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 3 - Approximation Structures
The objective of this chapter is to present and discuss several neural, fuzzy, and traditional approximation structures in a unifying framework. The presentation will make direct references to the approximator properties presented in Chapter 2. In addition to introducing the reader to these various approximation structures, this chapter will be referenced throughout the remainder of the text. Each section of this chapter discusses one type of function approximator, presents the motivation for the development of the approximator, and shows how the approximator can be represented in one of the standard nonlinearly and linearly parameterized forms:  where x ∈ D ⊂n, θ ∈ The ultimate objective is to adjust the approximator parameters θ and σ to encode information that will enable better control performance. Proper design requires selection of a family of function approximators, specification of the structure of the approximator, and estimation of appropriate approximator parameters. The latter process is referred to as parameter estimation, adaptation, or learning. Such processes are discussed in Chapter 4.
Figure 3.1: Simple pendulum. |
N, σ ∈ 
: D 
