Nonlinear System Theory

1 Nonlinear System Theory The Volterra/Wiener Approach by Wilson J. Rugh Originally published by The Johns Hopkins University Press, 1981 (ISBN O-8018-2549-0). Web version prepared in 2002. Contents PREFACE CHAPTER 1 Input/Output Representations in the Time Domain 1 Linear Systems 1 Homogeneous Nonlinear Systems 3 Polynomial and Volterra Systems 18 Interconnections of Nonlinear Systems 21 Heuristic and Mathematical Aspects 34 Remarks and References 37 Problems 42 Appendix Convergence Conditions for Interconnections of Volterra Systems 44 Appendix The Volterra Representation for Functionals 49 CHAPTER 2 Input/Output Representations in the Transform Domain 54 The Laplace Transform 54

2 Laplace Transform Representation of Homogeneous Systems 60 Response Computation and the Associated Transform 68 The Growing Exponential Approach 75 Polynomial and Volterra Systems 81 Remarks and References 85 Problems 87 CHAPTER 3 Obtaining Input/Output Representations from Differential-Equation Descriptions 93 Introduction 94 A Digression on Notation 103 The Carleman Linearization Approach 105 The Variational Equation Approach 116 The Growing Exponential Approach 124 Systems Described by thN Order Differential Equations 127 Remarks and References 131 Problems 135 Appendix Convergence of the Volterra Series Representation for Linear-Analytic State Equations 137 CHAPTER 4 Realization Theory 142 Linear Realization Theory 142 Realization of Stationary Homogeneous Systems 152 Realization of Stationary Polynomial and Volterra Systems 163 Properties of Bilinear State Equations 173 The Nonstationary Case 180 Remarks and References 183 Problems 191 Appendix Interconnection Rules for the Regular Transfer

3 Function 194 CHAPTER 5 Response Characteristics of Stationary Systems 199 Response to Impulse Inputs 199 Steady-State Response to Sinusoidal Inputs 201 Steady-State Response to Multi-Tone Inputs 208 Response to Random Inputs 214 The Wiener Orthogonal Representation 233 Remarks and References 246 Problems 250 CHAPTER 6 Discrete-Time Systems 253 Input/Output Representations in the Time Domain 253 Input/Output Representations in the Transform Domain 256 Obtaining Input/Output Representations from State Equations 263 State-Affine Realization Theory 269 Response Characteristics of Discrete-Time Systems 277 Bilinear Input/Output Systems 287 Two-Dimensional Linear Systems 292 Remarks and References 298 Problems 301 CHAPTER 7 Identification 303 Introduction 303 Identification Using Impulse Inputs 305 Identification Based on Steady-State Frequency Response 308 Identification Using Gaussian White Noise Inputs 313 Orthogonal Expansion of the Wiener Kernels 322 Remarks and References 326 Problems

4 329 PREFACEWhen confronted with a Nonlinear systems engineering problem, the first approachusually is to linearize; in other words, to try to avoid the Nonlinear aspects of the is indeed a happy circumstance when a solution can be obtained in this way. When itcannot, the tendency is to try to avoid the situation altogether, presumably in the hope thatthe problem will go away. Those engineers who forge ahead are often viewed as foolish,or worse. Nonlinear systems engineering is regarded not just as a difficult and confusingendeavor; it is widely viewed as dangerous to those who think about it for too skepticism is to an extent justifiable.

5 When compared with the variety oftechniques available in linear System Theory , the tools for analysis and design of nonlinearsystems are limited to some very special categories. First, there are the relatively simpletechniques, such as phase-plane analysis , which are graphical in nature and thus of limitedgenerality. Then, there are the rather general (and subtle) techniques based on the theoryof differential equations, functional analysis , and operator Theory . These provide alanguage, a framework, and existence/uniqueness proofs, but often little problem-specificinformation beyond these basics.

6 Finally, there is simulation, sometimes ad nauseam, onthe digital do not mean to say that these techniques or approaches are useless. Certainlyphase-plane analysis describes Nonlinear phenomena such as limit cycles and multipleequilibria of second-order systems in an efficient manner. The Theory of differentialequations has led to a highly developed stability Theory for some classes of nonlinearsystems. (Though, of course, an engineer cannot live by stability alone.) Functionalanalysis and operator theoretic viewpoints are philosophically appealing, and undoubtedlywill become more applicable in the future.

7 Finally, everyone is aware of the occasionalsuccess story emanating from the local computer I do mean to say is that a Theory is needed that occupies the middle ground ingenerality and applicability. Such a Theory can be of great importance for it can serve as astarting point, both for more esoteric mathematical studies and for the development ofengineering techniques. Indeed, it can serve as a bridge or communication link betweenthese two the early 1970s it became clear that the time was ripe for a middle-of-the-roadformulation for Nonlinear System Theory .

8 It seemed that such a formulation should usesome aspects of differential- (or difference-) equation descriptions, and transformrepresentations, as well as some aspects of operator-theoretic descriptions. The questionwas whether, by making structural assumptions and ruling out pathologies, a reasonably1simple, reasonably general, Nonlinear System Theory could be developed. Hand in handwith this viewpoint was the feeling that many of the approaches useful for linear systemsought to be extensible to the Nonlinear Theory . This is a key point if the Theory is to beused by practitioners as well as by considerations led me into what has come to be called the Volterra/Wienerrepresentation for Nonlinear systems.

9 Articles on this topic had been appearingsporadically in the engineering literature since about 1950, but it seemed to be time for aninvestigation that incorporated viewpoints that in recent years proved so successful inlinear System Theory . Thefirst problem was to specialize the topic, both to avoid thevagueness that characterized some of the literature, and to facilitate the extension of linearsystem techniques. My approach was to consider those systems that are composed offeedback-free interconnections of linear dynamic systems and simple static course, a number of people recognized the needs outlined above.

10 About the sametime that I began working with Volterra/Wiener representations, others achieved a notablesuccess in specializing the structure of Nonlinear differential equations in a profitable was shown that bilinear state equations were amenable to analysis using many of thetools associated with linear state equations. In addition, the Volterra/Wiener representationcorresponding to bilinear state equations turned out to be remarkably topics,interconnection-structured systems, bilinearstateequations,Volterra/Wiener representations, and their various interleavings form recurring themes inthis book.