In parallel with developments in adaptive nonlinear control, there has been a tremendous amount of activity in neural control and adaptive fuzzy approaches. In these studies, neural networks or fuzzy approximators are used to approximate unknown nonlinearities. The input/output response of the approximator is modified by adjusting the values of certain parameters, usually referred to as weights. From a mathematical control perspective, neural networks and fuzzy approximators represent just two classes of function approximators. Polynomials, splines, radial basis functions, and wavelets are examples of other function approximators that can be used and have been used in a similar setting. We refer to such approximation models with adaptivity features as adaptive approximators, and control methodologies that are based on them as adaptive approximation based control. Adaptive approximation based control encompasses a variety of methods that appear in the literature: intelligent control, neural control, adaptive fuzzy control, memory-based control, knowledge-based control, adaptive nonlinear control, and adaptive linear control. |
During the last few years there have been significant developments in the control of highly uncertain, nonlinear dynamical systems. For systems with parametric uncertainty, adaptive nonlinear control has evolved as a powerful methodology leading to global stability and tracking results for a class of nonlinear systems. Advances in geometric nonlinear control theory, in conjunction with the development and refinement of new techniques, such as the backstepping procedure and tuning functions, have brought about the design of control systems with proven stability properties. In addition, there has been a lot of research activity on robust nonlinear control design methods, such as sliding mode control, Lyapunov redesign method, nonlinear damping, and adaptive bounding control. These techniques are based on the assumption that the uncertainty in the nonlinear functions is within some known, or partially known, bounding functions.
TABLE OF CONTENTS Chapter 2 - Approximation Theory
This chapter formulates the numeric data processing issues of interpolation and function approximation, and then discusses function approximator properties that are relevant to the use of adaptive approximation for estimation and feedback control. Our interest in function approximation is derived from the hypothesis that online control performance could be improved if unknown nonlinear portions of the model are more accurately modeled. Although the data to improve the model may not be available a priori, additional data can be accumulated while the system is operating. Appropriate use of such data to guarantee performance improvement requires that the designer understand the areas of function approximation, control, stability, and parameter estimation. This chapter focuses on several aspects of approximation theory. The discussion of function approximation is subdivided into offline and online approximation. Offline function approximation is concerned with the questions of selecting a family of approximators and parameters of a particular approximator to optimally fit a given set of data. The issue of the design of the set of data is also of interest when the acquisition of the data is under the control of the designer. An understanding of offline function approximation is necessary before delving into online approximation. The discussion of online approximation builds on the understanding of offline approximation, and also raises new issues motivated by the need to guarantee stability of the dynamic system and estimation process, the possible need to forget old stored information at a certain rate, and the inability to control the data distribution. Section 2.1 presents an easy-to-understand (and replicate) example in order to motivate, in the context of online approximation based control, a few important issues that will be discussed through the remainder of this chapter. Section 2.2 discusses the problem of function interpolation. Section 2.3 discusses the problem of function approximation. Section 2.4 discusses function approximator properties in the context of online function approximation. |
