Volterra Models

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).