Professor Kang-Zhi Liu, Dept. of Electrical and Electronic Engineering, Chiba University, Japan. Professor Liu achieved his Ph.D. degree in 1991 from Chiba University, Japan.  His areas of expertise include Control Theory, Control and Operation of Power Systems, and System Integration of Smart-Grid, and he has worked in these related areas for 27 years (4 years as a professor, 13 years as an associate professor, 5 years as an assistant professor, and 5 years as a graduate student). He is currently Associate Editor of both the International Journal of Control Theory and Applications, and the International Journal of Systems Science.  He is the author of 6 books (two in Chinese and four in Japanese).Dr. Yu Yao is a Cheng Kong Scholar Chair Professor at the Harbin Institute of Technology, China. He also serves as Vice President of Harbin University of Engineering, China. His research interests include nonlinear systems, robust control and flight control. He has published over 100 journal papers.

• Preface xviiList of Abbreviations xixNotations xxi1 Introduction 11.1 Engineering Background of Robust Control 11.2 Methodologies of Robust Control 41.3 A Brief History of Robust Control 82 Basics of Linear Algebra and Function Analysis 102.1 Trace, Determinant, Matrix Inverse, and Block Matrix 102.2 Elementary Linear Transformation of Matrix and Its Matrix Description 122.3 Linear Vector Space 142.4 Norm and Inner Product of Vector 182.5 Linear Subspace 222.6 Matrix and Linear Mapping 232.7 Eigenvalue and Eigenvector 282.8 Invariant Subspace 302.9 Pseudo-Inverse and Linear Matrix Equation 342.10 Quadratic Form and Positive Definite Matrix 352.11 Norm and Inner Product of Matrix 372.12 Singular Value and Singular Value Decomposition 402.13 Calculus of Vector and Matrix 432.14 Kronecker Product 442.15 Norm and Inner Product of Function 45Exercises 53Notes and References 563 Basics of Convex Analysis and LMI 573.1 Convex Set and Convex Function 573.2 Introduction to LMI 723.3 Interior Point Method 81Exercises 83Notes and References 844 Fundamentals of Linear System 854.1 Structural Properties of Dynamic System 854.2 Stability 1004.3 Lyapunov Equation 1084.4 Linear Fractional Transformation 114Exercises 117Notes and References 1185 System Performance 1195.1 Test Signal 1205.2 Steady-State Response 1225.3 Transient Response 1305.4 Comparison of Open-Loop and Closed-Loop Controls 140Exercises 146Notes and References 1476 Stabilization of Linear Systems 1486.1 State Feedback 1486.2 Observer 1606.3 Combined System and Separation Principle 167Exercises 170Notes and References 1727 Parametrization of Stabilizing Controllers 1737.1 Generalized Feedback Control System 1747.2 Parametrization of Controllers 1787.3 Youla Parametrization 1847.4 Structure of Closed-Loop System 1867.5 2-Degree-of-Freedom System 188Exercises 193Notes and References 1968 Relation between Time Domain and Frequency Domain Properties 1978.1 Parseval’s Theorem 1978.2 KYP Lemma 200Exercises 214Notes and References 2149 Algebraic Riccati Equation 2159.1 Algorithm for Riccati Equation 2159.2 Stabilizing Solution 2189.3 Inner Function 223Exercises 224Notes and References 22410 Performance Limitation of Feedback Control 22510.1 Preliminaries 22610.2 Limitation on Achievable Closed-loop Transfer Function 22810.3 Integral Relation 23110.4 Limitation of Reference Tracking 237Exercises 244Notes and References 24411 Model Uncertainty 24511.1 Model Uncertainty: Examples 24511.2 Plant Set with Dynamic Uncertainty 24811.3 Parametric System 25311.4 Plant Set with Phase Information of Uncertainty 26411.5 LPV Model and Nonlinear Systems 26611.6 Robust Stability and Robust Performance 269Exercises 270Notes and References 27112 Robustness Analysis 1: Small-Gain Principle 27212.1 Small-Gain Theorem 27212.2 Robust Stability Criteria 27612.3 Equivalence between H∞ Performance and Robust Stability 27712.4 Analysis of Robust Performance 27912.5 Stability Radius of Norm-Bounded Parametric Systems 282Exercises 283Notes and References 28713 Robustness Analysis 2: Lyapunov Method 28813.1 Overview of Lyapunov Stability Theory 28813.2 Quadratic Stability 29013.3 Lur’e System 29613.4 Passive Systems 307Exercises 310Notes and References 31114 Robustness Analysis 3: IQC Approach 31214.1 Concept of IQC 31214.2 IQC Theorem 31414.3 Applications of IQC 31614.4 Proof of IQC Theorem* 319Notes and References 32115 H2 Control 32215.1 H2 Norm of Transfer Function 32215.2 H2 Control Problem 32915.3 Solution to Nonsingular H2 Control Problem 33115.4 Proof of Nonsingular Solution 33215.5 Singular H2 Control 33515.6 Case Study: H2 Control of an RTP System 337Exercises 342Notes and References 34516 H∞ Control 34616.1 Control Problem and H∞ Norm 34616.2 H∞ Control Problem 34816.3 LMI Solution 1: Variable Elimination 34916.4 LMI Solution 2: Variable Change 35116.5 Design of Generalized Plant and Weighting Function 35216.6 Case Study 35416.7 Scaled H∞ Control 355Exercises 358Notes and References 35917 μ Synthesis 36017.1 Introduction to μ 36017.2 Definition of μ and Its Implication 36417.3 Properties of μ 36517.4 Condition for Robust H∞ Performance 36817.5 D–K Iteration Design 36917.6 Case Study 371Exercises 373Notes and References 37418 Robust Control of Parametric Systems 37518.1 Quadratic Stabilization of Polytopic Systems 37518.2 Quadratic Stabilization of Norm-Bounded Parametric Systems 37918.3 Robust H∞ Control Design of Polytopic Systems 37918.4 Robust H∞ Control Design of Norm-Bounded Parametric Systems 382Exercises 38219 Regional Pole Placement 38419.1 Convex Region and Its Characterization 38419.2 Condition for Regional Pole Placement 38719.3 Composite LMI Region 39219.4 Feedback Controller Design 39419.5 Analysis of Robust Pole Placement 39619.6 Robust Design of Regional Pole Placement 402Exercises 405Notes and References 40620 Gain-Scheduled Control 40720.1 General Structure 40720.2 LFT-Type Parametric Model 40820.3 Case Study: Stabilization of a Unicycle Robot 41420.4 Affine LPV Model 42220.5 Case Study: Transient Stabilization of a Power System 428Exercises 434Notes and References 43521 Positive Real Method 43621.1 Structure of Uncertain Closed-Loop System 43621.2 Robust Stabilization Based on Strongly Positive Realness 43821.3 Robust Stabilization Based on Strictly Positive Realness 44121.4 Robust Performance Design for Systems with Positive Real Uncertainty 44221.5 Case Study 445Exercises 448Notes and References 449References 450Index 455

Professor Kang-Zhi Liu, Dept. of Electrical and Electronic Engineering, Chiba University, Japan. Professor Liu achieved his Ph.D. degree in 1991 from Chiba University, Japan.  His areas of expertise include Control Theory, Control and Operation of Power Systems, and System Integration of Smart-Grid, and he has worked in these related areas for 27 years (4 years as a professor, 13 years as an associate professor, 5 years as an assistant professor, and 5 years as a graduate student). He is currently Associate Editor of both the International Journal of Control Theory and Applications, and the International Journal of Systems Science.  He is the author of 6 books (two in Chinese and four in Japanese).Dr. Yu Yao is a Cheng Kong Scholar Chair Professor at the Harbin Institute of Technology, China. He also serves as Vice President of Harbin University of Engineering, China. His research interests include nonlinear systems, robust control and flight control. He has published over 100 journal papers.

• Preface xviiList of Abbreviations xixNotations xxi1 Introduction 11.1 Engineering Background of Robust Control 11.2 Methodologies of Robust Control 41.3 A Brief History of Robust Control 82 Basics of Linear Algebra and Function Analysis 102.1 Trace, Determinant, Matrix Inverse, and Block Matrix 102.2 Elementary Linear Transformation of Matrix and Its Matrix Description 122.3 Linear Vector Space 142.4 Norm and Inner Product of Vector 182.5 Linear Subspace 222.6 Matrix and Linear Mapping 232.7 Eigenvalue and Eigenvector 282.8 Invariant Subspace 302.9 Pseudo-Inverse and Linear Matrix Equation 342.10 Quadratic Form and Positive Definite Matrix 352.11 Norm and Inner Product of Matrix 372.12 Singular Value and Singular Value Decomposition 402.13 Calculus of Vector and Matrix 432.14 Kronecker Product 442.15 Norm and Inner Product of Function 45Exercises 53Notes and References 563 Basics of Convex Analysis and LMI 573.1 Convex Set and Convex Function 573.2 Introduction to LMI 723.3 Interior Point Method 81Exercises 83Notes and References 844 Fundamentals of Linear System 854.1 Structural Properties of Dynamic System 854.2 Stability 1004.3 Lyapunov Equation 1084.4 Linear Fractional Transformation 114Exercises 117Notes and References 1185 System Performance 1195.1 Test Signal 1205.2 Steady-State Response 1225.3 Transient Response 1305.4 Comparison of Open-Loop and Closed-Loop Controls 140Exercises 146Notes and References 1476 Stabilization of Linear Systems 1486.1 State Feedback 1486.2 Observer 1606.3 Combined System and Separation Principle 167Exercises 170Notes and References 1727 Parametrization of Stabilizing Controllers 1737.1 Generalized Feedback Control System 1747.2 Parametrization of Controllers 1787.3 Youla Parametrization 1847.4 Structure of Closed-Loop System 1867.5 2-Degree-of-Freedom System 188Exercises 193Notes and References 1968 Relation between Time Domain and Frequency Domain Properties 1978.1 Parseval’s Theorem 1978.2 KYP Lemma 200Exercises 214Notes and References 2149 Algebraic Riccati Equation 2159.1 Algorithm for Riccati Equation 2159.2 Stabilizing Solution 2189.3 Inner Function 223Exercises 224Notes and References 22410 Performance Limitation of Feedback Control 22510.1 Preliminaries 22610.2 Limitation on Achievable Closed-loop Transfer Function 22810.3 Integral Relation 23110.4 Limitation of Reference Tracking 237Exercises 244Notes and References 24411 Model Uncertainty 24511.1 Model Uncertainty: Examples 24511.2 Plant Set with Dynamic Uncertainty 24811.3 Parametric System 25311.4 Plant Set with Phase Information of Uncertainty 26411.5 LPV Model and Nonlinear Systems 26611.6 Robust Stability and Robust Performance 269Exercises 270Notes and References 27112 Robustness Analysis 1: Small-Gain Principle 27212.1 Small-Gain Theorem 27212.2 Robust Stability Criteria 27612.3 Equivalence between H∞ Performance and Robust Stability 27712.4 Analysis of Robust Performance 27912.5 Stability Radius of Norm-Bounded Parametric Systems 282Exercises 283Notes and References 28713 Robustness Analysis 2: Lyapunov Method 28813.1 Overview of Lyapunov Stability Theory 28813.2 Quadratic Stability 29013.3 Lur’e System 29613.4 Passive Systems 307Exercises 310Notes and References 31114 Robustness Analysis 3: IQC Approach 31214.1 Concept of IQC 31214.2 IQC Theorem 31414.3 Applications of IQC 31614.4 Proof of IQC Theorem* 319Notes and References 32115 H2 Control 32215.1 H2 Norm of Transfer Function 32215.2 H2 Control Problem 32915.3 Solution to Nonsingular H2 Control Problem 33115.4 Proof of Nonsingular Solution 33215.5 Singular H2 Control 33515.6 Case Study: H2 Control of an RTP System 337Exercises 342Notes and References 34516 H∞ Control 34616.1 Control Problem and H∞ Norm 34616.2 H∞ Control Problem 34816.3 LMI Solution 1: Variable Elimination 34916.4 LMI Solution 2: Variable Change 35116.5 Design of Generalized Plant and Weighting Function 35216.6 Case Study 35416.7 Scaled H∞ Control 355Exercises 358Notes and References 35917 μ Synthesis 36017.1 Introduction to μ 36017.2 Definition of μ and Its Implication 36417.3 Properties of μ 36517.4 Condition for Robust H∞ Performance 36817.5 D–K Iteration Design 36917.6 Case Study 371Exercises 373Notes and References 37418 Robust Control of Parametric Systems 37518.1 Quadratic Stabilization of Polytopic Systems 37518.2 Quadratic Stabilization of Norm-Bounded Parametric Systems 37918.3 Robust H∞ Control Design of Polytopic Systems 37918.4 Robust H∞ Control Design of Norm-Bounded Parametric Systems 382Exercises 38219 Regional Pole Placement 38419.1 Convex Region and Its Characterization 38419.2 Condition for Regional Pole Placement 38719.3 Composite LMI Region 39219.4 Feedback Controller Design 39419.5 Analysis of Robust Pole Placement 39619.6 Robust Design of Regional Pole Placement 402Exercises 405Notes and References 40620 Gain-Scheduled Control 40720.1 General Structure 40720.2 LFT-Type Parametric Model 40820.3 Case Study: Stabilization of a Unicycle Robot 41420.4 Affine LPV Model 42220.5 Case Study: Transient Stabilization of a Power System 428Exercises 434Notes and References 43521 Positive Real Method 43621.1 Structure of Uncertain Closed-Loop System 43621.2 Robust Stabilization Based on Strongly Positive Realness 43821.3 Robust Stabilization Based on Strictly Positive Realness 44121.4 Robust Performance Design for Systems with Positive Real Uncertainty 44221.5 Case Study 445Exercises 448Notes and References 449References 450Index 455

Professor Kang-Zhi Liu, Dept. of Electrical and Electronic Engineering, Chiba University, Japan. Professor Liu achieved his Ph.D. degree in 1991 from Chiba University, Japan.  His areas of expertise include Control Theory, Control and Operation of Power Systems, and System Integration of Smart-Grid, and he has worked in these related areas for 27 years (4 years as a professor, 13 years as an associate professor, 5 years as an assistant professor, and 5 years as a graduate student). He is currently Associate Editor of both the International Journal of Control Theory and Applications, and the International Journal of Systems Science.  He is the author of 6 books (two in Chinese and four in Japanese).Dr. Yu Yao is a Cheng Kong Scholar Chair Professor at the Harbin Institute of Technology, China. He also serves as Vice President of Harbin University of Engineering, China. His research interests include nonlinear systems, robust control and flight control. He has published over 100 journal papers.

• Preface xviiList of Abbreviations xixNotations xxi1 Introduction 11.1 Engineering Background of Robust Control 11.2 Methodologies of Robust Control 41.3 A Brief History of Robust Control 82 Basics of Linear Algebra and Function Analysis 102.1 Trace, Determinant, Matrix Inverse, and Block Matrix 102.2 Elementary Linear Transformation of Matrix and Its Matrix Description 122.3 Linear Vector Space 142.4 Norm and Inner Product of Vector 182.5 Linear Subspace 222.6 Matrix and Linear Mapping 232.7 Eigenvalue and Eigenvector 282.8 Invariant Subspace 302.9 Pseudo-Inverse and Linear Matrix Equation 342.10 Quadratic Form and Positive Definite Matrix 352.11 Norm and Inner Product of Matrix 372.12 Singular Value and Singular Value Decomposition 402.13 Calculus of Vector and Matrix 432.14 Kronecker Product 442.15 Norm and Inner Product of Function 45Exercises 53Notes and References 563 Basics of Convex Analysis and LMI 573.1 Convex Set and Convex Function 573.2 Introduction to LMI 723.3 Interior Point Method 81Exercises 83Notes and References 844 Fundamentals of Linear System 854.1 Structural Properties of Dynamic System 854.2 Stability 1004.3 Lyapunov Equation 1084.4 Linear Fractional Transformation 114Exercises 117Notes and References 1185 System Performance 1195.1 Test Signal 1205.2 Steady-State Response 1225.3 Transient Response 1305.4 Comparison of Open-Loop and Closed-Loop Controls 140Exercises 146Notes and References 1476 Stabilization of Linear Systems 1486.1 State Feedback 1486.2 Observer 1606.3 Combined System and Separation Principle 167Exercises 170Notes and References 1727 Parametrization of Stabilizing Controllers 1737.1 Generalized Feedback Control System 1747.2 Parametrization of Controllers 1787.3 Youla Parametrization 1847.4 Structure of Closed-Loop System 1867.5 2-Degree-of-Freedom System 188Exercises 193Notes and References 1968 Relation between Time Domain and Frequency Domain Properties 1978.1 Parseval’s Theorem 1978.2 KYP Lemma 200Exercises 214Notes and References 2149 Algebraic Riccati Equation 2159.1 Algorithm for Riccati Equation 2159.2 Stabilizing Solution 2189.3 Inner Function 223Exercises 224Notes and References 22410 Performance Limitation of Feedback Control 22510.1 Preliminaries 22610.2 Limitation on Achievable Closed-loop Transfer Function 22810.3 Integral Relation 23110.4 Limitation of Reference Tracking 237Exercises 244Notes and References 24411 Model Uncertainty 24511.1 Model Uncertainty: Examples 24511.2 Plant Set with Dynamic Uncertainty 24811.3 Parametric System 25311.4 Plant Set with Phase Information of Uncertainty 26411.5 LPV Model and Nonlinear Systems 26611.6 Robust Stability and Robust Performance 269Exercises 270Notes and References 27112 Robustness Analysis 1: Small-Gain Principle 27212.1 Small-Gain Theorem 27212.2 Robust Stability Criteria 27612.3 Equivalence between H∞ Performance and Robust Stability 27712.4 Analysis of Robust Performance 27912.5 Stability Radius of Norm-Bounded Parametric Systems 282Exercises 283Notes and References 28713 Robustness Analysis 2: Lyapunov Method 28813.1 Overview of Lyapunov Stability Theory 28813.2 Quadratic Stability 29013.3 Lur’e System 29613.4 Passive Systems 307Exercises 310Notes and References 31114 Robustness Analysis 3: IQC Approach 31214.1 Concept of IQC 31214.2 IQC Theorem 31414.3 Applications of IQC 31614.4 Proof of IQC Theorem* 319Notes and References 32115 H2 Control 32215.1 H2 Norm of Transfer Function 32215.2 H2 Control Problem 32915.3 Solution to Nonsingular H2 Control Problem 33115.4 Proof of Nonsingular Solution 33215.5 Singular H2 Control 33515.6 Case Study: H2 Control of an RTP System 337Exercises 342Notes and References 34516 H∞ Control 34616.1 Control Problem and H∞ Norm 34616.2 H∞ Control Problem 34816.3 LMI Solution 1: Variable Elimination 34916.4 LMI Solution 2: Variable Change 35116.5 Design of Generalized Plant and Weighting Function 35216.6 Case Study 35416.7 Scaled H∞ Control 355Exercises 358Notes and References 35917 μ Synthesis 36017.1 Introduction to μ 36017.2 Definition of μ and Its Implication 36417.3 Properties of μ 36517.4 Condition for Robust H∞ Performance 36817.5 D–K Iteration Design 36917.6 Case Study 371Exercises 373Notes and References 37418 Robust Control of Parametric Systems 37518.1 Quadratic Stabilization of Polytopic Systems 37518.2 Quadratic Stabilization of Norm-Bounded Parametric Systems 37918.3 Robust H∞ Control Design of Polytopic Systems 37918.4 Robust H∞ Control Design of Norm-Bounded Parametric Systems 382Exercises 38219 Regional Pole Placement 38419.1 Convex Region and Its Characterization 38419.2 Condition for Regional Pole Placement 38719.3 Composite LMI Region 39219.4 Feedback Controller Design 39419.5 Analysis of Robust Pole Placement 39619.6 Robust Design of Regional Pole Placement 402Exercises 405Notes and References 40620 Gain-Scheduled Control 40720.1 General Structure 40720.2 LFT-Type Parametric Model 40820.3 Case Study: Stabilization of a Unicycle Robot 41420.4 Affine LPV Model 42220.5 Case Study: Transient Stabilization of a Power System 428Exercises 434Notes and References 43521 Positive Real Method 43621.1 Structure of Uncertain Closed-Loop System 43621.2 Robust Stabilization Based on Strongly Positive Realness 43821.3 Robust Stabilization Based on Strictly Positive Realness 44121.4 Robust Performance Design for Systems with Positive Real Uncertainty 44221.5 Case Study 445Exercises 448Notes and References 449References 450Index 455