Modeling and Adaptive Nonlinear Control of Electric Motors [1st ed.] 978-3-540-00936-8;978-3-662-08788-6

Table of contents :
Front Matter ....Pages N1-x
Introduction (Farshad Khorrami, Prashanth Krishnamurthy, Hemant Melkote)....Pages 1-15
Dual-Axis Linear Stepper (Sawyer) Motors (Farshad Khorrami, Prashanth Krishnamurthy, Hemant Melkote)....Pages 17-25
Modeling of Stepper Motors (Farshad Khorrami, Prashanth Krishnamurthy, Hemant Melkote)....Pages 27-45
Stepping (Farshad Khorrami, Prashanth Krishnamurthy, Hemant Melkote)....Pages 47-54
Feedback Linearization and Application to Electric Motors (Farshad Khorrami, Prashanth Krishnamurthy, Hemant Melkote)....Pages 55-76
Robust Adaptive Control of a Class of Nonlinear Systems (Farshad Khorrami, Prashanth Krishnamurthy, Hemant Melkote)....Pages 77-108
Robust Adaptive Control of Stepper Motors (Farshad Khorrami, Prashanth Krishnamurthy, Hemant Melkote)....Pages 109-153
Current Control of Stepper Motors Using Position Measurements Only (Farshad Khorrami, Prashanth Krishnamurthy, Hemant Melkote)....Pages 155-177
Voltage Control of Stepper Motors Using Position and Velocity Measurements (Farshad Khorrami, Prashanth Krishnamurthy, Hemant Melkote)....Pages 179-214
Voltage Control of PM Stepper Motors Using Position Measurement Only (Farshad Khorrami, Prashanth Krishnamurthy, Hemant Melkote)....Pages 215-239
Brushless DC Motors (Farshad Khorrami, Prashanth Krishnamurthy, Hemant Melkote)....Pages 241-273
Induction Motor: Modeling and Control (Farshad Khorrami, Prashanth Krishnamurthy, Hemant Melkote)....Pages 275-314
Adaptive Control of Induction Motors (Farshad Khorrami, Prashanth Krishnamurthy, Hemant Melkote)....Pages 315-353
Passivity-Based Control of Electric Motors (Farshad Khorrami, Prashanth Krishnamurthy, Hemant Melkote)....Pages 355-382
Torque Ripple Reduction for Step Motors (Farshad Khorrami, Prashanth Krishnamurthy, Hemant Melkote)....Pages 383-400
Friction Compensation in Servo-Drives (Farshad Khorrami, Prashanth Krishnamurthy, Hemant Melkote)....Pages 401-442
Back Matter ....Pages 443-523

Citation preview

Power Systems F. Khorrami . P. Krishnamurthy . H. Melkote Modeling and Adaptive Nonlinear Control of Electric Motors

Springer-Verlag Berlin Heidelberg GmbH

Engineering

ONLINE LlBRARY

http://www.springer.de/engine/

F. Khorrami . P. Krishnamurthy . H. Melkote

Modeling and Adaptive Nonlinear Control of Electric Motors With 184 Figures

Springer

Prof. Farshad Khorrami Prashanth Krislmamurthy Dr. Hemant Melkote ControllRobotics Research Laboratory (CRRL) Department ofElectrical and Computer Engineering Polytechnic University Six Metrotech Center Brooklyn, NY 11201 USA

E-mail: [email protected] [email protected] [email protected]

ISBN 978-3-642-05667-3 ISBN 978-3-662-08788-6 (eBook) DOI 10.1007/978-3-662-08788-6 Cataloging-in-Publication Data applied for Bibliographie information published by Die Deutsche Bibliothek. Die Deutsche Bibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographie data is available in the Internet at

This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on mierofilm or in other ways, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag Berlin Heidelberg GmbH. Violations are liable for prosecution under the German Copyright Law.

http://www.springer.de © Springer-Verlag Berlin Heidelberg 2003

Originally published by Springer-Verlag Berlin Heidelberg New York in 2003 Softcover reprint of the hardcover 1st edition 2003 The use of general descriptive names, registered names, trademarks, 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. Typesetting: Camera ready by authors Cover-design: deblik, Berlin 62/3020 hu - 5 43 2 1 0 Printed on acid-free paper

-

To my wife, Brenda Farshad Khorrami

To my parents Prashanth Krishnamurthy

To my parents H emant M elkote

Preface Electromechanical actuators have been utilized in many applications from horne appliances to sophisticated guidance and control systems. Various electromechanical actuators such as electric motors, hydraulic and pneumatic actuators, smart materials (e.g., piezoceramics, magnetorestrictive materials, shapememory alloys, electrorheological fluids, etc.) have been considered. Modeling and control design for such actuators have been and are being pursued to achieve a higher level of performance. In this book, we consider modeling and control design of electric motors; namely step motors, brushless DC motors, and induction motors. These electrical motors are used in many applications; some requiring a high level of accuracy and performance such as machines used in the electronics industry for assembly or semiconductor wafer probing and inspection. It is the intent of this book to focus on recent advances on feedback control designs for various types of electric motors, with a slight emphasis on stepper motors. For this purpose, we explore modeling of these devices to the extent needed to provide a high-performance controller but at the same time amenable to model-based nonlinear designs. We will also focus on more recent efforts on nonlinear and adaptive controllers to derive robust and high-performance feedback controllers, which are essential for applications that require high performance and accuracies. It should also be pointed out that once good models of actuators are developed, controllers should utilize the model as much as possible to achieve high performance and then for robustness purposes, adaptive robust nonlinear controllers should be added to the controlled system. As will be shown, in many cases, the adaptive robust nonlinear controller on its own achieves a reasonably good performance without requiring the exact knowledge of motor parameters. Therefore, the designer needs to consider that the proposed robust adaptive nonlinear controllers need not be augmented with an inner-Ioop controller that utilizes a tight knowledge of motor parameters. Another important point is that although we have considered re cent adaptive and nonlinear control design methodologies, by no means is it implied that if in certain applications, linear or classical designs (such as PID controllers augmented by notch filters and feedforward terms) perform weIl, one should still pursue the techniques presented in this book. It is needless to say that many machines and systems do operate and achieve the required performance via well-designed and

11

carefully tuned classical controllers. But eventually, it is hoped that the advocated robust and adaptive designs will become standard "universal" controllers with minimal need for fine tuning of control parameters before and after release of a product. A design feature that we pursue in the book is sensorless (output feedback) control of various motors. In this vein, we consider controllers which utilize various subsets of states as, for instance, the position and velocity, only position, position and currents, etc. A particular case that we have not covered in this book is having only current measurements to achieve either position or velocity control. Various approaches including utilization of back emf for position estimation, zero crossing techniques, open-loop integration of motor dynamics, etc., have been proposed in the literature and some references to these works are provided in the appropriate chapters in this book.

Organization of the book Chap. 1 provides an introduction to various types of motors and the basic practical considerations while employing them in applications. Detailed explanation of the Sawyer motor (dual-axis linear stepper motor) is deferred to Chap. 2. Appendix A contains some of the fundamental AC machine concepts. Chap. 3 develops detailed mathematical models for stepper motors. Various nonidealities and perturbations that might be encountered in practice are characterized. Modeling of brushless DC motors and induction motors are postponed to Chaps. 11 and 12. Furthermore, the Lagrangian approach for deriving models of various motors is also provided in Chap. 14 where passivity-based results are given. The Direct-Quadrature (DQ) transformation, a common tool used to simplify the dynamics of electromechanical systems is introduced. A more detailed explanation of the DQ transformation and its extensions is contained in Appendix B. Although the modeling has been pursued to a reasonable detail for various motors, it was done with control design in mind; therefore, yielding mathematical models amenable to control design. The remaining chapters present control design issues for various motors. Chap. 4 describes the popular open-loop control techniques of stepping and micro-stepping utilized for step motors. Thereafter, feedback controllers are considered. To this extent, state feedback and output feedback (i.e., partial state measurement) solutions are sought. To make the book somewhat self-contained, Chaps. 5 and 6 and Appendices D, E, F, and G contain important nonlinear design tools utilized throughout this book. Chap. 5 introduces the concepts of relative degree, feedback linearization, system inversion, and zero dynamics. Furthermore, algorithms for stable system inversion are given in Appendix D. Application of feedback linearization to stepper motors are given in this chapter and results for brushless DC and induction motors are postponed to Chaps. 11 and 12. In Chap. 6 and Appendices Fand G, the backstepping technique and its

iii

variants (i.e., tuning functions, robust backstepping, and adaptive backstepping) and the nonlinear small gain results in conjunction with input-ta-state stability are presented. For completeness, Lyapunov stability and results are given in Appendix E. The aforementioned chapters are intended as a quick tutorial for the reader and proofs are omitted (only constructive proofs are included) and several worked examples are provided. The aforementioned robust adaptive nonlinear control design methodologies are presented in Chap. 7 for various types of stepp er motors under the assumption that state variables (i.e., rotor position, rotor velocity, and currents) are available for feedback. Output feedback solutions are presented in Chaps. 8-10. Chap. 8 includes current level control (the fast electrical dynamics are neglected) utilizing only position measurements. Chap. 9 presents voltagelevel control using position and velo city measurements. Chap. 10 provides a technique to eliminate velocity measurements. Control of brushless DC motors whose behavior is similar to that of the stepper motors is briefly explained in Chap. 11. Chaps. 12 and 13 are exclusively devoted to modeling and control of inducti on motors. Various control design approaches such as Field Oriented Control (FOC), a nonlinear extension of FOC, input-output decoupling (and dynamic feedback linearization), and Direct Torque Control (DTC) are given in Chap. 12. Furthermore, since flux measurements are not normally available in induction motors, flux observers and output feedback design based on these observers are given in Chap. 12. The adaptive nonlinear designs for induction motors are pursued in Chap. 13. These designs include both state and output feedback solutions. Another important approach to control design for various motors is the nation of passivation. Therefore, energy-based derivation of models of electric motors and the concept of passivity and some of the theoretical nonlinear results are introduced in Chap. 14. The passivity-based control designs are then applied to the general Euler-Lagrange systems that include motor dynamics. Applications to all types of electric motors are then presented in Chap. 14. Finally, two important issues in motor control designs are tackled in the last two chapters as generic items on their own and may be incorporated in other control designs with some effort. These two issues are: torque ripple minimization (Chap. 15) and friction compensation (Chap. 16). Appendix C treats field weakening, a method for designing current references to maximize torque in the presence of current and voltage constraints.

***************************************** The first author would like to acknowledge the effort and prior work by former students on these topics and the two students who are the coauthors of this book. I also would like to thank several of my colleagues at Brooklyn Poly that I have interacted with over the years: Profs. J.J. Bongiorno, Jr., P. Sarachik, D. Youla,

iv B. Friedland (currently at NJIT), L. Shaw, Z.P. Jiang, Z. Pan, V. Kapila, M. Tai, C. Georgakis, W. Blesser, S. Nourbakhsh, and P. Riseborough. I also would like to thank Profs. Ü. Özgüner, P.V. Kokotovic, H. Khalil, M.W. Spong, T.J. Tarn, and Dr. D. Repperger for their support over the years. I would like to thank Profs. A. Iftar, L. Acar, E. Barbieri, A. Tzes, J. Rastegar, and Dr. D. Schoenwald for their friendship and intellectual discussions over the years. Finally, I also would like to thank Profs. B. Paden, D. Dawson, R. Ortega, M. Bodson, and J. Chiasson for looking over a draft of this book, and for being the originators of several motor control results treated in this book. The first author also acknowledges support of the National Science Foundation, Army Research Office, and several corporations that have made this work possible. I would also like to especially thank Drs. G. Anderson and J. Chandra of the Army Research Office and Drs. N. Coleman and M. Mattice of the Picatinny Army Arsenal for their interest and support of our efforts. Lastly, but not least, I also would like to acknowledge my dear wife, Brenda, who has been a constant source of inspiration and encouragement to get things done and putting up with my many late night hours. Without her, many things would not have been possible. I also would like to thank my sister, Fereshteh, my brothers, Jamshid and Farshid, and most importantly, my mother, Mahin, and my late father, Hadi, who have supported and helped me throughout my life. Their help and support has been and continues to be invaluable. The second author wishes to express his deep gratitude to his parents, Krishnamurthy and Usha Murthy, and to his brother Ravi Murthy, for all their support and encouragement through the years. He would also like to thank Z. Wang and R.S. Chandra for many enlightening discussions. The third author would like to thank the members of the ControljRobotics Research Laboratory, namely, S. Jain, I. Zeinoun, J. Lewinsohn, S. Sankaranarayanan, N. Ahmad, Z. Wang, and I. Cherepinsky with whom I had the opportunity to interact and participate in stimulating discussions.

Brooklyn, NY, April 2003

Farshad Khorrami Prashanth Krishnamurthy Hemant Melkote

Contents 1 Introduction

1.1

Stepper Motors and Classification .

2

1.1.1

Variable Reluctance Stepper Motor .

3

1.1.2

Permanent Magnet Stepper Motor

4

1.1.3

Hybrid Stepper Motor

6

1.1.4

Sawyer Motors . . . .

7

1.2

Brushless DC (BLDC) Motors.

7

1.3

Induction Motors . . . .

8

1.4

Practical Considerations

9

1.4.1

Power Losses ..

9

1.4.2

Thermal Effects .

10

1.4.3

Current and Voltage Constraints

10

1.4.4

Power Electronics and Drive Circuits .

11

1.4.5

Finite Word Length and Noise . . . .

12

1.5

2

1

Nonlinear Control Design Applications to Electric Motors and Open Problems . . . . . . . . . . . . . . . . . . . . . . . . . . ..

13

Dual-Axis Linear Stepper (Sawyer) Motors

17

2.1

A Dynamical Model for Sawyer Motor

20

2.2

Position Sensing and Control Issues .

21

3 Modeling of Stepper Motors 3.1

27

Variable Reluctance Stepper Motor.

30

3.1.1

32

Models with Saturation . . .

vi

CONTENTS

3.2

Permanent Magnet Stepper Motor

33

3.3

Sawyer Motor . . . . . . . . . . . .

38

3.3.1

Modeling of Hysteresis and Eddy Current Losses

43

3.3.2

Disturbance due to Umbilical and Load Force.

43

3.3.3

Cogging and Ripple Force

43

3.4

4

Resonance .

47

Stepping 4.1

4.2

Full-Stepping

48

4.1.1

VR Motors

48

4.1.2

PM Motors

49

4.1.3

Sawyer Motors

50

Micro-Stepping . .

50

4.2.1

VR Motors

51

4.2.2

PM Motors

52

4.2.3

Sawyer Motors

54

5 Feedback Linearization and Application to Electric Motors 5.1

44

55

Feedback Linearization Theory

55

5.1.1

SISO Systems ..

55

5.1.2

MIMO Systems.

65

5.2

System Inversion . . . .

68

5.3

Feedback Linearizing Designs for Step Motors

70

5.3.1

VR Motors

70

5.3.2

PM Motors

72

5.3.3

Sawyer Motors

74

6 Robust Adaptive Control of a Class of Nonlinear Systems

77

6.1

Robust Control Under Strict Matching . . . . . . . . . . . . .

77

6.2

Nonlinear Systems of the Special Strict Feedback Form: Characterization . . . . . . . . . . . .

83

6.2.1 6.3

Robust Tracking Design .

Robust Output Feedback Design

85 92

vii

CONTENTS

6.3.1 6.4

The Design Procedure . . . . . . . . . . . . .

A Robust Control Design for Multi-Output Systems

7 Robust Adaptive Control of Stepper Motors 7.1

Previous Work . . . . . . .

93 102 109 109

7.1.1

VR Stepper Motors

111

7.1.2

PM Stepp er Motors

111

7.1.3

Sawyer Motors . . .

112

7.2

Problem Formulation for Various Stepper Motors

112

7.3

Control of VR Stepp er Motors

114

7.3.1 7.4 7.5

Simulation Results . . .

123

Control of PM Stepper Motors

130

7.4.1

Simulation Results

136

Control of Sawyer Motors

141

7.5.1

148

Simulation Results

8 Current Control of Stepper Motors Using Position Measurements Only 155 8.1

8.2

Current Controllers for VR and PM Stepper Motors

155

8.1.1

Commutation . . .

161

8.1.2

Simulation Results

161

Sawyer Motors . . . . . .

168

8.2.1

174

Simulation Results

9 Voltage Control of Stepper Motors Using Position and Velo city Measurements 179 9.1

9.2

PM Stepper Motors . . .

180

9.1.1

Simulation Results

194

Sawyer Motors . . . . . .

197

9.2.1

208

Simulation Results

10 Voltage Control of PM Stepper Motors Using Position Measurement Only 215 10.1 Feedback Design Assuming Knowledge of Electrical Time Constant215

CONTENTS

Vlll

10.1.1 Simulation Results . . . . . . . . . . . . . . . . . . . . . . 226 10.2 Feedback Design Assuming Known Bounds on Motor Parameters 228 10.2.1 Simulation Results 11 Brushless OC Motors

235 241

11.1 Electro-Mechanical Model

242

11.2 Feedback Linearizing Design.

249

11.3 Robust Adaptive Control Design

251

11.3.1 Application to Multi-Link Manipulators

256

11.3.2 Simulation Results . . . . . . . . . . . .

257

11.4 Current-Level Control of BLDC Motors Using Position Measurements Only . . . . . . . . . . . . . . . . . .

264

11.5 Torque Ripple Reduction for BLDC Motors

265

12 Induction Motor: Modeling and Control

275

12.1 Modeling . . . . . . . . . . . . . . . . . .

277

12.2 Field Oriented Control (FOC): Asymptotic Input-Output Linearization . . . . . . . . . . . . . . . . . . . . . .

283

12.2.1 A Nonlinear ISS-Based Extension of FOC

284

12.2.2 Simulation Results

286

12.3 Input-Output Decoupling

287

12.4 Dynamic Feedback Linearization

289

12.4.1 Integrator in the q-axis

289

12.4.2 Integrator in the d-axis

292

12.5 Direct Torque Control (DTC)

294

12.5.1 Simulation Results

299

12.6 Flux Observers . . . . . .

299

12.6.1 Observer Based on Position and Currents

299

12.6.2 Observer Based on Velocity and Currents

304

12.7 Output Feedback Control Design . . . . 13 Adaptive Control of Induction Motors

13.1 Adaptive State Feedback Control . . . .

305 315

. 316

CONTENTS

ix

13.1.1 All Parameters Unknown . . . . . . . . . . . . . . . . . . 316 13.1.2 A Singularity-Free Adaptive Controller Assuming Lower Bound on aL sr . . . . . . . . . . . . . . . . . . . . . . . . 320 13.1.3 A Lower-Order Dynamic Controller for Practical Tracking 326 13.2 Adaptive Output Feedback Control . . . . . . . . . . . . . . . . . 329 13.2.1 Only Rotor Velo city Measured: Mechanical Parameters Unknown . . . . . . . . . . . . . . . . . . . . . . . . ..

329

13.2.2 Load Torque, Rotor Resistance, and Friction Unknown .

333

13.2.3 Mechanical Parameters and Rotor Resistance Unknown

343

13.2.4 Simulation Results . . . . . . . . . .

351

14 Passivity-Based Control of Electric Motors

355

14.1 Dissipativity and Passivity .

355

14.2 Euler-Lagrange Systems. .

362

14.3 Electric Motors as Euler-Lagrange Systems

366

14.4 Passivity-Based Control of Electric Motors.

369

14.4.1 VR Stepper Motors

377

14.4.2 PM Stepper Motors

378

14.4.3 Brushless DC Motors

379

14.4.4 Induction Motors. . .

380

14.5 Adaptive Passivity-Based Control .

381

15 Torque Ripple Reduction for Step Motors

383

15.1 Torque Ripple in Open-Loop Operation

383

15.2 Schemes for Torque Ripple Reduction .

384

15.3 Adaptive Variable Structure Control Under Strict Matching

385

15.4 Application to PM Stepper Motors

389

15.4.1 Simulation Results . . . . .

390

15.5 Force Ripple Reduction for Sawyer Motors. 15.5.1 Simulation Results . . . . . . . . 16 Friction Compensation in Servo-Drives

16.1 Dynamic Friction Models: General Properties

391 397 401

. 403

CONTENTS

x

16.1.1 Existing Dynamic Friction Models . . . . . . . . .

405

16.2 Model-Independent Friction Compensation Methodology .

408

16.2.1 System Model. . . . . . . . . . . .

408

16.2.2 Control Design via Adaptive VSC

408

16.3 Simulation Results with Different Models

411

16.4 Experimental Results. . . . . . . . . . . .

420

16.4.1 Experiments on Low-Velo city Sinusoidal Trajectory Tracking . . . . . . . . . . . . . . . . . . . . . . 428 16.4.2 Experiments on Point-to-Point Maneuver

429

16.5 Compliant Transmission .

430

16.5.1 Simulation Results

436

A Fundamentals of AC Machines

443

B Floquet Frame: Extensions to DQ Transformation

449

B.l Case of Position-Only Dependent Transformations .

. 453

C Torque Maximization with Current and Voltage Constraints (Field Weakening) 459 C.l Low-Speed Range.

460

C.2 High-Speed Range

460

C.3 Intermediate-Speed Range .

461

C.4 Transition Speeds . . .

461

D Stahle System Inversion

463

E Lyapunov Stahility Theorems

473

F Backstepping

477

G Input-to-State Stahility and Nonlinear Small Gain

491

Bibliography

501

Index

519

Chapter 1 Introduction Electric motors have been widely used in various industrial applications (e.g., machine tools, compressors, fans, robots, etc.) and horne appliances. Due to progress in power electronics and drives, rare-earth magnets, and control systems, utilization of various types of electric motors is widespread. Among electric motors, brushed DC motors are easiest to control due to their reasonably simple dynamics. However, advances in nonlinear control system theory have made control of AC motors exhibiting nonlinear dynamics feasible especially in high-performance applications. In this book, recent advances on nonlinear feedback system designs for various types of electric motors are considered. Specifically, the following types of electric motors 1 are emphasized and covered throughout the book with some extra emphasis on stepper motors:

• Stepper motors Variable reluctance motors Permanent magnet motors Hybrid motors Sawyer motors: dual-axis linear motors (special case of hybrid motors) • Brushless DC motors • Induction motors 1 Motors may be rotary or linear. Although the notation used may imply rotary motors, the indicated algorithms do similarly apply to linear motors.

2

1.1

1. Introduction

Stepp er Motors and Classification

Stepper motors [2, 38, 122, 125] are incremental motion devices that are widely used because of their precise open-loop positioning capability. Their applications include probers, plotters, robots, computer peripherals, and manufacturing systems. Stepper motors can be regarded as polyphase synchronous motors with salient stator poles. The name "stepp er" is derived from their stepping action wherein a desired motion is accomplished through discrete increments by generating aseries of current pulses. Although both rotary and linear motion type stepper motors exist, the basic principle of operation is the same for both these types of motors. The advantages of these motors include low cost, ruggedness, and high reliability. Furthermore, they provide high torque at low speeds (up to twice that of an equivalent frame-sized brushless motor). They can work in a variety of environments and work fairly well even in open-loop stepping mode. However, when operated open-loop, they do not realize their full potential and are prone to missing steps, resonance effects (especially at low velocities), and long settling times. Therefore, for high-performance systems, closed-loop control of stepp er motors is desirable and nonlinear controls have become viable with the advent of nonlinear control design tools in recent years. Broadly speaking, stepp er (or step) motors are of three types: variable reluctance (VR), permanent magnet (PM), and hybrid type. Variable reluctance stepp er motors are characterized by the absence of permanent magnets both on the stator (i.e., the immovable part of the motor) and on the rotor (the moving part). As the name suggests, a permanent magnet stepper motor incorporates a permanent magnet rotor. Hybrid motors use both the variable reluctance and the permanent magnet principles for torque development and typically contain two variable reluctance stacks rotated by half a tooth pitch and separated by an axial permanent magnet. The stepping action is a common feature shared by all types of stepp er motors. As mentioned earlier, both rotary and linear stepper motors are commercially available. Linear stepp er motors may be thought of as rotary type motors with the stator and rotor cut along the motor axis and "opened" out (see Fig. 1.1). The majority of applications requiring linear motion utilize linear stepp er motors that are of the single-axis type. In applications requiring motion along two axes (such as in machine tool cutting applications or semiconductor wafer testing), dual-axis stepp er motors are utilized. Dual-axis linear stepp er motors possess, in addition to their two translational degrees-of-freedom (in the two axial directions), a small rotational (yaw) degree of freedom that must be controlled for proper functioning of the motor. Due to this phenomenon, dualaxis motors have their own peculiar problems associated with the control design and are deseribed in greater detail in Chap. 2.

In the following sections, the various types of stepp er motors and their principles of operation are outlined.

1.1 Stepper Motors and Classification

3

Figure 1.1: Linear motor visualized as cutting and opening out a rotary motor (photos courtesy of Baldor).

1.1.1

Variable Reluctance Stepper Motor

Variable reluctance (VR) motors have attracted much attention over the past few years due to their simplicity of construction and low cost. Compared to DC motors, variable reluctance motors have advantages such as no mechanical brushes and no permanent magnets either on the rotor or on the stator. However, the drawbacks are complex drive electronics and nonlinear torquecurrent-position characteristics, which in turn constitute a more complicated control problem than its brushed counterparts. Due to the absence of me chanical brushes, the commutation scheme must be electronically implemented to enable proper functioning of the motor. The motor operates in and out of the magnetic saturation region depending on the commanded acceleration. Mutual coupling between the phases is made small by careful design. The VR stepp er motor is constructed of a toothed rotor and a toothed stator part that carries the windings. A typical VR stepp er motor is shown in Fig. 1.2. There are two types of VR stepp er motors: the single-stack and the multi-stack types. The multi-stack VR stepp er motor consists of isolated sections called stacks, each of which is excited by aseparate winding. The multi-stack VR stepper motor may be viewed as three or more single-stack motors on a common shaft with their magnetic axes displaced from each other. The stator teeth in each stack have different relative orientations with respect to each other, but the rotor teeth in each stack have the same orientation as seen from the axis of the rotor shaft. The stator and rotor have equal number of teeth. Hence, exciting aphase causes alignment of rotor and stator teeth in that stack. When the excitation is changed to a different phase, the rotor teeth align with the stator teeth of that stack thus producing motion of one full step. Proper sequencing of the phase currents produce the required motion. Fig. 1.3 is a cross section of a 6:4 (Le., 6 stator poles and 4 rotor poles) single-stack VR motor with phase "A"

4

1. Intmduction

Figure 1.2: VR stepper motor from Elbtalwerk. excited. The windings on opposite stator teeth are connected together to form one phase, with the direction of one winding being opposite to that of the other. Hence, the direction of the ftux when aphase is excited is radially outwards in one tooth and inwards in the opposite tooth.

1.1.2

Permanent Magnet Stepper Motor

Permanent magnet (PM) stepper motors are widely used in accurate positioning applications due to their small step size (typically 1.8°). The rotor shaft consists of a permanent magnet with toothed end caps and the stator is a salient pole structure, with each pole having between two to six teeth (Fig. 1.4). The stator poles are provided with windings that are energized sequentially to produce the stepping action of the motor. A typical PM stepp er motor is shown in Fig. 1.5. The ftux path of the permanent magnet ftux lies from the north pole of the magnet, into the end cap and out radially across the air gap, axially into the stator back iron, across the air-gap, and radially into the other end cap of the motor into the south pole of the magnet. The rotor end caps are completely misaligned with the stator in sections X and Y as shown in Fig. 1.4. The windings of each phase are placed on alternate stator poles in opposing senses; for the above example, the windings of phase "A" are placed on poles 1, 3, 5, and 7 while those of phase "B" are placed on poles 2, 4, 6, and 8. Therefore, if phase

5

1.1 Stepper Motors and Classijication

Rotor Bo.ck Iron

l"Iinding

-""1-;7"---_ Sto. tOr Tooth

Figure 1.3: Cross section of a 6:4 single-stack VR motor with phase "A" excited.

Figure 1.4: Cross section of the end caps of a PM stepp er motor.

1. Introduction

6

Figure 1.5: PM stepper motor (photos courtesy

01 Danaher Motion).

Table 1.1: Illustrating the relation between flux and current direction. Winding Current direction Radially outward Radially inward magnetic field magnetic field 3,7 1,5 Positive A Negative 3,7 1,5 A Positive 4,8 2,6 B Negative 2,6 4,8 B

"A" is excited with positive current, the magnet flux flows radially outwards so that poles 3 and 7 of one end cap are aligned with the stator teeth, while poles 1 and 5 are aligned in the other end cap. This is depicted in Table 1.1.

1.1.3

Hybrid Stepper Motor

As the name suggests, hybrid stepp er motors incorporate principles from both the VR and PM types of stepper motors. A permanent magnet with field directed axially along the rotor is mounted on the rotor shaft. Both the stator poles and rotor end-caps are toothed. The Sawyer motor, a dual-axis linear stepper motor, which is discussed in Chap. 2, is an example of a hybrid stepper motor. In Sawyer motors, the permanent magnet and the electromagnets are located on the rotor.

1.2 Brushless DC (BLDC) Motors

1.1.4

7

Sawyer Motors

Sawyer 2 motors are dual-axis linear stepper motors. As alluded to above, Sawyer motors do possess magnets in the moving part (puck) of the motor (rotor) as well as the coils. Depending on the construction, two or four phase Sawyer motors are available. A description of this motor is given in detail in Chap. 2.

1.2

Brushless DC (BLDC) Motors

Brushless DC motors (or permanent magnet synchronous motors) are used in direct-drive applications mainly due to their large torque producing capability. Compared to conventional DC motors, they have no brushes or mechanical commutators, which eliminate the problems due to mechanical wear [68, 74J. In addition, the better heat dissipation characteristics and the ability to operate at high speeds [66] render them superior to the brushed DC motor. However, the brushless DC motor constitutes a more difficult problem than its brushed counterpart in terms of modeling and control system design due to its multiinput nature and ~oupled nonlinear dynamics. The BLDC motor consists of a permanent magnet rotor and stator windings that are sinusoidally distributed. A commutation scheme is provided that uses position information. Chap. 11 exclusively treats the operation and control of brushless DC motors. A typical brushless DC motor is shown in Fig. 1.6.

Figure 1.6: Brushless DC motor (photo courtesy of Danaher Motion).

2The original inventor of these motors [180].

8

1.3

1. Introduction

Induction Motors

Induction motors are utilized in many industrial applications due to the simplicity of their physical construction/design since they do not have brushes, commutator, permanent magnet or windings on the rotor. The name, induction motor, is due to the fact that the rotor voltage is induced in the rotor windings rather than being supplied by an external voltage source. No DC field current is required to run the machine. Induction motors have either squirrel-cage rotors or wound rotors. A squirrel-cage induction motor rotor consists of aseries of conducting bars (made of copper or aluminum) short-circuited together at both ends of the rotor. In wound rotors, the rotor windings are brought out to slip rings that can be short-circuited through external, variable resistances. For detailed operation of induction motors, refer to Chap. 12. A typical induction motor is shown in Fig. 1.7.

Figure 1. 7: Induction motor (photos courtesy of Baldor).

1.4 Pmctical Considemtions

1.4

9

Practical Considerations

There are a number of other practical issues, some of which are alluded to in this section that need to be included in the detailed modeling of the motor if deemed necessary (especially for high-performance applications).

1.4.1

Power Losses

The windings of the stepper motor are made up of several turns of copper coil. The two parameters characterizing the coil are the resistance and inductance, which also limit the motor performance. A major part of the power losses in the motor is due to the resistance of the coil, which causes heat dissipation (also known as i 2 R loss). To enable efficient operation of the motor, it should be operated to minimize power dissipation. Similarly, the inductance of the winding opposes changing current, which limits high-speed operation. To overcome this problem, the time constant of the windings (i.e., the ratio of inductance to resistance) must be decreased. However, increasing the resistance leads to increased power losses, and hence a better option to increase performance is to reduce the inductance of the windings. Other losses in the motor include eddy current losses and losses due to hysteresis. The rotor and stator of some electric motors are made of toothed ferromagnetic material. The magnetic flux in the air-gap of the motor is proportional to the applied magnetomotive force (i.e., the product of the number of turns and the winding current); however, because of magnetic hysteresis, the present flux is a function of the present winding current and the flux history. Fig. 1.8 depicts a typical hysteresis curve for a ferromagnetic material. When the magnetomotive force (mmf) is increased from zero, the magnetic flux increases with the applied mmf, sharply at first and then somewhat slower (due to saturation) until the point P is reached. Decreasing the applied mmf leads to the flux following the path PQR. When the applied mmf is zero, the value of the magnetic flux in the material is nonzero (point Q), and this is known as the remanent flux. To reduce the flux to zero, the direction of the mmf has to be reversed until the point R is reached, and this value of the mmf is the coercive mmf. Further reduction and increase of the applied mmf leads to the flux following the path RSTUP. Viewed from a microscopic scale, ferromagnetic materials are divided into magnetic domains with aligned magnetic dipoles. The dipoles are aligned within a magnetic domain, but differ in direction from one domain to another. When subjected to extern al magnetic fields, the domains enlarge thus increasing the flux in the material. However, energy is required to increase the size of the magnetic domains. This energy is equal to the area enclosed by the hysteresis loop, and the power associated with this energy is known as hysteresis loss [l11J. When a ferromagnetic material is subjected to alternating magnetic flux,

10

1. Introduction

B (Tesla) p

H (Amp-lumS/m)

Figure 1.8: Typical hysteresis curve for a ferromagnetic material.

the alternating flux induces eddy current that circulates in closed loops perpendicular to the direction of the inducing flux. This causes power loss due to heat dissipation. Common approaches to minimizing the hysteresis and eddy current losses are to laminate the rotor and stator providing insulation from each other. Due to the reduced area, the intensity of the eddy currents is also reduced, thereby reducing the overall heat losses.

1.4.2

Thermal Effects

Another important factor affecting position accuracy of motors is the thermal effects due to heating of the windings. This heating will cause the stator or rotor or parts of them to expand and therefore lose the mechanical symmetry in the motor mechanical structure. Furthermore, continuing cooling and heating effeets will induce dimension and shape hysteresis, which ultimately degrade the position accuracy of the motor. Although thermal effects have slower time constants, their effects need to be considered in high-performance applications. This may be achieved through real-time finite-element models of heat flow through the structure.

1.4.3

Current and Voltage Constraints

Physicallimits always exist on the currents and voltages that can be applied to the windings of a stepper motor. The technique of field weakening explained in Appendix C is used to maximize the generated torque in the presence of these constraints. The voltage saturation caused at high speeds due to the back-emf is alleviated by introducing a negative direct-axis current. Furthermore, rate saturation on current and voltages need to be considered.

11

1.4 Pmctical Considemtions

1.4.4

Power Electronics and Drive Circuits

The drive circuit of the stepp er motor plays an important role in the overall performance of the motor. The major function of the drive circuit is to control the magnitude and direction of the current in the motor windings. Two types of drive circuits are used: the bipolar drive and the unipolar drive. Bipolar drives are usually used for permanent magnet steppers, while unipolar drives are used for VR stepper motors where the direction of the winding current is not relevant to torque production. The current direction in bipolar drives is reversed by changing the voltage polarity across the winding terminals. A total of four switches arranged in a H-bridge configuration are needed to implement this drive (Fig. 1.9). One winding per phase is required to implement this scheme.

+,--------------,

+,-------------,

\

Figure 1.9: Bipolar drive circuit. On the other hand, a unipolar drive requires either a winding with a center tap or two separate windings per phase if the direction of flux is important to the functioning of the motor (such as in PM stepp er motors). In this case, the direction of the flux is reversed by switching the currents from one half of the winding to the other (Fig. 1.10). Two switches are required to implement this

+-------,----

Figure 1.10: Unipolar drive circuit. scheme. However, when used for VR stepp er motors, only the magnitude of the

12

1. Introduction

current needs to be controlled, and therefore each phase winding is excited by a drive circuit that utilizes a single switch. The windings of most PM stepper motors are bifilar. This means that there are two identical sets of windings on each pole wound in opposing senses (Fig. 1.11). A bidirectional field is achieved by passing unidirectional current

•

Winding 2

• +

--~-

---------

Figure 1.11: Bifilar winding arrangement. in either of the two coils, Le., rat her than reversing the direction of current to reverse the direction of flux, the field is reversed by transferring current to the second coil wound in the opposite sense. This type of winding arrangement often simplifies drive requirements since a unipolar drive can be used to drive the PM stepp er motor. Similar remarks hold for other types of motors. In some cases, especially when high performance is required, inclusion of the power electronics and driving circuit dynamics into the dynamical model used for the control purposes is essential. The model of the power electronics is largely ignored throughout this book by making sure that the voltages and currents generated through the control algorithm and the dynamics are within reasonable limits of existing power circuitry.

1.4.5

Finite Word Length and Noise

In most applications, controllaws are implemented through a dedicated microprocessor or DSP based processor. Therefore, discretization of the controller and interfaces to a microprocessor through a data acquisition board should also be considered. Fortunately, with the progress in the last decade, many controllers of higher order may be easily implemented on the processors and speed is not as critical (although in certain applications and sometimes due to cost constraint, speed does come into play). However, the finite word length in the processor, the resolution and accuracy of the sensors, noise present in various signals, and the finite word length available in the data acquisition system need

1.5 Nonlinear Control Design Applications to Electric Motors and Open Problems 13

to be considered and modeled in the simulation studies especially in conjunction with a choice for discretizing the controller dynamics. These considerations are especially important when high-performance applications are considered.

1.5

Nonlinear Control Design Applications to Electric Motors and Open Problems

Throughout the book, recent advances in nonlinear control system design are introduced and applied to control of the aforementioned electric motors. The designs may be categorized as: 1) full state measurement and 2) partial state measurement (output feedback). Full states correspond to having the full knowledge of rotor position and velo city and the phase currents (also ftuxes in the case of induction motors). Table 1.2 provides the states and parameters for various electric motors (to be defined in the upcoming chapters) considered in this book. Furthermore, the control designs may be furt her categorized as non-adaptive and adaptive (or robust adaptive) ones. Table 1.2: State variables and motor parameters. Motors I State variables I Parameters Brushless DC motor 0, w, id, i q J, D, if, R, La, L g , L m1 Induction motor J, D, R r , R s, Ln L s, L sr O,w, 'l/Ja, 'l/Jb, i a, ib VR stepp er motor O,w,i1,.·.,iN J,D,R,Lo,L 1 PM stepper motor 0, w, id, i q J, D, if, R, Lo, Lf4, L m1 Sawyer motor x,x,y,y,O,O,i 1, . .. ,i 8 M, I, r,p, i f , R, L o, Ga Different designs require different assumptions and knowledge of motor parameters. Table 1.3 summarizes the needed parameters for available states for various nonlinear control designs. The available state variables are indicated in the first row and various motors are listed on the first column. The entry in each row and column corresponds to the needed parameters for that particular motor (given by the row) when the indicated state variables (given by the column) are available. The sections in the book containing the indicated results are also mentioned in Table 1.3. For instance, the second column in Table 1.3 refers to the case when all the states are available for feedback purposes. In this case, it may be observed that when all states are available, knowledge of motor parameters is not required for any of the motors under consideration. This is also the case for permanent magnet stepper motors when rotor position and velocity are available for feedback. However, if only rotor position is measured, the knowledge of either or bounds on all parameters is needed for PM stepper motors. Thus, Table 1.3 summarizes all the available results to date and indicates what problems remain open as of now. It should be pointed out that some of the designs do also consider dynamic uncertainties in the motor model.

fo

14

1.

Introduction

These control designs are called robust adaptive nonlinear designs in this book. Table 1.3 emphasizes the position control problem. However, the velocity control problem can be tackled in a similar fashion and the techniques presented in this book can be, in a straightforward way, applied to the velocity control problem. In the case of the induction motor, since the practical applications usually involve velocity control, Chap. 13 treats the adaptive state feedback and output feedback problems for velocity tracking. While not all cases involving specific combinations of measured states and known parameters shown in Table 1.3 are covered explicitly in the book, various representative cases are treated in detail so that applications to the other cases follow easily using the advocated techniques. For instance, the control of Sawyer motors when only the rotor position is measured is similar to the corresponding case in Sects. 10.1 and 10.2 for PM stepper motors.

lt:l .--
Go (i.e., the permeance of the iron path is much larger than the air gap permeance). Utilizing the same assumption, the electrical dynamics can be derived as

. dcPA. diA zAR A + dt r::::; zAR A + NG0Tt

+ N,Gaz.j cos(-yx)x.

1

NG (VA - iARA - N,Gaij cos(-yx)x) o . R B + dt dcPB r::::; zB . R B + NG 0Tt diB + N , G' . (,X )'x 2B a2j sm 1 NG (VB - iBR B - N,Gai j sin(-yx)x). o

(3.59)

To account for the yaw rotation of the motor housing, x in (3.58) and (3.59) must be replaced by Xl = X + r sin e where e is the yaw rotation and r is the distance from the center of the motor to the forcer. (x, y) is the position of the center of the motor on the platen (in Cartesian coordinates). For convenience

41

3.3 Sawyer Motor

in controller design, a coordinate transformation is carried out as Cx 2

0 0

0 0

Cyl

Syl

Cy2

Sy2

-Syl

Cyl

-Sy2

-Cx 2

-Sx2

Cyl

Syl

-Cy2

Cy2 -Sy2

ic iD iE

-Cx 2 -Sx2

-Syl

-Cy2

~F

-Cyl

Cyl -Syl

Sy2

Sxl

Sx2 -Cx 2

Cy2

Sy2

~G

Cxl

Sx2

-C x 2

Syl

-Cyl

-Sy2

Cy 2

iH

Cxl

Sxl

i2 i3 i4 i5

-Sxl

0 0

Cxl 0 0

Cxl

Sxl

~6

-Sxl

Cxl

Cxl -Sxl

~l

i7 is

where Cxl

Cx 2 -Sx2

Sx2

0 0

0 0

0 0

0 0

iA iB

(3.60)

= COS(-yXl), Sxl = sin(-yxI), etc., with Xl = x+r sinO, X2 = x - r sinO,

ih = Y +r sin 0, and ih = Y - r sin O. This transformation can be interpreted as a generalized version of the popular DQ transformation. Applying this coordinate transformation, the overall model with four pairs of forcers can be written as 4 x

1

M (Fx

- Fdx),ii

1

= M (Fy -

F dy ),

..

1

0 = /(T - Tl)

r;,i l , F y = r;,i 3 , T = r;,ri 5

1

+ 1.(i6 + is)r cos Oe + L

,i2X

2

..

, (.

-,~IX -

~5

-

2

0

is)r cos

2

..

-'~3Y

- -,

2

(.

')

~5 - ~7

1

L

o

oe + L1 (V3 0

nlJ 1 (V4 r cos uu + -L

~ (i6 + is)x + ~ (i6 +

(Vl -

0

is)iJ

. R)

- ~4

.

+ ~7' ) X.

,- ( ~5 . - ~7 ' ) Y. - , ( ~l .

-

+ ~s. )'X

-

, (. - ~6 -

22

is

i3R - 2r;,iJ)

+ ,(i2 + i 4 )r cos oe

22

, (. ~6

. R)

~2

(V5 - i 5 R - 4r;,r cos (0)

. - ,- ( ~5 -

ilR - 2r;,x)

nlJ 1 (V2 + ~7. ) rcosuu + -L

+ 1. (i6 -

,i4iJ

0

') . ~8 Y

+, ( t2. -

nlJ + -L 1 (V6 + ~3' ) r cos uu 0

') ~4 r

11;' 1 (V7 cos uu + -L

0

-

. R)

~6

-

. R)

~7

- ~(i5 +i 7 )x + ~(i5 - i 7 )iJ - ,(i l - i 3 )r cos Oe 1

+L

o

.

(vs - ~sR)

(3.61)

where r;, = ,NGaiJ, L o = NG o, and R j = R, j = A, ... , H, for simplicity. (VI,"" Vs) is the image of (VA, ... , VH) under the transformation (3.60). A simplified equivalent circuit model for one phase of the motor is shown in Fig. 3.3. In this circuit, the permanent magnet is represented by a constant 4To simplify the dynamics of the motor, it can be assumed that alliengths are expressed as multiples of so that I can be taken as 1. In this case, the coefficients I appearing in (3.61) can be eliminated.

I."

42

3. Modeling of Stepper Motors

ePj . - - - - - - - ( .... 1 - - - - - ,

Figure 3.3: Simplified equivalent circuit of one phase. flux source with magnitude ePj. NiA represents the magnetomotive force due to winding "A", and R ai , i = 1, ... ,4, are reluctances of the air gaps between the platen and the motor teeth5 . From the geometry of the motor, R al (x) =

R a3 (x),R a2 (x) = R a4 (x) = Ral(x - ~),Ra5(X) = R a7 (x),R a6 (x) = Ras(x) = R al (x+ ~). Approximating the reluctances as sinusoidal functions

Ra5(X-~) =

of position (as done for the permeances earlier),

(3.62) where R o and Ra are positive constants 6 (R o ;::: Ra). Solving for the flux through coil A (cPA = cP2

+ cP3),

ePj(R al - Rd + 2NiA R a1 + R a2

(3.63)

The force expression is given by

l. T aeP(x)

-}

2

--

ax

(3.64)

where

.]T , }• = [.ZA ZB

(3.65)

with iB and ePB being the current and flux linkage of phase "B." Therefore, substituting (3.62) into (3.63) and using the result in (3.64), the following expression is obtained for the force produced by the first forcer pair: (3.66) 5The reluctance functions for the air gaps under the second phase of the forcer are given by Rai,i = 5, ... ,8. 6Saturation effects may be modeled by making Ro and Ra functions of the flux.

43

3.3 Sawyer Motor

where '" = R2~~' Similar expressions may be derived for the force produced by the other forcer pairs. The model does not predict the reluctance force, and neither model captures the cogging force of the motor.

3.3.1

Modeling of Hysteresis and Eddy Current Losses

When the Sawyer motor is accelerated to high velocities, platen los ses come into play, creating a drag force. As the forcer moves along the platen, the changing fiux field induces eddy currents in the platen leading to losses and ultimately to reduced output force. With reference to [122], if the platen thickness is t p and the surface area directly under the motor is A, then the drag force can be calculated as 1

v

P

P

n 22 Fd = [",B m - +~Bmtp2] x A x tp

(3.67)

where '" is the hysteresis coefficient, B m is the peak fiux density, ~ is the eddy current coefficient, v is the speed, and p is the platen pitch. The values of", and n depend on the platen material. The value of n varies between 1.5 to 2.5 and is known to be a function of B m in a given platen material [138]. For a fixed platen material, the saturation value of magnetic fiux is a fixed constant.

3.3.2

Disturbance due to Umbilical and Load Force

The disturbance due to the umbilical cord will, in general, depend on the position of the motor on the platen surface as weH as mass per unit length of the umbilical and the cord fiexibility. A dynamical model of the disturbance force due to the umbilical on the motor housing may be complicated. For control design, T o (i.e., the tension in the umbilical) may be bounded above with a constant, i.e., (3.68) ITol :::; Hreak where

3.3.3

Hreak

may be interpreted as the breaking strength of the umbilical.

Cogging and Ripple Force

The Sawyer motor suffers from an inherent cogging force and a current-dependent force ripple that is periodic with aperiod of one fuH step (i.e, one-fourth of the tooth pitch). An exact model of these forces is not known and may be approximated with a sinusoidal function (of position) for purposes of control design. For example, the cogging force and ripple force in the X-direction may be approximated by

Fe

>:::;

"'2

F rip

>:::;

.x {i r1 + i r2 } sin( 4')'x)

sin( 4')'x)

(3.69) (3.70)

44

3. Modeling

0/ Stepper Motors

where K:2 and Aare suitable constants, and i r1 and i r2 are the amplitudes of current in the X-forcer pairs given by

(3.71)

3.4

Resonance

Step motors suffer from poorly damped step response. This is because the motor manufacturers try to minimize friction as much as possible since friction leads to wear and tear of the moving parts. In practice, a small amount of friction exists and therefore the rotor eventually comes to rest at a small distance away from the commanded step position. The difference between the commanded and actual step position is known as the static step accuracy of the motor. The frequency of oscillation can be obtained from the static torque-position characteristic of the motor. The frequency of oscillation is given by 10 =

~ fE 21f V]

(3.72)

where k is the stiffness constant and J is the total mechanical inertia of the rotor and load. The frequency of oscillation can also be expressed in terms of the step length and holding torque. Negleeting non-ideal effeets, the statie torque-position characteristic is sinusoidal, i.e., (3.73) where Th is the holding torque and N r is the number of rotor teeth. The stiffness constant is the slope of the torque-position characteristic around the equilibrium point: (3.74)

Hence, using (3.72) 10

=

81f JO step

where

()step

=

2~r

(3.75)

is the size of eaeh fuH step.

Operation of the motor at stepping rates that are multiples or fractions of the resonant frequency (Le., ... , 10/3, 10/2, 10, 210, 310,"') can cause problems

3.4 Resonance

45

leading to 10ss of synchronization, with frequencies dosest to resonance being the most potentially damaging. A sudden 10ss of torque is experienced by the motor at these frequencies. Viscous or electromagnetic dampers are used in order to increase the damping and reduce the oscillatory behavior of the motor and therefore its settling time [2].

Chapter 4 Stepping The torque expressions for the various types of stepper motors were derived in Chap. 3 as: VR:

~'2 sm . (N B

T

K- ~Zj

(j - N1)27r)

(4.1)

+ i 2 cos(NrB)) + iB cos(NrB))

(4.2)

r

-

j=l

where

K-

=

N r2L 1

.

PM: T

K-( -i 1

sin(NrB)

K-(iA sin(NrB)

neglecting cogging torque with

K,

=

ifLmlNn iA

=

-i 1 ,

and i B = i 2 .

Sawyer:

Fx1

=

K-(iA cos(rx)

+ iB sin(rx))

for one X-axis forcer neglecting cogging force with

K-

(4.3)

= "fNGaif.

Applying a set of currents causes the motor to settle at a particular equilibrium position that is a function of the currents applied (and the initial position). In the absence of any load torques, the motor settles at a point at which the torque (or force in linear motors) is zero. Thus, using the torque express ions above, we can find the equilibrium positions that will be attained by applying any set of currents. Applying a sequence of current values will cause the stepp er motor to step through a sequence of equilibrium positions. A typical stepping action proceeds as folIows: when the user issues a desired position command, the command is translated to the number of current steps to be given to the motor. These current steps are then generated by the power amplifier driving

4.

48

Stepping

the motor. The motor can be operated in the full-stepping, half-stepping, or micro-stepping modes.

4.1 4.1.1

Full-Stepping VR Motors

To achieve bidirectional rotation with a VR motor, the minimum number of phases required is three. This is because the torque generated in VR motors is dependent not on the signs of the currents but only on the magnitudes. For VR motors with three phases, the excitation sequence for clockwise rotation of the motor is A,B,C,A,B,C ... while for anticlockwise motion, the phases are energized according to the sequence A,C,B,A,C,B (Fig. 1.3). This scheme is referred to as one-phase-on excitation since only one phase is energized at any time. The step size of the motor is related to the number ofrotor teeth NT) and the number of phases N, as folIows: for an N-phase motor, excitation of each phase in sequence produces N steps of rotor motion and at the end of this cycle, the excitation returns to the original phase. The rotor is once again aligned with the stator pole at the end of this sequence, except that it has moved by one rotor tooth pitch, i.e., 3t~O. Hence, the step size is given by

()step

=

3600 NrN·

(4.4)

This relation holds for both single-stack and multi-stack VR stepp er motors. The absence of permanent magnets implies that there is only attraction (and no repulsion) between the rotor poles (magnetically soft iron pieces) and the energized stator poles. Therefore, the direction of the torque produced by an excited phase is dependent only on the rotor position and not on the sign of the phase current. For this reason, the motor is usually driven by means of unipolar power amplifiers. A full-stepping operation can also be achieved with a two-phases-on excitation scheme using the sequence AB, BC, CA, ... for clockwise and CA, BC, AB, ... for anticlockwise rotation. The motor can also be operated in the halfstepping mode with an alternate one- and two-phase-on excitation as A, AB, B, BC, C, CA, ... for cIockwise and A, CA, C, BC, B, AB, ... for anticlockwise rotation. In this mode of operation, each excitation change produces an incremental movement that is half the length of a step in full-stepping. Fulland half-stepping can also be carried out with energization of greater number of phases for multi-phase motors. It can be shown [2] that the holding torque is maximized by energizing half the phases in a multi-phase machine, i.e., to

49

4,1 Full-Stepping

maximize the peak static torque, N umber of phases excited

4.1.2

{

~+l2 or

N-l

2

for N even for N odd.

(4.5)

PM Motors

Unlike VR motors, the torque generated in PM motors depends on the signs of the currents. Thus, bidirectional rotation can be achieved using two phases. Clockwise rotation may be obtained by exciting the phases in the sequence A+,B+,A-,B-,A+, .... For anticlockwise motion, the excitation sequence would be A+,B-,A-,B+,A+, .... This is illustrated in Table 4.1. Table 4.1: Excitation sequence for anticlockwise motion in full-stepping with one phase on (PM motors). ;;::=;:;;====;:=r===:==:===;;

Phase A

Phase B

+ 0

0

0

11

0 +

A full cycle of excitation produces four steps of rotor movement; at the end of this cycle, the rotor teeth assume the same alignment with the stator poles. Hence, four steps of rotor motion correspond to one tooth pitch (= 3t~O) and therefore the step size of the motor is given by (4.6) This sort of energizing scheme where the rotor moves by a fourth of a tooth pitch for each excitation change is known as jull-stepping. Although, in Table 4.1, full-stepping was achieved with only one phase energized at a time, the same motion may be achieved by exciting both phases at a time and is the normal mode of full-stepping. This is depicted in Table 4.2. In this case, greater torque is produced since all the stator poles interact with the rotor. The torque required to dis pi ace the rotor from its stable position is known as holding torque, and is dependent on the energizing scheme (i.e., the phase currents) used. In either case, the stator flux is rotated by 90° for every step of rotor motion. To increase resolution and smoothness of motion, half-stepping is often employed. In this case, the windings are excited alternately one and two at a time (Table 4.3). The step length is reduced to half of that given by (4.6), and the stator flux rotates by 45° for every change in excitation.

4.

50

Stepping

Table 4.2: Excitation sequence for anticlockwise motion in fuIl-stepping with both phases on (PM motors).

~P;:;=ha=s=e=;:A==;=P~h;=a=se=B~

ii=11

+ +

+ +

Table 4.3: Excitation sequence for anticlockwise motion in half-stepping (PM motors). Phase A Phase B 11 0 +

+ 0

0 0

+ 4.1.3

+ + +

Sawyer Motors

The stepping behavior of Sawyer motors is similar to that of the PM stepper motors. The stepping is illustrated in Fig. 4.1. Typical step sizes are fuIl steps (a quarter tooth pitch resolution) and half steps (an eighth of a tooth pitch resolution). For even greater resolutions, micro-stepping schemes are utilized.

4.2

Micro-Stepping

Micro-stepping is a technique by which the position resolution of the motor may be increased by aIlowing the phase currents to take on a large number of possible current values, thereby increasing the number of equilibrium states. UsuaIly, the current values are chosen such that the step lengths are integer fractions of the fuH step length. Micro-stepping results in reduced vibrations and noise and permits smoother motion compared to fuIl- or half-stepping. This (open-Ioop) method is applicable to both rotary as weIl as linear stepper motors.

51

4.2 Micro-Stepping

1)

2)

3)

4) Figure 4.1: Stepping sequence in full-stepping for Sawyer motors.

4.2.1

VR Motors

Applying the currents i j given by

.

-2 m COS

(N B

r d -

(j - N1)27r)

+ y"22m + 'Y2 , J. --

1, 2, ... ,

N(4.7)

where im and 'Y are positive constants, and Bd is the desired location to step to, the torque expression reduces to

T

=

-K.

~.

~2mCOS

(

NrB d

-

(j - 1)27r) . ( (j - 1)27r) N sm NrB N

52

4.

Stepping

(4.8) The equilibrium positions are given by

Nr(Od - 0) = mr, n = 0, ±1, ±2 ....

(4.9)

Of these, the ones with even n are the stable equilibrium positions. By choosing a stepping sequence of positions Od and applying currents i j as given in (4.7), micro-stepping can be achieved.

4.2.2

PM Motors

The torque expression for a PM stepper motor is given by (4.10) where K is a constant, T is the torque produced, N r is the number of rotor teeth, and 0 is the position of the rotor. In micro-stepping, the phase currents i A and i Bare given by im cos( k + 1. In this case, z(t) grows unbounded as time approaches

_I_log ( x(O)z(O) ) k +1 x(O)z(O) - k - 1 .

(5.23)

Although the zero dynamics is globally exponentially stable and the x subsystem is made globally exponentially stable, the interconnection is only locally exponentially stable. If the x subsystem is replaced by the double integrator

62

5. Feedback Linearization and Application to Electric Motors

x = u, then a linear state feedback of the form U = - 2ax -

> 0) stabilizes the x subsystem. However, it can be shown that this system still has a finite escape time and as one attempts to speed up the rate of decay of x by increasing a, the escape time shrinks. This is known as the peaking phenomenon [107]. a 2 x (a

Global asymptotic stabilization when the zero dynamics is asymptotically stable has been studied extensively and various results have been produced [83, 84, 181]. One such approach is through the notion of input-state stability [92, 185] which is covered in Appendix G. Another approach is through the concept of forwarding [141, 181, 194] which starts with stabilizing the x dynamics with saturated inputs so that the rate of decay of the x subsystem is controlled in a fashion so that aredesign for forwarding to the next subsystem would be possible. If the relative degree of system (5.1) is equal to the order of the system, there are no zero dynamics and input-state linearization can be achieved. In fact, this can be achieved if any dummy output can be found such that the system x = f(x) + g(x)u with the dummy output has relative degree n. A system for which such a dummy output can be found is said to be input-state feedback linearizable or simply feedback linearizable.

Definition 5.4 A system is said to be (input-state) feedback linearizable around Xo if a local change of coordinates ~ = ~(x) and an input transformation u = a(x) + ß(x)v defined in a neighborhood of Xo can be found such that, in the new coordinates, the system dynamics can be written as a chain of n integrators

v.

(5.24)

In (5.24), 6 is the dummy output with relative degree n. The existence of such a dummy output can be determined through certain geometric conditions on the vector fields fand g. Before we present these conditions, we need a few definitions.

Definition 5.5 Given a set of smooth vector fields, h, ... , fd, the span ofthese vector fields (referred to as a distribution) written as ß(x) = span{h (x), ... , fd(X)}

(5.25)

is defined as the set of all linear combinations of h, ... , fd over the ring of smooth real-valued functions, i.e., ß(x) is the set of all vectors that are of the form al(x)h (x) + ... + ad(x)fd(X) where al, ... , ad are smooth real-valued functions.

63

5.1 Feedback Linearization Theory

A distribution is a smooth assignment of a subspace of Rn to each point of Rn. A vector field 7 is said to belong to a distribution .6. if 7(X) E .6.(x) for each x ERn.

Definition 5.6 The Lie bracket of two smooth vector fields ft and 12 denoted as [ft,12] is defined as

(5.26) The notation adh12 is sometimes used instead of [ft, 12] to represent the Lie bracket. To denote repeated Lie brackets, the notation ad1112 is used, which is defined recursively as ad1J2(X) = [ft, ad~;l 12](x), k ;::: 1 with ad~J2(x) =

12(x).

Definition 5.7 A distribution .6. is involutive if the Lie bracket h, 72] of any pair of vector fields 71 and 72 belonging to .6. is a vector field belonging to.6.. If ft, ... , fd are smooth vector fields (locally) spanning .6., .6. is involutive if and only if

(5.27) Theorem 5.1 System x = f(x) + g(x)u can be input-state feedback linearized around a point Xo if and only if a function ),(x) defined in a neighborhood of Xo exists such that the system

y

f(x) ),(x)

+ g(x)u (5.28)

has relative degree n at xo, i.e., a) LgL~)'(x) = 0 for all x in a neighborhood of Xo and 0 ~ i ~ n - 2, (5.29) b) LgLj-l ),(xo) :j:. o. Furthermore, such a ), (x) exists if and only if 1) the n x n matrix with columns g(xo), adfg(xo), ... , adT 1g(xo) has rank n, 2) the distribution D = span{g, adfg, ... , adj-2 g} is involutive in a neighborhood of xo. 0 Condition (5.29) which implies a relative degree n for the dummy output )', generates the following partial differential equation

:~[g,adfg, ... ,adT 2 g](x) =

O.

(5.30)

64

5.

Feedback Linearization and Application to Electric Motors

Remark 5.4 Equations (5.30) can be derived from (5.29) by using recursively the identity (5.31) For instance, using the relations LgA(x) = LgLfA(X) == 0 (for all x in a neighborhood of xo) from (5.29), we derive, utilizing (5.31), that ~~adfg(x) == O. Continuing the recursion, (5.30) is obtained. Through the use of the Frobenius theorem [83], the integrability of the partial differential equations (5.30) can be guaranteed by the involutivity of the distribution D. The sufficiency proof of the Frobenius theorem yields a constructive algorithm for solving the partial differential equation (5.30) for A(x) by solving ordinary differential equations. Condition 1 in Theorem 5.1 is simply a controllability condition at Xo that corresponds to controllability of the linearization of the system at xo. If one specializes to a linear system, Le., f(x) = Ax and g(x) = B, the matrix in Condition 1 reduces to the controllability matrix [B, AB, ... , An-l B] of the linear system. Having found A(x), the input-state feedback linearization is achieved by choosing ~ = (x) = [A(X), LfA(x), ... , U;-l A(x)jT and

u

=

LgLf~l A(x) [v -

LjA(x)].

(5.32)

Example 5.4 Consider the third-order system eX1u

Xl

+ X2 + eX1u Xl - X2 + x~. X~

X2 X3

For this system,

f(x)

[

0 x~ +X2 x2 + x5

Xl -

g(x)

1

[ ," 1 eX1

0

adfg(x)

[ -(3xl

ad}g(x)

[ (3xl

~ 1)," 1

~+1),,, 1

-(3x~

1)e X1

(5.33)

5.1 Feedback Linearization Theory

65

[ -(3xl

[g, adJg(x)[

(3x~

+6~' + 1)e'" 1

+ 6X2 + l)e 2 3X 2

+1

XI

d () a fg x .

(5.34)

It is easily seen from (5.34) that the conditions of Theorem 5.1 are satisfied. Thus, an output A(X) can be found such that the system (5.33) with this output has relative degree 3. By inspection, from (5.33), A(X) can be chosen as A(X) = X3, which satisfies (5.30) for this example. The resulting change of coordinates is given by

A(X) = X3 LfA(X) = Xl - X2

~l

+ x~ + 2X3(Xl -

L}A(X) = -X2 - x~

X2

+ x~).

(5.35)

In the new coordinates, the system dynamics can be written as

~3

v

(5.36)

where a control input transformation has been introduced as

=

U

-e

-Xl

(X2

3

+ X2) -

3

e- XI 2

X2

+

1 V,

(5.37)

with v being the new control input.

5.1.2

MIMO Systems

We now consider square 1 multi-input multi-output (MIMO) systems of the form

x Yl

Ym

j(x)

+ gl(X)Ul + ... + gm(x)u m

hl(x) (5.38)

where U = [Ul, ... , umV E n m represents the inputs, Y = [Yl, ... , Ym]T E n m the outputs, and X E nn the state. j, gl, ... , gm are smooth vector fields and h 1 , ... , h m are smooth real-valued functions. 1A

system is called square if the number of inputs is equal to the number of outputs.

66

5. Feedback Linearization and Application to Electric Motors

The notion of relative degree can be generalized to the multivariable case. As in the S1SO case, each output is differentiated until an input appears. The condition in the 8180 case that LgL7lh(xO) i- 0 can be generalized to a matrix nonsingularity condition. This process yields a set of m relative degrees (collected in a vector as a vector relative degree). This not ion is made precise in the definition below.

Definition 5.8 8ystem (5.38) is said to have vector relative degree {rl,"" rm} at a point Xo if 1) LgjL7hi(x) = 0 for all x in a neighborhood of Xo and 1 ~ i,j ~ m, 0 ~ k ~ ri -

2,

2) the following m x m matrix, namley A(xo), is nonsigular:

LgIL?-lhl(XO) A(xo)

[

L g=L?-lh 1 (xo) ]. (5.39)

L9ILr/-lhm(xo)

Lg=Lj=-lhm(xo)

1t can be shown that rl + ... + T m ~ n if the vector relative degree is defined. Analogous to the 8180 case, the system can be transformed into a normal form at any point Xo at which the relative degree is defined. This is achieved by using as coordinates, 6,1 = hl(x), 6,1 = Ljhl(x), ... , ~rl,l = L?-lh l (x), .,. ,6,m = hm(x),6,m = Ljhm(x), ... , ~r=,m = Lj",-lhm(x), and n - (Tl + ... + Tm) additional coordinates z chosen to make the mapping x f--* (~, z) a local diffeomorphism around Xo where (5.40)

The existence of such a z follows, through application of the Frobenius theorem, from the linear independence of the differentials dL~hi(X),O ~ j ~ Ti -1, 1 ~ i ~ m. Unlike the S180 case, it is not, in general, possible to choose Z with the additional property that Lgjzi(x) == 0, 1 ~ j ~ m, 1 ~ i ~ n - (Tl + ... + Tm), Le., that the z dynamics does not directly involve the control inputs. A special case in which this can be achieved is if the distribution spanned by {gI, ... ,gm} is involutive in a neighborhood of Xo. Expressing the system in the new coordinates,

6,1

6,1 m

~rl,l

al(~, z)

+ Lßj,l(~' z)Uj j=l

~l,m

6,m

67

5.1 Feedback Linearization Theory

m

am(~, z)

+L

ßj,m(~, z)Uj j=l q(t z) + p(~, z)u

~r17lJm .:i:

Y1

6,1

Ym

(5.41)

~l,m'

In (5.41), the terms ai(~,z) and ßj,i(~,Z) are Lj'hi(x) and L gj Lj'- l hi (x), respectively, expressed in the (~, z) coordinates. The dynamics in (5.41) can be further simplified by introducing a control space transformation m

+ Lßj,l(~,Z)Uj

a1(~,z)

j=l

V1

m

am(~, z)

+L

j=l

ßj,m(~, z)Uj

(5.42)

where v = [V1, ... ,vmV represents the new inputs. The equations in (5.42) can be written in vector form as a(~,z)+ß(~,z)U

= v

(5.43)

where a(~, z) = [a1(~, z), ... , am(~, z)V and ß(~, z) is a m x m matrix with (i,j)th entry heing ßj,i(~,Z). The matrix ß(~,z) is simply the matrix A(x) (defined in (5.39)) expressed in the (~, z) coordinates. Since, by the definition ofrelative degree, A(xo) is nonsingular, the control space transformation (5.43) is (locally) invertible. Using (5.41) and (5.42),

~1,1

6,1

~l,m

~2,m

~rrn,m

Vm

68

5. Feedback Linearization and Application to Electric Motors

i Y1

Ym

q(~,z)+p(~,z)u

6,1

6,m.

(5.44)

The dynamies in (5.44) is partially linear with the linear input-output relations yYi) = Vi. Notice that the above coordinate and control space transformations have achieved decoupling, Le., the i th output Yi is only affected by the i th input As in the SISO case, the zero dynamics can be defined as the internal dynamies when all the outputs are held at zero by applying the appropriate output-zeroing input, v = 0, Le., U = -[ß(O, z)]-lo:(O, z). We assurne that f(xo) = (xo) = and h 1 (xo) = ... = hm(xo) = 0. With application of the output-zeroing input, the z dynamics can be written as

° i

= q(O, z) - p(O, z)[ß(O, z)t 1o:(0, z)

(5.45)

°

which is the zero dynamics. The system (5.38) is said to be asymptotically (exponentially) minimum-phase if the equilibrium point z = of the zero dynamics (5.45) is asymptotically (exponentially) stable. If r1 +.. .+rm = n, there are no z states in (5.41) and input-state linearization is achieved. If a set of dummy outputs A1(X), ... , Am(X) can be chosen such that the system j; = f(x)+ 2:::1 gi(X)Ui with these outputs has a well-defined relative degree at Xo and the sum of the entries of the vector relative degree is n, then the system is said to be input-state feedback linearizable (or simply feedback linearizable) around Xo. The following theorem provides geometrie conditions for a system to be feedback linearizable. j; = f(x) + 2:::1 gi(X)Ui in which the matrix g(xo) = g2(XO), . .. ,gm(XO)] has rank m is (input-state) feedback linearizable if and only if 1) the distributions Gi = span{ad1gj(x) : S; k S; i; 1 S; j S; m}, i = 0, 1, ... , n - 1, have constant dimension in a neighborhood of Xo, 2) the distribution G n - 1 has dimension n, 3) the distributions Gi, S; i S; n - 2, are involutive. 0

Theorem 5.2 System [91 (xo),

°

°

5.2

System Inversion

In this section, we outline the procedure of system inversion to obtain feedforward control inputs. The system inversion problem illustrated in Fig. 5.1 attempts to obtain ure! so that the system when driven by input ure! yields output equal to a given reference trajectory Yre!. If the initial state is not

69

5.2 System Inversion

Yrej

Ure!

Inverse System

System

Y

= Yrej

Figure 5.1: System inversion.

Yrej

Inverse System

System

Y

Feedback Controller

Figure 5.2: Inverse system augmented with feedback controller. as prescribed by the reference trajectory, or if the system model is not exact, the open-loop feedforward must be augmented with a closed-loop feedback controller as illustrated in Fig. 5.2. Fig. 5.2 can be redrawn as in Fig. 5.3 to show explicitly the feedforward and the feedback components of the control input. If the desired output trajectory form (5.7) yields

6 6 ~r

Yrej(t)

~lref 6ref

~

is given, the structure of the normal

= Yrej = Yre!

_ (r-l) rref - Y re !

(5.46)

where Y~~! denotes the i th derivative of Yrej(t). It is assumed that Yrej and its first r derivatives are bounded functions of time. Thus, the dynamics of z reduce to (5.47) where ~re! = [6 ref ,··. ,~rreflT. The initial conditions of the states 6,··· ,~r

70

5. Feedback Linearization and Application to Electric Motors

Feedforward

Y - Yref

Feedback

System

Y

+

Figure 5.3: Controller with feedforward and feedback components (Fig. 5.2 redrawn). are determined by (5.46) while the initial conditions for the z states is arbitrary. If a stable solution zref can be obtained for (5.47), then the control input u

=

y~:j

- a( ~ref' zref)

ß(~ref' zref)

(5.48)

achieves exact tracking of the desired output trajectory. In cases where the solution to (5.47) is not bounded for certain initial conditions (as, for instance, in nonminimum-phase systems), a stable inversion scheme (Appendix D) [48, 49, 100, 174, 195J needs to be utilized to obtain a bounded solution for (5.47).

5.3

Feedback Linearizing Designs for Step Motors

In this section, we apply the feedback linearization theory developed so far in this chapter to the various stepp er motors. Feedback linearization designs for brushless DC motors and induction motors are presented in their corresponding chapters (namely, Chaps. 11 and 12, respectively).

5.3.1

VR Motors

The dynamics of a VR stepp er motor is given by (3.1), (3.25), and (3.26). The friction is modeled as Dw, where D is the viscous damping factor, and the load

71

5.3 Feedback Linearizing Designs for Step Motors

torque Tl is assumed constant for simplicity. The system can be written as = f(x) + g(x)u with x = [e,w, i 1 , ... , in]T, u = [V1, ... , vnV,

:i;

. (Nr -tN R - LN· 1 rtNwsm

1

{.

LNN(B)

e

-

(N-1)27r)}

N

o o

0

1

0

0

L22(B)

Ll1(B)

g(x)

1

0 0 0

0

1

0

L(N-l)(N_l)(B)

0

0

0

LNN(B)

(5.49)

1

The feedback linearizability of the system given by (5.49) can be checked using Theorem 5.2. Here, the distribution Co = span{gl, ... , gN} is involutive and of constant dimension N. However, it can be shown that the dimension of the distribution Cl = span{gl, ... , gN, adfgl,.··, adfgN } is not constant around any point at which (5.50) Thus, using Theorem 5.2, it is seen that the dynamics cannot be feedback linearized around such a point. However, it can be shown by checking the remaining conditions in Theorem 5.2 that the system can be (locally) feedback linearized around a point at which i k sin( Nre - (k - 1) ~) "I 0 for some phase k. Feedback linearization can be achieved using the change of coordinates

e

6,1 6,1

w

6,1

--

6,j

Nr L 1 2J

L. N

j=l

ij-1 {

ij

2 .

t· J

sm(N

r

e - (J. -

if j - 1 < k if j ~ k

j

271" - -w D - -Tl 1 1)-) N J J

= 2,3, ... ,N.

(5.51)

Note that the transformation (5.51) is not globally injective (i.e., one-to-one) and, hence not a global change of coordinates. This is because the sign of ik is not determined by ~ = [6,1,6,1,6,1, 6,2,6,3, ... , 6,NV. However, since VR motors are typically powered by unipolar power converters, ik is positive

72

5.

Feedback Linearization and Application to Electric Motors

during operation of the motor, thus making (5.51) practically a global change of coordinates. Using the change of coordinates (5.51), and the control input transformation Vj =

27r ijR + L 1N rijw sin(NrB - (j - 1) N)

Lkk(B) vk=2' . (NB-(k-1)~) Zk sm r N

+

t

-t. j=l

L

+ Ljj (B)wj

-

(j -

1)~)) sin(NrB -

Dw + Tl J

}

(j -

1)~)

1)~ )N.w }

+ ik N r L 1sin(NrB- (k-1) ~) 2T ~ Zj sm -

27r

JJ

{NrL1~'2'

Lkk(B)D

.

~ L .. (B)v sm(Nr B-(J-1)N)

j=l,jlk

2~(jB) (ijR + L1Nrijw sin(NrB (j -

i: k

{~2ij, j

JJ

ij ooo(N.O -

, j

(NB (' 1)27r) r - J- N

Lkk(B)J

+ ikNrL1sin(NrB-(k-1)~)Wk

(5.52)

where W1, . .. , WN are the new control inputs, the system dynamics can be written as

6,1 Wk Wj-1 if j - 1 < k { Wj if j ~ k + 1

j = 2,3, ... ,N.

(5.53)

The system in (5.53) is a controllable linear system. The controllability index of

Wk is 3. The controllability indices of the other inputs Wj, j i: kare 1. A linear controller can be designed that uses Wk to control the (6,l,6,l,6,t) states and each of the Wj, j i: k to control one of the 6,j, 2 ::; j ::; N, states. Physically, this

corresponds to using the voltage of one of the phases whose contribution to the torque is nonzero to control the position (6,1), velocity (6,t), and acceleration (6,1)' The voltages of the other phases are used to control their corresponding currents.

5.3.2

PM Motors

Consider a two-phase PM stepp er motor with no saliency and a sinusoidal fiux distribution in the air gap. Here, we neglect the cogging torque, and assume for

73

5.3 Feedback Linearizing Designs for Step Motors

simplicity that the load torque Tl is constant. The friction is modeled as Tf = Dw with D being the viscous damping factor. Using the electrical dynamics given in (3.51) and the torque expression in (3.52), the dynamics of the motor can be written in the form x = f(x) + 9(X)U, where x = [8, w, i 1 , i 2 U = [VI, V2]T,

V,

w

f(x)

9(X)

[

kj

[

0 0 1

La

0

-lJw - ~Tl

+ i 2 cos(Nr8)] -..!li + La 1 kmwsin(N La r 8) -..!li2 La - ~w La cos(Nr 8)

[ _ i 1 sin(Nr 8)

0 0 0 1

La

1 (5.54)

1

This is a square MIMO system with two inputs and two outputs. The feedback linearizability of this system can be verified by using Theorem 5.2. Here, the distribution Go = span {g1, 92} is easily seen to be of constant dimension and involutive. Computing the Lie brackets,

[ [

0 jf- sin(Nr 8) a

R

:q 0

1

0 -~cos(Nr8) 0 R

:q

1'

(5.55)

the distribution GI = {91, 92, ad f 91, ad f 92} is also of constant dimension and involutive. Continuing, it can be shown that G 2 is also of constant dimension and involutive, and that G 3 is of constant dimension 4. Thus, the dynamics of the PM stepp er are globally feedback linearizable. Feedback linearization can be achieved by picking any dummy outputs such that the sum of the entries in the vector relative degree is 4. For instance, 8 and any 6,2(X) with relative degree 1 can be chosen as dummy outputs. Defining 6,1,6,1, and 6,1 to be 8 and its first two derivatives, respectively, 6,1

8

6,1

w

(5.56)

6,1

it is seen that the resulting coordinate transformation is very similar to the DQ transformation. Indeed, the terms and ~Tl in the definition of 6,3 are not needed to feedback linearize the system (the presence of these terms

lJw

74

5. Feedback Linearization and Application to Electric Motors

only enables us to transform the dynamics into the normal form). Thus, the DQ transformation is seen to be a special case of feedback linearization [214]. Choosing 6,2 as given by the DQ transformation (3.41),

(5.57) and introducing the control space transformation

(5.58) where (1

~~o W1 + L o{ Nrw [i 1 cos(NrO) + i2 sin(NrO)] + ~: w +~ [ -

i 1 sin(NrO)

DL [ + j O

(2

+ i 2cos(NrO)] }

.

i1 sm(NrO)

+ i2 cos(NrO) -

D km W

-

1]

km Tl

L ow 2 + R [i1 cos(NrO) + i2 sin(NrO)] - Nrw [ - i 1 sin(NrO)

+ i 2 cos(NrO)]

(5.59)

with W1 and W2 being the new inputs, the dynamics of the PM stepp er motor can be written as

6,1

6,1

6,1

6,1

~3,1 ~1,2

W1 W2·

(5.60)

The system in (5.60) is a controllable linear system. A linear controller can be designed utilizing W1 to control the (6,1,6,1,6,1) states and W2 to control 6,2. Physically, this corresponds to using the quadrature-axis voltage to control the mechanical motion of the motor and the direct-axis voltage to control the direct-axis current.

5.3.3

Sawyer Motors

The dynamics of a Sawyer motor are given in (3.61). Assume, for simplicity, that Fdx = Fdy = Tl = O. The system can be feedback linearized by introducing the state variables 6,1

X

6,1

X

75

5.3 Feedback Linearizing Designs for Step Motors

6,1 6,2 6,2 6,2 ~1,3

6,3 6,3 6,4

"'.t1

M Y

Y

"'.t3

M (J

e

",r .

yt 5

i2

6,5

i4

6,6 6,7

t6

6,s

is

i7

(5.61)

and the control input transformation

..X L 0 ( ,t2

M Lo + ',2 (t6. + ts') r cos (J(J') + t1. R + 2' "'x + -",-W1

VI

-

v2

-Lo ( - ,i1x -

V3

-

V4

- L o ( - ,i3 y -

V5

-Lo (~(i6

~(i5 + i 7)r cos (Je) + i 2R + Low2

.. + ',2 (t6 . - ts ') r cos (J(J') L 0 ( ,t4Y

~ (i 5 -

M Lo + t3. R + 2' "'Y + -",-W3

i 7 )r cos (Je)

+ i 4R + Low4

+ is)x + ~(i6 - is)Y + ,(i 2 + i 4)rcos(Je) + i 5 R

. IL o +4",r cos (J(J + -W5 ",r

v6

-Lo ( -~(i5

+ i 7)x - ~(i5 - i 7)y - ,(il + i 3 )r cos Be) +i6R+ LOW6

V7

-L o G(i 6 + is)x - ~(i6 - is)Y + ,(i 2

Vs

-L o (-~(i5

-

i 4)r cos Be)

+ hR + Low7

+ i 7)x + ~(i5 - i 7 )y - ,(i1 - i 3 )r cos Be) + isR (5.62)

+Lows

where w1, ... ,ws are new control inputs. In the new coordinates, the system dynamics can be written as

tl,1

6,1

6,1

6,1

t3,1

W1

76

5. Feedback Linearization and Application to Electric Motors 6,2

6,2

~2,2 ~3,2 ~1,3

6,2

6,3

6,3

6,3

W5

~1,4

W2

6,5

W4

6,6

W6

~1,7

W7

6,8

W8·

W3

6,3

(5.63)

The system in (5.63) is a controllable linear system and linear control design techniques can be utilized to construct a stabilizing feedback. The feedback linearization scheme provides a ni ce tool to transform a nonlinear dynamics to a linear one via a nonlinear change of coordinates and a nonlinear feedback. Therefore, in the new coordinates, linear designs may be pursued. To further improve robustness and performance of the closed-loop system, one may utilize robust outer-loop controllers or resort to adaptive controls or a combination thereof. Furthermore, availability of states is required in feedback linearizing controllers since the solution is astate feedback one. Therefore, partial state feedback (i.e., output feedback) solutions are also pursued in the remainder of the book.

Chapter 6 Robust Adaptive Control of a Class of Nonlinear Systems In this chapter, we present some theoretical design tools for robust adaptive control of nonlinear systems [84, 120, 164] for application to various motor control problems. The development follows the lines of [87, 88]. First, the control design for the case that the strict matching condition is satisfied is outlined. Next, the design is extended using a combination of backstepping, nonlinear damping, and tuning functions [120] to a dass of nonlinear systems that do not satisfy the strict matching condition, namely systems of the special strict feedback form [87, 88]. These systems are characterized using the notion of the degree of mismatch that illustrates the tradeoff between the state equations where the uncertainty can appear, and the state variables on which the uncertain terms depend. For the case where only the outputs are measured, the adaptive design procedure applies to systems transformable to the output feedback form [89, 136] where the unknown terms are output-dependent. Results in both cases are presented for state/output tracking and regulation problems. For the case where the objective is tracking, global uniform boundedness of the tracking error is attained with respect to a compact set whose size can be made arbitrarily small (practical tracking) based on an appropriate choice of control gains. In the regulation case, global asymptotic regulation of all the states is achieved. Before reading this chapter, Appendices Fand G provide a good basis for some of the required nonlinear design tools such as backstepping, tuning functions, and nonlinear small gain.

6.1

Robust Control Under Strict Matching

The strict matching condition was introduced in [4, 193] for adaptive control of a dass of nonlinear systems. Adaptation was utilized to maintain robustness to

78

6. Robust Adaptive Control of a Class of Nonlinear Systems

parametric uncertainties. Here, we show that for the dass of systems satisfying the strict matching requirement, a robust controller can be designed that can handle both parametric and dynamic uncertainties in the terms lying in the span of the input. In addition, the controller can reject any bounded, unmeasured disturbances entering the system. In the subsequent sections, this result is extended to the case where the uncertainties and disturbance terms are not strictly matched with the input. We consider nonlinear systems of the form

( = lo(() + ~(() + p(()w(t) + {go(() + g1 ((, u)} u

(6.1)

where ( ERn, U E R is the control input, and w(t) E R is a bounded unmeasurable disturbance. The vector fields 10, go, p, ~, and g1 are smooth with 10(0) = ~(O) = p(O) = 0, go(O) =1= o. Note that the input uncertainty g1(0,U) need not appear affinely in u. The vector fields 10 and go are completely known and form the nominal system. We make the following assumption about the nominal system.

Assumption 6.1.1 The nominal system is globally feedback linearizable, i.e., there exists aglobai diffeomorphism z = p>-l

(6.6)

l§(z,v)1 :::; p

(6.7)

with p being unknown. Without loss of generality, we consider the following reference model to be tracked (6.8) im = AZm + B {K Zm + r} where A and Bare as in (6.4), K is a 1 x N matrix, and r is an auxiliary tracking signal. The state tracking error (i = z - zm) system can then be written as

i=

Ai + B ["J;(z)

+ ß(z)w(t) + {I + g(z, v)} v -

KZ m - r] .

(6.9)

Since the terms "J;(z) and ß(z)w(t) are not known, v cannot be chosen to feedback linearize (6.9). Nevertheless, utilizing the available information about the uncertainties given by (6.5), nonlinear damping can be injected through v to stabilize (6.9). Under the Assumption 6.1.2 (bounds (6.5)-(6.7)), the following robust adaptive controller is proposed for the nonlinear system (6.4):

(6.10) where a is a positive constant, P is the symmetrie positive definite solution of the Algebraic Riccati Equation (ARE)

A T p+PA-2&PBBT p+Q=0

(6.11)

with & = a(l + p) > 0, and Q a symmetrie positive definite matrix. ß is the adaptation gain to counter the effects of the uncertainties. The adaptation law for ß is given by

ß= r- l (l + p)IIBT Pi11 2 (1 + IlzI1 2(p-I)) L .

p

k(zT Pz)(k-l) - r-1aß

(6.12)

k=l 1 For simplicity, in this chapter, we focus on polynomial-bounded uncertainties. Appendix F contains a description of control design techniques in the presence of uncertainties bounded by general non linear functions of the states. See also Appendix G for a discussion of the notion of input-to-state stability, which is useful in the treatment of interconnected systems and systems with non linear dynamic uncertainties. 2This assumption can be relaxed to Ig(z, v)1 ::; pg(z, v) with g(z, v) being a known function. In fact, even this rest riet ion can be removed if the term K Zm + r is omitted from the control input. This feedforward term, however, greatly improves tracking performance and is desirable to retain.

80

6.

Robust Adaptive Control

0/ a Class 0/ Nonlinear Systems

where a > 0 incorporates the "sigma-modification" and

r

is a positive constant.

The stability properties of the closed-loop system can be stated in the following Lemma:

Lemma 6.1:Robust Adaptive Nonlinear Damping Matched (RANDM) For the perturbed system (6.1) satisfying the strict matching condition, and Assumptions 6.1.1, and 6.1.2, the control law (6.10) along with the adaptation law (6.12) results in global uniform boundedness of the errar system [z, ß] with respect to a compact set around the origin and any bounded unmeasumble disturbance can be asymptotically rejected. Furthermore, for the case where the objective is regulation to the origin (i. e., Z = 0), global asymptotic regulation is achieved if in (6.5) (0 = Bo = O.

Proof: Consider the Lyapunov function p

V = l)zT pz)k + r(ß - ß*)2

(6.13)

k=l

where ß* is the desired value of the control gain to be specified later. Differentiating V along the trajectories of (6.9), utilizing (6.5), (6.7), (6.10), (6.11), and (6.12), along with the inequalities (6.14)

(6.15) we obtain p

V:::; -

L {kA~i~(P)Amin(Q) - Cl ((k + Bklwmaxl)2} IIzl1

2k

k=l

p

-2ß*(1

+ p)IIB T pz112(1 + IlzI1 2(p-l)) L

kA~i~(P)llzI12(k-l)

k=l

+ d2 + 2p2 + 2& + ,ö2d2)IIBT pzl12 L P

+(~d2

+llzI1 4(k-l)) -

A~~;l)(p)(llzI12(k-l)

k=l

a(ß - ß*)2

+ aß*2 + (p~! + dd

(6.16)

where Iwmaxl

sup Iw(t)1 tER

(6.17)

81

6.1 Robust Control Under Strict Matching (0

+ 80 Iw max I

2p

2 + l p3

(6.18)

1

L k2 = 6P4 (P + 1)(2p + 1)22P+1, p

(6.19)

k=l

p 2 +1 (max {(k,8klwmaxl}) (max {SUp IZm(t)I ... sup l::;k::;p

tER.

tER.

+IK Zm + rl

IZ~(t)I}) (6.20)

and , is introduced as a design freedom. Choosing, as ((k + 8kl wmaxl)2 } ,= l::;k::;p max { -co + k)..':ni~(P»\min(Q)

(6.21)

where CO is a positive constant smaller than mink=l, ... ,n(k)..':ni~(P) .. min(Q)) and ß* as ß* = Al + A2 where

rr = min {co)..;;!ax(P), ... ,co)..;:;-'~x(P), r-l(J} ,

I}!

= (Jß*2 + (ptd + dl ) 2.

It thus, follows that V decreases monotonically until the solution reaches a compact set

(6.23) where Vf = rr-ll}!. Therefore, the solutions [z, ßl are globally ultimately bounded with respect to the bound Vj . For the regulation case with

(J

= 0, under the conditions given,

I}!

= 0 and

rr = min {co)..;;!ax(P), . .. ,co)..;:;-'~x(P)} , from which global asymptotic regulation of all the states is easy to establish. Remark 6.1 Although the above treatment considers the single input case, it is not hard to conceive analogous results for multi-input systems with additional notational complexity. In this case, the uncertainty g(z) multiplying the control input v in (6.4) is a matrix for which the following assumption is assumed to hold in place of (6.7):

).. . [g(z) + gT(z)]

mm

)..

max

2

[g(z) + gT(Z)] 2

> p>-1

(6.24)



(6.112) k and the existence of desired control gain ß* is established such that the resultant of the last two terms in (6.109) is negative, i.e., (6.113)

6.4 A Robust Control Design for Multi-Output Systems

105

where

(6.114) Global uniform boundedness of all states of the closed-loop system follows easily as in the proof of Lemma 6.1 (RANDM).

Remark 6.10 The choice of Q satisfying the above stated condition is reflected in the virtual control in Step 0 through P. This propagates to subsequent virtual controls, and finally to the actual control in Step p. The magnitude of Amin(Q) and hence that of Amin(P) is proportional to the magnitude of the bounds on the uncertainties and disturbance. In essence, a higher control gain is required to counteract higher bounds on the uncertainties.

106

6.

Robust Adaptive Control

0/ a Class 0/ Nonlinear Systems

2~------r-------r-------.-------,-------~------~

-O.5L----'----~---~--~----'----

o

2

4

6

8

10

12

Time (seconds)

_2L---~---~

o

2

4

___ 6

~

__

~

___

~

10

__

~

12

Time (seconds)

Figure 6.3: Model reference tracking performance for states Yl and Y2.

107

6.4 A Robust Control Design foT' Multi-Output Systems

1 ,/\, , , "

\

IXT'\

0'

,

-1

-2

.3L-----~------~------~----~------~----~

o

2

4

6

10

12

Time (seconds)

Figure 6.4: Model reference tracking performance for state

Xl.

1.4 ,--------~------~------,__----~------~----___,

1.2

,

0.6

-,

.

.' "

\".",/ \"""".,,"" \,./" "\'",',/

:'

.',

"

"'..'

'.'

,',

0.4

0.2

oL---~----~----~--~~~==~----~ o 2 4 6 10 12

Time (seconds)

Figure 6.5: Adaptation ofparameters ß(solid),

e(dashed), and N(dash-dotted).

108

6. Robust Adaptive Control of a Class of Nonlinear Systems

Y

Yref 1.5

5 Time (seconds)

Figure 6.6: Output tracking performance under the adaptive output feedback control.

0.9.-----~----~----~------~----~----~----__.

0.8

0.4 0.3 0.2 0.1

o~--~----~----~~~=======---~--~ o 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Time (seconds)

Figure 6.7: Adaptation of ß.

Chapter 7 Robust Adaptive Control of Stepper Motors As seen in Chap. 3, the differential equations governing the behavior of stepper motors are nonlinear and several uncertainties are present in the dynamics. In accurate positioning applications, ignoring modeling of these uncertainties is detrimental to the motor performance. Many of the problems arising in the operation of the stepp er motors, such as poor settling times, resonance phenomena, torque ripple, and positioning inaccuracy due to friction can be overcome by utilizing feedback control. In this chapter, the robust control designs advocated in the previous chapter are applied to various types of stepp er motors to ensure good tracking performance. Some modifications to the control design methodology outlined in the previous chapter are made in order to account for unknown virtual and actual position-dependent control coefficients that arise in the dynamics of the stepper motors.

7.1

Previous Work

The abundance of literature on nonlinear control of electric motors can be broadly classified into three distinct methodologies: exact linearization designs, backstepping based designs, and energy shaping designs. In exact linearization based designs, the motor parameters and dynamics are assumed to be known exactly and an inner feedback loop to cancel the nonlinearities is developed to render the closed-loop system linear under a change of coordinates. In the new co ordinate system thus obtained, various objectives such as tracking, pole placement, etc., may be carried out using linear control techniques. The design is applicable to nonlinear systems that are minimum-phase, a condition satisfied by most electric machines. The drawback of this method is that the parameters and dynamics need to be known exactly. A discussion of exact linearization

110

7.

Robust Adaptive Contral of Stepper Motors

designs for stepp er motors has been presented in Sect. 5.3. Exact linearization designs for BLDC motors and for induction motors are given in Chaps. 11 and 12, respectively. In many cases, however, the motor parameters are unknown (or partially known). In such cases, adaptive feedback linearization has been utilized [13, 132, 135] to compensate for modeling uncertainty and yield a closed-loop system that is robust to parametric uncertainties. However, most of the schemes based on adaptive feedback linearization are restricted to linear parametrization of the uncertainties, i.e., the unknown parameters appear linearly in the vector fields describing the nonlinear system. Moreover, the schemes do not guarantee robustness to unmodeled dynamics. Another nonlinear design is based on integrator backstepping and nonlinear damping tools [120]. The technique is applicable to systems with parametric uncertainty. The essential idea of the design procedure is to select a pseudo control input for a lower dimensional subsystem of the original system and design a control law for this system 1 . Since this is not the true control input, the dynamics of the error between the desired and actual values of the pseudo control input are formulated. When this is done, the actual control inputs may or may not appear, in which case the procedure is repeated until the actual control inputs appear. The true control inputs are designed via a final Lyapunov function formed by summing up Lyapunov functions for the individual stages. A unified robust adaptive control design for stepp er motors using backstepping was presented in [143]. Robust adaptive output feedback control was considered in [148]. Another recently developed approach to control of electric motors is the socalled energy shaping design [158, 163, 164]. The energy shaping approach is based on physical properties of the system, such as energy conservation and passivity. The key step in energy shaping designs is to express a certain matrix associated with the Euler-Lagrange equations in skew-symmetric form. The terms of this matrix represent the system's workless forces, and since they do not affect the stability properties of the system, they need not be canceled or offset with feedback, which somewhat simplifies the control design. An advantage of this method is that for electromechanical systems (which may be viewed as interconnections of electrical and mechanical subsystems), the energy shaping design may be applied to only the electrical subsystem and the mechanical subsystem may be treated as a passive perturbation. This method will be described in Chap. 14. A discussion of available results on nonlinear control of the different types of stepper motors is presented next.

1 For

electric motors, this pseudo control input is typically the phase current.

7.1 Previous W ork

7.1.1

111

VR Stepper Motors

Feedback linearizing controllers for the VR stepp er motor have been proposed in [153] using the flux linkage model as given by (3.28) and (3.29). The commutation scheme designed in [153] switches the currents according to the sign of the required torque and a position dependent logic. However, only a single phase is assigned the task of producing the required torque. Such a commutation scheme, however, has the disadvantage of requiring high currents (since all the torque is produced by a single phase) and knowledge of electromechanical system parameters. Another treatment of feedback linearization of variable reluctance motors may be found in [191], where the feedback linearization is performed with respect to a reduced-order model of the system (i.e., the mechanical subsystem) since the electrical dynamics are much fast er compared to the mechanical dynamics. The flux linkage model used is as given by (3.31)-(3.33). Using a time-scale separation technique, separate nested loops for feedback of electrieal and mechanical variables are designed.

In [192], an adaptive feedback linearization technique based on the extended matching condition [95] is applied to a nonlinear model of the motor with linear magnetic materials. Parameter update laws and an asymptotic stability result are arrived at using standard Lyapunov arguments. In addition to adaptive feedback linearization, model reference adaptive control of the variable reluctance motor is addressed in [106], where the flux linkage model used was the one given in [153]. Spline functions were used to model the nonlinear torque--currentposition characteristie, with adaptation for the spline function coefficients. The order of the resulting dynamic compensator is somewhat large since each spline coefficient needs to be estimated. Moreover, the electrical dynamics are not considered in the development of the controller. In [23], an adaptive nonlinear voltage-Ievel controller based on the integrator backstepping technique [120] was developed under the assumption oflinear magnetic materials. A robust adaptive voltage-Ievel controller was designed in [152]. A robust adaptive current-Ievel controller utilizing only position measurement was proposed in [148].

7.1.2

PM Stepp er Motors

In [214], position control is achieved using feedback linearization relying on exact knowledge of motor dynamics and parameters. Parameter estimation was considered in [12]. The feedback linearization design was extended to handle certain parametric uncertainties through adaptations in [13, 132, 189]. An adaptive controller robust to parametric and dynamic uncertainties such as nonsinusoidal flux density due to geometrie imperfections in the motor construction was presented in [150]. To obviate rotor velocity measurement, a nonlinear speed observer utilizing position and current measurements with linear error dynamics was constructed in [16, 35]. These results were extended to the adaptive case in [200]. In [148],

112

7. Robust Adaptive Control

0/ Stepper Motors

a current-level output feedback controller robust to parametric and dynamic uncertainties such as friction, load torque, and cogging torque was designed. The output feedback voltage-level control problem without current measurements has also been investigated. In [188], under the assumption that lower and upper bounds for all motor parameters are known, a controller utilizing rotor position and velocity was proposed. A controller that requires only position measurement was designed in [21]. However, all parameters were required to be known. A robust adaptive controller using rotor position and velocity measurements was proposed in [114]. Robust adaptive controller design using only position measurement was considered in [115] under the assumption that the electrical time constant is known and in [116] under the assumption that upper and lower bounds on motor parameters are known. Torque ripple reduction for PM stepper motors was considered in [31, 32, 150].

7.1.3

Sawyer Motors

Closed-loop control of Sawyer motors has recently gained attention due to the development and availability of suitable sensors [18, 22, 82] and high speed digital hardware. Closed-loop control of Sawyer motors using classicallinear control techniques was reported in [172, 173] and the application of nonlinear control techniques to Sawyer motors was considered in [101, 146, 151). A nonlinear robust adaptive output feedback current-level controller utilizing only position and yaw angle measurements was proposed in [149). A robust adaptive voltage-level controller using position and velocity measurements was designed in [113).

7.2

Problem Formulation for Various Stepper Motors

The dynamics of the mechanical subsystem consisting of a motor actuating a load (such as a rigid link in the vertical plane) are the same regardless of the type of motor actuating the load; hence, the first step of the robust controller design consisting of the design of the torque profile is common to all types of motors considered. However, the torque-current-position characteristics and the dynamics of the electrical subsystem vary with the type of motor under consideration. In the second step, the design is propagated to the voltage inputs using the advocated control design methodology and a suitable commutation scheme. Here, an example of a rigid link with an attached payload rotating in the vertical plane is used for purposes of control design without loss of generality, although any other (possibly time-varying) load mayaiso be used. In this case, the load torque function is given by (7.1)

113

7.2 Problem Formulation for Various Stepper Motors

~ NI sin(O)

(7.2)

where m p is the payload mass, ml is the link mass, l is the link length, and is the acceleration due to gravity. Furthermore, the torque due to friction is considered to be due to viscous damping:

g

(7.3)

TI =Dw

with D being the viscous damping factor. The total inertia of the system is the sum of the inertias due to rotor and the load:

(7.4) with J r being the rotor inertia. The dynamics of the mechanical subsystem can be written as

[1]

(7.5)

or, in compact form as, z

=

Az + ~ [-Dw - NI sin((J)

+ Tl

(7.6)

where

z

[

~]

(7.7)

A

[~~ ]

(7.8)

[~].

(7.9)

B

=

The load torque and viscous friction terms in (7.6) may be bounded by a firstorder polynomial in the mechanical subsystem states: (7.10)

with (1 = V2max(D/J, Nt/J). The reference model to be tracked is given by im

with Zm

= [Om

wm]T, K

= AZm + B{Kzm + r}

= [K l K 2],

(7.11)

and r being the reference signal.

Having the mechanical dynamics, the model for uncertainties, and the reference model, one may proceed with voltage level control for various stepper motors by considering the electrical dynamics of the motors. The robust adaptive design for various types of stepper motors are given in the following sections.

114

7.3

7.

Robust Adaptive Control of Stepper Motors

Control of VR Stepper Motors

In this section, the application of the robust control design to VR stepp er motors is presented. Our design follows [152J. The flux-linkage-current-position relationship for the motor used for control design is based on the model in (3.31)-(3.33) with in = 1 and Ti = 1. Therefore, '2 may be bounded as

T

2B Pi{ i 1>'1 + i 2>'2 }

~ 211BTPil1 2d + ii + i~.

(7.119)

Substituting (7.119) in (7.118) and manipulating yields

V2 ~ -(>'min(Q) - 0 by Assumption 9.1.1. e3

satisfies the dynamics

e3

=

+ v q + r' (X1)X~ + r(X1)[a1 (Xd X3 + Q2(X1)X4 + Tnl + NrX4X2 - a4(x1)x2 - v q -"(e3 + NrX4X2 + ("(r(x1) - Q4(X1))X2 +r' (Xl)X~ + r(xl)a2(xdx4 + r(x1)Tn . -"(q

+k1X3

Remark 9.5 r(x1) and r'(x1) satisfy the bounds, \lx1 E 1:. 1:.'

< Ir(x1)1::; r < Ir'(x1)1::; Ti

(9.17)

n, (9.18)

184 9. Voltage Control of Stepper Motors Using Position and Velocity Measurements where [

['

0,

(9.19)

We will need the Lyapunov function (9.20)

Differentiating,

Vo

e3e3 + X4 X4 < e3{ - ,e3 + NrX4X2

+ (rr(xt) - CX4(Xt))X2 + r'(x1)x~ +r(x1) cx 2(X1)X4 +r(x1)rn } + X4 { - k 1X4 + Nrqx2+Vd +Nrr(x1)x~ - NrX2e3 + CX5(X1)X2} 2 2 I, - k 1 a2 IX 4 e31 < -,e3 - k 1 X 4 + 1

fh

+ r'(x1)x~ + r(x1)rn } +X4 {Nrqx2 + Vd + Nrr(x1)x~ + CX5(Xt)X2} .

+e3 {(rr(x1) - CX4(X1))X2

(9.21)

Using Assumption 9.1.2 and Remark 9.3,

Vo

-E( e~

::;

+ x~) + e3 {(rr(x1) - CX4(X1) )X2 + r' (xt)x~ + r(x1)rn }

+X4 {NrQx 2 + Vd

+ Nrr(x1)x~ + CX5(X1)xd

< -~ (e~ + x~) + c min(l1, l2, 1)(91 (xi) + 92(X~) + 1\;1) +(NrQx2

+ Vd)X4

(9.22)

where c

. 4 [,21'2 Emm(h,l2,1)

91 (xi)

P1 (xi)

92(X~)

x~

+ -:;:;2 + 1\;1'2 + N;'1' 2 + a; + a~]

+ xi + P2(X~),

(9.23)

and L1 and L2 are two positive design parameters to be chosen later. Remark 9.6 Note that :1;3, which involves uncertain parameters, namely r(x1), is not available for use in control design. However, q, which is essentially a lowpass filtered version of v q , is available and is utilized in the controller design.

185

9.1 PM Stepper Motors

Controller Design: From (9.22),

Vd

is designed as (9.24)

From (9.22) and (9.24), it is seen that the (e3, X4) subsystem is Input-to-Statepractically-Stable (ISpS) with (Xl, X2) as inputs. Remark 9.7 X3 = q+r(xt}x2 can be interpreted, though in a limited sense, as an 'estimate' for X3. e3 then appears as the estimation error. The 'estimate' for X4 can be taken as zero. With Vd chosen as in (9.24), the observer error system is ISpS with (Xl, X2) as inputs. If the control input v q can be designed to make (X1,X2) small, q behaves as an 'estimate' for X3. (9.24) can be interpreted as a certainty equivalence control law to cancel terms involving X3 in the dynamics of X4. The quadrature-axis voltage, v q , is designed using the robust adaptive backstepping technique [87, 120] applied to the subsystem Xl

X2

X2

01(X1)(q + r(X1)X2 - e3) -"(q

+ Vq.

+ 02(X1)X4 + T n (9.25)

The backstepping design proceeds by considering lower-dimensional subsystems and designing virtual control inputs (or equivalently, state transformations). The virtual control inputs in the first and second steps are X2 and q, respectively. In the third step, the control input v q appears and can be designed.

Step 1: For the first step, a Lyapunov function 1

2

2111 (zl)

(9.26)

is used where Zl = Xl - Brei and TJ1 is a dass K oo function to be chosen later in the design process to satisfy an inequality (9.55) to guarantee enough damping. A simple quadratic function would not suffice for V1 because of the presence of higher order nonlinearities in g2 and possibly P1 and P2. Differentiating (9.26), 2

.

TJ~(Zl)Zl(X2 - Brei)

-TJ~ (Zi)Zl/11 (zt)

+ TJ~ (Zi)Zl Z2

(9.27)

where Z2 = X2 - Örei + /11 (Zl) and /11 (S) = Cl S with Cl being a positive constant. Here, in the first step of backstepping, the one-dimensional system Xl = X2 is considered and X2 is regarded as the virtual control input. Asymptotic tracking can be achieved by choosing X2 = Örei - /11 (Zl)' However, since X2 is not the actual control input, the error between X2 and the desired X2 is formulated as

186 9. Voltage Control 01 Stepper Motors Using Position and Velocity Measurements

Z2. In the next step of backstepping, q will be regarded as the virtual control input and will be designed to make Z2 smalI, i.e., X2 converges to the virtual controllaw designed for X2, and hence, Xl converges to Brei' However, since q is also not the actual control input, the process is repeated by introducing an error Z3 which is the difference between q and the desired q. At the third step of backstepping, the control input v q appears and the controllaw that is designed at that step can actually be implemented. Step 2: A new Lyapunov function is defined as

(9.28)

This Lyapunov function satisfies

(9.29)

Using elementary algebra, the following inequalities can be derived:

Z2 r (xd x 2 < -z2 e 3 G2 (Xl)

Z2-(-)X4 GI Xl

-2 r

(1' + "4

+

-2 r 4h

2'2

)Z2

1

2 < -z22 + Ele3

4EI -2 G2

2

2

< --2-z2 + EIX4 4~hEl

1 Z2-(-)Tn < GI Xl

4

2 Ql

.

2 2 2 (l l 1)Z2+h[PI(4zd+P2(9vdzd)]

'"

mm

1, 2,

+l2P2(9z2 ) + PI (4B rei ) + P2(9B rei ) + "'1 1 2 "2 < 4"2Z2 + Brei 2

1 " -Z2-(-) Brei GI Xl

vi (Zl) . Z2-(-) (X2 - Brei) GI Xl

Gl (Xl)

2

2

+ Brei + h VI (zd

Ql


lt[z~ + 6l/r(zd + 3p1(4zr) + 3p2(9I/r(Zl)) + 28I/t(Zl)] {i6(Z~) > l2[4z~ + 3p2(9z~) + 27zi] {i5(Z~)

d4

2

d3 + P1 (40 ref )

+ P2(90·2ref ) + 30·2ref + 270·4ref + K1·

(9.51)

The inequality signs in the definition of {i5 and {i6 are to allow the possibility of simplifying the corresponding expressions. Before proceeding, we need the following lemma.

Lemma 9.1 Given any dass K function (s), it is possible to find a dass K co function ~(s) = O[s] as s -> 0+, and a positive constant, 0", to satisfy (s) ~ ~(s) Prao!

0"

+ 0"

vs;::: o.

(9.52)

= sUPs:::) (s) and

~(s)

=

(I)s { (s) + (s - 1)

for for

s ~ 1 s> 1

(9.53)

satisfy (9.52). Thus, using Lemma 9.1, class K co functions, fL5 and fL6, and positive constants, 71"5 and 71"6, can be found to satisfy fLiCS) = O[s] as s -> 0+ and

{ii(S)

~

fLiCS)

+ 71"i

Finally, smooth functions "T/1 E K co and

C1"T/~(Zr)zr Z21/2(Z2)

,

i

1/2 (s)

= 5,6. = sii2(s2)

(9.54)

are chosen to satisfy

> C1"T/1(Zr);::: 2fL5(zi) >

2fL6(Z~).

(9.55)

Remark 9.10 In this remark, we provide a proof that smooth functions "T/1 E K co and 1/2 can be chosen such that (9.55) is satisfied. By construction, fL5(S) and fL6(S) are O[s] as s -> 0+. Thus, smooth positive nondecreasing functions 71i(s) can be found such that S71i(S) ;::: fLiCS), i = 5,6, for sE [0,00). A smooth positive nondecreasing function 715c can be chosen such that 715c(S) ;::: 71~(s). Choosing any Cl > 0 and "T/1 (s) = (2s/ Cl) [715 (0) + 715Aw)dw], the inequalities Cl"T/l (s) ;::: 2fL5(S) and C1"T/~ (s)s = Cl"T/l (s) + 2s 2715c(S) ;::: C1"T/l (s) are satisfied for s E [0,00).

J;

Furthermore, "T/l(S) and 1h(s) ~ (1/2)7]~(s)s are smooth class K co functions. Choosing ii2(Z~) = 2(716(Z~) + 1), we have Z21/2(Z2) = 2z~716(zD + 2z~ ;::: 2fL6(Z~) for IZ21 E [0,00). By construction, sii2(s) is a smooth class K co function.

Remark 9.11 II can possibly be chosen to eliminate the dependence of 7]~ (zr) on Cl. This will make the tuning of the parameter Cl easier.

192 9. Voltage Contml of Stepper Motors Using Position and Velocity Measurements This concludes the contral design which can be summarized as folIows: 1. Pick positive constants Cl, ll, and l2 (see Remark 9.11 and the discussion in Sect. 9.1.1 for guidelines on picking these parameters). 2. Implement the input filter dynamics, q = -,q + v q with , > 0 chosen to satisfy (9.9). 3. Evaluate ""5 and ""6 from (9.51) and use (9.54) to find ii5 and ii6. Pick ""1 E K oo and V2 to satisfy (9.55). Choose any V3 to satisfy SV3(S) ~ C3s2 with C3 being a positive constant. 4. Design the control inputs as

(b)

[".,~ (z~) + 2",,~(z~)Z~](X2 - Öre!) - ßZ2 -[ß + 3z~ + v~(z2)l (Cl (X2 - Öre!) - Öre! ) -

,q ,

2

'

'

,

2

Z2 22

-[ß + 3z2 + v~(z2)ln1q - ßZ3[ß + 3z2 + v~(Z2)l (q + 1) -V3(Z3) (9.56) where

Xl - Bre ! X2 - Öre!

+ C1 Z 1

q +".,~ (Zr)Zl + ßZ2 + z~ + V2(Z2).

(9.57)

5. Implement the parameter update laws as ,

- CT ln1

'

2

+ b1z3[ß + 3z2 + v~(Z2)lq

-CT2ß + b2 { Z~ + Z5[ß + 3z~ +

V~(Z2W(q2 + 1) }. (9.58)

6. Compute actual stator phase voltage inputs (V1, V2) by inverting the DQ transformation in (3.41), Le.,

cos(NrO) [ sin(NrO)

- sin(NrO) ] [ vv d cos(NrB) q

] .

(9.59)

Stability Analysis: Utilizing (9.55), (9.50) reduces to (9.60)

193

9.1 PM Stepper Motors

Y(X)

=

(9.61)

From (9.55), it follows that Y(X) ~ E/2). Thus,

rr

V where

rr =

min(C1, 81 2Q1, 2C3, 0"1, 0"2, (9.62)

Applying the Comparison Lemma [99],

V(t)

::;

V(O)e- I1t

+

l

t

e- I1 (t-s)d 5(s)ds.

(9.63)

Since the reference signal Bref(t) and its first two derivatives are bounded, d5 (t) is a bounded time-varying signal. V is a smooth, positive definite, and radially unbounded function of X. Thus, from (9.63), V and hence X are bounded along the trajectories of the system. By routine signal chasing, it is seen that boundedness of X implies boundedness of all closed-Ioop signals. Furthermore, the bound in (9.63) implies that solutions te nd to the compact set in which (9.64)

where d5 = limsupt-+ood5(t) is a finite positive constant since d5 is bounded. In the set (9.64), the tracking error Zl satisfies the inequality

"21 171 (Zl)2

::;

rr- 1-d5.

(9.65)

The practical tracking property can be inferred from (9.65). Let an arbitrary te > 0 be given. 171 will be designed to regulate the tracking error Zl to [-te, tel. First, Cl > 0 and a smooth class K oo function r(s) are chosen to satisfy C1Sr'(S) ~ C1r(S) ~ 2ji,5(S). The discussion in Remark 9.10 guarantees that such a choice is possible. Choose Cl = max(l, 2rr-ld,5/r(t;)) and choose 171(S) = C1r(S). With this choice of Cl and 171, (9.55) is satisfied which ensures, as proved by the Lyapunov analysis presented above, that the solutions tend asymptotically to a compact set in which (9.64) and (9.65) are satisfied. From (9.65), (9.66)

Theorem 9.1 The designed dynamic controller guarantees global boundedness of all closed-Ioop signals and achieves practical stabilization of the rotor position tracking error.

194 9. Voltage Contral of Stepper Motors Using Position and Velocity Measurements

Theorem 9.2 If K:1 = 0 and pi(S) = O[s] as S --> 0+ for i = 1,2, the designed dynamic compensator with 0'1 = 0'2 = 0 achieves global asymptotic stabilization ((Jref == 0) ofthe rotor position to the origin, while keeping all closed-loop signals globally bounded. Proa! Under the conditions of Theorem 9.2, d4 statement of the theorem follows from (9.60). 0

= 7r5 =

7r6

= d5 = 0 and

the

Theorem 9.3 Under the assumption of sinusoidal ftux distribution, if K:1 = 0 and P1(Xi) and P2(X~) are identically zero, the proposed dynamic controller with 0'1 = 0'2 = 0 achieves global asymptotic regulation to any constant reference (Jref' while keeping all closed-loop signals globally bounded. Praa! Since Q1d = Bref = Öref = 7r5 = 7r6 = 0, using Remark 9.8, it follows that d5 = O. From (9.60), it is seen that Zl = Xl - (Jref asymptotically approaches

O.

0

9.1.1

Simulation Results

The efficacy of the proposed controller is demonstrated using simulation for a PM stepp er motor with the following parameters:

R= 1 0,

Tf

L m1 = 5 mH, L m4 = 0.0625 mH, i f = 1 A, + Tl = Dw + N 1 sin((J).

1 = 0.0733

L m2 = 0.5

kg-m 2 , mH,

L f4 = 1.766 mH, D = 0.002 kg-m 2 /s,

L o = 0.7 mH, L m3 = 0.166 mH, N r = 50,

N1

= 1.7201 Nm, (9.67)

The desired trajectory for the rotor position is a smooth-start sinusoid, (1 - e- O. 2t2 ) sin(10t). Initial conditions for all states including the parameter estimates are zero. The design parameters in the controller are chosen as 'Y

= 1500,

0'1

/l1(S) = 100s,

=

0'2

/l2(S)

= 1000,

b1 = b2 = 105 ,

= 0.076(s + i), /l3(S)

7]l(S)

= 5.2(s

= 2000s.

1

+ 2' s 2), (9.68)

The form of 7]1 and /12 follows from (9.55). The constants in 7]1, /11, and /12 were picked using an approximate linearization of the system around the origin assuming fast electrical, observer error, and Z3 dynamics to place the poles of the linearized mechanical subsystem at a natural frequency of 40 Hz with a 0.6 damping ratio. While these assumptions are only approximately true, the consideration of these pole locations provides a convenient method to pick the design parameters. The (approximate) characteristic polynomial of the secondorder mechanical subsystem is s2 + ~2S + ~1 where ~l

N 1

J+

(C1C2

+ clp)ifNrLml 1Lo

195

9.1 PM Steppe.,. Moto.,.s

(9.69) with C1 p S and C2S being the linear parts of 771(S) and 112(S), respectively. C3 is picked large enough to ensure fast convergence of Z3 to zero. The sigmamodification terms (Ti aid in preventing parameter drift instability. Too small a value of (Ti would not be effective in preventing large excursions of the parameter estimates. Furthermore, with a small value of (Ti, the response of the parameter estimates to achanging environment is sluggish. Too large a value of (Ti would result in the parameter estimates becoming small, unless bi are also made very large. Increasing (Ti would increase the bandwidth of the parameter estimator dynamics, increasing sensitivity to high frequency noise, and w0l!ld potentially have a destabilizing effect because of the presence of the term ßZ2 in the expression for v q . Typically, (Ti can be chosen to be two or three times the operating frequency (Nr SUPt Brej(t)). The simulation results are shown in Fig. 9.1. It is seen that the rotor position tracks the desired trajectory with a maximum error of 0.018° and an RMS error of less than 0.014°. To illustrate the robustness of the proposed controller, a simulation was carried out with time-varying perturbations in the plant parameters using the same controller parameters as above. Specifically, the values of L m1 , R, D, L o, and ] were modified to

L m1

5(1 +0.5sin(2t))mH

R

(1+0.5sin(t))f2

D

0.002(1 + 0.5 cos(15t)) kg-m 2 /s

Lo

0.7(1 + 0.5 cos(3t)) mH

]

0.0733(1 + 0.5 cos(7t)) kg-m 2 .

(9.70)

This represents a maximum change of 50% in each of the parameters. In the presence ofthese time-varying perturbations, the motor dynamics can be written as W

1

.

-(T-Tj-T/-]W) ]

(9.71)

196 9. Voltage Control of Stepper Motors Using Position and Velocity Measurements Even in the presence of severe time-varying parameter perturbations, the performance is degraded to a maximum tracking error of 0.036° and an RMS error of 0.016°. Furthermore, incorporating 16-bit quantizations on sensors and actuators degraded the maximum and RMS errors to 0.039° and 0.018°, respectively. The increase in the maximum tracking error when the quantization is taken into account corresponds to half a bit, indeed the least possible degradation 2 • Since the simulation plots in these two cases are virtually identical, only the plot including the perturbations and quantization effects is presented in Fig. 9.2. To provide a benchmark for motor performance under a typical controllaw, simulations were also performed using a feedback linearizing controller [16]. The feedback linearizing controller is based on the nominal sinusoidal flux distribution model since it is not practically reasonable to assume knowledge of coefficients of higher order inductance terms. The current references are designed including field weakening (see Appendix C) as

(9.72) where km = i! NrL m1 is the torque constant and Dw is friction. The control inputs are

(9.73) where

V qr

di qr L 0Tt

.

.

-kll(id -

i dr )

-k 22 (i q - i qr ) - k 23 (B -k 25

.

.

+ RZ qr + Nr(}re!LOz dr + km(}re!

J( ) -

(}re! )dt.

- Bre !) -

k 24 ((} - (}re!)

(9.74)

Note that an integral feedback term has been incorporated in the controller to furt her reduce the steady-state error. The gains kij in (9.74) are chosen as kll = 20000, k 22 = 376.4, k 23 = 1.525 X 105 , k 24 = 2.824 X 107 , and k 25 = 1.374 x 108 . This choice results in similar bandwidth and damping to that provided by the parameters chosen in the simulations with the robust adaptive 2The resolution to achieve 0.018° corresponds to about 15 bits on the position. Even if we degrade the quantization on the velocity measurement to 12 bits, the peak and RMS errors are 0.04° and 0.018°, respectively (a minor reduction in performance).

197

9.2 Sawyer Motors

output feedback controller. Furthermore, the control input magnitudes are also comparable with this choice of control gains (compare Fig. 9.1 with Fig. 9.3, and Fig. 9.2 with Fig. 9.4). Note that the feedback linearizing controller requires fullstate feedback and a knowledge of nominal plant parameters. The simulation results with the nominal plant parameters are shown in Fig. 9.3. The peak error is roughly 0.04° and the RMS error is around 0.015°. Simulations with the time-varying parameter perturbations are shown in Fig. 9.4. The peak and RMS errors are seen to be 0.17° and 0.05°, respectively. The quantization effects were not included in the simulations with the feedback linearizing controller since this controller was intended to be a baseline design.

9.2

Sawyer Motors

Electromechanical Model and Problem Statement: The system model is given in (3.61). The control objective is practical stabilization and tracking for the Sawyer motor, i.e., we seek to design dynamic control laws for VI, ... , Vs to make the tracking errors x - xre/' Y - Yre/, and B - BreI converge to [-fe!' f e 1], [-f e 3' f e 3], and [-f e 5' f e 5], respectively, for any arbitrary prespecified positive constants fej,j = 1,3,5, while keeping all closed-loop signals bounded. xre/' Yre/, and BreI are given twice-differentiable bounded reference signals with bounded first and second order derivatives. BreI is usually zero since it is desirable to avoid yaw rotation. This control objective is to be achieved utilizing only the position and velocity measurements, Le., x, X, y, iJ, B, and iJ are available for measurement. The currents are not measured. VI, ... , Vs are used as the control inputs. The actual voltage inputs, VA, ... , VH can be obtained by inverting the transformation (3.60). For convenience of notation, we introduce Zn = x, Z13 = y, Z15 = B, Z21 = X, Z23 = iJ, and Z25 = iJ. The uncertain terms,

F dx , Fdy,

and

(lFdxl + IFdyl + hl)2
c::

:;;: u

jg -0.5

0

5

10

15

20

5

10

15

20

5

10

15

20

Figure 9.3: Simulation results for PM stepper with feedback linearizing controller. x 10-'

4

'S

'S 2 ['!

!'! C

.,0

~ t:

";;;

Ql

0

0>

a.

:;;:

~

5

~

0

e::

10

15

-2 -4

20

100

20

50

10

0

0

5

10

15

20

0

5

10

15

20

~

>" -10

>"

-50

-20 -100 0

5

10

15

20

-30

Figure 9.4: Simulation results for PM stepp er with feedback linearizing controller and perturbed plant parameters.

213

9.2 Sawyer Motors

0.1

0.1

0.05

0.05

:[

~ß

0

>


-1 0

2

er

0

- 0.02 -0.04 20

6

0

I(S)

0.04 0.02

4

0.02

'"

u

0

-0.02

2

2

4

6

l(s)

15

4

6

0 X 10-3 1

..,

u

2

4

6

l(s)

5 0,05 l(s)

0,1

0.05 0

0

-2

.",N -1

·.J_4

10 ocQ.

-2

0

I(S)

0.05

~v

-0.04

2

0

I(S)

0.04

~ 0

-1

2

4 I(S)

6

-6

X

0

10- 3

0.1

I(S)

0.05

0.1

l(s)

Figure 9.5: Simulation results for Sawyer motor with robust adaptive output feedback controller.

214 9. Voltage Gontmi of Stepper Motors Using Position and Velocity Measurements

0.1 r------.,......,...--r-n

0.1 ,..--------,:-r-r""T'1

10,..----------,

~ :w~~_____-' ~

-0 1 L-_ _ _----'--'-..L-.L.J -0.1 L-_ _ _ _--'-..L-L.l 02460 2 4 6 l(s) 1(5)

o

0.1 l(s)

0.2

20

10 ~OH

>

_1L-_ _ _ _ _ _- J

o

2

4

_1L-_ _ _ _ _ _- J

6

1(5)

o

2

o

2

4

__

-10 -20 L -_ _ _ _ _- - '

6

1(5)

o

2

o

2

4

6

1(5)

20 10

~UJ >

0

-10 -20 L -_ _ _ _ _--.J

o

4

2

6

4 6 1(5) 0.04 , . . - - - - - - - . . . . ,

I(S)

0.04 r - - - - - - - - - - ,

0.02~

ö':~ -0.02

N

0 being a design freedom. From (10.9), it is seen that the observer error states (e2, e3 + ~, e4) are rendered Input-to-State-pmctically-Stable (ISpS)[92] with respect to Xl.

Controller Design: The control inputs, VI and V2, are designed using robust adaptive observer backstepping technique [87, 120] applied to the following subsystem:

+ Q2Xl

Xl

kl

:1;2

-Ql:1;2 - :1;3

:1;3

-1':1;3

(:1;2

+ VI

- e2)

sin(Nrxd

+ :1;4 cos(Nrxd (10.11)

Step 1: For the first step, a Lyapunov function VI

is used where Zl = Xl the design process.

-

=

1 2k

2

l

1]1(Zl)

(10.12)

Brei and 1]1 is a dass K oo function to be chosen later in

Differentiating (10.12), (10.13) Straightforward algebra yields the inequalities

Q21]~(zi)ZlXl


0 Vs E R\{O} ,i

(10.30)

= 3,4.

Using (10:30), (10.28) simplifies to

V3 :S -1]1 ZIVI I

1 - 2Z2V2 - z3 V3 - z4 V4 - 1 (2 z3

+ z42)

+ ß2 + ß3 - ß - b4)(b2 + b8) 2 2 2 +(ß + aß)b4 + 6lz 1 + 3lv1 (ZI) + 4Ee2 + d3. +(ßl A

The dynamics for

A

(10.31 )

ßare designed as (10.32)

Stability Analysis: To analyze the stability of the overall closed-loop system, a composite Lyapunov function is defined as (10.33) yielding

10.1 Feedback Design Assuming Knowledge

0/ Electrical

Time Constant

223

where (10.35)

Applying Lemma 9.1, we find a dass K oo function jj(s) and a positive constant cp such that jj(s) = O[s] as s ---- 0+ and (10.36)

where

Finally, '1]1

E /(00

and VI are chosen to ensure that (10.39)

with 'I]~ (zi), 'l]nzi) , and vHzl) being positive for all ZI E (-00,00), and with 'l]i(zi)zIVl(Zt) being a positive radially unbounded function of ZI.

Remark 10.4 In this remark, we provide a proof that functions '1]1 E K oo and VI can be chosen such that (10.39) is satisfied with 'I]~ (zi), 'l]nzr), and vi (ZI) being positive for all Zl E (-00,00). Choosing, for instance, VI (Zl) = clzl with Cl being a positive constant, and defining

J-t(zi)

=

2l[(8 + 3ci)zi

+ jj(4zi)],

(10.40)

224

10.

Voltage Control of PM Stepper Motors Using Position Measurement Only

(10.39) reduces to (10.41 )

Note that J1.(s) is continuous and J1.(s) = O[sJ as s -> 0+. Hence, a smooth positive nondecreasing function ii can be chosen such that sMs) ~ J1.(s) for s E [0,00). Choosing 171(S) = ji(t)~t+1dt, the inequality Cl17Hs)s ~ J1.(s) is satisfied for s E [0,00). The integral in the definition of 171 is well-defined by the continuity of ii. With this choice of 171, the function 17i (s) = ji(S)~S+1 is positive for all s E [0, 00 ). Moreover, since 17i is a monotonically increasing function, 17n s) is positive for positive s. Furthermore, note that with this choice of 17i and V1, the term 17i(zI)zlV1(Zl) = Cl17i(zI)zr is a class K oo function of zr.

J;

Remark 10.5 The parameter estimate /3 is initialized as a positive real number. From Remarks 10.3 and 10.4, b1 is seen to be positive. Also, b2 and bs are positive which implies that /3 and hence b1 remain positive. With 171 and V1 chosen to satisfy (10.39), (10.37) reduces to

. V

1

1

2

2

< -2171Z1V1 - 2Z2V2 - Z3V3 - Z4V4 - 'Y(z3 + z4) I

17 ' 2- 4Ee 2 2 -32E [ (e3+-) k 3 2+e42] --(ß-ß1-ß2-ß3) 2 2 a1 Nr (10.42) +d5 · Theorem 10.1 The proposed dynamic controller guarantees global boundedness of all closed-loop signals and achieves practical stabilization of the tracking error. Proof Equation (10.42) can be written as

V :::;

-y(X)

+ d5

(10.43)

y(X) =

is a smooth, positive definite, and radially unbounded function of X. Note that V is also a smooth, positive definite, and radially unbounded function of X. From (10.38) and (10.43), using the boundedness of f}re/(t) and its derivative, it is inferred using standard Lyapunov arguments that X is bounded along the trajectories of the solutions of the system. By routine signal chasing, it is seen that the boundedness of X implies the boundedness of all closed-loop signals. Furthermore, the design freedoms 171 and V1 can be picked to make the tracking error arbitrarily small asymptotically. 0

10.1 Feedback Design Assuming Knowledge

01 Electrical

Time Constant

225

Theorem 10.2 If BreI == 0, Po = 0, and p(s) = O[s] as s ....-.0+, the proposed dynamic controller with a = 0 achieves asymptotic stabilization of the rotor position to zero.

Praaf: Under the hypothesis of the theorem, po = cp = a = d5 = O. Hence, from (10.43), X and hence Zl = Xl te nd to 0 asymptotically as t ....-. 00. 0 The control design procedure can be summarized as follows: 1. Find p to satisfy (10.36) with p(s)

= O[sJ as s....-. 0+.

K. oo and VI (s) = CIS, Cl > 0, to satisfy (10.39) with 'TI~(s) and 'TIns) being positive for positive s, and 'TI~ (s) S being a dass K. oo function.

2. Pick a positive constant land functions 'TII E

3. Pick a positive constant C2 and define V2(S) = C2S. Pick functions V3 and V4 such that SVi(S) > 0 Vs E R\{O}, i = 3,4. For instance, Vi can be picked as Vi(S) = CiS with Ci being positive constants. 4. Pick positive numbers O!l and a. 5. Implement the observer dynamics

-0!15: 2 - 5: 3 sin(NrXI)

5: 2

-,5: 3 + VI

5: 3

+ 5: 4 cOS(NrXI) (10.45)

6. Design the control inputs as

VI

= Z2 sin(NrXI) + ,b3 sin(NrXI) + b5 sin(Nrxt) -((3 + b4)z3b~[1 + (32(1 + 'TIi2 )] - V3(Z3) -Z2 cOS(NrXI) - ,b3 cOS(NrXI) - b5 cOS(NrXI)

V2

-((3 + b4)z4b~[1

+ (32(1 + 'TIi2 )J -

V4(Z4)

where Zl

Xl - BreI

Z2

5: 2 + (3(1

Z3

5: 3

Z4

5: 4

bl b2

b3

-

+ 'TI~(Z~))ZI + VI(ZI)

b3 sin(NrXI)

+ b3 cos(Nrxt) (3(1 + 'TI~ (z~) + 2z~'TI~(z~)) + v~ (zt) ('TIi + 'TI~2)z~ + [bI + bi + bi (32 (1 + 'TID2Jz~ ' b I B' rel+'TI1'(I + 'TI ')2 'TII, ZI-0!IX2I Zl3 2 +Z2[(3 + (1 + 'TIDZIZ2] [bI + bi + bi(32(1 + 'TID ]

(10.46)

226

10.

Voltage Control of PM Stepper Motors Using Position Measurement Only

-0",8(1

+ 1]DZ1 + l/2(Z2)

z2(1 + 1]~)Zl 8b3 ~ 8b 3 · -,ß - -e ref 8ß 8z1

+X4 cOS(NrX1)

+88~3

x2

+ -8b3 { 8z2

, , Cl:1 X2 - x3 sin(Nr X1)

+ ,8(1 + 1]DZ1 -

{-Cl:1X2 - X3 sin(Nrxt)

b1Bref }

+ ~b3

8e ref

+ X4 cos(Nrxt)}

8b 3 8b3) ( 8z 1 + b18z2 sin(NrX1) + b3Nr cOS(NrX1) 8b 3 8b3) . ( 8z1 + b18z 2 cos(Nrxt) - b3Nr sm(Nr X1) z~b~[l + ,82(1 + 1]~2)] + z~b~[l + ,82(1 + 1]?)].

bs

Öref

(10.47)

7. Implement the parameter update law (10.48)

Remark 10.6 While the condition (10.39) imposes a constraint on the form of 1]1 and l/1, these functions can be scaled by multiplication with arbitrary constants by choosing 1 appropriately. This freedom can be exploited to avoid unwanted high gains in the controller. Furthermore, 1 can be made a function of Cl such that the choice of 1]1 to satisfy (10.41) is independent of Cl. This will make the tuning of the parameter Cl easier.

10.1.1

Simulation Results

The efficacy of the proposed controller is demonstrated using simulation for a PM stepp er motor with the following parameters: N r = 50, J = 0.0733 kg_m 2, L o = 0.7mH, Lm1 = 5mH, L f4 = 1.766mH, R = 10, if = lA,D = 0.002kg-m 2/s, Tl = 1.7201sin(x1)Nm. (10.49) The desired trajectory for the rotor position is a smooth-start sinusoid, (1 - e- O.2t2 ) sin(4t). Initial conditions for all states including the parameter estimates are zero. Abounding function for the torque uncertainties is assumed to be available as p(xi} = xi. The design parameters in the controller are chosen as:

= 500, 0" = 2000, l/i(S) = CiS, 1::; i ::; 4, 5 Cl = 10- , C2 = 800, C3 = C4 = 1500, 1 = 10- 5 , 1]l(Zn = 18zr· Cl:1

(10.50)

10.1 Feedback Design Assuming Knowledge of ElectTical Time Constant

227

The simulation results are shown in Fig. 10.1. It is seen that the rotor position tracks the desired trajectory with a peak error of 0.09° and an RMS error of less than 0.05°. The choice of the form of T/l and 1/1 is dictated by the structure of the bounding function for the torque uncertainties. The constants in T/l, 1/1, and 1/2 were picked using an approximate linearization of the system around the origin assuming fast electrical and observer error dynamics to place the poles of the linearized mechanical subsystem at a natural frequency of 40 Hz with a 0.6 damping ratio. C3, C4, and 0:1 are chosen large enough to ensure fast electrical and observer error dynamics. The sigma-modification term a aids in preventing parameter drift instability. Too small a value of a would not be effective in preventing large excursions of the parameter estimate. Furthermore, with a small value of a, the response of the parameter estimate to achanging environment is sluggish. Increasing a would increase the bandwidth of the parameter estimator dynamics increasing sensitivity to high frequency noise. Typically, a can be chosen to be one order of magnitude higher than the operating frequency (Nr SUPt Brej(t)). To illustrate the robustness of the proposed controller, a simulation was carried out with time-varying perturbations in the plant parameters using the same controller parameters as above. Specifically, the values of Rand L m1 were modified to R = (1 + 0.5sin(3t)) n and L m1 = 5(1 + 0.5sin(t)) mH. This represents a maximum change of 50% in each of the perturbed parameters. Note that in this case, our assumption that , is constant and known is violated. The simulation results are shown in Fig. 10.2. Even in the presence of severe timevarying parameter perturbations, the performance is degraded to a maximum steady state tracking error of 0.12° and an RMS error of 0.06°.

Remark 10.7 The simulation results indicate that the designed controller is robust to considerable time-varying variation of ,. Though the theoretical stability analysis assurnes that , is constant and known, some robustness to timevarying uncertainty in , can be inferred through a linearization study. In the case that only a nominal value 1 for the time-varying parameter, is available, an observer can be implemented with the dynamics in (10.5) with 1 appearing in place of,. The controller development is identical to the design above except that , is everywhere replaced by 1. Thus, the effect of amismatch between the real, and the available nominal value 1 is the introduction of additive terms (r -1)x3 and (r -1)x4 in the expressions for E3 and E4, respectively, in (10.7). The resulting additional terms in the derivative of the observer Lyapunov functi on Vo can be overbounded by quadratics and generate a term of the form d(x~ + x~) with d being a positive constant. Thus, in this case, the subsystem with states (e2,e3 + ~,e4) is ISpS with respect to the inputs (Xl,X3,X4). A composite Lyapunov function argument can be carried out to demonstrate c1osed-loop stability if an inequality of the form

228

10.

Voltage Control

0/ PM Stepper

Motors Using Position Measurement Only

(10.51 )

holds with a1 and a2 being positive constants. The presence of high order nonlinearities in the definition of Z2, Z3, and Z4 prevents (10.51) from holding globally. However, taking Brei == 0 and linearizing around the origin, we obtain Zl = Xl> Z2 = X2 + C1 X 1, Z3 = X3, and Z4 = X4 + (C1p + C1 C2)X1 + (C2 - at) X 2 where C1 p S is the linearization of 1]l(S) around o. Denoting by Q the transformation matrix that maps X n = (Xl, X2, X3, X4) into Z = (Zl, Z2, Z3, Z4), Q is obviously invertible, and hence QT Q is positive definite. With 1]~ (0) being positive, the right-hand side of (10.51) is locally bounded below by plzI 2 2': P>'min(QTQ)lx n I2 2': p>'min(QTQ)(x~ +x~) with p being a positive constant and >'min(QTQ) being the smallest eigenvalue of QTQ. Thus, (10.51) holds with a1 > 0, a2 = 0, hence implying local robustness to perturbation in "(.

10.2

Feedback Design Assuming Known Bounds on Motor Parameters

In this section, we design a controller to achieve global practical stabilization for the position tracking error of a voltage-fed permanent-magnet stepper motor. The controller utilizes measurement of only the rotor position and requires availability of upper and lower bounds on the electromechanical parameters of the motor. The controller proposed achieves global stability with practical stabilization of the tracking error. Furthermore, if the torque disturbances vanish at the origin, the designed controller guarantees asymptotic stabilization. Electromechanical Model and Problem Statement: The dynamics of the PM stepper motor in the state coordinates (10.1) are given in (10.2). Here, since the electrical time constant "( is unknown, k4 = "( is an additional uncertain parameter. Our design will be carried out under the following assumptions: Assumption 10.2.1 k 1 , k2 , k3 , and k4 are unknown constants bounded above and below by known positive constants, i.e., for i = 1, ... ,4, k < k1. < k·t _

(10.52)

-l-

with known positive

k.i

Assumption 10.2.2

and k i .

Tn

can be bounded as (10.53)

where '" and Po are unknown non-negative constants. Upper bounds on '" and Po are assumed to be known.

7{

and Po

229

10.2 Feedback Design Assuming Known Bounds on Motor Parameters

Remark 10.8 From the definition of 7 n in (10.3), it is seen that 7 n satisfies the bound in (10.53) if 7l does. This implies that time-varying position dependent load torques (e.g., a load depending on gravity such as N sin(O)) can be handled.

In the remainder of the section, we design a controller to make 0 track a given thrice-differentiable bounded reference signal Orej(t) with bounded derivatives up to third order, while keeping all other signals globally bounded, given that only 0 is available for feedback and only upper and lower bounds on the parameters are available. Observer Design: The following observer is utilized to generate estimates of the unmeasured states:

:1: 2

-01:1;2 - :1;3 sin(Nrxt) +:1;4 cOS(NrX1)

:1;3

-..y:1;3

+ V1 (10.54)

where..y and 01 are positive design parameters with ..y satisfying ~ ~ ..y ~ k4 . The observer errors are defined as

1

A

X2 - k X2 1 :1;3 - x3 -

+ 02x1

~ cOS(NrX1)

. (N k3 sm X4 - X4 - N rX1 ) r

(10.55)

A

where 02 = 0IIk~k2. The observer error dynarnics are e2

-01 e 2 - e3

e3

-k4e 3 -

e4C

sin(Nr X1) + e4 cOS(NrX1) + °1 0 2X1 + :17n k 4 X3 -

k 3 k4 N cOS(NrX1) r

-k4e4 - (..y - k 4 ):1;4 -

k~4 sin(Nr X 1)'

hA

)

A

(10.56)

The stability of this observer can be analyzed using the Lyapunov function (10.57) Differentiating (10.57),

Vo

e 2 ( - 0le2 - e3sin(NrX1) + e4cos(NrX1) +01 0 2X1 + :1 7n ) +

k~~l [(e 3 + ~) ( -

k 4e 3

230

10.

Voltage Contral oi PM Stepper Motors Using Position Measurement Only

,

) , k3 k 4 cOS(NrXI) r

-(r - k4)X3 - N

~:4 sin(Nrxd) ]

+e4 ( - k4e4 - (1' - k 4)X4 -

(10.58)

Therefore, the observer error states (e2, e3 + ~3 ,e4) are rendered Input-toState-practically-Stable (ISpS)[92] with respect to (Xl, X3, X4).

Controller Design: The control inputs, VI and V2, are designed using the robust observer backstepping technique [120] applied to the following subsystem: k l (X2 + Ct2XI - e2) -CtIX2 - X3 sin(NrXI)

Xl

X2

-1'X3 -1' X4

+ VI + V2·

+ X4 cOS(NrXI) (10.59)

Step 1: For the first step, a Lyapunov function VI = Zl = Xl - Brei is the tracking error. Differentiating VI,

2t zr is used where (10.60)

Straightforward algebra yields the inequalities Ct2ZlXl -zle2 1 . - k Brelzl l

2

Ct2 2 + "TZl + 02rel 1 + Ee 22 < _z2 4E 1


0 is a design freedom.

Thus, (10.60) reduces to (10.62)

where

(10.63)

Define Z2 = :1: 2 + (T1Z1 where (Tl = ß1 + 1]1 with 1]1 being a positive design parameter. Therefore, (10.62) may be rewritten as (10.64)

Step 2: A new Lyapunov function is defined as V2 = V1 + ~2 z~ where positive constant that can be picked by the designer. Differentiating V2 , T·r v2

:::::


6E. Differentiating V,

V
0

2

- 282~2

>0

252

> 0,

'fIl - 7"(2 - f.l T}2"(2

+ a1'Y2 -

4"(3 -

-

1

4f.l

'fI3'Y3 -

(10.79)

global boundedness of all closed-Ioop signals can be inferred. Furt hermore , by using standard Lyapunov arguments, it is seen from (10.77) that by choosing appropriate values of the design freedoms, the tracking error can be rendered arbitrarily small. The above stability analysis yields the following theorem.

Theorem 10.3 The proposed dynamic controller given by (10.54) and (10.71) guarantees global boundedness of all closed-Ioop signals and achieves practical stabilization of the tracking error. Theorem 10.4 If Brei == 0 and Po = 0, the proposed dynamic controller achieves asymptotic stabilization of the rotor position to zero. Praof Under the hypothesis of the theorem, Po = Xl --+ 0 as t --+ 00.

Zl

=

d5

=

o.

Hence, using (10.77),

10.2 Feedback Design Assuming Known Bounds on Motor Parameters

10.2.1

235

Simulation Results

The efficacy of the proposed controller is demonstrated using simulation for a PM stepp er motor with the parameters shown in (10.49). The desired trajectory for the rotor position is a smooth-start sinusoid, (1 - e- O.2t2 ) sin(8t). Initial conditions for all states including the parameter estimates are zero. The design parameters in the controller are chosen as: = 15 0'4 = 150 Q1 = 10

"(2

c=1

Cl

0'1

0'2 0'5

= 105

= 150

=1 = 7t

0'3

= 1000

"( = 1200 "(3

= 20

(10.80)

ft=O.l.

Note that this choice of parameters does not satisfy the conditions (10.79). Equation (10.79) which is a sufficient conservative condition generated by the Lyapunov analysis is used as a guideline to pick the control parameters that are then perturbed to obtain improved performance. The simulation results are shown in Fig. 10.3. It is seen that the rotor position tracks the desired trajectory with a peak error of 0.46° and an RMS error of less than 0.28°. To illustrate the robustness of the proposed controller, a simulation was carried out with time-varying perturbations in the plant parameters using the same controller parameters as above. Specifically, the values of R, L m1 , and J were modified to R = (1 + 0.25sin(3t)) n, L m1 = 5(1 + 0.25sin(t)) mH, and J = 0.0733(1 + 0.25 sin(5t)) kg-m 2 . This represents a maximum change of 25% in each of the perturbed parameters. The simulation results are shown in Fig. 10.4. Even in the presence of severe time-varying parameter perturbations, the performance is retained with similar peak and RMS errors as in the previous simulation.

236

10.

Vo ltage Control of PM Stepper Motors Using Position Measurement On ly

1 . 5.----~-------____,

-2~--~--~--~--~

o

5

5

10

10

15

15

l"!! ~

ß., >

15

20

20

20 1

5

~

10

5

5

10

15

20

5

10

15

20

x 10-'

0.8

oM

0.6 04

Si

-=0.2

~

-5

0

5

10

15

20

0

0

t

Figure 10.1: Simulation results for P M stepper with t he robust adaptive output feedback controller designed in Sect. 10.1.

237

10.2 Feedback Design Assuming Known Bounds on Motor Parameters

X 1.5.-------~--~--__,

10-3

3.--------~--~--_.

'Ö'

2

~

E Qj g>

:g -1 ~

- -2 _1

.5L---~--~--~------'

o

5

10

15

20

_3L-----~--------------~

o

5

10

15

20

10.----.----~------_,

-10L--~~------~--~

o

5

10

15

20

5

10

15

10r-----~----~--------__.

5

10

15

20

5.----~----------_,

x 10-' 1.2.--------~--~--_.

0.8 0.6 0.4

-5~----------~--~

o

5

10

15

20

Figure 10.2: Simulation results for PM stepper with the robust adaptive output feedback controller designed in Sect. 10.1 and perturbed plant parameters.

238

10.

0/ PM Stepper Motors

Voltage Control

Using Position Measurement Only

0.01 0.5

'0

~

0

~

-

gs 05

I-

U

W

~04 I-

W

U

m03

a:

w

u.

l :! 02 0.1

0 0

j 0.5

\ ) \ ) \ ) ~j \ 1.5

2 2.5 3 TIME (SECONDS)

3.5

4

4.5

5

Figure 11.2: Reference trajectory (in radians).

The parameters chosen for the controller are as follows: p = 1.0, r o = 0"0 = 10- 2 , r 1 = 0"1 = 10- 3 , a = 1.0, and the gains ci,i = 1, ... 4 in (11.74) and (11.75) were chosen to be equal to 10- 8 . The matrix Q in the ARE was chosen as Q = diag(5000, 3100). The initial rotor position was set to 0.1 rad and all other parameters were set to zero. The performance of the adaptive controller is given in Fig. 11.3. The peak tracking error after the initial transient is seen to be about 1.5 milli-rad. The phase currents and voltage inputs are shown in Figs. 11.4 and 11.5, respectively. The adaptation of /3 and .; are shown in Fig. 11.6. To illustrate the robustness of the controller to parametric uncertainties, the payload mass was increased by 40% and the other electromechanical system parameters were changed to those given in Table 11.2. The tracking performance is depicted in Fig. 11.7. The peak steady-state tracking error has increased slightly to about 1.6 milli-rad. The input voltages are depicted in Fig. 11.8. Next, to illustrate the robustness of the controller to dynamic uncertainties, magnetic saturation was incorporated into the motor model. The current interval was divided into five regions in which the following piecewise constant

259

11.3 Robust Adaptive Contml Design

-

01l\'

IOffi ffi

.

~

r

----------------------------------~

0

Cl

z

8 -0.05 ~

I-

-0.1 '--__..L.-_ _- ' -_ _- " -_ _---'_ _ _ _L -_ _..L.-_ _- ' -_ _- " -_ _---'_ _- - - ' 4 5 6 o 3 10

TIME (SECONDS)

6' J~~M~~~--,-----,-----o

1

2

3

4

5

6

7

8

9

10

o

1

2

3

4

5

6

7

8

9

10

o

1

2

3

4 5 6 TIME (SECONDS)

7

8

9

10

>.~

>~:

Figure 11.8: Phase voltages (in Volts) under the robust adaptive control. from the motor dynamics is depicted in Fig. 11.17. In the third case, the mechanical inertia and load torque were increased by 20% and the resistance of the windings was doubled. The tracking performance attained with the use of the robust adaptive controller is shown in Fig. 11.19. No appreciable change in the tracking performance is notieed. As in the earlier case, the performance of the adaptive controller is compared to that of a PID (with modified PI) controller whose parameters are tuned to the following values: k1 = 65.79, k 2 = 118.43, k 3 = 13.46, k 4 = 1, and k 5 = 4.618x 10 3 . The performance ofthe PID control for the corresponding cases is depicted in Figs. 11.20 and 11.21. A degradation in the transient performance with the PID controller is observed when magnetic saturation is included in the motor model. The settling time has increased from 1.5 s to about 2 s. The response of the PID controller when mechanical inertia and load torque terms are increased by 20% and the resistance doubled is shown in Fig. 11.22.

11.4

Current-Level Control ofBLDC Motors Using Position Measurements Only

The methodology applied for current-Ievel control of the different types of stepper motors using only position measurements considered in Chap. 8 mayaiso

265

11.5 Torque Ripple Reduction for BLDC Motors

i·ffiK

0.1..---,----r--r--,---,-----,--,-----,----,------,

~

ffi

0

Cl

z

---------------------------------~

8-0.05 Ci

I- -0.1 '--_...l-_--'-_ _L - ._

o

_'___---'--_

2

__'_ _-'--_

456

TIME (SECONDS)

_'__--'-_---1

8

10

~3

a: ~2

:i? ffi

0

~

-1

1

sz() -2 Ci -3

1-_4'--_ _~_ __'___ _. . . l __ __ ' __ _---'--_ ___'__ ___'~ 7 7.5 6 6.5 8.5 9

TIME (SECONDS)

Figure 11.9: Above: tracking error (in radians) under the robust adaptive control with magnetic saturation included in the motor model. Below: magnified view. be applied in the case of direct-drive BLDC motors. The direct-axis current is set to zero, as before, and hence the torque produced by the motor reduces to T

=

. t:, K 'r2Zq =

K,T

I

(11.94)

where K, = K'r2 and i q = T'. The design procedure for the robust torque profile proceeds along the same lines as given in Chap. 8. The details are rather straightforward and are omitted for brevity.

T'

11.5

Torque Ripple Reduction for BLDC Motors

An adaptive variable structure controller (see Chap. 15) may be used to reduce the torque ripple in BLDC motors [145]. For this purpose, a model of the torque ripple is necessary. To capture the phenomenon of torque ripple present in the motor, higher order harmonie terms in the inductance expression must be accounted for in the derivation of the torque expression. To this end, consider the direct-axis and quadrature-axis flux linkages given by

ePd

(11.95)

266

11. Brushless DC Motors

0.1,---..,---,----,--,.-----,------,--,---,-------r-----,

~

lo~~

ffi

----------------------------------~

0

(9

z

8-0.05 ii ~

-0.1 L - _ - ' -__-'-_---'_ _-'--_-'-_--'-_ _-'--_ 4 5 6 o 2 TIME (SECONDS)

___'___--'-_-----' 8 10

~2

!&1 er

~

er

0

w

(9 -1

Z

8-2

ii

~-3L-_-'-_ _ _~_ ____'___ ____'_ _ _~_ _--'-_ ______'

7.5 8 TIME (SECONDS)

6.5

8.5

Figure 11.10: Above: tracking error (in radians) under the robust adaptive control with magnetic saturation and time-varying uncertainties included in the motor model. Below: magnified view. (11.96)

cP q

where i j is the constant current in the fictitious winding corresponding to the permanent magnet. The transformed inductances are given by [128J 00

ldd

L

+

L ddO

L ddi cos( inpB)

(11.97)

i=6,12, ... 00

ldq

L

L dqi sin( inpB)

(11.98)

i=6,12, ... 00

ldj

L djO

L

+

L dji cos( inpB)

(11.99)

i=6,12, ... 00

lqq

L qqo +

L

[L qqi cos( inpB) + L qqi sin( inpB) J

(11.100)

i=6,12, ... 00

lqd

L

L qdi sin( inpB)

(11.101)

L qji sin( inpB).

(11.102)

i=6,12, ... 00

lqj

L

i=6,12, ...

267

11.5 To".que· Ripple Reduction Ja". BLDC M oto".s

1.5

6' !:S oa:

f\

f\

0, (12.32) reduces to (12.34) To design v q , the backstepping design for the subsystem with states (B, w, i q ) is commenced by defining Zll = B - Bref and Vll = ~Zfl. Differentiating Vll , Zll (w

-

-kllZrl

where kll

Defining V12 = ~ Zfl •

+ ZllZl2

(12.35)

> 0 is a design freedom and Zl2

V12

Bref )

=

+ ~ Zf2

2 -kllz ll

=

W

+ kllz ll - Bref ·

(12.36)

and differentiating,

. . + ZllZ12 + Zl2 [ P/1Pdreftq + f.J,ZOlt q -

(Tl

+J Tf)

-

2 kllz ll

+k n Z 12 - Öre! ] 2

< - k llZll

-

21 k12 Z 122 + Zl2 Zl3 + 2kf.J,2l2 ZOl2·2t q

(12.37)

where kl2 > 0 and zl3

= k 12 z 12

.

+ zll + f.J,'l/Jdreftq -

(TI+Tf)

J

Introducing the new Lyapunov function Vl3

-

2 kllz ll

+ k ll z l2 -

..

Bre !. (12.38)

= ~(Zfl + Zf2 + Zr3),

(12.39)

286

12. Induction Motor: Modeling and Control

Designing the q-axis voltage input Vq

=

--iP. 0, (12.39) simplifies to (12.41 ) Overall closed-loop stability can be inferred by considering the composite Lyapunov function

v

(12.42)

Differentiating V,

11

< -

p.2(a+kod 2 p. 2k03 2 2 2k k ZOl - 2k k Z02 - kn Zn 12 02

1

2

12 02

2

(12.43)

-"2k12Z12 - k 13 Z13 ·

Global boundedness of all closed-loop signals is inferred from (12.43). Furthermore, the tracking errors V;d - V;dref and () - ()ref go to zero exponentially.

12.2.2

Simulation Results

The performance of FOC is illustrated through simulation for an induction motor with electromechanical parameters shown in Table 12.l. The position reference is chosen to be ()ref (t) = lOO[t + (e- o.7t - 1)/0. 7J. Equivalently, the velocity reference is wref(t) = 100(1 - e- 0.7t ). The flux reference is V;dref(t) = 0.2 + 0.85(1 - e- 0.7t ).

°

The initial condition for fiux V;d at time t = is 0.2 Wb. All other states are assumed to be initially zero. Nonzero initial conditions for fiux and fiux reference at time t = are used to avoid the singularity in the dynamics (12.22) and the field oriented control (12.23). The application of FOC requires that the motor be started with a different control scheme or with open-loop contro!. FOC can be applied after a nonzero flux has been generated.

°

Linear controllers are designed to place the poles of the (V;d, id) subsystem at -14 and -124.2944 and the poles of the (J~()(s)ds,(),w,iq) subsystem (the position integrator added to reject the constant load disturbance) at -6 + 8i, -6 - 8i, -124.2944, and -1. The simulation results are shown in Fig. 12.2.

287

12.3 Input- Output Decoupling

Table 12.1: Electromechanical parameters of the induction motor used for simulation. 11 Parameter 1 Value 1 Units 0.007 kg-m~ /s D kg_m 2 J 0.0586 0.125 L sr H 0.145 L8 H Lr 0.145 H np 1 1.99 n Rr 3.15 Rs n 5 Nm Tl

100

1.5

~ 50

-2

-4

V 0

1500

5

t

10

-20

0

5 t

100

10

t

10

10

0

5

10

5

10

t

0

-300

-0.03

~

~

V

150

:::>"

100

-200 5

5

200

:::>

0

0

0.01

"E ~

~"C-loo

500

0

10

J-O.Ol :>I ;'-0.02

o

_1000

~

0

t

~

"C

0.

0

1\8"-10

"§

5

~.

U

0

I

0

10

! r a>

-50

10

I

0

5

t

10

50

0

t

Figure 12.2: Simulation results for induction motor with FOC.

12.3

Input-Output Decoupling

Field Oriented Control can be refined to achieve exact linearization and inputoutput decoupling (with position and square of the fiux norm being the outputs)

288

12. Induction Motor: Modeling and Control

by using the change of coordinates Xl

e

X2

w

X3

fL(1/Ja i b -1/Jb i a) -

X4 X5

1/J~ + 1/J~ -2a(1/J~

+ 1/J~) + 2aLsr (1/Ja i a + 1/Jb i b)

X6

arctan (

~:) .

(Tl +Tf) J

(12.44)

Note that (12.44) is not a global change of coordinates since it is invertible only if X4 > 0 and - ~ ~ X6 ~ ~. In this case, the inverse transformation is given by

e

Xl

w

X2 JX4 COS(X6) JX4sin(X6)

~

[COSX6

(X52:~:X4) -

; sin X6 (X3

+ (Tl

~ Tf))]

~ [sinX6 (X52:~:X4) +;COSX6(X3+ (Tl~Tf))] .(12.45) The dynamics (12.18) can be expressed in the new coordinates (12.44) as Xl

X2 X3 X4

X2 X3 Va X5 (12.46)

using the new inputs Va and Vb defined by

Va

Vb

= -fLßnpw(1/J~ + 1/J~) - fL(a + ,)(1/Ja i b -1/Jb i a) - fLnpw(1/Ja i a + 1/Jb i b) fL fL (+l++f) U U --L J a s 1/Jb a + -L a s 1/Ja b (4a 2 + 2a2ßLsr)(1/J~ + 1/J~) + 2aL sr npw(1/Jaib -1/Jbia) -(6a 2L sr + 2a,L sr )( 1/Jaia + 1/Jbib) + 2a 2L;r( i~ + i~) 2aL sr 2aL sr ua + -L-1/Jb U b. +-L-1/Ja a a s

s

(12.47)

289

12.4 Dynamic Feedback Linearization

Note that the input space transformation (12.47) is invertible if 1/J~ + 1/J~ > 0. In (12.46), the (Xl,X2,X3) subsystem is decoupled from the (X4,X5) subsystem. Hence, the design of the control inputs Va and Vb to make the rotor position Xl and the flux magnitude X4, respectively, track given reference trajectories can be carried out independently and using linear design techniques. The implementation of FOC or input-output linearizing control does require the use of the flux signal. Design of a flux observer to estimate the flux when the ftux is not directly measured is considered in Sect. 12.6. Note that the state feedback transformation (12.44) and (12.47) render the flux angle X6 unobservable from the outputs Xl, the rotor position and X4, the ftux magnitude. While the above feedback transformation achieves input-output linearization and decoupling, input-state linearization is not achieved. In fact, it can be shown via an easy application of Theorem 5.2 that (12.18) is not (static) feedback linearizable. However, as will be seen in Sect. 12.4, (12.18) can be input-state linearized through dynamic feedback linearization.

12.4

Dynamic Feedback Linearization

In Chap. 5, the concept of relative degree and the techniques of input-output linearization and input-state linearization were presented. In cases where uniform relative degree does not exist, input-state linearization is not possible through static state feedback and a nonlinear change of coordinates. However, introduction of dynamic nonlinear feedback may alleviate the difficulty [29, 30]. Dynamic feedback linearization of induction motor dynamics has been considered in [33, 34]. Dynamic feedback linearization of the DQ transformed dynamics (12.22) can be carried out by introducing an integrator either in the q-axis or the d-axis. While introducing a q-axis integrator achieves feedback linearization of the entire dynamics (12.22) including the position dynamics = w, introducing a d-axis integrator only achieves feedback linearization of the (w, 1/Jd, id, i q , p) dynamics. This is, however, practicallY sufficient since the control objective for an induction motor is typically formulated as a regulation or tracking problem for outputs wand 1/Jd. For notational uniformity and clarity, since it is trivial to add the position dynamics in the q-axis integrator case, we omit the position dynamics below.

e

12.4.1

Integrator in the q-axis

Introducing an integrator (with state X6 = v q ) in the q-axis of the model (12.24), (12.48)

290

12. Induction Motor: Modeling and Control

Induction Motor Dynamics

J

Vq

x

= X6

Figure 12.3: Dynamic feedback linearization with integrator in the q-axis. with vq being the new input, and defining (Xl, X2, X3, X4, X5) = (w, 1/Jd, id, i q , p), the dynamics (12.24) and (12.48) can be written as

h + DX1)

Xl

/1X2 X4 -

X2

X5

+ aL sr X3 -')'X3 + Vd -')'X4 + X6 X4 npXl + aL sr -

X6

vq

X3 X4

J

-ax2

X2

(12.49)

where the friction is modeled as Tf = Dw with D being the viscous damping constant and the load torque Tl is assumed to be constant for simplicity. The resulting system with enlarged state space is illustrated in Fig. 12.3. It can be easily checked by applying Theorem 5.2 that the extended dynamics (12.49) are feedback linearizable. Writing (12.49) in the form X = f(x) + gl(X)Vd + g2(X)V q and introducing the state variables

6,1 (Tl

+ DX1)

6,1

Lfch(x) = /1X2 X4 -

6,1

LJ

(Jks

+ 15)2

(13.59)

_Jk s

The parameter update laws are designed as the projection algorithms, i = 1,2,3

if f!.i if Q,i if Q,i if Q,i if Q,i

< Q,i :::; f!.i 2:: ai :::; f!.i 2:: ai

< ai and and and and

~i ~i ~i ~i

0 >0 00. Furthermore, from (13.74), -0a and -0b go to zero exponentially. Hence, the asymptotic speed and flux magnitude tracking objectives are achieved. The controller presented above provides a solution when both flux quantities and currents are unmeasured. Furthermore, by exploiting the natural damping in the system, a very low order compensator is obtained without the need for explicit current and flux observers. However, by the same token, since the controller relies on the natural damping in the system for current and flux tracking, the trajectories that can be successfully tracked must necessarily be slow enough. This constraint is more severe when the system does not have much natural damping (small "y and a). However, the simplicity of the controller makes it an attractive solution when the motor exhibits sufficient natural damping and the trajectories to be tracked are slow. The key to the above solution lies in the fact that the designed torque reference does not directly depend on the velo city. Hence, the derivatives of the torque reference and the current references involve only known quantities and the voltage inputs can be designed as in (13.71). However, since the torque reference uses a low-pass filtered version instead of the measured rotor velocity, an additional delay in the loop is introduced. This factor also degrades tracking performance for fast trajectories.

13.2.2

Load Torque, Rotor Resistance, and Friction Unknown

In this section, we design a controller that utilizes rotor velocity and stator current measurements and compensates for uncertainty in load torque, rotor resistance, and friction. The design in this section essentially follows [133]. However, while friction was neglected in [133], the design below incorporates a friction term with unknown viscous damping coefficient.

334

13. Adaptive Control o/Induction Motors

In the absence of position measurement, the transformation (12.76) can not be applied; However, as noted in Sect. 12.6.2, a position estimate can be ob-

e

tained as = wand the transformation (12.19) can be applied with T as in (12.76) with e replaced by yielding the DQ transformed dynamics (12.77). This however suggests an additional design freedom in choosing the dynamics of [133]. Defining

e,

e

i o = Wo

(13.79)

and applying the transformation (12.19) with sin(nPEo)] T = [ -cos(npEo) sin(npEo) cos(npEo) ,

(13.80)

the dynamics (12.18) are transformed into

1 -'yid + aß1/Jd + n pßw1/Jq + -L Ud + npwoiq a s

-,i q + aß1/Jq - n pßw1/Jd

+

LI u q - npwOi d a s -a1/Jd + np(wo - w)1/Jq + aLsrid

~d ~q

-a1/Jq - np(wo - W)1/Jd

where the friction

'Tf

is modeled as

'Tf

+ aLsri q

(13.81 )

= Dw.

The load torque, rotor resistance, and the viscous damping are assumed to be unknown. This implies that 'Tl, a, " and D are unknown parameters. However, since, = aßL sr +'1 with'1 = ..&..L ' we do not need to implement CT 8 separate estimates for a and,. A lower bound Q: > 0 on a is assumed to be known. The measured variables are the rotor velocity wand the stator currents i a and ib (equivalently, id and i q ). The control objective is to make wand the flux

J

magnitude 11/J1 = 1/J~ + 1/J~ track given trajectories wref and 1/Jref' respectively. To achieve asymptotic field orientation, the desired trajectories for 1/Jd and 1/Jq are chosen as 1/Jref and 0, respectively.

o bserver Design An observer is designed to estimate the unmeasured variables as

335

13.2 Adaptive Output Feedback Control

~d

,1 n pwo(1/Jq + 7./iq)

~q

,1 -npwo(1/Jd + (jeiJ

,., a1/Jrej

+ aLsrId -

k

i + (jeid

,.

1 - (jWi d + W..pd

k

i + aLsrIq + npw1/Jrej + (jeiq

1 - (jWi q (13.82)

+W..pq

where the observer errors are defined as iq

-

iq

, 1/Jd -1/Jd

+ (jeid

1

, 1/Jq -1/Jq

+ (jeiq

1

(13.83)

and k i > 0 is a positive design parameter. Wid' Wi q, W..pd' and w..pq will be picked during the controller design. & is a dynamic estimate for a and i' = &ßL sr +/'1. The estimation errors are defined as a = & - a and l' = i' - /'. The observer error dynamics can be written as eid ei q

-1'id + aß1/Jrej + aß(1/Jrej -1/Jd) - n pß w1/Jq - kieid -1'iq - aß1/Jq - npßw( 1/Jrej -1/Jd) - kiei q + Wi q

e..pd

npwoe..pq

+ Wid

+ W..pd

(13.84) Define a Lyapunov function

Va

2 2 )] = "21 [ eid2 + ei2q + /'O..p ( e..pd + e..pq

(13.85)

with /'O..p = a/'11f; and /'1..p being a positive design parameter.

Differentiating (13.85) along the trajectories of (13.84),

Va =

-ki(et + e;') + (eidwid + eiqWiq) - 1'(eidid + eiqiq) +&ßeid 1/Jrej + aßeid (1/Jrej - 1/Jd) - npßweid 1/Jq - aßei q1/Jq -npßwei q(1/Jrej -1/Jd)

+ /,o..p (e..pdw..pd + e..pq w..pq)'

(13.86)

Controller Design The tracking errors are defined as which satisfy the dynamics

w =

w= w -

Wrej, ,(J,d = 1/Jd -1/Jrej, and ,(J,q = 1/Jq,

(13.87)

336

13. Adaptive Contml

·-a'l/Jd + np(wo - w)'l/Jq -a'l/Jq - np(wo -

To eliminate the dependence of the rameter a, introduce

+ aLsrid W)'l/Jd + aLsri q.

0/ Induction

Motors

(13.88)

-J;ref

(13.89)

{;d and {;q dynamics on the unknown pa-

1 ' 'l/Jd - ßeid - 'l/Jref = 'l/Jd - e..pd - 'l/Jref 1 ' 'l/Jq - ßeiq = 'l/Jq - e..pq

(13.90)

whose dynamics are

(13.91 ) The dynamics of the velo city tracking error given in (13.87) suggests the following virtual controllaw for i q i qr

1

J-l'l/Jref

=

[

_.

- kww + wref

Tl

b]

+ J + JW

(13.92)

where k w is a positive design parameter, and Tl and bare dynamic estimates for Tl and D, respectively. The estimation errors are defined as Tl = Tl - Tl and D = b - D. Using (13.90) and (13.92), (13.87) can be rewritten as

w

=

.

.

J-l(Zd~q- Zq~d)

J-l.

+ ß(eid~q -

.

_

eiq~d) - kww+

(Tl

+ Dw) J

(13.93) where i q = i q - i qr . i d and Wo are utilized as virtual control inputs for Zd and Zq, respectively and the virtual controllaws are designed as -J;ref + ~ 6:L sr 6:L sr aLsriq Vq w+-----n p'l/Jre f np'l/Jref 'l/Jref L sr

Wo

with

Vd

=

+

(13.94)

and v q being auxiliary control inputs that will be designed later.

337

13.2 Adaptive Output Feedback Contral

Using (13.94), (13.91) can be rewritten as

(

Zq

-n pWo Zd

1) ki + -e· + v q + -e· ß ß 'q

1

(13.95)

- -Wo ß 'q

'd

Defining the Lyapunov function

v;1

TT

vO

=

2) + r1 YlW-2 + 12 (2 Zd + Zq

(13.96)

with 11 and 12 being positive design parameters, we have

l\ =

+ e;') + (eidwid + eiqWi q) + lo..p(e..pdw..pd + e..pq w..pq) -1'( eid id + eiqiq) + &.ßeid 1/Jref + aßeid (1/Jref - 1/Jd) - npßweid 1/Jq -aßei q1/Jq - npßwei q(1/Jref - 1/Jd) + 11 w [ - kww + J1-( Zdiq - zqi d ) -ki(et

J1- ( . .) 01. "' (Tl + Dw) ] +ß eidIq-eiqId +J1-'I'refIq+ J npWO +,2 Zd [ ßeiq

ki

+ Vd + ßeid

npwo +,2 Zq [ - ßeid

1 - ßWid

ki + v q + ßeiq -

+ aLsrId A",

]

1]

(13.97)

ßWiq .

Defining

(13.98)

=

Vq

VI

= -(ki + a)(et + e:q) _ +aßeid 1/Jref

11 k ww2 - 1'(eidid

+ eiqiq)

_ (Tl + Dw) _ + /1 W J - aß( 1/Jref - 1/Jd)eid A

338

13. Adaptive Control o/Induction Motors

+Etß~qeiq + l11/JreJlIWiq + (eidwidl + eiqWiqd "(2 -ß (ZdWid 1 + Zq Wiq1) + "(O,p (e,pd W,pd 1 + e,pq W,pq 1)

(13.99)

+"(2&ZdL sri d.

The dynamics of the current tracking errors

id

and

iq

can be written as

. ß.I. ß .1. 1 . -J;ref ~ref -"(Zd+ a o/d+np wo/q+ -L Ud+npWOZ q- - L - -:::--L a s sr a sr

&.

& + ~L Vd a sr

+~L 1/Jref

a

sr

1 [ ki ( _. -:::--L - -ß - "(Zd a sr

+aß(1/Jref -1/Jd) - n pß w 1/Jq - kieid

+ Wi d)

1 . . _ (Tl + Dw) J +fjl1(ei dZq - eiqZd) - kww +

-'TJW ( - "(iq + aß1/Jq - n pßw1/Jd

_

+ aß1/Jref

- 'TJiq (I1(Zdiq - Zqid) -: )

+ l11/JrefZq

+ a ~s Uq -

npWoid)

npß . • . ; -&(1/Jref -1/Jd) - &(1/Jref -1/Jd) - -w[-..yiq "(2

-aß1/J q - n pßW(1/Jref -1/Jd) - kiei q + Wi q] npß (11(ZdZq ' - -:y; + eTI +Jbw) I Ud a Ls

') ZqZd

+ 11fj ( eid'Zq -

') - keiqZd wW

+ l11/Jrefiq + Wref) ]

+ .L

'TJ L WUq + 1JdO

a sr a

s

+ae,pq1Jd4 + Tt1Jd5

+ Zd1Jdl + Zq1Jd2 + ae,pd 1Jd3

+ Et1Jd6 + D1Jd7 + ..y1Jd8

(13.100)

-"(i q + aß1/Jq- npßw1/Jd + LI u q - npwOid a s -J;ref [ k - . - ww+wref l1o/ re f

+~

-

111/J~ef (~ k Ww

+

iJ

(Tl

iJ] + JTl + JW

k w) (I1(Zd iq - zqi d ) +

+ Dw) + 110/.1. J

re

~(eidiq -

eiqid)

wref fl fZ-:) - -q l11/Jref l11/Jref J

iJ.

----w----w f l11/Jref J l11/Jref J re 1 -L uq + 1JqO + Zd1Jql + Zq1Jq2 a s

+ ae,pd1Jq3 + ae,pq1Jq4

13.2 Adaptive Output Feedback Contral

339 (13.101)

where 1]

cPd2

340

13. Adaptive Control of Induction Motors

-:Yi q + iiß(,(f;q + ~eiq)

1>qO

-J;ref ( + --::!.2

. -npwo2d

k -. wW + wref

-

f..L'Pref

- npweid - npßwV;ref

Tl D) + -J + -J W

__ l_(D _ k ) (!!.(e. i - e· q i d ) f..LV;ref J W ß td q t

-

k W W

+f..LV; f2'"' ) - -wref - - - - . . .Tl" , re q f..LV;ref J.1-V;ref J

-

D J.1-V;ref J

w-

D. J.1-V;ref J

-n ßw - _1 p V;ref _1 V;ref

(D _k

o -ß

- J.1-V;:ef J -ß(,(f;q

(DJ - k

)

w

J

W

re

f

)i

W

q

id

(~ - k w)

+ ~eiq)

(~ -

- f..LV;:ef J

kw)w

i q.

(13.102)

Defining the Lyapunov function

V with

')'3, ')'4, ')'5,

V =

-

V;

-

and

1

')'6

+ ')'3 (-:2 + -:2) + ')'4 Tl-2 + ')'5 D2 + ')'6-2 a

2

2d

2

w2 -

')'lk w

+a-ß eid .1.'Pref + ')'lW- (Tl +J Dw)

-

i(eidid

+ eiqiq)

a-ß(.I.'Pref

) - .7. 'Pd eid

+äß,(f;qei q + J.1-V;re!'Y1 Wi q + (eidWidl ')'2

-ß

(13.103)

being positive design parameters and differentiating,

+ a)(et + eU -

-(ki

2

2

Zq

(ZdWidl

+ eiqWiql) + ZqWiql) + ')'O,p( e,pd W,pd1 + e,pq W,pql) 1

+')'2 azd· u sr Zd + ')'3 Zd -L Ud

'f]+ Q'. L sr(J' L WU q s +1>dO + Zd1>dl + Zq1>d2 + ae,pd1>d3 + ae,pq1>d4 •

T

-:

'"'(

(j

+TI1>d5

s

+ ä1>d6 + D1>d7 + i1>d8 )

341

13.2 Adaptive Output Feedback Control

- ( 1 +'Y3 i q IJ L u q + 4>qO s

+ae1j;Aq3 +'Y4'hf l

+ zd4>ql + Zq4>q2

+ ae1j;q4>q4 + ;h4>q5+ a4>q6+ D4>q7+ i'4>q8)

+ 'Y5 D iJ + 'Y6 a &.

(13.104)

Designing the parameter estimate dynamics to be (13.105) (13.106)

and the design freedoms and the control inputs to be

1 'Yl..p

-

-

- - b 3 i d4>d3

+ 'Y3iq4>q3]

1 - - b 3 i d4>d4

+ 'Y3iq4>q4]

'Yl..p

(13.108)

-

(13.109)

~b3i'd4>dl + 'Y3i'q4>ql + 'Y2aLsri'd] 'Y2

ß

-

-b3 i d4>d2

'Y2

(13.110)

+ 'Y3iq4>d

IJ L s (-: - - k2~d

'Y3

'Y3'f/

(13.111) _

ß'Y3

+ a'L sr lJ L s wU q + -'Y2

ei d4>dl

+ 'Y34>do

+ ß'Y3 ei q4>d2

+ ßeid &.Lsr )

IJ L s ( -: - - k2~q

ß'Y3 + 'Y34>qO + -eid4>ql + -ß'Y3e iq4>q2

'Y2

'Y3

'Y2

(13.112) 'Y2

+f.L'l/Jref/1W)

(13.113)

with k 2 > 0 being a design parameter, (13.104) re duces to (13.114) Since the dynamic parameter estimate &. occurs in the denominator in several terms in the definitions of idr and Ud, it is essential to demonstrate that &. remains bounded away from zero to be able to infer any stability results from (13.114).

342

13. Adaptive Control o/Induction Motors

To ensure that a remains positive and bounded away from zero, a projection algorithm [169] can be used 1 . The dynamics of a in (13.107) is replaced by (13.115)

where

~

~( "16

ßeiAref

+ ß('l/Jref -

-0d)eid - ß-0qeiq - 'Y3 i d1>d6 - 'Y3iq1>q6

+ßL sr (eid id + ei qiq) - 'Y3idßLsr1>d8 - 'Y3iqßL sr1>q8)'

(13.116)

The projection function is defined as

Proj(~, a) = {

[1 -

;(&)]~

if p(a)

0 andjor otherwise

~

~ ~

0

(13.117)

where

p(&)

(13.118)

with ~ > 0 being a known lower bound on a and 0 being any positive constant less than~. & is initialized to be greater than~. The fo11owing properties are guaranteed by the projection algorithm. 1. &(t) ~ ~ - 0 1ft ~ 0

2.

&Proj(~,

&)

~ &~.

Thus, (13.114) is replaced by (13.119)

Equation (13.119) demonstrates the global boundedness of a11 closed-loop signals. Furthermore, by an application of LaSa11e's invariance principle, it is inferred from (13.119) that eid,eiq,w,id, and i q asymptotica11y go to zero. Furthermore, as shown in [133], under a persistency of excitation condition, which is always satisfied in practice, e1/Jd' e1/Jq, Zd, Zq, Tl, Ci, and jj also decay to zero asymptotica11y. Thus, the speed and ftux tracking objectives are satisfied asymptotica11y, asymptotica11y convergent ftux and parameter estimates are obtained, and asymptotic field orientation is achieved. 1 Alternatively, the projection algorithm defined in (13.38) can be used. The projection algorithm in (13.117) has the advantage that it is continuous.

13.2 Adaptive Output Feedback Control

13.2.3

343

Mechanical Parameters and Rotor Resistance Unknown

In this section, we design a controller that utilizes rotor velocity and stator current measurements and compensates for uncertainty in the mechanical parameters and the rotor resistance. The control objective is to track given speed and flux magnitude trajectories, Wrej and 1/Jrej, respectively. The design is based on [117J and combines the torque reference design procedure in Sect. 13.2.1 with the observer design methodology in Sect. 13.2.2. A similar technique was used in [6J to achieve position and flux magnitude tracking on the basis of position and current measurements. However, while [6J required lower and upper bounds on mechanical parameters, the development in [117J relaxes this assumption. However, in common with [6J and the controller in Sect. 13.2.2, a lower bound on the rotor resistance is needed. Applying the transformation (12.19) with T defined in (13.80), the dynamics (13.81) are obtained. In this section, the inertia J, the viscous damping D, the load torque Tl, and the rotor resistance R r are assumed to be unknown. The uncertainty in R r implies that a = &Lr and "( = a RLr L ;1j + Jl.... are unknown. As Lr S U Ls in Sect. 13.2.2, separate estimates do not need to be implemented for a and "( . npL sr J!:.Q. h npL sr smce "( = a ßL sr + "(1 where "(1 = Jl.... aLs· /-L = JL r = J W ere /-Lo = Lr is known. The availability of a lower bound on R r implies the knowledge of a lower bound Q. on a. Observer Design

The observer dynamics are designed as , ,1 ' ~d -ii d + &ß1/Jd + n pßwre j1/Jq + -L Ud + npwoiq - kieid a s +Uoid + ucid , ,1 ' -ii q + &ß1/Jq - n pßW re j1/Jd + aLs u q - npwOid - kiei q

+ Uciq -&-J;d + &Lsrid - npWrej-J;q + npwo-J;q + npwiq +Uo..pd + Uc..pd -&-J;q + &Lsri q + npWrej-J;d - npwo-J;d - npC;ji d

+Uoiq -J;d -J;q

+uo..pq

+ uc..pq

(13.120)

where k i > 0 is a design parameter, & is a dynamic estimate for a, i = &ßL sr + Ci = & - a, and l' = i - T Uoid, Ucid, Uoiq, Uciq, Uo..pd, Uc..pd, uo..pq, and uc..pq are terms that will be designed later. w = W-Wrej is the velocity tracking error.

"(1,

The observer errors are defined as

344

13. Adaptive Contral of Induction Motors e·'q

iq - iq

e,pd

,(f;d - Wd

e,pq

Wq - Wq·

(13.121)

We also introduce Zd

e,pd

1

+ ßeid 1

= e,pq + ßeiq'

Zq

(13.122)

The observer error dynamics are given by

-i'id + ß(a.,(f;d - 0Wd) + npß(Wrej,(f;q - WWq)

eid

+ Uoid + Ucid -i'id + ß(a,(f;d + oe,pJ + npß(Wreje,pq -

+ npWOei q

- k i eid

-kieid

WWq)

+ npWOei q

+ Uoid + Ucid

-i'iq + ß(a.,(f;q - OWq) - npß(Wrej,(f;d - WWd) - npwOeid

ei q

-kieiq

+ Uoiq + Uciq

-i'iq + ß(a,(f;q -kiei q + Uoiq

+ oe,pq) -

+ Uciq

-a.,(f;d + 0Wd + aLsrid

e,pd

+npwiq + Uo,pd e..pq

-a,(f;q

npß(Wreje,pd - WWd) - npWOeid

+ np(WWq -

Wrej,(f;q)

+ npwoe,pq

+ Uc,pd

+ OWq + aLsriq -

np(WWd - Wrej,(f;d) - npwOe..pd (13.123)

The dynamics of Zd and Zq can be written as

_~

npwozq + npwzq + Uo,pd _~

+ UC,pd -

-npwOzd - npwzd + Uo,pq

ki 1 (jei d + ßUOid

+ Uc,pq -

ki

(je;q

1

+ ßUCid

1

+ ßUO;q (13.124)

The auxiliary filter dynamics (d

npwo(q

+ npWrejeiq + npwiq + eid + Uc,pd (13.125)

are also introduced. Defining (13.126)

345

13.2 Adaptive Output Feedback Control

the dynamics of 7](d and

.

7](d

.

7](q

= n pw o7](q

7](q

can be written as

ki

- 7Jeid - eid - npWrefeiq

ki

= -np w o7](d +npWrefeid- 7Jeiq -

ei q

1

1

+ UO'l/ld + ßUOid + ßUCid 1

+ uo'l/lq+ ßUOiq+

1

ßUCiq.

(13.127)

The observer Lyapunov function is defined as

Differentiating (13.128) and simplifying,

(13.129) Designing A

a((d A

a((q -

1)

ßeid

1)

ßeiq

ki

1

+ 7Jeid

1

+ 7Jei q

-

ßUcid

-

ßUCiq

ki

Uoid

-ß&.((d -

~eiJ + ßnpWrefeiq

Uoiq

-ß&.( (q

~eiq) -

-

ßnpWrefeid'

(13.130)

346

13. Adaptive Contml o/Induction Motors

(13.129) reduces to

Vo

=

+ ei q1/Jd + iqZ d -

npw( -eid 1/Jq

" - Lsrei qi q + eid 1/Jd

1

r

)]

-eiq ( ':,q - ßeiq

+ eiq1/Jq -

idzq) + (i [

Lsreidid

-

1 eid (Cd - ßeid)

-~ k ( 2 2 ) + b1ex aa - 7f eid + ei q i

1

+ ß (eid Ucid + ei qUciq) + (ZdUc..pd + Zq Uc..pq).

(13.131 )

The parameter update law for (} is designed as the smooth projection algorithm (13.132) where the projection function Proj was defined in (13.117). Q. is the known lower bound on a and , , 1 ~ = - [ - Lsrei)d - Lsrei qiq + eid 1/Jd + ei q1/Jq - eid (Cd - ßeid) -eiq(Cq -

~eiq)].

(13.133)

Using (13.132), (13.131) simplifies to _

npw( -eid 1/Jq

0

+ ei q1/Jd + tqZd -

0

tdZq) -

ki

2

2

ß (eid + e iq )

1

+73 (eiducid + eiqUciq) + (ZdUc..pd + zquc..pq).

(13.134)

Controller Design Step 1: The desired torque is designed as T*

=

Tl

+ JWref + DWref + kWf

(13.135)

where Tl, J, and D are dynamic estimates for Tl, J, and D, respectively. The parameter estimation errors are defined as Tl = Tl-Tl, j = J-J, and D = D-D. W f is a filter state with dynamics Wf

= -kw - (k + kS)Wf

(13.136)

where k = k1 + k. k1 and ks are positive design parameters and k is adynamie estimate of k* = (J~J~D)2 with k = k - k* being the estimation error. s

347

13.2 Adaptive Output Feedback Control

A Lyapunov function is defined as (13.137) Differentiating (13.137),

V1 =

(w+wj)[r-rl-Dw-Jwrej-Jk(w+wj)-Jkswjl +wj[-k(w + Wj) - ksWjl

< (w+wj)(r-r*) - ~w; - Jk(w+Wj)2 +(w + Wj )(Tl

+ DWrej + ]wrej) + Jk*(w + Wj)2.

(13.138)

The parameter update laws are designed as

Tl

-br(w + Wj)

j)

-bD(w + Wj )wrej

j

-bj(w + Wj )wrej

k

bk (w+Wj)2

(13.139)

where bn bD, bj , and bk are positive design freedoms. A new Lyapunov function is defined by adding terms quadratic in the parameter estimate errors to V1 as (13.140) Differentiating (13.140), and using (13.138) and (13.139),

V1a

(w+Wj)(r-r*)_k;w;-Jk 1 (W+Wj)2.

::;

(13.141)

Step 2: The ftux magnitude is required to track 'frej. This can be achieved by choosing reference trajectories for 'fd and 'fq as 'fdr = 'frej and 'fqr = 0, respectively. A commutation strategy for the desired torque is obtained as

(13.142) where the desired current i qr is given by

i qr

=

1 -o-,,-(Tl /-La 'l-'rej

+ JWrej + DWrej + kWj). A

A

(13.143)

As in previous sections, we make he re the physically meaningful assumption that the desired ftux trajectory is bounded below by a positive constant.

348

13. Adaptive Control of Induction Motors

The flux tracking errors are defined as

(13.144) The dynamics of the flux tracking errors are 'l/J d 'l/J q

+ &Lsrid - npWrej.J;q + npwo.J;q + npibiq +Uo,pd + Uc,pd - -J;rej -&.J;q + &Lsriq + npWrej.J;d - npwo.J;d - npibi d -&.J;d

+uo,pq

+ Uc,pq·

(13.145)

Designing the auxiliary control inputs Uc,pd, Uc,pq, Ucid, and Uciq as Uc,pd

npwji q

Uc,pq

-npWjtd

Ucid

-1'hqeid - kn1'I/Jqeid

Uciq

-1'h dei q - knl'I/Jdeiq

~2

A2

~2

A2

(13.146)

where k n1 is a positive design parameter and {) is an adaptation state, (13.145) is written as ;fd

{;q

+ &Lsri d - npWrej;fq + npwo;fq + np(w + Wj )iq +&(d - npWjei q + eidOd - -J;rej -&{j;q + &Lsriq + npWrej{;d - npwo{;d - np(w + Wj )i d +&(q + npwjeid - npwo'I/Jrej + npWrej'I/Jrej + eiqOq (13.147) -&.J;d

where

(13.148) The direct-axis current reference idr and the auxiliary control input Wo are defined as

(13.149)

349

13.2 Adaptive Output Feedback Control

where k co , kcl, and k c2 are positive design parameters. Note that &, which is governed by the projection-based update law (13.132), is bounded below by Q. - J. The current tracking errors are defined as i d = i d - i dr and i q = i q - i qr . Substituting (13.149) into (13.147), 'lj.;d

-

~2

-

2 -

2 -

-kcO'lj.;d - {hq'lj.;d - kc1wj'lj.;d - k c2 n d'lj.;d +npwo;J;q -

-kco'lj.;q -

,-: + Ci.Lsr~d -

-

npWrej'lj.;q

+ np(w + Wj )iq - npWjeiq + eidnd '2 , {)id'lj.;q + npWrej'lj.;d - npwo'lj.;d - np(w + Wj )i d .

(13.150)

Step 3: The current tracking errors follow the dynamics

(13.151) where n id and n iq are given by

+ &ß~d + npßWrej~q + npwoiq +Uoid + Ucid - idr -ii q + &ß~q - npßWrej~d - npwoi d -

nid

-iid

n iq

+Uoiq

+ Uciq -

kieid kiei q

(13.152)

i qr .

Note that nid and niq are functions of only the measured quantities and do not involve any unknown parameters. This is due to the fact that i dr and i qr depend only on quantities whose derivatives are available. Note that idr and i qr do not depend directly on w. The control inputs Ud and Uq are designed as aLs [-n id - &Lsr;J;d - kciidl aLs[-n iq - /-Lo'lj.;rej(w

+ Wj)

- kciiq]

(13.153)

where k ci > 0 is a design parameter. The substitution of (13.153) into (13.151) yields the current tracking error dynamics

(13.154) A new Lyapunov function

v

/-Lo 1 -2 1 -2 1-:2 1-:2 = -V n 0 + V,l a + _ol'd 20/ + _01. 20/ q + -~d 2 +-~ 2 q p

(13.155)

350

13. Adaptive Control o/Induction Motors

is defined. Differentiating (13.155), and using (13.134), (13.141), (13.150), and (13.154), . V

-

~

110 ki

~

2

2

< 110W( -eid 'l/;q + eiq'l/;d + 1qZd - 1d Zq) - npß (eid + eiq ) 110 ( ~2 2 '2 2 ~2 2 '2 2 ) - npß 1hqeid + kn1 'l/; qe id + rhde iq + kn1 'l/; dei q +110 (Zd Wf1.q - ZqWf 1') d -Jk1(w +Wf? 2-

-kc2 n d'l/;d

+ (-W + Wf ) (7

+ -0d[ -

+ &Lsri- d -

k cü -0d

-

7

*)

-19i~-0d -

ks 2 "2wf

-

k cl wJ-0d

+ npwo'l/;q

-

npwref'l/;q

+np(w + wf)iq - npWfei q + eidnd]

19i~-0q + n pWre f-0d

+-0q [ - k co -0q -

-npwo-0d - np(w + wf )i d] + id[-kc;id - &L sr -0d] +iq[-kciiq -l1o'l/;ref(w + wf )].

(13.156)

Using the identity 110W( -eid'l/;q + ei q'l/;d + iqz d - idzq) + 110 (zdWfi q - zqwfi d ) +(w + Wf )(7 - 7*)

= 110(W

-,

-,

-

1

'

1

'

+ wf ) ('l/;diq - 'l/;qid + 'l/;refiq + fjeidi q - fjeiqid) (13.157)

-l1üWj(,(f;deiq - ,(f;qei.) ,

(13.156) is simplified to

V·

110ki

2

< - npß (eid

110 '2 2 - npßknl('l/;qeid -2

110 ~2 2 ~2 2 npß 19(1qei d + 1d eiq)

2

+ eiq ) -

'2 2

ks

'2 -2

'2 -2

2

+ 'l/;deiq) - 4 Wf

-2

Jk1 - T(W

+ Wf)

2

-2-2

-kCÜ('l/;d + 'l/;q) -19(i q'l/;d + id'l/;q) - kci(i d + i q)

+

3(n~

+ 115) -n1

Jk

1

-2~2

('l/;d q+'l/;q 1d)

3115

2

~2

2

~2

+ Jklß2(eid1q+eiqZd)

n~ 2 1 2 2115 '2 2 '2 2 + 4kc1 eiq + 4k c2 eid + T.('l/;dei q + 'l/;qeiJ·

(13.158)

The design parameters k i and k n1 are chosen as

1)

-npß [ max (n~ -4k ' - k +&i 110 cl 4 c2 [2 115 k] -npß 110 -+ k -n 1 s

] (13.159)

351

13.2 Adaptive Output Feedback Control

where ki > 0 and kn1

::::::

0 are design parameters.

iJ is designed as (13.160)

where fL is a non-negative design parameter and with the dynamics ~

iJ

=

[f-tO

~2 2

~2 2

J is an adaptation parameter ~2 -2

~2 -2 ]

-b" npß(Zqeid+Zdeiq)+(Zq'l/Jd+Zd'l/Jq)

(13.161)

with b" > 0 being a design freedom. Finally, the composite Lyapunov function Tl' Va

=

V

_1__02

+ 2b"

'U

(13.162)

is defined where (13.163) (13.164)

Differentiating (13.162),

Global boundedness of all closed-loop signals follows from (13.165). Furthermore, the signals eid,eiq,Wj, w,{;d,{;q,i d, and i q are seen to go to zero asymptotically by an application of LaSalle's invariance principle. Thus, asymptotic speed tracking is achieved. Furthermore, as in Sect. 13.2.2, under a physically meaningful persistency of excitation condition, Zd and Zq also decay to zero, providing asymptotic flux magnitude tracking.

13.2.4

Simulation Results

The efficacy of the proposed controller is demonstrated using simulation for an induction motor with the electromechanical parameters shown in Table 12.1. It is assumed that the lower bound 0.725n is known for the rotor resistance R r . The reference value for the rotor velocity is 100 rad/so A smooth trajectory was chosen as seen in Fig. 13.1 to smooth the transient. The reference trajectory for the flux magnitude is 'l/Jrej

=

0.25 + 0.75(1 - e- O.8t ).

(13.166)

352

13. Adaptive Control of Induction Motors

The initial conditions for all the states including the parameter estimates is zero. The controller parameters are designed as

k i = 2.5 , k s = 200 , k 1 = 800 , k n1 = 1 k co = 10 , k c1 = 5 , k c2 = 5 , k ci = 100 'J2. = 0 , {) = 2 ,

b-a

= 0.001

, b",

bT = bD = bJ = bk = 1

= 0.01.

(13.167)

The simulation results are shown in Fig. 13.1. As expected from the Lyapunov analysis, the speed tracking is superior to the fiux magnitude tracking. The maximum and RMS values of the speed tracking error inc1uding the transient are 0.6°/s and O.16°/s, respectively. At steady state, the maximum and RMS values of the speed tracking error are 0.1 0 /s and 0.05°/s, respectively. The peak and RMS values of the fiux tracking error both during the transient and at steady state are 0.37 Wb and 0.34 Wb, respectively.

353

13.2 Adaptive Output Feedback Contro l

j

100

=tI 50 ~

SO

-50

X

5

150

j

Wh\'*WIIHN,~~'MWHlH

0

-15 0

10-3

2

3

4

5

0

2

3

4

5

15

5 -5~--~----~--~--------~

o

-150

2

3

4

OL---------~--~----~--~

5

0 0

2

3

4

5

1.5

o

2

3

4

5

0

2

3

4

5

2

3

4

5

0.4

~

eGi 0.2

~

0>

c

:i: u

0

co

!>

,,-0.2

.öl u.

0

0

2

3

4

5

-0.4

0

Figure 13.1 : Simulation results .

Chapter 14 Passivity-Based Control of Electric Motors In this chapter, the passivity-based approach to control design for electromechanical systems is described and applied to electric motors. The notions of dissipativity and passivity are powerful tools for design and analysis of control systems [3, 71, 170,207,212, 213J. For completeness, we will start with definitions and some results on dissipativity and passivity.

14.1

Dissipativity and Passivity

A system is said to be dissipative if it dissipates energy, i.e., the rate of increase of energy in the system is less than or equal to the supply rate. The dissipativity notion is made precise in the following definition. Definition 14.1 The system

x

=

y

= h(x,u)

f(x,u) (14.1)

with x E Rn being the state, u E Rm being the input, and y E RP being the output is said to be dissipative, with respect to the supply rate q( u, y) if there exists a continuously differentiable function V (x) satisfying

Q(lIxiD ::; V(x)

::; a(llxlD 8V 8x f (x,u) ::;q(u,y)

(14.2) (14.3)

for all x E Rn and u E Rm with Q and a being dass K oo functions. The system is said to be strictly dissipative, if, instead of (14.3), the stricter inequality 8V 8x f(x,u) ::;

-a(llxID + q(u,y)

(14.4)

356

14. Passivity-Based Contral

0/ Electric Motors

holds with a being a dass K oo function. The function V(x) is called a storage junction for the system (14.1) and the inequalities (14.3) and (14.4) are referred to as the dissipation inequalities. The notion of dissipativity can also be extended to memoryless systems, i.e., nm --+ np is continuous and 0 2. the cascade H2 0 H1 is zero-state detectable and for some a > 0, M(a) is negative semidefinite, Sl = aST, and R 1 + aQ2 is nonsingular 3. both H 1 and H2 are zero-state detectable and M(a) is negative definite for some a > 0 where M : n

-t

n2mx2m is the matrix function

M(a)

=

(14.19)

361

14.1 Dissipativity and Passivity

U1

= -Y2

Yl

U2

Y2

= Y1

Figure 14.3: Feedback interconnection of dissipative systems. Theorem 14.3 yields as a simple corollary the small gain theorem. Corollary 14.1 If H 1 and H 2 are zero-state detectable and dissipative with respect to the supply rates qi(Ui, Yi)

·..dur Ui - yr Yi, i = 1,2,

=

(14.20)

with /1 and /2 being positive, then the negative feedback interconnection of H 1 and H 2 is globally asymptotically stable if (14.21)

/1/2< 1.

An important special case of Theorem 14.3 is for systems that are output feedback passive and input feedfürward passive, Le, für systems with supply rates of the form in (14.11). Corollary 14.2 If H 1 and H 2 are zero-state detectable and dissipative with respect to the supply rates qi(Ui, Yi)

=

ur Yi - PiY[ Yi - l/iUr Ui, i

= 1,2,

(14.22)

then the negative feedback interconnection of H 1 and H2 is globally asymptotically stable if

+ P2 > 0 1/2 + PI > o. 1/1

(14.23)

Detailed proofs of the theorems above and several more results along similar lines can be found in the books [84,99, 181].

362

14. Passivity-Based Control of Electric Motors

14.2

Euler-Lagrange Systems

An important dass of passive systems that is often encountered in applications is the dass of Euler-Lagrange (EL) systems [61, 123, 205] with the dynamics

d (8C .) 8C . dt 8q(q,q) - 8q(q,q)

=

Q

(14.24)

where C : n nX n nf--> R is the Lagrangian of the system and Q E n nrepresents the external forces. q E nn is a vector of generalized positions, and q = !fif is a vector of generalized velocities. These equations arise in physical situations that are governed by a variationallaw. In the case that Q = 0, they are obtained through a variational analysis [36, 70] minimizing

A=

l

b

(14.25)

C(q, q)dt

under terminal constraints. Our treatment here of the control of EL systems and applications to electric motors follows [158, 164]. We focus on a subclass of EL systems with the Lagrangian being of the form

C(q,q)

=

T(q,q)-V(q)

(14.26)

with T(q, q) being the kinetic energy (or the coenergy) and V(q) being the potential energy. Furthermore, we require that the potential energy should be lower-bounded, i.e., 3c ERsuch that V(q)

~

c Vq E Rn

(14.27)

and that the kinetic energy should be of the form (14.28)

with V(q) E nnxn being the symmetrie positive definite generalized inertia matrix. Without loss of generality (since an additive constant in the potential energy does not affect the EL equations (14.24)), we assume that V(q) is nonnegative, i.e., C = o. Using (14.28), the EL equations (14.24) can be rewritten as

V(q)q

+ C(q, q)q + ~~

=

Q

(14.29)

where the (i, k) th entry of the n x n matrix C (q, q) is given by n

Cik(q, q)

= I>ijk(q)qj j=l

(14.30)

363

14.2 Euler-Lagmnge Systems

with (14.31) being the Christoffel symbols of the first kind. In mechanical systems, V(q)ij captures the inertial forces and C(q, q)q models the Coriolis and centrifugal forces. The external forces can be classified int0 2 dissipative forces Qdissipation, disturbances Q(, and control forces Qcontro!. The dissipative forces can be modeled as

BF(q) Bq

---

Qdissipation

(14.32)

where F( q) is the Rayleigh dissipation function which, by definition, satisfies3

.TBF(q)

q

> o.

fiil

(14.33)

To capture the possibility that the system is not fuHy actuated, the control forces are modeled as

Qcontro!

=

Mu

(14.34)

where ME nnxn u models the way the control enters into the system. Without loss of generality, M is assumed to have fuH column rank. If n = n u , the EL system is said to be fully actuated. If n < n u , the EL system is underactuated. In this case, q can be partitioned into non-actuated (M.1.. f q and actuated MT q components where M.1.. denotes the perpendicular complement of M. The EL system is said to be fuHy damped if the Rayleigh dissipation function satisfies (14.35) with ai, i = 1, ... , n, being positive numbers. The EL system is said to be underdamped ifthere is no set ofpositive numbers (al, ... , an) such that (14.35) is satisfied. In most cases, the Rayleigh dissipation function is of the form

F(q)

(14.36)

with R being a symmetrie positive semidefinite matrix. In this case, the EL system is fuHy damped if R is positive definite. 2Forces due to gravity can be derived from a potential and are usually included in V(q).

a~(q) is often called the generalized gravity and is denoted by g(q). ~This condition ensures that the dissipative force dissipates energy.

364

14. Passivity-Based Control

0/ Electric Motors

Thus, an EL system is characterized by the quintuple

{T(q, q), V(q), F(q), M, Qd,

(14.37)

referred to as the Euler-Lagrange parameters. The total energy or the Hamiltonian of the EL system is given by the sum of the kinetic and the potential energy

H(q,q)

T(q, q) (

+ V(q)

8.c(q,q))T._.c( ') 8q q q, q .

(14.38)

Since D(q) is positive definite and V(q) is non-negative, His positive semidefinite. However, His not necessarily zero for zero argument. Differentiating the total energy, we have

it

[

:t (~~) Tl q+ (~~)

T

q_

(~~)

T

q_

(~~)

QT q = qTQ.

T

q (14.39)

Thus, the EL system (14.24) defines a passive map from Q to

q.

In the absence of disturbances Q" the passivity equation (14.39) can be rewritten as -LI IL

=

.TM

q

.T8F(q)

u-q---aq'

(14.40)

Ftom (14.40) and (14.33), it is seen that the system is passive from input u to output MTq. If qT8~~q) ~ 0:1IM Tql12 with some positive 0:, the map from u to MT q is output strictly passive. This holds, for instance, when the system is fully damped. Electric motors form a special dass of EL systems with the property that they can be represented as a negative feedback interconnection of two EL subsystems, the electrical and the mechanical subsystems. The following theorem will prove useful to establish this property.

Theorem 14.4 Consider an EL system with the Lagrangian (14.41) where q = [q;, q;;'lT with qe ERn. and qm E Rn m. Then, the EL system (14.24) can be represented as the negative feedback interconnection of two passive systems (Fig. 14.4)

[ -~:]

(r+Qm)

~ [~] ~

qm

(14.42)

365

14.2 Euler-Lagrange Systems

7

6

=

a.c e(qe,4e,qm) aqm

(14.43)

being the coupling signal between the subsystems, and Q = [QI, Q;;'V with Qe E Rn e and Qm E Rn m . The theorem follows by decomposing the EL equation in (14.24) into its components as (14.44) (14.45) While the Hm subsystem is directly seen from (14.45) to be passive from the input (7 + Qm) to 4m, the passivity of He follows by considering the energy function (14.46) which satisfies (14.47) o The Passivity Based Control (PBC) method attempts to obtain a passive (and, often, in particular EL) closed-loop system. In this context, a result that characterizes feedback interconnections of EL systems such that the closed-loop system is EL is useful. The following theorem provides a condition such that the feedback interconnection of two EL systems is EL with the further property that the EL parameters of the closed-loop are given by the sum of the EL parameters of the individual subsystems.

Theorem 14.5 Consider two EL systems H 1 and H2 with the EL parameters4 {Tl (q1l4d, VI (qd, F I (4d, MI, On and {T2(q2, 42), V2(ql, Q2), F 2(42), 0, on, respectively (where 0 denotes a zero matrix of appropriate dimensions). If the two systems are interconnected via (14.48) 40bserve that ql is an input to H2 that appears through the potential term.

366

14. Passivity-Based Control

0/ Electric

Motors

T

Figure 14.4: Feedback decomposition of an EL system. where u is the input of subsystem H 1 , then the closed-Ioop system is an EL system with EL parameters {T(q, q), V(q), F(q), 0, O} where q = [qf, qfV and

T(q, q) V(q) F(q)

T1(ql, qt) + T2(q2, q2) V1(q1) + V2(q1, q2) F 1(q1) + F2 (q2).

(14.49)

o

14.3

Electric Motors as Euler-Lagrange Systems

In Chap. 3, we developed general modeling techniques for electric motors. Here, we reinterpret the model from a variational perspective using the system Lagrangian. The magnetic-field coenergy was derived in Chap. 3 to be

w'f

(14.50)

and the flux was derived as N+1

q;i

=

L Lijij j=l

N

LLiji j j=l

+ Li(N+1h

'

i = 1, ... ,N

367

14.3 Electric Motors as Euler-Lagrange Systems N

LLj (N+1)i j

+ Lffi f

(14.51)

j=l

where i 1 , ... ,iN are the currents in the N phases, cPll ... ,cPN are the flux linkages in the N phases, iN+1 = i f is the fictitious current due to the permanent magnet (ifpresent), and cPN+1 is the flux linkage associated with the permanent magnet. Using (14.50) and (14.51), the magnetic coenergy can be rewritten as

W'f = 2qe-qe 1 'TL' +tf . LT . N+1qe

(14.52)

where L. is the symmetrie N x N matrix with (i,j)th element being L ij , LN+1 the N x 1 vector [L 1(N+1), ... ,L N(N+1)]T, and qe = [i 1, ... ,iN]T. The generalized state coordinates of the electrical subsystem are taken to be the total charge flows through any given points on the different phase windings, qe = [qel,'" ,qeNV· The mechanical kinetic energy is given by (14.53) where qm = () is the rotor angular position and J is the rotor inertia5 . The inductance matrix L. and the vector L N +1 are functions of qm' Neglecting the capacitive effects in the motor windings and assuming rigidity of the shaft, the potential energy of the system is only due to interactions between the magnetic materials in the stator and the rotor, i.e., V is only a function of the rotor position qm' Thus, the Lagrangian of the overall system is given by 1 'TL( )qe+tf . . LTN+1 (qm ) qe . 2qe-qm

+~Jq~ -

V(qm).

(14.54)

Observe that the Lagrangian does not depend on qe, though it depends on qe' State components such as qe are called cyclic coordinates. The external forces are given by 1. Qdissipation : due to resistances

Re in windings and mechanical friction

(with viscous damping constant D) 2. Qe : load torque

Tl

3. Qcontrol: phase voltages Meu

= [VI,'"

,VN]T.

5For a linear motor, qm is the position and J is the mass of the moving part of the motor.

368

14. Passivity-Based Control

0/ Electric

Motors

The matrix Me models the possibility that not all phase voltages may be individually controlled. For instance, in induction motors, only the stator windings are actuated, while the rotor windings are simply short-circuited. Me is assumed to be of the form (14.55) where qe = [q;, q;]T with qs E Rn s being the charges corresponding to the actuated windings (e.g., the stator windings in the case ofthe induction motor) and qr E Rn r being the charges corresponding to the non-actuated windings (e.g., the rotor windings in the case of the induction motor). Insxns is the n s x n s identity matrix and 0 is a zero matrix of appropriate dimensions. If n s = N, the motor is said to be fully actuated. If n s < N, the motor is said to be underactuated. Re is modeled as (14.56) with R s E Rn.xn s and R r E Rnrxn r being diagonal positive semidefinite matrices. Furthermore, to ensure damping of the nonactuated phases, R r is required to be positive definite. I!, is partitioned as [

Lu

(14.57)

L21

Using the Lagrangian in (14.54) and the characterization of the external forces above, the equations of motion for electric motors can be derived as

~ (012) _ 012 dt

:t

aqe

aqe

+ ifL N +!) W1qmqe + Üi.e + W 2 qm -Dqm -

Tl

(I!,qe

(14.58)

~ ( 012 ) _ 012 dt aqm aqm Jijm -

T

(14.59)

where al!,

aqm . aL N +! 2f--aqm and

T

(14.60)

is the generated torque T

=

(14.61 )

14.4 Passivity-Based Contrvl of Electric Motors

369

The dynamics of the fiux can be written as (14.62) Partitioning