I Introduction ............................................... 1
1 Basic Concepts ............................................... 3
1.1 Classes of hybrid and switched systems .................. 3
1.1.1 State-dependent switching ........................ 5
1.1.2 Time-dependent switching ......................... 6
1.1.3 Autonomous and controlled switching .............. 8
1.2 Solutions of switched systems ........................... 9
1.2.1 Ordinary differential equations .................. 9
1.2.2 Zeno behavior ................................... 10
1.2.3 Sliding modes ................................... 12
1.2.4 Hysteresis switching ............................ 14
II Stability of Switched Systems ............................ 17
2 Stability under Arbitrary Switching ......................... 21
2.1 Uniform stability and common Lyapunov functions ........ 21
2.1.1 Uniform stability concepts ...................... 21
2.1.2 Common Lyapunov functions ....................... 22
2.1.3 A converse Lyapunov theorem ..................... 24
2.1.4 Switched linear systems ......................... 26
2.1.5 A counterexample ................................ 28
2.2 Commutation relations and stability .................... 30
2.2.1 Commuting systems ............................... 30
2.2.2 Nilpotent and solvable Lie algebras ............. 34
2.2.3 More general Lie algebras ....................... 37
2.2.4 Discussion of Lie-algebraic stability
criteria ........................................ 41
2.3 Systems with special structure ......................... 42
2.3.1 Triangular systems .............................. 43
2.3.2 Feedback systems ................................ 45
2.3.3 Two-dimensional systems ......................... 51
3 Stability under Constrained Switching ....................... 53
3.1 Multiple Lyapunov functions ............................ 53
3.2 Stability under slow switching ......................... 56
3.2.1 Dwell time ...................................... 56
3.2.2 Average dwell time .............................. 58
3.3 Stability under state-dependent switching .............. 61
3.4 Stabilization by state-dependent switching ............. 65
3.4.1 Stable convex combinations ...................... 65
3.4.2 Unstable convex combinations .................... 68
III Switching Control ......................................... 73
4 Systems Not Stabilizable by Continuous Feedback ............. 77
4.1 Obstructions to continuous stabilization ............... 77
4.1.1 State-space obstacles ........................... 77
4.1.2 Brockett's condition ............................ 79
4.2 Nonholonomic systems ................................... 81
4.2.1 The unicycle and the nonholonomic
integrator ...................................... 83
4.3 Stabilizing an inverted pendulum ....................... 89
5 Systems with Sensor or Actuator Constraints ................. 93
5.1 The bang-bang principle of time-optimal control ........ 93
5.2 Hybrid output feedback ................................. 96
5.3 Hybrid control of systems with quantization ........... 100
5.3.1 Quantizers ..................................... 100
5.3.2 Static state quantization ...................... 103
5.3.3 Dynamic state quantization ..................... 108
5.3.4 Input quantization ............................. 116
5.3.5 Output quantization ............................ 121
5.3.6 Active probing for information ................. 124
6 Systems with Large Modeling Uncertainty .................... 129
6.1 Introductory remarks .................................. 129
6.2 First pass: basic architecture ........................ 131
6.3 An example: linear supervisory control ................ 134
6.4 Second pass: design objectives ........................ 137
6.5 Third pass: achieving the design objectives ........... 139
6.5.1 Multi-estimators ............................... 139
6.5.2 Candidate controllers .......................... 142
6.5.3 Switching logics ............................... 145
6.6 Linear supervisory control revisited .................. 154
6.6.1 Finite parametric uncertainty .................. 156
6.6.2 Infinite parametric uncertainty ................ 159
6.7 Nonlinear supervisory control ......................... 160
6.8 An example: a nonholonomic system with uncertainty .... 163
IV Supplementary Material .................................... 167
A Stability .................................................. 169
A.l Stability definitions ................................ 169
A.2 Function classes K, K∞, and KL ....................... 170
A.3 Lyapunov's direct (second) method .................... 171
A.4 LaSalle's invariance principle ....................... 173
A.5 Lyapunov's indirect (first) method ................... 174
A.6 Input-to-state stability ............................. 175
В Lie Algebras ............................................... 179
B.l Lie algebras and their representations ............... 179
B.2 Example: sl(2, ) and gl(2,) ......................... 180
B.3 Nilpotent and solvable Lie algebras .................. 181
B.4 Semisimple and compact Lie algebras .................. 181
B.5 Subalgebras isomorphic to sl(2,) .................... 182
B.6 Generators for gl(n,) ............................... 183
Notes and References .......................................... 185
Bibliography .................................................. 203
Notation ...................................................... 229
Index ......................................................... 231
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