Qualitative Behavior

Abstract

This chapter presents a brief overview of some of the most important qualitative characteristics of discrete-time Volterra models. In their simplest form, these models combine linear moving average models of dynamic order M with polynomial nonlinearities of degree NS, leading to the class of finite Volterra models defined in Section 2.1. Several important special cases of this general model family exist, including both the finite Hammerstein and Wiener model classes described briefly in Section 2.2. More generally, all finite Volterra models belong to the larger class of finite-dimensional nonlinear moving average (NMAX) models, and Section 2.3 presents a summary of results available concerning the qualitative behavior of this larger model class. Conversely, finite Volterra models are not capable of exhibiting certain types of qualitative behavior (e.g. subharmonic generation), and Section 2.3 also illustrates this point with a number of simple examples. Taking the limit as either the dynamic order M or the nonlinear degree N, or both parameters, become infinitely large, we obtain the class of infinite dimensional Volterra models; these models are described in Section 2.4, with particular emphasis on the behavioral consequences of their infinite-dimensionality. Section 2.5 then briefly considers two related issues: first, the effects of truncating infinite-dimensional Volterra models; second, the extent to which finite Volterra models are useful approximations of more general system dynamics. In particular, it has been shown (Boyd and Chua, 1985) that any member of the class of fading memory systems may be approximated arbitrarily well by a finite Volterra model of sufficiently high order and degree and this result is discussed further in Section 2.5. Finally, the chapter concludes with a brief summary in Section 2.6.