Control Systems. LMIs in. Guang-Ren Duan. Analysis, Design and Applications. Hai-Hua Yu. CRC Press. Taylor & Francis Croup

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1 LMIs in Control Systems Analysis, Design and Applications Guang-Ren Duan Hai-Hua Yu CRC Press Taylor & Francis Croup Boca Raton London New York CRC Press is an imprint of the Taylor & Francis Croup, an Informa business

2 Contents List of Figures List of Tables Preface Acknowledgments Authors List of Notations xv xvii xix xxiii xxv xxvii 1 Introduction What Are Linear Matrix Inequalities (LMIs)? General Form Standard Form Manipulations A Few Examples Involving LMIs Eigenvalue Minimization Matrix Norm Minimization Key Step in /x-analysis Schur Stabilization Brief History Seed Planted (1890) Rooting Period ( ) Growing Period ( ) PR Lemma and ARE Convex Optimization and Computational Algorithms LMIs in Control Systems Flourishing Period (2000-Present) General Search Record Record Related to Control Advantages Global Optimal Solution and Numerical Reliability Multiple Objective Design Mature Software Packages 22 vii

3 viii Contents 1.5 About the Book Structure Features Using the Book in Courses 25 Exercises 26 PART I PRELIMINARIES 2 Technical Lemmas Generalized Square Inequalities Restriction-Free Inequality Inequalities with Restriction Variable Elimination Lemma Schur Complement Lemma Schur Complements Matrix Inversion Lemma Schur Complement Lemma Elimination of Variables Variable Elimination in a Partitioned Matrix Projection Lemma Reciprocal Projection Lemma Some Other Useful Results Trace of an LMI Maximum Modulus Principle Parseval Lemma Notes and References 51 Exercises 52 3 Review of Optimization Theory Convex Sets Definitions and Properties Hyperplanes, Halfspaces, Polyhedrons, and Polytopes Convex Functions Definitions and Properties Criteria Mathematical Optimization Least-Squares Programming Linear Programming Quadratic Programming Convex Optimization Problem Local and Global Optima LMI Problems 81

4 Contents ix Convexity Extreme Result Standard Problems Feasibility Problem Convex Minimization Problem Generalized Eigenvalue Problem Notes and References About This Chapter Open Source Software CVX Counterexample for Numerical Reliability Numerical Examples in the Book 89 Exercises 91 PART II CONTROL SYSTEMS ANALYSIS 4 Stability Analysis Hurwitz and Schur Stability Hurwitz Stability Schur Stability Stability Special Cases General LMI Regions Generalized Lyapunov Theorem Quadratic Stability Family of Systems Ill Quadratic Hurwitz Stability Quadratic Schur Stability Quadratic ID-Stability Definition and Main Results Some Special Cases Time-Delay Systems Delay-Independent Condition Delay-Dependent Condition Notes and References Summary and References AfHne Quadratic Stability 132 Exercises Hoo/Hb Performance Hoo andh2 Indices Hoo Index H2 Index Hi Equivalent Definitions

5 x Contents Hoc Index H2 Index LMI Conditions for Hoc Index Basic Conditions Deduced Conditions LMI Conditions for H2 Index Basic Conditions Deduced Conditions Notes and References 161 Exercises Property Analysis 6.1 Hurwitz Stabilizability and Detectability Hurwitz Stabilizability Definition and the PBH Criterion LMI Conditions Hurwitz Detectability Schur Stabilizability and Detectability Schur Stabilizability Definition and the PBH Criterion LMI Conditions Schur Detectability Dissipativity Definition Equivalent Conditions Frequency Domain Condition Matrix Inequality Condition Passivity and Positive-Realness Definitions Positive-Real Lemma LMI Condition Nonexpansivity and Bounded-Realness Definitions Bounded-Real Lemma LMI Conditions Notes and References 191 Exercises PART 111 CONTROL SYSTEMS DESIGN 7 Feedback Stabilization State Feedback Stabilization 199

6 Contents xi Case of Continuous-Time Systems Case of Discrete-Time Systems Stabilization H(,^-Stabilization D(?,r)-Stabilization General D-Stabilization Quadratic Stabilization Family of Systems Quadratic Hurwitz Stabilization Problem Formulation Case of Interval Systems Case of Polytopic Systems Quadratic Schur Stabilization Problem Formulation Case of Interval Systems Case of Polytopic Systems Quadratic D-Stabilization Problem Formulation Solution Special Cases Case of Interval Systems Case of Polytopic Systems Insensitive Region Design Sensitivity of Matrix Eigenvalues Insensitive Strip Region Design Problem Solution Insensitive Disk Region Design Problem Solution Robust Stabilization of Second-Order Systems Stabilization Problem Formulation Solution Robust Stabilization Problem Formulation Solution Stabilization of Time-Delay Systems Case of Delay Independence Case of Delay Dependence Notes and References 244 Exercises 246

7 xii Contents 8 Hoo/H2 Control Hoo State Feedback Control Problem Solution Other Conditions H2 State Feedback Control Problem Solution Other Condition Robust Hoo/H2 State Feedback Control Problems Uncertain System Forms of Perturbation Problem Formulation Solution to the Robust Hoo Control Problem Solution to the Robust H2 Control Problem LQ Regulation via H2 Control Problem Description Relation to H2 Performance Solution Notes and References Summary and References Dissipative, Passive, and Nonexpansive Control 281 Exercises State Observation and Filtering Full- and Reduced-Order State Observers Full-Order State Observers Reduced-Order State Observer Design Problem Formulation Solution Full-Order Hoq/H2 State Observers Problem Formulation Solutions to Problems Hoo State Observer Design H2 State Observer Design Examples Hoo Filtering Problem Formulation Solution to Hoo Filtering H2 Filtering Problem Formulation Solution to H2 Filtering 309

8 Contents xiii 9.5 Notes and References 315 Exercises Multiple Objective Designs Insensitive Region Designs with Minimum Gain Insensitive Strip Region Designs with Minimum Gain 322 with Minimum Gain Insensitive Disk Region Designs 10.2 Mixed Hoo/H2 Designs with Desired Pole Regions Problem Solution to the Problem Condition for the Hqo Performance Condition for the H2 Performance Condition for the Closed-Loop Pole Location Solution Mixed Robust H00/H2 Designs with Desired Pole Regions Problem Solution to the Problem Condition for the Robust Hoo Performance Condition for the Robust H2 Performance Condition for the Robust Closed-Loop Pole Location Solution Notes and References Summary of Main Results Further Remarks 342 Exercises 343 PART IV APPLICATIONS 11 Missile Attitude Control Dynamical Model Models for Nonrotating Missiles Pitch Channel Model Yaw/Roll Channel Model Models for BTT Missiles Rolling Channel Model Pitch/Yaw Channel Model Attitude Control of Nonrotating Missiles Problem Solution Optimal Solution Feasible Solution Attitude Control of BTT Missiles 359

9 xiv Contents Problem Trajectory and Operating Points Rolling Rate cox Problem Quadratic Stabilization Family of Systems Quadratic Stabilization Numerical Results and Simulation Notes and References 369 Exercises Satellite Control System Modeling Second-Order System Form State-Space Form H2 and Hoo Feedback Control Hoo Control Problem and Solution Numerical Results H2 Control Problem and Solution Numerical Results Mixed H2/Hoo Feedback Control Problem and Solution Pole Region Assignment Disturbance Attenuation Numerical and Simulation Results Controller Numerical Simulation Notes and References Optimality versus Feasibility Problem of Space Rendezvous 387 Exercises 389 PART V APPENDICES Appendix A: Proofs oftheorems 393 Appendix B: Using the MATLAB LMI Toolbox 417 References 433 Index 451

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