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 . . . . . . . . . . . . .
