in [55] to be introduced in order to be consistent with magnetic flux conservation. Notice, however, that rrR ≈ 1 at the air gap and, hence, results obtained in the above procedure remain without change. Proceeding similarly for phases 2b and 3b we find (see Figs. 4.46 and 4.47):   N rR 1 rˆ , B2b (γ) = μ0 − I2b − i 2benclosed 9g g r ⎧ 0, 0◦ ≤ γ < 60◦ ⎪ ⎪ ⎪ ⎪ ⎨ −N I2b , 60 ≤ γ < 100◦ i 2benclosed = N I2b , 100 ≤ γ < 140◦ , ⎪ ⎪ −N I2b , 140 ≤ γ < 180◦ ⎪ ⎪ ⎩ 0, 180◦ ≤ γ < 360◦ and

(4.173)

(4.174)

4.7 Another Practical PM Synchronous Motor

191

Fig. 4.47 Computing the magnetic field produced at the air gap by the stator phase winding 3b

  N rR 1 rˆ , B3b (γ) = μ0 − I3b − i 3benclosed 9g g r ⎧ 0, 0 ≤ γ < 180◦ ⎪ ⎪ ⎪ ⎪ ⎨ −N I3b , 180 ≤ γ < 220◦ i 3benclosed = N I3b , 220 ≤ γ < 260◦ . ⎪ ⎪ −N I3b , 260 ≤ γ < 300◦ ⎪ ⎪ ⎩ 0, 300 ≤ γ < 360◦

(4.175)

(4.176)

Notice that (4.162), (4.164), (4.166), (4.172), (4.173), (4.175) can be rewritten as B1a (γ) = b1a (γ)I1a rˆ , B2a (γ) = b2a (γ)I2a rˆ ,

(4.177) (4.178)

B3a (γ) = b3a (γ)I3a rˆ , B1b (γ) = b1b (γ)I1b rˆ ,

(4.179) (4.180)

B2b (γ) = b2b (γ)I2b rˆ , B3b (γ) = b3b (γ)I3b rˆ ,

(4.181) (4.182)

where the scalar functions b1a (γ), b2a (γ) and b3a (γ), b1b (γ), b2b (γ) and b3b (γ), are obviously defined from (4.162), (4.164), (4.166), (4.172), (4.173), (4.175). Notice

192

4 Permanent Magnet Synchronous Motor

Fig. 4.48 Plots of b1 (γ), b2 (γ), and b3 (γ)

that I1 = I1a + I1b , I2 = I2a + I2b and I3 = I3a + I3b in this motor. Hence, assuming that I1a = I1b = 21 I1 , I2a = I2b = 21 I2 and I3a = I3b = 21 I3 we can define the following functions b1 (γ) = b1a (γ) + b1b (γ), b2 (γ) = b2a (γ) + b2b (γ), and b3 (γ) = b3a (γ) + b3b (γ), which are plotted in Fig. 4.48. According to these plots, if Ii > 0, i = 1, 2, 3, the magnetic fields: B1 (γ) = B1a (γ) + B1b (γ), B2 (γ) = B2a (γ) + B2b (γ), B3 (γ) = B3a (γ) + B3b (γ) are positive, i.e., radially directed toward the stator, at the first and third windings of each phase in both star connections, but they are negative, i.e., radially directed toward the rotor axis, at the second winding of each phase in both star connections. This is because polarity of the second winding of each phase is reversed with respect to the first and third windings, as explained at the beginning of this section (also see Fig. 4.43). We refer to Fig. 4.35 for a drawing showing how B1a (γ), defined in (4.177), distributes along the air gap. The reader can imagine that the magnetic field of the other phases also distribute similarly around their locations on stator. The total magnetic field produced by the six stator phases at the air gap is finally given as

4.7 Another Practical PM Synchronous Motor

193

B S (γ) = B1 (γ) + B2 (γ) + B3 (γ).

(4.183)

It is well known that in the case when the phase electric currents are sinusoidal

, I3 = cos ω S t + 2π , functions of time such as I1 = cos(ω S t), I2 = cos ω S t − 2π 3 3 the total magnetic field contributed by the six phases, i.e., B S (γ) defined in (4.183), rotates along the air gap as a function of time.12 This is verified in Fig. 4.49, where it is observed how the waveform located between 120◦ ≤ γ ≤ 240◦ when ω S t = 0, . moves to 240◦ ≤ γ ≤ 360◦ when ω S t = 2π 3 In Fig. 4.50, it is observed how the waveform located between 240◦ ≤ γ ≤ 360◦ when ω S t = 2π , moves to 0◦ ≤ γ ≤ 120◦ when ω S t = 4π . 3 3 Finally, in Fig. 4.51, is observed how the waveform located between 0◦ ≤ γ ≤ , moves to 120◦ ≤ γ ≤ 240◦ when to ω S t = 2π. 120◦ when ω S t = 4π 3 Notice, however, that we cannot see how these waveforms continuously move but we just observe that these waveforms appear in another segment of the stator when [rad] has elapsed in the variable ω S t. Moreover, if we observe an increment of 2π 3 carefully, we can see that several parts in the top plots in Figs. 4.49, 4.50, and 4.51 slightly move to the right as the bottom plots are reached. This is a feature which is more easy to observe in these figures than in Figs. 4.36, 4.37 and 4.38. Reason for this is explained by the use, in the present case, of two parallel connected stars which lay on the stator 180 mechanical degrees apart. These observations make us to expect torque generation with low harmonic content, i.e., a ripple-free torque response, resulting in smoother motion, which is consistent with the manufacturer’s predictions.

4.7.1.2

Magnetic Field Produced by Rotor

Consider Fig. 4.52. Suppose that rotor moves counter-clockwise. Let Q represent the rotor position defined as the counter-clockwise angle between the stator position where γ = 0 and the center of the S pole at rotor13 which is located at γ = 0 in Fig. 4.52. Since rotor has n p = 4 pole pairs, then the magnetic field produced by the permanent magnets at rotor distributes along the air gap according to B R (γ − Q) = −Bm cos(4(γ − Q))

rR rˆ , r

(4.184)

if assumed that this distribution is sinusoidal. The sign “−” arises from the fact that a S pole is used as reference to define the rotor position Q. Moreover, if γ = Q then B R (γ − Q) = −Bm cos(4(γ − Q)) rrR rˆ = −Bm rrR rˆ indicates that the magnetic field points toward the axis of motor at the stator position γ = Q. This can be verified in Fig. 4.39. The following cases can also be verified in Fig. 4.39. If γ − Q = 45◦

have defined I1a = I1b = 21 cos(ω S t), I2a = I2b = 21 cos ω S t − 2π 3 , I3a = I3b =

1 2π 2 cos ω S t + 3 to simulate the described situation. 13 Which has moved a counter-clockwise angle Q with respect to situation shown in Fig. 4.52. 12 We



194

4 Permanent Magnet Synchronous Motor

Fig. 4.49 Total magnetic field B S (γ), defined in (4.183) when I1 = cos(ω S t), I2 = cos ω S t −

2π 2π I3 = cos ω S t + 3 , and passing from ω S t = 0 to ω S t = 3

2π 3

,

Fig. 4.50 Total magnetic field B S (γ), defined in (4.183) when I1 = cos(ω S t), I2 = cos ω S t −

2π 4π I3 = cos ω S t + 2π 3 , and passing from ω S t = 3 to ω S t = 3

2π 3

,

4.7 Another Practical PM Synchronous Motor

195

Fig. 4.51 Total magnetic field B S (γ), defined in (4.183) when I1 = cos(ω S t), I2 = cos ω S t −

4π I3 = cos ω S t + 2π 3 , and passing from ω S t = 3 to ω S t = 2π

2π 3

Fig. 4.52 Computing the magnetic flux linkage of phase 1a at stator

,

196

4 Permanent Magnet Synchronous Motor

then B R (γ − Q) = −Bm cos(180◦ ) rrR rˆ = Bm rrR rˆ . Notice that a N pole is located on rotor at this value of γ at stator. Moreover, If γ − Q = 22.5◦ then B R (γ − Q) = −Bm cos(90◦ ) rrR rˆ = 0. Notice that the border between a S pole and a N pole is located on rotor at this value of γ at stator. Thus, the total magnetic field produced by both the stator phase windings and permanent magnet at rotor distributes along the air gap according to B(γ, Q) = B S (γ) + kB R (γ − Q),

(4.185)

where B S (γ) is defined in (4.183), B R (γ − Q) is defined in (4.184), and 0 ≤ k ≤ 1 is the coupling factor which is included to account for leakage.

4.7.2 Magnetic Flux Linkages Let Λ1a represent the magnetic flux linkage of phase 1a at stator. This flux linkage represents the magnetic flux through the stator windings located between γ = 120◦ and γ = 240◦ in Fig. 4.52. Define S A as the cylindrical surface subtended between these stator angular positions and the total length of stator. Thus,  Λ1a = N =N

B(γ, Q) · ds, SA  l1 0

 =N



240◦

120◦ l1  240◦ 120◦

0



l1

+N



0

l1 



0

l1 

+N +N 0

B(γ, Q) · ds, [B1a (γ) − k Bm cos(4(γ − Q))

240◦

120◦ 240◦



(4.186)

120◦ 240◦ 120◦

 B2a (γ) · ds + N

l1



240◦

120◦

0

rR rˆ ] · ds r

B3a (γ) · ds

B1b (γ) · ds  B2b (γ) · ds + N

Notice that r S is radius of stator and

0

l1



240◦ 120◦

B3b (γ) · ds.

4.7 Another Practical PM Synchronous Motor



l1 0



240◦

120◦

 B1a (γ) · ds = 0

l1

197



120◦



+ 

160◦

0

+

B1a (γ) · (r S dγdzrˆ )

l1 

200◦

l1 

160◦ 240◦

0

= μ0 r R l1

200◦

B1a (γ) · (−r S dγdzrˆ ) B1a (γ) · (r S dγdzrˆ ),

26N 2π I1a , 9g 9

where (4.162) and (4.163) have been employed. Also, recall that the phase 1a winding located at 160◦ ≤ γ ≤ 200◦ is wound in the opposite sense. Because of the same reason, we have that 

l1



240◦

120◦

0





l1

B2a (γ) · ds = 0

120◦



l1

+ 

0

+ 

l1



240◦ 120◦

0

160◦

200◦



l1

B3a (γ) · ds = 0

l1 

+ 

160◦ 120◦

 0

+

200◦

160◦ l1  240◦

0





B2a (γ) · (r S dγdzrˆ )

B2a (γ) · (r S dγdzrˆ ) = −l1 μ0

N 2π I2a r R , 9g 9

B3a (γ) · (r S dγdzrˆ )

200◦

160◦ 240◦

l1 

200◦

0

B2a (γ) · (−r S dγdzrˆ )

B3a (γ) · (−r S dγdzrˆ ) B3a (γ) · (r S dγdzrˆ ) = −l1 μ0

N 2π I3a r R , 9g 9

N N since B2a (γ) = −μ0 9g I2a rrR rˆ and B3a (γ) = −μ0 9g I3a rrR rˆ for 120◦ ≤ γ ≤ 240◦ . Furthermore,

 0

l1



240◦

120◦



l1

B1b (γ) · ds = 0



120◦



l1

+ 

160◦

0

+ 0



B1b (γ) · (r S dγdzrˆ )

200◦

160◦ l1  240◦ 200◦

B1b (γ) · (−r S dγdzrˆ ) B1b (γ) · (r S dγdzrˆ ) = −l1 μ0

where (4.172) and (4.169) have been employed. On the other hand,

N 2π I1b r R , 9g 9

198

4 Permanent Magnet Synchronous Motor

 0

l1



240◦ 120◦

 B2b (γ) · ds = 0

l1



120◦



l1 

+ 

160◦

0

+

B2b (γ) · (r S dγdzrˆ )

200◦

160◦ 240◦

l1 

200◦

0

B2b (γ) · (−r S dγdzrˆ ) B2b (γ) · (r S dγdzrˆ ),

N 2π rR π π π I2b r R − μ0 (N I2b − N I2b − (−N I2b )), 9g 9 g 9 9 9 11 N 2π = − l1 μ0 I2b r R , 2 9g 9  l1  240◦  l1  160◦ B3b (γ) · ds = B3b (γ) · (r S dγdzrˆ ) 0 120◦ 0 120◦  l1  200◦ + B3b (γ) · (−r S dγdzrˆ ) = −μ0



0 l1

+ 0

160◦ 240◦



200◦

B3b (γ) · (r S dγdzrˆ ),

N 2π rR π π π I3b r R − μ0 (−(−N I3b ) + (−N I3b ) + N I3b ), 9g 9 g 9 9 9 11 N 2π = − l1 μ0 I3b r R , 2 9g 9 = −μ0

where (4.173), (4.174), (4.175), (4.176) have been employed. Finally, in Sect. 4.6.2 has been already shown that 

l1

−N 0



240◦

120◦

k Bm cos(4(γ − Q))

rR rˆ · (r S dγdzrˆ ) r

5.6713 k N Bm l1r R cos(Θ), = 4 where Θ = 4Q, the trigonometric identity sin(A ± B) = sin A cos B ± cos A sin B has been employed several times, and we have used the surface differential (−r S dγ rˆ ) in the integral within the limits 160◦ and 200◦ because the winding in phase 1a laying between these angular values on the stator is connected in the opposite direction with respect to the other two windings of phase 1. Proceeding similarly for the remaining phases, we found that

4.7 Another Practical PM Synchronous Motor

199

Λ = L I + Γ,       Γa I Λa , Γ = , I = a , Λ= Λb Γb Ib ⎡ ⎤   sin(θ)

L 11 L 12 2π ⎦ ⎣ Γa = Γb = K E sin θ − 3 , L = , L 21 L 22 sin θ + 2π 3 ⎡ ⎤ 1 1 L s + L A − 26 L A − 26 LA 1 1 L 11 = L 22 = ⎣ − 26 L A L s + L A − 26 LA ⎦, 1 1 − 26 L A − 26 L A L s + L A ⎡ 1 ⎤ L − 5.5 L − 26 L A − 5.5 26 A 26 A 1 L 12 = L 21 = ⎣ − 5.5 L − 26 L A − 5.5 L ⎦, 26 A 26 A 5.5 5.5 1 − 26 L A − 26 L A − 26 LA 26 2π 5.6713 μ0 N 2 l1r R , K E = k N Bm l1r R , LA = 9 9g 4

(4.187)

where Ia = [I1a , I2a , I3a ] , Ib = [I1b , I2b , I3b ] , Λa = [Λ1a , Λ2a , Λ3a ] , Λb = [Λ1b , Λ2b , Λ3b ] , L s is the the leakage inductance, and it has been assumed that L B = 0, i.e., L 3 = 0, in (4.8) because the motor is considered to be a round rotor motor. Also recall that according to Sect. 4.1.4, the electrical rotor position θ and the mechanical rotor position q relate through θ = n p q, with n p = 4 for this motor.14 1 Since, 26  1 we can approximate ⎤ Ls + L A 0 0 ⎦. 0 0 Ls + L A =⎣ 0 0 Ls + L A ⎡

L 11 = L 22

(4.188)

Applying Kirchhoff’s Voltage Law, Faraday’s Law, and Ohm’s Law to each phase winding it follows that Λ˙ + R I = V, V =



 Va , Vb

where Va = [V1a , V2a , V3a ] , Vb = [V1b , V2b , V3b ] , represent voltages applied to the stator phase windings and R is a scalar standing for the electric resistance of each phase winding. Replacing (4.187) in the previous expression we find V = RI +

d (L I ) + Γ˙ , dt

which represents the three-phase motor model.

14 See

Sect. 4.6.2 for the relation between θ and Θ.

(4.189)

200

4 Permanent Magnet Synchronous Motor

4.7.3 The Motor dq Dynamical Model In this section, we obtain the dq dynamic model of motor under study following a procedure suggested in [260]. This is the generalization of the methodology presented in Sects. 4.1.3 and 4.1.4 in the present book to the case of multi-phase electric machines. This is required because the motor under study has more than three phases, i.e., six phases in two different star connections. Notice, however, that (a) these two star connections are parallel connected and (b) although phases 1a and 1b are located 180 mechanical degrees apart and they are identically affected by the rotor poles, as it is clear from Fig. 4.43. This also applies for phases 2a and 2b as well as for phases 3a and 3b. Thus, these observations (i) allow to define identical dq transformations for both star connections and (ii) result in a different expression for the inductance matrix (see matrix L defined above) with respect to that proposed in [260]. Define the following dq transformation matrix:  T =

Ta 03×3 03×3 Tb







 ⎡ cos(θ) cos θ − 2π cos θ + 2π ⎤ 3 3



2⎣ ⎦, sin θ + 2π sin(θ) sin θ − 2π , Ta = Tb = 3 3 3 1 1 √ √ √1 2

2

2

(4.190) where 03×3 stands for the 3 × 3 null matrix. Notice that Ta and Tb are identical to the transformation matrix defied in (4.14). Defining VN = T V = [Vqa , Vda , V0a , Vqb , Vdb , V0b ] , I N = T I = [Iqa , Ida , I0a , Iqb , Idb , I0b ] , replacing in (4.189) and proceeding as in Sect. 4.1.4, we find  VN = R I N + T

   ∂  ∂ L N I N q˙ + L N I˙N + T T Γ q. ˙ ∂q ∂q

(4.191)

By direct computation, which includes an exhaustive use of trigonometric identities, the reader can verify that the following matrices are obtained15 : L N = T LT  =



 Ta L 11 Ta Ta L 12 Tb , Tb L 21 Ta Tb L 22 Tb

Ta L 11 Ta = diag{L qa , L da , L 0a } = diag{L s + L A , L s + L A , L s + L A } = L N a ,  Tb L 22 Tb = diag{L qb , L db , L 0b } Ta L 12 Tb ΦM 15 The

= diag{L s + L A , L s + L A , L s + L A } = L N b , = Tb L 21 Ta = L N c = diag{0.1731L A + M1 , 0.1731L A + M1 , −0.4615L A },  3 n p K E > 0, = 2

expression in (4.188) has been employed for L 11 and L 22 .

4.7 Another Practical PM Synchronous Motor

201

,  ∂ ∂ Ta ∂q Ta L N a Ta ∂q Ta L N c ∂  LN = T T , (4.192) ∂ ∂ Tb ∂q Tb L N c Tb ∂q Tb L N b ∂q ⎡ ⎡ ⎤ ⎤ 0 L da 0 0 L db 0 ∂  ∂ T L N a = n p ⎣ −L qa 0 0 ⎦ , Tb Tb L N b = n p ⎣ −L qb 0 0 ⎦ , Ta ∂q a ∂q 0 0 0 0 0 0 ⎡ ⎤ 0 0.1731L A + M1 0 ∂  ∂  0 0⎦, T L N c = Tb Tb L N c = n p ⎣ −0.1731L A − M1 Ta ∂q a ∂q 0 0 0       ∂ 3 Γ = (4.193) T n K , 0, 0, 23 n p K E , 0, 0 , 2 p E ∂q 

where M1 is a leakage parameter that is suggested in [260] to be included when the stator is composed only by two different stars. Thus, replacing these expressions in (4.191), we obtain the following PM synchronous motor dynamical model: L qa I˙qa + (0.1731L A + M1 ) I˙qb =

(4.194)

−R Iqa − n p L da Ida q˙ − n p (0.1731L A + M1 )Idb q˙ − Φ M q˙ + Vqa , L da I˙da + (0.1731L A + M1 ) I˙db = −R Ida + n p L qa Iqa q˙ L 0a I˙0a − 0.4615L A I˙0b L qb I˙qb + (0.1731L A + M1 ) I˙qa −R Iqb − n p L db Idb q˙ L db I˙db + (0.1731L A + M1 ) I˙da

+ (0.1731L A + M1 )Iqb q˙ + Vda , = −R I0a + V0a

(4.195) (4.196)

= (4.197) − n p (0.1731L A + M1 )Ida q˙ − Φ M q˙ + Vqb , =

−R Idb + n p L qb Iqb q˙ + n p (0.1731L A + M1 )Iqa q˙ + Vdb , L 0b I˙0b − 0.4615L A I˙0a = −R I0b + V0b .

(4.198) (4.199)

On the other hand, according to D’Alembert’s principle, the generated torque (applied on the rotor) is given as the co-energy’s derivative with respect to rotor position. Thus, proceeding again as in Sect. 4.1.4 we have 

 1   I LI + Γ I , 2     ∂  ∂Γ   L N IN + T T IN . = I N T ∂q ∂q

∂ τ = ∂q

(4.200)

Hence, replacing (4.192), (4.193), and taking advantage from the fact that L qa = L da = L qb = L db because rotor has no saliency, we obtain τ = Φ M (Iqa + Iqb ). Notice that

(4.201)

202

4 Permanent Magnet Synchronous Motor

⎡

⎡ ⎤ ⎤ Iqa Vqa ⎢ ⎢ ⎥ ⎥  ⎢ Ida ⎥  ⎢ Vda ⎥   ⎢ ⎢ ⎥ I0a ⎥ V0a ⎥ Ta Ia Ta Va ⎥, , VN = T V = =⎢ =⎢ IN = T I = ⎢ ⎢ ⎥ ⎥ Tb Ib Tb Vb ⎢ Iqb ⎥ ⎢ Vqb ⎥ ⎣ Idb ⎦ ⎣ Vdb ⎦ I0b V0b and, hence, ⎡

⎤ ⎡ ⎤ Iqa + Iqb Iq Ta (Ia + Ib ) = Ta Ia + Tb Ib = ⎣ Ida + Idb ⎦ = ⎣ Id ⎦ , I0a + I0b I0 ⎡ ⎡ ⎤ ⎤ ⎡ ⎤ Vqa Vqb Vq Ta Va = ⎣ Vda ⎦ = Ta Vb = ⎣ Vdb ⎦ = ⎣ Vd ⎦ , V0a V0b V0 since Ta = Tb and Va = Vb , because of the parallel connection of the two stars at stator. Thus, we can write (4.201) as τ = Φ M Iq .

(4.202)

Moreover, adding (4.194) and (4.197), (4.195) and (4.198), (4.196) and (4.199), and defining L q = L qa = L qb = L d = L da = L db = L 0 = L 0a = L 0b , we can write (L q + 0.1731L A + M1 ) I˙q = −R Iq − n p (L d + 0.1731L A + M1 )Id q˙ − 2Φ M q˙ + 2Vqa , (L d + 0.1731L A + M1 ) I˙d = −R Id + n p (L q + 0.1731L A + M1 )Iq q˙ + 2Vdb , (L 0 − 0.4615L A ) I˙0 = −R I0 + 2V0 . (4.203) Notice that L q = L d = L 0 = L s + L A and, hence, L 0 − 0.4615L A > 0. Thus, (4.203) can be neglected because it is stable and I0 has no effect on the other equations. Moreover, defining L R = L q + 0.1731L A + M1 = L d + 0.1731L A + M1 , and inserting (4.202) in the motor mechanical dynamics, we finally find that the dq dynamical model of the motor is given as L R I˙q = −R Iq − n p L R Id q˙ − 2Φ M q˙ + 2Vq , L R I˙d = −R Id + n p L R Iq q˙ + 2Vd , J ω˙ + bω = Φ M Iq − τ L .

(4.204) (4.205) (4.206)

On the other hand, consider the following scalar function that represents the total energy stored in the motor: V (Id , Iq , ω) =

1 1 L R Id2 + L R Iq2 + J ω 2 , 2 2

4.7 Another Practical PM Synchronous Motor

203

where ω = q. ˙ Notice that two times the kinetic energy is considered. The time derivative of V is given as V˙ = Id L R I˙d + Iq L R I˙q + 2ω J ω˙ which, according to (4.204), (4.205), (4.206), can be written as V˙ = Id (−R Id + n p L R Iq ω + 2Vd ) + Iq (−R Iq − n p L R Id ω − 2Φ M ω + 2Vq ) +2ω(−bω + Φ M Iq − τ L ), = −R Id2 − R Iq2 − 2bω 2 + 2Id Vd + 2Iq Vq − 2τ L ω. Defining the input u = [2Vd , 2Vq , −2τ L ] and the output y = [Id , Iq , ω] , we can write ⎡ ⎤ R 0 0 V˙ = −y  Qy + y  u, Q = ⎣ 0 R 0 ⎦ . 0 0 2b Since Q is a positive definite matrix, this shows that model (4.204), (4.205), (4.206) is output strictly passive (see definition 2.42) for the output y and input u defined above. Thus, although model (4.204), (4.205), (4.206) is slightly different from the dq dynamic model of a round rotor PM synchronous motor given in (4.32), (4.33), (4.34), it still possesses the output strict passivity property that is exploited along this book for control design.

Chapter 5

Induction Motor

An advantage of induction motors with respect to PM synchronous motors is that the former have not the necessity to be set at a home position in order to operate. Moreover, induction motors are the workhorse in industrial applications. The main reason for this is that induction motors are cheaper than PM synchronous motors. As a matter of fact, some control strategies such as field-oriented control (FOC) and, recently, direct torque control (DTC) (see [43, 86, 201] and references therein, for instance) have developed thinking in induction motors and, after proving to be successful, these strategies were later adapted to other classes of AC-motors such as PM synchronous motors. Aside from their nonlinear and multivariable model, induction motors have the additional complication that they are underactuated machines: the rotor magnetic flux generation is produced indirectly (or induced) by voltages applied at the stator terminals, i.e., no voltage sources are dedicated exclusively to feed the rotor circuitry. Most of times, induction motors are controlled using a variant of FOC-denominated standard indirect field-oriented control (SIFOC). Despite the success of this control strategy in industrial applications, SIFOC is not provided with a global asymptotic stability proof. On the other hand, many previous controllers proposed in the literature for electric motors have not been adopted by practitioners because they are based on complex nonlinear ideas [222]. Motivated by this situation and the above-cited control challenges, in this chapter we present some SIFOC-based controllers for both velocity and position regulation in induction motors. These controllers are provided with formal stability proofs which, under mild assumptions, explain why SIFOC of inductor motors works well in practice providing tuning guidelines. This chapter is organized as follows. The dq model of an induction motor is derived in Sect. 5.1. As a result of this analysis, the SIFOC approach is straightforwardly explained for induction motors in this section. Paving the way for the main results in this chapter, in Sect. 5.2 is shown how energy naturally exchanges between the electrical and the mechanical subsystems of the motor. The velocity control problem is solved in Sect. 5.3 and the position control problem is solved in Sect. 5.4. Finally, a practical induction motor is modeled in Sect. 5.5. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 V. M. Hernández-Guzmán et al., Energy-Based Control of Electromechanical Systems, Advances in Industrial Control, https://doi.org/10.1007/978-3-030-58786-4_5

205

206

5 Induction Motor

Fig. 5.1 A one pole pair induction motor

5.1 Motor Modeling 5.1.1 The Working Principle An induction motor is a three-phase AC-motor provided with phase windings on both stator and rotor. The stator structure in an induction motor is identical to the stator structure of a PM synchronous motor. Having three-phase windings on rotor instead of a permanent magnet is what differentiates an induction motor from a PM synchronous motor. A one pole pair (n p = 1) induction motor is shown in Fig. 5.1. Its working principle is described in the following. The three phases on stator are symmetrical, i.e., they have the same structure but radians) they are wound on the stator at an orientation which differs by 120◦ ( 2π 3 the orientation of the other phase windings. If electric current through each phase is given as I1 = Im sin(ω S t), 2π ), 3 2π I3 = Im sin(ω S t + ), 3

I2 = Im sin(ω S t −

then a constant magnitude flux which rotates counter-clockwise, with angular velocity ω S , around the internal surface of stator is obtained (see [55], Chaps. 4, 7, for an analytical manner to verify this fact). This rotating magnetic flux, due to the stator electric currents, produces changes in flux linkages through phase windings on rotor. Hence, electromotive forces are induced in the rotor phase windings and, thus, electric currents flow through them. These electric currents at rotor generate a new magnetic flux which tends to align (see Remark 3.1) with the rotating magnetic flux due to electric currents through the stator forcing rotor to move.

5.1 Motor Modeling

207

In steady state, the angular velocity of rotor ω R is different from ω S , the angular velocity of the rotating magnetic field due to the stator electric currents. It is for this reason that induction motors are designated as asynchronous motors. The difference between these velocities is known as the slip velocity ωsli p = ω S − ω R . Moreover, if ωsli p = 0, i.e., ω R = ω S , then any torque is not exerted on rotor. This is because flux linkages through the rotor windings are constant if ω R = ω S . Under these conditions, Faraday’s Law establishes that any electromotive force is not induced in rotor windings and, hence, no electric current flow through them. This implies that the magnetic flux due to rotor electric currents is not generated in this case. Thus, the stator rotative magnetic flux does not produce any attractive force on rotor because there is not a rotor magnetic flux to align with the stator magnetic flux.

5.1.2 Three-Phase Dynamical Model Except A5, Assumptions A1 to A6 presented in Sect. 4.1.2 are considered again when modeling induction motors [204], Chap. 9, [60], Appendix C, [55], Chap. 7. Reasons to make these assumptions for induction motors are identical to those explained in Sect. 4.1.2 for PM synchronous motors. Assumption A5 is replaced by the following assumption: A7 . Three phases are wound on rotor. These windings are symmetrical and star connected. Each one of these windings is sinusoidally distributed on the rotor surface. This assumption means that turn densities of each phase on rotor, N Rφi , i = 1, 2, 3, are assumed to be given as (see Fig. 5.2) NR 2 NR N Rφ2 (γ) = 2 NR N Rφ3 (γ) = 2 N Rφ1 (γ) =

| sin(γ)|,     sin(γ − 2π ) ,  3      sin(γ + 2π ) ,  3 

where N R is the total number of turns in each rotor phase and γ is the angle defined in Fig. 5.2 to designate an angular position on the rotor surface. Thanks to this, it can be assumed that a magnetic field is produced by each rotor phase which is sinusoidally distributed around the internal side of stator (similar to the magnetic field distribution due to stator phases shown in Fig. 4.3). In practice, this assumption is approximated by using two layer windings as shown in Fig. 5.1 which produces a staircase magnetic field which is approximately sinusoidal [55], Chap. 7 (similar to the magnetic field distribution shown in Fig. 4.3). However, this magnetic field distribution rotates because of the rotor movement.

208

5 Induction Motor

Fig. 5.2 Sinusoidally distributed rotor phase windings 1 and 2

Notice that, because of the star connection of both, the stator phase windings and the rotor phase windings, electric currents through the stator phase windings I1 , I2 , I3 and electric currents through the rotor phase windings I R1 , I R2 , I R3 are balanced, i.e., (see Fig. 5.3): I1 (t) + I2 (t) + I3 (t) = 0, I R1 (t) + I R2 (t) + I R3 (t) = 0.

(5.1) (5.2)

Since the stator windings are star connected, stator must be fed as shown in Fig. 5.3. Let v1 Nˆ , v2 Nˆ , v3 Nˆ denote the phase to motor neutral voltages. In [55], pp. 423, it is shown that in the case when the source voltages are balanced, i.e., when V1 (t) + V2 (t) + V3 (t) = 0,

(5.3)

then V1 = v1 Nˆ , V2 = v2 Nˆ , V3 = v3 Nˆ , v Nˆ N = v Nˆ − v N = 0. Hence, applying Kirchhoff’s Voltage Law, Faraday’s Law, and Ohm’s Law to each phase winding on both, stator and rotor, it follows that ψ˙ 123s + Rs I123s = V123s , ψ˙ 123r + Rr I123r = V123r ,

(5.4) (5.5)

5.1 Motor Modeling

209

Fig. 5.3 Stator and rotor phases connection in an induction motor

where Rs and Rr are positive scalars representing resistance of the stator and rotor phase windings, respectively, V123s = [V1 , V2 , V3 ] , V123r = [V1r , V2r , V3r ] = [0, 0, 0] , I123s = [I1 , I2 , I3 ] , I123r = [I R1 , I R2 , I R3 ] . On the other hand, ψ123s = [ψ1s , ψ2s , ψ3s ] and ψ123r = [ψ1r , ψ2r , ψ3r ] stand, respectively, for flux linkages through the three-phase windings on stator and rotor, which are given as [60], Appendix C: 

ψ123s ψ123r



 =

L s L sr L sr L r



 I123s , I123r

(5.6)

where ⎡

L ls + L ms L s = ⎣ − 21 L ms − 21 L ms ⎡ L lr + L mr L r = ⎣ − 21 L mr − 21 L mr

⎤ − 21 L ms − 21 L ms L ls + L ms − 21 L ms ⎦ , − 21 L ms L ls + L ms ⎤ − 21 L mr − 21 L mr L lr + L mr − 21 L mr ⎦ , − 21 L mr L lr + L mr

(5.7)

210

5 Induction Motor

⎤ cos(θr ) cos(θr + 2π ) cos(θr − 2π ) 3 3 = L sl ⎣ cos(θr − 2π ) cos(θr ) cos(θr + 2π )⎦, 3 3 2π 2π cos(θr + 3 ) cos(θr − 3 ) cos(θr ) ⎡

L sr

and θr is defined as in Fig. 5.1, i.e., as the angular position of the magnetic axis of the rotor phase 1. Remark 5.1 The simple expressions in (5.7) are obtained, thanks to Assumptions A6 and A7, i.e., the stator phase windings are sinusoidally distributed on the internal side of stator and the rotor phase windings are sinusoidally distributed on the rotor surface. The reader is encouraged to consult [55], Chap. 7, for an analytical manner to obtain the expressions in (5.7). On the other hand, L s is the self inductance and mutual inductance matrix for the stator phase windings, whereas L r is the self inductance and mutual inductance matrix for the rotor phase windings. These matrices are explained exactly as matrix L 1 + L 2 in (4.8) was explained in Remark 4.4. In this order of ideas, it must be said that L ls , L ms stand, respectively, for the stator phase windings leakage inductance and magnetizing inductance, whereas L lr , L mr represent, respectively, the rotor phase windings leakage inductance and magnetizing inductance. L sl is the mutual inductance between the stator phase windings and rotor phase windings. Hence, L sr is the mutual inductance matrix between stator and rotor phase windings. It is clear that dependence of this matrix on θr is due to the relative movement existing between stator and rotor phase windings. Furthermore, according to Fig. 5.1, phase windings on stator are aligned with phase windings on rotor when θr = 0, i.e., mutual inductance between phases 1, between phases 2, and between phases 3 on stator and rotor is maximal when θr = 0. This explains function cos(θr ) appearing in the diagonal entries of matrix L sr and a similar reasoning explains the other entries of this matrix. The simple structure of matrix L sr is possible thanks to Assumptions A6 and A7, i.e., the sinusoidal distribution of stator and rotor phase windings. Moreover, under the assumption of an equal mutual inductance and A6, A7, it is obtained [60], Appendix C: L ms = L mr = L sl . Remark 5.2 The expressions in (5.4), (5.5) represent the three-phase model of the electrical subsystem of an induction motor. This is a set of six ordinary nonlinear differential equations (notice that products between electric currents and trigonometric functions of θr as well as their time derivatives appear when replacing (5.6) in (5.4), (5.5)). The main problem with (5.4), (5.5) is their structure; it is not clear how to design a controller from (5.4), (5.5). It is shown in the following that this problem can be simplified by using a suitable coordinate transformation.

5.1 Motor Modeling

211

Fig. 5.4 Stator and rotor phase windings in a three-phase induction motor

5.1.3 ab Transformation As in the case of PM synchronous motors, multiple pole pairs are included in induction motors in order to increase the generated torque. See Sect. 4.1.4 for an example explaining how the stator phases are wound in a two pole pairs motor. Rotor phases are wound similarly to match multiple pole pairs in stator. Again, all of the above discussion is still valid for a multiple pole pair induction motor by defining θr = n p q, where n p stands for the number of pole pairs, q is the mechanical position of rotor (given in mechanical radians), and θr is the electrical position of rotor (given in electrical radians). In order to simplify the induction motor’s three-phase dynamical model given in (5.4), (5.5) it is common to obtain an equivalent two-phase dynamical model by projecting the three-phase windings on both, stator and rotor, on two fictitious phase windings which are assumed to be fixed to stator. These fictitious phase windings are designated as “a” and “b”. Symbol “b” indicates that this phase is aligned with phase 1 on stator and “a” represents a phase winding which is orthogonal to phase “b”, i.e., 90◦ exist between phases “b” and “a”. This disposition of components is shown in Fig. 5.4. Using Fig. 5.4, it is found that projection of the stator three phases 1,2,3, on the stator fictitious phases “a” and “b” is given as

212

5 Induction Motor  xbs = x1s − x2s sin(30◦ ) − x3s sin(30◦ ), 1 1 = x1s − x2s − x3s , 2 2  = −x2s cos(30◦ ) + x3s cos(30◦ ), xas √ √ 3 3 =− x2s + x3s , 2 2

where xis represent either electric currents or voltages at phase i on stator. Introducing

2  x , 3 bs 2  x , = 3 as

xbs =

(5.8)

xas

(5.9)

the ab transformation for stator is defined as ⎡ ⎤ ⎤ ⎡ ⎤ 1 1 −√21 − x1s xbs 2 √ ⎥ ⎣ xas ⎦ = Ts ⎣ x2s ⎦ , Ts = 2 ⎢ ⎣ 0 − 23 23 ⎦ . 3 1 1 1 √ √ √ x0s x3s ⎡

2

2

(5.10)

2

Symbol x0s stands for either electric current, voltages, or magnetic flux linkages of the so-called zero phase of stator. We stress that x0s = 0 for electric currents and voltages because of (5.1) and (5.3). Moreover, in [55], Chap. 7, is shown that x0s = 0 also stands when x0s represents magnetic flux linkages if (5.1) and (5.3) are satisfied. Notice that the redefinition of coordinates introduced in (5.8) and (5.9) is intended to ensure that (Ts )−1 = (Ts ) for matrix Ts defined in (5.10) [60], Appendix C. Using again Fig. 5.4, the rotor three phases can be projected on the fictitious phases “a” and “b”, which are fixed to stator, to obtain  = x1r cos(θr ) − x2r cos(180◦ − (120◦ + θr )) − x3r cos(180◦ − (120◦ − θr )), xbr = x1r cos(θr ) + x2r cos(120◦ + θr ) + x3r cos(120◦ − θr ),

2π 2π + x3r cos −θr + , = x1r cos(−θr ) + x2r cos −θr − 3 3  xar = −x1r sin(θr ) − x2r sin(180◦ − (120◦ + θr )) + x3r sin(180◦ − (120◦ − θr )),

2π , = x1r sin(−θr ) − x2r sin(120◦ + θr ) + x3r sin −θr + 3

2π 2π + x3r sin −θr + . = x1r sin(−θr ) + x2r sin −θr − 3 3

Introducing

5.1 Motor Modeling

213



2  x , 3 br 2  x , = 3 ar

xbr =

(5.11)

xar

(5.12)

the ab transformation for rotor is defined as ⎡   ⎤  ⎤ ⎡ ⎤ 2π cos (−θr ) cos  −θr − 2π xbr x1r 3  cos  −θr + 3  2 ⎢ sin (−θ ) sin −θ − 2π sin −θ + 2π ⎥ ⎣ xar ⎦ = Tr ⎣ x2r ⎦ , Tr = r r r ⎣ 3 3 ⎦. 3 √1 √1 √1 x0r x3r 2 2 2 ⎡

(5.13)

Symbol x0r stands for either electric currents, voltages or magnetic flux linkages of the so-called zero phase of rotor. Using similar argument as for x0s , i.e., because of (5.1), (5.2), (5.3), and V123r = [V1r , V2r , V3r ] = [0, 0, 0] , it is always true that x0r = 0 [55], Chap. 7. Notice again that the redefinition of coordinates in (5.11) and (5.12) is intended to ensure that (Tr )−1 = (Tr ) for matrix Tr defined in (5.13) [60], Appendix C.

5.1.4 The ab Dynamical Model The three-phase dynamic model of an induction motor is given by (5.4), (5.5). In this section, model (5.4), (5.5) will be expressed in terms of the ab fictitious phases fixed to stator by applying the ab transformation defined in the previous section. For this goal, consider the case when (5.13) represents ⎡

⎤ ⎡ ⎤ ψbr ψ1r ⎣ ψar ⎦ = Tr ⎣ ψ2r ⎦ = Tr ψ123r . ψ0r ψ3r

(5.14)

Differentiating this expression with respect to time, replacing (5.5), using (5.13) in the form (5.14) and ⎡

⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ Vbr V1r Ibr I1r ⎣ Var ⎦ = Tr ⎣ V2r ⎦ , ⎣ Iar ⎦ = Tr ⎣ I2r ⎦ , V0r V3r I0r I3r yields ⎤ ⎤ ⎡ ψ˙ br Vbr − Rr Ibr + θ˙r ψar ⎣ ψ˙ar ⎦ = ⎣ Var − Rr Iar − θ˙r ψbr ⎦ .  ψ˙ 0r ⎡

(5.15)

214

5 Induction Motor

The third component in (5.15) does not matter since x0s = 0 and x0r = 0 (see Sect. 5.1.3). Applying (5.13) to the second row of (5.6): ψ123r = L  sr I123s + L r I123r ,  ψab0r = Tr ψ123r = Tr L  sr I123s + Tr L r I123r , ψab0r = [ψbr , ψar , ψ0r ] .

Using the definition of Tr , given in (5.13), and the definition of matrices L sr and L r , given in (5.7), and after a long but straightforward procedure which involves use of some trigonometric identities, it follows that ψabr =

3 3 L sl Iabs + (L lr + L mr )Iabr , 2 2

(5.16)

where Iabr = [Ibr , Iar ] ,

Iabs = [Ibs , Ias ] , ψabr = [ψbr , ψar ] .

Solving (5.16) for Iabr , replacing in (5.15), defining Me =

3 L ms , 2

L r = L lr + Me ,

L ms = L mr = L sl ,

and recalling V123r = [0, 0, 0] , yields  Lr

ψ˙ br ψ˙ar



 =

 −Rr ψbr + Rr Me Ibs + n p q˙ L r ψar , −Rr ψar + Rr Me Ias − n p q˙ L r ψbr

(5.17)

which defines dynamics of rotor flux. On the other hand, consider the case when (5.10) represents ⎡

ψab0s

⎤ ⎡ ⎤ ψbs ψ1s = ⎣ ψas ⎦ = Ts ⎣ ψ2s ⎦ . ψ0s ψ3s

Differentiating with respect to time: ψ˙ab0s = Ts ψ˙ 123s , and replacing (5.4) ψ˙ab0s = Ts (V123s − Rs I123s ) = Vab0s − Rs Iab0s . From the first row in (5.6): ψ123s = L s I123s + L sr I123r ,

(5.18)

5.1 Motor Modeling

215

and using again (5.10): ψab0s = Ts L s I123s + Ts L sr I123r . Employing the definition of matrix Ts , given in (5.10), and the definition of matrices L sr and L s , given in (5.7), and after a long but straightforward procedure which involves use of some trigonometric identities, it follows that ψabs

1 3 = (L ls + L ms ) + L ms Iabs + L sl Iabr , ψabs = [ψbs , ψas ] . 2 2

Differentiating with respect to time:

1 3 ψ˙abs = (L ls + L ms ) + L ms I˙abs + L sl I˙abr . 2 2 Using the first two rows in (5.18):

1 3 Vabs − Rs Iabs = (L ls + L ms ) + L ms I˙abs + L sl I˙abr , Vabs = [Vbs , Vas ] . 2 2

(5.19)

Solving (5.16) for Iabr and differentiating with respect to time: I˙abr =

1 3 (ψ˙abr − L sl I˙abs ). 2 L lr + 23 L mr

Replacing this in (5.19) and using (5.17), yields L I I˙bs = Vbs − R I Ibs + α1 ψbr − α2 qψ ˙ ar , ˙ L I Ias = Vas − R I Ias + α1 ψar + α2 qψ ˙ br ,

(5.20) (5.21)

where L s = L ls + Me , α1 =

L I = Ls −

Me2 , Lr

RI =

L r2 Rs + Rr Me2 , L r2

Me n p Rr Me , α2 = . L r2 Lr

  M2 Notice that L I = L s σ, where σ = 1 − L s Le r > 0 is the total leakage factor of the motor [204], i.e., L I > 0. The expressions in (5.20), (5.21) represent dynamics of the stator electric currents. On the other hand, magnetic co-energy due to stator and rotor phase windings is given, according to (5.6), as

216

5 Induction Motor

1 E= 2



I123s I123r

 

L s L sr L sr L r



 I123s . I123r

According to D’Alembert’s principle, the generated torque is computed as ∂E , ∂q  1 ∂    I123s L sr I123r + I123r L = sr I123s , 2 ∂q  1 ∂    I123s L sr I123r + I123s = L sr I123r , 2 ∂q

∂ ∂   (I L sr I123r ) = I123s L sr I123r , = ∂q 123s ∂q

τ =

  since the terms I123s L s I123s and I123r L r I123r do not depend on q or, equivalently, on θr . Using (5.10) and (5.13) it is obtained:

∂ L sr (Tr )−1 Tr I123r , τ = ((Ts ) Ts I123s ) ∂q

∂  − = Iab0s (Ts ) L sr (Tr )−1 Iab0r . ∂q −1





Using the definitions of matrices Ts and Tr , given in (5.10) and (5.13), as well as the definition of matrix L sr , given in (5.7), Me = 23 L ms , L ms = L sl and after a long but straightforward procedure involving use of several trigonometric identities, it is found that ⎡ ⎤

0 Me 0 ∂ L sr (Tr )−1 = n p ⎣ −Me 0 0 ⎦ , (Ts )− ∂q 0 0 0 i.e., τ =

n p Me (Ibs ψar − Ias ψbr ). Lr

(5.22)

Remark 5.3 It is stressed that the disposition of components in Fig. 5.4 is such that the stator phase 1 is wound horizontally. This means that the positive axis of this phase winding is that designated as “1s” in Fig. 5.4. Notice that “1s” also indicates the direction of magnetic flux produced when a positive electric current flows through the stator phase 1. This is determined by using the right-hand rule and by the definition a positive electric current through phase 1 as in Fig. 5.1.

5.1 Motor Modeling

217

The positive axis of the fictitious phase “bs” (which coincides with “1s”) indicates the positive sense of a magnetic flux produced by a positive electric current Ibs through phase “bs”. Similarly, the positive axis of the fictitious phase “as” represents the positive sense of a magnetic flux when a positive electric current Ias flows through this phase winding. Remark 5.4 According to Remark 3.1, the direction of torque produced by the interaction between two magnetic fluxes is such that these magnetic fluxes tend to align. Further, a maximal torque is produced when the magnetic fluxes are orthogonal. On the other hand, according to (5.22), torque exerted on rotor is positive when both Ibs and ψar are positive (or when both are negative). Both of the above arguments indicate that torque on rotor is positive when exerted in the positive sense defined for θr in Fig. 5.4. Moreover, according to (5.22) torque on rotor is also positive when Ias and ψbr have opposite signs, i.e., according to the above arguments when torque on rotor is exerted in the positive sense defined for θr in Fig. 5.4. Thus, since torque in (5.22) is positive when exerted in the positive sense of θr and q, the application of Newton’s Second Law to rotor yields J ω˙ + bω =

n p Me (Ibs ψar − Ias ψbr ) − τ L , Lr

(5.23)

where J and b are positive constants standing for rotor inertia and viscous friction coefficient, respectively, whereas τ L is the external load torque and ω = q˙ represents rotor velocity. The expression in (5.23) is the dynamical model of the motor mechanical subsystem. Finally, the ab dynamical model of the induction motor is given by expressions in (5.17), (5.20), (5.21), and (5.23).

5.1.5 Park’s Transformation or dq Transformation Park’s or dq transformation is used to express the two-phase ab dynamical model of the induction motor in terms of two new fictitious phases designated as “d” and “q” which rotate in the same sense as rotor at a velocity to be determined in the following. The main ideas behind this new transformation are explained also in the following. First, define the following coordinate transformation1 : xr = e−J n p q xab , xab = eJ n p q xr ,   cos(n p q) − sin(n p q) , eJ n p q = sin(n p q) cos(n p q)

(5.24) (5.25)

where x is a two-dimensional vector standing for magnetic flux linkages and electric currents. 1 This transformation can be obtained from Fig. 5.5 if subindex “r” is replaced by “d’q’ ”, with “d’ ” the first component of “r”, and assuming, only at this moment, that α0 = n p q.

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5 Induction Motor

Fig. 5.5 dq transformation

The coordinate change in (5.24) means that vector xr is obtained by rotating counter-clockwise vector xab an angle θr = n p q, i.e., vector xr is expressed in terms of two orthogonal phases which are fixed to rotor. Further, the reader is encouraged to consult [250], Chap. 2, to verify that matrix e−J n p q , whose inverse is defined in (5.25), represents such a rotation and possesses the following properties [204], Chap. 10:  J θr −1   e = e−J θr = eJ θr ,   0 −1 = eJ θr J , J = . 1 0

e−J θr J eJ θr = J ,

(5.26)

J eJ θr

(5.27)

Using the coordinate change in (5.24) together with properties in (5.26), (5.27), and assuming that from this point on xab = [xa , xb ] , it is found that the rotor flux dynamics (5.17) and the expression for torque in (5.22) can be written as Rr Rr Me Ir , ψ˙r = − ψr + Lr Lr n p Me  τ = I J ψr , Lr r where ψr = [ψr 1 , ψr 2 ] ,

Ir = [Ir 1 , Ir 2 ]

(5.28) (5.29)

5.1 Motor Modeling

219

stand for rotor flux and stator electric current, respectively. The relative velocity ˙ and rotor magnetic flux velocity, is between rotor electrical velocity, θ˙r = n p q, computed as

ψr 2 d , arctan ρ˙ = dt ψr 1 1 ˙ = ψ J ψr , ψr 2 r Rr = τ, n p ψr 2

(5.30)

where (5.28) and (5.29) have been employed. Now, following Fig. 5.5, define the new transformation: xd  q  = e−J α0 xab , xab = eJ α0 xd  q  ,

(5.31)

for some scalar α0 and a two-dimensional vector xd  q  representing voltages, fluxes and electric currents expressed in terms of a frame rotating fixed to rotor flux, i.e., such that ψd  q  = [ψd  , ψq  ] = [ψr , 0] . Using (5.31) and properties in (5.26), torque in (5.22) can be written as τ =

n p Me n p Me n p Me  I   J ψd  q  = (Iq  ψd  − Id  ψq  ) = I q  ψd  . Lr d q Lr Lr

(5.32)

Hence, according to (5.30) and (5.32), velocity of rotor flux (in electrical radians) with respect to a frame fixed to stator is α˙ 0 = n p ω +

Rr Rr Me Iq  τ = n pω + . 2 n p ψr  L r ψd 

(5.33)

Using the transformation (5.31) in (5.17), (5.20) and (5.21) it is obtained: L I I˙d  q  = −L I α˙ 0 J Id  q  − R I Id  q  + α1 ψd  q  − α2 ω J ψd  q  + Ud  q  , Rr Rr Me Id  q  + (n p ω − α˙ 0 )J ψd  q  , Ud  q  = e−J α0 Vab . ψ˙ d  q  = − ψd  q  + Lr Lr

(5.34)

Replacing (5.33) in the second row of ψ˙ d  q  in last expression yields Rr ψ˙q  = − ψq  . Lr This ensures ψq  → 0 because imply that

Rr Lr

> 0. Moreover, this fact and the first row of ψ˙ d  q 

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5 Induction Motor

Rr Rr Me Id  . ψ˙ d  = − ψd  + Lr Lr Hence, ψd  → ψd∗ =constant if Id  = it can be written τ = Choosing Iq  = bω = τ − τ L :

Lr n p Me ψd∗

ψd∗ Me

(5.35)

is chosen. Replacing these results in (5.32)

n p Me Iq  ψd∗ . Lr

(5.36)

τ ∗ it is obtained the following from (5.23), i.e., from J ω˙ + J ω˙ + bω = τ ∗ − τ L .

(5.37)

Thus, it is only the matter to design a suitable τ ∗ to stabilize the linear mechanical dynamics in (5.37). It must be stressed that the above procedure, from the beginning of the present section until (5.33), has been developed just to find a manner to compute α0 , which is required to perform the coordinate change in (5.31). It has been shown that the coordinate change (5.31) expresses the induction motor dynamics in terms of two fictitious phases which rotate in such a way that one of them (phase “d’ ”) is always parallel to rotor flux and the other one (phase “q’ ”) is always orthogonal to phase “d’ ”. Notice that, according to (5.35) and (5.36), electric current in phase “ d’ ”, Id  , can be designed specially to force rotor flux to reach a desired value ψd∗ . Once this is achieved, electric current in phase “q’ ”, Iq  , can be designed specially to generate a desired torque, i.e., τ = τ ∗ .2 According to these ideas, an induction motor can be controlled mimicking control of a PM brushed DC-motor (see Remark 3.2). (i) A desired constant rotor magnetic flux is produced using electric current in phase “d’ ”. (ii) A desired torque is generated as the product of a constant depending on the constant rotor magnetic flux (see (5.36)) and a suitable value for electric current in phase “q’ ”. This is the main idea behind the coordinate change in (5.31). However, expression in (5.33) cannot be computed because rotor flux (i.e., ψd  = ψr ) cannot be measured. This means that α0 cannot be computed and, hence, the coordinate change in (5.31) cannot be performed. In order to solve this problem, the variable ψd  is replaced in (5.33) by its desired value ψd∗ . Thus, the following new variable is defined: ε˙0 = n p ω0 = n p ω +

2 This

Rr Me Iq , L r ψd∗

(5.38)

means that electric currents are assumed to be the control inputs. This is achieved in practice using PI electric current controllers as explained later in Sect. 5.1.8

5.1 Motor Modeling

221

which is intended to be used instead of α0 . The positive constant ψd∗ represents the desired rotor flux magnitude. Also, the following coordinate change, in terms of the new fictitious “d” and “q” phase windings, is introduced: xdq = e−J ε0 xab , xab = eJ ε0 xdq ,

(5.39)

which has to be used instead of (5.31). The coordinate change in (5.39) is known as the dq transformation for the induction motor.

5.1.6 The dq Dynamical Model Following the coordinate change in (5.39) defines Idq = e−J ε0 Iab ,

Iab = eJ ε0 Idq ,

ψdq = e−J ε0 ψab , ψab = eJ ε0 ψdq , Udq = e−J ε0 Vab , Vab = eJ ε0 Udq , xdq = [xd , xq ] , xab = [xa , xb ] .

(5.40)

Differentiating Iab = eJ ε0 Idq with respect to time and using (5.20), (5.21), (5.26), (5.27) it is obtained: ˙ ψdq + Udq − ω0 n p L I J Idq . L I I˙dq = −R I Idq + α1 ψdq − α2 qJ Differentiating ψab = eJ ε0 ψdq with respect to time and using (5.17), (5.26), (5.27) it follows that L r ψ˙ dq = −Rr ψdq + Rr Me Idq + n p q˙ L r J ψdq − ω0 n p L r J ψdq . Finally, using Iab = eJ ε0 Idq , ψab = eJ ε0 ψdq and (5.26), (5.27), torque given in (5.22) can be written as τ =

n p Me  I J ψdq . L r dq

Thus, the dq dynamical model of an induction motor is the following:

(5.41)

222

5 Induction Motor J ω˙ + bω =

n p Me  I J ψdq − τ L , L r dq

˙ ψdq + Udq − ω0 n p L I J Idq , L I I˙dq = −R I Idq + α1 ψdq − α2 qJ L r ψ˙ dq = −Rr ψdq + Rr Me Idq + n p q˙ L r J ψdq − ω0 n p L r J ψdq , Rr Me Iq , ε˙ 0 = n p ω0 = n p ω + L r ψd∗ Idq = [Id , Iq ] , ψdq = [ψd , ψq ] , Udq = e−J ε0 Vab , Vab = [Va , Vb ] , (5.42)

where ψd∗ is a positive constant representing the desired rotor flux magnitude.

5.1.7 dq Decomposition of the Magnetic Flux In the following, we use the dq decomposition of the magnetic flux produced by stator and rotor to explain some issues on the induction motor operation.

5.1.7.1

The Open-Loop Working Principle

In Sect. 5.1.1, the open-loop operation of an induction motor is described. The corresponding situation is depicted in Fig. 5.6 for n p = 1. The angular velocity of magnetic flux due to rotor ψ equals the angular velocity of magnetic flux due to the stator λ in steady state. This means that ψ does not move with respect to λ. As a matter of fact, if this was not the case, the sign of torque exerted on rotor would change as λ goes from behind to ahead of ψ. This means that either rotor velocity could not be kept constant or rotor would move in the opposite sense. Hence, according to Sect. 5.1.5, velocities of ψ and λ equal ε˙0 defined in (5.38). Since the slip velocity is given as ωsli p = ω S − ω R , where ω S = ε˙0 is velocity of λ and ω R = n p ω is the rotor velocity, I we conclude that the slip velocity is given by ωsli p = RrLMr e ψq∗ (see (5.38)). d According to the dq coordinate transformation introduced in the previous sections, in an induction motor the original three-phase windings at stator are projected or decomposed in two fictitious windings designated as d and q. The d and q phase windings are assumed to rotate at the same velocity as the rotor magnetic flux ψ. Moreover, phase d is assumed to be parallel to ψ. Thus, since ψ does not move with respect to λ, flux due to winding q, i.e., λq = L I Iq , and flux due to winding d, i.e., λd = L I Id , are the orthogonal components of λ (see Fig. 5.6). Contrary to the situation in a PM synchronous motor, an induction motor does not possess a permanent magnet at rotor. This means that the very origin of the rotor magnetic flux ψ must be the stator electric currents. In particular Id , the electric current through the fictitious winding d, must be responsible of ψ since winding d is always parallel to ψ whereas winding q is always orthogonal to ψ. We conclude that, contrary to what happens in a PM synchronous motor, λ and ψ cannot be orthogonal in steady state, i.e., γ =90◦ , since it is necessary that λd = L I Id = 0 in order to magnetize rotor.

5.1 Motor Modeling

223

Fig. 5.6 Stator and rotor magnetic flux orientation in an induction motor

According to the dq coordinate transformation, the rotor flux ψ = ψdq has the orthogonal components ψd and ψq which have the same orientation as the fictitious windings d and q defined for stator. Since ψ is parallel to winding d, we conclude that ψ = ψd and, consequently, ψq = 0 at least in steady state. Using this fact and (5.42) n M  n M J ψdq = pL r e ψd Iq = bω + τ L , we find that the generated torque is given as pL r e Idq since the generated torque equals load torque and friction torque in steady state. I Finally, recall that ωsli p = RrLMr e ψq∗ , where it is assumed that ψd = ψd∗ . Hence, we d conclude that the slip velocity grows as the generated torque grows. However, a deeper study shows that in steady state this direct relation between the slip velocity and the generated torque is only valid until the generated torque reaches a critical value. After this critical value, the generated torque decreases as the slip velocity increases. In [184], pp. 21, it is explained that this behavior is due to the fact that in open-loop control of induction motors the peak value of the applied three-phase voltages and velocity of λ, i.e., ω S , are kept constant. Also see [55] for more details on this topic.

5.1.7.2

Closed-Loop Operation

According to the previous discussion, when performing closed-loop control of an induction motor the goal is to force ψq → 0, i.e., to align the rotor flux ψ to the d-axis, and use Id to generate a desired magnitude for ψd , i.e., ψd = ψd∗ . Recall that this implies that λ and ψ are not orthogonal in induction motors. Once this is n M  n M J ψdq = pL r e ψd∗ Iq the desired torque is generaccomplished, according to pL r e Idq ated just by forcing Iq to reach a suitable value. This is performed using a closed-loop controller to determine the required voltages Udq , which are present in (5.42), and using Vab = eJ ε0 Udq , with ε0 the integral of (5.38), and the inverse of transforma-

224

5 Induction Motor

tion in (5.10) to compute the three-phase voltages to be applied to the motor stator V123s = [V1 , V2 , V3 ] . Contrary to what happens in closed-loop control of PM synchronous motors, closed-loop control of induction motors does not require to move rotor to a suitable initial position or home position. This is because the rotor magnetic flux has not a defined orientation with respect to rotor position. Moreover, the rotor magnetic flux is not static with respect to rotor, i.e., the slip velocity is different from zero in order to generate a nonzero torque.

5.1.7.3

Field Weakening

Assuming steady-state operation and ψq = 0, ψd = ψd∗ , we can combine the second component of I˙dq and the first component of ψ˙ dq , given in (5.42), and (5.38) to obtain Uq = n p ω

Me LI + Lr Me



Rr ψd + R I + Lr

Iq ,

ψd = M e I d . We stress that the maximal value of Uq is constrained by the maximal available peak value for the three-phase voltages applied to stator V123s = [V1 , V2 , V3 ] , which has an upper bound imposed by the power amplifiers. Hence, according to the first of the above expressions, once the maximal value of Uq is reached the rotor magnetic flux can be decreased in order to further increase rotor velocity. This is known as field weakening for obvious reasons and, according to the second of the above expressions, it is achieved by decreasing Id . Notice, however, that the generated torque depends on the product ψd∗ Iq . This means that field weakening allows to reach larger velocities at the price of reducing the generated torque. See [184], pp. 20, and [55] for further details on this topic.

5.1.8 Standard Indirect Field-Oriented Control Two block diagrams are shown in Fig. 5.7 which represent standard indirect fieldoriented control of an induction motor. Block diagram in Fig. 5.7a presents how the ab and dq transformations, given in (5.10) and (5.40), respectively, have to be used to compute the fictitious electric currents Id , Iq , from the stator real three-phase electric currents I1 , I2 , I3 , and how the inverse ab and dq transformations (also defined in (5.10) and (5.40)) have to be used to compute the real three-phase voltages V1 , V2 , V3 , to be applied to motor from the fictitious voltages Udq = [Ud , Uq ] . On the other hand, Fig. 5.7b presents an equivalent block diagram which is more suitable for the exposition that follows.

5.1 Motor Modeling

225

Fig. 5.7 Standard indirect field-oriented control of an induction motor. ϒ =

Lr n p Me ψd∗ , 

= ψd∗ /Me

This control scheme has been very successful in high performance industrial applications because of simplicity and good performance. The rationale behind this control scheme is the following. Two classical PI electric current controllers3 : Ud = αdp (Id∗ − Id ) + αdi Uq = αq p (Iq∗ − Iq ) + αqi

 

t

0

0

t

(Id∗ (s) − Id (s))ds − L I n p ω0 Iq ,

(5.43)

(Iq∗ (s) − Iq (s))ds + L I n p ω0 Id

(5.44)

are used to force Id and Iq to reach their desired values Id∗ and Iq∗ , respectively, very fast. Constant scalars αdp , αdi stand for the proportional and integral gains for phase “d”, whereas αq p , αqi are defined correspondingly for phase “q”. In the control literature it is common to assume that power electronics technology is available ensuring that three-phase voltages, obtained from (5.43), (5.44), and the inverse dq transformation, are actually applied at the motor terminals. See [263] for an approach taking into account these practical aspects. This allows to neglect dynamics of the power electronics stage during the stability analysis. However, it is important to say that some proposals exist in the literature [144] taking into account the power electronics dynamics in the stability analysis.

3 Although

proportional-integral are the most employed controllers for electric current in induction motors, some other approaches also exist such as proportional-resonant controllers and hysteresis controllers. See [108] for instance.

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5 Induction Motor

Fig. 5.8 Standard indirect field-oriented control of an induction motor presented in Fig. 5.7 reduces to control of a simple linear mechanical system

As explained in Sect. 5.1.5, the dq transformation, defined in (5.39), is different from the d’q’ transformation defined in (5.31). This is because definition of ε0 in (5.38) differs from definition of α0 given in (5.33) by use of ψd∗ instead of ψd .4 However, despite this difference, the dq transformation is intended to achieve the same results obtained with the d’q’ transformation which are explained between (5.31) and (5.38). Hence, the desired values for the dq electric currents are chosen as for the d’q’ electric currents, i.e., ψd∗ , Me Lr τ ∗, Iq∗ = n p Me ψd∗

Id∗ =

(5.45) (5.46)

which, as explained between (5.31) and (5.38), are assumed to achieve ψd → ψd∗ and τ → τ ∗ once controllers (5.43) and (5.44) achieve Id = Id∗ and Iq = Iq∗ . We recall that the definition of ε0 in (5.38) is intended to force ψq → 0 as also explained in Sect. 5.1.5. This implies that block diagram in Fig. 5.7b can be simplified to block diagram in Fig. 5.8. Thus, the proportional and integral gains k p and ki of the classical PI velocity controller (where ω ∗ represents the desired velocity): ∗

∗



t

τ = k p (ω − ω) + ki

(ω ∗ (s) − ω(s))ds

(5.47)

0

can be chosen by taking into account only the mechanical dynamics, i.e., using the simplified block diagram in Fig. 5.8. Notice, however, that the computation of ε0 according to (5.38) requires the exact knowledge of the rotor resistance Rr . This represents a drawback of standard indirect field-oriented control (SIFOC) because the rotor resistance is uncertain and changes its value under normal operation conditions, which results in performance deterioration. This has motivated lots of works on rotor resistance estimation in induction motors. See [17, 131, 179, 181, 182, 184, 235, 271] and references therein for instance. Other problems related to low resolution of sensors are studied in [19]. A similar procedure is also valid for motor position control if the PI velocity controller in (5.47) is replaced by the classical PID position controller (see Fig. 5.9): that ψd is not measurable and this forces the introduction of (5.38) and (5.39). This is the main reason why the field-oriented control strategy shown in Fig. 5.7 is designated including the word indirect. 4 Recall

5.1 Motor Modeling

227

Fig. 5.9 Standard indirect field-oriented control for position control in induction motors. ϒ = Lr ∗ n p Me ψ ∗ ,  = ψd /Me d

d(q ∗ − q) + ki τ = k p (q − q) + kd dt ∗

∗



t

(q ∗ (s) − q(s))ds,

(5.48)

0

where q ∗ represents the desired position, and the additional state equation is considered: q˙ = ω. The integral action in controllers (5.47) and (5.48) is included in order to ensure that ω = ω ∗ and q = q ∗ are achieved in steady state when both q ∗ and ω ∗ are constant and despite the presence of some unknown but constant external torque disturbance τL . Although dynamics in (5.42) is nonlinear, controller gains in (5.43), (5.44), (5.47), and (5.48) are traditionally selected using approximate classical control arguments. An alternative design method is presented in this chapter to find out some criteria for selection of the controller gains in (5.43), (5.44), (5.47), (5.48). This new methodology takes into account the nonlinear character of the closed-loop dynamics and, contrary to the classical control techniques when applied to induction motors, does not rely on linear approximations. Finally, recall that the star connection assumption on the stator phase windings as well as the balanced assumption on the source voltages V1 , V2 , V3 , i.e., condition stated in (5.3), are fundamental for the dynamic modeling presented in Sect. 5.1.2 and, hence, in Sects. 5.1.4 and 5.1.6. Notice that the star connection assumption on the stator phase windings is satisfied by construction of motor. On the other hand,

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5 Induction Motor

the source voltages V1 , V2 , V3 , are computed using (5.10) and last expression in (5.42) from the fictitious phase voltages Udq = [Ud , Uq ] , which are directly given by controllers in (5.43), (5.44). We stress that (5.10) and last expression in (5.42) assume that the zero phase voltage is zero, i.e., that (5.3) is always satisfied.

5.2 Open-Loop Energy Exchange 5.2.1 The Velocity Model The dq dynamical model of an induction motor is given in (5.42) and it is rewritten here for the ease of reference: n p Me  I J ψdq − τ L , L r dq = −R I Idq + α1 ψdq − α2 qJ ˙ ψdq + Udq − ω0 n p L I J Idq ,

J ω˙ + bω =

L I I˙dq L r ψ˙ dq = −Rr ψdq + Rr Me Idq + n p q˙ L r J ψdq − ω0 n p L r J ψdq , Rr Me Iq ε˙0 = n p ω0 = n p ω + , L r ψd∗

(5.49)

(5.50)

Idq = [Id , Iq ] , ψdq = [ψd , ψq ] , Udq = e−J ε0 Vab , Vab = [Va , Vb ] , where ψd∗ is a positive constant representing the desired rotor flux magnitude, whereas ω, τ L , Idq , ψdq , Udq stand for motor velocity (i.e., ω = q˙ where q is the rotor angular position), load torque, electric currents flowing through the stator dq phase windings, rotor fluxes, and voltages applied to the stator dq phase windings, respectively. The reader is referred to Sect. 5.1 for a complete description of the several parameters present in model (5.49). The following scalar function represents the total energy stored in the motor: V (Idq , ψdq , ω) =

1 1  1  L I Idq Idq + ψdq ψdq + J ω 2 , 2 2L r 2

(5.51)

where the first and second terms represent the magnetic energy stored in the motor electrical subsystem, i.e., stator and rotor, respectively, whereas the third term stands for the kinetic energy stored in the mechanical subsystem. The time derivative of V is given as 1  ˙  L I I˙dq + ψ ψdq + ω J ω, ˙ V˙ = Idq L r dq which, according to (5.49), can be written as

5.2 Open-Loop Energy Exchange

229

   −R I Idq + α1 ψdq − α2 qJ ˙ ψdq + Udq − ω0 n p L I J Idq V˙ = Idq  1   + 2 ψdq −Rr ψdq + Rr Me Idq + n p q˙ L r J ψdq − ω0 n p L r J ψdq Lr

n p Me  I J ψdq − τ L . +ω −bω + L r dq Notice that

n p Me  I J ψdq ω L r dq

 + Idq (−α2 qJ ˙ ψdq ) = 0,    Idq −ω0 n p L I J Idq = 0,  1   ψ n p q˙ L r J ψdq − ω0 n p L r J ψdq = 0, L r2 dq

(5.52)

(5.53)

M n

since α2 = Le r p , ω = q˙ and J is a skew-symmetric matrix. Recall that V represents energy stored in motor, i.e., (5.52) represents the energy exchange between the motor electrical and mechanical subsystems. Hence    −R I Idq + α1 ψdq + Udq (5.54) V˙ = Idq  1   + 2 ψdq −Rr ψdq + Rr Me Idq + ω (−bω − τ L ) , Lr       R I I2 −α1 I2 Idq Idq 2 T + Idq Udq , P0 = , P0 = −bω − ωτ L − −α1 I2 LR2r I2 ψdq ψdq r    ≤ −bω 2 − λmin (P0 ) Idq 2 + ψdq 2 + Idq Udq − ωτ L ,  where I2 stands for the 2 × 2 identity matrix. Defining the input u = [Udq , −τ L ]   and the output y = [Idq , ω] , we can write

V˙ ≤ −y  Q 0 y + y  u,

 Q0 =

 λmin (P0 )I2 02×1 . 01×2 b

(5.55)

Matrix P0 is positive definite if and only if its four leading principal minors are positive, i.e., if and only if R I > 0,

Rr R I > α12 . L r2

(5.56)

Using the definitions for R I and α1 given after (5.21), it is not difficult to verify that both of these expressions are always true and, hence, λmin (P0 ) > 0. This means that Q 0 is a positive definite matrix. Thus, (5.55) shows that model (5.49) is output strictly passive (see Definition 2.42) for the output y and input u defined above. In

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5 Induction Motor

this chapter, it will be shown that this is the basic theoretical principle behind the common and successful industrial velocity control scheme depicted in Fig. 5.7.

5.2.2 The Position Model Suppose that load torque is given as a nonlinear function of position τ L = G(q), , where P(q) is a positive semidefinite scalar function. Using such that G(q) = d P(q) dq these ideas and ω = q, ˙ the dq dynamical model of an induction motor, which is given in (5.42), can be rewritten as n p Me  I J ψdq − G(q), L r dq ˙ ψdq + Udq − ω0 n p L I J Idq , L I I˙dq = −R I Idq + α1 ψdq − α2 qJ L r ψ˙ dq = −Rr ψdq + Rr Me Idq + n p q˙ L r J ψdq − ω0 n p L r J ψdq , Rr Me Iq ε˙0 = n p ω0 = n p q˙ + , L r ψd∗

J q¨ + bq˙ =

(5.57)

Idq = [Id , Iq ] , ψdq = [ψd , ψq ] , Udq = e−J ε0 Vab , Vab = [Va , Vb ] . The following scalar function represents the total energy stored in the motor: ˙ = V (Idq , ψdq , q, q)

1 1  1  L I Idq Idq + ψ ψdq + J q˙ 2 + P(q), 2 2L r dq 2

(5.58)

where the latter new term represents the potential energy stored in the mechanical subsystem. Notice that V is a positive semidefinite function since P(q) is assumed to have this property. Proceeding as in the previous section to compute V˙ , we find that (5.52)–(5.53) stand again. Moreover, a new cancellation exists between cross terms ±G(q)q˙ which represents exchange between kinetic and potential energies in the mechanical subsystem. Thus, we find that    Udq , V˙ ≤ −bq˙ 2 − λmin (P0 ) Idq 2 + ψdq 2 + Idq with matrix P0 defined in (5.54) and I2 is the 2 × 2 identity matrix. Since b > 0, selecting the input u = Udq and the output y = Idq , we can write V˙ ≤ −λmin (P0 )y  y + y  u.

(5.59)

Notice that λmin (P0 ) > 0 since (5.56) is always true. Thus, (5.59) shows that model (5.57) is output strictly passive (see Definition 2.42) for the output y and input u defined above. In this chapter, it will be shown that this is the basic theoretical principle behind the common and successful industrial control scheme depicted in Fig. 5.7 when a PID position controller is used instead of a PI velocity controller.

5.3 Velocity Control

231

5.3 Velocity Control One of the first manners that were known to control velocity in induction motors was to apply, at the stator windings, a three-phase alternating current with a variable frequency to adjust the motor desired velocity. This approach is also known as openloop induction motor control. However, it was soon recognized the drawbacks of this approach in practice. Furthermore, it was demonstrated later in [212, 270] that this control strategy is stable for lightly loaded machines, and performances achieved with this control strategy are also limited. The introduction of FOC [22] solved this problem achieving high performances in induction motor control. One variant of FOC, called standard indirect field-oriented control (SIFOC, see Sect. 5.1.8) is the preferred approach to control induction motors. Despite the practical success of SIFOC it is not provided with a global asymptotic stability proof. For instance, in [219] are presented some arguments to justify local asymptotic stability. The following result formalizes the stability properties of standard indirect fieldoriented control of induction motors. Proposition 5.5 Consider the standard dq induction motor model (5.49)–(5.50) in closed loop with the following control law: Udq = Vdq + L I n p ω0 J Idq ,  t ∗ Vdq = −α p I˜dq − αi , I˜dq (s)ds, I˜dq = Idq − Idq

(5.61)

0

 ψd∗ Lr τ ∗  ∗ , , ψdq = [ψd∗ , 0] , Me n p Me ψd∗  t ∗ τ = −k p ω˜ − ki ω(s)ds, ˜ ω˜ = ω − ω ∗ ,

∗ = Idq

(5.60)



(5.62) (5.63)

0

where ω ∗ is the constant desired motor velocity and ψd∗ is the constant desired rotor flux magnitude. There always exist positive scalars k p , ki , and 2 × 2 diagonal positive definite matrices α p , αi , such that the closed-loop dynamics has an unique ∗ . equilibrium point. At this equilibrium point we have that ω˜ = 0, I˜dq = 0, ψdq = ψdq ˜ Furthermore, this equilibrium point is locally asymptotically stable in ψdq = ψdq − ∗ and semiglobally asymptotically stable in the remaining states. ψdq Proof According to (5.62) and (5.27): ∗ ∗ Idq J ψdq =

Lr ∗ τ . Me n p

(5.64)

Adding and subtracting some convenient terms and using (5.64) it can be written

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5 Induction Motor

n p Me  I J ψdq = τc + τ ∗ , L r dq n p Me ˜ ˜ ∗  ∗ τc = [ Idq J ψdq + Idq J ψ˜ dq + I˜dq J ψdq ]. Lr

(5.65)

Using this and (5.63) in the first expression in (5.49), taking advantage from the fact that ω˙ ∗ = 0, adding and subtracting term bω ∗ and defining 

t

z=

ω(s)ds ˜ +

0

1 (τ L + bω ∗ ), ki

(5.66)

yields J ω˙˜ = τc − bω˜ − k p ω˜ − ki z.

(5.67)

Notice that using (5.66) it can be written τ ∗ = −k p ω˜ − ki z + τ L + bω ∗ .

(5.68)

On the other hand, replacing (5.60), (5.61), in the second expression in (5.49), using (5.62), (5.68), adding and subtracting some convenient terms and defining 

t

z1 = 0

I˜dq (s)ds 

∗ ∗ −(αi )−1 −α2 ω ∗ J ψdq + α1 ψdq − RI



(5.69)   ∗ , ∗ (τ L + bω )

ψd∗ Lr , M e n p M e ψd

yields L I I˙˜dq = −(R I I2 + α p ) I˜dq + α1 ψ˜ dq − α2 ωJ ˜ ψdq − α2 ω ∗ J ψ˜ dq ∗ L r τ˙ Lr  −L I [0, (k p ω˜ + ki z)] − αi z 1 , (5.70) ∗ ] + R I [0, n p M e ψd n p Me ψd∗ τ˙ ∗ = −k p ω˙˜ − ki ω, ˜ (5.71) ∗ where ψ˜ dq = ψdq − ψdq and I2 stands for the 2 × 2 identity matrix. Finally, adding ∗ ∗ ∗ /L r , Rr Me Idq /L r and (n p ω − n p ω0 )J ψdq in the third and subtracting terms Rr ψdq ∗ ˙ expression in (5.49), taking advantage from the fact that ψdq = 0, using (5.62) and the fourth expression in (5.49), the following is obtained:

L r ψ˙˜ dq = −Rr ψ˜ dq + Rr Me I˜dq + L r (n p ω − n p ω0 )J ψ˜ dq − [0, Rr Me I˜q ] , Lr τ ∗ . (5.72) I˜q = Iq − n p Me ψd∗

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233

Notice that the closed-loop dynamics, which is given by (5.66), (5.67), (5.69), (5.70), (5.72), can be rewritten as follows (recall that τ L is constant): n p Me ˜ ˜ I J ψdq − T L , L r dq = −(R I I2 + α p ) I˜dq + α1 ψ˜ dq − α2 ωJ ˜ ψdq + Udq ,

J ω˙˜ = −(b + k p )ω˜ +

(5.73)

(5.74) L I I˙˜dq L r ψ˙˜ dq = −Rr ψ˜ dq + Rr Me I˜dq + L r (n p ω − n p ω0 )J ψ˜ dq − [0, Rr Me I˜q ] , (5.75) z˙ = ω, ˜ z˙ 1 = I˜dq , (5.76) n p Me ∗ ˜  ∗ [Idq J ψdq + I˜dq J ψdq ] + ki z, Lr Udq = −α2 ω ∗ J ψ˜ dq L r τ˙ ∗  Lr −L I [0, (k p ω˜ + ki z)] − αi z 1 , ∗ ] + R I [0, n p M e ψd n p Me ψd∗ Rr Me Iq ε˙0 = n p ω0 = n p ω + . L r ψd∗ TL = −

The closed-loop dynamics (5.66), (5.67), (5.69), (5.70), (5.72), or equivalently,    , z 1 , ψ˜dq ] ∈ R8 . (5.73)–(5.76), is autonomous with state defined as ξ = [ω, ˜ z, I˜dq Equilibria are computed as follows. From (5.66) and (5.69) we have that ω˜ = 0 and I˜dq = 0 at the equilibrium point. Use of these values and evaluating (5.72) at the equilibrium condition yields 0 = −Rr ψ˜ dq + L r (n p ω − n p ω0 )J ψ˜dq . Using the fourth expression in (5.49), Iq = n pLMr τe ψ∗ , because I˜dq = 0, and definition d of τ ∗ given in (5.68) it is found that last expression becomes ∗

Lr τ ∗ J ψ˜ dq ψ˜ dq = − n p ψd∗2 which, according to (5.27), implies that ψ˜ dq = 0. Using all of these results in (5.67) yields z = 0. Finally, from (5.70), z 1 = 0 follows. Thus, ξ = 0 is the only equilibrium point of the closed-loop dynamics (5.66), (5.67), (5.69), (5.70), (5.72). On the other hand, notice that from (5.72) and the first component in (5.70) it is obtained: L r ψ˙˜ dq = −Rr ψ˜ dq + Rr Me [ I˜d , 0] + L r (n p ω − n p ω0 )J ψ˜ dq , L I I˙˜d = −(R I + α p1 ) I˜d + α1 ψ˜ d + α2 ω˜ ψ˜q + α2 ω ∗ ψ˜q − αi1 z 11 , z˙ 11 = I˜d ,

(5.77)

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5 Induction Motor

where I˜d = Id − Id∗ , ψ˜q = ψq − ψq∗ , ψq∗ = 0, z 1 = [z 11 , z 12 ] , α p = diag{α p1 , α p2 }, αi = diag{αi1 , αi2 } have been used. Notice that the following scalar function: V1 (ζ) =

1 ˜2 1 1 ˜T ˜ 2 L I Id + αi1 z 11 + ψ ψdq , 2 2 2L r dq

where ζ = [ I˜d , z 11 , ψ˜d , ψ˜q ] is positive definite and radially unbounded satisfying σ1 ζ2 ≤ V1 (ζ) ≤ σ2 ζ2 , 1 1 1 1 σ1 = min{L I , αi1 , }, σ2 = max{L I , αi1 , }. 2 Lr 2 Lr

(5.78)

The time derivative of V1 along the trajectories of (5.77) is given as Rr  ˜ V˙1 = −(R I + α p1 ) I˜d2 + I˜d (2α1 ψ˜ d + α2 ω˜ ψ˜q + α2 ω ∗ ψ˜q ) − 2 ψ˜ dq ψdq , Lr ≤ −η  Pη, η = [ |ψ˜q |, |ψ˜d |, | I˜d |] , ⎡ ∗ ⎤ ˜ | Rr 0 − α22|ω| − α2 |ω L r2 2 ⎥ ⎢ Rr P=⎣ 0 −α1 ⎦, L2 ˜ − α22|ω| −

α2 |ω ∗ | 2

r

−α1

R I + α p1

where the fact that ±x y ≤ |x| |y|, ∀x, y ∈ R, has been used. Matrix P is positive definite if and only if its three leading principal minors are positive, i.e., Rr > 0, L r2



Rr L r2

2 (R I + α p1 ) −

Rr L r2



α2 |ω| α2 |ω ∗ | ˜ + 2 2

2 − α12

Rr > 0. (5.79) L r2

The first condition is naturally satisfied by the motor parameter properties. The second condition can be rewritten as ˜ + c, α p1 > a ω˜ 2 + b|ω|

(5.80)

for some real numbers c and a > 0, b > 0, which only depend on the motor parameters and the desired velocity ω ∗ . Thus, there always exist a positive α p1 satisfying (5.79) as long as |ω| ˜ be bounded, i.e., if ξ ≤ ξ M < ∞ (ξ M stands for the supreme value of ξ over the norm). This implies that λmin (P) > 0 as long as (5.80) be satisfied and it can be written V˙1 ≤ −λmin (P)η2 ≤ 0. This and (5.78) mean that ζ = 0 is uniformly stable (see Theorem 2.26), i.e., that ψ˜ d , ψ˜q , I˜d , z 11 ∈ L∞ . Furthermore, the above expression can be written as

5.3 Velocity Control

235

V˙1 ≤ −λmin (P)φ2 ≤ 0, where φ is any of ψ˜q , ψ˜ d or I˜d . Time integration of the above inequality yields 

t

V1 (t) − V1 (t0 ) ≤ −λmin (P)

φ2 (t)dt,

t0

or, equivalently: 

t

V1 (t0 ) − V1 (t) ≥ λmin (P)

φ2 (t)dt,

t0



t

V1 (t0 ) ≥ λmin (P)

φ2 (t)dt,

t0

because V1 (t) ≥ 0, ∀t ≥ t0 . Thus  ∞>

V1 (t0 ) ≥ λmin (P)



t

φ2 (t)dt,

t0

which proves that ψ˜q , ψ˜d , I˜d ∈ L2 . From the fourth expression in (5.49), (5.61), (5.62), (5.68) it is clear that n p ω − n p ω0 is bounded if I˜q and Iq∗ are bounded, i.e., if the load torque τ L and the desired velocity ω ∗ are bounded and ξ ≤ ξ M < ∞. Hence, from (5.77) it can be seen that ψ˙˜ d , ψ˙˜ q , I˙˜d ∈ L∞ . Thus, Barbalat’s Lemma (see Corollary 2.32) can be invoked to conclude that limt→∞ [ψ˜q (t), ψ˜d (t), I˜d (t)] = [0, 0, 0] if (5.79) and ξ ≤ ξ M < ∞ are true. Now, proceed with the stability analysis of the equilibrium point ξ = 0 of the closed-loop dynamics (5.66), (5.67), (5.69), (5.70), (5.72) or, equivalently, (5.73)– (5.76). Notice that expressions in (5.73)–(5.76) are almost identical to the open-loop ˙ τ L , Udq , ψdq by I˜dq , ω, ˜ T L , Udq , ψ˜ dq , model given in (5.49) if we replace Idq , ω = q, respectively. An important difference is that electric resistance and viscous friction have been enlarged, i.e., we have R I I2 + α p and b + k p in (5.73)–(5.76) instead of R I and b in (5.49). Another important difference is the presence of two additional equations in (5.76) which represent the integral parts of both, the PI velocity controller and the PI electric current controllers. These observations motivate use of the following “energy” storage function for the closed-loop dynamics: 1 ˜ ˜ 1 1 ˜ ˜ L I Idq Idq + ˜ z), ψ ψdq + z 1 αi z 1 + Vω (ω, 2 2L r dq 2 1 1 Vω (ω, ˜ z) = J ω˜ 2 + (ki + βk p + bβ)z 2 + β J z ω. ˜ 2 2 V (ξ) =

(5.81)

Function Vω (ω, ˜ z) was introduced in Chap. 3 for the stability analysis of the standard velocity control scheme in PM brushed DC-motors where its composition is

236

5 Induction Motor

explained. Moreover, first terms in V (ξ) above are included because of similar reasons as in Chap. 3: to take into account “energy” stored in the stator electrical subsystem and the rotor magnetic subsystem. Taking advantage from the fact that J is positive, it is shown in Appendix A.1 ˜ z) is positive definite and radially unbounded if that Vω (ω, ki + β(b + k p ) − J β 2 > 0, β > 0.

(5.82)

Hence, since L I , L r are always positive (see Sect. 5.1), the Lyapunov function candidate V given in (5.81) is positive definite and radially unbounded if αi is a positive definite matrix and (5.82) is satisfied. After several straightforward cancellations (see Remark 5.6), and taking advantage from |y  w| ≤ y w, ∀y, w ∈ Rn , and ±xs ≤ |x| |s|, ∀x, s ∈ R, the time derivative of V along the trajectories of the closed-loop dynamics (5.73)–(5.76) can be upper bounded as n p Me ∗ |ω| ˜ Idq  ψ˜ dq  − λmin (R I I2 + α p ) I˜dq 2 V˙ ≤ −(k p + b)ω˜ 2 + Lr   L r τ˙ ∗  T 0, +α1  I˜dq  ψ˜ dq  + α2 |ω ∗ |  I˜dq  ψ˜dq  − L I I˜dq n p Me ψd∗ RI Lr Rr Rr Me ˜ +  I˜dq (k p |ω| ˜ + ki |z|) − 2 ψ˜ dq 2 + ψdq   I˜dq  n p Me ψd∗ Lr L r2 βn p Me βn p Me ∗ |z|  I˜dq  ψ˜ dq  + |z| Idq  ψ˜ dq  +β J ω˜ 2 + Lr Lr βn p Me ∗ + |z|  I˜dq  ψdq  − βki z 2 . Lr Using (5.71), (5.67), and  ∗ ψ  Lr ∗ ∗ Idq  ≤ Idq 1 ≤  d  + (k p |ω| ˜ + ki |z| + |τ L + bω ∗ |), Me n p Me ψd∗ it can be written ˜ |z|, ψ˜dq ,  I˜dq ] , V˙ ≤ −x  Qx, x = [|ω|, where entries of matrix Q are given as

5.3 Velocity Control

237

kp ˜ βki ˜ ψdq , ∗ ψdq , Q 22 = βki − ψd ψd∗ LI kp ˜ LI kp , = λmin (R I I2 + α p ) − ∗ ψdq  − J ψd J (ki + βk p ) ˜ = Q 21 = − ψdq , 2ψd∗

n p Me ψd∗ |τ L + bω ∗ | = Q 31 = − + L r , 2L r Me n p Me ψd∗

βn p Me ψd∗ |τ L + bω ∗ | = Q 32 = − + L r , 2L r Me n p Me ψd∗

Q 11 = b + k p − β J − Q 44 Q 12 Q 13 Q 23

Q 14 = Q 41 = −

L I L r k 2p 2n p Me (ψd∗ )2 J

ψ˜ dq  −

Q 33 =

Rr , L r2 (5.83)

L I L r k p (k p + b) L I L r ki − ∗ 2n p Me J ψd 2n p Me ψd∗

RI Lr k p , 2n p Me ψd∗ L I L r k p ki L I L r k p ki R I L r ki ˜ =− ∗ 2 ψdq  − ∗ − 2n p Me J (ψd ) 2n p Me J ψd 2n p Me ψd∗ ∗ βn p Me ψd βn p Me ˜ − ψdq  − , 2L r 2L r LI kp L I L r k p |τ L + bω ∗ | α2 |ω ∗ | − = −α1 − − . 2 2J Me 2n p Me J (ψd∗ )2 −

Q 24 = Q 42

Q 34 = Q 43

It is clear that, provided that ψ˜ dq  is small enough, the four leading principal minors of Q can always be rendered positive by choosing large enough positive values for k p , ki , and a large enough positive definite matrix α p , i.e., a large enough λmin (R I I2 + α p ). Hence, matrix Q can always be rendered positive definite and, thus, V˙ ≤ 0. As remarked after (5.76), the closed-loop dynamics (5.73)–(5.76) is autonomous and, hence, the LaSalle invariance principle (see Corollary 2.18) can be applied as follows. Define a set S as T T , z 1T , ψ˜dq ] ∈ D|V˙ = 0} S = {[ω, ˜ z, I˜dq

= {ω˜ = z = 0, ψ˜dq = I˜dq = [0, 0]T , z 1 ∈ D ∩ R2 }, where D is a domain in R8 such that ψ˜ dq  is “small enough” to ensure positive definiteness of matrix Q. Evaluating the closed-loop dynamics (5.73)–(5.76) in S we obtain

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5 Induction Motor

n p Me 0 = −(b + k p )(0) + [0, 0]J [0, 0] − T L , Lr   0 = −(R I I2 + α p )[0, 0] + α1 [0, 0] − α2 (0)J ψdq + Udq , 0   0 = −Rr [0, 0] + Rr Me [0, 0] + L r (n p ω − n p ω0 )J [0, 0] − [0, Rr Me (0)] , 0 z˙ = 0, z˙ 1 = [0, 0] , n p Me ∗T ∗ TL = − [Idq J [0, 0] + [0, 0]J ψdq ] + ki (0), Lr Udq = −α2 ω ∗ J [0, 0] L r τ˙ ∗  Lr −L I [0, ] + R I [0, (k p (0) + ki (0))] − αi z 1 , n p Me ψd∗ n p Me ψd∗ τ˙ ∗ = −k p ω˙˜ − ki ω˜ = −k p (0) − ki (0).   These expressions have, as the unique solution, [ω, ˜ z, I˜dq , z 1 , ψ˜dq ] = 01×8 because αi is a positive definite matrix. Thus, according to Corollary 2.18, this implies that   , z 1 , ψ˜dq ] = 01×8 is an asymptotically stable equilibrium point. [ω, ˜ z, I˜dq On the other hand, notice that Lyapunov function in (5.81) satisfies

(5.84) κ1 ξ2 ≤ V (ξ) ≤ κ2 ξ2 , 1 κ1 = min{L I , λmin (αi ), 1/L r , λmin (P1 )}, 2 1 κ2 = max{L I , λmax (αi ), 1/L r , λmax (P2 )},   2  J −β J J βJ , P2 = . P1 = −β J β(k p + b) + ki β J β(k p + b) + ki Suppose that the initial state is chosen such that ξ(0) < r for some r > 0. Then, according to (5.84) and using the fact that V˙ ≤ 0, it is true that ξ(t) ≤ ξ M , ∀t ≥ 0, where r=

κ1 ξ M . κ2

(5.85)

Hence, a value for ξ M can be ensured by selecting ξ(0) < r according to (5.85). Also notice that ψ˜ dq  can always be rendered small enough since it has been proven that limt→∞ [ψ˜q (t), ψ˜ d (t), I˜d (t)] = [0, 0, 0] if (5.79) and ξ ≤ ξ M < ∞ are true. This can be achieved by using a common industrial practice [219] as explained in the following. Before controlling the mechanical part of the motor (i.e., by using ω˜ = ω ∗ = 0, τ L = 0), the direct phase is excited alone in order to generate the necessary direct phase current to obtain the required flux. Since limt→∞ [ψ˜q (t), ψ˜ d (t), I˜d (t)] = [0, 0, 0] , it is only matter of time to satisfy the

5.3 Velocity Control

239

requirement on a small enough ψ˜dq . After that a nonzero ω ∗ can be commanded  and a nonzero τ L can be applied. Moreover, it is useful to know that [ I˜d , z 11 , ψ˜dq ]= ∗ [0, 0, 0, 0] is globally exponentially stable when ω˜ = ω = 0, τ L = 0. This can be proven evaluating (5.77) at ω˜ = ω ∗ = 0 (also Iq = Iq∗ = 0 since τ ∗ = 0, τ L = 0), using V2 (ζ) =

1 1 ˜ ˜ 1 2 + γ L I I˜d z 11 ψ ψdq + (αi1 + γ(R I + α p1 ))z 11 L I I˜d2 + 2 2L r dq 2

(5.86)

as Lyapunov function and invoking Theorem 2.28. This requires, additionally γ > 0, αi1 + γ(R I + α p1 ) > L I γ 2 , R I + α p1 > L I γ, det(P3 ) > 0, (5.87) where ⎡

⎤ R I + α p1 − γ L I 0 −α1 γα1 0 γαi1 − 2 ⎦ . P3 = ⎣ −α1 − γα2 1 LR2r r

On the other hand, according to (5.80), for large values of ξ M (5.79) can be approximately replaced by α p1 > aξ2M . Using this in (5.85) yields r
2 Lr



α2 ξ M α2 |ω ∗ | + 2 2

2 + α12

Rr , L r2

(5.89)

(5.87), β > 0, ki + bβ + βk p > J β 2 , αi is a positive definite 2 × 2 matrix, some large enough constants k p > 0, ki > 0 and a positive definite 2 × 2 matrix α p such that the four leading principal minors of matrix Q defined in (5.83) are positive. As stated earlier, the requirement for a small enough ψ˜dq  can be satisfied by starting the controller using ω˜ = ω ∗ = 0, τ L = 0, to command the desired velocity ω ∗ = 0 after a time when ψ˜ dq  becomes small enough. Once the controller gains are selected, the region of attraction can be computed using (5.85). If a larger r is required then propose a larger ξ M , use (5.89) to select an α p1 and check again the remaining stability conditions.

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5 Induction Motor

Remark 5.6 The cancellation of cross terms referred after (5.82) involves    n p Me  ˜ ˜   ∗ Idq J ψdq + I˜dq = 0, −α2 ωJ ˜ ψdq + ω˜ J ψdq I˜dq Lr   which includes some terms belonging to dtd 21 J ω˜ 2 , the time derivative of the “kinetic in the mechanical subsystem, and some terms belonging  energy” stored   ˜ Idq , the time derivative of the “magnetic energy” stored in the to dtd 21 L I I˜dq electrical subsystem. These cancellations are due to the natural energy exchange between the electrical and the mechanical subsystems. They are obvious consequences of the almost identical structure of the closed-loop dynamics in (5.73)– (5.76), and the open-loop dynamics in (5.49) (see (5.52)). Moreover, we also have  L r (n p ω − n p ω0 )J ψ˜ dq = 0, again, because of the skew-symmetric property that ψ˜ dq of matrix J (see (5.53)). Remark 5.7 It is stressed that the closed-loop dynamics of electric currents is not ∗ completed by feeding back I˙dq but, instead, we dominate the terms arising from such a function. Moreover, this is accomplished despite velocity measurements are ∗ ∗ , i.e., despite acceleration is included in I˙dq . This fact is important included in Idq because it is instrumental to maintain the simplicity of standard indirect field-oriented control. As a matter of fact, previous works in the literature [204], Chap. 10, Sect. 3.7, ∗ which do complete the closed-loop dynamics of electric currents by feeding back I˙dq results in more complex controllers despite they employ velocity filtering to avoid ∗ . See Remark 5.18. acceleration in I˙dq Remark 5.8 Using (5.65) and (5.41), it is possible to write the closed-loop mechanical subsystem dynamics given in (5.73) as J ω˙˜ = −(b + k p )ω˜ + τc − ki z, where n p Me ˜ ˜ ∗  ∗ [ Idq J ψdq + Idq J ψ˜ dq + I˜dq J ψdq ], Lr n p Me  = I J ψdq − τ ∗ L r dq

τc =

is the difference between actual and desired torques. Hence,    ω˜ Q 11 0 ˙ ˜ z] Vω ≤ −[ω, + (ω˜ + βz)τc , 0 Q 22 z where function Vω is defined in (5.81) and Q 11 = b + k p − β J,

Q 22 = βki .

(5.90)

5.3 Velocity Control

241

On the other hand, the time derivative of the first component of V in (5.81), i.e., Ve =

1 ˜ ˜ 1 ˜ ˜ 1 ψ ψdq + z 1 αi z 1 L I Idq Idq + 2 2L r dq 2

along the trajectories of the closed-loop electrical subsystem dynamics (5.74), (5.75),  (R I I2 + α p ) I˜dq and − LR2r ψ˜ dq 2 . Thus, contains the quadratic negative terms − I˜dq r when computing V˙ = V˙ω + V˙e , we realize that instrumental for the stability result in Proposition 5.5 are the following features, which are very similar to those in Remark 3.4: • The scalar function Vω is a strict Lyapunov function when the desired torque is the input of the closed-loop mechanical subsystem dynamics, given in (5.90) i.e., when τc = 0. • Coefficient of the negative term −(R I I2 + α p ) I˜dq 2 , appearing in V˙ , can be enlarged arbitrarily. This is important to dominate cross terms in V˙ depending on I˜dq when I˜dq = 0. • Although the coefficient of term − LR2r ψ˜ dq 2 , i.e., LR2r , cannot be arbitrarily r r enlarged, its negative sign is instrumental to render positive third minor of matrix Q, defined in (5.83), by enlarging both k p and ki . • Cancellation of several cross terms belonging to V˙ω and V˙e . See Remark 5.6. Notice that all of these features are possible thanks to the passivity properties, described in Sect. 5.2.1. Furthermore, the above proof contains the main ideas of procedure described in Sect. 2.4. As pointed out in Remark 3.4, τc is given as a nonlinear function of the electrical dynamics error for AC-motors (see (5.65)). In the case of induction motors, some third-order terms arising from the bilinear nature of the motor model represent and obstacle to prove global asymptotic stability and, thus, only a restricted asymptotic stability result is established in Proposition 5.5. As it is shown in Proposition 5.10, this obstacle can be overcome by introducing a simple additional nonlinear term Remark 5.9 Standard indirect field-oriented control of induction motors was presented in Sect. 5.1.8, i.e., in (5.43)–(5.47), which are repeated here for the ease of reference:  t ∗ (Id∗ (s) − Id (s))ds − L I n p ω0 Iq , (5.91) Ud = αdp (Id − Id ) + αdi 0  t (Iq∗ (s) − Iq (s))ds + L I n p ω0 Id , (5.92) Uq = αq p (Iq∗ − Iq ) + αqi ψ∗ = d, Me

0

Lr = τ ∗, n p Me ψd∗  t ∗ ∗ (ω ∗ (s) − ω(s))ds. τ = k p (ω − ω) + ki Id∗

Iq∗

(5.93)

0

This control strategy is depicted in Fig. 5.10, again for the ease of reference. It is stressed that controller in Proposition 5.5 is exactly the control scheme in (5.91), i.e.,

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5 Induction Motor

Fig. 5.10 Standard indirect field-oriented control of induction motors. ϒ =

Lr n p Me ψd∗ , 

= ψd∗ /Me

in Fig. 5.10. The main reason to include Proposition 5.5 is to present a stability proof for the unique equilibrium point of the closed-loop system. Finally, let us say that a tuning procedure is derived in [49] for the PI velocity controller in (5.93). This, however, is performed on the basis of the current-fed induction motor model. This means that the stator electrical dynamics is neglected, (5.91) and (5.92) are not taken into account and it is assumed that electric currents Id = Id∗ and Iq = Iq∗ are the motor inputs. Thus, a global asymptotic stability result for standard indirect field-oriented control of voltage-fed induction motors is still lacking. Stability results in Proposition 5.5 are, at most, semiglobal. The following result introduces a slightly modified standard field-oriented controller which is proven to achieve a more general stability result. Proposition 5.10 Consider the standard dq induction motor model (5.49)–(5.50) in closed loop with the following control law: Udq = Vdq + L I n p ω0 J Idq ,  t ∗ ˜ Vdq = −α p Idq − αi , I˜dq (s)ds − αd I˜dq , I˜dq = Idq − Idq 0

(5.94) (5.95)

 ψd∗ Lr τ ∗  ∗ , , ψdq = [ψd∗ , 0] , αd = diag{α3 β0 , 0}, (5.96) Me n p Me ψd∗  t ω(s)ds, ˜ ω˜ = ω − ω ∗ , (5.97) τ ∗ = −k p ω˜ − ki

∗ Idq =



0

5.3 Velocity Control

243

 β0 = max ω , 2

I˜q2 ,



t 0

2  ω(s)ds ˜ ,

t

I˜q (s)ds

2  ,

0

∗ where ω ∗ is the constant desired motor velocity and ψdq is the desired flux. There always exist positive scalars k p , ki , α3 , and 2 × 2 diagonal positive definite matrices α p , αi , such that the closed-loop dynamics has an unique equilibrium point. At ∗ . Furthermore, this this equilibrium point we have that ω˜ = 0, I˜dq = 0, ψdq = ψdq equilibrium point is asymptotically stable, semiglobally in the rotor flux error, the d current error and the integral of the d current error, and globally in the remaining state variables.

Remark 5.11 Aside from term −αd I˜dq in (5.95), controller in Proposition 5.10 is exactly standard indirect field-oriented control of voltage-fed induction motors as described in [219] and in proposition 5.5 in the present book (see Fig. 5.10). This is important to stress because closed-loop system in Proposition 5.10 is proven below to have a bounded state that globally converges to the desired equilibrium point. This means that the proposed controller is the closest controller proposed in the literature to standard indirect field-oriented control of voltage-fed induction motors but provided with a proof for global convergence to the desired equilibrium point. A sketch of proof of Proposition 5.10 is presented in the following. The main idea is to describe with simple arguments the rationale behind the complete proof which is formally presented in Appendix D.1. The initial steps in this proof are identical to those in proof of Proposition 5.5. Hence, the main ideas of procedure described in Sect. 2.4 are also employed in what follows. Sketch of proof of Proposition 5.10. Notice that controller in Proposition 5.10 only differs from controller in Proposition 5.5 by term −αd I˜dq appearing in (5.95). Thus, taking advantage from the property −α3 β0 Id2 ≤ 0, it is shown that the part of the proof of Proposition 5.5 where it is proven that the origin of    , z 1 , ψ˜dq ] ∈ R8 is the only equilibrium point and it is locally asympξ = [ω, ˜ z, I˜dq totically stable provided that ψ˜ dq remains inside a ball of small enough radius is still valid. However, all of the other components of the initial state ξ(t0 ) can be arbitrary large. Thus, the following statement is proven: B1 ξ ∈ L8∞ and limt→∞ ξ(t) = 0, from any initial condition ξ(t0 ), if ψ˜ dq (t) remains inside a ball whose radius is small enough.  ˜ , Id ) subsystem dynamics is analyzed. This will prove to be useful Next, the (ψ˜ dq to overcome the restriction on a small ψ˜ dq . Hence, the stability properties of the following subsystem are analyzed: L r ψ˙˜ dq = −Rr ψ˜dq + Rr Me [ I˜d , 0] + L r (n p ω − n p ω0 )J ψ˜ dq , (5.98) ˙ L I I˜d = −(R I + α p1 ) I˜d − α3 β0 I˜d + α1 ψ˜ d + α2 ω˜ ψ˜q + α2 ω ∗ ψ˜q − αi1 z 11 , z˙ 11 = I˜d ,

244

5 Induction Motor

where ψ˜ d = ψd − ψd∗ , ψ˜q = ψq − ψq∗ , ψq∗ = 0, z 1 =[z 11 , z 12 ] , α p =diag{α p1 , α p2 }, αi = diag{αi1 , αi2 } have been used. As a result, the following statement is proven: B2 [ψ˜q (t), ψ˜ d (t), I˜d (t)] is bounded, ψ˜q , ψ˜d , I˜d ∈ L2 and limt→∞ [ψ˜q (t), ψ˜ d (t), I˜d (t)] = [0, 0, 0] from any initial condition [ψ˜q (t0 ), ψ˜d (t0 ), I˜d (t0 ), z 11 (t0 )] , if ξ ∈ L8∞ . Moreover ψ˜ dq converges in finite time into a ball whose radius decreases to zero if ξ → ∞, from any initial condition [ψ˜q (t0 ), ψ˜d (t0 ), I˜d (t0 ), z 11 (t0 )] . Finally, results in B1 and B2 are combined to prove what it is stated in the proposition. With this aim, itis first proven that in the case where [ψ˜ d , ψ˜q ] is not small enough, κ3

(t−t )

then ξ(t) ≤ κκ21 ξ(t0 ) e 2κ1 0 < ∞ for any finite time. This means that ξ(t) has not a finite escape time. Thus, when starting from any bounded initial condition ξ(t0 ), the state ξ(t) remains bounded for any finite time t > t0 . Assume that the worst case is present, i.e., that ξ(t) tends to grow without limit. In such a case B2 ensures that [ψ˜q (t), ψ˜ d (t)] < ε, for any ε > 0, in finite time, i.e., before ξ(t) escapes. Thus, by virtue of B1, ξ(t) will converge asymptotically to zero. In the case when ξ(t) does not tend to grow without limit, then ξ(t) will remain bounded until, by virtue of B2, [ψ˜q (t), ψ˜ d (t)] reaches and stays within a small enough value and, hence, ξ(t) will converge asymptotically to zero. This proves boundedness of the state and global convergence to the origin. Moreover, asymptotic  ˜ , Id , z 11 ] and globally in the remaining stability of the origin, semiglobally in [ψ˜ dq  variables is also proven by choosing α3 in terms of [ψ˜ dq (0), I˜d (0), z 11 (0)]. See Appendix D.1.3. This completes the proof of proposition 5.10. Conditions for this result are summarized by (D.12), β > 0, ki + bβ + βk p > J β 2 , αi is a positive definite 2 × 2 matrix, some large enough constants k p > 0, ki > 0, and a positive definite 2 × 2 matrix α p such that the four leading principal minors of matrix Q, defined in (D.8), are positive. Remark 5.12 We stress that boundedness of the state and global convergence to the origin does not ensure global asymptotic stability of the origin. This is the reason  ˜ why asymptotic stability of the origin, semiglobally in [ψ˜ dq , Id , z 11 ] and globally in the remaining variables must be stressed. Remark 5.13 As stated in Remark 5.8, some nonlinear terms arising from the bilinear nature of the motor model are the main obstacle to prove global asymptotic stability of standard indirect field-oriented control. In this respect, the term −αd I˜dq , included in (5.95), and the cross term cancellations referred in Remark 5.6 are instrumental to prove Proposition 5.10.

∗ Remark 5.14 In [97], the authors were forced to add the term R I Idq to the PI electric current controllers, which represents a clear difference with respect to standard field oriented control of voltage-fed induction motors. This term has been avoided in Proposition 5.10 by following a procedure which is similar to that first proposed in [226].

5.3 Velocity Control

245

∗ Aside from the term R I Idq and the use of a saturated PI velocity controller, the proposal in [97] is exactly SIFOC. The main drawback of this work is that it relies on the viscous friction that the mechanical subsystem naturally possesses. This work was extended in [101] to control rigid robot manipulators equipped with induction motors. Although [82, 83] are also proposed to solve this control problem, the main difference is that these works only consider current-fed induction motors whereas [101] considers voltage-fed induction motors.

Remark 5.15 In [219] is presented an improved indirect field-oriented control scheme which proves in experiments to achieve a better performance than standard indirect field-oriented control. In particular, this control scheme is robust with respect to uncertainties in the rotor resistance, a parameter that is required to be exactly known to compute ε0 (see (5.50)). However, the electric current controller in [219] is not a simple PI controller; it only has an integral term on the q electric current and it has several additional terms which depend on several motor parameters. Another modified indirect field-oriented controller (IFOC) is introduced in [16] which is ensured to achieve global exponential rotor velocity/ rotor flux tracking. The modifications to the IFOC scheme, involve the injection of nonlinear terms into the current control input and the so-called desired rotor flux angle dynamics. According to proof of Proposition 5.10, a large α p may be required to render positive the fourth principal minor of matrix Q because Q 14 , Q 41 , Q 24 , Q 42 , Q 34 , Q 43 are normally large. One manner to relax this condition on the controller gain α p is, on one hand, using adaptive control to cancel some cross terms giving rise to Q 14 , Q 41 , Q 24 , Q 42 avoiding the necessity for the exact knowledge of some uncertain parameters. On the other hand, it can be taken advantage from the fact that ψ˜ dq has been proven to converge to zero after (D.12). These are the main ideas behind the following result. Proposition 5.16 Consider controller in Proposition 5.10 when the following modification is included: (5.99) Udq = Vdq + L I n p ω0 J Idq + Udq2 ,   0 , (5.100) Udq2 = t γ2 0 ω(s)ds ˜  γ1 ω˜ +   t ˜  γ˙ 2 = −2 I˜q ω(s)ds ˜  γ˙ 1 = −1 I˜q ω, 0    t

2  t

2 2 ˜2 2 2 ˜ ω(s)ds ˜ , γ1 , γ2 Iq (s)ds , β0 = max ω , Iq , 0

where  γ1 and  γ2 are the estimates of

0

246

5 Induction Motor

L I L r k p (k p + b) RI Lr k p L I L r ki − , ∗ ∗ − n p M e J ψd n p M e ψd n p Me ψd∗ βn p Me ψd∗ L I L r k p ki R I L r ki − − , γ2 = n p Me J ψd∗ n p Me ψd∗ Lr γ1 =

and 1 , 2 are two arbitrary positive constants. There always exist positive scalars k p , ki , α3 , and 2 × 2 diagonal positive definite matrices α p , αi , such that the whole   state of the closed-loop dynamics is bounded and limt→∞ (ω(t), ˜ I˜dq (t), ψ˜ dq (t)) = (0, 0, 0, 0, 0) from any initial condition. Proof of Proposition 5.16 is similar to proof of Proposition 5.10 but it is more involved. The interested reader is referred to [185] for the complete proof. Finally, the following is a corollary of Proposition 5.10. Corollary 5.17 Consider the standard dq induction motor model (5.49)–(5.50) in closed loop with the following control law: Udq = Vdq + L I n p ω0 J Idq , ∗ ∗ ∗ ∗ − α p I˜dq + α2 ω ∗ J ψdq − α1 ψdq − αd I˜dq , I˜dq = Idq − Idq , Vdq = R I Idq  ∗ ∗ T ψd Lr τ ∗ ∗ Idq = , , ψdq = [ψd∗ , 0]T , αd = diag{α3 β0 , 0}, Me n p Me ψd∗  t ω(s)ds, ˜ ω˜ = ω − ω ∗ , τ ∗ = −k p ω˜ − ki 0   t

2  t

2  2 ˜2 ˜ ω(s)ds ˜ , β0 = max ω , Iq , Iq (s)ds , 0

0

∗ where ω ∗ is the constant desired motor velocity and ψdq is the desired flux. There always exist positive scalars k p , ki , α3 , and 2 × 2 diagonal positive definite matrices α p , αi , such that the closed-loop dynamics has an unique equilibrium point which is globally asymptotically stable. At this equilibrium point we have that ω˜ = 0, I˜dq = 0, ∗ . ψdq = ψdq

Proof of Corollary 5.17 is presented in Remark D.2, Appendix D. Remark 5.18 Recall from Sect. 5.1.4 that RI =

L r2 Rs + Rr Me2 Rr Me , α1 = . L r2 L r2

∗ ∗ This means that term −α1 ψdq cancels with a part of term R I Idq , i.e., we can employ ∗ ∗ ∗ ∗ a simpler expression for R I Idq + α2 ω J ψdq − α1 ψdq (a simpler expression for Vdq in Corollary 5.17) which, however, is equivalent. Despite this observation let us assume that Vdq in Corollary 5.17 is employed. Notice that the only difference 5.17 is the fact  t between Proposition 5.10 and Corollary ∗ ∗ + α2 ω ∗ J ψdq − that the integral term −αi 0 I˜dq (s)ds in (5.95) is replaced by R I Idq

5.3 Velocity Control

247

∗ α1 ψdq . We stress that the main result on velocity control of voltage-fed induction motors that is presented in [204], Proposition 10.6,5 is also established under this condition. As a matter of fact, in [204], Chap. 10, Sect. 3.7, it is shown that result in [204], Proposition 10.6, admits the following dq implementation:



Udq

d ∗ Me 1 ∗ ∗ Idq + (n p ω0 J + γ I2 )Idq = σL I + (n p ωJ − I2 )ψdq dt σL r L I Tr −K 1 (ω) I˜dq ,

K 1 (ω) =

n 2p Me2 41

ω 2 + k1 , 0 < 1 ≤ Rr , k1 ≥ 0,

(5.101)

∗ ∗ , n p ω0 and Idq are defined almost identically as in Proposition 5.10 and where ψdq Corollary 5.17 above, I2 is a 2 × 2 identity matrix, γ is a constant depending on both stator and rotor resistances, σ is the total leakage factor and Tr depends on rotor resistance. ∗ above are very similar to terms −αd I˜dq Terms −K 1 (ω) I˜dq and σL I n p ω0 J Idq and L I n p ω0 J Idq in Corollary 5.17, whereas terms

∗ + σL I γ Idq

Me 1 ∗ (n p ωJ − I2 )ψdq σL r L I Tr

∗ ∗ ∗ in (5.101) are very similar to terms R I Idq + α2 ω ∗ J ψdq − α1 ψdq in Corollary 5.17. Thus, the main difference between result in [204], Proposition 10.6, and result in Corollary 5.17 is that the latter does not require to compute online the time derivative of the desired electric current. Moreover, in Corollary 5.17 is explicitly included a linear proportional-integral (PI) velocity controller, whereas the result in [204], Proposition 10.6, has to resort to design a disturbance estimator and to employ velocity filtering. On the other hand, controller in Proposition 5.10 includes a linear proportionalintegral (PI) electric current controller which is not included in (5.101). As explained above, the integral action of this PI controller is useful to compensate for several terms that are included in (5.101). In this respect, it is important to say that it is demonstrated in [204], Chap. 10, Sect. 3.4, that controller in (5.101) can be handled to include an integral action on the electric current errors. However, this is performed by still including all the terms in (5.101) and, thus, this integral action have not any to compensate, at least in theory.

Remark 5.19 In current-fed induction motors it is assumed that the electric cur∗ can be rent controller achieves I˜dq → 0 very fast, and hence I˜dq = 0 and Idq = Idq assumed to be true all the time. This means that the stator electrical dynamics can be neglected. Thus, Remark D.2 can be followed to prove that, under this condition, SIFOC for current-fed induction motors is globally asymptotically stable. 5 Works

in [63, 210] and [204], Chap. 11, are valid only for induction motor reduced order models, i.e for current-fed induction motors.

248

5 Induction Motor

5.3.1 Simulation Results In this section, we present a simulation example to appreciate the performance that is achievable with controller in Proposition 5.10. We employ the numerical parameters of the induction motor that was experimentally tested in [219], i.e., L ms = 0.91[H], Me = (3/2)L ms , L r = 0.95[H], L ls = 0.95[H], n p = 1, Rs = 11[Ohm], Rr = 5.6[Ohm], and we assume b = 0, J = 0.05[kg m2 ], instead of J = 0.005[kg m2 ] reported in [219]. The controller gains were also chosen as in [219], i.e., ki = 11250, αi = diag{122000, 122000}, α p = diag{700, 700}, excepting k p = 100 which was employed instead of k p = 150 proposed in [219], and α3 = 1 which is a new parameter in Proposition 5.10. These parameters were used to test the stability conditions established in Proposition 5.10 using β = 1, γ = 1 and ψ˜ dq  = 0 as the (negligible) value for the flux error to be used in the stability conditions. We found that all of the conditions are satisfied excepting det(Q) > 0, where matrix Q is defined in (D.8). This condition, however, is satisfied if we choose α p = diag{91 × 107 , 91 × 107 } or larger. Since such a solution is not practical, we preferred to employ the controller gain α p cited above which was used in practice in [219]. Moreover, in [185] has been demonstrated that using the above-cited controller parameters, i.e., avoiding the use of k p = 150 and α p = diag{91 × 107 , 91 × 107 }, all of the conditions are satisfied for controller in Proposition 5.16. Since controller in Proposition 5.16 has been tested through simulations in [185], we have decided to perform simulations in this section with controller in Proposition 5.10. In the simulation results shown in Figs. 5.11 and 5.12 we have used ψd∗ = 0.9[Wb]. For this, we have applied the corresponding constant desired electric current Id∗ at t = 0 (see Fig. 5.12) and we wait until ψd has reached ψd∗ = 0.9[Wb] (see Fig. 5.11), to apply the desired velocity ω ∗ . This velocity reference is commanded as a ramp with slope 50[(rad/s)/s] which is applied at t = 1[s] and remains constant for t ≥ 2[s] when ω ∗ = 50[rad/s]. A step load torque is also applied for t ≥ 2.5[s] with τ L = 2.5[N m]. We realize from Fig. 5.11 that the velocity error remains less that 0.2% of the desired velocity 50[rad/s]. These velocity errors appear when the ramp is applied, when the desired velocity begins to be constant and when the step load torque is applied. Aside from these isolated times the velocity error is zero. The Id electric current remains close to 0.7[A], whereas the Iq electric current remains close to 2[A] with isolated peaks remaining less than 4[A]. We consider that these electric current values are acceptable since it is reported in [219] that 2.4[A] is the motor rated electric current. In Fig. 5.12, we also show voltages Ud and Uq . Since these voltages present large isolated peak values we have saturated these signals at ±315[V]. The intention for this is twofold. (1) Large peak voltage values are not possible in practice because the power supplies have a limited range of values. (2) It is important to verify that performance does not deteriorate if such large voltage spikes are not applied. By performing several simulations which are not presented here, we have observed that,

5.3 Velocity Control

Fig. 5.11 Simulation results when controller in Proposition 5.10 is employed

249

250

5 Induction Motor

Fig. 5.12 Simulation results when controller in Proposition 5.10 is employed (cont.)

5.3 Velocity Control

251

in fact, performance remains the same whether the referred saturation is considered or not. We believe that this is because such voltage spikes are applied only during very short time intervals. In Fig. 5.11 we present the three-phase voltages V1 , V2 , V3 , that are applied at the motor stator terminals. These signals have been computed using (5.10) and: ε˙0 = n p ω0 = n p ω +

Rr Me Iq L r ψd∗

Udq = e−J ε0 Vab , Vab = [Va , Vb ] .

5.4 Position Control A number of strategies have been reported until now for position control of mechanical systems actuated by induction motors [39, 47, 82, 83, 112, 204, 215, 216]. The result by [39] only ensures local stability. Controllers by [82, 83] are only valid for current-fed (reduced-order model) induction motors. In [112] a complex control law is proposed which includes several high-order nonlinear terms. As remarked by [204] pp. 257, 395, 403, these features degrade performance in practice because they amplify noise and produce input saturation as well as numerical errors. Moreover, it is stressed by [222] that solutions based on abstract nonlinear ideas are difficult to understand for practitioners and this fact explains why several of the aforementioned strategies have not been adopted in practice. Motivated by this, several works have been proposed which exploit indirect fieldoriented control. In a series of works [204, 206, 215, 216] some indirect fieldoriented-based controllers were presented for voltage-fed induction motors when used to actuate either constant unknown loads or robot manipulators. However, these control schemes include some nonlinear terms which render them different from indirect field-oriented control. Finally, the approach in [97] was extended in [101] to regulate position, using a saturated PD controller, in n−degrees of freedom rigid robots actuated by induction motors. Global asymptotic stability was proven. It is interesting to remark that, until now, it has not been presented with any proof showing that indirect field-oriented control of voltage-fed induction motors achieves global asymptotic stability when used to regulate position in mechanical systems. In particular, the use of a linear PID position controller has not been proposed for the mechanical subsystem. In this section, the following induction motor model is considered: J q¨ + bq˙ =

n p Me  I J ψdq − g(q), L r dq

L I I˙dq = −R I Idq + α1 ψdq − α2 qJ ˙ ψdq + Udq − ω0 n p L I J Idq , ˙ L r ψdq = −Rr ψdq + Rr Me Idq + n p q˙ L r J ψdq − ω0 n p L r J ψdq , ε˙ 0 = n p ω0 = n p q˙ +

Rr Me Iq , L r ψd∗

(5.102)

252

5 Induction Motor Idq = [Id , Iq ] , ψdq = [ψd , ψq ] , Udq = e−J ε0 Vab , Vab = [Va , Vb ] ,

which is the same model in (5.57) with g(q) instead of G(q). The function g(q) represents a position-dependent mechanical load which is assumed to possess the following properties: |g(x) − g(y)| ≤ kg |x − y|, ∀x, y ∈ R, (Lipschitz condition) (5.103)    dg(q)  , (5.104) kg > max  q∈R dq  dU (q) g(q) = , |g(q)| ≤ k  , ∀q ∈ R, (5.105) dq with U (q) the potential energy and kg , k  , some positive constants. The controller in Proposition 5.10 can be extended to the case of position control as shown in the following proposition Proposition 5.20 Consider the standard dq induction motor model (5.102) in closed loop with the following control law: Udq = Vdq + L I n p ω0 J Idq ,  t ∗ Vdq = −α p I˜dq − αi , I˜dq (s)ds − αd I˜dq , I˜dq = Idq − Idq

(5.106) (5.107)

0

 T  ∗ ψd∗ Lr ψd ∗ ∗ , αd = diag{α3 β0 , 0}, = , τ , ψdq = (5.108) ∗ 0 M e n p M e ψd  t q(s)ds, ˜ q˜ = q − q ∗ , (5.109) τ ∗ = −k p q˜ − kd q˙ − ki 0    t

2  t

2 2 ˜2 ˜ q(s)ds ˙ , β0 = max q˙ , Iq , Iq (s)ds ,

∗ Idq



0

0

  where the constant n p ML re ψ∗ > 0 is an estimate of the constant n p ML re ψ∗ > 0, q ∗ d d ∗ is the constant desired motor position and ψdq is the desired flux. There always exist positive scalars k p , kd , ki , α3 , and 2 × 2 diagonal positive definite matrices α p , αi , such that the closed-loop dynamics has an unique equilibrium point. At this ∗ . Furthermore, this equilibrium point we have that q˜ = 0, q˙ = 0, I˜dq = 0, ψdq = ψdq equilibrium point is asymptotically stable, semiglobally in the rotor flux error, the d current error and the integral of the d current error, and globally in the remaining variables. The complete proof of Proposition 5.20 is presented in Appendix D.2 for the reader interested in more technical details. Here, let us just stress that excepting the fact that Proposition 5.20 is intended for position control, instead of velocity control, its proof follows the same steps as in the proof for Proposition 5.10. This means

5.4 Position Control

253

that by analyzing the complete closed-loop dynamics, the following is first proven, ∗ : where ξ is the closed-loop system state and ψ˜ dq = ψdq − ψdq B1 ξ ∈ L8∞ and limt→∞ ξ(t) = 0, from any initial condition ξ(t0 ), if ψ˜ dq (t) remains inside a ball whose radius is small enough.  ˜ , Id ) subsystem dynamics is analyzed. to prove the following: Next, the (ψ˜ dq

B2 [ψ˜q (t), ψ˜ d (t), I˜d (t)] is bounded, ψ˜q , ψ˜d , I˜d ∈ L2 and limt→∞ [ψ˜q (t), ψ˜ d (t), I˜d (t)] = [0, 0, 0] from any initial condition [ψ˜q (t0 ), ψ˜d (t0 ), I˜d (t0 ), z 11 (t0 )], if ξ ∈ L8∞ . Moreover ψ˜dq converges in finite time into a ball whose radius decreases to zero if ξ → ∞, from any initial condition [ψ˜q (t0 ), ψ˜d (t0 ), I˜d (t0 ), z 11 (t0 )]. Finally, results in B1 and B2 are combined to prove the proposition. With this aim, it is first proven that, in the case where [ψ˜ d , ψ˜q ] is not small enough, then ξ(t) ≤  κ2 κ1

κ3

ξ(t0 ) e 2κ1

(t−t0 )

< ∞ for any finite time. This means that ξ(t) has not a finite

escape time. Thus, when starting from any bounded initial condition ξ(t0 ), the state ξ(t) remains bounded for any finite time t > t0 . Assume that the worst case is present, i.e., that ξ(t) tends to grow without limit. In such a case B2 ensures that [ψ˜q (t), ψ˜ d (t)] < ε, for any ε > 0, in finite time, i.e., before ξ(t) escapes. Thus, by virtue of B1, ξ(t) will converge asymptotically to zero. In the case when ξ(t) does not tend to grow without limit, then ξ(t) will remain bounded until, by virtue of B2, [ψ˜q (t), ψ˜ d (t)] reaches and stays within a small enough value and, hence, ξ(t) will converge asymptotically to zero. This proves boundedness of the state and global convergence to the origin. Moreover, the origin  ˜ , Id , z 11 ] and globally in is proven to be asymptotically stable, semiglobally in [ψ˜ dq  the remaining variables, by choosing α3 in terms of [ψ˜ dq (0), I˜d (0), z 11 (0)]. See Appendix D.2. This completes the proof of Proposition 5.20. Conditions for this result are summarized by (D.12), (A.3), (A.4), (A.5), (A.6), αi is a positive definite matrix, and the five principal minors of matrix P defined in (D.40) are positive. Remark 5.21 Notice that excepting the nonlinear term −αd I˜dq and the respective computation of  β0 = max q˙ , I˜q2 ,



2

0

t

2  q(s)ds ˙ ,

t

I˜q (s)ds

2  ,

0

the controller in Proposition 5.20 is exactly standard indirect field-oriented control for position control in induction motors as depicted in Fig. 5.13. Moreover, it is shown in Proposition 5.20 that the constant ϒ = n p ML re ψ∗ is not required to be exactly d known. Thus, Proposition 5.20 is the closest result in the literature to standard indirect field-oriented control for position control of inductor motors provided with a global

254

Fig. 5.13 Standard indirect field-oriented control of induction motors. ϒ =

5 Induction Motor

Lr n p Me ψd∗ , 

= ψd∗ /Me

convergence proof. Instrumental for this result is the cross term cancellations referred in Remark 5.6. Remark 5.22 We stress that controller in proposition Proposition 5.20 does not require to feedback the time derivative of the desired electric current. This fact is instrumental to present such a simple control law. Let us underline that previous passivity-based controllers in the literature [204] are forced to feedback the time derivative of the desired electric current which results in an important number of additional computations. Moreover, this fact also constraints them to employ position filtering to avoid velocity measurements since the later would result in either an even more complex expression for the time derivative of desired current or acceleration measurements. This is important to say because this also complicates the task to propose simple PID position controllers. The main reason for this is that the design of PID position controllers achieving good performances requires velocity measurements. As a matter of fact, several works have been proposed on PID control of robot manipulators without velocity measurements [98, 172, 203, 252, 253] but they impose important constraints on the controller gains that can be employed and, hence, performance deteriorates. Furthermore, some of those works rely on the friction that is naturally present in the mechanical subsystem. The above arguments explain why simple PID position controllers have not been proposed to cope with the mechanical subsystem when the electrical dynamics of the actuators is taken into account in the stability proof. This also clarifies the merit of controller in Proposition 5.20 where a PID position controller is employed, a fundamental component of standard indirect field oriented control when employed for position control tasks.

5.4 Position Control

255

Finally, we stress that Remarks 5.12, 5.13, 5.14, 5.18, stated for the velocity controller in Proposition 5.10 also apply for the position controller in Proposition 5.20.

5.4.1 Simulation Results In this section, we present a simulation example to give some insight on performance that is achievable with controller in Proposition 5.20. We employ the numerical parameters used in Sect. 5.3.1. The only difference is that we assume now that a simple pendulum is fixed at the motor shaft such that, see (5.102), g(q) = mgl sin(q) where m = 0.2[kg], g = 9.81[m/s2 ], l = 0.14[m]. Moreover, the mechanism inertia is given as J = 0.05[kg m2 ]+ml 2 + 13 m(2l)2 , where 0.05[kg m2 ] is inertia of the motor rotor. The controller gains were chosen as k p = 3, ki = 3, kd = 0.7, αi = diag{10000, 10000}, α p = diag{700, 700}, and α3 = 1. It was verified that these parameters satisfy all of the conditions established in the proof of Proposition 5.20 when using β = 1, α = 2 and ψ˜dq  = 0 as the (negligible) value for the flux error to be used in conditions for the proof. The desired position was fixed to q ∗ = π2 [rad] and all of the initial conditions were set to zero. In the simulation results shown in Figs. 5.14 and 5.15 we have used ψd∗ = 0.9[Wb]. For this, we have applied the corresponding constant desired electric current Id∗ at t = 0 and we wait until ψd has reached ψd∗ = 0.9[Wb] to apply the desired position q ∗ as a step command. See Fig. 5.14. The position response has about 17% overshoot and 0.43[s] rise time, which represent a realistic transient response in electromechanical systems. The Id electric current remains close to 0.7[A], whereas the Iq electric current has a peak of about 3[A] remaining less than 0.5[A] in steady state. We consider that these electric current values are acceptable since it is reported in [219] that 2.4[A] is the motor rated electric current. In Fig. 5.15, we also show voltages Ud and Uq . Since these voltages present large isolated peak values we have saturated these signals at ±315[V]. The intention for this is twofold. (1) Large peak voltage values are not possible in practice because the power supplies have a limited range of values. (2) It is important to verify that performance does not deteriorate if such large voltage spikes are not applied. By performing several simulations which are not presented here, we have observed that, in fact, performance remains the same whether the referred saturation is considered or not. We believe that this is because such voltage spikes are applied only during very short time intervals. In Fig. 5.14, we present the three-phase voltages V1 , V2 , V3 , that are applied at the motor stator terminals. These signals have been computed using (5.10) and

256

5 Induction Motor

Fig. 5.14 Simulation results when controller in Proposition 5.20 is employed

ε˙0 = n p ω0 = n p ω +

Rr Me Iq L r ψd∗

Udq = e−J ε0 Vab , Vab = [Va , Vb ]T . Notice that V1 , V2 , V3 , do not become constant despite the motor does not move, i.e., in steady state the motor remains at a constant position. This is because at q = q ∗ = π2 motor is generating a constant torque to compensate the gravity effect on the pendulum fixed to the motor shaft. In an induction motor, this requires that the three-phase voltages applied at the motor stator change as sinusoidal functions of time in order to generate (i.e., to induce) the required electric currents in the rotor

5.4 Position Control

Fig. 5.15 Simulation results when controller in Proposition 5.20 is employed

257

258

5 Induction Motor

Fig. 5.16 One star connection of the stator phase windings

windings. This is to be contrasted with the constant three-phase voltages shown in Fig. 4.20 when a PM synchronous motor is generating a constant torque at a constant rotor position.

5.5 A Practical Induction Motor This is a three-phase motor equipped with 12 stator windings. This motor can be fed using two different voltage values: 230 VAC and 460 VAC. When it is fed with 460 VAC, the 12 stator windings are connected to form one star. In Fig. 5.16 is presented a sketch showing this situation. The neutral point of this star is labeled as N. Terminal of phases 1, 2, and 3, which are labeled as 1, 2, and 3, respectively. In Fig. 5.17 is presented a drawing showing the relative positions of the stator [rad] exists phase windings. The stator contains 36 slots. Hence, an angle of 10◦ = 2π 36 between adjacent slots. Phase 1 is composed by four series connected windings. However, two of these windings are wound in the opposite direction to form two N poles and two S poles. The end terminal of the last series connected windings connects to the neutral point labeled as N. This can be verified in Fig. 5.17 where the arrows represent the sense of a positive electric current I1 through these phase

5.5 A Practical Induction Motor

259

Fig. 5.17 The relative positions of the stator phase windings. Note Special thanks to Fernando Moreno, the first author’s father-in-law, for finding out how the stator phases are wound

windings. A similar description stands for phases 2 and 3. The phase electric currents are designated as I1 , I2 , I3 . From the above description it is not difficult to realize that this is a two pole pairs (n p = 2) motor, i.e., two N poles and two S poles lay alternatively along the stator for each phase. In the following, we will obtain the mathematical model of this motor using a procedure described in [55].

5.5.1 Magnetic Field at the Air Gap 5.5.1.1

Magnetic Field Produced by the Stator Windings

In order to compute the magnetic field produced at the air gap by the stator phase winding 1, consider Fig. 5.18. Recall that the stator has 36 slots with an angle of [rad] between adjacent slots. The symbol  means that electric current 10◦ = 2π 36 through phase 1 is coming out of the page at γ = 40◦ and γ = 210◦ , for instance, whereas the symbol ⊗ means that electric current is going into the page at γ = 100◦

260

5 Induction Motor

Fig. 5.18 Computing the magnetic field produced at the air gap by the stator phase winding 1

and γ = 150◦ , for instance, to be consistent with the sense of a positive electric current flowing through phase 1, defined in Fig. 5.17. Because of the large number of  and ⊗ symbols, let us study phase 1 in two parts: in the first part, we only consider what is called phase 1b in Fig. 5.18 and in the second part we only consider what is called phase 1a in Fig. 5.18. Hence, let us apply Ampère’s Law (2.33) to the oriented closed trajectory 1-2-34-1 shown in Fig. 5.18:  H1b · dl = i 1benclosed ,  2  = H1b · dl + 1

3

 H1b · dl +

2

4



3

= g H1b (0) − g H1b (γ),

1

H1b · dl +

H1b · dl,

4

(5.110)

where  2 1

 3 2

 4 3

H1b · dl =

 2 1

H1b (0)ˆr · (dl rˆ ) =

 2 1

H1b (0)dl =

 2 1

H1b (0)dr = g H1b (0),

H1b · dl = 0, H1b · dl =

 4 3

H1b (γ)ˆr · (−dl rˆ ) = −

 4 3

H1b (γ)dl =

 4 3

H1b (γ)dr = −g H1b (γ),

5.5 A Practical Induction Motor  1 4

261

H1b · dl = 0.

In the above computations, the following assumptions have been taken into account [55]: • The phase loops are located at the surface of the stator. • Wires forming the loops have no width. • The segments of the closed trajectory 1–2–3–4–1 laying in the stator and the rotor are just below the surface of the stator and rotor. • The space between stator and rotor, known as the air gap, is constant and its width is represented by g. • The magnetic field H1b is radially oriented at the air gap. • H1b = 0 inside a ferromagnetic material with high relative magnetic permeability, i.e., the stator and rotor. • dl = dr in the segment 1–2, whereas dl = −dr in the segment 3–4. On the other hand, i 1benclosed depends on γ and is given as (see Fig. 5.18):

i 1benclosed

⎧ 0, 0 ≤ γ < 40◦ ⎪ ⎪ ⎪ ⎪ N I1b , 40 ≤ γ < 50◦ ⎪ ⎪ ⎪ ⎪ 2N I1b , 50 ≤ γ < 60◦ ⎪ ⎪ ⎪ ⎪ 3N I1b , 60 ≤ γ < 100◦ ⎪ ⎪ ⎪ ⎪ 2N I1b , 100 ≤ γ < 110◦ ⎪ ⎪ ⎪ ⎪ 110 ≤ γ < 120◦ ⎨ N I1b , 120 ≤ γ < 130◦ , = 0, ⎪ ⎪ −N I1b , 130 ≤ γ < 140◦ ⎪ ⎪ ⎪ ⎪ −2N I1b , 140 ≤ γ < 150◦ ⎪ ⎪ ⎪ ⎪ −3N I1b , 150 ≤ γ < 190◦ ⎪ ⎪ ⎪ ⎪ −2N I1b , 190 ≤ γ < 200◦ ⎪ ⎪ ⎪ ⎪ −N I1b , 200 ≤ γ < 210◦ ⎪ ⎪ ⎩ 0, 210 ≤ γ < 360◦

(5.111)

where N stands for the number of turns composing each winding and I1b = I1 is electric current flowing through phase 1b which is assumed to be positive. Recall that only phase 1b is considered. This means that the phase windings placed between γ = 220◦ and γ = 30◦ are assumed not to be present. At the end of the procedure we will add the magnetic fields produced by phases 1b and 1a to obtain the total magnetic field produced by phase 1. It is important to stress that in order to be consistent with Ampère’s Law the right-hand rule has been taken into account. See Fig. 2.8. Thus, from (5.110) we have 1 H1b (γ) = H1b (0) − i 1benclosed . g Since H1b (γ) is defined at the air gap, where of course only air is present, then the relation B1b (γ) = μ0 H1b (γ) can be employed to write

262

5 Induction Motor



1 B1b (γ) = μ0 H1b (0) − i 1benclosed rˆ . g

(5.112)

In the following H1b (0) is computed applying Gauss’ Law for the magnetic field, i.e.,  B1b · ds = 0, to a Gaussian surface represented by a cylinder, enclosing the rotor, whose cylindrical surface (with radius r ) lies within the air gap. Hence, 







B1b · ds =

B1b · ds + B1b · ds + B1b · ds, S2 S3

 2π 1 H1b (0) − i 1benclosed dγ = 0, = l1r μ0 g 0 S1

(5.113)

where (5.112) has been employed and  B1b · ds = 0, 

S1



S2

B1b · ds = 0, 

l1

B1b · ds = S3

0



2π

B1b (γ)ˆr · (r dγdzrˆ ).

0

In the above computations, the following considerations have been taken into account [55]: • S1 and S2 are the disk-shaped surfaces at each end of the cylinder. At these surfaces B1b = 0. • S3 is the cylinder-shaped part of the Gaussian surface where ds = r dγdzrˆ , in cylindrical coordinates, with r the radius of S3 and l1 the length of rotor (laying on the z axis), i.e., the length of S3. From (5.113) and (5.111) we find  2π 1 H1b (0) = i 1benclosed dγ, 2πg 0  2π 2π 4 × 2π 2π 2π 1 N I1b + 2N I1b + 3N I1b + 2N I1b + N I1b = 2πg 36 36 36 36 36  2π 2π 4 × 2π 2π 2π − 2N I1b − 3N I1b − 2N I1b − N I1b , −N I1b 36 36 36 36 36 = 0.

5.5 A Practical Induction Motor

263

Fig. 5.19 Computing the magnetic field produced at the air gap by the stator phase winding 2

Finally, replacing this in (5.112), we find the expression for the magnetic field at the air gap produced by electric current through phase 1b: 1 rR B1b (γ) = −μ0 i 1benclosed rˆ , g r

(5.114)

where the factor rrR , with r R the rotor radius, is suggested in [55] to be introduced in order to be consistent with magnetic flux conservation. Notice, however, that rrR ≈ 1 at the air gap and, hence, the results obtained in the above procedure remain without change. Proceeding similarly for phases 1a, 2a ,3a, 2b, and 3b we find (see Figs. 5.18, 5.19 and 5.20): B1a (γ) = μ0

3 1 N I1a − i 1aenclosed g g

rR rˆ , r

(5.115)

264

5 Induction Motor

Fig. 5.20 Computing the magnetic field produced at the air gap by the stator phase winding 3

⎧ 0, 0 ≤ γ < 10◦ ⎪ ⎪ ⎪ ⎪ N I1a , 10 ≤ γ < 20◦ ⎪ ⎪ ⎪ ⎪ 2N I1a , 20 ≤ γ < 30◦ ⎪ ⎪ ⎪ ⎪ 3N I1a , 30 ≤ γ < 220◦ ⎪ ⎪ ⎪ ⎪ 4N I1a , 220 ≤ γ < 230◦ ⎪ ⎪ ⎪ ⎪ ⎨ 5N I1a , 230 ≤ γ < 240◦ i 1aenclosed = 6N I1a , 240 ≤ γ < 280◦ , ⎪ ⎪ 5N I1a , 280 ≤ γ < 290◦ ⎪ ⎪ ⎪ ⎪ 4N I1a , 290 ≤ γ < 300◦ ⎪ ⎪ ⎪ ⎪ 3N I1a , 300 ≤ γ < 310◦ ⎪ ⎪ ⎪ ⎪ 2N I1a , 310 ≤ γ < 320◦ ⎪ ⎪ ⎪ ⎪ N I1a , 320 ≤ γ < 330◦ ⎪ ⎪ ⎩ 0, 330 ≤ γ < 360◦ 1 rR B2b (γ) = −μ0 i 2benclosed rˆ , g r

(5.116)

(5.117)

5.5 A Practical Induction Motor

i 2benclosed =

B2a (γ) =

i 2aenclosed =

B3b (γ) =

i 3benclosed =

265

⎧ 0 ≤ γ < 160◦ ⎪ ⎪ 0, ⎪ ⎪ N I2b , 160 ≤ γ < 170◦ ⎪ ⎪ ⎪ ⎪ 2N I2b , 170 ≤ γ < 180◦ ⎪ ⎪ ⎪ ⎪ 3N I2b , 180 ≤ γ < 220◦ ⎪ ⎪ ⎪ ⎪ 2N I2b , 220 ≤ γ < 230◦ ⎪ ⎪ ⎪ ⎪ 230 ≤ γ < 240◦ ⎨ N I2b , 0, 240 ≤ γ < 250◦ , ⎪ ⎪ −N I2b , 250 ≤ γ < 260◦ ⎪ ⎪ ⎪ ⎪ −2N I2b , 260 ≤ γ < 270◦ ⎪ ⎪ ⎪ ⎪ −3N I2b , 270 ≤ γ < 310◦ ⎪ ⎪ ⎪ ⎪ −2N I2b , 310 ≤ γ < 320◦ ⎪ ⎪ ⎪ ⎪ −N I2b , 320 ≤ γ < 330◦ ⎪ ⎪ ⎩ 0, 330 ≤ γ < 360◦

rR 1 2 rˆ , μ0 − N I2a − i 2aenclosed g g r ⎧ N I2a , 0 ≤ γ < 40◦ ⎪ ⎪ ⎪ ⎪ 0, 40 ≤ γ < 50◦ ⎪ ⎪ ⎪ ⎪ −N I2a , 50 ≤ γ < 60◦ ⎪ ⎪ ⎪ ⎪ ⎪ −2N I2a , 60 ≤ γ < 70◦ ⎪ ⎪ ⎪ −3N I2a , 70 ≤ γ < 80◦ ⎪ ⎪ ⎨ −4N I2a , 80 ≤ γ < 90◦ , ⎪ −5N I2a , 90 ≤ γ < 130◦ ⎪ ⎪ ⎪ ⎪ −4N I2a , 130 ≤ γ < 140◦ ⎪ ⎪ ⎪ ⎪ −3N I2a , 140 ≤ γ < 150◦ ⎪ ⎪ ⎪ ⎪ −2N I2a , 150 ≤ γ < 340◦ ⎪ ⎪ ⎪ ⎪ −N I2a , 340 ≤ γ < 350◦ ⎪ ⎩ 0, 350 ≤ γ < 360◦ 1 rR −μ0 i 3benclosed rˆ , g r ⎧ 0, 0 ≤ γ < 100◦ ⎪ ⎪ ⎪ ⎪ N I3b , 100 ≤ γ < 110◦ ⎪ ⎪ ⎪ ⎪ 2N I3b , 110 ≤ γ < 120◦ ⎪ ⎪ ⎪ ⎪ 3N I3b , 120 ≤ γ < 160◦ ⎪ ⎪ ⎪ ⎪ 2N I3b , 160 ≤ γ < 170◦ ⎪ ⎪ ⎪ ⎪ 170 ≤ γ < 180◦ ⎨ N I3b , 0, 180 ≤ γ < 190◦ , ⎪ ⎪ −N I3b , 190 ≤ γ < 200◦ ⎪ ⎪ ⎪ ⎪ −2N I3b , 200 ≤ γ < 210◦ ⎪ ⎪ ⎪ ⎪ −3N I3b , 210 ≤ γ < 250◦ ⎪ ⎪ ⎪ ⎪ −2N I3b , 250 ≤ γ < 260◦ ⎪ ⎪ ⎪ ⎪ −N I3b , 260 ≤ γ < 270◦ ⎪ ⎪ ⎩ 0, 270 ≤ γ < 360◦

(5.118)

(5.119)

(5.120)

(5.121)

(5.122)

266

5 Induction Motor



1 rR 1 rˆ , B3a (γ) = μ0 − N I3a − i 3aenclosed g g r ⎧ −N I3a , 0 ≤ γ < 10◦ ⎪ ⎪ ⎪ ⎪ −2N I3a , 10 ≤ γ < 20◦ ⎪ ⎪ ⎪ ⎪ −3N I3a , 20 ≤ γ < 30◦ ⎪ ⎪ ⎪ ⎪ −4N I3a , 30 ≤ γ < 70◦ ⎪ ⎪ ⎪ ⎪ −3N I3a , 70 ≤ γ < 80◦ ⎪ ⎪ ⎨ −2N I3a , 80 ≤ γ < 90◦ i 3aenclosed = . −N I3a , 90 ≤ γ < 280◦ ⎪ ⎪ ⎪ ⎪ 0, 280 ≤ γ < 290◦ ⎪ ⎪ ⎪ ⎪ N I3a , 290 ≤ γ < 300◦ ⎪ ⎪ ⎪ ⎪ 2N I3a , 300 ≤ γ < 340◦ ⎪ ⎪ ⎪ ⎪ N I3a , 340 ≤ γ < 350◦ ⎪ ⎪ ⎩ 0, 350 ≤ γ < 360◦

(5.123)

(5.124)

Notice that (5.114), (5.115), (5.117), (5.119), (5.121), (5.123), can be rewritten as B1a (γ) = b1a (γ)I1a rˆ ,

(5.125)

B2a (γ) = b2a (γ)I2a rˆ , B3a (γ) = b3a (γ)I3a rˆ ,

(5.126) (5.127)

B1b (γ) = b1b (γ)I1b rˆ , B2b (γ) = b2b (γ)I2b rˆ ,

(5.128) (5.129)

B3b (γ) = b3b (γ)I3b rˆ ,

(5.130)

where the scalar functions b1a (γ), b2a (γ) and b3a (γ), b1b (γ), b2b (γ), and b3b (γ), are obviously defined from (5.114), (5.115), (5.117), (5.119), (5.121), (5.123). Now, we obtain the total magnetic flux produced by phases 1, 2, and 3. This is accomplished by noticing that I1 = I1a = I1b , I2 = I2a = I2b and I3 = I3a = I3b . Hence, B1 (γ) = B1a (γ) + B1b (γ) = b1 (γ)I1rˆ , B2 (γ) = B2a (γ) + B2b (γ) = b2 (γ)I2 rˆ ,

(5.131)

B3 (γ) = B3a (γ) + B3b (γ) = b3 (γ)I3rˆ , where b1 (γ) = b1a (γ) + b1b (γ), b2 (γ) = b2a (γ) + b2b (γ), b3 (γ) = b3a (γ) + b3b (γ). These latter functions are plotted in Fig. 5.21 where they are represented by the staircase lines. We realize that these functions are very similar to sinusoidal functions

5.5 A Practical Induction Motor

267

Fig. 5.21 Plots of b1 (γ), b2 (γ), and b3 (γ)

of γ. As a matter of fact, in Fig. 5.21, the smooth lines represent the functions: 3N g 3N b2 (γ) = μ0 g 3N b3 (γ) = μ0 g b1 (γ) = μ0

rR cos (2(γ + 10)) , r rR cos (2(γ + 10 − 60)) , r rR cos (2(γ + 10 − 120)) , r

(5.132)

when used to approximate the magnetic field produced by the stator phase windings. We stress that the arguments of the cosine functions are in degrees. According to these plots, if Ii > 0, i = 1, 2, 3, the magnetic fields in (5.131) are positive, i.e., radially directed toward the stator, and negative, i.e., radially directed toward the rotor axis, at different angular positions on the stator which coincide with the stator winding locations which are wound in opposite directions. See Fig. 5.17. We refer to Fig. 5.22 for a drawing showing how B1 (γ), defined in (5.131), (5.132), distributes along the air gap. The reader can imagine that the magnetic field of the other phases also distribute similarly around their locations on stator. Thus, the total magnetic field contributed at the air gap by the three stator phases is given as

268

5 Induction Motor

Fig. 5.22 Distribution of B1 (γ), defined in (5.131), (5.132), along the air gap

B S (γ) = B1 (γ) + B2 (γ) + B3 (γ).

(5.133)

On the other hand, it is well known that in the case when the phase   electric , currents are sinusoidal functions of time such as I1 = cos(ω S t), I2 = cos ω S t − 2π 3   I3 = cos ω S t + 2π , the total magnetic field contributed by the three phases, i.e., 3 B S (γ) defined in (5.133), rotates along the air gap as a function of time. This is , from 2π to 4π , verified in Figs. 5.23, 5.24, and 5.25 where ω S t changes from 0 to 2π 3 3 3 4π and from 3 to 2π, respectively. Notice that in these figures, we have employed the staircase approximations for b1 (γ), b2 (γ), b3 (γ). π Finally, defining the variable change β = 2(γ + 10) 180 ◦ we can write 3N r R cos (β) , g r 3N r R cos β − μ0 g r 3N r R cos β − μ0 g r b1 (β)I1 rˆ , b2 (β)I2 rˆ ,

b1 (β) = μ0 b2 (β) = b3 (β) = B1 (β) = B2 (β) =

B3 (β) = b3 (β)I3 rˆ .

(5.134)

2π , 3

4π , 3

5.5 A Practical Induction Motor

269

Total magnetic field  B S (γ), defined in (5.133) when I1 = cos(ω S t), I2 =   2π , I3 = cos ω S t + 2π 3 , and passing from ω S t = 0 to ω S t = 3

Fig. 5.23 cos ω S t −

2π 3

Fig. 5.24 cos ω S t −

2π 3

Total magnetic field  B S (γ), defined in (5.133) when I1 = cos(ω S t), I2 =   2π 4π , I3 = cos ω S t + 2π 3 , and passing from ω S t = 3 to ω S t = 3

270

Fig. 5.25 cos ω S t −

5 Induction Motor

Total magnetic field  B S (γ), defined in (5.133) when I1 = cos(ω S t), I2 =   4π , I3 = cos ω S t + 2π 3 , and passing from ω S t = 3 to ω S t = 2π

2π 3

Notice that the cosine function arguments are given in radians. These expressions are more convenient for the computations involved in the motor modeling that we present next. Thus, the total magnetic field produced by the three stator phases at the air gap is finally given as B S (β) = B1 (β) + B2 (β) + B3 (β).

5.5.1.2

(5.135)

Magnetic Field Produced by Rotor

Although the particular motor under study has a squirrel cage rotor, instead of a sinusoidally wound rotor, it is usual in the literature to model the rotor exactly as if it was a sinusoidally wound rotor [55]. Thus, the reader is encouraged to consult [55] to verify that, in such a case, the magnetic field produced by rotor is given as 3N R g 3N R B R2 (β, θ) = μ0 g 3N R B R3 (β, θ) = μ0 g B R1 (β, θ) = μ0

rR I R1 cos (β − θ) rˆ , r rR I R2 cos β − θ − r rR I R3 cos β − θ − r

(5.136)

2π rˆ , 3

4π rˆ , 3

5.5 A Practical Induction Motor

271

where N R is the number of turns in each winding of the rotor, I R1 , I R2 , I R3 are electric currents through the rotor phases and θ is the electric rotor position which relates to the mechanical rotor position q through θ = n p q where n p = 2. Recall that the rotor velocity is represented by ω = q. ˙ Hence, the total magnetic field produced at the air gap by the rotor is given as B R (β, θ) = B R1 (β, θ) + B R2 (β, θ) + B R3 (β, θ).

(5.137)

Finally, the total magnetic field produced by both the stator phase windings and the rotor phase windings distributes along the air gap according to B(β, θ) = B S (β) + kB R (β, θ),

(5.138)

where B S (γ) is defined in (5.135), B R (β, θ) is defined in (5.138) and 0 ≤ k ≤ 1 is the coupling factor which is included to account for leakage.

5.5.2 The Magnetic Flux Linkages Let ψ1b represent the magnetic flux linkage of phase 1b at stator (see Fig. 5.18). This flux linkage represents the magnetic flux through the stator windings of phase 1 located between γ = 40◦ and γ = 210◦ in Fig. 5.18 (also see Fig. 5.17). Notice that phase 1b has six different windings between these angular positions. Thus, 



ψ1b = −N



B(β, θ) · ds − N

B(β, θ) · ds − N  S2

 S1 B(β, θ) · ds + N

+N S4

B(β, θ) · ds  S3

B(β, θ) · ds + N S5

B(β, θ) · ds, S6

(5.139) where B(β, θ) is given in (5.138), S1 represents the cylindrical surface subtended between the stator angular positions 40◦ and 120◦ , with radius r equal to the stator radius, and with the same length as rotor. S2 is defined similarly between the stator angular positions 50◦ and 110◦ , S3 between 60◦ and 100◦ , S4 between 130◦ and 210◦ , S5 between 140◦ and 200◦ , and S6 between 150◦ and 190◦ . Furthermore, ds = r dγdzrˆ and N is the number of turns in each stator winding. We stress that the negative sign affecting the first three integrals are used because of the fact that a negative magnetic field through these windings (a magnetic field pointing toward the rotor, see Fig. 5.22) represent, in fact, a positive flux linkage in phase 1b because of the sense these windings are wound, see Fig. 5.17. According to (5.134), (5.135), (5.136), (5.137), (5.138), each integral in (5.139) receives a contribution of electric currents in the stator phases I1 , I2 , I3 , and the electric currents in the rotor phases I R1 , I R2 , I R3 . In the following, we will compute

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5 Induction Motor

each individual contribution of each one of these electric currents to the magnetic flux linkage of phase 1, ψ1s , and we will add them at the end. Moreover, we will adopt again the convention shown in Fig. 5.18 to separate phase 1 into phase 1b and phase 1a, and then adding the flux linkages in phases 1b and 1a to obtain the total flux linkage in phase 1. Let ψ1bs1 only represent the contribution to ψ1b of the electric current in phase 1, i.e., I1 . Then according to (5.134), (5.135), (5.138), the first integral in (5.139), when contributing only to ψ1bs1 , can be written as 



−N

120◦



l1

3N r R cos (β) I1rˆ · (dzr dγ rˆ ), g r 40◦ 0  120◦ 3N r R cos (β) I1 dγ. = −Nl1 μ0 g 40◦

B(β, θ) · ds = −N S1

μ0

Since similar expressions are also valid for the other integrals in (5.139), we can write 

 110◦ 3N 3N r R cos (β) I1 dγ − Nl1 r R cos (β) I1 dγ μ0 g g 40◦ 50◦  100◦  210◦ 3N 3N r R cos (β) I1 dγ + Nl1 r R cos (β) I1 dγ −Nl1 μ0 μ0 ◦ ◦ g g 60 130  200◦  190◦ 3N 3N r R cos (β) I1 dγ + Nl1 r R cos (β) I1 dγ. +Nl1 μ0 μ0 ◦ ◦ g g 140 150

ψ1bs1 = −Nl1

120◦

μ0

π Since β = 2(γ + 10) 180 ◦ , we have

 −Nl1

120◦ 40◦

3N 3 180◦ μ0 r R cos (β) I1 dγ = −N 2 l1 μ0 r R I1 g g 2π

260◦ π 3 180◦ 180◦ = −N 2 l1 μ0 r R I1 [sin (β)] 100 ◦π . g 2π 180◦



260◦ π 180◦ 100◦ π 180◦

cos (β) dβ

Proceeding similarly with the other integrals we can write

" 260◦ π 240◦ π 220◦ π 3 180◦ 180◦ 180◦ 180◦ − [sin (β)] 100 ψ1bs1 = N 2 l1 μ0 r R I1 ◦ π + [sin (β)] 120◦ π + [sin (β)] 140◦ π g 2π 180◦ 180◦ 180◦ # 440◦ π 420◦ π 400◦ π 180◦ 180◦ 180◦ , + [sin (β)] 280 ◦ π + [sin (β)] 300◦ π + [sin (β)] 320◦ π 180◦

180◦

180◦

3 180◦ I1 . = 9.9744N 2 l1 μ0 r R g 2π Finally, the flux linkage in phase 1 produced by the electric current in phase 1 is given as

5.5 A Practical Induction Motor

273

3 180◦ I1 , ψ1s1 = ψ1bs1 + ψ1as1 = 2ψ1bs1 = 19.9488N 2 l1 μ0 r R g 2π

(5.140)

since the flux linkage en phase 1a produced by electric current in phase 1, ψ1as1 , satisfies ψ1as1 = ψ1bs1 , which can be verified by proceeding as before to compute ψ1as1 . Proceeding similarly we can employ b2 (β), defined in (5.134), to compute the magnetic flux linkage in the stator phase 1b, ψ1bs2 , produced by electric current in the stator phase 2, I2 , to obtain 3 180◦ × ψ1bs2 = N 2 l1 μ0 r R I2 g 2π   ◦π ◦π ◦π 

 260

 240

 220   180◦ 180◦ 180◦ 2π 2π 2π + sin β − + sin β − − sin β − 100◦ π 120◦ π 140◦ π 3 3 3 180◦ 180◦ 180◦ ◦ ◦ ◦ 440 π 420 π 400 π

 180◦

 180◦

 180◦     2π 2π 2π + sin β − + sin β − + sin β − , 280◦ π 300◦ π 320◦ π 3 3 3 ◦ ◦ ◦ 180

180

3 180◦ = −4.9872N 2 l1 μ0 r R I2 . g 2π

180

Hence, the flux linkage in phase 1 produced by the electric current in phase 2 is given as 3 180◦ I2 . ψ1s2 = ψ1bs2 + ψ1as2 = 2ψ1bs2 = −9.9744N 2 l1 μ0 r R g 2π

(5.141)

Furthermore, the flux linkage in phase 1 produced by the electric current in phase 3 is given as 3 180◦ I3 . ψ1s3 = −9.9744N 2 l1 μ0 r R g 2π

(5.142)

Now, let ψ1br 1 represent only the contribution of electric current in phase 1 of rotor, i.e., I R1 , to ψ1b , then according to (5.139), (5.138), (5.137), (5.136), we can write 

120◦

3N R r R I R1 cos (β − θ) dγ g  110◦ 3N R r R I R1 cos (β − θ) dγ −k Nl1 μ0 g 50◦  100◦ 3N R r R I R1 cos (β − θ) dγ −k Nl1 μ0 g 60◦  210◦ 3N R r R I R1 cos (β − θ) dγ +k Nl1 μ0 g 130◦

ψ1br 1 = −k Nl1

40◦

μ0

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5 Induction Motor



200◦

3N R r R I R1 cos (β − θ) dγ g  190◦ 3N R r R I R1 cos (β − θ) dγ. +k Nl1 μ0 ◦ g 150

+k Nl1

140◦

μ0

π Since β = 2(γ + 10) 180 ◦ , we have



120◦

3N R r R I R1 cos (β − θ) dγ g 40◦  260◦◦π 180 3N R 180◦ r R I R1 = −k Nl1 μ0 cos (β − θ) dβ ◦π g 2π 100 180◦ −k Nl1

= −k Nl1 μ0

μ0

260◦ π 3N R 180◦ 180◦ r R I R1 [sin (β − θ)] 100 ◦π . g 2π 180◦

Proceeding similarly with the other integrals we can write 3N R 180◦ r R I R1 (5.143) ψ1br 1 = k Nl1 μ0 g 2π

" 260◦ π 240◦ π 220◦ π 180◦ 180◦ 180◦ × − [sin (β − θ)] 100 ◦ π + [sin (β − θ)] 120◦ π + [sin (β − θ)] 140◦ π 180◦ 180◦ 180◦ # 440◦ π 420◦ π 400◦ π 180◦ 180◦ 180◦ . + [sin (β − θ)] 280 ◦ π + [sin (β − θ)] 300◦ π + [sin (β − θ)] 320◦ π 180◦

180◦

180◦

Notice, for instance, that



260◦ π π 260◦ π 260◦ π π sin = − sin θ + − ( − θ = − sin θ − + ) 180◦ 180◦ 2 180◦ 2 ◦ ◦ 260 π π π 260 π π π = −[sin(θ + ) cos( + ) − cos(θ + ) sin( + )] 2 180◦ 2 2 180◦ 2 π π = −[0.9848 sin(θ + ) + 0.1736 cos(θ + )], 2 2 where the trigonometric identity sin(A ± B) = sin(A) cos(B) ± cos(A) sin(B) has been employed, and − sin

Thus,



π 100◦ π 100◦ π 100◦ π π = sin θ + − ( ) − θ = sin θ − + 180◦ 180◦ 2 180◦ 2 ◦ 100 π π 100◦ π π π π = sin(θ + ) cos( + ) − cos(θ + ) sin( + ) ◦ ◦ 2 180 2 2 180 2 π π = −0.9848 sin(θ + ) + 0.1736 cos(θ + ). 2 2

5.5 A Practical Induction Motor

275

260◦ π ◦

180 [sin (β − θ)] 100 ◦ π = −2 × 0.9848 sin(θ + 180◦

π ) = −2 × 0.9848 cos(θ). 2

Proceeding similarly with the other trigonometric functions in (5.143) we find ψ1br 1 = k Nl1 μ0

3N R 180◦ r R I R1 4 × (0.9848 + 0.866 + 0.6428) cos(θ). g 2π

On the other hand, following the same procedure it is possible to verify that ψ1ar 1 , the contribution of electric current in phase 1 of rotor to flux linkage in phase 1a of stator, satisfies ψ1ar 1 = ψ1br 1 . Thus, the total contribution of phase 1 of rotor to flux linkage in phase 1 of stator is given as ψ1r 1 = ψ1ar 1 + ψ1br 1 = 2ψ1b R1 , 3N R 180◦ = k Nl1 μ0 r R I R1 8 × (0.9848 + 0.866 + 0.6428) cos(θ), g 2π 3N R 180◦ r R I R1 cos(θ). (5.144) = 19.9488k Nl1 μ0 g 2π Using B R2 (β, θ) and B R3 (β, θ), defined in (5.136), and the procedure above it is found that the contribution of electric currents in the rotor phases 2 and 3 to the flux linkage in the stator phase 1, i.e., ψ1r 2 and ψ1r 3 are given as 3N R 180◦ 2π r R I R2 cos(θ + ), g 2π 3 4π 3N R 180◦ r R I R3 cos(θ + ). = 19.9488k Nl1 μ0 g 2π 3

ψ1r 2 = 19.9488k Nl1 μ0

(5.145)

ψ1r 3

(5.146)

Notice that cos(θ + phase 1 is given as

4π ) 3

= cos(θ −

2π ). 3

Finally, the total flux linkage in the stator

ψ1s = ψ1s1 + ψ1s2 + ψ1s3 + ψ1r 1 + ψ1r 2 + ψ1r 3 , with the expressions given in (5.140), (5.141), (5.142), (5.144), (5.145), (5.146). Thus, following the same procedure for the other stator phases we find that 

ψ123s ψ123r



 =

L s L sr L sr L r



 I123s , I123r

where ψ123s = [ψ1s , ψ2s , ψ3s ] represent the flux linkages in the stator phases, ψ123r = [ψ1r , ψ2r , ψ3r ] represent the flux linkages is the rotor phases, I123s = [I1 , I2 , I3 ] are electric currents through the stator phase windings, I123r = [I R1 , I R2 , I R3 ] are electric currents through the rotor phase windings, and

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5 Induction Motor

⎡

⎤ L ls + L ms − 21 L ms − 21 L ms L s = ⎣ − 21 L ms L ls + L ms − 21 L ms ⎦ , − 21 L ms − 21 L ms L ls + L ms ⎡ ⎤ L lr + L mr − 21 L mr − 21 L mr L r = ⎣ − 21 L mr L lr + L mr − 21 L mr ⎦ , − 1 L mr − 21 L mr L lr + L mr ⎤ ⎡2 cos(θr ) cos(θr + 2π ) cos(θr − 2π ) 3 3 L sr = L sl ⎣ cos(θr − 2π ) cos(θr ) cos(θr + 2π )⎦, 3 3 2π ) cos(θ ) cos(θr + 3 ) cos(θr − 2π r 3 ◦ 3 180 , L ms = 19.9488N 2 l1 μ0 r R g 2π 3 180◦ L sl = 19.9488k N R Nl1 μ0 r R , g 2π πμ0 l1r R N R2 L mr = , 4g where L ls and L lr are the stator and the rotor leakage inductances and θr = θ. As previously stated, it is usual in the literature to model a squirrel cage rotor as if it was a sinusoidally wound rotor. This implies that the procedure to obtain the expressions for the rotor flux linkages ψ123r = [ψ1r , ψ2r , ψ3r ] does not depend on considerations of the particular structure of rotor under study but it is rather a general procedure taking into account the general structure of a sinusoidally wound rotor. Because of these reasons, the rotor flux linkages ψ123r = [ψ1r , ψ2r , ψ3r ] are not computed here and the reader is referred to [55] for a complete procedure to compute these flux linkages. The parameter L mr is the rotor magnetizing inductance which is also computed in [55].

5.5.3 The Motor Dynamic Model Notice that the above expressions for the stator and rotor flux linkages are exactly the same as those presented in (5.6). Hence, applying the Kirchhoff’s Voltage Law, the Faraday’s Law, and the Ohm’s Law to each phase winding on both, stator and rotor, the expressions in (5.4), (5.5) are retrieved. Thus, we can follow exactly the procedure in Sects. 5.1.3, 5.1.4, 5.1.5, 5.1.6, to exactly retrieve the dq dynamic model given in (5.42).

Chapter 6

Switched Reluctance Motor

Switched reluctance motors (SRMs) are powerful driving actuators because of their unique torque producing characteristics [27, 259, 272]. SRMs produce very large torques at low velocities, they require very low maintenance and they produce much higher torques than brushless DC-motors. The principle of operation of SRMs was established in 1838. However, their applications were sparse because of the lack of automatic circuitry allowing switching-on and switching-off the motor phases in a coordinated manner. Moreover, SRMs produce acoustic noise, large torque ripple, and they are difficult to control because of their complex nonlinear dynamic model and their multi-input nature. SRM technology has developed recently because of the availability of the required digital and power electronics technologies to switch-on and switch-off the motor phases in a coordinated manner. There exists, at present, a large number of efficient heuristically based and experimentally validated control approaches for SRMs. However, articles on control of switched reluctance drives that include a rigorous stability analysis are rare [168]. The approach followed in the present chapter is to solve the velocity and position control problems in the original coordinates. This means that a coordinate change (such as the dq transformation in PM synchronous motors and induction motors), showing the SRM model to be independent of the rotor position in the new coordinates, is not employed. This chapter is organized as follows. The mathematical model of SRM is obtained in Sect. 6.1. Both, the unsaturated and the saturated models are presented. The torque sharing approach is also reviewed as well as the standard control scheme for SRM. Energy exchange between the electrical and the mechanical subsystems is studied in Sect. 6.2. Some controllers are presented in Sects. 6.3 and 6.4 for velocity control and position control, respectively. In Sect. 6.5 is modeled a practical SRM.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 V. M. Hernández-Guzmán et al., Energy-Based Control of Electromechanical Systems, Advances in Industrial Control, https://doi.org/10.1007/978-3-030-58786-4_6

277

278

6 Switched Reluctance Motor

6.1 Motor Modeling 6.1.1 The Working Principle A three-phase with four-rotor poles switched reluctance motor (SRM) is depicted in Fig. 6.1.1 Rotor has not neither permanent magnets nor windings and it simply consists of a piece of iron. Torque is generated by reluctance, i.e., by means of a torque production mechanism which is identical to that appearing when an electromagnet is placed close to a piece of iron. In Fig. 6.1a, the stator phase 1 is energized because it produces the maximum torque on the rotor (through rotor poles a and c). When the stator phase 1 and the rotor poles a and c are aligned, as in Fig. 6.1b, they contribute with a zero torque. Hence, stator phase 1 is disconnected and stator phase 2 is energized because it produces the maximum torque on the rotor (through rotor poles b and d). When the stator phase 2 and the rotor poles b and d are aligned, as in Fig. 6.1c, the stator phase 2 is disconnected and the stator phase 3 is energized. This sequence of actions repeats as rotor turns. Note that only positive electric currents are required to be applied to the stator windings even when torque has to be generated in the opposite direction, i.e., in Fig. 6.1 first energize the stator phase 2, then the stator phase 1, the stator phase 3, and so on. It is not difficult to realize that such a motor operation results in large torque variations which is an important drawback for SRM applications. Torque sharing [69, 119, 259] is a methodology which has shown to be very successful to reduce torque variations. Torque sharing is explained in Sect. 6.1.5.

6.1.2 Magnetic Circuits A magnetic circuit is shown in Fig. 6.2. It consists of a core which is composed of two separate pieces of iron. Two air gaps of width x exist between these pieces and n turns of conductor wire are wound on the core. An electric current i flows through the conductor under the effect of a voltage v applied at the conductor terminals. Because of the high permeability of iron, the magnetic flux λ produced by the electric current concentrates in the path defined by the iron core and the two air gaps. Since B¯ = μr μ0 H¯ , where μ0 is permeability of air and μr is the relative permeability of the material, Ampere’s Law (2.33) can be written as  C

1 Without loss of

1 B · dl = Ienc . μr μ0

generality and for the sake of clarity, only three-phase SRM’s are considered. The reader can verify that these ideas are easily extended to the case of SRM with an arbitrary number of phases.

6.1 Motor Modeling

279

Fig. 6.1 Sequence of actions that allow the rotor of a three-phase and four-rotor poles SRM to turn counter-clockwise. This rotor and stator configuration is known as doubly salient Fig. 6.2 Magnetic circuit

280

6 Switched Reluctance Motor

Applying this expression to the closed trajectory C represented by the mean length of the core we obtain  1 1 1 B · dl = Bi l + Bg (2x) = Ienc = ni, (6.1) μi μ0 μ0 C μr μ0 where l is the mean length of the iron core, Bi and Bg stand for the magnetic field in the iron core and the air gap, respectively, and we have employed the fact that the relative permeability of air is unity. Notice that the right-hand rule (see Fig. 2.8) must be taken into account to obtain this result. ¯ where the “dot” stands Since the magnetic flux in the core is given as λ = B¯ · A, ¯ for the standard scalar product of vectors, A is the surface vector that is orthogonal to the transversal section of the core, B¯ is orthogonal to the core section and uniform, and the magnetic flux is the same in both, the iron core and air gap, then λ = Bi A = Bg A. Using this result in (6.1):

2x λ 1 λ l+ = ni, μi μ0 A μ0 A

and, thus l 2x 2x + ≈ , μi μ0 A μ0 A μ0 A

λR = M, R =

M = ni,

(6.2)

where R is known as the reluctance of the magnetic circuit, we have taken advantage from the fact that the iron relative permeability is very large compared to unity, i.e., μi  1, and M is known as the magnetomotive force. . The flux linkage is defined as ψ = nλ and, according to Faraday’s Law, v = d(nλ) dt Using this and (6.2), the power w supplied by the electric circuit is given as w = iv =

M ˙ ˙ n λ = M λ˙ = E, n

where E is the energy supplied by the electric circuit which is stored in the magnetic circuit as 

t

E=

M 0

dλ dt = dt



λ(t)

Mdλ. λ(0)

Using (6.2), this can be written as  E=

λ(t) λ(0)

Rλdλ =

1 Rλ2 (t), 2

(6.3)

6.1 Motor Modeling

281

if λ(0) = 0. Notice that this represents the energy stored in the reluctance, i.e., reluctance stores energy. This must be stressed since most people think that reluctance is the magnetic analogue of electric resistance which is not true since electric resistance dissipates energy and reluctance stores energy. The stored energy E also represents the mechanical work performed by an external force F in order to create two gaps of width x. This means that F is applied in the opposite direction of the magnetic force f , i.e., F = − f and  E=

x



x

Fds = −

0

f ds,

f =−

0

∂E . ∂x

Using (6.3): 1 ∂R . f = − λ2 2 ∂x This and (6.2) imply that the magnetic force is produced by the reluctance and it is applied in the sense where reluctance, energy, and gap width decrease. On the other hand, the inductance L(x) depends on the gap width x as it is explained in the following. The flux linkage in the inductor is given as the product of inductance and electric current, i.e., ψ = L(x)i = nλ. Moreover, according to (6.2), if x decreases, flux λ increases if electric current is kept constant. According to nλ = L(x)i an increase of flux λ only can be produced by an increase of inductance L(x), i.e., inductance increases if x decreases and thus ∂ L(x) < 0. ∂x From (6.3) and (6.2) it follows that E=

1 ψ 1 1 1 Mλ = ni = iψ = L(x)i 2 . 2 2 n 2 2

(6.4)

Finally, according to f = − ∂∂xE : f =−

1 ∂ L(x) 2 i . 2 ∂x

(6.5)

It is stressed that f in (6.5), defined in Fig. 6.2 as applied in the sense where the air gap x decreases, is positive because ∂ L(x) < 0. This explains why the rotor poles ∂x tend to align to the phase saliencies in Fig. 6.1, as described in Sect. 6.1.1. The above ideas are based on [32].

6.1.3 SRM Unsaturated Dynamical Model It is widely accepted and experimentally proven the assumption that the three stator phases are magnetically decoupled. As a matter of fact, in Fig. 6.26 is shown how

282

6 Switched Reluctance Motor

magnetic flux produced by phase 1 distributes on the air gap in a practical SRM. From the situation described there, it is clear that the magnetic flux produced by each phase does not contribute to magnetic flux linkages in the other phases. This means that the three stator phases are magnetically decoupled. Hence, application of Kirchhoff’s Voltage Law, Faraday’s Law and Ohm’s Law to each stator phase winding, as well as Newton’s Second Law to the rotor, yields ψ˙ + r I = U,

J ω˙ + bω = τ − τ L ,

(6.6)

where ψ = [ψ1 , ψ2 , ψ3 ] , I = [I1 , I2 , I3 ] and U = [U1 , U2 , U3 ] represent, respectively, phase flux linkages, phase currents, and voltages applied at each phase. The positive constant scalars J , b, r represent rotor inertia, viscous friction coefficient, and stator winding resistance whereas τ , τ L , q, and ω = q˙ stand for generated torque, load torque, rotor angular position, and rotor angular velocity. Suppose that the stator phase 1 is energized, such that a constant electric current I1 flows through this phase winding, and rotor is moving counter-clockwise. Notice that flux linkage in phase 1, ψ1 , is maximal when the air gap is minimal, i.e., when the rotor poles a and c are aligned with phase 1 and when the rotor poles b and d are aligned with phase 1. Also notice that ψ1 is minimal when the air gap is maximal, i.e., when the space between the rotor poles a and b (or, equivalently, the space between the rotor poles c and d) is aligned with phase 1. If unsaturated magnetic circuits are assumed, one manner to approximate these variations of ψ1 is the following: ψ1 = L 1 (q)I1 ,

L 1 (q) = l0 − l1 cos(Nr q),

where, hence, L 1 (q) is the position dependent inductance of phase 1. This dependence of inductance on rotor position is produced by the variable gap at each phase as rotor turns (see Fig. 6.1). The function cos(·) is introduced to approximate the abovedescribed variations, constants l0 and l1 are such that magnetic flux is always positive (flux direction does not change), the number of rotor poles Nr is included to take into account that the above-described variations repeat Nr times within one complete rotor revolution and it is assumed that Nr q = π when rotor poles a and c are aligned with phase 1. Thus, assuming unsaturated magnetic circuits, the motor flux linkages are given as ψ = D(q)I, where the inductance matrix D(q) depends on the rotor angular position q as D(q) = diag{L 1 (q), L 2 (q), L 3 (q)},   2π , L j (q) = l0 − l1 cos Nr q − ( j − 1) 3

(6.7) j = 1, 2, 3,

(6.8)

6.1 Motor Modeling

283

and the phase angle 2π in radians is introduced to take into account that phases are 3 wound differing in orientation by 120◦ . The numerical values of constants l0 and l1 are such that they make L j (q), j = 1, 2, 3, strictly positive functions, i.e., matrix D(q) is positive definite. Replacing (6.7) and (6.8) in (6.6) we find that the dynamical model of a SRM is given as D(q) I˙ + K (q)I q˙ + R I = U,

(6.9)

J ω˙ + bω = τ − τ L ,

(6.10)

where R = diag{r, r, r } is a positive definite matrix whereas matrix K (q) represents the phase inductance variation with respect to the rotor angular position and is given as K (q) = diag{K 1 (q), K 2 (q), K 3 (q)},   ∂ L j (q) 2π = l1 Nr sin Nr q − ( j − 1) , K j (q) = ∂q 3

j = 1, 2, 3.

(6.11)

According to (6.4) and taking into account that SRM under study has three phases, the stored magnetic energy is given as 21 I T D(q)I . Since the magnetic circuits are assumed to be linear, then this expression also stands for the magnetic co-energy. Thus, according to D’Alembert principle, the generated torque is given as [27, 69, 259] τ=

∂ ∂q



1  I D(q)I 2



1 1 T I K (q)I = K j (q)I j2 . 2 2 j=1 3

=

(6.12)

Finally, let us say that the negative sign in (6.5) is only intended to indicate that, in that example, the magnetic force is applied in the direction where the air gap decreases, i.e., in the opposite direction where inductance decreases. In this respect, torque produced by phase 1 in Fig. 6.1a and 6.1c is in opposite directions but always in the sense where the air gap decreases. Thus, we have to define the positive sense for torque which we have already chosen to be counter-clockwise. This is the same direction of the positive rotor angular position. Moreover, torque τ in (6.10) is affected by a positive sign which means that τ is positive when applied counter-clockwise, i.e., the same direction as the angular velocity ω. Recall that ω = q. ˙ Notice that, according to (6.12), contribution of phase 1 to torque τ is positive if K 1 (q) > 0. This is consistent with the fact that, according to (6.11) and the two paragraphs 1 (q) > 0 if Nr q is a little less than π, i.e., as in Fig. 6.1a. before (6.7), K 1 (q) = ∂ L∂q Furthermore, the above reasoning explains why a clockwise (negative) contribution 1 (q) < 0, i.e., if Nr q is of phase 1 to generated torque is produced when K 1 (q) = ∂ L∂q a little greater than π as in Fig. 6.1c. This explains why any negative sign does not appear in (6.12).

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6 Switched Reluctance Motor

6.1.4 SRM Saturated Dynamical Model For the sake of simplicity and without loss of generality, we consider again a SRM with three phases at the stator (see Fig. 6.1). It is well known that flux saturation is common in SRM’s. In such a case, [269] suggests to model this phenomenon in phase j as ψ j (q, I j ) = ψs arctan(β L j (q)I j ), for j = 1, 2, 3, where q represents rotor position whereas ψ j (q, I j ) and I j , stand for flux linkage and electric current in phase j. L j (q) is a function which stands for phase j, it is periodic on position q and strictly positive, and it is assumed to be given as L j (q) = l0 +

    ∞   2π 2π ln cos n Nr q − ( j − 1) + cn sin n Nr q − ( j − 1) , 3 3 n=1

j = 1, 2, 3, Nr is the number of rotor poles and l0 , ln , cn , are real constants. Finally, ψs and β, are positive constants which have to be obtained experimentally. Applying Kirchhoff’s Law and Faraday’s Law to the three-phase windings on the stator we find ψ˙ + r I = U,

(6.13)

where ψ = [ψ1 , ψ2 , ψ3 ] , I = [I1 , I2 , I3 ] and U = [U1 , U2 , U3 ] represent, respectively, phase flux linkages, phase currents and voltages applied at each phase, whereas r > 0 represents the stator winding resistance. Replacing: ∂ψ j ˙ ∂ψ j q˙ + Ij, ψ˙ j = ∂q ∂Ij

j = 1, 2, 3,

in (6.13) we find that the dynamical model of such a SRM is given as [58, 68] D(q, I ) I˙ + C(q, I )I ω + R I = U, J ω˙ + bω = τ − τ L ,

(6.14) (6.15)

where the matrix R = diag{r, r, r } is positive definite, whereas the inductance matrix = diag{D1 (q, I1 ), D2 (q, I2 ), D3 (q, I3 )} is positive definite and diagD(q, I ) = ∂ψ ∂I onal, where D j (q, I j ) =

∂ψ j ψs β L j (q) = , j = 1, 2, 3, ∂Ij 1 + β 2 L 2j (q)I j2

(6.16)

and C(q, I )I = [C1 (q, I1 )I1 , C2 (q, I2 )I2 , C3 (q, I3 )I3 ] , where C j (q, I j )I j =

d L j (q) ∂ψ j ψs β I j = , j = 1, 2, 3. 2 2 2 ∂q 1 + β L j (q)I j dq

(6.17)

6.1 Motor Modeling

285

The equation in (6.15) results from the application of Newton’s Second Law to the rotor, i.e., ω = q˙ stands for rotor angular velocity, τ L is the load torque, whereas the scalars J > 0, b > 0, represent the rotor inertia and the viscous friction coefficient. Finally, the generated torque by phase j is given, according to the D’Alembert principle, as the derivative of the magnetic co-energy with respect to rotor position, i.e., ∂ τj = ∂q



Ij

ψjdIj =

0

d L j (q) ψs ln(1 + β 2 L 2j (q)I j2 ). 2β L 2j (q) dq

Since the total torque generated by the motor is the addition of torques produced by the three phases, we have τ=

3  j=1

τj =

3  j=1

d L j (q) ψs ln(1 + β 2 L 2j (q)I j2 ). 2 2β L j (q) dq

(6.18)

6.1.5 The Torque Sharing Approach Torque sharing is an approach which is intended to reduce torque variations by means of motor phases collaboration for torque production. This is achieved as follows. When rotor is oriented as in Fig. 6.3a, the generated torque has to be produced by using the stator phase 1 alone because this stator phase produces the maximum torque. However, in Fig. 6.3b, as the rotor poles a and c align with the stator phase 1, torque produced by the stator phase 1 decreases. This is because a zero torque will be produced when this stator phase and these rotor poles will be completely aligned. Hence, the stator phases 1 and 2 can be used together, in Fig. 6.3b, to produce the total torque to be generated by the SRM. This idea exploits the fact that torque generated by the stator phase 2 and the rotor poles b and d increases as the stator phase 1 and the rotor poles a and c align. Note that the stator phase 3 must not be energized at this moment since it would contribute with a torque in the opposite direction, i.e., according to (6.12) K 1 (q) > 0, K 2 (q) > 0 and K 3 (q) < 0, or vice versa, at this rotor angular position. Then, in Fig. 6.3c, phase 2 is energized alone because it produces the maximal torque attracting the rotor poles b and d. In Fig. 6.3d, torque produced by phase 2 decreases because the rotor poles b and d tend to align to phase 2. Thus, at that moment, phase 3 can also be energized to collaborate with phase 2 producing additional torque by attracting the rotor poles a and c. Phase 1 must not be energized at this point because it would produce some torque in the opposite direction. This means that, according to (6.12), K 1 (q) < 0, K 2 (q) > 0, and K 3 (q) > 0, or vice versa, at this rotor angular position. The above ideas were rendered precise in [69, 119, 259] as follows. Given a desired torque τ ∗ this torque has to be generated by the contribution of the three phases of the motor:

286

6 Switched Reluctance Motor

Fig. 6.3 One phase may work alone or two phases may collaborate in a three-phase and four-rotor poles SRM

τ ∗ = m 1 (q)τ ∗ + m 2 (q)τ ∗ + m 2 (q)τ ∗ ,

(6.19)

which implies that 3j=1 m j (q) = 1. The scalar functions m j (q), j = 1, 2, 3, are known as the sharing functions and they represent the fraction of the total torque to be generated by motor that each phase has to contribute with. The following two sets define rotor positions where phase j can produce a positive or a negative torque [69, 119, 259]:



 − Θ+ j = q : K j (q) ≥ 0 , Θ j = q : K j (q) < 0 .

(6.20)

Choose m +j and m −j as bounded functions with bounded and continuous first derivatives such that

6.1 Motor Modeling



287

3  m +j (q) > 0, ∀q ∈ Θ + j ; m +j (q) = 1, ∀q ∈ R, m +j (q) = 0, ∀q ∈ Θ − j

(6.21)

3  m −j (q) > 0, ∀q ∈ Θ − j ; m −j (q) = 1, ∀q ∈ R, m −j (q) = 0, ∀q ∈ Θ + j

(6.22)

j=1

j=1

and assign m j (q) =

m +j (q), τ ∗ ≥ 0 . m −j (q), τ ∗ < 0

(6.23)

Notice that (6.23) together with (6.21) and (6.22) ensure (6.19) and that m j (q) and K j (q) become zero simultaneously. For the purposes of the sharing functions approach, i.e., only for the purposes of this section, in the case of the saturated model we define again K j (q) =

∂ L j (q) , ∂q

j = 1, 2, 3,

but the second equality in (6.11) is not necessarily true.

6.1.6 Standard Control A block diagram of standard control of SRMs is presented in Fig. 6.4 [150]. It is important to stress that, contrary to PM synchronous motors and induction motors, standard control of SRMs is performed in original coordinates, i.e., a dq transformation is not required in SRMs. Given a desired torque τ ∗ to generate, the desired current command is obtained using a torque constant K t which is computed from the linearized inductance versus rotor position characteristics for a particular value of current [150]. The electric current error is applied to a hysteresis controller, although a proportional controller is also employed [64, 115, 173], which determines the voltage to be applied to

Fig. 6.4 Standard control of a SRM. Electrical subsystem refers to (6.9) whereas mechanical subsystem refers to (6.10)

288

6 Switched Reluctance Motor

Fig. 6.5 Hysteresis nonlinearity

motor. This voltage command, however, is applied by taking into account the socalled rise and fall angles, i.e., the rotor angles when the electric current must flow and stop at each stator phase [150]. Hence, I j ≈ I j∗ and τ ≈ τ ∗ . As stated above this is accomplished by using, both proportional and hysteresis electric current controllers at each phase [64, 150], i.e., U = H (I ∗ − I ) + α(I ∗ − I ),

(6.24)

where it is defined the vectorial function H (I ∗ − I ) = [h(I1∗ − I1 ), h(I2∗ − I2 ), h(I3∗ − I3 )]T with h(·) the hysteresis nonlinearity depicted in Fig. 6.5, and α is a positive scalar. The desired torque τ ∗ is given as the output of a linear PI velocity controller (where ω ∗ represents the desired velocity): τ ∗ = k p (ω ∗ − ω) + ki



t

(ω ∗ (s) − ω(s))ds.

(6.25)

0

Since τ ≈ τ ∗ , the block diagram in Fig. 6.4 can be simplified to block diagram shown in Fig. 6.6. Under this assumption, the proportional and integral gains k p and ki of the classical PI velocity controller can be chosen by taking into account only the mechanical dynamics. This can be accomplished using classical linear control design tools since control system in Fig. 6.6 is linear. In Fig. 6.7 is presented the standard control scheme for position control in a SRM. Notice that a classical PID position controller is employed in the outer loop (q ∗ represents the desired position):

6.1 Motor Modeling

289

Fig. 6.6 Standard control of a SRM presented in Fig. 6.4 reduces to control of a simple linear mechanical system

Fig. 6.7 Standard position control of a SRM

τ ∗ = k p (q ∗ − q) + kd

d(q ∗ − q) + ki dt



t

(q ∗ (s) − q(s))ds

(6.26)

0

and the additional state equation is considered: q˙ = ω. The integral action in controllers (6.25) and (6.26) is included in order to ensure that ω = ω ∗ and q = q ∗ are achieved in steady state when both q ∗ and ω ∗ are constant and despite the presence of some unknown but constant external torque disturbance τL . However, notice that the electrical dynamics in (6.9) is nonlinear and, thus, the complete control system is nonlinear. This implies that nonlinear control techniques must be used in order to design the controller gains in (6.24)–(6.26). Such a methodology is presented in the subsequent sections in this chapter.

6.2 Open-Loop Energy Exchange 6.2.1 The Unsaturated Velocity Model According to Sect. 6.1.3, the unsaturated dynamic model of a three-phase switched reluctance motor (SRM) is given by (6.7)–(6.12), which is rewritten here for the ease of reference:

290

6 Switched Reluctance Motor

D(q) I˙ + K (q)I q˙ + R I = U, J ω˙ + bω = τ − τ L , τ =

(6.27) (6.28)

1 T I K (q)I = 2

3  j=1

1 K j (q)I j2 , 2

D(q) = diag{L 1 (q), L 2 (q), L 3 (q)},   2π , L j (q) = l0 − l1 cos Nr q − ( j − 1) 3 K (q) = diag{K 1 (q), K 2 (q), K 3 (q)},   ∂ L j (q) 2π = l1 Nr sin Nr q − ( j − 1) , K j (q) = ∂q 3 R = diag{r, r, r },

(6.29)

(6.30)

where, j = 1, 2, 3, I = [I1 , I2 , I3 ] and U = [U1 , U2 , U3 ] represent, respectively, phase currents and voltages applied at each phase. The positive constant scalars J , b, r represent rotor inertia, viscous friction coefficient and stator winding resistance whereas τ , τ L , q and ω = q˙ stand for generated torque, load torque, rotor angular position, and rotor angular velocity. The numerical values of constants l0 and l1 are such that they make L j (q), j = 1, 2, 3, strictly positive functions, i.e., matrix D(q) is positive definite, and Nr is the number of rotor poles. The following scalar function represents the total energy stored in the motor: V (I, ω) =

1 1  I D(q)I + J ω 2 , 2 2

(6.31)

where the first term represents the magnetic energy stored in the electrical subsystem whereas the second term stands for the kinetic energy stored in the mechanical subsystem. The time derivative of V is given as 1 ˙ V˙ = I  D(q) I˙ + I  K (q)I q˙ + ω J ω, 2 where (6.30) has been used. According to (6.27)–(6.29) this can be written as   1  1   ˙ V = I (−K (q)I q˙ − R I + U ) + I K (q)I q˙ + ω −bω + I K (q)I − τ L . 2 2 Notice that −I  K (q)I q˙ + 21 I  K (q)I q˙ + 21 ω I  K (q)I = 0, because ω = q. ˙ Since V represents the energy stored in the motor, these cancellations (appearing in V˙ ) represent the energy exchange between the motor electrical and mechanical subsystems. Hence, it can be written: V˙ = −I  R I + I  U − bω 2 − ωτ L .

(6.32)

6.2 Open-Loop Energy Exchange

291

Defining the input u = [U  , −τ L ] and the output y = [I  , ω] , we can write (6.32) as ⎡ ⎤ r 000 ⎢0 r 0 0⎥ ⎥ V˙ = −y  Qy + y  u, Q = ⎢ (6.33) ⎣0 0 r 0⎦. 000b Since Q is a positive definite matrix, (6.33) shows that the model (6.27)–(6.29) is output strictly passive (see Definition 2.42) for the output y and input u defined above. It will be shown in this chapter that this property is fundamental to design a simple velocity controller for SRMs.

6.2.2 The Saturated and Unsaturated Position Models 6.2.2.1

The Unsaturated Model

Suppose that the load torque is given as a nonlinear function of position τ L = G(q), , where P(q) is a positive semidefinite scalar function. Using such that G(q) = d P(q) dq these ideas and ω = q, ˙ the unsaturated dynamic model of a three-phase four-rotor poles switched reluctance motor (SRM) is given as in (6.27)–(6.30) with the following change instead of (6.28): J q¨ + bq˙ = τ − G(q).

(6.34)

The following scalar function represents the total energy stored in motor: V (I, q, q) ˙ =

1 1  I D(q)I + J ω 2 + P(q), 2 2

(6.35)

where the latter new term represents the potential energy stored in the mechanical subsystem. Notice that V is a positive semidefinite scalar function since P(q) is assumed to posses this property. Following a similar procedure as in the previous section to compute V˙ , we find that the cross term cancellations referred before (6.32) appear again. Notice that an additional cancellation appears between cross terms ±G(q)q, ˙ which represents the exchange between kinetic and potential energies in the mechanical subsystem. Thus, we find that V˙ = −I  R I + I  U − bq˙ 2 .

(6.36)

Since b > 0, we can define the input u = U and the output y = I , to write (6.36) as V˙ ≤ −y  Ry + y  u.

(6.37)

292

6 Switched Reluctance Motor

Thus, (6.37) shows that the model (6.27), (6.29), and (6.34) is output strictly passive (see Definition 2.42) for the output y and input u defined above. It will be shown in this chapter that this property is fundamental to design a simple position controller for SRMs.

6.2.2.2

The Saturated Model

Assume again that the load torque is given as a nonlinear function of position τ L = , where P(q) is a positive semidefinite scalar function. G(q), such that G(q) = d P(q) dq Using these ideas and ω = q, ˙ the saturated dynamic model of a three-phase switched reluctance motor (SRM) is given as in (6.14)–(6.18) with (6.34) instead of (6.15). The following scalar function represents the total “energy” stored in the motor: V (I, q, q) ˙ =

1 1  I D(q, I )I + J q˙ 2 + P(q). 2 2

(6.38)

Mimicking the unsaturated case, the “magnetic energy” stored in the electrical subsystem of the SRM is proposed to be given by 21 I  D(q, I )I . However, the use of this “energy” function complicates the stability analysis, i.e., when computing the time derivative of such an energy function, because of the multiple nonlinearities in D(q, I ) and the dependence of this function on I . This can be verified by considering (6.38) and taking its time derivative to find that the simplification obtained in (6.32) is far from achieved in this case. Hence, we propose to use instead: V (I, q, q) ˙ = P(q) +

1 1  I L(q)I + J q˙ 2 , 2 2

which is positive semidefinite. Notice that, according to (6.16), L(q) is a component of D(q, I ). Thus 1 d L(q) d P(q) q˙ + q˙ J q¨ + I  I q˙ + I  L(q) I˙, V˙ = dq 2 dq 1 d L(q) I q˙ = G(q)q˙ + q[−b ˙ q˙ + τ − G(q)] + I  2 dq + I  L(q)D −1 (q, I )[−C(q, I )I q˙ − R I + U ] 1 d L(q) I q˙ = −bq˙ 2 + qτ ˙ − I 2 dq − r I  L(q)D −1 (q, I )I + I  L(q)D −1 (q, I )U,

(6.39)

I q˙ where (6.16) and (6.17) have been employed. Notice that the term − 21 I  d L(q) dq results from a cancellation of cross terms which represents the energy exchange in the system. It is clear that in the case of the saturated model, this term cannot cancel with +qτ ˙ , because of the nonlinearities appearing in the definition of τ . Also notice that,

6.2 Open-Loop Energy Exchange

293

since L(q)D −1 (q, I ) is a positive definite matrix, the term −r I  L(q)D −1 (q, I )I is negative and quadratic in I and the term I  L(q)D −1 (q, I )U indicates that electric current feedback can still be employed to improve the passivity properties of the system. These are the properties of the saturated model of SRM that will be exploited in this chapter for control design purposes.

6.3 Velocity Control Motivated by the challenges presented by control of SRM as well as the advances in digital and power electronics technologies, in [27, 60, 62, 119, 142, 191, 214, 236, 237, 259, 272] were proposed several control schemes for SRMs. Recent proposals [57, 69, 168–171] present passivity-based controllers which exploit and extend the approach introduced in [204]. However, as it is explained below in Remark 6.6, these controllers have a singularity introduced by the time derivative of the desired electric current when the desired torque crosses by zero. This situation is common when a change on the sense of rotation is commanded, for instance. In [168] is pointed out that there exists a large number of efficient heuristically based and experimentally validated control approaches for SRMs. However, articles on control of switched reluctance drives that include a rigorous stability analysis are rare. We have verified this affirmation since [170], which appeared in 2015, is the most recent formal work on SRM control, related in somehow to the approach adopted in the present paper, that we have found after a recent search in the literature. Thus, solving the above-cited singularity problem becomes important. In this section we present a velocity controller which avoids such a drawback. Assumption A1 Functions m i+ (q) and m i− (q) are chosen such that m i (q) → 0 as fast as (q − qi0 )r → 0 for an integer r ≥ 4, were qi0 is the rotor position such that K i (qi0 ) = 0. Proposition 6.1 Consider the SRM unsaturated model in (6.27)–(6.30), with τ L an unknown constant, in closed-loop with the following controller: U = R I ∗ − α[I − I ∗ ] + K (q)I ∗ ω ∗ + D(q)h,

I ∗ = [I1∗ , I2∗ , I3∗ ] ,    ∗ 2m i (q) β1 2mKi i(q)τ + β f (|τ ∗ |), if K i (q) = 0 ∗ 2 (q) K i (q)sign(τ ∗ ) Ii = , 0, if K i (q) = 0 τ ∗ = K v σ2 (ϑ) − K p σ1 (z), ζ˙ = −Aσ2 (ϑ), ϑ = ζ − Bz,

(6.40)

(6.41) (6.42)

294

6 Switched Reluctance Motor

 t z = q − q ∗, q ∗ = ω ∗ dt, 0     K i (q)τ ∗ ∂m i (q) 1 m i (q) ∂ K i (q) ∗ ω β1 2m i (q) , if K i (q) = 0 − 2 ∂q K i (q) K i (q) ∂q hi = , 0, if K i (q) = 0 or m i (q) = 0 1, if τ ∗ ≥ 0 ∗ , h = [h 1 , h 2 , h 3 ]T , sign(τ ) = −1, if τ ∗ < 0 where i = 1, 2, 3, ω ∗ is the constant desired velocity, τ ∗ represents the desired torque and z is the position error. We use the normalized rotor angular position q which is adjusted to the workspace [(n 1 − 2)π, n 1 π), where n 1 is some positive or negative constant integer or a zero constant. σ1 and σ2 are strictly increasing linear saturation functions,  ∗ as those described in Definition 2.34, whose saturation limits are  L  M1 > L 1 >  bωK+τ  > 0 and M2 > L 2 > 0, respectively. Furthermore, σ1 and σ2 p are required to be continuously differentiable satisfying 1≥

∂σ(ς) > 0, ∀ ς ∈ R. ∂ς

(6.43)

Functions m i (q) are defined in (6.23) and f (|τ ∗ |) = α f (1 − cos(ω f |τ ∗ |)), ∀ |τ ∗ | ≤ T ∗ ,

(6.44)

where ω f is the smallest positive number satisfying 1 − cos(ω f T ∗ ) = 2T ∗ , ω f sin(ω f T ∗ )

(6.45)

which can be solved numerically, α f > 0 is chosen as √

αf =

T∗ , 1 − cos(ω f T ∗ )

(6.46)

and T ∗ is an arbitrarily small positive constant. Finally, we have that β1 = 1 and β2 = 0, if |τ ∗ | > T ∗ , β1 = 0 and β2 = 1, if |τ ∗ | ≤ T ∗ . There always exist positive scalars K v , K p , A, B, and a 3 × 3 diagonal positive definite matrix α, such that for any initial condition, the whole state (which includes the velocity error ω˜ = ω − ω ∗ ) remains bounded and it has an ultimate bound which can be rendered arbitrarily small by using a suitable choice of controller gains. Remark 6.2 Since the motor position q ∈ R, we can assume that (n 1 − 2)π ≤ q(0) < n 1 π. Hence, q ∈ R can be forced to stay in the range [(n 1 − 2)π, n 1 π).

6.3 Velocity Control

295

This can be accomplished by assigning q − 2π → q each time q arrives to n 1 π or q + 2π → q the first time q < (n 1 − 2)π. Hence, q ∗ must also be adjusted each time q folds from n 1 π to (n 1 − 2)π (or from (n 1 − 2)π to n 1 π) by assigning q ∗ − 2π → q ∗ (or q ∗ + 2π → q ∗ ). It is not difficult to verify that this ensures that z is continuous and, moreover, continuously differentiable, i.e., z˙ = ω − ω ∗ is continuous, with ω = q˙ the motor angular velocity. We stress that velocity measurements are not required to implement controller in Proposition 6.1. Moreover, the filter (6.42) has been implemented to construct an estimate of the angular velocity error, i.e., ϑ is the estimate of ω˜ = ω − ω ∗ . Also notice that the integral term K p σ1 (z) is intended to compensate for constant but unknown load torques which is possible since motor position q can be adjusted without affecting motor velocity in steady state. Notice that L 1 can be always computed from an approximate of the maximal load torques to be allowed. Recall that, in practical applications, a motor is selected by taking into account an estimate of the maximal load torques to be applied.  ∗ ∗ i (qi ) Remark 6.3 The term K i (q2m ∗ f (|τ |) in the definition of Ii is introduced to √ ∗ i )sign(τ ) ∗ replace the function |τ | with f (|τ |) when |τ ∗ | ≤ T ∗ (see Fig. E.1 in Appendix E.1) and this is done in order to render possible of I˙i∗ in the stability √ ∗ the domination 1 analysis: recall that the time derivative of |τ | depends on √|τ ∗ | which is unbounded when τ ∗ = 0. Moreover, this substitution is continuous up to the first derivative: (a) (6.41) appears together with function f (|τ ∗ |) notice that sign(τ ∗ ) in the first row of d f (|τ ∗ |) ∗ which is zero when τ = 0, (b) d|τ ∗ | = 0 when τ ∗ = 0, according to the definition of f (|τ ∗ |) in (6.44). Thus, any discontinuity is not produced by sign(τ ∗ ). Remark 6.4 Finding ω f as the numeric solution of the expression in (6.45) is not to be performed online. Moreover, since T ∗ is a known constant, a simple computer program can be written to increase ω f from zero to choose the smallest positive value satisfying such an expression. This can be easily performed graphically since it is not to be done online. Hence, any online optimization procedure is not required. Remark 6.5 Notice that the definition of h i includes the expression:  ∗

ω β1

K i (q)τ ∗ 2m i (q)



 ∂m i (q) 1 m i (q) ∂ K i (q) − 2 . ∂q K i (q) K i (q) ∂q

(6.47)

According to (6.30) and assumption A1, m i (q) → 0 as fast as (q − q0 )r → 0 for an integer r ≥ 4, whereas K i (q) → 0 as fast as Nr (q − q0 ) →  0 because     K i (q) d 2π 2π sin Nr q − (i − 1) 3 → Nr as sin Nr q − (i − 1) 3 → 0. Thus, 2m →0 dq i (q)

as fast as (q − q0 )− 2 → 0 whereas ∂m∂qi (q) K i1(q) → 0 as fast as (q − q0 )r −2 → 0. Hence, the product of these terms tends to zero as fast as (q − q0 )μ → 0 where μ = −1.5 + r/2 > 0 since r ≥ 4. Thus, there is not any singularity when computing the product of these terms in (6.47) as m i (q) → 0. On the other hand, r −1

296

6 Switched Reluctance Motor

m i (q) ∂ K i (q) K i2 (q) ∂q

→ 0 as fast as (q − q0 )r −2 → 0. Thus, there is not either any singu K i (q) larity when computing the product of this term and 2m . i (q) The complete proof of Proposition 6.1 is presented in Appendix E.2. In the following we just present a sketch of such a proof in order to highlight the rationale behind the proof and to illustrate how energy ideas are exploited. Sketch of proof of Proposition 6.1 Let us obtain the closed-loop dynamics of the closed-loop system described in Proposition 6.1. Replacing (6.40) in (6.27), defining ξ = I − I ∗ , ω˜ = q˙ − ω ∗ , α¯ = α + R and adding and subtracting some convenient terms, we have that D(q)ξ˙ = −αξ ¯ − K (q)ξ ω˜ + K (q)ξω ∗ − K (q)I ∗ ω˜ + D(q)(h − I˙∗ ),

(6.48)

where I˙∗ is continuous according to the above definitions of m i (q) and f (|τ ∗ |). On the other hand, replacing I = ξ + I ∗ in (6.29), we find that τ = τc + τc =

1 ∗ I K (q)I ∗ , 2

(6.49)

1  ξ K (q)ξ + ξ  K (q)I ∗ . 2

Using (6.19) and (6.41), we obtain 1 ∗ I K (q)I ∗ = τ ∗ + β2 F(τ ∗ ), 2 F(τ ∗ ) = f 2 (|τ ∗ |)sign(τ ∗ ) − τ ∗ ,

(6.50) (6.51)

where function |β2 F(τ ∗ )| is upper bounded, and this bound decreases to zero as T ∗ > 0 approaches to zero since, according to (6.44), f (|τ ∗ |) → 0 for all |τ ∗ | ≤ T ∗ if T ∗ → 0 and β2 = 0 if |τ ∗ | > T ∗ . According to (6.42) and (6.50), (6.49) can be written as τ = τc + K v σ2 (ϑ) − K p σ1 (z) + β2 F(τ ∗ ). Hence, (6.28) can be written as J ω˙˜ + bω˜ = τc + K v σ2 (ϑ) − K p χ + β2 F(τ ∗ ), where χ = σ1 (z) +

bω ∗ + τ L . Kp

(6.52)

(6.53)

Thus, the closed-loop dynamics is given by (6.48), (6.52) and ϑ˙ = −Aσ2 (ϑ) − B ω, ˜ z˙ = ω. ˜

(6.54)

6.3 Velocity Control

297

This closed-loop dynamics can be rewritten as D(q)ξ˙ = −αξ ¯ − K (q)ξ ω˜ + U, 1 J ω˙˜ + bω˜ = ξ  K (q)ξ − T L , 2 ϑ˙ = −Aσ2 (ϑ) − B ω, ˜ z˙ = ω, ˜ ∗ ∗ U = K (q)ξω − K (q)I ω˜ + D(q)(h − I˙∗ ),

(6.55) (6.56) (6.57)

T L = −ξ  K (q)I ∗ − K v σ2 (ϑ) + K p χ − β2 F(τ ∗ ). The state of this closed-loop dynamics is [ω, ˜ ϑ, z, ξ  ] ∈ R6 . Notice that (6.55)– (6.57) are almost identical to the open-loop model given in (6.27)–(6.29) if I and ω = q˙ are replaced by ξ and ω. ˜ An important difference is that the electric resistance has been enlarged, i.e., instead of R in (6.27), we have α¯ in (6.55). Moreover, aside from term bω˜ in (6.56) we also have the filter term K v σ2 (ϑ) which is intended to improve the closed-loop system damping. Another important difference is the presence of two additional differential equations in (6.57) which stand for the filter dynamics and the integral part of the velocity controller. These observations motivate us to propose the following energy function to analyze the stability of the closed-loop system: 1 ˜ z d , ϑ) + ξ  D(q)ξ, (6.58) V (ω, ˜ z d , ϑ, ξ1 , ξ2 , ξ3 ) = V1 (ω, 2  zd −d 1 1 ˜ z d , ϑ) = J [ω˜ + γχ]2 − J γ 2 χ2 + K p [σ1 (r ) + d] dr V1 (ω,    2 2 −d 

ϑ

+ 0

Kv σ2 (r )dr + B



χ

z d −d −d

γb [σ1 (r ) + d] dr,   

(6.59)

χ

where γ > 0 is a constant scalar, d = [bω ∗ + τ L ]/K p and we have introduced the variable change z d = z + d. Notice that the terms 21 ξ  D(q)ξ and 21 J ω˜ 2 are included to take into account the “magnetic energy” and the “kinetic energy” stored in the electrical and the mechanical subsystems, respectively. The cross term J γ ωχ ˜ is included to obtain a negative quadratic term in the function χ whereas the term − 21 J γ 2 χ2 is included to cancel an undesired cross term arising from the quadratic term 21 J [ω˜ + γχ]2 . The first and the second integrals in (6.59) are included to consider the contribution to the “system energy” of the integral of velocity z and the filter variable ϑ. Finally, the third integral in (6.59) is included to cancel the undesired cross term −γbωχ, ˜ arising from the time derivative of J γ ωχ. ˜ In Appendix A.5 is shown that V (y), y = [ω, ˜ z d , ϑ, ξ1 , ξ2 , ξ3 ] ∈ R6 , defined in (6.58) satisfies

298

6 Switched Reluctance Motor

α1 (y) ≤ V (y) ≤ α2 (y), ∀ y ∈ R6 , c1 y2 , y < 1 , ∀y ≥ 0. α1 (y) = c1 y, y ≥ 1

(6.60)

α2 (y) = c2 y2 , for some small enough constant c1 > 0 and c2 > 0 a constant defined in Appendix A.5, if K v and B are positive and (A.19) is true. In Appendix E.2 it is shown that the time derivative of V (y), defined in (6.58), along the trajectories of the closed-loop dynamics (6.55)–(6.57) satisfies V˙ ≤ −(1 − Θ)λmin {Q}μ2 , ∀ y ≥

β2 Γ1 Γ2 , Θλmin {Q}

(6.61)

where μ = [ω, ˜ χ, σ2 (ϑ), ξ1 , ξ2 , ξ3 ] , and some positive constants 0 < Θ < 1, λmin {Q}, Γ1 , Γ2 . Then using (6.61) and (6.60) and invoking Theorem 2.29, it is concluded in Appendix E.2 that the closed-loop system state is bounded and it has an ultimate bound which cannot be reduced to zero but can be rendered arbitrarily small by a suitable choice of controller gains. This completes the proof of Proposition 6.1. Finally, we emphasize that the conditions to guarantee Proposition 6.1 are summarized by (A.19), (E.5)–(E.8), K v and B positive constants and some constant T ∗ > 0. Remark 6.6 Following the same steps in the proof of Proposition 6.1, it is not difficult to verify that the only obstacle to achieve global asymptotic stability using the following desired electric current:  ∗

I =

[I1∗ , I2∗ , I3∗ ] ,

Ii∗

=

0,

2m i (q)τ ∗ , K i (q)

if K i (q) = 0 , i = 1, 2, 3, (6.62) if K i (q) = 0

is the fact that the corresponding time derivative is not bounded when τ ∗ = 0 and τ˙ ∗ = 0. This can be seen from the product of the second term between brackets and the factor at the left in the following expression:  I˙i∗

=

K i (q) 2m i (q)τ ∗



 ∂m i (q) τ ∗ m i (q) ∗ m i (q)τ ∗ ∂ K i (q) ω+ τ˙ − ω , ∂q K i (q) K i (q) ∂q K i2 (q)

where i = 1, 2, 3, if K i (q) = 0. This problem was pointed out by the first time in [92] where the authors have introduced the function f (|τ ∗ |),defined in (6.44), to replace √ ∗ |τ | when |τ ∗ | ≤ T ∗ , for some T ∗ > 0. This avoids τ1∗ to appear in I˙∗ which is responsible of such a singularity. We stress that neither m i (q) and τ ∗ nor τ ∗ and τ˙ ∗ necessarily become zero simultaneously. See (6.23) and think, for instance, the case when τ ∗ crosses zero due to a velocity command which changes the sense of rotation,

6.3 Velocity Control

299

i.e., τ˙ ∗ = 0 because motor needs to be accelerated, and recall that m i (q) = 0 only at certain intervals of motor positions q. This fact is important to stress since several authors [57, 69, 168–171] seem not to be aware of this problem. Moreover, in Chap. 10, in the present book, is shown that magnetic levitation systems represent the one phase and one pole case of SRMs. It is interesting to recall that in [204], Chap. 8, Remark 8.5, it is pointed out that when the passivity-based approach introduced in [204] (where it was introduced the fundamental theory of the approaches in the above cite papers) is applied to a magnetic levitation system, a singularity exists when the desired force to be exerted by the electromagnet crosses through zero. Remark 6.7 Instrumental to obtain (E.1) and, subsequently, (6.61) are the can˜  K (q)ξ = 0, where cellations of cross terms −ξ  K (q)ξ ω˜ + 21 ξ  K (q)I ξ ω˜ + 21 ωξ d 1  ( ξ D(q)ξ), the time derivative of the “magnetic energy” stored in the elecdt 2 tric subsystem, and dtd ( 21 J ω˜ 2 ), the time derivative of the “kinetic energy” stored in the mechanical subsystem, are involved. These cancellations are due to the natural energy exchange between the electrical and the mechanical subsystems. They are obvious consequences of the almost identical structure of the closed-loop dynamics in (6.55)–(6.57) and the open-loop dynamics in (6.27)–(6.29). Remark 6.8 The closed-loop mechanical subsystem dynamics given in (6.52) can be written as J ω˙˜ + bω˜ = τe + K v σ2 (ϑ) − K p χ,

(6.63)

if we define 1  ξ K (q)ξ + ξ  K (q)I ∗ + β2 F(τ ∗ ). 2 F(τ ∗ ) = f 2 (|τ ∗ |)sign(τ ∗ ) − τ ∗ . τe =

(6.64)

Notice that τe represents the difference between the actual and the desired torques. Hence, taking into account (6.43) it is possible to write ⎡

⎤⎡ ⎤ b − γJ 0 0 ω˜ γ K p − γ K2 v ⎦ ⎣ χ ⎦ + (ω˜ + γχ)τe (6.65) ˜ χ, σ2 (ϑ) ] ⎣ 0 V˙1 ≤ −[ ω, σ2 (ϑ) 0 − γ K2 v KBv A where function V1 is defined in (6.59). On the other hand, the time derivative of the last component of V in (6.58), i.e., Ve =

1  ξ D(q)ξ, 2

along the trajectories of the closed-loop electrical subsystem dynamics (6.48), con¯ Thus, when computing V˙ = V˙1 + V˙e , we tains the quadratic negative term −ξ  αξ.

300

6 Switched Reluctance Motor

realize that instrumental for the stability result in Proposition 6.1 are the following features, which are very similar to those in Remark 3.4: • The scalar function V1 is a strict Lyapunov function when the desired torque is the input of the closed-loop mechanical subsystem dynamics, given in (6.63), i.e., when τe = 0. ¯ appearing in V˙ can be enlarged • The coefficients of the negative term −ξ  αξ, arbitrarily. This is important to dominate the cross terms in V˙ depending on ξ when ξ = 0. • Cancellation of several cross terms belonging to V˙1 and V˙e which was explained in Remark 6.7. Notice that all of these features are possible thanks to the passivity properties of the open-loop motor model described in Sect. 6.2.1. Furthermore, the above proof contains the main ideas of the procedure described in Sect. 2.4. As pointed out in Remark 3.4, τe is given as a nonlinear function of the electrical dynamics error for electric motors different from PM brushed DC-motors (see (6.64)). In the case of switched reluctance motors we have resorted to the introduction of a saturated desired torque, defined in (6.42), which renders χ and I ∗ saturated functions. This is important to dominate some higher order cross terms in the expression for V˙ (see (E.1)), such as 21 γχξ  K (q)ξ and γχξ  K (q)I ∗ , which is instrumental to achieve the global stability results presented in Proposition 6.1. Notice that higher order cross terms such as 21 γχξ  K (q)ξ and γχξ  K (q)I ∗ arise in the stability analysis as a consequence of the nonlinear nature of the motor model. Remark 6.9 It is clear from the proof of Proposition 6.1 that the different from zero ultimate bound appears only if |τ ∗ | ≤ T ∗ in steady state for some arbitrarily small T ∗ > 0. Hence, if |τ ∗ | > T ∗ > 0 in steady-state global asymptotic stability follows from (6.61) because μ = 0 in such a case. It is clear from (6.28), (6.49), (6.50), that |τ ∗ | = bω ∗ + τ L in steady state and, hence, μ = β2 = 0 if bω ∗ + τ L > T ∗ . Thus, global asymptotic stability is accomplished if a large enough desired velocity is commanded or a large enough load torque is present. Remark 6.10 In the traditional control scheme for switched reluctance motors, both hysteresis and proportional electric current controllers are employed because the bandwidth is not constant, i.e., since the inductance is not constant either [64]. Furthermore, the non-constant shape of the desired electric current also explains why integral electric current controllers are not employed and why the term R I ∗ is employed in (6.40). In Fig. 6.8 is depicted the traditional control scheme for switched reluctance motors which was presented in Sect. 6.1.6, whereas the control scheme in Proposition 6.1 is depicted in Fig. 6.9. Notice that controller in Proposition 6.1 employs the electric current proportional controller but not a hysteresis controller. Moreover, the additional terms R I ∗ + K (q)I ∗ ω ∗ + D(q)h are included. Also notice that instead of the simple relation used in the traditional control scheme to define the desired  current I ∗ , in Proposition 6.1 is used (6.41) which includes the additional term β2

2m i (q) K i (q)sign(τ ∗ )

f (|τ ∗ |) introduced to avoid singularities when the desired torque

6.3 Velocity Control

301

Fig. 6.8 Standard control of a SRM

Fig. 6.9 Control scheme in Proposition 6.1

becomes zero, as explained in Remark 6.6. It is important to say that thanks to (6.41), the computation of rise and fall angles required in the traditional control scheme of Fig. 6.8 is avoided. We stress that thanks to these modifications it is possible to present global stability results in Proposition 6.1. In this respect it is convenient to stress that global stability results do not exist until now for the traditional control scheme depicted in Fig. 6.8. A drawback of controller in Proposition 6.1 is that it relies on the viscous friction that the mechanical subsystem naturally possesses.

6.3.1 Simulation Results In this section we present a simulation example intended to give some insight on the performance achievable with controller in Proposition 6.1. For this, we use the SRM whose numerical parameters are given in [69, 168]: Nr = 8, R = 5 [Ohms], l0 = 0.03 [H], l1 = 0.02 [H], J = 0.001 [kg m2 ], b = 0.02 [Nm/(rad/s)]. Inspired by [286] we choose the following saturation function for σ1 (·) and σ2 (·): " ς+L # ⎧ ⎨ −L + (M − L) tanh M−L , if ς < −L if |ς| ≤ L , σ(ς) = ς, " ς−L # ⎩ , if ς > L L + (M − L) tanh M−L where L takes the values of L 1 and L 2 whereas M takes the values M1 , M2 . Sharing functions m i (q) were designed similarly as described by [69, 92], i.e., they are composed of a constant unitary value, a constant zero value, a raising polynomial pr (x), and a falling polynomial p f (x) = 1 − pr (x). As the raising polynomial we used pr (x) = 35x 4 − 84x 5 + 70x 6 − 20x 7 , where x = h/qmi (see Sect. 12.4 in the

302

6 Switched Reluctance Motor

present book and [69, 92] for definitions of h and qmi and further details). This ensures that Assumption A1 is satisfied. The controller gains were set to K p = 10, K v = 0.55, α = diag{400, 400}, A = 200, B = 100, L 1 = 10.9, M1 = 10.95, L 2 = 40, M2 = 40.1, T ∗ = 0.1, α f = 0.1632, ω f = 27.85. It was verified that these controller gains satisfy all of the conditions established in the proof of Proposition 6.1. For this, we were required to use γ = 10−5 , τ L = −4 [Nm] and ω ∗ = 50 [rad/s]. All of the initial conditions were set to zero. The desired velocity was designed as follows. A ramp takes the desired value from 0 [rad/s] at t = 0 [s] to 50 [rad/s] at t = 0.15 [s]. It remains constant at 50 [rad/s] from t = 0.15 [s] to t = 0.4 [s]. Then, a ramp takes it from 50 [rad/s] at t = 0.4 [s] to −50 [rad/s] at t = 0.7 [s]. Finally, the desired velocity remains constant at −50 [rad/s] for t ≥ 0.7 [s]. See Fig. 6.10. The load torque was selected to be constant at τ L = −4 [Nm] from t = 1 [s] to t = 1.4 [s] and τ L = 0 elsewhere. This allows to study performance when step changes in the load torque are applied. We stress that the above-specified load torque is unknown for the controller and, despite this, its effect is successfully rejected by the integral part of the controller, i.e., K p σ1 (z), as observed in Fig. 6.10. We consider that this feature represents an important improvement with respect to results in [58, 168] where any information is not given on the load torque that they have considered. Moreover, a zero load torque due to friction is considered in [168]. On the other hand, although a nonzero constant (for all time) load torque is reported in simulations presented in [171], performance during the transient response cannot be evaluated from such a test. The simulation results obtained when using controller in Proposition 6.1 are presented in Figs. 6.10 and 6.11. We can see that the velocity error ω˜ tends to zero when the desired velocity becomes constant. The velocity error reaches important differences from zero values only when step changes in the load torque appear. However, such differences disappear very fast which proves the good disturbance rejection properties of controller in Proposition 6.1. We stress that a zero steady-state velocity error is achieved when the desired velocity is constant despite Proposition 6.1 only ensures an ultimate bound in velocity error. Reason for this is that the presence of friction and a load torque ensure that |τ ∗ | > T ∗ > 0 in steady state, which converts the global ultimate boundedness result into a global asymptotic stability result. See Remark 6.9. In Figs. 6.10 and 6.11, we also present the applied voltages and electric currents through all the three motor phases. Note that these variables have important changes when step changes in the load torque appear and when the desired velocity is described by ramps. We stress that the desired torque τ ∗ becomes zero at some point of time which is close to time when velocity is zero. This is because the only load torque applied before t = 1 [s] is that due to friction, and this load torque smoothly passes through zero when velocity changes direction. We realize that, despite this, any voltage (nor electric current) does not become large when velocity passes through zero. This proves that singularity referred in Remark 6.6 is not present in controller in Proposition 6.1.

6.3 Velocity Control

Fig. 6.10 Simulation results when controller in Proposition 6.1 is employed

303

304

6 Switched Reluctance Motor

Fig. 6.11 Simulation results when controller in Proposition 6.1 is employed (cont.)

6.3 Velocity Control

305

We observe that large negative voltage spikes appear all the time, specially when load torque is increased. These voltage spikes are naturally produced by the controller as a means to reduce to zero electric current through those stator phases which were being required to produce a nonzero torque but, then, they are suddenly required not to produce torque. This happens when sign of K i (q) changes for those phases, which would generate torque in the opposite direction if electric currents through such phases were not forced to be zero very fast. SRMs are provided with electronic drivers which are able to provide such negative voltage values as described in [1]. These voltage spikes are also present in [168], for instance. In this respect, in order to perform realistic simulations, we have saturated the voltages U1 , U2 , U3 , to the range ±200[V]. The reason for this is twofold. (1) The power amplifiers only deliver a restricted range of voltages in practice. (2) It is important to verify that performance will not deteriorate when limited voltages be applied. In this respect, after several simulations, we have observed that performance remains the same whether the applied voltages are saturated or not.

6.4 Position Control Recent works in the SRM control literature [57, 69, 168–171] only consider the case of velocity control, where the mechanical load is linear. In [92] was presented a novel passivity-based approach to regulate position in n-degrees-of-freedom rigid robots actuated by n direct-drive SRMs. Notice that in this control problem the mechanical load is nonlinear and highly coupled which represents the merit of [92]. In the following section we present the approach in [92] when applied to control a simple pendulum that is actuated by a single SRM.

6.4.1 Position Control Without Velocity Measurements The unsaturated dynamic model of a three-phase four-rotor poles SRM was described in Sect. 6.1.3. This is the model that is employed in the present section and it is rewritten here for the ease of reference: D(q) I˙ + K (q)I q˙ + R I = U, J q¨ + bq˙ = τ − g(q), 3  1 1 T K j (q)I j2 , τ = I K (q)I = 2 2 j=1 D(q) = diag{L 1 (q), L 2 (q), L 3 (q)},   2π , L j (q) = l0 − l1 cos Nr q − ( j − 1) 3

(6.66) (6.67) (6.68)

306

6 Switched Reluctance Motor

K (q) = diag{K 1 (q), K 2 (q), K 3 (q)},   ∂ L j (q) 2π = l1 Nr sin Nr q − ( j − 1) , K j (q) = ∂q 3 R = diag{r, r, r }.

(6.69)

The function g(q) represents a position dependent mechanical load which is assumed to possess the following properties: |g(x) − g(y)| ≤ k g |x − y|, ∀x, y ∈ R, (Lipschitz condition)    dg(q)  ,  k g > max  q∈R dq  dU (q) g(q) = , |g(q)| ≤ k  , ∀q ∈ R dq

(6.70) (6.71) (6.72)

with U (q) the potential energy and k g , k  , some positive constants. Assumption A2 Functions m +j and m −j are chosen such that m j → 0 as (q − q j0 )3 → 0 or faster were q j0 are the rotor positions such that K j (q j0 ) = 0. Proposition 6.11 Consider the unsaturated model (6.66)–(6.69), in closed loop with the following control law: U = R I ∗ − Ra I, I ∗ = [I1∗ , I2∗ , I3∗ ] , (6.73)    ∗ 2m j (q)τ 2m j (q) ∗ β1 + β2 K j (q)sign(τ ∗ ) f (|τ |), if K j (q) = 0 K j (q) I j∗ = , j = 1, 2, 3, 0, if K j (q) = 0 ˜ + g(q ∗ ) + K V σ2 (ϑ), z˙ = −Aσ2 (ϑ), ϑ = z − Bq, (6.74) τ ∗ = −K P σ1 (q) q˜ = q − q ∗ , R = Ra + R,

(6.75)

where q ∗ is the constant desired position, sign(τ ∗ ) = 1, when τ ∗ ≥ 0, and sign(τ ∗ ) = −1, when τ ∗ < 0. On the other hand, σ1 and σ2 are strictly increasing linear saturation functions (see Definition 2.34) for some large enough 0 < L 1 < M1 and (L 2 , M2 ) satisfying 0 < L 2 < M2 .

(6.76)

Further, we also require functions σ1 and σ2 to be continuously differentiable satisfying 1≥

∂σ(ς) > 0, ∀ ς ∈ R. ∂ς

(6.77)

6.4 Position Control

307

Functions m j (q) are defined in (6.23) and f (|τ ∗ |) = α f (1 − cos(ω f |τ ∗ |)), ∀|τ ∗ | ≤ T ∗ ,

(6.78)

where ω f is the smallest positive number satisfying 1 − cos(ω f T ∗ ) = 2T ∗ ω f sin(ω f T ∗ )

(6.79)

which can be solved numerically, α f > 0 is chosen as √

αf =

T∗ 1 − cos(ω f T ∗ )

(6.80)

and T ∗ is an arbitrarily small positive constant. √ According to Appendix E.1, this choice of parameters ensures that by replacing |τ ∗ | with f (|τ ∗ |), ∀|τ ∗ | ≤ T ∗ , we obtain a function which is continuous up to the first derivative. Finally β1 = 1 and β2 = 0, if |τ ∗ | > T ∗ . β1 = 0 and β2 = 1, if |τ ∗ | ≤ T ∗ There always exist constant positive scalars K V , K P , A, B, Ra , such that for any initial condition the whole state remains bounded and has an ultimate bound which can be rendered arbitrarily small by using a suitable choice of controller gains. In the following we present a sketch of the proof of Proposition 6.11 in order to highlight the rationale behind the proof and to illustrate how energy ideas are exploited. The reader interested in the complete proof is referred to [92] where the more general problem of controlling position in n-degrees-of-freedom rigid robots is solved.

6.4.1.1

Closed-Loop Dynamics

Replacing (6.73) in (6.66), using (6.75), defining ξ = I − I ∗ and adding and subtracting some convenient terms we find D(q)ξ˙ = −Rξ − K (q)ξ q˙ − K (q)I ∗ q˙ − D(q) I˙∗ .

(6.81)

Notice that, according to Assumption A3 on functions m j (q) and the definition of f (|τ ∗ |) in Proposition 6.11, respectively, the function I˙∗ is continuous. Adding and subtracting some convenient terms in (6.68) we find that τ = τc +

1 ∗ 1 I K (q)I ∗ , τc = ξ  K (q)ξ + ξ  K (q)I ∗ . 2 2

(6.82)

308

6 Switched Reluctance Motor

On the other hand, using (6.19) and the definition of I ∗ in Proposition 6.11, it is not difficult to verify that 1 ∗ I K (q)I ∗ = τ ∗ + β2 F(τ ∗ ), 2 F(τ ∗ ) = ( f (|τ ∗ |))2 sign(τ ∗ ) − τ ∗ ,

(6.83)

where function |β2 F(τ ∗ )| is bounded and this bound decreases to zero as T ∗ > 0 approaches to zero since, according to (6.78), f (|τ ∗ |) → 0 for all |τ ∗ | ≤ T ∗ if T ∗ → 0 and β2 = 0 if |τ ∗ | > T ∗ . Thus, according to (6.74) and (6.83), (6.82) can be written as ˜ + g(q ∗ ) + K V σ2 (ϑ) + β(τ ∗ ) τ = τc − K P σ1 (q) β(τ ∗ ) = β2 F(τ ∗ ). Thus, (6.67) can be written as ˜ + g(q ∗ ) − g(q) + K V σ2 (ϑ) + τc + β(τ ∗ ). (6.84) J q¨ + bq˙ = −K P σ1 (q) Notice that the closed-loop dynamics is given by (6.81), (6.84) and ˙ ϑ˙ = −Aσ2 (ϑ) − B q.

(6.85)

This closed-loop dynamics can be written as D(q)ξ˙ = −Rξ − K (q)ξ q˙ + U, 1 J q¨ + bq˙ = ξ T K (q)ξ + G, 2 ϑ˙ = −Aσ2 (ϑ) − B q, ˙ U = −K (q)I ∗ q˙ − D(q) I˙∗ ,

(6.86) (6.87) (6.88)

G = ξ T K (q)I ∗ − K P σ1 (q) ˜ + g(q ∗ ) − g(q) + K V σ2 (ϑ) + β(τ ∗ ). The state of the closed-loop dynamics is y = [q, ˜ q, ˙ ϑ, ξ  ] ∈ R6 . If I, q, U, g(q) are replaced by ξ, q, ˜ U, G, (6.86)–(6.88) is almost identical to the open-loop model (6.66)–(6.69). An important difference is that a larger resistance R is present in (6.86) instead of R in (6.66). Another important difference is the additional filter equation in (6.88) which is intended to introduced suitable damping without requiring velocity measurements, i.e., through the term K V σ2 (ϑ) in the definition of G. These observations motivate to propose the following scalar function for stability analysis purposes:

6.4 Position Control

309

1 V (y) = V1 (q, ˜ q, ˙ ϑ) + ξ  D(q)ξ, (6.89) 2  ϑ  q˜ KV 1 σ2 (r )dr + V1 (q, ˜ q, ˙ ϑ) = J q˙ 2 + γ J σ1 (q) ˜ q˙ + Ucl (q) ˜ + γbσ1 (r )dr, 2 B 0 0  q˜ Ucl (q) ˜ = [g(r + q ∗ ) + K P σ1 (r ) − g(q ∗ )]T dr, (6.90) 0

where γ ∗ > γ > 0 is a constant scalar. Notice that the “magnetic energy” stored in the electrical subsystem and the “kinetic energy” stored in the mechanical subsystem are considered through the terms 21 ξ  D(q)ξ and 21 J q˙ 2 , respectively. The cross term ˜ q˙ is included to obtain a negative quadratic term in the function σ1 (q) ˜ when γ J σ1 (q) computing V˙ . The first integral in the definition of V1 is included to take into account the filter dynamics whereas the second integral is useful to cancel the undesired cross ˜ q˙ arising in the time derivative of γ J σ1 (q) ˜ q. ˙ Finally, the function term −γbσ1 (q) ˜ represents a convenient definition of the “potential energy” in the mechanical Ucl (q) subsystem.

6.4.1.2

A Positive Definite Decrescent Function

Suppose that K P = K P∗ + K P , where K P∗ and K P are positive constants satisfying the following three conditions: (1)

G(q) ˜ ≥

γ ∗2 J H (q), ˜ ∀q˜ ∈ R, 2

(6.91)

where γ ∗ > 0 is a constant and

G(q) ˜ =

⎧ ⎪ ⎨ ⎪ ⎩

KP 2 q˜ , 2 KP 2 L1 + 2 KP 2 L1 − 2

(2)

|q| ˜ ≤ L1 K P L 1 (q˜ − L 1 ), q˜ > L 1 , K P L 1 (q˜ + L 1 ), q˜ < −L 1

K P∗ > k g , (3)

H (q) ˜ =

˜ ≤ M1 q˜ 2 , |q| . M12 , |q| ˜ > M1

2k  < L 1 < M1 . K P∗

(6.92)

In [92] it is shown that there exist some small enough constant c1 > 0 and some large enough constant c2 > 0 such that the scalar function defined in (6.89) satisfies

310

6 Switched Reluctance Motor

Fig. 6.12 Comparing functions in (6.91). λ M =

γ ∗2 2

J

α1 (y) ≤ V (y) ≤ α2 (y), ∀y ∈ R6 c1 y2 , y < 1 , ∀y ≥ 0, α1 (y) = c1 y, y ≥ 1

(6.93)

α2 (y) = c2 y2 , if K V and B are positive constants and (6.91), (6.92), are true. It is clear from Fig. 6.12 that there always exist a large enough K P and a small enough γ ∗ > 0 such that (6.91) is satisfied. We do not present the detailed procedure to obtain this result since it is rather involved. The reader is referred to [92] for detailed information on how to arrive at the above conclusions.

6.4.1.3

Stability Analysis

In [130], pp. 105–107, is proven that |g(q) − g(q ∗ )| ≤ khg σ1 (|q|), ˜ khg ≥

σ1

2k     , ∀q˜ ∈ R. 2k kg

Using this fact, (2.32), (6.77), Definition 2.34, A = maxi |λi (A)|, where λi (A) stands for eigenvalues of matrix A, and 1 ˙ = 0, qτ ˙ c + ξ  K (q)ξ q˙ + ξ  [−K (q)ξ q˙ − K (q)I ∗ q] 2

(6.94)

6.4 Position Control

311

it is straightforward to find that the time derivative of V , defined in (6.89), along the trajectories of the closed-loop dynamics (6.81), (6.84), (6.85), is bounded as KV A 2 σ (ϑ) − λmin (R)ξ2 ˜ − V˙ ≤ −bq˙ 2 − γ(K P − khg )σ12 (q) B 2 γ + γ J q˙ 2 + M1 max |λk (K (q))| ξ2 k 2 + γ|σ(q)| ˜ max |λk (K (q))| ξ I ∗  + γ K V |σ1 (q)| ˜ |σ2 (ϑ)| k

+ λmax (D(q))ξ  I˙∗  + |q| ˙ |β2 F(τ ∗ )| + γ|σ1 (q)| ˜ |β2 F(τ ∗ )|. Using (6.77), the fact that x ≤ x1 and the triangle inequality we can write  I˙∗  ≤  I˙∗ a +  I˙∗ b ,       & % 3  m (q)   ∂m (q) τ ∗   m (q)τ ∗ ∂ K (q)   K j (q) j j  j   ∗    j | q| ˙ + | τ ˙ | q| ˙ ,  I˙∗ a ≤ | +        K j (q)   K 2 (q) 2m j (q)τ ∗  ∂q K j (q)  ∂q  j j=1

 I˙∗ b ≤

3  j=1

+



 3  j=1

K j (q)sign(τ ∗ ) 2m j (q)

   & %  ∂m (q) 1   m (q) ∂ K (q)  j j   j   ˙ + 2 ˙ f (|τ ∗ |)   |q|  |q|  ∂q K j (q)   K (q) ∂q 

(6.95)

j

   ∂ f (|τ ∗ |)  ∗ 2m j (q)   |τ˙ |, K j (q)sign(τ ∗ )  ∂|τ ∗ | 

(6.96)

|τ˙ ∗ | ≤ K P |q| ˙ + K V (A|σ2 (ϑ)| + B|q|). ˙

According to Assumption A2 on the sharing functions m j (q) and the definition of ˙ all terms in (6.95) and (6.96) are bounded. f (|τ ∗ |) in Proposition 6.11, aside from |q| Hence, we conclude that there always exist finite positive constants d1 and d2 (which are independent of R and Ra ) such that we can write V˙ ≤ −ζ  Qζ + (1 + γ)|β2 F(τ ∗ )| ζ,

(6.97)

˜ |σ2 (ϑ)|, ξ] and the entries of matrix Q are given as where ζ = [|q|, ˙ |σ1 (q)|, Q 11 = b − γ J, Q 22 = γ(K P − khg ),

Q 14 = Q 41 Q 24 = Q 42

KV A , B

γ M1 max |λk (K (q))|, k 2 = Q 13 = Q 31 = 0, 1 = − λmax (D(q))d1 , 2 γ = − max |λk (K (q))| I ∗  M , 2 k

Q 44 = λmin (R) − Q 12 = Q 21

Q 33 =

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6 Switched Reluctance Motor

1 Q 34 = Q 43 = − λmax (D(q))d2 , 2 γ Q 23 = Q 32 = − K V . 2 Symbol  ·  M stands for the supreme value over the norm and, in this respect, notice that I ∗ is a bounded vector, i.e., Q 24 and Q 42 are bounded. Matrix Q is positive definite if and only if all of its leading principal minors are positive, i.e., b − γ J > 0, γ(K P − khg ) > 0,

(6.98) (6.99)

KV A Q 11 Q 22 − Q 23 Q 11 Q 32 > 0, (6.100) B d4 Q 44 − Q 14 Q 41 [Q 22 Q 33 − Q 23 Q 32 ] + Q 24 Q 11 [Q 32 Q 43 − Q 33 Q 42 ] − Q 34 Q 11 [Q 22 Q 43 − Q 23 Q 42 ] > 0. (6.101) d4 =

The expressions in (6.100) and (6.101) represent, respectively, the third and fourth leading principal minors of  third and fourth columns, respec Q solved through the   ∂ f (|τ ∗ |)  tively. Notice that  ∂|τ ∗ |  is bounded and so is τ1∗ as long as T ∗ > 0 (see Remark 6.13), i.e., according to (6.95)–(6.97), the entries Q 14 , Q 41 , Q 34 and Q 43 are bounded. Also notice that Q 44 is the only entry of Q depending on R. Hence, we only have to enlarge this value until the remaining terms in (6.101) be dominated. Thus, provided that T ∗ > 0 all of conditions (6.98)–(6.101) are always satisfied by some small γ > 0, a large enough positive constant scalar K P , some suitable positive numbers K V , A, B and a large enough positive value for λmin (R), i.e., a large positive value for λmin (Ra ). This ensures that matrix Q is positive definite. Hence, defining c3 = λmin (Q) > 0 and using some constant 0 < Θ < 1 we can write (6.97) as ˙ σ1 (q), ˜ σ2 (ϑ), ξ  ] , V˙ ≤ −c3 ζ ∗ 2 + (1 + γ)|β2 F(τ ∗ )| ζ ∗ , ζ ∗ = [q, ≤ −(1 − Θ)c3 ζ ∗ 2 − Θc3 ζ ∗ 2 + (1 + γ)|β2 F(τ ∗ )| ζ ∗ , 1+γ ≤ −(1 − Θ)c3 ζ ∗ 2 , ∀ζ ∗  ≥ |β2 F(τ ∗ )|. Θc3 According to paragraph after (6.83), |β2 F(τ ∗ )| decreases to zero as T ∗ > 0 |β2 F(τ ∗ )| can always be forced approaches to zero. Hence, the inequality ζ ∗  ≥ 1+γ Θc3 to be satisfied within the linear part of all of the functions composing ζ ∗ . Thus, it is always possible to write 1+γ |β2 F(τ ∗ )| > 0, V˙ ≤ −(1 − Θ)c3 v2 , ∀y ≥ Θc3 v = [q, ˙ σ1 (q), ˜ σ2 (ϑ), ξ  ] .

(6.102)

6.4 Position Control

6.4.1.4

313

Proof of Proposition 6.11

Based on (6.93) and (6.102) we can invoke Theorem 2.29 to conclude that for any initial state y(t0 ) ∈ R6 the closed-loop system state y satisfies y(t) ≤ δ(y(t0 ), t − t0 ), ∀t0 ≤ t ≤ t0 + T,    1+γ |β2 F(τ ∗ )| , ∀t ≥ t0 + T, y(t) ≤ α1−1 α2 Θc3

(6.103)

where δ(·, ·) is a KL function and T ≥ 0 depends on y(t0 ) and 1+γ |β2 F(τ ∗ )|. Recall Θc3 that |β2 F(τ ∗ )| decreases to zero as T ∗ > 0 approaches to zero. Since α1−1 (α2 (·)) is a K∞ function, this means that the ultimate bound in (6.103) approaches to zero as T ∗ > 0 approaches to zero. However, as remarked in the paragraph after (6.101), finite controller gains satisfying (6.98)–(6.101), i.e., to keep (6.102) valid, exist only if T ∗ > 0. Thus, we conclude that the closed-loop system has an ultimate bound which, although cannot be reduced to zero, can be rendered arbitrarily small by a suitable choice of controller gains. Moreover, this result stands globally. This completes the proof of Proposition 6.11. We stress that conditions for result in this proposition are summarized by (6.91), (6.92), (6.76), (6.98)–(6.101), and positive constants K V , B, γ, T ∗ . Remark 6.12 Instrumental for result in Proposition 6.11 is the cancellation of cross terms shown in (6.94). Notice that terms in qτ ˙ c belong to the time derivative of the “kinetic energy” of the mechanical subsystem whereas terms 21 ξ  K (q)ξ q˙ + ˙ belong to the time derivative of the “magnetic energy” ξ  [−K (q)ξ q˙ − K (q)I ∗ q] stored in the electrical subsystem. These cancellations are due to the natural energy exchange between the mechanical and the electrical subsystems. We stress that this property is a consequence of the fact that the closed-loop dynamics in (6.86)–(6.88) is almost identical to the open-loop dynamics (6.66)–(6.69) and the latter has been proven to be output strictly passive in Sect. 6.2.2. This cancellation of cross terms is very important since it involves several thirdorder terms which either would represent an obstacle to achieve global results if not cancelled or force the designer to include additional terms in the controller. For instance, in [69] they are forced to include some additional nonlinear terms, referred ˙ d in that paper (equivalent to K (q)I ∗ q˙ in the present section), and to include as C(θ)θi a velocity-dependent inner electric current loop.  2m j (q) ∗ Remark 6.13 As in Proposition 6.1, the term K j (q)sign(τ ∗ ) f (|τ |) in the definition √ of I ∗ is introduced to replace the function |τ ∗ | with f (|τ ∗ |) when |τ ∗ | ≤ T ∗ (see Fig. E.1 in Appendix E.1) and this is done in order to render possible the domination √ of I˙∗ in the stability analysis: recall that time derivative of |τ ∗ | depends on √|τ1 ∗ | which is unbounded when τ ∗ = 0. Although this solution does not ensure asymptotic stability, it is instrumental to present a formal stability analysis proving that for any initial condition the state has an ultimate bound which can be rendered arbitrarily

314

6 Switched Reluctance Motor

small. Furthermore, in this case also applies the observation in Remark 6.9, i.e., that global asymptotic stability is accomplished if the steady-state torque is not zero. Notice that the only obstacle to achieve global asymptotic stability using the desired current:  2m j (q)τ ∗ , if K j (q) = 0 ∗ ∗ ∗ ∗  ∗ K j (q) , j = 1, 2, 3, (6.104) I = [I1 , I2 , I3 ] , I j = 0, if K j (q) = 0 is the fact that the corresponding time derivative is not bounded when τ ∗ = 0 because (see the product of second term between brackets and factor at the left):  I˙∗  ≤

3  j=1



K j (q) 2m j (q)τ ∗

     & %  m (q)   m (q)τ ∗ ∂ K (q)   ∂m (q) τ ∗  j j  j   ∗   j  | q| ˙ + | τ ˙ | q| ˙ . | +        K j (q)   K 2 (q)  ∂q K j (q)  ∂q  j

Moreover, this feature is not taken into account by [69]2 since they use (6.104) as the desired current but they seem not to be aware of singularity present in the time derivative of the desired current when the desired torque is zero. As a matter of fact, they feedback the time derivative of the desired current and, despite this, they present some simulations where changes in the direction of rotation are commanded, i.e., this requires the desired torque to be zero at some isolated points of time. This observation explains the large voltage spike appearing at t = 0.25[s] in Fig. 8 reported by [69]. Remark 6.14 The rationale behind controller in Proposition 6.11 is the following. Since we are allowed to use a large positive definite matrix Ra such that Ra  R, we can assume that R ≈ Ra . Then, the controller in Proposition 6.11 can be represented as in Fig. 6.13. This block diagram shows that our proposal has the three main components of the standard control scheme for switched reluctance motors shown in Fig. 6.14: (i) a high-gain inner current loop, driven by a linear proportional controller (without the hysteresis controller), is used to force the electric current to reach the desired electric current, i.e., to achieve I ≈ I ∗ , (ii) since the desired current generator satisfies (6.83) and |β2 F(τ ∗ )| can be rendered as small as desired, the generated torque approximately reaches the desired torque, i.e., τ ≈ τ ∗ (recall (6.29) and I ≈ I ∗ ), (iii) hence, the desired torque τ ∗ , defined in (6.74) as an output feedback PD position controller (instead of a linear PID controller), is designed to control the motor alone. Finally, it is important to say that the rise and fall angles required in the standard position control of SRM are not required because of the definition of I j∗ in Proposition 6.11. A drawback of controller in Proposition 6.11 is that it relies on the viscous friction that the mechanical subsystem naturally possesses. 2 In Chap. 10, in the present book, is shown that magnetic levitation systems represent the one phase and one pole case of SRMs. It is interesting to recall that in [204], Chap. 8, Remark 8.5, it is pointed out that when the passivity-based approach in [204] (i.e., where it is introduced the fundamental theory for the approach in [69]) is applied to a magnetic levitation system, a singularity exists when the desired force to be exerted by the electromagnet crosses through zero.

6.4 Position Control

315

Fig. 6.13 Block diagram of controller in Proposition 6.11. The desired current generator block refers to definition of I ∗j in Proposition 6.11

Fig. 6.14 Standard position control of a SRM

Remark 6.15 The closed-loop mechanical subsystem dynamics given in (6.87) can be written as ˜ + g(q ∗ ) − g(q) + K V σ2 (ϑ), J q¨ + bq˙ = τe − K P σ1 (q)

(6.105)

where τe =

1  ξ K (q)ξ + ξ  K (q)I ∗ + β(τ ∗ ). 2

(6.106)

Notice that τe represents the difference between the actual and the desired torques. Hence, taking into account (6.77) it is possible to write ⎤⎡ ⎤ b − γJ 0 0 q˙ ˙ σ1 (q), ˜ σ2 (ϑ) ] ⎣ 0 ˜ ⎦ + (ω˜ + γσ1 (q))τ γ(K p − khg ) − γ K2 v ⎦ ⎣ σ1 (q) V˙1 ≤ −[ q, ˜ e, γ Kv Kv A (ϑ) σ 0 − 2 2 B ⎡

where the function V1 is defined in (6.89). On the other hand, the time derivative of the last component of V in (6.89), i.e., Ve =

1  ξ D(q)ξ, 2

along the trajectories of the closed-loop electrical subsystem dynamics (6.86), con¯ Thus, when computing V˙ = V˙1 + V˙e , we tains the quadratic negative term −ξ  Rξ. realize that instrumental for the stability result in Proposition 6.11 are the following features, which are very similar to those in Remark 3.4: • The scalar function V1 is a strict Lyapunov function when the desired torque is the input of the closed-loop mechanical subsystem dynamics, given in (6.105), i.e., when τe = 0.

316

6 Switched Reluctance Motor

¯ appearing in V˙ can be enlarged arbi• Coefficients of the negative term −ξ  Rξ, trarily. This is important to dominate the cross terms in V˙ depending on ξ when ξ = 0. • The cancellation of several cross terms belonging to V˙1 and V˙e which was explained in Remark 6.12. Notice that all of these features are possible thanks to the passivity properties of the open-loop motor model described in Sect. 6.2.2. Furthermore, the above proof contains the main ideas of procedure described in Sect. 2.4. As pointed out in Remark 3.4, τe is given as a nonlinear function of the electrical dynamics error for electric motors different from PM brushed DC-motors (see (6.106)). In the case of switched reluctance motors we have resorted to the introduction of a saturated desired torque, defined in (6.74), which renders I ∗ a saturated function. This is important to dominate some higher order cross terms in the expression for ˜  K (q)ξ and γσ1 (q)ξ ˜  K (q)I ∗ , V˙ (see the expression after (6.94)), such as 21 γσ1 (q)ξ which is instrumental to globally dominate them and to achieve the global stability results presented in Proposition 6.11. Notice that the above-cited higher order cross terms arise in the stability analysis as a consequence of the nonlinear nature of the motor model.

6.4.2 Position Control Taking into Account Magnetic Saturation The previous works [57, 69, 92, 168–171] only consider the case when magnetic saturation does not exist. Moreover, although [69] studies the magnetic saturation case, any explicit controller is presented for such a case but only for the unsaturated case. Furthermore, the simulation study that they present is only for the unsaturated model of SRMs. Contrary to those works, in [96] is presented a velocity controller, which takes into account the magnetic saturation and semiglobal stability results are proven. In this section we present a position regulator which takes into account the magnetic flux saturation that is common under normal operation conditions of SRM. The following assumption is taken into consideration in the present section to ensure that I ∗ and I˙∗ are continuous and bounded [92].

Assumption A3 Functions m +j (q) and m −j (q) are chosen such that m j (q) → 0 as fast as (q − d L (q)

q j0 )ρ → 0, were q j0 is the rotor position such that dqj |q=q j0 = 0 and ρ > 0 is some integer which depends on the particular form of L j (q). This assumption ensures continuity and boundedness of both I j∗ and I˙j∗ as q → q j0 [92]. In particular, this assumption ensures that whenever the quotient of two

6.4 Position Control

317

functions depending on q − q j0 appears, it is the function at the numerator who tends to zero faster. See Sect. 12.4 for a numerical example. In the present section we will consider the SRM saturated dynamical model introduced in Sect. 6.1.4 which is rewritten here for the ease of reference: D(q, I ) I˙ + C(q, I )I q˙ + R I = U, J q¨ + bq˙ = τ − g(q), 3  d L j (q) ψs ln(1 + β 2 L 2j (q)I j2 ), τ= 2 dq 2β L (q) j j=1

(6.107) (6.108) (6.109)

where D(q, I ) and C(q, I ) are diagonal matrices defined in Sect. 6.1.4, whereas the function g(q) represents a position dependent mechanical load which is assumed to possess the properties in (6.70)–(6.72). Proposition 6.16 Consider the SRM saturated model in (6.107)–(6.109), in closedloop with the following controller:   U = −αξ + −N − K q q˙ 2 − K f |q| ˙ − K d ξ1 eσ SIGN(ξ) + R I ∗ − k1 |q|ξ ˙ + C(q, I )I ∗ q, ˙  ' d L j (q) β1 j ζ j + β2 j f j (ζ j ), if dq = 0 I j∗ = , d L (q) 0, if dqj = 0 ( ( d L(q) ( eσ j − 1 ψs β ( ( ( , , k1 > ζj = 2 2 2 ( dq ( β L (q)

(6.110) (6.111) (6.112)

M

j

τ ∗ = −k p h(q) ˜ − kd q˙ − ki sat (z),      t  γk p εγkd ε 1+ h(q) ˜ + 1+ q˙ dt, z= ki ki 0 2β L 2j (q)m j (q)τ ∗ σj = , eσ = diag{eσ1 , eσ2 , eσ3 }, d L (q) ψs dqj

(6.113) (6.114)

where j = 1, 2, 3, I ∗ = [I1∗ , I2∗ , I3∗ ] , L(q) = diag{L 1 (q), L 2 (q), L 3 (q)}, with L j (q), C(q, I ), ψs , β, defined in Sect. 6.1.4, q˜ = q − q ∗ , with q ∗ the constant desired position, ξ = I − I ∗ , τ ∗ represents the desired torque, the symbol  ·  M stands for the supreme value over the norm and ξ1 = |ξ1 | + |ξ2 | + |ξ3 |. We define the vectorial function SIGN(ξ) = [sign(ξ1 ), sign(ξ2 ), sign(ξ3 )]T , where sign(x) = +1, if x ≥ 0, and sign(x) = −1, if x < 0. h(q) ˜ = s(q), ˜ and sat (z) = s(z), where s(·) is a strictly increasing linear saturation function for some M ∗ > L ∗ (see Definition 2.34). Furthermore, it is also required that the function s(·) be continuously differentiable such that 0
T j∗ , β1 j = 0 and β2 j = 1, if ζ j ≤ T j∗ , where we assign: T j∗ = ζ j , when |τ ∗ | = Td∗ and

d|τ ∗ | < 0, dt

with Td∗ an arbitrarily small positive constant. At t = 0 assign T j∗ = Td∗ . There always exist positive scalars k p , kd , ki , ε, γ, N , α, K q , K f , K d , such that the whole state (which includes the position error q) ˜ remains bounded and it has an ultimate bound which can be rendered arbitrarily small by using a suitable choice of controller gains. This result stands when starting from any initial condition, i.e., globally. Remark 6.17 Notice that I j∗ is well-posed since m j (q)τ ∗ /

d L j (q) dq

is always positive d L j (q) dq

and τ ∗ have or zero, and hence ζ j ≥ 0, because a m j (q) > 0 is used only if the same sign (see (6.20) and (6.23)). See Remark 6.19 for an interpretation of I j∗ . Furthermore, according to Assumption A3, m j (q) → 0 faster than

d L j (q) dq

→ 0 [92].

The control scheme in Proposition 6.16 is depicted in Fig. 6.15 and it contains the following main components: (1) A nonlinear proportional-integral-derivative (NPID) external loop is intended to regulate position. This part of the controller improves the passivity properties of the mechanical subsystem without relying on friction that the mechanical part naturally possesses, i.e., resulting in a Lyapunov function time derivative that has negative quadratic terms in joint velocities and positions as well as in the integral ˙ , that are present in (6.39), and terms of the controller. In fact, the terms −bq˙ 2 + qτ previous results in robot control [93] represent the evidence that such a solution is possible.

6.4 Position Control

319

Fig. 6.15 Control scheme in Proposition 6.16. Υ is defined in (6.111) and Ψ is defined in (6.109). Recall that ω = q˙

(2) An inner electric current loop is driven by both a sign controller and a proportional controller; this internal loop is intended to force the motor to generate the desired torque defined as the output of the NPID position controller.3 Furthermore, the proportional parts of these controllers are intended to improve the passivity properties of electrical subsystem as suggested in Sect. 6.2.2. On the other hand, the electrical dynamics must be expressed in terms of the electric current error, a necessary step to force the generated torque to reach the desired torque. This requires to complete the electric current error dynamics with the time derivative of the desired electric current. This step is performed in [57, 69, 168–171] by online computing and feeding back these large quantity of terms, which results in performance deterioration because of numerical errors and noise amplification [204]. Instead of doing that, in Proposition 6.16 we employ the sign controller to dominate the terms composing the time derivative of the desired electric current. (3) Contrary to what happens in unsaturated SRM models, the terms qτ ˙ − 1  d L(q) I I q ˙ in (6.39) do not cancel in the saturated SRM model. This represents 2 dq one of the difficulties when considering magnetic saturation. However, the term sim˙ plifications shown in (6.39) are useful in this case. In this respect, the terms −k1 |q|ξ and C(q, I )I ∗ q˙ that appear in (6.110) are intended to dominate an equivalent term I q. ˙ Finally, the term equivalent to qτ ˙ will be dominated using some to − 21 I  d L(q) dq negative sign defined terms depending on the controller gains. The complete proof of Proposition 6.16 is presented in Appendix E.3. In the following we present just a sketch to render clear how the energy ideas are exploited. Sketch of proof of Proposition 6.16. The closed-loop dynamics is first obtained which can be written as

3 Hysteresis and proportional controllers are commonly used in practice [64] for SRM. We stress that a sign controller can be seen as the limit case of a hysteresis controller when the width of the hysteresis window is zero.

320

6 Switched Reluctance Motor

D(q, I )ξ˙ = −C(q, I )ξ q˙ − (r + α)ξ + U,

(6.118)

J q¨ = −(b + kd )q˙ + Φ − G, (6.119) ∗ ˙ U = −k1 |q|ξ ˙ − D(q, I ) I   + −N − K q q˙ 2 − K f |q| ˙ − K d ξ1 eσ SIGN(ξ), ⎡ ⎤ 3  ˜ + g(q) − g(q ∗ ) + ki x(z) − τ ∗ ⎣ β1 j m j (q) − 1⎦ − φ, G = k p h(q) j=1

x(z) = sat (z) +

1 g(q ∗ ), ki

(6.120)

where I˙∗ is defined in (E.11), the torque error Φ is defined in (E.13) and φ is defined in (E.12). Notice that (6.118) and (6.119) have the same structure as the open-loop model (6.107) and (6.108), if we replace I, U, g(q) by ξ, U, G. An important difference is that friction and electrical resistance have been increased in (6.118) and (6.119). Thus, these equations possess the same passivity properties that were described in Sect. 6.2.2 for the saturated model. We will take advantage from these properties in the present proof. This is the main reason to propose the following scalar function as Lyapunov function candidate for stability analysis purposes: V (q, ˜ q, ˙ z + g(q ∗ )/ki , ξ  ) = Vq (q, ˜ q, ˙ z + g(q ∗ )/ki ) = ˙ q) ˜ = V1 (q, P(q) ˜ = V2 (q, ˙ z + g(q ∗ )/ki ) =

1  ξ L(q)ξ + Vq (q, ˜ q, ˙ z + g(q ∗ )/ki ), (6.121) 2 V1 (q, ˙ q) ˜ + P(q) ˜ + V2 (q, ˙ z + g(q ∗ )/ki ),  q˜ 1 2 J q˙ + εJ h(q) ˜ q˙ + ε(b + kd ) h(r )dr, 4 0  q˜ ∗ kp h(r )dr + U (q) − U (q ∗ ) − qg(q ˜ ), 0  z 1 2 J q˙ + εγ J x(z)q˙ + ki x(r )dr, 4 −g(q ∗ )/ki

where ε and γ are some positive constants. As explained in Sect. 6.2.2, the function 1  ξ L(q)ξ is employed instead of the “magnetic energy” stored in the electrical 2 ˜ q, ˙ z + g(q ∗ )/ki ) is composed of three terms which subsystem. The function Vq (q, constitute the “potential energy” stored in the mechanical subsystem, P(q), ˜ and two cross functions of the kinetic energy and the “potential energy” stored in the ˙ q), ˜ and the kinetic energy and the “energy” stored in mechanical subsystem, V1 (q, ˙ z + g(q ∗ )/ki ). the integral term of the controller, V2 (q, ∗ This choice for V (q, ˜ q, ˙ z + g(q )/ki , ξ T ) allows to obtain a time derivative of V that can be upper bounded as

6.4 Position Control

321

dh(q) ˜ 2 V˙ ≤ −(b + kd )q˙ 2 + εJ q˙ − εh(q)(g(q) ˜ − g(q ∗ )) d q˜ − εk p h 2 (q) ˜ + εγ J [(ε + εγk p /ki )h(q) ˜ + (1 + εγkd /ki )q] ˙

(6.122) d x(z) q˙ dz

− εγx(z)(g(q) − g(q ∗ )) − εγki x 2 (z) − εγbx(z)q, ˙ ⎛ ⎤ ⎞ ⎡ 3  ∗ [εh(q) ˜ + q˙ + εγx(z)] ⎝Φ + τ ⎣ β1 j m j (q) − 1⎦ + φ⎠ -

j=1

-

.

.

˙ − K d ξ1 eσ SIGN(ξ) , + ξ T Δ −(r + α)ξ − D(q, I ) I˙∗ + −N − K q q˙ 2 − K f |q|

thanks to several straightforward cross term cancellations (see Remark 6.18) including   1  d ξ L(q) ξ − ξ  L(q)D −1 (q, I )[k1 |q|ξ ˙ + C(q, I )ξ q] ˙ ≤ 0, 2 dt which is due to the energy exchange in the system. Notice that V˙ has negative quadratic terms on all of the state variables. This is one important reason to define V (q, ˜ q, ˙ z + g(q ∗ )/ki , ξ  ) as above. The function V (q, ˜ q, ˙ z + g(q ∗ )/ki , ξ  ) is also shown to be positive definite, radially unbounded and decrescent. On the other hand, I˙∗ is ensured to be bounded thanks to the introduction of the functions f j (ζ j ) in Proposition 6.16 and Assumption A3 before Proposition 6.16. Hence, it is shown that a suitable selection of the gains N , K q , K f , K d together with the properties of the sign function ensure that     ˙ − K d ξ1 eσ SIGN(ξ) < 0, ξ  Δ −D(q, I ) I˙∗ + −N − K q q˙ 2 − K f |q| ∀y ∈ R6 and, thus V˙ ≤ −(1 − Θ)λm (Q) y¯ 2 , ∀ y ≥ μ0 =

φ¯ 1 , Θλm (Q)

(6.123)

where the entries of matrix Q are defined in (E.27) whereas φ¯ 1 , which is defined in (E.28), can be rendered arbitrarily small by choosing a Td∗ > 0 arbitrarily small. The proof is completed by taking into consideration the decrescent properties of V (q, ˜ q, ˙ z + g(q ∗ )/ki , ξ  ), (6.123), and invoking Theorem 2.29. The conditions to guarantee Proposition 6.16 are summarized by (A.8)–(A.10), (A.14),4 (E.22)–(E.25), the four leading principal minors of matrix Q defined in (E.27) are positive and some small constant Td∗ > 0. Remark 6.18 Notice that the term simplifications in (6.39) are very useful to solve this control problem. This is due to the fact that the closed-loop system (6.118), (6.119) have the same structure as the open-loop model (6.107), (6.108), as explained in the paragraph after (6.120). 4 Notice

that k p , kd , ki , L ∗ , must be employed instead of k p , kd , ki , L.

322

6 Switched Reluctance Motor

Fig. ' 6.16 Approximation of ζ j with f j (ζ j ) for all ζ j ≤ T j∗

Remark 6.19 Suppose that β1 j = 1 and β2 j = 0, j = 1, 2, 3, in I j∗ defined in (6.111). It is not difficult to verify from this, (6.112), and the definition of σ j after (6.114), that assuming I j = I j∗ , j = 1, 2, 3, in (6.109), yields τ = m 1 (q)τ ∗ + m 2 (q)τ ∗ + m 3 (q)τ ∗ = τ ∗ , by virtue of 3j=1 m j (q) = 1. Thus, the desired current I j∗ is defined as in (6.111) in order to ensure that the desired torque τ ∗ is generated if the motor electric current I j reaches its desire value I j∗ . This means that the definition of I j∗ given in (6.111) is determined by the structure of the motor model. The functions f j (ζ j ), in the definition of I j∗ given in (6.111), are introduced to ' replace the functions ζ j when 0 ≤ ζ j ≤ T j∗ (see Fig. 6.16) and this is done in order ' to avoid I˙j∗ to become unbounded: recall that the time derivative of ζ j depends on √1 which is unbounded when τ ∗ = 0, i.e., when ζ j = 0. Moreover, this substitution ζj

is continuous up to the first derivative. This is ensured because the coefficients given in (6.117) for the polynomial defined in (6.116) are computed such that f j (0) = 0, √     d ζj  d f j (ζ j )  ∗ ∗ d f j (ζ j )  = 0, f j (T j ) = T j , dζ j  = dζ j  . Since I˙j∗ is ensured to dζ j ζ =0 ζ j =T ∗ ∗ j j

ζ j =T j

be bounded and continuous it is possible to'dominate it in the above stability analysis. According to the above discussion, if ζ j is not replaced in some manner in the definition of I j∗ then a singularity appears which is intrinsic to the SRM model. Furthermore, it appears whenever the desired electric currents are defined as a means to generate a desired torque by forcing the actual motor phase electric currents to reach their desired values. This problem appears in the passivity-based approaches presented in [57, 69, 168–171] where the time derivative of the desired electric current must be cancelled to be able to exploit the system passivity properties to achieve exponential stability, i.e., the fundamental result in those papers. This singularity has been pointed out in [92, 269]. Moreover, in Chap. 10, in the present book, is shown that magnetic levitation systems represent the one phase and one pole case of SRMs. It is interesting to recall that in [204], Chap. 8, Remark 8.5, it is pointed out that when the passivity-based approach in [204] (i.e., where it is introduced the

6.4 Position Control

323

fundamental theory for the approach in [57, 69, 168–171]) is applied to a magnetic levitation system, a singularity exists when the desired force to be exerted by the electromagnet crosses through zero. Remark 6.20 One difference with respect to [92] is that Proposition 6.16 includes an external loop driven by a PID controller intended to regulate the robot position. This allows to design the controller without depending on the friction that the mechanical part naturally possesses. It is important to stress that the simplicity and the robustness properties of a nonlinear PID position controller must not be underestimated. Moreover, in order to design a PID position controller without relying on the natural friction that is present in the mechanical subsystem it is important to allow velocity measurements.5 Otherwise, performance is far from satisfactory (i.e., try to simulate the controllers in [98, 172, 203, 252, 253], for instance). This problem seems to be the reason why the passivity-based approaches in [57, 69, 168–171] are devoted to velocity control and the PID control of position problem has not been studied: velocity measurements are avoided in those works and high-pass position filtering is employed instead. Although those authors argue that position filtering is intended to reduce the effects of noise that is present in velocity measurements, it is worth to say such a solution is also intended to reduce the complexity of the resulting desired electric current time derivative that they have to compute online. Thus, it is important to stress that we allow velocity measurements and, despite this, we avoid the computation of the desired electric current time derivative. We also consider both proportional and sign controllers6 for electric current. In this respect, we point out that both, hysteresis control and high-gain proportional control of electric current, are usually employed for SRM control in practice [64]. Hence, we present a formal proof explaining why these electric current controllers work well together in practice. Remark 6.21 The closed-loop mechanical subsystem dynamics given in (6.119) can be written as ˜ − g(q) + g(q ∗ ) − ki x(z) J q¨ = −(b + kd )q˙ + τe − k p h(q) ⎡ ⎤ 3  + τ∗ ⎣ β1 j m j (q) − 1⎦ + φ,

(6.124)

j=1

x(z) = sat (z) +

5 We

1 g(q ∗ ), ki

stress that even [92] and the result in Proposition 6.11 rely on the friction that is naturally present in the mechanical subsystem. 6 A sign controller can be seen as a hysteresis controller with a zero hysteresis window. In Chap. 12 is presented an extended version of controller in Proposition 6.16 to the case of PID Position regulation in n-degrees-of-freedom robots actuated by n direct-drive SRMs. There, hysteresis electric current controllers are employed instead of sign controllers but semiglobal stability results are obtained instead.

324

6 Switched Reluctance Motor

if we define τe = Φ. Notice that τe represents the difference between the actual ˜ q, ˙ z + g(q ∗ )/ki ), and the desired torques. The time derivative of the function Vq (q, defined in (6.121), along the trajectories of the mechanical subsystem dynamics (6.124), is given as dh(q) ˜ 2 q˙ − εh(q)(g(q) ˜ − g(q ∗ )) V˙q ≤ −(b + kd )q˙ 2 + εJ d q˜ − εk p h 2 (q) ˜ + εγ J [(ε + εγk p /ki )h(q) ˜ + (1 + εγkd /ki )q] ˙

d x(z) q˙ dz

− εγx(z)(g(q) − g(q ∗ )) − εγki x 2 (z) − εγbx(z)q, ˙ ⎛ ⎤ ⎞ ⎡ 3  [εh(q) ˜ + q˙ + εγx(z)] ⎝Φ + τ ∗ ⎣ β1 j m j (q) − 1⎦ + φ⎠ . j=1

This can be written as ⎡

⎤ |q| ˙ ˜ ⎦ V˙q ≤ −[|q|, ˙ |h(q)|, ˜ |x(z)|]Q  ⎣ |h(q)| |x(z)| ⎛ ⎡ + [q˙ + εh(q) ˜ + εγx(z)] ⎝τe + τ ∗ ⎣

(6.125) 3 

⎤

⎞

β1 j m j (q) − 1⎦ + φ⎠ ,

j=1

⎡

⎤ Q 11 Q 12 Q 13 Q  = ⎣ Q 21 Q 22 Q 23 ⎦ . Q 31 Q 32 Q 33

The entries of matrix Q  are identical to the entries in the first three rows and columns of matrix Q defined in (E.27). The matrix Q  is positive definite if and only if: ⎛

Q 11 > 0,

Q 11 Q 22 − Q 12 Q 21

⎞ Q 11 Q 12 Q 13 > 0, det ⎝ Q 21 Q 22 Q 23 ⎠ > 0, (6.126) Q 31 Q 32 Q 33

which ensures λmin (Q  ) > 0. On the other hand, the time derivative of the first component of V , defined in (6.121), i.e., Ve =

1  ξ L(q)ξ, 2

along the trajectories of the closed-loop electrical subsystem dynamics (6.118) contains the negative term −(r + α)ξ  Δξ. Thus, when computing V˙ = V˙q + V˙e to obtain (6.122), we realize that instrumental for the stability result in Proposition 6.16 are the following features, which are similar to those in Remark 3.6:

6.4 Position Control

325

• The scalar function Vq is a strict Lyapunov function when the desired torque is the input of the closed-loop mechanical subsystem dynamics, given in (6.124), i.e., when τe = 0, 3j=1 β1 j m j (q) = 1, and the disturbance function φ = 0. See (6.126) and (6.125). • The coefficient of the negative term −(r + α)ξ  Δξ in V˙ can be enlarged arbitrarily. This is important to dominate cross terms in V˙ , given in (6.122), when ξ = 0. • The cancellation of cross terms referred after (6.122). • The domination of terms composing I˙∗ using the sign function in the electric current controller. Notice that the first three of these features are possible thanks to the passivity properties inherited from the open-loop motor model described in Sect. 6.2.2. In this respect, the above proof contains the main ideas of procedure described in Sect. 2.4. As pointed out in Remark 3.6, τe is given as a nonlinear function of the electrical dynamics error for SRM (see (E.13)). As it has been shown above, we have resorted to the domination of cross terms existing between the electrical and the mechanical subsystem dynamics to cope with the nonlinear function defining τe .

6.4.2.1

Simulation Results

In this section we present a simulation example intended to give some insight on the performance that is achievable with controller in Proposition 6.16. For this, we use the parameters of the SRM that was described in Sect. 6.3.1. We have assumed, additionally, that a simple pendulum is fixed to the motor shaft such that, see (6.108), g(q) = mgl sin(q) where m = 0.8[kg], l = 0.25[m], and g = 9.81[m/s2 ], whereas inertia of the mechanism is given as J = 0.001[kg m2 ] + ml 2 + 13 m(2l)2 , where 0.001[kg m2 ] is the rotor inertia. Moreover, since the SRM model is assumed to have saturated magnetic fluxes we consider that ψs = 0.5 and β = 1.8, parameters that have been identified experimentally in [269]. Inspired by [286] we choose the following saturation function for s(·): " ς+L # ⎧ ⎨ −L + (M − L) tanh M−L , if ς < −L if |ς| ≤ L , s(ς) = ς, " ς−L # ⎩ , if ς > L L + (M − L) tanh M−L where L = 2.9 and M = 3. Sharing functions m i (q) were designed similarly as described by [69, 92], i.e., they are composed of a constant unitary value, a constant zero value, a raising polynomial pr (x), and a falling polynomial p f (x) = 1 − pr (x). As the raising polynomial we used pr (x) = 35x 4 − 84x 5 + 70x 6 − 20x 7 , where x = h/qmi (see [69, 92] for definitions of h and qmi and further details). This ensures that Assumption A3 is satisfied. See Sect. 12.4 for further details.

326

6 Switched Reluctance Motor

The controller gains were set to k p = 29, kd = 13.3, ki = 30, k1 = 10, α = 50, Td∗ = 0.1, N = 10, K q = 10, K f = 10, K D = 10, ε = γ = 9. It was verified that these controller gains satisfy all of the conditions established in the proof of Proposition 6.16, excepting (E.22)–(E.25). All of the latter conditions, however, are satisfied using very large values for N , K q , K f , K D (in the order of 106 ). Since such values are not realistic we have decided to employ the above-listed values for these constants. Despite this, our approach must not be underestimated. In this respect, the passivity-based approaches in [57, 69, 168–171] would be obliged, instead, to compute online a very large number of terms whose numerical values are in the order of 4 × 106 (four times 106 because of the four constants N , K q , K f , K D ) since they have to exactly compute (the time derivative of the desired electric current) what our approach is required just to dominate. On the other hand, formal studies abound which establish stability conditions that result in very large controller gains which prohibits their use in successful simulations or experiments. See, for instance, the formal works on robot control [48, 127]. All of the initial conditions were set to zero. The desired position was designed as follows. A ramp with slope 15[rad/s] takes the desired value from 0[rad/s] at t = 0[s] until q ∗ = π2 [rad/s] is reached and remains constant at this value. See Fig. 6.17. This desired position was designed instead of using a step command in order to maintain the applied voltages and the resulting electric currents within the admissible ranges of this motor, i.e., ±200[V] and about 10[A]. We can see in Fig. 6.17 that the position reaches the desired value in steady

Fig. 6.17 Simulation results when controller in Proposition 6.16 is employed

6.4 Position Control

327

Fig. 6.18 Simulation results when controller in Proposition 6.16 is employed (cont.)

state, the rise time is about 2[s] and the transient response is very damped, i.e., without overshoot. We consider that these features represent a good response for the mechanism at hand. Moreover, convergence to the desired position is achieved despite Proposition 6.16 only ensures the ultimate boundedness of the state error. This is because, in steady state at q = q ∗ = π2 [rad] the desired torque is far from zero, i.e., it is larger than Td∗ and, thus, asymptotic stability is obtained in such a case. See Remark 6.9. In Fig. 6.17 we can also see the three-phase electric currents. Notice that these variables remain below 9[A] and the peak currents increase as the maximum torque position, i.e., q = π2 [rad] is approached. In Fig. 6.18 we present voltages applied at each phase. Notice that these voltages never exceed ±100[V]. Moreover, we can observe the switching nature of these voltages due to the sign function that is present

328

6 Switched Reluctance Motor

in the electric current controller. Notice that this switching behavior remains at a minimum in steady state. It is good news that the switching behavior only appears in the transient response but not in the steady-state response.

6.5 A Practical Switched Reluctance Motor This motor is commonly employed in washing machines. In Fig. 6.19 are presented two pictures of the stator windings of this motor. In Fig. 6.20 is presented a drawing showing the relative positions of the stator phase windings and the rotor saliencies or poles. Recall that rotor in a switched reluctance motor (SRM) has neither permanent magnets or windings. It is composed of a simple piece of iron having a number of saliencies which are called, perhaps incorrectly, poles by people. The motor under ◦ = 45◦ = study has eight-rotor saliencies or poles (see Fig. 6.21), i.e., an angle of 360 8 π [rad] exists between adjacent rotor saliencies or poles. 4 Also, the motor under study has three phases on stator. Each phase is composed of two parallel circuits, each one of them having two series connected windings. Hence, each motor phase is composed of four windings, i.e., there are 12 windings on stator. ◦ = 30◦ = π6 [rad] exists between adjacent windings. The stator Thus, an angle of 360 12 windings locations are indicated in Fig. 6.20 by their corresponding magnetic cores on stator. Also in Fig. 6.20, two windings are labeled with a N and two are labeled with an S in each phase, i.e., 1N, 1S, 2N, 2S, 3N, 3S. We have adopted this convention in order to identify the magnetic polarity at each stator winding. This polarity has been identified by observing the sense each winding is wound. In phase 1, one winding labeled with 1N is series connected to a winding labeled with 1S. The other 1N and 1S windings in phase 1 are series connected in the other parallel circuit composing phase 1. The same situation applies for phases 2 and 3. Each stator winding is wound on a strip-shaped core which lies on stator parallel to the rotor axis (see Fig. 6.19). The symbols I1 , I2 , I3 represent the electric current flowing through phases 1, 2, 3, respectively, when assumed to be positive. Hence, a current I1a = I1b = 21 I1 flows through each winding in phase 1. A similar situation is valid for phases 2 and 3. In the following we will obtain the mathematical model of this motor using a procedure described in [55].

6.5.1 Magnetic Field at the Air Gap In order to compute the magnetic field produced at the air gap by the stator phase winding 1, consider Fig. 6.22. There, the symbol  means that electric current through phase 1 is coming out of the page, whereas the symbol ⊗ means that electric current is going into the page. These symbols are employed such that they are consistent

6.5 A Practical Switched Reluctance Motor

329

Fig. 6.19 Two views of the stator windings

with the magnetic polarities shown in Fig. 6.20. Let us apply Ampère’s Law (2.33) to the oriented closed trajectory 1-2-3-4-1 shown in Fig. 6.22:  H1 · dl = i 1enclosed ,  2  = H1 · dl + 1

3 2

 H1 · dl +

= g0 (q)H1 (0) − gγ (q)H1 (γ), where

4 3

 H1 · dl +

1

H1 · dl,

4

(6.127)

330

6 Switched Reluctance Motor

Fig. 6.20 Relative positions of the stator phase windings and the rotor saliencies or poles



2

1



 H1 · dl = 1

2

 H1 (0)ˆr · (dl rˆ ) = 1

2



2

H1 (0)dl =

H1 (0)dr = g0 (q)H1 (0),

1

3

H1 · dl = 0,  4  4  4  4 H1 · dl = H1 (γ)ˆr · (−dl rˆ ) = − H1 (γ)dl = H1 (γ)dr = −gγ (q)H1 (γ), 3 3 3 3  1 H1 · dl = 0. 2

4

In the above computations, the following assumptions have been taken into account [55]: • Wires forming the loops have no width. • The space between stator and rotor, known as the air gap, is not constant but depends on the rotor position q, and its width is represented by g0 (q), for γ = 0, or gγ (q), for γ = 0. • The magnetic field H1 is radially oriented at the air gap.

6.5 A Practical Switched Reluctance Motor

331

Fig. 6.21 Two views of the rotor

• H1 = 0 inside a ferromagnetic material with high relative magnetic permeability, i.e., the stator and rotor. • dl = dr in the segment 1-2, whereas dl = −dr in the segment 3-4. On the other hand, i 1enclosed depends on γ and is given as (see Fig. 6.22):

i 1enclosed

⎧ 0, ⎪ ⎪ ⎪ ⎪ N I1a , ⎪ ⎪ ⎪ ⎪ 2N I1a , ⎪ ⎪ ⎪ ⎪ N I1a , ⎨ = 0, ⎪ ⎪ N I1a , ⎪ ⎪ ⎪ ⎪ 2N I1a , ⎪ ⎪ ⎪ ⎪ N I ⎪ 1a , ⎪ ⎩ 0,

0≤γ 0 is constant standing for the phase windings resistance. Adding these expressions and defining ψ = ψa + ψb , I = Ia + Ib = [I1 , I2 , I3 ] : ψ˙ + r I = 2U.

340

6 Switched Reluctance Motor

Fig. 6.28 Dependence of the air gap on rotor position (cont)

Replacing (6.141), (6.143)–(6.145) we retrieve (6.9), i.e., D(q) I˙ + K (q)I q˙ + R I = 2U, where R = diag{r, r, r } and K (q) = diag{K 1 (q), K 2 (q), K 3 (q)},   ∂ L j (q) 2π = l2 Nr sin Nr q − ( j − 1) , K j (q) = ∂q 3

j = 1, 2, 3.

Finally, according to the D’Alembert principle, the generated torque is given as in (6.12), i.e., ∂ τ= ∂q



1  I D(q)I 2



1 1  I K (q)I = K j (q)I j2 . 2 2 j=1 3

=

6.5 A Practical Switched Reluctance Motor

341

Thus, inserting the generated torque in the motor mechanical dynamics we finally find that the SRM dynamical model is given as in (6.9), (6.10), i.e., D(q) I˙ + K (q)I q˙ + R I = 2U, J ω˙ + bω = τ − τ L , where ω = q˙ is the rotor velocity. Notice that this is the unsaturated motor model described in Sect. 6.1.3. Reason for this is that the procedures presented in Sects. 6.5.1, 6.5.2, only stand if unsaturated magnetic circuits are considered.

Chapter 7

Synchronous Reluctance Motor

Although the synchronous reluctance motor has a long history [190], the interest in this machine for variable-speed applications had a maximum in 1960s in textile industry [166]. This was because the synchronism of the machine allowed a better velocity control than that achieved with DC-motors and induction motors at that point of time. Thus, a higher grade of synthetic fibers could be produced. However, the machine was operated in open loop and its performance was not as good as that achieved with induction motors when vectorial control appeared. This was because performance of synchronous reluctance machines strongly depends on the saliency ratio which was very low at that time. More than 20 years ago, the synchronous reluctance motor was regarded as inferior to other types of AC machines due to its lower average torque and larger torque pulsation. Recently, however, researchers have proposed many methods to improve the characteristics of the motor as well as the drive system [160]. For instance, significant research has been conducted in the recent years and the saliency ratio has now reached large enough values. It has now been shown that a synchronous reluctance motor under closed-loop control has some advantages compared to induction motor, which is its powerful industrial competitor, when variable-speed operation with high efficiency is demanded [192, 225]. (1) Their stator is identical to that of an induction motor, so the stator can be constructed in the already available assembly line. (2) They can achieve high power densities in regard to their size. (3) They can function without a starting cage, since they can start in synchronous mode. (4) Compared to the induction motor, the synchronous reluctance motor achieves high torque output in regard to iron losses and higher efficiency. Motivated by these facts, several works on control of synchronous reluctance motors have been presented [120, 160, 192, 194, 229, 285]. However, although field-oriented control of these motors is a well-known control strategy [265, 268], the literature on formal studies on this control scheme for these motors is almost null. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 V. M. Hernández-Guzmán et al., Energy-Based Control of Electromechanical Systems, Advances in Industrial Control, https://doi.org/10.1007/978-3-030-58786-4_7

343

344

7 Synchronous Reluctance Motor

Fig. 7.1 Stator and rotor magnetic flux orientation in a synchronous reluctance motor

In this chapter, we present two controllers for this class of machines. The proposals are almost identical to standard field-oriented control when used to regulate velocity and position. The main advantage of our proposals is that they are provided with formal global asymptotic stability proofs. This chapter is organized as follows. In Sect. 7.1, we present the working principle, the dynamical model, and the standard field-oriented control of synchronous reluctance motors. In Sect. 7.2, we study the passivity properties of the motor model. In Sect. 7.3, we present our main result for the case of velocity control. Finally, Sect. 7.4 is devoted to present our main result for position control.

7.1 Motor Modeling 7.1.1 The Working Principle A synchronous reluctance motor (SYRM) is a three-phase motor whose stator is identical to the stator of a three-phase PM synchronous motor. The rotor of a SYRM has neither permanent magnets nor windings: it consists of a simple piece of iron which is not round but it possess a high degree of saliency. In Fig. 7.1 is shown a onepole SYRM where the single-salient property of these motors is appreciated, contrary to the double-salient property of SRM observed in Fig. 6.1. Hence, when balanced three-phase electric currents flow through the stator windings, a constant magnitude magnetic flux appears which rotates counter clockwise with angular velocity ω S around the internal surface of the stator (see Sect. 4.1.1). Since the rotor is made in a ferromagnetic material (iron), then this rotating magnetic flux exerts an attractive

7.1 Motor Modeling

345

force on the rotor by the effect of the reluctance at the gap between the stator and rotor. Thus, producing the rotor movement. When no load torque exists, the salient part of the rotor aligns to the rotating magnetic field produced by the stator windings because the generated torque tries to minimize the reluctance, i.e., the rotating air gap [268]. When a load torque is present, the rotor lags the rotating magnetic flux produced by the stator windings producing a misalignment between them. The electromagnetic torque is produced to minimize the reluctance trying to maintain the alignment. When this torque is equal and opposite to the load torque on the rotor, the lagging rotor will again rotate at the same velocity as the rotating flux produced by the stator windings [268]. The word synchronous is used to designate this class of reluctance motor to stress the fact that, in steady state, the angular velocity of rotor ω R equals the angular velocity of the magnetic flux due to the stator windings, i.e., ω R = ω S . Since contrary to the PM synchronous motor any magnetic flux is not produced by the rotor in a SYRM, the electric current through the stator fictitious phase d must not be zero in order to create a magnetic flux at the rotor which will interact with the magnetic flux produced by the stator fictitious phase q to produce suitable torque on the rotor. Thus, the salient part of the rotor moves parallel to the magnetic flux produced by the stator phase d. In Fig. 7.1 are depicted the vectors representing the magnetic flux produced by the stator windings λ and the dq frame attached to rotor. Notice that the d phase is attached at the salient part of the rotor. The stator fluxes produced by phases d and q, i.e., λd = L d Id and λq = L q Iq , respectively, are the only contributors to λ which leads the rotor saliency, i.e., the d-axis, by a constant angle δ under nonzero load torque conditions [120, 268]. What we have just described is also known as the open-loop operation of a SYRM.

7.1.2 dq Dynamical Model Because of the above description, the Park’s transformation or dq transformation in Sect. 4.1.3 is identically applied to SYRMs. Hence, the mathematical model of a SYRM is identical to the dynamical model of a salient rotor PM synchronous motor given in (4.29)–(4.31). The only difference is that the two terms −Φ M q˙ and Φ M Iq are eliminated since they were introduced because of the presence of a permanent magnet at the rotor of a PM synchronous motor which does not exist in a SYRM. Thus, the dq model of a SYRM is given by L q I˙q = −R Iq − n p L d Id q˙ + Vq L d I˙d = −R Id + n p L q Iq q˙ + Vd

(7.1) (7.2)

J ω˙ + bω = n p (L d − L q )Id Iq − τ L .

(7.3)

346

7 Synchronous Reluctance Motor

Fig. 7.2 Standard field-oriented control of a SYRM

Notice that according to (4.25), L d > L q . Moreover, the efficiency of a SYRM increases as the saliency ratio L d /L q and the difference L d − L q increase [268]. The torque generated by a SYRM is given as τ = n p (L d − L q )Id Iq and, thus, contrary to a PM synchronous motor, a nonzero electric current Id must flow through the fictitious phase d of the stator to produce suitable torque.

7.1.3 Standard Field-Oriented Control of a SYRM In Fig. 7.2 is shown the standard velocity control scheme for a SYRM [268]. Because of the reasons exposed above, a suitable nonzero constant electric current Id∗ must be proposed as the desired current through phase d. A classical linear proportionalintegral (PI) controller is employed to force Id → Id∗ . A classical linear PI velocity controller is used to compute the required torque τ ∗ to be generated by the motor to force the measured velocity ω to reach its constant desired value ω ∗ . Using the relationship τ = n p (L d − L q )Id Iq the electric current that is required to flow through the phase q is computed as Iq∗ = τ ∗ /kt , where kt = n p (L d − L q )Id∗ . Finally, a classical linear PI electric current controller is used to force the measured electric current Iq to reach its desired value Iq∗ . Thus, the equations of the standard control scheme for a SYRM are summarized as

7.1 Motor Modeling

347

Fig. 7.3 Standard field-oriented control of a SYRM presented in Fig. 7.2 reduces to control of a simple linear mechanical system

Vd = Vq =

αdp (Id∗ αq p (Iq∗

 − Id ) + αdi 

t

0 t

− Iq ) + αqi 0

(Id∗ − Id )dr, (Iq∗ − Iq )dr,

1 ∗ τ , kt = n p (L d − L q )Id∗ , kt  t (ω ∗ − ω)dr, τ ∗ = k p (ω ∗ − ω) + ki Iq∗ =

(7.4)

0

with ω ∗ and Id∗ > 0 some known constants. The internal PI electric current loops are intended to achieve Id → Id∗ and Iq → Iq∗ very fast. This means that n p (L d − L q )Id Iq = τ → τ ∗ = n p (L d − L q )Id∗ Iq∗ very fast and, hence, the block diagram in Fig. 7.2b can be simplified to block diagram in Fig. 7.3. Thus, the proportional and integral gains k p and ki of the classical PI velocity controller (7.4) can be chosen by taking into account only the mechanical dynamics, i.e., using the simplified block diagram in Fig. 7.3. A similar procedure is valid for position control. In such a case, we only have to replace the PI velocity controller by a proportional-integral-derivative (PID) position controller. Thus, in the above equations only the desired torque expression must be replaced by d(q ∗ − q) + ki τ = k p (q − q) + kd dt ∗

∗



t

(q ∗ − q)dr,

0

where q stands for the rotor position, i.e., ω = q, ˙ and q ∗ is a known constant standing for the rotor desired position.

7.2 Open-Loop Energy Exchange 7.2.1 The Velocity Model According to Sect. 7.1.2, the dynamic model of a synchronous reluctance motor (SYRM) is given by (7.1)–(7.3), which is rewritten here for the ease of reference:

348

7 Synchronous Reluctance Motor

L q I˙q = −R Iq − n p L d Id q˙ + Vq , L d I˙d = −R Id + n p L q Iq q˙ + Vd , J ω˙ + bω = n p (L d − L q )Id Iq − τ L .

(7.5) (7.6) (7.7)

Variables Id , Iq , ω ∈ R represent the electric currents through the d and q phase windings as well as the motor angular velocity, respectively, with ω = q, ˙ being q the rotor angular position. Symbols Vd and Vq stand for voltages applied at phases d and q, respectively. The positive constants n p , L d , L q , R, J, b, represent the number of pole pairs, the stator inductances, the stator winding resistance, the rotor moment of inertia, and the viscous friction coefficient. Finally, τ L is the load torque. Notice that the only difference between the model of a salient rotor PM synchronous motor, given in (4.29)–(4.31), and the model of a SYRM, given in (7.5)– (7.7), is the absence in the latter of the terms involving the torque constant Φ M which is due to the fact that permanent magnets are not present in the rotor of a SYRM (see Sect. 7.1). Thus, analysis and controller design for SYRM is very similar to analysis and controller design for salient rotor PM synchronous motors. As a matter of fact, proceeding as in Sect. 4.2.1, it is easy to show that natural energy exchange exists between the electrical and the mechanical subsystems using the energy function: V (Id , Iq , ω) =

1 1 1 L d Id2 + L q Iq2 + J ω 2 . 2 2 2

Moreover, this energy exchange is quantified by the following cancellation of cross terms which appear in V˙ : n p L q Iq Id ω − Iq n p L d Id ω + n p (L d − L q )Id Iq ω = 0. Thus, it is not difficult to show that V˙ = −R Id2 − R Iq2 − bω 2 + Id Vd + Iq Vq − τ L ω, and that defining the input u = [Vd , Vq , −τ L ] and the output y = [Id , Iq , ω] , we can write ⎡ ⎤ R 0 0 V˙ = −y  Qy + y  u, Q = ⎣ 0 R 0 ⎦ . 0 0 b Since Q is a positive definite matrix, this expression shows that the model (7.5)–(7.7) is output strictly passive (see Definition 2.42) for the output y and input u defined above. In this chapter, it will be shown that this is the basic theoretical principle behind the common and successful industrial velocity control scheme depicted in Fig. 7.2.

7.2 Open-Loop Energy Exchange

349

7.2.2 The Position Model If the rotor position is designated by q then ω = q. ˙ Assume that the load torque is given as a nonlinear function of position, i.e., τ L = G(q), which is given as the . Using these ideas gradient of a positive semidefinite function P(q), G(q) = d P(q) dq in the dynamic model of a SYRM given in (7.5)–(7.7), we can write L q I˙q = −R Iq − n p L d Id q˙ + Vq , L d I˙d = −R Id + n p L q Iq q˙ + Vd , J q¨ + bq˙ = n p (L d − L q )Id Iq − G(q).

(7.8) (7.9) (7.10)

Hence, we can proceed as in Sect. 4.2.2 to show that given the following energy function: ˙ = V (Iq , Id , q, q)

1 1 1 L q Iq2 + L d Id2 + J q˙ 2 + P(q), 2 2 2

it is found that V˙ = −R Id2 − R Iq2 − bq˙ 2 + Id Vd + Iq Vq . Instrumental to find this expression is the cancellation of cross terms: ˙ + qG(q) ˙ = 0, n p L q Iq Id ω − Iq n p L d Id ω + n p (L d − L q )Id Iq ω − qG(q) which represents the natural energy exchange between the electrical and the mechanical subsystems and between the kinetic and the potential energies in the mechanical subsystem recall that b > 0. Hence, defining the input u = [Vd , Vq ] and the output y = [Id , Iq ] , we can write V˙ ≤ −Ry  y + y  u. Since R > 0, this expression shows that the model (7.8)–(7.10), is output strictly passive (see Definition 2.42) for the output y and input u defined above. In this chapter, it will be shown that this is the basic theoretical principle behind the common and successful industrial control scheme described in Sect. 7.1.3.

7.3 Velocity Control Standard field-oriented control of a SYRM has been described in Sect. 7.1.3. This successful control scheme is depicted again in Fig. 7.4 for the ease of reference. In the following proposition a control scheme, which will be shown to be very similar

350

7 Synchronous Reluctance Motor

Fig. 7.4 Standard field-oriented control of a SYRM

to standard field-oriented control, is introduced. Contrary to standard field-oriented control, the proposed control scheme is provided with a formal global asymptotic stability proof, i.e., velocity and the whole state converge to their desired values from any initial condition. Moreover, this result relies on energy ideas. Proposition 7.1 Consider the dq model of a SYRM given in (7.5)–(7.7), in closed loop with the controller: Vd = −αd I˜d − αdi

 

t

I˜d (s)ds − K d ω˜ 2 I˜d ,

(7.11)

0 t

I˜q (s)ds − K q ω˜ 2 I˜q − K f | I˜d | I˜q , Vq = −αq I˜q − αqi 0   t  1 ω(s)ds ˜ , Iq∗ =  −k p ω˜ − ki σ kt 0

(7.12) (7.13)

where I˜q = Iq − Iq∗ , I˜d = Id − Id∗ and ω˜ = ω − ω ∗ with ω ∗ and Id∗ > 0 real constants which stand for the desired velocity and the desired current through the phase d, respectively, kt > 0 represents the estimate of kt = n p (L d − L q )Id∗ > 0, and σ(·) is a strictly increasing linear saturation function (see Definition 2.34) for some ∗ | which we additionally require to be continu(L , M) such that M > L > | τL +bω ki ously differentiable such that 0
0 such that kt = εkt , k p = k p /ε, ki = ki /ε. Hence, it is possible to write Iq∗

1 = kt

  t    −k p ω˜ − ki σ ω(s)ds ˜ ,

(7.15)

0

and the closed-loop dynamics is given as  t 1 ∗ z(x) = σ(x) +  (τ L + bω ), x = ω(s)ds, ˜ ki 0    t n p L q ω∗ 1 ∗ ∗ ˜ R Id − (τ L + bω ) , zd = Id (s)ds + αdi kt 0    t R 1 ∗ ∗ ∗ ˜ (τ L + bω ) + n p L d Id ω , Iq (s)ds + zq = αqi kt 0 L I˙˜ = −(R + α ) I˜ − n L ω˜ I˜ + V , q q

q

q

p

d

d

q

L d I˙˜d = −(R + αd ) I˜d + n p L q ω˜ I˜q + Vd , J ω˙˜ = −(b + k p )ω˜ + n p (L d − L q ) I˜d I˜q − T L ,

(7.16) (7.17) (7.18)

T L = −n p (L d − L q ) I˜d Iq∗ − n p (L d − L q )Id∗ I˜q + ki z(x), Vd = n p L q ω˜ Iq∗ + n p L q ω ∗ I˜q − αdi z d −

n p L q ω ∗ k p kt

ω˜ −

n p L q ω ∗ ki z(x) kt

−K d ω˜ I˜d , Rk p Rki Vq = ω˜ + z(x) − n p L d ω ∗ I˜d − n p L d ω˜ Id∗ − αqi z q − L q I˙q∗ kt kt −K q ω˜ 2 I˜q − K f | I˜d | I˜q . 2

The state of this closed-loop dynamics is ξ = [ω, ˜ x + τL +bω , I˜d , z d , I˜q , z q ] ∈ R6 ki and ξ = 0 is its only equilibrium point if ki > 0, αdi > 0, αqi > 0. Moreover, this closed-loop dynamics is autonomous. ∗

352

7 Synchronous Reluctance Motor

The following Lyapunov function candidate is proposed for stability analysis purposes: V (ξ)

  ∗ ˜ x + τ L +bω , = 21 L q I˜q2 + 21 L d I˜d2 + 21 αdi z d2 + 21 αqi z q2 + Vω ω, (7.19)  ki   ∗ ˜ x + τ L +bω = 21 J ω˜ 2 + [ki + β(b + k p )] x τ L +bω∗ z(r )dr + β J z(x)ω. Vω ω, ˜ k −

i

ki

This function can always be rendered positive definite and radially unbounded by suitable selection of the controller gains. Taking advantage from cross term cancellations described above, it is found that the time derivative of V along the trajectories of the closed loop dynamics (7.16)–(7.18) can be upper bounded as V˙ ≤ −ζ T Qζ +

L q k p n p |L d − L q |

n p L d k p

J kt

| I˜d | I˜q2 − K f | I˜d | I˜q2

(7.20)

| I˜d |ω˜ 2 − K d1 ω˜ 2 I˜d2 − k p2 ω˜ 2 kt L q k p n p |L d − L q |k p + | I˜d | | I˜q | |ω| ˜ − K q ω˜ 2 I˜q2 − K d2 ω˜ 2 I˜d2 − αd2 I˜d2 − αq2 I˜q2 , J kt2 +

where ζ = [|ω|, ˜ |z|, | I˜d |, | I˜q |] , k p = k p1 + k p2 , with k p1 > 0, k p2 > 0, K d = K d1 + K d2 , with K d1 > 0, K d2 > 0, αd = αd1 + αd2 , and αq = αq1 + αq2 , with αd1 , αd2 , αq1 , αq2 positive constants and the entries of matrix Q are given as Q 11 = b + k p1 − β J,

Q 23 = Q 32 Q 14 = Q 41 Q 24 = Q 42 Q 34 = Q 43 If

L q k p n p |L d

− Lq |

Q 33 = R + αd1 ,

Id∗ , Q 12 = Q 21 = 0, (7.21) 2J kt n p L q |ω ∗ |k p n p L d ki M n p L d |τ L + bω ∗ | =− − − , 2kt kt 2kt n p L q |ω ∗ |ki =− , 2kt



Rk p + L q ki L q k p (b + k p )

− n p L d + |L d − L q | I ∗ , = −

− d

2kt 2J kt 2



Rk L q k p ki

, = −

i − 2kt 2J kt



n p (L d + L q )ω ∗ n p |L d − L q |

− = −

(2ki M + |τ L + bω ∗ |).

2 2J kt2

Q 44 = R + αq1 − Q 13 = Q 31

Q 22 = βki ,

7.3 Velocity Control

353

Fig. 7.5 Block diagram of controller in proposition 7.1. yq = −K q ω˜ 2 I˜q − K f | I˜d | I˜q , y = −K d ω˜ 2 I˜d and the PI velocity controller has a saturated integral part. ktd = n p (L d − L q )Id

Kf > k p2

L q k p n p |L d − L q | 

>

.

(7.22)

1 . K d1

(7.23)

J kt n p L d k p kt

2

αq2 > 0 and αq2 αd2 > a=

a4 , 4K q K d2

L q k p n p |L d − L q |k p J kt2

(7.24)

,

then we can write V˙ ≤ −ζ  Qζ ≤ 0, ∀ξ ∈ R6 . Matrix Q can always be rendered positive definite if its four leading principal minors are positive, i.e., by a suitable selection of the controller gains. Thus, the origin is stable. Global asymptotic stability of ξ = 0 is proven invoking the LaSalle invariance principle. This completes the proof of Proposition 7.1. Conditions for this stability result are summarized by (A.7), (C.13), (7.22), (7.23), (7.24) and the four principal minors of matrix Q defined in (7.21) are positive. These conditions ensure that V is positive definite and radially unbounded and V˙ is negative semidefinite. Remark 7.2 Since the closed-loop model in (7.16)–(7.18) is almost identical to model in (7.5)–(7.7), it is clear that the cancellations of cross terms explained in Sect. 7.2 appear again and they are instrumental for the proof of Proposition 7.1. Remark 7.3 Standard field-oriented control of a SYRM was presented in Sect. 7.1.3 and it is repeated here for the ease of reference:

354

7 Synchronous Reluctance Motor

Vd = αdp (Id∗ − Id ) + αdi Vq =

αq p (Iq∗

 

t

0 t

− Iq ) + αqi 0

(Id∗ − Id )dr, (Iq∗ − Iq )dr,

1 = τ ∗ , kt = n p (L d − L q )Id∗ , kt  t ∗ ∗ (ω ∗ − ω)dr, τ = k p (ω − ω) + ki Iq∗

0

where Id∗ > 0 is a constant. This control strategy is depicted in Fig. 7.4, whereas controller in Proposition 7.1 is depicted in Fig. 7.5. Excepting the addition of the terms in −K d ω˜ 2 I˜d , and −K q ω˜ 2 I˜q − K f | I˜d | I˜q , in (7.11), (7.12), and the use of a saturated integral part of the PI velocity controller, controller in Proposition 7.1 is exactly the control scheme in Fig. 7.4. The advantage of controller in Proposition 7.1 is that it is provided with a global asymptotic stability proof. Furthermore, notice that controller in Proposition 7.1 does not require the exact knowledge of any motor parameter. It is not difficult to realize that an adaptive controller similar to that presented in Proposition 4.23 is also useful for SYRM. This is important to remark since the adaptive versions impose less restrictive conditions to the controller gains. Remark 7.4 In [204] is claimed that the passivity-based approach introduced in that book can be also applied to SYRM. Moreover, it is stated that this controller can be derived from the application that in that book is presented for PM synchronous motors (see (4.72) in this book). This is reasonable since the only difference between the models of SYRM and PM synchronous motor is that the former has not a back electromotive force term. Thus, when applying the approach in [204] to the SYRM, the resulting controller is very similar as that presented in (4.72) for the PM synchronous motors. The only difference is that the term [Φ M q˙m , 0, 0]T does not appear for SYRM. Thus, the same comments that were presented in Remark 4.11 are also valid for the case of SYRM when the approach in [204] is used.

7.4 Position Control In this section, the following SYRM model is considered: L q I˙q = −R Iq − n p L d Id q˙ + Vq , L d I˙d = −R Id + n p L q Iq q˙ + Vd , J q¨ + bq˙ = n p (L d − L q )Id Iq − g(q),

(7.25) (7.26) (7.27)

where g(q) represents a position-dependent mechanical load which is assumed to possess the following properties:

7.4 Position Control

355

|g(x) − g(y)| ≤ kg |x − y|, ∀x, y ∈ R, (Lipschitz condition)



dg(q)

,

kg > max

q∈R dq

dU (q) g(q) = , |g(q)| ≤ k  , ∀q ∈ R, dq

(7.28) (7.29) (7.30)

with U (q) the potential energy and kg , k  , some positive constants. Controller in Proposition 7.1 can be extended to the case of position control as shown in the following Proposition Proposition 7.5 Consider the SYRM model (7.25), (7.26), (7.27) together with the following controller: Vd = −αdp I˜d − αdi Vq = −αq p I˜q − αqi

 

t

I˜d (s)ds − K d q˙ 2 I˜d ,

(7.31)

I˜q (s)ds − K q q˙ 2 I˜q − K f | I˜d | I˜q ,

(7.32)

0 t 0

1  ˜ − kd q˙ − ki sat (z) ,  −k p h(q) kt      t  βk p αβkd α 1+ h(q) ˜ + 1+ q˙ ds, z= ki ki 0

Iq∗ =

(7.33)

where q˜ = q − q ∗ and I˜d = Id − Id∗ , with q ∗ and Id∗ > 0 real constants standing for the desired position and the desired current in the phase d, respectively, kt is a positive constant representing an estimate of kt = n p (L d − L q )Id∗ > 0, I˜q = Iq − Iq∗ , whereas K d , K q , K f , k p , kd , ki , αdp , αdi , αq p , αqi , are constant scalars, and h(q) ˜ = σ(q), ˜ sat (z) = σ(z), where σ(·) is a strictly increasing linear saturation function for some (L , M) (see Definition 2.34). Furthermore, it is also required that function σ(·) be continuously differentiable such that 0


k . ki

(7.39)

This closed-loop dynamics is autonomous because it can be written as ξ˙ = f (ξ) for some nonlinear f (·) ∈ R7 . The following Lyapunov function candidate is proposed for stability analysis purposes: (7.40) W (q, ˜ q, ˙ z + g(q ∗ )/ki , I˜q , I˜d , z d , z q ) = 1 1 ˜2 1 ˜2 1 L q Iq + L d Id + αqi z q2 + αdi z d2 + V (q, ˜ q, ˙ z + g(q ∗ )/ki ), 2 2 2 2 where

7.4 Position Control

V (q, ˜ q, ˙ z + g(q

357 ∗

)/ki )

 q˜ 1 2  = J q˙ + αJ h(q) ˜ q˙ + α(b + kd ) h(r )dr 2 0  q˜ ∗ +k p h(r )dr + U (q) − U (q ∗ ) − qg(q ˜ ) 0  z +ki s(r )dr + αβ J s(z)q. ˙ −g(q ∗ )/ki

The function W (q, ˜ q, ˙ z + g(q ∗ )/ki , I˜q , I˜d , z d , z q ) is positive definite and radially unbounded and it is found that W˙ can be upper bounded as W˙ ≤ −x  Qx − K f | I˜d | I˜q2 +L q kd n p |L d

−

(7.41)

L q | | I˜d | I˜q2 /(J kt )

 −q˙ 2 | I˜d |(K d1 | I˜d | − n p L d kd /kt ) − kd2 q˙ 2 −K q q˙ 2 I˜q2 − K d2 q˙ 2 I˜d2

+L q (kd )2 n p |L d − L q | |q| ˙ | I˜q | | I˜d |/(J kt2 ) −αq p2 I˜q2 − αdp2 I˜d2 , x  = [|q|, ˙ |h(q)|, ˜ |s(z)|, | I˜q |, | I˜d |],   , kd2 , αq p1 , αq p2 , αdp1 , αdp2 are positive constant scalars such where K d1 , K d2 , kd1   + kd2 = kd , αq p1 + αq p2 = αq p , and αdp1 + αdp2 = αdp . that K d1 + K d2 = K d , kd1 The entries of matrix Q as given as  − αJ − αβ J (1 + αβkd /ki ), Q 11 = b + kd1

Q 22 =

α(k p

− khg ),

Q 44 = R + αq p1 −

Q 15

Q 24

L q kd n p |L d J kt

− Lq |

(7.42)

Q 55 = R + αdp1 , Id∗ ,

Q 13 = Q 31 = −αβb/2,

α2 β αβ (1 + βk p /ki )J, Q 23 = Q 32 = − khg , 2 2 1 1 1 = Q 41 = − Rkd /kt − L q k p /kt − L q ki (1 + αβkd /ki )/kt 2 2 2 1 1 1   − L q kd (b + kd )/(J kt ) − n p |L d − L q |Id∗ − n p L d Id∗ , 2 2 2 1 1   = Q 51 = − n p L d k p M/kt − n p L d ki M/kt − n p L d k  /kt 2 2 α   − n p kd |L d − L q | M/kt − αβkd n p |L d − L q |M/kt , 2 1 1 1 = Q 42 = − L q kd k p /(J kt ) − L q ki α(1 + βk p /ki )/kt − Rk p /kt 2 2 2 αn p |L d − L q | ∗ 1  Id , − L q kd khg /(J kt ) − 2 2

Q 12 = Q 21 = − Q 14

Q 33 =

αβki ,

358

7 Synchronous Reluctance Motor

α Q 25 = Q 52 = − n p k p |L d − L q |M/kt − αn p ki |L d − L q |M/kt 2 α  − n p k |L d − L q |/kt , 2 1 1 Q 34 = Q 43 = − Rki /kt − L q kd ki /(J kt ) − αβn p |L d − L q |Id∗ /2, 2 2 αβ Q 35 = Q 53 = − n p k p |L d − L q |M/kt − αβn p ki |L d − L q |M/kt 2 αβ n p k  |L d − L q |/kt , − 2 1 Q 45 = Q 54 = − L q kd n p |L d − L q |/(k p M + 2Mki + k  ) 2J kt2 1 − n p |L d − L q |(αM + 2αβ M), 2 where the following result, taken from [130], pp. 105–107, has been used: |g(q) − g(q ∗ )| ≤ khg h(|q|), ˜ khg ≥

2k  ∀q˜ ∈ R,  , h( 2k ) kg

as well as the facts that h(|q|) ˜ = |h(q)| ˜ and ±uv ≤ |u| |v|, ∀q, ˜ u, v ∈ R. The five leading principal minors of matrix Q can always be rendered positive by choosing small enough α > 0, β > 0 and large enough positive definite matrices kd , k p , ki , αdp , αq p . Hence, matrix Q is positive definite. On the other hand, if the following conditions are satisfied: K f > L q kd n p |L d − L q |/(J kt ),  kd2 >

η12

, K d1 1 αq p2 αdp2 > [L q (kd )2 n p |L d − L q |/(J kt2 )]2 η42 , 4 η1 = n p L d kd /kt , η4 =

(7.43) (7.44) (7.45)

L q (kd )2 n p |L d − L q |/(J kt2 )

, 4K q K d2

then it is possible to write W˙ ≤ −x  Qx, ∀ξ ∈ R7 . Hence, W˙ ≤ 0, with W˙ = 0 if (q, ˙ q, ˜ s(z), I˜q , I˜d ) = (0, 0, 0, 0, 0). This means that the origin is stable. Since the closed-loop dynamics is autonomous, global asymptotic stability of the origin is shown by invoking the LaSalle invariance principle. This completes the proof of Proposition 7.5.

7.4 Position Control

359

Fig. 7.6 Control scheme in Proposition 7.5, yq = −K q q˙ 2 I˜q − K f | I˜d | I˜q , and y = −K d q˙ 2 I˜d . Recall that ktd = n p (L d − L q )Id

Conditions for this stability result are summarized by (7.39), (7.43), (7.44), (7.45), (A.8), (A.9), (A.10), (A.14), αqi and αdi are positive constants and the five principal minors of matrix Q defined in (7.42) are positive. These conditions ensure positive definiteness of W and negative semidefiniteness of W˙ . Remark 7.6 Since the closed-loop model in (7.35)–(7.37) is almost identical to model in (7.8)–(7.10), it is clear that the cancellations of cross terms explained in Sect. 7.2 appear again and they are instrumental for the proof of Proposition 7.5. Remark 7.7 The control scheme in Proposition 7.5 is depicted in Fig. 7.6. Notice that excepting the three simple terms −K d q˙ 2 I˜d and −K q q˙ 2 I˜q − K f | I˜d | I˜q , and the fact that Iq∗ represents a nonlinear PID position controller, the control scheme in Proposition 7.5 exactly represents standard field-oriented control of SYRM (see Sect. 7.1.3) when used to regulate position. Thus, Proposition 7.5 introduces the closest control scheme to standard field-oriented control of SYRM in the literature provided with a formal global asymptotic stability proof. Notice that the control scheme in Proposition 7.5 does not require to feedback the time derivative of the desired electric current. Instead of that, the terms composing that function are dominated using the simple nonlinear terms −K d q˙ 2 I˜d and −K q q˙ 2 I˜q − K f | I˜d | I˜q . This also allows to dominate the terms in the time derivative of the desired electric current arising from the fact that velocity measurements are allowed. We stress that measuring velocity is an important step to design a PID position controller. Hence, this represents another contribution in Proposition 7.5. It is not difficult to realize that an adaptive controller similar to that presented in Proposition 4.18 is also useful for SYRM. This is important to remark since the adaptive versions impose less restrictive conditions to the controller gains.

Chapter 8

Bipolar Permanent Magnet Stepper Motor

Permanent magnet (PM) stepper motors are nonlinear electromechanical actuators widely used as positioning devices. They have the ability to deliver higher peak torque per unit weight than brushed DC-motors, they have higher torque to inertia ratio and they are more reliable requiring less maintenance since they have no brushes. Moreover, PM stepper motors are especially suitable as direct drives in robotics and aerospace applications where motor weight is a crucial factor [34, 180, 198, 291]. These features have attracted the attention of the international control community who has found challenging the controller design task for PM stepper motors because of the nonlinear and multivariable nature of the corresponding dynamical model. Although this has resulted in lots of works on the subject [23, 34, 138, 140, 147, 180, 198, 244, 291, 292], the complex nature of the PM stepper motor model has forced authors of these works to design complex controllers which, although formally supported, have not been widely accepted by practitioners. Moreover, the drives community prefers simple controllers such as commutation. Commutation is a standard technique used in closed-loop control of PM stepper motors [148, 149]. Although a simple, well known, and successful idea in practice, any formal stability analysis has not been presented until now justifying the use of commutation for velocity and position regulation purposes. Unfortunately, the recent literature on this subject is sparse. However, as in the case of field-oriented control of PM synchronous motor and induction motors, this does not mean that the problem is solved or it has no importance (see the recent paper [205], for instance). It is important to say that another approach for control of PM stepper motors is microstepping [135–137, 139, 141, 175]. The main difference between commutation and microstepping is that in the former the desired electric currents are chosen in terms of torque that is desired to be generated by motor. In this chapter we present two controllers, one for velocity and the other for position, which are very similar to commutation. The advantage of our proposals is that they are provided with formal global stability proofs. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 V. M. Hernández-Guzmán et al., Energy-Based Control of Electromechanical Systems, Advances in Industrial Control, https://doi.org/10.1007/978-3-030-58786-4_8

361

362

8 Bipolar Permanent Magnet Stepper Motor

This chapter is organized as follows. In Sect. 8.1 we explain the working principle, derive the dynamical model, and describe the standard control scheme for PM stepper motors. In Sect. 8.2 we study the passivity properties of the dynamical model. In Sect. 8.3 we present our proposal for velocity control whereas the position control problem is solved in Sect. 8.4. Finally, a practical PM stepper motor is modeled in Sect. 8.5.

8.1 Motor Modeling There are several kinds of PM stepper motors. In this book, only the so-called bipolar PM stepper motor is considered and it will be referred as PM stepper motor for short in the sequel.

8.1.1 The Working Principle A PM stepper motor is shown in Fig. 8.1 [59]. It consists of two-phase windings, called a and b, on the stator and a permanent magnet at the rotor. In this case, the permanent magnet at the rotor has six poles (i.e., three pole pairs) which alternate in polarity. On the stator, phase winding a produces two magnetic poles, one at the top of Fig. 8.1 and the other at the bottom. These poles can be either N or S depending on the sense of electric current through phase a. Phase winding b, on stator, produces two poles, one on the left of Fig. 8.1 and the other at the right. Again, these poles can be either N or S depending on the sense of electric current through phase b. A counter-clockwise movement of rotor can be achieved using the following sequence of actions [59]. • In Fig. 8.1, a zero electric current is applied to phase b and, hence, no magnetic poles are produced at the left nor at the right of stator. On the other hand, a positive electric current is applied to phase a producing a S pole at the top and a N pole at the bottom of stator. Thus, rotor remains oriented as shown in Fig. 8.1. • Let electric current through phase a be zero and allow a positive electric current to flow through phase b. This produces no magnetic poles at the top nor at the bottom on stator but a S pole is produced at the right and a N pole is produced at the left on stator. Hence, rotor moves counter-clockwise until the rotor poles N2 and S2 align with phase windings b at the right and at the left, respectively. • Let electric current through phase b be zero and allow a negative electric current to flow through phase a. This produces no magnetic poles at the left nor at the right of stator but a N pole is produced at the top and a S pole is produced at the bottom of stator. Hence, rotor moves counter-clockwise until the rotor poles S3 and N3 align with phase windings a at the top and at the bottom, respectively.

8.1 Motor Modeling

363

Fig. 8.1 A bipolar PM stepper motor

• Let electric current through phase a be zero and allow a negative electric current to flow through phase b. This produces no magnetic poles at the top nor at the bottom of stator but a S pole is produced at the left and a N pole is produced at the right of stator. Hence, rotor moves counter-clockwise until the rotor poles N1 and S1 align with phase windings b at the left and at the right, respectively. These actions can be accomplished using the sequence of applied voltages shown in Table 8.1. Using these ideas is not difficult to verify that a clockwise movement of rotor can be accomplished using the sequence of applied voltages shown in Table 8.2. The above-described operation is known as open-loop control of a PM stepper motor and it is not difficult to realize that it results in strong torque variations. Furthermore, position response exhibits large overshoots and oscillations each time one step is commanded, i.e., each time that one of the voltage changes shown in Table 8.1 or in Table 8.2 is commanded. Performance improvement is the main reason

364

8 Bipolar Permanent Magnet Stepper Motor

Table 8.1 Sequence of applied voltages required to make rotor to move counter-clockwise [59] Applied Positive Zero Negative Zero Positive voltage at phase a Applied voltage at phase b

Zero

Positive

Zero

Negative

Zero

Table 8.2 Sequence of applied voltages required to make rotor to move clockwise Applied Positive Zero Negative Zero Positive voltage at phase a Applied voltage at phase b

Zero

Negative

Zero

Positive

Zero

to design a closed-loop controller for a PM stepper motor. This chapter is devoted to analyze and design closed-loop controllers for PM stepper motors.

8.1.2 Dynamical Model Flux linkages produced in the stator phases by the rotor permanent magnet are also shown in Fig. 8.1. On the stator windings, however, we indicate the positive sense of electric current. Rotor position in Fig. 8.1 is zero q = 0. Let λa and λb represent the contribution to flux linkages in phases a and b, respectively, of the rotor permanent magnet. Notice that flux linkage in phase a, λa , is maximal and positive whereas flux linkage in phase b, λb , is zero. It is not difficult to realize that flux linkages behave as shown in Table 8.3 as rotor moves counter-clockwise, i.e., in the sense where q grows. Using Table 8.3 and assuming that the stator phase windings1 as well as the magnetic flux due to the rotor permanent magnet are sinusoidally distributed, it is concluded that λa = λ0 cos(n p q), λb = λ0 sin(n p q), n p = 3,

(8.1)

where n p and λ0 stand, respectively, for the number of pole pairs and the maximal flux linkage. On the other hand, flux linkages due to both the rotor permanent magnet and the electric currents through the stator phases are given as

1 Compare

the two layer windings distribution in Fig. 8.1 and that in Fig. 4.1 which produces a staircase magnetic field as shown in Fig. 4.3.

8.1 Motor Modeling

365

Table 8.3 Flux linkages behavior as rotor moves counter-clockwise q = 0 [rad] q = π6 [rad] q = 2π q= 6 [rad]

3π 6

[rad]

q=

4π 6

[rad]

λa

Maximal (positive)

Zero

Minimal (negative)

Zero

Maximal (positive)

λb

Zero

Maximal (positive)

Zero

Minimal (negative)

Zero

ψa = L Ia + λa , ψb = L Ib + λb ,

(8.2)

where L is the phase windings inductance which is assumed to be the same for both phases, and Ia , Ib , ψa , ψb are electric currents and flux linkages in phases a and b, respectively. Since each phase is constituted by a series RL circuit placed in a magnetic field produced by permanent magnets, use of Kirchhoff’s Voltage Law, Faraday’s Law and Ohm’s Law allows to write ψ˙a + R Ia = va , ψ˙ b + R Ib = vb ,

(8.3)

where va and vb are voltages applied at phases a and b, respectively, and it is assumed that resistance R of both phase windings are the same. Using (8.1) and (8.2), expressions in (8.3) are equivalent to L I˙a − km ω sin(n p q) + R Ia = va , L I˙b + km ω cos(n p q) + R Ib = vb , where the fact that q˙ = ω has been used and km = n p λ0 has been defined. On the other hand, according to D’Alembert principle, the generated torque (applied on the rotor in the sense of rotor velocity ω) is given as the co-energy’s derivative with respect to rotor position, i.e.,   ∂ 1 2 1 2 L Ia + L Ib + Ia λa + Ib λb , ∂q 2 2 = −km Ia sin(n p q) + km Ib cos(n p q),

τ =

(8.4)

since L, Ia and Ib are independent of the rotor position. Finally, according to Newton’s Second Law: J ω˙ = −bω + τ − τ L , where the positive constants J and b represent rotor inertia and viscous friction coefficient, respectively, τ L represents load torque (supposed to be applied in the opposite direction of the rotor velocity ω). Notice that in this expression the generated torque τ , given in (8.4), appears affected by a positive sign which is due to the fact

366

8 Bipolar Permanent Magnet Stepper Motor

Fig. 8.2 Standard closed-loop control of a PM stepper motor. Electrical subsystem refers to second and third expressions in (8.5), whereas mechanical subsystem refers to first expression in (8.5)

that τ is applied in the same direction of velocity ω. Thus, the PM stepper motor dynamical model is given by J ω˙ + bω = −km Ia sin(n p q) + km Ib cos(n p q) − τ L ,

L I˙a − km ω sin(n p q) + R Ia = va , L I˙b + km ω cos(n p q) + R Ib = vb .

(8.5)

Let us stress that in some other bipolar PM stepper motors, as that modeled in [55], the rotor only has one pole pair. However, these poles are placed one over the other concentric to the rotor axis and each pole has N R teeth, which are out of alignment by the width of one tooth. This disposition of pieces creates the effect of n p pole pairs such that n p = N R . This is why in some literature N R is used instead of n p in the model (8.5).

8.1.3 Standard Control A standard technique used for closed-loop control of PM stepper motors is shown in Fig. 8.2 [134]. It is important to stress that, contrary to PM synchronous motors and induction motors, standard control of PM stepper motors can be performed in original coordinates, i.e., a dq transformation is not required in PM stepper motors. As shown in Fig. 8.2, the following desired electric currents are proposed and this is known as commutation [134, 149] Ia∗ = − Ib∗ =

τ∗ sin(n p q), km

τ∗ cos(n p q). km

(8.6) (8.7)

The phase currents Ia and Ib are forced to reach such desired values using electric current loops driven by a high-gain proportional-integral controllers [134, 217], i.e.,

8.1 Motor Modeling

367

Fig. 8.3 Standard control of a PM stepper motor, presented in Fig. 8.2, reduces to control of a simple linear mechanical system

va =

αap (Ia∗



t

− Ia ) + αai

vb = αbp (Ib∗ − Ib ) + αbi



0 t

0

(Ia∗ (s) − Ia (s))ds,

(8.8)

(Ib∗ (s) − Ib (s))ds,

(8.9)

where Ia∗ and Ib∗ are given in (8.6), (8.7) and αap , αai are positive constants representing the proportional and integral controller gains for phase a. The positive constants αbp , αbi are defined accordingly for phase b. These controller gains are chosen to be large such that Ia = Ia∗ and Ib = Ib∗ are achieved very fast. Once this is true, the generated torque is given as (see (8.4)) τ = −km Ia∗ sin(n p q) + km Ib∗ cos(n p q), which, replacing (8.6) and (8.7), yields  ∗  ∗   τ τ cos(n p q) cos(n p q), τ = −km − sin(n p q) sin(n p q) + km km km = τ ∗ sin2 (n p q) + τ ∗ cos2 (n p q), = τ ∗. This means that use of the electric current loops in (8.8) and (8.9) forces motor to generate the desired torque τ ∗ . Hence, block diagram in Fig. 8.2 reduces to block diagram in Fig. 8.3. Since plant to be controlled in Fig. 8.3 is linear, the desired torque can be designed as a classical PI velocity controller, i.e., τ ∗ = k p (ω ∗ − ω) + ki



t

(ω ∗ (s) − ω(s))ds,

(8.10)

0

where the positive constants k p and ki are the proportional and integral controller gains, respectively, whereas ω ∗ is the desired velocity. A similar procedure is also valid for motor position control if the PI velocity controller in (8.10) is replaced by the classical PID position controller (where q ∗ represents the desired position): d(q ∗ − q) τ = k p (q − q) + kd + ki dt ∗

∗



t 0

(q ∗ (s) − q(s))ds,

(8.11)

368

8 Bipolar Permanent Magnet Stepper Motor

Fig. 8.4 Standard closed-loop position control of a PM stepper motor

where the positive constants k p , kd and ki stand for the proportional, derivative and integral controller gains, and the additional state equation must be considered: q˙ = ω. This control scheme is depicted in Fig. 8.4. The integral action in controllers (8.10) and (8.11) is included in order to ensure that ω = ω ∗ and q = q ∗ are achieved in steady state when both q ∗ and ω ∗ are constant and despite the presence of some unknown but constant external torque disturbance τL .

8.2 Open-Loop Energy Exchange 8.2.1 The Velocity Model According to Sect. 8.1, the dynamical model of a bipolar PM stepper motor is given in (8.5), which is rewritten here for the ease of reference: J ω˙ + bω = −km Ia sin(n p q) + km Ib cos(n p q) − τ L , L I˙a − km ω sin(n p q) + R Ia = va , (8.12) ˙ L Ib + km ω cos(n p q) + R Ib = vb , where L, R, are the phase windings inductance and resistance, Ia , Ib , va , vb are electric currents through stator phase windings and applied voltages, respectively, whereas J , b, km , n p are the rotor inertia, viscous friction coefficient, torque constant, and number of pole pairs. Finally, τ L , ω, q, are load torque, rotor angular velocity, and angular position, i.e., ω = q. ˙ The following scalar function represents the total energy stored in the motor: V (Ia , Ib , ω) =

1 2 1 2 1 2 L I + L Ib + J ω , 2 a 2 2

(8.13)

8.2 Open-Loop Energy Exchange

369

where the first two terms represent the magnetic energy stored in the electrical subsystem whereas the third term stands for the kinetic energy stored in the mechanical subsystem. The time derivative of V is given as ˙ V˙ = Ia L I˙a + Ib L I˙b + ω J ω. According to (8.12) this can be written as V˙ = Ia (km ω sin(n p q) − R Ia + va ) + Ib (−km ω cos(n p q) − R Ib + vb ) +ω(−bω − km Ia sin(n p q) + km Ib cos(n p q) − τ L ). Notice that several cross terms cancel in this expression. Since V represents the energy stored in motor, these cancellations (appearing in V˙ ) represent the energy exchange between the motor electrical and mechanical subsystems. Hence, it can be written: V˙ = −R Ia2 + Ia va − R Ib2 + Ib vb − bω 2 − ωτ L .

(8.14)

Defining the input u = [va , vb , −τ L ] and the output y = [Ia , Ib , ω] , we can write the above expression as ⎡

V˙ = −y  Qy + y  u,

⎤ R 0 0 Q = ⎣ 0 R 0⎦. 0 0 b

(8.15)

Since Q is a positive definite matrix, (8.15) shows that the model (8.12) is output strictly passive (see Definition 2.42) for the output y and input u defined above. It will be shown in this chapter that this property is fundamental to design some simple velocity controllers for PM stepper motors which have a similar structure to the successful industrial control scheme shown in Fig. 8.2.

8.2.2 The Position Model Suppose that the load torque is given as a nonlinear function of position τ L = G(q), , where P(q) is a positive semidefinite scalar function. Using such that G(q) = d P(q) dq these ideas and ω = q, ˙ the dynamical model of a bipolar PM stepper motor is given as J q¨ + bq˙ = −km Ia sin(n p q) + km Ib cos(n p q) − G(q), ˙ L Ia − km q˙ sin(n p q) + R Ia = va , (8.16) ˙ L Ib + km q˙ cos(n p q) + R Ib = vb .

370

8 Bipolar Permanent Magnet Stepper Motor

The following scalar function represents the total energy stored in motor: ˙ q) = V (Ia , Ib , q,

1 2 1 2 1 2 L I + L Ib + J q˙ + P(q), 2 a 2 2

(8.17)

where the last new term represents the potential energy stored in the mechanical subsystem. Proceeding as in the previous section to compute V˙ , we find that the cancellation of cross terms referred before (8.14) appears again. Moreover, an additional cancellation exists between terms ±G(q)q, ˙ which represents exchange between the kinetic and potential energies in the mechanical subsystem. Thus, we find that V˙ = −R Ia2 + Ia va − R Ib2 + Ib vb − bq˙ 2 .

(8.18)

Since b > 0, defining the input u = [va , vb ] and the output y = [Ia , Ib ] , we can write (8.18) as V˙ ≤ −Ry  y + y  u.

(8.19)

Since R is a positive constant, (8.19) shows that model (8.16) is output strictly passive (see Definition 2.42) for the output y and input u defined above. It will be shown in this chapter that this property is fundamental to design some simple position controllers for PM stepper motors which have a similar structure to the successful industrial control scheme shown in Fig. 8.2.

8.3 Velocity Control Proposition 8.1 Consider the dynamical model in (8.12), where τ L is an unknown constant, together with the following control law: σ1 sin(n p q) + qτ ˙ ∗ σ2 cos(n p q), I˜a = Ia − Ia∗ , va = −αa I˜a + σ3 cos(n p q) + qτ ˙ ∗ σ4 sin(n p q), I˜b = Ib − Ib∗ , vb = −αb I˜b + ∗ ∗ τ τ Ia∗ = −  sin(n p q), Ib∗ =  cos(n p q), km km  t τ ∗ = −k p ω˜ − ki ω(s)ds, ˜ ω˜ = ω − ω ∗ ,

(8.20) (8.21) (8.22) (8.23)

0

σ˙ 1 = −Γ1 I˜a sin(n p q), σ˙ 2 = −Γ2 I˜a qτ ˙ ∗ cos(n p q), σ˙ 3 = −Γ3 I˜b cos(n p q), σ˙ 4 = −Γ4 I˜b qτ ˙ ∗ sin(n p q),

(8.24) (8.25)

where km > 0 is an estimate of km > 0, Γi , for i = 1, . . . , 4, are arbitrary positive constants and ω ∗ is a real constant standing for the desired velocity. There always

8.3 Velocity Control

371

exist positive constants k p , ki , αa , αb , such that the whole state is bounded and limt→∞ [ω(t), ˜ I˜a (t), I˜b (t)] = [0, 0, 0] when starting from any initial condition. Proof of Proposition 8.1 Since km > 0 and km > 0, there always exist a positive constant ε such that km = εkm . Hence, using (8.22), (8.23), it is possible to write Ia∗ = (k p ω˜ + ki z − bω ∗ − τ L ) sin(n p q),

(8.26)

Ib∗ = −(k p ω˜ + ki z − bω ∗ − τ L ) cos(n p q),  t 1 ω(s)ds ˜ +  (bω ∗ + τ L ), z= ki 0

(8.27) (8.28)

where k p = εk p and ki = εki . Adding and subtracting some convenient terms in the first expression of (8.12), taking advantage from the fact that ω˙ ∗ = 0, and using (8.26)–(8.28) is found J ω˙˜ + bω˜ = −km I˜a sin(n p q) + km I˜b cos(n p q) − k p ω˜ − ki z.

(8.29)

Adding and subtracting some convenient terms in the second and third expressions of (8.12) is found Rk p L I˙˜a = −(R + αa ) I˜a + km ω˜ sin(n p q) − ω˜ sin(n p q) km   Rki R ∗ ∗ z sin(n p q) + (bω + τ L ) + km ω + σ1 sin(n p q) − km km Ln p L )qτ ˙ ∗ cos(n p q) + (−k p ω˙˜ − ki ω) ˜ sin(n p q), +(σ2 + εkm km Rk p ω˜ cos(n p q) L I˙˜b = −(R + αb ) I˜b − km ω˜ cos(n p q) + km   Rki R ∗ ∗ z cos(n p q) + − (bω + τ L ) − km ω + σ3 cos(n p q) + km km Ln p L  ˙ )qτ ˙ ∗ sin(n p q) + (k ω˜ + ki ω) ˜ cos(n p q). +(σ4 + εkm km p Define: R (bω ∗ + τ L ) − km ω ∗ , km Ln p σ2 − σ2∗ , σ2∗ = − , σ˜ 2 = εkm R σ3 − σ3∗ , σ3∗ = (bω ∗ + τ L ) + km ω ∗ , σ˜ 3 = km Ln p σ4 − σ4∗ , σ4∗ = − . σ˜ 4 = εkm σ1 − σ1∗ , σ1∗ = − σ˜ 1 =

372

8 Bipolar Permanent Magnet Stepper Motor

Then, it is possible to write L I˙˜a = −(R + αa ) I˜a + km ω˜ sin(n p q) −

Rk p

ω˜ sin(n p q)

(8.30)

Rki z sin(n p q) + σ˜ 1 sin(n p q) km L ˙ ∗ cos(n p q) + (−k p ω˜˙ − ki ω) ˜ sin(n p q), +σ˜ 2 qτ km Rk p ω˜ cos(n p q) L I˙˜b = −(R + αb ) I˜b − km ω˜ cos(n p q) + km Rk  + i z cos(n p q) + σ˜ 3 cos(n p q) km L  ˙ ˙ ∗ sin(n p q) + (k ω˜ + ki ω) ˜ cos(n p q). +σ˜ 4 qτ km p

(8.31)

km

−

Thus, the closed-loop dynamics is given by (8.29), (8.30), (8.31) and: ˙ ∗ cos(n p q), σ˙˜ 1 = −Γ1 I˜a sin(n p q), σ˙˜ 2 = −Γ2 I˜a qτ ˙ ∗ sin(n p q). σ˙˜ 3 = −Γ3 I˜b cos(n p q), σ˙˜ 4 = −Γ4 I˜b qτ

(8.32) (8.33)

Notice that this closed-loop dynamics can be written as J ω˙˜ = −(b + k p )ω˜ − km I˜a sin(n p q) + km I˜b cos(n p q) − T L , L I˙˜a = −(R + αa ) I˜a + km ω˜ sin(n p q) + Va , L I˙˜ = −(R + α ) I˜ − k ω˜ cos(n q) + V , b

b

b

m

p

b

˙ ∗ cos(n p q), σ˙˜ 1 = −Γ1 I˜a sin(n p q), σ˙˜ 2 = −Γ2 I˜a qτ ˙ ∗ sin(n p q), σ˙˜ 3 = −Γ3 I˜b cos(n p q), σ˙˜ 4 = −Γ4 I˜b qτ TL =

(8.34) (8.35) (8.36) (8.37) (8.38)

ki z,

Rk p

Rki z sin(n p q) + σ˜ 1 sin(n p q) km km L ˙ ∗ cos(n p q) + (−k p ω˙˜ − ki ω) ˜ sin(n p q), +σ˜ 2 qτ km Rk p Rki ω˜ cos(n p q) + z cos(n p q) + σ˜ 3 cos(n p q) Vb = km km L  ˙ ˙ ∗ sin(n p q) + (k ω˜ + ki ω) ˜ cos(n p q), +σ˜ 4 qτ km p

Va = −

ω˜ sin(n p q) −

and the state is given as ξ = [ω, ˜ z, I˜a , I˜b , σ˜ 1 , σ˜ 2 , σ˜ 3 , σ˜ 4 ] ∈ R8 . Notice that the closed-loop dynamics (8.34)–(8.38) is almost identical to the open-loop dynam˜ I˜a , I˜b , T L , Va , Vb . An important ics (8.12) if we replace ω, Ia , Ib , τ L , va , vb , by ω,

8.3 Velocity Control

373

advantage is that viscous friction and electric resistance have been rendered larger, i.e., instead of b and R in (8.12) we have b + k p , R + αa and R + αb in (8.34)– (8.36). Moreover, we have four additional equations in (8.37), (8.38), which can be seen as the integral parts of two nonlinear PI electric current controllers. This observations motivate us to propose the following Lyapunov function candidate for stability analysis purposes: 1 ˜2 1 ˜2 L I + L Ib + Vσ (σ˜ 1 , σ˜ 2 , σ˜ 3 , σ˜ 4 ) + Vω (ω, ˜ z), 2 a 2 (8.39) 1 2 1  Vω (ω, ˜ z) = J ω˜ + [ki + γ(b + k p )]z 2 + γ J z ω, ˜ (8.40) 2 2 i=4

1 1 2 Vσ (σ˜ 1 , σ˜ 2 , σ˜ 3 , σ˜ 4 ) = σ˜ , (8.41) 2 Γi i i=1

˜ z, σ˜ 1 , σ˜ 2 , σ˜ 3 , σ˜ 4 ) = V ( I˜a , I˜b , ω,

where γ is a positive constant. Reason for the first two terms in V defined in (8.39) is to take into account the “energy” stored in the electrical subsystem which is composed of the “magnetic energy” in both motor phases. Function Vσ is included to take into account the “energy” stored in the nonlinear integral terms of the PI electric current ˜ z), intended to take into account the “energy” stored in controller. Function Vω (ω, the mechanical subsystem, was introduced in (3.44) and the reader is referred to that part of Chap. 3 for a complete explanation of each one of the terms that compose it. ˜ z) is positive definite and radially unbounded In Appendix A.1 is proven that Vω (ω, if: J > 0, ki + γ(b + k p ) − J γ 2 > 0.

(8.42)

Thus, V defined in (8.39) is positive definite and radially unbounded if (8.42) is satisfied and L > 0, Γi > 0, for i = 1, . . . , 4. Taking advantage of several straightforward cancellations which include those reflecting the energy exchange between the mechanical and the electrical subsystems referred in Sect. 8.2.1, it is not difficult to find that the time derivative of V , defined in (8.39), along the trajectories of the closed-loop system (8.34)–(8.38) is given as Rk p

Rki ˜ Lk  I˜a ω˜ sin(n p q) − Ia z sin(n p q) − i I˜a ω˜ sin(n p q) km km km   Lk p I˜a −bω˜ − km I˜a sin(n p q) + km I˜b cos(n p q) − k p ω˜ − ki z sin(n p q) − J km Rk p Rki ˜ Lk  I˜b ω˜ cos(n p q) + Ib z cos(n p q) + i I˜b ω˜ cos(n p q) −(R + αb ) I˜b2 + km km km   Lk p I˜b −bω˜ − km I˜a sin(n p q) + km I˜b cos(n p q) − k p ω˜ − ki z cos(n p q) + J km

V˙ = −(R + αa ) I˜a2 −

374

8 Bipolar Permanent Magnet Stepper Motor

−(b + k p )ω˜ 2 + γ J ω˜ 2 − γkm I˜a z sin(n p q) + γkm I˜b z cos(n p q) − γki z 2 . (8.43) Using the facts that ±uv ≤ |u| |v|, for all u, v ∈ R, and | sin(y)| ≤ 1, | cos(y)| ≤ 1, for all y ∈ R, it is possible to write ˜ |z|, | I˜a |, | I˜b |] , V˙ ≤ −x  Qx, x = [|ω|,

(8.44)

where entries of matrix Q are defined as Lk p Lk p Q 11 = b + k p − γ J, Q 22 = γki , Q 33 = R + αa − , Q 44 = R + αb − , J J



Rk p Lk p (b + k p ) Lki

, − + Q 12 = Q 21 = 0, Q 13 = Q 31 = −

2km 2J km 2km



Rk Lk p ki γkm

, Q 23 = Q 32 = Q 24 = Q 42 = −

i − + 2km 2J km 2



Rk p Lk p (b + k p ) Lk p Lki

Q 14 = Q 41 = − , Q 34 = Q 43 = − − − .

2km 2J km 2km J Matrix Q is positive definite if and only if its four leading principal minors are positive which is always possible by selecting large enough positive controller gains k p , ki , αa , αb and a small enough γ > 0. Hence, λmin (Q) > 0 and (8.44) implies that the closed-loop is stable, i.e., [ω, ˜ z, I˜a , I˜b , σ˜ 1 , σ˜ 2 , σ˜ 3 , σ˜ 4 ] ∈ L8∞ . Using this in (8.34)–(8.38), we conclude that [ω, ˜˙ z˙ , I˙˜a , I˙˜b ] ∈ L4∞ . On the other hand, integrating (8.44) it is found 

V (0) ≥ λmin (Q)



t

x(r ) 2 dr ,

0

which implies that [ω, ˜ z, I˜a , I˜b ] ∈ L42 . Thus, we can invoke Corollary 2.32 to conclude that limt→∞ [ω(t), ˜ z(t), I˜a (t), I˜b (t)] = [0, 0, 0, 0]. This completes the proof of Proposition 8.1. We stress that conditions that ensure this result are summarized by (8.42), L > 0, Γi > 0, for i = 1, . . . , 4, and the four leading principal minors of matrix Q defined in (8.44) are positive. Remark 8.2 Useful to obtain (8.43) are the cancellations of cross terms d 1 ˜ ˜ ω(−k ˜ ˜ 2 ), the time derivative of m Ia sin(n p q) + km Ib cos(n p q)) appearing in dt ( 2 J ω the “kinetic energy” stored in the mechanical subsystem, and terms km I˜a ω˜ sin(n p q) − km I˜b ω˜ cos(n p q) appearing in dtd ( 21 L I˜a2 + 21 L I˜b2 ), the time derivative of the “magnetic energy” stored in the electrical subsystem. These cancellations are due to the natural energy exchange between the electrical and the mechanical subsystems. They are

8.3 Velocity Control

375

obvious consequences of the almost identical structure of the closed-loop dynamics in (8.34)–(8.38) and the open-loop dynamics (8.12). Remark 8.3 The closed-loop mechanical subsystem dynamics given in (8.34) can be written as J ω˙˜ = −(b + k p )ω˜ + τe − ki z,

(8.45)

τe = −km I˜a sin(n p q) + km I˜b cos(n p q).

(8.46)

if we define

Notice that τe represents the difference between the actual and the desired torques. Hence, it is possible to write  ˜ |z| ] Vω ≤ −[ |ω|,

b + k p − γ J 0 0 γki



 |ω| ˜ + (ω˜ + γz)τe , |z|

(8.47)

where function Vω is defined in (8.40). On the other hand, the time derivative of Ve = 21 L I˜a2 + 21 L I˜b2 + Vσ (σ˜ 1 , σ˜ 2 , σ˜ 3 , σ˜ 4 ) along the trajectories of the closed-loop electrical subsystem dynamics (8.35)–(8.38) contains the quadratic negative terms −(R + αa ) I˜a2 − (R + αb ) I˜b2 . Thus, when computing V˙ = V˙ω + V˙e , we realize that instrumental for the stability result in Proposition 8.1 are the following features, which are very similar to those in Remark 3.4: • The scalar function Vω is a strict Lyapunov function when the desired torque is the input of the closed-loop mechanical subsystem dynamics, given in (8.45), i.e., when τe = 0. • The coefficients of the negative terms −(R + αa ) I˜a2 − (R + αb ) I˜b2 , appearing in V˙ can be enlarged arbitrarily. This is important to dominate the cross terms in V˙ depending on I˜a and I˜b when I˜a = 0 and I˜b = 0. • Cancellation of several cross terms belonging to V˙ω and V˙e explained in Remark 8.2. Notice that all of these features are possible thanks to the passivity properties of the open-loop motor model described in Sect. 8.2.1. Furthermore, the above proof contains the main ideas of procedure described in Sect. 2.4. As pointed out in Remark 3.4, τe is given as a nonlinear function of the electrical dynamics error for stepper motors. However, the involved nonlinearities are easy to handle in the stability analysis that we have presented above. This is an important advantage of the energy-based design that we have presented in this section. Remark 8.4 In Sect. 8.1.3 was presented the standard control scheme for PM stepper motors which is depicted again in Fig. 8.5 for the ease of reference. In this control scheme the PI blocks stand for classical PI controllers. In Fig. 8.6 is depicted the control scheme introduced in Proposition 8.1. We stress that aside from the nonlinear integral terms of the PI electric current controller in

376

8 Bipolar Permanent Magnet Stepper Motor

Fig. 8.5 Standard closed-loop control of a PM stepper motor

Fig. 8.6 Control scheme introduced in Proposition 8.1. NPI blocks refer to (8.20), (8.21), (8.24), (8.25)

both phases a and b of the motor, controller in Proposition 8.1 is identical to standard control of PM stepper motors. The advantage of our proposal is that it is provided with a formal global stability proof. We stress that nonlinear PI electric current controllers (in the form of adaptive terms) are required because of the non-constant character of the desired electric current.

8.3.1 Simulation Results In this section we present a numerical example to give some insight on the achievable performance when controller in Proposition 8.1 is employed. To this aim, we use the numerical values of the bipolar PM stepper motor model 4018X– 07 from Lin Engineering: R = 5 [Ohm], L = 0.006 [H], km = 0.1 [Nm/A], J = 187.2 × 10−6 [kg m2 ], b = 0.002 [N m/(rad/s)], n p = 50. The controller gains were chosen to be k p = 0.1, ki = 5, ε = 1.2, αa = 200, αb = 200, Γ1 = Γ2 = Γ3 = Γ4 = 1 × 104 . These controller gains were found to satisfy all of the stability conditions established in the proof of Proposition 8.1 using γ = 10. The desired velocity ω ∗ was chosen as a ramp with slope 1000 [(rad/s)/s], starting with a zero value at t = 0 and becoming constant when reaching 8 [rad/s]. The load torque τ L = 0.25 [Nm] was considered to be a step signal which appears for t ≥ 0.15 [s]. In order to perform realistic simulations, both of these values, i.e., ω ∗ and τ L , were chosen to satisfy the capabilities of the H bridge LMD18200 from Texas Instruments which we have found in a previous work [266] to be well suited as a driver for this motor. In this respect, the H bridge LMD18200 delivers voltages in the range ±60 [V] and a maximal electric current of 3 [A]. In Fig. 8.8 we verify that Ia and Ib remain within this range. Moreover, in Fig. 8.7 we also realize that va and vb remain within the range ±60 [V]. This, however, is possible because va and vb

8.3 Velocity Control

377

Fig. 8.7 Simulation results when controller in Proposition 8.1 is employed

are saturated at ±60 [V]. Otherwise, a very large initial voltage spike appears. Using additional simulations we have verified that performance remains the same whether va and vb are saturated or not. We believe that this is because the peak values that are observed are applied during very short time intervals. On the other hand, notice that the motor velocity tracks very close the ramp and overlaps the desired velocity when the latter is constant. On the other hand, the torque disturbance is well rejected despite this requires a electric current that is close to the maximal limit of 3 [A]. Finally, in Fig. 8.8 we also present the evolution of the adaptive variables σ2 , σ3 , σ4 . Notice that these variables have a slower rate of convergence despite σ1 , the large adaptive gains Γ1 = Γ2 = Γ3 = Γ4 = 1 × 104 that are employed. This slow rate of convergence, however, has not any effect on the velocity performance.

378

8 Bipolar Permanent Magnet Stepper Motor

Fig. 8.8 Simulation results when controller in Proposition 8.1 is employed (cont.)

8.4 Position Control In this section we consider the PM stepper motor model presented in (8.16) with G(q) replaced by g(q), i.e., J q¨ + bq˙ = −km Ia sin(n p q) + km Ib cos(n p q) − g(q), ˙ L Ia − km q˙ sin(n p q) + R Ia = va , (8.48) ˙ L Ib + km q˙ cos(n p q) + R Ib = vb . The function g(q) represents a position dependent mechanical load which is assumed to possess the following properties:

8.4 Position Control

379

|g(x) − g(y)| ≤ k g |x − y|, ∀x, y ∈ R, (Lipschitz condition)

dg(q)

,

k g > max q∈R dq dU (q) g(q) = , |g(q)| ≤ k  , ∀q ∈ R, dq

(8.49) (8.50) (8.51)

with U (q) the potential energy and k g , k  , some positive constants. Proposition 8.5 Consider the dynamical model in (8.48) together with the following control law: va = −αa I˜a + σ1 sin(n p q) + qτ ˙ ∗ σ2 cos(n p q), I˜a = Ia − Ia∗ , σ3 cos(n p q) + qτ ˙ ∗ σ4 sin(n p q), I˜b = Ib − Ib∗ , vb = −αb I˜b + ∗ ∗ τ τ Ia∗ = −  sin(n p q), Ib∗ =  cos(n p q), km km  t τ ∗ = −k p q˜ − kd q˙ − ki q(s)ds, ˜ q˜ = q − q ∗ ,

(8.52) (8.53) (8.54) (8.55)

0

σ˙ 1 = −Γ1 I˜a sin(n p q), σ˙ 2 = −Γ2 I˜a qτ ˙ ∗ cos(n p q), σ˙ 3 = −Γ3 I˜b cos(n p q), σ˙ 4 = −Γ4 I˜b qτ ˙ ∗ sin(n p q),

(8.56) (8.57)

where km > 0 is an estimate of km > 0, Γi , for i = 1, . . . , 4, are arbitrary positive constants and q ∗ is a real constant standing for the desired position. There always exist positive constants k p , kd , ki , αa , αb , such that the whole state is bounded and limt→∞ [q(t), ˜ q(t), ˙ I˜a (t), I˜b (t)] = [0, 0, 0, 0] when starting from any initial condition. Proof of Proposition 8.5 The proof is very similar to the proof of Proposition 8.1. For the sake of completeness we present the main steps. Defining km = εkm , ε > 0, k p = εk p , kd = εkd , ki = εki , k p = k p + ki , ki = ki /α, α > 0: R g(q ∗ ), km Ln p σ2 − σ2∗ , σ2∗ = − , σ˜ 2 = εkm R σ3 − σ3∗ , σ3∗ = g(q ∗ ), σ˜ 3 = km Ln p σ4 − σ4∗ , σ4∗ = − . σ˜ 4 = εkm σ˜ 1 = σ1 − σ1∗ , σ1∗ = −

and

380

8 Bipolar Permanent Magnet Stepper Motor



t

1 ∗  g(q ), k 0 i ˙ ∗ cos(n p q), σ˙˜ 1 = −Γ1 I˜a sin(n p q), σ˙˜ 2 = −Γ2 I˜a qτ ˙ ∗ sin(n p q), σ˙˜ 3 = −Γ3 I˜b cos(n p q), σ˜˙ 4 = −Γ4 I˜b qτ z=

(αq˜ + q)ds ˙ + q(0) ˜ +

(8.58) (8.59) (8.60)

it is found J q¨ = −(b + kd )q˙ − km I˜a sin(n p q) + km I˜b cos(n p q) − G, L I˙˜a = −(R + αa ) I˜a + km q˙ sin(n p q) + Va L I˙˜ = −(R + α ) I˜ − k q˙ cos(n q) + V , b

G=

k p q˜

b ki z

b

m

p

b

(8.61) (8.62) (8.63)

∗

+ + g(q) − g(q ),  R   −k p q˜ − kd q˙ − ki z sin(n p q) + σ˜ 1 sin(n p q) Va = km L ˙ ∗ cos(n p q) + (−k p q˙ − kd q¨ − ki (αq˜ + q)) ˙ sin(n p q), +σ˜ 2 qτ km  R   k q˜ + kd q˙ + ki z cos(n p q) + σ˜ 3 cos(n p q) Vb = km p L  ˙ ∗ sin(n p q) + (k q˙ + kd q¨ + ki (αq˜ + q)) ˙ cos(n p q). +σ˜ 4 qτ km p For stability analysis purposes, the following Lyapunov function candidate is proposed: 1 1 V ( I˜a , I˜b , q, ˜ q, ˙ z, σ˜ 1 , σ˜ 2 , σ˜ 3 , σ˜ 4 ) = L I˜a2 + L I˜b2 + Vσ (σ˜ 1 , σ˜ 2 , σ˜ 3 , σ˜ 4 ) + Vq (q, ˜ q, ˙ z), 2 2 Vq (q, ˜ q, ˙ z) = V1 (q, ˜ q) ˙ + V2 (z, q) ˙ + V3 (q, ˜ z), (8.64) 1 α 1  2 2 2 2  2 V1 (q, ˜ q) ˙ = J (q˙ + 2αq) ˜ − α J q˜ + (b + kd )q˜ + k p q˜ 4 2 2 ∗ +U (q) − U (q ∗ ) − qg(q ˜ ), 1 αβ  2 1  2 V2 (z, q) ˙ = J (q˙ + 2αβz)2 − α2 β 2 J z 2 + k z + ki z , 4 2 d 2 β   2 V3 (z, q) ˜ = (k p − αkd )(z − q) ˜ , 2

where α and β are two positive constants. The function Vq (q, ˜ q, ˙ z) was introduced in (3.58) and the reader is referred to that part of Chap. 3 for an explanation of the components of this function. On the other hand, the function Vσ is identical to that introduced in (8.41). The time derivative of function V , defined in (8.64), along the trajectories of the closed-loop dynamics (8.58), (8.59), (8.60), (8.61), (8.62), and (8.63) is found to be bounded as ˙ |q|, ˜ |z|, | I˜a |, | I˜b |] , V˙ ≤ −x  Q  x, x = [|q|,

(8.65)

8.4 Position Control

381

where the first three rows and columns of matrix Q  are identical to the three first rows and columns of matrix Q defined in (B.32) and the remaining entries are given as Lk  Lk  Q 44 = R + αa − d , Q 55 = R + αb − d J J

    

L(k Rk (b + k ) Lk p + ki )

d d d  

− + Q 14 = Q 41 = −

, 2km 2J km 2km



Rk p Lkd k p αLki

Lkd k g αkm − Q 24 = Q 42 = −

+ − + , 2k 2J km 2 2km 2J km

m

αβkm Rki Lk  k  + − d i

, Q 34 = Q 43 = −

2 2km 2J km

  

Lk Rk Lk Lk  (b + kd )

p Q 51 = Q 15 = −

d + + i − d

. 2km 2km 2km 2J km

αkm Lkd k p Lkd k g Rk p αLki

− Q 52 = Q 25 = −

+ − . + 2 2km 2km 2J km 2J km

αβkm Rki Lk  k  + − d i

. Q 53 = Q 35 = −

2 2km 2J km  Lk Q 54 = Q 45 = − d . J It is always possible to find some positive constants k p , kd , ki , α, β, αa , αb such that the five leading principal minors of matrix Q  are positive and, hence, λmin (Q  ) > 0 is ensured. Thus, we can proceed as at the end of the proof of Proposition 8.1 to con˜ q(t), ˙ z(t), I˜a (t), I˜b (t)] = [0, 0, 0, 0, 0] . This completes clude that limt→∞ [q(t), the proof of Proposition 8.5. We stress that conditions that ensure this result are summarized by k p > k g , α > 0,

1 (b + kd ) > αJ, 2

αβ  1 k + k  > α2 β 2 J, β > 0, k p > αkd , 2 d 2 i ⎞ ⎛ Q 11 Q 12 Q 13 Q 11 > 0, Q 11 Q 22 − Q 12 Q 21 > 0, det ⎝ Q 21 Q 22 Q 23 ⎠ > 0, Q 31 Q 32 Q 33 where Q i j , for i, j = 1, 2, 3, are entries of matrix Q defined in (B.32), L > 0, Γi > 0, for i = 1, . . . , 4, and the five leading principal minors of matrix Q  defined in (8.65) are positive. Remark 8.6 The reader can verify that similar observations as those in Remarks 8.2 and 8.3 also apply in the result in Proposition 8.5.

382

8 Bipolar Permanent Magnet Stepper Motor

Fig. 8.9 Standard closed-loop position control of a PM stepper motor

Fig. 8.10 Control scheme introduced in Proposition 8.5. NPI blocks refer to (8.52), (8.53), (8.56), (8.57)

On the other hand, in Sect. 8.1.3 was presented the standard position control scheme for PM stepper motors which is depicted again in Fig. 8.9 for the ease of reference. The control scheme introduced in Proposition 8.5 is depicted in Fig. 8.10 Notice that the only difference between these two control schemes is the nonlinear PI electric current controllers employed in Proposition 8.5 instead of the classical PI electric current controllers employed in the standard position control scheme. The advantage of our proposal is that it is provided with a formal global stability proof. This is the main contribution in this part of the book. Remark 8.7 Notice that the controllers in Propositions 8.1 and 8.5 are designed on the basis of original coordinates, i.e., a coordinate transformation similar to the dq transformation in PM synchronous motors and induction motors is not required. This is important to stress since a control method based on such a coordinate transformation also exists, see [291] for instance.

8.4.1 Simulation Results In this section we present a numerical example to give some insight on the achievable performance when controller in Proposition 8.5 is employed. We use the numerical values of the bipolar PM stepper motor described in Sect. 8.3.1. The only difference is that we assume now that a simple pendulum is fixed at the motor shaft such that, see (8.48), g(q) = mgl sin(q) where m = 0.1 [kg], g = 9.81 [m/s2 ], l = 0.1 [m]. Moreover, the mechanism inertia is given as J = 187.2 × 10−6 [kg m2 ]+ml 2 + 13 m(2l)2 , where 187.2 × 10−6 [kg m2 ] is inertia of the motor rotor.

8.4 Position Control

383

Fig. 8.11 Simulation results when controller in Proposition 8.5 is employed

The controller gains were chosen to be k p = 0.7, kd = 0.1, ki = 1, ε = 1.2, αa = 450, αb = 450, Γ1 = Γ2 = Γ3 = Γ4 = 1 × 104 . These controller gains were found to satisfy all of the stability conditions established in the proof of Proposition 8.5 using α = 2.5 and β = 4. The desired position q ∗ = π2 [rad] was commanded as a step signal which changes the desired position from 0.55 [rad] to π2 [rad] at t = 0 [s]. The initial position was set to 0.55 [rad]. These initial values were chosen in order to perform realistic simulations, i.e., to satisfy the electrical capabilities of the H bridge LMD18200 from Texas Instruments which we have found in a previous work [266] to be well suited as a driver for this motor. In this respect, the H bridge LMD18200 delivers voltages in the range ±60 [V] and a maximal continuous electric current of 3 [A] with maximal isolated peaks of 6 [A]. In Figs. 8.11 and 8.12 we verify that Ia and Ib have initial

384

8 Bipolar Permanent Magnet Stepper Motor

Fig. 8.12 Simulation results when controller in Proposition 8.5 is employed (cont.)

isolated peaks that remain within this range and continuous values that are within ±3 [A]. Since the voltages va and vb have very large initial peaks, we have saturated them to the range ±60 [V] and it is because of this that these voltages remain within the range ±60 [V] in Figs. 8.11 and 8.12. Using additional simulations, we have verified that performance remains the same whether such voltage saturation is employed or not. We believe that this is because the peak values observed in va and vb are applied only during very short time intervals. On the other hand, notice that the position response has about 8.7% overshoot and 0.32 [s] rise time, which we consider is a fast transient response. Finally, in Fig. 8.12 we also present the evolution of the adaptive variables σ2 , σ3 , σ4 , which, we observe, converge to constant values when σ1 ,  tmotor stops. All initial conditions were chosen to be zero, excepting q = 0.55 [rad] 0 q˜ dt = −0.06, Ia = −0.485 [A] Ib = 0.1 [A].

8.5 A Practical PM Stepper Motor

385

Fig. 8.13 Two views of the stator windings. The coil shown at the right was wound in the empty space in the stator to form one of the stator windings

8.5 A Practical PM Stepper Motor This is a bipolar PM stepper motor with two-phase windings at stator. These phase windings are designated as a and b. Each one of these windings is circularly wound defining two identical tori. The axis of these toroidal windings is the same as the rotor axis and they are placed one on the other to compose the cylindrical stator, see Fig. 8.13. It is important to say that a ring made in a ferromagnetic material separates these phase windings in the stator. This allows each one of these windings to possess its own magnetic circuit which is isolated from the magnetic circuit of the other phase winding. Thus, the phase windings a and b can be considered to be magnetically decoupled. The cylindrical internal surface of stator is divided into two cylindrical surfaces. One of these surfaces constitutes the core of phase winding a and the other surface constitutes the core of phase winding b. Both of these surfaces are made in a ferromagnetic material which is divided into 24 triangular pieces which are embedded in plastic. See Fig. 8.14 to observe how these triangular pieces lay on the stator surface. ◦ = 15◦ angle exists between adjacent triangular pieces in each phase. Hence, a 360 24 Notice that adjacent triangular pieces in each phase have opposite orientation.

386

8 Bipolar Permanent Magnet Stepper Motor

Fig. 8.14 The cylindrical internal surface of stator

The rotor is composed of a cylindrical permanent magnet possessing 12 pole pairs (n p = 12). Hence, 12 N poles and 12 S poles are alternatively placed on the rotor 360◦ = 15◦ angle exists between adjacent N and S poles. surface. This means that a 2×12 Rotor is cylindrically shaped and each magnetic pole has the form of a strip which lays parallel to the rotor axis. These poles are buried in rotor and they cannot be seen (see Fig. 8.15). However, the above description is concluded by using another permanent magnet to discover the poles shapes and locations. In Fig. 8.14 is shown a developed view of the relative positions of the rotor permanent magnet poles and the triangular ferromagnetic pieces in the stator when the mechanical rotor position is q = 0 (the electrical position θ = 0, since θ = n p q). Since the rotor poles lay parallel to the rotor axis, the magnetic flux is oriented orthogonal to rotor axis along the internal stator surface for phase b as it is shown in Fig. 8.14. This means that the flux linkage produced by rotor in phase b is zero λb = 0 in this figure. Recall that axis of the a and b phase windings is the rotor axis. On the contrary, the orientation of the triangular ferromagnetic pieces at stator for phase a together with the relative position of the rotor poles produce a deviation of the magnetic flux at the air gap toward a direction which is parallel to the rotor axis

8.5 A Practical PM Stepper Motor

387

Fig. 8.15 Two views of the rotor

(see Fig. 8.14). This means that flux linkage produced by rotor in phase a is positive, i.e., λa > 0. Notice that such a magnetic flux deviation is not produced for phase b in Fig. 8.14 because the relative position of the rotor poles and the triangular pieces (the same N or S pole lays on both sides of the plastic gap between adjacent triangular pieces) cannot deviate the magnetic flux toward a direction parallel to the rotor axis. Suppose that a positive electric current flows through phase a (Ia > 0) and a zero electric current flows through phase b (Ib = 0). This produces a positive flux L Ia > 0 through the a phase winding in the sense shown in Fig. 8.14. Hence, magnetic fluxes produced by rotor λa and by stator L Ia are in the same sense which forces the rotor to stay at q = 0. Any magnetic torque is not generated by phase b since L Ib = 0. Still consider Fig. 8.14. If Ia = 0 and Ib > 0, a positive flux L Ib > 0 through the phase winding b is produced. This forces rotor to move to configuration shown in Fig. 8.16 because fluxes L Ib and λb are orthogonal under these conditions in Fig. 8.14 and these fluxes have the same sense in Fig. 8.16.2 Thus, rotor moves a single step. Since, the ferromagnetic triangular pieces in phase b lay at intermediate

2 In

Remark 3.1 is explained that a maximal torque is produced when two magnetic fields are orthogonal and a zero torque is produced when the magnetic fields are parallel.

388

8 Bipolar Permanent Magnet Stepper Motor

Fig. 8.16 The cylindrical internal surface of stator after one step

◦

positions with respect to triangular pieces in phase a, then q = 152 = 7.5◦ (θ = 90◦ ) in Fig. 8.16 and 7.5◦ represents the angular increment per step for this motor. Motor configurations when q = 2 × 7.5◦ = 15◦ (θ = 180◦ , Ia < 0, Ib = 0) and q = 3 × 7.5◦ = 22.5◦ (θ = 270◦ , Ia = 0, Ib < 0) are shown in Figs. 8.17 and 8.18. When q = 4 × 7.5◦ = 30◦ (θ = 360◦ , Ia > 0, Ib = 0) we retrieve configuration in Fig. 8.14. Observing the variations of λa and λb shown in Figs. 8.14, 8.15, 8.16, 8.17, and 8.18, we conclude that the expressions in (8.1), for flux linkages produced by permanent magnet at rotor, still stand for this PM stepper motor with n p = 12, i.e., λa = λ0 cos(n p q), λb = λ0 sin(n p q), n p = 12,

(8.66)

where λ0 > 0 is the maximal flux linkage. On the other hand, as explained above, the phase windings a and b are magnetically decoupled. Also notice that they are identical. Furthermore, since rotor is an uniform cylinder made in a ferromagnetic material and the stator triangular ferromagnetic pieces have not relative movement with respect to the a and b phase windings, it is reasonable to consider that L Ia and L Ib , with L a positive constant, are the flux

8.5 A Practical PM Stepper Motor

389

Fig. 8.17 The cylindrical internal surface of stator after two steps

linkages produced at the air gap by the stator phase windings (as correctly guessed in the above description). Thus, the total flux linkages produced by the rotor permanent magnet and the electric currents through the stator phases are given as in (8.2), i.e., ψa = L Ia + λa , ψb = L Ib + λb .

(8.67)

Since each phase is constituted by a series RL circuit placed in a magnetic field produced by permanent magnets, use of Kirchhoff’s Voltage Law and Faraday’s Law allows to retrieve (8.3), i.e., ψ˙a + R Ia = va , ψ˙ b + R Ib = vb ,

where va and vb are voltages applied at phases a and b, respectively, and it is assumed that resistance R of both phase windings are the same. Hence, replacing (8.66) and (8.67) in the previous expression we retrieve

390

8 Bipolar Permanent Magnet Stepper Motor

Fig. 8.18 The cylindrical internal surface of stator after three steps

L I˙a − km ω sin(n p q) + R Ia = va , L I˙b + km ω cos(n p q) + R Ib = vb , where the fact that q˙ = ω has been used and km = n p λ0 has been defined. On the other hand, according to D’Alembert’s principle, the generated torque (applied on the rotor in the sense of rotor velocity ω) is given as in (8.4), i.e.,   ∂ 1 2 1 2 L I + L Ib + Ia λa + Ib λb , τ = ∂q 2 a 2 = −km Ia sin(n p q) + km Ib cos(n p q), since L, Ia and Ib are independent of the rotor position. Finally, according to Newton’s Second Law: J ω˙ = −bω + τ − τ L , where the positive constants J and b represent rotor inertia and viscous friction coefficient, respectively, τ L represents load torque (supposed to be applied in the

8.5 A Practical PM Stepper Motor

391

opposite direction of the rotor velocity ω). Thus, the PM stepper motor dynamic model is given as in (8.5), i.e., J ω˙ + bω = −km Ia sin(n p q) + km Ib cos(n p q) − τ L , ˙ L Ia − km ω sin(n p q) + R Ia = va , L I˙b + km ω cos(n p q) + R Ib = vb .

Chapter 9

Brushless DC-Motor

Brushless DC (BLDC) motors are an attractive alternative to induction motors because the employment of rare earth permanent magnets improves the motor performance and increases the power density. Moreover, BLDC motors can be controlled at a reduced cost with respect to induction motors [80]. On the other hand, BLDC motors have not any brushes reducing the operation cost with respect to use of standard PM brushed DC-motors and its construction is simpler and cheaper than that of PM synchronous motors [77]. In this respect, although PM synchronous motors and BLDC motors have many similarities; they both have permanent magnets on the rotor and require alternating stator currents to produce constant torque, for instance, for application considerations these two motor drives have to be differentiated [224]. The standard control scheme for BLDC motors, which is conceived under steadystate motor operation conditions,1 is based only on intuitive ideas and any formal stability study has not been presented to explain why it works well and, in particular, how to select the gains of the PI electric current controllers. Moreover, an attractive feature of the traditional control scheme is that it requires simple Hall effect sensors (instead of encoders) to switch-on and switch-off the desired current at each phase. However, such an approach only works if velocity is controlled and position control tasks demand the use of encoders. On the other hand, several recent control schemes have been proposed in the literature [85, 153, 213, 233, 238, 278, 284] which, however, are not provided with formal stability proofs taking into account the complete motor model (see [151] for a practical point of view on BLDC motor control). A remarkable exception to this trend is the work in [80]. In that nice paper, the passivity-based control approach introduced in [204] is extended to control velocity in BLDC motors providing a global asymptotic stability proof. However, some important details have been for1 See

[55] and Sect. 9.1.2 in the present chapter. Also see [199] for a practical point of view.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 V. M. Hernández-Guzmán et al., Energy-Based Control of Electromechanical Systems, Advances in Industrial Control, https://doi.org/10.1007/978-3-030-58786-4_9

393

394

9 Brushless DC-Motor

gotten. (i) The time derivatives of the desired electric currents that are proposed are assumed to exist. However, because of the trapezoidal shape of torque generated by each phase, the time derivatives of the desired electric currents that are proposed by [80] are not defined at some specific rotor positions. (ii) The control scheme proposed by [80] requires to feedback the complete expression for the time derivative of the desired electric currents. This is an important drawback because the number of required computations is very large which commonly results in performance deterioration because of numerical errors and noise amplification, as remarked in [204]. Furthermore, it is pointed out in [222] that the electric drives community is not enthusiastic with complex controllers. Thus, it is important to design controllers that are simple to implement but provided with formal stability proofs resulting in stability conditions useful to understand how the controller works. (iii) Aside from point (i), the expressions presented in [80] for the time derivatives of the desired currents are incomplete, i.e., the required computations are even more complex. (iv) The viscous friction coefficient and the load torque are assumed to be known in [80]. These are, perhaps, consequences of the facts that velocity measurements are not allowed and that a proportional velocity controller is considered instead of a more robust PI velocity controller. (v) Velocity control of a single motor is considered, i.e., the position control problem is not solved. Motivated by [80], in the present chapter we introduce two controllers: one for velocity regulation and another for position regulation. We present formal stability proofs which ensure global boundedness of the state and global convergence to a ball which can be rendered arbitrarily small. This chapter is organized as follows. In Sect. 9.1 we study the dynamic model of BLDC motors. As a result, the standard control scheme for BLDC motors is explained. In Sect. 9.2 is demonstrated, from the open-loop model of BLDC motors, that energy exchange exists between the electrical and the mechanical subsystem, a previous step to apply our novel passivity-based approach. In Sects. 9.3 and 9.4 are presented two controllers, for velocity and position regulation, which are provided with formal global stability proofs Finally, in Sect. 9.5 is modeled a practical BLDC motor.

9.1 Motor Modeling

BLDC motors are three-phase synchronous motors with a permanent magnet fixed at the rotor which does not present any saliency. The stator three phases are star connected. The back electromotive force in BLDC motors has a trapezoidal shape because of uniformly distributed phase windings on the stator [55, 150]. This is the main difference with respect to PM synchronous motors, studied in Sect. 4.1, whose back electromotive force is sinusoidal because of sinusoidally distributed phase windings on the stator. This fact explains why BLDC motors are also called trapezoidal back electromotive force PM synchronous motors.

9.1 Motor Modeling

395

Fig. 9.1 A one pole pair BLDC motor

9.1.1 Dynamic Model In Fig. 9.1 is depicted the stator and the rotor of a one pole pair BLDC motor [55]. The rotor is a cylinder made in soft iron. The permanent magnet is fixed on the surface of this cylinder in the form of a uniform layer. One half of this layer is the north pole and the other half of the layer is the south pole. Although the two halves of the layer are physically separated on the rotor, this separation is small enough to obtain a very good approximation from the modeling point of view if it is assumed that this separation does not exist and, hence, that the rotor has not any saliency. This construction results in a uniform distribution of the rotor magnetic flux in the air gap. In Fig. 9.1can be seen that a 120◦ separation exists between the stator phase windings. Each phase winding is composed of the same number of turns N which are uniformly distributed on the sections of the stator that are shown in Fig. 9.1. This means that the density of turns (turns/angle increment) is constant for all phases within the sections of the stator shown in Fig. 9.1. It is important to stress that each loop of each phase winding is placed on a plane passing through the motor axis. For instance, for phase 1, this plane forms an angle of 2π/3[rad], with the positive horizontal axis, for the first loop and forms an angle of π/3[rad] for the last loop in phase 1. This is indicated in Fig. 9.1 by the points where the electric current I1 enters and exits phase 1. The uniform distribution of the phase windings as well as the rotor non-salient structure allow to show that the phases flux linkages are rotor position independent and that are given as [55] ⎡

λ = L I,

⎤ L s −M −M L = ⎣ −M L s −M ⎦ , −M −M L s

I = [I1 , I2 , I3 ] ,

(9.1)

where the symbols Ii , for i = 1, 2, 3, represent the electric currents flowing through the phases i = 1, 2, 3, and L s and M are the self-inductance and the mutual induc-

396

9 Brushless DC-Motor

tance of the phases and they relate through L s = (7/3)M. The 3 × 3 symmetric matrix L is known as the inductance matrix and using the given relationship between L s and M it is easy to verify that L is a positive definite matrix. The phases flux linkages due to the rotor magnetic field are given as [55]      2π 2π  , λR q + Γ (q) = E p λ R (q), λ R q − , 3 3 ⎧ 6 2 5π − π6 ≤ q < π6 ⎪ ⎪ −π q +  6 , π 1 ⎨ 2 π3 − q + π3 , ≤ q < 5π 6 6 λ R (q) = , 6 5π 5π 2 (q − π) − 6 ,  ≤ q < 7π 2⎪ ⎪ 6 6 ⎩ π π −2 3 − (q − π) − π3 , 7π ≤ q < 11π 6 6 i.e., λ R (q) is the function plotted in Fig. 9.2a, q stands for the rotor position, and E p is a constant depending on the rotor magnetic field, the number of turns in each Fig. 9.2 Functions λ R (q) and E(q), where R (q) E(q) = − dλdq

9.1 Motor Modeling

397

phase and the rotor dimensions. Thus, the flux linkages in each phase are given as ψ = L I + Γ (q).

(9.2)

According to Kirchhoff’s and Faraday’s laws, the electrical dynamics of the stator is given as ψ˙ + R I = U,

(9.3)

where U = [U1 , U2 , U3 ] are the three-phase applied voltages and R is the phase windings resistance. Replacing (9.2) in (9.3): L I˙ +

dΓ (q) q˙ + R I = U, dq

(9.4)

where the back electromotive force is given as [55, 80] dΓ (q) q˙ = −E p E R (q)q. ˙ dq The vectorial function E R (q) is given as       2π dλ q + dλ R (q) dλ R q − 2π R 3 3 E R (q) = − , , dq dq dq      2π  2π ,E q+ = E(q), E q − , 3 3

(9.5)

(9.6)

where (see Fig. 9.2b) ⎧ 6q , − π6 ≤ q < π6 ⎪ π ⎪ ⎨ π dλ R (q) 1, ≤ q < 5π 6 6 = . E(q) = − 6(q−π) 5π ⎪ − π , 6 ≤ q < 7π dq ⎪ 6 ⎩ 7π −1, ≤ q < 11π 6 6

(9.7)

The fact that each component of the vector −E p E R (q)q˙ has a trapezoidal shape when the motor velocity is constant, is the reason why this motor is also called trapezoidal back electromotive force PM synchronous motor [55]. On the other hand, according to D’Alembert’s principle [60], torque produced by the three phases together is given as   ∂ 1   I L I + Γ (q)I , τ = ∂q 2   dΓ (q)  = I, dq = −τ p E  R (q)I,

(9.8)

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9 Brushless DC-Motor

according to (9.5), where τ p = E p but it is usual to use different symbols for these constants. Since torque given by (9.8) is applied in the direction where q increases, then use of Newton’s Second Law yields J q¨ = τ − bq˙ − τ L , = −τ p E  R (q)I − bq˙ − τ L ,

(9.9)

where J and b are positive constants representing the rotor inertia and the viscous friction constant whereas τ L stands for the load torque. Recalling (9.4), (9.5) and (9.9), we conclude that the BLDC motor model is given as L I˙ + R I = E p E R (q)q˙ + U, J q¨ + bq˙ =

−τ p E  R (q)I

− τL .

(9.10) (9.11)

Finally, replacing ω = q, ˙ the velocity model is obtained [55]: L I˙ + R I = E p E R (q)ω + U, J ω˙ + bω = −τ p E  R (q)I − τ L .

(9.12) (9.13)

9.1.2 Standard Control The generated torque is given according to (9.8). The functions that compose E R (q) are plotted in Fig. 9.3. Suppose that it is desired that the electric currents flowing through the stator phases be switched-on and switched-off in function of the rotor position as indicated in Fig. 9.3. Notice that the values that the desired phase currents can take only have three different possibilities [55, 150]: Ii∗ = I ∗ > 0, or Ii∗ = −I ∗ < 0, or Ii∗ = 0, for i = 1, 2, 3.

(9.14)

Hence, replacing the desired currents in (9.8) we obtain an expression for the desired generated torque τ ∗ :       2π 2π I2∗ + E q + I3∗ , τ ∗ = −τ p E(q)I1∗ + E q − 3 3 = 2τ p I ∗ ,

(9.15)

according to Fig. 9.3, (9.7), and (9.14). In order to force the actual phase currents Ii to reach their desired values Ii∗ , for i = 1, 2, 3, the following proportional-integral (PI) electric current controllers are employed [55]: U1 = α1 p (I1∗ − I1 ) + α1i



t 0

(I1∗ (s) − I1 (s))ds,

(9.16)

9.1 Motor Modeling

399

Fig. 9.3 Switching of the desired electric currents taking into account the waveforms of functions defining E R (q). Adapted from [55, 150]

U2 = α2 p (I2∗ − I2 ) + α2i U3 = α3 p (I3∗ − I3 ) + α3i

 

t

0 t 0

(I2∗ (s) − I2 (s))ds,

(9.17)

(I3∗ (s) − I3 (s))ds,

(9.18)

where αkp and αki , k = 1, 2, 3, are the proportional and integral gains, respectively. Switching of the desired electric currents at the required rotor positions shown in Fig. 9.3 is performed using Hall effect sensors. The desired current I ∗ is computed from (9.15) as I∗ =

1 ∗ τ , 2τ p

(9.19)

where the desired torque is given as the output of either a proportional-integral (PI) velocity controller [55]: ∗

∗



τ = k p (ω − ω) + ki

t

(ω ∗ (s) − ω(s))ds,

(9.20)

0

or a proportional-integral-derivative (PID) position controller [55]: τ ∗ = k p (q ∗ − q) + kd

d ∗ (q − q) + ki dt

 0

t

(q ∗ (s) − q(s))ds,

(9.21)

400

9 Brushless DC-Motor

Fig. 9.4 Standard control of a BLDC motor Fig. 9.5 Standard control of a BLDC motor, presented in Fig. 9.4, reduces to control of a simple linear mechanical system

if either velocity or position is to be controlled. In Fig. 9.4 is depicted a block diagram of this control scheme which represents the standard control scheme for BLDC motors. Notice that if the PI electric current controllers in (9.16)–(9.18), accomplish their control objective, i.e., that Ii → Ii∗ , for i = 1, 2, 3, then, by virtue of (9.8) and (9.15), the generated torque reaches its desired value, i.e., τ → τ ∗ and the block diagram in Fig. 9.4 reduces to the block diagram in Fig. 9.5.

9.2 Open-Loop Energy Exchange 9.2.1 The Velocity Model According to Sect. 9.1.1, the dynamic model of a BLDC motor is given by (9.12), (9.13), which is rewritten here for the ease of reference: L I˙ + R I = E p E R (q)ω + U, J ω˙ + bω = −τ p E  R (q)I − τ L ,

(9.22) (9.23)

where I, U ∈ R3 stand for the three-phase electric currents and the three-phase applied voltages and L is a 3 × 3 symmetric positive definite matrix. The reader is referred to Sect. 9.1 for a definition of the variables and parameters involved in this model. The fact that E p = τ p is equivalent to the property indicated in (3.28) for a PM brushed DC-motor, i.e., that the torque constant equals the back electromotive force constant ke = km . Also notice the presence of the functions E R (q) and E RT (q) in (9.22) and (9.23), respectively. It is shown in what follows that these functions appear because of a reason: they are instrumental for the natural energy exchange between the motor electrical and mechanical subsystems.

9.2 Open-Loop Energy Exchange

401

The following scalar function represents the total energy stored in the motor: V (I, ω) =

1 1  I L I + J ω2 , 2 2

(9.24)

where the first term represents the magnetic energy stored in the electrical subsystem whereas the second term stands for the kinetic energy stored in the mechanical subsystem. The time derivative of V is given as ˙ V˙ = I  L I˙ + ω J ω, which, according to (9.22), (9.23), can be written as V˙ = I  (−R I + E p E R (q)ω + U ) + ω(−bω − τ p E  R (q)I − τ L ). Notice that the terms I  E p E R (q)ω and −ωτ p E  R (q)I cancel in this expression. Since V represents the energy stored in the motor, we conclude that these cancellations (appearing in V˙ ) represent the energy exchange between the motor electrical and mechanical subsystems. Hence, this yields V˙ = −R I  I + I  U − bω 2 − ωτ L .

(9.25)

Defining the input u = [U  , −τ L ] and the output y = [I  , ω] , we can write (9.25) as ⎡ ⎤ R 0 0 0 ⎢ 0 R 0 0⎥ ⎥ V˙ = −y  Qy + y  u, Q = ⎢ (9.26) ⎣ 0 0 R 0⎦. 0 0 0 b Since Q is a positive definite matrix, (9.26) shows that the model (9.22), (9.23), is output strictly passive (see Definition 2.42) for the output y and input u defined above. In this chapter we will exploit this basic theoretical principle to design a velocity control scheme for BLDC motors.

9.2.2 The Position Model If the rotor position is designated by q then ω = q. ˙ Assume that the load torque is given as a nonlinear function of position, i.e., τ L = G(q), which is given as the . Using these ideas gradient of a positive semidefinite function P(q), G(q) = d P(q) dq and the dynamic model of a BLDC motor given by (9.10), (9.11), in Sect. 9.1.1, we can write

402

9 Brushless DC-Motor

L I˙ + R I = E p E R (q)q˙ + U, J q¨ + bq˙ =

−τ p E  R (q)I

− G(q).

(9.27) (9.28)

The following scalar function represents the total energy stored in motor and load: V (I, q, q) ˙ =

1 1  I L I + J q˙ 2 + P(q), 2 2

(9.29)

where the last (new) term represents the potential energy stored in the load. Notice that V , given in (9.29), is a positive semidefinite function because P(q) is assumed to have this property. Proceeding as in the previous section it is found that the cross ˙ p E terms cancellation I  E p E R (q)q˙ − qτ R (q)I = 0 appears again when computing ˙ V . Notice, however, that a new cancellation exists between cross terms ±G(q)q, ˙ which represents exchange between the kinetic and the potential energies in the mechanical subsystem. Thus, we find that V˙ = −R I  I − bq˙ 2 + I  U.

(9.30)

Recall that b > 0. Hence, defining the input u = U and the output y = I , we can write (9.30) as V˙ ≤ −Ry  y + y  u.

(9.31)

Since R > 0, (9.31) shows that the model (9.27), (9.28), is output strictly passive (see Definition 2.42) for the output y and input u defined above. In this chapter we will exploit this basic theoretical principle to design a position control scheme for BLDC motors.

9.3 Velocity Control Proposition 9.1 Consider the dynamical model (9.22), (9.23), together with the following controller: U = R I ∗ − α p ξ − K q ω˜ 2 ξ − K d ξ ξ, τ∗ E (q), I∗ = E p E R (q)2 R  t  τ ∗ = k p ω˜ + ki σ ω(s)ds ˜ ,

(9.32) (9.33) (9.34)

0

where I ∗ = [I1∗ , I2∗ , I3∗ ] , ω˜ = ω − ω ∗ , with ω ∗ a constant scalar standing for the desired velocity, ξ = I − I ∗ , and τ ∗ represents the desired torque. E p > 0 is an estimate of E p > 0 whereas E R (q) is an approximate of the function E R (q), introduced

9.3 Velocity Control

403

in (9.6), which is defined as      2π 2π  , E q + , E R (q) = E (q), E q − 3 3

(9.35)

where  ⎧  2 ⎪ −1 + r − r 2 − − π6 − a − q , − π6 ≤ q < − π6 − a + b ⎪ ⎪ ⎪ 6q ⎪ ⎪ , − π6 − a + b ≤ q < π6 + a − b ⎪ π  ⎪ ⎪  2 ⎪ π ⎪ ⎪ 1 − r + r 2 − π6 + a − q , + a − b ≤ q < π6 + a ⎪ 6 ⎪ ⎪ π ⎪ + a ≤ q < 5π −a ⎪ 6 6 ⎨ 1,    2 5π 5π 5π , E (q) = 1 − r + r 2 − −a−q , − a ≤ q < 6 − a + b (9.36) 6 6 ⎪ 6(q−π) ⎪ 5π 7π ⎪ ⎪ − π ,  −a+b ≤q < 6 +a−b ⎪ 6 ⎪ ⎪  7π 2 7π ⎪ 2 ⎪ −1 + r − r − 6 + a − q , 6 + a − b ≤ q < 7π +a ⎪ ⎪ 6 ⎪ ⎪ 7π 11π ⎪ +a ≤q < 6 −a ⎪ 6  ⎪ −1, ⎪  11π 2 11π ⎩ 2 −1 + r − r − 6 − a − q , 6 − a ≤ q < 11π 6   π  π δ2 , b = r cos(δ1 ), δ2 = − δ1 , a = r tan . δ1 = arctan 6 2 2 The function σ(·) is a strictly increasing linear saturation function for some M ∗ > L ∗ (see Definition 2.34). Furthermore, it is also required that function σ(·) be continuously differentiable such that 0
0 is different from zero. According to (9.38), E R (q)2 also satisfies all of these properties, which is easy to verify numerically.

404

9 Brushless DC-Motor

Fig. 9.6 Definition of the function E (q) and comparison with the function E(q). Continuous: E (q). Dashed: E(q)

Remark 9.3 In Fig. 9.7 we depict the control scheme in Proposition 9.1. Notice that it is composed of a nonlinear proportional-integral (NPI) external loop intended to regulate velocity. An inner electric current loop is driven by a proportional controller intended to dominate some cross terms existing between the mechanical and the electrical subsystems. The term R I ∗ is included to compensate for the presence of the windings electric resistances. Finally, the terms −K q ω˜ 2 ξ − K d ξ ξ are included to dominate some terms involved in I˙∗ , i.e., for stability proof purposes. Contrary to what happens in the standard control scheme for BLDC motors described in Sect. 9.1.2, the desired electric currents defined in Proposition 9.1 are not stepwise (see Remark 9.2). Hence, the use of electric current controllers with integral parts is not recommended. This is the main reason why the terms R I ∗ − α p ξ are considered in (9.32) instead of a PI electric current controller.

9.3 Velocity Control

405

Fig. 9.7 Control scheme in Proposition 9.1. Υ is defined in (9.33) and Ψ is defined in (9.8). ζ = R I ∗ − K q ω˜ 2 ξ − K d ξ ξ.

It is concluded that the proposed control scheme is simple. One important factor contributing to this feature is that we are not required to feedback the expression for I˙∗ , which includes a large number of computations, but we just dominate such terms. Furthermore, notice that this is accomplished despite that velocity measurements are required by the PI velocity controller. The complete proof of Proposition 9.1 is presented in Appendix F.1. In the following we present just a sketch of such a proof to highlight how energy ideas are exploited. Sketch of Proof of Proposition (9.1) The closed-loop dynamics is found to be given as L ξ˙ = E p E R (q)ω˜ − (R + α p )ξ + U,

(9.39)

J ω˙˜ + (k p + b)ω˜ = −τ p E  R (q)ξ − T L , ∗

(9.40) ˙∗

U = E p E R (q)ω − K q ω˜ ξ − K d ξ ξ − L I , 2

T L = Φ(k p ω˜ + ki x(z) − bω ∗ − τ L ) + ki x(z), bω ∗ + τ L x(z) = σ(z) + , z˙ = ω, ˜ ki

(9.41)

where k p = εk p , and ki = εki for some constant ε > 0 and φ = E R (q) − E R (q),

(9.42)

φ E R (q) . E R (q)2

(9.43)

Φ=

T

Recall that I˙∗ is continuous and defined for all q ∈ R, according to the definition of E R (q) in Proposition 9.1. Notice that the closed-loop dynamics in (9.39)–(9.41) is almost identical to the open-loop BLDC motor dynamical model in (9.22), (9.23) if we replace I, U, ω, τ L by ξ, U, ω, ˜ T L . Two important differences are that resistance and viscous friction

406

9 Brushless DC-Motor

have been increased from R and b in (9.22), (9.23) to R + α p and b + k p in (9.39)– (9.41). Also a new differential equation appears in (9.41) which is due to the integral part of the PI velocity controller. These observations are important to propose the following scalar function for stability analysis purposes:   τ L + bω ∗ 1  ˜ z+ , (9.44) V (y) = ξ Lξ + Vω ω, 2 ki    z τ L + bω ∗ 1 2 ˜ z+ = J ω ˜ Vω ω, + [k + β(b + k )] x(r )dr + β J x(z)ω, ˜ i p τ +bω ∗ ki 2 − L ki

∗

, ξ  ] . The first term in V (y) is included to take into where y = [ω, ˜ z + τL +bω ki account the “magnetic energy” stored in the electrical subsystem. The first two terms in Vω stand for the “kinetic energy” stored in the mechanical subsystem and “energy” stored in the integral of velocity. Since the integral term of the PI velocity controller is nonlinear, a nonlinear integral of velocity is used as a kind of nonlinear “energy” function. Finally, the third term in Vω is a cross term intended to provide V˙ω with a negative quadratic term on the nonlinear function x(z). The main reason for this is to obtain a strict Lyapunov function in order to prove ultimate boundedness of z the state. The term ki − τL +bω∗ x(r )dr is included in V (y) because its time derivative ki z cancels with term −ki x(z)ω˜ arising from −ωT ˜ L . Finally, β(b + k p ) − τL +bω∗ x(r )dr ki

is included to cancel an undesired cross term arising from time derivative of β J x(z)ω. ˜ The function V (y) given in (9.44) is proven to satisfy α1 (y) ≤ V (y) ≤ α2 (y), ∀y ∈ R5 ,  c1 y2 , y < 1 , α2 (y) = c2 y2 , α1 (y) = c1 y, y ≥ 1

(9.45)

where c1 > 0 is some small enough constant and c2 > 0 is some large enough con∗ | are true. Notice that α1 (·) and α2 (·) are two class stant, if (A.7) and L ∗ > | τL +bω ki K∞ functions, Defining:        d  E R (q)  E R (q)  ,   , Λ2 = sup  Λ1 = sup  dq E p E R (q)2  E p E R (q)2  and choosing: K q > λ M (L)Λ1 k p , K d > λ M (L)Λ2 τ p E R (q) M

(9.46) k p J

,

(9.47)

9.3 Velocity Control

407

we find that the time derivative of V (y) defined in (9.44) along the trajectories of the closed-loop system (9.39)–(9.41) can be upper bounded as V˙ ≤ − y¯  Q y¯ + γ1  y¯  + γ2  y¯ ,  k p ki M ∗ ∗ |Φ| M γ1 = 2(1 + β)(bω + τ L ) + Λ2 J γ2 = |ω ∗ |(Λ1 ki M ∗ + E p E R (q) M ),

(9.48)

y¯ = [|ω|, ˜ |x(z)|, ξ] , if: ξ >

1 λ M (L)Λ1 k p , Kq

(9.49)

ξ >

k p 1 λ M (L)Λ2 τ p E R (q) M , Kd J

(9.50)

where the entries of matrix Q are defined in (F.15). It is important to stress that thanks to the introduction of the function E R (q) in Proposition 9.1, the constant Λ1 is defined for all q ∈ R. This is opposite to the problem appearing in some previous works such as [80]. This, however, is the reason for the nonzero constant γ1 which will prohibit asymptotic stability and force an ultimate boundedness result, instead. We also stress that both constants Λ1 and Λ2 appear as a part of I˙∗ . Some of these terms are dominated by choosing (9.46) and (9.47) and some other terms are included in the entries of matrix Q. These observations are instrumental to design the simple controller in Proposition 9.1 because they allow to dominate I˙∗ instead of cancelling it, a drawback that is present in [80]. The three leading principal minors of matrix Q can always be rendered positive by choosing small enough β > 0, r > 0, and large enough k p > 0, ki > 0, α p > 0. Hence, matrix Q is positive definite and λm (Q) > 0 is ensured. Using some constant 0 < Θ < 1, we can rewrite (9.48) as γ1 + γ2 . V˙ ≤ −(1 − Θ)λm (Q) y¯ 2 , ∀ y ≥ μ0 = Θλm (Q)

(9.51)

Taking into consideration (9.45) and (9.51), we can invoke Theorem 2.29 to conclude that the closed-loop system state y is bounded and it has an ultimate bound which can be rendered arbitrarily small by choosing suitable controller gains. This is possible because μ0 > 0 can be rendered arbitrarily small by choosing suitable controller gains. Furthermore, this result stands when beginning from any initial condition. This completes the proof of Proposition 9.1. The conditions to guarantee Proposition 9.1 ∗ |, (A.7), (F.10), (F.11), the three principal minors are summarized by L ∗ > | τL +bω ki of matrix Q defined in (F.15) are positive, and some small constant r > 0.

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9 Brushless DC-Motor

Remark 9.4 Instrumental to obtain V˙ (see (F.9)) is cancellation of the terms E p ξ  E R (q)ω˜ belonging to dtd 21 ξ  Lξ , the time derivative of the“magnetic energy”   d 1 ˜ 2 , the stored in the electrical subsystem, and −τ p ω˜ E  R (q)ξ belonging to dt 2 J ω time derivative of the “kinetic energy” stored in the mechanical subsystem. These cancellations are due to the natural energy exchange between the electrical and the mechanical subsystems. They are obvious consequences of the almost identical structure of the closed-loop dynamics in (9.39)–(9.41), and the open-loop dynamics in (9.22), (9.23). Remark 9.5 Notice that, excepting the electric resistance, controller in Proposition 9.1 does not require the exact knowledge of any motor parameter. Only a rough estimate for the motor torque constant τ p = E p and the maximal allowable load torque are required. In this respect, we stress that an estimate of load torque to be compensated is always taken into account in practice when selecting a motor. Moreover, only approximate values of the motor parameters are required to verify the stability conditions summarized above. Remark 9.6 The closed-loop mechanical subsystem dynamics given in (9.40) can be written as J ω˙˜ + bω˜ = τe − Φ(k p ω˜ + ki σ(z)) − k p ω˜ − ki x(z),

(9.52)

τe = −τ p E  R (q)ξ,

(9.53)

if we define

which represents the difference between the actual and the desired torques. Hence,   |ω| ˜ Q 11 Q 12 ˜ |x(z)|] V˙ω ≤ −[|ω|, + βx(z)τe + (ω˜ + βx(z))Φ(bω ∗ + τ L ), Q 21 Q 22 |x(z)| where function Vω is defined in (9.44) and Q 11 and Q 22 , Q 12 = Q 21 are entries of matrix Q defined in (F.15). On the other hand, the time derivative of the first component of V in (9.44), i.e., Ve =

1  ξ Lξ, 2

along the trajectories of the closed-loop electrical subsystem dynamics (9.39), contains the quadratic negative term −(R + α p )ξ  ξ. Thus, when computing V˙ = V˙ω + V˙e , we realize that instrumental for the stability result in Proposition 9.1 are the following features, which are very similar to those in Remark 3.4: • Aside from the terms (ω˜ + βx(z))Φ(bω ∗ + τ L ), the scalar function Vω is a strict Lyapunov function when the desired torque is the input of the closed-loop mechanical subsystem dynamics, given in (9.52), i.e., when τe = 0.

9.3 Velocity Control

409

• Coefficient of the negative term −(R + α p )ξ  ξ appearing in V˙ can be enlarged arbitrarily. This is important to dominate some cross terms in V˙ depending on ξ when ξ = 0. • Cancellation of some terms belonging to V˙ω and V˙e which is explained in Remark 9.4. Notice that all of these features are possible thanks to the passivity properties of the open-loop motor model described in Sect. 9.2.1. Furthermore, the above proof contains the main ideas of procedure described in Sect. 2.4. As pointed out in Remark 3.4, τe is given as a nonlinear function of the electrical dynamics error for AC-motors (see (9.53)). In the case of BLDC motors we have resorted to introduction of the terms −K q ω˜ 2 ξ − K d ξ ξ, in (9.32), which are important to dominate some third-order terms appearing in V˙ . These nonlinear terms arise from the bilinear nature of the motor model, a feature that has traditionally complicated controller design for AC-motors. Thanks to introduction of the above terms, controller in Proposition 9.1 is very simple, i.e., we are not required to feedback the expression for I˙∗ but we just dominate it using the above-cited nonlinear terms and we use a nonlinear integral part for the PI velocity controller.

9.3.1 Simulation Results In this section we present a numerical example to give some insight on the achievable performance when controller in Proposition 9.1 is employed. To this aim, we use the numerical values of the BLDC motor reported in [80]: τ p = E p = 0.5128(Nm/A), J = 0.2 × 10−3 (kg m2 ), b = 2 × 10−3 (Nm/(rad/s)), R = 7(Ohm), M = 0.0015(mH), L s = 0.0027(mH). This motor has an input voltage range of ±200(V) and a peak electric current of 10(A), see [80]. The controller gains were chosen to be k p = 0.05, ki = 2.5, ε = 1.0011, α p = 500, K q = K d = 1. These controller gains were found to satisfy all of the stability conditions established in the proof of Proposition 9.1 using β = 4. These also requires |Φ| M = 0.01 which is accomplished using a r = 0.01. See (9.43). All of the initial conditions were set to zero. The desired velocity ω ∗ was chosen as a ramp with slope 105 ((rad/s)/s), starting with a zero value at t = 0 and becoming constant when reaching 157(rad/s) (i.e., 1500(rpm)). The load torque τ L = 2(Nm) was considered to be a step signal which appears for t ≥ 0.15(s). In order to perform realistic simulations, both of these values, i.e., ω ∗ and τ L , were chosen to satisfy the above-cited ranges of input voltage and electric currents. In Fig. 9.8 we verify that the three electric currents I1 , I2 , I3 remain within this range. Moreover, we also realize that the three applied voltages U1 , U2 , U3 remain within the range of ±100(V). In this case, voltages do not exhibit very large voltage peaks because the reference of velocity is not applied as a step but as a ramp. Notice that velocity reaches the desired value in steady state and this response has an

410

9 Brushless DC-Motor

Fig. 9.8 Simulation results when controller in Proposition 9.1 is employed

about 0.1(s) settling time. On the other hand, the torque disturbance is well rejected requiring about 3(A) maximum.

9.4 Position Control In this section, the following modified version of the BLDC motor model presented in (9.27), (9.28), is considered: L I˙ + R I = E p E R (q)q˙ + U, J q¨ + bq˙ = −τ p E  R (q)I − g(q),

(9.54) (9.55)

9.4 Position Control

411

where g(q) represents a position dependent mechanical load which is assumed to possess the following properties: |g(x) − g(y)| ≤ kg |x − y|, ∀x, y ∈ R, (Lipschitz condition)    dg(q)  , kg > max  q∈R dq  dU (q) g(q) = , |g(q)| ≤ k , ∀q ∈ R, dq

(9.56) (9.57) (9.58)

with U (q) the potential energy and kg , k , some positive constants. Controller in Proposition 9.1 can be extended to the case of position control as shown in the following result. Proposition 9.7 Consider the dynamical model (9.54), (9.55), together with the following controller: U = R I ∗ − α p ξ − K q q˙ 2 ξ − K d ξ ξ, τ∗ E (q), I∗ = E p E R (q)2 R τ ∗ = k p h(q) ˜ + kd q˙ + ki sat (z),      t  kp kd z= ε 1+γ h(q) ˜ + 1 + εγ q˙ dt, ki ki 0

(9.59) (9.60) (9.61) (9.62)

where I ∗ = [I1∗ , I2∗ , I3∗ ] , q˜ = q − q ∗ , with q ∗ ∈ R the constant desired rotor position, ξ = I − I ∗ , and τ ∗ represents the desired torque. E p > 0 is an estimate of E p > 0 whereas E R (q) is an approximate of the function E R (q), introduced in (9.6), which is defined in (9.35), (9.36), see Fig. 9.6. We define the functions h(q) ˜ = s(q), ˜ and sat (z) = s(z), where s(·) is a strictly increasing linear saturation function for some M ∗ > L ∗ (see Definition 2.34). Furthermore, it is also required that function s(·) be continuously differentiable such that 0
0. Recall that I˙∗ is continuous and defined for all q ∈ R, according to the definition of E R (q) in Proposition 9.1. Notice that the closed-loop dynamics (9.64)–(9.66) is almost identical to the openloop dynamics in (9.54), (9.55), if we replace I, U, g(q) by ξ, U, G. An important difference is that electric resistance and viscous friction have been increased from R and b in (9.54), (9.55), to R + α p and b + kd in (9.64)–(9.66). An improvement that has been accomplished thanks to the output strict passivity property of the openloop model that has been studied in Sect. 9.2.2. Finally, an additional equation is introduced in (9.66) which is due to the integral part of the PID position controller that is employed. Taking into account the above observations, the following scalar function is proposed for stability analysis purposes: 1  ξ Lξ + Vq (q, ˜ q, ˙ z + g(q ∗ )/ki ), (9.67) 2 Vq (q, ˜ q, ˙ z + g(q ∗ )/ki ) = V1 (q, ˙ q) ˜ + P(q) ˜ + V2 (q, ˙ z + g(q ∗ )/ki ),  q˜ 1 V1 (q, ˙ q) ˜ = J q˙ 2 + εJ h(q) ˜ q˙ + ε(b + kd ) h(r )dr, 4 0  q˜ ∗ P(q) ˜ = kp h(r )dr + U (q) − U (q ∗ ) − qg(q ˜ ), 0  z 1 V2 (q, ˙ z + g(q ∗ )/ki ) = J q˙ 2 + εγ J x(z)q˙ + ki x(r )dr, 4 −g(q ∗ )/ki

V (q, ˜ q, ˙ z + g(q ∗ )/ki , ξ) =

where ε and γ are some positive constants. The first term in V represents the “magnetic energy” stored in the electrical subsystem. On the other hand, the function Vq includes the kinetic energy and the closed-loop “potential energy”: P(q) ˜ =

k p



q˜

∗ h(r )dr + U (q) − U (q ∗ ) − qg(q ˜ ),

0

as well as “energy” stored in the integral of position through an integral of x(·). The cross terms εJ h(q) ˜ q˙ and εγ J x(z)q˙ are required to provide V˙ with negative quadratic terms on both h(q) ˜ and x(z). In this respect, negative quadratic terms on both h(q) ˜ and x(z) are required to ensure ultimate boundedness of the state, a result which is initially motivated by the function Φ appearing in (9.65). Finally, the integral term  q˜ α(b + kd ) 0 h(r )dr is intended to cancel an undesired cross term appearing in time derivative of εJ h(q) ˜ q. ˙ It is concluded that there exist some small enough constant c1 > 0 and some ˜ q, ˙ z + g(q ∗ )/ki , ξ), large enough constant c2 > 0 such that the scalar function V (q, defined in (9.67) satisfies

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9 Brushless DC-Motor

α1 (y) ≤ V (y) ≤ α2 (y), ∀y = [q, ˜ q, ˙ z + g(q ∗ )/ki , ξ  ] ∈ R6 , (9.68)  c1 y2 , y < 1 , α2 (y) = c2 y2 , α1 (y) = c1 y, y ≥ 1 if (A.8), ( A.9), (A.10), and (A.14) are true. Notice that α1 (·) and α2 (·) are two class K∞ functions. Defining:         d  E R (q)  , Λ2 = sup  E R (q)  , Λ1 = sup   dq E E (q)2   E E (q)2  p p R R and choosing: K q > λ M (L)Λ1 kd ,

k K d > λ M (L)Λ2 τ p E R (q) M d , J

(9.69) (9.70)

we find that the time derivative of V (y), defined in (9.67), along the trajectories of the closed-loop system (9.64)–(9.66), can be upper bounded as V˙ ≤ − y¯  Q y¯ + β y¯ ,  kd |Φ| M k , β = 3(1 + ε + εγ) + Λ2 J y¯ = [|q|, ˙ |h(q)|, ˜ |x(z)|, ξ] ,

(9.71)

where Φ is defined in (9.43), if: 1 λ M (L)Λ1 kd , Kq k 1 ξ > λ M (L)Λ2 τ p E R (q) M d , Kd J ξ >

(9.72) (9.73)

where the entries of matrix Q are defined in (F.30). It is important to stress that thanks to the introduction of the function E R (q) in Proposition 9.1, the constant Λ1 is defined for all q ∈ R. This is opposite to the problem appearing in some previous works such as [80]. This, however, is the reason for the nonzero constant β which will prohibit asymptotic stability and force an ultimate boundedness result, instead. We also stress that both constants Λ1 and Λ2 appear as a part of I˙∗ . Some of these terms are dominated by choosing (9.69) and (9.70) and some others are included in the entries of matrix Q. These observations are instrumental to design the simple controller in Proposition 9.7 because they allow to dominate I˙∗ instead of cancelling it, a drawback that is present in [80]. The four leading principal minors of matrix Q can always be rendered positive by choosing small enough ε > 0, γ > 0, r > 0, and large enough positive controller

9.4 Position Control

415

gains kd , kd , ki , α p . Hence, matrix Q is positive definite and λm (Q) > 0 is ensured. Using some constant 0 < Θ < 1, we can rewrite (9.71) as V˙ ≤ −(1 − Θ)λm (Q) y¯ 2 , ∀ y ≥ μ0 =

β . Θλm (Q)

(9.74)

Taking into consideration (9.68) and (9.74), we can invoke Theorem 2.29 to conclude that the closed-loop state is bounded and has an ultimate bound which can be rendered arbitrarily small because μ0 > 0 can be rendered arbitrarily small by choosing a small enough r > 0. Furthermore, this result stands when starting from any initial condition. This completes the proof of Proposition 9.7. Finally, we emphasize that the conditions to guarantee Proposition 9.7 are summarized by (A.8), ( A.9), (A.10), (A.14), (F.25), (F.26), the four leading principal minors of matrix Q defined in (F.30) are positive, and some small constant r > 0 is chosen. Remark 9.10 Instrumental to obtain V˙ , see (F.24), is the cancellation of the terms E p ξ  E R (q)q˙ belonging to dtd 21 ξ  Lξ , the time derivative of the “magnetic energy”   d 1 2 stored in the electrical subsystem, and −τ p E  R (q)ξ q˙ belonging to dt 2 J q˙ , the time derivative of the “kinetic energy” stored in the mechanical subsystem. These cancellations are due to the natural energy exchange between the electrical and mechanical subsystems. They are obvious consequences of the almost identical structure of the closed-loop dynamics in (9.64)–(9.66), and the open-loop dynamics in (9.54), (9.55). Remark 9.11 Although the mathematical expressions are more complex in this case, the same observations as in Remark 9.6 are also instrumental to prove global asymptotic stability in the case of position control, i.e., • Aside from some simple terms, the scalar function Vq , intended for the mechanical subsystem, is a strict Lyapunov function when the desired torque is the input of the closed-loop mechanical subsystem dynamics, i.e., when ξ = 0. • The coefficient of the negative term −(R + α p )ξ  ξ appearing in V˙ can be enlarged arbitrarily. This is important to dominate some cross terms in V˙ depending on ξ when ξ = 0. • Cancellation of several cross terms which is explained in Remark 9.10. Notice that all of these features are possible thanks to the passivity properties of the open-loop motor model described in Sect. 9.2.2. Furthermore, the above proof contains the main ideas of procedure described in Sect. 2.4. As stressed in Remark 9.6, we have succeeded to design a simple controller, provided with a global stability proof, for BLDC motors thanks to the fact that we avoid to cancel the terms composing I˙∗ . Instead we dominate such complex terms. Moreover, notice that, excepting the electric resistance, controller in Proposition 9.7 does not require the exact knowledge of any motor parameter. Only a rough estimate for the motor torque constant τ p = E p is required. Moreover, only approximate values of the motor parameters are required to verify the stability conditions summarized above.

416

9 Brushless DC-Motor

Remark 9.12 Our main contribution with respect to [80] is described in the following items. (1) The desired electric currents that we propose are continuously differentiable for all rotor positions. (2) We dominate the time derivatives of the desired electric currents, i.e., we do not feedback the time derivatives of the desired electric currents. This results in a much simpler control law. (3) The time derivatives of the desired currents that we present are complete. (4) We do not require the knowledge of the viscous friction coefficients. We must stress at this point, however, that the work in [80] has the advantage of not requiring velocity measurements. (5) We do not require either to know the load torque due to gravity. This is because we employ PID position controllers in the external loop. Additionally, the result in Proposition 9.7 has been extended in [94] to regulate position in n-degrees of freedom rigid robots actuated by n direct-drive BLDC motors. This means that a complex nonlinear and highly coupled mechanical load is considered. We consider that it is important to remark the difference between our proposal and that by [98]. The latter paper is concerned with robots actuated by PM synchronous motors instead of BLDC motors. In this respect, let us stress that it is common in the control literature [60] to call BLDC motors also to PM synchronous motors. Thus, care must be put to differentiate both classes of motors when searching for related literature.

9.4.1 Simulation Results In this section we present a numerical example to give some insight on the achievable performance when controller in Proposition 9.7 is employed. To this aim, we use the numerical values of the BLDC motor presented in Sect. 9.3.1. We additionally assume that a simple pendulum is fixed at the motor shaft such that, see (9.55), g(q) = mgl sin(q) where m = 0.8(kg), g = 9.81(m/s2 ), l = 0.25(m) whereas the mechanism inertia is given as J = 0.2 × 10−3 (kg m2 )+ml 2 + 13 m(2l)2 where 0.2 × 10−3 (kg m2 ) is the inertia of the motor rotor. The parameters of g(q) were selected such that the resulting voltages and electric currents remain within the limits of the motor, i.e., ±200(V) and ±10(A). The controller gains were chosen to be k p = 5, ki = 10, kd = 1.5,  = 1.0011, α p = 8500, K q = K d = 1. These controller gains were found to satisfy all of the stability conditions established in the proof of Proposition 9.7 using γ = ε = 1.5. These also requires |Φ| M = 0.01 which is accomplished using a r = 0.01. See (9.43). All of the initial conditions were set to zero. The desired position q ∗ was chosen as a ramp with slope 10(rad/s), starting with a zero value at t = 0 and becoming constant when reaching π2 (rad). In Fig. 9.10 we verify that the three electric currents I1 , I2 , I3 , remain within ±3(A). Moreover, we also realize that the three applied voltages U1 , U2 , U3 remain within the range ±25(V). In this case, voltages do not exhibit very large voltage peaks because the reference of position is not applied as a step but as a ramp. Notice that position reaches the desired value in steady state and this response has an about 3(s) settling time. We

9.4 Position Control

417

Fig. 9.10 Simulation results when controller in Proposition 9.7 is employed

observe that the position response is very damped. In this respect, we have performed several additional simulations  whichhave made us to arrive at similar conclusions as in Sect. 4.4.1: the term 1 + εγ kkdi q˙ appearing in (9.62) is responsible for such damped response.

9.5 A Practical BLDC Motor In Figs. 9.11 and 9.12 are presented some pictures of the stator windings of this motor. In Fig. 9.13 is presented a drawing showing the relative positions of the stator phase ◦ = 60◦ = π3 (rad) apart. Each winding windings. There are six windings which are 360 6

418

9 Brushless DC-Motor

Fig. 9.11 Some views of the stator windings

is wound on a strip-shaped core which lies on stator parallel to the rotor axis (see Figs. 9.11 and 9.12). In the following we describe how these windings compose three phases labeled as 1,2,3 which are star connected. The symbols I1 , I2 , I3 represent the electric current flowing through phases 1,2,3, respectively, when assumed to be positive. These symbols I1 , I2 , I3 are also used in Fig. 9.13 to label the external terminals of the stator phases. Each phase is composed of two windings which are series connected and placed 180 mechanical degrees apart on the stator. Each one of the two windings composing phase 1 has one point labeled as 1a which are connected together to render possible the series connection of these two windings. The points 2a and 3a in phases 2 and 3 play the same role in the corresponding phases. The three-phase terminals labeled with a N connect together and constitute the neutral point of the star connection which is isolated. Only terminals labeled I1 , I2 , I3 are accessible to user. This is a four permanent magnet poles motor, i.e., two N poles and two S poles are alternatively fixed on the surface of rotor. These poles are buried in the rotor and cannot be observed (see Fig. 9.14). However, using another permanent magnet we realize that each one of these poles have the shape of a strip laying parallel to the ◦ = 90◦ = π2 (rad) apart on the stator. As we rotor axis. Adjacent poles are placed 360 4

9.5 A Practical BLDC Motor

419

Fig. 9.12 Some views of the stator windings (cont.)

can see in Fig. 9.13, this allows for the two windings composing each stator phase to be identically excited by the rotor poles. In the following we will obtain the mathematical model of this motor using a procedure described in [55].

9.5.1 Magnetic Field at the Air Gap 9.5.1.1

Magnetic Field Produced by the Stator Windings

In order to compute the magnetic field produced at the air gap by the stator phase winding 1, consider Fig. 9.15. There, the symbol  means that electric current through phase 1 is coming out of the page at γ = 180◦ and γ = 0◦ whereas the symbol ⊗ means that electric current is going into the page at γ = 120◦ and γ = 300◦ , to be consistent with the sense of a positive electric current flowing through phase 1, defined in Fig. 9.13. Let us apply Ampère’s Law (2.33) to the oriented closed trajectory 1-2-3-4-1 shown in Fig. 9.15:

420

9 Brushless DC-Motor

Fig. 9.13 Relative positions of the stator phase windings

 H1 · dl = i 1enclosed ,  2  = H1 · dl + 1

3 2



4

H1 · dl +

= g H1 (0) − g H1 (γ),

3

 H1 · dl +

1

H1 · dl,

4

(9.75)

where  2  2  2  2 H1 · dl = H1 (0)ˆr · (dl rˆ ) = H1 (0)dl = H1 (0)dr = g H1 (0), 1 1 1 1  3 H1 · dl = 0, 2  4  4  4  4 H1 · dl= H1 (γ)ˆr · (−dl rˆ ) = − H1 (γ)dl = H1 (γ)dr = −g H1 (γ), 3 3 3 3  1 H1 · dl = 0. 4

9.5 A Practical BLDC Motor

421

Fig. 9.14 Tow views of the rotor permanent magnets

In the above computations, the following assumptions have been taken into account [55]: • The phase loops are located at the surface of the stator. • Wires forming the loops have no width. • The segments of the closed trajectory 1-2-3-4-1 laying in the stator and the rotor are just below the surface of the stator and rotor. • The space between stator and rotor, known as the air gap, is constant and its width is represented by g. • The magnetic field H1 is radially oriented at the air gap. • H1 = 0 inside a ferromagnetic material with high relative magnetic permeability, i.e., the stator and rotor. • dl = dr in the segment 1–2, whereas dl = −dr in the segment 3–4. On the other hand, i 1enclosed depends on γ and is given as (see Fig. 9.15): ⎧ 0, 0 ≤ γ < 120◦ ⎪ ⎪ ⎪ ⎪ ⎨ −N I1 , 120 ≤ γ < 180◦ 180 ≤ γ < 300◦ i 1enclosed = 0, , (9.76) ⎪ ◦ ⎪ −N I , 300 ≤ γ < 360 ⎪ 1 ⎪ ⎩ 0, 360 ≤ γ < 360◦ + 120◦

422

9 Brushless DC-Motor

Fig. 9.15 Computing the magnetic field at the air gap produced by phase 1

where N stands for the number of turns composing each winding and I1 is electric current flowing through phase 1 which is assumed to be positive. It is important to stress that in order to be consistent with Ampère’s Law the right-hand rule has been taken into account. See Fig. 2.8. Thus, from (9.75) we have 1 H1 (γ) = H1 (0) − i 1enclosed . g Since H1 (γ) is defined at the air gap, where of course only air is present, then the relation B1 (γ) = μ0 H1 (γ) can be employed to write   1 B1 (γ) = μ0 H1 (0) − i 1enclosed rˆ . g

(9.77)

In the following H1 (0) is computed applying Gauss’ Law for the magnetic field, i.e.,  B1 · ds = 0,

9.5 A Practical BLDC Motor

423

to a Gaussian surface represented by a cylinder, enclosing the rotor, whose cylindrical surface (with radius r ) lies within the air gap. Hence, 

  B1 · ds + B1 · ds + B1 · ds, S1 S2 S3   2π  1 H1 (0) − i 1enclosed dγ = 0, = l1r μ0 g 0



B1 · ds =

(9.78)

where (9.77) has been employed and  B1 · ds = 0,  S1 B1 · ds = 0, 

S2



l1

B1 · ds = S3

0



2π

B1 (γ)ˆr · (r dγdzrˆ ).

0

In the above computations, the following considerations have been taken into account [55]: • S1 and S2 are the disk-shaped surfaces at each end of the cylinder. At these surfaces B1 = 0. • S3 is the cylinder-shaped part of the Gaussian surface where ds = r dγdzrˆ , in cylindrical coordinates, with r the radius of S3 and l1 the length of rotor (laying on the z axis), i.e., the length of S3. From (9.78) and (9.76) we find  2π 1 i 1enclosed dγ, 2πg 0 π π 1  −N I1 × − N I1 × , = 2πg 3 3 N H1 (0) = − I1 . 3g H1 (0) =

Finally, replacing this in (9.77), we find the expression for the magnetic field at the air gap produced by electric current through phase 1:   N rR 1 rˆ , B1 (γ) = μ0 − I1 − i 1enclosed 3g g r

(9.79)

where the factor rrR , with r R the rotor radius, is suggested in [55] to be introduced in order to be consistent with magnetic flux conservation. Notice, however, that rrR ≈ 1 at the air gap and, hence, results obtained in the above procedure remain without change.

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9 Brushless DC-Motor

Fig. 9.16 Computing the magnetic field at the air gap produced by phase 2

Proceeding similarly for phases 2 and 3 we find (see Figs. 9.16 and 9.17):   N rR 1 rˆ , B2 (γ) = μ0 − I2 − i 2enclosed 3g g r ⎧ −N I2 , 0◦ ≤ γ < 60◦ ⎪ ⎪ ⎨ 0, 60 ≤ γ < 180◦ i 2enclosed = , −N I2 , 180 ≤ γ < 240◦ ⎪ ⎪ ⎩ 0, 240◦ ≤ γ < 360◦

(9.80)

(9.81)

and   N rR 1 rˆ , B3 (γ) = μ0 − I3 − i 3enclosed 3g g r ⎧ 0, 0 ≤ γ < 60◦ ⎪ ⎪ ⎪ ⎪ ⎨ −N I3 , 60 ≤ γ < 120◦ 120 ≤ γ < 240◦ . i 3enclosed = 0, ⎪ ⎪ −N I3 , 240 ≤ γ < 300◦ ⎪ ⎪ ⎩ 0, 300 ≤ γ < 360◦

(9.82)

(9.83)

9.5 A Practical BLDC Motor

425

Fig. 9.17 Computing the magnetic field at the air gap produced by phase 3

Notice that (9.79), (9.80), (9.82), can be rewritten as B1 (γ) = b1 (γ)I1 rˆ , B2 (γ) = b2 (γ)I2 rˆ ,

(9.84) (9.85)

B3 (γ) = b3 (γ)I3 rˆ ,

(9.86)

where the scalar functions b1 (γ), b2 (γ), and b3 (γ) are obviously defined from (9.79), (9.80), (9.82), and they are plotted in Fig. 9.18. We refer to Fig. 9.19 for a drawing showing how B1 (γ), defined in (9.84), distributes along the air gap. The reader can imagine that the magnetic field of the other phases also distributes similarly around their locations on stator. Thus, the total magnetic field produced by the three stator phases at the air gap is finally given as B S (γ) = B1 (γ) + B2 (γ) + B3 (γ).

(9.87)

426

9 Brushless DC-Motor

Fig. 9.18 Plots of b1 (γ), b2 (γ), and b3 (γ)

Fig. 9.19 Distribution along the air gap of the magnetic flux due to phase 1

9.5 A Practical BLDC Motor

427

Fig. 9.20 Some definitions required to write (9.88)

9.5.1.2

Magnetic Field Produced by Rotor

The rotor position q is defined in Fig. 9.20 as the counter-clockwise angle between the center of one of the N poles of rotor and the stator position where γ = 0◦ . Since the the magnetic field produced by the rotor permanent magnets is assumed to be uniformly distributed along the air gap, we have that ⎧ Bm , ⎪ ⎪ ⎪ ⎪ ⎨ −Bm , B R (γ − q) = Bm , ⎪ ⎪ ⎪ −Bm , ⎪ ⎩ Bm ,

−45◦ ≤ γ − q < 45◦ 45◦ ≤ γ − q < 3 × 45◦ 3 × 45◦ ≤ γ − q < 5 × 45◦ . 5 × 45◦ ≤ γ − q < 7 × 45◦ 7 × 45◦ ≤ γ − q < 8 × 45◦

(9.88)

In Fig. 9.21 we present a drawing where it is shown how B R (γ − q) distributes along the air gap. Thus, the total magnetic field produced by both the stator phase windings and permanent magnet at rotor distributes along the air gap according to B(γ, q) = B S (γ) + kB R (γ − q),

(9.89)

428

9 Brushless DC-Motor

Fig. 9.21 Distribution of the rotor magnetic flux along the air gap

where B S (γ) is defined in (9.87), B R (γ − q) is defined in (9.88) and 0 ≤ k ≤ 1 is the coupling factor which is included to account for leakage.

9.5.2 Magnetic Flux Linkages Let ψ1 represent the magnetic flux linkage of phase 1 at stator. This flux linkage represents the addition of magnetic flux through the stator windings located in the ranges 120◦ ≤ γ < 180◦ and 300◦ ≤ γ < 360◦ in Fig. 9.20. Notice that the rotor magnets exert an identical effect on the stator windings located at these two locations. Define S A as the cylindrical surface subtended between the stator angular positions 120◦ ≤ γ < 180◦ and the total length of stator. Thus  ψ1 = 2N

B(γ, q) · ds, SA l1



0

120◦ l1  180◦

 = 2N



+2N 0

180

(9.90) 

◦

l1 

B1 (γ) · ds + 2N

120◦

0



B3 (γ) · ds + 2N k 0

180

◦

B2 (γ) 120◦  l1 180◦ 120◦

· ds

B R (γ − q) · ds.

9.5 A Practical BLDC Motor

429

Notice that r S is radius of stator and 

l1



180◦

2N 120◦

0



l1

B1 (γ) · ds = 2N



120◦ 2

0

=

180◦

B1 (γ) · (r S dγdzrˆ ),

4 πN μ0 r R l1 I1 , 9 g

where (9.79) and (9.76) have been employed. Also notice that 

l1



180◦

2N 0

120◦



l1

B2 (γ) · ds = 2N



180◦ 120◦

0

B2 (γ) · (r S dγdzrˆ ),

πN2 2 I2 , = − μ0 r R l1 9 g  l1  180◦  l1  180◦ 2N B3 (γ) · ds = B3 (γ) · (r S dγdzrˆ ), 0

120◦

0

120◦

πN2 2 = − μ0 r R l1 I3 , 9 g where (9.80), (9.81), (9.82), and (9.83) have been employed. Furthermore, from Figs. 9.20, 9.22, 9.23, and 9.24 we conclude that 

l1



180◦



l1



180◦

2N k B R (γ − q) · ds = 2N k B R (γ − q) · (r S dγdzrˆ ), ◦ 0 120◦ ⎧ 0 120 −1, 45◦ ≤ q < 75◦ ⎪ ⎪ ⎪ 3 ◦ ⎪ −[(60 − (q − 75)) − (q − 75)] × 180◦ , 75◦ ≤ q < 135◦ ⎪ ⎪ ⎪ ⎪ 1, 135◦ ≤ q < 165◦ ⎪ ⎪ ⎨ 3 ◦ +[(60 − (q − 165)) − (q − 165)] × 180◦ , 165◦ ≤ q < 225◦ = Ep , −1, 225◦ ≤ q < 255◦ ⎪ ⎪ ⎪ 3 ◦ ◦ ◦ ⎪ −[(60 − (q − 255)) − (q − 255)] × 180◦ , 255 ≤ q < 315 ⎪ ⎪ ⎪ ⎪ 1, 315◦ ≤ q < 345◦ ⎪ ⎪ ⎩ 3 ◦ +[(60 − (q − 345)) − (q − 345)] × 180◦ , 345◦ ≤ q < 360◦ + 45◦ π (9.91) = E p λ R (q), E p = 2N kl1r R Bm , 3 π π where the factor 180 ◦ is introduced to convert to radians. After that, 3 is used as a π 3 3 common factor which, however requires to multiply 180◦ × π = 180◦ .

9.5.3 Mathematical Model Proceeding similarly for the remaining phases, we find that

430

9 Brushless DC-Motor

Fig. 9.22 Computing the magnetic flux linkages due to rotor

ψ = L I + Γ (q), ⎡ ⎤ L s −M −M L = ⎣ −M L s −M ⎦ , −M −M L s

(9.92)

2 πN2 πN2 4 μ0 r R l1 , M = μ0 r R l1 , 9 g 9 g    π π  Γ (q) = E p λ R (q), λ R q − , λR q + , 3 3 Ls =

where λ R (q) is obviously defined from (9.91), ψ=[ψ1 , ψ2 , ψ3 ] and I = [I1 , I2 , I3 ] . Applying Kirchhoff’s Voltage Law, Faraday’s Law, and Ohm’s Law to each phase winding it follows that ψ˙ + R I = U, where U = [V1 , V2 , V3 ] , represents voltages applied to the stator phase windings and R is a scalar standing for the electric resistance of each phase winding. Replacing (9.92) in the previous expression we retrieve (9.4), i.e.,

9.5 A Practical BLDC Motor

431

Fig. 9.23 Computing the magnetic flux linkages due to rotor (cont.)

L I˙ +

dΓ (q) q˙ + R I = U, dq

dΓ (q) q˙ = −E p E R (q)q, ˙ dq   dλ R (q) dλ R (q − π3 ) dλ R (q + π3 ) E R (q) = − , , , dq dq dq    π  π ,E q+ . = E(q), E q − 3 3 R (q) Since in the derivative dλdq the angular position q must be in radians, the situation 3 3 is solved by replacing the factor 180 ◦ by π in (9.91) to perform the derivative with respect to q (in radians). Thus ⎧ 0, 45◦ ≤ q < 75◦ ⎪ ⎪ ⎪ 3 ⎪ −2 × π , 75◦ ≤ q < 135◦ ⎪ ⎪ ⎪ ⎪ 0, 135◦ ≤ q < 165◦ ⎪ ⎪ ⎨ 3 dλ R (q) 2 × π , 165◦ ≤ q < 225◦ = . E(q) = − 0, 225◦ ≤ q < 255◦ ⎪ dq ⎪ ⎪ 3 ◦ ◦ ⎪ −2 × π , 255 ≤ q < 315 ⎪ ⎪ ⎪ ⎪ 0, 315◦ ≤ q < 345◦ ⎪ ⎪ ⎩ 3 2 × π , 345◦ ≤ q < 360◦ + 45◦

432

9 Brushless DC-Motor

Fig. 9.24 Computing the magnetic flux linkages due to rotor (cont.)

In Figs. 9.25 and 9.26 we present the plots of functions λ R (q), λ R (q − π3 ), λ R (q + π3 )     and E(q), E q − π3 , E q + π3 , respectively. On the other hand, according to Sect. 9.1.1, torque produced by the three phases together is given according to the D’Alembert’s principle: τ = −τ p E  R (q)I, where τ p = E p . Hence, the motor dynamical model is given by (9.12), (9.13), i.e., L I˙ + R I = E p E R (q)ω + U, J ω˙ + bω = −τ p E  R (q)I − τ L ,

(9.93)

where ω = q˙ is the rotor velocity. Comparing the motor models in (9.93) and in (9.12), (9.13) we conclude that they only differ in the following: D1 Functions λ R (q) and E(q) plotted in Figs. 9.25 and 9.26 are not identical to functions λ R (q) and E(q) plotted in Fig. 7.2. Function λ R (q) in Fig. 9.25 is flat instead of having the parabolic shape that λ R (q) has in Fig. 7.2 at its minimal and maximal values. This results in a zero constant value for E(q) in Fig. 9.26, whereas E(q) plotted in Fig. 7.2 has a ramp shape at these rotor position values.

9.5 A Practical BLDC Motor

Fig. 9.25 Plots of the flux linkages produced by the rotor permanent magnet

Fig. 9.26 Plots of the counter-electromotive force functions

433

434

9 Brushless DC-Motor

D2 The inductance matrix L, defined in (9.92), is not positive definite but positive semidefinite. This is because L s = 2M in (9.92) instead of L s = 73 M defined for (9.1). The origin of the above differences is the fact that, in the motor under study in this section, the phase windings are not uniformly distributed on the stator but they are lumped at specific stator positions. Compare, for instance, the uniform phase windings distribution of motor studied in [55], Chap. 10, to the phase windings distribution in Fig. 9.13 in this section.

Chapter 10

Magnetic Levitation Systems and Microelectromechanical Systems

Magnetic levitation systems and microelectromechanical systems are commonly used as benchmark problems to test novel control approaches [2, 9, 14, 67, 76, 195, 239, 246]. Moreover, several passivity-based approaches are applied in [75, 202, 204, 211, 234]. Among these approaches, only that in [204] is capable to be designed possessing a PID controller to cope with the mechanical subsystem. However, an internal PI electric current loop is not designed. Moreover, the IDA passivity-based control approaches [75, 202, 211, 234] cannot be designed possessing an integral action for either the mechanical subsystem nor the electrical subsystem. In this respect, although the recent passivity-based approaches in [12, 288] are devoted to include an integral action in the controller, this depends on finding a particular output which renders passive the open-loop plant and it is not clear whether position is such an output. An integral action and, in particular PID control, is expected to be employed to regulate the system position because uncertainties in either the gravitational term (in magnetic levitation systems) or the system stiffness (in microelectromechanical systems) result in a different from zero position steady-state error. An internal PI controller is intended to render robust the electrical dynamics of the system. In this chapter we introduce a position controller for magnetic levitation systems and another for microelectromechanical systems. These controllers possess a PID position regulator intended for the mechanical subsystems and an internal PI controller driven by (1) the electric current error in magnetic levitation systems and (2) the capacitor voltage error in microelectromechanical systems. Thus, these controllers are provided with the same components as the traditional control scheme for both of these electromechanical systems. Formal asymptotic stability proofs are presented for each controller. It is interesting to realize that both the magnetic levitation system model and the microelectromechanical system model can be seen as the non-rotative versions, or the one phase and one pole versions, of the SRM unsaturated model studied in Sect. 6.1.3. In this respect, an internal PI controller is well suited in this case because both the desired electric current (in a magnetic levitation system) and the desired voltage at © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 V. M. Hernández-Guzmán et al., Energy-Based Control of Electromechanical Systems, Advances in Industrial Control, https://doi.org/10.1007/978-3-030-58786-4_10

435

436

10 Magnetic Levitation Systems and Microelectromechanical Systems

capacitor (in a microelectromechanical system) become constant in steady state. The verification of these ideas is the main reason to include the present chapter. This chapter is organized as follows. The problem of controlling magnetic levitation systems is studied in Sect. 10.1. The mathematical model is derived in Sect. 10.1.1. In Sect. 10.1.2 is shown that energy exchange exists between the electrical and the mechanical subsystems. Finally, the proposed controller is introduced in Sect. 10.1.3. The problem of controlling microelectromechanical systems is studied in Sect. 10.2. The mathematical model is derived in Sect. 10.2.1. In Sect. 10.2.2 is shown that energy exchange exists between the electrical and the mechanical subsystems. Finally, the proposed controller is introduced in Sect. 10.2.3.

10.1 Magnetic Levitation Systems 10.1.1 Mathematical Model A magnetic levitation system is depicted in Fig. 10.1. It consists of a ball with mass m, made in a ferromagnetic material, which receives an upwards magnetic force F from an electromagnet. This force must cancel the downwards ball weight mg in order to levitate the ball in the space. The electromagnet is basically a ferromagnetic core with a conductor wire wound around it. An electric voltage u is applied at the electromagnet terminals which forces an electric current i to flow through the electromagnet winding and this current produces the attractive magnetic force F on the ball. Also see [113, 114, 243].

Fig. 10.1 A magnetic levitation system

10.1 Magnetic Levitation Systems

437

Fig. 10.2 Electromagnet inductance as a function of the ball position

The working principle of a levitation system is identical to that of a switched reluctance motor (SRM, see Sect. 6.1.3), i.e., the magnetic force of the electromagnet on the ball is produced by reluctance in the air gap between these components. Moreover, the electromagnet inductance L(y) is dependent on the ball position y measured from the bottom surface of the electromagnet (see Fig. 10.1). The magnetic flux linkage in the electromagnet winding is given as ψ = L(y)i = N λ, where N is the number of turns and λ is the magnetic flux in the core. If i is kept constant and the ball approaches the electromagnet then the air gap decreases which reduces its reluctance and, hence, the flux linkage ψ increases. Since i was kept constant, then according to ψ = L(y)i the increment on ψ must be due to an increment in the inductance L(y). This also shows that inductance increases as y → 0 and inductance decreases as y → ∞. Furthermore, when y → ∞ is the case when the ball is not present which implies that L(y) becomes constant and finite. This means that the inductance L(y) changes with y as depicted in Fig. 10.2. According to (6.12), a reluctance produced force is computed as (also see [106] for a detailed procedure to obtain this expression): ∂ F= ∂y



1 L(y)i 2 2

 =

1 d L(y) 2 i . 2 dy

(10.1)

Notice that, according to the above arguments and Fig. 10.2: d L(y) < 0, ∀y ≥ 0. dy

(10.2)

Applying Faraday’s, Kirchhoff’s and Newton’s Second laws, the following expressions are found which are similar to those in (6.6): ψ˙ + Ri = u, m y¨ = F + mg,

(10.3)

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10 Magnetic Levitation Systems and Microelectromechanical Systems

where R is the copper resistance in the electromagnet winding. Replacing ψ = L(y)i and (10.1): L(y)

d L(y) 1 d L(y) 2 di + i y˙ + Ri = u, m y¨ = i + mg. dt dy 2 dy

(10.4)

The expressions in (10.4) are identical to those in (6.9), (6.10), if q is replaced by y and b = 0. This means that magnetic levitation systems represent the one phase and one pole case of a SRM. Notice that b = 0 is the main reason to introduce magnetic levitation systems, i.e., because a system with zero friction is obtained. In the second expression in (10.4), both terms on the right hand are affected by “+” signs. This is because both of these forces are assumed to be applied on the direction where y increases. However, the first of these terms is negative because of (10.2) and, thus, the magnetic force opposes to the ball weight mg. Thus, both expressions in (10.4) represent the mathematical model of the magnetic levitation system depicted in Fig. 10.1.

10.1.2 Open-Loop Energy Exchange Consider the following slightly modified version of the mathematical model in (10.4), i.e., L(y)

d L(y) 1 d L(y) 2 di + i y˙ + Ri = u, m y¨ = i − G(y), dt dy 2 dy

(10.5)

with P(y) a positive semidefinite scalar function. The total where G(y) = d P(y) dy energy stored in the system is given as V (y, y˙ , i) =

1 1 L(y)i 2 + m y˙ 2 + P(y), 2 2

where first term stands for the magnetic energy stored in the electrical subsystem and the second and third terms represent the kinetic and potential energies stored in the mechanical subsystem. The time derivative of V along the trajectories of system in (10.5) is given as d P(y) di 1 d L(y) 2 y˙ i + i L(y) + y˙ m y¨ + y˙ , V˙ = 2 dy dt dy     1 d L(y) 2 1 d L(y) 2 d L(y) = y˙ i + i − i y˙ − Ri + u + y˙ i − G(y) + G(y) y˙ , 2 dy dy 2 dy = −Ri 2 + iu.

(10.6)

10.1 Magnetic Levitation Systems

439

Notice that cancellation of the cross terms: 1 d L(y) 2 d L(y) 2 1 d L(y) 2 y˙ i − i y˙ + y˙ i = 0 2 dy dy 2 dy

(10.7)

represents the natural energy exchange between the electrical and the mechanical subsystems whereas cancellation of the terms ±G(y) y˙ represents the exchange between kinetic and potential energies. Hence, if we define the input u and the output i, then the expression in (10.6) proves that the model in (10.5) is output strictly passive. This property is exploited in the next section to design a position controller for the magnetic levitation system. In this respect, it is important to stress that including the function −G(y) in (10.5) instead of +mg in (10.4) is to be accomplished using feedback.

10.1.3 Position Control Lots of control schemes have been proposed in the literature for magnetic levitation systems. However, we are only interested in those based on energy ideas or passivity based. This is the case of controllers introduced in [204, 211, 234]. The idea of these approaches is to take advantage of the plant structure to design simple controllers achieving good performances. However, although the control scheme in [204] is capable to include a PID position controller, it is not capable to include an internal PI controller to cope with the electrical subsystem.1 Moreover, the proposals in [211, 234] do not include any of these components. In the following proposition we introduce a control scheme which possesses both of these ingredients which are common in the standard control scheme in current practice for magnetic levitation systems (see [106], Chap. 13, for instance). Proposition 10.1 Consider the mathematical model in (10.4) in closed-loop with the following controller:  t ˜ ˜ i|, ˜ (10.8) i(s)ds − kq i˜ y˙ 2 − k f i| i˜ = i − i ∗ , u = −α p i˜ − αi 0   2  ∗

F ∗ , F ∗ = k p h( y˜ ) + kd y˙ + ki sat (z), y˜ = y − y ∗ , (10.9) i =

d L(y)

dy

     t  βk p αβkd α 1+ h( y˜ ) + 1 + y˙ ds, (10.10) z= ki ki 0 where y ∗ > 0 is a real constant standing for the desired position, h( y˜ ) = σ( y˜ ), sat (z) = σ(z), where σ(·) is a strictly increasing linear saturation function for some 1 In

[204], Chap. 10, it is shown that such a PI controller can be included. However, this controller, in fact, has nothing to compensate since the term Ri ∗ still needs to be used in voltage to be applied.

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10 Magnetic Levitation Systems and Microelectromechanical Systems

(L , M) (see Definition 2.34). Furthermore, it is also required that function σ(·) be continuously differentiable such that dσ(ς) ≤ 1, ∀ς ∈ R. dς

0
γ1 , F > γ2 , | y˜ | ≤ L , for some γ1 > 0 and γ2 > 0. Under these conditions there always exist constant scalars α, β, kq , k f , k p , kd , ki , α p , αi , such that the closed-loop system has an unique equilibrium point which is asymptotically stable. At this equilibrium point y˜ = 0. Poof of Proposition 10.1 Replacing (10.8) in the first expression in (10.4), defining: 

t

zi = 0

  2mg R  ˜

, i(s)ds +

αi

d L(y ∗ )

(10.13)

dy

where

d L(y ∗ ) d L(y)

= , dy dy y=y ∗ and adding and subtracting the terms Ri ∗ , R



2mg∗ ,

d L(y )

dy

∗

L(y) didt ,

d L(y) ∗ i y˙ , dy

it is found

⎞   d i˜ ⎜  2mg ⎟ d L(y) ˜ − L(y) = −(α p + R)i˜ − R ⎝i ∗ −

i y˙

d L(y ∗ ) ⎠ dt dy

dy

⎛

−

di ∗ d L(y) ∗ ˜ i|, ˜ i y˙ − L(y) − αi z i − kq i˜ y˙ 2 − k f i| dy dt

where ⎞−1/2 ⎛ ∗ di ∗ 2F

⎠ = ⎝

d L(y)

dt

dy



   d

d L(y)

−1 × y˙ k p h( y˜ ) + kd y˙ + ki s(z) + mg dy dy

(10.14)

10.1 Magnetic Levitation Systems

441





d L(y) −1 dh( y˜ ) dsat (z)



kp y˙ + kd y¨ + ki z˙ , +

dy

d y˜ dz s(z) = sat (z) −

1 mg. ki

(10.15) (10.16)

˜ 1 d L(y) i ∗2 , and replacing i ∗ Adding and subtracting the terms 21 d L(y) i ∗ i, 21 d L(y) i ∗ i, dy dy 2 dy ∗ and F from (10.9), the second expression in (10.4) becomes m y¨ =

1 d L(y) ˜2 d L(y) ˜ ∗ i + ii − k p h( y˜ ) − kd y˙ − ki s(z). 2 dy dy

(10.17)

The closed-loop dynamics is given by (10.13)–(10.17) and (10.10). The equilibria of this dynamics are found as follows. From the state equation y˙˜ = y˙ = 0 it is concluded that y˙ = 0. Using this result in z˙ = 0 (from (10.10)) yields y˜ = 0. From z˙ i = 0 (see (10.13)), we find i˜ = 0. Then, from m y¨ = 0 in (10.17) we find z = k1i mg if: L>

1 mg. ki

(10.18)

˜

With these values in ddti = 0, from (10.14), and in (10.15) it is found that z i = 0. Thus, the only equilibrium point of the closed-loop dynamics is ξ = [ y˜ , y˙ , z − 1 ˜ z i ] = [0, 0, 0, 0, 0] . Notice that this closed-loop dynamics is autonomous mg, i, ki because it can be written as ξ˙ = f (ξ) for some nonlinear f (·) ∈ R5 . The closed-loop dynamics (10.13)–(10.17), (10.10), can be rewritten as d L(y) ˜ d i˜ = −(α p + R)i˜ − i y˙ + U, (10.19) dt dy 1 d L(y) ˜2 (10.20) i − G, m y¨ = 2 dy     ˜ z˙ = α 1 + βk p h( y˜ ) + 1 + αβkd y˙ , (10.21) z˙ i = i, ki ki d L(y) ˜ ∗ G=− ii + k p h( y˜ ) + kd y˙ + ki s(z), dy ⎞ ⎛   2mg d L(y) ∗ di ∗ ⎜  ˜ i| ˜

⎟ − i y˙ − L(y) − kq i˜ y˙ 2 − k f i| U = −R ⎝i ∗ −

⎠ ∗

d L(y )

dy dt



L(y)

dy

−αi z i . Notice that (10.19)–(10.21) are almost identical to the open-loop model in (10.5) if we ˜ U, G. One important difference is that the resistance replace y, y˙ , i, u, G, by y˜ , y˙ , i, R in (10.5) has been enlarged to α p + R in (10.19). Moreover, we can see that suitable damping can be introduced thanks to the term kd y˙ in the definition of G.

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10 Magnetic Levitation Systems and Microelectromechanical Systems

Another important difference is the two new equations in (10.21) which represent the integral terms of the PI electric current controller and the PID position controller, which are intended to compensate for the effects of the gravity term mg. These observations motivate the use of the following “energy” storage function for the closed-loop dynamics: ˜ z i ) = 1 L(y)i˜2 + 1 αi z i2 + V ( y˜ , y˙ , z − mg/ki ), (10.22) W ( y˜ , y˙ , z − mg/ki , i, 2 2

where  y˜ 1 2 V ( y˜ , y˙ , z − mg/ki ) = m y˙ + αmh( y˜ ) y˙ + αkd h(r )dr 2 0  y˜  z +k p h(r )dr + ki s(r )dr + αβms(z) y˙ . 0

mg/ki

Notice that the function V ( y˜ , y˙ , z − mg/ki ) defined in (10.22) is very similar to the function analyzed in Appendix A.4. Thus, we can proceed similarly to prove that V ( y˜ , y˙ , z − mg/ki ) is positive definite and radially unbounded if (A.8), (A.9), (A.10), (A.14), are satisfied with y˜ = q, ˜ y˙ = q, ˙ y ∗ = q ∗ , k p = k p , kd = kd , ki = ki , b = 0, k  = 0, kg = 0, g(q ∗ ) = mg, m = J . Thus, the function W ( y˜ , y˙ , z − mg/ ˜ z i ) qualifies as a Lyapunov function candidate because it is positive definite and ki , i, radially unbounded. The first two terms in W represent the “magnetic energy” stored in the electrical subsystem and the “energy” stored in the integral term of the PI electric current controller. On the other hand, function V includes the kinetic energy and the closedloop “potential energy”:  P( y˜ ) = k p

y˜

h(r )dr,

0

as well as the “energy” stored in the integral of position through an integral of s(·). The cross terms αmh( y˜ ) y˙ and αβms(z) y˙ are required to provide W˙ with negative quadratic terms on both h( y˜ ) and s(z). In this respect, it is easy to verify that2 : d dt



 1 1 L(y)i˜2 + m y˙ 2 + P( y˜ ) = 2 2

(10.23)

d L(y) ˜ ∗ y˙ ii − ki s(z) y˙ . −(R + α p )i˜2 − kd y˙ 2 + i˜ U + dy

2 This implies that cross term cancellations due to the natural energy exchange between the electrical

and the mechanical subsystems of the open-loop system, described in (10.7), are also present in the closed-loop system.

10.1 Magnetic Levitation Systems

443

Since U depends on both h( y˜ ) and s(z), negative quadratic terms on both h( y˜ ) and ˜ as well as in i˜ and s(z) and s(z) are required to dominate some cross terms in h(i)  ˜i (the quadratic term −(R + α p )i˜2 already exists). The integral term αkd y˜ h(r )dr 0 is intended to cancel an undesired cross term appearing in the time derivative of αmh( y˜ ) y˙ . Finally, notice that because of the product i˜ U above, some third-order terms involving i ∗ appear. These terms can be dominated by quadratic negative terms in y˙ and i˜ because |h( y˜ )| and |s(z)| are bounded by finite constants. This is the reason to employ a PID position controller with saturated proportional and integral actions. Some other third-order terms must be dominated by the nonlinear terms ˜ i| ˜ included in (10.8). −kq i˜ y˙ 2 − k f i| Taking into account (G.9)–(G.13) in Appendix G.1 it is found that W˙ can be upper bounded as   1 ˜3 krr kd kσ |i| (10.24) W˙ ≤ −x Qx − k f − 2m 1 −kq1 y˙ 2 i˜2 + krr kd kσ kδ i˜2 | y˙ | − α p2 i˜2 2m ˜ y˙ 2 − kd2 y˙ 2 , −kq2 y˙ 2 i˜2 + (kr kd + Rkt )|i| ˜ x = [| y˙ |, |h( y˜ )|, |s(z)|, |i|], where kd1 , kd2 , kq1 , kq2 , α p1 , α p2 , are positive constant scalars such that kd1 + kd2 = kd , kq1 + kq2 = kq , and α p1 + α p2 = α p . The entries of matrix Q are defined as  αβkd , Q 22 = αk p , Q 33 = αβki , = kd1 − αm − αβm 1 + ki α αβ 1 kσ M − krr kd kσ I ∗ (0), = R + α p1 − kσ M − 2 2 m   βk p α2 βm , 1+ = Q 21 = − 2 ki = Q 13 = Q 23 = Q 32 = 0, krr kd2 kr k p M Rk ∗ kr mg krr k p − kr k i M − − − − = Q 41 = − 2 2 2m 2   2 αkσ kδ M αβkd 3Rkt M ki krr − 1+ − αβ Mkδ kσ − , − 2 ki 2 2   βk p krr kd k p αkσ I ∗ (0) αki krr Rk ∗ − − 1+ − , = Q 24 = − 2m 2 ki 2 2 αβkσ I ∗ (0) Rk ∗ krr kd ki − − , = Q 34 = − 2m 2 2 

Q 11 Q 44 Q 12 Q 31 Q 14

Q 42 Q 43 where

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10 Magnetic Levitation Systems and Microelectromechanical Systems

⎧ ⎨

⎛

⎧ ⎨

⎛

⎞−1/2

d 2F ⎠



kr = max L(y) ⎝



d L(y)

dy ⎩



krr

∗

dy

⎫



d L(y) −1

⎬



dy

⎭ ,

⎫ ⎞−1/2



∗

d L(y) −1 ⎬ 2F



⎠ = max L(y) ⎝

dy ⎭ .

d L(y)

⎩

dy

and constants kσ , kδ , k ∗ , kt , I ∗ (0), are defined in Appendix G.1. Notice that the four leading principal minors of matrix Q can always be rendered positive definite by suitable selection of the controller gains kd1 , k p , ki , α p1 , and hence, λmin (Q) > 0. From: −kq1 y˙ 2 i˜2 +

1 krr kd kσ kδ i˜2 | y˙ | < 0, 2m

we find | y˙ | >

1 krr kd kσ kδ . 2mkq1

Using this in the following expression: 1 krr kd kσ kδ i˜2 | y˙ | − α p2 i˜2 < 2m



1 krr kd kσ kδ 2m

2

1 ˜2 i − α p2 i˜2 < 0, kq1

we find  α p2 >

1 krr kd kσ kδ 2m

2

1 . kq1

(10.25)

Hence, (10.25) ensures that the second row in (10.24) is always negative. On the other hand, from ˜ y˙ 2 < 0, −kq2 y˙ 2 i˜2 + (kr kd + Rkt )|i| we find ˜ > |i|

kr kd + Rkt . kq2

Using this in the following expression: 2 ˜ y˙ 2 − kd2 y˙ 2 < (kr kd + Rkt ) y˙ 2 − kd2 y˙ 2 < 0, (kr kd + Rkt )|i| kq2

10.1 Magnetic Levitation Systems

445

we find kd2 >

(kr kd + Rkt )2 . kq2

(10.26)

Hence, (10.26) ensures that the third row in (10.24) is always negative. Thus, if additionally: kf >

1 krr kd kσ , 2m

(10.27)

it is concluded that W˙ ≤ 0 for all ξ ∈ D where D is a subset of R5 satisfying (10.12). Thus, stability of the origin is concluded. Since the closed-loop system is autonomous, the LaSalle invariance principle (see Corollary 2.18) can be used to try to prove asymptotic stability. Define a set S as S = {ξ ∈ D|W˙ = 0} = { y˜ = 0, y˙ = 0, z − mg/ki = 0, i˜ = 0, z i ∈ R}. Evaluating the closed-loop dynamics (10.19)–(10.21) in S we obtain d L(y) d i˜ = −(α p + R)(0) − (0)(0) + U, dt dy 1 d L(y) 2 (0) − G, m y¨ = 2 dy     βk p αβkd (0) + 1 + (0), z˙ i = (0), z˙ = α 1 + ki ki d L(y) G=− (0)i ∗ + k p (0) + kd (0) + ki (0), dy d L(y) ∗ i (0) − L(y)(0) − kq (0)(0)2 − k f (0)|0| U = −R(0) − dy −αi z i .

L(y)

˜

These expressions have, as the unique solution, ξ = 0 because ddti = 0, since the set S is invariant, and αi > 0. Thus, according to Corollary 2.18, this implies that ξ = 0 is an asymptotically stable equilibrium point. This completes the proof of Proposition 10.1. Conditions for this stability result are summarized by (A.8), (A.9), (A.10), (A.14), with y˜ = q, ˜ y˙ = q, ˙ y ∗ = q ∗ , k p = k p , kd = kd , ki = ki , b = 0, k  = ∗ 0, kg = 0, g(q ) = mg, the four leading principal minors of matrix Q defined in (10.24) are positive and (10.25), (10.26), (10.27). Remark 10.2 The control scheme in Proposition 10.1 is depicted in Fig. 10.3. Notice ˜ i|, ˜ and the fact that F ∗ represents that excepting the two simple terms −kq i˜ y˙ 2 − k f i| ∗ a nonlinear PID position controller and i introduces and additional nonlinearity, the

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10 Magnetic Levitation Systems and Microelectromechanical Systems

Fig. 10.3 Control scheme in Proposition 10.1. The block i ∗ stands for (10.9) and η = −kq i˜ y˙ 2 − ˜ i|. ˜ k f i|

Fig. 10.4 Traditional control scheme for magnetic levitation systems

control scheme in Proposition 10.1 is very similar to the traditional control scheme for magnetic levitation systems depicted in Fig. 10.4. The advantage of controller in Proposition 10.1 is that it is provided with a (local) asymptotic stability proof based on the complete nonlinear model of the plant. The traditional control scheme is based on a linear approximate model of the magnetic levitation system and, hence, kt represents an approximate force constant for the electromagnet, i.e., it is assumed that F = kt i. Notice that both controllers in Figs. 10.3 and 10.4 employ a negative definition of the position error. This is to compensate for the negative value of d L(y) dy appearing in the definition of the magnetic force. Remark 10.3 It is stressed that, contrary to previous results in the literature [204], controller in Proposition 10.1 does not require to use feedback to complete an isolated error equation for the electrical subsystem (as a matter of fact, the time derivative of the desired electric current is not fed back). This is a fundamental step in [204] since it allows to prove exponential stability of the electrical subsystem which, in turn, allows to ensure stability of the mechanical subsystem. These are some of the novelties of our approach that render simpler the control law for the electrical subsystem. However, the price to be payed for this simplicity is that some problematic terms arise because they are not cancelled. Moreover, some other terms appear because velocity measurements are allowed, which is instrumental for PID position control design in nonlinear systems (see Remark 10.4). In this ˜ i| ˜ are introduced to dominate all of respect, the two simple terms −kq i˜ y˙ 2 − k f i| these problematic terms and to ensure stability. Thus, simplicity of controller in Proposition 10.1 is due to this novel solution. Remark 10.4 A PID position controller which does not require velocity measurements is designed in [204], Chap. 8, for a magnetic levitation system. This, however, is possible because the mechanical dynamics is linear and, hence, linear tools such as the concept of Hurwitz matrices can be employed to ensure stability.

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447

On the other hand, it is important to stress that controller in [204], Chap. 8, becomes singular when the desired force crosses through zero. This is opposite to controller in Proposition 10.1 because the time derivative of the desired electric current is not fed back. In Proposition 10.1 we constrain the operation region such that the desired force does not become zero. However, this is just because in such a case stability is not ensured, which does not mean that the closed-loop system becomes unstable because the stability conditions are just sufficient. Also notice that problems ∗ )F ∗ might arise when F ∗ < 0. These problems would be avoided if using −sign(F d L(y) instead of

∗

F

d L(y)

dy

dy

in the expression for i ∗ in (10.9). It is important to stress that in such

a case i ∗ remains continuous because −sign(F ∗ )F ∗ is also continuous. Furthermore, although the derivative of −sign(F ∗ )F ∗ might be discontinuous (only at F ∗ = 0), it is bounded and can be dominated as what happens with the time derivative of i ∗ . Nevertheless, F ∗ ≤ 0 is an undesirable situation that must be avoided. The arguments given above are just to stress that our controller is free of singularities. We also stress √ that the trick of including a function f (|F ∗ |), as in Chap. 6 for SRMs, to replace F ∗ when F ∗ is close to zero also works in the case of magnetic levitation systems and, since F ∗ = 0 in steady state in this case, (local) asymptotic stability is still achieved. Another PID control scheme without velocity measurements was introduced by the authors in [105] which is based on the proposal in [203]. Experience with this controller has taught that it is very difficult to find controller gains achieving good performances. Remark 10.5 Instrumental to obtain (10.24) are some cancellationsof cross terms ˜ ∗ y˙ belonging to d 1 L(y)i˜2 the y˙ i˜2 − d L(y) i˜2 y˙ − d L(y) ii which involve 21 d L(y) dy dy dy dt 2 time derivative of the“magnetic energy” stored in the electrical subsystem and ! " 1 d L(y) ˜2 y˙ i + d L(y) i ∗ i˜ y˙ belonging to dtd 21 m y˙ 2 , the time derivative of the “kinetic 2 dy dy energy” stored in the mechanical subsystem. These cancellations are due to the natural energy exchange between the electrical and mechanical subsystems. They are obvious consequences of the almost identical structure of the closed-loop dynamics in (10.19)–(10.21), and the open-loop dynamics in (10.5). Remark 10.6 The closed-loop mechanical dynamics in (10.20) can be written as m y¨ = Fe − k p h( y˜ ) − kd y˙ − ki s(z),     βk p αβkd h( y˜ ) + 1 + y˙ , z˙ = α 1 + ki ki

(10.28)

where Fe =

1 d L(y) ˜2 d L(y) ˜ ∗ i + ii , 2 dy dy

(10.29)

represents the difference between the desired and the actual magnetic force on the ball. The time derivative of V ( y˜ , y˙ , z − mg/ki ), defined in (10.22), along the trajectories of (10.28) is given as

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10 Magnetic Levitation Systems and Microelectromechanical Systems

⎤⎡ ⎤ | y˙ | Q 11 Q 12 Q 13 V˙ ≤ −[| y˙ |, |h( y˜ )|, |s(z)|] ⎣ Q 21 Q 22 Q 23 ⎦ ⎣ |h( y˜ )| ⎦ + Fe ( y˙ + αh y˜ + αβs(z)), |s(z)| Q 31 Q 32 Q 33 ⎡

where Q i j , i, j = 1, 2, 3, are entries of matrix Q defined in (10.24). On the other hand, the time derivative of Ve = 21 L(y)i˜2 along the trajectories of the closed-loop electric dynamics in (10.19) includes the term −(R + α p )i˜2 . Thus, instrumental to prove asymptotic stability in the proof of Proposition 10.1 are the following observations: • The scalar function V , intended for the mechanical subsystem, is a strict Lyapunov function when the desired magnetic force is the input of the closed-loop mechanical subsystem dynamics, i.e., when i˜ = 0. • The coefficient of the negative term −(R + α p )i˜2 appearing in V˙ + V˙e can be enlarged arbitrarily. This is important to dominate cross terms in W˙ depending on ˜ when i˜ = 0. i, • Cancellation of several cross terms which is explained in Remark 10.5. Notice that all of these features are possible thanks to the passivity properties of the open-loop magnetic levitation system model described in Sect. 10.1.2. Furthermore, the above proof contains the main ideas of procedure described in Sect. 2.4. The magnetic force error is given as a nonlinear function of the electrical dynam˜ i|, ˜ in ics error (see (10.29)). This has forced to introduce the terms −kq i˜ y˙ 2 − k f i| (10.8), which are important to dominate some third-order terms appearing in W˙ . These nonlinear terms arise from the nonlinear nature of the electromagnet model, a feature that has traditionally complicated the controller design task for magnetic levitation systems. Thanks to the introduction of the above terms, the controller in Proposition 10.1 is very simple, i.e., only differs from standard control for magnetic ˜ i|, ˜ which are zero at the desired levitation systems by the simple terms −kq i˜ y˙ 2 − k f i| equilibrium point, and a nonlinear PID position controller.

10.1.4 Simulation Results In this section we present a numerical example to give some insight on the achievable performance when controller in Proposition 10.1 is employed. To this aim, we use the numerical values of the magnetic levitation system that has been tested experimentally in [106]. In that work, the electromagnet inductance is modeled as L(y) = k0 +

k , 1 + ay

where k0 = 36.3 × 10−3 [H], k = 3.5 × 10−3 [H], a = 5.2 × 10−3 [m], and the remaining parameters are R = 2.72[Ohm], m = 0.018[kg], g = 9.81[m/s2 ]. The

10.1 Magnetic Levitation Systems

449

Fig. 10.5 Simulation results when controller in Proposition 10.1 is employed

practical range of input voltages is [0, +12][V] and the maximal electric current is 3[A]. The controller gains were chosen to be k p = 2.5, kd = 0.8, ki = 2, α p = 470, αi = 1000, kq = 100, k f = 11, α = β = 2. We used the saturation function presented in (4.73) with L = 0.1059, M = 0.1165. These controller gains were proven to satisfy all of the stability conditions in the proof of Proposition 10.1, excepting (10.26) and det(Q) > 0, with Q the matrix defined in (10.24). These conditions are satisfied, however, using α p = 2 × 107 and kq = 1 × 1010 . Since these values are not realistic, we have decided not to employ them in the simulations that follow. In Sect. 10.2.4 is given an explanation on why such large controller gains are obtained and an important justification to use in simulations, and in practice, controller gains that not satisfy the Lyapunov-based stability conditions.

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10 Magnetic Levitation Systems and Microelectromechanical Systems

On the other hand, despite the above situation, our approach must not be underestimated since it offers a number of advantages. Controller in Proposition 10.1 does not require to feedforward the term Ri ∗ . Furthermore, the exact value of the electric resistance R is not required at all. This is very important because the large values of electric current that are common in magnetic levitation systems result in important changes in the electric resistance under normal operation conditions. Moreover, we have learned from experiments with magnetic levitation systems (see [106]) that an electric current loop (driven by a linear PI controller) is important to ensure a good performance in experiments. In this respect, in [204] Chap. 8 is proposed a passivitybased controller which requires to feedforward a term depending on the desired flux and the exact value of electric resistance. It is claimed there that this is enough to avoid the necessity for an internal loop in the magnetic flux. Although avoiding such a magnetic flux loop is fortunate because of the difficulties that poses magnetic flux measurements, this internal loop is in fact necessary to achieve good performances in experiments, as stressed above. Moreover, although controller in [204] Chap. 8 has the possibility to employ a flux internal loop, only a proportional controller is allowed in such an approach. As a matter of fact, in Chap. 10 of [204] it is shown that an integral action can be included in the internal loop of such a passivity-based approach. However, this integral action has nothing to compensate for since the term depending on the electric resistance and the desired magnetic flux is still required in the formal proof. Instrumental to include an internal PI electric current loop in Proposition 10.1 is the fact that we do not require to ensure that the electric current error converges to zero exponentially. In order to ensure this in [204] Chap. 8, an isolated electric current closed-loop dynamics is required to be completed by feeding back the time derivative of the desired magnetic flux. Our approach does not require this because we are capable to dominate the cross terms existing between the electrical and the mechanical subsystems. On the contrary, the isolated electric current closed-loop dynamics that is required [204] Chap. 8 is an obstacle to include the integral action in the magnetic flux controller. After discussing the above justification for controller in Proposition 10.1, let us continue with our simulations. The initial conditions were set as follows y(0) = 0 ˜ = −4.24362 × 10−3 , z(0) = 0.0885. 0.006[m], y˙ (0) = 0, i(0) = 1.56[A], −∞ idt The desired position y ∗ , in meters, was chosen as ⎧ ⎨ 0.006, 0 ≤ t < 0.5 y ∗ = 0.008, 0.5 ≤ t < 2.75 . ⎩ 0.004, 2.75 ≤ t In Fig. 10.5 we verify that the electric i and the applied voltage u remain below 3[A] and within [0, +12][V], respectively. We have saturated the applied voltage at 0[V] since a peak value of −6[V] appeared at t = 0.5. Notice that the ball position y reaches the desired value y ∗ in steady state and this response has an about 2[s] settling time. We observe that the position response is very damped. In this respect,

10.1 Magnetic Levitation Systems

451

we have performed several additional simulations which have made us to arrive to  similar conclusions as in Sect. 4.4.1: the term 1 + αβ kkdi y˙ appearing in (10.10) is responsible for such a damped response.

10.2 Microelectromechanical Systems 10.2.1 Mathematical Model A popular microelectromechanical system (MEMS) is the electrostatic micromirror depicted in Fig. 10.6. It consists basically of a capacitor with capacitance Ca which possesses one mobile plate and the other is fixed. The outer surface of the mobile plate is a mirror and the main purpose is to change the tilt angle θ to modify the incidence angle of light on the mirror. The movable plate has an inertia J , a viscous friction coefficient b, and a stiffness constant K . Since capacitance depends on the separation distance between the plates, it is not difficult to realize that Ca depends on the angle θ, i.e., Ca = Ca (θ). Since the plates of the capacitor contain equal but opposite sign charges Q a , the fixed plate exerts an attractive electrostatic torque on the mobile plate which is given as τ=

∂ ∂θ



 1 2 1 dCa (θ) va Ca (θ) = va2 , 2 2 dθ

(10.30)

where 21 va2 Ca (θ) is the electric energy stored in the capacitor and va is the voltage at the terminals of Ca . Hence, using Newton’s Second Law, the mathematical model of the mechanical subsystem is found to be given as 1 dCa (θ) . J θ¨ + bθ˙ + K θ = va2 2 dθ

(10.31)

a (θ) Notice that, according to Fig. 10.6, the torque τ = 21 va2 dCdθ is applied in the direction where θ increases. Moreover, since capacitance always increases as the distance between plates decreases, it is not difficult to realize that

dCa (θ) > 0. dθ

(10.32)

The electric diagram of the system is depicted in Fig. 10.7. vs is the voltage applied at the circuit and it is used as the control signal, C p represents the capacitance at the output circuit of the high-voltage amplifier used to actuate on the capacitor Ca whereas R is the input resistance of the circuit. Using Kirchhoff’s current Law:

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10 Magnetic Levitation Systems and Microelectromechanical Systems

Fig. 10.6 An electrostatic micromirror

Fig. 10.7 Electric diagram of the system

1 Is = I p + Ia = Q˙ p + Q˙ a = (vs − va ). R Defining Q = Q p + Q a : 1 Q˙ = (vs − va ), R and given the total capacitance of the system C T (θ) = C p + Ca (θ): Qa Q = va = , Ca (θ) C T (θ) we have d dCa (θ) ˙ 1 Q˙ = (va C T (θ)) = θva + C T (θ)v˙a = (vs − va ), dt dθ R where C p is assumed to be constant. Hence, C T (θ)v˙a =

1 dCa (θ) ˙ (vs − va ) − θva . R dθ

(10.33)

10.2 Microelectromechanical Systems

453

Fig. 10.8 Electrostatic micromirror

Thus, the mathematical model of the system is given by (10.31) and (10.33). Notice that excepting the differences in the mechanical structure, i.e., that a inertia-springdamper system is present in (10.31), the mathematical models for an electrostatic micromirror, given in (10.31), (10.33), and for a magnetic levitation system, given in (10.4), are identical3 if we replace va by i, θ by y, and L(y) by C T (θ), recall that dC T (θ) a (θ) = dCdθ because C p is assumed to be constant. dθ The capacitance Ca (θ) can be computed by assuming that the mobile plate (a rectangular one with length L and width W ) is divided in many strips of area W d x where the area of the complete mobile plate is A = W L. Recall that the capacitance of , where ε is the dielectric permittivity a parallel plates capacitor is given as C = εA d of the material between the plates, A is the area of the plates and d is the plates separation. Hence, the complete capacitor can be seen as the parallel connection of many parallel plates capacitors each one with capacitance (see Fig. 10.8): dCa (θ) =

εW d x . d − x sin(θ)

Thus, Ca (θ) is found as the addition of these capacitances: 

 L εW d x dCa (θ) = , 0 d − x sin(θ)  L dx , = εW 0 d − x sin(θ)  L εW − sin(θ)d x . = − sin(θ) 0 d − x sin(θ)

Ca (θ) =

Defining v = d − x sin(θ) and dv = − sin(θ)d x:

3 Thus,

an electrostatic micromirror represents the electrostatic analogue of a one phase and one pole SRM.

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10 Magnetic Levitation Systems and Microelectromechanical Systems

 d−L sin(θ) dv εW , Ca (θ) = − sin(θ) d v εW sin(θ) = , [ln(v)]d−L d − sin(θ) to finally find [3]:   d εW ln . Ca (θ) = sin(θ) d − L sin(θ)

(10.34)

In Fig. 10.9 is presented the plot of Ca (θ) which corroborates (10.32). Notice that Ca (θ) → ∞ as θ → arcsin(d/L), i.e., when the plates are in contact. This singularity is avoided on [3] by designing the micromirror to possess a comb structure at the outer side of the mobile plate. This prohibits the mobile plate to continue moving before the singularity appears (see [3] for further details). Finally, let us say that a singularity does not exist at θ = 0 as can be demonstrated using L’Hopital’s ˆ rule: lim Ca (θ) = εW lim

θ→0

θ→0

= εW lim

d dθ

'  ( ln d−L dsin(θ) d sin(θ) dθ

d−L sin(θ) d

'

, (

d − (d−L sin(θ)) 2 (−L cos(θ))

θ→0

cos(θ)

,

L cos(θ) d−L sin(θ)

, cos(θ) L = εW lim , θ→0 d − L sin(θ) εW L . = d = εW lim

θ→0

Ca (θ)|θ=0

This represents the capacitance of the parallel plates capacitor, with plates area given as W L and d as the distance between the plates, retrieved when θ = 0. See Fig. 10.6.

10.2.2 Open-Loop Energy Exchange Consider the mathematical model in (10.31), (10.33) and the following scalar function: ˙ = 1 C T (θ)v 2 + 1 J θ˙2 + 1 K θ2 . V (va , θ, θ) a 2 2 2

10.2 Microelectromechanical Systems

455

Fig. 10.9 Ca (θ) vs θ, where ) θ = arcsin(d/L)

˙ represents the energy stored in the complete system since Notice that V (va , θ, θ) 1 2 C (θ)va is the electric energy stored in the electric subsystem, whereas 21 J θ˙2 and 2 T 1 K θ2 are the kinetic and the potential energies, respectively, stored in the mechan2 ˙ along the trajectories of (10.31), ical subsystem. The time derivative of V (va , θ, θ) (10.33), is given as 1 dCa (θ) ˙ 2 ˙ V˙ = θva + va C T v˙a + θ˙ J θ¨ + K θθ, 2 dθ   1 1 dCa (θ) ˙ 2 dCa (θ) ˙ (vs − va ) − = θva + va θva 2 dθ R dθ   1 dC (θ) a ˙ + K θθ, +θ˙ −bθ˙ − K θ + va2 2 dθ 1 1 = − va2 − bθ˙2 + vs va , R R where the fact that been used:

dC T (θ) dθ

=

dCa (θ) dθ

(10.35)

and the following cancellation of cross terms have

1 dCa (θ) ˙ 2 dCa (θ) ˙ 2 1 2 dCa (θ) ˙ θva − θva + va θ = 0. 2 dθ dθ 2 dθ

(10.36)

This cancellation of terms represents the natural energy exchange between the electrical and the mechanical subsystems whereas the cancellation of the terms ±K θθ˙ represents the exchange between the kinetic and the potential energies in the mechanical subsystem. Hence, since b > 0, if we define the input u = vs /R and the output y = va , then from (10.35) we can write 1 V˙ ≤ − y 2 + uy. R

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10 Magnetic Levitation Systems and Microelectromechanical Systems

This expression proves that the model in (10.31), (10.33), is output strictly passive. This property is exploited in the next section to design a position controller for the electrostatic micromirror system.

10.2.3 Position Control Closed-loop control of electrostatic micromirrors has been motivated by the so-called pull-in phenomenon [61, 283]. In static equilibrium, the electrostatic force/torque and the stiffness mechanical force/torque are equal, resulting in a stable condition of the actuator. As the voltage is increased, the electrostatic force/torque increases and eventually overcomes the mechanical force/torque, resulting in instability or a collapse condition, where a contact between the two plates is formed (see Fig. 10.11). This limits the stable open-loop operation to a small portion of the whole physically available range. Closed-loop control is considered to be a viable solution for extending the stable operation range and further enhancing the performance of the electrostatic MEMS [3]. In [177, 178], the micromirror is controlled through the regulation of the electrical charge. A feedback linearization controller is employed accompanied by capacitive stabilization in the former paper and passivity-based stabilizing control law in the latter. A backstepping-based controller is proposed in [290] to render robust the closed-loop system with respect to modeling errors. Another feedback linearization controller accompanied by a passivity-based stabilizing controller is proposed in [276] when the electric charge is controlled. A feedback linearization controller accompanied by a pole assignment stabilizing control law is presented in [3]. In [247] is introduced a LPV-based gain scheduled output-feedback controller. We believe that the stability proof of this controller is involved and the design procedure is complex. Finally, a controller based on IDA-PBC is proposed in [202] whereas a power-based controller is introduced in [75]. We stress that any of these works do not present a controller with integral action, i.e., PID position control. Moreover, any internal PI loop is not included to cope with the electrical dynamics of the MEMS. Even worst, most of these works consist in complex control laws requiring the computation of many terms to exactly cancel the system nonlinearities. This means that such control schemes are in fact not robust. In the following proposition we introduce a control scheme which possesses an internal loop driven by a PI controller to cope with the electrical dynamics and an external loop driven by a PID position controller. Thus, this control law is quite simple and robust. Proposition 10.7 Consider the mathematical model in (10.31), (10.33), in closedloop with the following controller:

10.2 Microelectromechanical Systems

457

 t vs = −α p ξ − αi ξ(s)ds − kq ξ θ˙2 − k f ξ|ξ|, ξ = va − va∗ , (10.37) 0 * 2 ∗ ˜ − kd θ˙ − ki sat (z), θ˜ = θ − θ∗ , (10.38) va = dC (θ) τ ∗ , τ ∗ = −k p h(θ) a

dθ

     t  βk p αβkd ˙ ˜ α 1+ h(θ) + 1 + z= θ ds, ki ki 0

(10.39)

˜ = σ(θ), ˜ where θ∗ > 0 is a real constant standing for the desired position, h(θ) sat (z) = σ(z), where σ(·) is a strictly increasing linear saturation function for some (L , M) (see Definition 2.34). Furthermore, it is also required that the function σ(·) be continuously differentiable such that 0
γ1 , τ > γ2 , Ca (θ) < γ3 , |θ| ≤ L , for some finite γ1 > 0, γ2 > 0, γ3 > 0. Under these conditions there always exist constant scalars α, β, kq , k f , k p , kd , ki , α p , αi , such that the closed-loop system has an unique equilibrium point which is asymptotically stable. At this equilibrium point θ˜ = 0. Poof of Proposition 10.7 Replacing (10.37) in (10.33), defining:  zξ = 0

t

1 ξ(s)ds + αi

*

2K θ∗ dCa (θ∗ ) dθ

,

(10.42)

where

dCa (θ)

dCa (θ∗ ) = , dθ dθ θ=θ∗ and adding and subtracting the terms

1 R

+

αp αi 1 C T (θ)ξ˙ = − ξ − z ξ − R R R −

2K θ∗

dCa (θ∗ ) dθ

, C T (θ)

 va∗ −

*

dva∗ , dt

2K θ∗ dCa (θ∗ ) dθ

a (θ) ∗ ˙ R dCdθ va θ, it is found

 −

dCa (θ) ˙ ξθ dθ

kq kf dCa (θ) ∗ ˙ va θ − C T (θ)v˙a∗ − ξ θ˙2 − ξ|ξ|, dθ R R

(10.43)

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10 Magnetic Levitation Systems and Microelectromechanical Systems

where  v˙a∗

=

−1/2

2 dCa (θ) dθ



τ

∗

  ( dCa (θ) −1 ˙ ' ˜ − kd θ˙ − ki s(z) + K θ∗ × θ −k p h(θ) dθ - −1 ,  ˜ dCa (θ) dh(θ) dsat (z) z˙ , (10.44) + −k p θ˙ − kd θ¨ − ki dθ dz d θ˜ d dθ

s(z) = sat (z) +



1 K θ∗ . ki

(10.45)

a (θ) ∗ a (θ) ∗ a (θ) ∗2 va va , 21 dCdθ va ξ, 21 dCdθ va , and replacing Adding and subtracting the terms 21 dCdθ va∗ and τ ∗ from (10.38), the expression in (10.31) becomes

1 dCa (θ) 2 dCa (θ) ∗ ˜ − kd θ˙ − ki s(z).(10.46) ξ + ξva − k p h(θ) J θ¨ + bθ˙ + K θ˜ = 2 dθ dθ The closed-loop dynamics is given by (10.42)–(10.46) and (10.39). The equilibria of this dynamics are found as follows. From the state equation θ˙˜ = θ˙ = 0 it is concluded that θ˙ = 0. Using this result in z˙ = 0 (from (10.39)) yields θ˜ = 0. From z˙ ξ = 0 (see (10.42)), we find ξ = 0. Then, from J θ¨ = 0 in (10.46) we find z = − k1i K θ∗ if: L>

1 K θ∗ . ki

(10.47)

With these values in ξ˙ = 0, from (10.43), and in (10.44) it is found that z ξ = 0. ˜ θ, ˙ z+ Thus, the only equilibrium point of the closed-loop dynamics is ζ = [θ, 1 ∗



K θ , ξ, z ξ ] = [0, 0, 0, 0, 0] . Notice that this closed-loop dynamics is ki autonomous because it can be written as ζ˙ = f (ζ) for some nonlinear f (·) ∈ R5 . The closed-loop dynamics (10.42)–(10.46), (10.39), can be rewritten as αp dCa (θ) ˙ 1 ξ θ + Vs , C T (θ)ξ˙ = − ξ − R dθ R

˜ + 1 dCa (θ) ξ 2 − G, J θ¨ = −(b + kd )θ˙ − K θ˜ − k p h(θ) 2 dθ     βk p ˜ + 1 + αβkd θ, ˙ h(θ) z˙ ξ = ξ, z˙ = α 1 + ki ki dCa (θ) ∗ ξva + ki s(z), G=− dθ

(10.48) (10.49) (10.50)

10.2 Microelectromechanical Systems

 Vs = −

* va∗

−

−αi z ξ .

2K θ∗ dCa (θ∗ ) dθ

459

 −R

dCa (θ) ∗ ˙ va θ − RC T (θ)v˙a∗ − kq ξ θ˙2 − k f ξ|ξ| dθ

Notice that (10.48)–(10.50) are almost identical to the open-loop model in (10.31), ˜ θ, ˙ ξ, Vs . One important difference is that 1/R ˙ va , vs , by θ, (10.33), if we replace θ, θ, in (10.33) has been enlarged to α p /R in (10.48) and the viscous friction coefficient has been enlarged from b in (10.31) to (b + kd ) in (10.49). Moreover, the stiffness constant K in (10.31) has been enhanced with the constant k p in (10.49). Another important difference is the two new equations in (10.50) which represent the integral terms of the PI electric current controller and the PID position controller, which are intended to compensate for the effects of the spring in steady-state K θ∗ . Finally, the new term G appears in (10.49). These observations motivate the use of the following “energy” storage function for the closed-loop dynamics: ˜ θ, ˙ z + K θ∗ /ki ), ˜ θ, ˙ z + K θ∗ /ki , ξ, z ξ ) = 1 C T (θ)ξ 2 + 1 αi z 2 + V (θ, W (θ, ξ 2 2R where  θ˜ ˜ θ, ˙ z + K θ∗ /ki ) = 1 J θ˙2 + αJ h(θ) ˜ θ˙ + α(b + kd ) V (θ, h(r )dr (10.51) 2 0  θ˜ 1 1 ˜ θ∗ +k p h(r )dr + K θ2 − K θ∗2 − θK 2 2 0  z ˙ +ki s(r )dr + αβ J s(z)θ. −K θ∗ /ki

˜ θ, ˙ z + K θ∗ /ki ) defined in (10.51) is very similar to Notice that the function V (θ, the function analyzed in Appendix A.4 and also to function defined in (10.22). Thus, ˜ θ, ˙ z + K θ∗ /ki ) is positive definite and we can proceed similarly to prove that V (θ, radially unbounded if (A.8), (A.9), (A.10), and (A.14) are satisfied with θ˜ = q, ˜ θ˙ =   ∗ ∗  ∗ ∗ q, ˙ θ = q , k p = k p , kd = kd , ki = ki , kg = K , g(q ) = K θ , m = J and k  = K θ M , where θ M is the maximal absolute value of θ in the set D defined in (10.41). ˜ θ, ˙ z + K θ∗ /ki , ξ, z ξ ) qualifies as a Lyapunov function canThus, the function W (θ, didate because it is positive definite and radially unbounded. The first two terms in W represent the “electric energy” stored in the electrical subsystem and the “energy” stored in the integral term of the PI voltage controller. On the other hand, the function V includes the kinetic energy and the closed-loop “potential energy”: ˜ = kp P(θ)

 0

θ˜

h(r )dr +

1 2 1 ∗2 ˜ ∗ K θ − K θ − θK θ , 2 2

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10 Magnetic Levitation Systems and Microelectromechanical Systems

as well as the “energy” stored in the integral of position through an integral of s(·). ˜ θ˙ and αβ J s(z)θ˙ are required to provide W˙ with negative The cross terms αJ h(θ) ˜ and s(z). In this respect, it is easy to verify that4 : quadratic terms on both h(θ) d dt



 1 1 ˙2 2 ˜ C T (θ)ξ + J θ + P(θ) = (10.52) 2 2 αp dCa (θ) ˙ ∗ ˙ − ξ 2 − (b + kd )θ˙2 + ξ Vs /R + θξva − ki s(z)θ. R dθ

˜ and s(z), negative quadratic terms on both h(θ) ˜ Since Vs depends on both h(θ) ˜ and s(z) are required to dominate some cross terms in h(θ) as well as in ξ and s(z) and ξ (the quadratic term −(α p /R)ξ 2 already exists). The integral term α(b +  θ˜ kd ) 0 h(r )dr is intended to cancel two undesired cross terms appearing in the time ˜ θ. ˙ Finally, notice that because of the product ξ Vs /R above, some derivative of αJ h(θ) third-order terms involving va∗ appear. These terms can be dominated by quadratic ˜ and |s(z)| are bounded by finite constants. negative terms in θ˙ and ξ because |h(θ)| This is the reason to employ a PID position controller with saturated proportional and integral actions. Some other third-order terms must be dominated by the nonlinear terms −kq ξ θ˙2 − k f ξ|ξ| included in (10.37). ˜ = |θ|. ˜ Taking into account this and According to the constraint in (10.41), |h(θ)| ˙ (G.9)–(G.13) in Appendix G.2 it is found that W can be upper bounded as  kf 1 − krr kd kσ |ξ|3 R 2J kq1 ˙2 2 1 ˙ − α p2 ξ 2 − θ ξ + krr kd kσ kδ ξ 2 |θ| R J R kq2 ˙2 2 2 − θ ξ + (kr kd + kt /R)|ξ|θ˙ − kd2 θ˙2 , R ˙ |h(θ)|, ˜ |s(z)|, |ξ|], x = [|θ|, W˙ ≤ −x Qx −



(10.53)

where kd1 , kd2 , kq1 , kq2 , α p1 , α p2 , are positive constant scalars such that kd1 + kd2 = kd , kq1 + kq2 = kq , and α p1 + α p2 = α p . The entries of matrix Q are defined as  αβkd , Q 22 = αk p − αK , = b + kd1 − αJ − αβ J 1 + ki α 1 = α p1 /R − kσ M − αβkσ M − krr kd kσ V ∗ (0), 2 J   βk p α2 β J , 1+ = Q 21 = − 2 ki 

Q 11 Q 44 Q 12

Q 33 = αβki ,

4 This implies that cross term cancellations due to the natural energy exchange between the electrical

and the mechanical subsystems of the open-loop system, described in (10.36), are also present in the closed-loop system.

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αβb αβ K , Q 23 = Q 32 = − , 2 2 krr k p kr k p M krr kd (kd + b) k∗ kr K θ ∗ − kr k i M − − − − = Q 41 = − 2 2 2J 2R  2  αβkd αkσ kδ M 3kt M krr ki 1+ − αβ Mkδ kσ − , − − 2 ki 2 2R   βk p krr kd k p αkσ V ∗ (0) αkrr ki k∗ krr kd K − − 1+ − − , = Q 24 = − 2J 2 ki 2 2R 2J αβkσ V ∗ (0) k∗ krr kd ki − − , = Q 34 = − 2J 2 2R

Q 31 = Q 13 = − Q 14

Q 42 Q 43

where ⎧

⎫  −1/2 

d dC (θ) −1 ⎬ ⎨ 2τ ∗ a



kr = max C T (θ) dC (θ)

, a

dθ

⎭ ⎩ dθ dθ ⎧ ⎫  −1/2   ⎨ 2τ ∗ dCa (θ) −1 ⎬ krr = max C T (θ) dC (θ) , a ⎩ ⎭ dθ dθ

kσ , kδ are defined in (G.12), k ∗ , kt are defined in (G.13) and V ∗ (0) is defined in (G.8). Notice that the four leading principal minors of matrix Q can always be rendered positive definite by suitable selection of the controller gains kd1 , k p , ki , α p1 , and hence, λmin (Q) > 0. Proceeding as in the proof of Proposition 10.1 we find that  α p2 >

1 krr kd kσ kδ 2J

2

R2 . kq1

(10.54)

ensures that the second row in (10.53) is always negative and kd2 >

(kr kd + Rkt )2 R . kq2

(10.55)

ensures that the third row in (10.53) is always negative. Thus, if additionally: kf >

R krr kd kσ , 2J

(10.56)

it is concluded that W˙ ≤ 0 for all ζ ∈ D where D is a subset of R5 satisfying (10.41). Thus, stability of the origin is concluded. Since the closed-loop system is autonomous, the LaSalle invariance principle (see Corollary 2.18) can be used to try to prove asymptotic stability.

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Define a set S as S = {ζ ∈ D|W˙ = 0} = {θ˜ = 0, θ˙ = 0, z + K θ∗ /ki = 0, ξ = 0, z ξ ∈ R}. Evaluating the closed-loop dynamics (10.48)–(10.50) in S we obtain dCa (θ) (0)(0) + Vs /R, dθ 1 dCa (θ) 2 (0) − G, J θ¨ = −(b + kd)(0) − K (0) − k p (0) + 2 dθ     βk p αβkd (0) + 1 + (0), z˙ ξ = (0), z˙ = α 1 + ki ki dCa (θ) (0)va∗ + ki (0), G=− dθ dCa (θ) ∗ va (0) − RC T (θ)(0) − kq (0)(0)2 − k f (0)|0| Vs = −(0) − R dθ −αi z ξ .

C T (θ)ξ˙ = −(α p /R)(0) −

These expressions have, as the unique solution, ζ = 0 because dξ = 0, since the dt set S is invariant, and αi > 0. Thus, according to Corollary 2.18, this implies that ζ = 0 is an asymptotically stable equilibrium point. This completes the proof of Proposition 10.7. Conditions for this stability result are summarized by (A.8), (A.9), (A.10), (A.14), with θ˜ = q, ˜ θ˙ = q, ˙ θ∗ = q ∗ , k p = k p , kd = kd , ki = ki , kg = ∗ ∗  K , g(q ) = K θ , and k = K θ M , where θ M is the maximal absolute value of θ in D, the four leading principal minors of matrix Q defined in (10.53) are positive and (10.54), (10.55), (10.56). Remark 10.8 As the reader can realize from the above proof, the control scheme in Proposition 10.7, for an electrostatic micromirror, is very similar to control scheme in Proposition 10.1, for a magnetic levitation system. Hence, the same remarks presented at the end of the proof of Proposition 10.1 apply for controller in Proposition 10.7. The control scheme in Proposition 10.7 is depicted in Fig. 10.10. Notice that the controller in Proposition 10.7 employs an error position defined as the negative of a (θ) the error position for control scheme in Proposition 10.1. This is because dCdθ

Fig. 10.10 Control scheme in Proposition 10.7. The block va∗ stands for (10.38) and η = −kq ξ θ˙ 2 − k f ξ|ξ|. The block NPID stands for the nonlinear PID controller defined in (10.38)

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and d L(y) have opposite signs. We stress that control scheme in Fig. 10.10 is very dy simple compared to the exact feedback linearization schemes previously proposed in several works in the literature as well as some passivity-based controllers which rely on the solution of partial differential equations. This is one important advantage of the energy-based design proposed in this book. Finally, we stress that contrary to [75, 202] our proposal is provided with both a PID position controller and an internal PI controller for voltage at the capacitor. Although velocity is required to compute the derivative term of the PID controller, in [3] is shown experimentally that velocity can be computed accurately using numerical differentiation of position.

10.2.4 Simulation Study In this section we employ the numerical parameter values of the micromirror designed, built, and tested experimentally in [3], i.e., W = 250 × 10−6 [m], W = 250 × 10−6 [m], L = 300 × 10−6 [m], d = 12 × 10−6 [m], R = 100[Ohm], C p = 2 × 10−7 [F], ε = 8.85 × 10−12 [F/m], and Ca (θ) is given as in (10.34). From the following values ζ = 0.06, ωn = 11 × 103 [rad/s], K = 2.1 × 10−7 [Nm/rad] is defined J = K /ωn2 and b = 2J ζωn . The comb structure referred in the paragraph after (10.34) fixes a maximal (or contact) angle at θC = 1.96◦ . Our first simulation is intended to give some insight on the pull-in phenomenon that appears when the MEMS is controlled in open loop. In the paragraph after (10.34) is explained that a singularity does not exist in the definition of Ca (θ) at θ = 0. However, in the simulations presented in Fig. 10.11 we have chosen the initial conditions θ(0) = 0.1◦ (π/180◦ )[rad] and va (0) = 30[V]. A voltage vs is applied which is defined by a ramp with slope 100/0.05[V/s] starting at vs (0) = 30[V] and remaining constant when vs = 71[V] is reached. a (θ) overcomes the In Fig. 10.11, we realize that the electrostatic torque 21 va2 dCdθ stiffness torque K θ for approximately t > 0.021[s]. At that moment, va has approximately reached 71[V] and remains there because vs remains constant at that value. We also observe that, despite this, the mirror position θ continues to grow until the contact angle θ = θC = 1.96◦ is reached. Since, va remains constant at 71[V], the only explanation for θ to continue growing is that instability has appeared. This is what is called the pull-in phenomenon and closed-loop control is suggested to solve the situation. This is presented in what follows. In order to test controller in Proposition 10.7, we used the following controller parameters. k p = 1 × 10−4 , kd = 1 × 10−7 , ki = 2 × 10−6 , α = β = 100, α p = 100, αi = 2000, kq = 10, and k f = 10. We used the saturation function presented in (4.73) with L ∗ = 10.1, M = 10.11. These controller gains were proven to satisfy all of the stability conditions in the proof of Proposition 10.7, excepting (10.54), (10.55), (10.56) and det(Q) > 0

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10 Magnetic Levitation Systems and Microelectromechanical Systems

Fig. 10.11 Pull-in phenomenon in the MEMS

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465

Fig. 10.12 Simulation results when controller in Proposition 10.7 is employed

with matrix Q defined in (10.53). These conditions are satisfied, however, using α p = 1 × 1027 , kq = 1 × 1027 and k f = 1 × 107 . The reason why these values are very large (unrealistic) is that in the Lyapunov stability analysis we have to consider the worst case to ensure that all the sign undefined terms are dominated by negative terms which can be enlarged by enlarging the above controller gains. This implies that every sign undefined term must be considered to be positive and, then, added to all the other of these terms. Moreover, the maximal possible values in the region under consideration must be accounted for each term despite such a maximal value is not reached in the particular experiment or simulation that is considered. This criterion results in large controller gains that must satisfy very conservative stability conditions. This is the reason why many controllers designed using Lyapunov methods do work in practice despite the, too conservative, stability conditions are not satisfied. We have proceeded like that in the proof of Proposition 10.7 in order to formally show that controller gains satisfying all of the stability conditions do exist. However, to show the conservativeness of such stability conditions, in Fig. 10.12a we plot the instantaneous value of W˙ obtained during the simulations in Fig. 10.13. In Fig. 10.12b we present a zoom-in in Fig. 10.12a. From these figures we realize that, aside from some isolated points of time, W˙ is negative and tends to zero despite α p = 100, αi = 2000, kq = 10, k f = 10 are employed instead of α p = 1 × 1027 , kq = 1 × 1027 , k f = 1 × 107 . In Fig. 10.13 we present the simulation results when controller in Proposition 10.7 is employed. All of the initial conditions were set to zero excepting

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10 Magnetic Levitation Systems and Microelectromechanical Systems

Fig. 10.13 Simulation results when controller in Proposition 10.7 is employed (cont.)

θ(0) = 0.1◦ π/180◦ [rad] and va (0) = 30[V]. In [3], a supply voltage in the range [−300, +300][V] is employed in experiments. In order to perform realistic simulations, we have designed the desired position θ∗ such that this range of voltages is not violated. The desired position θ∗ is designed as three ramps each one with a slope of 30[rad/s]. The first ramp starts at θ∗ = 0.1◦ π/180◦ [rad] and t = 0.005[s] and remains constant when θ∗ = 0.7◦ π/180◦ [rad] is reached. The second ramp takes the desired position from the latter value at t = 0.02[s] to θ∗ = 1.3◦ π/180◦ [rad]. The last ramp takes the desired position from the latter value at t = 0.035[s] to θ∗ = 1.9◦ π/180◦ [rad]. We observe that the mirror position reaches the constant desired position with a settling time of about 0.005[s]. This time is comparable with the settling times reported experimentally in [3]. The supplied voltage vs has large spikes, at isolated points of

10.2 Microelectromechanical Systems

467

time, reaching values far above ±300[V]. Hence, in order to investigate whether good performances are still achieved if vs is forced to remain within [−300, +300][V], we have saturated vs within this voltage limits. We observe that va , the voltage at the MEMS capacitor, is a smooth signal that remains within the range [30, 80][V]. We remark that, aside from the isolated spikes, vs and va overlap all the time.

Chapter 11

Trajectory Tracking for Robot Manipulators Equipped with PM Synchronous Motors

It is widely recognized at present that PM synchronous motors have several important advantages which motivate their application in robotics. For instance, since no windings are present on rotor, normal operation is allowed without requiring mechanical brushes nor commutators. Hence, expensive maintenance procedures are not required and, thus, operation cost is importantly reduced. Moreover, PM synchronous motors combine high torque and high power production with small size, they have reduced friction, high reliability, and good heat dissipation characteristics [26, 111, 222]. One of the most celebrated approaches for motor control is that of passivitybased control introduced in [204]. Moreover, it is claimed by those authors that extensions to trajectory tracking control in rigid robot manipulators equipped with PM synchronous motors can be obtained from ideas reported in [204]. One goal that was pursued in [204] was that of proving global asymptotic stability of standard field-oriented control for voltage-fed AC-motors (see Fig. 11.1). However, the nested-loop passivity-based control scheme introduced in [204] has several model-dependent terms that render this approach quite different and less robust than standard field-oriented control. For instance, (i) time derivative of the desired electric current has to be fed back; this is an important drawback because of the important amount of additional terms that have to be computed online which, besides, are model-dependent in robot control, (ii) bilinear terms in the motor electrical model must be exactly feedback cancelled, i.e., requiring the exact knowledge of motor inductances; this step is instrumental in nested-loop passivity-based control [204] to force electric current error to converge exponentially to zero which is a necessary step to treat electrical dynamics as a vanishing disturbance for the mechanical subsystem, (iii) the counter-electromotive force term must be exactly feedback cancelled. See Chap. 9 in [204]. Despite the above problems remain open until now, interest of the control and robotics communities on formal studies on robot control when taking into account the © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 V. M. Hernández-Guzmán et al., Energy-Based Control of Electromechanical Systems, Advances in Industrial Control, https://doi.org/10.1007/978-3-030-58786-4_11

469

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11 Trajectory Tracking for Robot Manipulators Equipped …

Fig. 11.1 Standard field-oriented control of a PM synchronous motor (PMSM)

electric dynamics of electric motors in the stability proof has drastically diminished after the books [60, 204] were published. According to the previous paragraph, it is the authors belief that this is not because there are no more problems to solve but it seems that it is just because no proposals have been presented until now. Moreover, it was stated in [204] that proving stability of standard field-oriented control must not be underestimated since formal studies give the user additional confidence in the design and provide useful guidelines in the difficult task of commissioning the controller. This affirmation is corroborated by the recent work [205] which is still devoted to such a control problem. Furthermore, although theoretically well justified high-performance motor controllers have been proposed [26, 89, 133, 289], it is remarked in [222] that the electric drives community is not enthusiastic with such controllers because of their complexity. Hence, standard field-oriented control of AC-motors is still the preferred control scheme industrial applications because of its simplicity, good performance and reliability [157, 217]. In the present chapter we introduce a controller for trajectory tracking in ndegrees-of-freedom robot manipulators actuated by PM synchronous motors. We take into account the electrical dynamics of PM synchronous motors during the design stage. We formally prove that system error remains bounded, it has an ultimate bound which depends on the maximal bound of trajectory to be tracked and its time derivative, and this result stands when starting from any initial condition. This controller has two internal electric current loops driven by proportional-integral (PI) controllers and one external loop driven by a proportional-derivative (PD) plus feedforward trajectory tracking controller. Aside from: (1) saturation in the proportional part of the PD plus feedforward controller, (2) saturation in the integral part of the PI electric current controllers, and (3) three simple nonlinear terms added at output of the PI electric current controllers, our proposal is identical to standard field-oriented control. This means that drawbacks remarked in items (i), (ii), (iii), above are eliminated in our proposal. Finally, it is important to stress that proving global ultimate boundedness instead of global uniform asymptotic stability is consistent with current practice as we explain in Remark 11.2 and in the simulations and experiments section. Moreover, this is an important observation that explains why tracking error does not converge to zero in several previously reported experimental results [230, 240]. On the other hand, this recalls attention to the problem of suitable tuning of electric current loops: most people neglect tuning of electric current loops assuming that commercial motor drives are supplied with controller gains that are suitable for all applications.

11.1 Dynamical Model of Robot Manipulators Equipped with PM Synchronous Motors

471

11.1 Dynamical Model of Robot Manipulators Equipped with PM Synchronous Motors The dynamical model of an n-degree-of-freedom rigid robot manipulator equipped with a direct-drive PM synchronous motor at each joint, and equipped only with revolute joints, can be obtained form (4.29)–(4.31), rewriting these expressions to consider n motors simultaneously, i.e., L q I˙q + R Iq + N p L d I D q˙ + Φ M q˙ = Vq , L d I˙d + R Id − N p L q I Q q˙ = Vd ,

(11.1) (11.2)

where Vq = [Vq1 , Vq2 , . . . , Vqn ] ∈ Rn , Vd = [Vd1 , Vd2 , . . . , Vdn ] ∈ Rn , Iq = [Iq1 , Iq2 , . . . , Iqn ] ∈ Rn , Id = [Id1 , Id2 , . . . , Idn ] ∈ Rn , I Q = diag{Iq1 , Iq2 , . . . , Iqn } ∈ Rn×n , I D = diag{Id1 , Id2 , . . . , Idn } ∈ Rn×n , Φ M = diag{Φ M1 , Φ M2 , . . . , Φ Mn } ∈ Rn×n ,

(11.3)

whereas L q , L d , R, N p , and Φ M are constant, diagonal, positive definite matrices representing phase windings inductance and resistance, number or pole pairs and torque constant of each motor, respectively. Also assume that the motors load is a robot manipulator, i.e., τ L is now a ndimensional vector such that ˙ q˙ + g(q), τ L = [M(q) − J ]q¨ + C(q, q)

(11.4)

in (4.31), with all motor mechanical models included in the robot mechanical model1 M(q)q¨ + C(q, q) ˙ q˙ + g(q). Notice that, using nomenclature in (11.1)–(11.3), torque generated simultaneously by all motors can be expressed as τ = [N p (L d − L q )I D + Φ M ]Iq .

(11.5)

Hence, replacing (11.4) and (11.5) in (4.31), and assuming that no friction exists at any robot joint, yields [26, 60, 222]

J is a n × n diagonal matrix containing each motor inertia at its diagonal entries, matrices M(q), C(q, q), ˙ and vector g(q) also suitably contain the mechanical contributions of all motors, i.e., effects of both, rotors, and stators. 1 Although

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11 Trajectory Tracking for Robot Manipulators Equipped …

M(q)q¨ + C(q, q) ˙ q˙ + g(q) = [N p (L d − L q )I D + Φ M ]Iq ,

(11.6)

where q = [q1 , . . . , qn ] ∈ Rn , with qi , i = 1, . . . , n, standing for angular position at joint i, C(q, q) ˙ q˙ is the Coriolis and centripetal effects term whereas g(q) is the gravity effects term which is given as g(q) = ∂U∂q(q) where U (q) is potential energy due to gravity. The n × n inertia matrix M(q) is symmetric and positive definite. On the other hand, as it is by now well known, the following are some important properties of the mechanical part M(q)q¨ + C(q, q) ˙ q˙ + g(q) when all joints are revolute. Property 1 (See Refs. [125] and [130], pp. 96) Matrices M(q) and C(q, q) ˙ satisfy 0 < λm (M(q)), ∀q ∈ Rn , and q˙





 1 ˙ M(q) − C(q, q) ˙ q˙ = 0, ∀q, q˙ ∈ Rn , 2

˙ M(q) = C(q, q) ˙ + C  (q, q), ˙

(11.7)

∀q, q˙ ∈ Rn ,

where latter expression only stands if C(q, q) ˙ is defined using Christoffel symbols. Furhtermore, we use the symbols λm (A) and λ M (A) to represent, respectively, the smallest and the largest eigenvalues of the symmetric matrix A. Property 2 (See Ref. [130], pp. 101, 102) There exists a positive constant kg such that    ∂g(q)  n    ∂q  ≤ kg , ∀q ∈ R . Property 3 (See Ref. [130], pp. 107) The so-called residual dynamics is defined as ˙˜ = [M(qd ) − M(q)]q¨d + [C(qd , q˙d ) − C(q, q)] ˙ q˙d + g(qd ) − g(q). h(t, q, ˜ q) There always exist two positive constants kh1 , kh2 , depending only on robot parameters as well as on the desired trajectory and its first and second time derivatives, such that ˙˜ + kh2  tanh(q), ˙˜ ≤ kh1 q ˜ ∀q, ˜ q˙˜ ∈ Rn , t ∈ R, h(t, q, ˜ q)

(11.8)

where q˜ = qd − q = [q˜1 , . . . , q˜n ] with qd = qd (t) ∈ Rn the time varying desired trajectory which is assumed to be continuous up to the third time derivative and tanh(q) ˜ = [tanh(q˜1 ), . . . , tanh(q˜n )] , with tanh(·) representing the tangent hyperbolic function, if argument is a scalar, or a vector, if argument is a vector.

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473

11.2 Main Result Proposition 11.1 Consider the dynamical model (11.1), (11.2), (11.6), when all of the PM synchronous motors used as actuators have a salient rotor, i.e., L d = L q , together with the following controller: ˙˜ 2 Id , (11.9) Vd = −αdp Id − αdi satd (wd ) − K d q 2 ˙˜ I˜q − K f Id  I˜q , (11.10) Vq = −αq p I˜q − αqi satq (wq ) − K q q −1 ∗ ˜ + kd q˜˙ + F(qd , q˙d , q¨d )], I = Φ [k p tanh(q) q

M

F(qd , q˙d , q¨d ) = M(qd )q¨d + C(qd , q˙d )q˙d + g(qd ),

(11.11)

where q˜ = qd − q is the tracking error with qd = qd (t) ∈ Rn the desired trajectory t which is assumed to be continuous up to its second time derivative, wd = 0 Id dt, t wq = 0 I˜q dt, I˜q = Iq − Iq∗ , whereas satd (x) = [σd (x1 ), . . . , σd (xn )] , satq (x) = [σq (x1 ), . . . , σq (xn )] , with σd (xi ), σq (xi ), i = 1, . . . , n, strictly increasing linear saturation functions for some ( L¯ d , Md ) and ( L¯ q , Mq ) (see Definition 2.34). Furthermore, we also require functions σd (xi ), σq (xi ) to be continuously differentiable such that 0
0 and large enough positive definite diagonal matrices k p and kd such that (A.28) is satisfied and (ii) large enough positive definite matrices αqi + L q−1 (R + αq p ) and αdi + L −1 d (R + αdp ) such that (A.29) and (A.30) are satisfied. ˙˜ contains the robot “kinetic energy” given as 1 q˙˜  M(q)q, ˙˜ Notice that V1 (t, q, ˜ q) 2 whereas V2 ( I˜q , Id , wq , wd ) contains the “magnetic energy” stored in the electrical subsystem given as 21 I˜q L q I˜q + 21 Id L d Id . Thus, when computing V˙ = V˙1 + V˙2 , terms I˜q N p L d I D q˙˜ + I˜q Φ M q˙˜ − Id N p L q Q˙˜ I˜q − Id N p L q Q˙˜ Iq∗ , belonging to   d 1 ˜ 1  ˜ , and −q˙˜  N p (L d − L q )I D I˜q − q˙˜  Φ M I˜q + q˙˜  N p L L + I L I I I q q d d dt 2 q 2 d   d 1 ˙ ∗ ˙ q I D Iq , belonging to dt 2 q˜ M(q)q˜ , cancel. Recall that this cancellation of cross terms represents energy exchange between the mechanical and the electrical subsystems.  q˜ Integral 0 tanh (r )k p dr represents the closed-loop system “potential energy” ˜ which exchanges with the “kinetic energy” through cancellation of terms ± tanh (q)  q˜  ˙ ˜ Moreover, integral γ 0 tanh (r )kd dr is introduced to achieve cancellation of k p q. term −γ tanh (q)k ˜ d q˙˜ appearing in the time derivative of γ tanh (q)M(q) ˜ q˙˜ function

476

11 Trajectory Tracking for Robot Manipulators Equipped …

which is included in order to force term −γ tanh (q)k ˜ p tanh(q) ˜ to appear in V˙ . This term is important in order to ensure the global ultimate boundedness result claimed in Proposition 11.1. Integrals appearing in V2 are required to take into account “energy” stored in the integral parts of the PI electric current controllers whereas terms I˜q satq (wq ) and Id satd (wd ) are instrumental for negative quadratic terms in both satq (wq ) and satd (wd ) to appear in V˙ which, again, is important to ensure the global ultimate boundedness result claimed in Proposition 11.1. Using (11.7), and taking advantage of the above-described cross term cancellations, it is straightforward to verify that time derivative of V along trajectories of the closed-loop system (11.13)–(11.15) is given as in Appendix A.6.2, ˙ d , q˙d , q¨d , qd(3) ) = d (F(qd , q˙d , q¨d )) has been defined. Hence, using (11.8), where F(q dt (11.12), we can group common terms of V˙ given in (A.32) to upper bound as V˙ ≤ −1 −x  Qx − K f Id   I˜q 2 + L q Φ M kd M −1 (q)N p (L d − L q ) M Id   I˜q 2 (11.17)

√ −1 −1 2 ˙˜ Id  K d1 Id  − λ M (N p L d Φ kd ) − n Md K d λ M (L ) − λm (kd2 )q ˙˜ 2 −q M d ˙˜ 2  I˜q  K q1  I˜q  − L q Φ −1 kd M −1 (q) M kc1 −q M

√ ˙˜ 2 − n Mq K q λ M (L q−1 ) − λm (kd3 )q

˙˜ 2  I˜q 2 − K d2 q −K q2 q ˜˙ 2 Id 2 + γ M √ −λm (αq p2 ) I˜q 2 − λm (αdp2 )Id 2 + n Mq K f λ M (L q−1 )Id   I˜q  −1 + I˜q k G + k F λ M (N p L q Φ M )q˙d  M Id  + satq (wq )k G λ M (L q−1 ) −1 +k F λ M (L −1 d N p L q Φ M )q˙d  M satd (wd ),  ˜˙ βe  tanh(q), ˜  I˜q , Id , βq satq (wq ), βd satd (wd )], x = [q, −1 −1 ˙˜  I˜q  Id , kd M −1 (q)N p (L d − L q )Φ M kd  M q γ M = L q Φ M

where symbol  ·  M stands for the supreme value of a matrix or a vector over the −1 −1 F(qd , q˙d , q¨d ) + Φ M q˙d + L q Φ M norm, F(qd , q˙d , q¨d ) M ≤ k F and RΦ M (3) ˙ d , q˙d , q¨d , qd ) ≤ k G , where k F and k G are two finite positive constants. MoreF(q

, and αdi = over, βe , βq , βd are positive scalars such that k p = βe2 k p , αqi = βq2 αqi 2





βd αdi , whereas kd1 , kd2 , kd3 , αq p1 , αq p2 , αdp1 , αdp2 are diagonal, positive definite matrices and K q1 , K q2 , K d1 , K d2 , are positive constants such that K q1 + K q2 = K q ,



+ kd2 + kd3 = kd , αq p1 + αq p2 = αq p , and αdp1 + αdp2 = K d1 + K d2 = K d , kd1 αdp . Entries of matrix Q are defined in Appendix A.6.3. The remaining of the proof is devoted to establish conditions ensuring that V˙ is negative and to determine region where this is true. Assume that we choose: √ nk H M M > βq Mq > βq L¯ q > , θλm (Q) √ nk H , M M > βe > βe > θλm (Q)

M M > βd Md > βd L¯ d >

−1 k H = max{k G , k F λ M (N p L q Φ M )q˙d  M ,

√

nk H , θλm (Q)

(11.18) kG kF −1 λ M (L q−1 ), λ M (L −1 d N p L q Φ M )q˙d  M }, βq βd

11.2 Main Result

477

for some finite constants M M > 0 and 0 < θ < 1. The six leading principal minors of matrix Q can always be rendered positive as follows. For first leading principal minor choose a large enough positive definite kd1 and small values for γ > 0 and Mq > 0, Md > 0 which, according to (11.18), is always possible by choosing large βq and βd , i.e., large αqi and αdi . For second leading principal minor choose a large enough positive definite k p and a small value for γ > 0. For the remaining leading principal



, αdi . minors, simply choose large enough positive definite matrices αq p1 , αdp1 , αqi Hence, matrix Q is positive definite, i.e., λm (Q) > 0. On the other hand, first row in (11.17) is negative if: −1 kd M −1 (q)N p (L d − L q ) M . K f > L q Φ M

(11.19)

First expressions on second and third rows in (11.17) are negative for 1 K d1 1  I˜q  > K q1

Id  >

 √ η1 −1 λ M (N p L d Φ M kd ) + n Md λ M K d (L −1 , d ) = K d1  √ η2 −1 L q Φ M kd M −1 (q) M kc1 + n Mq λ M K q (L q−1 ) = , K q1

where η1 , η2 are defined in a obvious manner. Hence, if: λm (kd2 ) >

η12 η2 , λm (kd3 ) > 2 , K d1 K q1

(11.20)

then second and third rows in (11.17) are negative. Recall that Mq and Md decrease as both αqi and αdi are larger. We recall that, according to (11.18), Md and Mq can be rendered arbitrarily small by choosing large values for βd and βq , respectively. Finally, fourth row in (11.17) is negative if: ˙˜ > q

−1 −1 kd M −1 (q)N p (L d − L q )Φ M kd  M L q Φ M

= η3 . 4K q2 K d2

Hence, fourth plus fifth rows are negative if: λm (αq p2 )λm (αdp2 ) > 2 √ 1 −1 −1 L q Φ M kd M −1 (q)N p (L d − L q )Φ M kd η3 + n Mq K f λ M (L q−1 ) . 4 (11.21) We conclude that we can write

478

11 Trajectory Tracking for Robot Manipulators Equipped … −1 )q˙d  M Id  V˙ ≤ −x  Qx +  I˜q k G + k F λ M (N p L q Φ M

−1 −1 +k F λ M (L −1 d N p L q Φ M )q˙d  M satd (wd ) + satq (wq )k G λ M (L q ),

≤ −x  Qx + k H x1 , √ √ ≤ −x  Qx + nk H x ≤ −λm (Q)x2 + nk H x, or, adding and subtracting term θλm (Q)x2 , for a constant 0 < θ < 1: √ V˙ ≤ −λm (Q)(1 − θ)x2 − θλm (Q)x2 + nk H x, √ nk H . ≤ −λm (Q)(1 − θ)x2 , ∀x > θλm (Q)

(11.22)

Notice that last inequality in (11.22) can be satisfied thanks to (11.18). In [127] is proven that   tanh(q) ˜ ≥ δ(q) ˜ :=

αq, ˜ if q ˜ 1,

βq >

√  nk H , : ξ ≤ θλm (Q)

˙˜ q, where ξ  = [q, ˜  I˜q , Id , wq , wd ]. Notice that ξ = y. Thus, it is possible to write V˙ ≤ −λm (Q)(1 − θ)x2 , ∀y >

√ nk H . θλm (Q)

(11.23)

Using this and (A.31) (see Appendix A.6), we can invoke Theorem 2.29, to conclude that the state remains bounded when starting from any initial condition and it has an ultimate bound which depends on the bound of F(qd , q˙d , q¨d ) and its time derivative (see definition of k H , k F and k G ). This completes the proof of Proposition 11.1. We remark that conditions ensuring result in Proposition 11.1 are summarized as follows: • A small enough γ > 0 and large enough positive definite diagonal matrices k p and kd such that conditions in (A.28) are satisfied. • Large enough positive definite matrices αqi + L q−1 (R + αq p ) and αdi + L −1 d (R + αdp ) such that conditions in (A.29) and (A.30) are satisfied. • The six leading principal minors of matrix Q, introduced in (11.17), are positive. • Conditions in (11.18), (11.19), (11.20), (11.21), are satisfied. Remark 11.2 Result in Proposition 11.1 establishes that the system error has an ultimate bound instead of proving global uniform asymptotic stability. We stress, however, that this is not a situation arising from a inadequate procedure in the stability proof but it is a fundamental limitation when using PI controllers for the internal electric current loops. Recall that integral action in controllers are intended for regulation, not for trajectory tracking. In this respect, in the control task this chapter is concerned with, PI electric current controllers have to track a desired current Iq∗ which includes the time varying function F(qd , q˙d , q¨d ). This function is responsible ˙ d , q˙d , q¨d , qd(3) ) to appear as for several terms involving both F(qd , q˙d , q¨d ) and F(q time varying disturbances in (11.14) and (11.15), through I˙q∗ , which integral action of PI electric current controllers should compensate for. The problem is that integral action does not compensate for time varying disturbances but only for constant disturbances, as it is well known. Furthermore, it is also well known from classical control theory that error decreases as the time varying disturbance is either smaller or slower, which is consistent with the fact that the ultimate bound in Proposition ˙ d , q˙d , q¨d , qd(3) ). Thus, result in 11.1 depends on norms of both F(qd , q˙d , q¨d ) and F(q Proposition 11.1 is consistent with current practice.

2 Because

L¯ q , L¯ d are assumed to be small and

√ nk H θλm (Q)

is assumed to be large.

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11 Trajectory Tracking for Robot Manipulators Equipped …

Remark 11.3 It is clear from the controller equations in Proposition 11.1 that we do not need to feedback additional terms intended to complete the closed-loop system error equations. In particular, we do not need to feedback I˙q∗ , the time derivative of the desired electric current, because we simply dominate such terms. In this respect, the ˙˜ 2 Id and −K q q ˙˜ 2 I˜q − K f Id  I˜q are introduced to domnonlinear terms −K d q inate some third-order terms appearing in I˙q∗ and some other terms due to bilinear nature of motors model. It is important to stress, however, that this domination does not require the exact knowledge of any motor nor robot parameter because it does not rely on the exact cancellation of undesired terms. This fact enhances robustness of the proposed control scheme. Remark 11.4 Although the result in Proposition 11.1 is stated for the more challenging salient rotor motor case, i.e., when L d = L q , this result also stands for the round rotor motor case and the case when some motors have salient rotors and other motors have round rotors. This can be easily verified just by considering that L d = L q and that L di = L qi only for some few i in proof of Proposition 11.1. It is important to stress that contrary to control strategies in the literature (see [65, 222], for instance), our controller is much simpler for both, the salient and round rotor cases. Finally, also note that aside from Φ M the exact value of any electric parameter of motors is not required to perform the control task described in Proposition 11.1. This is an important robustness property of our proposal.

11.3 Simulation and Experimental Results 11.3.1 Simulation Study In this section we present a simulation study as an application example of results in Proposition 11.1. We use the mathematical model of robot manipulator reported in [230], i.e.,  2.351 + 0.168 cos(q2 ) 0.102 + 0.084 cos(q2 ) , M(q) = 0.102 0.102 + 0.084 cos(q2 )   −0.168 sin(q2 )q˙2 −0.084 sin(q2 )q˙2 C(q, q) ˙ = , 0.084 sin(q2 )q˙1 0   3.921 sin(q1 ) + 0.186 sin(q1 + q2 ) . g(q) = 9.81 0.186 sin(q1 + q2 ) 

(11.24)

This is a two-degrees-of-freedom rigid robot with rotative joints. For simulation purposes we assume that this mechanical system is actuated by two PM synchronous motors having parameters reported in [37], i.e., N p = diag{120, 120}, L d = diag{0.00636, 0.00636} [Hy], L q = diag{0.00672, 0.00672}[Hy], Φ M = √ 3/2n p K B , K B = diag{0.106, 0.0106}[Wb], Rs = diag{1.9, 1.9} [Ohm]. Inertias

11.3 Simulation and Experimental Results

481

and masses of these motors are considered to be included in the robot model (11.24). The desired trajectory is given as qd1 = b1 (1 − e−2t ) + c1 (1 − e−2t ) sin(ω1 t), 3

3

qd2 = b2 (1 − e−1.8t ) + c2 (1 − e−1.8t ) sin(ω2 t), 3

3

where b1 = π/4[rad], c1 = π/18[rad], ω1 = 15[rad/s], b2 = π/3[rad], c2 = 25π/36[rad], ω2 = 3.5[rad/s]. This desired trajectory, depicted in Fig. 11.3a, was used in experiments reported in [230] and in simulations reported in [129] using robot parameters in (11.24). The saturation functions σd (xi ) = σq (xi ) are defined as ⎧

xi + L¯ ⎪ ¯ ¯ ¯ ⎪ ⎨ − L + [M − L] tanh M− L¯ , if xi < − L σd (xi ) = σq (xi ) = xi , if |xi | ≤ L¯ ⎪ ¯ ⎪ x − L i ⎩ L¯ + [M − L] ¯ ¯ tanh , if xi > L, M− L¯

where L¯ = L¯ d = L¯ q = 49 and M = Md = Mq = 50. The controller parameters were chosen to be k p = diag{2000, 1000}, kd = diag{150, 15},

(11.25)

K d = K q = K f = 5, and αdp = diag{100, 100}, αdi = diag{1000, 1000},

(11.26)

αq p = diag{100, 100}, αqi = diag{1000, 1000}. Controller gains in (11.25) were chosen because of comparison purposes since they have been tested experimentally in [230], using the same robot parameters and the same desired trajectory. Hence, these controller gains were not proven to satisfy the stability conditions in the proof of Proposition 11.1. No information is presented in [230] on values of gains αdp , αdi , αq p , αqi used in their experiments. All initial conditions were fixed to zero. In Fig. 11.3b we present tracking errors when using controller gains in (11.25), (11.26). Note that tracking error does not converge asymptotically to zero but its norm remains within a maximal value which can be estimated from Fig. 11.3c. We conclude that this behavior has been correctly predicted in Proposition 11.1 (see Remark 11.2). It is important to remark that the simulation results in [129], performed using the same robot parameters and the same desired trajectory but smaller proportional gains, do report asymptotic convergence of tracking errors to zero. We stress that the simulation results in [129] were performed by neglecting the motors electric dynamics, i.e., assuming that perfect torque tracking is accomplished and, hence,

482

11 Trajectory Tracking for Robot Manipulators Equipped …

that torque is the control input. Contrary to that simulation study, the simulation results that we report in Fig. 11.3b have been obtained taking into account motors electric dynamics. Hence, we conclude that tracking errors do not converge to zero because the difference between generated torque and desired torque, in our more realistic simulations, neither converge to zero as can be seen in Fig. 11.3e. As stated in Remark 11.2 this is due to the fact that electric current PI controllers cannot produce a zero electric current error when the desired current is time varying, i.e., because of the feedforward terms defining F(qd , q˙d , q¨d ). As a matter of fact we remark that the experiments in [230] also report similar tracking errors as those in Fig. 11.3b in the present paper, instead of asymptotic convergence to zero. We stress that experiments in [230] use the same robot parameters, the same desired trajectory, and the same controller gains (11.25), as those used in Fig. 11.3b in the present chapter. We encourage the reader to consult [230] to verify this observation. Authors of [230] attribute such nonzero tracking errors to a wrong friction compensation and bad high velocity computation via backwards difference algorithm. We stress that such tracking errors are also present in our simulation study in Fig. 11.3b despite any of such implementation errors do not exist in our simulations. Thus, we conclude that tracking errors do not converge to zero because PI electric current controllers do not accomplish zero torque errors. From (11.23) it is clear that the ultimate bound of error can be rendered smaller if λm (Q) > 0 is rendered larger. According to Theorem 2.13, an important requirement for this is that all of the controller gains contained in the diagonal matrices k p , kd , αq p , αqi , αdp , αdi , must be rendered larger. However, in the problem at hand some of the out of diagonal entries of matrix Q also grow as entries of k p , kd , αq p , αqi , αdp , αdi , grow. This is unfortunate since, according to Theorem 2.13, Ri (A) will not grow for some i = 1, . . . , n. Despite this technical limitation, this observation suggests us to enlarge controller gains to reduce the tracking error. Moreover, classical control results indicate that the effect of time varying disturbances can be decreased enlarging the controller gains when a PI controller is employed. In order to verify this affirmation, we performed another set of simulations using the controller gains in (11.25) and: αdp = diag{698, 698}, αdi = diag{4000, 4000},

(11.27)

αq p = diag{698, 698}, αqi = diag{4000, 4000}. In Fig. 11.3e we realize that norm of torque error is rendered smaller, which results in smaller tracking errors in Fig. 11.3d and a smaller norm of tracking error in Fig. 11.3c. Note that this performance improvement has been accomplished just by redesigning the PI electric current controller gains whereas gains of the controller intended for the mechanical subsystem, i.e., the PD plus feedforward controller, remain the same. Thus, this simulation study allows us to recall the paper [258] where it has been found that neglecting the actuators’ electric dynamics may result in performance deterioration. In this respect, our conclusion is that attention must be payed to gain

11.3 Simulation and Experimental Results

Fig. 11.3 Simulation results when using controller in Proposition 11.1

483

484

11 Trajectory Tracking for Robot Manipulators Equipped …

Fig. 11.4 Simulation results. Tracking errors when using sinusoidal and ramp functions of time as desired trajectories

selection of the electric current controllers. We consider that this is important to stress since it is usual in the control community to worry only for gain selection of the part of the controller concerned with the mechanical subsystem and it is incorrectly assumed that (commercial) electric current controllers are always suitably tuned for all applications, i.e., see conclusions in [230] for instance. We have also performed simulations using K d = K q = K f = 0, i.e., when using standard field-oriented control for both PM synchronous motors, and we have found that tracking errors overlap all the time with responses in Fig. 11.3b and d. Thus, we conclude that performance achieved with controller in Proposition 11.1 is identical to the performance achieved with standard field-oriented control and, because of that,

11.3 Simulation and Experimental Results

485

Fig. 11.5 CICESE robot

our proposal may explain to some extent why standard field-oriented control works well in practice. This is one of our contributions that we want to stress. In order to further study the performance of controller in Proposition 11.1, we present next some additional simulations when the desired trajectory is composed of harmonic waves and ramps. Controller gains in (11.25), (11.26), have been employed. The corresponding tracking errors are shown in Fig. 11.4. It is observed that larger tracking errors are obtained for the harmonic wave. We stress that frequency of these sinusoidal waves is larger than the frequency for desired trajectory in Fig. 11.3a. Thus, a larger tracking error is explained using arguments in Remark 11.2.

11.3.2 Experimental Results We have used the CICESE robot located at the Automatic Control Laboratory of Instituto Tecnológico de La Laguna as experimental platform (see Fig. 11.5). This robot is a 2-dof robot manipulator with two revolute joints whose dynamic model has been presented previously in [129, 230] (see (11.24)). Both links are made of 6061 aluminum and 0.45[m] long. The CICESE robot is equipped with the Yokogawa PM synchronous motors DM1200-A and DM1015-B for the shoulder and elbow joints, respectively. Motor DM1200-A is capable to deliver a 200[Nm] maximal torque and it is equipped with an incremental encoder with resolution of 1,024,000[p/r] for position measurement. Motor DM1015-B delivers a maximal torque of 15[Nm] and it is equipped with an incremental encoder with resolution 655,360[p/r] for position measurement. Joint velocities are computed by applying the standard backwards difference algorithm to position measurements. The control algorithm was programmed using the software WinMechLab [36] and executed in a personal computer Pentium IV, 3GHz, Windows Xp, 32 bits. The electronic board MultiQ-PCI Model 626 from Sensoray was used as interface. Sample period was fixed to 2.5 [ms]. Both motors are operated in torque mode. This means that two linear PI electric current controllers are enabled to control electric currents in motor phases d and

486

11 Trajectory Tracking for Robot Manipulators Equipped …

Fig. 11.6 Experimental results when using controller in Proposition 11.1

q, i.e., standard field-oriented control is employed to control both of these motors. However, since this is not an open-architecture platform, any information is not given on gains of these PI electric current controllers and any electric variable cannot be recorded nor manipulated. Thus, we have just chosen to following controller gains: k p = diag{2000, 1000}, kd = diag{150, 15}, and we have fixed all of the remaining parameters as for study in Fig. 11.3. The obtained experimental results are shown in Fig. 11.6. In order to compare with a controller previously reported in the literature [129, 230], in Fig. 11.7 we present some experimental results when using a controller with an unsaturated proportional −1 [k p q˜ + kd q˙˜ + F(qd , q˙d , q¨d )] is used in controller tracking error, i.e., when Iq∗ = Φ M in Proposition 11.1. We realize that results in Figs. 11.6 and 11.7 are almost identical, which can be explained by the fact that tracking error q˜ is small and, hence, a saturated

11.3 Simulation and Experimental Results

487

Fig. 11.7 Experimental results when using controller in Proposition 11.1 with an unsaturated proportional tracking error, i.e., −1 Iq∗ = Φ M [k p q˜ + kd q˙˜ + F(qd , q˙d , q¨d )]

proportional part of the controller makes not any difference with respect to use of an unsaturated proportional part of the controller. Also note that tracking error in Figs. 11.6a and 11.7a is similar to tracking error in Fig. 11.3b and d in the sense that tracking error does not converge to zero but it just remains small. As it was explained in the simulation study above, this behavior of error is produced by the fact that zero electric current errors cannot be achieved by linear PI electric current controllers in tracking tasks. Thus, these experimental results validate arguments in Remark 11.2.

Chapter 12

PID Control of Robot Manipulators Equipped with SRMs

In recent formal works on SRM control [57, 168–171] only a single SRM is controlled, the mechanical load is linear, position control is not considered (only velocity control is studied) and only the unsaturated model of SRMs is considered. In [92], the authors have employed the unsaturated SRM model to formally present global stability results when simultaneously controlling the position of n SRMs which actuate on a complex, highly coupled, and nonlinear mechanical load: the output feedback PD position regulation is achieved in n-degrees-of-freedom rigid robots equipped with n direct-drive SRMs. Compared to [57, 69, 168–171] the approach in [92] has the advantage of requiring much less online computations despite the more complex plant that is controlled. At this point it is important to recall that it is stressed in [204] that implementing control laws including a large number of computations goes beyond the arguments of availability of cheap and fast numerical processors, it pertains instead to poor numerical robustness of complicated arithmetic operations. Furthermore, it is pointed out in [222] that the electric drives community is not enthusiastic with such complex controllers. Thus, the efforts in [92] to reduce the number of online computations, described above, in a formally supported control scheme for SRMs are motivated by these observations. Moreover, there is also the intention to provide stability conditions useful to understand how the controller works and how to tune its gains. On the other hand, another source of difficulty to control SRMs is magnetic flux saturation which appears under normal operation conditions. This has motivated some works on SRM control by taking into account magnetic flux saturation which are summarized as follows. Exact feedback linearization is presented in [119], which results in very complex control laws and relies on the exact knowledge of many motor parameters. A backstepping approach is presented in [269], which requires lots of computations deteriorating performance because of numerical errors. A passivity-based control scheme is proposed in [69], but any explicit control law was not presented and the corresponding simulation results were only presented for the unsaturated model case. In [96] the problem of controlling SRMs under saturation conditions has been studied. However, [96] is devoted to control a single SRM © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 V. M. Hernández-Guzmán et al., Energy-Based Control of Electromechanical Systems, Advances in Industrial Control, https://doi.org/10.1007/978-3-030-58786-4_12

489

490

12 PID Control of Robot Manipulators Equipped with SRMs

actuating on a simple linear mechanical load and only velocity control is considered, instead of position control of n-degrees-of-freedom robot manipulators. In the present chapter we extend the approaches in [92, 96] to the case when n-degrees-of-freedom rigid robot manipulators are actuated by n direct-drive SRMs and all of these motors are assumed to operate under magnetic flux saturation conditions. This problem must not be underestimated since the saturated SRM model and the position control of n-degrees-of-freedom robot manipulators pose a number of difficulties when considered together. Another difference with respect to [92] is that our proposal includes an external loop driven by a PID controller intended to regulate the robot position. It is important to stress that the simplicity and the robustness properties of a nonlinear PID position controller must not be underestimated. Moreover, in order to design a PID position controller without relying on the natural friction that is present in the mechanical subsystem it is important to allow velocity measurements. This problem seems to be the reason why the passivity-based approaches in [57, 69, 168–171] are devoted to velocity control and the PID control of position problem has not been studied: velocity measurements are avoided in those works and position filtering is employed instead. Although those authors argue that position filtering is intended to reduce the effects of noise that is present in velocity measurements, it is worth to say that such a solution is also intended to reduce the complexity of the resulting desired electric current time derivative. At this point, it is important to stress that such works require the online computation of the desired electric current time derivative. Thus, it is important to stress that we allow velocity measurements and, despite this, we avoid the computation of the desired electric current time derivative among other features that reduce the number of online computations. We also consider both proportional and hysteresis controllers for electric current. In this respect, we point out that both, hysteresis control and high-gain proportional control of electric current, are usually employed for SRM control in practice [64]. Hence, presenting a formal proof explaining why these electric current controllers work well together in practice is one of our main contributions.

12.1 Dynamical Model We refer to Chap. 6 for an explanation on how SRMs work. The dynamical model of SRMs with magnetic saturation has been presented in Sect. 6.1.4. However, we consider it necessary to review such a material in order to define the corresponding expressions for the case when n SRMs are going to be controlled simultaneously. Moreover, the presence of the dynamic model of a n degrees-of-freedom robot manipulator must also be taken into account. For the sake of simplicity and without loss of generality, we consider that all of the SRMs have Nir rotor poles, i.e., rotor saliencies, and three phases. It is well known that flux saturation is common in SRMs. In such a case, [269] suggests to model this phenomenon in phase j of motor at joint i as ψi j (qi , Ii j ) = ψis arctan(βi L i j (qi )Ii j ),

12.1 Dynamical Model

491

for j = 1, 2, 3, where ψi j (qi , Ii j ) and Ii j stand for flux linkage and electric current in phase j of motor at joint i, respectively, whereas qi is the angular position of motor at joint i. L i j (qi ) is a function which stands for phase j and motor i, it is periodic on position qi and strictly positive. It is assumed to be given as L i j (qi ) = li0 +

  ∞   2π lin cos n Nir qi − ( j − 1) 3 n=1   2π , +cin sin n Nir qi − ( j − 1) 3

for j = 1, 2, 3, where li0 , lin , cin , are real constants. Finally, ψis and βi are positive constants which have to be obtained experimentally. The dynamical model of such a SRM is given as [68]: Di (qi , Ii ) I˙i + Ci (qi , Ii )Ii q˙i + ri Ii = Ui ,

(12.1)

where Ii = [Ii1 , Ii2 , Ii3 ] and Ui = [Ui1 , Ui2 , Ui3 ] represent, respectively, the phase currents and voltages applied at each phase, whereas ri > 0 is a scalar standing for the windings electric resistance, the 3 × 3 inductance matrix Di (qi , Ii ) is diagonal and positive definite with the following diagonal entries: Di j (qi , Ii j ) =

ψis βi L i j (qi ) , j = 1, 2, 3, 1 + βi2 L i2j (qi )Ii2j

(12.2)

and Ci (qi , Ii ) is a diagonal matrix with the following diagonal entries: Ci j (qi , Ii j ) =

d L i j (qi ) ψis βi , j = 1, 2, 3. 1 + βi2 L i2j (qi )Ii2j dqi

(12.3)

Using the above expressions, the dynamical model of an n-degrees-of-freedom rigid robot equipped with a direct-drive SRM at each joint is given as M(q)q¨ + C(q, q) ˙ q˙ + g(q) = τ , Di (qi , Ii ) I˙i + Ci (qi , Ii )Ii q˙i + ri Ii = Ui , 3  d L i j (qi ) ψis τi = ln(1 + βi2 L i2j (qi )Ii2j ), 2 dq 2β L (q ) i i i ij j=1

(12.4) (12.5) (12.6)

where i = 1, . . . , n, q = [q1 , . . . , qn ] is the joint positions vector, the n × n inertia matrix M(q) is symmetric and positive definite, C(q, q) ˙ q˙ is the Coriolis and centripetal effects term whereas g(q) is the gravity effects term which is given as g(q) = ∂U (q) where U (q) is potential energy due to gravity, and, finally, τ = [τ1 , . . . , τn ] ∂q is the applied torques vector. The mechanical models of the n direct-drive SRMs are

492

12 PID Control of Robot Manipulators Equipped with SRMs

assumed to be included in the mechanical model of the robot. We will distinguish ˙ by their arguments and the use of subindex between matrices Ci (qi , Ii ) and C(q, q) in the former. On the other hand, as it is by now well known, aside from Properties 1, 2, 3, introduced in Chap. 11, the following are some important properties of the mechanical part of the robot when all joints are revolute. Property 4 (See reference [130], pp. 101, 102) There exists a positive constant K  such that for all q ∈ Rn , we have that g(q) ≤ K  . This means that every element of the gravity effects vector, i.e., gi (q), i = 1, . . . , n, satisfies |gi (q)| ≤ K i , ∀q ∈ Rn , for some positive constants K i , i = 1, . . . , n. Property 5 ([125], [130] pp. 98, [261]) There exists a positive constant kc such that for all w, y, z ∈ Rn , we have C(w, y)z ≤ kc yz.

(12.7)

12.2 The Torque Sharing Approach Although the torque sharing approach has already been introduced in Sect. 6.1.5, we consider it necessary to present a review in order to define the corresponding functions for the case when n different SRMs are going to be controlled simultaneously. Given a desired torque τi∗ to be generated by motor at joint i, the idea is to generate this torque by the contribution of the three phases of the motor, i.e., τi∗ = m i1 (qi )τi∗ + m i2 (qi )τi∗ + m i3 (qi )τi∗ ,

(12.8)

 which implies that 3j=1 m i j (qi ) = 1, i = 1, . . . , n. The scalar functions m i j (qi ), j = 1, 2, 3, are known as the sharing functions and they represent the fraction of the total torque to be generated by motor i that each phase has to contribute with. The ideas behind the torque sharing approach have been rendered precise in [69, 119, 259] as follows. The following two sets define the rotor positions where phase j can produce a positive or a negative torque [69, 119, 259]:     d L i j (qi ) d L i j (qi ) ≥ 0 , Θi−j = qi : 0, ∀qi ∈ Θi+j  + ; m i j (qi ) = 1, ∀qi ∈ R, m i+j (qi ) = 0, ∀qi ∈ Θi−j



3

(12.10)

j=1

m i−j (qi ) > 0, ∀qi ∈ Θi−j  − ; m i j (qi ) = 1, ∀qi ∈ R, m i−j (qi ) = 0, ∀qi ∈ Θi+j 3

(12.11)

j=1

and assign:  m i j (q) =

m i+j (qi ), τi∗ ≥ 0 . m i−j (qi ), τi∗ < 0

(12.12)

Notice that (12.12) together with (12.10) and (12.11) ensure (12.8) and that m i j (qi ) d L i j (qi ) and dq become zero simultaneously. i

12.3 Main Result Assumption A1 Functions m i+j (qi ) and m i−j (qi ) are chosen such that m i j (qi ) → 0 as fast as (qi − qi j0 )ρ → 0, where qi j0 is the rotor position of motor i such that d L i j (qi ) |qi =qi j0 = 0 and ρ > 0 is some integer which depends on the particular dqi form of L i j (qi ). This assumption ensures continuity and boundedness of both Ii∗j and I˙i∗j as qi → qi j0 [92]. In particular, this assumption ensures that whenever the quotient of two functions depending on qi − qi j0 appears, it is the function at the numerator who tends to zero faster. See Sect. 12.4 for a numerical example. Proposition 12.1 Consider the dynamical model (12.4)–(12.6) together with the following controller: n   Ui = Ni + K ie h(q) ˜ + K iq q ˙ 2 + K i f q ˙ + K id ξi 1 S AT (ei σ )H(−ξi ) i=1

Ii∗j =

−αi ξi + ri Ii∗ − ki1 |q˙i |ξi + Ci (qi , Ii )Ii∗ q˙i ,

 d L i j (qi ) βi1 j ζi j + βi2 j f i j (ζi j ), i f dq = 0 i d L (q )

ij i i f dq =0 i σi j e −1 ψis βi d L i (qi ) , , ki1 > ζi j = 2 2 2 dqi M β L i j (qi )

0,

(12.13) ,

τ ∗ = −K P h(q) ˜ − K D q˙ − K I sat (z),

t       ε In×n + γ K P K I−1 h(q) ˜ + In×n + εγ K D K I−1 q˙ dt, z= 0

(12.14) (12.15) (12.16) (12.17)

494

12 PID Control of Robot Manipulators Equipped with SRMs

Fig. 12.1 Some useful functions

σi j =

2βi L i2j (qi )m i j (qi )τi∗ ψis

d L i j (qi ) dqi

, S AT (ei σ ) = diag{Sat (eσi1 ), Sat (eσi2 ), Sat (eσi3 )},

where j = 1, 2, 3, i = 1, . . . , n, Ii∗ = [Ii1∗ , Ii2∗ , Ii3∗ ] , L i (q) = diag{L i1 (qi ), L i2 (qi ), L i3 (qi )}, with L i j (q), Ci (qi , Ii ), ψis , βi , defined in Sect. 12.1, q˜ = q − q ∗ , with q ∗ ∈ Rn the vector of constant desired joint positions, ξi = Ii − Ii∗ , τ ∗ = [τ1∗ , . . . , τn∗ ] represents the vector of desired torques, and In×n is the n × n identity matrix. We define the vectorial function H(−ξi ) = [H (−ξi1 ), H (−ξi2 ), H (−ξi3 )] , where H (−ξi j ) is the standard hysteresis function defined in Fig. 12.1a. There, we assume that the slope sm is large but finite to ensure that a Lipschitz condition is satisfied. h(q) ˜ = [s(q˜1 ), . . . , s(q˜n )] , and sat (z) = [s(z 1 ), . . . , s(z n )] , where s(·) is a strictly increasing linear saturation function for some M ∗ > L ∗ (see Definition 2.34). Furthermore, it is also required that function s(·) be continuously differentiable such that 0
0 is a finite constant. Functions m i j (qi ) are defined in (12.12) and (see Fig. 12.1b):

12.3 Main Result

495

f i j (ζi j ) = aζi3j + bζi2j , ∀0 ≤ ζi j ≤ Ti∗j , (12.19) ⎛ ⎞ ⎞ ⎛ Ti∗j 1 ⎝ ⎠, b = 1 ⎝  1  a = −2 − 3aTi∗j ⎠ . ∗ 3 − ∗ ∗ (Ti j ) 2 2 T ∗T ∗ 4 Ti j (Ti j )2 ij ij (12.20) Finally, we have that βi1 j = 1 and βi2 j = 0, if ζi j > Ti∗j , βi1 j = 0 and βi2 j = 1, if ζi j ≤ Ti∗j , where we assign Ti∗j = ζi j , when |τi∗ | = Td∗ and

d|τi∗ | 0 is used the same sign (see (12.9) and (12.12)). Furthermore, according to Assumption A1, d L i j (q) m i j (qi ) → 0 faster than dq → 0 [92]. See Remark 6.19 for an interpretation of i ∗ Ii j and for the rationale behind the functions f i j (ζi j ), defined in (12.19) and depicted in Fig. 12.1b.

12.3.1 The Rationale Behind Controller in Proposition 12.1 The control scheme in Proposition 12.1 exploits the passivity observations that we have pointed out in (6.39), Chap. 6, but this time applied to robot control. In Fig. 12.2 we depict the control scheme in Proposition 12.1. This control scheme has three main components. The rationale behind these three main components can be explained in the same manner as in the three items listed after Remark 6.17 in Chap. 6. Just replace the use of a sign controller in Chap. 6 by a hysteresis controller in the present chapter.

496

12 PID Control of Robot Manipulators Equipped with SRMs

Fig. 12.2 Control scheme in Proposition 12.1. Υ is defined in (12.14) and Ψ is defined in (12.6). The symbol NL in the hysteresis block refers to the nonlinear function that modulates the hysteresis function in (12.13)

12.3.2 Sketch of Proof of Proposition 12.1 The proof is divided into several steps. (1) The closed-loop dynamics is found. During this step, the mechanical subsystem is shown to be directly affected by the NPID position controller, i.e., the desired torque, by defining a variable representing the torque error, i.e., the difference between the generated torque and the desired torque. In unsaturated models, the term containing this torque error, belonging to the mechanical subsystem, cancels with the term representing the back electromotive force, belonging to the electrical subsystem. This cancellation of cross terms represents the energy exchange between the electrical and the mechanical subsystems. In the case of magnetically saturated SRMs these terms do not cancel and they must be dominated later in the stability proof. (2) In order to exploit the passivity properties of the system, a Lyapunov function candidate is proposed that is composed of the total energy of the closed-loop system, i.e., the “magnetic energy” stored in the electrical subsystem and “the energy” stored in the mechanical subsystem. The latter is represented by the kinetic energy, the “potential energy” as well as “the energy” stored in the integral terms of the NPID position controller. This Lyapunov function candidate is shown to positive definite, radially unbounded, and decrescent. (3) The time derivative of the Lyapunov function candidate is computed. An important feature of this part, and a large amount of space is devoted to that, is to show that the time derivative of the desired electric current is dominated. (4) Using the results in items (2) and (3), Theorem 2.29 is invoked to complete the proof of Proposition 12.1.

12.3.3 Closed-Loop Dynamics Replacing (12.13) in (12.5), and adding and subtracting some convenient terms, we find that

12.3 Main Result

497

Di (qi , Ii )ξ˙i = −ri ξi − αi ξi − ki1 |q˙i |ξi − Ci (qi , Ii )ξi q˙i − Di (qi , Ii ) I˙i∗ (12.21) n   ˜ + K iq q ˙ 2 + K i f q ˙ + K id ξi 1 S AT (ei σ )H(−ξi ), + Ni + K ie h(q) 

i=1

−1/2

βi1 j eσi j − 1 I˙i∗ja = 2 βi2 L i2j (qi )    ∂σi j ∂σi j ∂σi j 2(eσi j − 1)q˙i d L i j (qi ) eσi j × − 2 3 + 2 2 q˙i + q¨i + z˙ i , dqi ∂ q˙i ∂z i β L i j (qi ) β L i j (qi ) ∂qi d f i j (ζi j ) I˙i∗jb = βi2 j dζi j    σi j ∂σi j ∂σi j ∂σi j 2(e − 1)q˙i d L i j (qi ) eσi j × − 2 3 + 2 2 q˙i + q¨i + z˙ i , dqi ∂ q˙i ∂z i β L i j (qi ) βi L i j (qi ) ∂qi where I˙i∗j = I˙i∗ja + I˙i∗jb , for j = 1, 2, 3 and i = 1, . . . , n. This can be verified by trying to cancel in (12.21) the terms not appearing in (12.5) to retrieve the latter equation. Recall that I˙i∗ is continuous and bounded according to definition of m i j (qi ) and f i j (ζi j ) in Assumption A1 and in Proposition 12.1, respectively. See Remark 6.19. Hence, (12.21) represents the closed-loop dynamics corresponding to the electrical subsystem. Let us obtain the closed-loop dynamics corresponding to the mechanical subsys d L i j (qi ) tem. Adding and subtracting the term 3j=1 2β Lψ2is(q ) dq ln(1 + βi2 L i2j (qi )Ii∗2 j ) in i (12.6) and replacing Ii∗j from (12.14), we have τi =

3  j=1

−

j=1

φi =

j=1

ij

i

d L i j (qi ) ψis ln(1 + βi2 L i2j (qi )Ii2j ) 2βi L i2j (qi ) dqi

3 

3 

i

 d L i j (qi ) ψis 2 2 ∗2 ∗ ln(1 + β L (q )I ) + τ βi1 j m i j (qi ) + φi , i i i j i j i 2βi L i2j (qi ) dqi j=1

βi2 j

3

d L i j (qi ) ψis ln(1 + βi2 L i2j (qi ) f i2j (ζi j )), 2 2βi L i j (qi ) dqi

(12.22)

where |φi | ≤ φ¯ i for some φ¯ i > 0 which approaches to zero as Td∗ > 0 approaches to  d L i j (qi ) zero. The expression in (12.22) can be verified by noticing that 3j=1 2β Lψ2is (q ) dq i i

ij

i

∗ ln(1 + βi2 L i2j (qi )Ii∗2 j ) = τi , if βi1 j = 1, and taking into account the two possible values that βi1 j and βi2 j can take. Let us define the torque error:

498

12 PID Control of Robot Manipulators Equipped with SRMs

Φi =

3  j=1

−

d L i j (qi ) ψis ln(1 + βi2 L i2j (qi )Ii2j ) 2β L i2j (qi ) dqi

3  j=1

d L i j (qi ) ψis ln(1 + βi2 L i2j (qi )Ii∗2 j ). 2βi L i2j (qi ) dqi

Hence, using the mean value Theorem 2.14 we can write Φi =

3  j=1

=

3 

d L i j (qi ) ψis 2βi L i2j (qi ) dqi



  ∂ [ln(1 + βi2 L i2j (qi )Ii2j )]  (Ii j − Ii∗j ), ∂ Ii j Ii j = I¯i j

I¯i j Ci j (qi , I¯i j )ξi j , ξi = [ξi1 , ξi2 , ξi3 ]T ,

(12.23)

j=1

where I¯i j is a point belonging to the line joining Ii j and Ii∗j . Notice that there always exists a positive constant K C I¯i j such that | I¯i j Ci j (qi , I¯i j )| ≤ K C I¯i j , ∀qi ∈ R,

I¯i j ∈ R.

(12.24)

˜ − K D q˙ − Thus, from (12.22) and replacing (12.16) we have that τ = Φ − K P h(q) K I sat (z) + τ¯ ∗ + φ, where Φ = [Φ1 , . . . , Φn ] , φ = [φ1 , . . . , φn ] , and: ⎡

⎡

τ¯ ∗ = ⎣τ1∗ ⎣

3 

⎤

⎡

β11 j m 1 j (q1 ) − 1⎦ , . . . , τn∗ ⎣

j=1

3 

⎤⎤ βn1 j m n j (qn ) − 1⎦⎦ .

j=1

Finally, replacing τ in (12.4) it is obtained: M(q)q¨ + C(q, q) ˙ q˙ + g(q) = −K P h(q) ˜ − K D q˙ − K I x(z) + g(q ∗ ) + Φ + τ¯ ∗ + φ, x(z) = sat (z) + K I−1 g(q ∗ ).

(12.25) (12.26)

The closed-loop dynamics is given by (12.21), (12.25), (12.26), (12.17).

12.3.4 A Positive Definite and Decrescent Function Consider the following scalar function: V (q, ˜ q, ˙ z + K I−1 g(q ∗ ), ξ1 , . . . , ξn ) =

n 1  ξi L i (qi )ξi + Vq (q, ˜ q, ˙ z + K I−1 g(q ∗ )), 2 i=1

12.3 Main Result

499

Vq (q, ˜ q, ˙ z + K I−1 g(q ∗ )) = V1 (q, ˙ q) ˜ + P (q) ˜ + V2 (q, ˙ z + K I−1 g(q ∗ )), (12.27)

q˜ 1 V1 (q, ˙ q) ˜ = q˙  M(q)q˙ + εh  (q)M(q) ˜ q˙ + ε h  (r )K D dr, 4 0

q˜ P (q) ˜ = h  (r )K P dr + U (q) − U (q ∗ ) − q˜  g(q ∗ ), 0

z 1 V2 (q, ˙ z + K I−1 g(q ∗ )) = q˙  M(q)q˙ + εγx  (z)M(q)q˙ + x  (r )K I dr, 4 −K I−1 g(q ∗ )

where ε and γ are some positive constants. Remark 12.3 As explained in Sect. 6.2.2, a typical Lyapunov function that is proposed for passivity-based control design is the system In this respect, it is n energy. ξi L i (qi )ξi is proposed to also explained in the Sect. 6.2.2 that the function 21 i=1 replace the magnetic energy stored in the electrical subsystem. On the other hand, ˜ q, ˙ z + K I−1 g(q ∗ )) has three components. (1) P(q) ˜ can be seen the function Vq (q, ˙ q) ˜ can be seen as as the “potential energy” of the mechanical subsystem. (2) V1 (q, a cross function of the kinetic energy and the “potential energy” of the mechanical ˙ z + K I−1 g(q ∗ )) can be seen as a cross function of the kinetic subsystem. (3) V2 (q, energy of the mechanical subsystem and the “energy” stored by the integral part of ˜ q, ˙ z + K I−1 g(q ∗ )) can be seen as the “energy” the position controller. Thus, Vq (q, stored in the mechanical subsystem whereas V (q, ˜ q, ˙ z + K I−1 g(q ∗ ), ξ1 , . . . , ξn ) is the total “energy” of the closed-loop system. ˜ q, ˙ z + K I−1 g(q ∗ )) was analyzed in [93]. From that study and The function Vq (q, proceeding as in [92] (also see [95]) it is concluded that there exist some small enough constant c1 > 0 and some large enough constant c2 > 0 such that the scalar function V (q, ˜ q, ˙ z + K I−1 g(q ∗ ), ξ1 , . . . , ξn ), defined in (12.27) satisfies α (y) ≤ V (y) ≤ α2 (y), ∀y ∈ R6n , 1 c y2 , y < 1 , α2 (y) = c2 y2 , α1 (y) = 1 c1 y, y ≥ 1

(12.28)

where y = [q˜  , q˙  , (z + K I−1 g(q ∗ )) , ξ1 , . . . , ξn ] , if (27), (28), (32) in [93] are true1 and   Ki ∗ , i = 1, . . . , n, (12.29) L > max i KIi where K I i is the ith diagonal entry of matrix K I .

1 The

formulation of these conditions requires the space consuming definition of several functions. Thus, we refer the reader to [93].

500

12 PID Control of Robot Manipulators Equipped with SRMs

12.3.5 Time Derivative of V ( y) First notice that, because of the energy exchange in the system (see (6.39)), we have that ! n n  1  d ξ ξi L i (qi )Di−1 (qi , Ii )[ki1 |q˙i |ξi + Ci (qi , Ii )ξi q˙i ] ≤ 0. L i (qi ) ξi − 2 i=1 i dt i=1 Hence, it is possible to verify, after some straightforward algebraic manipulations and the use of both expressions in (11.7), that the time derivative of V (y), defined in (12.27), along the trajectories of the closed-loop system (12.21), (12.25), (12.26), (12.17), can be upper bounded as dh(q) ˜ M(q)q˙ − εh  (q)(g(q) ˜ − g(q ∗ )) − εh  (q)K ˜ P h(q) ˜ V˙ ≤ −q˙  K D q˙ + εq˙  d q˜ d x(z) M(q)q˙ ˜ + (In×n + εγ K D K I−1 )q] ˙  +εγ[ε(In×n + γ K P K I−1 )h(q) dz −εγx  (z)(g(q) − g(q ∗ )) − εγx  (z)K I x(z) +[εh(q) ˜ + εγx(z)] C  (q, q) ˙ q˙ + [εh(q) ˜ + q˙ + εγx(z)] (Φ + τ¯ ∗ + φ) n    ξi Δi −(ri + αi )ξi − Di (qi , Ii ) I˙i∗ + Γi , (12.30) + i=1



Γi = Ni + K ie h(q) ˜ + K iq q ˙ 2 + K i f q ˙ + K id

n 

ξi 1 S AT (ei σ )H(−ξi ),

i=1

where Δi , i = 1, . . . , n, are the diagonal positive definite matrices defined as Δi = L i (q)Di−1 (qi , Ii ), 1+β 2 L 2 (qi )I 2

(12.31)

i ij ij whose diagonal entries are given as , j = 1, 2, 3. Notice that the torque βi ψis  error Φ = [Φ1 , . . . , Φn ] , defined in (12.23), does not cancel and still appears in (12.30) through the terms [εh(q) ˜ + q˙ + εγx(z)] Φ. Although this is one of the complications that arise in the case of magnetically saturated SRMs, we show in the following that such terms can successfully be dominated. Now, let us find an upper bound for the last terms in (12.30). First notice that  • If βi1 j = 1, for j = 1, 2, 3, then 3j=1 βi1 j m i j (qi ) = 1 and:

⎡ τi∗ ⎣

3 

⎤ βi1 j m i j (qi ) − 1⎦ = 0.

j=1

• If βi1 j = 0, for some j = 1, 2, 3, and |τi∗ | ≤ Td∗ then

12.3 Main Result

501

  ⎡  ⎤     3 3     ∗   ∗  τ ⎣ ⎦ βi1 j m i j (qi ) − 1  ≤ Td  βi1 j m i j (qi ) − 1 < ϕ,  i     j=1 j=1 with Td∗ > 0 and ϕ > 0 some small numbers. Notice that ϕ can be rendered arbitrarily small by choosing a Td∗ arbitrarily small. • Suppose that βi11 = 1, βi12 = 1, βi13 = 0, for instance. Since βi13 = 0 then ζi3 < Ti3∗ for some small Ti3∗ > 0 and, hence, |Ii3∗ | is also small. This means that 3 d L i j (qi ) ψis ∗ ∗ ln(1 + βi2 L i2j (qi )Ii∗2 j=1 2β L 2 (q ) dqi j ) ≈ [m i1 (qi ) + m i2 (qi )]τi ≈ τi . Then i

ij

i

 ⎤  3   βi1 j m i j (qi ) − 1⎦ < ϕ0 ,   j=1 for some small ϕ0 > 0. Notice that ϕ0 can be rendered arbitrarily small by choosing Ti3∗ arbitrarily small, i.e., Td∗ arbitrarily small. Finally, also notice that this result stands for any combination of 0’s and 1’s that βi1 j , j = 1, 2, 3, can take. Recall that for i = 1, . . . , n, |q˙i | ≤ q ˙ and |q¨i | ≤ q ¨ = M(q)−1 [−C(q, q) ˙ q˙ − ˜ − K D q˙ − K I x(z) − g(q) + g(q ∗ ) + Φ + τ¯ ∗ + φ]. Hence, using the K P h(q) above discussion, we can upper bound the following function that is defined in (12.21) as ˙ σi j + 1) + λi3 eσi j {λi4 (K Di q ˙ + λi5 )q ˙ | I˙i∗ja | ≤ λi1 [λi2 q(e   n  ˙ 2 + max{K C I¯i j } ξi 1 +λi6 K Di kc q i, j

i=1

  ˙ + λi7 + λi8 h(q) ˜ + λi9 h(q) ˜ + λi10 K I i q}] ˙ +λi6 K Di λ M (K D )q n  ≤ Λi1 q ˙ 2 eσi j + Λi2 qe ˙ σi j + Λi3 eσi j ξi 1 i=1

+Λi4 h(q)e ˜ σi j + Λi5 eσi j + Λi6 q, ˙ where λik > 0, k = 1, . . . , 10, are constants standing for the bounds of functions appearing in (12.21), whereas Λik > 0, k = 1, . . . , 6, are constants defined as straightforward combinations of the constants λik > 0, k = 1, . . . , 10, in the upper expression. It is easy to see that a similar expression is also valid for | I˙i∗jb |, i.e., ˙ 2 eσi j + Λi8 qe ˙ σi j + Λi9 eσi j | I˙i∗jb | ≤ Λi7 q

n  i=1

+Λi10 h(q)e ˜ σi j + Λi11 eσi j + Λi12 q, ˙

ξi 1

502

12 PID Control of Robot Manipulators Equipped with SRMs

where Λik > 0, k = 7, . . . , 12 are some finite constants. Notice that given a particular i = 1, . . . , n, each one of the constants Λik > 0, k = 1, . . . , 12, are identically defined for any of the three phases j = 1, 2, 3. The above expressions mean that we can upper bound: −ξi j Δi j Di j (qi , Ii j ) I˙i∗j n     < |ξi j |Δi j Ni + K ie h(q) ˜ + K iq q ˙ 2 + K i f q ˙ + K id ξi 1 eσi j , i=1

∀y ∈ R6n , i = 1, . . . , n, j = 1, 2, 3, where subindex j stands for either the jth entry if a vector or the jth diagonal entry if a matrix, and we define the following constants: Ni > βi ψis max{L i j (qi )}(Λi5 + Λi11 ),

(12.32)

K ie > βi ψis max{L i j (qi )}(Λi4 + Λi10 ),

(12.33)

 K iq > βi ψis max{L i j (qi )}(Λi1 + Λi7 ),

(12.34)

K i f > βi ψis max{L i j (qi )}(Λi2 + Λi6 + Λi8 + Λi12 ),

(12.35)

 K id > βi ψis max{L i j (qi )}(Λi3 + Λi9 ).

(12.36)

qi qi qi qi qi

On the other hand, since −ξi j H (−ξi j ) = |ξi j |, for |ξi j | ≥ δ > 0, see Fig. 12.1a, then if we choose   , K i f > K i f , K id > K id , i = 1, . . . , n, Ni > Ni , K ie > K ie , K iq > K iq

(12.37)  we have that ξi j Δi j −Di j (qi , Ii j ) I˙i∗j + Γi j < 0, ∀y ∈ R6n such that eσi j < kσ , if |ξi j | ≥ δ, i.e., the effects of the desired current time derivative are dominated if |ξi j | ≥ δ, and   ξi j Δi j −Di j (qi , Ii j ) I˙i∗j + Γi j   < δΔi j Ni + Ni + (K ie + K ie )h(q) ˜ + (K iq + K iq )q ˙ 2  +(K i f + K i f )q ˙ + (K id + K id )

n 

ξi 1 Sat (eσi j ),

i=1

∀y ∈ R6n such that eσi j < kσ , if |ξi j | < δ. These remaining terms have to be dominated only in the case when |ξi j | < δ. Notice that, from the definition of σi j in Proposition 12.1, it is clear that satisfying the condition eσi j ≤ Sat (eσi j ) < kσ only ˙ < kv . depends on fixing a finite bound kv > 0 for the joints velocities, i.e., that q Thus, we can write (12.30) as

12.3 Main Result

503

dh(q) ˜ V˙ ≤ −q˙  (K D1 + K D2 )q˙ + εq˙  M(q)q˙ − εh  (q)K ˜ P h(q) ˜ d q˜ d x(z) ˜ + (In×n + εγ K D K I−1 )q] ˙  +εγ[ε(In×n + γ K P K I−1 )h(q) M(q)q˙ dz ˜ + εγx(z)] C  (q, q) ˙ q˙ −εγx  (z)(g(q) − g(q ∗ )) − εγx  (z)K I x(z) + [εh(q) √ ∗ ¯ +[εh(q) ˜ + q ˙ + εγx(z)](Φ + ϕ0 τ  + nϕ + φ) (12.38) −

n  ri + αi ξi 2 − εh  (q)(g(q) ˜ − g(q ∗ )) βi ψis i=1

+

n 

   ξi∗ Δi Ni + Ni + (K ie + K ie )h(q) ˜ + (K iq + K iq )q ˙ 2

i=1  +(K i f + K i f )q ˙ + (K id + K id )

n 

ξi 1 [Sat (eσi1 ), Sat (eσi2 ), Sat (eσi3 )] ,

i=1

where K D = K D1 + K D2 , with K D1 , K D2 two diagonal positive definite matrices, ξi∗ is a three-dimensional row vector whose jth entry equals either 0, if |ξi j | ≥ δ, or δ, if |ξi j | < δ, and φ¯ = [φ¯ 1 , . . . , φ¯ n ].

(12.39)

Moreover, we can still go further. Notice that    n  n 3    3    ≤ ¯ ¯ Φ ≤ Φ1 = |Φi | = C (q , I )ξ max {K } |ξi j | I ¯ i j i j i i j i j CIi j    i=1 j∈{1,2,3} i=1 i=1  j=1 j=1 n 

≤ max{K C I¯i j } i, j

3 n   i=1 j=1

√ |ξi j | ≤ max{K C I¯i j }ξ1 ≤ max{K C I¯i j } 3nξ, i, j

i, j

where ξ is the column vector obtained by piling up"ξi , i #= 1, . . . , n, i.e., ξ = n ri +αi i ξ2 , and [εh(q) ξi 2 ≤ − mini rβi i+α ˜ + col(ξi ). Also notice that − i=1 βi ψis ψis εγx(z)] C  (q, q) ˙ q˙ ≤ [εh(q) ˜ M + εγx(z) M ]kc q ˙ 2 . We stress that we can always choose a small enough δ > 0 such that n 

  ξi∗ Δi Ni + Ni + (K ie + K ie )h(q) ˜ + (K iq + K iq )q ˙ 2 + (K i f + K i f )q ˙

i=1  +(K id + K id )

n  i=1

−q˙  K D2 q˙ −

ξi 1 [Sat (eσi1 ), Sat (eσi2 ), Sat (eσi3 )]

  ri + αi 1 ξ2 < min 2 i βi ψis

504 n 

12 PID Control of Robot Manipulators Equipped with SRMs

ξi∗

i=1

1   N + Ni + (K ie + K ie )M ∗ βi ψis i

√  +(K i f + K i f )kω + (K id + K id )n 3kξ [kσ2 , kσ2 , kσ2 ] = δ0 , (12.40)

√ √ where ξi 1 ≤ 3ξi  ≤ 3kξ has been employed with ξi  < kξ , i = 1, . . . , n, ˙ < kω , where kξ > 0 is a small constant (recall the discussion after (12.37)), and q kω > 0 is a small constant. The term −q˙  K D2 q˙ is intended to dominate the term  + K iq )q ˙ 2 for all q ˙ and the term containing (K i f + K i f )q ˙ for containing (K iq " # ri +αi 1 2 large values of q. ˙ Also the term − 2 mini βi ψis ξ is intended to dominate the n  + K id ) i=1 ξi 1 for large values of ξi 1 . In this respect, term containing (K id notice that Δi j grows with Ii2j = (ξi j + Ii∗j )2 which is proportional to eσi1 for large values of Ii∗j because |ξi j | < δ is small. Hence, the terms on the left-hand side of the above inequality grow with Sat 2 (eσi j ) ≤ kσ2 . Recall that eσi j ≤ Sat (eσi j ) is assumed. Thus, there always exists a small enough δ > 0 such that the above inequality is true. It is important to realize that δ0 > 0 can be rendered arbitrarily small by choosing an arbitrarily small δ > 0. Also notice that δ0 > 0 represents the effects of the desired current time derivative when |ξi j | < δ, which will be dominated in the following. According to the above ideas, we can write (12.38) as √ ¯ + ϕ0 (ϕ1 + εK  + εγ K  )] y¯  + δ0 , V˙ ≤ − y¯  Q y¯ + [3(1 + ε + εγ)( nϕ + φ) (12.41) √ ∗  ϕ1 = n M (λ M (K P ) + ελ M (K D ) + 2εγλ M (K D ) + 2λ M (K I ) + K ), where y¯ = [|q|, ˙ |h(q)|, ˜ |x(z)|, ξ] , the symbol λ M (A) stands for the largest eigenvalue of matrix A, and the entries of matrix Q are given as Q 11 = λm (K D1 ) − ϕ0 λ M (K D ) − ελ M (M(q))

Q 23 = Q 22 = Q 44 = Q 12 = Q 14 =

−εγλ M (M(q))λ M (In×n + εγ K D K I−1 ) −(εh(q) ˜ M + εγx(z) M )kc , εγ khg ε(γλ M (K P ) + λ M (K I ))ϕ0 , (12.42) Q 32 = − − 2 ka 2   khg , Q 33 = εγ(λm (K I ) − ϕ0 λ M (K I )), ε λm (K P ) − ϕ0 λ M (K P ) − ka   √ ri + αi 1 εγ , Q 34 = Q 43 = − max{K C I¯i j } 3n, min 2 i βi ψis 2 i, j 2 ε γ λ M (In×n + γ K P K I−1 )λ M (M(q)), Q 13 = Q 31 = 0, Q 21 = − 2 √ √ 1 ε Q 41 = − max{K C I¯i j } 3n, Q 24 = Q 42 = − max{K C I¯i j } 3n, 2 i, j 2 i, j

12.3 Main Result

where khg ≥

505

2K   (see s( 2K kg )

[130]) and we have used s(q) ˜ ≤

h(q) ˜ ka

(see [104]) where

ka is identical to parameter α introduced in [127]. The symbol λm (A) stands for the smallest eigenvalue of matrix A. The four leading principal minors of matrix Q can always be rendered positive by choosing small enough ε > 0, γ > 0, Td∗ > 0, large enough positive definite matrices K D1 , K P , K I , and large enough scalars αi > 0, i = 1 . . . , n. Hence, matrix Q is positive definite and λm (Q) > 0 is ensured. Achieving positive definiteness of matrix Q is what we mean when stating along the chapter that we dominate the undesired terms instead of cancelling them. Using some constant 0 < Θ < 1, we can rewrite (12.41) as 1 Θλm (Q) y¯ 2 + φ¯ ∗1  y¯  2 1 Θλm (Q) y¯ 2 + δ0 , 2 $ 2δ0 2φ¯ ∗1 2 and  y¯  > , ≤ −(1 − Θ)λm (Q) y¯  , ∀  y¯  ≥ Θλm (Q) Θλm (Q) (12.43) ¯ + ϕ0 (ϕ1 + εK  + εγ K  ). φ¯ ∗1 = 3(1 + ε + εγ)(ϕ + φ) 1 V˙ ≤ − (1 − Θ)λm (Q) y¯ 2 − 2 1 − (1 − Θ)λm (Q) y¯ 2 − 2

Since φ¯ ∗1 > 0 approaches to zero as Td∗ → 0 and δ0 approaches to zero as δ → 0, then the last inequalities in (12.43) can be forced to be satisfied by the linear parts of functions in y¯ . Hence, it is always possible to choose a small Td∗ > 0 and a small δ > 0 such that we can finally write

2φ¯ ∗1 , V˙ ≤ −(1 − Θ)λm (Q) y¯ 2 , ∀ y ≥ μ0 = max Θλm (Q)

$

% 2δ0 . Θλm (Q) (12.44)

12.3.6 Proof of Proposition 12.1 Taking into consideration (12.28) and (12.44), we can invoke Theorem 2.29 to conclude that given an initial state y(t0 ) ∈ R6n , such that eσi j < kσ (i.e., such that q ˙ < kv ) is not violated for2 t ≥ t0 , we can always find controller gains such that the closed-loop system state y satisfies y(t) ≤ β0 (y(t0 ), t − t0 ), ∀ t0 ≤ t ≤ t0 + T, y(t) ≤ α1−1 (α2 (μ0 )) , ∀ t ≥ t0 + T,

(12.45)

is possible because we can define a domain where V˙ < 0 on its outer border = {y ∈ R6n |V (y) < kv } with V defined in (12.27) and kv can be computed such that q ˙ < kv for all t ≥ t0 .

2 This

506

12 PID Control of Robot Manipulators Equipped with SRMs

where β0 (·, ·) is a KL function and T ≥ 0 depends on y(t0 ) and μ0 . We stress that kσ (and kv ) can be chosen to be larger by using a smaller δ such that δ0 , defined in (12.40), is rendered arbitrarily small. Thus, kσ (and kv ) can always be rendered arbitrarily large and, hence, the above result stands semiglobally. On the other hand, φ¯ ∗1 → 0 as Td∗ > 0 approaches to zero. Recall that δ0 → 0 as δ > 0 approaches to zero. Hence, μ0 > 0 can be rendered arbitrarily small by choosing some small enough Td∗ > 0 and δ > 0. Notice that this means that the effects of the desired current time derivative when |ξi j | < δ, i.e., δ0 > 0, are dominated. Since α1−1 (α2 (·)) is a K∞ function, then the ultimate bound in (12.45) tends to zero as both Td∗ > 0 and δ > 0 approach to zero. Notice that Td∗ = 0 would imply that some of the Λk , k = 1, . . . , 10, are infinitely large and, thus, (12.37) could not be satisfied. Thus Td∗ > 0 cannot be zero and, hence, φ¯ ∗1 > 0 cannot be zero either. On the other hand, since the slope of the hysteresis function sm is assumed to be finite, then δ cannot be zero and, thus, δ0 cannot be zero either. This means that the closed-loop system has an ultimate bound which cannot be reduced to zero but can be rendered arbitrarily small by a suitable choice of controller gains. Recall that this result stands semiglobally. This completes the proof of Proposition 12.1. Finally, we emphasize that the conditions to guarantee Proposition 12.1 are summarized by (27), (28), (32) in [93], (12.29), (12.37), the four principal minors of matrix Q defined in (12.42) are positive, and some small constants Td∗ > 0 and δ > 0. Remark 12.4 We stress that ultimate boundedness of the state that is formally proven in Proposition 12.1 is consistent with current practice [64] where the use of hysteresis controllers for electric current do not achieve convergence to zero of the electric current error but, in fact, ultimate boundedness of the electric current error is accomplished. Remark 12.5 Notice that any singularity due to an ill definition of the desired current Ii∗j is not present in controller in Proposition 12.1. We stress that such a problem appears in the passivity-based approaches presented in [57, 69, 168–171]. See Remark 6.19 for further details.

12.4 Simulation Study In this section we present a numerical example using the model and parameters of the two degrees-of-freedom (n = 2) robot manipulator reported in [36], which is equipped with two rotative joints. Both links of this robot move on a vertical plane and point downwards when q = [0, 0] . For simulation purposes we assume that this robot is equipped with two direct-drive  SRMs whose parameters are given as , Nir = 8, ri = 5[Ohm], li0 = 0.03[H], L i j (qi ) = li0 + li1 cos Nir qi − ( j − 1) 2π 3 li1 = 0.02[H], where i = 1, 2, and j = 1, 2, 3. Additionally, since we are considering SRMs with saturated magnetic fluxes, we have chosen ψs = 0.5[Wb], β = 1.8, which were experimentally identified in [269].

12.4 Simulation Study

507

The sharing functions m i j (qi ) were designed as follows. Consider the expressions for I˙i∗ja and I˙i∗jb given in (12.21). There, consider the term: βi1 2



 ∂σi j eσi j q˙i β 2 L i2j (qi ) ∂qi $   2 2 ∂σi j eσi j βi1 βi L i j (qi ) = q˙i , (12.46) 2 eσi j − 1 β 2 L i2j (qi ) ∂qi

eσi j − 1 βi2 L i2j (qi )

−1/2 

where σi j is defined in Proposition 12.1. According to this definition, σi j → 0 when m i j (qi ) → 0 which happens as qi → qi j0 (see Assumption A1). Hence, we must ensure that a singularity is not present in (12.46) as qi → qi j0 . For this, let us replace the approximation eσi j ≈ 1 + σi j , when σi j ≈ 0, in (12.46) to obtain βi1 2

$

βi2 L i2j (qi ) σi j



 ∂σi j eσi j q˙i . β 2 L i2j (qi ) ∂qi

According to Assumption A1, let us propose that m i j (qi ) → 0 as fast as (qi − d L i j (qi ) d L i j (qi ) qi j0 )4 → 0, i.e., ρ = 4. Since dq is a sine function, then dq → 0 as (qi − i i ∂σ

qi j0 )1 → 0. Hence, according to the definition of σi j in Proposition 12.1, ∂qiij → 0 √ as (qi − qi j0 )2 → 0. Also notice that σi j → 0 as (qi − qi j0 )3/2 → 0. Thus βi1 2

$

βi2 L i2j (qi ) σi j



∂σi j eσi j q˙i β 2 L i2j (qi ) ∂qi

 → 0 as (qi − qi j0 )1/2 → 0. (12.47)

This means that a singularity does not exist in (12.46) as qi → qi j0 . Carefully analyzing the remaining terms in the expressions for I˙i∗ja and I˙i∗jb given in (12.21), we conclude that (12.46) represents the worst case and, hence, any singularity does not exist in I˙i∗ja nor I˙i∗jb , i.e., I˙i∗ is free from singularities for i = 1, . . . , n. In this respect, we stress that although the control scheme in Proposition 12.1 does not require to compute I˙i∗ , for i = 1, . . . , n, it does require that these functions be bounded in order to be dominated. Hence, such a requirement is ensured to be satisfied. We stress that the singularity problem that we have just ensured not to appear is a different singularity problem to that described in Remark 6.19. There, it is explained that a singularity appears when the desired torque becomes zero, i.e., when τi∗ = 0 for some i = 1, . . . , n, if it is not employed the function f i j (ζi j ) in the definition of Ii∗j given in (12.14). We propose m i j (qi ) as composed of a constant unit value, a constant zero value, a raising polynomial pr (x) and a falling polynomial p f (x) = 1 − pr (x). In order to satisfy the analysis in the previous paragraphs, as the rising polynomial we used π , h = qi − αμ qmi , pr (x) = 35x 4 − 84x 5 + 70x 6 − 20x 7 , where x = qhmi , qmi = 24

508

12 PID Control of Robot Manipulators Equipped with SRMs

Fig. 12.3 Simulation results when controller in Proposition 12.1 is employed

αμ = int(qi /qmi ). The coefficients of pr (x) were computed by requiring p f (1) = dp (x)

d 2 p (x)

d 3 p (x)

1 − pr (1) = 0, df x |x=1 = 0, d xf 2 |x=1 = 0 and d xf 3 |x=1 = 0. The controller parameters were chosen as K P = diag{75, 75}, K D = diag{30, 30}, K I = diag{55, 55}, ε = 2.9, γ = 2, α1 = 50, α2 = 50, Td∗ = 0.1, δ = 0.5, k11 = 10, k21 = 10, kσ = 10, N1 = 10, N2 = 10, , K 1q = 10, K 2q = 10, K 1 f = 10, K 2 f = 10, K 1d = 10, K 2d = 10, K 1e = 10, K 2e = 10. The linear saturation function that has been employed is  x+L ∗  ⎧ ⎨ −L ∗ + (M ∗ − L ∗ ) tanh M ∗ −L ∗ , x < −L ∗ |x| ≤ L ∗ s(x) = x,  x−L ∗  ⎩ ∗ ∗ ∗ L + (M − L ) tanh M ∗ −L ∗ , x > L ∗ with M ∗ = 3 and L ∗ = 2.9. These controller gains were verified to satisfy all of the stability conditions in Proposition 12.1 excepting (12.37). Although this latter condition is satisfied using values for the constants N1 , N2 , K 1q , K 2q , K 1 f , K 2 f , K 1d , K 2d , K 1e , K 2e that are in the order of 108 , such values are not realistic and we have preferred to employ the above-cited numerical values. Despite this, our approach must not be underestimated. In this respect, the passivity-based approaches in [57, 69, 168–171] would be obliged, instead, to compute online a very large number of terms whose numerical values are in the order of 5 × 108 (five times 108 because of the five constants N , K q , K f , K d , K e ) since they have to exactly compute (the time derivative of the desired electric current) what

12.4 Simulation Study

509

Fig. 12.4 Simulation results when controller in Proposition 12.1 is employed (cont.)

our approach is required just to dominate. On the other hand, formal studies abound which establish stability conditions that result in very large controller gains which prohibits their use in successful simulations or experiments. See, for instance, the formal works on robot control [48, 127]. Finally, the desired positions were set as follows. q1∗ is given as a ramp with slope 10[rad/s] such that q1∗ = 0 at t = 0. When q1∗ = π2 [rad] the reference remains at this value for all future time. q2∗ is given as a ramp with slope 50[rad/s] such that q2∗ = 0 at t = 0. When q2∗ = π4 [rad] the reference remains at this value for all future time. All of the initial conditions were set to zero. In Figs. 12.3, 12.4 and 12.5 we present the corresponding simulation results. We observe that both steady-state position errors are zero, despite Proposition 12.1 only ensures convergence of the state error to a small ball. This is because the required torques to maintain both joints at the desired positions are far from zero, as explained in Remark 6.9. However, asymptotic stability is not achieved in this case because of the hysteresis electric current controllers. Also notice that the steady state is reached in about 3 s for both joints. This is a good time response for this robot. See [130]. The required torques remain below 14[Nm], for joint 1, and 4[Nm], for joint 2, which are the maximal torques that can be applied to this robot in experiments. See [36]. This requires electric currents in the order of 20[A]. These large values, however, are due to the torque capabilities of the SRMs that we have selected, i.e., it

510

12 PID Control of Robot Manipulators Equipped with SRMs

Fig. 12.5 Simulation results when controller in Proposition 12.1 is employed (cont.)

is not a problem of the control algorithm. As a matter of fact, notice that about 20[A] are required in the first motor to maintain the joint at its desired position. We can see that the maximal values of electric currents in the phases of motor 1 increase as the joint arrives at its desired position ( π2 [rad]) which is the joint position that requires the motor to generate the maximum torque.

Appendix A

Energy Functions

A.1 Velocity Control Consider the following scalar function: 1 2 1  J ω˜ + [ki + γ(b + k p )]z 2 + γ J z ω˜ 2 2   1 J γJ ω˜ ˜ z] = [ω, . γ J ki + γ(b + k p ) z 2

˜ z) = Vω (ω,

Notice that this quadratic form is positive definite and radially unbounded if the matrix involved is positive definite, which is true if and only if its two leading principal minors are positive, i.e., J > 0, ki + γ(b + k p ) − J γ 2 > 0. Both conditions can always be satisfied using large enough values for ki > 0 and k p > 0 if it is chosen a constant γ > 0, since J is always positive.

A.2 Position Control Consider the following scalar function:

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 V. M. Hernández-Guzmán et al., Energy-Based Control of Electromechanical Systems, Advances in Industrial Control, https://doi.org/10.1007/978-3-030-58786-4

511

512

Appendix A: Energy Functions

Vq (q, ˜ q, ˙ z) = V1 (q, ˜ q) ˙ + V2 (z, q) ˙ + V3 (q, ˜ z), 1 α 1 ˜ q) ˙ = J (q˙ + 2αq) V1 (q, ˜ 2 − α2 J q˜ 2 + (b + kd )q˜ 2 + k p q˜ 2 4 2 2 ∗ +U (q) − U (q ∗ ) − qg(q ˜ ), 1 αβ  2 1  2 k z + ki z , ˙ = J (q˙ + 2αβz)2 − α2 β 2 J z 2 + V2 (z, q) 4 2 d 2 β   2 ˜ = (k p − αkd )(z − q) ˜ , V3 (z, q) 2

(A.1)

where α and β are two constants. Notice that it can be written: α  1 J (q˙ + 2αq) ˜ 2+ (b + kd ) − α2 J q˜ 2 + h(q), ˜ 4 2 1 ∗ h(q) ˜ = k p q˜ 2 + U (q) − U (q ∗ ) − qg(q ˜ ), 2

V1 (q, ˜ q) ˙ =

(A.2)

with dU (q) d(q˜ + q ∗ ) dh(q) ˜ = k p q˜ + − g(q ∗ ), d q˜ dq d q˜ = k p q˜ + g(q) − g(q ∗ ), d 2 h(q) ˜ dg(q) d(q˜ + q ∗ )  = k + , p d q˜ 2 dq d q˜ dg(q) , = k p + dq where q = q˜ + q ∗ and (3.54) have been used. Using (3.53), it is clear that ∀q ∈ R, i.e., ∀q˜ ∈ R, if: k p > kg .

d 2 h(q) ˜ d q˜ 2

> 0,

(A.3)

It is stressed that, although the first term in (A.2) becomes zero on the line q˙ = −2αq, ˜ the second and third terms in the same expression are positive whenever q˜ = 0 if (A.3) and the following are required: α > 0,

1 (b + kd ) > αJ. 2

(A.4)

Hence, V1 (q, ˜ q) ˙ is positive definite and radially unbounded if J > 0 and (A.3), (A.4), are satisfied. ˙ is positive definite and radially Using similar arguments as before, V2 (z, q) unbounded if J > 0, α > 0, and

Appendix A: Energy Functions

513

αβ  1 kd + ki > α2 β 2 J, β > 0. 2 2

(A.5)

Finally, V3 (z, q) ˜ is positive semidefinite if α > 0, β > 0, and k p > αkd .

(A.6)

˜ q, ˙ z), defined in (A.1) is a positive definite and radially unbounded scalar Thus, Vq (q, function if J > 0 and (A.3), (A.4), (A.5), (A.6) are true.

A.3 Velocity Control with Saturated Integral of Velocity Consider the mechanical part of Lyapunov function introduced in (C.12), i.e., Vω (ω, ˜ x+

τ L + bω ∗ 1 ) = J ω˜ 2 + [ki + β(b + k p )]  ki 2



x −

τ L +bω ∗ ki

z(r )dr + β J z(x)ω. ˜

This scalar function can be written as ˜ x+ Vω (ω,

τ L + bω ∗ 1 1 ) = J (ω˜ + βz(x))2 − J β 2 z 2 (x) ki 2 2  x +[ki + β(b + k p )] z(r )dr. ∗ −

τ L +bω ki

According to Definition 2.34:  |σ(x)| ≥

|x|, |x| ≤ L , L , |x| > L

and, hence, by direct integration we find that (see Fig. A.1): 

x −

τ L +bω ∗ ki

z(r )dr ≥ G(x),

where ⎧  2 ⎪ τ L +bω ∗ 1 ⎪ x + , |x| ≤ L  ⎪ ki ⎪ ⎨2    ∗ 2 ∗ G(x) = 21 L + τL +bω (x − L), x>L . + L + τL +bω ki ki ⎪ ⎪     2 ⎪ ∗ ∗ ⎪1 ⎩ −L + τL +bω (x + L), x < −L + −L + τL +bω 2 k k i

i

514

Appendix A: Energy Functions

Also notice that ⎧  2 ⎪ τ L +bω ∗ 1 ⎪ x + , |x| ≤ M  ⎪ ki ⎪ ⎨2  ∗ 2 H (x) = 21 M + τL +bω , x>M . ki ⎪ ⎪   ⎪ ∗ 2 ⎪ ⎩ 1 −M + τL +bω , x < −M 2 k

1 1 − J β 2 z 2 (x) ≥ − J β 2 H (x), 2 2

i

Hence, it is clear that it is always possible to find large enough constants ki > 0 and k p > 0 and a small enough constant β > 0 such that [ki + β(b + k p )]



x

τ +bω ∗ − L k i

z(r )dr −

≥ [ki + β(b + k p )]G(x) −

1 2 2 J β z (x) ≥ 2

1 2 J β H (x) > 0, 2

(A.7)

which can be verified graphically (see Fig. A.1), i.e., that [ki + β(b + k p )]



x

τ +bω ∗ − L k i

z(r )dr −

1 2 2 J β z (x) 2

is a positive definite radially unbounded function in x +

τ L +bω ∗ . ki

Thus, since term

J (ω˜ + βz(x)) ≥ 0 is zero only when ω˜ + βz(x) = 0, property in (A.7) ensures ∗ ∗ that Vω (ω, ˜ x + τL +bω ) is zero only when both ω˜ and x + τL +bω are zero, i.e., when ki ki z(x) = 0. This proves that Vω is positive definite and radially unbounded. 1 2

2

A.4 PID position Control with Saturated Proportional and Integral Actions The mechanical part of function introduced in (C.31), i.e., function V (q, ˜ q, ˙ z+ g(q ∗ )/ki ), can be rewritten as ˙ q) ˜ + P(q) ˜ + V2 (q, ˙ z + g(q ∗ )/ki ), V (q, ˜ q, ˙ z + g(q ∗ )/ki ) = V1 (q,  q˜ 1 V1 (q, ˙ q) ˜ = J q˙ 2 + αJ h(q) ˜ q˙ + α(b + kd ) h(r )dr, 4 0  q˜ ∗ P(q) ˜ = k p h(r )dr + U (q) − U (q ∗ ) − qg(q ˜ ), 0  z 1 V2 (q, ˙ z + g(q ∗ )/ki ) = J q˙ 2 + αβ J s(z)q˙ + ki s(r )dr. 4 −g(q ∗ )/ki

Appendix A: Energy Functions

515

Fig. A.1 Graphical verification of (A.7)

Notice that V1 (q, ˙ q) ˜ can be written as ˙ q) ˜ = V1 (q,

1 J (q˙ + 2αh(q)) ˜ 2 − α2 J h 2 (q) ˜ + α(b + kd ) 4



q˜

h(r )dr.

0

On the other hand, according to Definition 2.34 and by direct integration:

516

Appendix A: Energy Functions

α(b +

kd )



q˜ 0

h(r )dr ≥ G d (q), ˜ ⎧ α(b+k  ) 2 ⎪ |q| ˜ ≤L ⎨ 2 d q˜ , α(b+kd ) 2  G d (q) ˜ = L + α(b + k )L( q ˜ − L), q˜ > L . d 2  ⎪ ⎩ α(b+k  2 d) L − α(b + kd )L(q˜ + L), q˜ < −L 2

Invoking again Definition 2.34: α2 J h 2 (q) ˜ ≤ α∗2 J h 2 (q) ˜ ≤ H (q), ˜  2 |q| ˜ ≤M q˜ , H (q) ˜ = α∗2 J |q| ˜ >M M 2, for some α∗ > α. According to Fig. A.2, it is always possible to choose a large enough kd > 0 such that ˜ ≥ H (q), ˜ ∀ q˜ ∈ R, G d (q)

(A.8)

i.e., such that α(b + kd )



q˜

h(r )dr ≥ G d (q) ˜ ≥ α∗2 J h 2 (q) ˜ ≥ α2 J h 2 (q). ˜

0

Thus 1 J (q˙ + 2αh(q)) ˜ 2 − α2 J h 2 (q) ˜ + α∗2 J h 2 (q), ˜ 4 1 = J q˙ 2 + αJ h(q) ˜ q˙ + α∗2 J h 2 (q), ˜ 4  

˙ q) ˜ ≥ V1 (q,

= J ζ

1 4 α 2

α 2 ∗2

α

ζ,

where ζ = [q, ˙ h(q)] ˜  , and the above matrix is positive definite because α∗ > α. Hence, V1 is globally positive definite. On the other hand, dP(q) ˜ = k p h(q) ˜ + g(q) − g(q ∗ ). d q˜ If: k p L > 2k  , then it is possible to write

(A.9)

Appendix A: Energy Functions

517

Fig. A.2 Graphical verification of (A.8)

dP(q) ˜ = k p q˜ + g(q) − g(q ∗ ). d q˜ Hence,

dP(q) ˜ d q˜

= 0 has q˜ = 0 as unique solution if: k p > kg ,

see [130]. Moreover: dg(q) d(q˜ + q ∗ ) d 2 P(q) ˜ dh(q) ˜ + , q˜ = q − q ∗ , = k p 2 d q˜ d q˜ dq d q˜ dg(q) dh(q) ˜ + . = k p d q˜ dq If (A.9) is true, there are two cases: • |q| ˜ ≤ L. In this case Definition 2.34 can be invoked to write d 2 P(q) ˜ dg(q) > 0, = k p + 2 d q˜ dq if (A.10) is true (see [130] for instance). • |q| ˜ > L. In this case (A.9) can be employed to conclude dP(q) ˜ = k p h(q) ˜ + g(q) − g(q ∗ ) > 0, d q˜

(A.10)

518

Appendix A: Energy Functions

if q˜ > L and dP(q) ˜ ˜ + g(q) − g(q ∗ ) < 0, = k p h(q) d q˜ if q˜ < −L. Thus, P(q) ˜ is positive definite and radially unbounded. Function V2 (q, ˙ z + g(q ∗ )/ki ) can be rewritten as ˙ z + g(q ∗ )/ki ) = V2 (q,

1 J (q˙ + 2αβs(z))2 4 

−α2 β 2 J s 2 (z) + ki

z

−g(q ∗ )/ki

s(r )dr.

Defining d = g(q ∗ )/ki , z  = z + d and by direct integration, it is found that (see Fig. A.3): ki



z

−g(q ∗ )/ki

s(r )dr =

ki



z  −d −d

s(r )dr ≥ Γ (z  ),

(A.11)

⎧ k  2 ⎪ |z| ≤ L ⎨ 2i (z ) , ki  Γ (z ) = 2 (L + d)2 + ki (z  − (L + d))(L + d), z>L . ⎪ ⎩ ki   2 (L − d) + ki (z − (−L + d))(−L + d), z < −L 2 Moreover: α2 β 2 J s 2 (z) ≤ α∗2 β ∗2 J s 2 (z  − d) ≤ F(z  ),

(A.12)

for some α∗ > α, β ∗ > β and ⎧ |z| ≤ M ⎨ (z  )2 ,  ∗2 ∗2 F(z ) = α β J (M + d)2 , z > M . ⎩ (−M + d)2 , z < −M

(A.13)

From (A.11)–(A.13), it is concluded that there always exists a large enough ki > 0 such that (see Fig. A.3): g(q ∗ ) , ki

(A.14)

s(r )dr > (α∗2 β ∗2 − α2 β 2 )J s 2 (z) > 0

(A.15)

Γ (z  ) ≥ F(z  ),

L>

and hence − α2 β 2 J s 2 (z) + ki



z −g(q ∗ )/ki

Appendix A: Energy Functions

519

Fig. A.3 Graphical verification of (A.14)

for all z + g(q ∗ )/ki = 0. Thus, V (q, ˜ q, ˙ z + g(q ∗ )/ki ) is positive definite and radially unbounded if (A.8)–(A.10), and (A.14) are satisfied.

A.5 Velocity Sensorless Regulator for Switched Reluctance Motors Consider the following scalar function:

520

Appendix A: Energy Functions

1 1 V1 (ω, ˜ z d , ϑ) = J [ω˜ + γχ]2 − J γ 2 χ2 + 2 2 

ϑ

+ 0

Kv σ2 (r )dr + B



z d −d −d



z d −d

−d

K p [σ1 (r ) + d] dr   χ

γb [σ1 (r ) + d] dr,  

(A.16)

χ

where γ > 0 is a constant scalar, d = [bω ∗ + τ L ]/K p and we have introduced the variable change z d = z + d. According to Definition 2.34 and by direct integration, we have that (see [92] for a similar procedure): 

z d −d

(A.17) [σ1 (r ) + d] ≥ G(z d ), ⎧ Kp 2 ⎪ if − L 1 ≤ z d − d ≤ L 1 ⎨ 2 [z d ] , Kp 2 G(z d ) = . [L + d] + K [z − d − L ][L + d], if z d − d > L 1 1 p d 1 1 2 ⎪ ⎩ K p [−L + d]2 + K [z − d + L ][−L + d], if z − d < −L 1 p d 1 1 d 1 2 Kp

−d

Notice that G(z d ) ≥ 0 if L 1 > |d|. Also notice that 1 ∗2 J γ [σ1 (z d − d) + d]2 ≤ H (z d ), 2 ⎧ ∗2 ⎪ J γ2 (z d )2 , if − M1 ≤ z d − d ≤ M1 ⎨ ∗2 H (z d ) = J γ2 (M1 + d)2 , if z d − d > M1 ⎪ ⎩ J γ ∗2 2 (−M1 + d) , if z d − d < −M1 2

(A.18)

for some γ ∗ > γ. From (A.17) and (A.18), we obtain Fig. A.4, which can be used to conclude that there always exist a large enough positive constant K p and a small enough positive constant γ ∗ such that G(z d ) ≥ H (z d ), ∀ z d ∈ R,

(A.19)

and therefore  Kp

z d −d

−d z d −d

[σ1 (r ) + d] dr −

1 2 J γ [σ1 (z d − d) + d]2 ≥ 2

[σ1 (r ) + d] dr −

1 ∗2 J γ [σ1 (z d − d) + d]2 ≥ 0. 2

 Kp

−d

(A.20)

Thus, we have that V1 (ω, ˜ z d , ϑ) defined in (A.16) is positive definite and radially unbounded if (A.19) is satisfied and K v , B are positive constants. Now, consider the following scalar function: 1 ˜ z d , ϑ) + ξ  D(q)ξ, V (ω, ˜ z d , ϑ, ξ1 , ξ2 , ξ3 ) = V1 (ω, 2

(A.21)

Appendix A: Energy Functions

521

Fig. A.4 Comparing functions in (A.19)

where function V1 (ω, ˜ z d , ϑ) was defined in (A.16). From (A.20), we can find the following lower bound:  zd −d 1 2 2 1 2 J (ω˜ + γχ) − J γ χ + K p χdr ≥ 2 2 −d 1 1 ≥ J [ω˜ 2 − 2γ|ω||χ| ˜ + γ 2 χ2 ] + J (γ ∗ 2 − γ 2 )χ2 , 2 2      1 1 −γ |ω| ˜ |ω| ˜ = J , |χ| |χ| −γ γ ∗ 2 2   P

1 ≥ J λmin {P}[ω˜ 2 + χ2 ], 2

(A.22)

where λmin {P} > 0 because γ ∗ > γ. Using (A.16), (A.22) and by direct integration  z −d ϑ of 0 KBv σ2 (r )dr and −dd γb [σ1 (r ) + d] dr , it can be proven that there always exists a small enough c1 > 0 such that V (y) ≥ α1 (|ω|) ˜ + α1 (|z d |) + α1 (|ϑ|) +

3 

α1 (|ξi |),

i=1

where y = [ω, ˜ z d , ϑ, ξ1 , ξ2 , ξ3 ] ∈ R6 and  α1 (|ψ|) =

c1 |ψ|2 , |ψ| < 1 , ∀ψ ∈ R. c1 |ψ|, |ψ| ≥ 1

(A.23)

522

Appendix A: Energy Functions

Based on (A.23) and following similar steps as those reported in [92], we conclude that V (y) is lower bounded as α1 ( y ) ≤ V (y), ∀ y ∈ R6 . Moreover, consider the following: 1 γ∗2 2 1 2 J ω˜ + γ J χω˜ ≤ J ω˜ 2 + γ J χω˜ + Jχ , 2 2 2      1 1γ ω˜ ω˜ ≤ J , χ χ γ γ∗2 2   P

1 ≤ J λmax {P  }[ω˜ 2 + χ2 ], 2 1 ≤ J λmax {P  }[ω˜ 2 + z d2 ], 2

(A.24)

where we have used the fact that |χ| = |σ1 (z) + d| ≤ |z + d + d| = |z d |,

(A.25)

and λmax {P  } > 0 because γ ∗ > γ. Also, note that, according to Definition 2.34, there always exist two hard saturation functions: ⎧ ⎨ x, |x| ≤ k j SAT j (x) = k j , x > k j , ⎩ −k j , x < −k j

j = 1, 2,

with k j > 0, such that |σ1 (x)| ≤ |SAT1 (x)| and |σ2 (x)| ≤ |SAT2 (x)|, ∀ x ∈ R. Therefore, we have that 

z d −d −d



 K p [σ1 (r ) + d]dr ≤  0

z d −d

−d

ϑ

σ2 (r )

Kv dr ≤ B

γ[σ1 (r ) + d]bdr ≤

z d −d

−d  ϑ



SAT2 (r )

0 z d −d −d

where, by direct integration, we obtain that

K p [SAT1 (r ) + d]dr, Kv dr, B

γ[SAT1 (r ) + d]bdr,

Appendix A: Energy Functions

 

z d −d −d ϑ

[SAT1 (r ) + d]dr = W1 (z d ) ≤

523

1 2 z , 2 d

1 SAT2 (r )dr = W2 (ϑ) ≤ ϑ2 , (A.26) 2 0 ⎧1 if |z d − d| ≤ k1 ⎨ 2 [z d ]2 , W1 (z d ) = 21 [k1 + d]2 + [z d − d − k1 ][k1 + d], if z d − d > k1 , ⎩1 2 [−k + d] + [z − d + k ][−k + d], if z d − d < −k1 1 d 1 1 ⎧ 12 2 if |ϑ| ≤ k2 ⎨ 2 [ϑ] , W2 (ϑ) = 21 k22 + k2 [ϑ − k2 ], if ϑ > k2 . ⎩1 2 k − k2 [ϑ + k2 ], if ϑ < −k2 2 2

Then, using (A.24), (A.26), and Theorem 2.12, we can obtain the following inequality: 1 Kv 2 1 [J λmax {P  } + K p + γb]z d2 + J λmax {P  }ω˜ 2 + ϑ 2 2 2B 1 + λmax {D(q)} ξ 2 , 2 ≤ α2 ( y ) = c2 y 2 ,

V (y) ≤

with c2 = 21 max{J λmax {P  } + K p + γb, KBv , λmax {D(q)}}. Thus, V (y) defined in (A.21) satisfies (A.27) α1 ( y ) ≤ V (y) ≤ α2 ( y ), ∀ y ∈ R6 , if K v and B are positive and (A.19) is true.

A.6 Trajectory Tracking in Robots Equipped with PM Synchronous Motors ˙˜ I˜q , Id , wq , wd ) defined in This appendix is concerned with function V (t, q, ˜ q, (11.16).

A.6.1 Positive Definiteness and Radially Unboundedness ˙˜ which can be rewritten as Let us first study function V1 (t, q, ˜ q),

524

Appendix A: Energy Functions

 q˜ 1 ˙  ˙ ˙ ˙ V1 (t, q, ˜ q) ˜ = q˜ M(q)q˜ + γ tanh (q)M(q) ˜ q˜ + tanh (r )(k p + γkd )dr, 2 0 1 = (q˜˙ + γ tanh(q)) ˜  M(q)(q˜˙ + γ tanh(q)) ˜ 2  q˜ γ2 − tanh (q)M(q) ˜ tanh(q) ˜ + tanh (r )(k p + γkd )dr. 2 0 Notice that 

q˜

tanh (r )k p dr =

0

i=n 

G i (q˜i ), G i (q˜i ) = k pi ln(cosh(q˜i )),

i=1

where k pi , i = 1, . . . , n, are the diagonal entries of matrix k p , q˜i is the i−th component of vector q˜ ∈ Rn , ln(·) is the natural logarithm function and cosh(·) is the cosine hyperbolic function. Function G i (q˜i ) is plotted in Fig. A.5. On the other hand, use of Theorem 2.12 yields  γ2 γ2 tanh (q)M(q) λmax (M(q)) tanh(q)

˜ tanh(q) ˜ ≤ ˜ 2= Hi (q˜i ), 2 2 i=1 i=n

Hi (q˜i ) =

γ2 λmax (M(q)) tanh(q˜i )2 . 2

Function Hi (q˜i ) is plotted in Fig. A.5. It is clear from Fig. A.5 that there always exists large enough constants k pi > 0 and a small enough constant γ > 0 such that G i (q˜i ) ≥ Hi (q˜i ), ∀q˜i ∈ R, ∀i = 1, . . . , n.

(A.28)

This condition can always be verified graphically as in Fig. A.5. Thus, it is possible to write  q˜ γ2  − tanh (q)M(q) ˜ tanh(q) ˜ + tanh (r )(k p + γkd )dr ≥ 2 0  q˜ i=n i=n   − Hi (q˜i ) + G i (q˜i ) + γ tanh (r )kd r ≥ i=1 q˜

 γ

i=1

0

tanh (r )kd r,

0

 q˜  i=n since − i=n i=1 Hi (q˜i ) + i=1 G i (q˜i ) ≥ 0 because of (A.28). Notice that, γ 0 ˙˜ is positive definite and tanh (r )kd r is a positive definite function, i.e., V1 (t, q, ˜ q) radially unbounded if (A.28) is satisfied.

Appendix A: Energy Functions

525

Fig. A.5 Graphical verification of (A.28)

A similar procedure can be used to show positive definiteness and radial unboundedness of function: 1 1 V2 ( I˜q , Id , wq , wd ) = I˜q L q I˜q + Id L d Id 2 2 wq + satq (r )[αqi + L q−1 (R+ αq p )]dr 0  wd satd (r )[αdi + L −1 + d (R+ αdp )]dr 0

+ I˜q satq (wq ) + Id satd (wd ). Notice that it is possible to write V2 ( I˜q , Id , wq , wd ) =

1 ˜ ( Iq + L q−1 satq (wq )) L q ( I˜q + L q−1 satq (wq )) 2 1 − satq (wq )L q−1 satq (wq ) 2 wq

+ 

0

+ 0

wq

satq (r )L q−1 (R + αq p )dr satq (r )αqi dr

1 −1  ˜ + ( I˜d + L −1 d satd (wd )) L d ( Id + L d satd (wd )) 2

526

Appendix A: Energy Functions

1 − satd (wd )L −1 d satd (wd ) 2 wd satd (r )L −1 + d (R + αdp )dr 0  wd satd (r )αdi dr. + 0

On the other hand, 

wq

0

satq (r )L q−1 (R + αq p )dr =

Fi (wqi ) =  0

wd

Ri + αq pi L qi

Ri + αdpi L di

Fi (wqi ),

i=1



wqi

satqi (ri )dri ,

0

satd (r )L −1 d (R + αdp )dr =

Di (wdi ) =

i=n 



i=n 

Di (wdi ),

i=1 wdi

satdi (ri )dri ,

0

and   i=n  1 1  1 −1

satq (wq ) 2 = satq (wq )L q satq (wq ) ≥ max E i (wqi ), 2 2 i L qi i=1   1 1 E i (wqi ) = max satqi2 (wqi ), 2 i L qi   i=n  1 1  1 −1

satd (wd ) 2 = satd (wd )L d satd (wd ) ≥ max Bi (wdi ), 2 2 i L di i=1   1 1 2 satdi Bi (wdi ) = max (wdi ). 2 i L di Functions Fi (wqi ), Di (wdi ), E i (wdi ), and Bi (wdi ) are plotted in Fig. A.6 where it is clear that there always exist large enough constants αq pi and αdpi such that Fi (wqi ) ≥ E i (wqi ), ∀wqi ∈ R, ∀i = 1, . . . , n, Di (wdi ) ≥ Bi (wdi ), ∀wdi ∈ R, ∀i = 1, . . . , n. Similarly as in the case for V1 , this ensures that

(A.29) (A.30)

Appendix A: Energy Functions

527

Fig. A.6 Graphical verification of both (A.29) and (A.30)

 wq 1 − satq (wq )L q−1 satq (wq ) + satq (r )L q−1 (R + αq p )dr ≥ 0, 2 0 wd 1  −1 satd (r )L −1 − satd (wd )L d satd (wd ) + d (R + αdp )dr ≥ 0, 2 0 i.e., V2 ( I˜q , Id , wq , wd ) ≥

1 ˜ ( Iq + L q−1 satq (wq )) L q ( I˜q + L q−1 satq (wq )) 2 wq

+ 0

satq (r )αqi dr

1 −1  ˜ + ( I˜d + L −1 d satd (wd )) L d ( Id + L d satd (wd )) 2 wd

+ 0

satd (r )αdi dr.

This proves that V2 is positive definite and radially unbounded if (A.29) and (A.30) are satisfied. ˙˜ I˜q , Id , wq , wd ) defined in (11.16) is positive definite and radially Thus, V (t, q, ˜ q, unbounded if i) kdi > 0 for i = 1, . . . , n, and γ > 0, k pi > 0 for i = 1, . . . , n, are chosen such that (A.28) is satisfied and ii) αqii > 0, αdii > 0 and αq pi > 0, αdpi > 0 are chosen such that (A.29), (A.30), are satisfied. Finally, notice that: a) all saturation functions that are involved in the above procedure are straight lines, or close to straight lines, for small values of the saturated variable and they become constant, or close to a constant, for large values of

528

Appendix A: Energy Functions

the saturated variable, b) integrals of these saturation functions are quadratic functions, or close to quadratic functions, for small values of the saturated variable and they become straight lines, or close to straight lines, for large values of the saturated variables. Using these observations and the above procedure, we realize that ˙˜ I˜q , Id , wq , wd ) is a decrescent function which can be bounded as V (t, q, ˜ q, ˙˜ I˜q , Id , wq , wd ) ≤ α2 ( y ), ∀y ∈ R6n , α1 ( y ) ≤ V (t, q, ˜ q,  c1 y 2 , y < 1 α1 ( y ) = , c1 y , y ≥ 1

(A.31)

α2 ( y ) = c2 y 2 , where c1 > 0 is a small enough constant whereas c2 > 0 is a large enough constant. The interested reader is referred to [92] for a complete and formal procedure to demonstrate correctness of the above bound for a Lyapunov function which is very ˙˜ I˜q , Id , wq , wd ). Since this is a rather elaborate procedure, we similar to V (t, q, ˜ q, prefer to refer the reader to the above cited paper.

A.6.2 Expression for V˙ Along the Trajectories of the Closed-Loop System

˜ ˙˜ + γ q˙˜  ∂ tanh(q) M(q)q˙˜ ˜ q) V˙ = −q˙˜  N p L d I D Iq∗ − q˙˜  kd q˙˜ − q˙˜  h(t, q, ∂ q˜ +γ tanh (q)C ˜  (q, q) ˙ q˙˜ − γ tanh (q)N ˜ p (L d − L q )I D I˜q

(A.32) −1  ˜ ˜ ˜ p (L d − − γ tanh (q)Φ ˜ M Iq − Iq RΦ M k p tanh(q) ˜ −γ tanh (q)N    ˙ ˜ ˜ −γ tanh (q)k ˜ p tanh(q) ˜ + γ tanh (q)h(t, ˜ q, ˜ q) ˜ − Iq (R + αq p ) Iq 

L q )I D Iq∗



∂ tanh(q) ˜ ˙ −1 −1 − I˜q RΦ M kd q˙˜ − I˜q L q Φ M kp q˜ − I˜q N p L d I D q˙d − Φ M I˜q ∂ q˜ −1 − I˜q L q Φ M kd M −1 (q)[−C(q, q) ˙ q˙˜ − N p (L d − L q )I D I˜q − N p (L d − L q )I D Iq∗ ˙˜ 2 I˜q − I˜ K f Id I˜q ˙˜ − I˜ K q q

−k p tanh(q) ˜ − kd q˙˜ − h(t, q, ˜ q)] q q −1 −1 ˙ − I˜q [RΦ M F(qd , q˙d , q¨d ) + Φ M q˙d + L q Φ M F(qd , q˙d , q¨d , qd(3) )] ˙˜ 2 Id +I  N p L q Q˙ d I˜q + I  N p L q Φ −1 Q˙ d [k p tanh(q) ˜ + kd q] ˜˙ − I  K d q

d

d

d

M

−1 ˙ +Id N p L q Φ M Q d F(qd , q˙d , q¨d )

+

∂satq (wq ) ˜ I˜q Iq ∂wq

+

∂satd (wd ) Id Id ∂wd

˙˜ ∗ −1   ˙ ˜ −satd (wd )L −1 d N p L q Q Iq + satd (wd )L d N p L q Q d Iq − Id (R + αdp )Id ˙˜ − sat  (wd )L −1 N p L q Q˙˜ I˜q ˙ −1 +satd (wd )L −1 ˜ + kd q) d d N p L q Q d Φ M (k p tanh(q) d

Appendix A: Energy Functions

529

−1 ˙ +satd (wd )L −1 ¨d ) − satd (wd )L −1 ˜˙ 2 Id d N p L q Φ M Q d F(qd , q˙d , q d K d q

˙˜ −sat  (wq )L −1 [N p L d I D q˜˙ + Φ M q˙˜ − RΦ −1 k p tanh(q) ˜ − RΦ −1 kd q] q

q

M

M

−satq (wq )L q−1 αqi satq (wq ) − satd (wd )L −1 d αdi satd (wd ) ∂ tanh(q) ˜ ˙ q˜ − N p L d I D q˙d ] ∂ q˜ −1 kd M −1 (q)C(q, q) ˙ q˙˜ +satq (wq )Φ M ˙˜ 2 I˜q − K f Id I˜q ] −sat  (wq )L −1 [−K q q

−1 −satq (wq )L q−1 [L q Φ M kp

q

q

−1 −satq (wq )Φ M kd M −1 (q)

×[−N p (L d − L q )I D I˜q − N p (L d − L q )I D Iq∗ − Φ M I˜q ˙˜ −k p tanh(q) ˜ − kd q˙˜ − h(t, q, ˜ q)] −1 −1 ˙ −satq (wq )L q−1 [RΦ M F(qd , q˙d , q¨d ) + Φ M q˙d + L q Φ M F(qd , q˙d , q¨d , qd(3) )].

A.6.3 Entries of Matrix Q Introduced in (11.17) √ √ −1 Q 11 = λm (kd1 ) − kh1 − γλ M (M(q)) − γ nkc1 − kc1 n Mq Φ M kd M −1 (q) M √ −1 − n Md λ M (L −1 d N p L q Φ M kd ), 1 Q 22 = γλm (k p ) − 2 γkh2 , βe −1 Q 33 = λm (R + αq p1 ) − L q Φ M kd M −1 (q)Φ M M − 1,   Q 44 = λm (R + αdp1 ) − 1, Q 55 = λm (L q−1 αqi ), Q 66 = λm (L −1 d αdi ),   kh2 1 γkh1 γkc1 − , Q 12 = Q 21 = −

q˙d M − βe 2 2 2 1 1 −1 −1 Q 13 = Q 31 = − λ M (RΦ M kd ) − λ M (L q Φ M k p) 2 2 1 kc1 −1 −1 kd M −1 (q) M q˙d M − L q Φ M kd M −1 (q)kd M − L q Φ M 2 2 √ n Md kh1 −1 −1 λ M (L −1 − L q Φ M kd M (q) M − d N P L q ), 2 2  γ 1 1 1 −1 −1 −1 − λ M (Φ M ) − λ M (RΦ M k p ) − L q Φ M kd M −1 (q)k p M Q 23 = Q 32 = βe 2 2 2  kh2 −1 − L q Φ M kd M −1 (q) M , 2

530

Appendix A: Energy Functions

1 kF −1 −1

N p L d Φ M Q 14 = Q 41 = − N p L d Φ M k p M tanh(q)

˜ M−

M 2 2 √ γ n 1 −1 −1

N p (L d − L q )Φ M − kd M − λ M (N p L q Φ M kd ) q˙d M 2 2 √ √ n Mq n Mq −1 −1 − kd M −1 (q)N p (L d − L q )Φ M kd M , λ M (L q−1 N p L d ) −

Φ M 2 2  √ γ n 1 γ −1 −1 −

N p (L d − L q )Φ M Q 24 = Q 42 = k p M − N p (L d − L q )Φ M

M k F βe 2 2  1 −1 − λ M (N p L q Φ M k p ) q˙d M , 2 √ γ n 1

N p (L d − L q ) M − λ M (N p L d ) q˙d M Q 34 = Q 43 = − 2 2 √ n −1 −1 −1

L q Φ M kd M (q)N p (L d − L q )Φ M − k p M , 2 1 kF −1 −1 kd M −1 (q)N p (L d − L q )Φ M

M − λ M (N p L q ) q˙d M − L q Φ M 2 2 √ n Mq −1

Φ M − kd M −1 (q)N p (L d − L q ) M , 2  1 1 1 −1 − λ M (L q−1 N p L d Φ M ) − λ M (L q−1 RΦ M kd ) Q 15 = Q 51 = βq 2 2 kc1 1 −1 1 −1 −1

Φ M k p) − kd M −1 (q) M q˙d M − Φ M kd M −1 (q)kd M − λ M (Φ M 2 2 2  kh1 −1 −1 − Φ M kd M (q) M , 2  1 1 1 −1 −1 − λ M (L q−1 RΦ M Q 25 = Q 52 = k p ) − Φ M kd M −1 (q)k p M βe βq 2 2  kh2 −1 −1 − Φ M kd M (q) M , 2 1 −1 Q 35 = Q 53 = −

Φ M kd M −1 (q)Φ M M , 2βq  1 1 − λ M (L q−1 N p L d ) q˙d M Q 45 = Q 54 = βq 2 √ n −1 −1

Φ M kd M −1 (q)N p (L d − L q )Φ M − k p M 2  kF −1 −1 − Φ M kd M −1 (q)N p (L d − L q )Φ M

M , 2  √ n 1 kF −1 −1 Q 16 = Q 61 = − λ M (L −1 λ M (L −1 d N p L q ΦM k p ) − d N p L q ΦM ) βd 2 2  1 −1 −1 − λ M (L d N p L q Φ M kd ) q˙d M , 2

Appendix A: Energy Functions

1 −1 λ M (L −1 d N p L q Φ M k p ) q˙d M , 2βe βd 1 =− λ M (L −1 d N p L q ) q˙d M , 2βd = Q 65 = Q 56 = 0.

Q 26 = Q 62 = − Q 36 = Q 63 Q 46 = Q 64

531

Appendix B

Proofs of Propositions for Brushed DC-Motors

B.1 Proof of Proposition 3.3 First notice that there always exists a positive constant ε such that kme = εkm

(B.1)

with kme = km if ε = 1. Also note that expressions in (3.38) can be written as v = −α p i˜ − αi



t

˜ i(s)ds,

(B.2)

0

i∗ =

1 ∗ τ , τ ∗ = −k p ω˜ − ki kme



t

ω(s)ds, ˜

(B.3)

0

where i˜ = i − i ∗ and ω˜ = ω − ω ∗ . Dynamics of the closed-loop connection of (3.26), (3.27) and (B.2), (B.3), is obtained as follows. Defining two positive constants k p and ki such that k p = εk p , ki = εki , Equations (B.3) and (B.1) can be used to obtain i∗ =

   t 1 −k p ω˜ − ki ω(s)ds ˜ . km 0

(B.4)

Now, adding and subtracting terms km i ∗ and bω ∗ , taking advantage from the fact that ω˙ ∗ = 0 (ω ∗ is constant), and using (B.4), expression in (3.27) can be written as J ω˙˜ = −bω˜ + km i˜ − k p ω˜ − ki



t 0

 bω ∗ + τ L . ω(s)ds ˜ + ki

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 V. M. Hernández-Guzmán et al., Energy-Based Control of Electromechanical Systems, Advances in Industrial Control, https://doi.org/10.1007/978-3-030-58786-4

533

534

Appendix B: Proofs of Propositions for Brushed DC-Motors

Finally, defining:  z=

t

ω(s)ds ˜ +

0

bω ∗ + τ L , ki

(B.5)

yields J ω˙˜ = −bω˜ + km i˜ − k p ω˜ − ki z.

(B.6)

On the other hand, replacing (B.2) in (3.26) and adding and subtracting term Ri ∗ we have  t di ˜ = −R(i − i ∗ ) − Ri ∗ − ke ω − α p i˜ − αi i(s)ds. L dt 0 Subtracting I˙∗ on both sides, adding and subtracting ke ω ∗ , and using (B.4), (B.5), yields L

   bω ∗ + τ L R d i˜ −k p ω˜ − ki z − = −(R + α p )i˜ − ke ω˜ − ke ω˜ ∗ − dt km ki  t ˜ −αi i(s)ds − L i˙∗ . 0

Finally, defining: 

t

zi = 0

1 ˜ i(s)ds + αi

 R

bω ∗ + τ L + ke ω ∗ km

 (B.7)

and using (B.4) again we find L

 Rk p Rki d i˜ L   ˙ = −(R + α p )i˜ − ke ω˜ + −k p ω˜ − ki ω˜ (B.8) ω˜ + z − αi z i − . dt km km km

Expressions in (B.5)–(B.8) represent the closed-loop dynamics which can be rewritten as follows (recall that τ L is constant): L

d i˜ = −(R + α p )i˜ − ke ω˜ + V, dt J ω˙˜ = −(b + k p )ω˜ + km i˜ − TL , z˙ = ω, ˜ ˜ z˙ i = i, TL = ki z, V =

Rk p km

 Rki L   ˙ −k p ω˜ − ki ω˜ . ω˜ + z − αi z i − km km

(B.9) (B.10) (B.11) (B.12)

Appendix B: Proofs of Propositions for Brushed DC-Motors

535

It is stressed that the closed-loop dynamics (B.9)–(B.12) is autonomous because it ˜ ω, can be written as x˙ = f (x), where x = [i, ˜ z i , z] , and f (x) ∈ R4 is some function of x. Furthermore, using the definition f (x ∗ ) = 0 for an equilibrium point, it is not difficult to verify that x ∗ = [0, 0, 0, 0] is the only equilibrium point. Consider the following “energy” storage function for the closed-loop dynamics: 1 ˜2 1 ˜ z), L i + αi z i2 + Vω (ω, 2 2 1 1 Vω (ω, ˜ z) = J ω˜ 2 + [ki + γ(b + k p )]z 2 + γ J z ω, ˜ 2 2

˜ z i , ω, ˜ z) = V (i,

(B.13)

where γ is a positive constant. Taking advantage from the fact that J is positive, it is shown in Appendix A.1 that Vω (ω, ˜ z) is positive definite and radially unbounded if: ki + γ(b + k p ) − J γ 2 > 0, γ > 0.

(B.14)

Hence, since inductance L is always positive, the Lyapunov function candidate V given in (B.13) is positive definite and radially unbounded if: αi > 0,

(B.15)

and (B.14) are satisfied. After several straightforward cancellations, it is found that the time derivative of V along the trajectories of the closed-loop dynamics (B.9)(B.12) is given as Rk p

Rki ˜ L  ˙ ˜ Lki ˜ zi + k ω˜ i + ω˜ i i˜ω˜ + km km km p km −(b + k p )ω˜ 2 + γ J ω˜ 2 + γkm z i˜ − γki z 2 .

V˙ = −(R + α p )i˜2 +

Replacing ω˙˜ from (B.10) and using the fact that ±r s ≤ |r | |s|, ∀r, s ∈ R we can write ˜ , ˜ |z|, |i|] V˙ ≤ −ξ  Qξ, ξ = [|ω|,

(B.16)

where the entries of matrix Q are given as Lk p , Q 11 = b + k p − γ J, Q 22 = γki , Q 33 = R + α p − J    Rk  bLk p L(k p )2  Lki  p Q 13 = Q 31 = −  + − − ,  2km 2km 2km J 2km J      Rki Lk p ki  γkm ,  Q 23 = Q 32 = −  − + 2km 2 2km J  Q 12 = Q 21 = 0.

(B.17)

536

Appendix B: Proofs of Propositions for Brushed DC-Motors

Matrix Q is positive definite if and only if all of its three leading principal minors are positive, i.e., b + k p > γ J, γ > 0, ki > 0, det(Q) > 0.

(B.18)

Notice that the first three conditions can be always satisfied using a large enough value for k p > 0, a small γ > 0 and some positive ki . On the other hand, the last condition det(Q) > 0 is always rendered true by using a large enough α p > 0, since Q 33 is the only entry of matrix Q which depends on α p . Thus, we conclude that ˜ ω, V˙ ≤ −ξ  Qξ ≤ 0, ∀[i, ˜ z i , z] ∈ R4 . As remarked after (B.12), the closed-loop dynamics (B.9)–(B.12) is autonomous and, hence, the LaSalle invariance principle (see Corollary 2.19) can be applied as follows. Define a set S as ˜ ω, S = {[i, ˜ z i , z] ∈ R4 |V˙ = 0} = {i˜ = 0, ω˜ = 0, z = 0, z i ∈ R}. Evaluating the closed-loop dynamics (B.9)–(B.12) in S we obtain 0 = −(R + α p )(0) − ke (0) + V, 0 = −(b + k p )(0) + km (0) − TL , z˙ = 0, z˙ i = 0, TL = ki (0), V =

Rk p km

(0) +

 Rki L   −k p (0) − ki (0) . (0) − αi z i − km km

These expressions have, as the unique solution, i˜ = 0, ω˜ = 0, z = 0, z i = 0 because ˜ ω, ˜ z i , z] = [0, 0, 0, 0] αi > 0. Thus, according to Corollary 2.19, this implies that [i, is a globally asymptotically stable equilibrium point. This completes the proof of Proposition 3.3. Conditions for this stability result are summarized by (B.14), (B.15), (B.18).

B.2 Proof of Proposition 3.5 There always exists a positive constant ε such that kme = εkm , with kme = km if ε = 1. Using this, (3.51) and defining:

Appendix B: Proofs of Propositions for Brushed DC-Motors

537

k p = εk p , kd = εkd , ki = εki , it is obtained: i∗ =

1 km

   t −k p q˜ − kd q˙ − ki q(s)ds ˜ ,

(B.19)

0

where q˜ = q − q ∗ . Defining: 1 k p = k p − ki , ki = ki , α  t 1 (αq(s) ˜ + q(s))ds ˙ + q(0) ˜ +  g(q ∗ ) z= ki 0

(B.20)

for some positive constant α, and recalling that 

t

q(t) ˜ − q(0) ˜ =

˙˜ q(s)ds =

0



t

q(s)ds, ˙

0

because q˙ ∗ = 0, it can be written: −

k p q˜

−

kd q˙

−

ki



t

0

q(s)ds ˜ = −k p q˜ − kd q˙ − ki z + g(q ∗ ).

(B.21)

Adding and subtracting term km i ∗ in (3.50) and using (B.19), (B.21) yields J q¨ = km i˜ − bq˙ − k p q˜ − kd q˙ − ki z + g(q ∗ ) − g(q),

(B.22)

where i˜ = i − i ∗ . On the other hand, replacing (3.51) in (3.49), adding and subtracting terms Ri ∗ ∗ and L didt , using (B.19), (B.21) and defining:  zi = 0

t

1 ˜ i(s)ds + αi



 R g(q ∗ ) , km

(B.23)

it is obtained: L

Rk p Rkd Rki d i˜ = −(R + α p )i˜ − ke q˙ + q˜ + q˙ + z − αi z i dt km km km  L   −k p q˙ − kd q¨ − ki (αq˜ + q) ˙ . − km

(B.24)

Expressions in (B.20), (B.22)–(B.24), represent the closed-loop dynamics which can be rewritten as follows:

538

L

Appendix B: Proofs of Propositions for Brushed DC-Motors

d i˜ = −(R + α p )i˜ − ke q˙ + V, dt J q¨ = −(b + kd )q˙ + km i˜ − G, ˜ z˙ = αq˜ + q, ˙ z˙ i = i, G= V=

k p q˜ + ki z + g(q) Rk p Rkd km

q˜ +

km

q˙ +

(B.25) (B.26) (B.27)

∗

− g(q ),  Rki L   −k p q˙ − kd q¨ − ki (αq˜ + q) z − αi z i − ˙ . km km

It is stressed that the closed-loop dynamics (B.20), (B.22)–(B.24), or equivalently (B.25)–(B.27), is autonomous because it can be written as x˙ = f (x), for x = ˜ z i ] , and some f (x) ∈ R5 . Furthermore, using the definition f (x ∗ ) = 0 [q, ˜ q, ˙ z, i, for an equilibrium point, it is not difficult to verify that x ∗ = [0, 0, 0, 0, 0] is the only equilibrium point. Consider the following “energy” storage function for the closed-loop dynamics: ˜ z i ) = 1 L i˜2 + 1 αi z i2 + Vq (q, ˜ q, ˙ z), (B.28) V (q, ˜ q, ˙ z, i, 2 2 Vq (q, ˜ q, ˙ z) = V1 (q, ˜ q) ˙ + V2 (z, q) ˙ + V3 (q, ˜ z), 1 α 1 ˜ 2 − α2 J q˜ 2 + (b + kd )q˜ 2 + k p q˜ 2 ˜ q) ˙ = J (q˙ + 2αq) V1 (q, 4 2 2 ∗ +U (q) − U (q ∗ ) − qg(q ˜ ), 1 αβ  2 1  2 k z + ki z , ˙ = J (q˙ + 2αβz)2 − α2 β 2 J z 2 + V2 (z, q) 4 2 d 2 β   2 ˜ = (k p − αkd )(z − q) ˜ , V3 (z, q) 2 where α and β are two positive constants. Taking advantage from the fact that J is positive, it is shown in Appendix A.2 that the Vq given in (B.28) is positive definite and radially unbounded if: k p > kg , α > 0,

1 (b + kd ) > αJ, 2

(B.29)

αβ  1 k + k  > α2 β 2 J, β > 0, k p > αkd . 2 d 2 i

Thus, since inductance is positive L > 0, the Lyapunov function candidate V given in (B.28) is positive definite and radially unbounded if (B.29) and αi > 0

(B.30)

are true. After several straightforward cancellations, it is found that the time derivative of V along the trajectories of the closed-loop dynamics (B.25)–(B.27) is given as

Appendix B: Proofs of Propositions for Brushed DC-Motors

539

∗ ˜ ) − g(q)) V˙ = −(b + kd )q˙ 2 + αJ q˙ 2 + αkm q˜ i˜ − αk p q˜ 2 + αq(g(q

+α2 β J q˙ q˜ + αβ J q˙ 2 − αβbz q˙ + αβkm z i˜ − αβki z 2 + αβz(g(q ∗ ) − g(q)) Rk p q˜ i˜ −βαk p q˜ 2 + βα2 kd q˜ 2 − (R + α p )i˜2 + km Rki ˜ Lk p ˜ Lkd ˜ Lk  ˜ Rk  Lk  zi + q˙ i + + d q˙ i˜ + i q¨ + α i q˜ i˜ + i q˙ i. km km km km km km Replacing q¨ from (B.26), using (3.52), (3.53), and the fact that ±r s ≤ |r | |s|, ∀r, s ∈ R, we can write ˜ , ˙ |q|, ˜ |z|, |i|] V˙ ≤ −ξ  Qξ, ξ = [|q|,

(B.31)

where the entries of matrix Q are given as Q 11 = b + kd − αJ − αβ J, Q 33 = αβki ,

Q 22 = αk p − αkg + αβk p − α2 βkd ,

Q 44 = R + α p −

Lkd , J

(B.32)

α2 β J αβb , Q 13 = Q 31 = − , 2 2 αβkg , =− 2     Rkd + L(k p + ki ) Lk  (b + kd )  = −  − d , 2km 2J km    αkm Rk p Lkd k p  Lkd kg αLki −  + , = − + − 2 2km 2km 2km J  2km J    αβkm Rki Lk  k   + = −  − d i  . 2 2km 2J km

Q 12 = Q 21 = − Q 23 = Q 32 Q 14 = Q 41 Q 24 = Q 42 Q 34 = Q 43

Matrix Q is positive definite if and only if all of its four leading principal minors are positive, i.e., ⎛

⎞ Q 11 Q 12 Q 13 Q 11 > 0, Q 11 Q 22 − Q 12 Q 21 > 0, det ⎝ Q 21 Q 22 Q 23 ⎠ > 0, det(Q) > 0. Q 31 Q 32 Q 33 (B.33) The first condition can be always satisfied using a large enough value for kd > 0 and some small α > 0 and β > 0. With these values, the second condition can be rendered true with a large enough k p > 0. Once this is achieved, third condition is always satisfied using a large enough ki > 0 (which can be verified by computing the indicated determinant and noticing that Q 33 is the only entry of this determinant which depends on ki ). Finally, and proceeding as before, the fourth condition is rendered true using a large α p > 0. Thus, we conclude that

540

Appendix B: Proofs of Propositions for Brushed DC-Motors

˜ z i ] ∈ R5 . ˜ q, ˙ z, i, V˙ ≤ −ξ  Qξ ≤ 0, ∀[q, As remarked after (B.27), the closed-loop dynamics (B.25)–(B.27) is autonomous and, hence, the LaSalle invariance principle (see Corollary 2.19) can be applied as follows. Define a set S as ˜ z i ] ∈ R5 |V˙ = 0} = {q˜ = 0, q˙ = 0, z = 0, i˜ = 0, z i ∈ R}. S = {[q, ˜ q, ˙ z, i, Evaluating the closed-loop dynamics (B.25)–(B.27) in S we obtain 0 = −(R + α p )(0) − ke (0) + V, 0 = −(b + kd )(0) + km (0) − G, z˙ = α(0) + (0), z˙ i = (0), G = k p (0) + ki (0) + g(q) − g(q ∗ ), V=

Rk p

(0) +

Rkd Rki (0) + (0) − αi z i km km

km  L   −k p (0) − kd (0) − ki (α(0) + (0)) . − km

These expressions have, as the unique solution, q˜ = 0, q˙ = 0, z = 0, i˜ = 0, z i = 0 ˜ zi ] = ˜ q, ˙ z, i, because αi > 0. Thus, according to Corollary 2.19, this implies that [q, [0, 0, 0, 0, 0] is a globally asymptotically stable equilibrium point. This completes the proof of Proposition 3.5. Conditions for this stability result are summarized by (B.29), (B.30), (B.33).

B.3 Proof of Proposition 3.8

The time derivative of the positive definite and radially unbounded scalar function V (s) = 21 s 2 , along the trajectories of (3.63) is    ∗ ∗   1 di di |s| 1 di   − E < 0, (B.34) −υ − L − ≤ + E V˙ = s s˙ = s dt dt L  dt 2  2 

  ∗ where (3.71) has been used, if −υ − L didt + 21 E  − 21 E < 0. By considering the ∗ ∗ two possibilities −υ − L didt + 21 E > 0 and −υ − L didt + 21 E < 0, it is not difficult to show that (B.34) implies (3.75). From the sliding condition s˙ = 0, (3.63) and (3.75) we find that the equivalent control satisfies the following bound: 0 < u eq

  di ∗ 1 υ+L < 1, = E dt

Appendix B: Proofs of Propositions for Brushed DC-Motors

541

which means that the sliding regime is possible. On the other hand, (B.34) ensures that the sliding surface s = i − i ∗ = 0, is reached, i.e., that i = i ∗ is reached. Thus, it is only required to study the stability of the dynamics (3.64)–(3.66) in closed-loop with (3.72)–(3.74) when evaluated at i = i ∗ . , 1 (Bω d + Using i = i ∗ and (3.72) in (3.64), adding and subtracting terms i a , C dυ dt km TL ), using i a given in (3.73), and defining k p2 = k p /km , ki2 = ki /km , where k p , ki are positive constants, it is found:   ki dυ 1 + k p1 e − ki1 ζ + ea + ξ+C , C e˙ = − R km dt  t 1 ζ= e(τ )dτ − (Bω d + TL ), km ki1 0  t 1 ω(τ ˜ )dτ − (Bω d + TL ), ξ= ki 0

(B.35) (B.36) (B.37)

where C

  f kp ˙ Ra k i dυ = C −ra e˙a + ω˜ − γea + ω˜ . dt km km

(B.38)

On the other hand, adding and subtracting terms υ, ke ωd , L a didta in (3.65), using (3.73), ωd (t) = ωd∗ (t) + ω d , k p2 = k p /km , ki2 = ki /km , and defining r = Ra + ra , ea = ρ + σ, it can be written: 1 1 L a ρ˙ = − e − r ρ − γz 1 + ke ω˜ − ke ωd∗ , 2 2 1 1 L a σ˙ = − e − r σ − γz 2 + f 1 ω˜ − ke ωd∗ , 2 2  t 1 ke ω d , ρ(τ )dτ + z1 = 2γ 0  t 1 ke ω d , z2 = σ(τ )dτ + 2γ 0 kp ki f1 = f − La > 0. km km

(B.39) (B.40) (B.41) (B.42) (B.43)

Finally, adding and subtracting terms J ω˙ d , km i a , Bωd in (3.66), using the definition of i a in (3.73), ωd (t) = ωd∗ (t) + ω d and ea = ρ + σ it is found: J ω˙˜ = −km ρ − km σ − ki ξ − B ω˜ + J ω˙ d + Bωd∗ ,

(B.44)

with ξ defined in (B.37). Hence, the closed-loop dynamics on the sliding surface s = 0 is given by (B.35)–(B.44) and the state of this dynamics is given as ys = [ω, ˜ ξ, σ, ρ, z 2 , z 1 , e, ζ] ∈ R8 . Consider the following scalar function:

542

Appendix B: Proofs of Propositions for Brushed DC-Motors

W (ys ) = W1 (ω, ˜ ξ) + W2 (ρ, z 1 ) + W3 (e, ζ) + W4 (σ, z 2 ), 1 2 1 W1 (ω, ˜ ξ) = J ω˜ + αJ ωξ ˜ + ki ξ 2 , 2 2 1 1 2 W2 (ρ, z 1 ) = L a ρ + pL a z 1 ρ + γz 12 , 2 2 1 2 1 W3 (e, ζ) = Ce + δCeζ + ki1 ζ 2 , 2 2 1 km 2 1 km W4 (σ, z 2 ) = L a σ + β L a z 2 σ + γ z 22 . 2 f1 2 f1

(B.45)

It is not difficult to verify that each one of functions Wi , i = 1, . . . , 4, is positive definite and, hence, W (ys ) is positive definite and radially unbounded if: α > 0, k i = ki − α2 J > 0, k i1 = ki1 − δ 2 C > 0, β > 0,

p > 0, γ = γ − p 2 L a > 0, δ > 0, p f 1 = km . (B.46) β

After some straightforward cancellations, some of them similar to those√present in (3.68), and using the facts that ±qw ≤ |q| |w|, ∀q, w ∈ R, h 1 ≤ n h 2 , ∀h ∈ Rn , it is found that the time derivative of W , given above, along the trajectories of the closed-loop dynamics on the sliding surface s = 0, i.e., (B.35)–(B.44), can be bounded as W˙ ≤ −y  Qy + y |x|,

(B.47)

where y = [|ω|, ˜ |ξ|, |σ|, |ρ|, |z 2 |, |z 1 |, |e|, |ζ|] , x is a bounded scalar function of ωd∗ and ω˙ d which is zero when both ωd∗ = 0 and ω˙ d = 0, and Q is a 8 × 8 symmetric matrix whose entries are given as km Q 33 = r − β L a , Q 44 = r − pL a , (B.48) f1   Cra 1 + k p1 − = βγ, Q 66 = pγ, Q 77 = − δC, Q 88 = δki1 , R La = Q 31 = Q 41 = Q 14 = Q 52 = Q 25 = Q 62 = Q 26 = Q 43 = Q 34 = 0, = Q 45 = Q 63 = Q 36 = Q 65 = Q 56 = 0, αB β f1 pke , Q 15 = Q 51 = − , Q 16 = Q 61 = − , = Q 21 = − 2  2 2 C f kp B C Ra k i Cra f k p L a ki − = Q 71 = − + ke + − , 2L a km km 2km 2J km αkm , = Q 81 = δ Q 71 , Q 32 = Q 23 = Q 42 = Q 24 = − 2   C f kp ki 1+ , Q 82 = Q 28 = δ Q 27 , = Q 27 = − 2km J

Q 11 = B − αJ, Q 55 Q 13 Q 54 Q 12 Q 17 Q 18 Q 72

Q 22 = αki ,

Appendix B: Proofs of Propositions for Brushed DC-Motors

543

γC βr km Cra r 1 C f kp − , Q 53 = Q 35 = − , Q 73 = Q 37 = − − − − 2 4 f1 2L a 2 2J 2   km pr Q 83 = Q 38 = δ Q 73 + , Q 64 = Q 46 = − , 4 f1 2   km 1 1 , , Q 84 = Q 48 = δ Q 47 + Q 74 = Q 47 = Q 37 − + 4 4 f1 4 Cra γ β Cra γδ Q 75 = Q 57 = − − , Q 85 = Q 58 = − , 2L a 4 2L a Cra γ p Cra γδ − , Q 86 = Q 68 = − , Q 76 = Q 67 = − 2L a 4 2L a   Cra δ δ 1 + k p1 − . Q 78 = Q 87 = − 2 R 2L a The eight leading principal minors of Q can always be rendered positive as follows. The first leading principal minor is rendered positive by choosing a small enough α > 0 since B > 0. The second leading principal minor is rendered positive by means of a large enough ki > 0. A large enough r > 0 and a small enough β > 0 suffice to render positive the third leading principal minor. Similarly, a large enough r > 0 and a small enough p > 0 suffice to render positive the fourth leading principal minor. Given any f 1 > 0 a large enough γ > 0 suffices to render positive the fifth and the sixth leading principal minors. The seventh leading principal minor is always rendered positive by choosing a large enough k p1 > 0 and the eighth leading principal minor is positive if we choose large enough ki1 > 0 and a small enough δ > 0. Thus, it can always be ensured that λmin (Q) > 0 to write (B.47) as W˙ ≤ −λmin (Q) ys 2 + ys |x|, = −(1 − Θ)λmin (Q) ys 2 − Θλmin (Q) ys 2 + ys |x|, |x| ≤ −(1 − Θ)λmin (Q) ys 2 , ∀ ys ≥ Θλmin (Q) for some 0 < Θ < 1. Notice that, since functions W j (·, ·), for j = 1, . . . , 4, given in (B.45), are quadratic forms, it is clear that there always exist two class K ∞ functions α1 ( ys ), α2 ( ys ), satisfying α1 ( ys ) ≤ W (ys ) ≤ α2 ( ys ). Thus, according to Theorem 2.29, this means that ys ∈ R8 is bounded and converges to a ball whose radius depends on the upper bound of the scalar function of time |x|. Moreover, since ωd (t f ) = ω d , ∀t ≥ t f , where t f > 0 is a finite constant, then ωd∗ (t) = 0 and ω˙ d (t) = 0, ∀t ≥ t f . This implies that |x(t)| = 0, ∀t ≥ t f , as explained before (B.48). This ensures that ys converges to zero as t → ∞ and completes the proof of Proposition 3.8. Recall that conditions ensuring this result are (B.46) and a suitable choice for α > 0, ki > 0, r > 0, β > 0, r > 0, p > 0, γ > 0, k p1 > 0, ki1 > 0 and δ > 0, such that the eight leading principal minors of matrix Q, defined in (B.48), are positive.

544

Appendix B: Proofs of Propositions for Brushed DC-Motors

Finally, although it might seem to be cumbersome to verify that all the leading principal minors of matrix Q are positive (see (B.48)), it is stressed that it is rather straightforward to check this. The reader can realize that positiveness of the leading principal minors of Q is determined by the diagonal elements of Q. Thus the i−th leading principal minor can always be rendered positive if the (i − 1)−th leading principal minor is positive and Q ii is rendered large enough just by choosing a large enough value for the controller gain appearing in Q ii . This process is repeated until the 8−th leading principal minor is rendered positive.

Appendix C

Proofs of Propositions for PM Synchronous Motors

C.1 Proof of Proposition 4.5 Notice that there always exists a positive constant ε such that  = εΦ M . ΦM

Hence, defining: k p = k p /ε, ki = ki /ε, it can be written: Iq∗ =

1 ΦM



−k p ω˜ − ki σ



t

 ω(s)ds ˜

.

(C.1)

0

Adding and subtracting the terms bω ∗ , n p (L d − L q )Id Iq∗ , Φ M Iq∗ in (4.41), taking advantage from the fact that ω˙ ∗ = 0 and using (C.1) it is obtained: J ω˙˜ = −bω˜ + n p (L d − L q )Id I˜q + n p (L d − L q )Id Iq∗  t  +Φ M I˜q − τ L − bω ∗ − k p ω˜ − ki σ ω(s)ds ˜ , 0

which, defining: z(x) = σ(x) +

1 (τ L + bω ∗ ), x = ki



t

ω(s)ds, ˜

(C.2)

0

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 V. M. Hernández-Guzmán et al., Energy-Based Control of Electromechanical Systems, Advances in Industrial Control, https://doi.org/10.1007/978-3-030-58786-4

545

546

Appendix C: Proofs of Propositions for PM Synchronous Motors

becomes J ω˙˜ = −bω˜ + n p (L d − L q )Id I˜q + n p (L d − L q )Id Iq∗ + Φ M I˜q − k p ω˜ − ki z(x).

(C.3)

On the other hand, replacing (4.51) in (4.40), adding and subtracting terms n p L q ω ∗ Iq , n p L q ω˜ Iq∗ , n p L q ω ∗ Iq∗ , using (C.1), (C.2), and defining:  zd =

t

Id (s)ds −

0

1 n p L q ω∗ (τ L + bω ∗ ), αdi Φ M

(C.4)

yields L d I˙d = −(R + αd )Id + n p L q ω˜ I˜q + n p L q ω˜ Iq∗ + n p L q ω ∗ I˜q − αdi z d −

n p L q ω ∗ k p ΦM

ω˜ −

n p L q ω ∗ ki z(x) − K d ω˜ 2 Id . ΦM

(C.5)

Finally, replacing (4.52) in (4.39), adding and subtracting the terms R Iq∗ , n p L d ω ∗ Id , Φ M ω ∗ , L q I˙q∗ , using (C.1), (C.2), and defining:  zq =

t

0

1 I˜q (s)ds + αqi



 R (τ L + bω ∗ ) + Φ M ω ∗ , ΦM

(C.6)

it is found Rk p Rki L q I˙˜q = −(R + αq ) I˜q + ω˜ + z − n p L d ω˜ Id − n p L d ω ∗ Id ΦM ΦM L q k p L q ki dσ(x) −Φ M ω˜ − αqi z q + ω˙˜ + ω. ˜ ΦM ΦM d x −K q ω˜ 2 I˜q − K f |Id | I˜q .

(C.7)

Notice that the closed-loop dynamics is given by (C.2)–(C.7), which can be rewritten as follows (recall that τ L is constant): L q I˙˜q = −(R + αq ) I˜q − n p L d ω˜ Id − Φ M ω˜ + Vq , L d I˙d = −(R + αd )Id + n p L q ω˜ I˜q + Vd , J ω˙˜ = −(b + k p )ω˜ + n p (L d − L q )Id I˜q + Φ M I˜q − T L , x˙ = ω, ˜ z˙ d = Id , z˙ q = I˜q , T L = −n p (L d −

L q )Id Iq∗

+

−K d ω˜ Id ,

(C.9) (C.10) (C.11)

ki z(x),

Vd = n p L q ω˜ Iq∗ + n p L q ω ∗ I˜q − αdi z d − 2

(C.8)

n p L q ω ∗ k p ΦM

ω˜ −

n p L q ω ∗ ki z(x) ΦM

Appendix C: Proofs of Propositions for PM Synchronous Motors

Vq =

Rk p

547

L q k p Rki L q ki dσ(x) z(x) − n p L d ω ∗ Id − αqi z q + ω˙˜ + ω˜ ΦM ΦM ΦM ΦM d x −K q ω˜ 2 I˜q − K f |Id | I˜q . ω˜ +

The state of the closed-loop dynamics (C.2)–(C.7), or equivalently (C.8)–(C.11), is ∗ , Id , z d , I˜q , z q ] ∈ R6 . It is not difficult to verify that ξ = 0 is ξ = [ω, ˜ x + τL +bω ki the only equilibrium point of the closed-loop dynamics if ki > 0, αdi > 0, αqi > 0. Moreover, this closed-loop dynamics is autonomous since it can be written as ξ˙ = f (ξ) for some nonlinear f (·) ∈ R6 . Consider the following Lyapunov function candidate for the closed-loop dynamics:   1 1 1 1 τ + bω ∗ L q I˜q2 + L d Id2 + αdi z d2 + αqi z q2 + Vω ω, ˜ x+ L  ,(C.12) 2 2 2 2 ki    x τ L + bω ∗ 1 2  + β(b + k  )] J ω ˜ Vω ω, + [k z(r )dr + β J z(x)ω. ˜ ˜ x+ = p i τ +bω ∗ 2 ki − L  V (ξ) =

ki

Taking advantage from the fact that J is positive, it is shown in Appendix A.3 ∗ ˜ x + τL +bω ) is positive definite and radially unbounded if ki > 0, k p > 0, that Vω (ω, ki are chosen large enough and β > 0 small enough such that (A.7) is satisfied. Hence, since inductances L d , L q , are always positive, the Lyapunov function candidate V given in (C.12) is positive definite and radially unbounded if: αdi > 0, αqi > 0,

(C.13)

and (A.7) are satisfied. The time derivative of V along the trajectories of the closedloop dynamics (C.8)–(C.11) is given as V˙ = −(R + αq ) I˜q2 + −

L q (k p )2

I˜q ω˜ −

J ΦM n p L q ω ∗ k p

Rk p

ω˜ I˜q +

ΦM L q k p ki

L q k p b Rki z(x) I˜q − n p L d ω ∗ Id I˜q − ω˜ I˜q ΦM J ΦM

I˜q z(x) +

L q ki dσ(x) I˜q ω˜ − (R + αd )Id2 ΦM d x

(C.14)

J ΦM n p L q ω ∗ ki dσ(x) 2 − ω˜ Id − z(x)Id − bω˜ 2 − k p ω˜ 2 + β J ω˜ ΦM ΦM dx L q k p 2 I˜ + n p L q ω ∗ I˜q Id + βΦ M I˜q z(x) −βki z 2 (x) + J q n p L d k p − Id ω˜ 2 + βn p (L d − L q )Id I˜q z(x) ΦM βn p (L d − L q )k p βn p (L d − L q )ki n p L d ki Id ωz(x) ˜ − Id z(x)σ(x) − ω˜ Id σ(x) − ΦM ΦM ΦM    L q k p n p (L d − L q ) L q k p n p (L d − L q )k p −K d ω˜ 2 Id2 + Id I˜q2 − Id I˜q ω˜ 2 J ΦM J ΦM

548

Appendix C: Proofs of Propositions for PM Synchronous Motors −

L q k p n p (L d − L q )ki 2 J ΦM

Id I˜q σ(x) − K q ω˜ 2 I˜q2 − K f |Id | I˜q2 ,

which, taking advantage from ±vy ≤ |v| |y|, ∀v, y ∈ R, (4.54) and |σ(x)| < M, ∀x ∈ R, and defining ζ = [|ω|, ˜ |z|, |Id |, | I˜q |] , can be upper bounded as V˙ ≤ −ζ T Qζ +

L q k p n p |L d − L q | J ΦM

n p L d k p

|Id | I˜q2 − K f |Id | I˜q2

(C.15)

|Id |ω˜ 2 − K d1 ω˜ 2 Id2 − k p2 ω˜ 2 ΦM L q k p n p |L d − L q |k p + |Id | | I˜q | |ω| ˜ − K q ω˜ 2 I˜q2 − K d2 ω˜ 2 Id2 − αd2 Id2 − αq2 I˜q2 , 2 J ΦM +

where k p = k p1 + k p2 , with k p1 > 0, k p2 > 0, K d = K d1 + K d2 , with K d1 > 0, K d2 > 0, αd = αd1 + αd2 , and αq = αq1 + αq2 , with αd1 , αd2 , αq1 , αq2 positive constants and entries of matrix Q are given as Q 11 = b + k p1 − β J, Q 12 = Q 21 = 0,

Q 14 = Q 41 Q 24 = Q 42 Q 34 = Q 43

n p L q |ω ∗ |k p

Q 33 = R + αd1 ,

Q 44 = R + αq1 −

L q k p

, J (C.16)

βn p |L d − L q |k p M n p L d ki M − , 2Φ M 2Φ M ΦM n p L q |ω ∗ |ki βn p |L d − L q |ki M =− − , 2Φ M 2Φ M     Rk p + L q ki L q k p (b + k p )  , = −  −  2Φ M 2J Φ M      Rki L q k p ki  βΦ M , − = −  + 2Φ M 2 2J Φ M     n p (L d − L q )ω ∗  L k  n |L − L q |ki M  − βn p |L d − L q |M − q p p d  = − .  2 2 2J Φ M

Q 13 = Q 31 = − Q 23 = Q 32

Q 22 = βki ,

−

Matrix Q is positive definite if and only if all of its four leading principal minors are positive, i.e., ⎛

⎞

Q 11 Q 12 Q 13 b + k p1 > β J, β > 0, ki > 0, det ⎝ Q 21 Q 22 Q 23 ⎠ > 0, det(Q) > 0. Q 31 Q 32 Q 33

(C.17)

Notice that Q 33 and Q 44 are the only entries in the 3rd and 4th leading principal minors, respectively, which depend on αd > 0 and αq > 0. Hence, it is only matter to choose these gains large enough to render positive 3rd and 4th leading principal minors. Also note that the first three conditions in (C.17) are always rendered true using suitable positive values for k p , β and ki . This ensures that λmin (Q) > 0. Thus,

Appendix C: Proofs of Propositions for PM Synchronous Motors

549

first row in (C.15) is negative if: Kf >

L q k p n p |L d − L q | J ΦM

.

(C.18)

On the other hand, notice that n p L d k p ΦM

|Id |ω˜ 2 − K d1 ω˜ 2 Id2 < 0, if |Id | >

In the case when |Id | ≤

n p L d k p Φ M K d1

n p L d k p ΦM

n p L d k p Φ M K d1

.

we have 

|Id |ω˜ 2 ≤

n p L d k p

2

ΦM

1 2 ω˜ . K d1

Thus, the second row in (C.15) is always negative if: k p2

 >

n p L d k p

2

ΦM

1 . K d1

Also notice that L q k p n p |L d − L q |k p 2 J ΦM

|Id | | I˜q | |ω| ˜ − K q ω˜ 2 I˜q2 − K d2 ω˜ 2 Id2 < 0

is equivalent to 

 ˜ − a2 K q |ω| w > 0, w − a2 K d2 |ω| ˜ L q k p n p |L d − L q |k p , w = [| I˜q |, |Id |] . a= 2 J ΦM 

This is satisfied if and only if:  ˜ > 0, and |ω| ˜ > K q |ω|

a2 . 4K q K d2

In the case when  |ω| ˜ ≤ then

a2 = μ, 4K q K d2

(C.19)

550

Appendix C: Proofs of Propositions for PM Synchronous Motors

a|Id | | I˜q | |ω| ˜ − αd2 Id2 − αq2 I˜q2 ≤ aμ|Id | | I˜q | − αd2 Id2 − αq2 I˜q2 . This is equivalent to w





 αq2 − aμ 2 w > 0, − aμ αd2 2

which is satisfied if and only if: αq2 > 0 and αq2 αd2 >

a4 , 4K q K d2

(C.20)

and, hence, third row in (C.15) is always negative. Thus, we can write V˙ ≤ −ζ  Qζ ≤ 0, ∀ξ ∈ R6 , which means that the origin is stable. As remarked after (C.11), the closed-loop dynamics (C.8)-(C.11) is autonomous and, hence, the LaSalle invariance principle (see Corollary 2.19) can be applied as follows. Define a set S as τ L + bω ∗ = 0, I˜q = 0, Id = 0, z d ∈ R, z q ∈ R}. S = {ξ ∈ R6 |V˙ = 0} = {ω˜ = 0, x + ki Evaluating the closed-loop dynamics (C.8)-(C.11) in S we obtain 0 = −(R + αq )(0) − n p L d (0)(0) − Φ M (0) + Vq , 0 = −(R + αd )(0) + n p L q (0)(0) + Vd , 0 = −(b + k p )(0) + n p (L d − L q )(0)(0) + Φ M (0) − T L , x˙ = 0, z˙ d = 0, z˙ q = 0, T L = −n p (L d − L q )(0)Iq∗ + ki (0), Vd = n p L q (0)Iq∗ + n p L q ω ∗ (0) − αdi z d −

n p L q ω ∗ k p ΦM

(0) −

n p L q ω ∗ ki (0) ΦM

−K d (0) (0), Rk p L q k p Rki L q ki dσ(x) (0) (0) + (0) − n p L d ω ∗ (0) − αqi z q + (0) + Vq = ΦM ΦM ΦM ΦM d x −K q (0)2 (0) − K f |(0)|(0). 2

These expressions have, as the unique solution, ξ = 0 because αdi > 0 and αqi > 0. Thus, according to Corollary 2.19, this implies that ξ = 0 is a globally asymptotically stable equilibrium point. This completes the proof of Proposition 4.5. Conditions for this stability result are summarized by (A.7), (C.13), (C.17)–(C.20).

Appendix C: Proofs of Propositions for PM Synchronous Motors

551

C.2 Proof of Proposition 4.12  Since Φ M is a positive estimate of Φ M , there always exists a positive ε such that  = εΦ M . ΦM

(C.21)

Hence, defining k p = εk p , kd = εkd and ki = εki , adding and subtracting g(q ∗ ) and defining s(z) = sat (z) + k1 g(q ∗ ), it is possible to write i

Iq∗ =

1 [−k p h(q) ˜ − kd q˙ − ki s(z) + g(q ∗ )]. ΦM

(C.22)

Adding and subtracting terms n p (L d − L q )Id Iq∗ , Φ M I˜q∗ , and replacing (C.22), it is found that (4.76) can be written as J q¨ = −(b + kd )q˙ + n p (L d − L q )Id I˜q + n p (L d − L q )Id Iq∗ + Φ M I˜q −k p h(q) ˜ − g(q) + g(q ∗ ) − ki s(z).

(C.23)

Replacing (4.80) in (4.75), and adding and subtracting n p L q q˙ Iq∗ it is possible to write L d I˙d = −(R + αdp )Id + n p L q q˙ I˜q + n p L q q˙ Iq∗ − K d q˙ 2 Id − αdi z d , (C.24)  t Id dt. zd = 0

On the other hand, replacing (4.81) in (4.74), adding and subtracting I˙q∗ , R Iq∗ and replacing (C.22): L q I˙˜q = −(R + αq p ) I˜q − n p L d Id q˙ − Φ M q˙ − K q q˙ 2 I˜q + Rk  + i s(z) − αqi z q − L q I˙q∗ − K f |Id | I˜q , ΦM  t R zq = g(q ∗ ). I˜q dt + α Φ qi M 0

Rk p ΦM

h(q) ˜ +

Rkd q˙ ΦM (C.25)

Thus, the closed-loop dynamics is given by (C.23)–(C.25), (4.82). Notice that ξ = [q, ˜ q, ˙ z + g(q ∗ )/ki , I˜q , Id , z d , z q ] ∈ R7 is the state of this dynamics and ξ = 0 is its unique equilibrium point if: L>

k . ki

(C.26)

552

Appendix C: Proofs of Propositions for PM Synchronous Motors

This closed-loop dynamics is autonomous because it can be written as ξ˙ = f (ξ) for some nonlinear f (·) ∈ R7 . Dynamics (C.23)–(C.25), (4.82), can also be written as L q I˙˜q = −(R + αq p ) I˜q − n p L d Id q˙ − Φ M q˙ + Vq , (C.27) ˜ ˙ (C.28) L d Id = −(R + αdp )Id + n p L q q˙ Iq + Vd ,  ˜ ˜ (C.29) J q¨ = −(b + kd )q˙ + n p (L d − L q )Id Iq + Φ M Iq − G,     βk αβk p d h(q) ˜ + 1+ q, ˙ (C.30) z˙ q = I˜q , z˙ d = Id , z˙ = α 1 + ki ki Rk p Rkd Rki Vq = h(q) ˜ + q˙ + s(z) − αqi z q − L q I˙q∗ − K q q˙ 2 I˜q − K f |Id | I˜q , ΦM ΦM ΦM Vd = n p L q q˙ Iq∗ − K d q˙ 2 Id − αdi z d , G = −n p (L d − L q )Id Iq∗ + k p h(q) ˜ + g(q) − g(q ∗ ) + ki s(z). Consider the following “energy” storage function for closed-loop dynamics: (C.31) W (q, ˜ q, ˙ z + g(q ∗ )/ki , I˜q , Id , z d , z q ) = 1 1 1 ˜2 1 L q Iq + L d Id2 + αqi z q2 + αdi z d2 + V (q, ˜ q, ˙ z + g(q ∗ )/ki ), 2 2 2 2 where V (q, ˜ q, ˙ z + g(q ∗ )/ki ) =

 q˜ 1 2 J q˙ + αJ h(q) ˜ q˙ + α(b + kd ) h(r )dr 2 0  q˜  ∗ +k p h(r )dr + U (q) − U (q ∗ ) − qg(q ˜ ) 0  z +ki s(r )dr + αβ J s(z)q. ˙ −g(q ∗ )/ki

V (q, ˜ q, ˙ z + g(q ∗ )/ki ) is positive definite and radially unbounded if (A.8)–(A.10), (A.14), are satisfied (see Appendix A.4). Thus, it is clear that W (q, ˜ q, ˙ z + g(q ∗ )/ki , I˜q , Id , z d , z q ) qualifies as a Lyapunov function candidate because it is positive definite and radially unbounded if αqi , αdi , are chosen to be positive since L q , L d , are positive constants by definition. It is straightforward to verify that the time derivative of W along trajectories of the closed-loop system (C.23)–(C.25), (4.82), is given as dh(q) ˜ 2 W˙ = n p L d Id q˙ Iq∗ − (b + kd )q˙ 2 + αJ ˜ − g(q ∗ )) q˙ − αh(q)(g(q) d q˜ ˜ p (L d − L q )Id Iq∗ + αh(q)Φ ˜ M I˜q +αh(q)n ˜ p (L d − L q )Id I˜q + αh(q)n −αk p h 2 (q) ˜ + αβ J [(α + αβk p /ki )h(q) ˜ + (1 + αβkd /ki )q] ˙

ds(z) q˙ dz

−αβs(z)(g(q) − g(q ∗ )) + αβn p (L d − L q )Id s(z) I˜q + αβn p (L d − L q )Id s(z)Iq∗

Appendix C: Proofs of Propositions for PM Synchronous Motors

553

+αβΦ M s(z) I˜q − αβki s 2 (z) − (R + αq p ) I˜q2 − K q q˙ 2 I˜q2 + Rk p h(q) ˜ I˜q /Φ M   +Rk q˙ I˜q /Φ M + Rk s(z) I˜q /Φ M d

i

dh(q) ˜ ds(z) q˙ + ki {(α + αβk p /ki )h(q) ˜ + (1 + αβkd /ki )q}]/Φ ˙ M d q˜ dz 2  ∗ 2 −K f |Id | I˜q − L q k (g(q) − g(q )) I˜q /(J Φ M ) − (R + αd p )I − I˜q L q [−k p

d

d

+L q kd n p (L d − L q )Id I˜q2 /(J Φ M ) + L q kd n p (L d − L q )Id Iq∗ I˜q /(J Φ M ) ˜ I˜q /(J Φ M ) −K d q˙ 2 Id2 + L q kd I˜q2 /J − L q kd k p h(q) ˙ −L q kd ki s(z) I˜q /(J Φ M ) − L q kd (b + kd )q˙ I˜q /(J Φ M ) − αβbs(z)q,

where k p /ki = k p /ki and kd /ki = kd /ki have been used. Hence, W˙ can be upper bounded as W˙ ≤ −x  Qx − K f |Id | I˜q2

(C.32)

+L q kd n p |L d − L q | |Id | I˜q2 /(J Φ M )  −q˙ 2 |Id |(K d1 |Id | − n p L d kd /Φ M ) − kd2 q˙ 2 −K q q˙ 2 I˜q2 − K d2 q˙ 2 Id2 2 +L q (kd )2 n p |L d − L q | |q| ˙ | I˜q | |Id |/(J Φ M ) 2 2 −αq p2 I˜q − αdp2 Id ,

x  = [|q|, ˙ |h(q)|, ˜ |s(z)|, | I˜q |, |Id |],   , kd2 , αq p1 , αq p2 , αdp1 , αdp2 are positive constant scalars such where K d1 , K d2 , kd1   + kd2 = kd , αq p1 + αq p2 = αq p , and αdp1 + αdp2 = αdp . that K d1 + K d2 = K d , kd1 The entries of matrix Q are given as  ) − αJ − αβ J (1 + αβkd /ki ), Q 11 = (b + kd1  Q 22 = α(k p − khg ), Q 33 = αβki ,

Q 44 = R + αq p1 − L q kd /J,

Q 15

Q 24

Q 13 = Q 31 = −αβb/2,

α β αβ (1 + βk p /ki )J, Q 23 = Q 32 = − khg , 2 2 1 1 1 = − Rkd /Φ M − L q k p /Φ M − L q ki (1 + αβkd /ki )/Φ M 2 2 2

Q 12 = Q 21 = − Q 14 = Q 41

Q 55 = R + αdp1 ,

(C.33)

2

1 − L q kd (b + kd )/(J Φ M ), 2 1 1 1 = Q 51 = − n p L d k p |h(q)| ˜ M /Φ M − n p L d ki |s(z)| M /Φ M − n p L d k  /Φ M 2 2 2 α αβ  k n p |L d − L q | |s(z)| M /Φ M , − n p kd |L d − L q | |h(q)| ˜ M /Φ M − 2 2 d 1 1 1 = Q 42 = − L q kd k p /(J Φ M ) − L q ki α(1 + βk p /ki )/Φ M − Rk p /Φ M 2 2 2

554

Q 25

Q 34 Q 35

Q 45

Appendix C: Proofs of Propositions for PM Synchronous Motors

1 α − L q kd khg /(J Φ M ) − Φ M , 2 2 α α  = Q 52 = − n p k p |L d − L q | |h(q)| ˜ M /Φ M − n p ki |L d − L q | |s(z)| M /Φ M 2 2 α  − n p k |L d − L q |/Φ M , 2 αβ 1 1 = Q 43 = − Φ M − Rki /Φ M − L q kd ki /(J Φ M ), 2 2 2 αβ αβ n p k p |L d − L q | |h(q)| n p ki |L d − L q | |s(z)| M /Φ M = Q 53 = − ˜ M /Φ M − 2 2 αβ n p k  |L d − L q |/Φ M , − 2 α αβ n p |L d − L q | |s(z)| M = Q 54 = − n p |L d − L q | |h(q)| ˜ M− 2 2 1 2 − L q kd k p n p |L d − L q | |h(q)| ˜ M /(J Φ M ) 2 1 1 2 2 − L q kd ki n p |L d − L q | |s(z)| M /(J Φ M ) − L q kd k  n p |L d − L q |/(J Φ M ), 2 2

where the following result, taken from [130], pp. 105–107, has been used: |g(q) − g(q ∗ )| ≤ khg h(|q|), ˜ khg ≥

2k  ∀q˜ ∈ R,  , h( 2k ) kg

as well as the facts that h(|q|) ˜ = |h(q)| ˜ and ±uv ≤ |u| |v|, ∀q, ˜ u, v ∈ R. The symbol | · | M stands for the supremum value over the absolute value. The five leading principal minors of matrix Q can always be rendered positive by choosing small enough α > 0, β > 0 and large enough positive definite matrices kd , k p , ki , αdp , αq p . Hence, matrix Q is positive definite. On the other hand, first row plus second row in (C.32) is negative if: K f > L q kd n p |L d − L q |/(J Φ M ).

(C.34)

First expression on third row in (C.32) are negative for |Id | >

1 η1 n p L d kd /Φ M = . K d1 K d1

Hence, if:  λm (kd2 )>

η12 , K d1

(C.35)

then third row in (C.32) is negative. Finally, expression −K q q˙ 2 I˜q2 − K d2 q˙ 2 Id2 + 2 L q (kd )2 n p |L d − L q | |q| ˙ | I˜q | |Id |/(J Φ M ) is negative if:

Appendix C: Proofs of Propositions for PM Synchronous Motors

|q| ˙ >

555

2 ) L q (kd )2 n p |L d − L q |/(J Φ M = η4 . 4K q K d2

Hence, the above expression plus the last row in (C.32) is negative if: αq p2 αdp2 >

1 2 2 2 [L q (kd )2 n p |L d − L q |/(J Φ M )] η4 . 4

(C.36)

Then, W˙ ≤ 0, with W˙ = 0 if [q, ˙ q, ˜ s(z), I˜q , Id ] = [0, 0, 0, 0, 0] . This means that the origin is stable. Thus, the LaSalle invariance principle (see Corollary 2.19) can be employed because the closed-loop dynamics (C.27)–(C.30) is autonomous. Define a set S as: S = {ξ ∈ R7 |W˙ = 0}, = {q˙ = 0, q˜ = 0, z +

1 g(q ∗ ) = 0, I˜q = 0, Id = 0, z d ∈ R, z q ∈ R}. ki

Evaluating the closed-loop dynamics (C.27)–(C.30) in S it is obtained: 0 = −(R + αq p )(0) − n p L d (0)(0) − Φ M (0) + Vq , 0 = −(R + αdp )(0) + n p L q (0)(0) + Vd , 0 = −(b + kd )(0) + n p (L d − L q )(0)(0) + Φ M (0) − G,     βk p αβkd (0) + 1 + (0), z˙ q = (0), z˙ d = (0), 0 = α 1 + ki ki Rk p Rkd Rki Vq = (0) + (0) + (0) − αqi z q − L q (0) − K q (0)2 (0) − K f |0|(0), ΦM ΦM ΦM Vd = n p L q (0)Iq∗ − K d (0)2 (0) − αdi z d , G = −n p (L d − L q )(0)Iq∗ + k p (0) + g(q ∗ ) − g(q ∗ ) + ki (0). These expressions have, as the unique solution, ξ = 0 because αdi > 0 and αqi > 0. Thus, according to Corollary 2.19, this implies that ξ = 0 is a globally asymptotically stable equilibrium point. This completes the proof of Proposition 4.12. Conditions for this stability result are summarized by (C.26), (C.34)–(A.8), (A.9)–(A.14), αqi and αdi are positive constants and the five principal minors of matrix Q defined in (C.33) are positive.

C.3 Proof of Proposition 4.18 First, notice that there always exists a positive constant ε such that

556

Appendix C: Proofs of Propositions for PM Synchronous Motors  ΦM = εΦ M .

Hence, defining: k p = k p /ε, kd = kd /ε, ki = ki /ε, we can write Iq∗ =

1 ΦM

   t −k p q˜ − kd q˙ − ki q(s)ds ˜ .

(C.37)

0

Moreover, defining: 1   k , k = k p − ki , α i p  t 1 (αq(s) ˜ + q(s))ds ˙ + q(0) ˜ +  g(q ∗ ), z= ki 0

ki =

(C.38)

and recalling that 

t

q(t) ˜ − q(0) ˜ =

˙˜ q(s)ds =



0

t

q(s)ds, ˙

0

because q˙ ∗ = 0, yields Iq∗ =

 1   −k p q˜ − kd q˙ − ki z + g(q ∗ ) . ΦM

(C.39)

Using this expression and adding and subtracting terms n p (L d − L q )Id Iq∗ , Φ M Iq∗ in (4.76) results in: J q¨ = −bq˙ + n p (L d − L q )Id I˜q + n p (L d − L q )Id Iq∗ + Φ M I˜q −k p q˜ − kd q˙ − ki z + g(q ∗ ) − g(q).

(C.40)

On the other hand, replacing (4.91) in (4.75) yields L d I˙d = −(R + αd )Id − αdi z 1 + n p L q q˙ I˜q + n p L q q˙ Iq∗ + h,  t Id (s)ds. z1 = 0

Now, define γ˜ i = γi − ! γi , for i = 1, . . . , 4, where

(C.41)

Appendix C: Proofs of Propositions for PM Synchronous Motors

γ1 = n p L d , γ2 = n p (L d − L q ), γ3 = αβ(q(0) ˜ + γ4 =

L q kd n p (L d − L q ). J ΦM

557

1 g(q ∗ ))n p (L d − L q ), ki

Hence, according to (4.95) the following is found: γ˙˜ 1 = −Γ1 Id q˙ Iq∗ , γ˙˜ 2 = −Γ2 G 1 Id , γ˙˜ 3 = −Γ3 Id Iq , γ˙˜ 4 = −Γ4 Id Iq I˜q . (C.42) Thus, adding and subtracting some convenient terms in (4.94): h = γ˜ 1 q˙ Iq∗ + γ˜ 2 G 1 + γ˜ 3 Iq + γ˜ 4 Iq I˜q − γ1 q˙ Iq∗ − γ2 G 1 − γ3 Iq − γ4 Iq I˜q . Finally, replacing (4.92) in (4.74), defining: 

t

z2 =

I˜q (s)ds +

0

R g(q ∗ ), αqi Φ M

(C.43)

using (C.39), and adding and subtracting terms R Iq∗ , L q I˙q∗ , yields R  L q I˙˜q = −(R + αq ) I˜q + (k q˜ + kd q˙ + ki z) − αqi z 2 − n p L d q˙ Id − Φ M q˙ ΦM p L q  + (k q˙ + kd q¨ + ki (αq˜ + q)). ˙ (C.44) ΦM p The closed-loop dynamics is given by (C.38), (C.40)–(C.44), which can be rewritten as follows: L d I˙d = −(R + αd )Id + n p L q q˙ I˜q + Vd , L q I˙˜q = −(R + αq ) I˜q − n p L d q˙ Id − Φ M q˙ + Vq , J q¨ = −(b + kd )q˙ + n p (L d − L q )Id I˜q + Φ M I˜q − G, z˙ = αq˜ + q, ˙ z˙ 1 = Id , z˙ 2 = I˜q , Vd =

n p L q q˙ Iq∗

(C.45) (C.46) (C.47) (C.48)

− αdi z 1 + h,

L q  R  (k q˜ + kd q˙ + ki z) − αqi z 2 + (k q˙ + kd q¨ + ki (αq˜ + q)), ˙ ΦM p ΦM p G = −n p (L d − L q )Id Iq∗ + k p q˜ + ki z − g(q ∗ ) + g(q).

Vq =

The state of the closed-loop dynamics (C.38), (C.40)–(C.44), or equivalently (C.45)– (C.48), is ξ = [q, ˜ q, ˙ z, Id , z 1 , I˜q , z 2 , γ˜ 1 , . . . , γ˜ 4 ] ∈ R11 . It is not difficult to verify that ξ = 0 is always an equilibrium point of the closed-loop dynamics. In this respect notice that −γ2 G 1 − γ3 Iq = 0 if z = q˜ = 0. Consider the following “energy” storage function for the closed-loop dynamics:

558

Appendix C: Proofs of Propositions for PM Synchronous Motors

 1 1 ˜2 1 1 1 ˜ q, ˙ z) + γ˜ i2 , L q Iq + L d Id2 + αdi z 12 + αqi z 22 + Vq (q, 2 2 2 2 2Γ i i=1 4

V (ξ) =

(C.49) Vq (q, ˜ q, ˙ z) = V1 (q, ˜ q) ˙ + V2 (z, q) ˙ + V3 (q, ˜ z), 1 α 1 ˜ 2 − α2 J q˜ 2 + (b + kd )q˜ 2 + k p q˜ 2 ˜ q) ˙ = J (q˙ + 2αq) V1 (q, 4 2 2 ∗ +U (q) − U (q ∗ ) − qg(q ˜ ), 1 αβ  2 1  2 ˙ = J (q˙ + 2αβz)2 − α2 β 2 J z 2 + V2 (z, q) k z + ki z , 4 2 d 2 β  ˜ = (k p − αkd )(z − q) ˜ 2. V3 (z, q) 2 Taking advantage from the fact that J is positive, it is shown in Appendix A.2 that the Vq given in (C.50) is positive definite and radially unbounded if: k p > kg , α > 0,

1 (b + kd ) > αJ, 2

(C.50)

αβ  1 kd + ki > α2 β 2 J, β > 0, k p > αkd . 2 2

Thus, since the inductance is positive L d > 0, L q > 0, the Lyapunov function candidate V given in (C.50) is positive definite and radially unbounded if (C.50) and αdi > 0, αqi > 0,

(C.51)

are true. The time derivative of V along the trajectories of the closed-loop dynamics (C.45)–(C.48) can be obtained replacing q¨ from (C.47), and employing (4.77)–(4.79). Using the fact that ±r s ≤ |r | |s|, ∀r, s ∈ R, we can write ˙ |q|, ˜ |z|, |Id |, | I˜q |] , V˙ ≤ −ζ  Mζ, ζ = [|q|,

(C.52)

where the entries of matrix M are given as follows. The entries in the first three rows and columns are the same as those belonging to the first three rows and columns of the 4 × 4 Q matrix defined in (B.32). The remaining entries are defined as M41 = M42 = M43 = M14 = M24 = M34 = M54 = M45 = 0, L q kd , M44 = R + αd , M55 = R + αq − J     Rkd Lq k p L q ki L q kd (b + kd )  M51 = M15 = −  + + − , 2Φ M 2Φ M 2Φ M 2J Φ M    Rk p L q kd k p  L q kd kg L q αki αΦ M −  M52 = M25 = −  − + + , 2Φ M 2Φ M 2 2J Φ M  2J Φ M    Rk  L q kd ki  αβΦ M . − M53 = M35 = −  i + 2Φ M 2 2J Φ M 

Appendix C: Proofs of Propositions for PM Synchronous Motors

559

Hence, matrix M is positive definite if and only if all of its five leading principal minors are positive. Thus, the first three conditions given in (B.33) must be satisfied and M44 > 0, det(M) > 0.

(C.53)

As explained in Appendix B.2, the first condition in (B.33) can always be satisfied using a large enough value for kd > 0 and some small α > 0 and β > 0. With these values, the second condition in (B.33) can be rendered true with a large enough k p > 0. Once this is achieved, the third condition in (B.33) is always satisfied using a large enough ki > 0. Finally, conditions in (C.53) can be always satisfied using large enough positive values for αd and αq . This ensures that λmin (M) > 0. Notice that we can write V˙ ≤ −ζ  Mζ ≤ −λmin (M)(q˙ 2 + q˜ 2 + z 2 + Id2 + I˜q2 ) ≤ −λmin (M)q˜ 2 ≤ 0, ∀ξ ∈ R11 , which means that the origin is stable implying that the state is bounded, i.e., ξ ∈ L11 ∞ and q, ˜ q˙ ∈ L∞ . Further, from the previous expression it follows that 

t 0

dV ds = V (t) − V (0) ≤ −λmin (M) ds



t

q˜ 2 (s)ds,

0

i.e.,  V (0) ≥ −V (t) + V (0) ≥ λmin (M)

t

q˜ 2 (s)ds,

0

which proves that q˜ ∈ L2 . Thus, invoking Corollary 2.32 we conclude that limt→∞ q(t) ˜ = 0 from any initial condition. This completes the proof of Proposition 4.18. We stress that conditions for this result are summarized by (C.50), (C.51), (C.53), and the first three conditions in (B.33).

Appendix D

Proofs of Propositions for Induction Motors

D.1 Proof of Proposition 5.10 Notice that ω = ω˜ + ω ∗ . This means that ω can be written in terms of a constant plus the velocity error  t below as a part of the closed-loop state. Also  t which is defined ˜ and 0 I˜q (s)ds represent the integral parts of the PI note that integrals 0 ω(s)ds velocity and electric current controllers. As it will be shown below, this means that two closed-loop states have to be written as the difference between each one of these integrals and a constant, i.e., the steady-state value of each one of these integrals. Hence, β0 , defined in proposition 5.10, is a function of the closed-loop state and, thus β0 is not a time varying function but a nonlinear function of state. Controller in Proposition 5.10 only differs from controller in Proposition 5.5 by the term −αd I˜dq appearing in (5.95). Thus, closed-loop dynamics for Proposition 5.10 is the same as in (5.73)–(5.76) but including the above cited term, i.e., J ω˙˜ = −(b + k p )ω˜ +

n p Me  ˜ I˜ J ψdq − T L , L r dq

L I I˙˜dq = −(R I I2 + α p ) I˜dq + α1 ψ˜ dq − α2 ωJ ˜ ψdq + Udq , L r ψ˙˜ dq = −Rr ψ˜ dq + Rr Me I˜dq + L r (n p ω − n p ω0 )J ψ˜ dq − [0, Rr Me I˜q ] , z˙ = ω, ˜ z˙ 1 = I˜dq ,

(D.1) (D.2) (D.3) (D.4)

n p Me ∗ ˜  J ψ ∗ ] + k z, [Idq J ψdq + I˜dq i dq Lr ∗ ˜ ˜ Udq = −α2 ω J ψdq − αd Idq L r τ˙ ∗  Lr −L I [0, ] + R I [0, (k p ω˜ + ki z)] − αi z 1 , n p Me ψd∗ n p Me ψd∗ TL = −

ε˙ 0 = n p ω0 = n p ω +

Rr Me Iq . L r ψd∗

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 V. M. Hernández-Guzmán et al., Energy-Based Control of Electromechanical Systems, Advances in Industrial Control, https://doi.org/10.1007/978-3-030-58786-4

561

562

Appendix D: Proofs of Propositions for Induction Motors

This closed-loop dynamics is autonomous (variables defining β0 can be expressed as  , z 1 , the addition of one state and a constant) with the state defined as ξ = [ω, ˜ z, I˜dq   ] ∈ R8 . Since term −αd I˜dq is the only difference between (D.1)–(D.4) and ψ˜ dq (5.73)–(5.76) it is easy to verify that ξ = 0 is the only equilibrium point of the closed-loop dynamics (D.1)–(D.4). Thus, using the same arguments as in the proof of Proposition 5.5, the following energy storage function is proposed, which is identical to that introduced in (5.81): 1 ˜ ˜ 1 1 ˜ ˜ L I Idq Idq + ˜ z), ψdq ψdq + z 1 αi z 1 + Vω (ω, 2 2L r 2 1 1 Vω (ω, ˜ z) = J ω˜ 2 + (ki + βk p + bβ)z 2 + β J z ω. ˜ 2 2 V (ξ) =

(D.5)

Taking advantage from the fact that J is positive, it is shown in Appendix A.1 that ˜ z) is positive definite and radially unbounded if: Vω (ω, ki + β(b + k p ) − J β 2 > 0, β > 0.

(D.6)

Hence, V given in (D.5) is positive definite and radially unbounded if αi is a positive definite matrix and (D.6) is satisfied. After several straightforward cancellations, which include those representing the energy exchange pointed out in Remark 5.6, considering the facts that |v  w| ≤ v w , ∀v, w ∈ Rn , ±x y ≤ |x| |y|, ∀x, y ∈ R, and −α3 β0 I˜d2 ≤ 0, where I˜d = Id − Id∗ , it is found that the time derivative of V along the trajectories of the closed-loop dynamics (D.1)–(D.4) can be upper bounded as n p Me ∗ |ω| ˜ Idq

ψ˜ dq − λmin (R I I2 + α p ) I˜dq 2 V˙ ≤ −(k p + b)ω˜ 2 + Lr   L r τ˙ ∗  T 0, +α1 I˜dq ψ˜ dq + α2 |ω ∗ | I˜dq ψ˜dq − L I I˜dq n p Me ψd∗ RI Lr Rr Rr Me ˜ +

I˜dq (k p |ω| ˜ + ki |z|) − 2 ψ˜ dq 2 +

ψdq I˜dq

n p Me ψd∗ Lr L r2 βn p Me βn p Me ∗ |z| I˜dq ψ˜ dq + |z| Idq

ψ˜ dq

+β J ω˜ 2 + Lr Lr βn p Me ∗ + |z| I˜dq ψdq

− βki z 2 . Lr ∗ ∗ Using (D.1), τ˙ ∗ = −k p ω˙˜ − ki ω, ˜ and Idq

≤ Idq

1 = |ψd∗ /Me | + L r (k p |ω| ˜ + ∗ ∗ ki |z| + |τ L + bω |)/(n p Me ψd ), it is possible to write

V˙ ≤ −x  Qx, x = [|ω|, ˜ |z|, ψ˜dq , I˜dq ] , where the entries of matrix Q are given as

(D.7)

Appendix D: Proofs of Propositions for Induction Motors

kp ˜ βki ˜

ψdq , ∗ ψdq , Q 22 = βki − ψd ψd∗ LI kp ˜ LI kp , = λmin (R I I2 + α p ) − ∗ ψdq − J ψd J (ki + βk p ) ˜ = Q 21 = −

ψdq , 2ψd∗   n p Me ψd∗ |τ L + bω ∗ | = Q 31 = − + L r , 2L r Me n p Me ψd∗   βn p Me ψd∗ |τ L + bω ∗ | = Q 32 = − + L r , 2L r Me n p Me ψd∗

Q 11 = b + k p − β J − Q 44 Q 12 Q 13 Q 23

Q 14 = Q 41 = −

L I L r k 2p 2n p Me (ψd∗ )2 J

ψ˜ dq −

563

Q 33 =

Rr , L r2 (D.8)

L I L r k p (k p + b) L I L r ki − ∗ 2n p Me J ψd 2n p Me ψd∗

RI Lr k p , 2n p Me ψd∗ L I L r k p ki L I L r k p ki R I L r ki ˜ = Q 42 = − ∗ 2 ψdq − ∗ − 2n p Me J (ψd ) 2n p Me J ψd 2n p Me ψd∗ ∗ βn p Me ψd βn p Me ˜ −

ψdq − , 2L r 2L r LI kp L I L r k p |τ L + bω ∗ | α2 |ω ∗ | − = Q 43 = −α1 − − . 2 2J Me 2n p Me J (ψd∗ )2 −

Q 24

Q 34

Notice that provided that ψ˜ dq remains inside a ball of small enough radius, the four leading principal minors of Q can be rendered positive by choosing large enough positive values for k p , ki , and a large enough positive definite matrix α p , i.e., a large enough λmin (R I I2 + α p ). Hence, matrix Q can be rendered positive definite and, thus, V˙ ≤ 0. Asymptotic stability follows invoking the LaSalle invariance principle (see Corollary 2.18), i.e., proceeding exactly as in the proof of Proposition 5.5 between (5.83) and (5.84). This result means that aside from ψ˜ dq , which must be small enough, all of the other components of the initial state ξ(t0 ) can be arbitrarily T , I˜d ) subsystem dynamics is analyzed. This will prove large. In the following the (ψ˜ dq to be useful to overcome the restriction on a small ψ˜dq .

˜  , I˜d ) Subsystem Dynamics D.1.1 Stability of the (ψ dq From (D.3) and the first component in (D.2):

564

Appendix D: Proofs of Propositions for Induction Motors

L r ψ˙˜ dq = −Rr ψ˜dq + Rr Me [ I˜d , 0] + L r (n p ω − n p ω0 )J ψ˜ dq , (D.9) ˙ ∗ L I I˜d = −(R I + α p1 ) I˜d − α3 β0 I˜d + α1 ψ˜ d + α2 ω˜ ψ˜q + α2 ω ψ˜q − αi1 z 11 , z˙ 11 = I˜d , where ψ˜d = ψd − ψd∗ , ψ˜q = ψq − ψq∗ , ψq∗ = 0, z 1 = [z 11 , z 12 ] , α p = diag{α p1 , α p2 }, αi = diag{αi1 , αi2 } have been used. Notice that the following scalar function: V1 (ζ) =

1 ˜2 1 1 ˜ ˜ 2 L I Id + αi1 z 11 + ψ ψdq , 2 2 2L r dq

where ζ = [ I˜d , z 11 , ψ˜d , ψ˜q ] , is positive definite and radially unbounded satisfying σ1 ζ 2 ≤ V1 (ζ) ≤ σ2 ζ 2 , 1 1 1 1 σ1 = min{L I , αi1 , }, σ2 = max{L I , αi1 , }. 2 Lr 2 Lr

(D.10)

The time derivative of V1 along the trajectories of (D.9) is given as Rr  ˜ V˙1 = −(R I + α p1 ) I˜d2 − α3 β0 I˜d2 + I˜d (2α1 ψ˜ d + α2 ω˜ ψ˜q + α2 ω ∗ ψ˜q ) − 2 ψ˜ dq ψdq , Lr ≤ −η  Pη, η = [ |ψ˜q |, |ψ˜ d |, | I˜d |] , ⎤ ⎡ R r 0 − α22|ω| L r2 ⎥ ⎢ Rr P=⎣ 0 −α1 ⎦, L r2 − α22|ω| −α1 R I + α p1 + α3 β0 where the fact that ±x y ≤ |x| |y|, ∀x, y ∈ R has been used. Matrix P is positive definite if and only if its three leading principal minors are positive, i.e., Rr > 0, L r2



Rr L r2

2 (R I + α p1 + α3 β0 ) −

Rr Rr α22 ω 2 − α12 2 > 0. 2 Lr 4 Lr

(D.11)

The first condition is naturally satisfied by the motor parameter properties, whereas the second condition is satisfied if: α3 >

L r2 α22 L2 , α p1 > r α12 − R I , 4Rr Rr

(D.12)

since β0 ≥ ω 2 . This implies that λmin (P) > 0 as long as (D.12) be satisfied and, hence, it is possible to write V˙1 ≤ −λmin (P) η 2 ≤ 0.

(D.13)

Appendix D: Proofs of Propositions for Induction Motors

565

This and (D.10) mean that ζ = 0 is uniformly stable (see Theorem 2.26), i.e., that ψ˜ d , ψ˜q , I˜d , z 11 ∈ L∞ . Furthermore, since η 2 = ψ˜q2 + ψ˜ d2 + I˜d2 , then V˙1 ≤ −λmin (P)φ2 ≤ 0, where φ is any of ψ˜q , ψ˜ d or I˜d . Time integration of the above inequality yields 

t

V1 (t) − V1 (t0 ) ≤ −λmin (P)

φ2 (t)dt,

t0

or, equivalently: 

t

V1 (t0 ) − V1 (t) ≥ λmin (P)

φ2 (t)dt,

t0



t

V1 (t0 ) ≥ λmin (P)

φ2 (t)dt,

t0

because V1 (t) ≥ 0, ∀t ≥ t0 . Thus  ∞>

V1 (t0 ) ≥ λmin (P)



t

φ2 (t)dt.

t0

This proves that ψ˜q , ψ˜d , I˜d ∈ L2 . From (5.50), (5.96), (5.68), it is clear that n p ω − n p ω0 is bounded if I˜q and Iq∗ are bounded which is ensured if the load torque τ L and the desired velocity ω ∗ are bounded and ξ ∈ L∞ . Hence, from (D.9) it can be seen that ψ˙˜ d , ψ˙˜ q , I˙˜d ∈ L∞ . Thus, Corollary 2.32 (Barbalat’s Lemma) can be invoked to conclude that limt→∞ [ψ˜q (t), ψ˜ d (t), I˜d (t)] = [0, 0, 0] , from any initial [ψ˜q (t0 ), ψ˜ d (t0 ), I˜d (t0 )], if (D.12) is true and ξ ∈ L∞ . Moreover in Appendix D.1.2 is proven that ψ˜ dq converges in finite time into a ball whose radius decreases to zero if ξ → ∞.

˜  , I˜d ) D.1.2 Additional Stability Properties of the (ψ dq Subsystem Dynamics Consider the following dynamics: L r ψ˙˜ dq = −Rr ψ˜ dq + Rr Me [ I˜d , 0] + L r (n p ω − n p ω0 )J ψ˜ dq ,

(D.14)

which is present in (D.9), together with the following positive definite and radially unbounded scalar function:

566

Appendix D: Proofs of Propositions for Induction Motors

V2 (ψ˜ dq ) =

1 ˜ ˜ ψ ψdq . 2L r dq

Taking advantage from the fact that matrix J is skew symmetric and using |y  w| ≤

y w , ∀y, w ∈ Rn , it is found that the time derivative of V2 along the trajectories of (D.14) can be bounded as Rr  ˜ Rr Me ˜

ψdq | I˜d |. ψdq + V˙2 ≤ − 2 ψ˜ dq Lr L r2 Adding and subtracting term

Rr  ˜ Θ ψ˜ dq ψdq , L r2

for some constant 0 < Θ < 1, yields

Rr Me ˜  ˜ | Id |. V˙2 ≤ − 2 (1 − Θ)ψ˜ dq ψdq , ∀ ψ˜ dq ≥ Lr Θ Recall that I˜d was proven to be bounded after (D.12). Notice that c1 ψ˜dq 2 ≤ V2 (ψ˜ dq ) ≤ c2 ψ˜ dq 2 for some 0 < c1 ≤ 2L1 r ≤ c2 . Thus, Theorems 2.29 and 2.28 can be invoked to conclude that for any ψ˜ dq (t0 ) ∈ R2 there exists a finite T0 ≥ 0 (depending on ψ˜ dq (t0 ) and Me | I˜d |) such that Θ

(

c c2 ˜ − 3 (t−t )

ψdq (t0 ) e 2c2 0 , ∀ t0 ≤ t ≤ t0 + T0 c1 ( c 2 Me ˜

ψ˜ dq (t) ≤ | Id |, ∀ t ≥ t0 + T0 , c1 Θ

ψ˜ dq (t) ≤

where c3 =

Rr (1 L r2

(D.15)

− Θ) > 0. On the other hand, consider the following dynamics:

L I I˙˜d = −(R I + α p1 ) I˜d − α3 β0 I˜d + α1 ψ˜d + α2 ω˜ ψ˜q + α2 ω ∗ ψ˜q − αi1 z 11(D.16) , which is also present in (D.9), together with the following positive definite and radially unbounded scalar function: 1 V3 ( I˜d ) = L I I˜d2 . 2 √ Using the facts that ±wv ≤ |w| |v|, for all w, v ∈ R, and y 1 ≤ n y , for all y ∈ Rn , the triangle inequality, and adding and subtracting term Θ(R I + α p1 + α3 β0 ) I˜d2 , 0 ≤ Θ ≤ 1, it is found that the time derivative of V3 along the trajectories of (D.16) can be upper bounded as √ max(α1 , α2 |ω|, αi1 ) 3 ˜ 

[ψdq , z 11 ] = μ0 . V˙3 ≤ −(1 − Θ)(R I + α p1 ) I˜d2 , ∀| I˜d | ≥ Θ(R I + α p1 + α3 β0 )

(D.17)

Appendix D: Proofs of Propositions for Induction Motors

567

 Recall that μ0 is bounded since [ψ˜ dq , z 11 ] was proven to be bounded after (D.12) 2 and β0 ≥ ω . Notice that it is possible to write d1 I˜d2 ≤ V3 ( I˜d ) ≤ d2 I˜d2 for some 0 < d1 ≤ 21 L I ≤ d2 . Thus, Theorems 2.29 and 2.28 can be invoked again to conclude that for any I˜d (t0 ) ∈ R there exists a finite T1 ≥ 0 (depending on I˜d (t0 ) and μ0 ) such that  d d2 ˜ − 3 (t−t ) | Id (t0 )|e 2d2 0 , ∀ t0 ≤ t ≤ t0 + T1 (D.18) | I˜d (t)| ≤ d1  d2 μ0 , ∀ t ≥ t0 + T1 , | I˜d (t)| ≤ d1

where d3 = (1 − Θ)(R I + α p1)) > 0. Expressions in (D.18) indicate that I˜d (t) con-

verges into a ball of radius dd21 μ0 in finite time. Note that radius of this ball decreases to zero as ξ tends to increase without limit (ξ is defined in paragraph after (D.4)) because β0 (which is greater or equal than ω 2 ) increases if t t ˜ 2 ˜ ˜ 0 ω(s)ds]

˜ increases1 . Once this accomplished, expressions

[ Iq , 0 Iq (s)ds, ω, ) in (D.15) indicate that ψdq (t) converges in finite time into a ball of radius c2 Me | I˜d | which decreases to zero as ξ tends to grow without limit.

c1 Θ

D.1.3 Boundedness of the State and Global Convergence to the Origin in the Complete Closed-Loop System

According to the above analysis the following is the scenario: B1 ξ ∈ L8∞ and limt→∞ ξ(t) = 0, from any initial condition ξ(t0 ), if ψ˜ dq (t) remains inside a ball whose radius is small enough. B2 [ψ˜q (t), ψ˜ d (t), I˜d (t)] is bounded, ψ˜q , ψ˜d , I˜d ∈ L2 and limt→∞ [ψ˜q (t), ψ˜ d (t), I˜d (t)] = [0, 0, 0] from any initial condition [ψ˜q (t0 ), ψ˜d (t0 ), I˜d (t0 ), z 11 (t0 )], if ξ ∈ L8∞ . Moreover ψ˜dq converges in finite time into a ball whose radius decreases to zero if ξ → ∞, from any initial condition [ψ˜q (t0 ), ψ˜d (t0 ), I˜d (t0 ), z 11 (t0 )]. Assume that [ψ˜d , ψ˜q ] is not small enough, then from (D.7) we have that

   , z ] remains bounded and  t I˜ (s)ds  → |z | as |z | grows since that [ I˜d , ψdq  0 q  11 12 12  * ∗ +  ψd Lr ∗ + α ψ∗ − R ∗ (αi )−1 −α2 ω ∗ J ψdq is finite and, because of simi1 dq I Me , n p Me ψ ∗ (τ L + bω ) d    t  lar reasons,  0 ω(s)ds ˜ ˜ as |ω| ˜ grows  → |z| as |z| grows and |ω| → |ω|

1 Recall

568

Appendix D: Proofs of Propositions for Induction Motors

V˙ ≤ −x T Qx ≤ λmax (−Q) x 2 ≤ κ3 ξ 2 , where κ3 = λmax (−Q) > 0.(D.19) On the other hand, notice that Lyapunov function in (D.5) satisfies (D.20) κ1 ξ 2 ≤ V (ξ) ≤ κ2 ξ 2 , 1 κ1 = min{L I , λmin (αi ), 1/L r , λmin (P1 )} > 0, 2 1 κ2 = max{L I , λmax (αi ), 1/L r , λmax (P2 )} > 0,   2  J −β J J βJ , P2 = . P1 = −β J β(k p + b) + ki β J β(k p + b) + ki Hence, it is possible to proceed as in the proof of the exponential ) stability theorem (see κ3 (t−t ) Theorem 2.28) to prove, using (D.19) and (D.20), that ξ(t) ≤ κκ21 ξ(t0 ) e 2κ1 0
t0 . Assume that the worst case is present, i.e., that ξ(t)

tends to grow without limit. In such a case B2 ensures that [ψ˜q (t), ψ˜ d (t)] < ε, for any ε > 0, in finite time, i.e., before ξ(t) escapes. Thus, by virtue of B1, ξ(t) will converge asymptotically to zero. In the case when ξ(t) does not tend to grow without limit, then ξ(t) will remain bounded until, by virtue of B2, [ψ˜q (t), ψ˜ d (t)]

reaches and stays within a small enough value and, hence, ξ(t) will converge asymptotically to zero. This proves boundedness of the state and convergence to the origin globally. On the other hand, it is clear from (D.17) that α3 can always be chosen large enough to satisfy simultaneously (D.12) and 

[ψ˜dq (0), I˜d (0), z 11 (0)]

α3

< μ0 ,

 ˜ with μ0 > 0 arbitrarily small. Then, since the origin of [ψ˜ dq , Id , z 11 ] is uniformly stable and β0 ≥ ω 2 , we can always choose a large enough α p1 to simultaneously render positive the fourth leading principal minor of matrix Q defined in (D.7) and such that μ0 , defined in (D.17), be arbitrarily small for all t ≥ 0. According to (D.15) and (D.18), this ensures that ψ˜dq reaches an arbitrarily small value in finite time    and, since ξ = [ω, ˜ z, I˜dq , z 1 , ψ˜dq ] ∈ R8 has not a finite escape time, this ensures   that the origin of ξ is asymptotically stable, semiglobally in [ I˜d , z 11 , ψ˜dq ] and globally in the remaining variables. This completes the proof of Proposition 5.10. Conditions for this result are summarized by (D.12), β > 0, ki + bβ + βk p > J β 2 , αi is a positive definite 2 × 2 matrix, some large enough constants k p > 0, ki > 0, and a positive definite 2 × 2 matrix α p such that the four leading principal minors of matrix Q, defined in (D.8), be positive.

Appendix D: Proofs of Propositions for Induction Motors

569

Remark D.1 Notice that aside from ψ˜dq all entries of matrix Q are expressed in terms of constants. Recall that ψ˜ dq is always bounded even if ξ(t) tends to grow without limit (ζ = 0 is proven to be uniformly stable after (D.12) and, thus, ψ˜ dq is bounded). Hence, all entries of matrix Q are bounded. According to Theorem 2.13, this means that all of the eigenvalues of matrix −Q are bounded. Thus, κ3 defined in (D.19) always exists. Remark D.2 Suppose that the following expression is employed instead of (5.95) in Proposition 5.10: ∗ ∗ ∗ − α p I˜dq + α2 ω ∗ J ψdq − α1 ψdq − αd I˜dq , Vdq = R I Idq

(D.21)

t ∗ ∗ + α2 ω ∗ J ψdq − i.e., the integral term −αi 0 I˜dq (s)ds in (5.95) is replaced by R I Idq ∗ α1 ψdq , and all of the remaining variables are defined to be the same as in Proposition 5.10. Furthermore, recall from Sect. 5.1.4 that L r2 Rs + Rr Me2 Rr Me , α1 = . 2 Lr L r2

RI =

∗ ∗ This means that term −α1 ψdq cancels with a part of term R I Idq , i.e., we can employ a simpler expression for (D.21) which, however, is equivalent. Despite this fact, let us suppose that (D.21) is employed. Then, proceeding as in Sect. D.1.1 we find that V˙1 given in (D.13) is negative definite in this case and, hence, [ψ˜ d , ψ˜q , I˜d ] = [0, 0, 0] is globally uniformly asymptotically stable (see Theorem 2.26). Moreover, from Theorem 2.28 we conclude that [ψ˜ d , ψ˜q , I˜d ] = [0, 0, 0] is globally exponentially stable. Furthermore, since λmax (−Q) in (D.7), (D.19), is positive only for large enough ψ˜ dq , this also means that λmax (−Q) decreases exponentially to some negative value as ψ˜ dq → 0. Since it has been proven that  ˜  the closed-loop state has not finite escape time, this means that [ω, ˜ z, ψ˜dq , Idq ] →  [0, 0, 0, 0, 0, 0] from any initial condition. These ideas also imply that for each   (0), I˜dq (0)] < δ ⇒ ε > 0 there is some δ = δ(ε) > 0 such that [ω(0), ˜ z(0), ψ˜dq   

[ω(t), ˜ z(t), ψ˜ dq (t), I˜dq (t)] < ε for all t ≥ 0, i.e., that the origin is stable according to Definition 2.5. Thus, we conclude that the origin is ensured to be globally asymptotically stable. This completes the proof of Proposition 5.17.

D.2 Proof of Proposition 5.20  First notice that, since both positive scalar ε such that

Lr n p Me ψd∗

 ε



and

Lr n p Me ψd∗



Lr n p Me ψd∗

=

are positive, there always exists a

Lr . n p Me ψd∗

570

Appendix D: Proofs of Propositions for Induction Motors

Hence, defining k p = εk p , kd = εkd and ki = εki , for some positive scalars k p , kd , ki , (5.108) and (5.109) can be written as ∗ Idq

 =

ψd∗ 1 L r , τ∗ Me ε n p Me ψd∗



τ ∗ = −εk p q˜ − εkd q˙ − εki

,



t

q(s)ds, ˜

0

i.e., 

∗ Idq

=

ψd∗ Lr , τ Me n p Me ψd∗

τ  = −k p q˜ − kd q˙ − ki

 

, t

q(s)ds. ˜

(D.22) (D.23)

0

According to (D.22) and (5.27): ∗ ∗ J ψdq = Idq

Lr  τ. Me n p

(D.24)

Adding and subtracting some convenient terms and using (D.24) it can be written: n p Me  I J ψdq = τc + τ  , L r dq n p Me ˜ ˜ ∗  ∗ τc = [ Idq J ψdq + Idq J ψ˜ dq + I˜dq J ψdq ]. Lr

(D.25)

On the other hand, defining k p = k p + ki , ki = ki /α, with k p , ki , α positive constants, and  z=

t

(αq˜ + q)ds ˙ + q(0) ˜ +

0

1 g(q ∗ ), ki

(D.26)

it is possible to write (D.23) as τ  = −k p q˜ − kd q˙ − ki z + g(q ∗ ).

(D.27)

Using (D.25)–(D.27), in the first expression in (5.102) yields J q¨ + bq˙ =

n p Me ˜ ˜ ∗  ∗ [ Idq J ψdq + Idq J ψ˜ dq + I˜dq J ψdq ] Lr −k p q˜ − kd q˙ − ki z + g(q ∗ ) − g(q).

(D.28)

Appendix D: Proofs of Propositions for Induction Motors

571

Now, replacing (5.106), (5.107), in the second expression in (5.102), using (D.22), (D.27), adding and subtracting some convenient terms and defining: 

t

z1 =

 I˜dq (s)ds − (αi )

−1

0

∗ α1 ψdq

 − RI

ψd∗ Lr , g(q ∗ ) Me n p Me ψd∗

 

, (D.29)

yields  L I I˙˜dq = −(R I I2 + α p ) I˜dq + α1 ψ˜ dq − α2 qJ ˙ ψdq + L I 0,  +R I 0,

Lr (k  q˜ + kd q˙ + ki z) n p Me ψd∗ p ˙ τ˙ ∗ = k p q˙ + kd q¨ + ki (αq˜ + q),



L r τ˙ ∗ n p Me ψd∗

− αi z 1 − αd I˜dq ,



(D.30) (D.31)

∗ and I2 stands for the 2 × 2 identity matrix. Finally, adding where ψ˜ dq = ψdq − ψdq ∗ ∗ ∗ /L r , Rr Me Idq /L r and (n p q˙ − n p ω0 )J ψdq in the third and subtracting terms Rr ψdq ∗ expression in (5.102), taking advantage from the fact that ψ˙ dq = 0 and using (D.22) and the fourth expression in (5.102), the following is obtained:

L r ψ˙˜ dq = −Rr ψ˜ dq + Rr Me I˜dq + L r (n p q˙ − n p ω0 )J ψ˜ dq − [0, Rr Me I˜q ] , Lr τ  . (D.32) I˜q = Iq − n p Me ψd∗ Notice that the closed-loop dynamics, which is given by (D.26), (D.28)–(D.32), can be rewritten as follows: n p Me ˜ ˜ I J ψdq − G(q), L r dq = −(R I I2 + α p ) I˜dq + α1 ψ˜ dq − α2 qJ ˙ ψdq + Udq ,

J q¨ + (b + kd )q˙ =

L I I˙˜dq L r ψ˙˜ dq = −Rr ψ˜ dq + Rr Me I˜dq + L r (n p q˙ − n p ω0 )J ψ˜dq −[0, Rr Me I˜q ] ,

(D.33) (D.34)

(D.35)

(D.36) z˙ = αq˜ + q, ˙ z˙ 1 = I˜dq , n p Me ∗ ˜  ∗ G(q) = − [Idq J ψdq + I˜dq J ψdq ] + k p q˜ + ki z − g(q ∗ ) + g(q), Lr     L r τ˙ ∗  Lr    + R I 0, (k q˜ + kd q˙ + ki z) Udq = L I 0, n p Me ψd∗ n p Me ψd∗ p −αi z 1 − αd I˜dq , ε˙0 = n p ω0 = n p q˙ +

Rr Me Iq . L r ψd∗

(D.37)

572

Appendix D: Proofs of Propositions for Induction Motors

The closed-loop dynamics (D.26), (D.28)–(D.32), or equivalently (D.33)-(D.37), is    , z 1 , ψ˜dq ] ∈ R9 . Equilibria autonomous with the state defined as ξ = [q, ˜ q, ˙ z, I˜dq are computed as follows. From q˙˜ = q, ˙ and (D.36) we have that q˙ = 0, q˜ = 0 and I˜dq = 0 at the equilibrium point. The use of these values and evaluating (D.35) at the equilibrium condition yields 0 = −Rr ψ˜ dq + L r (−n p ω0 )J ψ˜dq . Using (D.37) with Iq = becomes

Lr τ  , n p Me ψd∗

because I˜dq = 0, it is found that last expression

Lr τ ∗ J ψ˜ dq ψ˜ dq = − n p ψd∗2 which, according to (5.27), implies that ψ˜ dq = 0. Using all of these results in (D.33) yields z = 0. Finally, from (D.34), z 1 = 0 follows. Thus, ξ = 0 is the only equilibrium point of the closed-loop dynamics (D.33)–(D.37). Notice that (D.33)–(D.37), are almost identical to (5.57) if we replace Idq , ψdq , g(q), Udq by I˜dq , ψ˜ dq , G, Udq , respectively. An important difference is that the electric resistance and the viscous friction have been enlarged in the closed-loop dynamics, i.e., we have R I I2 + α p and b + kd in (D.34) and (D.33) instead of R I and b in (5.57). Another important difference is the presence of two additional equations in (D.36) which represent the integral parts of both, the PID position controller and the PI electric current controller, respectively. These observations motivate the use of the following “energy” storage function for the closed-loop dynamics: V (ξ) = Vq (q, ˜ q, ˙ z) = ˜ q) ˙ = V1 (q,

˙ = V2 (z, q) ˜ = V3 (z, q)

1 ˜ ˜ 1 1 ˜ ˜ L I Idq Idq + ˜ q, ˙ z), (D.38) ψdq ψdq + z 1 αi z 1 + Vq (q, 2 2L r 2 V1 (q, ˜ q) ˙ + V2 (z, q) ˙ + V3 (q, ˜ z), 1 α 1 J (q˙ + 2αq) ˜ 2 − α2 J q˜ 2 + (b + kd )q˜ 2 + k p q˜ 2 4 2 2 ∗ +U (q) − U (q ∗ ) − qg(q ˜ ), 1 αβ  2 1  2 J (q˙ + 2αβz)2 − α2 β 2 J z 2 + k z + ki z , 4 2 d 2 β  (k − αkd )(z − q) ˜ 2. 2 p

The reason for the three first terms in (D.38) is to take into account the magnetic “energy” in the electrical subsystem of both the stator and the rotor and “energy” ˜ q, ˙ z) was due to the integral part of the PI electric current controller. Function Vq (q, introduced in (3.58) and the reader is referred to that part of Chap. 3 for an explanation ˜ q, ˙ z). Conditions and an energy interpretation of the terms that are present in Vq (q, ˜ q, ˙ z) is positive definite and radially unbounded are given in ensuring that Vq (q, Appendix A.2. Notice that, additionally, L I > 0, L r > 0, and αi a diagonal positive

Appendix D: Proofs of Propositions for Induction Motors

573

definite matrix are required for positive definiteness and radially unboundedness of V (ξ) in (D.38). After several straightforward cancellations, which include energy exchange cancellations similar to those presented in (5.52)–(5.53) and a procedure similar to that used to obtain (3.59), it is found that the time derivative of V (ξ) in (D.38) along the trajectories of the closed-loop dynamics (D.33)-(D.37) can be upper bounded as + n p Me ∗ ˜ n p Me * ˜ ˜ ∗  ∗ Idq J ψdq + Idq Idq J ψdq + αq˜ J ψ˜ dq + I˜dq J ψdq V˙ ≤ −ρ Qρ + q˙ Lr Lr + n p Me * ˜ ˜ ∗  ∗ Idq J ψdq + Idq J ψ˜dq + I˜dq J ψdq +αβz L , r   Lr     (k q ˜ + k q ˙ + k z) + I˜dq −(R I I2 + α p ) I˜dq + α1 ψ˜ dq + R I 0, d i n p Me ψd∗ p   Lr    +L I 0, (k q˙ + kd q¨ + ki (αq˜ + q)) ˙ − α3 β0 I˜d2 n p Me ψd∗ p   Rr ˜ 1 ˜ Rr Me ˜  + ψdq − ψdq + [ Id , 0] , Lr Lr Lr where ρ = [|q|, ˙ |q|, ˜ |z|] and entries of matrix Q are given as Q 11 = b + kd − αJ − αβ J, Q 33 = αβki ,

Q 22 = αk p − αkg + αβk p − α2 βkd ,

Q 12 = Q 21 = −

Q 23 = Q 32 = −

αβkg . 2

α2 β J , 2

Q 13 = Q 31 = −

αβb , 2

(D.39)

Considering the facts that |v  w| ≤ v w , ∀v, w ∈ Rn , ±x y ≤ |x| |y|, ∀x, y ∈ R, ∗ , q, ¨ it is −α3 β0 I˜d2 ≤ 0, where I˜d = Id − Id∗ , and replacing the expressions for Idq ˙ found that V can be upper bounded as V˙ ≤ −x  P x, x = [|q|, ˙ |q|, ˜ |z|, ψ˜dq , I˜dq ] ,

(D.40)

where the entries of matrix P are given as P11 = Q 11 − P44 =

Rr , L r2

kd ˜

ψdq , ψd∗

P22 = Q 22 −

P55 = λm (R I I2 + α p ) −

P12 = P21 = Q 12 −

(αkd + k p ) 2ψd∗

αk p ψd∗

ψ˜ dq ,

P33 = Q 33 −

n p Me f ˜ ∗ ( ψdq + ψdq

), Lr

ψ˜ dq ,

f =

L I L r kd , J n p Me ψd∗

αβki ˜

ψdq , ψd∗ (D.41)

574

Appendix D: Proofs of Propositions for Induction Motors

(αβkd + ki ) ˜

ψdq , 2ψd∗ (αβk p + αki ) P23 = P32 = Q 23 −

ψ˜ dq , 2ψd∗

P13 = P31 = Q 13 −

P14 = P41 = P24 = P42 = P34 = P43 = −

n p Me ˜ ∗ − ( ψdq + ψdq

) ∗ 2n p Me ψd 2L r α2 f L r kd ˜ f (kd + b) −

ψdq , − 2 2n p Me ψd∗ L r (R I k p + αL I ki ) αn p Me ∗ P52 = − − ( ψ˜ dq + ψdq

) 2n p Me ψd∗ 2L r α2 f L r k p f (k p + kg ) −

ψ˜dq , − 2 2n p Me ψd∗ L r R I ki αβn p Me ˜ ∗ P53 = − ( ψdq + ψdq

) ∗ − 2n p Me ψd 2L r α2 f L r ki ˜ f k 

ψdq , − i − 2 2n p Me ψd∗ f α2 ψd∗ α2 f L r k  P54 = −α1 − − . 2Me 2n p Me ψd∗

P15 = P51 = −

P25 =

P35 =

P45 =

L r (R I kd + L I (k p + ki ))

ψd∗ Lr k − , 2Me 2n p Me ψd∗

Notice that provided that ψ˜ dq remains inside a ball of small enough radius, the five leading principal minors of P can be rendered positive by choosing large enough positive values for k p , kd , ki , and a large enough positive definite matrix α p , i.e., a large enough λmin (R I I2 + α p ). Hence, matrix P can be rendered positive definite and, thus, V˙ ≤ 0. Asymptotic stability follows invoking the LaSalle invariance principle (see Corollary 2.18), i.e., proceeding similarly as in the proof of Proposition 5.5 between (5.83) and (5.84). This result means that aside from ψ˜ dq , which must be small enough, all of the other components of the initial state ξ(t0 ) can be arbitrary large. Thus, it is concluded that B1 ξ ∈ L8∞ and limt→∞ ξ(t) = 0, from any initial condition ξ(t0 ), if ψ˜ dq (t) remains inside a ball whose radius is small enough.  ˜ Notice that the dynamics (ψ˜ dq , Id ) in (D.33)–(D.37) is identical to (5.98). Thus, the analysis presented in Sects. D.1.1 and D.1.2 is also valid in the present case which allows to conclude that

B2 [ψ˜q (t), ψ˜ d (t), I˜d (t)] is bounded, ψ˜q , ψ˜d , I˜d ∈ L2 and limt→∞ [ψ˜q (t), ψ˜ d (t), I˜d (t)] = [0, 0, 0] from any initial condition [ψ˜q (t0 ), ψ˜d (t0 ), I˜d (t0 ), z 11 (t0 )], if ξ ∈ L8∞ . Moreover ψ˜dq converges in finite time into a ball whose radius decreases to zero if ξ → ∞, from any initial condition [ψ˜q (t0 ), ψ˜d (t0 ), I˜d (t0 ), z 11 (t0 )].

Appendix D: Proofs of Propositions for Induction Motors

575

Thus, the reasoning in Sect. D.1.3 is still valid in this case and this proves boundedness of the state and global convergence to the origin. Moreover, this also ensures that    , z 1 , ψ˜dq ] ∈ R9 is asymptotically stable, semiglobally the origin of ξ = [q, ˜ q, ˙ z, I˜dq   in [ I˜d , z 11 , ψ˜dq ] and globally in the remaining variables. This completes the proof of Proposition 5.20. Conditions for this result are summarized by (D.12), (A.3)–(A.6), αi is a positive definite matrix, and the five leading principal minors of matrix P defined in (D.40) are positive.

Appendix E

Proofs of Propositions for Switched Reluctance Motors

E.1 A Function to Replace

√

|τ ∗ |, ∀|τ ∗ | ≤ T ∗

√ In Fig. E.1 we compare function f (|τ ∗ |) defined in (6.44) to |τ ∗ | for |τ ∗ | ≤ T ∗ . In order to render continuous up to the first derivative this substitution, the positive constants ω and α have to be chosen such that the following conditions be satisfied: α(1 − cos(ωT ∗ )) =

√

T ∗ , i.e., f (|τ ∗ |) = 1 d f (|τ ∗ |) = αω sin(ωT ∗ ) = √ , i.e., d|τ ∗ | 2 T∗

|τ ∗ | at τ ∗ = T ∗ , √ d |τ ∗ | at τ ∗ = T ∗ . d|τ ∗ |

From these expressions we retrieve (6.45) and (6.46).

E.2 Proof of Proposition 6.1 After some algebraic manipulations, it is found that the time derivative of V (y), y = [ω, ˜ z d , ϑ, ξ1 , ξ2 , ξ3 ] ∈ R6 , defined in (6.58), along the trajectories of the closedloop system (6.55)–(6.57) is given as 1 V˙ = −bω˜ 2 + β2 F(τ ∗ )ω˜ + γχξ T K (q)ξ + γχξ T K (q)I ∗ + γ K v χσ2 (ϑ) − γ K p χ2 2 ∂σ1 (z) 2 K v A 2 ω˜ − σ (ϑ) − ξ T αξ +γχβ2 F(τ ∗ ) + γ J ¯ + ξ T D(q)(h − I˙∗ ) ∂z B 2 1 − ω ∗ ξ T K (q)ξ. (E.1) 2

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 V. M. Hernández-Guzmán et al., Energy-Based Control of Electromechanical Systems, Advances in Industrial Control, https://doi.org/10.1007/978-3-030-58786-4

577

578

Appendix E: Proofs of Propositions for Switched Reluctance Motors

Fig. E.1 Comparison of the function √ f (|τ ∗ |) defined in (6.44) to |τ ∗ |.

It is important to say that several straightforward cancellations are involved in the above referred algebraic procedure which include the cancellations due to the energy exchange referred in Sect. 6.2.1. Using (2.32) and upper bounding each term in (E.1), it is not difficult to find that V˙ is bounded as γ V˙ ≤ −(b − γ J )ω˜ 2 + β2 F(τ ∗ )ω˜ + K (q) (M1 + d) ξ 2 2 +γ K (q)

I ∗ |χ| ξ + γ K v |χ||σ2 (ϑ)| Kv A 2 |σ2 (ϑ)|2 − λmin {α} ξ

¯ −γ K p |χ|2 + γβ2 |χ|F(τ ∗ ) − B 1 + ω ∗ K (q)

ξ 2 + ξ

D(q)

(h − I˙∗ ) . 2 Using (6.43), the fact that x ≤ x 1 and the triangle inequality, we can write

D(q)

(h − I˙∗ ) ≤ ≤ λmax {D(q)}(ga + β2 gb f (|τ ∗ |)), (E.2) ,     3 ∗   ∂m i (q) τ   m i (q) ∂ K i (q) ∗  K i (q)     ga ≤  ∂q K (q)  +  K 2 (q) ∂q τ  ∗ 2m (q)τ i i i i=1    ∂ f (|τ ∗ |) 2m i (q)sign(τ ∗ ) + max (K v B + K p ) |ω| ˜ τ∗ K i (q) ∂τ ∗    3  ∂ f (|τ ∗ |) 2m i (q)sign(τ ∗ ) K v A|σ2 (ϑ)|, max + τ∗ K i (q) ∂τ ∗ i=1      3  K i (q)sign(τ ∗ )  ∂m i (q) 1   m i (q) ∂ K i (q)  ∗ gb ≤  ∂q K (q)  +  K 2 (q) ∂q  |ω |, 2m (q) i i i i=1

Appendix E: Proofs of Propositions for Switched Reluctance Motors

579

where we have used: ∂σ2 (ϑ) ∂σ1 (z) [−Aσ2 (ϑ) − B ω] ω, ˜ ˜ − Kp σϑ ∂z |τ˙ ∗ | ≤ K v [A|σ2 (ϑ)| + B|ω|] ˜ + K p |ω|. ˜ τ˙ ∗ = K v

(E.3)

According to Assumption A1 on the sharing functions m i (q), introduced  before the   f (|τ ∗ |)  Proposition 6.1, and the definition of f (|τ ∗ |) in Proposition 6.1,  ∂ ∂|τ ∗ |  is bounded ) and so is τ1∗ provided that T ∗ > 0. Hence, aside from |ω|, ˜ all terms in (E.2) are bounded. Hence, there always exist finite positive constants d1 and d2 , which are independent of α¯ and α, such that we can write V˙ ≤ − y¯  Q y¯ + β2 Γ1 Γ2 y¯ , y¯ = [|ω|, ˜ |χ|, |σ2 (ϑ)|, ξ ] , . / Γ1 = max F(τ ∗ ), f (|τ ∗ |) , Γ2 = (1 + γ + λmax {D(q)}gb ),

(E.4)

where the elements of Q are given as Kv A , Q 22 = γ K p , Q 33 = B  γ |ω ∗ | , λmin {α} ¯ − K (q) M1 + d + 2 γ Q 21 = Q 13 = Q 31 = 0, 1 Q 41 = − λmax {D(q)}d1 , 2 γ Q 42 = − max |λi {K (q)}| I ∗ M , 2 i 1 Q 43 = − λmax {D(q)}d2 , 2 γ Q 32 = − K v . 2

Q 11 = (b − γ J ), Q 44 = Q 12 = Q 14 = Q 24 = Q 34 = Q 23 =

Symbol I ∗ M represents the supreme value over the norm of I ∗ and, therefore, Q 24 and Q 42 are bounded since I ∗ is a bounded vector. Then Q is positive definite if and only if all of their leading principal minors are positive, that is

580

Appendix E: Proofs of Propositions for Switched Reluctance Motors

b − γ J > 0,

(E.5)

γ K p > 0, Kv A Q 11 Q 22 − Q 23 Q 11 Q 32 > 0, d3 = B d3 Q 44 − Q 14 Q 41 [Q 22 Q 33 − Q 23 Q 32 ] +Q 24 Q 11 [Q 32 Q 43 − Q 33 Q 42 ] − Q 34 Q 11 [Q 22 Q 43 − Q 23 Q 42 ] > 0.

(E.6) (E.7) (E.8)

Third and fourth leading principal minors of Q are represented by (E.7) and (E.8), respectively. We recall that all entries of matrix Q are bounded if T ∗ > 0. Also notice ¯ Therefore, we have only to enlarge α¯ that Q 44 is the only entry of Q depending on α. until the remaining terms in (E.8) be dominated. Then, provided that T ∗ > 0, all of conditions (E.5)–(E.8) are always satisfied by some small γ, some suitably selected ¯ that is, a positive numbers K p , K v , A, B, and a large enough positive value for α, large positive value for α. This ensures that Q is positive definite, i.e., λmin (Q) > 0. Using some constant 0 < Θ < 1, we can rewrite (E.4) as ˜ χ, σ2 (ϑ), ξ1 , ξ2 , ξ3 ] , V˙ ≤ −λmin {Q} μ 2 + β2 Γ1 Γ2 μ , μ = [ω, ≤ −(1 − Θ)λmin {Q} μ 2 − Θλmin {Q} μ 2 + β2 Γ1 Γ2 μ , β2 Γ1 Γ2 . ≤ −(1 − Θ)λmin {Q} μ 2 , ∀ μ ≥ Θλmin {Q} According to paragraph after (6.51), |F(τ ∗ )| and f (|τ ∗ |) are upper bounded and these bounds approach to zero as T ∗ > 0 approaches to zero. Hence, it is always possible β2 Γ1 Γ2 to lay in the linear parts of all components to render the inequality μ ≥ Θλ min {Q} of vector μ. Thus, it is always possible to write V˙ ≤ −(1 − Θ)λmin {Q} μ 2 , ∀ y ≥

β2 Γ1 Γ2 . Θλmin {Q}

(E.9)

Now, we are ready to prove Proposition 6.1. Taking into consideration (6.60) and (E.9), we can invoke Theorem 2.29 to conclude that for any initial state y(t0 ) ∈ R6 , the closed-loop system state y = [ω, ˜ z d , ϑ, ξ1 , ξ2 , ξ3 ] ∈ R6 satisfies

y(t) ≤ δ( y(t0 ) , t − t0 ), ∀ t0 ≤ t ≤ t0 + T,    β2 Γ1 Γ2 , ∀ t ≥ t0 + T,

y(t) ≤ α1−1 α2 Θλmin {Q} where δ(·, ·) is a KL function and T ≥ 0 depends on y(t0 ) and β2 Γ1 Γ2 Θλmin {Q}

β2 Γ1 Γ2 . Θλmin {Q} ∗

(E.10) According

to (6.44) and paragraph after (6.51), decreases to zero as T > 0 approaches −1 to zero. Since α1 (α2 (·)) is a K∞ function, then the ultimate bound in (E.10) tends to zero as T ∗ > 0 approaches to zero. Notice that, as it was explained in paragraph after (E.8), finite controller gains satisfying (E.5)–(E.8), that is, to keep (E.9) valid, exist only if T ∗ > 0. This means that the closed-loop system has an ultimate bound

Appendix E: Proofs of Propositions for Switched Reluctance Motors

581

which cannot be reduced to zero but it can be rendered arbitrarily small by a suitable choice of controller gains. This completes the proof of Proposition 6.1. Finally, we emphasize that the conditions to guarantee Proposition 6.1 are summarized by (A.19), (E.5)–(E.8), K v and B positive constants and some constant T ∗ > 0.

E.3 Proof of Proposition 6.16 E.3.1 Closed-Loop Dynamics Replacing (6.110) in (6.107), and adding and subtracting some convenient terms, we have that ˙ − C(q, I )ξ q˙ − D(q, I ) I˙∗ D(q, I )ξ˙ = −Rξ − αξ − k1 |q|ξ 0 1 + −N − K q q˙ 2 − K f |q| ˙ − K d ξ 1 eσ SIGN(ξ), (E.11) ∗ ∗ ∗ I˙j = I˙ja + I˙jb ,  −1/2 σj β e − 1 1 j ∗ I˙ja = 2 β 2 L 2j (q) ,  ∂σ j ∂σ j ∂σ j 2(eσ j − 1)q˙ d L j (q) eσ j × − 2 3 + 2 2 q˙ + q¨ + z˙ , dq ∂ q˙ ∂z β L j (q) β L j (q) ∂q d f (ζ j ) ∗ = β2 j I˙jb dζ j ,  ∂σ j ∂σ j ∂σ j eσ j 2(eσ j − 1)q˙ d L j (q) + 2 2 q˙ + q¨ + z˙ × − 2 3 , dq ∂ q˙ ∂z β L j (q) β L j (q) ∂q for j = 1, 2, 3. Recall that I˙∗ is continuous according to definition of m j (q) and f j (ζ j ) in Assumption A3 and in Proposition 6.16, respectively. See Remark 6.19. The expression in (E.11) represents the closed-loop dynamics of the electrical subsystem. Let us obtain the closed-loop dynamics corresponding to the mechanical sub d L (q) system. Adding and subtracting the term 3j=1 2β Lψ2s (q) dqj ln(1 + β 2 L 2j (q)I j∗2 ) in (6.18) and replacing I j∗ from (6.111), we have

j

582

Appendix E: Proofs of Propositions for Switched Reluctance Motors

τ =

3  j=1

−

d L j (q) ψs ln(1 + β 2 L 2j (q)I j2 ) 2β L 2j (q) dq

3  j=1

φ=

3 

 d L j (q) ψs β1 j m j (q) + φ, ln(1 + β 2 L 2j (q)I j∗2 ) + τ ∗ 2 2β L j (q) dq j=1

β2 j

j=1

3

d L j (q) ψs ln(1 + β 2 L 2j (q) f 2 (ζ j )), 2β L 2j (q) dq

(E.12)

where |φ| ≤ φ¯ for some φ¯ > 0 which approaches to zero as Td∗ > 0 approaches to d L (q) zero. The expression in (E.12) can be verified by noticing that 2β Lψ2s (q) dqj ln(1 + j

β 2 L 2j (q)I j∗2 ) = β1 j m j (q)τ ∗ , if β1 j = 1, and taking into account the two possible values that β1 j and β2 j can take. Let us define the torque error: Φ=

3  j=1

−

d L j (q) ψs ln(1 + β 2 L 2j (q)I j2 ) 2β L 2j (qi ) dq

3  j=1

d L j (q) ψs ln(1 + β 2 L 2j (q)I j∗2 ). 2 2β L j (q) dq

Hence, using the mean value theorem (Theorem 2.14) we can write Φ=

3  j=1

=

3 

d L j (q) ψs 2 2β L j (q) dq



  ∂ 2 2 2 [ln(1 + βi L j (q)I j )]  (I j − I j∗ ), ∂Ij I j = I¯j

I¯j C j (q, I¯j )ξ j , ξ = [ξ1 , ξ2 , ξ3 ]T ,

(E.13)

j=1

where I¯j is a point belonging to the line joining I j and I j∗ . Notice that there always exists a positive constant K C I¯ j such that | I¯j C j (q, I¯j )| ≤ K C I¯ j , ∀q ∈ R,

I¯j ∈ R.

(E.14)

˜ − kd q˙ − ki Thus, from (E.12) we have that τ = Φ − k p h(q) * and replacing (6.113) + 3 ∗ sat (z) + τ j=1 β1 j m j (q) − 1 + φ. Finally, replacing τ in (6.108) it is obtained: J q¨ = −(b + kd )q˙ − k p h(q) ˜ − g(q) + g(q ∗ ) − ki x(z) + Φ ⎤ ⎡ 3  +τ ∗ ⎣ β1 j m j (q) − 1⎦ + φ, j=1

(E.15)

Appendix E: Proofs of Propositions for Switched Reluctance Motors

583

where x(z) = sat (z) +

1 g(q ∗ ). ki

(E.16)

Thus, the closed-loop dynamics is given by (E.11), (E.15), (E.16), and (6.114).

E.3.2

A Positive Definite and Decrescent Function

Consider the following scalar function: 1  ξ L(q)ξ + Vq (q, ˜ q, ˙ z + g(q ∗ )/ki ), (E.17) 2 Vq (q, ˜ q, ˙ z + g(q ∗ )/ki ) = V1 (q, ˙ q) ˜ + P(q) ˜ + V2 (q, ˙ z + g(q ∗ )/ki ),  q˜ 1 V1 (q, ˙ q) ˜ = J q˙ 2 + εJ h(q) ˜ q˙ + ε(b + kd ) h(r )dr, 4 0  q˜ ∗ P(q) ˜ = kp h(r )dr + U (q) − U (q ∗ ) − qg(q ˜ ), 0  z 1 V2 (q, ˙ z + g(q ∗ )/ki ) = J q˙ 2 + εγ J x(z)q˙ + ki x(r )dr, 4 −g(q ∗ )/ki

V (q, ˜ q, ˙ z + g(q ∗ )/ki , ξ  ) =

where ε and γ are some positive constants. The function Vq (q, ˜ q, ˙ z + g(q ∗ )/ki ) is analyzed in Appendix A.4. From that study and proceeding as in [92] (also see [95]), it is concluded that there exist some small enough constant c1 > 0 and some ˜ q, ˙ z + g(q ∗ )/ki , ξ), large enough constant c2 > 0 such that the scalar function V (q, defined in (E.17) satisfies ˜ q, ˙ z + g(q ∗ )/ki , ξ  ] ∈ R6 , (E.18) α1 ( y ) ≤ V (y) ≤ α2 ( y ), ∀y = [q,  c1 y 2 , y < 1 , α2 ( y ) = c2 y 2 , α1 ( y ) = c1 y , y ≥ 1 if (A.8)–(A.10), and (A.14) are satisfied.

E.3.3 Time Derivative of V ( y) First notice that, because of the energy exchange in the system, we have that   1  d ξ L(q) ξ − ξ  L(q)D −1 (q, I )[k1 |q|ξ ˙ + C(q, I )ξ q] ˙ ≤ 0. 2 dt

584

Appendix E: Proofs of Propositions for Switched Reluctance Motors

It is possible to verify, after some straightforward algebraic manipulations, that the time derivative of V (y), defined in (E.17), along the trajectories of the closed-loop system (E.11), (E.15), (E.16), and (6.114), can be upper bounded as dh(q) ˜ 2 V˙ ≤ −(b + kd )q˙ 2 + εJ ˜ − g(q ∗ )) q˙ − εh(q)(g(q) d q˜ −εk p h 2 (q) ˜ + εγ J [(ε + εγk p /ki )h(q) ˜ + (1 + εγkd /ki )q] ˙

d x(z) q˙ dz

˙ −εγx(z)(g(q) − g(q ∗ )) − εγki x 2 (z) − εγbx(z)q, ⎛ ⎤ ⎞ ⎡ 3  ∗ [εh(q) ˜ + q˙ + εγx(z)] ⎝Φ + τ ⎣ β1 j m j (q) − 1⎦ + φ⎠ *

j=1

*

+

+

+ξ  Δ −(r + α)ξ − D(q, I ) I˙∗ + −N − K q q˙ 2 − K f |q| ˙ − K d ξ 1 eσ SIGN(ξ) ,

(E.19)

where (6.112) has been employed and Δ is the diagonal positive definite matrix defined as Δ = L(q)D −1 (q, I ),

(E.20)

1+β 2 L 2 (q)I 2

j j , j = 1, 2, 3. Notice that the torque whose diagonal entries are given as βψs error Φ, defined in (E.13), does not cancel and still appears in (E.19) through the terms [εh(q) ˜ + q˙ + εγx(z)]Φ. Although this is one of the complications that arise in the case of magnetically saturated SRMs, we show in the following that such terms can successfully be dominated. Now, let us find an upper bound for the last terms in (E.19). First notice that  • If β1 j = 1, for j = 1, 2, 3, then 3j=1 β1 j m j (q) = 1 and

 ⎡ ⎤   3  ∗   τ ⎣ β1 j m j (q) − 1⎦ = 0.    j=1

(E.21)

• If β1 j = 0, for some j = 1, 2, 3, and |τ ∗ | ≤ Td∗ then    ⎡ ⎤     3 3     ∗  ∗  τ ⎣ ⎦ β1 j m j (q) − 1  ≤ Td  β1 j m j (q) − 1 < ϕ     j=1  j=1 with Td∗ > 0 and ϕ > 0 some small numbers. Notice that ϕ can be rendered arbitrarily small by choosing a Td∗ arbitrarily small. • Suppose that β11 = 1, β12 = 1, β13 = 0, for instance. Since β13 = 0 then ζ3 < T3∗ for some small T3∗ > 0 and, hence, |I3∗ | is also small. This means that  d L j (q) 3 ψs ∗2 2 2 ∗ ∗ j=1 2β L 2 (q) dq ln(1 + β L j (q)I j ) ≈ [m 1 (q) + m 2 (q)]τ ≈ τ . Then j

Appendix E: Proofs of Propositions for Switched Reluctance Motors

585

 ⎤  3   β1 j m j (q) − 1⎦ < ϕ0 ,   j=1 for some small ϕ0 > 0. Notice that ϕ0 can be rendered arbitrarily small by choosing T3∗ arbitrarily small, i.e., Td∗ arbitrarily small. Finally, also notice that this result stands for any combination of 0’s and 1’s that β1 j , j = 1, 2, 3, can take. Recall ¨ = |1/J ˜ − (b + kd )q˙ − ki x(z) − g(q) + g(q ∗ ) + Φ + * that |q| + [−k p h(q) 3 τ∗ j=1 β1 j m j (q) − 1 + φ]|. Hence, using the above discussion, we can upper bound the following function that is defined in (E.11) as  2 ∗ | ≤ λ1 λ2 |q|(e ˙ σ j + 1) + λ3 eσ j λ4 (kd |q| ˙ + λ5 )|q| ˙ + λ6 kd max{K C I¯ j } ξ 1 | I˙ja j 3 0 1 +λ6 kd λ11 |q| ˙ + λ7 + λ8 |h(q)| ˜ + λ9 |h(q)| ˜ + λ10 ki |q| ˙ ˙ 2 eσ j + Λ2 |q|e ˙ σ j + Λ3 eσ j ξ 1 + Λ4 eσ j + Λ5 |q|, ˙ ≤ Λ1 |q| where λ1 = sup λ3 = sup

⎧ ⎨β

1j

⎩ 2 7



7 8 −1/2 ⎫ ⎬ d L j (q) 2 e −1 , λ2 = sup , ⎭ β 2 L 2j (q) β 2 L 3j (q) dq 8 σj

1 , β 2 L 2j (q)

∂σ

the constants λ4 and λ5 arise from the factor ∂qj by noticing that this factor can be written as the addition of two terms: one of them is bounded and the other is ∂σ linear in q. ˙ The constants λ6 , λ7 , λ8 , λ11 arise from the function ∂ q˙j q¨ by noticing that

∂σ j ∂ q˙

is a bounded function and taking into account the bound of |q| ¨ that has been ∂σ

∂σ

defined above. Finally, the constants λ9 , λ10 arise from ∂zj z˙ by noticing that ∂zj is a bounded function whereas z˙ depends linearly on h(q) ˜ and q. ˙ The constants Λk > 0, k = 1, . . . , 5, are defined as straightforward combinations of the constants λk > 0, k = 1, . . . , 11, in the upper expression. Recall that |h(q)| ˜ is bounded. It is easy to ∗ |, i.e., see that a similar expression is also valid for | I˙jb ∗ | I˙jb | ≤ Λ6 |q| ˙ 2 eσ j + Λ7 |q|e ˙ σ j + Λ8 eσ j ξ 1 + Λ9 eσ j + Λ10 |q|, ˙

where Λk > 0, k = 6, . . . , 10 are some finite constants. Notice that each one of the constants Λk > 0, k = 1, . . . , 10, are identically defined for any of the three phases j = 1, 2, 3. Using the above expressions, and taking advantage from the properties of the sign function, we realize that if we choose N , K q , K f , K d such that,

586

Appendix E: Proofs of Propositions for Switched Reluctance Motors

N > βψs max{L j (q)}(Λ4 + Λ9 ),

(E.22)

K q > βψs max{L j (q)}(Λ1 + Λ6 ),

(E.23)

K f > βψs max{L j (q)}(Λ2 + Λ7 + Λ5 + Λ10 ),

(E.24)

K d > βψs max{L j (q)}(Λ3 + Λ8 ),

(E.25)

q q q q

then 0 0 1 1 ξ  Δ −D(q, I ) I˙∗ + −N − K q q˙ 2 − K f |q| ˙ − K d ξ 1 eσ SIGN(ξ) < 0, ∀y ∈ R6 . Thus, we can write dh(q) ˜ 2 q˙ − εh(q)(g(q) ˜ − g(q ∗ )) V˙ ≤ −(b + kd )q˙ 2 + εJ d q˜ ˜ + εγ J [(ε + εγk p /ki )h(q) ˜ + (1 + εγkd /ki )q] ˙ −εk p h 2 (q)

d x(z) q˙ dz

−εγx(z)(g(q) − g(q ∗ )) − εγki x 2 (z) − εγbx(z)q˙ √ ¯ +[ε|h(q)| ˜ + |q| ˙ + εγ|x(z)|]( 3 max{K C I¯ j } ξ + ϕ0 |τ ∗ | + ϕ + φ) j

r +α −

ξ 2 , βψs i.e., √ ¯ + ϕ0 (ϕ1 + εk  + εγk  )] y¯ , V˙ ≤ − y¯  Q y¯ + [ 3(1 + ε + εγ)(ϕ + φ) y¯ = [|q|, ˙ |h(q)|, ˜ |x(z)|, ξ ] , ∗ ϕ1 = M (k p + εkd + 2εγkd + 2ki + k  ).

(E.26)

The entries of matrix Q are given as Q 11 = b + kd (1 − ϕ0 ) − εJ − εγ J (1 + εγkd /ki ),

(E.27)

Q 22 = ε[k p (1 − ϕ0 ) − khg ], Q 33 = εγki (1 − ϕ0 ), r +α , Q 13 = Q 31 = −εγb/2, Q 44 = βψs ε(γk p + ki )ϕ0 ε2 γ εγ (1 + γk p /ki )J, Q 23 = Q 32 = − khg − , Q 12 = Q 21 = − 2 2 2 1√ ε√ Q 14 = Q 41 = − 3 max{K C I¯ j }, Q 24 = Q 42 = − 3 max{K C I¯ j }, j j 2 2 εγ √ 3 max{K C I¯ j }, Q 34 = Q 43 = − j 2 where the following result, taken from [130], pp. 105–107, has been used:

Appendix E: Proofs of Propositions for Switched Reluctance Motors

|g(q) − g(q ∗ )| ≤ khg h(|q|), ˜ khg ≥

587

2k  ∀q˜ ∈ R,  , h( 2k ) kg

as well as the facts that h(|q|) ˜ = |h(q)| ˜ and ±uv ≤ |u| |v|, ∀q, ˜ u, v ∈ R. The four leading principal minors of matrix Q can always be rendered positive by choosing small enough ε > 0, γ > 0, Td∗ > 0, and large enough positive definite matrices kd , k p , ki , α. Hence, matrix Q is positive definite and λm (Q) > 0 is ensured. Using some constant 0 < Θ < 1, we can rewrite (E.26) as V˙ ≤ −λm (Q) y¯ 2 + φ¯ 1 y¯ ,

≤ −(1 − Θ)λm (Q) y¯ 2 − Θλm (Q) y¯ 2 + φ¯ 1 y¯ , φ¯ 1 , ≤ −(1 − Θ)λm (Q) y¯ 2 , ∀ y¯ ≥ Θλm (Q) √ ¯ + ϕ0 (ϕ1 + εk  + εγk  ). φ¯ 1 = 3(1 + ε + εγ)(ϕ + φ)

(E.28)

Since φ¯ 1 > 0 approaches to zero as Td∗ → 0, then the latter inequality in (E.28) can be forced to be satisfied by the linear parts of functions in y¯ . Hence, it is always possible to choose a small Td∗ > 0 such that we can write V˙ ≤ −(1 − Θ)λm (Q) y¯ 2 , ∀ y ≥ μ0 =

φ¯ 1 . Θλm (Q)

(E.29)

E.3.4 Proof of Proposition 6.16 Taking into consideration (E.18) and (E.29), we can invoke Theorem 2.29 to conclude that given any initial state y(t0 ) ∈ R6n , we can always find controller gains such that the closed-loop system state y satisfies

y(t) ≤ β0 ( y(t0 ) , t − t0 ), ∀ t0 ≤ t ≤ t0 + T,

y(t) ≤ α1−1 (α2 (μ0 )) , ∀ t ≥ t0 + T,

(E.30)

where β0 (·, ·) is a KL function and T ≥ 0 depends on y(t0 ) and μ0 . On the other hand, φ¯ 1 → 0 as Td∗ > 0 approaches to zero. Hence, μ0 > 0 can be rendered arbitrarily small by choosing some small enough Td∗ > 0. Since α1−1 (α2 (·)) is a K∞ function, then the ultimate bound in (E.30) tends to zero as Td∗ > 0 approaches to zero. Notice that Td∗ = 0 would imply that some of the Λk , k = 1, . . . , 10, are infinitely large and, thus, some of (E.22)–(E.25) could not be satisfied. Thus Td∗ > 0 cannot be zero and, hence, φ¯ 1 > 0 cannot be zero either. This means that the closedloop system has an ultimate bound which cannot be reduced to zero but can be rendered arbitrarily small by a suitable choice of controller gains. Recall that this result stands globally. This completes the proof of Proposition 6.16.

588

Appendix E: Proofs of Propositions for Switched Reluctance Motors

Finally, we emphasize that the conditions to guarantee Proposition 6.16 are summarized by (A.8)–(A.10), (A.14),2 (E.22)–(E.25), the four leading principal minors of matrix Q defined in (E.27) are positive and some small constant Td∗ > 0.

2 Notice

that k p , kd , ki , L ∗ , must be employed instead of k p , kd , ki , L.

Appendix F

Proofs for BLDC Motors

F.1 Proof of Proposition 9.1 F.1.1 Closed-Loop Dynamics Notice that there always exist a constant ε > 0 such that E p = εE p . Hence, defining k p = εk p , and ki = εki we find that 

I∗ =

τ∗ E  (q), E p E R (q) 2 R

(F.1)



τ ∗ = k p ω˜ + ki σ(z),  t ω(s)ds. ˜ z=

(F.2)

0

On the other hand, from (9.8) we have  ∗ τ = −τ p E  R (q)ξ − τ p E R (q)I ,    ∗   ∗ = −τ p E  R (q)ξ − τ p [E R (q) − (E R ) (q)]I − τ p (E R ) (q)I ,

and replacing (F.1):  ∗  ˜ − ki σ(z), τ = −τ p E  R (q)ξ − τ p φ I − k p ω

where φ = E R (q) − E R (q).

(F.3)

Replacing again (F.1): © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 V. M. Hernández-Guzmán et al., Energy-Based Control of Electromechanical Systems, Advances in Industrial Control, https://doi.org/10.1007/978-3-030-58786-4

589

590

Appendix F: Proofs for BLDC Motors  τ = −τ p E  ˜ + ki σ(z)) − k p ω˜ − ki σ(z), R (q)ξ − Φ(k p ω

Φ=

φ E R (q) .

E R (q) 2

(F.4)

Hence, replacing this expression for τ in (9.23) (recall that −τ p E  R (θ)I = τ ), adding and subtracting bω ∗ and taking advantage from the fact that ω˙ ∗ = 0, it is found:  ˜ + ki σ(z)) − k p ω˜ − ki x(z), J ω˙˜ + bω˜ = −τ p E  R (q)ξ − Φ(k p ω bω ∗ + τ L x(z) = σ(z) + . ki

(F.5)

On the other hand, replacing (9.32) in (9.22), and adding and subtracting the terms L I˙∗ and E p E R (q)ω ∗ , we have that L ξ˙ = E p E R (q)ω˜ + E p E R (q)ω ∗ − (R + α p )ξ − K q ω˜ 2 ξ − K d ξ ξ − L I˙∗ ,    E R (q) ˙I ∗ = d ω[k ˜ p ω˜ + ki σ(z)] (F.6) dq E p E R (q) 2    E R (q) d ω ∗ [k p ω˜ + ki σ(z)] + dq E p E R (q) 2   E R (q)  ˙  dσ(z) ω ˜ + k + k ω ˜ . i E p E R (q) 2 p dz Recall that I˙∗ is continuous and defined for all q ∈ R, according to the definition of E R (q) in Proposition 9.1. The closed-loop dynamics is given by (F.5), (F.6), (F.2).

F.1.2 A Positive Definite and Decrescent Function The following scalar function is proposed for stability analysis purposes: τ L + bω ∗ 1  ξ Lξ + Vω (ω, ˜ z+ ), (F.7) 2 ki  z τ L + bω ∗ 1 2   J ω ˜ Vω (ω, ˜ z+ ) = + [k + β(b + k )] x(r )dr + β J x(z)ω, ˜ i p τ +bω ∗ ki 2 − L  V (y) =

ki

∗

∗

, ξ  ] . The function Vω (ω, ˜ z + τL +bω ) in (F.7) is studied where y = [ω, ˜ z + τL +bω ki ki in Appendix A.3. From that analysis, and proceeding as in [92] (also see [95]), it is concluded that there always exists some small enough constant c1 > 0 and some large enough constant c2 > 0 such that the scalar function V (y) defined in (F.7) satisfies

Appendix F: Proofs for BLDC Motors

591

α1 ( y ) ≤ V (y) ≤ α2 ( y ), ∀y ∈ R5 ,  c1 y 2 , y < 1 , α2 ( y ) = c2 y 2 , α1 ( y ) = c1 y , y ≥ 1

(F.8)

∗

if (A.7) and L ∗ > | τL +bω | are true. Notice that α1 (·) and α2 (·) are two class K∞ ki functions.

F.1.3 Time Derivative of V ( y) It is possible to verify, after some straightforward algebraic manipulations (see Remark 9.4), that the time derivative of V (y), defined in (F.7), along the trajectories of the closed-loop system (F.5), (F.6), (F.2), can be upper bounded as 0 1 V˙ ≤ −(R + α p )ξ  ξ + ξ  E p E R (q)ω ∗ + ξ  −K q ω˜ 2 ξ − K d ξ ξ − L I˙∗ ∗ ˜ + ωΦ(bω ˜ + τ L ) + β J ω˜ 2 −(k p + b − k p |Φ|)ω˜ 2 − ki Φ ωx(z)  −βτ p x(z)E  ˜ − (βki − βki |Φ|)x 2 (z) R (q)ξ − βk p Φx(z)ω

+βΦ(bω ∗ + τ L )x(z).

(F.9)

On the other hand, according to (F.6) and (F.5) we can upper bound:

I˙∗ ≤ Λ1 [k p ω˜ 2 + ki |ω|M ˜ ∗ ] + Λ1 |ω ∗ |[k p |ω| ˜ + ki M ∗ ] ,  kp   T  ∗   +Λ2 [(b + k p + |Φ|k p )|ω| ˜ + |τ p E R (q)ξ| + |Φ|ki M + ki |x(z)|] + ki |ω| ˜ , J

where 9 9 9  9 9 9 d 9 E R (q) 9 E R (q) 9 9, 9 9 , Λ2 = sup 9 Λ1 = sup 9 dq E p E R (q) 2 9 E p E R (q) 2 9 |τ p E  R (q)ξ| ≤ τ p E R (q) M ξ . Thus, if we choose K q > λ M (L)Λ1 k p , K d > λ M (L)Λ2 τ p E R (q) M we can write

(F.10) k p J

,

(F.11)

592

Appendix F: Proofs for BLDC Motors

V˙ ≤ − y¯  Q y¯ + γ1 y¯ + γ2 y¯ ,   k p ki M ∗ ∗ |Φ| M γ1 = 2(1 + β)(bω + τ L ) + Λ2 J γ2 = |ω ∗ |(Λ1 ki M ∗ + E p E R (q) M ),

(F.12)

y¯ = [|ω|, ˜ |x(z)|, ξ ] , if:

ξ >

1 λ M (L)Λ1 k p , Kq

(F.13)

ξ >

k p 1 λ M (L)Λ2 τ p E R (q) M . Kd J

(F.14)

The entries of matrix Q are given as Q 11 = b + k p − k p |Φ| M − β J, Q 22 = βki − βki |Φ| M , Q 33 = R + α p , (F.15)  ,  - kp 1 (b + k p + |Φ|k p ) + ki Q 13 = Q 31 = − λ M (L) Λ1 (ki M ∗ + |ω ∗ |k p ) + Λ2 , 2 J 1 1 Q 12 = Q 21 = − ki |Φ| M − βk p |Φ| M , 2 2 β 1 Λ2 λ M (L)k p ki . Q 23 = Q 32 = − τ p E R (q) M − 2 2J

The three leading principal minors of matrix Q can always be rendered positive by choosing small enough β > 0, r > 0, and large enough k p > 0, ki > 0, α p > 0. Hence, matrix Q is positive definite and λm (Q) > 0 is ensured. Using some constant 0 < Θ < 1, we can rewrite (F.12) as V˙ ≤ − y¯  Q y¯ + γ1 y¯ + γ2 y¯ , ≤ −(1 − Θ)λm (Q) y¯ 2 − Θλm (Q) y¯ 2 + γ1 y¯ + γ2 y¯ , γ1 + γ2 ≤ −(1 − Θ)λm (Q) y¯ 2 , ∀ y¯ ≥ . Θλm (Q)

(F.16)

Notice that |Φ| M → 0, i.e., γ1 → 0, as r → 0 (see (9.38), (F.3), (F.4), (F.12)). Also notice that, according to Theorem 2.13, λm (Q) > 0 can be enlarged arbitrarily if r > 0 is chosen arbitrarily small, k p > 0, ki > 0 are chosen arbitrarily large and α p > 0 is chosen to grow faster than (k p )3 and (ki )3 . Then the last inequality in (F.16) can always be forced to be satisfied by the linear parts of functions in y¯ . Moreover, according to (F.13) and (F.14), we can always choose some large enough K q and K d such that ξ is arbitrarily small. Hence, it is always possible to choose controller gains such that we can write γ1 + γ2 . V˙ ≤ −(1 − Θ)λm (Q) y¯ 2 , ∀ y ≥ μ0 = Θλm (Q)

(F.17)

Appendix F: Proofs for BLDC Motors

593

F.1.4 Proof of Proposition 9.1 Taking into consideration (F.8) and (F.17), we can invoke Theorem 2.29 to conclude that given an arbitrary initial state y(t0 ) ∈ R5 , we can always find controller gains such that the closed-loop system state y satisfies

y(t) ≤ β0 ( y(t0 ) , t − t0 ), ∀ t0 ≤ t ≤ t0 + T,

y(t) ≤ α1−1 (α2 (μ0 )) , ∀ t ≥ t0 + T,

(F.18)

where β0 (·, ·) is a KL function and T ≥ 0 depends on y(t0 ) and μ0 . On the other hand, according to the arguments in the previous discussion, μ0 > 0 can be rendered arbitrarily small by choosing suitable controller gains. Since α1−1 (α2 (·)) is a K∞ function, then the ultimate bound in (F.18) decreases arbitrarily to zero. Recall that this result stands when starting from any initial condition. This completes the proof of Proposition 9.1. Finally, we emphasize that the conditions to ∗ |, (A.7), (F.10), (F.11), guarantee Proposition 9.1 are summarized by L ∗ > | τL +bω ki the three leading principal minors of matrix Q defined in (F.15) are positive, and some small constant r > 0.

F.2 Proof of Proposition 9.7 F.2.1 Closed-Loop Dynamics Notice that there always exist an ε > 0 such that E p = εE p . Hence, defining k p = εk p , kd = εkd , and ki = εki we find that 

τ∗ I = E  (q), E p E R (q) 2 R ∗

(F.19)



τ ∗ = k p h(q) ˜ + kd q˙ + ki sat (z). On the other hand, proceeding as between (F.2)–(F.5), i.e., (F.3) and (F.4) also stand in this case, we find that (9.55) can be written as  ˜ + kd q˙ + ki x(z) − g(q ∗ )), J q¨ + bq˙ + g(q) = −τ p E  R (q)ξ − (1 + Φ)(k p h(q)

x(z) = sat (z) + g(q ∗ )/ki .

(F.20)

On the other hand, replacing (9.59) in (9.54), and adding and subtracting the term L I˙∗ , we have that

594

Appendix F: Proofs for BLDC Motors

L ξ˙ = E p E R (q)q˙ − (R + α p )ξ − K q q˙ 2 ξ − K d ξ ξ − L I˙∗ ,    E R (q) ˙I ∗ = d ˜ + kd q˙ + ki sat (z)] q[k ˙ p h(q) dq E p E R (q) 2   ˜ E R (q)  dh(q)   d sat (z) q˙ + kd q¨ + ki z˙ . + k E p E R (q) 2 p d q˜ dz

(F.21)

Recall that I˙∗ is continuous and defined for all q ∈ R, according to the definition of E R (q) in Proposition 9.1. The closed-loop dynamics is given by (F.20), (F.21), (9.62).

F.2.2 A Positive Definite and Decrescent Function The following scalar function is proposed for stability analysis purposes: 1  ξ Lξ + Vq (q, ˜ q, ˙ z + g(q ∗ )/ki ), (F.22) 2 ∗  ∗  ˜ q, ˙ z + g(q )/ki ) = V1 (q, ˙ q) ˜ + P(q) ˜ + V2 (q, ˙ z + g(q )/ki ), Vq (q,  q˜ 1 V1 (q, ˙ q) ˜ = J q˙ 2 + εJ h(q) ˜ q˙ + ε(b + kd ) h(r )dr, 4 0  q˜ ∗ P(q) ˜ = k p h(r )dr + U (q) − U (q ∗ ) − qg(q ˜ ), 0  z 1 V2 (q, ˙ z + g(q ∗ )/ki ) = J q˙ 2 + εγ J x(z)q˙ + ki x(r )dr, 4 −g(q ∗ )/ki

V (q, ˜ q, ˙ z + g(q ∗ )/ki , ξ) =

˜ q, ˙ z + g(q ∗ )/ki ) where ε and γ are some positive constants. The function Vq (q, is analyzed in Appendix A.4. From that study and proceeding as in [92] (also see [95]) it is concluded that there exist some small enough constant c1 > 0 and some ˜ q, ˙ z + g(q ∗ )/ki , ξ), large enough constant c2 > 0 such that the scalar function V (q, defined in (F.22) satisfies ˜ q, ˙ z + g(q ∗ )/ki , ξ] ∈ R6 , (F.23) α1 ( y ) ≤ V (y) ≤ α2 ( y ), ∀y = [q,  c1 y 2 , y < 1 , α2 ( y ) = c2 y 2 , α1 ( y ) = c1 y , y ≥ 1 if (A.8)–(A.10) and (A.14) are true. Notice that α1 (·) and α2 (·) are two class K∞ functions.

Appendix F: Proofs for BLDC Motors

595

F.2.3 Time Derivative of V ( y) It is possible to verify, after some straightforward algebraic manipulations (see Remark 9.10), that the time derivative of V (y), defined in (F.22), along the trajectories of the closed-loop system (F.20), (F.21), (9.62), can be upper bounded as dh(q) ˜ 2 q˙ − εh(q)(g(q) ˜ − g(q ∗ )) V˙ ≤ −(b + kd )q˙ 2 + εJ d q˜ −[εh(q) ˜ + q˙ + εγx(z)]Φ(k p h(q) ˜ + kd q˙ + ki x(z) − g(q ∗ )) −εk p h 2 (q) ˜ + εγ J [ε(1 + γ

k p ki

)h(q) ˜ + (1 + εγ

kd d x(z) q˙ )q] ˙ ki dz

−εγx(z)(g(q) − g(q ∗ )) − εγki x 2 (z) + [εh(q) ˜ + εγx(z)](−τ p E  R (q)ξ) 0 1   2 ∗ ˙ −(R + α p )ξ ξ + ξ −K q q˙ ξ − K d ξ ξ − L I , (F.24) where the facts that k p /ki = k p /ki and kd /ki = kd /ki have been employed. On the other hand, according to (F.21) and the first expression in (F.20) we can upper bound: ˙ p M ∗ + kd |q| ˙ + ki M ∗ ]

I˙∗ ≤ Λ1 |q|[k 11 0  0 +Λ2 k p |q| ˙ + ki ε(1 + γk p /ki )|h(q)| ˜ + (1 + εγkd /ki )|q| ˙ ,  1  +Λ2 kd [khg |h(q)| ˜ + (1 + |Φ| M )k p |h(q)| ˜ J

1 +(1 + |Φ| M )kd |q| ˙ + (1 + |Φ| M )ki |x(z)| + |Φ| M k  + |τ p E  R (q)ξ|] ,

where Φ is defined in (9.43), and 9 9 9 9  9 9 d 9 E R (q) 9 E R (q) 9 9, 9 9 , Λ2 = sup 9 Λ1 = sup 9 dq E p E R (q) 2 9 E p E R (q) 2 9 |τ p E  R (q)ξ| ≤ τ p E R (q) M ξ . Thus, if we choose K q > λ M (L)Λ1 kd , K d > λ M (L)Λ2 τ p E R (q) M

(F.25) kd J

,

(F.26)

we can write V˙ ≤ − y¯  Q y¯ + β y¯ ,   k β = 3(1 + ε + εγ) + Λ2 d |Φ| M k  J y¯ = [|q|, ˙ |h(q)|, ˜ |x(z)|, ξ ] ,

(F.27)

596

Appendix F: Proofs for BLDC Motors

if: 1 λ M (L)Λ1 kd , Kq k 1

ξ > λ M (L)Λ2 τ p E R (q) M d . Kd J

ξ >

(F.28) (F.29)

The entries of matrix Q are given as Q 11 = b + kd − εJ − εγ J (1 + εγkd /ki ) − |Φ| M kd ,   Q 22 = ε k p − |Φ| M k p − khg , Q 33 = εγ(ki − |Φ| M ki ), εγ 1 Q 44 = R + α p , Q 13 = Q 31 = − |Φ| M kd − |Φ| M ki , 2 2 ε2 γ 1 ε J (1 + γk p /ki ) − |Φ| M k p − |Φ| M kd , Q 12 = Q 21 = − 2 2 2 εγ εγ ε |Φ| M k p − |Φ| M ki , Q 23 = Q 32 = − khg − 2 2 2 1 1 Q 14 = Q 41 = − Λ1 k p M ∗ − Λ1 ki M ∗ 2 2  1 1 k  1 0 − Λ2 k p + ki 1 + εγkd /ki − Λ2 d (1 + |Φ| M )kd , 2 2 J   ε ε Q 24 = Q 42 = − τ p E R (q) M − Λ2 ki 1 + γk p /ki 2 2 1 − Λ2 kd [khg + (1 + |Φ| M )k p ], 2J εγ 1 Q 34 = Q 43 = − τ p E R (q) M − Λ2 kd (1 + |Φ| M )ki , 2 2J where khg ≥

2k   (see s( 2k kg )

(F.30)

[130]) and we have used s(|q|) ˜ ≤ |h(q)|. ˜ The four leading

principal minors of matrix Q can always be rendered positive by choosing small enough ε > 0, γ > 0, r > 0, and large enough positive controller gains kd , kd , ki , α p . Hence, matrix Q is positive definite and λm (Q) > 0 is ensured. Using some constant 0 < Θ < 1, we can rewrite (F.27) as V˙ ≤ − y¯  Q y¯ + β y¯ , ≤ −(1 − Θ)λm (Q) y¯ 2 − Θλm (Q) y¯ 2 + β y¯ , β . ≤ −(1 − Θ)λm (Q) y¯ 2 , ∀ y¯ ≥ Θλm (Q)

(F.31)

Since |Φ| M → 0, i.e., β → 0, as r → 0 (see (9.38), (F.3), (F.4)), then the last inequality in (F.31) can always be forced to be satisfied by the linear parts of functions in y¯ . Moreover, according to (F.28) and (F.29), we can always choose some large enough K q and K d such that ξ is arbitrarily small. Hence, it is always possible to choose

Appendix F: Proofs for BLDC Motors

597

a small radius r > 0 such that we can write V˙ ≤ −(1 − Θ)λm (Q) y¯ 2 , ∀ y ≥ μ0 =

β . Θλm (Q)

(F.32)

F.2.4 Proof of Proposition 9.7 Taking into consideration (F.23) and (F.32), we can invoke Theorem 2.29 to conclude that given an arbitrary initial state y(t0 ) ∈ R6 , we can always find controller gains such that the closed-loop system state y satisfies

y(t) ≤ β0 ( y(t0 ) , t − t0 ), ∀ t0 ≤ t ≤ t0 + T,

y(t) ≤ α1−1 (α2 (μ0 )) , ∀ t ≥ t0 + T,

(F.33)

where β0 (·, ·) is a KL function and T ≥ 0 depends on y(t0 ) and μ0 . On the other hand, β → 0 as r > 0 approaches to zero, hence, μ0 > 0 can be rendered arbitrarily small by choosing a small enough r > 0. Since α1−1 (α2 (·)) is a K∞ function, then the ultimate bound in (F.33) tends to zero as r > 0 approaches to zero. Notice that, r = 0 would imply that E R (q) is not differentiable and, hence I˙∗ is not continuously differentiable, i.e., Λ1 is not defined. Thus, r > 0 cannot be zero and, hence, β > 0 cannot be zero either. This means that the closed-loop system has an ultimate bound which cannot be reduced to zero but can be rendered arbitrarily small by a suitable choice of controller gains. Recall that this result stands when starting from any initial condition. This completes the proof of Proposition 9.7. Finally, we emphasize that the conditions to guarantee Proposition 9.7 are summarized by (A.8)– (A.10), (A.14), (F.25), (F.26), the four leading principal minors of matrix Q defined in (F.30) are positive, and some small constant r > 0 is chosen.

Appendix G

Derivation of Some Expressions for the Proofs in Chap. 10

G.1 Proof of Proposition 10.1 Notice the following: : ; 2 ;  |i ∗ | ≤ <   d L(y)   dy 

(k p M + kd | y˙ | + ki M) = I ∗ (| y˙ |),

(G.1)

min

where (10.9) has been employed. On the other hand, according to the mean value theorem (Theorem 2.14):   d I ∗ (| y˙ |)  d I ∗ (| y˙ |)  ∗ ∗ 0 ≤ I (| y˙ |) − I (0) = (| y˙ | − 0), > 0, ∀ζ > 0, d| y˙ | | y˙ |=ζ d| y˙ | | y˙ |=ζ i.e., I ∗ (| y˙ |) =

 d I ∗ (| y˙ |)  | y˙ | + I ∗ (0), d| y˙ | | y˙ |=ζ

(G.2)

for some ζ > 0 belonging to the line joining the points | y˙ | and 0. Hence, from (G.1) and (G.2) we find |i ∗ | ≤ kδ | y˙ | + I ∗ (0),  ∗  d I (| y˙ |) . kδ = max | y˙ |>0 d| y˙ |

(G.3)

Notice that kδ and I ∗ (0) are positive and finite. This allows to write

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 V. M. Hernández-Guzmán et al., Energy-Based Control of Electromechanical Systems, Advances in Industrial Control, https://doi.org/10.1007/978-3-030-58786-4

599

600

Appendix G: Derivation of Some Expressions for the Proofs in Chap. 10

   d L(y) 2 ∗   ˜ i  ≤ kσ i˜2 (kδ | y˙ | + I ∗ (0)), i  dy      d L(y) α ˜ ∗  ≤ αkσ kδ M|i| ˜ | y˙ | + αkσ |i| ˜ |h( y˜ )|I ∗ (0), h( y ˜ ) ii   dy    d L(y)  αβ ˜ ∗  ≤ 2αβ Mkσ kδ |i| ˜ | y˙ | + αβkσ |i| ˜ |s(z)|I ∗ (0), s(z) ii   dy    d L(y)   . kσ = max  dy 

(G.4) (G.5) (G.6)

Notice that, according to Fig. 10.2, kσ > 0 is always finite. On the other hand, we have that at the equilibrium point i ∗ = i e∗ , where i e∗

: ; 2mg ; . = 0 d|θ| Notice that kδ , kσ , and V ∗ (0) are positive and finite. On the other hand, define:

602

Appendix G: Derivation of Some Expressions for the Proofs in Chap. 10

 ∗ vae

=

2K θ∗ dCa (θ∗ ) dθ

,

as the value of va∗ at the equilibrium point and, hence:  va∗

=

˜ θ, ˙ s(z)) va∗ (h(θ),

=

2 dCa (θ) dθ

˜ − kd θ˙ − ki s(z) + K θ∗ ], [−k p h(θ)

∗ = va∗ (0, 0, 0). Since these expressions are very similar to those after (G.6) is and vae not difficult to realize that we can write     2K θ∗   ∗ ˙ ˜ + |θ| ˙ + |s(z)|], θ)| (G.13) va − dCa (θ∗ )  ≤ (k ∗ + kt |θ|)[|h(   dθ

where the constants k ∗ and kt are defined from the norm of the following vector: −1/2  2τ ∗ ∂va∗ (x) = dC (θ) a ∂x dθ 7     + ˜ d θ˜ * d dCa (θ) −1 d(θ∗ + θ) ˜ + ki s(z) + K θ∗ k p h(θ) × [1, 0, 0] ˜ dθ dθ d θ˜ dh(θ) 8   1 dCa (θ) −1 0 + k p , kd , ki dθ    −1/2   ˜ d θ˜ 2τ ∗ d dCa (θ) −1 d(θ∗ + θ) ˙ + dC (θ) [1, 0, 0] k θ, a ˜ d dθ dθ d θ˜ dh(θ) dθ

 ∗ −1/2 dCa (θ) d θ˜ 2τ recalling that, according to (10.41), dh( is bounded, > 0, and dCa (θ) ˜ dθ θ) dθ  −1 dCa (θ) d ˜ ≤ M, |s(z)| ≤ 2M. Thus, k ∗ and kt are finite. are bounded and |h(θ)| dθ dθ

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Index

A Ab fictitious phase windings, 211 Adaptive control, 142, 245, 354, 359 Ampère’s Law, 5, 30, 168, 187, 260, 278, 329, 419 Asymptotically stable equilibrium point, 16 Asynchronous motors, 207 Autonomous systems, 15, 75, 79, 122, 135, 233, 351, 356, 441, 458

E Edison, T.A., 1 Eigenvalues of a symmetric matrix, 19 Electrostatic micromirror, 452 Electrostatic torque, 451 Equilibrium point, 15 Euclidean and spectral norm properties, 29 Euclidean norm, 29 Exponential stability, 26 Extended space, 28

B Backstepping, 2 Balanced currents and voltages, 101, 208 Barbalat’s Lemma, 27, 28, 235

F Faraday’s Law, 50, 53, 102, 199, 207, 208, 280, 282, 365, 437 Field weakening, 113, 224 Final value theorem, 65 Finite-gain L-stability, 28 Flux linkages, 102 Forced response, 9 Frequency response method, 12

C Classical control, 7 Class K functions, 24 Class KL functions, 25 Commutation, 361, 366

D D’Alembert’s Principle, 53, 110, 201, 216, 283, 340, 365 DC to DC Buck converter driven DC-motor system, 84 Decrescent Lyapunov function, 25, 528 Dissipative system, 32 Double-salient, 344 Doubly salient, 279 Dq fictitious phase windings, 105, 217

G Gauss’ Law for the magnetic field, 169, 189, 262, 332, 422 General linear state space representation, 12 General nonlinear state space representation, 12 General response of a linear system, 8 Geršgorin Theorem, 20 Global asymptotic stability Theorem, 21

H High-pass filter, 65

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 V. M. Hernández-Guzmán et al., Energy-Based Control of Electromechanical Systems, Advances in Industrial Control, https://doi.org/10.1007/978-3-030-58786-4

617

618

Index

Home position in PM synchronous motors, 114, 205 Hysteresis controller, 287 Hysteresis nonlinearity, 288

Output, 7 Output strictly passive, 72, 73, 118, 119, 203, 230, 291, 292, 348, 349, 369, 370, 401, 402, 439, 456

I Induction motor modeling assumptions, 207 Input, 7 Input-output stability, 27 Inverse dq transformation, 107

P Parseval’s Theorem, 28, 160 Passive circuit elements, 31 Passive network, 33 Passive system, 32 Passivity, 31, 38, 86 Passivity-based control with total energy shaping, 4, 45 Passivity properties, 77, 82, 125, 139, 144, 241, 300, 316, 325, 375, 409, 415, 448 Phases and poles in a SRM, 279 PM synchronous motor modeling assumptions, 99 Poles, 7 Positive definite and positive semidefinite matrices eigenvalues, 19 leading principal minors, 19 Positive definite function, 18 Positive definite matrix, 19 Positive Real (PR) function, 33 Positive semidefinite function, 18 Positive semidefinite matrix, 19 Pull-in phenomenon, 456, 464

K Kalman–Yakubovich–Popov Lemma, 35, 37 Kirchhoff’s Current Law, 451 Kirchhoff’s Laws, 84, 437 Kirchhoff’s Voltage Law, 52, 53, 102, 199, 208, 282, 365

L LaSalle Invariance Principle, 22, 76, 81, 124, 137, 237, 353, 358, 445, 461 Leading principal minors, 19 Leakage inductance, 103, 210 Lenz’s Law, 51 Level surfaces, 16 Lipschitz condition, 39 L-stability, 28 L2-stability, 39, 160 Lyapunov function candidate, 16, 17 Lyapunov Theorem, 16

Q Quadratic form, 19 M Magnetizing inductance, 103, 210 Mean Value Theorem, 20 Mutual inductance, 103, 210

N Natural response, 9 Negative definite function, 18 Negative semidefinite function, 18 Nested-loop passivity-based control, 4, 44 Newton’s Second Law, 5, 52, 111, 217, 282, 365, 437, 451 Nonautonomous systems, 23 1-norm, 29

O Ohm’s Law, 102, 199, 208, 282, 365

R Radially unbounded function, 21 Rayleigh–Ritz Theorem, 19 Reluctance, 278, 280, 437 Reluctance and magnetic force, 281, 437 Residual dynamics, 472 Right-hand rule, 30 Rigid robot model properties, 472, 492 Root locus method, 11, 55 Rotor, 49, 98 Rotor resistance estimation, 226 Round rotor PM synchronous motor, 100

S Saliency ratio, 343, 346 Salient rotor PM synchronous motor, 100

Index Self inductance, 210 Sharing functions, 286, 301, 492 Single-salient, 344 Sinusoidally distributed rotor magnetic flux, 99 Sinusoidally distributed windings, 99 Sliding modes control, 87 Slip velocity, 207, 222 Spectral norm, 29 Stability in the sense of Lyapunov, 15 Stability of an arbitrary linear system, 9 Stable equilibrium point, 16 Standard passivity-based control, 2 Stator, 49, 98 Storage function, 32 Strict Lyapunov function, 19, 77, 82, 90, 125, 139, 144, 241, 300, 315, 325, 375, 408, 448 Strictly increasing linear saturation, 29 Strictly Positive Real (SPR) function, 35 Supply rate, 32

T Tesla, N., 1

619 Three phase model, 104, 210 Transfer function, 7 Trapezoidal back electromotive force, 394, 397 Truncation, 28 Typical transient responses of linear systems, 9

U Ultimate bound, 27, 298, 313, 403, 407, 411, 473, 479, 495, 506 Underactuated machine, 205 Uniformly distributed windings, 394 Uniform stability and uniform asymptotic stability, 24, 25

V Vector norms, 14

Z Zeros, 7 Zero-state detectability, 38