Uncertain Dynamical Systems: Stability and Motion Control

✍ Scribed by A. A. Martynyuk, Yu. A. Martynyuk-Chernienko

• Publisher: CRC Press
• Year: 2012
• Tongue: English
• Leaves: 311
• Series: Pure and Applied Mathematics
• Category: Library

✦ Synopsis

This self-contained book provides systematic instructive analysis of uncertain systems of the following types: ordinary differential equations, impulsive equations, equations on time scales, singularly perturbed differential equations, and set differential equations. Each chapter contains new conditions of stability of unperturbed motion of the above-mentioned type of equations, along with some applications. Without assuming specific knowledge of uncertain dynamical systems, the book includes many fundamental facts about dynamical behaviour of its solutions. Giving a concise review of current research developments, Uncertain Dynamical Systems: Stability and Motion Control Details all proofs of stability conditions for five classes of uncertain systems Clearly defines all used notions of stability and control theory Contains an extensive bibliography, facilitating quick access to specific subject areas in each chapter Requiring only a fundamental knowledge of general theory of differential equations and calculus, this book serves as an excellent text for pure and applied mathematicians, applied physicists, industrial engineers, operations researchers, and upper-level undergraduate and graduate students studying ordinary differential equations, impulse equations, dynamic equations on time scales, and set differential equations.

✦ Table of Contents

Contents......Page 8
Preface......Page 12
Acknowledgments......Page 14
1. Introduction......Page 15
1.1 Parametric Stability......Page 17
1.2 Stability with Respect to Moving Invariant Sets......Page 19
2.1 Problem Setting and Auxiliary Results......Page 21
2.2.1 Matrix-valued Lyapunov functions......Page 25
2.2.2 Comparison functions......Page 26
2.2.3 Properties of matrix-valued functions......Page 27
2.2.4 Vector Lyapunov functions......Page 29
2.2.5 Scalar Lyapunov functions......Page 30
2.3 Theorems on Stability and Uniform Stability......Page 31
2.4 Exponential Convergence of Motions to a Moving Invariant Set......Page 47
2.5 Instability of Solutions with Respect to a Given Moving Set......Page 53
2.6 Stability with Respect to a Conditionally Invariant Moving Set......Page 58
3.1 Problem Setting......Page 69
3.2 Synthesis of Controls......Page 70
3.3 Convergence of Controlled Motions to a Moving Set......Page 76
3.4 Stabilization of Rotary Motions of a Rigid Body in an Environment with Indefinite Resistance......Page 79
3.5 Stability of an Uncertain Linear System with Neuron Control......Page 82
3.6 Conditions for Parametric Quadratic Stabilizability......Page 85
4.1 Uncertain Quasilinear System and Its Transformation......Page 93
4.2 Application of the Canonical Matrix-Valued Function......Page 95
4.3 Isolated Quasilinear Systems......Page 99
4.4 Quasilinear Systems with Nonautonomous Uncertainties......Page 102
4.5 Synchronizing of Motions in Uncertain Quasilinear Systems......Page 106
5.1 Description of a Large-Scale System......Page 113
5.2 Stability of Solutions with Respect to a Moving Set......Page 115
5.3 Application of the Hierarchical Lyapunov Function......Page 121
5.4 Stability of a Class of Time Invariant Uncertain Systems......Page 127
6.1 Conditions for the Stability of a Quasilinear System (Continued)......Page 133
6.2 Interval Stability of a Linear Mechanical System......Page 138
6.3 Parametric Stability of an Uncertain Time Invariant System......Page 142
7.1 Problem Setting......Page 155
7.2 Principle of Comparison with a Block-Diagonal Matrix Function......Page 158
7.3 Conditions for Strict Stability......Page 160
7.4 Application of the Vector Approach......Page 162
7.5 Robust Stability of Impulsive Systems......Page 165
7.6 Concluding Remarks......Page 171
8.1 Elements of the Analysis on a Time Scale......Page 173
8.2 Theorems of the Direct Lyapunov Method......Page 180
8.3 Applications and the Discussion of the Results......Page 190
9.1 Structural Uncertainties in Singularly Perturbed Systems......Page 197
9.2.1 Non-uniform time scaling......Page 200
9.2.2 Uniform time scaling......Page 208
9.3.1 Non-uniform time scaling......Page 214
9.3.2 Uniform time scaling......Page 215
9.4.1 Non-uniform time scaling......Page 216
9.4.2 Uniform time scaling......Page 222
10. Qualitative Analysis of Solutions of Set Differential Equations......Page 229
10.1 Some Results of the General Theory of Metric Spaces......Page 230
10.2 Existence of Solutions of Set Differential Equations......Page 232
10.3 The Matrix-Valued Lyapunov Function and Its Application......Page 238
10.4 Stability of a Set Stationary Solution......Page 240
10.5 Theorems on Stability......Page 242
10.6 The Application of the Strengthened Lyapunov Function......Page 247
10.7 Boundedness Theorems......Page 251
11.1 Preliminary Results......Page 255
11.2 Comparison Principle......Page 256
11.3 Estimates of Funnel for Solutions......Page 259
11.4 Test for Stability......Page 262
12.1 Auxiliary Results......Page 271
12.2 Heterogeneous Lyapunov Function......Page 272
12.3 Sufficient Stability Conditions......Page 275
12.4 Impulsive Equations with Delay under Small Perturbations......Page 277
13. Comments and References......Page 285
Appendix......Page 289
Bibliography......Page 301
M......Page 309
V......Page 1