List of Figures .............................................. xiii
1 Modeling of hybrid systems ................................... 1
1.1 Introduction ............................................ 1
1.2 Towards a definition of hybrid systems .................. 3
1.2.1 Continuous and symbolic dynamics ................. 3
1.2.2 Hybrid automaton ................................. 6
1.2.3 Features of hybrid dynamics ...................... 9
1.2.4 Generalized hybrid automaton .................... 12
1.2.5 Hybrid time evolutions and hybrid behavior ...... 14
1.2.6 Event-flow formulas ............................. 18
1.2.7 Simulation of hybrid systems .................... 25
1.2.8 Representations of hybrid systems ............... 31
1.3 Notes and References for Chapter 1 ..................... 34
2 Examples of hybrid dynamical systems ........................ 35
2.1 Introduction ........................................... 35
2.2 Examples ............................................... 35
2.2.1 Hysteresis ...................................... 35
2.2.2 Manual transmission ............................. 37
2.2.3 Bouncing ball ................................... 37
2.2.4 Temperature control system ...................... 38
2.2.5 Water-level monitor ............................. 40
2.2.6 Multiple collisions ............................. 42
2.2.7 Variable-structure system ....................... 44
2.2.8 Supervisor model ................................ 46
2.2.9 Two carts ....................................... 47
2.2.10 Coulomb friction ................................ 49
2.2.11 Systems with piecewise linear elements .......... 50
2.2.12 Railroad crossing ............................... 51
2.2.13 Power converter ................................. 52
2.2.14 Constrained pendulum ............................ 54
2.2.15 Degenerate Van der Pol oscillator ............... 55
2.3 Notes and References for Chapter 2 ..................... 56
3 Variable-structure systems .................................. 57
3.1 Discontinuous dynamical systems ........................ 57
3.2 Solution concepts ...................................... 59
3.3 Reformulations ......................................... 62
3.4 Systems with many regimes .............................. 63
3.5 The Vidale-Wolfe advertizing model ..................... 65
3.6 Notes and references for Chapter 3 ..................... 67
4 Complementarity systems ..................................... 71
4.1 Examples ............................................... 73
4.1.1 Circuits with ideal diodes ...................... 73
4.1.2 Mechanical systems with unilateral
constraints ..................................... 74
4.1.3 Optimal control with state constraints .......... 75
4.1.4 Variable-structure systems ...................... 77
4.1.5 A class of piecewise linear systems ............. 79
4.1.6 Projected dynamical systems ..................... 81
4.1.7 Diffusion with a free boundary .................. 84
4.1.8 Max-plus systems ................................ 89
4.2 Existence and uniqueness of solutions .................. 90
4.3 The mode selection problem ............................. 91
4.4 Linear complementarity systems ......................... 96
4.4.1 Specification ................................... 96
4.4.2 A distributional interpretation ................ 100
4.4.3 Well-posedness ................................. 102
4.5 Mechanical complementarity systems .................... 104
4.6 Relay systems ......................................... 109
4.7 Notes and references for Chapter 4 .................... 110
5 Analysis of hybrid systems ................................. 1ll
5.1 Correctness and reachability .......................... 11l
5.1.1 Formal verification ............................ 11l
5.1.2 An audio protocol .............................. 113
5.1.3 Algorithms for verification .................... 117
5.2 Stability ............................................. 118
5.2.1 Lyapunov functions and Poincaré mappings ....... 118
5.2.2 Time-controlled switching ...................... 122
5.2.3 State-controlled switching ..................... 124
5.3 Chaotic phenomena ..................................... 129
5.4 Notes and references for Chapter 5 .................... 132
6 Hybrid control design ...................................... 133
6.1 Safety and guarantee properties ....................... 134
6.1.1 Safety and controlled invariance ............... 135
6.1.2 Safety and dynamic game theory ................. 138
6.2 Switching control ..................................... 140
6.2.1 Switching logic ................................ 140
6.2.2 PWM control .................................... 143
6.2.3 Sliding mode control ........................... 144
6.2.4 Quadratic stabilization by switching control ... 146
6.3 Hybrid feedback stabilization ......................... 147
6.3.1 Energy decrease by hybrid feedback ............. 148
6.3.2 Stabilization of nonholonomic systems .......... 151
6.3.3 Set-point regulation of mechanical systems
by energy injection ............................ 156
6.4 Notes and References for Chapter 6 .................... 158
Bibliography .................................................. 159
Index ......................................................... 171
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