ESSENTIALS OF 目 ROBUST E CONTROLI e11n∠101 WITH JOHN C. DOYLE PRENTICE HALL
ESSENTIALS OF ROBUST CONTROL Kemin Zhou May25,1999
ESSENTIALS OF ROBUST CONTROL Kemin Zhou May 25, 1999
Preface Robustness of control systems to disturbances and uncertainties has always been the central issue in feedback control. Feedback would not be needed for most control systems if there were no disturbances and uncertainties. Developing multivariable robust control methods has been the focal point in the last two decades in the control community. The state-of-the-art Hoo robust control theory is the result of this effort. another book by this author, John C. Doyle, and Keith Glover, entitled Robust and Optimal Control, which has been extensively class-tested in many universities around the world. Unlike that book, which is intended primarily as a comprehensive reference of robust and Hoo control theory, this book is intended to be a text for a graduate course in multivariable control. It is also intended to be a reference for practicing control engineers who are interested in applying the state-of-the-art robust control techniques in their applications. With this objective in mind, I have streamlined the presentation added more than 50 illustrative examples, included many related MATLAB commands and more than 150 exercise problems, and added some recent developments in the area of robust control such as gap metric, v-gap metric, model validation, and mixed u problem. In addition, many proofs are os completely rewritten and some advanced topics are either deleted completely or do not get an in-depth treatment The prerequisite for reading this book is some basic knowledge of classical control theory and state-space theory. The text contains more material than could be covered in detail in a one-semester or a one-quarter course. Chapter 1 gives a chapter-by-chapter summary of the main results presented in the book, which could be used as a guide for the selection of topics for a specific course. Chapters 2 and 3 can be used as a refresher for some linear algebra facts and some standard linear system theory. A course focusing on Hoo control should cover at least most parts of Chapters 4-6, 8, 9, 11-13, and Sections 14.1 and 14.2. An advanced Hoo control course should also include the rest of Chapter 14, Chapter 16, and possibly Chapters 10, 7, and 15. A course focusing on robustness and model uncertainty should cover at least Chapters 4, 5, and 8-10. Chapters 17 and 18 can be added to any advanced robust and Hoo control course if time permits I have tried hard to eliminate obvious mistakes. It is, however, impossible for me to make the book perfect. Readers are encouraged to send corrections, comments, and MATLAB is a registered trademark of The Math Works, Inc
Preface Robustness of control systems to disturbances and uncertainties has always been the central issue in feedback control. Feedback would not be needed for most control systems if there were no disturbances and uncertainties. Developing multivariable robust control methods has been the focal point in the last two decades in the control community. The state-of-the-art H∞ robust control theory is the result of this effort. This book introduces some essentials of robust and H∞ control theory. It grew from another book by this author, John C. Doyle, and Keith Glover, entitled Robust and Optimal Control, which has been extensively class-tested in many universities around the world. Unlike that book, which is intended primarily as a comprehensive reference of robust and H∞ control theory, this book is intended to be a text for a graduate course in multivariable control. It is also intended to be a reference for practicing control engineers who are interested in applying the state-of-the-art robust control techniques in their applications. With this objective in mind, I have streamlined the presentation, added more than 50 illustrative examples, included many related Matlab R commands1 and more than 150 exercise problems, and added some recent developments in the area of robust control such as gap metric, ν-gap metric, model validation, and mixed µ problem. In addition, many proofs are completely rewritten and some advanced topics are either deleted completely or do not get an in-depth treatment. The prerequisite for reading this book is some basic knowledge of classical control theory and state-space theory. The text contains more material than could be covered in detail in a one-semester or a one-quarter course. Chapter 1 gives a chapter-by-chapter summary of the main results presented in the book, which could be used as a guide for the selection of topics for a specific course. Chapters 2 and 3 can be used as a refresher for some linear algebra facts and some standard linear system theory. A course focusing on H∞ control should cover at least most parts of Chapters 4–6, 8, 9, 11–13, and Sections 14.1 and 14.2. An advanced H∞ control course should also include the rest of Chapter 14, Chapter 16, and possibly Chapters 10, 7, and 15. A course focusing on robustness and model uncertainty should cover at least Chapters 4, 5, and 8–10. Chapters 17 and 18 can be added to any advanced robust and H∞ control course if time permits. I have tried hard to eliminate obvious mistakes. It is, however, impossible for me to make the book perfect. Readers are encouraged to send corrections, comments, and 1Matlab is a registered trademark of The MathWorks, Inc. vii
PREFACE suggestions to me, preferably by electronic mail, at kemin@ee. Isu. edu I am also planning to put any corrections, modifications, and extensions on the Internet so that they can be obtained either from the following anonymous ftp ftp gate. ee. lsu. edu cd pub/kemin/books/ essentials or from the author's home page http://kilo.ee.lsuedu/kemin/books/essentials/ This book would not be possible without the work done jointly for the previous book with Professor John C. Doyle and Professor Keith Glover. I thank them for their influence on my research and on this book. Their serious attitudes toward scientific research have been reference models for me. I am especially grateful to John for having me as a research fellow in Caltech, where I had two very enjoyable years and had opportunities to catch a glimpse of his"BIG PICTURE of control I want to thank my editor from Prentice Hall, Tom Robbins, who originally proposed he idea for this book and has been a constant source of support for me while writing it Without his support and encouragement, this project would have been a difficult one It has been my great pleasure to work with him. I would like to express my sincere gratitude to Professor Bruce A. Francis for giving me many helpful comments and suggestions on this book. Professor Francis has also kindly provided many exercises in the book. I am also grateful to Professor Kang-Zhi liu and Professor Zheng-Hua Luo, who have made many useful comments and suggestions I want to thank Professor Glen Vinnicombe for his generous help in the preparation of Chapters 16 and 17. Special thanks go to Professor Jianqing Mao for providing me the opportunity to present much of this material in a series of lectures at Beijing University of aeronautics and astronautics in the summer of 1996 In addition, I would like to thank all those who have helped in many ways in making this book possible, especially Professor Pramod P Khargonekar, Professor Andre Tit Professor Andrew Packard, Professor Jie Chen, Professor Jakob Stoustrup, Professor Hans Henrik Niemann, Professor Malcolm Smith, Professor Tryphon Georgiou, Profes- sor Tongwen Chen, Professor Hitay Ozbay, Professor Gary Balas, Professor Carolyn Beck, Professor Dennis S Bernstein, Professor Mohamed Darouach, Dr. Bobby Boden- heimer, Professor Guoxiang Gu, Dr. Weimin Lu, Dr. John Morris, Dr. Matt Newlin Professor Li Qiu, Professor Hector P. Rotstein, Professor Andrew Teel, Professor Ja- gannathan Ramanujam, Dr. Linda G. Bushnell, Xiang Chen, Greg Salomon, Pablo A Parrilo, and many other people for supporting my research: National Air Force of scientific Research. and the board of regents in the state of louisiana nally, I would like to thank my wife, Jing, and my son, Eric, for their generous support, understanding, and patience during the writing of this book Kemin zhou
viii PREFACE suggestions to me, preferably by electronic mail, at kemin@ee.lsu.edu I am also planning to put any corrections, modifications, and extensions on the Internet so that they can be obtained either from the following anonymous ftp: ftp gate.ee.lsu.edu cd pub/kemin/books/essentials/ or from the author’s home page: http://kilo.ee.lsu.edu/kemin/books/essentials/ This book would not be possible without the work done jointly for the previous book with Professor John C. Doyle and Professor Keith Glover. I thank them for their influence on my research and on this book. Their serious attitudes toward scientific research have been reference models for me. I am especially grateful to John for having me as a research fellow in Caltech, where I had two very enjoyable years and had opportunities to catch a glimpse of his “BIG PICTURE” of control. I want to thank my editor from Prentice Hall, Tom Robbins, who originally proposed the idea for this book and has been a constant source of support for me while writing it. Without his support and encouragement, this project would have been a difficult one. It has been my great pleasure to work with him. I would like to express my sincere gratitude to Professor Bruce A. Francis for giving me many helpful comments and suggestions on this book. Professor Francis has also kindly provided many exercises in the book. I am also grateful to Professor Kang-Zhi Liu and Professor Zheng-Hua Luo, who have made many useful comments and suggestions. I want to thank Professor Glen Vinnicombe for his generous help in the preparation of Chapters 16 and 17. Special thanks go to Professor Jianqing Mao for providing me the opportunity to present much of this material in a series of lectures at Beijing University of Aeronautics and Astronautics in the summer of 1996. In addition, I would like to thank all those who have helped in many ways in making this book possible, especially Professor Pramod P. Khargonekar, Professor Andr´e Tits, Professor Andrew Packard, Professor Jie Chen, Professor Jakob Stoustrup, Professor Hans Henrik Niemann, Professor Malcolm Smith, Professor Tryphon Georgiou, Professor Tongwen Chen, Professor Hitay Ozbay, Professor Gary Balas, Professor Carolyn ¨ Beck, Professor Dennis S. Bernstein, Professor Mohamed Darouach, Dr. Bobby Bodenheimer, Professor Guoxiang Gu, Dr. Weimin Lu, Dr. John Morris, Dr. Matt Newlin, Professor Li Qiu, Professor Hector P. Rotstein, Professor Andrew Teel, Professor Jagannathan Ramanujam, Dr. Linda G. Bushnell, Xiang Chen, Greg Salomon, Pablo A. Parrilo, and many other people. I would also like to thank the following agencies for supporting my research: National Science Foundation, Army Research Office (ARO), Air Force of Scientific Research, and the Board of Regents in the State of Louisiana. Finally, I would like to thank my wife, Jing, and my son, Eric, for their generous support, understanding, and patience during the writing of this book. Kemin Zhou
PREFACE Here is how Hoo is pronounced in Chi 愛趣無窮 It means "The joy of love is endless
PREFACE ix Here is how H∞ is pronounced in Chinese: It means “The joy of love is endless
Contents Preface Notation and Symbols List of acronyms XvII 1 Introduction 1.1 What Is This book about? 1.2 Highlights of This Book 3 1. 3 Notes and references 9 2 Linear Algebra 2.1 Linear Subspaces 2.2 Eigenvalues and eigenvectors 2.3 Matrix Inversion Formulas 2.4 Invariant Subspaces 2.5 Vector Norms and Matrix Norms 6 Singular Value Decomposition 2.7 Semidefinite Matrices 2. 8 Notes and References 2.9 Problems 3 Linear Systems 3.1 Descriptions of Linear Dynamical Systems 3.2 Controllability and Observability 3.3 Observers and Observer-Based Controllers 31 3.5 State-Space Realizations for Transfer Matrices 3.6 Multivariable System Poles and Zeros 3.7 Notes and References
Contents Preface vii Notation and Symbols xv List of Acronyms xvii 1 Introduction 1 1.1 What Is This Book About? ......................... 1 1.2 Highlights of This Book ........................... 3 1.3 Notes and References ............................. 9 1.4 Problems ................................... 10 2 Linear Algebra 11 2.1 Linear Subspaces ............................... 11 2.2 Eigenvalues and Eigenvectors ........................ 12 2.3 Matrix Inversion Formulas .......................... 13 2.4 Invariant Subspaces ............................. 15 2.5 Vector Norms and Matrix Norms ...................... 16 2.6 Singular Value Decomposition ........................ 19 2.7 Semidefinite Matrices ............................ 23 2.8 Notes and References ............................. 24 2.9 Problems ................................... 24 3 Linear Systems 27 3.1 Descriptions of Linear Dynamical Systems ................. 27 3.2 Controllability and Observability ...................... 28 3.3 Observers and Observer-Based Controllers ................. 31 3.4 Operations on Systems ............................ 34 3.5 State-Space Realizations for Transfer Matrices .............. 35 3.6 Multivariable System Poles and Zeros ................... 38 3.7 Notes and References ............................. 41 3.8 Problems ................................... 42 xi
CONTENTS 4 H2 and Hoo Spaces 4.1 Hilbert Spaces 45 4.3 Computing L2 and H2 Norms 4.4 Computing Loo and Ho Norm 4.5 Notes and references 4. 6 Problems 5 Internal Stability 5.1 Feedback Struct 5.2 Well-Posedness of Feedback Loop 5.3 Internal Stability 5.4 Coprime Factorization over RH 5.5 Notes and references 5.6 Problems 6 Performance Specifications and Limitations 6.1 Feedback Properties 6.2 Weighted H2 and Hoo Performance 6.3 Selection of Weighting Functions 6.4 Bode s Gain and phase relation 6.5 Bode's Sensitivity Integral 6.6 Analyticity Constraints 6.7 Notes and References 6.8 Problems 102 7 Balanced model reduction 1 7.1 Lyapunov Equations 7.2 Balanced realizations 7.3 Model Reduction by Balanced Truncation 117 7.4 Frequency-Weighted Balanced Model Reduction 7.5 Notes and references 126 7. 6 Problems 8 Uncertainty and robustness 129 8.1 Model Uncertainty 8.2 Small Gain Theorem 8.3 Stability under Unstructured Uncertainties 8. 4 Robust Performance 147 8.5 Skewed Specifications 150 8.6 Classical Control for MIMO Systems 8. 7 Notes and references 157 8.8 Problems
xii CONTENTS 4 H2 and H∞ Spaces 45 4.1 Hilbert Spaces ................................ 45 4.2 H2 and H∞ Spaces .............................. 47 4.3 Computing L2 and H2 Norms ........................ 53 4.4 Computing L∞ and H∞ Norms ....................... 55 4.5 Notes and References ............................. 61 4.6 Problems ................................... 62 5 Internal Stability 65 5.1 Feedback Structure .............................. 65 5.2 Well-Posedness of Feedback Loop ...................... 66 5.3 Internal Stability ............................... 68 5.4 Coprime Factorization over RH∞ ...................... 71 5.5 Notes and References ............................. 77 5.6 Problems ................................... 77 6 Performance Specifications and Limitations 81 6.1 Feedback Properties ............................. 81 6.2 Weighted H2 and H∞ Performance ..................... 85 6.3 Selection of Weighting Functions ...................... 89 6.4 Bode’s Gain and Phase Relation ...................... 94 6.5 Bode’s Sensitivity Integral .......................... 98 6.6 Analyticity Constraints ........................... 100 6.7 Notes and References ............................. 102 6.8 Problems ................................... 102 7 Balanced Model Reduction 105 7.1 Lyapunov Equations ............................. 106 7.2 Balanced Realizations ............................ 107 7.3 Model Reduction by Balanced Truncation ................. 117 7.4 Frequency-Weighted Balanced Model Reduction .............. 124 7.5 Notes and References ............................. 126 7.6 Problems ................................... 127 8 Uncertainty and Robustness 129 8.1 Model Uncertainty .............................. 129 8.2 Small Gain Theorem ............................. 137 8.3 Stability under Unstructured Uncertainties ................ 141 8.4 Robust Performance ............................. 147 8.5 Skewed Specifications ............................ 150 8.6 Classical Control for MIMO Systems .................... 154 8.7 Notes and References ............................. 157 8.8 Problems ................................... 158
CONTENTS 9 Linear Fractional Transformation 165 9.1 Linear fractional Transformations 9.2 Basic Principle 9.3 Redheffer Star Products 9. 4 Notes and references 10 u and u Synthesis 10.1 General Framework for System Robustness 10.2 Structured Singular value 10.3 Structured Robust Stability and Performance 10.4 Overview of u Synthesis 213 10.5 Notes and references 216 10.6 Problems 11 Controller parameterization 221 11.1 Existence of Stabilizing Controllers 11.2 Parameterization of All Stabilizing Controllers 11.4 Notes and references 11.5 Proble 12 Algebraic Riccati Equations 233 12.1 Stabilizing Solution and Riccati Operator 234 245 12.3 Notes and references 246 12.4 Problems 13 H2 Optimal Control 13.1 Introduction to Regulator Problem 253 13.2 Standard LQR Problem 13.3 Extended LQR Problem 13.4 Guaranteed Stability Margins of LQR 13.5 Standard H2 Problem 13.6 Stability Margins of H2 Controllers 265 13.7 Notes and References 13.8 Problems 14H。 Control 269 14.1 Problem formulation 14.2 A Simplified Hoo Control Problem 270 14.3 Optimality and Limiting Behavior 14.4 Minimum Entropy Controller 14.5 An Optimal Controller
CONTENTS xiii 9 Linear Fractional Transformation 165 9.1 Linear Fractional Transformations ..................... 165 9.2 Basic Principle ................................ 173 9.3 Redheffer Star Products ........................... 178 9.4 Notes and References ............................. 180 9.5 Problems ................................... 181 10 µ and µ Synthesis 183 10.1 General Framework for System Robustness ................ 184 10.2 Structured Singular Value .......................... 187 10.3 Structured Robust Stability and Performance ............... 200 10.4 Overview of µ Synthesis ........................... 213 10.5 Notes and References ............................. 216 10.6 Problems ................................... 217 11 Controller Parameterization 221 11.1 Existence of Stabilizing Controllers ..................... 222 11.2 Parameterization of All Stabilizing Controllers .............. 224 11.3 Coprime Factorization Approach ...................... 228 11.4 Notes and References ............................. 231 11.5 Problems ................................... 231 12 Algebraic Riccati Equations 233 12.1 Stabilizing Solution and Riccati Operator ................. 234 12.2 Inner Functions ................................ 245 12.3 Notes and References ............................. 246 12.4 Problems ................................... 246 13 H2 Optimal Control 253 13.1 Introduction to Regulator Problem ..................... 253 13.2 Standard LQR Problem ........................... 255 13.3 Extended LQR Problem ........................... 258 13.4 Guaranteed Stability Margins of LQR ................... 259 13.5 Standard H2 Problem ............................ 261 13.6 Stability Margins of H2 Controllers ..................... 265 13.7 Notes and References ............................. 267 13.8 Problems ................................... 267 14 H∞ Control 269 14.1 Problem Formulation ............................. 269 14.2 A Simplified H∞ Control Problem ..................... 270 14.3 Optimality and Limiting Behavior ..................... 282 14.4 Minimum Entropy Controller ........................ 286 14.5 An Optimal Controller ............................ 286
CONTENTS 14.6 General Ho Solutions 14.7 Relaxing Assumptions 14.8 H2 and Hoo Integral Control 294 14.9 Hoo Filtering 14.10Notes and references 14.11 Problems 15 Controller reduction 15.1 Hoo Controller Reductions 15.2 Notes and references 12 15.3 Problems 16 Hoo Loop Shaping 16.1 Robust Stabilization of Coprime Factors 315 16.2 Loop-Shaping Design 325 16.3 Justification for Hoo Loop sha 16.4 Further Guidelines for Loop Shaping 16.5 Notes and references 341 16.6 Problems 342 17 Gap Metric and v-Gap Metric 349 17.1 Gap Metric 350 17.2 v-Gap Metric 17.3 Geometric Interpretation of v-Gap Metric 370 17.4 Extended Loop-Shaping Design 17.5 Controller Order Reduction 375 17.6 Notes and References 17.7 Problems 18 Miscellaneous Topics 377 18.1 Model Validation 18.2 Mixed u Analysis and Synthesis 18.3 Notes and References 18.4 Problems biography 391 Index
xiv CONTENTS 14.6 General H∞ Solutions ............................ 288 14.7 Relaxing Assumptions ............................ 291 14.8 H2 and H∞ Integral Control ........................ 294 14.9 H∞ Filtering ................................. 297 14.10Notes and References ............................. 299 14.11Problems ................................... 300 15 Controller Reduction 305 15.1 H∞ Controller Reductions .......................... 306 15.2 Notes and References ............................. 312 15.3 Problems ................................... 313 16 H∞ Loop Shaping 315 16.1 Robust Stabilization of Coprime Factors .................. 315 16.2 Loop-Shaping Design ............................. 325 16.3 Justification for H∞ Loop Shaping ..................... 328 16.4 Further Guidelines for Loop Shaping .................... 334 16.5 Notes and References ............................. 341 16.6 Problems ................................... 342 17 Gap Metric and ν-Gap Metric 349 17.1 Gap Metric .................................. 350 17.2 ν-Gap Metric ................................. 357 17.3 Geometric Interpretation of ν-Gap Metric ................. 370 17.4 Extended Loop-Shaping Design ....................... 373 17.5 Controller Order Reduction ......................... 375 17.6 Notes and References ............................. 375 17.7 Problems ................................... 375 18 Miscellaneous Topics 377 18.1 Model Validation ............................... 377 18.2 Mixed µ Analysis and Synthesis ....................... 381 18.3 Notes and References ............................. 389 18.4 Problems ................................... 390 Bibliography 391 Index 407
Notation and Symbols R and fields of real and complex numbers F field. either R or C_ and C open and closed left-half plane C+ and open and closed right-half plane ∈ belong to subset end of proof end of remark ≈and态 asymptotically greater and less than 》and< much greater and less than complex conjugate of aE C absolute value of a∈C Re(a) real part of a∈C n× n identity matrix A and 4*y/ a matrix with ai; as its ith row and jth column element diag(a1, an n x n diagonal matrix with ai as its ith diagonal element transpose and complex conjugate transpose of A A- and a inverse and pseudoinverse of A shorthand for(A-) det(a) determinant of A
Notation and Symbols R and C fields of real and complex numbers F field, either R or C C− and C− open and closed left-half plane C+ and C+ open and closed right-half plane jR imaginary axis ∈ belong to ⊂ subset ∪ union ∩ intersection ✷ end of proof ✸ end of remark := defined as ' and / asymptotically greater and less than and much greater and less than α complex conjugate of α ∈ C |α| absolute value of α ∈ C Re(α) real part of α ∈ C In n × n identity matrix [aij ] a matrix with aij as its ith row and jth column element diag(a1,...,an) an n × n diagonal matrix with ai as its ith diagonal element AT and A∗ transpose and complex conjugate transpose of A A−1 and A+ inverse and pseudoinverse of A A−∗ shorthand for (A−1)∗ det(A) determinant of A trace(A) trace of A xv