Adaptive Estimation and Control for Systems with Parametric and Nonparametric Uncertainties

Adaptive control has been developed for decades, and now it has become a rigorous and mature discipline which mainly focuses on dealing parametric uncertainties in control systems, especially linear parametric systems. Nonparametric uncertainties were seldom studied or addressed in the literature of adaptive control until new areas on exploring limitations and capability of feedback control emerged in recent years. Comparing with the approach of robust control to deal with parametric or nonparametric uncertainties, the approach of adaptive control can deal with relatively larger uncertainties and gain more flexibility to fit the unknown plant because adaptive control usually involves adaptive estimation algorithms which play role of “learning” in some sense. This chapter will introduce a new challenging topic on dealing with both parametric and nonparametric internal uncertainties in the same system. The existence of both two kinds of uncertainties makes it very difficult or even impossible to apply the traditional recursive identification algorithms which are designed for parametric systems. We will discuss by examples why conventional adaptive estimation and hence conventional adaptive control cannot be applied directly to deal with combination of parametric and nonparametric uncertainties. And we will also introduce basic ideas to handle the difficulties involved in the adaptive estimation problem for the system with combination of parametric and nonparametric uncertainties. Especially, we will propose and discuss a novel class of adaptive estimators, i.e. information-concentration (IC) estimators. This area is still in its infant stage, and more efforts are expected in the future for gainning comprehensive understanding to resolve challenging difficulties. Furthermore, we will give two concrete examples of semi-parametric adaptive control to demonstrate the ideas and the principles to deal with both parametric and nonparametric uncertainties in the plant. (1) In the first example, a simple first-order discrete-time nonlinear system with both kinds of internal uncertainties is investigated, where the uncertainty of non-parametric part is characterized by a Lipschitz constant L, and the nonlinearity of parametric part is characterized by an exponent index b. In this example, based on the idea of the IC estimator, we construct a unified adaptive controller in both cases of b = 1 and