Adaptive control for a discrete-time first-order nonlinear system with both parametric and non-parametric uncertainties

A simple first-order discrete-time nonlinear system, which has both parametric uncertainty and non parametric uncertainty, is studied in this paper. 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. An adaptive controller is constructed for this model in both cases of b = 1 and b > 1, and its closed-loop stability is established under some conditions. When b = 1, the conditions given reveal the magic number 3/2 + √2 which appeared in previous study on capability and limitations of the feedback mechanism.

I. Introduction

Feedback, a fundamental concept in automatic control, aims to reduce the effects of the plant uncertainty on the desired control performance. Because of the essence of the feedback control, there has been much effort devoted to the control of uncertain dynamical systems in the history of control, particularly in the areas of adaptive control and robust control (e.g. [1], [2], [3], [4], [5], [6]). In these areas, considerable progress has been made in dealing with uncertainties of dynamical systems; however, systematic and quantitative characterization of the maximum capability and limitation of the whole feedback mechanism only appeared in the last decade. The term “feedback mechanism” refers to all possible feedback control laws and hence it is not restricted in a class of special control laws. A brief survey on this challenging topic can be found in the plenary lecture [7] by Guo in International Congress of Mathematicians, 2002.