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 1.1 Linear Systems 1 1.2 Homogeneous Nonlinear Systems 3 1.3 Polynomial and Volterra Systems 18 1.4 Interconnections of Nonlinear Systems 21 1.5 Heuristic and Mathematical Aspects 34 1.6 Remarks and References 37 1.7 Problems 42 Appendix 1.1 Convergence Conditions for Interconnections of Volterra Systems 44 Appendix 1.2 The Volterra Representation for Functionals 49 CHAPTER 2 Input/Output Representations in the Transform Domain 54 2.1 The Laplace Transform 54 2.2 Laplace Transform Representation of Homogeneous Systems 60 2.3 Response Computation and the Associated Transform 68 2.4 The Growing Exponential Approach 75 2.5 Polynomial and Volterra Systems 81 2.6 Remarks and References 85 2.7 Problems 87 CHAPTER 3 Obtaining Input/Output Representations from Differential-Equation Descriptions 93 3.1 Introduction 94 3.2 A Digression on Notation 103 3.3 The Carleman Linearization Approach 105 3.4 The Variational Equation Approach 116 3.5 The Growing Exponential Approach 124 3.6 Systems Described by Nth − Order Differential Equations 127 3.7 Remarks and References 131 3.8 Problems 135 Appendix 3.1 Convergence of the Volterra Series Representation for Linear-Analytic State Equations 137 CHAPTER 4 Realization Theory 142 4.1 Linear Realization Theory 142 4.2 Realization of Stationary Homogeneous Systems 152 4.3 Realization of Stationary Polynomial and Volterra Systems 163 4.4 Properties of Bilinear State Equations 173 4.5 The Nonstationary Case 180 4.6 Remarks and References 183 4.7 Problems 191 Appendix 4.1 Interconnection Rules for the Regular Transfer Function 194 CHAPTER 5 Response Characteristics of Stationary Systems 199 5.1 Response to Impulse Inputs 199 5.2 Steady-State Response to Sinusoidal Inputs 201 5.3 Steady-State Response to Multi-Tone Inputs 208 5.4 Response to Random Inputs 214 5.5 The Wiener Orthogonal Representation 233 5.6 Remarks and References 246 5.7 Problems 250 CHAPTER 6 Discrete-Time Systems 253 6.1 Input/Output Representations in the Time Domain 253 6.2 Input/Output Representations in the Transform Domain 256 6.3 Obtaining Input/Output Representations from State Equations 263 6.4 State-Affine Realization Theory 269 6.5 Response Characteristics of Discrete-Time Systems 277 6.6 Bilinear Input/Output Systems 287 6.7 Two-Dimensional Linear Systems 292 6.8 Remarks and References 298 6.9 Problems 301 CHAPTER 7 Identification 303 7.1 Introduction 303 7.2 Identification Using Impulse Inputs 305 7.3 Identification Based on Steady-State Frequency Response 308 7.4 Identification Using Gaussian White Noise Inputs 313 7.5 Orthogonal Expansion of the Wiener Kernels 322 7.6 Remarks and References 326 7.7 Problems 329 PREFACE When confronted with a nonlinear systems engineering problem, the first approach usually is to linearize; in other words, to try to avoid the nonlinear aspects of the problem. It is indeed a happy circumstance when a solution can be obtained in this way. When it cannot, the tendency is to try to avoid the situation altogether, presumably in the hope that the 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 confusing endeavor; it is widely viewed as dangerous to those who think about it for too long. This skepticism is to an extent justifiable. When compared with the variety of techniques available in linear system theory, the tools for analysis and design of nonlinear systems are limited to some very special categories. First, there are the relatively simple techniques, such as phase-plane analysis, which are graphical in nature and thus of limited generality. Then, there are the rather general (and subtle) techniques based on the theory of differential equations, functional analysis, and operator theory. These provide a language, a framework, and existence/uniqueness proofs, but often little problem-specific information beyond these basics. Finally, there is simulation, sometimes ad nauseam, on the digital computer. I do not mean to say that these techniques or approaches are useless. Certainly phase-plane analysis describes nonlinear phenomena such as limit cycles and multiple equilibria of second-order systems in an efficient manner. The theory of differential equations has led to a highly developed stability theory for some classes of nonlinear systems. (Though, of course, an engineer cannot live by stability alone.) Functional analysis and operator theoretic viewpoints are philosophically appealing, and undoubtedly will become more applicable in the future. Finally, everyone is aware of the occasional success story emanating from the local computer center. What I do mean to say is that a theory is needed that occupies the middle ground in generality and applicability. Such a theory can be of great importance for it can serve as a starting point, both for more esoteric mathematical studies and for the development of engineering techniques. Indeed, it can serve as a bridge or communication link between these two activities. In the early 1970s it became clear that the time was ripe for a middle-of-the-road formulation for nonlinear system theory. It seemed that such a formulation should use some aspects of differential- (or difference-) equation descriptions, and transform representations, as well as some aspects of operator-theoretic descriptions. The question was whether, by making structural assumptions and ruling out pathologies, a reasonably 1 simple, reasonably general, nonlinear system theory could be developed. Hand in hand with this viewpoint was the feeling that many of the approaches useful for linear systems ought to be extensible to the nonlinear theory. This is a key point if the theory is to be used by practitioners as well as by researchers. These considerations led me into what has come to be called the Volterra/Wiener representation for nonlinear systems. Articles on this topic had been appearing sporadically in the engineering literature since about 1950, but it seemed to be time for an investigation that incorporated viewpoints that in recent years proved so successful in linear system theory. The first problem was to specialize the topic, both to avoid the vagueness that characterized some of the literature, and to facilitate the extension of linear system techniques. My approach was to consider those systems that are composed of feedback-free interconnections of linear dynamic systems and simple static nonlinear elements. Of course, a number of people recognized the needs outlined above. About the same time that I began working with Volterra/Wiener representations, others achieved a notable success in specializing the structure of nonlinear differential equations in a profitable way. It was shown that bilinear state equations were amenable to analysis using many of the tools associated with linear state equations. In addition, the Volterra/Wiener representation corresponding to bilinear state equations turned out to be remarkably simple. These topics, interconnection-structured systems, bilinear state equations, Volterra/Wiener representations, and their various interleavings form recurring themes in this book. I believe that from these themes will be forged many useful engineering tools for dealing with nonlinear systems in the future. But a note of caution is appropriate. Nonlinear systems do not yield easily to analysis, especially in the sense that for a given analytical method it is not hard to find an inscrutable system. Worse, it is not always easy to ascertain beforehand when methods based on the Volterra/Wiener representation are appropriate. The folk wisdom is that if the nonlinearities are mild, then the Volterra/Wiener methods should be tried. Unfortunately, more detailed characterization tends to destroy this notion before capturing it, at least in a practical sense. So, in these matters I ask some charity from the reader. My only recommendation is the merely obvious one to keep all sorts of methods in mind. Stability questions often will call for application of methods based on the theory of differential equations. Do not forget the phase plane or the computer center, for they are sure to be useful in their share of situations. At the same time I urge the reader to question and reflect upon the possibilities for application of the Volterra/Wiener methods discussed herein. The theory is incomplete, and likely to remain so for some time. But I hope to convince that, though the sailing won’t be always smooth, the wind is up and the tide fair for this particular passage into nonlinear system theory - and that the engineering tools to be found will make the trip worthwhile. This text represents my first attempt to write down in an organized fashion the nonlinear system theory alluded to above. As such, the effort has been somewhat frustrating since the temptation always is to view gaps in the development as gaps, and not as research opportunities. In particular the numerous research opportunities have forced 2 certain decisions concerning style and content. Included are topics that appear to be a good bet to have direct and wide applicability to engineering problems. Others for which the odds seem longer are mentioned and referenced only. As to style I eschew the trappings of rigor and adopt a more mellifluous tone. The material is presented informally, but in such a way that the reader probably can formalize the treatment relatively easily once the main features are grasped. As an aid to this process each chapter contains a Remarks and References section that points the way to the research literature.