Introduction and Overview

Transcription

1 Chapter 1 Introduction and Overview 1.1 Statement of Purpose The theory of integral equations, transforms and models is well developed and dense in the literature and applications, since they have been found to provide efficient ways of solving a variety of problems arising in engineering, physical sciences and economics. However we have been unable to find a systematic treatment of integral models of casual and evolving nonlinear dynamical systems. Therefore this text is a modest attempt to partly fill that gap. The topic dynamical systems is the study of the long standing behavior of evolving systems, of which systematic theoretical studies have been initiated at the end of XIX century concerning the fundamental questions of the evolution and stability of the solar system. These studies led to the development of diversified and powerful fields with various applications in economics, ecology, biology, energetics, meteorology, astronomy, and other areas. By way of introduction let us consider the following integral equation b a K(t, s)x(s) ds = f(t). (1) With known function f(t) andkernelk(t, s) this equation is known as the Fredholm equation of the first kind with respect to x(s). On the other hand, the relation (1) can be considered as a general integral transform, e.g., we can consider the following case a =, b=+,k(t, s) = 1 2π e its, the functions x and f are correspondingly input and output signals, we get as is well known in digital signal processing, the Fourier integral transform + 1 e its x(s) ds = ˆf. (2) 2π 1

2 2 Integral Dynamical Models: Singularities, Signals and Control The solution of the equation (2) would give the inverse Fourier transform. Similarly we can define the Fourier cosine transform 2 cos(ts)x(s) ds = f(t), π which is the well-known image and signal processing tool for lossy compression. The Laplace transform, Hilbert transform and other widely used integral transforms can be defined similarly. If we consider the case of a =, b= t we obtain a general representation for finite-dimension deterministic linear dynamical system which is well known in system control theory as the input-output explicit integral dynamical model with infinite delay K(t, s)x(s) ds = f(t), where kernel matrix K(t, s) is the impulse response function, x(t) is input vector, f(t) is vector of outputs. It is to be noted that many real-life inputoutput systems involve feedback. Most production and economic systems are also systems with memory. A real-life input-output dynamic system involves a feedback A(t, s) that connects inputs x and outputs f, wherea specifies the feedback type and intensity, then K(t, s)a(t, s, f(s)) ds = f(t). Fig Feedback control. These models take into account the memory/aftereffect of a dynamical system when its past impacts its future evolution. The memory is implemented in the existing technological and financial structure of physical capital (equipment). The memory duration is determined by the age of the oldest capital unit (e.g., equipment) still employed. When applying these ideas to a problem in economics it is convenient to write A(t, s) asλ(t, s)y (s), where Y is distribution of products and the

3 Introduction and Overview 3 intensity control function Λ(t, s) reflects the varying load of different equipment types in the production of various products. The intensity control Λ could be specific, e.g., it has to be capable to model the case when production systems use the newest, most efficient facilities and scrap the oldest when it becomes obsolete. Taking into account the control function A, the input-output explicit integral dynamical model becomes the linear Volterra model K(t, s)x(s) ds = f(t), t [t,t] (3) t with special kernels that have jump discontinuities on delay control curves. Here readers may refer to the article of [Hritonenko and Yatsenko (25)] As we see now the Volterra (evolutionary) equations with jump discontinuous kernels play an important role in the theory of evolving dynamical systems in economics, ecology and energetics. The origins on the integral models studied in this book are from the class of the macroeconomic models with technical progress, integral models of [Solow (1969)] are examples of this kinds of models. Such kinds of models were initially proposed by [Kantorovich and Gor kov (1959)]. [Kantorovich and Gor kov (1959)] proposed a new integral macroeconomic model which is a development of the Solow model. Independently of Kantorovich, [Glushkov et al. (1983)] proposed a two-sector integral model described by a system of the Volterra equations which are connected with the Volterra equation of the first kind studied in this book. Volterra integral equations are used to model causal dynamical systems with memory. Causal essentially means that the output signal at each time instant t depends only on the history of the input signal up to time t. Memory means that the output at time t does not depend exclusively on the instant value of the input at time t, but rather it depends on the entire history of the input from some initial time t,wheret<t. In industrial applications and applications in other fields there is a large demand to apply nonlinear algorithms to control nonlinear dynamical systems with memory. For example, the evolution of an epidemic depends on the entire history of the epidemic; the number of individuals who will contract a disease at time t does not depend only on the number of infected individuals at time t, but also depend on the history of the infected numbers, because the disease conventionally progresses in several stages, and patients at different stages of infection have different symptoms. In economics the effectiveness of a strategy depends not only on the current state of the relevant economic system but also on the history of the system, e.g.,

4 4 Integral Dynamical Models: Singularities, Signals and Control the effectiveness of advertising, the adaptation of consumers to new products or new prices, and other relevant processes are not memoryless effects. Other examples of causal systems with memory include hereditary continuum mechanics, namely materials with memory, flow of water in porous media and other generally nonlinear systems. With algorithms considering the process nonlinearities, better control performance is expected in the whole operating range compared with conventional linear control algorithms. To cope with this demand the significant part of this book is devoted to the generalizations of the continuous-time Volterra models for casual nonlinear dynamical systems feed-back control. Such Volterra models appear from the generalization of the Volterra integral-functional series which is the conventional tool in nonlinear casual dynamical systems modelling introduced in classic book of [Volterra (25)]. In the proposed generalized Volterra models the transfer functions K j (t, s 1,...s j,u(s 1 ),...u(s j )) are assumed to be known. It is to be noted here that the Volterra series belongs to one of the best understood nonlinear system representations in signal processing. Namely we address the following nonlinear equation w.r.t. the continuous solution u(t) u as t ( Φ K 1 (t, s, u(s))ds, K 2 (t, s, s 1,s 2,u(s 1 ),u(s 2 ))ds 1 ds 2,... ) K n (t, s 1,...s n,u(s 1 ),...u(s n ))ds 1...ds n,u(t),t =, where t [,T), and Φ : E 1 E 1 R 1 E 2 is nonlinear continuous operator of n + 1 variables ω 1,...,ω n,u, which are abstract continuous functions of real variable t with values in the Banach space E 1, Φ(,...,,u, ) =, K i : R 1...R }{{} 1 i+1 E 1 E 1 }{{} i E 2 are nonlinear continuous operators depending on u(s) = (u(s 1 ),...,u(s n )) and t, s 1,...s n are real variables. Special attention is given to the main solution of such equations in sense of L. V. Kantorovich. Here readers may refer to classic book of [Kantorovich et al. (195)]. A special case of such a nonlinear integral equation is the following polynomial equation with convolution kernels N m=1... K m (t s 1,...,t s m ) m x(s i )ds i f (t) =, t T. The method for construction of the generalized solutions (i.e., impulse control) to such an equation is also presented in this text. It is to be i=1

5 Introduction and Overview 5 noted that such models are known in the signal processing community as continuous-time analogue to generalized polynomial kernel regression models, here readers may refer to book of [Mathews and Sicuranza (2)] and to [Franz and Schölkopf (26)]. The application of autoregressive models in power engineering for unstable oscillations detection is also studied in this text. Among other nonlinear equations studied in this book is the following Hammerstein integral equation u(x) = b a K(x, s)(u(s)+f(s, u, λ)) ds, and an abstract nonlinear Volterra equation with non-invertible operator. In this book we further develop and employ the theory of nonlinear parameter-dependent mathematical models introduced in book of [Sidorov et al. (22)]. The proofs of the theorems on asymptotic behavior of solutions contain (rather simple) elements of functional analysis and were written in a way that one may omit the proofs since they are not of prejudice to the understanding of the main ideas of the book. The principal part of the text is made up of various results that are mostly available in research papers, some in English or only in Russian. A number of results presented in this text have not appeared in print before, and previously published results have been substantially modified with examples and also improved. 1.2 The Scope of This Book It is assumed that the reader has some background in integral and differential equations, functional analysis, some knowledge of digital signal processing is also expected. For some parts of the text the reader will need knowledge of the theory of distributions. The book consists of three parts. Part 1 (Chapters 2 5) of this monograph presents the detailed overview of the theory behind the continuoustime Volterra models of evolving dynamical systems. For didactical reasons we start this book with single Volterra Integral Equations (VIEs), then we address systems of VIEs with piecewise continuous kernels. Finally we generalize our results on abstract Volterra operator equations with piecewise continuous kernels and consider solutions in the distributions space. Thereby, in Chapter 2 we introduce the VIEs of the first kind with

6 6 Integral Dynamical Models: Singularities, Signals and Control piecewise continuous kernels, describe the structure of solutions and prove the existence theorem. In Chapters 3 and 4 we address a system of VIE with piecewise continuous kernels and the abstract integro-functional Volterra operator equation. In Chapter 5 we examine the special case when the VIE has no continuous solution and the generalized solutions are constructed in the Sobolev-Schwartz distributions space. In Part 2 (Chapters 6 11) of the text we systematically study nonlinear continuous-time integral models starting from the Hammerstein integral equation, which is a well-known tool in control theory. In Chapter 7 we employ the similar technique (presented in Chapter 6) for the nonlinear Volterra equation of the second kind with non-invertible operator in the main part. In Chapter 8 solutions to nonlinear differential equations in the neighborhood of branching points are studied. In Chapters 9, 1 and 11 we correspondingly study the classical and generalized solutions to nonlinear Volterra equations arising in the theory of nonlinear continuous-time casual dynamical systems control based on the Volterra integral-functional series. Chapter 1 presents the theoretical results on generalized solution of the special class of nonlinear Volterra equations important for solution of the integral equations discussed in Chapters 9 and 11. In Section we review state-of-the-art identification algorithms for Volterra models. The material of the book is organized so that as continuous-time methods for the Volterra models are presented, some corresponding discrete-time methods are also presented. Part 3 (Chapters 12 14) is devoted to applications of the integral dynamical models. In Chapter 12 Volterra series based models are applied to nonlinear heat-exchange dynamics modelling. As applied to Subsection 12.2 we outline the elements of the economics theory of evolving dynamical systems, and describe the numerical method for solution of discrete-time VIEs of the first kind with piecewise continuous kernels. Although most of the examples and applications in this book are concerned with energetics and electrical systems, some examples and applications utilize dynamical systems from other fields, e.g., motion picture restoration. This is done to illustrate the diversity of applications for methods of signal processing and integral models. Indeed, Chapter 13 presents the practical example of using the Fourier transform in motion picture restoration. The book is concluded with Chapter 14 where the problem of electrical power systems parameters forecasting is addressed using the Hilbert integral transform and artificial neural networks. In Chapter 14, Subsection we also discuss the unifying view on the Volterra theory and polynomial modelling

7 Introduction and Overview 7 as a kernel regression and present the recent results concerning application of non-stationary autoregressive models for on-line detection of inter-area oscillations in power systems. 1.3 Acknowledgments This book benefitted from the support of the Russian Scientific Foundation, project No , the Russian Ministry of Education & Science Project Singular Integral Models and Transforms: Theory & Applications (State Contract No. 14.B ), Seventh Framework EU Programme (ICOEUR Project), Sixth Framework EU Programme (BRAVA Project) under IST research and technological development programme and German Academic Exchange Service (DAAD) Grant No. A12665, Russian Foundation of Basic Research Grant No The author is grateful for discussions to Prof. Alfredo Lorenzi on the Volterra equations during his visit to the University of Milano and during the conference on Inverse and Ill-Posed Problems of Mathematical Physics in Novosibirsk. Also the author expresses his deepest gratitude to Prof. Hans-Jürgen Reinhardt for his hospitality during my stay in Siegen and to Prof.-Ing. Christian Rehtanz for kind invitation to present part of these lectures at the University of Dortmund during the research stay in Germany. The author is particularly grateful to Prof. Nikolai A. Sidorov and Prof. Vladilev A. Trenogin for their deep co-operation and friendly advice of long-standing. The author is indebted to his collaborators and colleagues Prof. Anatoly S. Apartsyn, Prof. Karen Egiazarian, Prof. Mihail V. Falaleev, Dr. Alexander V. Kiselev, Prof. Anil C. Kokaram, Prof. Victor G. Kurbatsky, Dr. Paul Leahy, Dr. Eugenia V. Markova, Mr. Daniil A. Panasetsky, Prof. Alexei V. Savvateev, Dr. Inna V. Sidler, Dr. Vaclav Šmídl, Prof. Valery S. Sizikov, Dr. Svetlana V. Solodusha, Mr. Vadim A. Spiryaev and Dr. Nikita V. Tomin who deserve much credit for these notes. The author gratefully acknowledges Prof. Malcolm Brown, Prof. Vladimir K. Gorbunov, Prof. Alfredo Lorenzi and Prof. Hans-Jürgen Reinhardt for valuable discussions of the manuscript. I m grateful to my wife, Aliona, and my children Lev Ryan and Alisa for their patience and love.