Analysis and design of descriptor linear systems

Advances in Mechanics and Mathematics

Volume 23

Series Editors:

David Y. Gao, Virginia Polytechnic Institute and State University

Ray W. Ogden, University of Glasgow

Romesh C. Batra, Virginia Polytechnic Institute and State University

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For more titles in this series, go to

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Guang-Ren Duan

Analysis and Design

of Descriptor Linear Systems

ABC

Guang-Ren Duan

Harbin Institute of Technology

Center for Control Theory and Guidance Technology

Harbin, 150001

P. R. China

[email protected]

ISSN 1571-8689 e-ISSN 1876-9896

ISBN 978-1-4419-6396-3 e-ISBN 978-1-4419-6397-0

DOI 10.1007/978-1-4419-6397-0

Springer New York Dordrecht Heidelberg London

Library of Congress Control Number: 2010933873

Mathematics Subject classification (2010): 58E25, 93B52, 93C05, 93C35

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To Shichao and Jiefu

Preface

Descriptor linear systems theory is an important part in the general field of control

systems theory, and has attracted much attention in the last two decades. In spite

of the fact that descriptor linear systems theory has been very rich in content, there

have been only a few comprehensive books on this topic, e.g., Campbell (1980),

Campbell (1982), and Dai (1989b). There do exist some other books and some PhD

thesises related to descriptor systems, but they are all focused on very special topics.

This book aims to provide a relatively systematic introduction to the basic results

in descriptor linear systems theory. The whole book has 11 chapters, and focuses

on the analysis and design problems on continuous-time descriptor linear systems.

Materials about analysis and design of discrete-time descriptor linear systems are

not included. Besides most of the fundamental context, it also contains some of the

author’s research work, which are reflected in the topics of response analysis, regu￾larization, dynamical order assignment, eigenstructure assignment, and parametric

approaches for observer design, etc.

Many researchers in the world have made great contribution to descriptor linear

systems theory. Owing to length limitation and the structural arrangement of the

book, many of their published results are not included or even not cited. I would

extend my apologies to these researchers.

Most of the materials of the book have been lectured by the author himself in the

spring terms of 20022005 in a postgraduate course at Harbin Institute of Tech￾nology. My colleagues, Prof. Zhi-Bin Yan and Dr Cang-Hua Jiang, assisted me

in lecturing this course in the spring terms of 20062008, respectively, and have

helped a lot in proofreading the manuscripts. Prof. Zhi-Bin Yan, Prof. Xian Zhang

and Dr Ai-Guo Wu have all coauthored with me a few papers, which have been in￾cluded in this book. Here, I would like to express my heartfelt appreciation of their

contribution.

All my graduate and PhD students and those who took the graduate course

“Descriptor Linear Systems” at Harbin Institute of Technology in the spring terms

of 20022008 have offered tremendous help in finding the errors and typos in the

manuscripts. Their help has greatly improved the quality of the manuscripts, and is

indeed very much appreciated. Dr Hai-Hua Yu, Dr Ai-Guo Wu, Dr Bing Liang,

Dr Yan-Ming Fu, Dr Ying Zhang, Dr Liu Zhang, Dr Yong-Zheng Shan, and

Dr Hong-Liang Liu, who were really my PhD students years ago, have helped me

vii

viii Preface

with the indices, the references, and the parts of the revision of the book. My present

Ph.D students, Mr. Da-Ke Gu, Mr. Shi-Jie Zhang, Ms. Ling-Ling Lv, Mr. Yan-Jiang

Li, Ms. Shi Li, and Mr. Guang-Bin Cai, all helped me with the examples of the book.

Particularly, Dr Hai-Hua Yu, besides all the above, has helped me with the whole

formatting of the book. I would extend my great thanks to all of them. My thanks

would also go to my colleague, Prof. Hui-Jun Gao, who once was in 2003 a student

in my class of the course, has proofread several chapters of the book as well.

I would also like to thank my wife, Ms Shi-Chao Zhang, for her continuous sup￾port in every aspect. Sincere thanks also go to my secretary, Ms Ming-Yan Liu, for

helping me in typing a few chapters of the manuscripts. Part of the book was written

when I was with the Queen’s University of Belfast, UK, from September 1998 to

October 2002. I would like to thank Professor G. W. Irwin and Dr S. Thompson

for their help, suggestions, and support. The reviewers of the book have given some

real valuable and helpful comments and suggestions, which are indeed very much

appreciated.

The author would like to gratefully acknowledge the financial support kindly pro￾vided by the many sponsors, including NSFC, the National Natural Science Foun￾dation of China (National Science Fund for Distinguished Young Scholar’s Grant

No.60474015), the Ministry of Education (Program of The New Century Excellent

Talents in University and the Chang Jiang Scholars Program), and also EPSRC, the

UK Engineering and Physical Science Research Council (GR/K83861/01).

At the last, let me thank in advance all the readers for choosing to read this

book. I would be indeed very grateful if readers could possibly provide, via email:

[email protected], feedback about any problems found. Your help will certainly

make any future editions of the book much better.

Harbin Institute of Technology, Guang-Ren Duan

12 December 2009

Contents

Preface.............................................................................. vii

List of Notation .................................................................... xvii

1 Introduction .................................................................. 1

1.1 Models for Descriptor Systems....................................... 1

1.1.1 State Space Representation.................................. 1

1.1.2 Time-Invariant Descriptor Linear Systems ................. 3

1.2 Examples of Descriptor Linear Systems ............................. 5

1.2.1 Electrical Circuit Systems................................... 5

1.2.2 Large-Scale Systems with Interconnections................ 7

1.2.3 Constrained Mechanical Systems........................... 8

1.2.4 Robotic System–A Three-Link Planar Manipulator ....... 12

1.3 Problems for Descriptor Linear Systems Analysis and Design ..... 18

1.3.1 Feedback in Descriptor Linear Systems .................... 18

1.3.2 Problems for Descriptor Linear Systems Analysis......... 22

1.3.3 Problems for Descriptor Linear Systems Design........... 24

1.4 Overview of the Book ................................................. 28

1.5 Notes and References ................................................. 29

Part I Descriptor Linear Systems Analysis

2 Equivalence and Solutions of Descriptor Linear Systems............... 35

2.1 Restricted System Equivalence ....................................... 35

2.1.1 The Definition ............................................... 36

2.1.2 Common Properties ......................................... 38

2.2 Canonical Equivalent Forms.......................................... 40

2.2.1 Dynamics Decomposition Form ............................ 40

2.2.2 The Kronecker Form ........................................ 43

2.2.3 Canonical Equivalent Forms for Derivative Feedback ..... 44

ix

x Contents

2.3 Solutions of Descriptor Linear Systems.............................. 50

2.3.1 System Decomposition Based on the

Kronecker Form ............................................. 50

2.3.2 Solution to the Basic Types of Equations .................. 51

2.4 Notes and References ................................................. 55

3 Regular Descriptor Linear Systems ....................................... 57

3.1 Regularity of Descriptor Linear Systems ............................ 57

3.1.1 The Definition and Its Relation with Solutions ............ 57

3.1.2 Criteria for Regularity ....................................... 61

3.2 Equivalence of Regular Descriptor Linear Systems ................. 64

3.2.1 Standard Decomposition Form.............................. 65

3.2.2 The Inverse Form ........................................... 70

3.3 Transfer Function Matrices ........................................... 72

3.3.1 The Definition ............................................... 73

3.3.2 Properties .................................................... 74

3.4 State Responses of Regular Descriptor Linear

Systems: Distributional Solutions .................................... 75

3.4.1 Solutions of Slow and Fast Subsystems .................... 76

3.4.2 The Distributional Solutions ................................ 80

3.4.3 Examples..................................................... 81

3.5 State Responses of Regular Descriptor Linear

Systems: Classical Solutions ......................................... 83

3.5.1 Consistency .................................................. 84

3.5.2 The Classical Solutions ..................................... 85

3.5.3 The Example ................................................. 88

3.6 Generalized Eigenvalues and Eigenvectors .......................... 89

3.6.1 Finite Eigenvalues and Eigenvectors ....................... 90

3.6.2 Infinite Eigenvalues and Eigenvectors ...................... 95

3.7 Eigenstructure Decomposition with Relation to

Standard Decomposition .............................................. 98

3.7.1 Eigenstructure Decomposition .............................. 98

3.7.2 Relation with Standard Decomposition..................... 102

3.7.3 The Deflating Subspaces .................................... 103

3.8 Stability ................................................................ 107

3.8.1 The Definition ............................................... 107

3.8.2 The Direct Criterion ......................................... 108

3.8.3 Criterion via Lyapunov Equation ........................... 110

3.8.4 Examples..................................................... 111

3.9 Admissibility: Stability plus Impulse-Freeness...................... 113

3.9.1 The Definition ............................................... 113

3.9.2 The Criterion................................................. 114

3.9.3 The Example ................................................. 117

3.10 Notes and References ................................................. 117

Contents xi

4 Controllability and Observability.......................................... 121

4.1 State Reachable Subsets .............................................. 121

4.1.1 The Definition ............................................... 122

4.1.2 Characterization of Rt Œ0 and Rt ........................... 124

4.1.3 Two Examples ............................................... 130

4.2 Controllability ......................................................... 131

4.2.1 C-Controllability............................................. 132

4.2.2 R-Controllability............................................. 135

4.2.3 I-Controllability and S-Controllability ..................... 136

4.3 Observability .......................................................... 142

4.3.1 C-Observability .............................................. 143

4.3.2 R-Observability .............................................. 147

4.3.3 I-Observability and S-Observability ........................ 148

4.4 The Dual Principle .................................................... 153

4.4.1 The Dual System ............................................ 153

4.4.2 The Dual Principle........................................... 154

4.5 Direct Criteria ......................................................... 156

4.5.1 C-Controllability and C-Observability ..................... 157

4.5.2 R-Controllability and R-Observability ..................... 160

4.5.3 I-Controllability and I-Observability ....................... 161

4.5.4 S-Controllability and S-Observability ...................... 165

4.6 Criteria Based on Equivalent Forms.................................. 165

4.6.1 Criteria Based on the Dynamics

Decomposition Form ........................................ 165

4.6.2 Criteria Based on the Inverse Form ......................... 168

4.6.3 Criteria Based on Equivalent Form

for Derivative Feedback ..................................... 173

4.7 System Decomposition................................................ 177

4.7.1 The General Structural Decomposition ..................... 178

4.7.2 Special Cases ................................................ 181

4.8 Transfer Function Matrix and Minimal Realization ................. 185

4.8.1 Transfer Function Matrix.................................... 185

4.8.2 Minimal Realization ......................................... 187

4.9 Notes and References ................................................. 194

Part II Descriptor Linear Systems Design

5 Regularization of Descriptor Linear Systems ............................ 199

5.1 Regularizability under P- (D-) Feedback............................. 199

5.1.1 Proportional Feedback....................................... 199

5.1.2 Derivative Feedback ......................................... 203

5.2 Regularizability under P-D Feedback ................................ 207

5.2.1 Problem Formulation ........................................ 207

5.2.2 Regularizability Conditions ................................. 209

xii Contents

5.3 Regularizing Controllers .............................................. 212

5.3.1 Problem Formulation ........................................ 212

5.3.2 Preliminaries ................................................. 213

5.3.3 The Conclusion .............................................. 214

5.4 Proof of Theorem 5.7 ................................................. 218

5.4.1 Preliminary Results.......................................... 218

5.4.2 Proof of Theorem 5.7 ........................................ 220

5.5 Notes and References ................................................. 224

6 Dynamical Order Assignment and Normalization....................... 227

6.1 Assignable Dynamical Orders ........................................ 227

6.1.1 Full-State Derivative Feedback ............................. 228

6.1.2 Partial-State Derivative Feedback........................... 232

6.2 Dynamical Order Assignment via Full-State Derivative Feedback . 235

6.2.1 Problem Formulation ........................................ 236

6.2.2 Preliminary Results.......................................... 237

6.2.3 Solution to the Problem ..................................... 238

6.3 Dynamical Order Assignment via State Derivative

Feedback with Minimum Norm ...................................... 240

6.3.1 Problem Formulation ........................................ 240

6.3.2 A Preliminary Result ........................................ 241

6.3.3 Solution to the Problem ..................................... 243

6.4 Dynamical Order Assignment via Partial-State

Derivative Feedback .................................................. 247

6.4.1 Problem Formulation ........................................ 247

6.4.2 Preliminary Results.......................................... 248

6.4.3 Solution to the Problem ..................................... 251

6.4.4 The Example ................................................. 254

6.5 Normalization of Descriptor Linear Systems ........................ 256

6.5.1 Normalizability .............................................. 256

6.5.2 Normalizing Controllers..................................... 259

6.6 Notes and References ................................................. 262

7 Impulse Elimination ......................................................... 265

7.1 The Impulse-Free Property ........................................... 265

7.1.1 Basic Criteria ................................................ 266

7.1.2 Criteria Based on Equivalent Forms ........................ 268

7.2 Impulse Elimination by State Feedback .............................. 272

7.2.1 Solution Based on Dynamics Decomposition Forms ...... 272

7.2.2 Solution Based on Standard Decomposition ............... 276

7.2.3 Solution Based on Canonical Equivalent

Form for Derivative Feedback .............................. 279

7.3 Impulse Elimination by Output Feedback............................ 281

7.3.1 Problem Formulation ........................................ 281

7.3.2 The Solution ................................................. 282

Contents xiii

7.4 I-Controllablizability and I-Observablizability ...................... 284

7.4.1 Basic Criterion ............................................... 285

7.4.2 Criteria Based on Equivalent Forms ........................ 289

7.5 Impulsive Elimination by P-D Feedback ............................. 293

7.5.1 Method I ..................................................... 294

7.5.2 Method II..................................................... 298

7.6 Notes and References ................................................. 302

8 Pole Assignment and Stabilization......................................... 305

8.1 Pole Assignment by State Feedback.................................. 305

8.1.1 Problems Formulation ....................................... 305

8.1.2 Pole Assignment under R-Controllability .................. 307

8.1.3 Pole Assignment under S-Controllability .................. 310

8.2 Pole Assignment by P-D Feedback................................... 315

8.2.1 Problem Formulation ........................................ 315

8.2.2 The Solution ................................................. 315

8.3 Stabilizability and Detectability ...................................... 318

8.3.1 Stabilizability ................................................ 319

8.3.2 Detectability ................................................. 321

8.4 Stabilizing Controller Design ......................................... 324

8.4.1 Design Based on Standard Decomposition ................. 324

8.4.2 Design Based on Controllability Canonical Forms ........ 327

8.4.3 Design Based on Lyapunov Theory ........................ 330

8.5 Notes and References ................................................. 333

9 Eigenstructure Assignment................................................. 337

9.1 The Problem ........................................................... 338

9.1.1 The Problem ................................................. 339

9.1.2 Interpretations of Requirements ............................ 340

9.1.3 Problem Decomposition ..................................... 344

9.2 The Parametric Solution .............................................. 345

9.2.1 Solution of Closed-Loop Eigenvectors ..................... 345

9.2.2 Solution of the Gain Matrix K .............................. 347

9.2.3 The Algorithm for Problem 9.1 ............................. 350

9.3 The Left Eigenvector Matrix.......................................... 352

9.3.1 Preliminaries ................................................. 352

9.3.2 The Parametric Expressions................................. 353

9.4 Response of the Closed-Loop System................................ 356

9.4.1 The Canonical Form for the Closed-Loop System ......... 356

9.4.2 The Closed-Loop Response ................................. 358

9.5 An Example ........................................................... 361

9.5.1 The General Solutions....................................... 362

9.5.2 Special Solutions ............................................ 363

9.6 Notes and References ................................................. 365

xiv Contents

10 Optimal Control ............................................................. 369

10.1 Introduction ............................................................ 369

10.2 Optimal Linear Quadratic State Regulation .......................... 371

10.2.1 Problem Formulation ........................................ 371

10.2.2 The Conversion .............................................. 372

10.2.3 The Optimal Regulator ...................................... 375

10.2.4 An Illustrative Example ..................................... 377

10.3 Time-Optimal Control ................................................ 379

10.3.1 Problem Formulation ........................................ 379

10.3.2 Time-Optimal Control of the Slow and Fast Subsystems.. 380

10.3.3 The Solution ................................................. 382

10.4 Notes and References ................................................. 385

11 Observer Design ............................................................. 389

11.1 Introduction ............................................................ 389

11.1.1 State Observers .............................................. 390

11.1.2 Function Kx Observers ..................................... 391

11.2 Descriptor State Observers............................................ 392

11.2.1 Existence Condition ......................................... 392

11.2.2 Design Methods ............................................. 394

11.3 Eigenstructure Assignment Design ................................... 396

11.3.1 Eigenstructure Assignment Result .......................... 397

11.3.2 The Algorithm and Example ................................ 399

11.4 Observer Design with Disturbance Decoupling ..................... 402

11.4.1 Problem Formulation ........................................ 403

11.4.2 Preliminaries ................................................. 403

11.4.3 Constraints for Disturbance Decoupling ................... 406

11.4.4 The Example ................................................. 407

11.5 Normal Reduced-Order State Observers ............................. 409

11.5.1 Normal RankE-Order State Observers ..................... 409

11.5.2 Normal-State Observers of Order n m ................... 412

11.6 Normal Function Kx Observers...................................... 417

11.6.1 Conditions for Normal Function Kx Observers ........... 417

11.6.2 Parametric Design for Normal Function Observers........ 419

11.7 Notes and References ................................................. 424

Part III Appendices

A Some Mathematical Results ................................................ 429

A.1 Delta Function ı./ .................................................... 429

A.2 Laplace Transform .................................................... 432

A.3 Determinants and Inverses of Block Matrices ....................... 434

A.4 Nilpotent Matrices..................................................... 437

A.5 Some Operations of Linear Subspaces ............................... 441

A.6 Kernels and Images of Matrices ...................................... 443

A.7 Singular Value Decomposition ....................................... 445