Advanced Direct Thrust Force Control of Linear Permanent Magnet Synchronous Motor (Power Systems) 9783030403249, 9783030403256, 3030403246

Table of contents :
Preface
Acknowledgements
Contents
List of Figures
List of Tables
1 Introduction
1.1 Motivation and Scope
1.2 Types of Linear Permanent Magnet PMSMs
1.3 A Review of Developments in Direct Thrust Force Control of Linear PMSM
1.4 Description of Experimental Setup
1.5 Book Outline
1.6 Conclusions
References
2 Mathematical Modeling of Surface-Mount Linear Permanent Magnet Synchronous Motor
2.1 Introduction
2.2 Construction of Tubular Surface-Mount Linear PMSM
2.3 Dynamic Modeling of Surface-Mount Linear PMSM in 3-Phase Stationary abc-Reference Frame
2.3.1 Mapping of Three-Phase Machine Variables to Complex Space Vectors
2.4 Two-Axis Dynamic Models of Linear PMSM
2.4.1 Dynamic Model of Linear PMSM in the dq-Reference Frame
2.4.2 Dynamic Model of Linear PMSM in αβ-Reference Frame
2.4.3 Dynamic Model of Linear PMSM in xy-Reference Frame
2.5 Estimation of Stator Flux Magnitude and Thrust Force Base on Dq-Axes Current Model
2.6 Conclusion
References
3 Direct Thrust Force Control Based on Duty Ratio Control
3.1 Introduction
3.2 Basic Principle of Direct Thrust Force Control
3.2.1 Selection of Reference Stator Flux Magnitude λref
3.2.2 Operational Range of Load Angle δ for Low Inductance Surface-Mount Linear PMSM
3.3 Stability Analysis of Direct Thrust Force Control
3.3.1 Lyapunov Stability Analysis of Conventional DTFC
3.4 Effect of Inverter Voltage Vectors on Thrust Force and Flux Variation
3.5 Experimental Results for Conventional DTFC
3.6 Duty Ratio Control
3.7 Review of Classical Duty Ratio Control Methods
3.8 State of the Art Duty Ratio Control Method
3.8.1 Tuning of Static Gains CT and CF
3.9 Proposed Duty Ratio Control Method
3.9.1 Derivation of Expression for FT
3.9.2 Derivation of Expression for λs
3.9.3 Derivation of Expressions for dF and dλ
3.10 Experimental Results for Duty Ratio Controlled DTFC
3.10.1 Start-Up Performance with Speed Loop Closed
3.10.2 Speed Reversal and Steady-State Performance
3.10.3 Analysis of Steady State Error in Force for DTFC1
3.10.4 Flux Trajectory
3.10.5 Transient Response of Force with Outer Speed Loop Disabled
3.11 Conclusions
References
4 SV-PWM Based Direct Thrust Force Control of a Linear Permanent Magnet Synchronous Motor
4.1 Introduction
4.2 Stator Flux and Thrust Force Regulation in xy-Reference Frame
4.3 Analysis of Thrust Force Control in Surface-Mount Linear PMSM
4.3.1 Selection of Reference Stator Flux Magnitude λref
4.3.2 Characteristics of the Co-efficient K for Surface-Mount Linear PMSM with Low Stator Inductance and Short Pole-Pitch
4.4 Derivation of Transfer Function for Stator Flux Regulation
4.5 Derivation of Transfer Function for Thrust Force Regulation
4.6 PI Controller Based Direct Thrust Force Control (PI-DTFC)
4.6.1 Stator Flux Control Loop
4.6.2 Thrust Force Control Loop
4.6.3 Discrete Time Design of Stator Flux and Force PI Controllers
4.7 Linear Quadratic Regulator Based Direct Thrust Force Control of Linear PMSM (Optimal-DTFC1)
4.7.1 Formulation of a Novel State Space Model of the Linear PMSM
4.7.2 Controllability of the Novel State Space Model
4.8 Linear Quadratic Regulator Based State Feedback Control with Integral Action
4.8.1 Mathematical Formulation of Error Dynamics
4.8.2 Optimal Linear Quadratic Regulator Design
4.9 Novel LQR Based Direct Thrust Force Control of the Linear-PMSM
4.9.1 Controllability Analysis of the Error Dynamics
4.9.2 Choice of Gain Matrix for State Feedback Law
4.10 Experimental Validation of Proposed Control Scheme
4.10.1 Dynamic Response with Outer Speed Loop Disabled
4.10.2 Steady-State Regulation with Outer Speed Loop Disabled
4.10.3 Start-Up Speed Response with Outer Speed Loop Enabled
4.10.4 Speed Reversal with Outer Speed Loop Enabled
4.10.5 Steady-State Response with Outer Speed Loop Enabled
4.11 Conclusions
References
5 Optimal, Combined Speed and Direct Thrust Force Control of a Linear Permanent Magnet Synchronous Motors
5.1 Introduction
5.2 State Space Model of the Linear PMSM in the xy-Reference Frame for Combined Flux, Thrust and Speed Dynamics
5.2.1 Controllability Analysis of the Novel State Space Model
5.3 Linear Quadratic Regulator Based State Feedback Control with Integral Action
5.3.1 Mathematical Formulation of Error Dynamics
5.3.2 Formulation of the Linear Quadratic Regulator
5.4 Linear Quadratic Regulator Based Combined Speed and Direct Thrust Control
5.4.1 Controllability Analysis of the Error Dynamics
5.4.2 Choice of Gain Matrix for State Feedback Law
5.5 Experimental Validation of Proposed Optimal-DTFC2
5.5.1 Start-Up Speed Response
5.5.2 Speed Reversal and Steady-State Response
5.5.3 Effect of Parameter Variation on the Performance
5.6 Conclusions
References
6 Sliding Mode Based Combined Speed and Direct Thrust Force Control of a Linear Permanent Magnet Synchronous Motors
6.1 Introduction
6.2 Dynamic Model of the Linear PMSM in xy-Reference Frame
6.3 Sliding Mode Control
6.3.1 Fundamentals of Sliding Mode Control
6.3.2 Variable Structure Based Direct Torque Control
6.4 Sliding Mode Control with Augmented Integral Action
6.4.1 Sliding Surface for Stator Flux Regulation
6.4.2 Sliding Surface for Speed Regulation
6.4.3 Control Law for Stator Flux Regulation
6.4.4 Control Law for Combined Speed and Thrust Force Regulation
6.4.5 Chattering Reduction
6.5 Experimental Validation of Proposed SM-DTFC1
6.5.1 Start-Up Speed Response
6.5.2 Speed Reversal and Steady-State Response
6.5.3 Evaluation of Robustness to the Parameter Variation
6.6 Conclusions
References
7 Sensorless Control of a Linear Permanent Magnet Synchronous Motors Using a Combined Sliding Mode Adaptive Observer
7.1 Introduction
7.2 Dynamic State Space Model of Linear PMSM in xy-Reference Frame
7.3 Combined Speed and Direct Thrust Force Control of Linear PMSM Based on Integral Sliding Mode Control
7.3.1 Sliding Surface for Stator Flux Regulation
7.3.2 Sliding Surface for Speed Regulation
7.3.3 Control Law for Stator Flux Regulation
7.3.4 Control Law for Thrust Force Regulation
7.4 A Novel Combined Sliding Mode State Observer
7.4.1 Stability Analysis of the Proposed Observer
7.4.2 Gain Selection for the Proposed Observer
7.4.3 Adaption Scheme for Speed Estimation
7.4.4 Estimation of Stator Flux Magnitude and Thrust Force
7.5 Experimental Results
7.6 Experimental Evaluation of the Control Performance of SM-DTFC2
7.6.1 Start-Up Response
7.6.2 Speed Reversal Response
7.6.3 Robustness to Parameter Variation
7.7 Experimental Evaluation of the SM-Observer
7.7.1 Speed Reversal Response
7.7.2 Speed Reversal Response Without the Improved SM Function
7.7.3 Steady State Response
7.7.4 Position Estimation
7.7.5 Flux Estimation
7.8 Conclusion
References
8 Conclusions and Future Work
8.1 Conclusions
8.2 Main Contributions of the Book
8.3 Future Work
Appendix A Description of the Experimental Setup
A.1 Description of the Prototype Tubular Surface-Mount Linear PMSM
A.2 Description of 3-Phase Voltage Source Inverter
A.3 Description of Voltage Sensing Board
A.4 Description of Current Sensing Board
A.5 Description of dSPACE© DS 1104 R&D Controller Board
Appendix B Derivation of Expressions for dFT dt and dλs dt
B.1 Derivation of Expression for dFT dt in Terms of Inverter Voltages
B.2 Derivation of Expression for dλs dt in Terms of Inverter Voltages

Citation preview

Power Systems

Muhammad Ali Masood Cheema John Edward Fletcher

Advanced Direct Thrust Force Control of Linear Permanent Magnet Synchronous Motor

Power Systems

Electrical power has been the technological foundation of industrial societies for many years. Although the systems designed to provide and apply electrical energy have reached a high degree of maturity, unforeseen problems are constantly encountered, necessitating the design of more efficient and reliable systems based on novel technologies. The book series Power Systems is aimed at providing detailed, accurate and sound technical information about these new developments in electrical power engineering. It includes topics on power generation, storage and transmission as well as electrical machines. The monographs and advanced textbooks in this series address researchers, lecturers, industrial engineers and senior students in electrical engineering. ** Power Systems is indexed in Scopus**

More information about this series at http://www.springer.com/series/4622

Muhammad Ali Masood Cheema John Edward Fletcher

•

Advanced Direct Thrust Force Control of Linear Permanent Magnet Synchronous Motor

123

Muhammad Ali Masood Cheema Research and Special Design Northern Transformer Corporation Maple, ON, Canada

John Edward Fletcher The University of New South Wales Sydney, NSW, Australia

ISSN 1612-1287 ISSN 1860-4676 (electronic) Power Systems ISBN 978-3-030-40324-9 ISBN 978-3-030-40325-6 (eBook) https://doi.org/10.1007/978-3-030-40325-6 © Springer Nature Switzerland AG 2020 This work is subject to copyright. All rights are reserved 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

To “My Old Man” Muhammad Masood Cheema “Baba! we did it together. It is your immense love and spiritual bond I have with you that enabled me to accomplish this great achievement.” To Dr. Farrukh Humayun Cheema “For your selfless love that always gave me strength.” Dr. Muhammad Ali Masood Cheema To “To those who walked before me for making this research possible, and for those who will follow for their dedication and pursuit of understanding.” Dr. John Edward Fletcher

Preface

The application areas for linear permanent magnet synchronous motors (linear PMSMs) include, but not limited to, servo-mechanisms, automation of manufacturing processes, transportation, renewable energy devices and pumping machinery. The broadening scope of the linear PMSMs implies the importance of developing robust, simple and fast control mechanisms for the linear drives to suit the essential needs of these emerging industries. This book is focused on the direct thrust force control (DTFC) of tubular surface-mount linear PMSMs. The concept of DTFC is derived from direct torque control scheme which was first introduced in the 1980s initially for rotational induction machines, and in later studies this scheme was also applied to rotational permanent magnet synchronous motors (PMSMs). The simple structure of the direct torque control scheme provides it with inherent robustness to parameter variation and a fast transient response. These attributes make this control scheme a prime candidate when robustness and fast transient response are of vital significance. In recent years, direct torque control scheme has also been extended to linear PMSMs as DTFC; however, a little is published so far in this regard. The DTFC scheme is analogous to direct torque control and has a similar switching table-based control structure in conjunction with two multilevel hysteresis comparators to regulate stator flux and thrust force. These hysteresis comparators are simplest form of a scaled relay controller with unity gain and are robust to parameter variation; therefore, DTFC is variable structure control scheme in nature. However, the main disadvantage of DTFC, due to the variable structure nature of the control scheme, is the presence of ripple in the stator flux and thrust force and a variable switching frequency. The implementation of DTFC for linear PMSMs is substantially challenging compared to direct torque control of rotational PMSMs. In general, most of the linear PMSMs have a low value of stator inductance due to their larger air gap compared to rotational PMSMs. Also, due to the linear motion, the pole pitch of the linear PMSMs also affects the thrust force regulation. Therefore, classical switching table-based DTFC-controlled linear PMSMs with low inductance and short pole pitch exhibit large thrust force ripple which is typically beyond acceptable limits. vii

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This book aims at in-depth formulation of nonlinear dynamic models for linear permanent magnet synchronous motors from a control system perspective and subsequently utilizes these models to develop several advanced DTFC schemes based on optimal control and sliding mode control approaches. A detailed chapter is also dedicated to formulation of sliding mode stator flux observer for sensorless speed estimation of linear PMSM. It is worth mentioning that extensive experimental results based on a laboratory prototype for validation of the presented control schemes along with exhaustive discussions are included in the text as well. Chapters 1 and 2 of the book are based on a detailed literature review and comprehensive mathematical modelling of linear PMSMs in various reference frames. The conventional switching table-based DTFC is analysed at length in Chap. 3. In order to reduce the ripple and steady-state error in the stator flux and thrust force response of the linear PMSM, a duty ratio control-based control is proposed and studied to adapt switching table-based DTFC as provided in Chap. 3 of the book. Another approach to effectively reduce steady-state thrust force ripple is to use space vector pulse-width modulation approach. In Chaps. 4–7, several advanced DTFC schemes utilizing space vector pulse-width modulation are proposed and analysed. Finally, a stator flux observer for sensorless speed estimation comprising a linear state observer and an improved sliding mode component is also proposed in Chap. 7. The key feature of this book is formulation of a nonlinear second-order state-space model of the linear PMSM in a synchronously rotating stator flux vector xy-reference frame for combined dynamics of speed and thrust force as system states. The proposed nonlinear model is used to formulate the sliding mode control law with space vector pulse-width modulation for combined control of speed and thrust force. In view of this fact that this text is written in way that the development of advanced concepts stems from a detailed mathematical foundation of the subject matter, this book is a suitable choice to be used as textbook or reference for undergraduate/graduate students for the subject of advanced electrical drives. It is important to note that, although this book is set in the context of linear PMSMs, however the concepts and mathematical formulations presented here can conveniently be adapted or generalized for direct torque control for rotational multiphase PMSMs with or without saliency. This can be of great help for graduate research students and professional engineers in the field of electrical drives to expand their understanding of advanced nonlinear dynamics and control schemes for linear/rotational PMSMs. Maple, Canada Sydney, Australia

Muhammad Ali Masood Cheema John Edward Fletcher

Acknowledgements

The core of this book is based on my doctoral and postdoctoral research under the supervision of Prof. John Edward Fletcher and co-supervision of Prof. Muhammad Fazlur Rahman. First and foremost, I want to express my gratitude to my supervisors, Prof. John Edward Fletcher and Prof. Muhammad Fazlur Rahman of the Energy Systems Research Group at the University of New South Wales, for their guidance and support throughout the course of my research for this book. I will always cherish my time that I spent under supervision of Prof. John Edward Fletcher who has always been kind to me and provided me the opportunity for this research. I also thank the University of New South Wales and the Faculty of Engineering for providing the scholarship that enabled the research for this book and also for conferring on me the title of adjunct lecturer that allowed me to further advance the research for this book. I also wish to acknowledge valuable support and financial assistance from the management of Northern Transformer Corporation (NTC) who facilitated me in writing this manuscript. I wish to express my gratitude to Alexei Miecznikowski, CEO of NTC, who has been very supportive throughout the course of this book and encouraged me to push the envelope in research. I would also like to thank the technical staff in the energy system research group of UNSW for their support in logistics. Special thanks to Dr. Dan Xiao and Mr. Gamini Liyadipitiya for their assistance in the experimental work vital to conclude this book. I would also like to thank Mr. Merlin Chai, Mr. Kazi Ahsanullah and Mr. David Tan for their insightful discussion and suggestions. I wish to pay special thanks to my friend Dr. Mohammad Farashadnia for his valuable support throughout the duration of this research. I am also grateful to my brother Muhammad Ammar Masood Cheema for his love and support during my studies and always being my strength to fight my forward in the life. I am greatly indebted to my parents and sister for their endless love, continuous encouragement, spiritual and financial support during my studies leading to this book. My father Engr. Muhammad Masood Cheema and mother Dr. Fozia have

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played a pivotal role throughout in my life, and it is their immense love and faith in myself that enabled me to accomplish this task and made me the person I am today. I am very fortunate to have Dr. Farrukh Humayun Cheema and Najma Farrukh also as my parents who selflessly loved me, supported me and stood strong by me throughout my life; without them, this book would have been a far-fetched dream. I wish to express my love and gratitude to my wife Sunbal for standing by my side during my Ph.D. studies. Wify! you have always been there for me. Lastly, I wish to express love for my sons Muhammad Ali Murtaza and Muhammad Ali Mujtaba who have always been a source of great motivation and joy in my life. Dr. Muhammad Ali Masood Cheema

Contents

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Motivation and Scope . . . . . . . . . . . . . . . . . 1.2 Types of Linear Permanent Magnet PMSMs . 1.3 A Review of Developments in Direct Thrust Force Control of Linear PMSM . . . . . . . . . . 1.4 Description of Experimental Setup . . . . . . . . 1.5 Book Outline . . . . . . . . . . . . . . . . . . . . . . . 1.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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2 Mathematical Modeling of Surface-Mount Linear Permanent Magnet Synchronous Motor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Construction of Tubular Surface-Mount Linear PMSM . . . . . . 2.3 Dynamic Modeling of Surface-Mount Linear PMSM in 3-Phase Stationary abc-Reference Frame . . . . . . . . . . . . . . . . . . . . . . . 2.3.1 Mapping of Three-Phase Machine Variables to Complex Space Vectors . . . . . . . . . . . . . . . . . . . . . . 2.4 Two-Axis Dynamic Models of Linear PMSM . . . . . . . . . . . . . 2.4.1 Dynamic Model of Linear PMSM in the dq-Reference Frame . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.2 Dynamic Model of Linear PMSM in ab-Reference Frame . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.3 Dynamic Model of Linear PMSM in xy-Reference Frame . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Estimation of Stator Flux Magnitude and Thrust Force Base on Dq-Axes Current Model . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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3 Direct Thrust Force Control Based on Duty Ratio Control . . . . 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Basic Principle of Direct Thrust Force Control . . . . . . . . . . . 3.2.1 Selection of Reference Stator Flux Magnitude kref . . 3.2.2 Operational Range of Load Angle d for Low Inductance Surface-Mount Linear PMSM . . . . . . . . 3.3 Stability Analysis of Direct Thrust Force Control . . . . . . . . . 3.3.1 Lyapunov Stability Analysis of Conventional DTFC 3.4 Effect of Inverter Voltage Vectors on Thrust Force and Flux Variation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Experimental Results for Conventional DTFC . . . . . . . . . . . . 3.6 Duty Ratio Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.7 Review of Classical Duty Ratio Control Methods . . . . . . . . . 3.8 State of the Art Duty Ratio Control Method . . . . . . . . . . . . . 3.8.1 Tuning of Static Gains CT and CF . . . . . . . . . . . . . 3.9 Proposed Duty Ratio Control Method . . . . . . . . . . . . . . . . . . 3.9.1 Derivation of Expression for DFT . . . . . . . . . . . . . . 3.9.2 Derivation of Expression for Dks . . . . . . . . . . . . . . 3.9.3 Derivation of Expressions for dF and dk . . . . . . . . . 3.10 Experimental Results for Duty Ratio Controlled DTFC . . . . . 3.10.1 Start-Up Performance with Speed Loop Closed . . . . 3.10.2 Speed Reversal and Steady-State Performance . . . . . 3.10.3 Analysis of Steady State Error in Force for DTFC1 . 3.10.4 Flux Trajectory . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.10.5 Transient Response of Force with Outer Speed Loop Disabled . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.11 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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4 SV-PWM Based Direct Thrust Force Control of a Linear Permanent Magnet Synchronous Motor . . . . . . . . . . . . . . . . . . . . 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Stator Flux and Thrust Force Regulation in xy-Reference Frame . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Analysis of Thrust Force Control in Surface-Mount Linear PMSM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.1 Selection of Reference Stator Flux Magnitude kref . . . 4.3.2 Characteristics of the Co-efficient K for Surface-Mount Linear PMSM with Low Stator Inductance and Short Pole-Pitch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Derivation of Transfer Function for Stator Flux Regulation . . . 4.5 Derivation of Transfer Function for Thrust Force Regulation . . 4.6 PI Controller Based Direct Thrust Force Control (PI-DTFC) . .

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Stator Flux Control Loop . . . . . . . . . . . . . . . . . . . . . Thrust Force Control Loop . . . . . . . . . . . . . . . . . . . . Discrete Time Design of Stator Flux and Force PI Controllers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.7 Linear Quadratic Regulator Based Direct Thrust Force Control of Linear PMSM (Optimal-DTFC1) . . . . . . . . . . . . . . . . . . . . 4.7.1 Formulation of a Novel State Space Model of the Linear PMSM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.7.2 Controllability of the Novel State Space Model . . . . . 4.8 Linear Quadratic Regulator Based State Feedback Control with Integral Action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.8.1 Mathematical Formulation of Error Dynamics . . . . . . 4.8.2 Optimal Linear Quadratic Regulator Design . . . . . . . . 4.9 Novel LQR Based Direct Thrust Force Control of the Linear-PMSM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.9.1 Controllability Analysis of the Error Dynamics . . . . . 4.9.2 Choice of Gain Matrix for State Feedback Law . . . . . 4.10 Experimental Validation of Proposed Control Scheme . . . . . . . 4.10.1 Dynamic Response with Outer Speed Loop Disabled . 4.10.2 Steady-State Regulation with Outer Speed Loop Disabled . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.10.3 Start-Up Speed Response with Outer Speed Loop Enabled . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.10.4 Speed Reversal with Outer Speed Loop Enabled . . . . 4.10.5 Steady-State Response with Outer Speed Loop Enabled . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.11 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Optimal, Combined Speed and Direct Thrust Force Control of a Linear Permanent Magnet Synchronous Motors . . . . . . . . . 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 State Space Model of the Linear PMSM in the xy-Reference Frame for Combined Flux, Thrust and Speed Dynamics . . . . 5.2.1 Controllability Analysis of the Novel State Space Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Linear Quadratic Regulator Based State Feedback Control with Integral Action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.1 Mathematical Formulation of Error Dynamics . . . . . 5.3.2 Formulation of the Linear Quadratic Regulator . . . .

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5.4

Linear Quadratic Regulator Based Combined Speed and Direct Thrust Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.1 Controllability Analysis of the Error Dynamics . . . . . 5.4.2 Choice of Gain Matrix for State Feedback Law . . . . . 5.5 Experimental Validation of Proposed Optimal-DTFC2 . . . . . . 5.5.1 Start-Up Speed Response . . . . . . . . . . . . . . . . . . . . . 5.5.2 Speed Reversal and Steady-State Response . . . . . . . . 5.5.3 Effect of Parameter Variation on the Performance . . . 5.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6 Sliding Mode Based Combined Speed and Direct Thrust Force Control of a Linear Permanent Magnet Synchronous Motors . . 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Dynamic Model of the Linear PMSM in xy-Reference Frame 6.3 Sliding Mode Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.1 Fundamentals of Sliding Mode Control . . . . . . . . . . 6.3.2 Variable Structure Based Direct Torque Control . . . . 6.4 Sliding Mode Control with Augmented Integral Action . . . . . 6.4.1 Sliding Surface for Stator Flux Regulation . . . . . . . . 6.4.2 Sliding Surface for Speed Regulation . . . . . . . . . . . 6.4.3 Control Law for Stator Flux Regulation . . . . . . . . . . 6.4.4 Control Law for Combined Speed and Thrust Force Regulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.5 Chattering Reduction . . . . . . . . . . . . . . . . . . . . . . . 6.5 Experimental Validation of Proposed SM-DTFC1 . . . . . . . . . 6.5.1 Start-Up Speed Response . . . . . . . . . . . . . . . . . . . . 6.5.2 Speed Reversal and Steady-State Response . . . . . . . 6.5.3 Evaluation of Robustness to the Parameter Variation 6.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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166 169 169 170 172 176 177 178

7 Sensorless Control of a Linear Permanent Magnet Synchronous Motors Using a Combined Sliding Mode Adaptive Observer . . . . 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Dynamic State Space Model of Linear PMSM in xy-Reference Frame . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3 Combined Speed and Direct Thrust Force Control of Linear PMSM Based on Integral Sliding Mode Control . . . . . . . . . . . 7.3.1 Sliding Surface for Stator Flux Regulation . . . . . . . . . 7.3.2 Sliding Surface for Speed Regulation . . . . . . . . . . . . 7.3.3 Control Law for Stator Flux Regulation . . . . . . . . . . . 7.3.4 Control Law for Thrust Force Regulation . . . . . . . . .

. . 181 . . 181 . . 182 . . . . .

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184 185 185 185 186

Contents

xv

7.4

A Novel Combined Sliding Mode State Observer . . . . . . 7.4.1 Stability Analysis of the Proposed Observer . . . . 7.4.2 Gain Selection for the Proposed Observer . . . . . 7.4.3 Adaption Scheme for Speed Estimation . . . . . . . 7.4.4 Estimation of Stator Flux Magnitude and Thrust Force . . . . . . . . . . . . . . . . . . . . . . . 7.5 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.6 Experimental Evaluation of the Control Performance of SM-DTFC2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.6.1 Start-Up Response . . . . . . . . . . . . . . . . . . . . . . 7.6.2 Speed Reversal Response . . . . . . . . . . . . . . . . . 7.6.3 Robustness to Parameter Variation . . . . . . . . . . 7.7 Experimental Evaluation of the SM-Observer . . . . . . . . . 7.7.1 Speed Reversal Response . . . . . . . . . . . . . . . . . 7.7.2 Speed Reversal Response Without the Improved SM Function . . . . . . . . . . . . . . . . . . . . . . . . . . 7.7.3 Steady State Response . . . . . . . . . . . . . . . . . . . 7.7.4 Position Estimation . . . . . . . . . . . . . . . . . . . . . . 7.7.5 Flux Estimation . . . . . . . . . . . . . . . . . . . . . . . . 7.8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

8 Conclusions and Future Work . . . . . 8.1 Conclusions . . . . . . . . . . . . . . . 8.2 Main Contributions of the Book . 8.3 Future Work . . . . . . . . . . . . . . .

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Appendix A: Description of the Experimental Setup . . . . . . . . . . . . . . . . 215 Appendix B: Derivation of Expressions for

dFT dt

and

dks dt .

..............

221

List of Figures

Fig. 1.1

Fig. 2.1 Fig. 2.2

Fig. 2.3

Fig. 2.4

Fig. 2.5

Fig. 2.6 Fig. 2.7

Fig. 2.8

Fig. 2.9

Schematic diagram of experimental setup showing the prototype linear PMSM, voltage source inverter and the control circuitry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Schematic diagram of a tubular surface-mount linear PMSM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Schematic diagram of the three-phase winding configuration for the linear PMSM. The orientation of the windings is arbitrarily shown and does not correspond to the physical magnetic axes of the machine as in the usual case of rotational machine [9, 10] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Space vector representation of stator phase voltages, a voltage space vector (green) and phase voltage vectors (blue), b synthesis of voltage space vector according to Eq. (2.18) . . Illustration and angular orientations and of stator current, permanent magnet flux linkage, and stator flux linkage space vectors stationary abc-reference frame. All the angles are measured from a-axis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Steady-state space vector diagram of the linear PMSM in stationary abc-reference frame according to Eq. (2.49). The length of all the space vectors is arbitrarily shown . . . . . The stator and mover’s flux linkages in various reference frames . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Equivalent circuit of the tubular surface-mount linear PMSM, a d-axis Equivalent circuit according to Eq. (2.72), b q-axis Equivalent circuit according to Eq. (2.73) . . . . . . . . . . . . . . . Steady-state space vector diagram of the linear PMSM in dq-reference frame as Eq. (2.80). The length of all the space vectors are arbitrarily shown . . . . . . . . . . . . . . . . . . . . . . . . . Steady-state space vector diagram of the linear PMSM in xy-reference frame according to Eq. (2.131). The length of all the space vectors is arbitrarily shown . . . . . . . . . . . . . .

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36 xvii

xviii

Fig. 2.10

Fig. 3.1

Fig. 3.2 Fig. 3.3

Fig. 3.4

Fig. 3.5

Fig. 3.6

Fig. 3.7

Fig. 3.8

Fig. 3.9

Fig. 3.10

Fig. 3.11

List of Figures

Steady-state space vector diagram of the linear PMSM I s is neglected. in xy-reference frame when the resistive drop Rs~ The length of all the space vectors is arbitrarily shown . . . . . Voltage space vectors generated by a 2-level VSI [5, 6] and control of magnitude and rotation of stator flux vector. Note the magnitude variation of stator flux is kept within the hysteresis band (diagram is not to scale) . . . . . . . . . . . . . Block diagram of the conventional DTFC scheme . . . . . . . . . Rate of change of stator flux for active (V1 to V6) and zero (V0 and V7) voltage vectors (vm ¼ 600 mm=s and operating thrust force F0 ¼ 53 N) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Rate of change of thrust force for active (V1 to V6) and zero (V0 and V7) voltage vectors (vm ¼ 600 mm=s and operating thrust force F0 ¼ 53 N) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Variation in stator flux for active (V1 to V6) and zero (V0 and V7) voltage vectors during one sampling period Ts ¼ 100 ls (vm ¼ 600 mm=s and operating thrust force F0 ¼ 53 N) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Variation in thrust force for active (V1 to V6) and zero (V0 and V7) voltage vectors during one sample time Ts ¼ 100 ls (vm ¼ 600 mm=s and operating thrust force F0 ¼ 53 N) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Variation in peak value of the negative rate of change of thrust force dFdtT , caused by any of the zero voltage vectors (V0 and V7), at different speed for various values of operating thrust force F0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Variation in the peak value of positive rate of change of thrust force dFdtT , caused by any of the active voltage vectors (V1 to V6), at different speed for various values of operating thrust force F0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . From top to bottom, speed response, thrust force response, flux response, and stator phase a current response under conventional DTFC. a Start-up from 0 to 200 mms−1, b speed reversal from −600 to 600 mm/s (experiment) . . . . . . . . . . . . Illustration of duty ratio control, the active voltage vector is applied at the beginning of the sample time for a duration t1 followed by a zero vector applied for the rest of the sampling period Ts . s1 is the thrust force slope (rate of change) when active voltage vector is applied and s2 the thrust force slope (rate of change) when zero voltage vector is applied (diagram not to scale). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Block Diagram of the duty ratio controlled DTFC scheme, duty ratio is computed using (3.52) for the state of the art [39] and for proposed method (3.56) is used . . . . . . . . . . . . . . . . .

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List of Figures

Fig. 3.12

Fig. 3.13

Fig. 3.14

Fig. 3.15

Fig. 3.16

Fig. 3.17

Fig. 3.18

Fig. 3.19

Fig. 3.20

Fig. 3.21

Fig. 3.22

Fig. 3.23 Fig. 3.24

Illustration of proposed duty ratio control, the active voltage vector is applied at the beginning of the sample time for duration t1 followed by a zero vector applied for the rest of the sampling period Ts . Dotted line shows the variation in thrust force if the active vector would have been applied for full sampling period (diagram not to scale) . . . . . . . . . . . . . . . . . Start-up performance from 0 to 80 mm/s with outer speed loop closed. Speed, force, flux, duty ratio and stator phase currents responses are shown from top to bottom respectively for both the DTFC1 (CT = 120 N, CF = 3 Wb) and the novel DTFC (experiment) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Magnified view of the speed response during start-up. a DTFC1 (CT = 120 N, CF = 3 Wb), b novel DTFC (experiment) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Error plots for speed and thrust force during startup transient. a DTFC1 (CT = 120 N, CF = 3 Wb) and, b novel DTFC (experiment) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Speed reversal from −600 to 600 mm/s and steady-state response at 600 mm/s with outer speed loop closed. Speed, force, flux, duty ratio and stator phase a current responses are shown from top to bottom respectively for both the DTFC1 (CT = 120 N, CF = 3 Wb) and novel DTFC (experiment) . . . Magnified view of the speed reversal transient illustrating the rise times. a DTFC1 (CT = 120 N, CF = 3 Wb) and, b novel DTFC (experiment) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Error plots for speed during speed reversal transient. a DTFC1 (CT = 120 N, CF = 3 Wb) and, b novel DTFC (experiment) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Thrust force response during steady-state at 600 mm/s, a DTFC1 (CT = 120 N, CF = 3 Wb) and, b novel DTFC (experiment) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Duty-ratio dF , force error eF , CT for DTFC 1 (CT = 120 N, CF = 3 Wb) and DFT for novel DTFC (experiment). a Whole duration of steady-state from 0.3 to 0.7 s. b Magnification to 10 sampling periods from 0.35 to 0.351 s, each time division is equal to 2.5 sampling periods . . . . . . . . . . . . . . . . . . . . . . . ab-axes stator voltages, currents and flux components during steady-state at 600 mm/s, a DTFC1 (CT = 120 N, CF = 3 Wb) and, b novel DTFC (experiment) . . . . . . . . . . . . . . . . . . . . . . Variation in average steady-state error and percent ripple in force with speed for both DTFC1 (CT = 120 N, CF = 3 Wb) and the novel DTFC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Flux trajectories at various speeds (experiment) . . . . . . . . . . . Transient response of force under the DTFC1 (experiment) . .

xix

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xx

Fig. 3.25 Fig. 4.1

Fig. 4.2 Fig. 4.3 Fig. 4.4 Fig. 4.5

Fig. 4.6 Fig. 4.7 Fig. 4.8 Fig. 4.9 Fig. 4.10 Fig. 4.11 Fig. 4.12 Fig. 4.13

Fig. 4.14

Fig. 4.15 Fig. 4.16 Fig. 4.17 Fig. 4.18

List of Figures

Transient response of force under the novel DTFC (experiment) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Thrust force versus load angle for the surface-mount linear PMSM of Table 1.1, the exact curve is according to (4.5) and the linearized curve is according to (4.6) with ks ¼ 0:0846 Wb, according to (4.7) under (MFPA) . . . . . . . . . . . . . . . . . . Characteristics of K according to (4.10) with ks ¼ 0:0846 Wb . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Block diagram of stator flux regulation according to (4.11) . . Block diagram of thrust force regulation according to (4.20) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . PI controller based direct thrust force control (PI-DTFC) of the linear PMSM with space vector pulse width modulation (SV-PWM) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Block diagram of the stator flux control loop with transport delay and disturbance cancellation (4.11) and (4.26) . . . . . . . Block diagram of the thrust force control loop with transport delay and disturbance cancellation (4.20) and (4.50) . . . . . . . Discrete time stator flux control loop for the surface-mount linear PMSM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The root locus and the selected closed loop poles for the stator flux controller loop . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Discrete time thrust force control loop for the linear PMSM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The root locus and the selected closed loop poles for the thrust force control loop . . . . . . . . . . . . . . . . . . . . . . . Open loop frequency response of the thrust force control loop . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Force step response with corresponding error plots for different values of the damping ratio nd : a nd ¼ 0:95, b nd ¼ 0:9, c nd ¼ 0:707, d nd ¼ 0:50 (experiment, reference force is shown in red and estimated force is shown in green) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Open loop frequency response of thrust force control loop for various values of stator resistance Rs . (nd ¼ 0:95, kp _F ¼ 0:0480, and ki0 _F ¼ 0:0174) . . . . . . . . . . . PI controller with output limiter . . . . . . . . . . . . . . . . . . . . . . . PI controller with limited output and anti-windup scheme for the integrator [11] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Implementation of the voltage limiter for the PI controller . . . Type-0 servo system with full state feedback and integral action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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. . 111

. . 112 . . 113 . . 113 . . 114 . . 117

List of Figures

Fig. 4.19

Fig. 4.20 Fig. 4.21 Fig. 4.22

Fig. 4.23

Fig. 4.24 Fig. 4.25

Fig. 4.26 Fig. 4.27

Fig. 5.1 Fig. 5.2

Fig. 5.3

Fig. 5.4 Fig. 5.5 Fig. 5.6 Fig. 5.7

Proposed linear quadratic regulator based direct thrust control (Optimal-DTFC1) of linear PMSM, the integral action is added by state augmentation. . . . . . . . . . . . . . . . . . . . . . . . Force step response of PI-DTFC (experiment) . . . . . . . . . . . . Force step response of Optimal-DTFC1 . . . . . . . . . . . . . . . . . From top to bottom the force response, magnified force response, steady-state error in the force and steady-state stator flux response is shown, a PI-DTFC, and b Optimal-DTFC1 . Start-up performance from 0 to 200 mm/s with outer speed loop closed. Speed, force, stator flux, and stator phase currents responses are shown from top to bottom respectively for both a the PI-DTFC and b Optimal-DTFC1 (experiment) . . . . . . . Magnified view of the speed response during start-up, a PI-DTFC, and b Optimal-DTFC1 (experiment) . . . . . . . . . . Speed reversal from −600 to 600 mm/s and steady-state response at 600 mm/s with outer speed loop closed. Speed, force, stator flux, duty ratio and stator phase a current responses are shown from top to bottom respectively for both a the PI-DTFC and b Optimal-DTFC1 (experiment) . . . . . . . Magnified view of the speed reversal transient a PI-DTFC, and b Optimal-DTFC1 (experiment) . . . . . . . . . . . . . . . . . . . Steady-state performance at 600 mm/s. From top, speed, force, and stator flux responses are shown for both a the PI-DTFC and b Optimal-DTFC1 (experiment) . . . . . . . . . . . . . . . . . . . Type-0 servo system with full state feedback and integral action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Proposed linear quadratic regulator based combined speed and thrust control of linear PMSM, the integral action is added by state augmentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Start-up performance from 0 to 200 mm/s. Speed, thrust force, stator flux, and stator phase currents responses are shown from top to bottom respectively for both a the PI-DTFC and b optimal-DTFC2 (experiment) . . . . . . . . . . . . . . . . . . . . Magnified view of the speed response during start-up, a PI-DTFC, and b optimal-DTFC2 (experiment) . . . . . . . . . . Magnified view of the speed response during start-up, a PI-DTFC, and b optimal-DTFC2 (experiment) . . . . . . . . . . Magnified thrust force response during start-up, a PI-DTFC, and b optimal-DTFC2 (experiment) . . . . . . . . . . . . . . . . . . . . Speed reversal from −600 to 600 mm/s and steady-state response at 600 mm/s. Speed, force, flux, and stator phase “a” current responses are shown, a PI-DTFC, and b optimal-DTFC2 (experiment) . . . . . . . . . . . . . . . . . . . . . . .

xxi

. . 118 . . 125 . . 126

. . 127

. . 128 . . 128

. . 130 . . 131

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xxii

Fig. 5.8 Fig. 5.9 Fig. 5.10

Fig. 5.11

Fig. 6.1

Fig. 6.2

Fig. 6.3

Fig. 6.4 Fig. 6.5 Fig. 6.6 Fig. 6.7

Fig. 6.8 Fig. 6.9 Fig. 6.10

Fig. 7.1

Fig. 7.2 Fig. 7.3

List of Figures

Magnified view of the speed reversal transient, a PI-DTFC, and b optimal-DTFC2 (experiment) . . . . . . . . . . . . . . . . . . . . Error plots for the speed response during speed reversal transient, a PI-DTFC, and b Optimal-DTFC2 (experiment) . . Steady-state performance at 600 mm/s. Speed, thrust force, and stator flux responses are shown, a PI-DTFC, and b optimal-DTFC2 (experiment) . . . . . . . . . . . . . . . . . . . . . . . Effect of parameter variation on the transient performance of optimal-DTFC2 during speed reversal from −600 to 600 mm/s, a Rs is increased by 100%, b Ls is increased by 100% (experiment) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Schematics describing the sliding mode control with first order plus integral sliding condition for non-linear combined speed and thrust force dynamics proposed in (6.5) and (6.5a) . . . . . Proposed sliding mode based combined speed and thrust control of linear PMSM (SM-DTFC), the integral action is added by modification of the reachability condition . . . . . . Start-up performance from 0 to 200 mm/s. Speed, thrust force, stator flux, and stator phase currents responses are shown from top to bottom respectively for both a the PI-DTFC and b SM-DTFC1 (experiment) . . . . . . . . . . . . . . . . . . . . . . . Magnified view of the speed response during start-up, a PI-DTFC, and b SM-DTFC1 (experiment) . . . . . . . . . . . . . Magnified view of the speed error during start-up, a PI-DTFC, and b SM-DTFC1 (experiment) . . . . . . . . . . . . . . . . . . . . . . . Magnified thrust force response during start-up, a PI-DTFC, and b SM-DTFC1 (experiment) . . . . . . . . . . . . . . . . . . . . . . . Speed reversal from −600 to 600 mm/s and steady state response at 600 mm/s. Speed, force, flux, and stator phase “a” current responses are shown, a PI-DTFC, and b SM-DTFC1 (experiment) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Magnified view of the speed reversal transient, a PI-DTFC, and b SM-DTFC1 (experiment) . . . . . . . . . . . . . . . . . . . . . . . Error plots for the speed response during the speed reversal transient, a PI-DTFC, and b SM-DTFC1 (experiment) . . . . . Steady state performance at 600 mm/s. Speed, thrust force, and stator flux responses are shown, a PI-DTFC, and b SM-DTFC1 (experiment) . . . . . . . . . . . . . . . . . . . . . . . . . . Schematic illustration of the non-linear combined speed and thrust force dynamics proposed in (7.2) and (7.2a) and the proposed sliding mode control with integral action of (7.31) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Block diagram of proposed observer . . . . . . . . . . . . . . . . . . . Block diagram illustration of improved “sgnm ” function . . . .

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. . 189 . . 190 . . 191

List of Figures

Fig. 7.4

Fig. 7.5

Fig. 7.6

Fig. 7.7 Fig. 7.8

Fig. 7.9 Fig. 7.10

Fig. 7.11

Fig. 7.12 Fig. 7.13

Fig. 7.14

Fig. 7.15

Fig. 7.16 Fig. 7.17

Block diagram of the proposed combined sliding mode speed and direct thrust control of linear PMSM with the novel adaptive observer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Startup performance from 0 to 200 mm/s. Measured Speed, thrust force, stator flux, and stator phase currents responses are shown from top to bottom respectively. a PI-DTFC and b SM-DTFC2 (experiment) . . . . . . . . . . . . . . . . . . . . . . . Magnified view of the measured speed and thrust force response during start-up. a PI-DTFC and b SM-DTFC2 (experiment) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Error plots for speed start-up. a PI-DTFC and b SM-DTFC (experiment) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Speed reversal from −600 to 600 mm/s and steady-state response at 600 mm/s. Measured and estimated Speed, thrust force, stator flux, duty ratio and stator phase “a” current responses are shown from top to bottom. a PI-DTFC and b SM-DTFC2 (experiment) . . . . . . . . . . . . . . . . . . . . . . . Magnified view of the measured and estimated speed response during start-up. a PI-DTFC and b SM-DTFC (experiment) . . Magnified view of the measured and estimated speed response during steady-state. a PI-DTFC and b SM-DTFC2 (experiment) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Command voltage vy to control thrust force, generated by the PI-DTFC and the proposed SM-DTFC2. a Start-up and b Speed reversal (experiment) . . . . . . . . . . . . . . . . . . . . . Integral sliding surface sv according to (7.7). a Start-up and b Speed reversal (experiment) . . . . . . . . . . . . . . . . . . . . . Speed reversal response. From top speed response, speed error, thrust force and stator flux, a −200 to 200 mm/s, b −600 to 600 mm/s (experiment) . . . . . . . . . . . . . . . . . . . . . Speed reversal response without the sliding mode component. From top, speed response, speed error, and flux responses are shown, a −200 to 200 mm/s, b −600 to 600 mm/s (experiment) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Steady-state response. From top, speed response, speed error, thrust force and stator flux are shown, a 200 mm/s, b 600 mm/s (experiment) . . . . . . . . . . . . . . . . . . . . . . . . . . . . Position estimation. a 200 mm/s, b 600 mm/s (experiment) . . kd and kq estimation errors with sliding mode component (novel) and without sliding mode component (linear), a 200 mm/s, b 600 mm/s (experiment) . . . . . . . . . . . . . . . . .

xxiii

. . 195

. . 197

. . 198 . . 198

. . 199 . . 199

. . 200

. . 200 . . 201

. . 202

. . 203

. . 204 . . 205

. . 206

xxiv

Fig. Fig. Fig. Fig.

List of Figures

A.1 A.2 A.3 A.4

Experimental setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . Schematic circuit diagram of voltage sensing board . . . Schematic circuit diagram of current sensing board . . . Architectural overview of DS 1104 controller board . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

216 217 218 219

List of Tables

Table 1.1 Table 3.1 Table 3.2 Table 3.3 Table 3.4 Table 4.1

Table 4.2 Table 4.3

Table 4.4 Table 4.5

Table 4.6 Table 5.1 Table 5.2 Table 5.3 Table 6.1

Parameters of the prototype tubular surface-mount linear PMSM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Switching table to generate Vk for DTFC [13] . . . . . . . . . . . . ab-components of inverter voltage vectors . . . . . . . . . . . . . . Comparison of transient performance of DTFC1 and the novel DTFC using IAE index . . . . . . . . . . . . . . . . . . Comparison of steady-state performances of the DTFC1 for various values of CF and CT with the novel DTFC . . . . . Comparison of phase margin /m , crossover frequency, xc and closed loop bandwidth xb for various values of damping ratio nd . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Comparison of force step response for various values of the damping ratio nd . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Comparison of phase margin /m , crossover frequency xc and closed loop bandwidth xb for various values of stator resistance Rs ¼ 3:01 X . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Controller gains used in experiment for the performance comparison of Optimal-DTFC1 and PI-DTFC . . . . . . . . . . . . Comparison of rise time, percent overshoot, and percent steady-state ripple in the response of thrust force PI controller under PI-DTFC for various damping ratios . . . . . . . . . . . . . . Comparison of steady-state performance of PI-DTFC and the Optimal-DTFC1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . Controller gains used in experiment for the optimal-DTFC2 and PI-DTFC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Comparison of transient performance of PI-DTFC and the optimal-DTFC2 using IAE index . . . . . . . . . . . . . . . Comparison of steady-state performance of DTFC1 and the optimal DTFC at 600 mm/s . . . . . . . . . . . . . . . . . . . Comparison of transient performance of PI-DTFC and the SM-DTFC1 using IAE index . . . . . . . . . . . . . . . . . .

.. .. ..

7 46 54

..

70

..

73

. . 109 . . 111

. . 112 . . 123

. . 125 . . 131 . . 146 . . 148 . . 152 . . 172 xxv

xxvi

Table 6.2 Table 6.3 Table 6.4 Table 6.5 Table 7.1 Table 7.2 Table 7.3 Table 7.4 Table 7.5 Table 8.1

Table 8.2 Table A.1

List of Tables

Comparison of steady state performance of PI-DTC and the SM-DTFC1 at 600 mm/s, 52 N . . . . . . . . . . . . . . . . Comparison of rise time of PI-DTFC, optimal-DTFC2, and SM-DTFC1 with variation in Rs . . . . . . . . . . . . . . . . . . . Comparison of rise time of PI-DTFC, optimal-DTFC2, and SM-DTFC1 with variation in Ls . . . . . . . . . . . . . . . . . . . Comparison of steady-state performance of PI-DTFC, Optimal-DTFC2, and SM-DTFC1 with variation in Ls . . . . . Controller gains used for proposed SM-DTFC2. . . . . . . . . . . Controller gains used in experiment for PI-DTFC . . . . . . . . . Comparison of Transient Performance of PI-DTFC and the SM-DTFC Using IAE Index . . . . . . . . . . . . . . . . . . . Comparison of steady-state performance of PI-DTFC and the SM-DTFC2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Comparison of robustness to parameter variation, increase in speed ripple for PI-DTFC and SM-DTFC2 . . . . . Comparison of transient performance in terms of rise time of PI-DTFC, optimal-DTFC1, optimal-DTFC2, and SM-DTFC1 using IAE index . . . . . . . . . . . . . . . . . . . . . . . . . Comparison of steady-state performance of PI-DTFC, optimal-DTFC1, optimal-DTFC2 and SM-DTFC1 . . . . . . . . . Technical specifications of the DS1104 controller board . . . .

. . 175 . . 176 . . 176 . . 177 . . 196 . . 196 . . 198 . . 200 . . 201

. . 213 . . 213 . . 219

Chapter 1

Introduction

1.1 Motivation and Scope The application areas for linear permanent magnet synchronous motors (linear PMSMs) are increasing to include automation of manufacturing processes, transportation, renewable energy devices, and pumping applications. This ever-growing application scope for linear PMSMs implies the need to develop robust, simple and fast control mechanisms for linear PMSMs, to suit the essential requirements for its industrial applications. The application of linear PMSMs in industrial automation processes often involves servomechanisms that necessitate fast transient response of thrust force with minimal steady state ripple as an essential performance requirement for the control systems. A control system can be designed to have low damping to achieve a faster transient response of thrust force; however, deterioration in the steady state performance of thrust force response is caused by the low damping. Therefore, the control system is designed to achieve a balance between the transient and steady state performance. In general, linear PMSMs have a low value of stator inductance compared to their rotational counterparts due to a relatively larger air gap. Another important feature of the linear PMSMs that plays a pivotal role in thrust force regulation is a fractional value of pole-pitch. The low stator inductance and short pole-pitch results in a significantly lower electrical time constant of the thrust force regulation dynamics (described by a first order transfer function) for linear PMSMs compared to rotational PMSMs. This short time constant in the thrust force dynamics results in two main challenging problems which are key motivations for this book. First, linear PMSMs tend to exhibit higher ripple in the thrust force under conventional direct thrust force control schemes (conventional-DTFC) which is typically beyond acceptable limits. Secondly, under PI based DTFC, when the PI controller is designed to achieve a fast transient response in the thrust force, the overshoot in the thrust force response is unacceptably large due to the small damping ratio caused by the low inductance and short pole-pitch. © Springer Nature Switzerland AG 2020 M. A. M. Cheema and J. E. Fletcher, Advanced Direct Thrust Force Control of Linear Permanent Magnet Synchronous Motor, Power Systems, https://doi.org/10.1007/978-3-030-40325-6_1

1

2

1 Introduction

When the PI controller for thrust force regulation is designed to achieve a higher damping ratio, the transient response then becomes slow. This book primarily focuses on the improvements to DTFC of a tubular surfacemount linear PMSM and proposes and experimentally validates several novel DTFC based control schemes.

1.2 Types of Linear Permanent Magnet PMSMs There are various types of linear PMSMs based on their configurations. These configurations include flat single-sided, flat double-sided and tubular structures. In all these configurations the stator winding may constitute the mover or, more likely, the mover will hold the magnets. The core of the stator winding of a flat (single or double-sided) linear PMSM is made of longitudinal laminations with uniformly distributed slots, which house the windings. The windings are typically three-phase distributed windings and generally have one to four slots/pole per phase [1]. In tubular structures the laminations of the stator windings may be either longitudinal or disk-shaped as in linear induction actuators. The mover of a tubular linear machine is generally made of solid magnetic steel. Rare-earth permanent magnets (PMs) in the mover of the tubular linear PMSM are capable of producing adequate flux densities with ~1-mm effective air-gaps. Because of ease of assembly, tubular linear PMSMs with disk-shaped laminations are often preferred to flat geometries. In order to reduce eddy current losses, the disk laminations may be slitted radially [1]. As mentioned earlier, the mover of the tubular linear PMSM comprises permanent magnets placed back to back in a steel tube. If these magnets are not separated by any other material, a uniform air-gap is produced and the linear PMSM will not have any saliency and is regarded as tubular surface-mount linear PMSM. When the mover’s magnets are separated by any other nonmagnetic material, the non-uniform air-gap causes saliency in the machine which can be regarded as tubular interior-type linear PMSM. The prototype used in this book for experimental validation is a tubular surface-mount linear PMSM.

1.3 A Review of Developments in Direct Thrust Force Control of Linear PMSM The concept of DTFC for linear PMSMs is derived from direct torque control (DTC) scheme which was first introduced in 1980s for induction motors [2–4]. Since its introduction, it has become a popular control scheme for its simple structure and a fast dynamic response. The key advantage of DTC lies in its simple structure which makes the control performance robust to parameter variation resulting in increased

1.3 A Review of Developments in Direct Thrust Force Control …

3

robustness. Due to these advantages, the application of the DTC scheme has been extended to PMSMs and significant literature is available on the subject. An initial concept of closed loop torque control of PMSMs by regulation of stator currents is presented in [5]. The application of the DTC algorithm extended to PMSMs has been presented in [6, 7]. This research also analyses the torque regulation of PMSM with the DTC algorithm. A study of conventional-DTC of the PMSM considering the field-weakening (FW) operation has been detailed in [8]. An application of DTC with a modified switching table, excluding the zero voltage vectors in contrast to the switching table of [2, 3], is presented in [9]. This work illustrates the effect of zero voltage vectors on torque and flux regulation of PMSMs and proposes that stable DTC performance can be achieved without using zero vectors in the switching table. There are also some inherited disadvantages of conventional switching-tablebased direct torque control (ST-DTC) which include large torque ripple and variable switching frequency. A simulation based study on the stability of DTC of PMSMs shows that the ripple in torque response of ST-DTC can be reduced if the duty ratio of the DC-link voltage is controlled in proportion to the rotor speed [10]. However, a fixed duty ratio may reduce the ripple but it will also result in steady state torque error. The work in [10] does not account for the calculation of duty ratio for the DC-link voltage. The application of ST-DTC to PMSMs considering the maximum torque per ampere (MTPA) based trajectory control is given in [11]. A study related to stator flux estimation issues in ST-DTC of PMSMs is detailed in [12]. It is important to note that the literature [2–12] does not provide insight on selection of hysteresis bands for flux and torque controllers and how the parameters of the PMSM affect the selection of these bands. In addition, the effect of low stator inductance of the machine on the performance of ST-DTC has not been analysed. In order to reduce the ripple produced in torque and flux by ST-DTC and to achieve a fixed switching frequency, a number of methods have been proposed in the literature. These methods can be divided into different categories. Space vector modulation (SV-PWM) is one of the popular methods that can be employed in the implementation of DTC of PMSMs to reduce the ripple in both torque and flux response. The switching table of conventional ST-DTC comprises a limited number of voltage vectors with fixed amplitudes and positions, while SVPWM is capable of generating an arbitrary voltage vector with a variable length and angle. SV-PWM is capable of more accurate and moderate torque and flux regulation with fixed switching frequency. Another feature of SV-PWM is that the sampling frequency is significantly lower compared to that of conventional ST-DTC. The main point of focus in SV-PWM is how to generate the command voltage vector. Significant research is available on various methods that can be employed to generate the command voltage vector in SV-PWM based direct torque control of PMSMs. These methods include flux vector error compensation based DTC [13], variable structure based torque and flux control [14], decoupled control of torque and flux control using PI controllers [15], and torque error compensation based SV-PWM based DTC [16]. A method based on a combination of sliding mode control with PI based control for DTC of the induction machine has been discussed in [17]. In [18]

4

1 Introduction

the command voltage vector is computed from the magnitude of torque error, flux error and the sign of flux error, i.e. the modulation index is achieved by dividing the torque and flux error magnitudes by two constant gains. It is however, important to note that higher values of these gains may lead to steady state error in torque response. Another method reported in [19] employs a predictive torque controller to generate the command voltage vector for SV-PWM based DTC. The methods detailed in [13–19] are based on SV-PWM based DTC and present low ripple in torque and flux response, however these methods involve rotary co-ordinate transformation, extensive knowledge of machine parameters and high computational ability which is in contrast to the simplicity and robustness of conventional ST-DTC. A survey based on review of various DTC strategies for inverter-fed AC machine has been given by [20], however it does not include any details which explain the impact of machine inductance on the performance of digital controller based implementation of ST-DTC. Recently, model predictive based DTC (MP-DTC) has attracted the focus of academic and industrial communities. Model Predictive DTC (MP-DTC) [21–29] schemes are presented for PMSMs that effectively reduce the ripple in the stator flux and torque simultaneously. However, these control schemes are parameter dependent and a parameter mismatch can deteriorate the performance of these schemes. In SV-PWM based DTC, the inverter and modulation are considered as a constant gain for the purpose of controller design, while in MP-DTC, the inverter and machine are considered from a system point of view. In MP-DTC, a cost function is defined to evaluate the effect of every available voltage vector on torque and flux regulation and the vector with minimum value cost function is applied to the PMSM. The SV-PWM based DTC and MP-DTC provide excellent dynamic and steady state performance at a low switching frequency. However, for these methods, the system complexity, parameter dependency, and computational effort are also increased significantly. Therefore, performance improvement of DTC without compromising the simplicity and robustness of control structure is also of critical importance. In recent years, the concept of direct torque control has also been extended to linear PMSMs as direct thrust force control (DTFC). The ST-DTC and DTFC both are similar; however, the only difference is that, in DTFC, the electromagnetic thrust of the mover is controlled directly instead of torque as in the case of ST-DTC. It is also important to note that the same switching scheme for ST-DTC of [6, 7] can also be used for DTFC of the linear PMSM. DTFC of the linear PMSM has not been discussed significantly in the literature and therefore there is considerable room to explore the performance of conventional switching table based ST-DTFC. A study of ST-DTFC including the maximum force per ampere (MFPA) and field weakening (FW) operation has been presented in [30], however this research is based only on simulations and does not discusses the thrust force and force regulation of linear PMSM under ST-DTFC. Research based on ST-DTFC with experimental results is detailed in [31]. In this work the switching table of [6] is utilized, however this research is more focused on genetic algorithm based generation of gains for the speed PI controller, and also discusses a low pass filter based flux estimation of stator flux. The experimental results of the research

1.3 A Review of Developments in Direct Thrust Force Control …

5

presented in [31] do not include any results for the thrust force and flux regulation, moreover the ripple in thrust force has also not been discussed in this work. Another simulation based study of ST-DTFC is given by [32, 33], this research is a straight forward implementation of ST-DTC [6] for the linear PMSM, in addition simulation results exhibit significant ripple in thrust force and flux response. SV-PWM has also been utilized in DTFC of the linear PMSM in order to achieve a smooth thrust force and flux response. In [34] an SV-PWM based DTFC of the linear PMSM is presented, in this work the reference voltage vector is computed by utilizing the force error amplitude and the sign of the flux error. Apparently, this method is an extension of work in [18] to the linear PMSM. However, this research does not include experimental results. A SV-PWM based DTFC based on decoupled control of thrust force and torque is presented in [35]. Experimental results of this research show that excellent thrust force and flux response with reduced ripple can be achieved at a fixed switching frequency using SV-PWM. Another approach to generate the command voltage for SV-PWM based DTFC is based on variable structure control detailed in [36–38]. A fuzzy logic based implementation of DTFC is proposed in [39]. Simulation results indicate reduction in thrust force ripple. The methods presented in [30–39] demonstrate improvements in thrust force ripple, however their complex structures require more computational effort as compared to ST-DTFC and the robustness and simplicity of the control architecture is also compromised. A novel DTC scheme for PMSM with reduced ripple in torque response is given in [40], the proposed method utilizes carrier based PWM, and the command voltages are generated from torque and flux references. The torque reference is produced by using an anti-windup based PI controller, and the flux reference is computed from a look-up table based MTPA/field-weakening (FW) strategy. However, this approach is computationally intensive and involves several coordinate transformations. In the literature, considerable research investigates performance improvements of ST-DTC. The size of hysteresis band for the torque and flux controller affects the switching frequency of ST-DTC, i.e. the smaller the hysteresis band the larger the switching frequency. Recent research in [41] proposes a new method for the selection of optimal hysteresis bands according to the minimum distortion in motor current of PMSM criteria. This research provides an offline analytical method to compute these bands for ST-DTC of the PMSM. However, the offline calculation of hysteresis bands may deteriorate the control performance in the case of parameter variation, or the use of incorrect parameters. Moreover, this research does not provide any details on the minimum value of hysteresis bands that can be selected, for a particular machine, based on its design parameters particularly the machine inductance. It is important to note that the performance of the conventional ST-DTFC algorithm is affected by machine inductance. Whenever an active voltage is applied to linear PMSM it produces a change in thrust force, in order to keep the thrust force ripple within the hysteresis band. The hysteresis band must be greater than the maximum change in the thrust force that can be produced during a sample period. Otherwise the ST-DTFC will fail to keep the ripple within the hysteresis bands. Moreover, if the machine inductance is very low, an active voltage vector will yield a much higher

6

1 Introduction

change in the thrust force and the minimum possible value of the hysteresis band may exceed 10% of the rated thrust force of the machine. The literature [2–41] does not provide any details of the quantitative effect of the low machine inductance on the performance of DTC and its remedies. Another modification to DTC, which effectively reduces the torque ripple while retaining the robust control structure of basic DTC, is duty ratio control. In this method, one sample time is divided into several intervals and the duty ratio of the active voltage vector is determined by using various approaches reported in [42–49]. The method reported in [42] calculates the optimal duty ratio for DTC of induction machines by solving a quadratic equation such that the instantaneous torque should be equal to the reference torque at the end of each cycle, acting in a deadbeat fashion. The approach presented in [42] was developed in the stationary reference frame that was later revisited in the synchronously rotating stator flux reference frame [43]. Another approach that calculates the duty ratio such that the mean torque over a sample time should be equal to the reference torque is reported in [44] and [45] for induction machines and PMSMs respectively; this approach is referred to as direct mean torque control. The research detailed in [46] proposes a duty ratio calculation method for DTC controlled induction machines based on minimizing the RMS torque ripple value over one cycle and also ensures a fixed average switching frequency. The approach in [46] is significantly improved in [47] and guarantees the global minimization of the RMS torque ripple over one switching cycle by using a modified switching pattern. A study presented in [10] shows that the duty ratio in a DTC controlled PMSM should vary in proportion to the rotor speed. The duty ratio control based DTC schemes in [42–47] have been proven effective in torque ripple reduction however the duty ratio calculation procedures are complex and dependent on machine parameters. Recent research reported in [48] carefully examines the methods presented in [42–47] and proposes a relatively simple method to calculate the duty ratio for DTC controlled PMSMs. According to [48] the duty ratio for the active voltage vector is computed by dividing the absolute values of flux and torque error by two static gains and then adding them, in contrast to [10] the rotor speed is not considered in this method. The main disadvantage of this approach is the tuning of the static gains and the significant steady state error in the torque response. An improvement to [48] is reported in [49] for rotational PMSM, where the rotor speed is also taken into account for calculation of the duty ratio and steady-state error in torque is significantly reduced. In [50] various duty-ration based methods which also consider the ripple minimization of stator flux in addition to the torque ripple are proposed and compared. It is important to note that the duty ratio control methods of [42–48] have not been evaluated for DTFC of the linear PMSM. In addition, the only literature related to the duty ratio control of DTFC is [10]. In [10] the duty ratio based DTFC is briefly discussed, however no method to calculate the duty ratio is discussed. In general, most linear PMSMs have a low value of stator inductance due to their larger air gap compared to rotational PMSMs. Also, due to the linear motion, the pole pitch of the linear PMSMs also effects the force regulation. Therefore, DTFC controlled LPMSMs with low inductance and short pole-pitch exhibit large force ripple.

1.3 A Review of Developments in Direct Thrust Force Control …

7

The focus of this research book is to assess the impacts of these characteristics of linear PMSMs, investigate and propose potential solutions, and experimentally verify the solutions.

1.4 Description of Experimental Setup A block diagram of the experimental setup used in this research is illustrated in Fig. 1.1. The parameters of the prototype tubular surface-mount linear PMSM are listed in Table 1.1. The prototype linear PMSM is supplied by a three phase voltage source inverter and the DC bus voltage is maintained by three phase full bridge rectifier. The phase currents, mover’s position and the DC bus voltages are measured by appropriate sensors and are used as feedback for the controller. The digital controller is a dSPACE DS1104 research and development board. The relevant data sheets and circuit diagrams are provided in Appendix A. Rectifier

Voltage Source Inverter (VSI)

s1 vdc

s3

s5

+ -

Linear PMSM

s2

s4

s6

ia ib ic θr

s1-s6 (Gating Signals)

vdc

dSPACE DS1104 Controller

Fig. 1.1 Schematic diagram of experimental setup showing the prototype linear PMSM, voltage source inverter and the control circuitry

Table 1.1 Parameters of the prototype tubular surface-mount linear PMSM

Pole pairs

3

Max. current (A)

3.27

λ f PM Flux (mWb)

84.6

L s (mH)

1.95

Rs (ohms)

3.01

τ (m)

0.0256

Mover’s mass M (kg)

1.25

Rated cont. force (N)

52

Peak force (N)

312

8

1 Introduction

1.5 Book Outline This book comprises eight chapters. The chapters are titled: 1. Introduction This chapter provides the motivation, literature review of the state of the art direct thrust control schemes and describes the experimental setup used in this research. 2. Mathematical Modeling of Surface-Mount Linear PMSM A detailed mathematical modeling of the prototype linear PMSM in various reference frames is performed in this chapter. 3. Duty-ratio based Direct Thrust Force Control The concept of conventional direct thrust force control (conventional DTFC) is rigorously analysed and experimentally evaluated for the linear PMSM in this chapter. Linear PMSMs, especially those with low inductance and short pole pitch, exhibit significant ripple in flux and thrust force under conventional direct thrust force control. Moreover, a novel duty ratio control scheme for direct thrust force control of linear PMSM is also proposed. 4. SV-PWM Based Direct Thrust Control of a Linear Permanent Magnet Synchronous Motor This chapter proposes, experimentally validates and compares two schemes for direct thrust force control (DTFC) based on space vector pulse width modulation (SVPWM) for the prototype linear PMSM. The first direct thrust force control scheme is the PI controller-based regulation of the flux and thrust force and is referred to as “PI-DTFC”. The second direct thrust force control scheme is based on a linear quadratic regulator based control of the stator flux and thrust force and referred to as “Optimal-DTFC1”. 5. Optimal, Combined Speed and Direct Thrust Force Control of a Linear Permanent Magnet Synchronous Motors An optimal control scheme referred to as “Optimal-DTFC2” for combined speed and direct thrust force control based on SV-PWM is proposed and experimentally validated for the prototype linear PMSM. The combined speed and direct thrust force control is achieved by formulating the optimal linear state feedback control law using the linear quadratic regulator (LQR) based approach. 6. Sliding Mode based Combined Speed and Direct Thrust Force Control of a Linear Permanent Magnet Synchronous Motors This chapter proposes and experimentally validates a sliding mode control scheme for combined speed and direct thrust force control referred to as “SM-DTFC1” utilizing SV-PWM for the prototype linear PMSM.

1.5 Book Outline

9

7. Sensorless Control of a Linear Permanent Magnet Synchronous Motors Using a Combined Sliding Mode Adaptive Observer A combined sliding mode adaptive observer for flux and sensorless speed estimation of the direct thrust controlled, surface-mount linear PMSM is proposed. The observer comprises a linear state observer combined with a modified nonlinear sliding mode component. The sliding mode component is improved by using two boundary layers which reduces the chattering without compromising robustness. Moreover, an integral sliding mode control law to achieve combined control of the thrust force and mover’s speed is formulated. The proposed observer and the control law are also experimentally validated. 8. Conclusions and Future Work This chapter details the significance and contributions of this book and also provides the suggestions for future work.

1.6 Conclusions This chapter discusses the motivation for this book and provides the scope of the book. The type of linear PMSMs and developments pertinent to direct thrust force control of the linear PMSM are detailed. The experimental setup utilized in this book is introduced briefly. A brief book outline is also provided.

References 1. I. Boldea, S.A. Nasar, Linear Electric Actuaters and Gnerators (Cambridge University Press Inc., New York, 1997) 2. I. Takahashi, T. Noguchi, A new quick-response and high-efficiency control strategy of an induction motor. IEEE Trans. Ind. Appl. IA(22), 820–827 (1986) 3. I. Takahashi, Y. Ohmori, High-performance direct torque control of an induction motor. IEEE Trans. Ind. Appl. 25, 257–264 (1989) 4. U. Baader, M. Depenbrock, G. Gierse, Direct self control (DSC) of inverter-fed induction machine: a basis for speed control without speed measurement. IEEE Trans. Ind. Appl. 28, 581–588 (1992) 5. C. French, P. Acarnley, Direct torque control of permanent magnet drives. IEEE Trans. Ind. Appl. 32, 1080–1088 (1996) 6. M.F. Rahman, L. Zhong, W.Y. Hu, K.W. Lim, M.A. Rahman, A direct torque controller for permanent magnet synchronous motor drives, in Proceedings of the Electric Machines and Drives Conference Record (1997), pp. TD1/2.1–TD1/2.3 7. L. Zhong, M.F. Rahman, W.Y. Hu, K.W. Lim, Analysis of direct torque control in permanent magnet synchronous motor drives. IEEE Trans. Power Electron. 12, 528–536 (1997) 8. M.F. Rahman, L. Zhong, L. Khiang Wee, A direct torque-controlled interior permanent magnet synchronous motor drive incorporating field weakening. IEEE Trans. Ind. Appl. 34, 1246–1253 (1998)

10

1 Introduction

9. Y. Hu, C. Tian, Y. Gu, Z. You, L.X. Tang, M.F. Rahman, In-depth research on direct torque control of permanent magnet synchronous motor, in Proceedings of the Annual Conference of the Industrial Electronics Society (IECON), vol. 3 (2002), pp. 1060–1065 10. J. Linni, S. Liming, Stability analysis for direct torque control of permanent magnet synchronous motors, in Proceedings of the International Conference on Electrical Machines and Systems (ICEMS) (2005), pp. 1672–1675 11. M.E. Haque, M.F. Rahman, Incorporating control trajectories with the direct torque control scheme of interior permanent magnet synchronous motor drive. IET Electr. Power Appl. 3, 93–101 (2009) 12. M.F. Rahman, M.E. Haque, T. Lixin, Z. Limin, Problems associated with the direct torque control of an interior permanent-magnet synchronous motor drive and their remedies. IEEE Trans. Ind. Electron. 51, 799–809 (2004) 13. T. Lixin, Z. Limin, M.F. Rahman, H. Yuwen, A novel direct torque controlled interior permanent magnet synchronous machine drive with low ripple in flux and torque and fixed switching frequency. IEEE Trans. Power Electron. 19, 346–354 (2004) 14. X. Zhuang, M. Faz Rahman, Direct torque and flux regulation of an IPM synchronous motor drive using variable structure control approach. IEEE Trans. Power Electron. 22, pp. 2487–2498 (2007) 15. G. Foo, C.S. Goon, M.F. Rahman, Analysis and design of the SVM direct torque and flux controlled IPM synchronous motor drive, in Proceedings of the Australasian Universities Power Engineering Conference (AUPEC) (2009), pp. 1–6 16. Y. Cho, D.-H. Kim, K.-B. Lee, Y.I. Lee, J.-H. Song, Torque ripple reduction and fast torque response strategy of direct torque control for permanent-magnet synchronous motor, in Proceeding of the International Symposium on Industrial Electronics (ISIE) (2013), pp. 1–6 17. C. Lascu, A.M. Trzynadlowski, Combining the principles of sliding mode, direct torque control, and space-vector modulation in a high-performance sensorless AC drive. IEEE Trans. Ind. Appl. 40, 170–177 (2004) 18. Z. Yongchang, Z. Jianguo, X. Wei, G. Youguang, A simple method to reduce torque ripple in direct torque-controlled permanent-magnet synchronous motor by using vectors with variable amplitude and angle. IEEE Trans. Ind. Electron. 58, 2848–2859 (2011) 19. Z. Hao, X. Xi, L. Yongdong, Torque ripple reduction of the torque predictive control scheme for permanent-magnet synchronous motors. IEEE Trans. Ind. Electron. 59, 871–877 (2012) 20. G.S. Buja, M.P. Kazmierkowski, Direct torque control of PWM inverter-fed AC motors—a survey. IEEE Trans. Ind. Electron. 51, 744–757 (2004) 21. T. Geyer, G. Papafotiou, M. Morari, Model predictive direct torque control-part I-concept, algorithm, and analysis. IEEE Trans. Ind. Electron. 56, 1894–1905 (2009) 22. M. Preindl, S. Bolognani, Model predictive direct torque control with finite control set for PMSM drive systems, part 1: maximum torque per ampere operation. IEEE Trans. Ind. Informat. pp. 1–1 (2013) 23. M. Preindl, S. Bolognani, model predictive direct torque control with finite control set for PMSM drive systems, part 2: field weakening operation. IEEE Trans. Ind. Informat. 9, 648–657 (2013) 24. Z. Ma, S. Saeidi, R. Kennel, FPGA implementation of model predictive control with constant switching frequency for PMSM drives. IEEE Trans. Ind. Informat. 10, 2055–2063 (2014) 25. B. Boazzo, G. Pellegrino, Model-based direct flux vector control of permanent-magnet synchronous motor drives. IEEE Trans. Ind. Appl. 51, 3126–3136 (2015) 26. Y. Cho, K.B. Lee, J.H. Song, Y.I. Lee, Torque-ripple minimization and fast dynamic scheme for torque predictive control of permanent-magnet synchronous motors. IEEE Trans. Power Electron. 30, 2182–2190 (2015) 27. F. Wang, S. Li, X. Mie, W. Xie, J. Rodriguez, R.M. Kennel, Model-based predictive direct control strategies for electrical drives: an experimental evaluation of PTC and PCC methods. IEEE Trans. Ind. Informat. 11, 671–681 (2015) 28. M. Preindl, S. Bolognani, Optimal state reference computation with constrained MTPA criterion for PM motor drives. IEEE Trans. Power Electron. 30, 4524–4535 (2015)

References

11

29. W. Xie, X. Wang, F. Wang, W. Xu, R.M. Kennel, D. Gerling, R.D. Lorenz, Finite-control-set model predictive torque control with a deadbeat solution for PMSM drives. IEEE Trans. Ind. Electron. 62, 5402–5410 (2015) 30. M. Abroshan, K. Malekian, J. Milimonfared, B.A. Varmiab, An optimal direct thrust force control for interior permanent magnet linear synchronous motors incorporating field weakening, in Proceedings of the International Symposium on Power Electronics, Electrical Drives, Automation and Motion(SPEEDAM) (2008), pp. 130-135 31. S. Cheng-Chung, H. Yi-Sheng, Based on direct thrust control for linear synchronous motor systems. IEEE Trans. Ind. Electron. 56, 1629–1639 (2009) 32. J. Cui, C. Wang, J. Yang, L. Liu, Analysis of direct thrust force control for permanent magnet linear synchronous motor, in Proceedings of the World Congress on Intelligent Control and Automation(WCICA), vol. 5 (2004), pp. 4418–4421 33. J. Cui, C. Wang, J. Yang, D. Yu, Research on force and direct thrust control for a permanent magnet synchronous linear motor, in Proceedings of the Annual Conference of the Industrial Electronics Society (IECON) (2004), pp. 2269–2272 34. A. Mohammadpour, L. Parsa, SVM-based direct thrust control of permanent magnet linear synchronous motor with reduced force ripple, in Proceedings of the International Symposium on Industrial Electronics (ISIE) (2011), pp. 756–760 35. M.A.M. Cheema, J. Fletcher, M.F. Rahman, D. Xiao, Modified direct thrust control of linear permanent magnet motors with sensorless speed estimation, in Proceedings of the Annual Conference of the Industrial Electronics Society (IECON) (2012), pp. 1908–1914 36. L. Guan, J. Yang, J. Cui, Direct thrust control approach using adaptive variable structure for permanent magnet linear synchronous motor, in Proceedings of the International Conference on Control and Automation (ICCA) (2007), pp. 2217–2220 37. Y. Junyou, H. Guofeng, C. Jiefan, Analysis of PMLSM direct thrust control system based on sliding mode variable structure, in Proceedings of the International Power Electronics and Motion Control Conference (IPEMC) (2006), pp. 1–5 38. Y.S. Huang, C.C. Sung, Implementation of sliding mode controller for linear synchronous motors based on direct thrust control theory. IET Control Theory Appl. 4, 326–338 (2010) 39. Z. Jihao, Z. Shan’an, Fuzzy logic direct force control of surface permanent magnet linear synchronous motors without speed sensors, in Proceedings of the World Congress on Intelligent Control and Automation (WCICA) (2004), pp. 4491–4495 40. Y. Inoue, S. Morimoto, M. Sanada, Control method suitable for direct-torque-control-based motor drive system satisfying voltage and current limitations. IEEE Trans. Ind. Appl. 48, 970–976 (2012) 41. S. Mathapati, J. Bocker, Analytical and offline approach to select optimal hysteresis bands of DTC for PMSM. IEEE Trans. Ind. Electron. 60, 885–895 (2013) 42. T.G. Habetler, F. Profumo, M. Pastorelli, L.M. Tolbert, Direct torque control of induction machines using space vector modulation. IEEE Trans. Ind. Appl. 28, 1045–1053 (1992) 43. B.H. Kenny, R.D. Lorenz, Stator- and rotor-flux-based deadbeat direct torque control of induction machines. IEEE Trans. Ind. Appl. 39, 1093–1101 (2003) 44. E. Flach, R. Hoffmann, P. Mutschler, Direct mean torque control of an induction motor. Proc. EPE 3, 672–677 (1997) 45. M. Pacas, J. Weber, Predictive direct torque control for the PM synchronous machine. IEEE Trans. Ind. Electron. 52(5), 1350–1356 (2005) 46. K. Jun-Koo, S. Seung-Ki, New direct torque control of induction motor for minimum torque ripple and constant switching frequency. IEEE Trans. Ind. Appl. 35, 1076–1082 (1999) 47. S. Kuo-Kai, L. Juu-Kuh, P. Van-Truong, Y. Ming-Ji, W. Te-Wei, Global minimum torque ripple design for direct torque control of induction motor drives. IEEE Trans. Ind. Electron. 57, 3148–3156 (2010) 48. Z. Yongchang, Z. Jianguo, Direct torque control of permanent magnet synchronous motor with reduced torque ripple and commutation frequency. IEEE Trans. Power Electron. 26, 235–248 (2011)

12

1 Introduction

49. Y. Ren, Z.Q. Zhu, J. Lue, Direct torque control of permanent-magnet synchronous machine drives with a simple duty ratio regulator. IEEE Trans. Ind. Electron. 61, 5249–5258 (2014) 50. F. Niu, K. Li, Y. Wang, Direct torque control for permanent-magnet synchronous machines based on duty ratio modulation. IEEE Trans. Ind. Electron. 62, 6160–6170 (2015)

Chapter 2

Mathematical Modeling of Surface-Mount Linear Permanent Magnet Synchronous Motor

2.1 Introduction Precise mathematical modeling of the tubular surface-mount linear PMSM based on the physics of the machine is an essential requirement to formulate any type of control scheme for the machine. In this chapter, the dynamic model of tubular surface-mount linear PMSM in three-phase stationary abc-reference frame is formulated. In order to reduce mathematical complexity, the dynamic model of the surface-mount linear PMSM is developed under some standard assumptions which are often used in formulations of mathematical models for a vast variety of electric machines. In addition, the dynamic model of the surface-mount linear PMSM in three phase stationary abc-reference is transformed to two-axis reference frames by using appropriate transformations to further simplify the controller design process for the machine. The structure of a three phase tubular surface-mount linear PMSM is illustrated by a schematic diagram showing various components and parameters of the machine. The stator windings of the machine are represented by a three-phase equivalent circuit. The dynamics of this three-phase equivalent circuit are described mathematically by a set of differential equations in three phase stationary abc-reference frame variables. In addition, the winding inductance matrix for the linear PMSM is also computed taking into account machine parameters as illustrated by the schematic diagram. In order to simplify the controller design process, first the dynamic model of the machine in stationary abc-reference frame variables is expressed in terms of the mover’s flux vector dq-reference frame variables, then in the stationary αβ-reference frame variables, and lastly, in the stator flux vector xy-reference frame variables by using appropriate mathematical transformations. The dynamic models of the linear PMSM in dq-reference frame variables, stationary αβ-reference frame variable and xy-reference frame variables are then used in the following chapters of this thesis to perform the analysis and design of various control strategies for the prototype linear PMSM. © Springer Nature Switzerland AG 2020 M. A. M. Cheema and J. E. Fletcher, Advanced Direct Thrust Force Control of Linear Permanent Magnet Synchronous Motor, Power Systems, https://doi.org/10.1007/978-3-030-40325-6_2

13

14

2 Mathematical Modeling of Surface-Mount Linear …

2.2 Construction of Tubular Surface-Mount Linear PMSM The construction details of a tubular linear PMSM are provided in [1]. The word surface-mount, when used for a tubular linear PMSM, implies a tubular linear PMSM with no saliency. According to [1], the stator (or primary) of the tubular linear PMSM is constructed of co-axial coils which may be slotted or slot-less. The mover (or secondary) of the tubular linear PMSM is made of solid magnetic steel, composed of successive North-South poles. The stator winding completely surrounds the mover and results in geometric structure with cylindrical symmetry which assures a vector force on the mover with no radial component. Moreover, because of this cylindrical symmetry, the magnetic field distribution is distributed evenly resulting in optimal efficiency. In order to further clarify the above construction details of the tubular linear PMSM according to [1], the schematic diagram of a conceptual 4-pole tubular surface-mount linear PMSM is illustrated in Fig. 2.1. The stator of the machine consists of three-phase winding. It is clear from Fig. 2.1 that each phase winding comprises a number of coaxial solenoids. These co-axial solenoids are slot-less (air-cored) for the prototype linear PMSM used in this thesis. The coil-ends of co-axial solenoids corresponding to phase a, b, and c are denoted by aa’, bb’ and cc’ respectively. It is important to note that the coils-end denoted by a primed literal, e.g., a’, b’ and c’ shows the phase current’s direction into the plane of paper. The coils corresponding to one phase can be connected either in parallel or series. However, for the prototype linear PMSM used in this thesis, these coils are x (m) a

leff c'

b

N N a'

a’

c

S S c

b'

b'

a

c'

b

N N a

c'

b

a'

a'

c

S S c

b'

b' N

a

c'

N

b

PM Flux Density (fundamental)

lg τ

O

2π

3π

π

Fig. 2.1 Schematic diagram of a tubular surface-mount linear PMSM

4π θr

Dg

2.2 Construction of Tubular Surface-Mount Linear PMSM

15

connected in series. The solenoids for each phase are identical and symmetrically distributed along the stator. The mover of the linear PMSM is constructed of a number of permanent magnets (PM) that are placed back-to-back inside a steel tube. It is observed from Fig. 2.1 that there no separator between the consecutive permanent magnets and therefore a uniform air-gap is achieved which results in a non-salient tubular linear PMSM termed as tubular surface-mount linear PMSM. However, inclusion of non-magnetic separators between the consecutive permanent magnets produces a non-uniform air-gap which causes saliency. It should be noted that the prototype linear PMSM used in this research is a surface-mount tubular linear PMSM. It is important to note that high energy rare-earth permanent magnets with large values of remnant flux densities are used in the mover of the tubular linear PMSMs and are capable of producing adequate flux densities in approximately 1 mm air-gaps [1]. The three-phase winding of the linear PMSM are connected in star-configuration and supplied by a set of balanced three phase currents displaced by 120 electrical degrees from each other and create a moving magnetic field inside the stator. The interaction between the fundamental component of the stator field and the fundamental component of the PM flux density creates the linear thrust force that moves the mover inside the stator of the linear PMSM.

2.3 Dynamic Modeling of Surface-Mount Linear PMSM in 3-Phase Stationary abc-Reference Frame The stator windings for the linear PMSM can be described by a three phase equivalent circuit to perform the mathematical modeling as shown in Fig. 2.2, taking into account the following standard simplifying assumptions [2–8]: Fig. 2.2 Schematic diagram of the three-phase winding configuration for the linear PMSM. The orientation of the windings is arbitrarily shown and does not correspond to the physical magnetic axes of the machine as in the usual case of rotational machine [9, 10]

NT

Rs

ia

+ Rs

ib

+ ic

+

_

va NT

_

vb Rs

NT vc

_

16

2 Mathematical Modeling of Surface-Mount Linear …

• Only the fundamental component of stator magneto-motive force (MMF) is taken into account. • The machine has uniform air-gap and is magnetically symmetrical and the sloteffect is not present. • The effect of saturation on the inductances is neglected. • The hysteresis and eddy current losses are neglected, therefore no core-losses is present in the machine. • The self-inductances of all the phases are independent of the mover’s position. • The thermal effect on the stator resistance and permanent magnet flux is neglected. In case of rotational machines, it is often a common practice for convenient conceptual understanding that the windings are shown to be 120° apart in the schematic diagram and this orientation of the schematic windings corresponds to the magnetic axis of the machine [9, 10]. However, this is not the case for linear PMSM and the orientation of the windings in Fig. 2.2 is arbitrarily shown and does not correspond to the magnetic axes of the linear PMSM. The dynamic equation that describes the stator winding of the linear PMSM represented by the equivalent circuit of Fig. 2.2 is expressed in matrix form as: vabc = Rs iabc +

dλabc dt

(2.1)

where vabc is the 3 × 1 phase voltage matrix, iabc is the 3 × 1 phase current matrix, λabc is the 3 × 1 phase stator flux linkage matrix, and Rs is a 3 × 3 diagonal matrix describing the winding resistances: ⎡

vabc

⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎤ va ia λa Rs 0 0 = ⎣ vb ⎦, iabc = ⎣ i b ⎦, λabc = ⎣ λb ⎦, and Rs = ⎣ 0 Rs 0 ⎦ vc ic λc 0 0 Rs

(2.2)

The stator flux linkage vector comprised of the flux linkage due to the stator currents, λabcs , and the flux linkage due to the permanent magnet flux density, λpm , and can be written as: λabc = λabcs + λ pm

(2.3)

where λpm is a 3 × 1 a matrix containing mover’s flux linkage (produced by the permanent magnets) in the mover linked to each phase: ⎡

λ pm

⎤ λ pm,a = ⎣ λ pm,b ⎦ λ pm,c

And λabcs is the 3 × 1 phase flux linkage matrix due to the stator currents

(2.4)

2.3 Dynamic Modeling of Surface-Mount Linear PMSM in 3-Phase …

17

⎡

λabcs

⎤ λas = ⎣ λbs ⎦ λcs

(2.5)

In Eqs. (2.3) and (2.5), the flux linkage due to the stator currents, λabcs , is proportional to the currents flowing through the three-phase windings and is expressed using the winding inductances as: λabcs = Ls iabc

(2.6)

where Ls is the 3 × 3 stator inductance matrix and is described by: ⎡

⎤ L aa Mab Mac Ls = ⎣ Mba L bb Mbc ⎦ Mca Mcb L cc

(2.7)

In Eq. (2.7), L aa , L bb and L cc are the self-inductances associated to the phase windings, and M ab , M ac , M ba , M bc , M ca , and M cb are the mutual inductances between the corresponding phase windings. In a surface-mount linear PMSM, the self-inductance is constant with respect to the mover’s position and is expressed by [4, 7]: L aa = L bb = L cc = L s = L ls + L A

(2.8)

where L ls is the leakage-inductance, and L A is expressed as:  LA =

NT P

2

μ0 A lg

(2.9)

In Eq. (2.9), N T is the total number of turns in a phase winding, μ0 is the permeability of air, lg is the axial length of one solenoid coil in the phase winding as illustrated in Fig. 2.1, and A is the cross-sectional area of solenoid coil expressed as: A=

π Dg2

(2.10)

4

where, Dg is the inner diameter of the stator winding. Substitution of Eq. (2.10) into Eq. (2.9) yields the detailed expression for L A :  LA =

NT P

2

π μ0 Dg2 4l g

(2.11)

Since, the leakage inductance L ls is negligible compared to L A therefore, it can be inferred from Eqs. (2.8) and (2.11) that the self-inductance L s of the stator winding is inversely linked to the air-gap lg . Hence, in general, surface-mount linear PMSMs

18

2 Mathematical Modeling of Surface-Mount Linear …

tend to have low stator inductance owing to the larger air gap compared to their rotational counterparts. It is worth mentioning that low inductance is the key factor that makes surface-mount linear PMSMs difficult to be controlled compared to rotational PMSM as they exhibit higher ripple in thrust force, especially under conventional direct thrust force control scheme. This limitation and its effects on the control of surface-mount linear PMSMs are investigated in the later chapters of this thesis. The stator windings of the prototype surface-mount linear PMSM comprises of coaxial air-core solenoids corresponding to different phases as shown in Fig. 2.1. The permeability of the permanent magnets in the moving rod is close to that of air. Hence, the magnetic flux created by each solenoid, crosses a non-magnetic region with permeability equal to μ0 . This significantly increases the reluctance of the flux path. According to the theory of electromagnetics [4–7], the flux lines close their path through the route with the lowest reluctance. Hence, the flux of each solenoid closes its path without going through the extra length of the neighboring solenoids. Only a minimal amount of flux links the two neighboring solenoids due to the fringing effect of the flux lines and is negligibly small. The mutual inductance between phase a and b is defined as [4–7]:  λab  (2.12) Mab = i a ib =0 where λab is the flux linkage between phase a and b. According to the above explanation, since λab is negligible, hence, a zero mutual inductance is assumed between the phase windings in surface-mount linear PMSMs. Therefore, the inductance matrix simplifies to a diagonal form: ⎡

⎤ Ls 0 0 Ls = ⎣ 0 L s 0 ⎦ 0 0 Ls

(2.13)

From Eqs. (2.2), (2.6) and (2.13), the flux linkage due to the stator phase currents becomes: ⎤ ⎡ ⎤⎡ ⎤ ⎡ Ls 0 0 ia λas ⎣ λbs ⎦ = ⎣ 0 L s 0 ⎦⎣ i b ⎦ (2.14) λcs ic 0 0 Ls Now, from Eqs. (2.3), (2.4) and (2.15), the stator flux linkage is expressed as: ⎤ ⎡ ⎤⎡ ⎤ ⎡ ⎤ Ls 0 0 ia λ pm,a λa ⎣ λb ⎦ = ⎣ 0 L s 0 ⎦⎣ i b ⎦ + ⎣ λ pm,b ⎦ λc ic λ pm,c 0 0 Ls ⎡

(2.15)

2.3 Dynamic Modeling of Surface-Mount Linear PMSM in 3-Phase …

19

2.3.1 Mapping of Three-Phase Machine Variables to Complex Space Vectors The dynamic equations for the linear PMSM can be expressed using complex variable notion in a compact form. The key benefit of using complex variable notion is that it allows easy algebraic manipulation and simple graphic interpretations. Therefore, it provides a logical and direct development of the concepts which are difficult to understand using real variable analysis [4, 5]. The concept of this complex variable notion also known as “complex space vector notion” evolved from the pioneering works of Fortescue [11] and Lyon [12, 13] and later extensively applied to the analysis of electric machines [2–7]. The basic concept of complex space vector notion is that the instantaneous values of machine phase variables (voltage, current or flux linkage) are expressed by a complex variable to be referred as “complex space vector” [4]. The complex space vector corresponding to the machine phase variables is obtained from the vector sum of three fixed vectors in a complex plane which are displaced by 120 electrical degrees with their magnitudes equal to the instantaneous values of these phase variables. These vectors with fixed directions and variable magnitudes are termed as phase vectors. A rigourous mathematical discourse of AC machine analysis based on complex space vector notion is detailed in [2, 4, 5]. In order to understand the concept of complex space vector, first, consider the example of three-phase stator voltages. Assuming a sinusoidal and balanced threephase stator phase voltages, the phase voltage waveforms are expressed as: ⎧ ⎨ va = Vm cos(ω s t + ϕ)

vb = Vm cos ωs t − 2π + ϕ 3 ⎩ +ϕ vc = Vm cos ωs t + 2π 3

(2.16)

where, V m is the amplitude of the phase voltage waveform, and ϕ is the phase angle (in electrical degrees) for the voltage waveform. ωs is the synchronous speed in elec. rad/s and is expressed in terms of the frequency f (Hz) of the supply voltage as: ωs = 2π f

(2.17)

The following condition holds true for a balanced set of three-phase voltage [5]: va + vb + vc = 0

(2.18)

According to [4, 5], for a balanced set of three-phase voltages expressed as Eq. (2.16) such that the condition given by Eq. (2.18) is satisfied, the complex voltage space vector Vabc can be expressed as:

2 Vabc = va + avb + a 2 vc 3

(2.19)

20

2 Mathematical Modeling of Surface-Mount Linear …

where, a = ej

2π 3

(2.20)

and, a2 = e j

4π 3

= e− j

2π 3

(2.21)

It is important to note that the quantities expressed in the manner of Eq. (2.19) are called “complex space vectors” [4] or alternatively referred to as “space-phasors” by Krause [7], however, throughout this thesis they are referred to as “space vectors” to avoid confusion. Therefore, when current and flux phase variables are expressed according to Eq. (2.19), they are called as current space vector and flux space vector respectively. Moreover, it should be noted that space vectors should not be confused with complex phasors which are used to express sinusoidally alternating quantities [4]. It is clear from Eqs. (2.19)–(2.21) that the voltage space vector Vabc is a complex variable and can be graphically illustrated by a vector in a complex plane as shown in Fig. 2.3a. In this graphical interpretation, va is treated as vector fixed in direction of a-axis with variable magnitude (instantaneous value of phase “a” voltage) as shown in Fig. 2.3a. It is also clear from Fig. 2.3a that the a-axis is aligned with the real axis of the complex plane. It is important to note that a-axis is used as a reference and all the angles are measured with respect to a-axis. In Eq. (2.19), the terms avb , and a 2 vc are the vectors in direction of b-axis and c-axis respectively as shown in Fig. 2.3a.

(a)

(b)

Complex-Plane

Complex-Plane imaginary – axis

imaginary – axis

b – axis

b – axis

avb

Vabc

Vabc

v

2 3 a

real – axis

va

a – axis

2 3

avb

real – axis

a – axis 2 3

a 2 vc

a 2 vc

c – axis

c – axis

Fig. 2.3 Space vector representation of stator phase voltages, a voltage space vector (green) and phase voltage vectors (blue), b synthesis of voltage space vector according to Eq. (2.18)

2.3 Dynamic Modeling of Surface-Mount Linear PMSM in 3-Phase …

21

In addition, a and a 2 are the unit vectors along b-axis and c-axis respectively. Therefore, a multiplication by a and a 2 results in a rotation of 120° and 240° (−120°) with respect to a-axis respectively [2]. The synthesis of voltage space vector Vabc as vector sum of the phase vectors va , avb , and a 2 vc according to Eq. (2.19) is 2 in Eq. (2.19) is a scaling factor to maintain the illustrated in Fig. 2.3b. The 3   factor   length (absolute value) Vabc  of the voltage space vector equal to the amplitude of the phase voltage Vm when a balanced three phase supply to the machine   is applied   [2, 4], i.e. to achieve magnitude invariance expressed as Vabc  = Vm . By substitution of Eq. (2.16) into Eq. (2.19) and after simplification the voltage space vector Vabc is expressed as: Vabc = Vm e jϕ e jωs t = Vm e j (ωs t+ϕ)

(2.22)

It is clear from Eq. (2.22) that Vabc is a vector (complex number) with length (absolute value) Vm which is same as the amplitude of the phase voltage given in Eq. (2.18). In Eq. (2.22), the exponential function e jωs t is regarded as vector rotator operator that rotates whatever vector or scalar it multiplies by the angle ωs t in positive direction (counter clock wise) such that the speed of rotation is equal to the synchronous speed ωs [6, 7]. Therefore, Vabc is a vector of length Vm which rotates in positive direction with respect to a-axis at synchronous speed ωs such that its initial phase angle with respect to a-axis is ϕ electrical degrees at the start of rotation when the time t = 0. The conversion of phase voltages va , vb , and vc to the voltage space vector Vabc according to Eqs. (2.18) and (2.19) is a mathematical transformation and does not necessarily requires a graphical connotation. However, for a conceptual visualization, it is convenient to express the link between the phase voltages and the voltage space vector as trigonometric relations and for this purpose, the phase voltages are va , vb , and vc are considered to be the magnitudes of voltage phase vectors directed along three fixed axes namely a, b, and c each displaced by 120° in a complex plane as shown in Fig. 2.3 [7]. Therefore, the voltage space vector Vabc can be resolved along a, b, and c axes so that its projections on these axes represents the instantaneous values of phase voltages which can be expressed using trigonometric relations from Eqs. (2.16) and (2.22) as:   ⎧   ⎪ V v =  cos(ωs t + ϕ)  ⎪ a abc ⎪   ⎨

  vb = Vabc  cos ωs t − 2π +ϕ 3  ⎪  ⎪

⎪ ⎩ vc = Vabc  cos ωs t + 2π + ϕ 3

(2.23)

Another important benefit of the graphical illustration of the voltage, current and flux linkage space vector according to Fig. 2.3 is that the concepts from vector algebra can applied for the mathematical manipulation of these space vectors which will be illustrated later in this section.

22

2 Mathematical Modeling of Surface-Mount Linear …

It is clear from the above discussion and mathematical explanation that a, b, and c are three fixed axes in a complex plane and constitute the stationary abcreference frame. The a-axis is aligned with the real axis of the complex plane. It is important to note that the real and imaginary axes of the complex plane correspond to the stationary αβ-reference frame, with α and β axes aligned with the real and imaginary axes of the complex plane respectively. Therefore, when the voltage space vector is resolved into two orthogonal components along the real and imaginary axes of the complex plane then these components represent the α and β components of the voltage space vector thus providing the foundation for transforming variables in three phase stationary abc-reference frame to the two axes αβ-reference frame using the trigonometric relationships known as Clarke transformation which will be explained in a later section. In case of rotational machines, many texts [4, 7, 10] often consider that the stationary abc-reference frame is fixed in the stator windings and represent the magnetic axes for the stator winding implying that a, b, and c axes are thought to be directed along the magnetic axes (which are also 120° apart) of a conceptual two pole machine. It is important to note that this is an abstract superimposition of the complex plane of stationary abc-reference frame to the stator windings and there is no physical link between the stationary abc-reference frame and the magnetic axes of the stator winding. The only benefit of this superimposition is that it allows visualization of the magnetic field intensity space vector in the same reference frame/plane that contains the voltage, current and flux space vectors. However, it is important to note that in case of the tubular linear PMSM the axes of stationary abc-reference frame should not be thought of as the magnetic axes of the linear machine. As discussed previously, the axes of stationary abc-reference for tubular linear PMSMs are 120° apart, whereas the magnetic axes are collinear as observed from Fig. 2.1. Therefore, the concept of superimposing the axes of stationary abc-reference frame to the magnetic axes of a conceptual two pole tubular linear PMSM is not valid for tubular linear PMSM. When the linear PMSM is supplied by a balanced set of three phase voltages of Eq. (2.16), the instantaneous values of the resulting three phase current are expressed as: ⎧ ⎨ i a = Im cos(ω s t + γ2π)

(2.24) i = Im cos ωs t − 3 + γ ⎩ b + γ i c = Im cos ωs t + 2π 3 where, I m is the amplitude of the phase voltage waveform, and γ is the phase angle (in electrical degrees) for the current waveform. It is important to note that for linear PMSM being the inductive load the voltage waveform leads the current waveform and therefore, γ < ϕ. The following condition holds true for a balanced set of three-phase currents [5]: ia + ib + ic = 0

(2.25)

2.3 Dynamic Modeling of Surface-Mount Linear PMSM in 3-Phase …

23

Now, the current space vector Iabc can be expressed as:

2 Iabc = i a + ai b + a 2 i c 3

(2.26)

The current space vector Iabc is illustrated in Fig. 2.4. Substituting values of ia , ib , and ic from (2.24) into (2.26): 2 Iabc = 3



Im cos(ωs t + γ ) + a Im cos ωs t − + a 2 Im cos ωs t +

2π 3 2π 3

 + γ

+γ

(2.27)

Substituting the value of a and a2 from (2.20) and (2.21) into (2.27) and expressing the trigonometric cosine functions in complex form using Euler’s identity: ⎛ 2 ⎜ Iabc = Im ⎜ 3 ⎝

e j (ωs t+γ ) + e− j (ωs t+γ ) 2

 +e

j 2π 3

+ e− j

e j (ωs t−

 ⎞

2

 2π 3

2π +γ ) 2π 3 + e− j (ωs t− 3 +γ )

e

j (ωs t+ 2π 3 +γ )

+e 2

− j (ωs t+ 2π 3 +γ )

⎟ ⎟ ⎠

(2.28)

After mathematical simplification of (2.28) the current space vector Iabc is expressed as: Iabc = Im e jγ e jωs t = Im e j (ωs t+γ )

(2.29)

 abcs for the flux linkage produced by the stator currents is given The space vector λ as: Fig. 2.4 Illustration and angular orientations and of stator current, permanent magnet flux linkage, and stator flux linkage space vectors stationary abc-reference frame. All the angles are measured from a-axis

b – axis

abc abcs

I abc f r

s

a – axis

c – axis

Ls I abc

24

2 Mathematical Modeling of Surface-Mount Linear …

2 λas + aλbs + a 2 λcs 3

 abcs = λ

(2.30)

 abcs can be expressed as: Now, from Eqs. (2.14), (2.26) and (2.30), λ  abcs = L s Iabc λ

(2.31)

 abcs is shown in Fig. 2.4, and it is observed that λ  abcs is parallel The space vector λ   to the current space vector Iabc . The mover’s flux space vector λ f is the flux linkage produced by the permanent magnets in the mover is given using Eq. (2.4) as: f = λ

2 λ pm,a + aλ pm,b + a 2 λ pm,c 3

(2.32)

 abc is: The stator flux linkage space vector λ  abc = λ

2 λa + aλb + a 2 λc 3

(2.33)

From, Eqs. (2.15), (2.26), (2.31) and (2.31):  abc = L s Iabc + λ f λ

(2.34)

 abc is the vector sum of λ  abcs and λ  f as illustrated It is clear from Eq. (2.34) that λ   in Fig. 2.4. In Fig. 2.4, the angles of λabc and λ f with respect to the a-axis is θs and θr respectively both expressed in electrical radians. The difference between θs and  f can also be θr is the angle δ known as load angle. The mover’s flux space vector λ expressed as:  f = λ f e jθr λ

(2.35)

where λf is the amplitude of the permanent magnet flux linkage. Now from Eqs.  abc can be expressed as: (2.34) and (2.35), the stator flux linkage space vector λ  abc = L s Iabc + λ f e jθr λ

(2.36)

The instantaneous phase value of the permanent magnet flux linking with each  f along a, b, and c axes phase winding by resolving the mover’s flux space vector λ as [14]: ⎤ ⎤ ⎡ cos(θ λ pm,a r) ⎦ = ⎣ λ pm,b ⎦ = λ f ⎣ cos θr − 2π 3 λ pm,c cos θr + 2π 3 ⎡

λ pm

(2.37)

2.3 Dynamic Modeling of Surface-Mount Linear PMSM in 3-Phase …

25

In Eqs. (2.35) and (2.37), θ r corresponds to the movement of the mover. As shown in Fig. 2.1, one cycle of the permanent magnet flux linkage fundamental component is expressed by 2π electrical radians. This corresponds to two permanent magnets. The length of each permanent magnet equals to one pole pitch, τ (meters) and shown in Fig. 2.1. Hence, θ r (in electrical radians) can be expressed in terms of the mover’s displacement x (in meters) as follows [1, 15–28]: θr = P

π x τ

(2.38)

In Eq. (2.38) P is the number of pole pairs. It is important to note that, some authors omit pole pairs P in Eq. (2.38) by considering per pole model of the linear PMSM. From a mathematical point of view, this does not affect the modelling of machine, as long as same form of Eq. (2.38) for θ r is used in all the two-axes transformation and their respective inverses. In other words, the transformation equations (e.g. dq or xy) that are used to determine the parameters from experimental measurements must be the equations used in the controller/state observer/machine model. However, the number of pole pairs must always be included in the thrust force calculation. It is worth mentioning, that the mathematical modelling presented in this book will be valid either way. Substitution of θ r from Eq. (2.38) into Eq. (2.37), the permanent magnet flux linkage in a surface-mount linear PMSM becomes: π ⎤ cos πP τ x 2π ⎣ = λ f cos P τ x − 3 ⎦

cos P πτ x + 2π 3 ⎡

λ pm

(2.39)

where P is the number of pole pairs. Now from Eqs. (2.1), (2.15) and (2.37), the dynamic model of the tubular surface-mount linear PMSM can be expressed in stationary abc-reference frame as: ⎤ ⎡ ⎤⎡ ⎤ ⎡ ⎤ Rs 0 0 ia λa va d ⎣ vb ⎦ = ⎣ 0 Rs 0 ⎦⎣ i b ⎦ + ⎣ λb ⎦ dt vc ic λc 0 0 Rs

(2.40)

⎤ ⎡ ⎤⎡ ⎤ ⎡ ⎤ λa Ls 0 0 ia cos(θr ) ⎣ λb ⎦ = ⎣ 0 L s 0 ⎦⎣ i b ⎦ + λ f ⎣ cos(θr − 2π/3) ⎦ cos(θr + 2π/3) λc ic 0 0 Ls

(2.41)

⎡

where, ⎡

26

2 Mathematical Modeling of Surface-Mount Linear …

Substitution of Eq. (2.39) into Eq. (2.41) yields: ⎤ ⎤ ⎡ ⎤⎡ ⎤ ⎡ Ls 0 0 ia cos(P(π/τ )x) λa ⎣ λb ⎦ = ⎣ 0 L s 0 ⎦⎣ i b ⎦ + λ f ⎣ cos(P(π/τ )x − 2π/3) ⎦ cos(P(π/τ )x + 2π/3) λc ic 0 0 Ls ⎡

(2.42)

The dynamic model of the tubular surface-mount linear PMSM can be expressed in space vector notation by multiplying the second and third row of Eq. (2.40) by a and a 2 respectively and adding them to the first row:  abc dλ Vabc = Rs Iabc + dt

(2.43)

The time derivative of Eq. (2.38) results in: π vm τ

(2.44)

ωr =

dθr dt

(2.45)

vm =

dx dt

(2.46)

ωr = P where

In Eq. (2.45), ωr is the electrical speed of the mover and represents the rotational  f in the stationary abc-reference frame with speed of the mover’s flux space vector λ respect to the a-axis and is expressed in elec. rad/s. In Eq. (2.46), vm is the mechanical speed of the mover and is expressed in m/s. It is noted that the number of pole pairs P in Eq. (2.44) is used to convert the mechanical speed of the mover into the electrical speed. From Eqs. (2.36), (2.43) and (2.45), the dynamic model of the tubular surface-mount linear PMSM can be expressed: d Iabc f + jωr λ Vabc = Rs Iabc + L s dt

(2.47)

Substituting Eq. (2.44) into Eq. (2.47) yields: π d Iabc f + j P vm λ Vabc = Rs Iabc + L s dt τ

(2.48)

During steady-state operation the derivative of the current space vector in Eq. (2.47) is zero, therefore Eq. (2.47) becomes:

2.3 Dynamic Modeling of Surface-Mount Linear PMSM in 3-Phase … Fig. 2.5 Steady-state space vector diagram of the linear PMSM in stationary abc-reference frame according to Eq. (2.49). The length of all the space vectors is arbitrarily shown

27

imaginary axis

b axis

RI abc

j

Vabc

abc abcs

r

f

Ls I abc

I abc f r

real axis

s

a axis

c axis

f Vabc = Rs Iabc + jωr λ

(2.49)

The space vector diagram for tubular surface-mount linear PMSM during steadystate according to Eq. (2.49) in stationary abc-reference frame is shown in Fig. 2.5. The instantaneous active power in the stationary abc-reference frame is described using the instantaneous three-phase voltages and currents as [3, 7]: p = va i a + vb i b + vc i c

(2.50)

This expression will be used in the later sections to derive the thrust force equation in the two-axis reference frames. Note that thrust force is defined as the active power divided by the mover’s speed.

2.4 Two-Axis Dynamic Models of Linear PMSM The three-phase voltage equations as shown in Eqs. (2.40)–(2.42) are complicated and position dependent. A simplification can be made in the machine equations if they are transformed into a suitable two-axis reference frame. In this section, the threephase mathematical model of the surface-mount linear PMSM will be transformed to the mover’s flux vector (dq) [8, 9, 29], stationary (αβ) [10, 30], and stator flux vector (xy) [31] frames of reference. These frames and their orientations with respect to each other are illustrated in Fig. 2.5. In this figure, the α-axis is aligned with the machine’s phase a-axis.

28

2 Mathematical Modeling of Surface-Mount Linear …

The three-phase machine model derived in Eqs. (2.40)–(2.50) will be transformed into the aforementioned reference frames in the following sections.

2.4.1 Dynamic Model of Linear PMSM in the dq-Reference Frame In 1920s, Park formulated a change of variables famously known as Park’s transformation which transforms the machine variables in the stationary abc-reference frame fixed in the stator to a two axes dq-reference frame fixed in the rotor [9]. Originally Park used this transformation to simplify the analysis of synchronous machines and later its application was extended to the analysis of PMSMs by Jahns [8, 29]. The dq-reference frame consists of two orthogonal axes namely q-axis or quadrature axis and d-axis or direct axis such that the q-axis leads d-axis by 90°. According to [8, 29] the dq-reference frame is fixed in the rotor which means that d-axis is superimposed on the rotor flux space vector and in case of linear PMSMs d-axis is aligned with the  f as illustrated in Fig. 2.6. mover’s flux space vector λ Therefore dq-axes can also be referred to as mover’s flux vector dq-reference frame. It is important to note that Lipo and Krause [4, 7] consider q-axis to be  f for linear aligned with the rotor flux space vector (or mover’s flux space vector λ PMSM) instead of d-axis which is also correct; however this approach is not adopted in this thesis. In the dq-reference frame, since the d-axis is fixed to the mover’s flux space vector, therefore, the dq-reference frame rotates at an angular speed ωr with respect to the stationary abc-reference frame. Park transformation matrix for transferring the three

Fig. 2.6 The stator and mover’s flux linkages in various reference frames

2.4 Two-Axis Dynamic Models of Linear PMSM

29

phase machine variables to the dq-reference frame is defined as [8, 9, 29]: ⎤ cos θr cos(θr − 2π/3) cos(θr + 2π/3) 2 = ⎣ − sin θr − sin(θr − 2π/3) − sin(θr + 2π/3) ⎦ 3 1/2 1/2 1/2 ⎡

Kdq0

(2.51)

The inverse park transformation matrix is obtained as: ⎡

−1 Kdq0

− sin θr cos θr = ⎣ cos(θr − 2π/3) − sin(θr − 2π/3) cos(θr + 2π/3) − sin(θr + 2π/3)

⎤ 1 1⎦ 1

(2.52)

By multiplying both sides of Eq. (2.40) by Ks , the machine variables in the dq reference frame can be obtained as: ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎞ ⎛ va Rs 0 0 id λd d −1 ⎣ −1 ⎣ Kdq0 ⎣ vb ⎦ = Kdq0 ⎣ 0 Rs 0 ⎦Kdq0 i q ⎦ + Kdq0 ⎝Kdq0 λq ⎦⎠ (2.53) dt vc i0 λ0 0 0 Rs In Eq. (2.53), i0 and λ0 are the zero-sequence components of the current and stator flux respectively. The following simplifications are used: ⎡

⎤ ⎤ ⎡ Rs 0 0 Rs 0 0 −1 = ⎣ 0 Rs 0 ⎦ Kdq0 ⎣ 0 Rs 0 ⎦Kdq0 0 0 Rs 0 0 Rs ⎡ ⎤ ⎡ ⎤⎞ ⎛ ⎡ ⎤ λd λd λd   d d −1 ⎣ −1 d ⎣ Kdq0 Kdq0 ⎝Ks−1 ⎣ λq ⎦⎠ = Kdq0 λq ⎦ + Kdq0 Kdq0 λq ⎦ dt dt dt λ0 λ0 λ0

(2.54)

(2.55)

It can be shown that: ⎡ − cos θr − sin θr d  −1  Kdq0 = ωr ⎣ − sin(θr − 2π/3) − cos(θr − 2π/3) dt − sin(θr + 2π/3) − cos(θr + 2π/3)

⎤ 0 0⎦ 0

(2.56)

Hence, ⎡ ⎤ 0 −1 0   d −1 Kdq0 = ωr ⎣ 1 0 0 ⎦ Kdq0 dt 0 0 0

(2.57)

30

2 Mathematical Modeling of Surface-Mount Linear …

Equation (2.55) now simplifies to ⎤⎞ ⎡ ⎤ ⎡ ⎤ λd −λq λ d d ⎣ d⎦ −1 ⎣ ⎦ ⎠ ⎣ ⎦ = ω Kdq0 ⎝Kdq0 + λq λd λq r dt dt λ0 0 λ0 ⎛

⎡

(2.58)

Substitution of Eqs. (2.54) and (2.57) into Eq. (2.53): ⎤ ⎡ ⎤⎡ ⎤ ⎡ ⎤ ⎡ ⎤ Rs 0 0 id −λq vd λd d ⎣ vq ⎦ = ⎣ 0 Rs 0 ⎦⎣ i q ⎦ + ωr ⎣ λd ⎦ + ⎣ λq ⎦ dt v0 i0 0 λ0 0 0 Rs ⎡

(2.59)

In Eq. (2.59), v0 is the zero-sequence component of the voltage. The last term of Eq. (2.59), the flux linkage vector in the dq reference frame, can be obtained by multiplying Eq. (2.41) by Kdq0 : ⎡

⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ λa Ls 0 0 id cos(θr ) Ks ⎣ λb ⎦ = Ks ⎣ 0 L s 0 ⎦Ks−1 ⎣ i q ⎦ + λ f Ks ⎣ cos(θr − 2π/3) ⎦ λc i0 0 0 Ls cos(θr + 2π/3)

(2.60)

Following simplifications are used: ⎡ ⎤ ⎤ Ls 0 0 Ls 0 0 Ks ⎣ 0 L s 0 ⎦Ks−1 = ⎣ 0 L s 0 ⎦ 0 0 Ls 0 0 Ls ⎡ ⎤ ⎡ ⎤ cos(θr ) 1 Ks ⎣ cos(θr − 2π/3) ⎦ = ⎣ 0 ⎦ cos(θr + 2π/3) 0 ⎡

(2.61)

(2.62)

Substitution of Eqs. (2.61) and (2.62) into Eq. (2.60) yields: ⎤ ⎡ ⎤⎡ ⎤ ⎡ ⎤ Ls 0 0 id λf λd ⎣ λq ⎦ = ⎣ 0 L s 0 ⎦⎣ i q ⎦ + ⎣ 0 ⎦ λ0 i0 0 0 Ls 0 ⎡

(2.63)

Now, the final equations of the surface-mount linear PMSM in the dq reference frame can be summarized as follows: vdq0 = Rs idq0 +

d λdq0 + ωr λqd dt

(2.64)

2.4 Two-Axis Dynamic Models of Linear PMSM

λdq0 = Ls idq0 + λ pm,dq where,

31

(2.65)

⎡

⎡ ⎤ ⎡ ⎡ ⎤ ⎤ ⎤ vd id λq λd vdq0 = ⎣ vq ⎦, idq0 = ⎣ i q ⎦, λqd = ⎣ −λd ⎦, λdq0 = ⎣ λq ⎦, λ pm,dq = v0 i0 0 λ0 ⎤ ⎤ ⎡ ⎡ ⎡ ⎤ Rs 0 0 Ls 0 0 λf ⎣ 0 ⎦, and Rs = ⎣ 0 Rs 0 ⎦, Ls = ⎣ 0 L s 0 ⎦ 0 0 0 Rs 0 0 Ls Considering a three-phase balanced system, the zero sequence components in Eqs. (2.64) and (2.65) becomes zero. Hence, the dynamic model of the surfacemount linear PMSM in the dq reference frame can be expressed in expanded form of Eqs. (2.64) and (2.65) as follows: vd = Rs i d +

d λd − ωr λq dt

(2.66)

vq = Rs i q +

d λq + ωr λd dt

(2.67)

where λd = L s i d + λ f

(2.68)

λq = L s i q

(2.69)

Substituting Eqs. (2.68) and (2.69) into both Eqs. (2.66) and (2.67) results in: di d − ωr L s i q dt

(2.70)



di q + ωr L s i d + λ f dt

(2.71)

vd = Rs i d + L s vq = Rs i q + L s

By substituting the value of ωr from Eq. (2.44) into Eqs. (2.70) and (2.71): π di d − P vm L s i q dt τ

(2.72)

di q π + P vm L s i d + λ f dt τ

(2.73)

vd = Rs i d + L s vq = Rs i q + L s

The dq-axes equivalent circuit of the tubular surface-mount linear PMSM according to Eqs. (2.72) and (2.73) is illustrated in Fig. 2.7. The voltage pace vector Vdq , current space vector Idq and the stator flux space  s can be expressed in terms dq-reference frame variables as: vector λ

32

2 Mathematical Modeling of Surface-Mount Linear …

(a)

id

(b)

Ls

Rs

+

_

+

P τπ vm Lsiq

vd

_ +

P τπ vm λf

_

Ls

Rs

iq

+ P τπ vm Lsid

vq

_

_

+

Fig. 2.7 Equivalent circuit of the tubular surface-mount linear PMSM, a d-axis Equivalent circuit according to Eq. (2.72), b q-axis Equivalent circuit according to Eq. (2.73)

Vdq = vd + jvq

(2.74)

Idq = i d + ji q

(2.75)

 s = λd + jλq λ

(2.76)

By substituting Eqs. (2.68) and (2.69) into Eq. (2.76), the stator flux space vector  s becomes: λ f  s = L s Idq + λ λ

(2.77)

The magnitude (or length) of the stator flux vector is denoted by λs is expressed as: −    → λs =  λs  = λ2d + λq2

(2.78)

The space vector model of the surface-mount linear PMSM in dq-reference frame is obtained by substituting Eqs. (2.70) and (2.71) into Eq. (2.74) and after simplification can be given as: ⎛

d Idq Vdq = Rs Idq + L s dt

⎞ s λ    ⎜ ⎟   ⎟ + jωr ⎜ ⎝ L s Idq + λ f ⎠ 



Vom

(2.79)



In Eq. (2.79), the term labeled as Vom is the maximum induced emf that can be generated in the machine. It is important to note that during steady state the derivative term vanishes and Eq. (2.79) can be expressed as:

2.4 Two-Axis Dynamic Models of Linear PMSM

33

Fig. 2.8 Steady-state space vector diagram of the linear PMSM in dq-reference frame as Eq. (2.80). The length of all the space vectors are arbitrarily shown

⎛

Vdq

⎞ s λ    ⎜ ⎟   ⎟ = Rs Idq + jωr ⎜ ⎝ L s Idq + λ f ⎠ 



Vom

(2.80)



The steady-state space vector diagram of the surface-mount linear PMSM according to Eq. (2.80) is shown in Fig. 2.8. It can be observed from Fig. 2.8 that the voltage space vector Vdq coincides with Vom if the resistive drop term Rs Idq is negligibly small. The instantaneous active power Eq. (2.50) can be expressed in the dq-reference frame using the park’s transformation as [3, 4, 7]: p=

3 vd i d + vq i q 2

(2.81)

Detailed expression of the instantaneous power in terms of current can be found by substituting the voltages vd and vq from Eqs. (2.66) and (2.67) into Eq. (2.81) as follows:  

3d 1

1 3π 3 2 2 2 2 Rs i d + Rs i q + L d id + L q iq + Pvm λd i q − λq i d (2.82) p= 2 2 dt 2 2 2 τ          Resistive losses

Rate of change of the stored energy

active power

The first term in Eq. (2.82) is the resistive losses in the machine conductors, the second term is the rate of change of stored energy in the magnetic field. The last term

34

2 Mathematical Modeling of Surface-Mount Linear …

in Eq. (2.82) indicates the power participating in the energy conversion process, and is called the electromechanical power: pem =



3π Pvm λd i q − λq i d 2τ

(2.83)

The electromechanical thrust force, F T , can be expressed in terms of pem and the mechanical speed, vm , as: FT =

pem vm

(2.84)

Substitution of Eq. (2.83) into Eq. (2.84) yields: FT = P

3π λd i q − λq i d 2τ

(2.85)

There are end effects on the thrust that are caused by the flux fringing at the ends of the stator and are quantified by a constant k F in the thrust force equation [32]. Therefore, Eq. (2.85) is modified as follows: FT = Pk F

3π λd i q − λq i d 2τ

(2.86)

Typically the value of k F is 0.9 [32], however, in this work the value of end-effect coefficient is taken as 1 which does not affect the experimental results. Substitution of λd and λq from Eqs. (2.68) and (2.69) into Eq. (2.85) gives the simplified equation of the total thrust force for surface-mount linear PMSM as: FT = Pλ f

3π iq 2τ

(2.87)

From Fig. 2.8, the load angle δ can also be expressed as: δ = tan

−1



L s iq L s id + λ f

 (2.88)

2.4.2 Dynamic Model of Linear PMSM in αβ-Reference Frame In 1930s Stanley [10] proposed a transformation of the machine variables in the stationary abc-reference frame to an arbitrary stationary αβ-reference frame which are co-planer with the former. The αβ-reference frame consists of two perpendicular

2.4 Two-Axis Dynamic Models of Linear PMSM

35

axes namely α-axis and β-axis such that β-axis leads α-axis by 90°. According to H. C. Stanley, αβ-reference frame is fixed with respect to the stationary abc-reference frame with α-axis is oriented at an angle θ with respect to the a-axis. It is convenient to set θ = 0 so that α-axis becomes aligned with a-axis as shown in Fig. 2.6. Now with θ set to zero, Clark transformation [30] is used to transform the machine variables from the stationary abc-reference frame to the stationary αβ-reference frame. The transformation matrix is given by: ⎤ 1 √ −1/2 −1/2 √ 2 = ⎣ 0 3/2 − 3/2 ⎦ 3 1/2 1/2 1/2 ⎡

K αβ

(2.89)

The inverse transformation matrix is given by: ⎡

−1 K αβ

⎤ 1 1 √0 ⎦ = ⎣ −1/2 √3/2 1 −1/2 − 3/2 1

(2.90)

Machine equations in the αβ-reference frame can be obtained by multiplying Eqs. (2.40) and (2.41) by Eq. (2.89). The conversion process from the three-phase system to the αβ-reference frame is similar to the process detailed for transformation of threephase machine variables to the dq-reference frame and will not be discussed here. The final dynamic model of surface-mount linear PMSM in stationary αβ-reference frame is given by: d λαβ dt

(2.91)

λαβ = Ls iαβ + λ pm,αβ

(2.92)

λ pm,αβ = λ f e jθr

(2.93)

vαβ = Rs iαβ +

In Eqs. (2.91)–(2.93), vαβ is the 2 × 1 αβ-voltage matrix, iαβ is the 2 × 1 αβcurrent matrix, λαβ is the 2 × 1 αβ-stator flux matrix, and λ pm,αβ is the 2 × 1 αβ-mover’s permanent magnet flux linkage matrix:        vα iα λα λ pm,α , iαβ = , λαβ = , λ pm,αβ = vαβ = vβ iβ λβ λ pm,β and ⎡ ⎡ ⎤ ⎤ Rs 0 0 Ls 0 0 Rs = ⎣ 0 Rs 0 ⎦, Ls = ⎣ 0 L s 0 ⎦ 0 0 Rs 0 0 Ls 

36

2 Mathematical Modeling of Surface-Mount Linear …

Equations (2.92)–(2.93) can be expanded as: vα = Rs i α +

dλα dt

(2.94)

vβ = Rs i β +

dλβ dt

(2.95)

λα = L s i α + λ pm,α

(2.96)

λβ = L s i β + λ pm,β

(2.97)

From Fig. 2.9 it is clear that the αβ-components of the mover’s permanent magnet  f along αβ-axes as: flux linkage space vector can be achieved by resolving λ λ pm,α = λ f cos θr

(2.98)

λ pm,β = λ f sin θr

(2.99)

The voltage pace vector Vαβ , current space vector Iαβ and the stator flux space  αβ can be expressed in terms dq-reference frame variables as: vector λ Vαβ = vα + jvβ

(2.100)

Fig. 2.9 Steady-state space vector diagram of the linear PMSM in xy-reference frame according to Eq. (2.131). The length of all the space vectors is arbitrarily shown

2.4 Two-Axis Dynamic Models of Linear PMSM

37

Iαβ = i α + ji β

(2.101)

 αβ = λα + jλβ λ

(2.102)

The space vector model of the surface-mount linear PMSM in αβ-reference frame is obtained by substituting Eqs. (2.94)–(2.99) into Eq. (2.102) and after simplification can be given as: d Iαβ f + jωr λ Vαβ = Rs Iαβ + L s dt

(2.103)

During steady-state operation the derivative of the current space vector in Eq. (2.103) is zero, therefore Eq. (2.103) becomes: f Vabc = Rs Iabc + jωr λ

(2.104)

The abc-reference frame and αβ-reference frame are stationary with respect to each such that when the space vector of machine variables in abc-reference can be trigonometrically resolved along α and β axes, therefore it can be shown mathematically [4–7]: Vαβ = Vabc

(2.105)

Iαβ = Iabc

(2.106)

 αβ = λ  abc λ

(2.107)

It is clear from Eqs. (2.103) to (2.107) that space vector dynamic model of the surface-mount linear PMSM remains the same for both the abc-reference frame  αβ can be and αβ-reference frame. The magnitude of the stator flux space vector λ expressed as: −→    λαβ  = λ2α + λ2β

(2.108)

The electromagnetic thrust force FT can be expressed in terms of αβ-components of the stator current and stator flux linkage for a surface-mount linear PMSM as: FT = Pk F

3π λα i β − λβ i α 2τ

(2.109)

38

2 Mathematical Modeling of Surface-Mount Linear …

2.4.3 Dynamic Model of Linear PMSM in xy-Reference Frame The transformation of machine variables in the stationary abc-reference frame to an arbitrary two axes reference frame rotating at synchronous speed with respect to the abc-reference frame is proposed by Krone [31]. In this thesis this arbitrary reference frame is termed as xy-reference frame consisting of two orthogonal axes namely x-axis and y-axis such that y-axis leads x-axis by 90°. The x-axis is oriented at angle θ s from the a-axis and is aligned with (or superimposed on) the stator flux  s as illustrated in Fig. 2.6. Therefore, the xy-reference frame rotates at space vector λ synchronous speed ωs with respect to the a-axis. The dynamic model of surface-mount linear PMSM in the dq-reference frame given by Eqs. (2.66)–(2.69) can be transformed into the xy-reference frame using the following transformation: 

fx fy



 =

cos δ sin δ − sin δ cos δ



fd fq

 (2.110)

With the inverse transformation being: 

fd fq



 =

cos δ − sin δ sin δ cos δ



fx fy

 (2.111)

where, f represents the voltage, current or flux linkage. From Fig. 2.7 it is evident that: sin δ =

λq λs

(2.112)

cos δ =

λd λs

(2.113)

and

Substituting the values of λq and λd from Eqs. (2.112) and (2.113) into Eq. (2.86) gives: FT = Pk F

3π λs i q cos δ − i d sin δ 2τ

(2.114)

From Eq. (2.110), it is evident that:

i y = i q cos δ − i d sin δ

(2.115)

2.4 Two-Axis Dynamic Models of Linear PMSM

39

Hence, Eq. (2.114) simplifies to FT = Pk F

3π λs i y 2τ

(2.116)

Using Eq. (2.110), the flux linkages can be converted from the dq-reference frame to xy- reference frame as: 

λx λy





cos δ sin δ = − sin δ cos δ



L s id + λ f L s iq

 (2.117)

Expansion of Eq. (2.117) gives 

λx λy





cos δ sin δ = − sin δ cos δ



L s id L s iq





cos δ sin δ + − sin δ cos δ



λf 0



Hence, 

λx λy





cos δ sin δ = − sin δ cos δ



Ls 0 0 Ls



id iq



 + λf

cos δ − sin δ



Therefore, 

λx λy



 =

Ls 0 0 Ls



ix iy



 + λf

cos δ − sin δ

 (2.118)

From Eq. (2.118), the xy-components of the stator flux vector can be given as: λx = L s i x + λ f cos δ

(2.119)

λ y = L s i y − λ f sin δ

(2.120)

− → Since the stator flux vector λs is aligned with x-axis as shown in Fig. 2.7 and λ y = 0, therefore, Eq. (2.120) becomes: iy =

λf sin δ Ls

(2.121)

Substituting Eq. (2.121) into Eq. (2.116): FT = Pk F

3π λs λ f sin δ 2τ

(2.122)

The dq-reference frame model of the surface-mount linear PMSM given by Eqs. (2.66)–(2.69) is converted to xy-reference frame using Eq. (2.110) as:

40

2 Mathematical Modeling of Surface-Mount Linear …

v x = Rs i x +

dλx − ωs λ y dt

(2.123)

v y = Rs i y +

dλ y + ωs λs dt

(2.124)

− → From Fig. 2.7, it is clear that λs is aligned with the x-axis, therefore, λx = λs and λ y = 0 so that Eq. (2.130) can be expressed in the xy-reference frame as: dλs dt

(2.125)

v y = Rs i y + ωs λs

(2.126)

v x = Rs i x +

The voltage pace vector Vs , current space vector Is and the stator flux space vector  s can be expressed in terms xy-reference frame variables as: λ Vs = vx + jv y

(2.127)

Is = i x + ji y

(2.128)

 s = λx + jλ y λ

(2.129)

The space vector model of the surface-mount linear PMSM in xy-reference frame is obtained by substituting Eqs. (2.123) and (2.124) into Eq. (2.127) and after simplification can be given as: − → − → d λs − → − → + jωs λs Vs = R Is + dt

(2.130)

During steady-state operation the derivative of the current space vector in Eq. (2.130) is zero, therefore Eq. (2.130) becomes: s Vs = Rs Is + jωs λ

(2.131)

The steady-state space vector diagram corresponding to Eq. (2.131) is shown in Fig. 2.9. It can be observed from Fig. 2.9 that the voltage space vector Vs coincides  s if the resistive drop term Rs Is is negligibly small as illustrated in Fig. 2.10. with jωs λ It is also important to note that the absolute value of the space vectors for voltage, current and stator flux remains the same in all reference frames, e.g.      − → −→ λs =  λs  = λαβ  = λ2d + λq2 = λ2α + λ2β

(2.132)

2.5 Estimation of Stator Flux Magnitude and Thrust Force Base on …

41

Fig. 2.10 Steady-state space vector diagram of the linear PMSM in xy-reference frame when the resistive drop Rs Is is neglected. The length of all the space vectors is arbitrarily shown

2.5 Estimation of Stator Flux Magnitude and Thrust Force Base on Dq-Axes Current Model The flux linkages for the surface-mount linear PMSM in dq-axes are given as (Fig. 2.8): λd = L s i d + λ f

(2.133)

λq = L s i q

(2.134)

As illustrated in Fig. 2.8, the dq-axes components of stator flux given by Eqs. (2.133) and (2.134) can be converted to αβ-axes flux linkages as: 

λα λβ





cos θr − sin θr = sin θr cos θr



λd λq

 (2.135)

As shown in Fig. 2.3, θr is the angle (in elec. rad.) between the mover’s flux vector (produced by mover’s permanent magnet) and the α-axis. The stator flux magnitude λs and angle θs (in elec. rad., as shown in Fig. 1.3) can be expressed as: λs =

 λ2α + λ2β

(2.136)

42

2 Mathematical Modeling of Surface-Mount Linear …

θs = tan−1



λβ λα

 (2.137)

The thrust force FT for a surface-mount linear PMSM can be expressed in terms of i q as: FT = Pk F

3π λ f iq 2τ

(2.138)

The estimation of the stator flux magnitude and the thrust force will be performed using Eqs. (2.133)–(2.138) and will be used in the various control schemes proposed in the later chapters.

2.6 Conclusion This chapter is dedicated to mathematical modeling of the tubular surface-mount linear PMSM. The structure of the machine was first analyzed and dynamic equations for the surface-mount linear PMSM in the three-phase abc-reference frame are developed. In order to reduce the complexity of the equations, machine model was transformed to the two-axes reference frames: the mover’s flux vector dq-reference frame, the stationary αβ-reference frame, and the stator flux vector xy-reference frame. The resultant machine equations in the two-axis reference frames are simpler compared with the three-phase system equations and are apt for controller design purposes.

References 1. I. Boldea, S. A. Nasar, Linear Electric Actuators and Generators (Cambridge University Press Inc., New York, 1997) 2. D. Gerling, Electrical Machines: Mathematical Fundamentals of Machine Topologies (Springer-Verlag, Berlin Heidelberg, 2015) 3. P. S. Chandana Perara, Sensorless Control of Permanent Magnet Synchronous Motor Drives, Ph.D. Dissertation, Faculty of Engineering and Sciences, Alborg Univ., Denmark, 2002 4. D.W. Novotny, T.A. Lipo, Vector Control and Dynamics of AC Drives (Oxford University Press Inc, New York, 1996) 5. T.A. Lipo, Analysis of Synchronous Machines (CRC Press, Taylor & Francis Group, New York, 2012) 6. C.M. Ong, Dynamic Simulation of Electric Machinery Using MATLAB/SIMULINK (Prentice Hall, PTR, Upper Saddle Rive, New Jersey, 1998) 7. P. Krause, O. Wasynczuk, S. Sudhoff, S. Pekarek, Analysis of Electric Machinery and Drive Systems (Wiley, Hoboken, New Jersey, 2013) 8. T. M. Jahns, Flux-weakening regime operation of an interior permanent-magnet synchronous motor drive. IEEE Trans. Ind. Appl. IA-23, 681–689 (1987)

References

43

9. R.H. Park, Two-reaction theory of synchronous machines generalized method of analysis-part I. AIEE Trans. 48, 716–727 (1929) 10. H.C. Stanley, An analysis of induction motor. AIEE Trans. 57, 751–755 (1938) 11. C.L. Fortescue, Method of symmetrical co-ordinates applied to the solution of polyphase networks. AIEE Trans. 37, 629–716 (1918) 12. W.V. Lyon, Transient conditions in electric machinery. AIEE Trans. 42, 159–179 (1923) 13. W.V. Lyon, Transient Analysis of Alternating Current Machinery (Wiley, New York, 1954) 14. M.S.W. Tam, N.C. Cheung, A high speed high precision linear drive system for manufacturing automation. Sixt. Annu. IEEE APEC 1, 440–444 (2001) 15. P. Famouri, Control of a linear permanent magnet brushless DC motor via exact linearization methods. IEEE Trans. Energy Convers. 7, 544–551 (1992) 16. F. Lin, C. Lin, C. Hong, Robust control of linear synchronous motor servodrive using disturbance observer and recurrent neural network compensator. IEE Electr. Power Appl. 147, 263–272 (2000) 17. F. Lin, R. Wai, Hybrid control using recurrent fuzzy neural network for linear induction motor servo drive. IEEE Trans. Fuzzy Syst. 9, 102–115 (2001) 18. R. Wai, W. Liu, Nonlinear decoupled control for linear induction motor servo-drive using the sliding-mode technique. IEE. Control. Theory Appl. 148, 217–231 (2001) 19. F. Lin, K. Shyu, C. Lin, Incremental motion control of linear synchronous motor. IEEE Trans. Aerosp. Electron. Syst. 38, 1011–1022 (2002) 20. T. Liu, Y. Lee, Y. Crang, Adaptive controller design for a linear motor control system. IEEE Trans. Aerosp. Electron. Syst. 40, 601–616 (2004) 21. J. Vittek, J. Michalik, V. Vavrus, V. Horvath, Design of control system for forced dynamics control of an electric drive employing linear permanent magnet synchronous motor. International Conference on Industrial Electronics and Control Applications, ICIECA (2005), pp. 1–6 22. C. Sung, Y. Huang, Based on direct thrust control for linear synchronous motor systems. IEEE Ind. Electron. 56, 1629–1639 (2009) 23. A.Y.M. Abbas, J.E. Fletcher, Synthetic loading applied to linear permanent magnet synchronous machines. IET Renew. Power Gener. 4, 221–231 (2010) 24. A.Y.M. Abbas, J.E. Fletcher, Implementation of sliding mode controller for linear synchronous motors based on direct thrust control theory. IET Control Theory Appl. 4, 326–338 (2010) 25. J. Linares-Flores, C. García-Rodríguez, H. Sira-Ramírez, O.D. Ramírez-Cárdenas, Robust backstepping tracking controller for low-speed PMSM positioning system: Design, analysis, and implementation. IEEE Ind. Informat. 11, 1130–1141 (2015) 26. W. Zhao, S. Jiao, Q. Chen, D. Xu, J. Ji, Sensorless control of a linear permanent-magnet motor based on an improved disturbance observer. IEEE Ind. Electron. 65, 9291–9300 (2018) 27. W. Zhao, A. Yang, J. Ji, Q. Chen, J. Zhu, Modified flux linkage observer for sensorless direct thrust force control of linear vernier permanent magnet motor. IEEE Trans. Power Electron. 34, 7800–7811 (2019) 28. P.C. Sen, Principles of Electric Machines and Power Electronics (Wiley, USA, 1997) 29. T. M. Jahns, G. B. Kliman, T. W. Neumann, Interior permanent-magnet synchronous motors for adjustable-speed drives. IEEE Trans. Ind. Appl. IA-22, 738–747 (1986) 30. W.C. Duesterhoeft, M.W. Schulz, E. Clarke, Determination of instantaneous currents and voltages by means of alpha, beta, and zero components. AIEE Trans. 70, 1248–1255 (1951) 31. G. Kron, Equivelent Circuits of Electric Machinery (Wiley, New York, 1951) 32. Y.S. Huang, C.C. Sung, Implementation of sliding mode controller for linear synchronous motors based on direct thrust control theory. IET Control Theory Appl. 4, 326–338 (2010)

Chapter 3

Direct Thrust Force Control Based on Duty Ratio Control

3.1 Introduction In this chapter the concept of conventional direct thrust force control (conventional DTFC) [1–17] is rigorously analysed and experimentally evaluated for the linear PMSM. Linear PMSMs, especially those with low inductance and short pole pitch, exhibit significant ripple in flux and thrust force under conventional direct thrust force control. Aimed at reducing ripple in the flux and the force, various methods, including space vector modulation [9, 18–28], model predictive control [29–31] and duty ratio control [32–39] can be used. This chapter proposes a novel duty ratio control scheme for direct thrust force control of linear PMSM. The effect of inverter voltage vectors on the flux and the thrust force are analysed. Subsequently, a precise expression for the on-line computation of the required duty ratio is derived. Compared to the state of the art duty ratio control, the proposed approach eliminates the need to tune gains. The proposed on-line duty ratio calculation is based on the selected voltage vector from the switching table and takes into account the mover’s speed. The proposed approach retains the characteristics of the conventional direct thrust force control in terms of fast transient and steady-state responses for both flux and thrust force. Experimental results demonstrate a faster transient response and negligible steady-state error for flux, force, and speed responses under the proposed approach when compared to the state of the art.

3.2 Basic Principle of Direct Thrust Force Control The basic principle of direct thrust force control (DTFC) is based on the relationship between electromagnetic thrust force FT and the load angle δ.

© Springer Nature Switzerland AG 2020 M. A. M. Cheema and J. E. Fletcher, Advanced Direct Thrust Force Control of Linear Permanent Magnet Synchronous Motor, Power Systems, https://doi.org/10.1007/978-3-030-40325-6_3

45

46

3 Direct Thrust Force Control Based on Duty Ratio Control

FT = Pk F

3 π λs λ f sin δ 2 τ Ls

(3.1)

Equation (3.1) relates the electromagnetic thrust force FT and the load angle δ and it is observed from (3.1) that the thrust force increases with the load angle δ if the magnitude of the stator flux vector λs is kept constant and δ is controlled within  the range − π2 , π2 . The maximum force is produced when δ is ± π2 radians. The time derivative of (3.1) is: d FT 3 π dδ = Pk F λs λ f cos δ dt 2 τ Ls dt

(3.2)

  In (3.2), ddtFT is always positive when δ is controlled within the range − π2 , π2 . From (3.2) the stator flux vector should be controlled such that its magnitude λs is kept constant (by restricting the magnitude variation within a small hysteresis band) − → and rotation speed of the stator flux vector λs controlled as fast as possible to achieve the maximum rate of change in actual thrust force. According to DTFC [13], the magnitude and rotation of the stator flux vector, and hence the force is controlled by applying a suitable inverter voltage vector from Table 3.1. The selection of voltage vector from Table 3.1 is based on the force and stator flux control signals ε F and ελ which are generated by force and stator flux hysteresis controllers respectively and are defined as: 

1, to increase FT ↑ if e F > +Δτ F −1, to decrease FT ↓ if e F < −Δτ F  1, to increase λs ↑ if eλ > +Δτλ ελ = sgn(eλ ) = −1, to decrease λs ↓ if eλ < −Δτλ

ε F = sgn(e F ) =

(3.3) (3.4)

In (3.3) and (3.4), e F and eλ are the force and flux error respectively, and 2Δτ F and 2Δτλ are the force and flux hysteresis bands respectively. It is important to note that in case of |e F | ≤ 2Δτ F or |eλ | ≤ 2Δτλ the flux and thrust force controllers of (3.3) and (3.4) will continue to apply the same voltage vector during the previous sampling time respectively as explained in [13].  6 ) of a voltage source inverter are shown in  1 to V The six active voltage vectors (V Fig. 3.1. The rotation of stator flux vector from sector I to sector II, when the inverter voltage vector Vk is selected from Table 3.1 according to (3.3) and (3.4), is illustrated in Fig. 3.1 when using a DTFC control system, shown in Fig. 3.2, and S a , S b , and Table 3.1 Switching table to generate V k for DTFC [13]

− → λs in sector number n (1–6) Stator flux magnitude λs

Thrust force FT ↑

↓

↑

Vn+1

Vn−1

↓

Vn+2

Vn−2

3.2 Basic Principle of Direct Thrust Force Control

47

β

y

V4 is applied C

D

Sector II

Δλ =V Ts

V3 (010)

λ

Sector III V2 (110)

λ λ

V4(011)

O

Sector IV

B A

λ

V3 is applied

V2 is applied x α

V1 (100)

Sector I

V5 (001)

Sector V

V6 (101)

Reference Flux

Sector VI

Hysteresis Band

Fig. 3.1 Voltage space vectors generated by a 2-level VSI [5, 6] and control of magnitude and rotation of stator flux vector. Note the magnitude variation of stator flux is kept within the hysteresis band (diagram is not to scale)

λref

+

-

vref

PI

+

-

vm

Fref +

-

Flux Controller Eq. (3.3) Force Controller Eq. (3.4)

λs FT

Switching Table 3.1 Sa Sb Sc

Inverter (VSI)

LinearPMSM

x θs Estimation of θs ,λs and FT Eq. (2.133)-(2.138)

abc

id iq

αβ

iα iβ

αβ dq

θr

P

π τ

d dt

Fig. 3.2 Block diagram of the conventional DTFC scheme

S c are the control signals to the inverter. The stator flux vector rotates from point A to D and variation in its magnitude remains within the hysteresis band as illustrated. The effect of inverter voltage vectors on flux and force variations is comprehensively analysed in the next section.

48

3 Direct Thrust Force Control Based on Duty Ratio Control

3.2.1 Selection of Reference Stator Flux Magnitude λ r e f The stator flux reference λr e f is selected according to maximum force per ampere (MFPA) and for a surface-mount linear PMSM can be expressed in terms of force reference Fr e f as:  λr e f =



λ2f +

2 τ Ls Fr e f 3 π Pk F

2 (3.5)

The prototype surface-mount linear PMSM used in this research has a peak force of ±312 N. According to (3.5), when Fr e f varies from 0 to ±312 N, the corresponding variation in λr e f is negligible, i.e. from 0.0846 to 0.0847 Wb because of low values of L s and τ for the prototype linear PMSM as given in Table 1.1. Therefore, λr e f is set to 0.0846 Wb.

3.2.2 Operational Range of Load Angle δ for Low Inductance Surface-Mount Linear PMSM According to (3.1) the maximum force occurs when δ is controlled at ± π2 radians. However a closer inspection of (3.1) suggests that, when λs is kept constant at λr e f (as explained previously), then for a linear machine with small value of pole-pitch τ and low stator inductance L s , the rated maximum force of the linear PMSM is achieved at much smaller values of δ than ± π2 radians. Therefore, a surface-mount linear PMSM with short pole pitch and low stator inductance has a small operational range of δ. Hence a small change in δ can result in a large change in force FT for a surface-mount linear PMSM which results in unacceptably large ripple in force when the linear PMSM is controlled using conventional DTFC. In general, surfacemount linear PMSMs tend to have low stator inductance owing to the larger air gap compared to their rotational counterparts. Also, for most linear PMSMs the pole-pitch is fraction of a meter resulting in a small operational range of δ. It is worth mentioning that low inductance and short pole pitch are the key factor that makes surface-mount linear PMSMs difficult to control under conventional DTFC compared to rotational machines. If δmax is the load angle corresponding to the rated maximum force Fmax , then from (3.1) δmax can be given as: δmax = sin−1



2τ L s Fmax 3π Pk F λs λ f

 (3.6)

As described, the stator flux for the prototype system is adjusted to λr e f = 0.0846 Wb, and from (3.6) δmax is given as: δmax = ±0.154 radians(i. e., ±8.8 degrees)

3.2 Basic Principle of Direct Thrust Force Control

49

Hence, the operational range of δ for the prototype linear PMSM according to  (3.6) is within [−0.1545, 0.1545] radians which is a small subset of − π2 , π2 . This analysis justifies the assumption that cos δ ≈ 1 for the small range of values for δ.

3.3 Stability Analysis of Direct Thrust Force Control From (2.125), the dynamics of stator flux in xy-reference frame are determined by: dλs = v x − Rs i x dt

(3.7)

Since from Fig. 2.9, δ = θs −θr , the time derivative of load angle δ can be obtained as: dθs dθr dδ = − dt dt dt Since

dθs dt

= ωs and

dθr dt

(3.8)

= ωr , therefore: dδ = ωs − ωr dt

(3.9)

v y − Rs i y λs

(3.10)

From (2.126), ωs is as: ωs =

Substituting the value of ωs from (3.10) into (3.9): v y − Rs i y dδ = − ωr dt λs

(3.11)

Now from (3.2) and (3.11), the expression for the time derivative of thrust force for surface-mount linear PMSM is given as:   v y − Rs i y 3 π d FT = Pk F λs λ f cos δ − ωr dt 2 τ Ls λs

(3.12)

It is evident from (3.7) and (3.12) that for the surface-mount linear PMSM, the magnitude of the stator flux space vector λs and thrust force FT can be controlled by vx and v y respectively. Therefore, it can be concluded that whenever a voltage vector (out of the six) is applied, its x-component will cause a change in the stator flux, and the y-component will change the thrust force. In order to examine the stability of thrust force and flux control mechanisms, we define the following error vector as:

50

3 Direct Thrust Force Control Based on Duty Ratio Control

T  e = eλ e F

(3.13)

eλ = λr e f − λs

(3.14)

e F = Fr e f − FT

(3.15)

where,

The error dynamics can be defined by taking the time derivative of (3.14) and (3.15): e˙λ = λ˙ r e f − λ˙ s ⇒ e˙λ = λ˙ r e f − vx + Rs i x

(3.16)

The error dynamics for the thrust force are: e˙ F = F˙r e f − F˙T

(3.17)

Substituting the value of F˙T from (3.12) into (3.17):   v y − Rs i y 3 π λs λ f cos δ − ωr e˙ F = F˙r e f − Pk F 2 τ Ls λs

(3.18)

After simplification, (3.18) becomes:  ⇒ e˙ F = F˙r e f − c1 v y − Rs i y − λs ωr

(3.19)

where, c1 = Pk F

3 π λ f cos δ 2 τ Ls

(3.20)

3.3.1 Lyapunov Stability Analysis of Conventional DTFC The inverter voltage vector, in a DTFC controlled linear PMSM, is selected from Table 3.1 based on the output of the hysteresis controllers defined in (3.3) and (3.4). As discussed in Sect. 3.4 that vx and v y of the applied inverter voltage vector are utilised to control the stator flux and thrust force respectively. The voltage components vx and v y can be expressed in terms of the hysteresis controllers of (3.3) and (3.4) as: vx = |vx |sgn(eλ )

(3.21)

3.3 Stability Analysis of Direct Thrust Force Control

51



v y = v y sgn(e F )

(3.22)

In order to ensure a globally stable operation of the conventional DTFC, a Lyapunov candidate function in terms of the error vector e can be defined as: V = eT e

(3.23)

Substituting the value of e from (3.13) into (3.23):   V = eλ2 e2F ≥ 0

(3.24)

The time derivative of the Lyapunov candidate function of (3.24) is given as: V˙ = e T e˙

(3.25)

 e˙ ⇒ V˙ = e T λ e˙ F

(3.26)

By substituting the values of e˙λ and e˙ F from (3.16) and (3.18) respectively into (3.26):

⇒ V˙ = e T

F˙r e f

λ˙ re f − vx + Rs i x  →

− − c1 v y − Rs i y − λs ωr

 (3.27)

In (3.27), λ˙ r e f and F˙r e f are the time derivatives of the arbitrary commanded values of flux and thrust force. In DTFC the flux command value is fixed, and also the thrust force command value is constant during any transient event. Therefore, the derivatives of flux and force command values are evidently zero. By putting the values of xy-components of the applied voltage vector from (3.21) and (3.22) in (3.27), the derivative of the Lyapunov candidate function can be expressed as: V˙ = e T



|sgn(e Rs i x − |v

λ ) x ≤0 c1 Rs i y + λs ωr − c1 v y sgn(e F ) 

(3.28)

If the stator 1. vx is large enough to establish

flux. 2. c1 is positive and v y > Rs i y + λs ωr . Considering (3.20), for LPMSM:   surface-mount π c1 > 0 implies δ ∈ −π , i.e. for stable operation of the machine, the oper2 2   π ational range of should be within −π . It is important to note that Lyapunov 2 2 stability analysis gives the same condition that has been discussed in Sect. 3.3 with reference to (3.1).

52

3 Direct Thrust Force Control Based on Duty Ratio Control

3.4 Effect of Inverter Voltage Vectors on Thrust Force and Flux Variation In order to analyse DTFC for the surface-mount linear PMSM it is important to formulate the effect of the inverter voltage vectors on the variation of thrust force FT and the stator flux magnitude λs . The time rate of change of the thrust force FT can be given in terms of αβ-reference as (see Appendix B for derivation):  Rs 3 π kF P  π d FT = − F0 + −vα λ pm,β + vβ λ pm,α − P λs λ f vm cos δ (3.29) dt Ls 2 τ Ls τ where, F0 on right hand side of (3.29) is the initial operating thrust force at the current instant of time. In order to formulate the effect of the inverter voltage vectors on the rate of change of stator flux, following expression is derived as (see Appendix B): Rs 1 dλs Rs vα λα + vβ λβ λ f cos δ + = − λs + dt Ls Ls λs

(3.30)

It is evident from (3.29) and (3.30) that the rate of change of stator flux and the thrust force are controlled by the applied inverter voltage vector and are also the functions of the operating point of the machine. It is observed from (3.29) that the rate of change of thrust force is dependent on the operating thrust force F0 and the mover’s speed vm in addition to the αβ-components of the stator voltage. It is clear from (3.30) that the rate of change of stator flux is a function of the operating flux of the machine in addition to stator voltage. It can be concluded that the rate of change of flux and thrust force will vary for different operating flux, thrust force and mover’s speed. Moreover it is also necessary to quantitatively analyze the relationship between the corresponding variations in stator flux and thrust force and the applied voltage vectors considering the machine operating conditions. Since DTFC is implemented using a digital controller, therefore, the change in flux during one sampling period Ts for a given voltage vector can be given by using (3.30) as: λs = −

Rs Rs Ts  vα λα + vβ λβ λs ∗ Ts + λ f cos δ ∗ Ts + Ls Ls λs

(3.31)

The change in thrust force caused by the applied voltage vector during one sampling time Ts can be expressed as: FT = F1 + F2 + F3

(3.32)

where, F1 = −

Rs F0 ∗ Ts Ls

(3.33)

3.4 Effect of Inverter Voltage Vectors on Thrust Force …

F2 = − F3 =

3 π 2 kF P2 λs λ f vm cos δ ∗ Ts 2 τ 2 Ls

3 π kF P  −vα λ pm,β + vβ λ pm,α ∗ Ts 2 τ Ls

53

(3.34) (3.35)

It is observed from (3.32) that the variation in FT in thrust forces comprises three components. The first two terms in (3.32), F1 and F2 are proportional to operating thrust force F0 and mover’s speed vm and are always negative. The third term F3 that contributes to thrust force variation reflects the effect of the applied inverter voltage vector. It is important to note that F3 is a sinusoidal component and has a zero average value over one period of variation of αβ-components of rotor flux. Therefore, the average negative rate of change of thrust force is always greater than the average positive rate of change of thrust force. In order to observe the effect of the applied inverter voltage vector on the rate of change of stator flux and thrust force, it is necessary to plot the rate of change of flux dλs and rate of change of thrust force ddtFT as function of θs under a given operating dt condition. The αβ-components of the stator flux vector are (according to Fig. 2.9): λα = λs cos θs

(3.36)

λβ = λs sin θs

(3.37)

Substituting (3.36) and (3.37) in (3.30):  Rs Rs dλs = − λs + λ f cos δ + vα cos θs + vβ sin θs dt Ls Ls

(3.38)

Since, in a DTFC controlled linear PMSM the stator flux magnitude is maintained at λr e f , (3.38) becomes:  dλs Rs Rs = − λr e f + λ f cos δ + vα cos θs + vβ sin θs dt Ls Ls

(3.39)

Substituting the value of λr α and λrβ from (1.60) and (1.61) respectively into (3.29):  Rs d FT 3 π kF P  π = − F0 + λ f −vα sin θr + vβ cos θr − P λs vm cos δ (3.40) dt Ls 2 τ Ls τ By substituting the value of θr from (1.49) into (3.40), and setting λs = λr e f , (3.40) becomes:

54

3 Direct Thrust Force Control Based on Duty Ratio Control

Rs d FT 3 π kF P  = − F0 + λ f −vα sin(θs − δ) + vβ cos(θs − δ) dt Ls 2 τ Ls  π −P λr e f vm cos δ τ

(3.41)

In Eqs. (3.39) and (3.41) the load angle δ is dependent on the operating thrust force F0 and λr e f . It is important to note that δ is fixed for a given operating condition and can be determined using (3.1) for given F0 and λr e f as: δ = sin−1



2τ L s F0 3π Pk F λr e f λ f

 (3.42)

Equations (3.39), (3.41), and (3.42) are to compute the rate of change of stator flux and thrust force as function of θs under any operating conditions for all the applied inverter voltage vectors, whereas, the αβ-components of the applied inverter voltage vector can be determined from Table 3.2. Figures 3.3 and 3.5 illustrates the rate of change of stator flux and thrust force as a function of θs for the prototype linear-PMSM of Table 3.2. The operating thrust force F0 is 53 N which is the rated continuous thrust force for the prototype. The mover’s speed is 600 mm/s. The reference value λr e f for the stator flux vector is 0.0846 Wb according to (3.5) as explained in Sect. 3.2. Table 3.2 αβ-components of inverter voltage vectors αβ-voltage

Inverter voltage vectors − → − → − → V0 V1 V2 (110) (000) (100)

− → V3 (010)

− → V4 (011)

− → V5 (001)

− → V6 (101)

− → V7 (111)

0.5Vdc

0

vα

0

Vdc

0.5Vdc

−0.5Vdc

−Vdc

−0.5Vdc

vβ

0

0

0.866Vdc

0.866Vdc

0

−0.866Vdc −0.866Vdc 0

100

V (000) 0

d

s

/dt (Wb/s)

75

V (111) 7

50

V (100)

79 Wb/s

25

1

V (110) 2

0

V (010) 3

-25

V (011)

-79 Wb/s

-50

4

V (001) 5

-75 -100 -30

V (101) 6

0

30

60

90

120

150 s

180

210

240

270

300

330

(degree)

Fig. 3.3 Rate of change of stator flux for active (V1 to V6 ) and zero (V0 and V7 ) voltage vectors (vm = 600 mm/s and operating thrust force F0 = 53 N)

3.4 Effect of Inverter Voltage Vectors on Thrust Force …

55

It can be observed from Fig. 3.3 that, at rated condition, zero voltage vectors do not change the stator flux significantly. Moreover, the positive and negative rate of change of flux has the same absolute value. However, it can be observed from Fig. 3.4 that the rate of change of thrust force for the prototype linear PMSM under the zero voltage vectors is −196.3 kN/s which will always decrease the thrust force. The maximum positive rate of change of thrust force for the active voltage vectors is 763 kN/s and the peak value for the negative rate of change of thrust force is −1150 kN/s. It is clear that under an active voltage vector the negative rate of change of thrust force is much higher in magnitude than the positive rate of change of thrust force. It can also be observed from Fig. 3.4 that the maximum peak for both positive and negative rate of change of thrust force is same for all the active vectors and this is valid for all the operating conditions. In order to analyse the performance of digitally implemented DTFC, it is important to consider the variations in stator flux and thrust force for under both zero and active voltage vectors when applied for the duration of one sampling period Ts . Figures 3.5 and 3.6 illustrates the change in stator flux and thrust force for the prototype linear PMSM for different positions of stator flux vector θs when the voltage vector is applied for a sampling period Ts = 100 μs according to (3.31) and (3.32) with the same operating conditions as described previously. It is important to note that while considering the variation in thrust force over a sampling period, the change in stator flux vector position θs is not considered and the initial stator flux vector position at the beginning of the sampling period is utilized. It can be observed from Fig. 3.6 that the magnitude variation in the thrust force is significantly larger when expressed as a percentage of the rated continuous force which explains the ripple in thrust force under DTFC. It can be concluded from (3.29) and (3.32) that the rate of change of thrust force d FT and the change in thrust force over one sampling period FT both depend on dt the machine inductance L s and the pole-pitch τ . Therefore the linear-PMSM with a

dFT /dt (N/s)

6

x 10 1 0.75 0.5 763000 N/s 0.25 0 -0.25 -0.5 -196300 N/s -1150000 N/s -0.75 -1 -1.25 -1.5 -30 0 30 60 90

V (000) 0

V (111) 7

V (100) 1

V (110) 2

V (010) 3

V (011) 4

V (001) 5

V (101) 6

120

150 s

180

210

240

270

300

330

(degree)

Fig. 3.4 Rate of change of thrust force for active (V1 to V6 ) and zero (V0 and V7 ) voltage vectors (vm = 600 mm/s and operating thrust force F0 = 53 N)

56

3 Direct Thrust Force Control Based on Duty Ratio Control 0.01

V (000) 0

0.0075

V (111) 7

s

(Wb)

0.005

0.0079 Wb

V (100) 1

0.0025

V (110) 2

0

V (010) 3

-0.0025

V (011)

-0.0079 Wb

-0.005

4

V (001) 5

-0.0075 -0.01 -30

V (101) 6

0

30

60

90

120

150 s

180

210

240

270

300

330

(degree)

FT (N)

Fig. 3.5 Variation in stator flux for active (V1 to V6 ) and zero (V0 and V7 ) voltage vectors during one sampling period Ts = 100 μs (vm = 600 mm/s and operating thrust force F0 = 53 N) 100 75 50 76.34 N 25 0 -25 -50 -115.6 N -75 -19.36 N -100 -125 -150 -30 0 30 60 90

V (000) 0

V (111) 7

V (100) 1

V (110) 2

V (010) 3

V (011) 4

V (001) 5

V (101) 6

120

150 s

180

210

240

270

300

330

(degree)

Fig. 3.6 Variation in thrust force for active (V1 to V6 ) and zero (V0 and V7 ) voltage vectors during one sample time Ts = 100 μs (vm = 600 mm/s and operating thrust force F0 = 53 N)

low value of L s and a short pole-pitch τ has large values of ddtFT and FT at any rated condition and exhibits significantly large ripple in thrust force under DTFC which is illustrated experimentally in the following section. The variation in the peak value of rate change of thrust force ddtFT with mover’s speed at different value of operating thrust force F0 under both zero and active voltage vectors is illustrated in Figs. 3.7 and 3.8. It can be observed from Fig. 3.7 that the peak negative rate of change of thrust force under the zero vector decreases linearly with speed for at a given operating force. From Fig. 3.8, it is seen that the peak positive rate of change of thrust force under active voltage vector decreases linearly with speed for a given operating force.

3.5 Experimental Results for Conventional DTFC

Peak dFT /dt (N/s)

0

x 10

57

5

0N 5N 10 N 15 N 20 N 25 N 30 N 35 N 40 N 53 N

-0.5 -1 -1.5 -2 -2.5 -3 -3.5 -4 0

200

400

600

800

1000

1200

Speed(mm/s) FT Fig. 3.7 Variation in peak value of the negative rate of change of thrust force ddt , caused by any of the zero voltage vectors (V0 and V7 ), at different speed for various values of operating thrust force F0

10

x 10

5

0N 5N 10 N 15 N 20 N 25 N 30 N 35 N 40 N 53 N

Peak dFT /dt (N/s)

9.5 9 8.5 8 7.5 7 6.5 6 0

200

400

600

800

1000

1200

Speed(mm/s) FT Fig. 3.8 Variation in the peak value of positive rate of change of thrust force ddt , caused by any of the active voltage vectors (V1 to V6 ), at different speed for various values of operating thrust force F0

3.5 Experimental Results for Conventional DTFC The conventional DTFC is digitally implemented for the prototype linear PMSM of Table 3.2 according to the block diagram of Fig. 3.2 using the switching Table 3.1. The sampling Ts set to 100 μs. The speed, thrust force, stator flux, and phase a current response are shown in Fig. 3.9a, b for start-up and speed reversal respectively. The parameters of the speed PI controller of Fig. 3.2 are proportional gain k p and integral gain ki which are set to 970 and 7 respectively. The gain for the anti-windup component is selected as unity. It is observed from Fig. 3.9a, b that a reasonable speed response is achieved during start-up and speed reversal transients. However, a significantly large amount of ripple is present in thrust force as well as the stator flux during both the start-up and thrust force transient and in the steady-state. The reason

58

3 Direct Thrust Force Control Based on Duty Ratio Control

Speed(mm/s)

400

Reference

0

Measured 1.55

250

1.6

1.65

0

Force(N)

Force(N)

Estimated 1.65

0.1

Estimated

Flux(Wb)

0.08

Reference 1.55

1.6

1.65

1.7

1.55

1.6

Time (s)

1.65

1.7

0.6

0.8

1

0.8

1

Estimated 0

0.2

0.4

0.6

Estimated

0.09 0.08 0.07

Current(A)

Current(A)

8 6 4 2 0 -2 -4 -6 -8 1.5

-300

0.1

0.09

0.07 1.5

0.4

0

-500

1.7

0.2

Reference

300

0

1.6

Measured

-600

500

Reference

1.55

Reference 0

1.7

150

-150 1.5

Speed Reversal Response

600

200

-200 1.5

Flux(Wb)

(b)

Start-up Response Speed(mm/s)

(a)

8 6 4 2 0 -2 -4 -6 -8

Reference 0

0.2

0.4

0.6

0.8

1

0

0.2

0.4

0.6

0.8

1

Time (s)

Fig. 3.9 From top to bottom, speed response, thrust force response, flux response, and stator phase a current response under conventional DTFC. a Start-up from 0 to 200 mms−1 , b speed reversal from −600 to 600 mm/s (experiment)

for the large ripple in the thrust force can be attributed to the low machine inductance and short pole-pitch for the prototype linear PMSM as discussed in Sects. 3.2 and 3.4. It is important to note that even though DTFC is globally stable in nature as proved in Sect. 3.4, the quality of thrust force control performance in terms of thrust force ripple is greatly dependent on the parameters of the linear PMSM.

3.6 Duty Ratio Control

59

3.6 Duty Ratio Control A modification to DTC, which effectively reduces the torque ripple while retaining the switching table based robust control structure of basic DTC, is duty ratio control. In this method, one sample time is divided into several intervals and the duty ratio of the active voltage vector is determined by using various approaches reported in [32–39]. The basic principle of duty ratio control is illustrated in Fig. 3.10. The active voltage vector selected from the Table 3.1 based on the output of the hysteresis controllers defined by (3.3) and (3.4) and applied at the beginning of the sample period at kTs for duration of t 1 which is then followed by a zero vector for the rest of the sampling period Ts . The duty ratio d for the active voltage vector can be defined as: d=

t1 and (0 ≤ d ≤ 1) Ts

(3.43)

In duty ratio control the duration for which the active voltage vector is applied is t1 as illustrated in Fig. 3.10, during t1 the rate of change of thrust force is given by the slope s1 . After t1 , a zero voltage vector is applied for the duration Ts − t1 , during which the rate of change of thrust force is given by the slope s2 . The slope s1 is given using (3.29) as:  d FT + Rs 3 π kF P  π −vα λ pm,β + vβ λ pm,α − P λs λ f vm cos δ = − F0 + dt Ls 2 τ Ls τ (3.44) FT (N)

s1 =

Slope s1 (Active vector applied)

Slope s2 (Zero vector applied)

Fref F0 t (s)

t1 kTs

Ts (one control cycle)

(k+1)Ts

Fig. 3.10 Illustration of duty ratio control, the active voltage vector is applied at the beginning of the sample time for a duration t 1 followed by a zero vector applied for the rest of the sampling period Ts . s1 is the thrust force slope (rate of change) when active voltage vector is applied and s2 the thrust force slope (rate of change) when zero voltage vector is applied (diagram not to scale)

60

3 Direct Thrust Force Control Based on Duty Ratio Control

The + sign in superscript indicates that s1 is a positive slope computed for the active voltage vector. The slope s2 for zero voltage vectors is computed from (3.29) by replacing vα = 0 and vβ = 0 as: s2 =

d FT − Rs 3 π 2 kF P2  λs λ f vm cos δ = − F0 − 2 dt Ls 2 τ Ls

(3.45)

The negative sign in the superscript indicates that s2 is a negative slope computed for the zero voltage vectors. It is important to note that the characteristics of s1 and s2 are same as that of the rate of change of thrust force ddtFT and have already been discussed in detail in Sect. 3.4. The values of s1 and s2 also depends on the parameters of the linear PMSM as well as the operating condition of the machine. The main objective of duty ratio control strategy is to formulate t1 in terms of the slopes s1 and s2 and thrust force error e F to reduce the ripple by decreasing the average/DC value of the applied voltage vector.

3.7 Review of Classical Duty Ratio Control Methods The method reported in [32] calculates the optimal duty ratio for DTC of induction machines by solving a quadratic equation such that the instantaneous torque should be equal to the reference torque at the end of each cycle, acting in a deadbeat fashion. The approach presented in [32] was developed in the stationary reference frame that was later revisited in the synchronously rotating stator flux reference frame [33]. The principle of [32] can analogously be extended to the linear PMSM. In this case torque is replaced by thrust force. Therefore, for linear PMSM, this principle can be expressed mathematically as: FT (k + 1) = Fr e f

(3.46)

By solving (3.46) for t1 in terms of s1 and s2 : t1 =

Fr e f − FT − s2 Ts s1 − s2

(3.47)

Another approach that calculates the duty ratio such that the mean torque over a sample time should be equal to the reference torque is reported in [34, 35] for induction machines and PMSMs respectively; this approach is referred to as direct mean torque control. For a linear PMSM, this method gives the following objective function:

Fri pp

1 = Ts

(k+1)T  s

kTs



FT − Fr e f dt → minimi ze

(3.48)

3.7 Review of Classical Duty Ratio Control Methods

61

where, Fri pp is the average value of thrust force ripple over one sampling period. Solving (3.48) for t1 :  t1 = Ts −

Ts   2 FT − Fr e f + s1 Ts s1 − s2

(3.49)

The research detailed in [36] proposes a duty ratio calculation method for DTC controlled induction machines based on minimizing the torque ripple RMS value over one cycle and also ensures a fixed average switching frequency. The approach in [36] is significantly improved in [37] and guarantees the global minimization of torque ripple RMS value over one switching cycle by using a modified switching pattern. The principle criteria for duty ratio determination can be analogously extended to the linear PMSM by using the following objective function:

Fri2 pp,r ms

1 = Ts

(k+1)T  s



FT − Fr e f

2

dt → minimi ze

(3.50)

kTs

where, Fri pp,r ms is the rms value of thrust force ripple over one sampling period. Solving (3.50) for t1 :  2 Fr e f − FT − s2 Ts t1 = 2s1 − s2

(3.51)

The duty ratio control based DTC schemes in [32–37] have been proven effective in torque ripple reduction in rotational machines, however the duty ratio calculation procedures are complex and dependent on machine parameters. A study presented in [38] suggests that the duty ratio in a DTC controlled PMSM should vary in proportion to the rotor speed. Recent research [39] carefully examines the methods presented in [32–36] and proposes a relatively simple method to calculate the duty ratio for DTC controlled PMSMs. According to [39] the duty ratio for the active voltage vector is computed by dividing the absolute values of flux and torque error by two static gains and then adding them, in contrast to [38] the rotor speed is not considered in this method. The main disadvantage of this approach is the tuning of the static gains and the significant steady-state error in the torque response. It is important to note that the duty ratio control methods of [32–39] have not been evaluated for DTFC of the linear PMSM. In addition the only literature related to the duty ratio control of DTFC is [12]. In [12] the duty ratio based DTFC is briefly discussed, however no method to calculate the duty ratio is discussed. In general most of the linear PMSMs have a low value of stator inductance due to their larger air gap compared to PMSMs. Also due to the linear motion, the pole pitch of the linear PMSMs also effects the force regulation. Therefore DTFC controlled LPMSMs with low inductance and short pole-pitch exhibit large force ripple.

62

3 Direct Thrust Force Control Based on Duty Ratio Control

3.8 State of the Art Duty Ratio Control Method In this research the application of the duty ratio control method of [39] has been extended to a prototype surface-mount linear PMSM and is used as a state of the art benchmark. The expression defining the duty ratio d according to [39] for the linear PMSM is given as: d = d F + dλ

(3.52)

where, d F = |eCFT | , and dλ = |eCλF| In (3.52), d is the duty ratio of the active voltage vector. C T and C F are constant gains. d F is the thrust force regulation duty ratio and dλ is linked to flux error as detailed in [39]. At the beginning of the kth sample time, the active voltage vector is selected according to (3.3) and (3.4) from Table 3.1 and applied to the linear PMSM (as shown in Fig. 3.11) while the duty ratio for that voltage vector is determined according to (3.52) which reduces the ripple by reducing the effective magnitude of the applied voltage vector.

3.8.1 Tuning of Static Gains C T and C F As reported in [39], the main limitation of this method is the tuning of gains C T and C F . A larger value of these gains reduces the ripple but increases the steady-state error significantly. In particular, large values of C T significantly reduce the ripple in the force of the linear PMSM but also introduce larger steady-state errors in the

λref vref

PI

+

-

vm

Fref

λs -

+

+

-

eλ eF

Flux Controller Eq. (3.3) Force Controller Eq. (3.4)

FT

λs

vm eF eλ Iαβ λ

Switching Table (Table 3.1)

Vk

θs

Duty ratio Calculation using Eq. (3.52) or Eq. (3.55) and Control

S a Sb S c

F

Vdc

θs

Estimation of dq θs ,λs and FT abc I Eq. (2.133)-(2.138) dq θr d dt

P

π τ

VSI

I abc x

Linear PMSM

Fig. 3.11 Block Diagram of the duty ratio controlled DTFC scheme, duty ratio is computed using (3.52) for the state of the art [39] and for proposed method (3.56) is used

3.8 State of the Art Duty Ratio Control Method

63

force response. Evidently the tuning of the gains is a trade-off between the transient performance and the steady-state performance. It is important to note that [39] does not provide any quantitative study of the steady-state error in torque response. In [39], C T and C F are chosen as 2 N m and 0.1 Wb respectively based on experimentation for a rotational PMSM. In this research the values of C T and C F are selected by experimentation to be 120 N and 3 Wb respectively such that the ripple in the force and flux during steady-state is the same as that of the novel duty ratio method. This provides a fair comparison between the techniques. It is found from experiment that by increasing C T beyond 120 N the ripple in force is not improved; however the steady-state error in the force increases significantly as described in Sect. 3.10.2. It is also found that the value of C F does not exhibit a significant impact on the thrust force ripple as well as the flux ripple when C T is 120 N and beyond.

3.9 Proposed Duty Ratio Control Method In this research a novel method for the duty ratio calculation is presented. The proposed method accurately determines the required duty ratio during each sample time for the voltage vector active during that sample time. The principle of the proposed method is illustrated in Fig. 3.12. The thrust force at the start of the sampling period is F0 . Consider the error in thrust force at the beginning of kth sample time is e F = Fr e f − F0 , which leads the hysteresis controllers of (3.3) and (3.4) to apply a voltage vector from the switching Table 3.1. If the selected active voltage vector is applied for the whole sampling period Ts , the change in thrust force will be FT . In order to reduce the ripple, it is desired that the active voltage vector needs to be

FT (N)

Active vector applied for full sampling period

D ∆ FT

Fref

C

B Zero vector applied after t1

eF F0

E

A

time (s)

t1 =dF.Ts Ts (one control cycle) kTs

(k+1)Ts

Fig. 3.12 Illustration of proposed duty ratio control, the active voltage vector is applied at the beginning of the sample time for duration t 1 followed by a zero vector applied for the rest of the sampling period Ts . Dotted line shows the variation in thrust force if the active vector would have been applied for full sampling period (diagram not to scale)

64

3 Direct Thrust Force Control Based on Duty Ratio Control

applied for a duration t 1 such that the change in thrust force equals the error e F instead of FT . The time t 1 is a fraction of the sampling period and is equal to d F Ts , such that d F is the required duty ratio of the active vector. Now, from Fig. 3.12; tan(∠ABC) = tan(∠D AE)

(3.53)

The above expression can be expressed as: FT eF = d F Ts Ts



eF



⇒ dF =

ΔF

⇒

(3.54)

T

where, d F is the duty ratio during the kth sample time for thrust force control. In (3.54), the use of absolute value ensures a positive duty ratio for thrust force control for all values of the error e F and FT . If the duty ratio is calculated only considering the force error according to (3.54), then small values of duty ratio d F will result in a flux decrease at low speeds when the resistive drop is not negligible in (3.7). The decrease in flux results in a thrust force decrease, this leads to an increase in duty ratio d F . However this compensation is sluggish and may result in flux oscillations. This situation can be avoided by taking into account the flux error. In order to account for flux ripple, the remainder of the duty cycle 1 − d F , will be utilized for flux regulation. The error in stator flux at the beginning of the kth sample time is eλ . The duty ratio dλ for flux can be defined using a similar approach as used for (3.54) and the duty ratio for flux control dλ is given as:



eλ

(3.55) dλ = (1 − d F )

Δλs

where, Δλs is change in stator flux caused by the applied voltage vector during the kth sample according to Table 3.1. Now the total duty ratio dT of the non-zero voltage vector for the thrust force and flux control is given as:  dT =

dλ + d F , if dλ + d F ≤ 1 1, if dλ + d F > 1

(3.56)

It is important to note that whenever d F exceeds unity, it will be limited to unity by the controller. It is clear from (3.56) that dT is also limited to unity. The analytical expressions for d F and dλ are derived in detail in the next section.

3.9 Proposed Duty Ratio Control Method

65

3.9.1 Derivation of Expression for ΔFT In order to evaluate d F , it is required to compute ΔFT during one sample time for the applied voltage vector. For this purpose, consider (3.12):   v y − Rs i y 3 π d FT = Pk F λs λ f cos δ − ωr dt 2 τ Ls λs

(3.57)

In discrete time, with a small sample time Ts , the change in thrust force during Ts can be computed as: ΔFT = Pk F

  v y − Rs i y 3 π π λs λ f cos δ − P vm Ts 2 τ Ls λs τ

(3.58)

Equation (3.58) can be simplified to: ΔFT =

 ϕ cos δ v y − Rs i y − σ λs vm Ts σ

(3.59)

where, ϕ = k F 2L fs and σ = P πτ The xy-components of voltages and currents in (3.59) can be found from the αβ-components using the following transformation: 3λ



Fx Fy



 =

cos θs sin θs − sin θs cos θs



Fα Fβ

(3.60)

Substituting (3.60) into (3.59), following expression is obtained:  ϕ cos δ −vα sin θs + vβ cos θs σ  −Rs −i α sin θs + i β cos θs − σ λs vm Ts

ΔFT =

(3.61)

In (3.61) the αβ-components are used and can be expressed in terms of the inverter switching states as [17]: 

vα = (2Sa −Sb3−Sc )Vdc √ c )Vdc vβ = (Sb −S 3

(3.62)

From (3.61) and (3.62), the following expression for ΔFT can be obtained in terms of inverter switching states:       ϕ (Sb − Sc )Vdc (2Sa − Sb − Sc )Vdc cos θs sin θs + ΔFT = cos δ − √ σ 3 3  (3.63) −Rs −i α sin θs + i β cos θs − σ λs vm Ts

66

3 Direct Thrust Force Control Based on Duty Ratio Control

As established in Sect. 3.2, the operational range of δ is small for the prototype surface-mount linear PMSM used in this research. If the operational range of δ is small, then cos δ ≈ 1 and (3.63) simplifies to:       ϕ (2Sa − Sb − Sc )Vdc (Sb − Sc )Vdc − sin θs + cos θs ΔFT = √ σ 3 3  −Rs −i α sin θs + i β cos θs − σ λs vm Ts (3.64) It is important to note that (3.64) is only valid for machines that have a small operational range of δ, otherwise (3.63) should be used and δ can be estimated from (1.49) or (3.42).

3.9.2 Derivation of Expression for Δλs In order to compute dλ , λs during Ts using the discrete time version of (3.7) given as: Δλs = (vx − Rs i x )Ts

(3.65)

The change in stator flux can also be expressed in terms of αβ-components of the voltage using (3.60) and (3.65): Δλs =

   vα cos θs + vβ sin θs −Rs i α cos θs + i β sin θs Ts

(3.66)

By using (3.62) and (3.66), Δλs can be expressed in terms of inverter switching states as:      (2Sa − Sb − Sc ) (Sb − Sc ) sin θs Vdc cos θs + Δλs = √ 3 3  −Rs i α cos θs + i β sin θs Ts (3.67)

3.9.3 Derivation of Expressions for d F and dλ The exact analytical expression to compute d F can be obtained from (3.54) and (3.64) after simple mathematical manipulation and is given as: dF =

σ |e F | ϕ|w|

(3.68)

3.9 Proposed Duty Ratio Control Method

67

where,       (2Sa − Sb − Sc )Vdc (Sb − Sc )Vdc cos θs sin θs + − √ 3 3  −Rs −i α sin θs + i β cos θs − σ λs vm

w=

The analytical expression to compute dλ can be derived from (3.55) and (3.67) and is given as: (1 − d F )|eλ |     dλ =  

(2Sa −Sb −Sc ) −Sc ) (Sb√ cos θ sin θ V i + − R cos θ + i sin θ

Ts s s dc s α s β s 3 3 (3.69)

3.10 Experimental Results for Duty Ratio Controlled DTFC The proposed duty ratio control method for the surface-mount linear PMSM is practically validated by extensive experimentation under various operating conditions. The experimental setup is illustrated in Fig. 1.1 and the parameters of the prototype surface-mount linear PMSM are provided in Table 1.1. The proposed duty ratio control method is digitally implemented according to block diagram of Fig. 3.11 using a dSPACE® DS-1104 controller and will be referred to as ‘novel DTFC’. The voltage source inverter used in the experiment was manufactured by SEMIKRON® . The sample time is chosen as 100 μs, and the DC bus voltage is 120 V. Experimental results prove the effectiveness of the novel DTFC when compared with the state of the art duty ratio control method of [39] to be referred to as ‘DTFC1’. The parameters of the speed PI controller of Fig. 3.11 are same for both the methods to compare their performance under similar conditions. The values of the speed controller proportional gain k p and integral gain ki are 970 and 7 respectively. The gain for the anti-windup component is selected as unity.

3.10.1 Start-Up Performance with Speed Loop Closed The start-up performance of the linear PMSM, with the outer speed loop closed, under DTFC1 (with C T = 120 N and, C F = 3 Wb) and novel DTFC are compared. The speed response, force response, stator flux response, duty ratio, and stator current respectively for both DTFC1 and novel DTFC are shown in Fig. 3.13. The magnified view of the speed response during the start-up transient for both the DTFC1 and the novel DTFC is shown in Fig. 3.14a, b respectively. It is clear from

68

3 Direct Thrust Force Control Based on Duty Ratio Control

Novel DTFC

Reference

Measured 3.2

3.25

Force (N)

200

3.3

3.35

Reference

100 0 -100 3.15

Estimated 3.2

3.25

0.095

3.3

0.075 3.15

Reference 3.2

3.25

3.3

-80 3.15

0.5

Estimated

0

Reference 3.2

3.25

3.3

3.35

Estimated

0.08 0.075 3.15

Reference 3.2

3.25

3.3

3.35

3.2

3.25

3.3

3.2

3.25

3.3

3.35

0 -3

Ic (Green) 3.2

3.25

Time (s)

3.3

0.5 0.25 0 3.15 5

3.35

Ia (Red)

Ib (Blue)

0.75

Current (A)

Current (A)

3.35

0.085

Duty-Ratio

Duty-Ratio

3.3

1

0.25

-5 3.15

3.25

100

1

3

3.2

0.09

3.35

0.75

0 3.15 5

Measured

0.095

0.085 0.08

0

-100 3.15

3.35

Estimated

0.09

80

200

Force (N)

-80 3.15

Flux (Wb)

Speed (mm/s)

80 0

Reference

160

Flux (Wb)

Speed (mm/s)

DTFC1 160

3.35

3

Ib (Blue)

Ia (Red)

0 -3 -5 3.15

Ic (Green) 3.2

3.25

3.3

3.35

Time (s)

Fig. 3.13 Start-up performance from 0 to 80 mm/s with outer speed loop closed. Speed, force, flux, duty ratio and stator phase currents responses are shown from top to bottom respectively for both the DTFC1 (C T = 120 N, C F = 3 Wb) and the novel DTFC (experiment)

3.10 Experimental Results for Duty Ratio Controlled DTFC

(a)

160

Measured

Reference

80 0 -80

35 ms 3.2

3.21

3.22

3.23

(b) Reference

Speed (mm/s)

Speed (mm/s)

160

69

80

Measured

0 -80

3.24

25 ms 3.2

3.21

3.22

3.23

3.24

Time (s)

Time (s)

Fig. 3.14 Magnified view of the speed response during start-up. a DTFC1 (C T = 120 N, C F = 3 Wb), b novel DTFC (experiment)

Fig. 3.14 that for the same parameters of the speed PI controller, the speed response under the novel DTFC is 28.6% faster than that of DTFC1. Figure 3.15a, b show the error plots for speed response and thrust force response during the start-up transient for both the DTFC1 and the novel DTFC respectively. It is observed that the novel DTFC results in faster convergence of the speed error to zero compared to DTFC1. In addition, for quantitative analysis, the integral of absolute error (IAE) indices for the speed error plots for both DTFC1 and novel DTFC during the start-up transient are computed and are given in Table 3.3. The IAE index for the speed error plot under the novel DTFC is reduced by 25.7% proving its effectiveness.

(b)

DTFC1

80 40

35 ms

0 -40

3.2

3.21

3.22

3.23

3.24

Speed error (mm/s)

Speed error (mm/s)

(a)

120

50 25 0

35 ms

-25 -50

80 40 0 -40

25 ms 3.2

3.21

3.22

3.23

3.24

75

Force error (N)

Force error (N)

75

Novel DTFC

120

3.2

3.21

3.22

Time (s)

3.23

3.24

50 25 0 -25 -50

25 ms 3.2

3.21

3.22

3.23

3.24

Time (s)

Fig. 3.15 Error plots for speed and thrust force during startup transient. a DTFC1 (C T = 120 N, C F = 3 Wb) and, b novel DTFC (experiment)

70

3 Direct Thrust Force Control Based on Duty Ratio Control

Table 3.3 Comparison of transient performance of DTFC1 and the novel DTFC using IAE index Type of transient phenomena

IAE (integral of absolute error) DTFC1 Speed

Novel DTFC Force

Start-up (0 to 80 mm/s)

1.52 ×

104

Speed reversal (−600 to 600 mm/s)

3.56 ×

105

Speed

1.31 ×

104

1.19 ×

105

Force

1.13 ×

104

3.16 × 103

3.20 ×

105

3.18 × 104

It is clear from Fig. 3.13 that when the speed command steps from 0 to 80 mm/s at 3.2 s the corresponding thrust force response under DTFC1 exhibits an under-shoot as the estimated force fails to track the reference in contrast to the novel DTFC. It can be observed from the force error plot of Fig. 3.15a that the force error does not converge to zero under DTFC1 during the start-up transient resulting in a steady-state error. However, it is clear from the force error plot of Fig. 3.15b that the force error converges to zero under the novel DTFC as the estimated force tracks the reference and results in a faster speed response. The force error plot of Fig. 3.15b also assumes negative values because of the overshoot in the force response under the novel DTFC as shown in Fig. 3.13. It can be observed from Table 3.3 that under the novel DTFC the IAE index for force error during start-up is significantly reduced demonstrating the superior dynamic performance of the novel DTFC during start-up. An average steady-state error of 36.4 N in thrust force can be observed in the case of DTFC1 as shown in Fig. 3.22a, when the speed settles to 80 mm/s after 3.25 s. It is also evident from Fig. 3.22a that for the novel DTFC, the average steady-state error is reduced to 3.8 N which is an indication of the improved steady-state performance of novel DTFC compared to DTFC1. The flux response in the case of the novel DTFC is also improved as it exhibits less oscillation at start-up compared to DTFC1. It can be observed from Fig. 3.13 that the duty ratio calculated by DTFC1 during the transient at 3.2 s is lower than the novel DTFC resulting in a relatively slower start-up response. Moreover, the stator current waveforms are relatively smooth during the start-up transient under novel DTFC when compared with DTFC1. The experimental results clearly validate the novel DTFC method.

3.10.2 Speed Reversal and Steady-State Performance The speed reversal and steady-state performance of the linear PMSM with the outer speed loop closed under both DTFC1 and novel DTFC is compared in Fig. 3.16. It is observed from Fig. 3.17a, b that the novel DTFC results in 7% faster speed response when speed is reversed from −600 to 600 mm/s. The speed error plots during the speed reversal transients for both DTFC1 and novel DTFC are shown in Fig. 3.18a, b respectively. It is clear from Fig. 3.18 that under novel DTFC the speed

3.10 Experimental Results for Duty Ratio Controlled DTFC

71

DTFC1

Novel DTFC 600

Speed(mm/s)

600

Reference

Reference 0

0 Measured

-600 0

0.2

500

0.6

0.8

0.2

500

Estimated 0

0.2

0.4

0.6

0.8

1

- 300 - 500

0

0.2

Flux(Wb)

0.6

0.8

1

0.085 Reference

Reference 0

0.2

0.4

0.6

0.8

1

0.075

1

1

0.75

0.75

0.5

0.5

0.25

0.25

Current(A)

0.4

Estimated

0.085

Duty-Ratio

1

Reference

Estimated

6 4 2 0 -2 -4 -6

0.8

0.095

0.095

0

0.6

0

-300

0.075

0.4

Estimated

300

0

500

0

1

Reference

300

Force(N)

0.4

Measured

- 600

0

0

0.2

0.2

0.4

0.4

0.6

0.6

Time (s)

0.8

0.8

1

1

0 6 4 2 0 -2 -4 -6

0

0.2

0.4

0.6

0.8

1

0

0.2

0.4

0.6

0.8

1

0

0.2

0.4

0.6

0.8

1

Time (S)

Fig. 3.16 Speed reversal from −600 to 600 mm/s and steady-state response at 600 mm/s with outer speed loop closed. Speed, force, flux, duty ratio and stator phase a current responses are shown from top to bottom respectively for both the DTFC1 (C T = 120 N, C F = 3 Wb) and novel DTFC (experiment)

72

3 Direct Thrust Force Control Based on Duty Ratio Control

DTFC1 Reference

Measured 600 0 -600 0.11

83.5 ms 0.13

0.15

0.17

Novel DTFC

(b)

0.19

0.21

Speed (mm/s)

Speed (mm/s)

(a)

Measured

0 -600 0.11

0.23

Reference

600

77 ms 0.13

0.15

0.17

0.19

0.21

0.23

Time (s)

Time (s)

Fig. 3.17 Magnified view of the speed reversal transient illustrating the rise times. a DTFC1 (C T = 120 N, C F = 3 Wb) and, b novel DTFC (experiment)

DTFC1

83.5 ms

(b) Speed error (mm/s)

Speed error (mm/s)

(a) 1200 1000 800 600 400 200 0 -200

1200 1000 800 600 400 200 0 -200

Novel DTFC

77 ms

0.11 0.13 0.15 0.17 0.19 0.21 0.23

0.11 0.13 0.15 0.17 0.19 0.21 0.23

Time (s)

Time (s)

Fig. 3.18 Error plots for speed during speed reversal transient. a DTFC1 (C T = 120 N, C F = 3 Wb) and, b novel DTFC (experiment)

error converges to zero at a higher rate providing a faster speed response during the speed reversal compared to DTFC1. It can be observed from Table 3.3 that IAE index for the speed error during speed reversal is reduced by 10% for the novel DTFC. The IAE indices of force error during speed reversal for both DTFC1 and novel DTFC are also given in Table 3.3. It is evident that the novel DTFC results in a significant reduction of IAE index for force error during speed reversal when compared to DTFC1. The steady-state performance of the prototype linear PMSM at 600 mm/s from 0.3 to 0.7 s under both DTFC1 and novel DTFC is also compared in Fig. 3.16, where the speed response, thrust force response, flux response, duty ratio and the phase current for both control schemes are shown during the steady-state. The quantitative results for steady-state performance of DTFC1 and the novel DTFC at 600 mm/s and 54 N (average force) in terms of percent flux ripple λrip (%), percent force ripple F rip (%), average steady-state force error ΔF ss and average switching frequency f av are summarized in Table 3.4. In this analysis λrip (%) and F rip (%) are given by (3.70) and (3.71),

3.10 Experimental Results for Duty Ratio Controlled DTFC

73

Table 3.4 Comparison of steady-state performances of the DTFC1 for various values of C F and C T with the novel DTFC 600 mm/s, 54 N

DTFC1

CF

0.065

Novel DTFC 0.087

3

3

0.027

CT

120

160

120

160

50

λrip (%)

1.65

1.62

1.63

1.60

2.20

F rip (%)

6.41

5.94

6.12

5.89

34.88

6.34

Δ F ss (N)

56.53

79.16

57.69

80.05

33.52

3.83

f av (kHz)

4.50

4.61

4.53

4.62

3.71

3.94

 λri p (%) =  Fri p (%) =

1 N

N

i=1 (λs (i)

− λav )2

λav 1 N

N

i=1 (FT (i)

Fav

− Fav )2

1.62

× 100

(3.70)

× 100

(3.71)

where, λav and Fav represent the average steady-state flux and force respectively and λs (i) and FT (i) are the instantaneous values of flux and thrust force. In this research, the steady-state performance of DTFC1 is evaluated for various combinations of C T and C F and is summarized in Table 3.4. In [39], it is suggested that a good compromise between dynamic and steady-state performance may be achieved if the ratio between C T and C F is half of the ratio between the rated force and the permanent magnet flux of the machine. In this research the ratio of the rated maximum force Fmax to the permanent magnet flux λf is 3688, half of which is 1844. It is found from extensive experimentation that setting C T to 120 N and C F to 3 Wb results in a force ripple of 6.1% which is very close to that of novel DTFC as observed from Table 3.4. However, when C F is changed from 3 to 0.065 Wb to maintain the ratio of 1844, the force ripple was increased to 6.4% while the average steady-state force error was reduced by 1.16 N. It is also found that when the value of C T is increased to 160 N while keeping C F to be 3 Wb, the reduction in the force ripple was negligible, however the average steady-state force error increased to 80.1 N. When C F is changed to 0.087 Wb while C T is set at 160 N to achieve the ratio of 1844, the force ripple is reduced to 6% whereas the average steady-state error is reduced by 0.89 N which is a negligible reduction. It is also found that reducing the value of C T to 50 N and C F to 0.027 Wb decreases the average steady-state error to 33.5 N while the force ripple increases to 35%, moreover the flux ripple also increases to 2.2%. It can be observed from Table 3.4 that by increasing C T the average switching frequency also increases. The value of C F does not have any significant impact on force ripple and the flux ripple for the prototype linear PMSM when the value of C T is 120 N and beyond.

74

3 Direct Thrust Force Control Based on Duty Ratio Control

DTFC1

(a)

(b)

Reference

200

Force(N)

Force(N)

200 100 0 -100

Estimated

-200 0.3

0.4

0.5

Time (s)

0.6

0.7

Novel DTFC Reference

100 0 -100 -200 0.3

Estimated

0.4

0.5

0.6

0.7

Time (s)

Fig. 3.19 Thrust force response during steady-state at 600 mm/s, a DTFC1 (C T = 120 N, C F = 3 Wb) and, b novel DTFC (experiment)

It is evident from Table 3.4 and the above discussion that the most suitable choices of C T and C F for benchmarking are 120 N and 3 Wb respectively and are underlined in Table 3.4. The comparison of the steady-state thrust force response at 600 mm/s for both DTFC1 and the novel DTFC is shown in Fig. 3.19, indicating a large steadystate error in the case of DTFC1. It is clear from Table 3.4 that under novel DTFC the average steady-state force error is 3.83 N which is negligible compared to 57.7 N caused by DTFC1. In addition, it is evident from Table 3.4 that the average switching frequency of novel DTFC during steady-state is 3.94 kHz which is less than 4.53 kHz with DTFC1 and C T = 120 N and C F = 3 Wb. The variation in average steadystate force error with speed is shown in Fig. 3.22a for both DTFC1 (C T = 120 N and C F = 3 Wb) and the novel DTFC. It is clear that average steady-state force error significantly increases with speed in the case of DTFC1 whereas for the novel DTFC it almost remains constant with speed which validates the superior steady-state performance of novel DTFC. The effect of mover’s speed on force ripple is illustrated in Fig. 3.22b for both DTFC1 (C T = 120 N and C F = 3 Wb) and the novel DTFC. It can be observed that both control schemes have a comparable force ripple, however at speeds higher than 600 mm/s the force ripple in the case of novel DTFC is slightly increased. The thrust force duty ratio d F and error in thrust force e F for both DTFC1 and the novel DTFC during steady-state are shown in Fig. 3.20. In addition C T and ΔFT for DTFC1 and the novel DTFC respectively are also shown in Fig. 3.20. The average steady-state force demand at 600 mm/s is 54 N and to produce the required duty ratio to meet that force demand under DTFC1 a large steady-state force error (57.7 N in this case) is inevitable, as the duty ratio d F is calculated by dividing that error by a constant value C T (120 N in this case) as seen from Fig. 3.20a. Figure 3.20b illustrates the magnified view of Fig. 3.20a for 10 sampling periods from 0.35 to 0.351 s. It can be observed from Fig. 3.20b that under DTFC1, the duty ratio varies between 0.4 and 0.5 with steady-state error in a proportional manner according to (3.52) with C F = 3 Wb. However, under novel DTFC, duty-ratio d F is computed according to (3.54) and is not proportional to the steady-state error as ΔFT is variable. It is observed that the duty ratio d F varies between 0.4 and 0.65. The thrust force error e F remains between 50 and 55 N under DTFC1 and is significantly larger compared

3.10 Experimental Results for Duty Ratio Controlled DTFC

DTFC1

(a)

Novel DTFC

0.75

0.75

dF

1

dF

1 0.5

0.6

0 0.3

0.7

eF (N)

0.5

0.4

0.5

0.6

10 8 6 4 2 0 -2

0.3

0.7

130

15

120

10

ΔFT (N)

T

C (N)

0.3

0.4

110 100 0.3

dF

(b)

0.4

0.5

Time (s)

0.6

0 0.3

0.7

0.7

0.6

0.6

0.5 0.4 0.3 0.35 70

0.3505

0.351

0.3 0.35 6

0.7

0.4

0.5

0.6

0.7

0.5

0.6

0.7

0.4

Time (s)

0.3505

0.351

0.3505

0.351

0.3505

0.351

5

e F (N)

e F (N)

40 0.3505

120 110 0.3505

Time (s)

0.351

4 3 2 0.35 9

0.351

Δ F T (N)

T

C (N)

0.6

0.5

60

100 0.35

0.5

0.4

50

30 0.35 130

0.4

5

0.7

dF

eF (N)

100 80 60 40 20 0 -20

0.5 0.25

0.25 0 0.3

75

8 7 6 0.35

Time (s)

Fig. 3.20 Duty-ratio d F , force error e F , C T for DTFC 1 (C T = 120 N, C F = 3 Wb) and ΔFT for novel DTFC (experiment). a Whole duration of steady-state from 0.3 to 0.7 s. b Magnification to 10 sampling periods from 0.35 to 0.351 s, each time division is equal to 2.5 sampling periods

76

3 Direct Thrust Force Control Based on Duty Ratio Control

0.4

0.5

0.6

-2

i 0.4

0.5

λ

0.1

β

0.6

0.4

-0.1

λ 0.5

Time (s)

0.5

β

0.6

0.7

0.6

i

0.7

α

0 -2

i 0.4

0.2 α

α

β

2

-4 0.3

0.7

-0

0.4

v

4 α

α

β

0.7

0

-0.2 0.3

v

β

Current(A)

i

0.2

α

β

2

-4 0.3

Novel DTFC

Current(A)

v

4

λ and λ

α

Voltage (V)

v

0 -40 -80 -120 0.3

(b) 120 80 40 0 -40 -80 -120 0.3

DTFC1

λ and λ

Voltage (V)

(a) 120 80 40

0.5

λ

0.1

β

0.6

0.7

α

-0

λ

-0.1 -0.2 0.3

0.4

0.5

β

0.6

0.7

Time (s)

Fig. 3.21 αβ-axes stator voltages, currents and flux components during steady-state at 600 mm/s, a DTFC1 (C T = 120 N, C F = 3 Wb) and, b novel DTFC (experiment)

to the force error e F under novel DTFC which varies from 3 to 6 N as observed from Fig. 3.20b. The αβ-axes voltages, currents and flux components’ waveforms during steadystate for DTFC1 and the novel DTFC are shown in Fig. 3.21. The αβ-axes currents for both duty ratio based schemes are distorted at a steadystate speed of 600 mm/s which is caused by low frequency steady-state oscillations in the required thrust force as observed from Fig. 3.19. These oscillations change with mover’s speed as shown in Fig. 3.22. It is important to note that, in contrast to rotational PMSM, due to complex nonlinear friction dynamics of linear PMSM there are low frequency speed oscillations even at steady-state. These speed oscillations also cause oscillations in the thrust force reference as shown in Fig. 3.19. The oscillatory force reference results in low frequency ripple in the current during steady-state causing distorted current waveforms shown in Figs. 3.16 and 3.21. In addition, high frequency ripple is also present in the αβ-axes currents for both DTFC1 and the novel DTFC which can be attributed to the low inductance of the prototype linear PMSM and structure of the control scheme being used.

3.10 Experimental Results for Duty Ratio Controlled DTFC

100

DTFC1

10 9

Novel DTFC

7.5

60

Frip (%)

ΔFss (N)

80

(a)

40

77

(b) DTFC1 Novel DTFC

6 4.5 3

20 6 0 0

1.5 200

400

600

800 1000 1200

Speed(mm/s)

0 0

200

400

600

800 1000 1200

Speed(mm/s)

Fig. 3.22 Variation in average steady-state error and percent ripple in force with speed for both DTFC1 (C T = 120 N, C F = 3 Wb) and the novel DTFC

The αβ-axes flux components for both DTFC1 and novel DTFC are not significantly different at 600 mm/s as confirmed by the quantitative analysis of flux ripple given in Table 3.4. The flux ripple is small and hence not visible from Fig. 3.19 because of the resolution of the figure.

3.10.3 Analysis of Steady State Error in Force for DTFC1 DTFC1 of [39] effectively reduces ripple in thrust force, however, this method exhibits large steady state error in the thrust force which can be attributed to the duty ratio determination principle used in this approach. The average force demand at steady state speed of 600 mm/s is 54 N. It can be observed from Fig. 3.20 that both DTFC1 and the novel DTFC approximately produce the same average force of 54 N during steady state. However in case of DTFC1, a large steady state error of 57.7 N is evident from Table 3.4. According to DTFC1, the average duty ratio for force is 57.7/120 = 0.48 which indicates that the active voltage vector is applied for 48% of the sampling period; therefore the steady state error apparently should converge to zero which is not the case. In DTFC1 the duty ratio for force d F is dependent on the steady state error. Therefore, under DTFC1, in order to maintain the average duty ratio at 0.48 to produce an average force of 54 N, the speed controller of Fig. 3.11 generates a force reference higher than 54 N such that an average steady state error of 57 N is required. The same explanation is valid for all other values of C T and C F . A smaller value of C T decrease the steady state error, however the ripple in thrust force increases significantly.

78

3 Direct Thrust Force Control Based on Duty Ratio Control

3.10.4 Flux Trajectory The stator flux trajectories at 80 and 600 mm/s for both DTFC1 and novel DTFC are shown in Fig. 3.23. It is observed from Fig. 3.23 that flux trajectories for both DTFC1 and novel DTFC follow a circular trajectory at both low and high speeds. However, at 600 mm/s, a small ripple in the stator flux trajectory can be observed for both DTFC1 and novel DTFC caused by the distorted αβ-axes currents. It is important to note that the steady state flux ripple at 600 mm/s for both control schemes is close to 1.6% which is small considering the distortion in αβ-axes currents as the low inductance of the linear PMSM reduces the effect of current distortion on the flux. DTFC1 (80 mm/s)

0.1

Novel DTFC (80 mm/s)

λ β (Wb)

λ β (Wb)

0.1

-0.1 -0.1

0.1

-0.1 -0.1

0.1

λ α (Wb)

λ α (Wb)

DTFC1 (600 mm/s)

Novel DTFC (600 mm/s)

0.1

λ β (Wb)

λ β (Wb)

0.1

-0.1 -0.1

0.1

-0.1 -0.1

λ α (Wb) Fig. 3.23 Flux trajectories at various speeds (experiment)

0.1

λ α (Wb)

Force(N)

3.10 Experimental Results for Duty Ratio Controlled DTFC

79

60 0

Estimated Reference

-60 1.49

1.51

1.53

1.57

1.55

1.585

Force(N)

Time (s)

60

Reference

0

Estimated

-60 1.51

Rise time = 2.87 ms

1.515

1.52

1.525

1.53

Time (s) Fig. 3.24 Transient response of force under the DTFC1 (experiment)

3.10.5 Transient Response of Force with Outer Speed Loop Disabled The transient performance of the novel DTFC is evaluated by comparing its force step response with that of DTFC1 while the outer speed loop is disabled. A force step command from −60 to +60 N is used as reference. The step response of DTFC1 is shown in Fig. 3.24 indicating a significant steady state error. The step response of the novel DTFC is shown in Fig. 3.25 and shows negligible error when the thrust force settles to steady state. It is also evident from these figures that the novel DTFC follows the step command of 60 N with a shorter rise time of 2.5 ms compared to DTFC1 2.87 ms. The comparison of step response without outer speed loop for both DTFC1 and novel DTFC clearly proves the improved force response of the novel DTFC under transient conditions.

3.11 Conclusions This research proposes and rigorously examines a novel approach for the calculation of duty ratio for DTFC that reduces the ripple and steady state error in the flux and thrust force response of the linear PMSM drive utilizing switching-table based direct thrust control. Exact analytical expressions for determination of the duty ratio, considering the machine parameters and the mover’s speed, have been derived.

3 Direct Thrust Force Control Based on Duty Ratio Control

Force(N)

80 60 0

Estimated Reference

-60 1.185

1.205

1.225

1.245

1.265

1.28

Force(N)

Time (s)

60

Estimated

0

Reference

-60 1.205

Rise time = 2.5 ms 1.21

1.215

1.22

1.225

Time (s) Fig. 3.25 Transient response of force under the novel DTFC (experiment)

Experimental results clearly indicate that the novel technique exhibits excellent control of flux and thrust force with lower ripple, faster transient response and reduced steady state error when compared to the prior duty-ratio based DTFC technique. Experimental assessment includes start-up performance, speed reversal and force transients. The proposed technique retains the switching table based control structure of conventional DTFC and has no requirement to arbitrarily select the control parameters or tune gains.

References 1. I. Takahashi, T. Noguchi, A new quick-response and high-efficiency control strategy of an induction motor. IEEE Trans. Ind. Appl. IA-22, 820–827 (1986) 2. I. Takahashi, Y. Ohmori, High-performance direct torque control of an induction motor. IEEE Trans. Ind. Appl. 25, 257–264 (1989) 3. U. Baader, M. Depenbrock, G. Gierse, Direct self control (DSC) of inverter-fed induction machine: a basis for speed control without speed measurement. IEEE Trans. Ind. Appl. 28, 581–588 (1992) 4. C. French, P. Acarnley, Direct torque control of permanent magnet drives. IEEE Trans. Ind. Appl. 32, 1080–1088 (1996) 5. M.F. Rahman, L. Zhong, W.Y. Hu, K.W. Lim, M.A. Rahman, A direct torque controller for permanent magnet synchronous motor drives, in Proceedings of the Electric Machines and Drives Conference Record (1997) 6. L. Zhong, M.F. Rahman, W.Y. Hu, K.W. Lim, Analysis of direct torque control in permanent magnet synchronous motor drives. IEEE Trans. Power Electron. 12, 528–536 (1997) 7. M.F. Rahman, L. Zhong, K.W. Lim, A direct torque-controlled interior permanent magnet synchronous motor drive incorporating field weakening. IEEE Trans. Ind. Appl. 34, 1246–1253 (1998)

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8. Y. Hu, C. Tian, Y. Gu, Z. You, L.X. Tang, M.F. Rahman, In-depth research on direct torque control of permanent magnet synchronous motor, in Proceedings of the IECON, vol. 2 (2002), pp. 1060–1065 9. M.F. Rahman, M.E. Haque, T. Lixin, Z. Limin, Problems associated with the direct torque control of an interior permanent-magnet synchronous motor drive and their remedies. IEEE Trans. Ind. Electron. 51, 799–809 (2004) 10. M.E. Haque, M.F. Rahman, Incorporating control trajectories with the direct torque control scheme of interior permanent magnet synchronous motor drive. IET Elect. Power Appl. 3, 93–101 (2009) 11. G.S. Buja, M.P. Kazmierkowski, Direct torque control of PWM inverter-fed AC motors—a survey. IEEE Trans. Ind. Electron. 51, 744–757 (2004) 12. K. Yoshida, Z. Dai, M. Sato, Sensorless DTC propulsion control of PM LSM vehicle, in Proceedings of the IPEMC, vol. 1 (2000), pp. 191–196 13. J. Cui, C. Wang, J. Yang, L. Liu, Analysis of direct thrust force control for permanent magnet linear synchronous motor, in Proceedings of the Fifth World Congress on Intelligent Control and Automation, WCICA, vol. 5 (2004), pp. 4418–4421 14. J. Cui, C. Wang, J. Yang, D. Yu, Research on force and direct thrust control for a permanent magnet synchronous linear motor, in Proceedings of the IECON (2004), vol. 3, pp. 2269–2272 15. M. Abroshan, K. Malekian, J. Milimonfared, B.A. Varmiab, An optimal direct thrust force control for interior permanent magnet linear synchronous motors incorporating field weakening, in Proceedings of the International Symposium on Power Electronics, Electrical Drives, Automation and Motion, SPEEDAM (2008), pp. 130–135 16. S. Cheng-Chung, H. Yi-Sheng, Based on direct thrust control for linear synchronous motor systems. IEEE Trans. Ind. Electron. 56, 1629–1639 (2009) 17. Y.S. Huang, C.C. Sung, Implementation of sliding mode controller for linear synchronous motors based on direct thrust control theory. IET Control Theory Appl. 4, 326–338 (2010) 18. T. Lixin, Z. Limin, M.F. Rahman, H. Yuwen, A novel direct torque controlled interior permanent magnet synchronous machine drive with low ripple in flux and torque and fixed switching frequency. IEEE Trans. Power Electron. 19, 346–354 (2004) 19. Y. Inoue, S. Morimoto, M. Sanada, Examination and linearization of torque control system for direct torque controlled IPMSM. IEEE Trans. Ind. Appl. 46, 159–166 (2010) 20. C. Lascu, A.M. Trzynadlowski, Combining the principles of sliding mode, direct torque control, and space-vector modulation in a high-performance sensorless AC drive. IEEE Trans. Ind. Appl. 40, 170–177 (2004) 21. X. Zhuang, M. Faz Rahman, Direct torque and flux regulation of an IPM synchronous motor drive using variable structure control approach. IEEE Trans. Power Electron. 22, 2487–2498 (2007) 22. G. Foo, C.S. Goon, M.F. Rahman, Analysis and design of the SVM direct torque and flux controlled IPM synchronous motor drive, in Proceedings of the AUPEC (2009), pp. 1–6 23. Z. Yongchang, Z. Jianguo, X. Wei, G. Youguang, A simple method to reduce torque ripple in direct torque-controlled permanent-magnet synchronous motor by using vectors with variable amplitude and angle. IEEE Trans. Ind. Electron. 58, 2848–2859 (2011) 24. Z. Hao, X. Xi, L. Yongdong, Torque ripple reduction of the torque predictive control scheme for permanent-magnet synchronous motors. IEEE Trans. Ind. Electron. 59, 871–877 (2012) 25. Y. Junyou, H. Guofeng, C. Jiefan, Analysis of PMLSM direct thrust control system based on sliding mode variable structure, in Proceeding of the IPEMC (2006), pp. 1–5 26. L. Guan, J. Yang, J. Cui, Direct thrust control approach using adaptive variable structure for permanent magnet linear synchronous motor, in Proceedings of the ICCA (2007), pp. 2217– 2220 27. A. Mohammadpour, L. Parsa, SVM-based direct thrust control of permanent magnet linear synchronous motor with reduced force ripple, in Proceedings of the IEEE International Symposium on Industrial Electronics, ISIE (2011), pp. 756–760 28. M.A.M. Cheema, J. Fletcher, M.F. Rahman, D. Xiao, Modified direct thrust control of linear permanent magnet motors with sensorless speed estimation, in Proceedings of the IECON (2012), pp 1908–1914

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29. T. Geyer, G. Papafotiou, M. Morari, Model predictive direct torque control—part I-concept, algorithm, and analysis. IEEE Trans. Ind. Electron. 56, 1894–1905 (2009) 30. M. Preindl, S. Bolognani, Model predictive direct torque control with finite control set for PMSM drive systems, part 1: maximum torque per ampere operation. IEEE Trans. Ind. Inf. 9, 1912–1921 (2013) 31. M. Preindl, S. Bolognani, Model predictive direct torque control with finite control set for PMSM drive systems, part 2: field weakening operation. IEEE Trans. Ind. Inf. 9, 648–657 (2013) 32. T.G. Habetler, F. Profumo, M. Pastorelli, L.M. Tolbert, Direct torque control of induction machines using space vector modulation. IEEE Trans. Ind. Appl. 28, 1045–1053 (1992) 33. B.H. Kenny, R.D. Lorenz, Stator- and rotor-flux-based deadbeat direct torque control of induction machines. IEEE Trans. Ind. Appl. 39, 1093–1101 (2003) 34. E. Flach, R. Hoffmann, P. Mutschler, Direct mean torque control of an induction motor. Proc. EPE 3, 672–677 (1997) 35. M. Pacas, J. Weber, Predictive direct torque control for the PM synchronous machine. IEEE Trans. Ind. Electron. 52(5), 1350–1356 (2005) 36. K. Jun-Koo, S. Seung-Ki, New direct torque control of induction motor for minimum torque ripple and constant switching frequency. IEEE Trans. Ind. Appl. 35, 1076–1082 (1999) 37. S. Kuo-Kai, L. Juu-Kuh, P. Van-Truong, Y. Ming-Ji, W. Te-Wei, Global minimum torque ripple design for direct torque control of induction motor drives. IEEE Trans. Ind. Electron. 57, 3148–3156 (2010) 38. J. Linni, S. Liming, Stability analysis for direct torque control of permanent magnet synchronous motors, in Proceedings of the ICEMS (2005), pp. 1672–1675 39. Z. Yongchang, Z. Jianguo, Direct torque control of permanent magnet synchronous motor with reduced torque ripple and commutation frequency. IEEE Trans. Power Electron. 26, 235–248 (2011)

Chapter 4

SV-PWM Based Direct Thrust Force Control of a Linear Permanent Magnet Synchronous Motor

4.1 Introduction In this chapter, two schemes for direct thrust force control (DTFC) based on space vector pulse width modulation (SV-PWM) are proposed, experimentally validated and compared for the prototype linear PMSM. The first direct thrust force control scheme is the PI controller-based regulation of the flux and thrust force and is referred to as “PI-DTFC”. The second direct thrust force control scheme is based on a linear quadratic regulator based control of the flux and thrust force and referred to as “Optimal-DTFC1”. It is important to note that Optimal-DTFC1 is based on a novel state space model which formulates the flux and thrust force dynamics for the linear PMSM. The PI-DTFC scheme is benchmarked as the state of the art. In PI-DTFC, the thrust force and the stator flux are controlled by two PI controllers instead of the conventional hysteresis controllers. The proposed control scheme is based on decoupled control of thrust force and stator flux in the stator flux xy-reference frame by two independent (decoupled) control loops. The thrust force regulation loop is concluded from a linearized relation between the thrust force and the load angle. The errors in the reference and estimated values of the thrust force and flux are the inputs to the two PI controllers to generate voltage commands in the stator flux xy-reference frame. These voltage commands are transformed to the stationary frame of reference and then used by a space vector pulse width modulation (SV-PWM) module to generate the reference voltage vector at a constant switching frequency. In this chapter, the control loops for the stator flux and thrust force regulation under PI-DTFC are rigorously analyzed using a frequency response technique and a detailed method for the design of the PI controllers based on discrete time root locus is also presented. Moreover, for Optimal-DTFC1 of the linear PMSM, a novel multiple-inputmultiple-output (MIMO) state space model, independent of the mover’s speed, having stator flux and thrust force as states, is formulated. An optimal linear state © Springer Nature Switzerland AG 2020 M. A. M. Cheema and J. E. Fletcher, Advanced Direct Thrust Force Control of Linear Permanent Magnet Synchronous Motor, Power Systems, https://doi.org/10.1007/978-3-030-40325-6_4

83

84

4 SV-PWM Based Direct Thrust Force Control …

feedback control scheme is then designed using the optimal linear quadratic regulator technique. Integral action is added to the designed control scheme by state augmentation to minimize the steady-state error and reduce the force ripple. Experimental results clearly prove that the proposed Optimal-DTFC1 results in a faster transient response of speed and thrust force with improved steady-state regulation of thrust force and stator flux when compared to the state of the art technique of PI-DTFC based direct thrust force control.

4.2 Stator Flux and Thrust Force Regulation in xy-Reference Frame Mathematically, the dynamic model of the surface-mount linear PMSM in the xyreference frame is given as: dλs dt

(4.1)

v y = Rs i y + ωs λs

(4.2)

v x = Rs i x +

The thrust force FT in terms of i y is expressed as: FT = Pk F

3π λs i y 2τ

(4.3)

Substituting the value of i y in terms of FT from (4.3) into (4.2) yields: vy =

2τ Rs FT + ωs λs 3π Pk F λs

(4.4)

It is important to note that (4.1) and (4.4) provide the basis to develop a decoupled approach for the control of thrust force and stator flux. Moreover, (4.1) is a clear illustration that stator flux can be controlled completely by vx assuming the xyreference frame is oriented correctly.

4.3 Analysis of Thrust Force Control in Surface-Mount Linear PMSM In PMSMs, the load angle δ is closely linked to torque generation as suggested by several studies [1–5]. These studies are also analogously valid for linear PMSMs. The relation between the electromagnetic thrust force FT and the load angle δ for a surface-mount linear PMSM from Chap. 2 is:

4.3 Analysis of Thrust Force Control in Surface-Mount Linear PMSM

FT = Pk F

3 π λs λ f sin δ 2 τ Ls

85

(4.5)

The relationship between the electromagnetic thrust force FT and the load angle δ for an interior permanent magnet (IPM) linear PMSM is given as [6]: FT = Pk F

   3π λs  2λ f L q sin δ − λs L d − L q sin 2δ 4τ L d L q

(4.5a)

In (4.5a), L d and L q are the d-axis and q-axis inductances (H) respectively for the IPM linear PMSM. In this research the end-effect co-efficient k F in (4.5) and (4.5a) is chosen to be unity for simplicity and it does not affect the experimental results. The studies in [4, 5] detail the linearization of the nonlinear relation between the load angle and torque for interior PMSMs. The linearization methods discussed in [4, 5] can also be applied to surface-mount/interior linear PMSM by linearizing (4.5) or (4.5a) using Taylor’s expansion at load angle δ0 as: FT = K (δ − δ0 ) + F0

(4.6)

where, K = d FT (δ)/dδ|δ=δ0 , and F0 = FT (δ0 ) In (4.6), K is the linearization co-efficient and for a linear PMSM it is evaluated at δ0 = 0 leading to F0 = 0. The characteristics of K for a linear PMSM (IPM or surface mounted) depend on the operational range of δ for that machine. It is important to note that (4.6) is valid for both surface-mounted and IPM linear PMSMs. However, in this book the prototype linear PMSM is surface-mounted with parameters given in Table 1.1 of Chap. 1. Therefore, for surface-mount linear PMSM, (4.6) will be achieved by linearization of (4.5) using the parameters of Table 1.1 of Chap. 1.

4.3.1 Selection of Reference Stator Flux Magnitude λ r e f The stator flux reference λr e f is selected according to maximum force per ampere (MFPA) and for a surface-mount linear PMSM can be expressed in terms of force reference Fr e f as:  λr e f =

λ2f +



2 τ Ls Fr e f 3 π Pk F

2 (4.7)

The prototype surface-mount linear PMSM being used in this research has a peak force of ±312 N. According to (4.7), when Fr e f varies from 0 to ±312 N, the corresponding variation in λr e f is negligible, i.e. from 0.0846 to 0.0847 Wb because of low values of L s and τ for the prototype linear PMSM as given in Table 1.1. Therefore, λr e f is set to 0.0846 Wb in the prototype system.

86

4 SV-PWM Based Direct Thrust Force Control …

4.3.2 Characteristics of the Co-efficient K for Surface-Mount Linear PMSM with Low Stator Inductance and Short Pole-Pitch According to (4.5) the maximum thrust force occurs when δ is controlled at ± π2 radians. However a closer inspection of (4.5) suggests that, when λs is kept constant at λr e f (as explained previously), then for a machine with small value of the pole-pitch τ and low stator inductance L s , the rated maximum thrust force of the linear PMSM is typically achieved at much smaller values of δ than ± π2 radians and therefore has a small operational range δ. The primary motivation for this research is that the low inductance and short pole pitch are the key factors that make surface-mount linear PMSMs difficult to control under DTFC compared to their rotational counterparts. If δmax is the load angle corresponding to the rated maximum thrust force Fmax , then from (4.5): δmax = sin−1



2τ L s Fmax 3π Pk F λs λ f

 (4.8)

For example, as described previously, the stator flux for the prototype system is λr e f = 0.0846 Wb, and from (4.8): δmax = ±0.1545 radians (i.e., ±8.85◦ ). Hence, the operational range of δ for the prototype linear PMSM according to   (4.8) is [−0.1545, 0.1545] radians which is a small subset of − π2 , π2 . The small operational range of δ justifies the assumption that cos δ ≈ 1. Therefore, using (4.5) and (4.6), the linearization coefficient K for the prototype linear PMSM with stator flux is set to λr e f can be approximated as: K = Pk F

3 π λr e f λ f , (∵ cos δ ≈ 1) 2 τ Ls

(4.9)

From the above discussion it is clear that the small operational range of load angle δ results in an almost constant linearization coefficient K for the linear PMSM when the reference flux is selected using (4.7). Therefore the thrust force varies linearly with the angle δ within its operational range as demonstrated in Fig. 4.1. It is also important to evaluate the effect of thrust force on the linearization coefficient K. In DTFC, the stator flux tracks the reference flux λr e f , and the thrust force FT tracks reference force Fr e f . Therefore, from (4.5) and (4.6), K is expressed in terms of λr e f and FT as:

2 

3 π 2τ L s

K = Pk F λr e f λ f 1 − FT 2 τ Ls 3π Pk F λr e f λ f 

  α

(4.10)

4.3 Analysis of Thrust Force Control in Surface-Mount Linear PMSM 300

87

Rated Peak Thrust Force 312 (N)

Force (N)

200 100 0 -100 -200

Exact Linearized 0.1 0.15

-300 -0.15

-0.1

-0.05

0

0.05

δ (elec. rad)

Fig. 4.1 Thrust force versus load angle for the surface-mount linear PMSM of Table 1.1, the exact curve is according to (4.5) and the linearized curve is according to (4.6) with λs = 0.0846 Wb, according to (4.7) under (MFPA)

K (N/elec. rad)

It is important to note that for the prototype linear PMSM α is vanishingly small because of the low inductance and short pole pitch and thus the effect of thrust force on K becomes negligible and (4.10) reduces to (4.9). According to (4.10), the variation in magnitude K for the prototype linear PMSM as reference force and hence the actual thrust force FT varies from 0 to 312 N (Fmax ) is less than 1.2% and therefore, K remains almost constant at the mid-point value which is 2020 N/elec. rad. as shown in Fig. 4.2. 2100 2060 2020 1980 1940 1900 1860 1820 1780 1740 1700

0

25

50

75

100

125

150

175

200

225

Force FT (N)

Fig. 4.2 Characteristics of K according to (4.10) with λs = 0.0846 Wb

250

275

300

88

4 SV-PWM Based Direct Thrust Force Control …

Rs ix

Fig. 4.3 Block diagram of stator flux regulation according to (4.11)

_

vx ( s )

+

Gλ(s)

λs ( s )

4.4 Derivation of Transfer Function for Stator Flux Regulation The transfer function for the stator flux regulation can be derived by taking the Laplace transform of (4.1) and re-arranging the terms: λs (s) =

1 [vx (s) − Rs i x (s)] s

(4.11)

where: s is the Laplacian variable. The resistive drop term Rs i x (s) can be assumed as a disturbance term [7] and therefore, the transfer function G λ (s) of the plant model for stator flux regulation is: G λ (s) =

1 s

(4.12)

The block diagram for stator flux regulation loop corresponding to (4.11) is shown in Fig. 4.3. It can be concluded from (4.12) that the plant model for stator flux regulation is a pure integrator with x-axis voltage vx as input and stator flux magnitude λs as output. Therefore, a closed loop regulation of the stator flux that maintains the value of λs at λr e f can be achieved by cascading a Proportional Integral (PI) controller with the stator flux regulation plant of (4.12).

4.5 Derivation of Transfer Function for Thrust Force Regulation The magnitude variation in stator flux λs under direct thrust force control is restricted within a small hysteresis band around the constant reference value for stator flux (as explained in Sect. 4.3.1) and can assumed constant [7]. Therefore, by taking the Laplace transform of (4.4) and by re-arranging: FT (s) =

 3π Pk F λs  v y (s) − ωs λs 2τ Rs

(4.13)

It can be observed from (4.13) that the transfer function of the plant for thrust force regulation is a constant term. In (4.13) the term “ωs λs ” is treated as a disturbance.

4.5 Derivation of Transfer Function for Thrust Force Regulation

89

However, because of the fact that the speed of stator flux vector ωs is dependent on variation of thrust force, the mathematical model of (4.13) does not represent the dynamics of thrust force regulation accurately. A further mathematical manipulation of (4.13) by expressing ωs as function of thrust force is necessary to formulate the precise dynamic model for thrust force regulation. According to Figs. 2.8 and 2.9, the stator flux vector angle θs can be expressed in terms of rotor flux vector angle θr and the load angle δ as: θs = θr + δ

(4.14)

Differentiating both sides of (4.14) with respect to time yields: ωs = ωr +

dδ dt

(4.15)

Now, differentiation of (4.6) with respect to time yields: d FT dδ =K dt dt Substitution of the value of

dδ dt

(4.16)

from (4.16) into (4.15) results:

ωs = ωr +

1 d FT K dt

(4.17)

It can be observed from (4.17) that ωs is dependent on the variation of thrust force FT . The Laplace transform of (4.17) can be given as: ωs = ωr +

s FT (s) K

(4.18)

Substituting that value of ωs from (4.18) into (4.13) yields: FT (s) =

  s 3π Pk F λs  v y (s) − λs ωr + FT (s) 2τ Rs K

(4.19)

Re-arranging (4.19) gives the following dynamic model for thrust force regulation: FT (s) =

  Kξ v y (s) − ωr λs ξ λs s + K

(4.20)

where: ξ = 3π2τPkRFs λs . In (4.20), ωr λs is treated as disturbance term and the transfer function of the G F (s) of the plant model for thrust force regulation is [6, 7]:

90

4 SV-PWM Based Direct Thrust Force Control …

Fig. 4.4 Block diagram of thrust force regulation according to (4.20)

G F (s) =

Kξ ξ λs s + K

(4.21)

The plant for thrust force regulation is illustrated in Fig. 4.4. It can be observed from (4.20) and (4.21) that the transfer function for the thrust force regulation represents a first order plant with y-axis voltage v y as input and thrust force FT as the output. Therefore, thrust force response can be compensated by cascading a PI controller with the thrust force regulation plant.

4.6 PI Controller Based Direct Thrust Force Control (PI-DTFC) The concept of PI-DTFC control scheme is based on the PI controller based direct torque control of interior permanent magnet motor (IPMSM) reported in [7, 8]. The transfer functions of plant models in (4.2) and (4.21) for stator flux and thrust force regulation, as shown in Figs. 4.3 and 4.4 respectively, provide the basis for PI control based decoupled stator flux and direct thrust force control of the linear PMSM. The block diagram illustration of the PI-DTFC utilizing the SV-PWM for the synthesis of the command voltages is shown in Fig. 4.5. The proposed scheme comprises two PI controllers; one for the thrust force control and second for the stator flux control. The thrust force reference value is generated by using another PI controller having speed error as an input. The stator flux and force errors are the inputs to the

λref

vref

Fref +

-

vm

PI

PI + anti-windup

+-

PI + anti-windup

+

-

FT

λs

vx vy

Voltage Limiter (Fig. 4.17)

v*x v

xy

* y

vα vβ

Inverter (VSI)+ SV-PWM

αβ

θs Estimation of λs ,θ s and FT Eq. (2.133) to (2.138)

Linear PMSM

x

abc αβ

iβ iα

id

αβ

iq

dq

θr

P

π τ

d dt

Fig. 4.5 PI controller based direct thrust force control (PI-DTFC) of the linear PMSM with space vector pulse width modulation (SV-PWM)

4.6 PI Controller Based Direct Thrust Force Control (PI-DTFC)

91

PI controllers to generate the reference voltages vx and v y respectively which are further transformed to the stationary (αβ) reference frame by using the αβ to xy transformation:      cos θs − sin θs vx vα = (4.22) vβ sin θs cos θs vy The reference voltage vector is generated from the output of a 3-phase 2-level voltage source inverter (shown in Fig. 4.5) based on the command values of vα and vβ by using space vector pulse width modulation (SV-PWM) to achieve a constant switching frequency. In case of large voltage values due to the integration operation of the PI controller the output command value of the voltage may exceed the maximum allowable limit. In order to avoid the over voltage output of the PI controller, a maximum and minimum limit on the output of the PI controller is included, and in addition, an anti-windup strategy (back calculation method) is also employed to compensate for large integration values. The maximum and minimum voltage limits for the output of the PI controller are given as: Vdc vmax ≤ √ 3

(4.23)

In (4.23), Vdc is the DC-link voltage for VSI. The stator flux magnitude λs , stator flux vector angle θs , and the thrust force FT for feedback signals are estimated by using the current model of stator flux linkages in dq-axes reference frame as explained in Sect. 2.5 of Chap. 2 using (2.133) to (2.138) as illustrated in Fig. 4.5.

4.6.1 Stator Flux Control Loop The control loop for the stator flux regulation can be synthesized by cascading a PI controller with the stator flux regulation plant of Fig. 4.3. The block diagram illustration of the stator flux control loop is given in Fig. 4.6. The PWM controlled inverter is modeled by a constant gain Vdc if the output of the PI controller is the modulation index for SV-PWM [9]. However, in this case the output of the PI controller is the x-axis voltage vx , therefore, the PWM controlled VSI can be modeled as a unity gain Fig. 4.6 Block diagram of the stator flux control loop with transport delay and disturbance cancellation (4.11) and (4.26)

92

4 SV-PWM Based Direct Thrust Force Control …

(Fig. 4.6). The transfer function of the PI controller for stator flux control (Fig. 4.6): G c_λ (s) = k p_λ +

ki_λ s

(4.24)

In (4.24), k p_λ and ki_λ are the proportional and integral gains of the stator flux PI controller respectively. The transfer function of PI controller of (4.24) can also be expressed as:  G c_λ (s) = k p_λ 1 +

1



τi_λ s

(4.25)

k

where; τi_λ = kp_λ is the integrator reset time. i_λ It is important to note that the implementation of the PI controller using a digital controller introduces a delay Td in the stator flux control loop. This delay Td in the control loop is caused by the PWM generation process, controller sampling/computation and analogue to digital conversion, and is called transport delay [6, 7, 9–11]. The effect of this transport delay Td should essentially be taken into account for accurate modeling of the flux control loop and can be modeled by introducing a delay block e−sTd [12], in series with the forward path transfer function of the plant as illustrated in Fig. 4.6. The open loop forward path transfer function of the stator flux control loop G λ_O L (s) with the delay, according to Fig. 4.6, can be given as:  G λ_O L (s) = k p_λ 1 +

  1 −sTd e τi_λ s s 1

(4.26)

In (4.26), Td is the transport delay and is usually assumed equal to the sampling time Ts of the digital controller [6, 7, 9–11, 13].  G λ_O L (s) = k p_λ

τi_λ s + 1 τi_λ s

  1 −sTd e s

(4.27)

The open loop frequency response of G λ_O L (s) can be computed by substituting s = jω in (4.27) and after simple mathematical manipulation, the expressions for the gain and phase in terms of frequency ω can be given as: k p_λ  2 2 ω τi_λ + 1 s= jω ω2 τi_λ    ∠ G λ_O L (s)s= jω = tan−1 τi_λ ω − π − ωTd   G λ_O L (s)

=

(4.28) (4.29)

According to linear control theory [9, 12], for the linear system of (4.27) the values   of k p_λ and ki_λ should be selected such that the open loop gain G λ_O L (s)s= jω  crosses unity as the phase ∠ G λ_O L (s)s= jω reaches the value of −π + φm . It is

4.6 PI Controller Based Direct Thrust Force Control (PI-DTFC)

93

important to note that φm is the target phase margin to achieve the desired damping in the closed loop step response of the system [12]. The open loop gain crosses the unity gain at frequency ωc called as cross-over frequency and is expressed in rad/s [10, 12]. Therefore, at cross-over frequency when ω = ωc , the phase angle given by (4.29) reaches −π + φm so that:   −π + φm = tan−1 τi_λ ωc − π − ωc Td

(4.30)

  The maximum value of tan−1 ωc τi_λ can be π2 for the maximum value of the cross-over frequency ωc(max) and can be given from (4.30) as [9, 10]: ωc(max) =

π 2

− φm Td

(4.31)

In (4.31) the value of Td can be set to the sampling time of the micro controller Ts to adequately account for all the delays caused by digital implementation   [6, 7]. The value of τi_λ should be selected to satisfy (4.31) such that tan−1 ωc τi_λ ≈ π2 , a good approximation is [9, 10]: τi_λ ≈

10 ωc(max)

(4.32)

The maximum possible value of k p_λ can be now determined by evaluating (4.28) at ωc(max) for unity value of gain and after simple mathematical manipulation the expression for k p_λ can be given as: 2 τi_λ ωc(max) k p_λ =  2 2 ωc(max) τi_λ +1

(4.33)

Apparently, from above discussion and (4.31)–(4.33), the selection of gains for the PI controller in the stator flux control loop is a straight forward process and the gains can be computed from (4.31) to (4.33) by selecting a suitable value of the phase margin φm . However, experimental evaluation of these gains deviates from the theoretical findings concluded from above discussion, and therefore, additional considerations are necessary while computing gains from (4.31) to (4.33). A typical value for the target phase margin φm is 40° which assures the closed loop stability [10, 12]. If the delay Td is set to be 200 µs (sampling time of the digital controller in this research Ts ), the following values of k p_λ and ki_λ can be computed from (4.31) to (4.33) for a phase margin of 40°:  k p_λ = 4340 Volts Wb , where, φm = 40◦ and ωc(max) = 2617rad/s (4.34) ki_λ = 1.89 × 106 Volts Wb s

94

4 SV-PWM Based Direct Thrust Force Control …

Although, the PI gains of (4.34) are theoretically correct, experimental evaluation of these gains for the actual prototype drive system demonstrates significantly large oscillations in the stator flux response during steady-state which consequently deteriorates the thrust force response. The reason for these oscillations in the stator flux response can be attributed to the fact that the PI gains of (4.34) do not provide adequate damping in the stator flux response due to a significantly large value of ki_λ compared to k p_λ in (4.34). Therefore, a further increase in the phase margin beyond 40° is necessary to eliminate these oscillations. The flux regulation plant model for the linear PMSM is an integrator with xaxis voltage vx as input. The contribution of vx to stator flux is limited by the low inductance of the prototype linear PMSM. In addition, the stator flux is largely set up by the permanent magnet flux of the mover. Therefore, the stator flux control loop needs to be over damped by targeting a phase margin close to 90° in order to eliminate the large oscillations during the steady-state. The PI gains to achieve a phase margin of 89.9° can be computed using (4.31)–(4.33):  k p_λ = 8.7 Volts Wb , where, φm = 89.9◦ and ωc(max) = 8.73 rad/s ki_λ = 7.6 Volts . Wb s

(4.35)

It is observed form the experimental evaluation of the PI gains of (4.35) that the steady-state oscillations in the stator flux response are significantly reduced and a satisfactory stator flux response in terms of minimal steady-state ripple is achieved. The stator flux regulation plant is an integrator as shown in (4.12). In addition, the value of the integral gain ki_λ in (4.35) is negligibly small compared to that of (4.34). Therefore, the stator flux regulation can be achieved by using a proportional controller only [8, 13]. The open loop transfer function of stator flux control loop with a proportional controller k p_λ is:   1 −sTd e G λ_O L (s) = k p_λ s

(4.36)

The expressions for the gain and phase of G λ_O L (s) in terms of frequency ω are:   G λ_O L (s)

k p_λ ω π = − − ωTd 2

s= jω

 ∠ G λ_O L (s)s= jω

=

(4.37) (4.38)

At a cross-over frequency where ω = ωc , the gain given by (4.37) crosses unity and the phase angle given by (4.40) reaches −π + φm so that the phase margin φm is computed from (4.38) at a crossover frequency ωc(max) [10, 12]: φm =

π − ωc(max) Td 2

(4.39)

4.6 PI Controller Based Direct Thrust Force Control (PI-DTFC)

95

The maximum value of the cross-over frequency ωc(max) is: ωc(max) =

π 2

− φm Td

(4.40)

The maximum possible value of k p_λ can be now determined by evaluating (4.37) at ωc(max) for unity gain and the expression for k p_λ is: k p_λ = ωc(max)

(4.41)

The value of k p_λ for a phase margin of 89.9° can be computed using (4.40) and (4.41): k p_λ = 8.73 Volts/Wb, where, φm = 89.9◦ , and ωc(max) = 8.7rad/s (4.42) It can be observed from (4.35) and (4.42) that the proportional gain k p_λ remains the same for both PI controller and proportional controller for a phase margin of 89.9°. In addition, experimental evaluation shows that the stator flux regulation performance of the proportional controller in terms of steady-state oscillation is the same as the PI controller when the phase margin is 89.9°. It is observed from (4.41) that the crossover frequency of the stator flux control loop with a proportional controller is equal to the gain k p_λ . A faster stator flux response in terms of a low rise time can be achieved by increasing the value of k p_λ which in turn will increase the crossover frequency ωc(max) . However, according to (4.40) an increase in ωc(max) will reduce the phase margin which results in increased steady-state oscillations. Therefore, the gain k p_λ should be tuned to achieve an adequate phase margin to ensure minimal steady-state oscillations. It is concluded from the above discussion that a proportional controller is sufficient for the stator flux regulation with satisfactory performance. However, in order to ensure a faster stator flux response a PI controller can be used with the integral gain selected such that the steady-state stator flux response is not compromised in terms of increased stator flux ripple as explained earlier. In this thesis the stator flux regulation is achieved by using a PI controller as shown in Figs. 4.5 and 4.6, and the design of PI controller in discrete time domain is detailed in later section.

4.6.2 Thrust Force Control Loop The thrust force control loop is synthesized by cascading a PI controller with the thrust force regulation plant shown in Fig. 4.4. The thrust force control loop is illustrated in Fig. 4.7. The transfer function of the PI controller for thrust force control is G c_F (s) and is given as (4.43):

96

4 SV-PWM Based Direct Thrust Force Control …

Fig. 4.7 Block diagram of the thrust force control loop with transport delay and disturbance cancellation (4.20) and (4.50)

G c_F (s) = k p_F +

ki_F s

(4.43)

In (4.43), k p_F and ki_F are the proportional and integral gains of the thrust force PI controller respectively. The transfer function of the PI controller of (4.43) can also be expressed as:  G c_F (s) = k p_F 1 +



1 τi_F s

(4.44)

k

where; τi_F = kp_F is the integrator reset time. i_F Since, under direct thrust force control, λs is fixed at λr e f which is calculated according to (4.7), therefore replacing λs by λr e f in (4.21) the transfer function of thrust force regulation can be given as (4.46):  G F (s) =

Kξ ξ λr e f s + K

 (4.45)

3π Pk λ

where, ξ = 2τ RF s r e f . The transfer function of thrust force regulation plant given in (4.45) can be expressed as:  G F (s) =

β αs + 1

 (4.46)

where: α=

ξ λr e f K

β=ξ

(4.47) (4.48)

It can be observed from (4.46) that the thrust force regulation plant is a first order system with time constant α and dc gain β. The value of α and β are expressed in terms of linear PMSM’s parameters in (4.47) and (4.48) respectively. According to (4.46), the plant G F (s) has a pole at − α1 . Moreover, the plant G F (s) is a first order type zero system which does not involve any pure integrator action in the transfer function unlike the stator flux regulation plant G λ (s) given by (4.12). Therefore,

4.6 PI Controller Based Direct Thrust Force Control (PI-DTFC)

97

according to linear control theory [12], the integral action provided by the thrust force PI controller is essential to eliminate the steady-state error in the thrust force response. The values of λr e f , K and ξ can be computed from (4.7), (4.9) and (4.45) for the prototype linear PMSM using the machine parameters given in Table 1.1. These values are then substituted in (4.47) and (4.48) to compute the values of α and β for the thrust force regulation plant of the prototype linear PMSM and are given as: α = 0.00055 s and β = 15.52

 (4.49)

It is observed from (4.49) that the thrust force regulation plant has a significantly small time constant that is attributed to the low inductance and short pole pitch of the prototype linear PMSM in accordance with (4.9) and (4.47). It is important to note that this small time constant of the thrust force regulation plant makes the output thrust force FT particularly sensitive to the input y-axis voltage v y making the thrust force control of linear PMSM difficult compared to their rotational counterparts. The precise design of the thrust force PI control is of critical importance considering the small time constant of the plant G F (s), to ensure a fast transient response with near zero overshoot and a steady-state response with minimal ripple. The thrust force PI controller is implemented using a digital controller and therefore, a transport delay is introduced in the thrust force control loop which is modeled by e−sTd with Td being the delay time as discussed previously. According to Fig. 4.7, using (4.44) and (4.46), the open loop transfer function of thrust force control loop G F_O L (s) with inclusion of transport delay is given as (4.50):  G F_O L (s) = k p_F

τi_F s + 1 τi_F s



 β e−sTd αs + 1

(4.50)

The expressions for the gain and phase of G F_O L (s) in terms of frequency ω are:   2 ω2 τi_F +1 βk p_F   = s= jω ωτi_F ω2 α 2 + 1   π = tan−1 ωτi_F − − tan−1 (ωα) − ωTd 2

  G F_O L (s)  ∠ G F_O L (s)s= jω

(4.51) (4.52)

According to linear control theory, [9, 12], for the open loop transfer function of thrust force control  loop of (4.50) the values of k p_λ and ki_λ are selected such that  the open loop gain G F_O L (s)s= jω intersects unity gain as the phase ∠ G F_O L (s)s= jω approaches the value of −π + φm . It is important to note that φm is the target phase margin to achieve the desired damping in the closed loop step response of the system [12]. The open loop gain crosses the unity gain at frequency ωc called as phase crossover frequency. Therefore, at the cross-over frequency when ω = ωc , the phase angle given by (4.52) reaches −π + φm so that:

98

4 SV-PWM Based Direct Thrust Force Control …

  π −π + φm = tan−1 ωc τi_F − − tan−1 (ωc α) − ωc Td 2

(4.53)

The phase margin φm can be given using (4.53) as: φm =

  π + tan−1 ωc τi_F − tan−1 (ωc α) − ωc Td 2

(4.54)

It is important to note that in (4.54), the term tan−1 (ωc α) cannot be simplified to due to the fact that the time constant α is very small. Therefore, the crossover frequency ωc cannot be computed explicitly from (4.54). However, if the phase margin φm and the crossover frequency ωc are predetermined then the integrator reset time τi_F for the thrust force PI controller can be given by using (4.54) and after simple mathematical manipulation: π 2

τi_F =

  1 π tan φm − + tan−1 (ωc α) + ωc Td ωc 2

(4.55)

The value of k p_F can be determined by evaluating (4.51) at ωc for unity gain and after simple mathematical manipulation the expression for k p_F is as (4.56): k p_F

ωc τi_F = β

   ωc2 α 2 + 1   2 +1 ωc2 τi_F

(4.56)

Equations (4.55) and (4.56) can be used to determine the parameters of thrust force PI controller for a given phase margin and crossover frequency. The values of phase margin φm and the cross over frequency ωc can be selected based on trial and error method so that a desired damping in the thrust force response is achieved. Extensive experimental evaluation of (4.55) and (4.56) suggests the PI gains computed for a phase margin of 66° and crossover frequency of 1.35 × 103 rad/s results in the dead-beat step response of thrust force with fast rise time for the prototype machine. It is important to note the time constant α of the thrust force regulation plant G F (s) is dependent on the linearization co-efficient K according to (4.47). The value of K is constant for the prototype linear PMSM as discussed in Sect. 4.3.2 because of the low inductance and short pole pitch. However, the value of K may vary as a function of operating thrust force FT according to (4.10) for machines with large inductance [5]. Therefore, the time constant α will also be variable. In this scenario of variable α, the gain scheduling of the thrust force PI controller can be achieved using (4.55) and (4.56) to ensure a fixed gain margin and the crossover frequency throughout the whole operational range of thrust force FT . In order to precisely design the thrust force PI controller it is necessary to analyze the thrust force control loop in discrete time and is reported next.

4.6 PI Controller Based Direct Thrust Force Control (PI-DTFC)

99

4.6.3 Discrete Time Design of Stator Flux and Force PI Controllers The design of PI controllers for stator flux and thrust force regulation is of key importance as it shapes the stator flux and thrust force response under the PI based direct thrust control scheme of Fig. 4.5. Previously the frequency response based analysis of stator flux and thrust force control loop in continuous time domain using Laplace transform is presented. The stator flux and thrust force PI controllers are implemented in digital controllers, therefore discrete time analysis of the stator flux and thrust control loop is necessary to design PI controllers for satisfactory control performance in terms of rise time and steady-state ripple. It is important to note that there is no literature available related to the design of PI controlled for SV-PWM based direct thrust control of LPMSM. Most of the available literature is related to the design of PI controllers for SV-PWM based direct torque control of interior permanent magnet synchronous motors (IPMSM) and can be analogously used for linear PMSM as explained in Chap. 1. In [8] the design of stator flux and torque PI controllers for an SV-PWM direct torque control of IPMSM based on discrete time root locus analysis is reported. However this work treats the torque transfer function as constant which does not represent the accurate dynamics for torque regulation as explained in [7]. Foo et al. [7] also discusses the design of stator flux and torque PI controllers in continuous time using the approximate model of transport delay. In [7], the design of the PI controllers is based on integral of time multiplied by the absolute of the error (ITAE) criteria. However, the main limitation of this method is that the gains of the PI controller cannot be computed for various damping ratios. In [14–22] various methods of designing PI controllers for first order plant are presented, however only [17] discusses the design of a dead-beat controller for direct torque control of the induction motor based on the discrete time root locus method. In this section, a detailed discrete time root locus analysis for both stator flux and thrust force control loops is performed. The detailed design of stator flux and thrust force PI controllers is carried out using discrete time root locus based on pole-zero cancellation and the gains of PI controllers corresponding to various damping ratios are computed. Design of Stator Flux PI Controller The stator flux regulation is a continuous time system with transfer function G λ (s). The discrete time transfer function of the stator flux regulation plant with the zeroorder hold effect is given as [11, 13, 23, 24]:     G λ (s) −1 G λ (z) = 1 − z Z trans s Substituting the value of G λ (s) from (4.12) into (4.57):

(4.57)

100

4 SV-PWM Based Direct Thrust Force Control …

  G λ (z) = 1 − z −1 Z trans From [23, 24], Z trans

1 s2



1 s2

 (4.58)

becomes:  Z trans

1 s2

 =

Ts z (z − 1)2

(4.59)

where, Ts = 200 µs is the sampling time of the digital controller. From (4.58) and (4.59), G λ (z) becomes: G λ (z) =

Ts z−1

(4.60)

The discrete time transfer function of the stator flux PI controller is [11, 23, 24]: G c_λ (z) = k p_λ +

ki_λ Ts 1 − z −1

(4.61)

where, k p_λ and ki_λ are the proportional and integral gains of the stator flux PI controller and are the same as in (4.24). In (4.61), the term ki_λ Ts can be replaced by  for simplicity, such that: ki_λ  ki_λ = ki_λ Ts

(4.62)

From (4.61) and (4.62): G c_λ (z) = k p_λ +

 ki_λ 1 − z −1

(4.63)

It is important to note that in this research the digital implementation of the stator  is related flux PI controller is based on (4.63) and the discrete time integral gain ki_λ to the continuous time integral gain ki_λ according to (4.62). Therefore, the integral  must be converted to ki_λ using (4.63) for the frequency response analysis gain ki_λ detailed in Sect. 4.6.1. The proportional gain k p_λ is the same for both the discrete time and continuous time case. The stator flux control loop in discrete time can be synthesized by cascading the digital stator flux PI controller of (4.63) with the discrete time stator flux regulation plant of (4.60) as illustrated in the block diagram of Fig. 4.8. In order to account for the delay caused by digital implementation, one sample delay represented by z −1 is Fig. 4.8 Discrete time stator flux control loop for the surface-mount linear PMSM

4.6 PI Controller Based Direct Thrust Force Control (PI-DTFC)

101

added in the stator flux control loop as shown in Fig. 4.8. The open loop discrete time transfer function of the stator flux control loop is: G λ_O L (z) = G c_λ (z)G λ (z)z −1

(4.64)

From (4.63), the stator flux PI controller in Pole-Zero form is [11]:

G c_λ (z) =

   z− k p_λ + ki_λ

k p_λ  k p_λ +ki_λ

 (4.65)

z−1

From (4.60), (4.64) and (4.65), G λ_O L (z) becomes: ⎛   k p_λ + ki_λ z− G λ_O L (z) = ⎝ z−1

k p_λ  k p_λ +ki_λ

⎞ ⎠



 Ts z −1 z−1

(4.66)

According to digital control theory [23, 24], the discrete time closed loop transfer function of the stator flux control loop of Fig. 4.8 is: G λ_O L (z) λs = λr e f 1 + G λ_O L (z)

(4.67)

G c_λ (z)G λ (z)z −1 λs = λr e f 1 + G c_λ (z)G λ (z)z −1

(4.68)

G λ_C L (z) = From (4.64) and (4.67) G λ_C L (z) =

From (4.66) and (4.67), (4.68) becomes: ⎛ ⎝ G λ_C L (z) =

λs = λr e f

 ) (k p_λ +ki_λ

 z− k

k p_λ  p_λ +ki_λ

⎞

z−1

⎛ 1+⎝



 (k p_λ +ki_λ )

z− k

z−1

k p_λ

 ⎠

 p_λ +ki_λ

Ts z−1

⎞

⎠





z −1

Ts z−1



(4.69) z −1

Simplifying (4.69):

G λ_C L (z) =

λs λr e f

   k  Ts k p_λ + ki_λ z − k p_λp_λ  +ki_λ  =   k  z(z − 1) + k p_λ + ki_λ z − k p_λp_λ  +k

(4.70)

i_λ

The characteristic equation of the closed loop stator flux control transfer function in (4.68) is:

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4 SV-PWM Based Direct Thrust Force Control …

1 + G c_λ (z)G λ (z)z −1 = 0

(4.71)

Substituting G c_λ (z) and G λ (z): 1+

   Ts k p_λ + ki_λ z−

k p_λ  k p_λ +ki_λ

z(z − 1)(z − 1)

 =0

(4.72)

Equation (4.72) can be expressed as: 1 + K λ H (z) = 0

(4.73)

   K λ = Ts k p_λ + ki_λ

(4.74)

where,

 H (z) =

z−

k p_λ  k p_λ +ki_λ



z(z − 1)(z − 1)

(4.75)

It is observed from (4.75) that H (z) has one pole z = 0 and two poles at z = 1. It is important to note that the stator flux PI controller introduces a zero in the discrete transfer function H (z) of (4.75) at z = z o_λ as: z = z o_λ =

k p_λ  k p_λ + ki_λ

(4.76)

The knowledge of z o_λ is essential to draw the root locus for the characteristic equation (4.72). The zero z o_λ can be selected such that it cancels one of the poles of H (z) at z = 1 to ensure a satisfactory performance for the stator flux controller [11, 23, 24]. Therefore selecting z o_λ as 1: z o_λ =

k p_λ =1  k p_λ + ki_λ

(4.77)

1 z(z − 1)

(4.78)

Substituting (4.77) into (4.75): H (z) =

From (4.78), the characteristic equation (4.73) becomes: 1+

K λ =0 z(z − 1)

(4.79)

4.6 PI Controller Based Direct Thrust Force Control (PI-DTFC)

103

After selecting the location of zero z o_λ as (4.77), the root locus for the characteristic equation (4.79) can be drawn and closed loop poles corresponding to the desired damping ratio can be selected on the plot. The value of the gain K λ corresponding to the selected closed loop poles is also obtained from the root locus. From knowledge of values of the gain K λ and zero z o_λ , the gains for the stator flux PI controller can be achieved from (4.74) and (4.77) as: K λ z o_λ Ts

(4.80)

 K λ  1 − z o_λ Ts

(4.81)

k p_λ =  ki_λ =

Substituting z o_λ = 1 from (4.77) into (4.80) and (4.81): k p_λ =

K λ Ts

ki_λ = 0

(4.82) (4.83)

Equations (4.82) and (4.83) suggest that only the proportional controller is adequate to control the stator flux and are in agreement with the frequency response analysis of the stator flux control loop detailed in Sect. 4.6.1. The stator flux regulation plant is an integrator. Therefore, the stator flux control loop PI controller becomes a proportional controller cascaded with the stator flux plant, when polezero cancelation is performed by selecting the zero z o_λ according to (4.77). Hence, the proportional controller for the stator flux control loop can regulate the stator flux with satisfactory transient performance and can eliminate the average steady-state error in the stator flux response. The root locus plot for (4.79) when K λ varies from 0.001 to 1 is shown in Fig. 4.9. The sample time Ts is 200 µs. The closed loop poles are selected to be 0.5 ± 0.0939i corresponding to a damping ratio of 0.95. The gain K λ corresponding to the selected closed loop poles is 0.259, substituting this value in (4.82): k p_λ = 1295

Volts Wb

(4.84)

The proportional gain k p_λ of (4.84) results in a crossover frequency ωc(max) of 1295 rad/s according to (4.40). From (4.40), the phase margin φm corresponding to this value of the crossover frequency is 75°. Experimental evaluation of the proportional gain k p_λ of (4.84) demonstrates significantly large oscillations in the steady-state stator flux response. The experimental evaluation is in agreement with the analysis detailed in Sect. 4.6.1 and suggests an increase in the phase margin to reduce the steady-state oscillations in the stator flux response. Therefore, the closed loop pole should be on the real axis (without the imaginary part) selected close to

104

4 SV-PWM Based Direct Thrust Force Control … Root Locus for Flux Control Loop

1

0.5π/T

0.6π/T

0.8

0.4π/T 0.1 0.3π/T

0.7π/T

0.2 0.3

0.6

0.8π/T

0.5 0.6

Imaginary Axis

0.4 0.2 0

0.2π/T

0.4

0.7 0.8

0.9π/T

Selected Closed Loop Poles

0.1π/T

0.9

π/T π/T

-0.2 0.9π/T

0.1π/T

-0.4 -0.6

0.8π/T

0.2π/T

0.7π/T

-0.8

0.3π/T 0.6π/T

-1 -1

-0.8

-0.6

-0.4

0.4π/T

0.5π/T

-0.2

0

0.2

0.4

0.6

0.8

1

Real Axis

Fig. 4.9 The root locus and the selected closed loop poles for the stator flux controller loop

the origin in the root locus plot to ensure a critically damped stator flux response with minimal steady-state oscillations [23, 24]. It is found from the root locus that the magnitude of the selected closed loop pole on the real axis approaches the gain K λ as the selected pole moves towards the origin. The closed loop pole 0.00174 on the real axis gives the value of K λ = 0.00174, substituting this value of K λ in (4.82): k p_λ = 8.7

Volts Wb

(4.85)

This value of the gain k p_λ is same as given in (4.42) and results in satisfactory performance of the stator flux response with minimal steady-state ripple. Experimental evaluation of various values of the gain k p_λ suggests that for the prototype linear PMSM, a satisfactory stator flux response can be achieved when the value of the gain k p_λ of the proportional controller is selected such that the phase margin φm remains within 88° to 89.9° according to (4.40)–(4.41). As discussed in Sect. 4.6.1, in order to ensure a fast transient stator flux response  can be added to the proportional an integral component with a small value of gain ki_λ controller resulting in a PI controller. Therefore, in this work a PI controller with  is used for the stator flux control. The value of the integral gain small integral gain ki_λ

4.6 PI Controller Based Direct Thrust Force Control (PI-DTFC)

105

 ki_λ is experimentally selected to be 0.0034 Volts , which is equivalent to ki_λ = 17 Volts Wb s Wb s according to (4.62).

Design of Thrust Force PI Controller The discrete time transfer function of the thrust force regulation plant with zero order hold effect is [11, 13, 23, 24]:     G F (s) G F (z) = 1 − z −1 Z trans s

(4.86)

From (4.45), G F (s) is given as:  G F (s) =

Kξ ξ λr e f s + K

 (4.87)

Re-arranging (4.87):  G F (s) =

a s+b

 (4.88)

where: K λr e f a b= ξ

a=

(4.89) (4.90)

Substituting (4.88) in (4.86):   G F (z) = 1 − z −1 Z trans



a s(s + b)

 (4.91)

Evaluating the Z-transformation in (4.91):    z 1 − z −1 1 − e−bTs   G F (z) = b(z − 1) z − e−bTs

(4.92)

Simplifying (4.92): G F (z) =

α1 z − β1

(4.93)

where: α1 =

 a 1 − e−bTs b

(4.94)

106

4 SV-PWM Based Direct Thrust Force Control …

Fig. 4.10 Discrete time thrust force control loop for the linear PMSM

Fref

+_

z-1

Gc_F (z)

GF (z)

β1 = e−bTs

FT ( z )

(4.95)

The discrete time transfer function of the thrust force PI controller is [11, 23, 24]: G c_F (z) = k p_F +

ki_F Ts 1 − z −1

(4.96)

where, k p_F and ki_F are the proportional and integral gains of the thrust force PI controller and are the same as (4.43). In (4.96), the term ki_F Ts can be replaced by  for simplicity, such that: ki_F  ki_F = ki_F Ts

(4.97)

From (4.96) and (4.97): G c_λ (z) = k p_F +

 ki_F 1 − z −1

(4.98)

The thrust force control loop in discrete time is illustrated in Fig. 4.10. The term z −1 is added to account for the transport delay caused by the digital implementation. The discrete time open loop transfer function of the thrust force control loop is: G F_O L (z) = G c_F (z)G F (z)z −1

(4.99)

The thrust force PI controller G c_F (z) is expressed in pole zero form as [11, 23, 24]: G c_F (z) =

   k p_F + ki_F z−

k p_F  k p_F +ki_F



z−1

(4.100)

Substituting G c_F (z), and G F (z) in (4.99) and re-arranging: ⎛   z− k p_F + ki_F G F_O L (z) = ⎝ z−1

k p_F  k p_F +ki_F

⎞   α1 ⎠ z −1 z − β1

(4.101)

The closed loop discrete time transfer function for the thrust force control loop is:

4.6 PI Controller Based Direct Thrust Force Control (PI-DTFC)

G F_C L (z) =

107

G c_F (z)G F (z)z −1 FT = Fr e f 1 + G c_F (z)G F (z)z −1

(4.102)

where, Fr e f is the thrust force reference. Substituting G c_F (z) from (4.100), and G F (z) into (4.102) and simplifying:    k  α1 k p_F + ki_F z − k p_Fp_F  +ki_F FT  G F_C L (z) = =   k  Fr e f z − k p_Fp_F z(z − 1)(z − β1 ) + α1 k p_F + ki_F  +k i_F

(4.103) The characteristic equation of the closed loop transfer function thrust force control loop in (4.102) is: 1 + G c_F (z)G F (z)z −1 = 0

(4.104)

From (4.93) and (4.100):

1+

   α1 k p_F + ki_F z−

k p_F  k p_F +ki_F

(z − 1)(z − β1 )

 =0

(4.105)

Equation (4.105) can be expressed as: 1 + K F HF (z) = 0

(4.106)

   K F = α1 k p_F + ki_F

(4.107)

where:

 HF (z) =

z−

k p_F  k p_F +ki_F



(z − 1)(z − β1 )

(4.108)

The transfer function HF (z) has two poles at z = 1, and z = β1 respectively. The thrust force PI controller introduces a zero in the discrete transfer function HF (z) at z = z o_F as: z = z o_F =

k p_F  k p_F + ki_F

(4.109)

In order to draw the root locus for the system, the zero of the transfer function HF (z) should be selected such that it cancels the pole at z = β1 to achieve satisfactory performance of the thrust force PI controller [11, 23, 24]. Therefore,

108

4 SV-PWM Based Direct Thrust Force Control …

z o_F =

k p_F = β1  k p_F + ki_F

(4.110)

Substituting (4.110) in (4.108): HF (z) =

1 (z − 1)

(4.111)

From (4.111) and (4.106), the characteristic equation becomes: 1 + K F

1 =0 (z − 1)

(4.112)

The root locus for the characteristic equation (4.112) can be drawn after selecting zero z o_F according to (4.110), and the closed loop poles corresponding to the desired damping ratio can be selected on the plot. The value of the gain K F associated with the selected closed loop poles of thrust force control loop is also obtained from the root locus. From knowledge of values of the gain K F and zero z o_F , the gains for the stator flux PI controller can be achieved from (4.107) and (4.110) as: K F β1 α1

(4.113)

K F (1 − β1 ) α1

(4.114)

k p_F =  ki_F =

The root locus plot for the characteristic equation (4.112) when K F varies from 0.1 to 1 is shown in Fig. 4.11. The sample time Ts is 200 µs. The closed loop poles are selected to be 0.5 ± 0.114i corresponding to a damping ratio ξd of 0.95 as shown in Fig. 4.11. The value gain K λ corresponding to the selected closed loop poles is 0.27. The gains of the thrust force PI controller can be computed using (4.113) and (4.114) as:  k p_F = 0.0480 Volts N , where ξd = 0.95, and K λ = 0.27  ki_F = 0.0174 Volts Ns

(4.115)

It is important to note that the performance of the thrust force control loop in terms of the rise time, overshoot and steady-state ripple is determined by the gains of the thrust force PI controller. In this research, the PI gains are required to ensure a fast transient response in terms of rise time with near zero overshoot and minimal steady-state ripple. Therefore, in order to validate above design process for the thrust force PI controller and to evaluate the effect of the gains on the thrust force response, the gains for the PI controller are computed corresponding to various damping ratios using the root locus plot of Fig. 4.11 and are summarized in Table 4.1. The open loop frequency responses of the thrust force control loop corresponding to various values damping ratios of Table 4.1 are shown in Fig. 4.12.

4.6 PI Controller Based Direct Thrust Force Control (PI-DTFC)

109

Root Locus for Thrust Force Control Loop 1

0.5π/T

0.6π/T

0.8

0.4π/T 0.1 0.3π/T

0.7π/T

0.2 0.3

0.6

0.8π/T

Imaginary Axis

0.4 0.2 0

0.2π/T

0.4 0.5 0.6 0.7

0.9π/T

0.1π/T

0.8 0.9

Selected Closed Loop Poles π/T π/T

-0.2 0.9π/T

0.1π/T

-0.4 0.8π/T

-0.6

0.2π/T

0.7π/T

-0.8

0.3π/T 0.6π/T

-1 -1

-0.8

-0.6

-0.4

0.4π/T

0.5π/T

-0.2

0

0.2

0.4

0.6

0.8

1

Real Axis

Fig. 4.11 The root locus and the selected closed loop poles for the thrust force control loop

Table 4.1 Comparison of phase margin φm , crossover frequency, ωc and closed loop bandwidth ωb for various values of damping ratio ξd Damping ratio ξd

k p_F

 ki_F

0.950

0.0480

0.900

0.0490

0.707 0.050

0.0606 0.0784

Phase margin φm (°)

Crossover frequency ωc (rad/s)

Closed loop bandwidth ωb (rad/s)

0.0174

66.7

1.35 × 103

2.65 × 103

0.0178

61.1

1.39 × 103

2.77 × 103

60.6

1.71 ×

103

4.03 × 103

51.7

2.22 ×

103

5.59 × 103

0.0219 0.0284

The open loop frequency response of Fig. 4.12 is computed from the discrete time open loop transfer function of (4.101) using MATLAB. The phase margin φm , crossover frequency ωc and closed loop bandwidth ωb for various values of damping ratio ξd determined from the open loop frequency responses of Fig. 4.12 are also summarized in Table 4.1. It is worth mentioning that the gains of Table 4.1 for thrust force PI controller and their respective phase margin φm and crossover frequency are in close agreement with (4.54)–(4.56). It is important to note from Table 4.1 that the phase margin φm decreases as the damping ratio ξd decreases from 0.95 to 0.50. The crossover frequency ωc and the

Magnitude (dB)

110

4 SV-PWM Based Direct Thrust Force Control …

30

Open Loop Frequency Response of Thrust Force Control Loop Damping Ratio 0.95 Damping Ratio 0.90 Damping Ratio 0.707 Damping Ratio 0.500

20 10 0 -10

Phase (deg)

-20 -90 -135 -180 -225 -270 2 10

3

10

4

10

Frequency (rad/sec)

Fig. 4.12 Open loop frequency response of the thrust force control loop

close loop bandwidth ωb increase as the damping ratio decreases. The increase in bandwidth results in faster transient response but the overshoot also increases due to reduced damping [23, 24]. The gains of thrust force PI controller are also validated experimentally by evaluating the thrust force response of the actual prototype to a step input using these gains. For this purpose a periodic square wave thrust force reference with amplitude of ±26.5 N and a period of 1.1 s is used. The sampling time of the digital controller is set to 200 µs. The step response of the thrust force with error plots for the PI gains corresponding to the damping ratios of 0.95, 0.900, 0.707 and 0.50 are given Fig. 4.13a–d respectively. The quantitative performance of these thrust force responses in terms of rise time, and percentage over shoot is summarised in Table 4.2. In addition, for quantitative analysis of the control performance for these PI gains, the integral of absolute error (IAE) [13] indices for the thrust force error plots during the step transient are computed and are also given in Table 4.2. It is observed from Fig. 4.13 that the rise time of the thrust force response decreases from 5 to 2.61 ms as the damping ratio decreases from 0.95 to 0.50. Moreover, it is clear from Table 4.2 that the IAE index for the thrust force response also decreases as the rise time decreases. It is important note from Table 4.2 that the reduction in rise time to achieve a faster transient response is at the cost of overshoot. The overshoot increases significantly as the integral gain is increased to reduce the damping and this behaviour can be attributed to the small time constant of the thrust force regulation plant given in (4.49). The overshoot increases to 22% for the damping ratio of 0.50 as observed from Table 4.2. The best thrust force response with near zero overshoot is achieved when the gains of thrust force PI controller are selected corresponding to the damping ratio of 0.95 and are underlined in Table 4.2. Therefore the gains

2.21

2.21

50 40 30 20 10 0 -10 2.195

2.2

2.205

2.21

2.21

50 40 30 20 10 0 -10 2.195

2.2

2.205

2.21

2.21

50 40 30 20 10 3. 0 -10 2.195

2.2

2.205

2.21

Force error (N)

-26.5

5 ms

2.205

2.2

50

Force error (N)

Force (N)

2.205

0

(b)

26.5 0 4.4 ms

-26.5 -50 2.195

(c) Force (N)

2.2

50 26.6

-50 2.195

2.2

2.205

50 26.5 0

-26.5 -50 2.195

(d) Force (N)

2.21

50 40 30 20 10 0 -10 2.195

Force error (N)

Force (N)

(a)

3.8 ms

2.2

2.205

50 26.5 0

-26.5 -50 2.195

2.61 ms

2.2

2.205

111

Force error (N)

4.6 PI Controller Based Direct Thrust Force Control (PI-DTFC)

Time (s)

Time (s)

Fig. 4.13 Force step response with corresponding error plots for different values of the damping ratio ξd : a ξd = 0.95, b ξd = 0.9, c ξd = 0.707, d ξd = 0.50 (experiment, reference force is shown in red and estimated force is shown in green) Table 4.2 Comparison of force step response for various values of the damping ratio ξd Damping-ratio ξd

PI controller gains

IAE index

Rise time (ms)

Over-shoot (%)

0.950

k p_F = 0.0480

396

5.0

0.7 (dead-beat)

386

4.4

6.9

324

3.8

9.1

266

2.61

22.4

ki_F 0.900

= 0.0174

k p_F = 0.0490  ki_F = 0.0178

0.707

k p_F = 0.0606  ki_F = 0.0219

0.500

k p_F = 0.0784  ki_F

= 0.0284

112

4 SV-PWM Based Direct Thrust Force Control …

of PI controllers corresponding to damping ratio of 0.95 are used for PI-DTFC for benchmarking the performance and these gains are underlined in Table 4.2. In order to evaluate the effect of the variation in the stator resistance Rs on the performance of the thrust force PI controller, with the benchmarked PI gains corresponding to the damping ratio of 0.95, the open loop frequency response computed from (4.101) using various values of stator resistance is shown in Fig. 4.14. The change in the phase margin φm , crossover frequency ωc and closed loop bandwidth ωb corresponding to the variation in the stator resistance is summarized in Table 4.3. It is concluded that increase in stator resistance increases the phase margin and therefore the transient response of the thrust force may become slower in terms of rise time as Rs increases with temperature.

Magnitude (dB)

Open Loop Frequency Response of Thrust Force Control Loop for Various Values of Stator Resistance 30

R

s

20

1.2 R

10

1.5 R

s s

2.0 R

s

0 -10

Phase (deg)

-20 -45 -90 -135 -180 -225 -270 2 10

3

4

10

10

Frequency (rad/sec)

Fig. 4.14 Open loop frequency response of thrust force control loop for various values of stator  resistance Rs . (ξd = 0.95, k p_F = 0.0480, and ki_F = 0.0174)

Table 4.3 Comparison of phase margin φm , crossover frequency ωc and closed loop bandwidth ωb for various values of stator resistance Rs = 3.01 Stator resistance Rs ( )

Phase margin φm (°)

Open loop crossover frequency ωc (rad/s)

Closed loop bandwidth ωb (rad/s)

Rs

66.7

1.35 × 103

2.65 × 103

1.2 Rs

74.2

1.20 ×

103

2.04 × 103

1.5 Rs

82.4

0.98 × 103

1.21 × 103

89.3

0.72 ×

0.73 × 103

2.0 Rs

103

4.6 PI Controller Based Direct Thrust Force Control (PI-DTFC)

4.6.3.1

113

Integrator Anti-windup for PI Controllers

It is important to note the output of the stator flux and thrust force PI controllers are limited by the maximum available voltage vmax from the voltage source inverter (VSI) as given in (4.23) [6, 7, 11]. The PI controller with limits is illustrated in the block diagram of Fig. 4.15. When the unconstrained output uuc exceeds the limit ulim , the output of the limiter uc saturates at ulim and the integrator keeps on integrating the error and the output of the integrator uci becomes unacceptably large. Consequently the output uuc unnecessarily remains at a large value and must be reduced to avoid any overshoot in the stator flux or thrust for response. This problem is solved by using an anti-windup strategy to limit the integrator term whenever the unconstrained output uuc exceeds the limit ulim . There are various anti-windup techniques proposed in the available literature, however in this research the anti-windup strategy used is detailed in [11] and illustrated in Fig. 4.16. It can be observed from Fig. 4.16 that an additional feedback path is created based on the error signal ei computed from the difference of unconstrained output uuc and 1 . output of the limiter uc . This error ei is fed back to the integrator through a gain kaw When the uuc exceeds the limit ulim , the additional feedback path rapidly reduces the input to the integrator to eliminate anti-windup. After extensive experimentation, the value of the anti-windup gain kaw is set to 1 to achieve satisfactory control performance of the stator flux and thrust force PI controllers. Fig. 4.15 PI controller with output limiter

rref

+_

error

rref

+_

rm

+

Limiter ulim ulim

+

uci ki 1-z-1 kp = Proportional gain ki = Integral gain

rm

Fig. 4.16 PI controller with limited output and anti-windup scheme for the integrator [11]

kp

uuc

error

+_

kp ki 1-z-1

uuc

+

Limiter ulim ulim

+

uci +

_

ei 1 kaw

uc

uc

114

4 SV-PWM Based Direct Thrust Force Control …

Fig. 4.17 Implementation of the voltage limiter for the PI controller

4.6.3.2

Voltage Limiter in the PI Controller Output

The implementation of the voltage limiter for the stator flux and thrust force PI controller with the disturbance cancelation terms is illustrated in Fig. 4.17 [7, 11]. The maximum voltage that is available from SV-PWM controller voltage source inverter is vmax as give in (4.23). The magnitude of the unconstrained voltage vector generated by the outputs of the stator flux and thrust force PI controller is v0 and expressed in terms of xy-reference frame components vx and v y as: v0 =



vx2 + v 2y

(4.116)

Whenever the magnitude of the unconstrained voltage v0 exceeds vmax , the magnitudes of vx and v y are computed without any change in the angle of the unconstrained voltage vector as: vmax v0 vmax v ∗y = v y v0

vx∗ = vx

(4.117) (4.118)

It is important to note that the dc-link voltage vdc is measured to compute vmax according to (4.23).

4.7 Linear Quadratic Regulator Based Direct Thrust Force Control …

115

4.7 Linear Quadratic Regulator Based Direct Thrust Force Control of Linear PMSM (Optimal-DTFC1) The optimal linear quadratic regulator (LQR) based control provides a systematic way of designing the control law for optimal performance of linear multivariable systems [25]. The application of optimal control to develop SVM based vector control of the PMSM using linear state feedback is reported in [26–28]. However the state feedback gains are speed dependent and have to be computed separately for every operating point making the process computationally intensive. It is important to note that there is no literature available related to the application of linear quadratic control theory to the direct thrust force control (DTFC) of the linear PMSM. In this work a DTFC scheme for the linear PMSM based on the optimal linear quadratic regulator is presented and referred to as “Optimal-DTFC1”. A novel multiple-input multiple-output (MIMO) state-space model, independent of the mover speed and having the stator flux and thrust force as states, is formulated for the linear PMSM. This model allows direct control of the thrust force and stator flux using linear state feedback law comprising of system states, i.e. stator flux and thrust force. Subsequently, an optimal linear state feedback control law is synthesized using the linear quadratic regulator approach. Integral action is added to the proposed optimal control scheme by state augmentation. Experimental results demonstrate the improvement in terms of steady-state and transient force and stator flux response of the proposed approach under various operating conditions when compared to the benchmark PI-DTFC of Sect. 4.6.

4.7.1 Formulation of a Novel State Space Model of the Linear PMSM The differential equation governing the thrust force regulation can be formulated by substituting (4.17) into (4.4): −K K d FT = FT + v y − K ωr dt ξ λs λs

(4.119)

where: ξ = 3π2τPkRFs λs same as in (4.20). Since, under direct thrust force control, λs is fixed at λr e f which is calculated according to (4.7), therefore replacing λs by λr e f in (4.119): d FT −K K = FT + v y − K ωr dt ξ λr e f λr e f The dynamics of stator flux regulation can be obtained from (4.1):

(4.120)

116

4 SV-PWM Based Direct Thrust Force Control …

dλs = −Rs i x + vx dt

(4.121)

Equations (4.120) and (4.121) can be combined in matrix form to formulate a novel state space model that governs the dynamics of force and stator flux regulation in the linear PMSM:            0 0 1 0 d λs λs vx −Rs 0 ix = + + (4.122) K 0 ξ−K 0 F v ω 0 −K dt FT T y r λr e f λr e f Evidently (4.122) presents a 2nd order state space model comprising stator flux λs and force FT as states. This state space model provides the linear time invariant governing dynamics of the stator flux and force regulation in the linear PMSM and provides the basis for formulating a linear state feedback control law forcing the states to track the desired trajectories. It is noteworthy that the state space model presents the force regulation dynamics as a type 0 servo system which does not include an integrator. Equation (4.122) can be expressed as: d x(t) = A p x(t) + B p u(t) + Ed(t) dt

(4.123)

where: 

u(t) = vx v y 

T

−Rs 0 E= 0 −K



, x(t) = λs FT



T

 , Ap =

 ix . and d(t) = ωr 

0 0 0 ξ−K λr e f



 , Bp =

1 0 0 λKr e f

 ,

The term d(t) in (4.123) can be regarded as a disturbance. The dynamic system of (4.123) has λs and FT as state variables. These states can be estimated using (2.133)– (2.138) as explained in Chap. 2, and are also the natural choices for the outputs of the system to be controlled. Therefore, the output equation of the system can be given as:      10 λs λs = (4.124) FT 0 1 FT Equation (4.124) can be expressed as: y(t) = C p x(t)  T where: y(t) = λs FT and C p is a 2 × 2 identity matrix.

(4.125)

4.7 Linear Quadratic Regulator Based Direct Thrust Force Control …

117

4.7.2 Controllability of the Novel State Space Model It is of critical importance to establish the controllability of the state space model of (4.123) to rationalize its formulation. The controllability matrix for the 2nd order dynamic system of (4.123) can be given as:   Pc = B p A p B p

(4.126)

Substitution of values of A p and B p from (4.123) into (4.126):  Pc =

1 0 0 0 2 0 λKr e f 0 λ−K 2 ξ

 (4.127)

ref

In (4.14), Pc is independent of system states, i.e. λs and FT and is independent of the mover’s speed. The rank of Pc is always 2 which is the same as the order of the system. Therefore, the system described by (4.123) is always controllable for all values of the inputs vx and v y as well as the disturbance d(t) [25].

4.8 Linear Quadratic Regulator Based State Feedback Control with Integral Action The novel state space model of linear PMSMs given by (4.123) is a type-0 MIMO servo system which does not involve integrator action in thrust force dynamics. The integral action needs to be incorporated in the system to ensure a zero steady-state error. Special steps are required to be taken when designing the state feedback law to include the integrator. The block diagram of full state feedback control of a general type-0 servo system utilizing the integral action is shown in Fig. 4.18. The state variables λs and FT according to the dynamics of (4.123) are also illustrated by Fig. 4.18. The block diagram of the proposed Optimal-DTFC1 is shown in Fig. 4.19.

Fig. 4.18 Type-0 servo system with full state feedback and integral action

118

4 SV-PWM Based Direct Thrust Force Control …

Fig. 4.19 Proposed linear quadratic regulator based direct thrust control (Optimal-DTFC1) of linear PMSM, the integral action is added by state augmentation

The integral action is achieved using state augmentation by inserting the integrator in the feed forward path between the error comparator and the plant which adds new states to the system. The mathematical formulation of the linear quadratic regulator (LQR) based state feedback control with integral action is detailed in the following subsections.

4.8.1 Mathematical Formulation of Error Dynamics The state space model of a general type-0 MIMO servo system utilizing integral action, using state augmentation, as shown in Fig. 4.18, can be expressed mathematically as: x˙ (t) = Ax(t) + Bu(t)

(4.128)

y(t) = Cx(t)

(4.129)

ξ˙ = r(t) − y(t) = r(t) − Cx(t)

(4.130)

where: x = [x1 x2 x3 . . . xn ]T y = [y1 y2 y3 . . . ym ]T u(t) = [u 1 u 2 u 3 . . . u m ]T T  r(t) = r1 r2 r3 . . . r p ξ = [ξ1 ξ2 ξ3 . . . ξm ]T A = (n × n)

is the (n × 1) state vector is the (m × 1) output vector is the (m × 1) input vector is the (p × 1) reference-input vector is the (p × 1) vector containing integral of the tracking errors (augmented states) Constant matrix (state transition matrix)

4.8 Linear Quadratic Regulator Based State Feedback Control …

B = (n × m) C = ( p × n)

119

Constant matrix (input co-efficient matrix) Constant matrix (output co-efficient matrix).

A linear state feedback control law comprising both the nominal and added states can be given as: u(t) = −Kx(t) + KI ξ (t)

(4.131)

where, K is a (m ×n) state feedback gain matrix and KI is a (m × p) integral feedback gain matrix. The state space dynamic model for the system obtained after state augmentation can be written in matrix form by combining (4.128)–(4.130): 

        x˙ (t) A 0 x(t) B 0 = + u(t) + r(t) ξ˙ (t) −C 0 ξ (t) 0 I

(4.132)

where, I is a ( p × p) identity matrix. The objective is to design the state feedback law such that the system becomes asymptotically stable so that x(∞), ξ (∞), and u(∞) approach constant values. In the steady-state (4.132) can be expressed as: 

        A 0 x(∞) B 0 x˙ (∞) + u(∞) + = r(∞) −C 0 ξ (∞) 0 ξ˙ (∞) I

(4.133)

Considering r(t) as a vector of step inputs, we have r(∞) = r(t) = r (constant) for t > 0. Now, by subtracting (4.133) from (4.132) the error dynamics can be obtained as:        B x˙ (t) − x˙ (∞) A 0 x(t) − x(∞) + = (4.134) [u(t) − u(∞)] ˙ξ (t) − ξ˙ (∞) 0 −C 0 ξ (t) − ξ (∞) Now, define: ⎧ ⎨ x(t) − x(∞) = xe (t) ξ (t) − ξ (∞) = ξ e (t) ⎩ u(t) − u(∞) = ue (t)

(4.135)

From (4.134) and (4.135), following expression can be obtained: 

      x˙ e (t) A 0 xe (t) B = + ue (t) ξ˙ e (t) −C 0 ξ e (t) 0

(4.136)

Now, define an ((n + m) × 1) order error vector as:  e(t) =

xe (t) ξ e (t)

 (4.137)

120

4 SV-PWM Based Direct Thrust Force Control …

By using (4.136) and (4.137) the error dynamics can be expressed in compact form as: e˙ (t) = Ae(t) + Bue (t)

(4.138)

where:     A 0 B A= , and B = , and from (4.131) and (4.136) it is evident that the −C 0 0 control law for error dynamics of (4.138) can be expressed as: ue (t) = −Kxe (t) + KI ξ e (t)

(4.139)

ue (t) = −Ke(t)

(4.140)

  where: K = K −KI , now from (4.138) and (4.140) the closed loop error dynamics are given as:   e˙ (t) = A − B K e(t)

(4.141)

It is evident from (4.141) that the problem of designing the state feedback law for (4.132) to track the reference values is now transformed to a regulator design problem. The gain matrix K is to be designed such that the error vector e(t) converges to zero at steady-state. The closed loop error dynamics of (4.141) also allow the utilization of the linear quadratic regulator design approach for computing the gain matrix K.

4.8.2 Optimal Linear Quadratic Regulator Design The linear quadratic regulator approach provides a systematic way of computing the feedback gain matrix K for making MIMO error dynamics of (4.138) asymptotically stable. The gain matrix K is computed by minimizing a quadratic performance index given [25]: ∞  J = ∫ xeT (t)Qxe (t) + ueT (t)Rue (t) dt

(4.142)

0

where: Q and R are both positive-definite matrices. Note that the second term on the right-hand side of (4.142) accounts for the expenditure of the energy of the control signals. The matrices Q and R determine the relative importance of the error and the expenditure of this energy. The feedback law designed is thus a compromise between the use of the control effort and therefore response speed, and at the same time guarantees a stable system.

4.8 Linear Quadratic Regulator Based State Feedback Control …

121

The optimal gain matrix that minimizes the performance index of (4.142) is given as [25]:   K = K −KI = R−1 BP

(4.143)

where P is a positive-definite matrix and is the solution of the following reducedmatrix Riccati equation: AP + PA − PBR−1 BP + Q = 0

(4.144)

From (4.140) and (4.143), the feedback control law for error dynamics can be computed as: ue (t) = −R−1 BP e(t)

(4.145)

From (4.136) and (4.145), the feedback control law for the dynamic system of (4.132) which drives the error dynamics of (4.138) to zero can be given as: 

x(t) u(t) = −R BP ξ (t) −1

 (4.146)

Equation (4.146) gives the state feedback control law with optimal gain matrix such that the error vector e(t) of (4.138) is driven to zero and the system states track their respective reference values.

4.9 Novel LQR Based Direct Thrust Force Control of the Linear-PMSM The objective of the Optimal-DTFC1 scheme proposed in this paper is that the states λs and FT of the novel state space model (4.123), which are also the outputs of the system, track the reference values λr e f and Fr e f . According to (4.123) and Figs. 4.18 and 4.19, the input reference vector r(t) can be given as:  r(t) =

λr e f Fr e f

 (4.147)

The novel augmented state space model of the linear PMSM, including integral action, can be formulated from (4.123), (4.132) and (4.147) after simple mathematical manipulation and is given as:   d 0 x¯ (t) = A p x¯ (t) + B p u(t) + E p d(t) + r(t) I dt

(4.148)

122

4 SV-PWM Based Direct Thrust Force Control …

T  where: u(t) = vx v y , x¯ (t) = [λs FT ξs ξT ]T , ⎡

⎡ ⎤ ⎤ ⎡ 0 0 00 1 0 −Rs 0 ⎢ 0 −K ⎢0 K ⎥ ⎥ ⎢ 00⎥ 0 −K ⎢ ⎢ ⎥ A p = ⎢ ξ λr e f ⎥, B p = ⎢ λr e f ⎥, E p = ⎢ ⎣ 00 ⎣ −1 0 0 0 ⎦ ⎣ 00 ⎦ 00 00 0 −1 0 0 ) )     λr e f − λs dt, and ξT = Fr e f − FT dt ξs =

⎤ ⎥ ⎥, ⎦

4.9.1 Controllability Analysis of the Error Dynamics The error dynamic for the system given by (4.148) can be formulated using (4.138) and (4.148) in compact form as: e˙ (t) = A p e(t) + B p ue (t)

(4.149)

where, e(t) = x¯ (t) − x¯ (∞) Since (4.149) presents 4th order error dynamics, therefore the controllability matrix is given as: + * 2 3 Pc = B p A p B p A p B p A p B p

(4.150)

It is easy to show that rank of Pc is 4 which is the same as the order of the system of (4.149). This clearly indicates that the error dynamics of (4.149) are always controllable for all values of the inputs vx and v y as well as the disturbance d(t).

4.9.2 Choice of Gain Matrix for State Feedback Law From (4.123) and (4.148) the stator flux and force dynamics are decoupled from each other, therefore, the linear state feedback law is: 

vx vy



 k 0 −kiλ 0 =− λ 0 k F 0 −ki F

⎡ ⎤ λs  ⎢ FT ⎥ ⎢ ⎥ ⎣ ξs ⎦ ξT

(4.151)

In (4.151), kλ and k F are state feedback gains, whereas kiλ and ki F are integral gains. The implementation of the control law of (4.151) for the system under study

4.9 Novel LQR Based Direct Thrust Force Control of the Linear-PMSM

123

Table 4.4 Controller gains used in experiment for the performance comparison of Optimal-DTFC1 and PI-DTFC Optimal-DTFC1 Stator flux controller Thrust force controller

PI- DTFC kλ

10

kiλ

0.004

kF

0.055

ki F

Stator flux PI controller Thrust force PI controller

0.023

k p_λ

8.7

 ki_λ

0.0034

k p_F

0.0480

 ki_F

0.0174

is illustrated by the diagram of Fig. 4.19. The feedback gain matrix for (4.151) is: 



K = K −KI =



kλ 0 −kiλ 0 0 k F 0 −ki F



The tuning of these gains is performed using the optimal linear quadratic regulator approach as explained in Sect. 4.8.2. From (4.146) and (4.151) the state feedback law of (4.141) can be expressed as: 

vx vy



⎡

⎤ λs ⎢ FT ⎥ ⎢ ⎥ = −R−1 B p P ⎣ ξs ⎦   Gain matri x K o f (4.151) ξT

(4.152)

where P is a positive-definite matrix and is the solution of the (4.144) when solved for the system of (4.148). The block diagram of the proposed linear quadratic regulator based DTFC is given in Fig. 4.19 and illustrates the structure of the optimal controller of (4.151). In addition, Fig. 4.18 shows the structure of the proposed controller of (54) in state space form. A comparison of Figs. 4.5 and 4.19 shows that under the proposed Optimal-DTFC1, vx and v y are obtained from the optimal state feedback law of (4.151) in contrast to the PI controllers of the state of the art method (PI-DTFC) detailed in Sect. 4.6. The controller gains of (4.151) are detailed in Table 4.4.

4.10 Experimental Validation of Proposed Control Scheme The proposed novel state space model of the linear PMSM and its optimal linear quadratic regulator based DTFC (‘Optimal-DTFC1’) has been practically validated on a prototype surface-mount linear PMSM control system in the laboratory. The main hardware components of the experimental setup are illustrated in Fig. 1.1. The parameters of the surface-mount linear PMSM are provided in Table 1.1. Experimental results indicate improved performance of the Optimal-DTFC1 method in terms of steady-state error and transient response of both stator flux and force when

124

4 SV-PWM Based Direct Thrust Force Control …

compared with the prior PI-DTFC method. The controller gains for both PI-DTFC and the proposed Optimal-DTFC1 are listed in Table 4.4. All the experimental waveforms for PI-DTFC and the Optimal-DTFC1 are based on the gains of Table 4.4. The proposed Optimal-DTFC1 is digitally implemented according to the block diagram of Fig. 4.19. The optimal state feedback control law of (4.151) as illustrated by Fig. 4.19 is digitally implemented using a dSPACE® DS1104 controller. The gains of thrust force and stator flux regulating PI controllers of PI-DTFC are tuned to achieve a damping ratio of 0.95 using the root locus method to ensure comparable conditions for benchmarking as described in Sect. 4.6.3. The integrators used in both the Optimal-DTFC1 and PI-DTFC have an anti-windup scheme as shown in Fig. 4.16 with the anti-windup gains for all the integrators set at unity. The proportional and integral gains for the speed PI controller (in Figs. 4.5 and 4.19) are selected as 970 and 6.5 respectively for both PI-DTFC and the Optimal-DTFC1 to ensure similar conditions for the comparison of the two methods. The speed PI controller also has an anti-windup scheme with the anti-windup gain set at unity.

4.10.1 Dynamic Response with Outer Speed Loop Disabled The improvements realized by the Optimal-DTFC1 are demonstrated by comparing its transient performance for thrust force regulation, without the outer speed control loop, to that of PI-DTFC. For this purpose a periodic square wave thrust force reference with amplitude of ±26.5 N and a period of 1.1 s is used. The transient and steady-state thrust force control performances of PI-DTFC are  of the thrust shaped by tuning the proportional gain k p_F and the integral gain ki_F force PI controller shown in Fig. 4.5. In this research, the tuning criteria adopted for these gains is to achieve a fast step response of the thrust force in terms of rise time with minimum possible or near zero percentage overshoot (i.e. a deadbeat response with a faster rise time). Moreover, a steady-state response of thrust force with minimal ripple is also of key interest while tuning the gains of the PI controller for thrust force regulation. The quantitative performance of the thrust force response in terms of rise time, percentage overshoot and percentage steady-state ripple Fri p for various values of the proportional gain k p_F and the integral gain kiF corresponding to the damping ratios of 0.95, 0.707, and 0.5 respectively are summarised in Table 4.5. The percentage steady-state ripple in thrust force, Fri p is defined by (4.154). It can be observed from Table 4.5 that tuning of the gains of thrust force PI controller for a higher damping ratio of 0.95 results in the minimum over shoot of 0.7% and a steadystate ripple of 3.0% with a rise time of 5 ms. It is important to note that when damping ratio is decreased to 0.707 the integral gain increases and results in further reduction of the rise time to 3.8 ms. The PI gains tuned for a damping ratio of 0.5 further reduce the rise time to 2.6 ms as observed from Table 4.5, however the percentage overshoot increases to 22.4% in this case, which is not acceptable. It can be concluded from Table 4.5 that a lower damping ratio results in a significantly larger overshoot and

4.10 Experimental Validation of Proposed Control Scheme

125

Table 4.5 Comparison of rise time, percent overshoot, and percent steady-state ripple in the response of thrust force PI controller under PI-DTFC for various damping ratios Damping ratio

PI gains

0.950

k p_F

0.0480

 ki_F

0.0174

k p_F

0.0606

 ki_F

0.0219

0.707 0.500

k p_F

0.0784

 ki_F

0.0284

Rise time (ms)

Overshoot (%)

Fri p (%)

5.0

0.7

3.00

3.8

9.1

3.14

2.6

22.4

3.59

Force (N)

an increase in the steady-state ripple in the thrust force. Therefore the gains of the PI controllers corresponding to a damping ratio of 0.95 are used for PI-DTFC for benchmarking the performance and these gains are underlined in Table 4.5 and also given in Table 4.4. The step response of the thrust force corresponding to the damping  = 0.0174) is shown in Fig. 4.20. ratio of 0.95 (k p_F = 0.0480 and ki_F The step response of thrust force under PI-DTFC is compared with that of optimalDTFC1 and is shown in Figs. 4.20 and 4.21 respectively. It can be observed from Fig. 4.20 that a reasonable transient response of thrust force is achieved under PIDTFC with a rise time of 5 ms. However, from Fig. 4.21, a superior transient response of thrust force under Optimal-DTFC1 with a rise time of 3.88 ms and a negligible overshoot of 3.1% is observed which is 28% faster than that of PI-DTFC in terms of rise time. It is clear from these figures that Optimal-DTFC1 results in a faster transient response of thrust force with lower steady-state ripple of 2.6% when compared to that of PI-DTFC. This clearly validates the superior control performance in terms of thrust force transient response of the Optimal-DTFC1. 50 40 30 20 10 0 -10 -20 -30 -40 -50 1.095

Reference

Estimated

5 ms

1.0975

1.1

1.1025

1.105

Time (s) Fig. 4.20 Force step response of PI-DTFC (experiment)

1.1075

1.11

1.1125

1.115

126

4 SV-PWM Based Direct Thrust Force Control … 50 Reference

40 30

Force (N)

20 10 Estimated

0 -10 -20 -30

3.88 ms

-40 -50 1.095

1.0975

1.1

1.1025

1.105

1.1075

1.11

1.1125

1.115

Time (s)

Fig. 4.21 Force step response of Optimal-DTFC1

4.10.2 Steady-State Regulation with Outer Speed Loop Disabled The steady-state regulation of stator flux and thrust force for both the PI-DTFC and Optimal- DTFC1 are presented in Fig. 4.22a, b respectively. In Fig. 4.22a, from top to bottom the steady-state response of thrust force, magnified view of the steady-state thrust force, steady-state error in thrust force and steady-state stator flux are shown and the same order is followed in Fig. 4.22b. The force reference is set at 26.5 N and the stator flux reference is fixed at 0.0846 Wb according to (4.7). It is noted from Fig. 4.22a, b that steady-state regulation of thrust force is much improved for the Optimal-DTFC1 as it has reduced steady-state ripple when compared to that of PI-DTFC. It can be observed from Table 4.5 that the steady-state ripple in the thrust force under PI-DTFC is 3%. However, under Optimal-DTFC1 the steady-state ripple is reduced to 2.6%. The magnified view of the thrust force steady-state response of Fig. 4.22a illustrates that under PI-DTFC the oscillation in thrust force is within a band of 4 N, while from Fig. 4.22b, it is evident that the thrust force oscillations under Optimal-DTFC1 remain within a band of less than 3 N which clearly validates the effectiveness of the Optimal-DTFC1. The steady-state error in thrust force under PI-DTFC has a peak of ±3.5 N whereas in case of the Optimal-DTFC1 it remains within ±2.5 N. In addition, it can be observed that the steady-state stator flux response is smoother under the OptimalDTFC1 when compared with that of PI-DTFC.

4.10 Experimental Validation of Proposed Control Scheme

(b) Reference (dashed)

Force (N)

35 30 25 20 15 10 5 0 1.25 29

Estimated (solid)

1.35

1.325

1.3

1.275

1.375

1.4

Force (N)

28 27 26

1.3175

1.3225

1.3275

1.3325

1.35

1.325

1.3

1.375

1.4

Reference (dashed)

27 26

5 4 3 2 1 0 -1 -2 -3 -4 -5 1.25 0.0855

1.275

1.3

1.325

1.35

1.375

1.4

Reference (dashed)

Flux (Wb)

Flux (Wb)

1.275

28

24 1.3125

1.3375

0.0853 0.085 0.0848

Estimated (solid) 0.0845 1.25

Estimated (solid)

25

Estimated (solid)

Force Error (N)

24 1.3125 5 4 3 2 1 0 -1 -2 -3 -4 -5 1.25 0.0855

Reference (dashed)

29

Reference (dashed)

25

35 30 25 20 15 10 5 0 1.25

Estimated (solid)

1.3175

1.3275

1.3225

1.3325

1.3375

Force Error (N)

Force (N)

Force (N)

(a)

127

1.275

1.3

1.325

Time (s)

1.35

1.375

1.4

1.275

1.3

1.325

1.35

1.375

1.4

1.375

1.4

Reference (dashed)

0.0853 0.085 0.0848 0.0845 1.25

Estimated (solid) 1.275

1.3

1.325

1.35

Time (s)

Fig. 4.22 From top to bottom the force response, magnified force response, steady-state error in the force and steady-state stator flux response is shown, a PI-DTFC, and b Optimal-DTFC1

4.10.3 Start-Up Speed Response with Outer Speed Loop Enabled The start-up response with the outer speed loop enabled for the surface-mount linear PMSM from zero speed to 200 mm/s under both the PI-DTFC and the OptimalDTFC1 is compared in Fig. 4.23 which shows the speed response, the corresponding thrust force, stator flux and the stator currents for both the PI-DTFC and the OptimalDTFC1 respectively. It can be observed from Fig. 4.24 that the speed response under Optimal-DTFC1 is 5.1 ms (17%) faster than that of PI-DTFC. The thrust force demand of the OptimalDTFC1 is comparable to that of PI-DTFC. However it is important to note that the transient response of thrust force under Optimal-DTFC1 is faster than that of PI-DTFC which consequently results in the faster speed response during the startup transient when the linear LPMSM is controlled under the Optimal-DTFC1. The stator flux response and the stator current demand for both the PI-DTFC and the Optimal-DTFC1 do not differ by much. It is important to note that the gains for a speed PI controller of both PI-DTFC and the Optimal-DTFC1 are the same as mentioned earlier. However the tuning of

128

4 SV-PWM Based Direct Thrust Force Control …

(b)

Reference

200 100 0 -100 0.95

Measured

1

150

1.05

1.15

1.1

1.2

Reference

Estimated

50 0 1

1.05

1.1

1.15

0.085 0.0845

3 2 1 0 -1 -2 -3 0.95

Reference

1

1.05

1.15

1.1

1.1

1.15

1.15

1.1

1.05

1

1.2

Reference Estimated

50 0 1.05

1

1.1

1.15

1.2

Estimated

0.0855 0.085 0.0845

1.2

0.084 0.95

1.2

3 2 1 0 -1 -2 -3 0.95

Ib (Blue)

Ia (Red)

1.05

Measured

100

0.95

Ic (Green)

1

0 -100 0.95

Current (A)

Current (A)

0.084 0.95

100

0.086

Estimated

0.0855

Reference

200

1.2

Flux (Wb)

Flux (Wb)

0.086

Optimal-DTFC1

150

100

0.95

300

Speed (mm/s)

PI-DTFC

300

Force (N)

Force (N)

Speed (mm/s)

(a)

Reference

1.15

1.1

1.05

1

1.2

Ib (Blue)

Ic (Green)

1

1.05

Ia (Red)

1.1

1.15

1.2

Time (s)

Time (s)

Fig. 4.23 Start-up performance from 0 to 200 mm/s with outer speed loop closed. Speed, force, stator flux, and stator phase currents responses are shown from top to bottom respectively for both a the PI-DTFC and b Optimal-DTFC1 (experiment)

(b)

PI-DTFC

300

300

Reference

Speed (mm/s)

Speed (mm/s)

(a) 200 100

Measured

0

35.2 ms

-100 0.95

0.975

1

Time (s)

1.025

1.05

Optimal-DTFC1 Reference

200 100 0 -100 0.95

Measured 30.1 ms 0.975

1

1.025

1.05

Time (s)

Fig. 4.24 Magnified view of the speed response during start-up, a PI-DTFC, and b Optimal-DTFC1 (experiment)

4.10 Experimental Validation of Proposed Control Scheme

129

these gains is performed to achieve a damping ratio of 0.95 for PI-DTFC and results in negligible overshoot in the speed response with a rise time of 35.2 ms, Fig. 4.24. However, the same set of the gains, when used for the speed PI controller of the Optimal-DTFC1, results in a comparatively larger overshoot in the speed response with a rise time of 30.1 ms. The larger over shoot in the speed response under Optimal-DTFC1 can be attributed to the faster transient control performance of the optimal inner thrust force controller loop.

4.10.4 Speed Reversal with Outer Speed Loop Enabled The experimental results comparing the performance of the PI-DTFC and the Optimal-DTFC1 during the speed reversal from −600 to +600 mm/s are shown in Fig. 4.25 which shows the speed response, the corresponding thrust force, stator flux and the stator currents for both the PI-DTFC and the Optimal-DTFC1. Figure 4.26 shows a magnified view of the speed response and demonstrates that under the Optimal-DTFC1 the speed response is 7.2 ms (12%) faster than PI-DTFC during the speed reversal transient. The thrust force response for the Optimal-DTFC1 is considerably less oscillatory when compared with PI-DTFC. The improved stator flux response is evident under the Optimal-DTFC1 with significantly reduced ripple. The stator current demand for both the control schemes is similar during the speed reversals.

4.10.5 Steady-State Response with Outer Speed Loop Enabled The steady-state performance of the prototype linear PMSM at 600 mm/s occurs from 0.95 to 1.15 s as observed from Fig. 4.25 under both PI-DTFC and OptimalDTFC1 and is also compared. In order to illustrate the comparison of steady-state performances of the two control schemes, the magnified view of the speed response, stator flux response and thrust force response, for DTFC1 and Optimal-DTFC1 during the steady-state are shown in Fig. 4.27. Figure 4.27 demonstrates that under OptimalDTFC1 the steady-state low frequency oscillations in speed, force and stator flux have noticeably reduced compared to PI-DTFC. The quantitative results for steady-state performance of PI-DTFC and the OptimalDTFC1 at 600 mm/s and 52 N (average force) in terms of percent stator flux ripple λrip (%), percent force ripple F rip (%), and percent speed ripple vri p (%) are summarized in Table 4.6. In this analysis λrip (%), F rip (%) and vri p (%) are given by (4.153), (4.154) and (4.155) respectively,

130

4 SV-PWM Based Direct Thrust Force Control … PI-DTFC

600 0

Reference

-600 0.7

Measured 0.8

1

0.9

0.9

0.087

1

1.1

1.2

1.3

Flux (Wb) 1

1.1

1.2

0.8

0.9

1

1.1

1.3

0 -3 1

1.1

1.2

Reference

0.084 0.083 0.7

Current (A)

Current (A)

1.2

Estimated

0.085

1.3

3

0.9

1.3

0 -100

5

0.8

1.2

Reference

0.086

Estimated 0.9

1.1

0.087

0.084 0.8

1

0.9

Reference

0.085

-5 0.7

Measured 0.8

100

-200 0.7

-

0.086

0.083 0.7

0.7 200

Estimated 0.8

Reference

-600

1.3

0 -100

0

Force (N)

Force (N)

1.2

100

-200 0.7

Flux (Wb)

1.1

600

Reference

200

Optimal-DTFC1

(b) Speed (mm/s)

Speed (mm/s)

(a)

1.3

Estimated 0.8

0.9

0.8

0.9

1

1.1

1.2

1.3

1

1.1

1.2

1.3

5 3 0 -3 -5 0.7

Time (s)

Time (s)

Fig. 4.25 Speed reversal from −600 to 600 mm/s and steady-state response at 600 mm/s with outer speed loop closed. Speed, force, stator flux, duty ratio and stator phase a current responses are shown from top to bottom respectively for both a the PI-DTFC and b Optimal-DTFC1 (experiment)

 λri p (%) =  Fri p (%) =  vri p (%) =

1 N

,N

i=1 (λs (i)

− λav )2

λav 1 N

,N

i=1 (FT (i)

− Fav )2

Fav 1 N

,N

i=1 (vm (i)

Fav

− vav )2

× 100

(4.153)

× 100

(4.154)

× 100

(4.155)

4.10 Experimental Validation of Proposed Control Scheme PI-DTFC

600

Measured

0 66.8 ms

-600 0.76

0.8

0.84

0.88

Optimal-DTFC1 Reference

(b)

Reference

0.92

600

Speed (mm/s)

Speed (mm/s)

(a)

131

Measured

0

59.6 ms

-600

0.96

0.76

0.8

0.84

Time (s)

0.88

0.92

0.96

Time (s)

Fig. 4.26 Magnified view of the speed reversal transient a PI-DTFC, and b Optimal-DTFC1 (experiment) PI-DTFC

(a) Speed (mm/s)

600 Measured

500 0.95

1

1.05

1.1

50

-50 0.95

Estimated

1

1.05

0.087

1.1 Reference

Flux (Wb)

Estimated 1

1.05

1.1

1

1.05

1.1

1.15

Reference

100 50 0

Estimated 1

1.05

1.1

1.15

1.1

1.15

0.087

0.085

0.083 0.95

Measured

500

-50 0.95

1.15

0.086

0.084

600

150

Reference

100

0

Reference

700

0.95

1.15

Force (N)

Force (N)

150

Flux (Wb)

Speed (mm/s)

Reference

700

Optimal-DTFC1

(b)

1.15

Reference

0.086 0.085 0.084

Estimated

0.083 0.95

1

1.05

Time (s)

Time (s)

Fig. 4.27 Steady-state performance at 600 mm/s. From top, speed, force, and stator flux responses are shown for both a the PI-DTFC and b Optimal-DTFC1 (experiment) Table 4.6 Comparison of steady-state performance of PI-DTFC and the Optimal-DTFC1

@ 600 mm/s, 52 N

PI-DTFC

Optimal-DTFC1

λrip (%)

0.34

0.238

F rip (%)

10.48

6.21

vrip (%)

1.92

1.13

132

4 SV-PWM Based Direct Thrust Force Control …

where, λav , Fav and vav represent the average steady-state stator flux, thrust force and mover’s speed respectively, λs (i), FT (i) and vm (i) are the instantaneous values of stator flux, thrust force and mover’s speed, and N the number of samples. It can be observed from Table 4.6 that the Optimal-DTFC1 reduces the percentage ripple in steady-state speed, stator flux and thrust force response compared to PI-DTFC which validates the superior steady-state performance of the proposed Optimal-DTFC1.

4.11 Conclusions In this chapter, the PI controller based direct thrust force control with space vector pulse-width modulation is rigorously analysed and its performance evaluated theoretically and experimentally for the surface mount-linear PMSM. A detailed approach for the design of the stator flux and thrust force PI controllers is also presented. Moreover, an optimal linear quadratic regulator-based, direct thrust control scheme utilizing space vector pulse-width modulation for the surface mount-linear PMSM is proposed. A multiple-input multiple-output (MIMO) state space model for the linear PMSM, comprising the stator flux and thrust force as states, is formulated which subsequently allows an optimal linear state feedback control law for direct thrust control to be synthesized using the optimal linear quadratic regulator approach. Integral action is incorporated in the control scheme by state augmentation of the novel state space model to reduce the steady-state error. Experimental results clearly indicate that the proposed control scheme exhibits excellent control of stator flux and thrust force with faster transient response and reduced steady-state error when compared to the state of the art controller. The novel state space model is independent of the mover’s speed and asymptotically state controllable over the whole speed range of the linear PMSM. The optimal state feedback control law involves static gains that are independent of mover’s speed. Therefore, only one set of gains is sufficient to provide optimal control performance for the whole speed range of the linear PMSM. Excellent steady-state and transient performance including speed/force reversals is achieved.

References 1. M.F. Rahman, L. Zhong, L.K. Wee, A direct torque-controlled interior permanent magnet synchronous motor drive incorporating field weakening. IEEE Trans. Ind. Appl. 34, 1246–1253 (1998) 2. J. Faiz, S.H. Mohseni-Zonoozi, A novel technique for estimation and control of stator flux of a salient-pole PMSM in DTC method based on MTPF. IEEE Trans. Ind. Electron. 50, 262–271 (2003) 3. J. Luukko, O. Pyrhonen, M. Niemela, J. Pyrhonen, Limitation of the load angle in a directtorque-controlled synchronous machine drive. IEEE Trans. Ind. Electron. 51, 793–798 (2004)

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4. Z. Jun, X. Zhuang, T. Lixin, M.F. Rahman, A novel direct load angle control for interior permanent magnet synchronous machine drives with space vector modulation, in Proceedings of the International Power Electronics and Drives Systems (PEDS) (2005), pp. 607–611 5. Y. Inoue, S. Morimoto, M. Sanada, Examination and linearization of torque control system for direct torque controlled IPMSM. IEEE Trans. Ind. Appl. 46, 159–166 (2010) 6. M.A.M. Cheema, J. Fletcher, M.F. Rahman, D. Xiao, Modified direct thrust control of linear permanent magnet motors with sensorless speed estimation, in Proceedings of the IEEE Industrial Electronics Conference (IECON) (2012), pp. 1908–1914 7. G. Foo, C.S. Goon, M. F. Rahman, Analysis and design of the SVM direct torque and flux controlled IPM synchronous motor drive, in Proceedings of the Australasian Universities Power Engineering Conference (AUPEC) (2009), pp. 1–6 8. D. Swierczynski, Direct torque control with space vector modulation (DTC-SVM) of inverterfed permanent magnet synchronous motor drives. Ph.D. dissertation. Faculty of Electrical Engineering, Warsaw University, Poland (2005) 9. D.G. Holmes, B.P. McGrath, S.G. Parker, Current regulation strategies for vector-controlled induction motor drives. IEEE Trans. Ind. Electron. 59, 3680–3689 (2012) 10. D.G. Holmes, T.A. Lipo, B.P. McGrath, W.Y. Kong, Optimized design of stationary frame three phase AC current regulators. IEEE Trans. Power Electron. 24, 2417–2426 (2009) 11. P.S.C. Perara, Sensorless control of permanent magnet synchronous motor drives. Ph.D. dissertation. Faculty of Engineering and Sciences, Alborg University, Denmark (2002) 12. N.S. Nise, Control Systems Engineering (Wiley, New York, 2011) 13. R. Kozio, J. Sawicki, Ludger Szklarski, Digital Control of Electric Drives (Polish Scientific Publishers, Warszawa, Poland, 1992) 14. T.M. Rowan, R.J. Kerkman, A new synchronous current regulator and an analysis of currentregulated PWM inverters. IEEE Trans. Ind. Appl. IA-22, 678–690 (1986) 15. F. Briz, M.W. Degner, R.D. Lorenz, Analysis and design of current regulators using complex vectors. IEEE Trans. Ind. Appl. 36, 817–825 (2000) 16. G.J. Silva, A. Datta, S.P. Bhattacharyya, PI stabilization of first-order systems with time delay. Automatica 37, 2025–2031 (2001) 17. L. Joong-Hui, K. Chang-Gyun, Y. Myung-Joong, A dead-beat type digital controller for the direct torque control of an induction motor. IEEE Trans. Power Electron. 17, 739–746 (2002) 18. B. Bon-Ho, S. Seung-Ki, A compensation method for time delay of full-digital synchronous frame current regulator of PWM AC drives. IEEE Trans. Ind. Appl. 39, 802–810 (2003) 19. W. Krajewski, A. Lepschy, U. Viaro, Designing PI controllers for robust stability and performance. IEEE Trans. Control Syst. Technol. 12, 973–983 (2004) 20. H. Kum-Kang, R.D. Lorenz, Discrete-time domain modeling and design for AC machine current regulation, in Industry Applications Conference, 2007. 42nd IAS Annual Meeting. Conference Record of the 2007 IEEE (2007), pp. 2066–2073 21. K. Hongrae, M.W. Degner, J.M. Guerrero, F. Briz, R.D. Lorenz, Discrete-time current regulator design for AC machine drives. IEEE Trans. Ind. Appl. 46, 1425–1435 (2010) 22. Y. Sun, P.W. Nelson, A.G. Ulsoy, Proportional-integral control of first-order time-delay systems via eigenvalue assignment. IEEE Trans. Control Syst. Technol. 21, 1586–1594 (2013) 23. B.C. Kuo, Digital Control Systems (Oxford University Press Inc, New York, 1992) 24. G.F. Franklin, J.D. Powell, M.L. Workman, Modelling and High-Performance Control of Electric Machines (Addison-Wesley Longman Publishing Co., Inc., Boston, MA, USA, 1997) 25. P.C. Young, J.C. Willems, An approach to the linear multivariable servomechanism problem. Int. J. Control 15, 961–979 (1972) 26. C. Kuan-Teck, L. Teck-Seng, L. Tong-Heng, An optimal speed controller for permanent-magnet synchronous motor drives. IEEE Trans. Ind. Electron. 41, 503–510 (1994) 27. T. Tarczewski, L.M. Grzesiak, State feedback control of the PMSM servo-drive with sinusoidal voltage source inverter, in Proceedings of the International Power Electronics and Motion Control Conference (EPE/PEMC) (2012) 28. L.M. Grzesiak, PMSM servo-drive control system with a state feedback and a load torque feed forward compensation. COMPEL Int. J. Comput. Math. Electr. Electron. Eng. 32, 18 (2013)

Chapter 5

Optimal, Combined Speed and Direct Thrust Force Control of a Linear Permanent Magnet Synchronous Motors

5.1 Introduction In this chapter, an optimal control scheme for combined speed and direct thrust force control based on SV-PWM is proposed and experimentally validated for the prototype linear PMSM. The combined speed and direct thrust force control is achieved by formulating the optimal linear state feedback control law using the linear quadratic regulator (LQR) based approach. The proposed control scheme is referred to as “Optimal-DTFC2”. In Chap. 4, the dynamic behaviour of the stator flux and thrust force is formulated in terms of a 2nd order linear state space model having the stator flux and thrust force as state variables. Since this model does not include the mechanical dynamics of the linear PMSM, a separate PI controller is necessary to close the speed control loop. This chapter is advancement to this controller since it proposes inclusion of the mechanical dynamics of the linear PMSM to the 2nd order linear state space model of previous chapter. For this purpose, a 3rd order linear multiple-input multiple output (MIMO) state-space model, having the stator flux, thrust force and mover’s speed as states, is formulated for the linear PMSM where the state transition matrix is independent of mover’s speed. This proposed model combines speed and direct thrust force control using a linear state feedback law comprising system states. An optimal linear state feedback control law is synthesized using the LQR technique. Integral action is added to the proposed optimal control scheme by state augmentation to eliminate the steady-state error. Experimental results demonstrate the improvement in terms of steady-state and transient speed, thrust force and stator flux responses of the proposed OptimalDTFC2 under various operating conditions when compared to the benchmark PI-DTFC described Chap. 4. The LQR based control is a systematic method of designing the control law ensuring optimal performance of linear multivariable systems [1]. The available

© Springer Nature Switzerland AG 2020 M. A. M. Cheema and J. E. Fletcher, Advanced Direct Thrust Force Control of Linear Permanent Magnet Synchronous Motor, Power Systems, https://doi.org/10.1007/978-3-030-40325-6_5

135

136

5 Optimal, Combined Speed and Direct …

literature pertinent to the application of LQR-based control to PMSM is detailed in [2–4]. In [2] LQR-based control is proposed which can achieve vector and speed control simultaneously for a rotational PMSM and the optimal linear state feedback law is formulated in terms of dq-reference frame currents and rotor speed considering them as state variables. Tarczewski and Grzesiak [3] presents a simulation based study that extends the application of LQR based vector control of [2] to the PMSM with the input LC filter to achieve a combined vector and rotor position control. In [4], an experimental study of LQR-based combined vector and rotor position control for PMSM along with feed forward compensation of load torque disturbance is presented. The limitation of the LQR-based vector control in [2–4] is that the feedback gains are speed dependent and must be computed for every operating point over the whole speed range making the control scheme computationally exhaustive. There is, so far, no literature available regarding the application of LQR-based control to linear-PMSM (or PMSM) in terms of combined speed and direct thrust force control (or direct torque control). Therefore the key contribution of this chapter is the formulation of a 3rd order novel state space model that allows formulating LQR based combined speed and direct thrust force control and experimental demonstration of its effectiveness.

5.2 State Space Model of the Linear PMSM in the xy-Reference Frame for Combined Flux, Thrust and Speed Dynamics As discussed in Chap. 4, the dynamic equations governing the stator flux and thrust force dynamics of linear PMSM are: dλs = −Rs i x + vx dt

(5.1)

d FT −K K = FT + v y − K ωr dt ξ λs λs

(5.2)

In (5.2), K is the linearization co-efficient. The characteristics of the linearization co-efficient K are already discussed in Chap. 4, and for the prototype surface-mount linear PMSM with low stator inductance and short pole-pitch the value of K remains constant at 2020 N/elec. rad. Since, under direct thrust force control λs is fixed at λr e f , therefore replacing λs by λr e f in (5.2): d FT −K K = FT + v y − K ωr dt ξ λr e f λr e f

(5.3)

In Chap. 4, the value of λr e f is selected to achieve maximum force per ampere (MFPA) according to (4.7) as:

5.2 State Space Model of the Linear PMSM in the xy-Reference …

 λr e f =



λ2f

+

2 τ Ls Fr e f 3 π Pk F

137

2 (5.4)

In (5.4), Fr e f is the reference force which is either selected arbitrarily or by the speed PI controller as explained in the previous chapter. However, in this chapter the combined speed and direct thrust force control does not produce a force reference and therefore the reference force Fr e f is replaced by operating thrust force FT and (5.4) becomes:  λr e f =

 λ2f

+

2 τ Ls FT 3 π Pk F

2 (5.5)

For the prototype surface-mount linear PMSM, the operating thrust force FT varies from 0 to ±312 N and therefore, λr e f is set to 0.0486 Wb as explained in Chap. 4. The angular speed of the mover’s flux space vector is ωr can be related to the mechanical speed of the mover vm as explained in Chap. 1. vm = P

π ωr τ

(5.6)

Substituting the value of ωr from (5.6) into (5.3): −K K K Pπ d FT = vm FT + vy − dt ξ λr e f λr e f τ

(5.7)

The mechanical dynamics of the linear PMSM can be expressed as: 1 B 1 dvm = FT − vm − FL dt M M M

(5.8)

where M is the mass of the mover (kg) and B(kg/s) is the friction constant, Now, (5.7) and (5.8) can be expressed in matrix form to formulate the combined speed and thrust force dynamics as:         −K  K − K τPπ d FT FT 0 ξ λr e f λr e f = + FL + v y 1 vm − M1 dt vm − MB 0 M

(5.9)

It is clear from (5.9) that the combined speed and direct thrust force control can be achieved by the y-axis voltage input v y considering the load torque FL as a disturbance. Since only one control input v y is available therefore vm is selected as output to be tracked. Whereas, FT is controlled implicitly and does not require a separate reference value to be tracked. Equations (5.1) and (5.9) can be combined into matrix form to formulate a novel 3rd order state space model that governs the dynamics of stator flux, thrust force and speed regulation in the linear PMSM:

138

5 Optimal, Combined Speed and Direct …

⎤⎡ ⎤ ⎡ ⎤ ⎤ ⎡ ⎤ ⎡ 0 0 0 1 0   −Rs 0   λs λs d ⎣ ⎦ ⎢ −K ix v ⎥ ⎢ K Pπ ⎥⎣ K x +⎣ 0 FT = ⎣ 0 ξ λr e f − τ ⎦ FT ⎦ + ⎣ 0 λr e f ⎦ 0 ⎦ v F dt y L vm vm 0 − M1 0 M1 − MB 0 0 (5.10) ⎡

Evidently (5.10) presents a 3rd order state space model comprising stator flux λs , thrust force FT and mover’s speed vm as states. This state space model provides the governing linear time invariant dynamics of the stator flux, thrust force and speed in the linear PMSM and thus provides the basis for formulating a linear state feedback control law forcing the states to track the desired trajectories. It is noteworthy that the state space model presents the combined speed and thrust force regulation dynamics as a “type-0” servo system which does not include an integrator. Equation (5.10) can be expressed as: d x(t) = A p x(t) + B p u(t) + Ed(t) (5.11) dt ⎤ ⎡ 0 0 0 T   T ⎢ − K τPπ ⎥ where: u(t) = vx v y , x(t) = λs FT vm , A p = ⎣ 0 ξ−K ⎦, B p = λr e f 1 B 0 M −M ⎤ ⎡ ⎡ ⎤   1 0 −Rs 0 ix ⎢0 K ⎥ 0 ⎦ and d(t) = ⎣ λr e f ⎦, E = ⎣ 0 FL 1 0 −M 0 0 The term d(t) in (5.11) can be regarded as a disturbance. Equation (5.11) has λs , FT and vm as state variables. Two control inputs vx and v y are available in the system of (5.11) and therefore only two outputs can be tracked. In order to achieve good tracking performance and maximum force per ampere control according to (5.6), λs and vm are selected as outputs. It is important to note that control of the thrust force FT is implicitly combined with that of vm according to (5.9). Since it is rational to assume that all states can be either estimated or measured, the output equation of the system can be formulated as: 

λs vm



⎡ ⎤  λs 100 ⎣ ⎦ = FT 001 vm 

(5.12)

Equation (5.12) can be expressed as: y(t) = C p x(t) 

where: y(t) = λs vm

T



100 and C p = 001



(5.13)

5.2 State Space Model of the Linear PMSM in the xy-Reference …

139

5.2.1 Controllability Analysis of the Novel State Space Model It is of critical importance to prove the controllability of the state space model of (5.11) for its validation. The controllability matrix for the 3rd order dynamic system of (5.11) is:   Pc = B p A p B p A2p B p

(5.14)

Substitution of values of A p and B p from (5.11) into (5.14) yields: ⎡

1 0 0

0

0



0



⎤

2 2 ⎥ ⎢ K 0 0 − ξ Kλ2 0 λKr e f ξ 2 Kλ2 − Pπ ⎥ Pc = ⎢ τ ref ⎣ λr e f  ref ⎦ K 0 0 0 MλKr e f 0 − λKr e f ξ Mλ + MB2 ref

(5.15)

It is clear from (5.15) that Pc is independent of system states, i.e. λs , FT and vm . The rank of Pc is always 3 which is the same as the order of the system. Therefore, the dynamic system described by (5.11) is always controllable for all values of the inputs vx and v y as well as the disturbance d(t) [1]. In addition, the controllability of the system is independent of the speed vm .

5.3 Linear Quadratic Regulator Based State Feedback Control with Integral Action The novel state space model of linear PMSMs given by (5.11) is a type-0 MIMO servo system which does not include integrator action in the combined speed and thrust force dynamics. The integral action needs to be incorporated in the system to ensure a zero steady-state error in the speed tracking. This inclusion of integral action requires special steps to be taken when designing the state feedback law. The block diagram of full state feedback control of a general type-0 servo system utilizing the integral action is shown in Fig. 5.1. The state variables λs , FT and vm according to the dynamics of (5.11) are also illustrated by Fig. 5.1. The block diagram of the proposed Optimal-DTFC2 is shown in Fig. 5.2. The stator flux magnitude λs , stator flux vector angle θs , and the thrust force FT for feedback signals are estimated by using the current model of stator flux linkages in the dq-axes reference frame as explained in Sect. 2.5 of Chap. 2 using (2.133) to (2.38) and as illustrated in Fig. 4.5. The integral action is included to the system dynamics of (5.11) by state augmentation, inserting the integrator in the feed forward path between the error comparator and the plant which adds new states to the system. The mathematical formulation of linear quadratic regulator (LQR) based state feedback control with integral action is detailed in the following subsections.

140

5 Optimal, Combined Speed and Direct …

r (t ) = ⎡⎣λref

vref ⎤⎦

T •

ξ

+_

u(t )= ⎡⎣vx v y ⎤⎦

ξ

∫

KI

B

+_

T

x(t ) = ⎡⎣λs FT v m ⎤⎦

++

T

C

∫

A

Integral Action

K

Plant

y (t ) = ⎡⎣λs vm ⎤⎦

T

Fig. 5.1 Type-0 servo system with full state feedback and integral action Proposed Control Law According to Equation (5.41)

λref

vref

+

Speed and Force Controller +

-

K iv ∫

+

+

Kv

KF

-

vm

-

K iλ

∫

+

vx

-

vα

xy

vβ

vy

Inverter (VSI)+ SV-PWM

-

Linear PMSM

x

abc

Kλ

FT

λs

id iβ iα

θs

θr

Estimation of

λs , θs and FT

Eq. (2.133) to (2.138)

iq

dq

P

π τ

d dt

Fig. 5.2 Proposed linear quadratic regulator based combined speed and thrust control of linear PMSM, the integral action is added by state augmentation

5.3.1 Mathematical Formulation of Error Dynamics The state space model of a general type-0 MIMO servo system utilizing integral action, using state augmentation, as shown in Fig. 5.1, can be expressed mathematically as: x˙ (t) = Ax(t) + Bu(t)

(5.16)

y(t) = Cx(t)

(5.17)

ξ˙ = r(t) − y(t) = r(t) − Cx(t)

(5.18)

where: x = [x1 x2 x3 . . . xn ]T is the (n × 1) state vector y = [y1 y2 y3 . . . ym ]T is the (m × 1) output vector u(t) = [u 1 u 2 u 3 . . . u m ]T is the (m × 1) input vector T  r(t) = r1 r2 r3 . . . r p is the ( p × 1) reference-input vector

5.3 Linear Quadratic Regulator Based State Feedback Control …

141

ξ = [ξ1 ξ2 ξ3 . . . ξm ]T is the ( p × 1) vector containing integral of the tracking errors (augmented states) A = (n × n) Constant matrix (state transition matrix) B = (n × m) Constant matrix (input co-efficient matrix) C = ( p × n) Constant matrix (output co-efficient matrix). A linear state feedback control law comprising both the nominal and augmented states is: u(t) = −Kx(t) + KI ξ (t)

(5.19)

where, K is a (m ×n) state feedback gain matrix and KI is a (m × p) integral feedback gain matrix. The state space dynamic model for the system obtained after state augmentation can be written in matrix form by combining (5.16) to (5.17): 

        x˙ (t) A 0 x(t) B 0 = + u(t) + r(t) ξ˙ (t) −C 0 ξ (t) 0 I

(5.20)

where, I is a ( p × p) identity matrix. The objective is to design the state feedback law such that the system becomes asymptotically stable so that x(∞), ξ (∞), and u(∞) approach constant values. In the steady-state (5.20) can be expressed as: 

        B 0 x˙ (∞) A 0 x(∞) + u(∞) + = r(∞) −C 0 ξ (∞) 0 ξ˙ (∞) I

(5.21)

Considering r(t) as a vector of step inputs, we have r(∞) = r(t) = r (constant) for t > 0. Now, by subtracting (5.21) from (5.20) the error dynamics can be obtained as:        B x˙ (t) − x˙ (∞) A 0 x(t) − x(∞) + (5.22) = [u(t) − u(∞)] ˙ξ (t) − ξ˙ (∞) 0 −C 0 ξ (t) − ξ (∞) Now, define: ⎧ ⎨ x(t) − x(∞) = xe (t) ξ (t) − ξ (∞) = ξ e (t) ⎩ u(t) − u(∞) = ue (t) From (5.22) and (5.23), following expression can be obtained:

(5.23)

142

5 Optimal, Combined Speed and Direct …



      x˙ e (t) A 0 xe (t) B = + ue (t) ξ˙ e (t) −C 0 ξ e (t) 0

(5.24)

Now, define an ((n + m) × 1) order error vector as: 

x (t) e(t) = e ξ e (t)

 (5.25)

By using (5.24) and (5.25) the error dynamics can be expressed in compact form as: e˙ (t) = Ae(t) + Bue (t)

(5.26)

where:     A 0 B A= , and B = , and from (5.19) and (5.24) it is evident that the −C 0 0 control law for error dynamics of (5.26) can be expressed as: ue (t) = −Kxe (t) + KI ξ e (t)

(5.27)

ue (t) = −Ke(t)

(5.28)

  where: K = K −KI , Now from (5.26) and (5.28) the closed loop error dynamics are given as:   e˙ (t) = A − BK e(t)

(5.29)

It is evident from (5.29) that the problem of designing the state feedback law for (5.20) to track the reference values is now transformed to a regulator design problem. The gain matrix K is to be designed such that the error vector e(t) converges to zero at steady-state. The closed loop error dynamics of (5.29) also allows the utilization of the linear quadratic regulator design approach for computing the gain matrix K.

5.3.2 Formulation of the Linear Quadratic Regulator The linear quadratic regulator approach provides a systematic method of computing the feedback gain matrix K for making MIMO error dynamics of (5.26) asymptotically stable. The gain matrix K is computed by minimizing a quadratic performance index given [1]:  J= 0

∞

  T xe (t)Qxe (t) + ueT (t)Rue (t) dt

(5.30)

5.3 Linear Quadratic Regulator Based State Feedback Control …

143

where: Q and R are both positive-definite matrices. Note that the second term on the right-hand side of (5.30) accounts for the expenditure of energy of the control signals. The matrices Q and R determine the relative importance of the error and the expenditure of this energy. The feedback law designed is thus a compromise between the use of the control effort and therefore response speed, and at the same time guarantees a stable system. The optimal gain matrix that minimizes the performance index of (5.30) is given as [1]:   K = K −KI = R−1 BP

(5.31)

where P is a positive-definite matrix and is the solution of the following reducedmatrix Riccati equation: AP + PA − PBR−1 BP + Q = 0

(5.32)

From (5.28) and (5.31), the feedback control law for error dynamics can be computed as: ue (t) = −R−1 BP e(t)

(5.33)

From (5.25) and (5.33), the feedback control law for the dynamic system of (5.20) which drives the error dynamics of (5.26) to zero can be given as: 

x (t) ue (t) = −R BP e ξ e (t) −1

 (5.34)

Now, from (5.23) and (5.34) the optimal state feedback control Law u(t) for the dynamic system of (5.20) is given as: 

x(t) u(t) = −R BP ξ (t) −1

 (5.35)

Equation (5.35) gives the state feedback control law with optimal gain matrix such that the error vector e(t) of (5.26) is driven to zero and the system states track their respective reference values.

5.4 Linear Quadratic Regulator Based Combined Speed and Direct Thrust Control As described earlier, the state space model of (5.11) has two control inputs, i.e. vx and v y , therefore only two states can be controlled directly. In order to achieve maximum

144

5 Optimal, Combined Speed and Direct …

force per ampere control with optimal speed response, the flux λs and the speed vm are selected to be controlled. The objective of the proposed optimal DTFC scheme is that the states λs and vm track the reference values λr e f and vr e f . Therefore, the input reference vector r(t) can be given as: 

λr e f r(t) = vr e f

 (5.36)

The novel augmented state space model of the linear PMSM, including integral action, can be formulated from (5.11), (5.20), and (5.36) after simple mathematical manipulation and is given as (5.37): ⎤ ⎡ 0 λs ⎢F ⎥ ⎢ 0 ⎢ T ⎥ d⎢ ⎢ ⎥ ⎢ ⎢ vm ⎥ = ⎢ 0 dt ⎢ ⎥ ⎢ ⎣ ξs ⎦ ⎣ −1 ξv 0 ⎡ 1 ⎢0 ⎢ ⎢ + ⎢0 ⎢ ⎣0 0 ⎡

⎤⎡ ⎤ 000 λs −K −K τ ⎥⎢ F ⎥ 0 0 ⎢ ⎥ T ⎥ ξ λr e f Pπ ⎥⎢ ⎥ 1 −B v ⎢ ⎥ ⎥ 0 0 M M ⎥⎢ m ⎥ ⎣ ⎦ ξ 0 000 s ⎦ ξv 0 −1 0 0 ⎡ ⎡ ⎤ ⎤ 0 −Rs 0 0 K ⎥ ⎢ 0 0 ⎥  ⎢ 0   ⎢ ⎢ ⎥ λr e f ⎥ ⎢ ⎢ ⎥ vx ⎥ ix + ⎢ 0 −1 + ⎢0 ⎥ 0 ⎥ M ⎥ F ⎢ ⎢ ⎥ vy L ⎣ ⎣1 ⎦ ⎦ 0 0 0 0 0 0 0 0

⎤ 0  0⎥ ⎥ ⎥ λr e f 0⎥ ⎥ vr e f 0⎦ 1

(5.37)

    where, ξs = λr e f − λs dt and ξv = vr e f − vm dt Equation (5.37) can also be expressed in compact form as:   d 0 x(t) = A p x(t) + B p u(t) + E p d(t) + r(t) I dt where,

 x(t) = λs FT vm ξs ⎡ 0 0 0 ⎢ 0 −K −K τ ⎢ ξ λr e f Pπ ⎢ A p = ⎢ 0 M1 −B M ⎢ ⎣ −1 0 0 0 0 −1

ξv 0 0 0 0 0

T

 , u(t) = vx ⎡ ⎤ 1 0 ⎢ ⎥ 0⎥ ⎢0 ⎢ ⎥ 0 ⎥ , Bp = ⎢ 0 ⎢ ⎥ ⎣0 0⎦ 0 0

vy

T

, ⎤

⎡

−Rs K ⎥ ⎢ 0 ⎢ λr e f ⎥ ⎢ ⎥ 0 ⎥, and E p = ⎢ 0 ⎢ ⎥ ⎣ 0 0 ⎦ 0 0 0

(5.38)

⎤ 0 0 ⎥ ⎥ −1 ⎥ M ⎥ ⎥ 0 ⎦ 0

5.4 Linear Quadratic Regulator Based Combined Speed and Direct …

145

5.4.1 Controllability Analysis of the Error Dynamics The error dynamic for the system given by (5.38) can be formulated using (5.26) and (5.38) in compact form as: e˙ (t) = A p e(t) + B p ue (t)

(5.39)

where, e(t) = x(t) − x(∞). Since (5.39) presents 5th order error dynamics, the controllability matrix is given as:   2 3 4 (5.40) Pc = B p A p B p A p B p A p B p A p B p It is easy to show that the rank of Pc is 5 which is same as the order of the system given by (5.38). This clearly indicates that the error dynamics of (5.39) are always controllable for all values of the inputs vx and v y as well as the disturbance d(t).

5.4.2 Choice of Gain Matrix for State Feedback Law It is clear from (5.1), (5.9) and (5.38) the flux and mover’s combined speed and thrust force dynamics are decoupled from each other, therefore, the linear state feedback law is: ⎤ ⎡ λs ⎥  ⎢   ⎢ FT ⎥ 0 kλ 0 0 −kiλ vx ⎥ ⎢ (5.41) =− ⎢ vm ⎥ ⎥ vy 0 k F kv 0 −kiv ⎢ ⎣ ξs ⎦ ξv The implementation of the control law of (5.41) for the system under study is illustrated by the block diagram of Fig. 5.2. The feedback gain matrix for (5.41) is: 



K = K −KI =



0 kλ 0 0 −kiλ 0 k F kv 0 −kiv

 (5.42)

   kλ 0 0 k 0 is state feedback gain matrix and KI = iλ is the 0 k F kv 0 kiv integral gain matrix. The tuning of these gains is performed using the optimal linear quadratic regulator approach as explained in the previous section. From (5.35) and (5.41) the state feedback law of (5.41) can be expressed as: 

where: K =

146

5 Optimal, Combined Speed and Direct …

Table 5.1 Controller gains used in experiment for the optimal-DTFC2 and PI-DTFC State of the art PI-DTFC

Flux PI controller

Proposed optimal-DTFC2

Thrust force PI controller

k p_λ

ki_λ

k p_F

ki_F

8.7

17

0.0480

0.0174

Flux controller

Speed and force controller

kλ

kiλ

kF

kv

kiv

10

20

0.155

99

0.8

⎡ 

vx vy

 =

−1

(R B p P)   

⎢ ⎢ ⎢ ⎢ ⎢ ⎣

Gain matri x K o f (5.41)

λs FT vm ξs ξv

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

(5.43)

where P is a positive-definite matrix and is the solution of the (5.32) when solved for the system of (5.38). The block diagram of the proposed linear quadratic regulator based combined speed and direct thrust force control is given in Fig. 5.2. The controller gains of (5.41) are detailed in Table 5.1.

5.5 Experimental Validation of Proposed Optimal-DTFC2 The proposed novel state space model of the linear PMSM given (5.11) and linear quadratic regulator based combined speed and direct thrust force control (‘OptimalDTFC2’) has been practically validated on a prototype surface-mount linear PMSM control system in the laboratory. The main hardware components of the experimental setup are illustrated in Fig. 1.1. The parameters of the surface-mount linear PMSM are provided in Table 1.1. Experimental results indicate improved performance of the Optimal-DTFC2 method in terms of steady-state error and transient response of the stator flux, thrust force and speed when compared with the prior PI-DTFC method of Chap. 4. The controller gains for both PI-DTFC and the proposed Optimal-DTFC2 are listed in Table 5.1. The tuning process and selection of the gains for the PI-DTFC is already explained in Chap. 4.

5.5.1 Start-Up Speed Response The start-up speed responses of the surface-mount linear PMSM under both the PIDTFC and the Optimal-DTFC2 are compared. The speed, flux, thrust force and stator currents for both the PI-DTFC and the Optimal-DTFC2 during the start-up transient are shown in Fig. 5.3a, b respectively.

5.5 Experimental Validation of Proposed Optimal-DTFC2

1.05

1.1

1

1.05

1.1

1.15

1.2

1.2

160 120 80 40 0 -40 0.95

1

1.05

1.1

1.15

1.2

Speed (mm/s) 1.15

Reference Estimated

1

1.05

1.1

0.086

Flux (Wb)

1.2

Measured 1

1.15

0.085 0.0845

Current (A)

0.084 0.95

1

3.5

Reference 1.1 1.15 1.2

Ic (Green)

1.5 0 -1.5 -3.5 0.95

1.05

Ia (Red) 1

1.05

1.1

Time (s)

Ib (Blue) 1.15

Reference

Measured

0.086

Estimated

0.0855

Optimal-DTFC2

300 200 100 0 -100 0.95

Force (N)

160 120 80 40 0 -40 0.95

(b)

PI-DTFC Reference

1.2

Flux (Wb)

300 200 100 0 -100 0.95

Estimated

0.0855 0.085 0.0845 0.084 0.95

Current (A)

Force (N)

Speed (mm/s)

(a)

147

Reference 1

3.5

1.1

I b (Blue)

1.5 0 -1.5 -3.5 0.95

1.05

1.15

1.2

I a (Red)

I c (Green) 1

1.05

1.1

1.15

1.2

Time (s)

Fig. 5.3 Start-up performance from 0 to 200 mm/s. Speed, thrust force, stator flux, and stator phase currents responses are shown from top to bottom respectively for both a the PI-DTFC and b optimal-DTFC2 (experiment)

It is clear from Fig. 5.3a that during the start-up transient under PI-DTFC, the speed response exhibits a speed dip immediately after reaching the speed reference of 200 mm/s. However under Optimal-DTFC2 the speed dip is noticeably reduced. Figure 5.3 clearly shows that the thrust force under Optimal-DFTC2 settles to steadystate faster compared to PI-DTFC. The magnified views of the start-up speed response for both PI-DTFC and Optimal-DTFC2 are shown in Fig. 5.4a, b respectively. It is clear from Fig. 5.4 a, b that the speed response under the optimal DTFC is 12% faster compared to that of the DTFC1. The error plots for speed response during the start-up transient for both the PIDTFC and the Optimal-DTFC2 are shown in Fig. 5.5a, b respectively. It is evident that the Optimal-DTFC results in faster convergence of the speed error to zero compared to PI-DTFC. In addition, for quantitative analysis, the integral of absolute error (IAE) indices for the speed error plots for both PI-DTFC and Optimal-DTFC2 during the start-up are computed and are given in Table 5.2. The IAE index for the speed error plot under the Optimal-DTFC is reduced by 17% compared to that of the DTFC1.

148

5 Optimal, Combined Speed and Direct …

300

Reference

200 100 0 -100 0.95

Measured 35.2 ms 0.975

1

1.025

Speed (mm/s)

Speed (mm/s)

(b)

PI-DTFC

(a)

1.05

300

Optimal-DTFC2 Reference

200 100

Measured

0

31 ms

-100 0.95

0.975

Time (s)

1

1.025

1.05

Time (s)

Fig. 5.4 Magnified view of the speed response during start-up, a PI-DTFC, and b optimal-DTFC2 (experiment) PI-DTFC 250 200 150 100 50 0 -50 0.97 0.98 0.99

1

Optimal-DTFC2

(b) Speed error (mm/s)

Speed error (mm/s)

(a)

1.01 1.02

250 200 150 100 50 0 -50 0.97 0.98 0.99

1.01 1.02

1

Time (s)

Time (s)

Fig. 5.5 Magnified view of the speed response during start-up, a PI-DTFC, and b optimal-DTFC2 (experiment)

Force (N)

(a)

160 120 80 139 N 40 0 -40 0.95 0.975

Type of transient phenomena

IAE index for speed error PI-DTFC

Optimal-DTFC2

Start-up (0 to 200 mm/s)

23,721

19,619

Speed reversal (−600 to 600 mm/s)

210,430

189,370

(b)

PI-DTFC Reference Estimated 61 ms 1

Time (s)

1.025

1.05

Force (N)

Table 5.2 Comparison of transient performance of PI-DTFC and the optimal-DTFC2 using IAE index

160 120 80 40 0 -40 0.95

Optimal-DTFC2 151 N

0.975

58 ms 1

1.025

1.05

Time (s)

Fig. 5.6 Magnified thrust force response during start-up, a PI-DTFC, and b optimal-DTFC2 (experiment)

The magnified views of the thrust force response during start-up for both the PIDTFC and Optimal-DTFC2 are shown in Fig. 5.6a, b respectively. It is evident from Figs. 5.4 and 5.6 that when speed command jumps from 0 to 200 mm/s at 0.975 s,

5.5 Experimental Validation of Proposed Optimal-DTFC2

149

the corresponding thrust force under the Optimal-DTFC2 reaches a peak value of 151 N whereas the thrust force under the PI-DTFC peaks to 139 N and therefore the Optimal- DTFC2 results in a faster speed response. Moreover, it is also evident from Fig. 5.6b that the thrust force under the Optimal-DTFC2 settles to steady-state 4 ms faster than that of PI-DTFC. It can be concluded from these experimental results that the combined control of the speed and thrust force achieved under the OptimalDTFC2 is capable of delivering faster speed response during start-up compared to PI-DTFC. The stator flux response for both the PI-DTFC and the Optimal-DTFC2 is shown in Fig. 5.3a, b respectively. It is clear from Fig. 5.3 that under Optimal-DTFC2 the flux response exhibits noticeably lower ripple compared to that of PI-DTFC.

5.5.2 Speed Reversal and Steady-State Response The speed reversal and steady-state performance of the linear PMSM under both PI-DTFC and Optimal-DTFC2 is compared. The speed, stator flux, thrust force and stator currents for both the PI-DTFC and the Optimal-DTFC2 during the speed reversal transient from −600 to 600 mm/s and steady-state at 600 mm/s are shown in Fig. 5.7a, b respectively. Figure 5.7a illustrates that the dip in speed response after the speed is reversed from −600 mm/s and reaches 600 mm/s is reduced under the Optimal-DTFC2. Moreover, a reduction in thrust force oscillations under OptimalDTFC2 can also be observed from Fig. 5.7a, b when the speed settles at 600 mm/s. The magnified views of the speed response of linear PMSM under PI-DTFC and the Optimal-DTFC2 during the speed reversal transient are compared in Fig. 5.8 a, b respectively. From Fig. 5.8, the speed response under Optimal-DTFC2 is 9% faster when compared to that of PI-DTFC. The speed error plots during the speed reversal transients for both PI-DTFC and Optimal-DTFC2 are shown in Fig. 5.9a, b respectively. From Fig. 5.9, under OptimalDTFC2 the speed error converges to zero at a higher rate providing a faster speed response during the speed reversal compared to PI-DTFC. It can be observed from Table 3.2 that IAE index for the speed error during speed reversal is reduced by 10% for the Optimal-DTFC2. The steady-state performance of the prototype linear PMSM at 600 mm/s from 0.95 to 1.15 s under both PI-DTFC and Optimal-DTFC2 is also compared. In order to illustrate the comparison of steady-state performances of the two control schemes, the magnified view of the speed response, flux response and thrust force response, for PI-DTFC and Optimal-DTFC2 during the steady-state are shown in Fig. 5.10a, b respectively. Figure 5.10 demonstrates that under Optimal-DTFC2 the steady-state low frequency oscillations in speed, thrust force and stator flux have noticeably reduced compared to PI-DTFC. The steady-state speed response under PI-DTFC exhibits low frequency oscillations due to complex friction dynamics of the linear PMSM. These steady speed oscillations under PI-DTFC also cause oscillations in the force response.

150

5 Optimal, Combined Speed and Direct …

PI-DTFC

600 0

Reference

-600 0.7

Measured 0.8

0.9

1

1.1

1.2

Force (N) 1.1 1.2

1.3

-

Reference

0.086 0.085 0.084 0.083 0.7

0.8

Estimated 0.9 1 1.1

1.2

1.1 1.2

Measured 1

1.1 1.2 1.3

200 100 0 -100 Estimated -200 0.7 0.8 0.9 1 1.1 1.2 1.3 0.087 Reference 0.086 0.085 0.084

Current(A)

Current(A)

0

1

Reference

-600

0.083 0.7

1.3

5 3 -3 -5 0.7 0.8 0.9

0

0.7 0.8 0.9

Flux (Wb)

Force (N) Flux (Wb)

0.087

600

1.3

Reference

200 100 0 -100 Estimated -200 0.7 0.8 0.9 1

Optimal-DTFC2

(b) Speed (mm/s)

Speed (mm/s)

(a)

1.3

Estimated 0.9

1

1.1

1.2

1.3

-3 -5 0.7 0.8 0.9

1

1.1 1.2

1.3

0.8

5 3 0

Time (s)

Time (s)

Fig. 5.7 Speed reversal from −600 to 600 mm/s and steady-state response at 600 mm/s. Speed, force, flux, and stator phase “a” current responses are shown, a PI-DTFC, and b optimal-DTFC2 (experiment)

PI-DTFC Reference

600

Measured

0 -600 0.76

0.8

66.8 ms 0.84 0.88 0.92 0.96

Time (s)

Optimal-DTFC2

(b) Speed (mm/s)

Speed (mm/s)

(a)

600

Reference Measured

0 -600 0.76

0.8

61 ms 0.84 0.88 0.92 0.96

Time (s)

Fig. 5.8 Magnified view of the speed reversal transient, a PI-DTFC, and b optimal-DTFC2 (experiment)

5.5 Experimental Validation of Proposed Optimal-DTFC2

1200 1000 800 600 400 200 0 -200 0.76

PI-DTFC

(b) 1200

Speed error (mm/s)

Speed error (mm/s)

(a)

0.8

0.84

1000 800 600 400 200 0 -200 0.76

0.88 0.92 0.96

151

Optimal-DTFC2

0.8

0.84 0.88 0.92 0.96

Time (s)

Time (s)

Fig. 5.9 Error plots for the speed response during speed reversal transient, a PI-DTFC, and b Optimal-DTFC2 (experiment) PI-DTFC

600 Measured

500 1

1.1

1.05

Reference

100 50 0 -50 0.95

Estimated 1

1.05

0.087

0.085 0.084 0.083 0.95

Estimated 1

1.05

500

1.05

1.1

1.15

1

1.05

1.1

1.15

100 50 0 -50 0.95

Reference

0.086 0.085 0.084 0.083 0.95

1.15

1.1

Measured 1

0.087

Reference

0.086

Reference

600

1.15

1.1

Optimal-DTFC2

700

0.95 150

1.15

Force (N)

Force (N)

Speed (mm/s)

700

0.95 150

Flux (Wb)

(b)

Reference

Flux (Wb)

Speed (mm/s)

(a)

Estimated 1

1.05

1.1

1.15

Time (s)

Time (s)

Fig. 5.10 Steady-state performance at 600 mm/s. Speed, thrust force, and stator flux responses are shown, a PI-DTFC, and b optimal-DTFC2 (experiment)

The quantitative results for steady-state performance of PI-DTFC and the OptimalDTFC2 at 600 mm/s and 52 N (average force) in terms of percent flux ripple λrip (%), percent force ripple F rip (%), and percent speed ripple vri p (%) are summarized in Table 5.3. In this analysis λrip (%), F rip (%) and vri p (%) are given by (5.44), (5.45) and (5.46) respectively,

λri p (%) =

1 N

!N

i=1 (λs (i)

λav

− λav )2

× 100

(5.44)

152

5 Optimal, Combined Speed and Direct …

Table 5.3 Comparison of steady-state performance of DTFC1 and the optimal DTFC at 600 mm/s

600 mm/s, 52 N (%)

PI-DTFC

λrip

0.34

0.21

F rip

10.48

5.83

vrip

1.92

1.11

Fri p (%) =

vri p (%) =

1 N

!N

i=1 (FT (i)

− Fav )2

Fav 1 N

!N

i=1 (vm (i)

− vav )2

Fav

Optimal-DTFC2

× 100

(5.45)

× 100

(5.46)

where, λav , Fav and vav represent the average steady-state flux, thrust force and mover’s speed respectively and λs (i), FT (i) and vm (i) are the instantaneous values of flux, thrust force and mover’s speed. It can be observed from Table 5.3 that the Optimal-DTFC2 reduces the percentage ripple in steady-state speed, stator flux and thrust force response compared to PIDTFC which validate the superior steady-state performance of the proposed OptimalDTFC2.

5.5.3 Effect of Parameter Variation on the Performance The effect of parameter variation on the Optimal-DTFC2 is also evaluated experimentally. Figure 5.11a, b demonstrates the effect of change in the stator resistance Rs and stator inductance L s on the transient response of optimal DTFC during speed reversal. It can be observed from the comparison of Figs. 5.8b and 5.11a that the rise time increases from 61 to 70.5 ms when Rs is increased by 100%. In order to experimentally simulate effect of 100% increase in machines’ resistance Rs and inductance L s , the controller gains were tuned using the half of the nominal values of Rs and L s given in Table 1.1 of Chap. 1. It is also evident from Figs. 5.8b and

Measured

0 -600 0.8

(b)

Reference

600

70.5 ms 0.85

0.9

Time (s)

0.95

Speed (mm/s)

Speed (mm/s)

(a)

600

Reference

Measured

0 -600 0.8

79.2 ms 0.85

0.9

0.95

Time (s)

Fig. 5.11 Effect of parameter variation on the transient performance of optimal-DTFC2 during speed reversal from −600 to 600 mm/s, a Rs is increased by 100%, b L s is increased by 100% (experiment)

5.5 Experimental Validation of Proposed Optimal-DTFC2

153

5.11b that the rise time during speed reversal increases from 61 to 79.2 ms when the value of L s is doubled. However, the increase in the values these parameters does not have any noticeable effect on the steady state speed response at 600 mm/s.

5.6 Conclusions In this chapter, an optimal linear quadratic regulator-based, combined speed and direct thrust force control scheme utilizing space vector modulation for the linear PMSM is proposed. A multiple-input multiple-output (MIMO) state space model for linear PMSM, comprising the stator flux, thrust force and mover’s speed as states, is formulated which allows an optimal linear state feedback law for combined speed and direct thrust control to be synthesized using the optimal linear quadratic regulator approach. Integral action is incorporated in the control scheme by state augmentation of the optimal state space model to reduce the steady-state error. The proposed OptimalDTFC2 has a simple linear state feedback control law that combines the speed and thrust force control and is also easy to implement compared to PI-DTFC. Experimental results clearly indicate that the proposed control scheme exhibits excellent control of flux and thrust force with faster transient response and reduced steady-state oscillations when compared to the state of the art controller. The state transition matrix for the state space model is independent of the mover’s speed and asymptotically state controllable over the whole speed range of the linear PMSM. The optimal state feedback control law involves static gains that are independent of the mover’s speed. Only one set of gains are sufficient to provide optimal control performance for the whole speed range of the linear PMSM. Excellent steady-state and transient performance including speed/force reversals is achieved. In addition, experimental results also prove that the proposed control scheme remains stable under parameter variation.

References 1. P.C. Young, J.C. Willems, An approach to the linear multivariable servomechanism problem. Int. J. Control 15, 961–979 (1972) 2. C. Kuan-Teck, L. Teck-Seng, L. Tong-Heng, An optimal speed controller for permanent-magnet synchronous motor drives. IEEE Trans. Ind. Electron 41, 503–510 (1994) 3. T. Tarczewski, L.M. Grzesiak, State feedback control of the PMSM servo-drive with sinusoidal voltage source inverter, in Proceedings of the International Power Electronics and Motion Control Conference (EPE/PEMC) (2012) 4. L.M. Grzesiak, PMSM servo-drive control system with a state feedback and a load torque feed forward compensation. COMPEL: Int. J. Comput. Math. Electr. Electron. Eng. 32, p. 18 (2013)

Chapter 6

Sliding Mode Based Combined Speed and Direct Thrust Force Control of a Linear Permanent Magnet Synchronous Motors

6.1 Introduction In this chapter, a sliding mode control scheme for combined speed and direct thrust force control utilizing SV-PWM is proposed and experimentally validated for the prototype linear PMSM. The proposed control scheme is referred to as “SM-DTFC1”. In the previous chapter, combined speed and thrust force dynamics are formulated as a second order single-input-single-output (SISO) linear system with the y-axis voltage as control input and the mover’s speed as output. This model uses thrust force and mover’s speed as system states. In this chapter, a sliding mode control law based on this model is formulated directly in terms of thrust force and mover’s speed to achieve a combined speed and direct thrust force control. The sliding surface is formulated in terms of tracking error in the mover’s speed. Moreover, in order to minimize the steady state error in state tracking a modified reachability condition is used to augment the pure integral action directly in the sliding mode control law. Similarly, using the above mentioned modified reachability condition, the tracking of stator flux under MFPA trajectory, is realised by a sliding model control law with augmented integral action. The sliding surface for stator flux regulation is formulated in terms of the tracking error in the stator flux. Experimental results demonstrate the improvement in terms of steady-state and transient speed, thrust force and stator flux responses of the proposed SM-DTFC1 under various operating conditions when compared to the benchmark PI-DTFC1 detailed in Chap. 4.

6.2 Dynamic Model of the Linear PMSM in xy-Reference Frame As discussed in Chaps. 4 and 5, the dynamic equations governing the stator flux dynamics of linear PMSM is: © Springer Nature Switzerland AG 2020 M. A. M. Cheema and J. E. Fletcher, Advanced Direct Thrust Force Control of Linear Permanent Magnet Synchronous Motor, Power Systems, https://doi.org/10.1007/978-3-030-40325-6_6

155

156

6 Sliding Mode Based Combined Speed …

dλs = −Rs i x + vx dt

(6.1)

From (6.1), the stator flux can be controlled by x-axis voltage input vx and the resistive drop Rs i x is assumed as a disturbance. Also, from Chap. 5, the combined speed and thrust force dynamics as:         −K  K − K τPπ d FT FT 0 ξ λr e f λr e f = + FL vy + 1 vm − M1 dt vm − MB 0 M

(6.2)

It is observed from the dynamic model of (6.2) that the combined speed and direct thrust force control can be achieved by the y-axis voltage input v y considering the load torque FL as a disturbance. It is important to note that since only one control input v y is available therefore vm is selected as an output to be tracked. Whereas, FT is controlled implicitly and does not require a separate reference value to be tracked. In (6.2), K is called the linearization co-efficient [1, 2]. The characteristics of the linearization constant K for the prototype surface-mount linear PMSM are already discussed in Chap. 4 and for a low stator inductance and a short pole-pitch the value of K remains constant for the whole operational range of the thrust force. However, in the case of high inductance or large pole pitch the value of K varies as function of thrust force according to (4.10): 3 π λr e f λ f K = K (FT ) = Pk F 2 τ Ls





1−

2τ L s FT 3π Pk F λr e f λ f

2 (6.3)

It is important to note that in (6.2) and (6.3) the reference stator flux value λr e f is selected according maximum force per ampere (MFPA) and is given as:  λr e f = λr e f (FT ) =

λ2f

 +

2 τ Ls FT 3 π Pk F

2 (6.4)

From (6.3) and (6.4), taking into account the nonlinear variation in K and λr e f as nonlinear functions of the operating thrust force FT , the dynamic model of (6.2) can be expressed in more generalised form as a second order nonlinear SISO system: X˙ = F(X ) + G(X )u + D(t)

(6.5)

where,      −K (FT ) F − K (FτT )Pπ vm FT f 1 (x1 , x2 ) ξ λr e f (FT ) T = , F(X ) = = , 1 f 2 (x1 , x2 ) vm F − MB vm M T   K (FT )     0 b(FT ) λr e f (FT ) = FL G(X ) = , u = v y and D(t) = 0 − M1 0 

x X= 1 x2





6.2 Dynamic Model of the Linear PMSM in xy-Reference Frame

157

The combined speed and thrust force dynamics for linear PMSM expressed in (6.5) represent a second-order nonlinear single-input–single-output (SISO) system such that the y-axis voltage v y is the only control input. Considering the speed control as the main control objective, vm is a natural choice for state tracking. The dynamic Eq. (6.5) clearly indicates that control of the thrust force FT is implicitly combined with that of vm . It can be assumed with sufficient rationality that all states can be either estimated or measured, therefore, vm can be expressed as the tracking output y in state-space form as:  

FT = H (X ) y = vm = 0 1 vm

(6.5a)

It is important to note that the dynamic model of (6.5 and 6.5a) is generally valid for all types of linear-PMSMs in the sense that it is based on the generalised characteristics of K and λr e f and does not restrict these quantities to be constant. Since the variation or uncertainty in the machine parameters (e.g. L s , Rs , M and B) due to measurement errors or atmospheric factors is bounded [3], therefore, estimation errors  f 1 and  f 2 in f 1 (x1 , x2 ) and f 2 (x1 , x2 ) are bounded by some know functions F1 (x1 , x2 ) and F2 (x1 , x2 ) respectively [3]:  f 1 ≤ F1 (x1 , x2 )

(6.5b)

 f 2 ≤ F2 (x1 , x2 )

(6.5c)

And

In (6.5), the control gain b(FT ) of the control input v y has also known bounds [3]: 0 < bmin ≤ b(FT ) ≤ bmax

(6.5d)

ˆ T ) of gain b(FT ) can be expressed as [3]: The estimated value b(F ˆ T) = b(F



bmin bmax

The above equation can also be expressed as [3]: ˆ

(6.5e)

T) ≤ ζ where, ζ = bbmax ζ −1 ≤ b(F b(FT ) min It is important to note that for constant values of K and λr e f , as in the case of prototype under study, (6.5) becomes a 2nd order linear differential equation. The differential Eq. (6.5) provide the basis to formulate the sliding mode control law in terms of the stator thrust force and mover’s speed as system state and to achieve combined control of the thrust force and the mover’s speed as explained in Sect. 6.4. The dynamic model (6.5) does not inherit integral action, therefore to eliminate the steady state error, the integral action must be provided by the control law. The sliding

158

6 Sliding Mode Based Combined Speed …

mode control for stator flux regulation is formulated using (6.1) such that stator flux λs tracks the reference flux value λr e f as determined by (6.4) under MFPA trajectory.

6.3 Sliding Mode Control The sliding mode or variable structure system is a type of control system that allows the control law or input to change its structure by switching at any instant from one to another member of a set of the possible continuous functions (linear or nonlinear) of the system states [4, 5]. The most attractive feature of the sliding mode control is its robustness to variations and uncertainties in the system parameters. Works of Itkis [4] and Utkin [5] brought sliding mode control into the spotlight in 1970s. Further developments in sliding mode control theory include the concept of equivalent control attributed to Utkin [5–9] and integral sliding mode control reported in [10–12]. Slotine [14–16] further advanced the sliding mode control theory by formulating sliding surfaces to achieve tracking control of a class of non-linear systems and introduced the η-reachability condition. The literature [4–16] (and the references contained therein) provides a systematic mathematical discourse of sliding mode control. However, in this chapter, only an abridged epistemology of sliding mode control will be provided for necessary understanding. The basic formulation of the sliding mode control problem without any exhaustive mathematical details and a review of applications of sliding mode control to electric machines is discussed in the following subsections.

6.3.1 Fundamentals of Sliding Mode Control The mathematical formulation of the sliding mode control problem presented in works by Utkin [4–9] starts by considering the design of sliding mode (variable structure) controller for an nth order single-input-single-output (SISO) linear system with uncertainties in the system parameters as described by (6.6) and (6.7) with the objective of zeroing of the output y = x1 : x˙i = xi+1 , for i = 1, . . . , n − 1 x˙n = −

n

(ai + ai (t))xi + (b0 + b0 (t))u + d(t)

(6.6) (6.7)

i=1

where, u is a scalar control input. d(t) is the disturbance in the system. ai and b0 are the nominal values of the system parameters. ai (t) and b0 (t) are the perturbations that constitute the uncertainty in the parameters, such that, for all t, these uncertainties are bounded [10]:

6.3 Sliding Mode Control

159

ki− < ai (t) < ki+ for i = 1, . . . , n

(6.8)

k0− < b0 (t) < k0+

(6.9)

where, ki− , ki+ , k0− , and k0+ are fixed known scalars. The time varying sliding surface s(X ; t) is formulated in terms of systems state as [4–9]: s(X ; t) =

n

ci xi , ci = ar bitrar y const., cn = 1

(6.10)

i=1

where, X = [x1 , x2 , . . . , xn ]T is the state vector. The variable control law u is selected as a function of systems states that undergoes discontinuities on the plane s(X ; t) = 0 and expressed as: u = us = −

k

Ψi xi − Ψ0 sgn(s), 1 ≤ k ≤ n − 1

(6.11)

⎧  +1, i f s > 0 ⎪ ⎪ ⎨ αi , i f sxi > 0 and sgn(s) = −1, i f s < 0 Ψi = ⎪ , i f sx < 0, where α , β and Ψ0 − const, β i i i ⎪ ⎩ i Ψ0 is small positive scalar.

(6.12)

i=1

where,

In [10], another choice for the variable structure control law known as scaled relay structure is given as: u = u s = −Ψ0 (X ; t)sgn(s)

(6.13)

where, Ψ0 (X ; t) can either be a function of state vector or a positive scalar. The sufficient condition for the existence of the sliding mode can be given as [4–9]: lim s˙ > 0 and lim s˙ < 0

s→−0

s→+0

(6.14)

The concept of equivalent control u eq is also attributed to Utkin [9, 10]. u eq is a state feedback law which computed by solving s˙ = 0 for u and assuming ai (t) and b0 (t) to be zero. Now the control law u comprises two components, i.e. equivalent control u eq and variable structure control u s and given as [9, 10]: u = u eq + u s

(6.15)

160

6 Sliding Mode Based Combined Speed …

where, u s is either selected according to (6.11) or (6.13) according to the nature of controller design problem. The above discussion is also applicable for an nth order multiple-input-multipleoutput (MIMO) nonlinear function described by [4]: x˙n = f (x, t, u)

(6.16)

where, x ∈ R n , u ∈ R m , and f ∈ R n . The control law u in now given in the form [4];  ui =

u i− (x, t), i f si (x) > 0 u i− (x, t), i f si (x) < 0

i = 1, . . . , m.

(6.17)

The control design problem is to compute continuous functions u i− (x, t) and and m-dimensional vector s (s ∈ R m ) with functions si (x) as components. Another advancement to sliding model control is the addition of the integral action [11–13]. The inclusion of integral action is achieved by formulating the sliding surface s(X ; t) as: u i− (x, t)

s(X ; t) =

n

ci xi + z, ci = ar bitrar y const., cn = 1

(6.18)

i=1

where, z is the integral of the tracking error given as: z = ∫(r − x1 )dt

(6.19)

where, r is the commanded reference value and may be selected as zero, if zeroing of the output y = x1 is the control objective as described previously. However, it is important to note that this integral action does not appear in the equivalent control u i and appears only in the variable structure component u s . If the variable structure component u s is the scaled relay type, then the integral action remains inside the “sgn” function and does not appear in the equivalent control law which reduces its effectiveness. Slotine [14–16] further advanced the concept of sliding mode control by formulation of the tracking control of the general SISO nonlinear system described as: x˙i = xi+1 , i = 1, . . . , n − 1 and x1 = x

(6.20)

x˙n = f (X ; t) + g(X ; t)u + d(t)

(6.21)



T where, X = x, x, ˙ . . . , x n−1 is the state vector. In (6.21), u is the control input. The function f (X ; t) (generally nonlinear) is not exactly known, however the imperfection  f in the estimated value fˆ(X ; t) is

6.3 Sliding Mode Control

161

bounded by a known continuous function Fb (X ; t). Similarly, the control gain g(X ; t) is not precisely known but is of constant sign and is bounded by a continuous function G b (X ; t) and the disturbance d(t) is also bounded. The control problem is that the state vector X should track the reference state vector X d with satisfactory tracking precision. The desired state vector X d is given as:

T X d = xd , x˙d , . . . , xdn−1

(6.22)

The tracking error vector between the desired state vector X d and the actual state vector X is X˜ and is: T  ˙˜ . . . , x˜ n−1 ˜ x, X˜ : X d − X = x,

(6.23)

The sliding surface to achieve the tracking is formulated in terms of the tracking error and is given as [14–16]:  sc (X ; t) =

n−1 d +λ xdt ˜ dt

(6.24)

where, λ is a strictly positive constant. The control law u for tracking is given as: u = u eq (X ; t) + ρ0 sgn(sc )

(6.25)

where, ρ0 is a positive constant gain. According to Slotine, u is selected to satisfy the η-reachability condition given as [10, 14–16]: 1 d 2 s (X ; t) ≤ η|sc (X ; t)| 2 dt c

(6.26)

where, η is a strictly positive constant. The equivalent control law u eq (X ; t) is computed by solving sc (X ; t) = 0 for the control input u and assuming no model uncertainty. In [16] an integral sliding surface formulated in terms of the integral of the tracking error is:  sc (X ; t) =

d +λ dt

n ∫ x˜ dt

(6.27)

It is important to note that (6.27) does not add the integral action directly in the equivalent control law u eq (X ; t). However, it can be observed from (6.25) that the sliding surface sc containing the integral term appears inside the discontinuous sgn(·) function in the control law and therefore, continuous pure integrator action is not available directly which leads to a steady-state error. Special steps are required to include a pure integrator action in the equivalent control law u eq (X ; t) to ensure a zero steady-state error as explained in Sect. 6.4.

162

6 Sliding Mode Based Combined Speed …

6.3.2 Variable Structure Based Direct Torque Control The conventional direct thrust force control discussed in Chap. 3 is a variable structure control scheme in nature. The stator flux and thrust force hysteresis controllers of (3.3) and (3.4) are the simplest form of the scaled relay controller of (6.13) with Ψ0 (X ; t) = −1. The inverter voltage vector is selected from a switching table (Table 3.1) based on the output of the stator flux and thrust force hysteresis controllers and the selected voltage vector is then applied to the linear PMSM for duration of a complete sampling period. However, as analyzed in Chap. 3, due to a low value of stator inductance and a short pole-pitch, an inverter voltage vector applied for the full length of the sampling period causes the prototype linear PMSM to exhibit an unacceptably large ripple in the thrust force response. Therefore, the relay type variable structure control produces large thrust force ripple in case of a low inductance linear PMSM despite the guaranteed stability of the controller. Several studies [17–22] extend the application of variable structure control theory discussed previously to the rotational machines based on the principal of direct torque control. In [17], a direct torque control scheme for induction motor based on variable structure control theory of [4, 5] is presented. This control scheme is capable of providing isolated direct torque or combined speed and direct torque control depending on the nature of the particular problem being considered. In this control scheme, the inverter voltage vector is selected based on the output of three relay type variable structure controllers of form (6.13) and is applied to the induction motor for the duration of whole sampling period. However, this control scheme is susceptible to unavoidable large ripple as demonstrated by the experimental results [17]. In [18–21], a variable structure torque control strategy for rotational PMSM based on the concept of vector control is formulated in dq-reference frame. The tracking error in the d-axis current and the torque are selected as the sliding surfaces such that the reference value of the d-axis current is zero and torque reference is selected by the outer speed control loop. It is important to note that although the torque tracking error is used as one of the sliding surfaces, the torque is not considered as a system state. In this research, the d and q axes variable structure control laws are scaled relay structures of type (6.13) with the sliding surfaces as inputs. The outputs of these controllers are then used to select the inverter voltage vector from a switching table which is then applied to the machine for a whole sampling period. It is important to note that all the variable structure control schemes discussed in [17–22] are based on scaled relay type controller of form (6.13). These control schemes are not suitable for linear PMSMs with low stator inductance and short pole pitch because of the fact that the selected inverter voltage vector is applied for full duration of the sampling period resulting in a large steady state ripple in the thrust force response as explained in Chap. 3. A variable structure (sliding mode) based direct torque control utilizing SV-PWM which reduces the ripple in torque is reported in [22] for rotational PMSMs and is extended to linear PMSM as DTFC in [23]. However, it should be noted that the

6.3 Sliding Mode Control

163

sliding mode control methods of [22, 23] do not account for the stator flux and torque (thrust force in case of linear PMSM) as direct system states, resulting in a possibly slower transient response for both the stator flux and torque/thrust force.

6.4 Sliding Mode Control with Augmented Integral Action The dynamic system of (6.20) and (6.21) can be further generalized by considering the fact that all the state derivatives can be expressed as nonlinear functions of two or more states instead of only one state, therefore (6.20) and (6.21) becomes: x˙i = f i (X ; t), i = 1, . . . , n − 1

(6.28)

x˙n = f n (X ; t) + g(X ; t)u + d(t)

(6.29)

The compact form of (6.28) and (6.29) is given as: X˙ = F(X ) + G(X )u + D(t) ⎡ ⎢ ⎢ F(X ) = ⎢ ⎣

f 1 (X ; t) f 2 (X ; t) .. .

⎡

⎤

⎢ ⎥ ⎢ ⎥ ⎥, G(X ) = ⎢ ⎣ ⎦

0 0 .. .

⎤

⎡

⎢ ⎥ ⎢ ⎥ ⎥, and D(t) = ⎢ ⎣ ⎦

g(X ; t)

f n (X ; t)

(6.30) 0 0 .. .

⎤ ⎥ ⎥ ⎥ ⎦

d(t)

It is important to note that (6.30) can be validly used to describe the dynamic model given in (6.5) of the prototype linear-PMSM. The sliding manifold sc (X ; t) for the dynamics of (6.29) is a function of the tracking error [12–14]:  sc (X ; t) =

d +λ dt

n−1 e(t)dt

(6.31)

where, e(t) = x − xd is the tracking error between the reference and actual values of system states, n is the order of the system and λ is a positive constant. A first order system has n = 1 [14–16]. In order to modify the η-reachability condition so that the pure integral action can be augmented in the control law, the following Lyapunov candidate function is defined [24, 25]. ⎞2 ⎛ t  1 2 1 2⎝ sc dt ⎠ V = sc + ωn 2 2 0

(6.32)

164

6 Sliding Mode Based Combined Speed …

By applying the Lyapunov stability criterion for stability, V˙ ≤ 0, the sliding condition is obtained as ⎛ ⎞ t sc .⎝s˙c + ωn2 sc dt ⎠ ≤ 0 (6.33) 0

In other words, s˙c + ωn2

t

sc dt ≥ 0, when sc < 0

0

and s˙c + ωn2

t

sc dt < 0, when sc ≥ 0

0

Now the modified the η-reachability condition can be expressed as: ⎛ sc ⎝s˙c + ωn2

t

⎞ sc dt ⎠ ≤ −η|sc |

(6.34)

0

where η is a positive constant. The equivalent control law u eq (X ; t) is now evaluated t by solving s˙c + ωn2 0 sc dt = 0 for the control input u [25]. In (6.34) ωn is called as the equivalent spring constant [25], the integral of the sliding surface sc in (6.34) provides and additional restoring effort and the dynamic behavior of the effect of the integral action can be tuned by ωn . It is important to note that the modified ηreachability condition adds an integral action directly into the equivalent control law u eq (X ; t). The damping of the system response is controlled by η and a higher value of η results in less oscillation in the system response; whereas a low value of η causes a lightly damped oscillation in the system response such that the frequency of the oscillation is determined by ωn [24, 25]. A detailed tuning mechanism for η and ωn is reported in [25].

6.4.1 Sliding Surface for Stator Flux Regulation The sliding surface sλ for the stator flux regulation dynamics of (6.1) can be given from (6.31) for n = 1 as: sλ = eλ where, eλ = λr e f − λs is the flux tracking error.

(6.35)

6.4 Sliding Mode Control with Augmented Integral Action

165

6.4.2 Sliding Surface for Speed Regulation The sliding surface sv for the combined speed and thrust force dynamics of (6.2) is computed from (6.31) for n = 2 as: devm (t) + λ1 evm (t) dt

sv =

(6.36)

where, evm = vr e f − vm is the speed tracking error, vr e f is the reference speed and λ1 is a strictly positive constant.

6.4.3 Control Law for Stator Flux Regulation Differentiating (6.35): " ! s˙λ = λ˙ r e f − λ˙ s

(6.37)

Since, λ˙ r e f = 0, and from (6.1), (6.37) becomes: s˙λ = Rs i x − vx

(6.38)

The control law for the stator flux to ensure s˙λ + ωn2

t 0 sλ dt

vx = vx∗ + η1 sgn(sλ )

= 0, is: (6.39)

where, η1 is a positive constant, and vx∗ = Rs i x + ωn2 0 sλ dt The control law (6.39) satisfies the sliding condition for global stability [25]: t

⎛ sλ ⎝s˙λ + ω12

t

⎞ sλ dt ⎠ ≤ −η1 |sλ |

(6.40)

0

In order to reduce chattering, “sgn(sλ )” in (6.39) is replaced by “sat(sλ /ε1 )”, with ε1 as boundary layer around sliding surface sλ = 0, (as explained in Sect. 6.4.5) yields: t v x = Rs i x +

ωn2

sλ dt + η1 sat(sλ /ε1 ) 0

(6.41)

166

6 Sliding Mode Based Combined Speed …

6.4.4 Control Law for Combined Speed and Thrust Force Regulation Taking the time derivative of (6.36): " ! " ! s˙v = v¨r e f − v¨m + λ1 v˙r e f − v˙m

(6.42)

The value of v¨m can be obtained from (6.2) as: v¨m =

1 ˙ B 1 ˙ FT − v˙m − FL M M M

(6.43)

In general, the load force FL is constant during steady-state and does not change abruptly, therefore, F˙L = 0, [26, 27] and (6.43) becomes: v¨m =

1 ˙ B FT − v˙m M M

(6.44)

Substituting the value of F˙T and v˙m from (6.2) into (6.44): v¨m =

1 M



   −K π K B 1 B 1 FT − vm − FL FT − K P vm + vy − ξ λr e f τ λr e f M M M M (6.45)

The reference speed vr e f is a constant during steady-state and, therefore: v¨r e f = v˙r e f = 0

(6.46)

Substituting (6.46) and the value of v˙m and v¨m from (6.2) and (6.45) respectively into (6.42) and re-arranging: s˙v = α FT + βvm + μFL + γ v y

(6.47)

where, # α=

$ # $   K B B λ −K B2 B π λ1 + 2 − 1 ,β = K P + 2 + λ1 − 2 , and γ = ,μ = Mξ λr e f M Mτ M M Mλr e f M M M

The equivalent control law v ∗y to achieve s˙v + ω22 v ∗y

t 0 sv dt

= 0 is defined from [25]:

  t 1 2 ˆ α FT + βvm + ω2 ∫ sv dt + μ FL = γ 0

(6.48)

The load force FL is usually an unknown quantity, therefore, in (6.48) the estimated value of load force FˆL is used. In order to achieve the sliding mode condition, the

6.4 Sliding Mode Control with Augmented Integral Action

167

discontinuous control sgn(sv ) is included in the equivalent control law of (6.47) as [14–16, 25]: v y = v ∗y +

η2 sgn(sv ) γ

(6.49)

where, γ, η2 are strictly positive constants. In (6.49), sgn is the sign function is defined as:  +1, i f sv > 0 sgn(sv ) = (6.50) −1, i f sv < 0 In order to rationalize and establish the stability of the controller of (6.49), the following Lyapunov candidate function [14] is defined: ⎞2 ⎛ t  1 2 1 2⎝ 1 ˜2 V = sv + ω2 sv dt ⎠ + F 2 2 2γ1 L

(6.51)

0

where, F˜L = FL − FˆL , γ1 > 0, and differentiating (6.51): ⎛ V˙ = sv ⎝s˙v + ω22

t

⎞ sv dt ⎠ −

0

1 ˙ˆ ˜ FL FL γ1

(6.52)

Substituting (6.47) into (6.52): ⎛ V˙ = sv ⎝α FT + βvm + μ FL − γ v y + ω22

t

⎞ sv dt ⎠ −

0

1 ˙ˆ ˜ FL FL γ1

(6.53)

Substituting (6.48) and (6.49) into (6.53), and simplifying: % & 1 V˙ = sv −η2 sgn(sv ) + μ F˜L − F˙ˆL F˜L γ1

(6.54)

Since, sv × sgn(sv ) = |sv |, therefore, (6.54) becomes:    1 V˙ = −η2 |sv | + F˜L μ sv − Fˆ˙L γ1

(6.55)

To ensure global stability of the controller, Lyapunov stability criteria suggests V˙ ≤ 0 and requires the following expression to estimate the load force FˆL :

168

6 Sliding Mode Based Combined Speed …

FˆL = γ1 μ

t sv dt

(6.56)

V˙ = −η2 |sv | ≤ 0

(6.57)

0

Hence, (6.55) reduces to:

Since, η2 is positive constants, therefore V˙ is strictly negative when the load force is estimated using (6.56). Now the substitution of (6.56) into (6.48) results: ⎞

⎛ ⎜ t ⎜ 1 ⎜ ∗ 2 v y = ⎜α FT + βvm + ω2 sv dt + γ⎜ ⎝ 0

⎟ ⎟ ⎟ γ1 μ2 sv dt ⎟ ⎟ ⎠ 0 ( )* + t

(6.58)

distur bance r ejection

In (6.58) ω2 determines the natural frequency of the damped oscillations in the speed response and can be used to tune the integral action [25]. It is important to note that the underlined term in (6.58) is the estimated load force according to (6.56) and acts as disturbance rejection which cancels out the effect of load disturbance in dynamic model given by (6.5). The two integral terms in (6.58) can be combined into a single term that provides the integral action as wells as disturbance rejection at the same time, Re-arranging terms in (6.58) and from (6.49) results in control law v y as: ⎞ ⎛ t " ! 2 1⎝ sv dt + η2 sgn(sv )⎠ vy = α FT + βvm + ω2 + γ1 μ2 γ

(6.59)

0

According to Sect. 6.4.5, in order to reduce chattering, “sgn(sv )” is replaced by “sat(sv /ε2 )”, (ε2 as boundary layer around sliding surface sv = 0), (6.59) becomes: ⎞ ⎛ t " ! 2 1⎝ vy = sv dt + η2 sat(sv /ε2 )⎠ α FT + βvm + ω2 + γ1 μ2 γ

(6.60)

0

The sliding mode control law with augmented integral action (6.60) to achieve combines speed and direct thrust control of dynamic system (6.5 and 6.5a) and (6.29) is illustrated by the schematic diagram provided in Fig. 6.1.

6.4 Sliding Mode Control with Augmented Integral Action

169

Sliding Mode Control Law vy for Speed Tracking According to (6.59) Combined Speed and Thrust Force Dynamics for Linear PMSM According to (6.5). (6.5-a) and (6.29)

Sliding Surface sv for Speed Tracking According to (6.35)

vref +-

ev m

λ1 Σ d dt

sv

D (t )

η2 sat ( s v /ε ) γ

Σ (ω 22 + γ1 μ2 )∫

u =v y

++-

∫

1

Σ

γ

Integral Action Including Disturbance Rejection (6.58)

T

H(X)

vm

F(X)

*

vy

X = ⎣⎡ FT vm ⎦⎤

G(X)

α FT + β vm

⎡⎣α

β ⎤⎦

Fig. 6.1 Schematics describing the sliding mode control with first order plus integral sliding condition for non-linear combined speed and thrust force dynamics proposed in (6.5) and (6.5a)

6.4.5 Chattering Reduction It is important to note that function “sgn(sc )” for a sliding manifold sc (x; t) = 0, introduces chattering in the tracking response. which can be reduced by introducing a boundary layer of thickness ε around the sliding manifold sc (x; t) = 0. This is achieved by replacing the “sgn” function with “sat” function defined as [3], [14–16]: sat(sc /ε) =

 sc

i f |sc | ≤ 0 ε sgn(sc /ε) i f |sc | > 0

(6.61)

6.5 Experimental Validation of Proposed SM-DTFC1 The proposed SM-DTFC1 control scheme has been practically validated on the prototype surface-mount linear PMSM control system. The block diagram of SMDTFC1 is shown in Fig. 6.2. The main hardware components of the experimental setup are illustrated in Fig. 1.1. The parameters of the surface-mount linear PMSM are provided in Table 1.1. Experimental results indicate improved performance of the SM-DTFC1 method in terms of steady-state error and transient response of the stator flux, thrust force and speed when compared with the prior PI-DTFC method, Chap. 4. The tuning process and selection of the gains for the PI-DTFC was described in Chap. 4.

170

6 Sliding Mode Based Combined Speed …

Fig. 6.2 Proposed sliding mode based combined speed and thrust control of linear PMSM (SMDTFC), the integral action is added by modification of the reachability condition

6.5.1 Start-Up Speed Response The start-up speed responses of the surface-mount linear PMSM under both the PIDTFC and the SM-DTFC1 are compared. The speed, flux, thrust force and stator currents for both the PI-DTFC and the SM-DTFC1 during the start-up transient are shown in Fig. 6.3a, b respectively. It is clear from Fig. 6.3a that during the start-up transient under PI-DTFC, the speed response exhibits a speed dip immediately after reaching the speed reference of 200 mm/s. However, under SM-DTFC1 the speed dip is noticeably reduced. Figure 6.3 clearly shows that the thrust force under SM-DTFC1 settles to steady state faster compared to PI-DTFC. The magnified views of the start-up speed response for both PI-DTFC and the SM-DTFC are shown in Fig. 6.4a, b respectively. It is clear from Fig. 6.4a, b that the speed response under the SM-DTFC is 15% faster compared to that of PI-DTFC. The error plots for speed response during the start-up transient for both the PIDTFC and the SM-DTFC1 are shown in Fig. 6.5a, b respectively. It is evident that the SM-DTFC1 results in faster convergence of the speed error to zero compared to PI-DTFC. In addition, for quantitative analysis, the integral of absolute error (IAE) indices for the speed error plots for both PI-DTFC and SM-DTFC1 during the start-up are computed, Table 6.1. The IAE index for the speed error plot under the SM-DTFC1 is reduced by 20% compared to that of the PI-DTFC. The magnified views of the thrust force response during start-up for both the PI-DTFC and SM-DTFC1 are shown in Fig. 6.6a, b respectively. It is evident from Figs. 6.4 and 6.5 that when the speed command jumps from 0 to 200 mm/s at 0.975 s, the corresponding thrust force under the SM-DTFC1 reaches a peak value of 160 N whereas the thrust force under the PI-DTFC peaks at 139 N, the additional force produced results in the faster speed response for the SM-DTFC1. Moreover, it is also evident from Fig. 6.6b that the thrust force under the SM-DTFC1 settles to steady state 4.4 ms faster than that of PI-DTFC. It can be concluded from these experimental results that the combined control of the speed and thrust force achieved under the

6.5 Experimental Validation of Proposed SM-DTFC1

PI-DTFC

(b)

Reference

1.2

300 200 100 0 -100 0.95

1

1.05

1.1

1.15

1.2

1.2

160 120 80 40 0 -40 0.95

1

1.05

1.1

1.15

1.2

Measured 1

1.1

1.05

Estimated

1.05

1.1

Flux (Wb)

Current (A)

1.15

Estimated

1

Reference 1.1 1.15 1.2

1.05

Ic (Green)

1.5 0 -1.5

Ib (Blue)

Ia (Red) 1

1.05 1.1 Time (s)

SM-DTFC1 Reference

Measured

0.086

1.15

Flux (Wb)

1

3.5

-3.5 0.95

Force (N)

Reference

0.086 0.0855 0.085 0.0845 0.084 0.95

1.15

Reference

0.0855 0.085 0.0845 0.084 0.95

Current (A)

160 120 80 40 0 -40 0.95

Force (N)

Speed (mm/s)

300 200 100 0 -100 0.95

Speed (mm/s)

(a)

171

1.1

1.05

1

3.5

1.15

1.2

Ia (Red) Ib (Blue)

1.5 0 -1.5 -3.5 0.95

1.2

Estimated

Ic (Green) 1

1.05 1.1 Time (s)

1.15

1.2

Fig. 6.3 Start-up performance from 0 to 200 mm/s. Speed, thrust force, stator flux, and stator phase currents responses are shown from top to bottom respectively for both a the PI-DTFC and b SM-DTFC1 (experiment)

PI-DTFC

300

200 100 0 -100 0.95

(b)

Reference Measured 35.2 ms 0.975

1

Time (s)

1.025

1.05

Speed (mm/s)

Speed (mm/s)

(a)

300

SM-DTFC1 Reference

200 100 0 -100 0.95

Measured 30 ms 0.975

1

1.025

1.05

Time (s)

Fig. 6.4 Magnified view of the speed response during start-up, a PI-DTFC, and b SM-DTFC1 (experiment)

SM-DTFC1 is capable of delivering faster speed response during start-up compared to PI-DTFC. The stator flux response for both the PI-DTFC and the SM-DTFC1 is shown in Fig. 6.3a, b respectively. It is clear from Fig. 6.3 that under SM-DTFC1 the average steady-state ripple in the stator flux response after settling to steady-state is reduced by 20% compared to that of PI-DTFC.

172

6 Sliding Mode Based Combined Speed …

PI-DTFC

250 200 150 100 50 0 -50 0.97 0.98 0.99

(b) Speed (mm/s)

Speed error (mm/s)

(a)

1

1.01 1.02

SM-DTFC1

250 200 150 100 50 0 -50 0.97 0.98 0.99

Time (s)

1

1.01 1.02

Time (s)

Fig. 6.5 Magnified view of the speed error during start-up, a PI-DTFC, and b SM-DTFC1 (experiment)

Table 6.1 Comparison of transient performance of PI-DTFC and the SM-DTFC1 using IAE index

Type of transient phenomena

IAE index for speed error PI-DTFC

Start-up (0 to 200 mm/s) Speed reversal (−600 to 600 mm/s)

PI-DTFC

(b)

Reference Estimated 61 ms 1

Time (s)

1.025

1.05

Force (N)

Force (N)

(a)

160 120 80 139 N 40 0 -40 0.95 0.975

160 120 80 40 0 -40 0.95

SM-DTFC1

23,721

18,975

210,430

188,775

SM-DTFC1 160 N

0.975

56.6 ms 1 1.025

1.05

Time (s)

Fig. 6.6 Magnified thrust force response during start-up, a PI-DTFC, and b SM-DTFC1 (experiment)

6.5.2 Speed Reversal and Steady-State Response The speed reversal and steady state performance of the linear PMSM under both PI-DTFC and SM-DTFC1 are compared. The speed, stator flux, thrust force and stator currents for both the PI-DTFC and the SM-DTFC1 during the speed reversal transient from −600 to 600 mm/s and steady state at 600 mm/s are shown in Fig. 6.7a, b respectively. Figure 6.7a illustrates that the dip in speed response after the speed is reversed from −600 mm/s and reaches 600 mm/s is reduced under the SM-DTFC1. Moreover, a reduction in thrust force oscillations under SM-DTFC1 can also be observed from Fig. 6.7a, b when the speed settles at 600 mm/s. The magnified views of the speed response of linear PMSM under PI-DTFC and the SM-DTFC1 during the speed reversal transient are compared in Fig. 6.8a, b

PI-DTFC

600 0

Reference

-600

Measured 0.8

0.9

1

Force (N)

1.2

0 -200 0.7

Estimated 0.8

1.1

1.2

0.084

0.7

1

1.1

1.2

1.3

0

0.8

0.9

1

1.1

1.1

1.2

1.3

0.8

1.2

1.3

0.9

1

1.1

1.2

1.3

Reference

0.085 0.084 0.083 0.7

5 3

-3 -5 0.7

1

Estimated

0.086

Current(A)

Current(A)

Estimated 0.9

0.9

0 -100

0.087

Reference

0.8

Measured 0.8

100

1.3

0.085

0.083 0.7

0.7

-

0.086

Reference

-600

-200 1

0.9

0.087

0

200

100 -100

SM-DTFC1

600

1.3

Reference

200

Flux (Wb)

1.1

Force (N)

0.7

173

(b) Speed (mm/s)

(a)

Flux (Wb)

Speed (mm/s)

6.5 Experimental Validation of Proposed SM-DTFC1

Estimated 0.8

0.9

1

1.1

1.2

1.3

0.8

0.9

1

1.1

1.2

1.3

5 3 0 -3 -5 0.7

Time(s)

Time (s)

PI-DTFC

(a)

Reference

600

Measured

0 66.8 ms

-600 0.76

0.8

0.84 0.88 0.92 0.96

Time (s)

Speed (mm/s)

Speed (mm/s)

Fig. 6.7 Speed reversal from −600 to 600 mm/s and steady state response at 600 mm/s. Speed, force, flux, and stator phase “a” current responses are shown, a PI-DTFC, and b SM-DTFC1 (experiment)

SM-DTFC1

(b)

Reference

600

Measured

0 -600 0.76

0.8

59.9 ms 0.84 0.88 0.92 0.96

Time (s)

Fig. 6.8 Magnified view of the speed reversal transient, a PI-DTFC, and b SM-DTFC1 (experiment)

174

6 Sliding Mode Based Combined Speed …

1200 1000 800 600 400 200 0 -200 0.76

(b)

PI-DTFC

Speed (mm/s)

Speed error (mm/s)

(a)

0.8

0.84

0.88 0.92 0.96

SM-DTFC1

1200 1000 800 600 400 200 0 -200 0.76

0.8

Time (s)

0.84 0.88 0.92 0.96

Time (s)

Fig. 6.9 Error plots for the speed response during the speed reversal transient, a PI-DTFC, and b SM-DTFC1 (experiment)

respectively. It is evident from Fig. 6.8 that the speed response under SM-DTFC1 is 10% faster when compared to that of PI-DTFC. The speed error plots during the speed reversal transients for both PI-DTFC and SM-DTFC1 are shown in Fig. 6.9a, b respectively. It is clear from Fig. 6.9 that under SM-DTFC1 the speed error converges to zero at a higher rate providing a faster speed response during the speed reversal compared to PI-DTFC. It can be observed from Table 6.1 that IAE index for the speed error during speed reversal is reduced by 10% for the SM-DTFC1. The steady state performance of the prototype linear PMSM at 600 mm/s from 0.95 to 1.15 s under both PI-DTFC and SM-DTFC1 is also compared. In order to illustrate the comparison of steady state performances of the two control schemes, the magnified view of the speed response, flux response and thrust force response, for PI-DTFC and SM-DTFC1 during the steady state are shown in Fig. 6.10a, b respectively. Figure 6.10 demonstrates that under SM-DTFC1 the steady-state low frequency oscillations in speed, thrust force and stator flux have significantly reduced compared to PI-DTFC. The steady state speed response under PI-DTFC exhibits low frequency oscillations due to complex friction dynamics of the linear PMSM. These steady speed oscillations under PI-DTFC also cause oscillations in the force response. The oscillations are not as significant with SM-DTFC1. The quantitative results for steady state performance of PI-DTFC and the SMDTFC1 at 600 mm/s and 52 N (average force) in terms of percent flux ripple λrip (%), percent force ripple F rip (%), and percent speed ripple vri p (%) are summarized in Table 6.2. In this analysis λrip (%), F rip (%) and vri p (%) are given by (6.62), (6.63) and (6.64) respectively, λri p (%) = Fri p (%) =

1 N

-N

i=1 (λs (i)

− λav )2

λav 1 N

-N

i=1 (FT (i)

Fav

− Fav )2

× 100

(6.62)

× 100

(6.63)

Speed (mm/s)

600 Measured

500 1

1.05

1.1 Reference

100 50 0

-50 0.95 0.087

Estimated

0.085 0.084 0.083 0.95

Estimated 1.1

1.05

1

700

Measured

600 500

Reference 1

1.1

1.15

1.1

1.15

1.1

1.15

50 0

Estimated

-50 0.95

1

1.05 Reference

0.086 0.085 0.084

Estimated

0.083 0.95

1.15

1.05

100

0.087

Reference

0.086

SM-DTFC1

(b)

1.15

1.1

1.05

1

175

0.95 150

1.15

Force (N)

Force (N)

Reference

700

0.95 150

Flux (Wb)

PI-DTFC

(a)

Flux (Wb)

Speed (mm/s)

6.5 Experimental Validation of Proposed SM-DTFC1

1

Time (s)

1.05

Time (s)

Fig. 6.10 Steady state performance at 600 mm/s. Speed, thrust force, and stator flux responses are shown, a PI-DTFC, and b SM-DTFC1 (experiment)

Table 6.2 Comparison of steady state performance of PI-DTC and the SM-DTFC1 at 600 mm/s, 52 N

Quantity (%)

PI-DTFC

SM-DTFC1

λrip

0.34

0.19

F rip

10.48

5.79

vrip

1.92

1.08

vri p (%) =

1 N

-N

i=1 (vm (i)

vav

− vav )2

× 100

(6.64)

where, λav , Fav and vav represent the average steady state flux, thrust force and mover’s speed respectively and λs (i), FT (i) and vm (i) are the instantaneous values of flux, thrust force and mover’s speed. It can be observed from Table 6.2 that the SM-DTFC1 reduces the percentage ripple in steady-state speed, stator flux and thrust force response compared to PIDTFC which validate the superior steady-state performance of the proposed SMDTFC1.

176

6 Sliding Mode Based Combined Speed …

6.5.3 Evaluation of Robustness to the Parameter Variation The performance of the proposed SM-DTFC1 subject to variations in stator resistance Rs and inductance L s is experimentally evaluated and compared to that of PI-DTFC and Optimal-DTFC2. Experimental results summarizing the comparison of the performance for these control schemes subject to parameter variation under various operating conditions are provided in Tables 6.3, 6.4 and 6.5. The effect of increase in the stator resistance on the rise time of speed response during start up and speed reversal transient for the three control schemes is compared in Table 6.3. In order to simulate experimentally the effect of 50 and 100% increase in the stator resistance, the controller gains for PI-DTFC, Optimal-DTC2 and the proposed SM-DTFC1 were tuned using the reduced values of 0.66 Rs and 0.5 Rs respectively for the stator resistance instead of the nominal value of Rs . An identical approach is adopted to experimentally evaluate the effect 50 and 100% increase in the stator inductance L s on the rise time of speed response for the three control schemes during start-up and speed reversals and results are detailed in Table 6.4. Table 6.3 Comparison of rise time of PI-DTFC, optimal-DTFC2, and SM-DTFC1 with variation in Rs Type of transient phenomena Start up 0 to 200 mm/s

Start up −600 to 600 mm/s

Parameter value

Rise time (s) PI-DTFC

Optimal-DTFC2

SM-DTFC1

Rs (nominal)

35.2

31.0

30.0

0.66 Rs

38

33.2

31.1

0.5 Rs

41

35.4

33.7

Rs (nominal)

66.8

61.0

59.9

0.66 Rs

69.1

65.8

63.9

0.5 Rs

72.6

70.0

67.3

Table 6.4 Comparison of rise time of PI-DTFC, optimal-DTFC2, and SM-DTFC1 with variation in L s Type of transient phenomena

Parameter value

Rise time (s) PI-DTFC

Optimal-DTFC2

SM-DTFC1

Start up 0 to 200 mm/s

L s (nominal)

35.2

31.0

30.0

0.66 L s

40.5

34.7

32.7

0.5 L s

44.1

37.9

35.3

L s (nominal)

66.8

61.0

59.9

0.66 L s

73.7

70.1

67.6

0.5 L s

82.4

79.2

73.0

Start up −600 to 600 mm/s

6.5 Experimental Validation of Proposed SM-DTFC1

177

Table 6.5 Comparison of steady-state performance of PI-DTFC, Optimal-DTFC2, and SM-DTFC1 with variation in L s 600 mm/s, 52 N (%)

Parameter value

Rise time (s) PI-DTFC

Optimal-DTFC2

SM-DTFC1

F rip

L s (nominal)

10.48

5.83

5.79

1.5 L s

12.3

6.01

5.85

2 Ls

14.8

6.62

5.98

vrip

L s (nominal)

1.92

1.11

1.08

1.5 L s

2.17

1.23

1.10

2 Ls

2.41

1.59

1.17

It can be observed from Table 6.3 that the rise time of the speed response during start up and speed reversal increases for PI-DTFC, Optimal-DTFC1 and the SMDTFC1, when the gains for these control schemes are tuned using reduced values of stator resistance compared to the nominal value. However, Table 6.3 demonstrates that SM-DTFC1 has the smallest variation in the rise time of speed response during both the start-up and reversal operations compared to the other two control schemes. Therefore, a superior robustness performance of SM-DTFC1 under variation in the stator resistance is evident when compared to that of PI-DTFC and Optimal DTFC2. It is also clear from Table 6.4, that SM-DTFC1 shows a superior robustness considering smaller variation in the rise time of the speed response compared to the other two benchmarked control schemes during start up and reversal when an increase of 50 and 100% in the stator inductance L s is experimentally simulated. The effect of variation in inductance on the steady state speed and thrust for response is also evaluated experimentally for the three control schemes and compared in Table 4.5. When the controller gains were tuned using a higher value of stator inductance, i.e. 1.5L s and 2L s , the steady state ripple in speed and thrust force response increased significantly for PI-DTFC and Optimal-DTFC2. However, SMDTFC1 demonstrates a minimal deterioration of the steady state speed and thrust force response in terms of percent ripple when compared to the benchmarks. Therefore, it can be concluded from the experimental results of Tables 6.3, 6.4 and 6.5 that SM-DTFC1 shows improved robustness under parameter variation when compared to PI-DTFC and Optimal DTFC2.

6.6 Conclusions In this chapter, a combined speed and direct thrust force control scheme is proposed based on sliding mode control utilizing space vector modulation for the linear PMSM. In order to eliminate the average steady state tracking error, pure integral action is augmented directly into the control laws by using a modified reachability condition.

178

6 Sliding Mode Based Combined Speed …

The combined dynamic of the mover’s speed and thrust force are expressed by a second order nonlinear SISO system considering the speed and thrust force as states. The sliding surface is formulated in terms of the tracking error in the speed. The formulation of the sliding mode control is performed using a modified reachability condition which allows direct inclusion of pure integral action in the control law to ensure zero average steady-state tracking error in thrust force. Similar approach is utilized to formulate the sliding mode control law to achieve the tracking of the stator flux under maximum force per ampere trajectory. Experimental results clearly indicate that the proposed SM-DTFC1 scheme exhibits excellent control of flux and thrust force with faster transient response, reduced steady-state oscillations and when compared to the PI-DTFC. SM-DTC1 also exhibit robustness to parameter variation during transient and steady state performance when compared to PI-DTFC and Optimal-DTFC2.

References 1. Z. Jun, X. Zhuang, T. Lixin, M.F. Rahman, A novel direct load angle control for interior permanent magnet synchronous machine drives with space vector modulation, in Proceedings of the International Power Electronics and Drives Systems (PEDS) (2005), pp. 607–611 2. Y. Inoue, S. Morimoto, M. Sanada, Examination and linearization of torque control system for direct torque controlled IPMSM. IEEE Trans. Ind. Appl. 46, 159–166 (2010) 3. T. Sharaf-Eldin, M.W. Dunnigan, J.E. Fletcher, B.W. Williams, Nonlinear robust control of a vector-controlled synchronous reluctance machine. IEEE Trans. Power Electron. 14, 1111– 1121 (1999) 4. U. Itkis, Control systems of variable structure (Wiley, New York, 1976) 5. V. Utkin, Variable structure systems with sliding modes. IEEE Trans. Autom. Control 22, 212–222 (1977) 6. V. Utkin, Equations of sliding mode in discontinuous systems. Automat. Remote Contr. I(12), pp. 1897–1907; II(2), pp. 211–219 (1972) 7. V. Utkin, Sliding mode control design principles and applications to electric drives. IEEE Trans. Ind. Electron 40, 23–36 (1993) 8. V. Utkin, J. Guldner, J. Shi, Sliding mode control in electro-mechanical systems (Taylor and Francis Group, LLC., New York, 2009) 9. V. Utkin, Sliding modes in control optimization (Springer, Berlin, 1992) 10. C. Edwards, S.K. Spurgeon, Sliding mode control: theory and applications (Taylor and Francis Group, LLC., New York, 1998) 11. T.L. Chern, Y.C. Wu, Design of integral variable structure controller and application to electrohydraulic velocity servosystems, in IEE Proceedings D Control Theory and Applications, vol. 138 (1991), pp. 439–444 12. T.L. Chern, Y.C. Wu, Design of brushless DC position servo systems using integral variable structure approach, in IEE Proceedings B Electric Power Applications, vol. 140 (1993), pp. 27– 34 13. V. Utkin, S. Jingxin, Integral sliding mode in systems operating under uncertainty conditions, in Proceedings of the IEEE Decision and Control, vol. 4 (1996), pp. 4591–4596 14. J.J.E. Slotine, S.S. Sastry, Tracking control of nonlinear systems using sliding surfaces, with application to robot manipulator. Int. J. Contr. 38(2), 465–492 (1983) 15. J.J.E. Slotine, Sliding controller design for nonlinear systems. Int. J. Contr. 40(2), 421–434 (1984)

References

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16. J.J.E. Slotine, Applied nonlinear control (Prentice Hall Inc, New Jersey, 1991) 17. A. Sabanovic, D.B. Izosimov, Application of sliding modes to induction motor control. IEEE Trans. Ind. Appl. IA(17), 41–49 (1981) 18. T.S. Low, K.J.K. Tseng, T.H. Lee, K.W. Lim, K.S. Lock, Strategy for the instantaneous torque control of permanent-magnet brushless DC drives, in IEE Proceedings B Electric Power Applications, vol. 137 (1990), pp. 352–363 19. T.S. Low, T.H. Lee, K.S. Lock, K.J. Tseng, DSP-based instantaneous torque control in permanent magnet brushless DC drives. Mechatronics 1, 203–229 (1991) 20. T.S. Low, K.W. Lim, M.F. Rahman, K.J. Binns, Comparison of two control strategies in development of high-torque electronically commutated drive, in IEE Proceedings B Electric Power Applications, vol. 139 (1992), pp. 26–36 21. L. Teck-Seng, L. Tong-Heng, K.J. Tseng, K.S. Lock, Servo performance of a BLDC drive with instantaneous torque control. IEEE Trans. Ind. Appl. 28, 455–462 (1992) 22. X. Zhuang, M. Faz Rahman, Direct torque and flux regulation of an IPM synchronous motor drive using variable structure control approach. IEEE Trans. Power Electron. 22, 2487–2498 (2007) 23. Y. Junyou, H. Guofeng, C. Jiefan, Analysis of PMLSM direct thrust control system based on sliding mode variable structure, in Proceedings of the Power Electronics and Motion Control Conference (IPEMC), vol. 1(5) (2006), pp. 1–1639 24. L.W. Chang, A MIMO sliding control with a first-order plus integral sliding condition. Automatica 27, 853–858 (1991) 25. L.W. Chang, Dynamics of a sliding control with a first-order plus integral sliding condition. Dyn. Control 2, 201–219 (1992) 26. K. Ohishi, M. Nakao, K. Ohnishi, K. Miyachi, Microprocessor-controlled DC motor for loadinsensitive position servo system. IEEE Trans. Ind. Electron. 34, 44–49 (1987) 27. J.S. Ko, J.H. Lee, S.K. Chung, Y. Myung Joong, A robust digital position control of brushless DC motor with dead beat load torque observer. IEEE Trans. Ind. Electron. 40, 512–520 (1993)

Chapter 7

Sensorless Control of a Linear Permanent Magnet Synchronous Motors Using a Combined Sliding Mode Adaptive Observer

7.1 Introduction In this chapter, a combined sliding mode adaptive observer for flux and speed estimation of the direct thrust controlled, surface-mount linear PMSM without using position sensors is proposed. The observer comprises a linear state observer combined with a modified nonlinear sliding mode component. The sliding mode component is improved by using two boundary layers which reduces the chattering without compromising robustness. The dynamic model of the linear PMSM, having stator flux, thrust force, and mover’s speed as state variables, has already been formulated in Chap. 6. This model is used here to synthesize an integral sliding mode control law to achieve combined control of the thrust force and mover’s speed to be referred as SM-DTFC2. The proposed observer and the control law are validated for a laboratory porotype. Experimental results show that the flux, speed, and position estimation errors are reasonably small resulting in reliable performance of the proposed observer and controller when compared to PI-DTFC scheme of Chap. 4. The application of the DTC concept to the linear PMSM, termed Direct Thrust Force Control (DTFC), is detailed in [1–6] and mostly deals with implementation of DTFC as an extension of the switching-table-based DTC of the linear PMSM and does not provide rigorous details of stator flux and thrust force regulation. The main disadvantage of DTC, due to the nature of its control structure, is the presence of large ripple in the torque and flux. Since DTFC is conceptually identical to DTC it inherits the same disadvantage resulting in larger ripple in the thrust force caused by the use of hysteresis-based flux and thrust force controllers and variable switching frequency. This variable switching is caused by the use of hysteresis-based controllers with a switching table to generate the applied voltage. Improvements to conventional DTFC have been suggested in [6–9], however most of the work is simulation based. Space vector modulation (SVM) is one of the popular methods that can be employed in the implementation of DTFC to reduce the ripple © Springer Nature Switzerland AG 2020 M. A. M. Cheema and J. E. Fletcher, Advanced Direct Thrust Force Control of Linear Permanent Magnet Synchronous Motor, Power Systems, https://doi.org/10.1007/978-3-030-40325-6_7

181

182

7 Sensorless Control of a Linear Permanent Magnet Synchronous …

in both torque and flux response and to achieve a fixed switching frequency. In [7], a simulation based study of SVM-based DTFC is reported. An SVM based DTFC utilizing decoupled control of thrust force and flux is reported in [9]. However, the inherent robustness of the conventional DTFC is compromised in SVM based DTFC using PI controllers. Sliding mode or variable structure control is a nonlinear control technique and is robust to parameter variation and uncertainties in the plant model. In [10], a variable structure (sliding mode) based direct torque control is reported for rotational PMSMs and is applied to linear PMSM as DTFC in [11]. The studies in [12–14] report an integral sliding mode control for the rotational PMSM which combines the speed and q-axis current control in one controller. It is important to note that the sliding mode control methods of [10–14] do not consider the stator flux and torque (thrust force in case of linear PMSM) as system state variables, and therefore, transient response may become slow. Another important concern which is also focused in this chapter for the satisfactory performance of the DTFC controlled linear PMSM is the stator flux estimation and mover’s speed estimation. In [1, 9], an open loop flux estimation technique is presented. However, the accuracy of this technique is sensitive to the integration error caused by the offset in the stator current measurement. The adaptive estimation of the stator flux based on the machine model is presented in [14–16]. The performance of the adaptive observer deteriorates in the case of parameter mismatch and therefore an adaptive flux observer with a sliding mode component to increase the robustness is proposed in [14, 15]. It is important to note that the sliding mode component causes chattering in the flux estimation leading to noise in the estimated speed and hence the position which affects the thrust force regulation under DTFC/DTC. In this chapter, this issue is solved here by formulating a modified sliding mode component with two boundary layers and a dead zone. A comprehensive mathematical modelling and stability analysis of the adaptive flux observer with improved sliding mode component is also provided. The SM-DTFC2 scheme, in conjunction with the proposed adaptive flux observer is validated for a prototype linear PMSM. Experimental results clearly demonstrate the superior control performance of the proposed control scheme when compared to the state of the art.

7.2 Dynamic State Space Model of Linear PMSM in xy-Reference Frame The dynamic equation governing the stator flux regulation as formulated (4.1) is reproduced as: dλs = −Rs i x + vx dt

(7.1)

7.2 Dynamic State Space Model of Linear PMSM in xy-Reference …

183

In (7.1), vx is control input, and the term “−Rs i x ” is regarded as disturbance. As already discussed at length in Chap. 6, the combined dynamics of speed vm , and thrust force FT given by (6.5) and (6.5a) are reproduced here as: X˙ = F(X ) + G(X )u + D(t)

(7.2)

where      −K (FT ) K (FT )Pπ F − v FT f 1 (x1 , x2 ) T m τ = , F(X ) = = ξ λr e f (FT1) , X= B f 2 (x1 , x2 ) vm F − v T M M m   K (FT )     b(FT ) 0 λr e f (FT ) = FL G(X ) = , u = v y and D(t) = 0 − M1 0     FT = H (X ) (7.2a) y = vm = 0 1 vm 

x1 x2





In (7.2),K (FT ), is called the linearization co-efficient [17, 18]. The characteristics of K (FT ) are given by (6.3). As discussed in Chap. 4, for a low stator inductance and a short pole-pitch the value of K remains constant for the whole operational range of the thrust force. However, in the case of high inductance or large pole pitch the value of K varies as function of thrust force according to (6.3). The reference flux λr e f (FT ) is a non-linear function of the operating thrust force and is selected according to maximum force per ampere (MFPA) criteria as given by (6.4). According to (6.4), when FT varies from 0 to ± 312 N (thrust force range of prototype), the corresponding variation in λr e f (FT ) is negligible for the prototype under study because of low values of L s and τ . Therefore, λr e f is set to 0.0846 Wb in the prototype system as discussed in Chap. 4. However, in the case of high inductance or large pole pitch the value of λr e f varies as a function of FT according to (6.4). The differential Eq. (7.2 and 7.2a) provide the basis to formulate the sliding mode control law to achieve combined control of the thrust force and the mover’s speed as explained in Sect. 7.3. In (7.2), only v y is the available a scalar control input, therefore only one state variable can be tracked. For this purpose, vm is a natural choice for tracking. It is important to note that control of the thrust force FT is implicitly combined with that of vm according to (7.2). The flux regulation model (7.1) evidently inherits the integral action as discussed in Chap. 4, Fig. 4.3. It should be noted that the dynamic model of (7.2) as rigorously analysed in Chap. 6, does not inherit integral action. Therefore, in order to eliminate the steady state error, the integral action must be provided by the control law. In this chapter the sliding surfaces for both stator flux and mover’s speed are defined in terms of integral of their respective tracking errors, moreover, a modified control law and reachability condition is applied to achieve the integral action in combined speed and thrust force control using (7.2).

184

7 Sensorless Control of a Linear Permanent Magnet Synchronous …

7.3 Combined Speed and Direct Thrust Force Control of Linear PMSM Based on Integral Sliding Mode Control Sliding mode control is a nonlinear control architecture that can tolerate bounded parameter variations parameter variation and imprecision in the plant model [19–24]. In sliding mode control the error dynamics for dynamic systems of form (7.1)–(7.2a) are formulated as a time varying sliding surface which is a function of the tracking error and is independent of the plant parameters [19–22]. As discussed by Slotine [23], In the integral sliding mode to ensure the elimination of the steady state error and to achieve robustness against matched (and unmatched in certain cases) model uncertainties, the sliding manifold sc (X ; t) is a function of the integral of the tracking error:  sc (X ; t) =

d +λ dt

n−1 ∫ x˜ dt

(7.3)

 T where, λ is a strictly positive constant, X = x, x, ˙ . . . , x n−1 is the state vector, x˜ is the instantaneous tracking error between the reference and actual values of system states and n is the order of the system relative to the integral of the tracking error ∫ x˜ dt. For example, a first order system has n = 2 for integral sliding mode control [23]. The control law u is to achieve the tracking for sc (X ; t) is reproduced from (6.25) and as [19–24]: u = u eq (X ; t) + ρ0 sgn(sc )

(7.4)

where, ρ0 is a positive constant gain. In (7.4), the concept of equivalent control u eq (X ; t) is also attributed to Utkin [19–24]. u eq (X ; t) is a state feedback law which is computed by solving s˙c (X ; t) = 0 for u and assuming no model uncertainty. “sgn(·)” is the sign function defined in (6.12). According to Slotine [21–23], the control law u is selected to satisfy the η-reachability condition (so called sliding condition) reproduced from (6.26): 1 d 2 s (X ; t) ≤ −η|sc (X ; t)| 2 dt c

(7.5)

where, η is a strictly positive constant. It can be observed from (7.4) that the sliding surface sc (X ; t) containing the integral term appears inside the discontinuous sgn(·) function in the control law and therefore, continuous pure integrator action is not available directly for the dynamics of (7.2), which will lead to a steady state error. However, (7.4) will provided sufficient tracking performance for stator flux regulation owing to inherent integrator in stator flux dynamics of (7.1).

7.3 Combined Speed and Direct Thrust Force Control of Linear …

185

7.3.1 Sliding Surface for Stator Flux Regulation The integral sliding surface sλ for the flux regulation dynamics of (7.1) can be found from (7.3) for n = 2 as: sλ = eλ + λ1 ∫ eλ dt

(7.6)

where, eλ = λr e f − λs is the flux tracking error and λ1 is a strictly positive constant.

7.3.2 Sliding Surface for Speed Regulation The integral sliding surface sv for the combined speed and thrust force dynamics of (7.2) is computed from (7.3) for n = 3 as: sv =

devm (t) + 2λ2 evm (t) + λ22 ∫ evm (t) dt

(7.7)

where, evm = vr e f − vm is the speed tracking error, vr e f is the reference speed and λ2 is a strictly positive constant.

7.3.3 Control Law for Stator Flux Regulation Differentiating (7.6): 



s˙λ = λ˙ r e f − λ˙ s + λ21 λr e f − λs

(7.8)

Since, λ˙ r e f = 0, and from (7.1), (7.8) becomes: s˙λ = +Rs i x + λ21 eλ − vx

(7.9)

The control law for the stator flux to ensure s˙λ = 0, is: vx = vx∗ + ρ1 sgn(sλ )

(7.10)

The control law (7.10) satisfies the sliding condition (7.5) for global stability [23, 24]: 1 dsλ2 ≤ −η1 |sλ | 2 dt where, ρ1 is a positive constant, and equivalent control law vx∗ is:

(7.11)

186

7 Sensorless Control of a Linear Permanent Magnet Synchronous …

vx∗ = Rs i x + λ21 eλ

(7.12)

In order to reduce chattering in tracking response, replace sgn(sλ ) in (7.10) with sat(sλ /ε1 ) (defined as 6.60 in Chap. 6) introduces as boundary layer of thickness ε1 around the sliding surface of (7.6) and from (7.12) the control law (7.10) becomes: 

vx = Rs i x + λ21 eλ + ρ1 sat sλ/ ε1

(7.13)

7.3.4 Control Law for Thrust Force Regulation Taking time derivative of (7.7): 





s˙v = v¨r e f − v¨m + 2λ2 v˙r e f − v˙m + λ22 vr e f − vm

(7.14)

The value of v¨m can be obtained from (7.2) as: v¨m =

1 ˙ B 1 ˙ FT − v˙m − FL M M M

(7.15)

In general, the load force FL is constant during steady-state and does not change abruptly [25, 26], therefore, F˙L = 0, and (7.15) becomes: v¨m =

1 ˙ B FT − v˙m M M

(7.16)

Substituting the value of F˙T and v˙m from (7.2) into (7.16): 1 v¨m = M



 B 1 −K π K B 1 FT − vm − FL (7.17) FT − K P vm + v y − ξ λs τ λs M M M M

The reference speed vr e f is a constant during steady-state and, therefore: v¨r e f = v˙r e f = 0

(7.18)

Substituting (7.18) and the value of v˙m and v¨m from (7.2) and (7.17) respectively into (7.14) and re-arranging: s˙v = α FT + βvm + μFL − λ22 evm − γ v y where α = MξKλr e f + γ=

K Mλr e f

B M2

−

2λ2 M

π , β = K P Mτ +

B2 M2

2 + 2λ2 MB , μ = 2λ − M

(7.19) B M2

 , and

7.3 Combined Speed and Direct Thrust Force Control of Linear …

187

The equivalent control law v ∗y to achieve the condition s˙v = 0 is defined from [23, 24]: v ∗y =

1 α FT + βvm + μ FˆL − λ22 evm γ

(7.20)

Since the load force FL is usually an unknown quantity, therefore, in (7.20) the estimated value of load force FˆL is used. In order to achieve the sliding mode condition, the discontinuous control sgn(sv ) is included in the equivalent control law of (7.20) as [20–24]: v y = v ∗y +

ρ2 sgn(sv ) γ

(7.21)

where, γ, ρ2 are strictly positive constants. In (7.21), sgn is the sign function is defined as:  1 i f sv > 0 sgn(sv ) = (7.22) −1 i f sv < 0 As discussed in Sect. 7.2, the combined speed and thrust force dynamic model of (7.2) does not include a pure integrator inherently, and the integral action should be provided by the control law to ensure the elimination of the steady state error in the speed response. For this purpose, an integral sliding surface sv given by (7.7) is formulated to add integral action to the control law of (7.21). However, it can be seen from (7.21) that the sliding surface sv is encapsulated by the discontinuous sgn(·) function in the control law and therefore, continuous pure integrator action is not available directly which leads to a steady-state error in the speed response. This issue is solved by including the sliding surface sv directly into the control law (7.21) to achieve a direct integral action in the control law [13, 27–29]: v y = v ∗y +

Ω ρ2 sv + sgn(sv ) γ γ

(7.23)

where, Ω is a strictly positive constant. According to [27–29], the main motivation behind adding the term Ωγ sv in (7.23) is to ensure a significant reduction in the “hitting time”. It is important to note that when the sliding surface sv is defined as (7.7), the control law (7.23) becomes capable of providing a direct integral action to eliminate the steady-state error. The modified sliding condition (reachability condition) for (7.23) is given as [27, 29]: Ω 1 d 2 sv (X ; t) ≤ − sv2 (X ; t) − η2 |sv (X ; t)| 2 dt γ

(7.23a)

In order to rationalize and establish the stability of the controller of (7.23), the following Lyapunov candidate function [13] is defined:

188

7 Sensorless Control of a Linear Permanent Magnet Synchronous …

V =

1 2 1 ˜2 sv + F 2 2γ1 L

(7.24)

where, F˜L = FL − FˆL , γ1 > 0, and differentiating (7.24): 1 V˙ = sv s˙v − F˙ˆL F˜L γ1

(7.25)

Substituting (7.19) into (7.25):

 1 V˙ = sv α FT + βvm + μ FL − λ22 evm − γ v y − F˙ˆL F˜L γ1

(7.26)

Substituting (7.20) and (7.23) into (7.26), and simplifying:

1 V˙ = sv −Ωsv − ρ2 sgn(sv ) + μ F˜L − Fˆ˙L F˜L γ1

(7.27)

Since, sv × sgn(sv ) = |sv |, therefore, (7.27) becomes:   1 V˙ = −Ωsv2 − ρ2 |sv | + F˜L μ sv − F˙ˆL γ1

(7.28)

To ensure global stability of the controller, V˙ < 0 suggests the following expression to estimate the load force FˆL : FˆL = γ1 μ ∫ sv dt

(7.29)

V˙ = −Ωsv2 − η2 |sv | < 0

(7.30)

Hence, (7.28) reduces to:

Since, Ω and η2 are positive constants, therefore V˙ is strictly negative when the load force is estimated using (7.30) and the condition (7.23a) In order 

is satisfied. to avoid chattering, sgn(sv ) in (7.30) is replaced with sat sv/ ε2 as explained in Sect. 6.4.5, and (6.60) and thus a boundary layer of thickness ε2 is introduced around the sliding surface of (7.7). The control law (7.30) becomes: v y = v ∗y +

 Ω ρ2 sv + sat sv/ ε2 γ γ

(7.31)

The integral sliding mode law (7.31) to achieve combines speed and direct thrust control of dynamic system (7.2) and (7.2a) is illustrated by the schematic diagram provided in Fig. 7.1.

7.4 A Novel Combined Sliding Mode State Observer

189

Sliding Mode Control Law vy for Speed Tracking According to (7.31) Combined Speed and Thrust Force Dynamics for Linear PMSM According to (7.2) and (7.2-a)

Sliding Surface sv for Speed Tracking According to (7.7)

vref +-

d (t )

Ω

λ22 ∫

ev m

2λ2

Σ

sv

ρ2 sat ⎛⎜ s v ε ⎞⎟ ⎝

d dt

2

1

Σ

⎠

μγ1∫

u =v y

++-

γ

∫

T

H(X)

vm

F(X)

*

vy

λ22

X = ⎣⎡ FT vm ⎦⎤

G(X)

Σ

α FT + β vm

⎡⎣α

β ⎤⎦

Fig. 7.1 Schematic illustration of the non-linear combined speed and thrust force dynamics proposed in (7.2) and (7.2a) and the proposed sliding mode control with integral action of (7.31)

7.4 A Novel Combined Sliding Mode State Observer The dynamic model of linear PMSM in dq-axis reference frame is given in Chap. 2 and can be expressed in state space form as: 

λ˙ d λ˙ q



    Rs − LRss p πτ vm λ λd vd f = + + Ls − p πτ vm − LRss λq vq 0       λf 1 0 λd id Ls = − Ls 1 iq λq 0 Ls 0 

(7.32)

(7.33)

Based on (7.32) and (7.33) considering i d and i q as outputs, a novel adaptive sliding mode observer for flux estimation in the dq-reference frame is formulated as: 

λ˙ˆ d λ˙ˆ q



 =

− LRss p πτ vˆm − p πτ vˆm − LRss 

iˆd iˆq



λˆ d λˆ q



=



1 Ls

0

 +

0 1 Ls

vd vq







λˆ d λˆ q

+

Rs λ Ls f

0 

−



λf Ls

0



+ KS + H sgn m (S) (7.34) (7.35)

where, the superscript ˆ denotes the estimated quantities, and K and H are the feedback gain matrices, while the improved nonlinear sliding mode feedback term “sgn m (S)” is introduced to increase the robustness of the observer to parameter variation. The block diagram of the proposed observer according to (7.34) and (7.35) is shown in Fig. 7.2. The “sgn m ” function which is used in (7.34) is defined as:

190

7 Sensorless Control of a Linear Permanent Magnet Synchronous …

Fig. 7.2 Block diagram of proposed observer

vαβ

SVM

αβ

vxy

xy

PWM duty cycles

Vdc

Vabc

3-Phase VSI

θ$ s

Linear PMSM

vxy from (7.13) and (7.23) and θ$ s from (7.59) iabc

abc

e

αβ

− jθˆr

PWM duty cycles

Transfer Matrix

Κ +

abc

e

∫

− jθˆr

vd , vq

i%d , i%q

H

+

αβ

Adaptive Observer

id , iq + _

Sgnm C Function % % ˆ Adaptive λd , λq Scheme (7.55) λˆd , λˆq Adaptive Model of iˆd , iˆq linear PMSM (7.34)

θˆr

DS1104 R&D Board

⎧ ⎪ 1 if ⎪ ⎪ ⎨ e−β1 i f F = sgn m (e) = emax ⎪0 if ⎪ ⎪ ⎩ −1 i f

e > β2 β1 ≤ |e| ≤ β2 |e| < β1 e < −β2

(7.36)

where    e  β2 = β1 + 1/ emax , and emax is the maximum error limit. In (7.36), β1 is a positive constant and is the width of first boundary layer and β2 is the variable width of the second boundary layer. It is evident from (7.36) that in the case of small estimation error e, the boundary layer width β2 expands and reduces the observer activity and vice versa. These two boundary layers result in reduced chattering. It is noteworthy that this improved sliding mode “sgn m ” function enhances the observer robustness towards parameter variation. Whenever the estimation error increases due to parameter variation, β2 shrinks to increase the observer activity and forces the error to remain within acceptable limits. The improved “sgn m ” function is illustrated in Fig. 7.3. The sliding surface S used in (7.34) is a time varying sliding surface defined as:  S=

S1 S2



 =

i d − iˆd i q − iˆq



=

1 Ls

0

0 1 Ls



λd − λˆ d λq − λˆ q

(7.37)

Subtracting (7.34) from (7.32) and after simple mathematical manipulation, the dynamic equation governing the estimation error can be derived as:

7.4 A Novel Combined Sliding Mode State Observer

191

Fig. 7.3 Block diagram illustration of improved “sgn m ” function

Sgnm(e) 1

− β2 − β1

β1

β2

-1



 where, A =  and C =

1 Ls

λ˙˜ d λ˙˜ q



  ˆ λ˜ ˜ λd − H sgn m (S) = (A − KC) ˜ d − A λq λˆ q

(7.38)

  − LRss p πτ vm 0 v˜m π ˜ , and v˜m = vm − vˆm , , A = p τ − p πτ vm − LRss −v˜m 0  0 . 1

0 Ls The superscript ~ denotes the estimation errors, and all the bold letters represent matrices.

7.4.1 Stability Analysis of the Proposed Observer In order to analyze the asymptotic stability of the proposed observer, a Lyapunov candidate function is defined as: V =

1 ˜ T ˜ v˜m2 (λ λ+ ) 2 ξ1

(7.39)

where, ξ1 is an arbitrary positive constant. The time derivative of (7.39) can be expressed as:

v˜m T v˙m − v˙ˆm V˙ = λ˜ λ˙˜ + ξ1

(7.40)

This proposed observer is implemented using a digital controller; therefore, it is rational to assume that during a very small sampling time Ts the mover’s speed vm is constant and v˙m = 0 and (7.40) can be written as:

192

7 Sensorless Control of a Linear Permanent Magnet Synchronous …

v˜m ˙ T vˆm V˙ = λ˜ λ˙˜ − ξ1

(7.41)

Substituting (7.38) in (7.40), simple mathematical manipulation yields:   ˙ˆm v π T V˙ = λ˜ (A − KC)λ˜ + p v˜m λˆ q λ˜ d − λˆ d λ˜ q − − λ˜ HC sgn m λ˜ τ ξ1 T

(7.42)

According to the Lyapunov direct method, for the system to be globally asymptotically stable V˙ < 0, this leads to the following conditions: λˆ q λ˜ d − λˆ d λ˜ q −

v˙ˆm =0 ξ1

(7.43)

A − KC < 0

(7.44)

T λ˜ HCsgn m λ˜ < 0

(7.45)

The feedback gain matrices K and H are selected to be positive diagonal matrices so that the inequalities (7.44) and (7.45) always hold true to ensure observer stability at any operating point. The positive diagonal feedback matrices are given as:  K=

k1 −k2 k2 k1



 and H =

h1 0 0 h2



The values of k1 , k2 , h 1 and h 2 are chosen such that the inequalities (7.44) and (7.45) will hold true for all the operating points.

7.4.2 Gain Selection for the Proposed Observer It is evident from (7.42) and (7.44) that the gain matrix K is selected such that the first term of (7.42) should always be negative semi-definite to ensure the stability of the observer. This clearly indicates that the eigenvalues of A − KC are in the left T half plane. Therefore, λ˜ (A − KC)λ˜ is also negative. The classical approach for the observer gains selection is to place the observer poles proportional to the motor poles such that the observer error converges to zero faster than the motor transients [14, 15]. However, this approach is vulnerable to noise. In this work a new pole-placement technique is used as detailed in [16, 17, 30]. The observer poles are placed by shifting the real part of the motor poles to the left by a constant “k” while the imaginary parts remain the same. Therefore, the contribution of the slower eigenvalues is amplified which in turn improves the error convergence rate.

7.4 A Novel Combined Sliding Mode State Observer

193

The poles of linear PMSM are the eigenvalues of A and can be given as the roots of the following characteristic equation: s2 +

2Rs R 2 π 2 s + s2 + p vm = 0 Ls Ls τ

(7.46)

The observer poles are the eigenvalues of A−KC and are the roots of the following characteristic equation (based on matrix K given in Sect. 7.4.1): s2 +

2(Rs + k1 ) (Rs + k1 )2 s+ + Ls L 2s



π k2 p vm + τ Ls

2 =0

(7.47)

By shifting the motor pole to left by k, mathematically it can be written as: Po1,2 (vm ) = Pm1,2 (vm ) − k

(7.48)

where, Po1,2 and Pm1,2 are the observer’s and motor’s poles respectively. It is evident from (7.48) that the sum of observer roots is 2 k smaller as compared to that of the motor; however, the imaginary parts are the same. Thus, the following expression is obtained: ( pm1 + pm2 ) − ( po1 + po2 ) = −

2Rs 2(Rs + k1 ) + = 2k Ls Ls

(7.49)

By equating the imaginary parts of the motor’s and observer’s poles: 

2Rs Ls

2

 2(Rs + k1 ) 2 Rs2 π 2 = −4 + p vm L 2s τ Ls    2 π k2 2 (Rs + k1 ) −4 + p vm + L 2s τ Ls 

(7.50)

After mathematical manipulation of (7.49) and (7.50) the following expressions for the gains k1 and k2 are determined as: k1 = k L s  1 k2 = sign(vm ) × 2

L 2s

(7.51)

   π 2 L 2s π 3 2 + k Rs L s − p L s vm p vm + 16 k 1 − τ 4 2 τ (7.52)

From (7.52) it is clear that the expression for k2 involves the motor speed, therefore, it is calculated online. In order to calculate the values of h 1 and h 2 (7.44) can be re-written as;

194

7 Sensorless Control of a Linear Permanent Magnet Synchronous …

        h 1 λ˜ d  + h 2 λ˜ q  > 0

(7.53)

It is evident from (7.53) that h 1 , h 2 > 0. The robustness of the observer can be enhanced by choosing moderately large values of h 1 , and h 2 . According to [14] in the case of the conventional sliding mode observer the large values of these gains can lead to chattering, however in this research a novel sliding mode function is defined by (7.36) to avoid chattering and hence increased robustness can be achieved even with larger values of gains.

7.4.3 Adaption Scheme for Speed Estimation From (7.43), the estimated mover speed can be expressed as: vˆm = ξ1 ∫(λˆ q λ˜ d − λˆ d λ˜ q )dt

(7.54)

In order to enhance the performance of the speed estimation scheme, the integrator in (7.54) is replaced by a PI controller and the speed estimation is represented as [15]:

vˆm = K p λˆ q λ˜ d − λˆ d λ˜ q + K i ∫(λˆ q λ˜ d − λˆ d λ˜ q )dt

(7.55)

where, K p and K i = ξ1 are the gains of the PI controller. The estimated mover position in electrical radians is θˆr and is determined by integrating vˆm as: θˆr = p

π ∫ vˆm dt τ

(7.56)

The output of the integrator is limited within the range of [−π , π ] to avoid any divergent error in position estimation and therefore, θˆr remains within [−π , π ] radians.

7.4.4 Estimation of Stator Flux Magnitude and Thrust Force The estimated dq-axes components of stator flux given by (7.34) can be converted to αβ-axes flux linkages as: 

λˆ α λˆ β



 =

cos θˆr − sin θˆr sin θˆr cos θˆr



λˆ d λˆ q

 (7.57)

7.4 A Novel Combined Sliding Mode State Observer

195

The stator flux magnitude λs and angle θs (in elec. rad.) can be expressed as: λˆ s =

 λˆ 2α + λˆ 2β 

θˆs = tan−1

λˆ β λˆ α

(7.58)

 (7.59)

The thrust force FT for a surface-mount linear PMSM can be expressed in terms of i q as: FT = Pk F

3π λ f iq 2τ

(7.60)

7.5 Experimental Results The integral sliding mode control scheme for the combined speed and direct thrust force control of the linear PMSM (to be referred as the SM-DTFC2) and the combined adaptive sliding mode observer of Fig. 7.2 (to be referred as the SM-observer) is validated by its practical application to the prototype surface-mount linear PMSM based drive system in the laboratory. The complete drive scheme comprising the SM-DTFC2 and the SM-observer for the linear PMSM is digitally implemented according to the block diagram of Figs. 7.1, 7.2 and 7.4. The PI-DTFC of Chap. 4 is also implemented in conjunction with SM-observer for benchmark purposes. In order to evidence the performance advantage of the improved sliding mode function sgn m (S) of (7.36), the SM-observer of (7.34) is also practically implemented as a linear state observer without including the improved sliding mode component sgn m (S) in (7.34) so that the performance improvement caused by sgn m (S) can be observed quantitatively.

Fig. 7.4 Block diagram of the proposed combined sliding mode speed and direct thrust control of linear PMSM with the novel adaptive observer

196

7 Sensorless Control of a Linear Permanent Magnet Synchronous …

Table 7.1 Controller gains used for proposed SM-DTFC2 Gain

λ1

λ2

ε1

ε2

ρ1

ρ2

Ω

Value

2

0.1

0.004

2.5

10

10,000

18,000

Table 7.2 Controller gains used in experiment for PI-DTFC

Flux PI controller Thrust force PI controller

k p_λ

8.7

ki_λ

17

k p_F

0.0480

ki_F

0.0174

The parameters of prototype linear PMSM are provided in Chap. 1, Table 1.1. The experimental setup is shown in Fig. 1.1. The experimental implementation is done using the dSPACE® ds1104 controller as per Fig. 7.4. The PI-DTFC in conjunction with SM-observer scheme is implemented in the same setup as described in Chap. 4 and Fig. 7.2. The experimental results are divided into two groups. First group of results compare the control performance of the proposed SM-DTFC2 with that of PIDTFC of Chap. 4 and proves the effectiveness of SM-DTFC2 over conventional PI based controllers. The second group validates the speed estimation performance of SM-observer under various conditions in conjunction with the SM-DTFC2. The basic control gains of (7.13) and (7.31) used in SM-DTFC2 are given in Table 7.1, the gains other than Table 7.1 used in (7.31) can also be computed from these values. The PI-DTFC1 control scheme uses three PI controllers to control speed, thrust force and flux respectively. The proportional and integral gains of PI controller for flux and thrust force are denoted by k p_λ , ki_λ , k p_F , and k p_i and are given in Table 7.2. The gains of thrust force and stator flux regulating PI controllers of PI-DTFC were tuned to achieve a damping ratio of 0.95 using the root locus method as explained in Chap. 4 to ensure comparable operating conditions for benchmarking.

7.6 Experimental Evaluation of the Control Performance of SM-DTFC2 7.6.1 Start-Up Response The start-up speed responses of the surface-mount linear PMSM under the PI-DTFC and the SM-DTFC2 are compared. The speed, flux, thrust force and stator currents for the PI-DTFC and the SM-DTFC during the start-up transient are shown in Fig. 7.5a, b respectively. It is important to note that, the SM-observer cannot be used during the start-up, therefore measured speed is used instead of the estimated speed. Figure 7.5

7.6 Experimental Evaluation of the Control Performance of SM-DTFC2 PI-DTFC

300

Measured 1.05

1.1

1.15

1.05

1.1

1.15

1.2

0.085 0.0845

3 2 1 0 -1 -2 -3 0.95

Reference 1

1.05

1.1

1.15

1

1.05

1.1

Time(Sec)

1.15

1.15

1.2

1

1.05

1.1

1.15

1.2

Estimated

0.085 0.084 0.95

1.2

3 2 1 0 -1 -2 -3 0.95

I a (Red)

1.1

Estimated

0.0845

1.2

I b (Blue)

I c (Green)

1.05

0.0855

Current(A)

Current(A)

0.084 0.95

Measured 1

0.086

Estimated

0.0855

0

200 150 100 50 0 -50 0.95

Estimated 1

100

1.2

Reference

Reference

200

Force (N)

200 150 100 50 0 -50 0.95

1

300

-100 0.95

Flux (Wb)

-100 0.95

0.086

Flux (Wb)

Speed (mm/s)

100 0

SM-DTFC

(b)

Reference

200

Force (N)

Speed (mm/s)

(a)

197

Reference 1

1.05

1.1

1.15

1.2

I c (Green)

I a (Red) 1

1.05

I b (Blue) 1.1

1.15

1.2

Time(Sec)

Fig. 7.5 Startup performance from 0 to 200 mm/s. Measured Speed, thrust force, stator flux, and stator phase currents responses are shown from top to bottom respectively. a PI-DTFC and b SM-DTFC2 (experiment)

clearly shows that the thrust force under SM-DTFC2 settles faster to steady-state compared to PI-DTFC. The magnified views of the start-up speed response for both PI-DTFC and the SM-DTFC2 are shown in Fig. 7.6a, b respectively. It is clear from Fig. 7.6a, b that the speed response under the SM-DTFC2 is 23% faster compared to that of PI-DTFC. The reason for the faster speed response under SM-DTFC is the higher value of the thrust force (180 N) produced by SM-DTFC compared to that of PI-DTFC (130 N) during the start-up which results in a higher acceleration. In addition, for quantitative analysis, the integral of absolute error (IAE) indices for the speed error plots for both PI-DTFC and SM-DTFC2 during the start-up are computed. These IAE indices are determined from the speed error plots for measured and reference speed during start-up for both PI-DTFC and SM-DTFC2 are shown in Fig. 7.7. The IAE indices for PI-DTFC and SM-DTFC2 are given in Table 7.3. It can be observed from Table 7.3, that the IAE index for the speed error plots under the SM-DTFC2 is reduced by 25% compared to that of the PI-DTFC. It can be concluded from these experimental results that the combined control of the speed and thrust force achieved under the SM-DTFC2 can deliver faster speed response during start-up compared to PI-DTFC.

198

7 Sensorless Control of a Linear Permanent Magnet Synchronous … PI-DTFC

200 100

Measured

0 -100 0.95

(b)

Reference

38 ms 1.025

1

0.975

200 150 100 130 N 50 0 -50 0.95 0.975

1.05

Reference

Estimated 1

1.025

Speed (mm/s)

300

Force (N)

Force (N)

Speed (mm/s)

(a)

1.05

SM-DTFC

300

Reference

200 100 0

Measured 29 ms

-100 0.95 0.975 200 150 100 180 N 50 0 -50 0.95 0.975

1

1.025

1.05

Estimated

1

1.025

1.05

Time (s)

Time (s)

Fig. 7.6 Magnified view of the measured speed and thrust force response during start-up. a PIDTFC and b SM-DTFC2 (experiment)

(b)

150 100 50 0 38 ms -50 0.96 0.97 0.98 0.99 1

Speed Error (mm/s)

Speed Error (mm/s)

(a) 200

1.01 1.02

200 150 100 50 0 29 ms -50 0.96 0.97 0.98 0.99 1

Time (s)

1.01 1.02

Time (s)

Fig. 7.7 Error plots for speed start-up. a PI-DTFC and b SM-DTFC (experiment)

Table 7.3 Comparison of Transient Performance of PI-DTFC and the SM-DTFC Using IAE Index

Type of transient phenomena

IAE index for speed error PI-DTFC

Start-up (0–200 mm/s) Speed reversal (−600 to 600 mm/s)

SM-DTFC2

24,221

18,100

211,230

188,103

7.6.2 Speed Reversal Response The experimental results comparing the performance of the PI-DTFC and the SMDTFC2 during the speed reversal from −600 to +600 mm/s are shown in Fig. 7.8 illustrating the speed response, the corresponding thrust force, stator flux and the stator currents for both the PI-DTFC and the SM-DTFC2. It is important to note that during speed reversal the estimated speed from the SM-observer is being used for both PI-DTFC and SM-DTFC, whereas the measured speed is just shown for comparison. Figure 7.9 shows a magnified view of the speed response and demonstrates that under the SM-DTFC2 the speed response is 11.4 ms (16%) faster than PI-DTFC during the

7.6 Experimental Evaluation of the Control Performance of SM-DTFC2

PI-DTFC

0

Measured

-600

Estimated 1

Force (N)

0.7 0.8 0.9

300 Reference 200 100 0 -100 Estimated -200 -300 0.7 0.8 0.9 1 1.1 1.2 1.3

Flux (Wb)

0.087 0.085 0.084

Reference 1

-600

1

1.1 1.2 1.3

300 200 100 0 -100 Estimated -200 -300 0.7 0.8 0.9 1 1.1 1.2 1.3 Estimated

0.086 0.085 0.084

1.1 1.2 1.3

Reference

0 -4

1

1.1 1.2 1.3

1

1.1 1.2 1.3

8

Current(A)

Current(A)

Estimated

0.083 0.7 0.8 0.9

4

1

Measured

0

0.7 0.8 0.9

8

-8 0.7 0.8 0.9

Reference

600

0.087

Estimated

0.086

0.083 0.7 0.8 0.9

SM-DTFC

(b)

1.1 1.2 1.3

Force (N)

600

Speed (mm/s)

Reference

Flux (Wb)

Speed (mm/s)

(a)

199

4 0 -4 -8 0.7 0.8 0.9

1.1 1.2 1.3

Time(Sec)

Time(Sec)

Fig. 7.8 Speed reversal from −600 to 600 mm/s and steady-state response at 600 mm/s. Measured and estimated Speed, thrust force, stator flux, duty ratio and stator phase “a” current responses are shown from top to bottom. a PI-DTFC and b SM-DTFC2 (experiment)

PI-DTFC

600 0

Estimated

-600 0.75

70 ms 0.8

0.85

Measured 0.9

Time (s)

SM-DTFC

(b)

Reference

Speed (mm/s)

Speed (mm/s)

(a)

0.95

1

Reference

600

Measured Estimated

0 -600 0.75

58.6 ms 0.8

0.85

0.9

0.95

1

Time (s)

Fig. 7.9 Magnified view of the measured and estimated speed response during start-up. a PI-DTFC and b SM-DTFC (experiment)

200

7 Sensorless Control of a Linear Permanent Magnet Synchronous …

(b)

PI-DTFC

700

Reference

Speed (mm/s)

Speed (mm/s)

(a)

650 600 Measured Estimated

550 500 0.95

1

1.05

1.1

1.15

SM-DTFC

700

Estimated

650 600 550 500 0.95

Reference

Measured

Time (s)

1

1.05

1.1

1.15

Time (s)

Fig. 7.10 Magnified view of the measured and estimated speed response during steady-state. a PIDTFC and b SM-DTFC2 (experiment)

speed reversal transient. In addition, it is clear from Table 7.3, that the IAE index for speed is also reduced by 11% under SM-DTFC2 which clearly proves the superior transient performance of the proposed SM-DTFC. It can also be observed from Fig. 7.8 that SM-DTFC2 results in a visibly smooth flux response. The steady state performances of the prototype linear PMSM, at 600 mm/s and rated load of 50 N, from 0.95 s to 1.15 s under the PI-DTFC and SM-DTFC2 are also compared in Fig. 7.10. Under SM-DTFC2 the steady-state low frequency oscillations in speed, thrust force and stator flux have significantly reduced compared to PI-DTFC. The quantitative results for steady state performance of PIDTFC and the SM-DTFC2 at 600 mm/s and 50 N (average force) in terms of percent flux ripple λrip (%), percent force ripple F rip (%), and percent speed ripple vri p (%) are summarized in Table 7.4. The control voltage v y to control the thrust force generated by PI-DTFC and SM-DTFC2 during start-up and speed reversal transient is compared in Fig. 7.11. It can be concluded that SM-DTFC2 generates a higher command voltage during the transient resulting in a faster rise time of thrust force and Table 7.4 Comparison of steady-state performance of PI-DTFC and the SM-DTFC2

600 mm/s, 50 N λrip (%)

0.34

0.24

8.48

6.01

vrip (%)

2.37

1.10

(b) 100 SM-DTFC

1

1.05

PI-DTFC 1.1 1.15 1.2

Time (s)

vy (volts)

vy (volts)

SM-DTFC2

F rip (%)

(a) 80 60 40 20 0 -20 0.95

PI-DTFC [32]

SM-DTFC

50 0 -50

PI-DTFC

-100 0.7 0.8 0.9

1

1.1 1.2 1.3

Time (s)

Fig. 7.11 Command voltage v y to control thrust force, generated by the PI-DTFC and the proposed SM-DTFC2. a Start-up and b Speed reversal (experiment)

7.6 Experimental Evaluation of the Control Performance of SM-DTFC2

(b) 300

(a) 150 100

2

50 0

Reaching -50 0.972 0.977

100 0

0.982

Sliding

200

Sliding

S

S2

201

0.987

-100 0.8

Reaching 0.81

0.82

0.83

Time (s)

Time (s)

Fig. 7.12 Integral sliding surface sv according to (7.7). a Start-up and b Speed reversal (experiment)

Table 7.5 Comparison of robustness to parameter variation, increase in speed ripple for PI-DTFC and SM-DTFC2

Inductance

% Speed ripple at 600 mm/s PI-DTFC

SM-DTFC2 1.10

L s (nominal)

2.37

1.5 L s

3.78

1.60

2 Ls

4.9

2

hence the speed when compared to that of PI-DTFC. The convergence of sliding surface for speed tracking during start-up and speed reversal under SM-DTFC2 is shown in Fig. 7.12, which proves the stability of the proposed SM-DTFC2 during the transient response.

7.6.3 Robustness to Parameter Variation The robustness of both the control schemes is also evaluated. For this purpose, the respective controller gains for both the control schemes were tuned using the wrong value of the inductance, and the effect on the speed ripple during steady state is recoded and is given in Table 7.5. The results summarized in Table 7.5 demonstrate the effectiveness of SM-DTFC2 under parameter variation when compared to PI-DTFC.

7.7 Experimental Evaluation of the SM-Observer 7.7.1 Speed Reversal Response The experimental results for the SM-observer during the speed reversals for two different speeds are shown in Fig. 7.13. The speed response, speed error, thrust force and flux response when the speed is reversed from −200 to 200 mm/s and −600 to 600 mm/s are shown in Fig. 7.13a, b respectively. In both the figures the speed estimated from the SM-observer is compared with the measured speed. It is

202

7 Sensorless Control of a Linear Permanent Magnet Synchronous …

Force (N)

0.8

0.85

Measured Reference Estimated 0.9 0.95

15 5 0 -5 -15

0.75 200 150 100 50 0 -50 -100 0.75

Speed (mm/s)

(b)

0.8

0.85

0.9

0.95

0.8

0.85

0.9

0.95

0.8

0.85

Measured Reference Estimated 0.9 0.95 1

0.8

0.85

0.9

0.95

1

0.8

0.85

0.9

0.95

1

0.85

Estimated Reference 0.9 0.95 1

-600

15 5 0 -5 -15

0.75 400 300 200 100 0 -100 -200 0.75 0.087

0.086 0.085 0.084 0.8

0.85

Estimated Reference 0.95 0.9

Time (s)

Flux (Wb)

Flux (Wb)

0

0.75

0.087

0.083 0.75

600

Speed error (mm/s)

Speed error (mm/s)

400 200 0 -200 -400 0.75

Force (N)

Speed (mm/s)

(a)

0.086 0.085 0.084 0.083 0.75

0.8

Time (s)

Fig. 7.13 Speed reversal response. From top speed response, speed error, thrust force and stator flux, a −200 to 200 mm/s, b −600 to 600 mm/s (experiment)

observed from Fig. 7.13a, during the speed reversal from −200 to 200 mm/s the estimated speed overshoots then settles to the measured speed. The speed error during the transient reaches a maximum value of −10 mm/s and reduces to zero as the estimated speed approaches the measured speed in steady-state. The thrust force peaks to 110 N during the speed reversal from −200 to 200 mm/s and settles to an average steady-state value 42 N after the transient. It is clear from Fig. 7.13b that the speed error during the speed reversal from −600 to 600 mm/s peaks at −15 mm/s during the transient. It is also observed from Fig. 7.13b that the thrust force peaks at 227 N during the transient and settles to an average steadystate value of 52 N. It can be concluded that the difference in the speed estimation error during the speed reversal for the two speeds is similar and this validates the performance of the SM-observer at different speeds during regular speed reversals. Moreover, from Fig. 7.13a, b the effectiveness of the SM-DTFC2 is also proved as it provides adequate performance for the combined control of the speed and the thrust force and ensures a small speed estimation error during the speed reversal for both the speed ranges. The flux response for both the speeds is also comparable. During the

7.7 Experimental Evaluation of the SM-Observer

203

0.8

100 50 0 -50 -100 -150 0.75

0.8

0.85

Measured Reference Estimated 0.9 0.95

Speed (mm/s)

(b)

400 200 0 -200 -400 0.75

Speed error (mm/s)

Speed error (mm/s)

Speed (mm/s)

(a)

0.85 0.9 Time (s)

0.95

600 0 -600 0.75 100 50 0 -50 -100 -150 0.75

Measured Reference Estimated 0.9 0.95 1

0.8

0.85

0.8

0.85 0.9 Time (s)

0.95

1

Fig. 7.14 Speed reversal response without the sliding mode component. From top, speed response, speed error, and flux responses are shown, a −200 to 200 mm/s, b −600 to 600 mm/s (experiment)

speed reversal from −600 to 600 mm/s, the flux response shows a disturbance which can be attributed to large thrust force demand. However, the magnitude variation in the estimated flux remains within a hysteresis band of 0.002 Wb for both the speeds.

7.7.2 Speed Reversal Response Without the Improved SM Function The speed estimation performance of the SM-observer without the improved sliding mode function (SM Function) “sgn m ” during the speed reversal transient from − 200 to 200 mm/s and 600 to 600 mm/s is shown in Fig. 7.14a, b respectively. It is observed from Fig. 7.14 that the speed estimation error peaks at −100 mm/s during the speed reversal for the two speeds ranges which is significantly larger than that in Fig. 7.13 and clearly shows the benefit of the improved sliding mode function “sgn m ” during the transient performance.

7.7.3 Steady State Response The steady-state speed response, speed error, thrust force, and the flux response of the SM-DTFC2 with the SM-observer for 200 and 600 mm/s are shown in Fig. 7.15a, b respectively. It is observed from Fig. 7.15, that the steady-state speed estimation error remains within ± 5 mm/s for the speeds of 200 and 600 mm/s. This clearly shows that the SM-observer gives satisfactory speed estimation performance at different speed ranges. Moreover, the percentage steady state speed ripple is 2.1 and 2.6% at 200 and 600 mm/s respectively. The percentage steady-state thrust force ripple is 4.3

204

7 Sensorless Control of a Linear Permanent Magnet Synchronous …

(b)

200 150 100 0.9 10

1.1

5 0 -5 1

1.1

1.2

1.3

1.4

Force (N)

-10 0.9 150 100 50 0

-50 0.9 0.0865

Flux (Wb)

1

Measured Estimated 1.2 1.3 1.4

1

1.1

1.2

1.1

Estimated Reference 1.2 1.3 1.4

1.3

1

Time (s)

700 650 600 550 500 1 10

Measured Estimated 1.4 1.3

1.1

1.2

1.1

1.2

1.3

1.4

1.1

1.2

1.3

1.4

5 0 -5 -10 1 100 50 0

-50 1 0.0865

1.4

0.085 0.0835 0.9

Speed (mm/s)

250

Speed error (mm/s)

300

Flux (Wb)

Force (N)

Speed error (mm/s)

Speed (mm/s)

(a)

0.085 0.0835 1

1.1

1.2

Estimated Reference 1.3 1.4

Time (s)

Fig. 7.15 Steady-state response. From top, speed response, speed error, thrust force and stator flux are shown, a 200 mm/s, b 600 mm/s (experiment)

and 4.8% for the two speeds respectively. Therefore, effective control performance of the SM-DTFC2 is also proved. It is also observed from Fig. 7.15 that the steady state flux oscillations are slightly decreased at 600 mm/s. It is important to note that, in contrast to rotational PMSM, due to complex nonlinear friction dynamics of the linear PMSM, there are low-frequency speed oscillations even at steady state as can be observed from Fig. 7.15.

7.7.4 Position Estimation The estimated mover’s position θˆr (elec. rad) computed by (7.56) is also compared with the actual measured electrical mover position in Fig. 7.16a, b for 200 and 600 mm/s respectively. The estimation error in the position estimation is also shown in Fig. 7.16 which clearly proves the effectiveness of the proposed SM-observer. The position estimation error lies within a band of ± 5 electrical degrees during both

7.7 Experimental Evaluation of the SM-Observer

205

8

0 -4 -8 0.6 0.8 1 8 4 0 -4 -8 0.6 0.8 1

Measured Estimated 1.2 1.4 1.6 1.8 2

1.2 1.4 1.6 1.8 2 Time (s)

θ r error (elec. deg.)

θ r error (elec. deg.)

4

θ r (elec. rad/s)

(b)

8

θ r (elec. rad/s)

(a)

4 0 -4

Measured Estimated 1.2 1.4 1.6 1.8 2

-8 0.6 0.8 1 8 4 0 -4 -8 0.6 0.8 1

1.2 1.4 1.6 1.8 2 Time (s)

Fig. 7.16 Position estimation. a 200 mm/s, b 600 mm/s (experiment)

the steady state and speed reversal operation for the speeds of 200 and 600 mm/s. The experimental results of Fig. 7.16 are a clear indication of the position estimation performance of the SM-observer is consistent as the position estimation error remains less than ± 5 degrees even during the regular zero crossings of the speed reference.

7.7.5 Flux Estimation The experimental results comparing the error in the estimation of dq-axis flux for both the SM-observer without sliding mode component (labelled as Linear in Fig. 7.17) and the SM-observer (labelled as Novel in Fig. 7.17) at 200 and 600 mm/s are shown in Fig. 7.17a, b respectively. In Fig. 7.17a, b, the speed reversal occurs at 0.8 s and the steady-state reached from 1 to 1.4 s. It is clear from Fig. 7.17 that the flux estimation error for the linear observer of (7.34) without improved sliding mode function “sgn m ” changes sign at the instants of speed reversal and maintains an average non-zero value during the steady-state. It can be observed from Fig. 7.17a, b that the SM-observer has reduced the flux estimation error thus rationalizes the inclusion of the improved sliding mode component. In addition, it is evident from these figures that the maximum flux estimation error for the SM-observer is of order ±1 × 10−4 which is less than 2% of the rated machine flux, i.e. 0.0846 Wb. This is clear experimental validation of the reliable performance and accuracy of the SM-observer.

206

7 Sensorless Control of a Linear Permanent Magnet Synchronous …

8

λ d error (Wb)

x 10

Linear 1

Novel 1.2 1.4 1.6 1.8 2

λ d error (Wb)

(b)

-4

x 10 5 4 3 2 1 0 -1 -2 -3 -4 -5 0.6 0.8

6 4 2 0 -2 -4 -6 -8 0.6 0.8

-4

x 10 5 4 3 2 1 0 -1 -2 -3 -4 -5 0.6 0.8

Linear 1

Novel 1.2 1.4 1.6 1.8 2

-4

-4

Linear

λ q error (Wb)

λ d error (Wb)

(a)

Novel

1

1.2 1.4 1.6 1.8

Time (s)

2

x 10 8 6 4 2 0 -2 -4 -6 -8 0.6 0.8

Linear Novel

1

1.2 1.4 1.6 1.8

2

Time (s)

Fig. 7.17 λd and λq estimation errors with sliding mode component (novel) and without sliding mode component (linear), a 200 mm/s, b 600 mm/s (experiment)

7.8 Conclusion In this chapter, a multiple-input-multiple-output state space model, having stator flux, thrust force, and mover’s speed as states, is utilized to formulate a robust sliding mode control law to achieve combined speed and direct thrust force control of the linear PMSM. The detailed Lyapunov stability analysis along with the experimental results validates the formulation of the proposed controller. Moreover, a novel flux and speed observer for a surface-mount linear PMSM has been proposed and implemented practically. The proposed observer is a combination of linear state observer and an improved sliding mode component to ensure accuracy and robustness. The proposed observer provides an effective method for speed estimation for the surface mount linear PMSM as a conventional signal injection based method cannot be employed for a surface mount linear PMSM. Experimental results have proved the accuracy of the proposed scheme for sensorless control of the linear PMSM. Stable performance is observed for the linear PMSM controlled using the combined speed and direct thrust force control at various speeds under both transient and steady-state conditions.

References 1. K. Yoshida, Z. Dai, M. Sato, Sensorless DTC propulsion control of PM LSM vehicle, in Proceedings of Power Electronics and Motion Control Conference (IPEMC), vol. 1 (2000), pp. 191–196

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Chapter 8

Conclusions and Future Work

8.1 Conclusions This book primarily focuses on the DTFC of tubular surface-mount linear PMSMs. It is important to note that key features of tubular surface-mount linear PMSMs include high acceleration, wide speed range and high precision, making them a prime candidate for industrial automation and servo-mechanisms. Therefore, development of high-performance control schemes for these linear PMSMs is of prime importance. This book proposes and rigorously analyses a number of novel DTFC schemes and experimentally validates these for a prototype tubular surface-mount linear PMSM. In addition, a combined sliding mode adaptive flux observer for sensorless speed estimation of the surface-mount linear PMSM is also proposed and experimentally validated. The mathematical analysis of surface-mount linear PMSMs demonstrates that these machines have a low value of stator inductance compared to their rotational counterparts due to a relatively larger air gap. Another important feature of the linear PMSMs that plays a pivotal role in thrust force regulation is a short polepitch. Consequently, linear PMSMs tend to exhibit higher ripple in thrust force under conventional direct thrust force control which is typically beyond acceptable limits. The duty ratio based DTFC (DTFC1 of Chap. 3) proposed in this book significantly reduces the ripple in thrust force and stator flux. In addition, DTFC1 ensures an improved transient and steady-state performance of thrust force and stator flux under various operating conditions when compared to the state of the art. As the duty ratio calculation is based on the knowledge of machine parameters, therefore the DTFC1 is most suitable when machines parameters are known and a fast transient response is desired. However, parameter variation or a long sampling period may deteriorate the performance of the controller. The variable switching frequency is another limitation of this approach caused by the selection of the inverter voltage vector by the hysteresis based thrust force and stator flux controllers.

© Springer Nature Switzerland AG 2020 M. A. M. Cheema and J. E. Fletcher, Advanced Direct Thrust Force Control of Linear Permanent Magnet Synchronous Motor, Power Systems, https://doi.org/10.1007/978-3-030-40325-6_8

209

210

8 Conclusions and Future Work

The PI-DTFC is a SV-PWM based control scheme. The transient and steady-state response of the stator flux and thrust force can be shaped by appropriate tuning of the gains for PI controllers. A detailed methodology is presented to compute these gains for a desired control performance of the PI controllers. PI-DTFC allows computation of the PI controllers’ gains to achieve any specified damping ratio and phase margin. A high value of integral gain for the thrust force PI controller results in a smaller rise time but due to low inductance of the linear PMSM, the overshoot in the transient response and steady-state ripple in thrust force increase. Therefore, under PI-DTFC the gains of PI controller are tuned to achieve a compromise between the transient and steady-state performance indicators such as steady-state error, overshoot and transient response. However it is important to note that the transient response of the thrust force under PI-DTFC may not be the fastest due to the above compromise. An improvement to PI-DTFC is realised in form of the Optimal-DTFC1. The control law under the Optimal-DTFC1 is formulated using the linear quadratic regulator approach and the state feedback gains are computed to achieve the optimal control performance in terms of transient response with no overshoot. It is observed that under Optimal-DTFC1 a significantly faster transient response of thrust force with no overshoot is achieved when compared to PI-DTFC. However, it is important to note that under both PI-DTFC, and Optimal-DTFC1 the speed control loop is closed by another PI controller to generate the required thrust force reference. The two methods are also suitable for the applications where independent thrust force reference is required to be tracked. The need of an additional PI controller for the speed control loop is eliminated in Optimal-DTFC2 (Chap. 5), SM-DTFC1 (Chap. 6). Optimal-DTFC2 provides a combined control of speed and thrust force based on the linear quadratic regulator approach. It is important to note that under Optimal-DTFC2 the thrust force is intrinsically controlled according to maximum force per ampere trajectory (MFPA) and no external thrust force reference is required. The control law in SM-DTFC1 is based on sliding mode control theory and the integral action is augmented by modification of the reachability condition. This control law provides robustness to electrical and mechanical parameter variation of the machine. It should be noted that under both the Optimal-DTFC2 and SM-DTFC1 a transient and steady-state speed response comparable to that of PI-DTFC and Optimal-DTFC2 can be achieved without any additional speed PI controller. The stator flux and sensorless speed estimation which is essential for the implementation of the abovementioned control schemes is performed by proposing an adaptive flux observer based on a linear state observer combined with a modified sliding mode observer. The modified sliding mode observer adds robustness in the overall flux and speed estimation schemes and also improves the observer’s performance during the speed reversal as validated by the experimental result. In this research this observer is implemented with the SM-DTFC2 control scheme. SMDTFC2 is similar to SM-DTFC1, however in SM-DTFC2 the sliding surface are defined in terms of the integral of tracking errors in speed and flux. Moreover the integral action is added by including the sliding surface corresponding to the speed error in the speed control law. In SM-DTFC2, this sliding surface adds a PID effect

8.1 Conclusions

211

in the control law which reduces the overshoot and steady-state ripple, however the transient response may become slower compared to SM-STFC1. SM-DTFC1 is a natural choice where faster transient response is desired.

8.2 Main Contributions of the Book The key contributions of this book: 1. Detailed analysis of conventional DTFC for linear PMSM. (Chap. 3) A rigorous analysis and experimental evaluation of switching table based conventional DTFC for the prototype linear PMSM is presented. The Lyapunov stability analysis of the switching table based conventional DTFC for the linear PMSM is performed. The effect of various inverter voltage vectors on the variations in stator flux and thrust force under different operating conditions is evaluated. This analysis concludes that due to the low stator inductance and short pole pitch of the prototype linear PMSM, whenever a voltage vector is applied to the machine for the whole duration of the sampling period the change in the thrust force is much higher than the required change. Therefore, significantly larger ripple in the stator flux and thrust force response under the conventional DTFC is observed for the machine under study. Despite the proven stability of conventional-DTFC for the surface-mount linear PMSMs, the quality of control performance in terms of thrust force and stator flux ripple is critically dependent on the stator inductance and pole pitch of the machine and necessitates the reduction in duty ratio of the applied voltage vector to reduce the ripple in thrust force. 2. A duty-ratio calculation method to reduce thrust force and stator flux ripple in conventional DTFC. (Chap. 3) A novel approach for the calculation of duty ratio for DTFC that reduces the ripple and steady-state error in the flux and thrust force response of the linear PMSM drive utilizing switching table based DTFC is proposed. Analytical expressions for determination of the duty ratio, considering the machine parameters and the mover’s speed, are derived. Experimental results, including start-up performance, speed reversal, and force transients, clearly indicate that the novel technique exhibits excellent control of flux and thrust force with lower ripple, faster transient response and reduced steady-state error when compared to the prior duty-ratio based DTFC technique. 3. A comprehensive procedure for tuning of PI controllers for SV-PWM based PI-DTFC schemes. (Chap. 4) An analysis and detailed experimental evaluation of PI controller based DTFC (PIDTFC) utilizing SV-PWM is provided. A detailed approach for the design of the stator flux and thrust force PI controllers is also presented. In addition, analytical expressions to compute the gains for stator flux and thrust force PI controllers are derived and experimentally validated. The PI-DTFC scheme is also set as a benchmark for comparison with other SV-PWM control schemes proposed in the book.

212

8 Conclusions and Future Work

4. Linear Quadratic Regulator based direct thrust force control of linear PMSM (Chap. 4) An optimal DTFC scheme referred to as Optimal-DTFC1 is proposed. For this purpose, a novel multiple-input multiple-output (MIMO) state space model for the linear PMSM, comprising the stator flux and thrust force as states, is formulated which subsequently allows an optimal linear state feedback control law for DTFC to be synthesized using the optimal linear quadratic regulator approach. Integral action is incorporated in the control scheme by state augmentation of the proposed model to reduce the steady-state error. Experimental results clearly indicate that Optimal-DTFC1 exhibits improved control of stator flux and thrust force with faster transient response and reduced steady-state error when compared to PI-DTFC. 5. Combined speed and direct thrust force control based on the linear quadratic regulator based technique. (Chap. 5) An optimal combined speed and direct thrust force control scheme referred to as Optimal-DTFC2 is proposed and experimentally validated. A 3rd order MIMO state space model for the linear PMSM, with the stator flux, thrust force and mover’s speed as states, is formulated. Importantly, the state transition matrix for the proposed state space model is independent of the mover’s speed and therefore, controllable over the whole speed range of the linear PMSM. An optimal linear state feedback law for combined speed and direct thrust force control is synthesized in terms of stator flux, thrust force, and mover’s speed using the optimal linear quadratic regulator approach. Integral action is incorporated in the control scheme by state augmentation of the optimal state space model to reduce the steady-state error. The optimal state feedback gains being independent of mover’s speed can achieve optimal combined speed and direct thrust force control of the linear PMSM for the whole speed range. 6. Sliding mode based combined speed and direct thrust force control (Chap. 6) A sliding mode control scheme (referred to as SM-DTFC1) that achieves the combined speed and direct thrust force control for linear PMSM is proposed. The combined dynamics of thrust force and speed are described by a general 2nd order non-linear state space model with y-axis voltage as input and thrust force and speed as system states. In order to eliminate the steady-state error and to ensure a fast transient response, the control law is augmented with integral action by using a modified reaching condition. The stability analysis of the proposed controller is discussed in detail. The SM-DTFC1 is experimentally validated and improvements in the transient and steady-state response are observed when compared with PI-DTFC. 7. A combined sliding mode based adaptive flux observer for sensorless speed estimation of linear PMSM (Chap. 7) A stator flux and speed observer comprising a linear state observer and an improved sliding mode component to ensure accuracy and robustness is proposed and experimentally validated. The proposed observer provides an effective method for speed

8.2 Main Contributions of the Book

213

Table 8.1 Comparison of transient performance in terms of rise time of PI-DTFC, optimal-DTFC1, optimal-DTFC2, and SM-DTFC1 using IAE index Type of transient phenomena

Rise time (ms) for speed response during the transient PI-DTFC

Optimal-DTFC1

Optimal-DTFC2

SM-DTFC1

Start-up (0 to 200 mm/s)

35.2

30.1

31.0

30.0

Speed reversal (−600 to 600 mm/s)

66.8

59.6

61.0

59.9

Table 8.2 Comparison of steady-state performance of PI-DTFC, optimal-DTFC1, optimal-DTFC2 and SM-DTFC1 600 mm/s, 52 N

PI-DTFC

Optimal-DTFC1

Optimal-DTFC2

SM-DTFC1

λrip (%)

0.34

0.238

0.21

0.19

F rip (%)

10.48

6.21

5.83

5.91

vrip (%)

1.92

1.13

1.11

1.08

estimation of the surface mount linear PMSM as a conventional signal injection based method cannot be employed for a surface-mount linear PMSM. Moreover, an integral sliding mode control (SM-DTFC2) to achieve combined speed and direct thrust force control of the linear PMSM is derived and experimentally validated. The sliding surfaces are formulated in terms of the integral of the tracking errors in the stator flux and speed. The comparison of transient response in terms of the rise time during start up and speed reversal for various control schemes proposed in this book is summarised in Table 8.1. The steady state performance comparison for these proposed control techniques is provided in Table 8.2.

8.3 Future Work In this book several novel control schemes for direct thrust force control of linear PMSMs are proposed. However these control schemes are only tested for the surfacemount linear PMSM. As future research, the application of these control schemes can be extended to interior permanent magnet linear motors. As discussed in Chap. 4, the linearization co-efficient is constant for the prototype surface-mount linear PMSM due to low stator inductance and short pole-pitch. However, in case of interior permanent magnet linear motors with larger saliency ratios, the linearization co-efficient may not be constant and will vary as a nonlinear function of operating thrust force and therefore one set of gains for thrust force PI in PI-DTFC will no longer be able to control the machine for the whole operational range of thrust force. It is important to perform gain scheduling of the thrust force PI

214

8 Conclusions and Future Work

controller using the analytical expression derived in Chap. 4 for interior permanent magnet linear motors and this should be validated experimentally. One of the key features of interior permanent magnet linear motors is wide speed operation under field-weakening control due to their saliency. Therefore, it is also of interest to evaluate control schemes proposed in the book for field-weakening control of interior permanent magnet linear motors. For this purpose the reference flux for the DTFC schemes needs be calculated under field-weakening control trajectory instead of maximum force per ampere as discussed in this book. It is important to note that the dynamic modelling of linear PMSM performed in this book does not consider the core loss resistance; therefore future studies may include the core loss resistance in the dynamic modelling of linear PMSM. The novel duty ratio calculation method proposed in Chap. 3 is parameter dependent and a parameter mismatch may deteriorate the performance of the proposed technique. Therefore, a parameter adaption algorithm can be added with duty ratio based DTFC to ensure the robustness of the control scheme. Although the novel DTFC schemes presented in this book are validated for the linear PMSM, these schemes are general in the sense that they can also be extended to rotational PMSMs. It is worth mentioning that, due to its inherent robustness, the sliding mode based combined speed, and direct thrust force control (Chap. 6) can be of great interest for field-weakening control of concentrated wound IPM machines where machines parameters depend on the operating conditions of the machines.

Appendix A

Description of the Experimental Setup

A.1 Description of the Prototype Tubular Surface-Mount Linear PMSM The prototype tubular surface-mount linear PMSM used in this research is Model No. STA 2504S manufactured by Dunker Motor Advanced Motion Solutions and is categorised as servo tube by the manufacture. The complete experimental setup is shown in Fig. A.1. The manufacturer website for the prototype linear PMSM is: https://www.dunkermotoren.com.

A.2 Description of 3-Phase Voltage Source Inverter The 3-phase 2-level voltage source inverter used for the prototype linear PMSM is manufactured by Semikron. The model no. “SKS 35F B6U+E1C1F+B6C1 21 V12”. The manufacturer website is: https://www.semikron.com.

A.3 Description of Voltage Sensing Board The schematic diagram for the voltage sensing board is provided in Fig. A.2.

A.4 Description of Current Sensing Board The schematic diagram for the current sensing board is provided in Fig. A.3.

© Springer Nature Switzerland AG 2020 M. A. M. Cheema and J. E. Fletcher, Advanced Direct Thrust Force Control of Linear Permanent Magnet Synchronous Motor, Power Systems, https://doi.org/10.1007/978-3-030-40325-6

215

216

Appendix A: Description of the Experimental Setup Braking Resistor

Load LPMSM

Prototype LPMSM Interface /Measurement Board DS1104 Controller Board

3-Phase Inverter

Fig. A.1 Experimental setup

A.5 Description of dSPACE© DS 1104 R&D Controller Board In this research, DS 1104 controller board is used for the implementation of the control algorithms, processing of the feedback signals from current, voltage and speed sensors and generation of PWM signals for the voltage source inverter. The DS1104 controller board is specifically designed for development of high speed multilevel digital controllers and real-time simulations in various fields. It is a complete realtime control system based on a 603 Power PC floating point processor of 250 MHz. For advance I/O purposes, the board includes a slave DSP subsystem which performs digital input and output along with generation of PWM signals. The heart of this subsystem is a TMS320F240 digital signal processor from Texas Instruments. The controller board can be directly programmed using MATLAB/SIMULINK or C program. An overview of the features provided by the DS1104 board and technical specifications of the board are given in the 2. The architectural overview of the DS 1104 board is provided in Fig. A.4. The technical specifications of the controller board are provided in Table A.1.

-15

2

1

2

1

+15

-15

1

-15

R7 100R

5K6 R8

R9 5K6

R1 5K6

R2 5K6

6

5

2

P3 20K

7

C4 10nF

U1B LF347

8

330pF

LF347

C3

R12 22K

9

10

10nF

C2 U1C

330pF

P1 20K

C1

R3 22K

2

3

-15

1

U1A LF347

LF347

U1D

+15

13

12

11

JUMPER

J2

2

-15

+15

R6 10K

J1

JUMPER

4

R11 10K

P4 2K

R10 10K

-15V

M

+15V

VT1

+15

-

+

LV25P

C14 10uF

C13 10uF

+15

R5 10K C15 10uF P2 2K

14

3

D3 1N4148 R13 10K

D2

9.1VZ

A4 Date: File:

+15

4

7

R16 10K

+5

V VW

1 6 2 7 3 8 4 9 5

+15

4

10-Feb-2004 C:\My Documents\appendix\3chvt.DDB

Number

4

Sheet 1 of Drawn By:

CN1 DB9 C

D

3 KVB

Revision R0

P5 10K

R14 10K

A

0-CROSS -V VW B

3 CH VOLTAGE SENSOR

R15 10K

3

2

U4 LM311

1

Size

Title

C5 47pF

D1

9.1VZ

R4 1K

3

6 8

1

Fig. A.2 Schematic circuit diagram of voltage sensing board

1

CN3 CON2

CN4 CON2

4

3

2

1

+5

2

5

A

47K 25W +V -V

+5V +15V 0V -15V

CN2 CON4

2

B

C

D

1

Appendix A: Description of the Experimental Setup 217

-15

U1B LF347

10nF

7 2

3

+15

3

6

10K

R17

4

IW

CN2 DB9

11

1 A4 Date: File:

11-Feb-2004 C:\My Documents\appendix\3chct.DDB

Number

4

Sheet 1 of Drawn By:

3 KVB

R0

Revision

1 6 2 7 3 8 4 9 5

3CH CURRENT SENSOR BOARD

R21 100K

CN1 DB15

Title

1N4148 D5

1 9 2 10 3 11 4 12 5 13 6 14 7 15 8

CN3

CURRENT LIMIT SIGNAL

4

Size

C6 27pF

7

I W(CL)

2

D2 9.1VZ

D1 9.1VZ

D4

5

1

-15

1

LF347 U1A

R4 1K

10K

R15

C5 27pF

-15

LF34 7

1

7

R20 4K7

0-CROSS-V VW

1

6

5

C2

2

U2B LF347

I V (FAULT)

2 3

+5

V VW

Fig. A.3 Schematic circuit diagram of current sensing board

2

4

R1 5K6

R2 5K6

10K R13 10K

R12

U2 A

D3

R19 10K

27pF

C7

I U (FAULT)

R18 10K

LM311

+15 U6

0-CROSS-V UV

A

1

2

CT3

+15

330pF

14

3

+15

-15

R23 10K R16 10K

P13 10K

R22 10K

V UV

IW

6

C1

13

12

U1D LF347

11

P3 50K

U1C LF347

8

10nF

C4 4

-15

R3 22K

9

10

330pF

C3

R14 10K

+15

5

R6 10K

1

R7 5K6

R8 5K6

R9 22K

P1 20K

C22 10uF

GND

2K

J1

C21 10uF

C20 10uF

1N4148

P4

2

JUMPER

J2

+15

JUMPER

-15

+5

4

B

R5 10K

R11 10K

2K

P2

R10 10K

4

3

2

1

3

V DC

+15

-15

+15

-15V

0V

+15V

CN4 CON4

2

6 8

C

D

+5V

1

A

B

C

D

218 Appendix A: Description of the Experimental Setup

1N4148

Appendix A: Description of the Experimental Setup

219

Fig. A.4 Architectural overview of DS 1104 controller board Table A.1 Technical specifications of the DS1104 controller board Manufacturer

dSPACE GmbH Technologiepark 25, 33100 Paderborn, Germany

Processor

• MPC 8240 with PPC630e core and on chip peripherals • 64-bit floating point processor • 250 MHz CPU

Memory

• Global memory: 32 MB SDRAM • flash memory: 8 MB

ADC

• 4 multiplexed channels 16 bit resolution, 2 µs conversion time • 4 A/D channels, 12 bit resolution and 800 ns conversion time

Incremental encoder interface

• • • • • •

2 channels Single ended TTL or differential RS422 input 4 fold subdivision Max 1.65 MHz 24-bit loadable position counter Rest on index

Slave DSP

• • • •

Texas instruments TMS320F240 DSP 20 MHz clock frequency 1X3-phase PWM output 4X1 phase PWM output

Appendix B

Derivation of Expressions for

B.1 Derivation of Expression for Voltages The values of i α , i β ,

dλα dt

and

dλβ dt

d FT dt

d FT dt

and

dλs dt

in Terms of Inverter

can be achieved from (2.94) to (2.97) as: dλα = vα − Rs i α dt

(B.1)

dλβ = vβ − Rs i β dt

(B.2)

iα =

λα − λ pm,α Ls

(B.3)

iβ =

λβ − λ pm,β Ls

(B.4)

By substituting (B.1) to (B.4) in (2.109) yields:    3π di β  di α d FT = k F P (vα − Rs i α )i β + λα − vβ − Rs i β i α − λβ dt 2τ dt dt

(B.5)

Equation (B.5) simplifies to:   3π di β di α d FT = k F P vα i β − vβ i sα + λα − λβ dt 2τ dt dt

(B.6)

di

Now the values of didtα and dtβ can be achieved by differentiation of (B.3) and di (B.4) respectively. Substituting these values of didtα and dtβ into (B.6): © Springer Nature Switzerland AG 2020 M. A. M. Cheema and J. E. Fletcher, Advanced Direct Thrust Force Control of Linear Permanent Magnet Synchronous Motor, Power Systems, https://doi.org/10.1007/978-3-030-40325-6

221

222

Appendix B: Derivation of Expressions for

d FT dt

and

dλs dt

     dλ pm,β dλ pm,α 3π λβ dλα d FT λα dλβ = k F P vα i β − vβ i α + − − − dt 2τ L s dt dt L s dt dt (B.7) The value of

dλr α dt

can be obtained by differentiating (2.98) as:   d λ f cos θr dλ pm,α dθr = = −λ f sin θr dt dt dt

(B.8)

From (2.99), it is clear that λ f sin θr = λ pm,β , therefore (B.8) becomes: dλ pm,α = −λ pm,β ωr dt Similarly, the values of

dλrβ dt

can be achieved by differentiating (2.98) as: dλ pm,β = λ pm,α ωr dt

Substituting of values of (B.7):

(B.9)

dλ pm,α dt

and

dλ pm,β dt

(B.10)

from (B.9) and (B.10) respectively, into

  3π d FT λα  = k F P vα i β − vβ i α + vβ − Rs i β − λ pm,α ωr dt 2τ Ls   λβ  vα − Rs i α + λ pm,β ωr − Ls

(B.11)

Substituting the values of i α and i β from (B.3) and (B.4) in the first two terms of (B.11) within the brackets results in:      λβ − λ pm,β λα − λ pm,α 3π d FT − vβ = k F P vα dt 2τ Ls Ls   λβ   λα  + vβ − Rs i β − λ pm,α ωr − vα − Rs i α + λ pm,β ωr Ls Ls

(B.12)

Simplifying (B.13) by re-arranging terms: 3 π kF P d FT = (vα λβ − vα λ pm,β − vβ λα + vβ λ pm,α + λα vβ − λα Rs i β dt 2 τ Ls  (B.13) −λα λ pm,α ωr − λβ vsα + λβ Rs i α − λβ λ pm,β ωr Equation (B.13) can be re-arranged to cancel the similar terms with opposite signs as:

Appendix B: Derivation of Expressions for

d FT dt

and

dλs dt

223

(B.14) The zero terms in (B.14) can be omitted to achieve the following expression for as:

d FT dt

  d FT 3 π kF P  = −Rs λα i β − λβ i α − vα λ pm,β + vβ λ pm,α dt 2 τ Ls   −ωr λα λ pm,α + λβ λ pm,β

(B.15)

In (B.15), λα λ pm,α +λβ λ pm,α represents the scalar product of the stator flux space − → − → vector λs and the rotor flux space vector λ f in αβ-reference frame. It is clear form − → − → Fig. 2.8 that the angle between λs and λ f is δ therefore: − →− → λsα λr α + λsβ λβα = λs . λr = λs λ f cos δ

(B.16)

From (B.15) into (B.16):    3 π kF P  d FT = −Rs λα i β − λβ i α − vα λ pm,β + vβ λ pm,α − ωr λs λ f cos δ dt 2 τ Ls (B.17) Equation (B.17) is simplified by re-arranging the terms: ⎛

⎞

⎟  Rs ⎜ 3 π d FT =− ⎜ k F P λα i β − λβ i α ⎟ ⎠ dt L s ⎝ 2 τ

 FT

 3 π kF P  −vα λ pm,β + vβ λ pm,α − ωr λs λ f cos δ + 2 τ Ls

(B.18)

  From (1.62), it is clear that in (B.18), 23 πτ P λα i β − λβ i α represent the thrust force FT , therefore (B.18) can be written as:  d FT Rs 3 π kF P  = − FT + −vα λ pm,β + vβ λ pm,α − ωr λs λ f cos δ dt Ls 2 τ Ls

(B.19)

Since, FT on right hand side of (B.19) is the initial operating thrust force at the current instant of time; therefore replacing FT by F0 results in:

224

Appendix B: Derivation of Expressions for

d FT dt

 Rs d FT 3 π kF P  = − F0 + −vα λ pm,β + vβ λ pm,α − ωr λs λ f cos δ dt Ls 2 τ Ls

and

dλs dt

(B.20)

Substituting the value of ωr from (2.44) into (B.20):  d FT 3 π kF P  π Rs −vα λ pm,β + vβ λ pm,α − P λs λ f vm cos δ (B.21) = − F0 + dt Ls 2 τ Ls τ The effect of the various voltage vectors applied by the voltage source inverter on the thrust force of the surface-mount linear PMSM can be evaluated by using (B.21). Equation (B.21) can be solved using the prototype linear PMSM parameters in Table 1.1 for various voltage vectors to compute the rate of change of thrust force.

B.2 Derivation of Expression for Voltages

dλs dt

in Terms of Inverter

In order to formulate the effect of the inverter voltage vectors on the rate of change of stator flux, first the magnitude of the stator flux space vector is expressed in terms of αβ-components by using (2.132) as: λs =

 λ2α + λ2β

(B.22)

Taking the time derivative of (B.22):   1 dλs dλα dλβ λα = + λβ dt dt dt λ2α + λ2β Substituting (2.94) to (2.97) in (B.23), the following expression for given as:  Rs Rs 1 dλs = − λs + vα λα + vβ λβ λ f cos δ + dt Ls Ls λs

(B.23)

dλs dt

can be

(B.24)

The rate of change in the stator flux magnitude is computed from (B.24) using the parameters of the prototype linear PMSM.