Stability and Stable Oscillations in Discrete Time Systems

By: Aristide Halanay, Vladimir Rasvan

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Hardcover | 31 October 2000 | Edition Number 1

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300 Pages

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22.86 x 15.24 x 1.75

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The expertise of a professional mathmatician and a theoretical engineer provides a fresh perspective of stability and stable oscillations. The current state of affairs in stability theory, absolute stability of control systems, and stable oscillations of both periodic and almost periodic discrete systems is presented, including many applications in engineering such as stability of digital filters, digitally controlled thermal processes, neurodynamics, and chemical kinetics. This book will be an invaluable reference source for those whose work is in the area of discrete dynamical systems, difference equations, and control theory or applied areas that use discrete time models.

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Series Editors' Preface	p. xi
Preface	p. xiii
Introduction	p. 1
Models with Discrete Storage of the Information	p. 2
Discrete Time Models Induced by the Impulses Occurring in Continuous-Time Systems	p. 7
Discrete Systems Occurring from Sampled Data Control Systems	p. 13
Numerical Treatment of Continuous-Time Systems	p. 16
About the Book	p. 19
Stability Theory	p. 21
Linear Discrete Time Systems with Constant Coefficients	p. 21
A stability criterion	p. 25
Example: stability of a digital filter	p. 26
Example: inherent stability of a digitally controlled thermal process	p. 28
The Liapunov matrix equation for the discrete case	p. 29
Robust stability and stability radii	p. 32
General Properties of Linear Systems	p. 37
Stability by the First Approximation	p. 43
A preliminary result	p. 44
Stability of equilibria for time-invariant systems	p. 44
Stability of motion for time varying systems	p. 47
LInear Discrete-Time Systems with Periodic Coefficients	p. 50
Multipliers and structure of solutions	p. 50
Stability results	p. 54
Liapunov Functions	p. 60
Stability concepts	p. 60
Stability via Liapunov functions	p. 62
Liapunov functions for linear systems. Exponential stability	p. 67
Stability under perturbations	p. 74
Invariance Principle. Barbashin-Krasovskii-La Salle Theorem	p. 79
Discrete dynamical systems	p. 79
The invariance principle	p. 82
Stability in Discrete Models of Chemical Kinetics	p. 84
Stability Results in Neurodynamics	p. 92
Stability via Input/Output Properties	p. 99
The Perron Condition	p. 99
Dissipativity and stability	p. 102
Stabilization of bilinear systems	p. 113
Absolute Stability of Control Systems	p. 119
The Simplest Absolute Stability Criterion of Tsypkin	p. 123
Quadratic Constraints of Yakubovich Type	p. 135
An Absolute Stability Criterion for the Case of Nondecreasing Nonlinear Functions	p. 140
The Brockett-Willems Type Criterion for Systems with Nondecreasing Nonlinearity	p. 143
Liapunov Functions and Frequency Domain Inequalities in Absolute Stability. The KSPY Lemma (Single-Input Case)	p. 153
For and against Liapunov functions in the study of nonlinear systems	p. 153
The Lemma for discrete systems	p. 155
Problem statement	p. 157
Proof of the lemma--the stable case	p. 159
Proof of the lemma--the general case	p. 168
A New Condition of Absolute Stability for Systems with Slope Restricted Nonlinearity	p. 169
System description. The class of nonlinearities. The associated system and the first integral	p. 171
The stability criterion	p. 173
An example	p. 181
Concluding remarks	p. 184
The KSPY Lemma (Multi-Input Case)	p. 184
Weak and strong frequency domain conditions	p. 185
An abstract minimization problem	p. 186
A linear quadratic minimization problem	p. 189
The KSPY lemma under the strong frequency domain inequality	p. 194
The KSPY lemma under the weak frequency domain inequality	p. 200
Applications of the Multi-Input KSPY Lemma	p. 203
Linear feedback disturbance attenuation	p. 203
A small gain result	p. 209
Absolute stability for systems with several sector and slope restricted nonlinearities	p. 210
Stable Oscillations	p. 221
Periodic Solutions of Forced Linear Systems with Periodic Coefficients (Forced Oscillations)	p. 221
Almost Periodic Sequences	p. 227
Motivation	p. 228
Standard theory of almost periodic sequences	p. 229
A new Bochner theory for almost periodic sequences	p. 234
Asymptotic almost periodic sequences	p. 235
Almost Periodic Linear Oscillations	p. 239
Periodic and Almost Periodic Oscillations in Nonlinear Systems	p. 249
Periodic oscillations	p. 249
Almost periodic oscillations	p. 253
Averaging	p. 256
References	p. 267
Index	p. 279
Table of Contents provided by Syndetics. All Rights Reserved.

Stability and Stable Oscillations in Discrete Time Systems

At a Glance

Hardcover

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