Robust Stability and Convexity : An Introduction

By: Jacob Kogan

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Paperback | 1 January 1994

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196 Pages

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A fundamental problem in control theory is concerned with the stability of a given linear system. The design of a control system is generally based on a simplified model. The true values of the physical parameters may differ from the assumed values.
Robust Stability and Convexity addresses stability problems for linear systems with parametric uncertainty. The application of convexity techniques leads to new computationally tractable stability criteria for families of characteristic functions with nonlinear dependence on the parameters. Stability results as well as stability criteria for time-delay systems with uncertainties in coefficients and delays are reported.

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Introduction and Motivation	p. 1
Illustration	p. 1
Stability and Convexity	p. 2
Stability of Box Polynomial Families	p. 7
Kharitonov's Theorem	p. 7
Other root location regions: [Omega] stability	p. 10
Principal Vertices and Stability	p. 11
Principal Directions and Principal Vertices	p. 13
Principal Vertices and Testing Family	p. 18
Examples: Schur, Hurwitz, and Sector Stability	p. 20
Hurwitz Stability of Weighted Diamonds	p. 23
Extreme Point Results	p. 29
Stability Radii and Convex Analysis	p. 37
Stability Radii and Degree Dropping	p. 38
Constrained Optimization Problem	p. 44
Complex Stability Radius	p. 44
Stability Radius	p. 46
Examples: Stability Radii for Schur and Hurwitz Polynomials	p. 49
Real Stability Radius: Examples	p. 52
Link to Structured Matrix Perturbations	p. 55
Matrix Perturbations	p. 56
Multiple Matrix Perturbations and [mu]	p. 60
Multiaffine Polynomial Families	p. 63
Orientation Preserving Multiaffine Transformation	p. 65
Image of a Box Under Multiaffine Transformation	p. 73
Robust Stability of Multiaffine Polynomial Families	p. 75
Cascade of Uncertain Blocks	p. 76
Interval Plants with Interval Controllers	p. 84
Numerical Example	p. 93
Multidimensional Systems and Systems with Commensurate Delays	p. 99
Stability of linear systems and zeros of characteristic polynomials	p. 100
Stability of a Bivariate Polynomial	p. 102
Stability Radius	p. 103
Reduction to the Polynomial Problem	p. 104
Evaluation of Stability Radius	p. 107
Example: Hurwitz Stability of Interval Bivariate Polynomials	p. 109
Quasipolynomials with Commensurate Delays and Zero Exclusion Criterion	p. 110
Diamond Quasipolynomials: Eight Edges Theorem	p. 113
Uncertain Time-Delay Systems	p. 117
Time-Delay Systems and Zero Exclusion Criterion	p. 118
Real Interval Quasipolynomials	p. 121
Kharitonov Rectangles and Robust Stability Conditions	p. 121
Image of a Box Under Nonlinear Mapping	p. 124
Principal Segments	p. 127
Numerical Example	p. 129
Eight Edges Theorem	p. 131
Convexity of Frequency Response Arcs Associated with Hurwitz Quasipolynomials	p. 137
Convexity for Entire Functions	p. 138
Angular Derivative Formalism	p. 144
Computation of Angular Derivatives	p. 146
Inner Frequency Response Set	p. 148
Epilogue	p. 153
Lyapunov Criterion	p. 154
Simultaneous Stabilization	p. 155
Bibliography	p. 157
List of Special Symbols	p. 170
Author Index	p. 172
Subject Index	p. 175
Table of Contents provided by Blackwell. All Rights Reserved.

Robust Stability and Convexity : An Introduction

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