STABLE ADAPTIVE CONTROL AND ESTIMATION FOR NONLINEAR SYSTEMS Neural and Fuzzy Approximator Techniques

Approximator-based control, primarily using neural networks and/or fuzzy systems as the main tool for function approximation, had been regarded as non-rigorous, but sold under the fashionable name of intelligent control. To some extent, this view point has some elements of true in it as historically it was indeed the case where we only knew the existence of a stabilizing controller but lacked the techniques to construct it. As the approximator-based control matures, we cannot only design stable controllers constructively, but also be able to quantify closed-loop performance. From the view point of control system design, we have been talking about building models that are complex enough to capture the dominant dynamics of the systems, yet simple enough for us to do control system design. In almost all cases, we have to simplify the model, under some assumptions, so that control system design can be carried out. Sometimes, we have to linearize the simpli=ed model as well in order to tap the rich linear system theory for control system design. Though our ultimate objective is to control the original systems, “approximation” has been accepted in the control community except for that it was introduced at the modeling stage, and mathematical rigor was emphasized for the subsequent analysis. By assuming that the systems took certain simpli=ed forms, such as linear-inthe-parameters, then carrying out the design rigorously with ease and elegance, it does not mean that practical systems are indeed in the simpli=ed forms. Modeling errors do exist, and robust control is supposed to handle this very fact. On the other hands, approximator-based control takes a diAerent approach. Recognizing the very fact that realistic model building itself might be more diBcult for complex systems in practice than controller design, many researchers have devoted to function approximator-based control design with guaranteed closed-loop stability and control performance using neural networks (Ge, Lee, & Harris, 1998; Ge, Hang, Lee, & Zhang, 2001; Lewis, Jagannathan, & Yesildirek, 1999; Narendra & Lewis, 2001; Poznyak, Sanchez, & Yu, 2001) and fuzzy systems (Wang, 1994) as the main parametrization tools, for unknown nonlinear systems without requiring the parametrization form, such as linear-in-the-parameters, as an attractive alternative though using approximators may raise the complexity of computing. Though many developed designs utilize adaptive control techniques and neural/fuzzy parametrization in the linear-in-the-parameters form, the residue approximation errors have to be dealt with explicitly in controller design. Though a small change in approach, it reduces the workload on modeling dramatically, and speeds up the development of a working control system. Approximator-based control deserves due recognition in the context of advanced control system design. Thanks to the collective eAorts of many researchers, many remarkable and fundamental contributions have been made in approximator-based control with rigorous mathematical treatments, and detailed analysis of stability, robustness and convergence of the closed-loop systems (Ge et al., 1998, 2001; Lewis et al., 1999; Lewis & Parisini, 1998; Narendra & Lewis, 2001; Poznyak et al., 2001; Wang, 1994).