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J. A. Richards Communications and Control Engineering Series Editors: A. Fettweis· J. L. Massey· M. Thoma J. A. Richards Analysis of Periodically Time¥arying Systems With 73 Figures Springer-Verlag Berlin Heidelberg New York 1983 J. A. RICHARDS School of Electrical Engineering and Computer Science University of New South Wales P.O. Box 1 Kensington, N.S.W. 2033, Australia ISBN-13:978-3-642-81875-2 e-ISBN-13:978-3-642-81873-8 DOl: 10.1007/978-3-642-81873-8 Library of Congress Cataloging in Publication Data Richards, John Alan, 1945-. Analysis of periodically time-varying systems. (Communications and control engineering series) Bibliography: p. Includes index. 1. System analysis. I. Title. II. Series. QA402.R47 1983 003 82-5978 AACR2 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to .Verwertungsgesellschaft Wort«, Munich. © Springer-Verlag Berlin, Heidelberg 1983 Softcover reprint of the hardcover 1st edition 1983 The use of registered names, trademarks, etc. in this publication does not imply, even in the absence ofa specific statement, that such names are exempt from tbe relevant protective laws and regulations and therefore free for general use. Typesetting: Asco Trade Typesetting Ltd., Hong Kong 206113020-543210 To Dick Huey Preface Many of the practical techniques developed for treating systems described by periodic differential equations have arisen in different fields of application; con­ sequently some procedures have not always been known to workers in areas that might benefit substantially from them. Furthermore, recent analytical methods are computationally based so that it now seems an opportune time for an applications-oriented book to be made available that, in a sense, bridges the fields in which equations with periodic coefficients arise and which draws together analytical methods that are implemented readily. This book seeks to ftll that role, from a user's and not a theoretician's view. The complexities of periodic systems often demand a computational approach. Matrix treatments therefore are emphasized here although algebraic methods have been included where they are useful in their own right or where they establish properties that can be exploited by the matrix approach. The matrix development given calls upon the nomenclature and treatment of H. D'Angelo, Linear Time­ Varying Systems: Analysis and Synthesis (Boston: Allyn and Bacon 1970) which deals with time-varying systems in general. It is recommended for its modernity and comprehensive approach to systems analysis by matrix methods. Since the present work is applications-oriented no attempt has been made to be complete theoretically by way of presenting all proofs, existence theorems and so on. These can be found in D'Angelo and classic and well-developed treatises such as McLachlan, N. W.: Theory and application of Mathieu functions. Clarendon: Oxford V.P. 1947. Reprinted by Dover, New York 1964. Arscott, F. M.: Periodic differential equations. Oxford: Pergamon 1964. Magnus, W.; Winkler, S.: Hill's equation. New York: Wiley 1966. Instead, this book relates theory to applications via analytical methods that are, in the main, computationally-based. The book is presented in two parts. The first deals with theory, and techniques for applying that theory in the analysis of systems. A highlight of this (chapter five) is the development of modelling procedures that allow intractable periodic differential equations to be handled. This is regarded as significarit since the great majority of differential equations with periodic coefficients cannot be treated by closed form methods of analysis. The second part presents an overview of the applications of periodic equations. The particular applications chosen have been done so to illustrate the variety of ways periodic differential equation descriptions arise and to demonstrate that the modelling procedures of Part I can be useful in determining system properties. VIII Preface The developments in Part I and the applications of Part II are all related to a standard form for the equations, which in the case of second order systems is the canonical Hill equation used by McLachlan, viz. x + (a - 2ql{l(/»x = O,I{I(/) = I{I(t + n) where I{I(/) is a general periodic coefficient. Adopting such a canonical form is of value if the results of the theory and techniques chapters are to be used directly. Original system equations for particular applications presented in Part II are thus transformed into the appropriate canonical form before drawing upon the material of Part I. The treatment is not intended to provide a text for the study of periodic differ­ ential equations but could be used for a single semester senior undergraduate or graduate level subject in systems with periodic parameters, particularly if applica­ tions are to be emphasised. The encouragement and assistance of others is of course essential in producing a book. It is the author's pleasure to acknowledge the inspiration of his mentor and friend Professor Dick Huey to whom this book is dedicated; over the years of their association he has done much to encourage in a quiet, yet effective, manner the completion of this work. The manuscript was typed by Mrs. Gellisinda Galang to whom the author is grateful for the competent and patient manner in which she undertook the task. Kensington, Australia, March 1982 J. A. Richards Contents List of Symbols Part I Theory and Techniques Chapter 1 Historical Perspective ........................................ 3 1.1 The Nature of Systems with Periodically Time-Varying Parameters .. 3 1.2 1831-1887 Faraday to Rayleigh-Early Experimentalists and Theorists ................................................... 7 1.3 1918-1940 The First Applications .............................. 9 1.4 Second Generation Applications ............................... 10 1.5 Recent Theoretical Developments .............................. 12 1.6 Commonplace Illustrations of Parametric Behaviour .............. 12 References for Chapter 1 ..................................... 14 Problems ................................................... 16 Chapter 2 The Equations and Their Properties ............................. 17 2.1 Hill Equations .............................................. 17 2.2 Matrix Formulation of Hill Equations .......................... 18 2.3 The State Transition Matrix ................................... 19 2.4 Floquet Theory ............................................. 20 2.5 Second Order Systems ........................................ 22 2.6 Natural Modes of Solution .................................... 23 2.7 Concluding Comments ....................................... 24 References for Chapter 2 ..................................... 25 Problems ................................................... 25 Chapter 3 Solutions to Periodic Differential Equations ...................... 27 3.1 Solutions Over One Period of the Coefficient .................... 27 3.2 The Meissner Equation ....................................... 28 3.3 Solution at Any Time for a Second Order Periodic Equation ....... 29 3.4 Evaluation of 4>(n, o)m, m Integral .............................. 30 3.5 The Hill Equation with a Staircase Coefficient ................... 32 3.6 The Hill Equation with a Sawtooth Waveform Coefficient ......... 32 3.6.1 The Wronskian Matrix with z Negative ......................... 35 3.6.2 The Wronskian Matrix with z Zero ............................. 35 3.6.3 The Case of f3 Negative ....................................... 36 3.7 The Hill Equation with a Positive Slope, Sawtooth Waveform Coefficient .................................................. 36 3.8 The Hill Equation with a Triangular Coefficient .................. 37 x Contents 3.9 The Hill Equation with a Trapezoidal Coefficient ................. 38 3.10 Bessel Function Generation . 38 3.11 The Hill Equation with a Repetitive Exponential Coefficient . 39 3.12 The Hill Equation with a Coefficient in the Form of a Repetitive Sequence ofImpulses ......................................... 40 3.13 Equations of Higher Order..................................... 41 3.14 Response to a Sinusoidal Forcing Function .................... ;.. 41 3.15 Phase Space Analysis. .. ... ... .. ... ... ... ... ... 44 3.16 Concluding Comments........................................ 46 References for Chapter 3 ...................................... 48 Problems. 49 Chapter 4 Stability. 50 4.1 Types of Stability . 50 4.2 Stability Theorems for Periodic Systems ......................... 51 4.3 Second Order Systems. 52 4.3.1 Stability and the Characteristic Exponent ........................ 52 4.3.2 The Meissner Equation. 53 4.3.3 The Hill Equation with an Impulsive Coefficient .................. 56 4.3.4 The Hill Equation with a Sawtooth Waveform Coefficient. 57 4.3.5 The Hill Equation with a Triangular Waveform Coefficient. 57 4.3.6 Hill Determinant Analysis.... .. .... ... ... ... .. 57 4.3.7 Parametric Frequencies for Second Order Systems. ... ... .. .. 62 4.4 General Order Systems . 63 4.4.1 Hill Determinant Analysis for General Order Systems. 63 4.4.2 Residues of the Hill Determinant for q ..... 0 ...................... 66 4.4.3 Instability and Parametric Frequencies for General
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