Search Results

As noted In Chapter 1, the general class of NARMAX models is extremely broad and includes almost all of the other discrete-time model classes discussed in this book, linear and nonlinear. Because of its enormity, little can be said about the qualitative behavior of the NARMAX family in general, but it is surprising how much can be said about the behavior of some important subclasses. In particular, there are sharp qualitative distinctions between nonlinear moving average models (NMAX models) and nonlinear autoregressive models (NARX models). Since these representations are equivalent for linear models (c.f. Chapter 2), this observation highlights one important difference between linear and nonlinear discrete-time dynamic models. Consequently, this chapter focuses primarily on the structure and qualitative behavior of various subclasses of the NARMAX family, particularly the NMAX and NARX classes. More specifically, Sec. 4.1 presents a brief discussion of the general NARMAX class, defining five important subclasses that are discussed in subsequent sections: the NMAX class (Sec. 4.2), the NARX class (Sec. 4.3), the class of (structurally) additive NARMAX models (Sec. 4.4), the class of polynomial NARMAX models (Sec. 4.5), and the class of rational NARMAX models (Sec. 4.6). More complex NARMAX models are then discussed briefly in Sec. 4.7 and Sec. 4.8 concludes the chapter with a brief summary of the NARMAX class. The general class of NARMAX models was defined and discussed in a series of papers by Billings and various co-authors (Billings and Voon, 1983, 1986a; Leontartis and Billings, 1985; Zhu and Billings, 1993) and is defined by the equation . . . y(k) = F(y(k – 1), . . . , y(k – p), u(k), . . . , u(k – q), e(k – 1), . . . , e(k – r)) + e(k). (4.1) . . . Because this book is primarily concerned with the qualitative input/output behavior of these models, it will be assumed that e(k) = 0 identically as in the class of linear ARMAX models discussed in Chapter 2.

One of the main points of Chapter 4 is that nonlinear moving-average (NMAX) models are both inherently better-behaved and easier to analyze than more general NARMAX models. For example, it was shown in Sec. 4.2.2 that if ɡ(· · ·) is a continuous map from Rq+1 to R1 and if ys = ɡ (us,..., us), then uk → us implies yk → ys. Although it is not always satisfied, continuity is a relatively weak condition to impose on the map ɡ(· · ·) . For example, Hammerstein or Wiener models based on moving average models and the hard saturation nonlinearity represent discontinuous members of the class of NMAX models. This chapter considers the analytical consequences of requiring ɡ(·) to be analytic, implying the existence of a Taylor series expansion. Although this requirement is much stronger than continuity, it often holds, and when it does, it leads to an explicit representation: Volterra models. The principal objective of this chapter is to define the class of Volterra models and discuss various important special cases and qualitative results. Most of this discussion is concerned with the class V(N,M) of finite Volterra models, which includes the class of linear finite impulse response models as a special case, along with a number of practically important nonlinear moving average model classes. In particular, the finite Volterra model class includes Hammerstein models, Wiener models, and Uryson models, along with other more general model structures. In addition, one of the results established in this chapter is that most of the bilinear models discussed in Chapter 3 may be expressed as infinite-order Volterra models. This result is somewhat analogous to the equivalence between finite-dimensional linear autoregressive models and infinite-dimensional linear moving average models discussed in Chapter 2. The bilinear model result presented here is strictly weaker, however, since there exist classes of bilinear models that do not possess Volterra series representations. Specifically, it is shown in Sec. 5.6 that completely bilinear models do not exhibit Volterra series representations. Conversely, one of the results discussed at the end of this chapter is that the class of discrete-time fading memory systems may be approximated arbitrarily well by finite Volterra models (Boyd and Chua, 1985).

The primary objective of this book has been to present a reasonably broad overview of the different classes of discrete-time dynamic models that have been proposed for empirical modeling, particularly in the process control literature. In its simplest form, the empirical modeling process consists of the following four steps: 1. Select a class C of model structures 2. Generate input/output data from the physical process P 3. Determine the model M ∊ C that best fits this dataset 4. Assess the general validity of the model M. The objective of this final chapter is to briefly examine these four modeling steps, with particular emphasis on the first since the choice of the model class C ultimately determines the utility of the empirical model, both with respect to the application (e.g., the difficulty of solving the resulting model-based control problem) and with respect to fidelity of approximation. Some of the basic issues of model structure selection are introduced in Sec. 8.1 and a more detailed treatment is given in Sec. 8.3, emphasizing connections with results presented in earlier chapters; in addition, the problem of model structure selection is an important component of the case studies presented in Secs. 8.2 and 8.5. The second step in this procedure—input sequence design—is discussed in some detail in Sec. 8.4 and is an important component of the second case study (Sec. 8.5). The literature associated with the parameter estimation problem—the third step in the empirical modeling process—is much too large to attempt to survey here, but a brief summary of some representative results is given in Sec. 8.1.1. Finally, the task of model validation often depends strongly on the details of the physical system being modelled and the ultimate application intended for the model. Consequently, detailed treatment of this topic also lies beyond the scope of this book but again, some representative results are discussed briefly in Sec. 8.1.3 and illustrated in the first case study (Sec. 8.2). Finally, Sec. 8.6 concludes both the chapter and the book with some philosophical observations on the problem of developing moderate-complexity, discrete-time dynamic models to approximate the behavior of high-complexity, continuous-time physical systems.