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 7 - Adaptive Approximation Based Control: General Theory
Chapter 6 motivated the use of adaptive approximation based control methods and discussed some of the key issues involved in the use of such methods for feedback control. In order to allow the reader to focus on the crucial issues without the distraction of mathematical complexities that occur while considering high-order systems, the design and analysis of that chapter was carried out on a class of scalar nonlinear systems. In this chapter, the design and analysis is extended to higher-order systems. The objective of this chapter is to illustrate the design of adaptive approximation based control schemes for certain classes of n-th order nonlinear systems and to provide a rigorous stability analysis of the resulting closed-loop system. Although the mathematics become more involved as compared to Chapter 6, several important aspects of adaptive approximation extend directly from that previous analysis. These issues – such as stability analysis, control robustness, ensuring that the state remains in the region D, and robustness modifications in the adaptive laws – are highlighted so that the reader is able to extract useful intuition for why various components of the control design follow a certain structure. A key objective is to help the reader obtain a sufficiently deep understanding of the mathematical analysis and design so that the results herein can be extended to a larger class of nonlinear systems or to a specific application whose model does not exactly fit within a standard class of nonlinear systems. The design and analysis of adaptive approximation based control in this chapter is applied to two general classes of nonlinear systems with unknown nonlinearities: (i) feedback linearizable systems (Section 7.2); and (ii) triangular nonlinear systems that allow the use of the backstepping control design procedure (Section 7.3). For each class of nonlinear systems, we first consider the ideal case where the uncertainties can be approximated exactly by the selected approximation model within a certain operating region of interest (i.e., the Minimum Function Approximation Error (MFAE) is zero within a certain domain D), and then we consider the case that includes the presence of residual approximation errors and disturbances. The latter case is referred to as robust adaptive approximation based control. As we will see, to achieve robustness, we utilize a modification in the adaptive laws for updating the weights of the adaptive approximation. This modification in the adaptive laws is based on a combination of projection and dead-zone – techniques that have been introduced in Chapter 4, and also used in Chapter 6. It is important to note that this chapter follows a structure parallel to that of Chapter 5 where we introduced various design and analysis tools for nonlinear systems under the assumption that the nonlinearities were known. In this chapter, we revisit these techniques (e.g., feedback linearization, backstepping), with adaptive approximation models representing the unknown nonlinearities. |
