Global Analysis Of Piecewise Linear Dynamical Systems

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Piecewise Linear Control Systems

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 . . . . . . . . . . . . .
Nonlinear Dynamics Of Piecewise Constant Systems And Implementation Of Piecewise Constant Arguments

Piecewise constant systems exist in widely expanded areas such as engineering, physics, and mathematics. Extraordinary and complex characteristics of piecewise constant systems have been reported in recent years. This book provides the methodologies for analyzing and assessing nonlinear piecewise constant systems on a theoretically and practically sound basis. Recently developed approaches for theoretically analyzing and numerically solving the nonlinear piecewise constant dynamic systems are reviewed. A new greatest integer argument with a piecewise constant function is utilized for nonlinear dynamic analyses and for establishing a novel criterion in diagnosing irregular and chaotic solutions from the regular solutions of a nonlinear dynamic system. The newly established piecewise constantization methodology and its implementation in analytically solving for nonlinear dynamic problems are also presented.