This book presents a computational approach to the analysis of nonlinear and uncertain systems. The main focus is systems with piecewise linear dynamics. The class of piecewise linear systems examined has nonlinear, possibly discontinuous dynamics, and allows switching rules that incorporate memory and logic. These systems may exhibit astonishingly complex behaviors. Some aspects of the successful theory of linear systems and quadratic criteria are extended here to piecewise linear systems and piecewise quadratic criteria. The book also describes numerical procedures for assessing stability, computing induced gains, and solving optimal control problems for piecewise linear systems. These developments enable researchers to analyze a large and practically important class of control systems that are not easily dealt with when using other techniques. 2. Piecewise Linear Modeling . . . . . . . . . . . . . . . . . . . . . 9 2. 1 Model Representation . . . . . . . . . . . . . . . . . . . . . 9 2. 2 Solution Concepts . . . . . . . . . . . . . . . . . . . . . . . 2. 3 Uncertainty Models . . . . . . . . . . . . . . . . . . . . . . 2. 4 Modularity and Interconnections . . . . . . . . . . . . . . 26 2. 5 Piecewise Linear Function Representations . . . . . . . . . 28 2. 6 Comments and References . . . . . . . . . . . . . . . . . . 30 3. Structural Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 32 3. 1 Equilibrium Points and the Steady State Characteristic . . 32 3. 2 Constraint Verification and Invariance . . . . . . . . . . . 35 3. 3 Detecting Attractive Sliding Modes on Cell Boundaries 37 3. 4 Comments and References . . . . . . . . . . . . . . . . . . 39 4. Lyapunov Stability . . . . . . . . . . . . . . . . . . . . . . . . . . 41 4. 1 Exponential Stability . . . . . . . . . . . . . . . . . . . . . . 41 4. 2 Quadratic Stability . . . . . . . . . . . . . . . . . . . . . . . 42 4. 3 Conservatism of Quadratic Stability . . . . . . . . . . . . . 46 4. 4 From Quadratic to Piecewise Quadratic . . . . . . . . . . . 48 4. 5 Interlude: Describing Partition Properties . . . . . . . . . 51 4. 6 Piecewise Quadratic Lyapunov Functions . . . . . . . . . 55 4. 7 Analysis of Piecewise Linear Differential Inclusions . . . . 61 4. 8 Analysis of Systems with Attractive Sliding Modes . . . . 63 4. 9 Improving Computational Efficiency . . . . . . . . . . . . 66 4. 10 Piecewise Linear Lyapunov Functions . . . . . . . . . . . 72 4. 11 A Unifying View . . . . . . . . . . . . . . . . . . . . . . . . 77 4. 12 Comments and References . . . . . . . . . . . . . . . . . . 82 5. Dissipativity Analysis . . . . . . . . . . . . . . . . . . . . . . . . 85 5. 1 Dissipativity Analysis via Convex Optimization . . . . . . 86 21 14 Contents Contents 5. 2 Computation of £2 induced Gain . . . . . . . . . . . . . . 88 5. 3 Estimation of Transient Energy . . . . . . . . . . . . . . . . 89 5. 4 Dissipative Systems with Quadratic Supply Rates . . . . . 91 5. 5 Comments and References . . . . . . . . . . . . . . . . . . 95 Controller Design . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 6. 1 Quadratic Stabilization of Piecewise Linear" Systems . . . 97 6. 2 Controller Synthesis based on Piecewise Quadratics . . . 98 6. 3 Comments and References . . . . . . . . . . . . . . . . . . 105 7. Selected Topics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 7. 1 Estimation of Regions of Attraction . . . . . . . . . . . . Lecture Notesin Control and Information Sciences......Page 2 Piecewise LinearControl Systems......Page 4 Contents......Page 9 1 Introduction......Page 11 Piecewise Linear Systems......Page 12 Computational Analysis of Dynamical Systems......Page 13 Philosophy of this Book......Page 16 About This Manuscript......Page 17 Model Representation......Page 19 Solution Concepts......Page 24 Uncertainty Models......Page 31 Modularity and Interconnections......Page 36 Piecewise Linear Function Representations......Page 38 Comments and References......Page 40 Equilibrium Points and the Steady-State Characteristic......Page 42 Constraint Verification and Invariance......Page 45 Detecting Attractive Sliding Modes on Cell Boundaries......Page 47 Comments and References......Page 49 Exponential Stability......Page 51 Quadratic Stability......Page 52 Conservatism of Quadratic Stability......Page 56 From Quadratic to Piecewise Quadratic......Page 58 Interlude: Describing Partition Properties......Page 61 Piecewise Quadratic Lyapunov Functions......Page 65 Analysis of Piecewise Linear Differential Inclusions......Page 71 Analysis of Systems with Attractive Sliding Modes......Page 73 Improving Computational Efficiency......Page 76 Piecewise Linear Lyapunov Functions......Page 82 A Unifying View......Page 87 Comments and References......Page 92 5. Dissipativity Analysis......Page 95 Dissipativity Analysis via Convex Optimization......Page 96 Computation of L2-induced Gain......Page 98 Estimation of Transient Energy......Page 100 Dissipative Systems with Quadratic Supply Rates......Page 101 Comments and References......Page 105 6. Controller Design......Page 106 Quadratic Stabilization of Piecewise Linear Systems......Page 107 Controller Synthesis based on Piecewise Quadratics......Page 108 Comments and References......Page 115 7. Selected Topics......Page 117 Estimation of Regions of Attraction......Page 118 Rigorous Analysis of Smooth Nonlinear Systems......Page 125 Automated Partition Refinements......Page 127 Fuzzy Logic Systems......Page 130 Comments and References......Page 136 8. Linear Hybrid Dynamical Systems......Page 139 Linear Hybrid Dynamical Systems......Page 140 Analysis Using Discontinuous Lyapunov Functions......Page 142 Stability Analysis Using Lyapunov Functionals......Page 145 Comments and References......Page 149 A. Computational Issues......Page 152 C Bibliography......Page 193
