Systems with Delays E-Book

A. V. Kim

• Herausgeber: John Wiley & Sons
• Kategorie: Wissenschaft und neue Technologien
• Sprache: Englisch

Seitenzahl: 129

Veröffentlichungsjahr: 2015

Beschreibung

The main aim of the book is to present new constructive methods of delay differential equation (DDE) theory and to give readers practical tools for analysis, control design and simulating of linear systems with delays. Referred to as "systems with delays" in this volume, this class of differential equations is also called delay differential equations (DDE), time-delay systems, hereditary systems, and functional differential equations. Delay differential equations are widely used for describing and modeling various processes and systems in different applied problems At present there are effective control and numerical methods and corresponding software for analysis and simulating different classes of ordinary differential equations (ODE) and partial differential equations (PDE). There are many applications for these types of equations, because of this progress, but there are not as many methodologies in systems with delays that are easily applicable for the engineer or applied mathematician. there are no methods of finding solutions in explicit forms, and there is an absence of generally available general-purpose software packages for simulating such systems. Systems with Delays fills this void and provides easily applicable methods for engineers, mathematicians, and scientists to work with delay differential equations in their operations and research.

Leseprobe

Contents

Cover

Half Title page

Title page

Copyright page

Preface

Chapter 1: Linear Time-Delay Systems

1.1 Introduction

1.2 Conditional Representation of Differential Equations

1.3 Initial Value Problem. Notion of Solution

1.4 Functional Spaces

1.5 Phase Space H. State of Time-Delay System

1.6 Solution Representation

1.7 Characteristic Equation and Solution Expansion into a Series

Chapter 2: Stability Theory

2.1 Introduction

2.2 Lyapunov-Krasovskii Functionals

2.3 Positiveness of Functionals

2.4 Stability via Lyapunov-Krasovskii Functionals

2.5 Coefficient Conditions of Stability

Chapter 3: Linear Quadratic Control

3.1 Introduction

3.2 Statement of the Problem

3.3 Explicit Solutions of Generalized Riccati Equations

3.4 Solution of Exponential Matrix Equation

3.5 Design Procedure

3.6 Design Case Studies

Chapter 4: Numerical Methods

4.1 Introduction

4.2 Elementary One-Step Methods

4.3 Interpolation and Extrapolation of the Model Pre-History

4.4 Explicit Runge-Kutta-Like Methods

4.5 Approximation Orders of ERK-Like Methods

4.6 Automatic Step Size Control

Chapter 5: Appendix

5.1 i-Smooth Calculus of Functionals

5.2 Derivation of Generalized Riccati Equations

5.3 Explicit Solutions of GREs (Proofs of Theorems)

5.4 Proof of Theorem 1.1. (Solution Representation)

Bibliography

Index

Systems with Delays

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Library of Congress Cataloging-in-Publication Data:

ISBN 978-1-119-11758-2

Preface

At present there are elaborated effective control and numerical methods and corresponding software for analysis and simulating different classes of ordinary differential equations (ODE) and partial differential equations (PDE). The progress in this direction results in wide application of these types of equations in practice. Another class of differential equations is represented by delay differential equations (DDE), also called systems with delays, time-delay systems, hereditary systems, functional differential equations.

Delay differential equations are widely used for describing and mathematical modeling of various processes and systems in different applied problems [3, 5, 1, 27, 32, 33, 34, 40, 50, 62, 63, 183, 91, 107, 108, 111, 127, 183].

Delay in dynamical systems can have several causes, for example: technological lag, signal transmission and information delay, incubational period (infection diseases), time of mixing reactants (chemical kinetics), time of spreading drugs in a body (pharmaceutical kinetics), latent period (population dynamics), etc.

Though at present different theoretical aspects of time-delay theory (see, for example, [3, 1, 27, 32, 34, 50, 62, 63, 67, 72, 73, 183, 91, 107, 111, 127] and references therein) are developed with almost the same completeness as the corresponding parts of ODE theory, practical implementation of many methods is very difficult because of infinite dimensional nature of systems with delays.

Also it is necessary to note that, unlike ODE, even for linear DDE there are no methods of finding solutions in explicit forms, and the absence of generally available general-purpose software packages for simulating such systems cause a big obstacle for analysis and application of time-delay systems.

In this book we try to fill up this gap.

The main aim of the book is to present new constructive methods of DDE theory and to give readers practical tools for analysis, control design and simulating of linear systems with delays1.

The main outstanding features of this book are the following:

1. on the basis of i-smooth analysis we give a complete description of the structure and properties of quadratic Lyapunov-Krasovskii functionals2;

2. we describe a new control design technique for systems with delays, based on an explicit form of solutions of linear quadratic control problems;

3. we present new numerical algorithms for simulating DDE.

Acknowledgements

N.N.Krasovskii, A. B. Lozhnikov, Yu.F.Dolgii, A. I. Korotkii, O. V. Onegova, M. V. Zyryanov, Young Soo Moon, Soo Hee Han.

Research was supported by the Russian Foundation for Basic Research (projects 08-01-00141, 14-01-00065, 14-01-00477, 13-01-00110), the program “Fundamental Sciences for Medicine” of the Presidium of the Russian Academy of Sciences, the Ural-Siberia interdisciplinary project.

1 The present volume is devoted to linear time-delay system theory. We plan to prepare a special volume devoted to analysis of nonlinear systems with delays.

2 Including properties of positiveness, and constructive presentation of the total derivative of functionals with respect to time-delay systems.

Chapter 1

Linear time-delay systems

1.1 Introduction

1.1.1 Linear systems with delays

In this book we consider methods of analysis, control and computer simulation of linear systems with delays

(1.1)

where A(t), Aτ(t) are n × n matrices with piece-wise continuous elements, G(t, s) is n × n matrix with piece-wise continuous elements on R × [−τ, 0], u is a given n-dimensional vector-function, τ(t) : R → [−τ, 0] is a continuous function, τ is a positive constant.

Much attention will be paid to the special class of linear systems