Identification and Control Using Volterra Models

This book is primarily concerned with the class of discrete-time, a class of mathematical models that is suitable for use in a variety of computer-based control strategies. The term Volterra model ultimately derives from the work of Vito Volterra at the end of the nineteenth century on the class of integral equations that now bears his name. Our primary motivation for considering discrete-time Volterra models is that they represent an extension of the linear convolution model on which linear model predictive control (MPC) strategies are based. As discussed further in the next section of this chapter, an extremely important open problem in practice is that of how to develop the models required for nonlinear MPC (NMPC) schemes to address the significant dynamic nonlinearities frequently encountered in practice. The discussion presented in this chapter is primarily limited to basic ideas, including a further discussion of motivations for considering discrete-time Volterra models, some very brief historical remarks about Volterra, his work, and some closely related work of others, a few remarks about the continuous-time Volterra models that emerge directly from Volterra's work, and ultimately a brief introduction to the problem of developing discrete-time dynamic models for computer control. The chapter then concludes with an overview of how the rest of the book is organized.

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.

Chapter 2 introduced the finite Volterra model class V(_N,M), its infinite-dimensional extensions, and four important special cases: the Hammerstein models, the Wiener models, the Uryson models, and the projection-pursuit models. This chapter presents a brief overview of some typical applications of these models, focusing on the particular model structure chosen and the underlying reasons for this choice. Initially, this chapter focuses on the following four special cases:

This chapter is primarily concerned with the problem of estimating unknown Volterra model parameters from input-output data, although a brief discussion is also given of the approximation of continuous-time nonlinear models by discrete-time Volterra models. As noted in Chapter 3, the following four special cases of the general Volterra model seem to arise most commonly in practice

In practice, the empirical model-building process usually consists of the following steps:

The first half of this book has been primarily concerned with characterizing Volterra models and identifying such models from input/output plant data. Beginning with this chapter, and for the next four chapters, we shift attention to the problem of how these models are used for designing effective controllers.

Within the IMC framework, synthesis results for several advanced control methodologies, such as feedforward control and constrained control (anti-windup control), have been proposed for linear systems. The notion of incorporating constraints into nonlinear IMC (NLIMC) has been explored by a number of investigators. Li and Biegler (1988) developed a single-step method that incorporated state and input constraints into the NLIMC framework. The control action was calculated using a specialized sequential quadratic programming problem and assumed that the model was perfect (plant = model). In that case, the incorporation of constraints into NLIMC resulted in a controller that preserved the structure of the partitioned nonlinear inverse (a linear dynamic controller with an additive nonlinear correction); however, it did not yield a direct synthesis nonlinear controller. In addition, compensation for measured disturbances, standard for linear IMC approaches (LIMC), was not considered. Wassick and Camp (1988) applied NLIMC to an industrial extruder and detailed several important considerations for application of nonlinear control in industry. Advanced issues such as nonlinear feedforward control and manipulated variable constraints were addressed in an effective, but sub-optimal, manner.

The preceding chapters have addressed the synthesis of controllers from Volterra series models. In Chapter 6, the concept of a partitioned inverse was introduced in the IMC framework to allow the explicit use of a Volterra series model in the controller design. Extensions were sketched for problematic dy¬namics, such as nonminimum phase behavior, although the derivations were carried out for the single-input-single-output (SISO) problem. In Chapter 7, a procedure was introduced to compensate for constraints, using classical techniques from constrained linear control theory. In the most general case, where multivariable interactions, problematic dynamics, and constraints are present, an MPC approach is the natural choice. As indicated in Chapter 6, there are natural extensions of the IMC algorithm to the MPC framework.

In this chapter, five application case studies are considered, consisting of various chemical and biochemical processes. These examples are chosen to reflect the experience of the authors in the identification and control of Volterra models, and they also reflect a range of challenging nonlinear dynamical systems. The tools from the preceding eight chapters will be explored in these case studies, highlighting their relevance and effectiveness for the particular problem. These include identification, analysis (stability), and controller synthesis.

The principal focus of this book has been on the topic of discrete-time Volterra models, motivated first by the need for discrete-time dynamic models as a basis for nonlinear model-based computer control applications, and second by the analytical and practical advantages of the Volterra model class relative to other nonlinear model classes. In particular, the class of discrete-time Volterra models may be viewed as a generically well-behaved extension of the class of linear FIR models on which typical linear MPC applications are based. One general feature of finite Volterra models discussed in Chapter 2 is that these models preserve many important qualitative characteristics of the input sequence, including boundedness (i.e. all V(_N,_M) models are BIBO stable), asymptotic constancy, and periodicity. These observations stand in marked contrast to even the simplest of polynomial models involving nonlinear autoregressive terms, which generally exhibit more strongly nonlinear behavior, including nonperiodic responses to periodic inputs, input-dependent stability, and responses to asymptotically constant inputs that do not settle out to constant values (e.g. chaotic step responses). In addition, since stable linear models that include autoregressive terms also preserve these input characteristics, many of the generic characteristics of the V(N,M) model class carry over to extensions like the AR-Volterra model class, which include linear autoregressive terms.