System identification, as a particular process of statistical inference, exploits two types of information. The first is experiment; the other, called a priori, is known before making any measurements. In a wide sense, the a priori information concerns the system itself and signals entering the system. Elements of the information are, for example:

• the nature of the signals, which may be random or nonrandom, white or correlated, stationary or not, their distributions can be known in full or partially (up to some parameters) or completely unknown,

• general information about the system, which can be, for example, continuous or discrete in the time domain, stationary or not,

• the structure of the system, which can be of the Hammerstein or Wiener type, or other,

• the knowledge about subsystems, that is, about nonlinear characteristics and linear dynamics.

In other words, the a priori information is related to the theory of the phenomena taking place in the system (a real physical process) or can be interpreted as a hypothesis (if so, results of the identification should be necessarily validated) or can be abstract in nature.

This book deals with systems consisting of nonlinear memoryless and linear dynamic subsystems, for example, Hammerstein and Wiener systems and other related structures.

In this chapter, we discuss the problem of identification of a class of semiparametric block-oriented systems. This class of block-oriented systems can be restricted to a parameterization that includes a finite-dimensional parameter and nonlinear characteristics that run through a nonparametric class of mostly univariate functions. The parametric part of a semiparametric model defines characteristics of linear dynamical subsystems and low-dimensional projections of multivariate nonlinearities. The nonparametric part of the model comprises all static nonlinearities defined by functions of a single variable. A general methodology for identifying semiparametric block-oriented systems is developed. This includes a semiparametric version of least squares and a direct method using the concept of the average derivative of a regression function. These general approaches are applied in cases of semiparametric versions of Wiener, Hammerstein, and parallel systems. Section 14.2 gives examples of semiparametric block-oriented systems. This includes the multivariate version of Hammerstein and Wiener systems. In Section 14.3, we give a general approach to semiparametric inference. Section 14.4 is devoted to an important case study concerning the semiparametric Wiener system. Sections 14.5 and 14.6 provide similar considerations for semiparametric Hammerstein and parallel systems. In Section 14.7, we derive direct estimation methods for semiparametric nonlinear systems.

Introduction

In all of the preceding chapters, we have examined various fully nonparametric block-oriented systems.

Thus far we have examined block-oriented systems of the cascade form, namely the Hammerstein and Wiener systems. The main tool that was used to recover the characteristics of the systems was based on the theory of nonparametric regression and correlation analysis. In this chapter, we show that this approach can be successfully extended to a class of block-oriented systems of the series-parallel form as well as systems with nonlinear dynamics. The latter case includes generalized Hammerstein and Wiener models as well as the sandwich system. We highlight some of these systems and present identification algorithms that can use various nonparametric regression estimates. In particular, Section 12.1 develops nonparametric algorithms for parallel, series-parallel, and generalized nonlinear block-oriented systems. Section 12.2 is devoted to a new class of nonlinear systems with nonlinear dynamics. This includes the important sandwich system as a special case.

Series-parallel, block-oriented systems

The cascade nonlinear systems presented in the previous chapters define the fundamental building blocks for defining general models of series-parallel forms. Together, all of these models may create a useful class of structures for modeling various physical processes. The choice of a particular model depends crucially on physical constraints and needs.

In this section, we present a number of nonlinear models of series-parallel forms for which we can relatively easily develop identification algorithms based on the regression approach used throughout the book.

In all of the preceding chapters, we have examined the identification problem for block-oriented systems of various forms, that are characterized by a one-dimensional input process. In numerous applications, we confront the problem of identifying a system that has multiple inputs and multiple interconnecting signals. The theory and practical algorithms for identification of multivariate linear systems have been thoroughly examined in the literature [332]. On the other hand, the theory of identification of multivariate nonlinear systems has been far less explored. This is mainly due to the mathematical and computational difficulties appearing in multivariate problems. In this chapter, we examine some selected multivariate nonlinear models that are natural generalizations of the previously introduced block-oriented connections. An apparent curse of dimensionality that takes place in high-dimensional estimation problems forces us to focus on low-dimensional counterparts of the classical block-oriented structures. In particular, we examine a class of additive models, which provides a parsimonious representation for multivariate systems. Indeed, we show that the additive systems provide simple and interpretable structures, which also give a reasonable trade-off between the systematic modeling error and the estimation error of an identification algorithm. The theory of finding an optimal additive model is examined.

Multivariate nonparametric regression

As in all of the previous chapters, we will make use of the notion of a regression function.

The aim of this book is to show that the nonparametric regression can be applied successfully to nonlinear system identification. It gathers what has been done in the area so far and presents main ideas, results, and some new recent developments.

The study of nonparametric regression estimation began with works published by Cencov, Watson, and Nadaraya in the 1960s. The history of nonparametric regression in system identification began about ten years later. Such methods have been applied to the identification of composite systems consisting of nonlinear memoryless systems and linear dynamic ones. Therefore, the approach is strictly connected with so-called block-oriented methods developed since Narendra and Gallman's work published in 1966. Hammerstein and Wiener structures are most popular and have received the greatest attention in numerous applications. Fundamental for nonparametric methods is the observation that the unknown characteristic of the nonlinear subsystem or its inverse can be represented as regression functions.

In terms of the a priori information, standard identification methods and algorithms work when it is parametric, that is, when our knowledge about the system is rather large; for example, when we know that the nonlinear subsystem has a polynomial characteristic. In this book, the information is much smaller, nonparametric. The mentioned characteristic can be, for example, any integrable or bounded or, even, any Borel function.