An adaptive controller that solves the problem of rendering a system passive is presented for a special class of systems with parametric uncertainty. The proposed control uses techniques of speed-gradient methodology from the Russian literature. Stability results can be given that do not require commonly used assumptions such as linearity in the parameters.
Algorithms that render the system strictly passive clarify some already known in the control of mechanical systems.
1. Introduction
We consider nonlinear adaptive control (NLAC) in the sense of adaptive control of plants having unknown parameters and known nonlinearities. Some important contributions to this field were surveyed in Kokotovic (1991b), where, roughly speaking, two main directions of research can be identified. On the one hand, adaptive schemes are presented for nonlinear systems in special structures or canonical forms that can be obtained under certain geometric conditions. On the other hand, some solutions are based on the Lyapunov approach, assuming the existence of a Lyapunov function for the system. This last approach has also been taken by Fradkov (1979Fradkov ( , 1990) ) and Fomin et al. (1981). These works provided adaptive controllers based on the so-called speed-gradient (SG) method, which ultimately relies upon the Lyapunov approach to stabilization. The essential idea of this scheme is to regulate the motion of adjustable parameters in the direction of the gradient of the rate of change of a functional that measures the desired performance of the system. This functional is required to be convex in the unknown parameters. Thus the main advantage of SG is to extend the allowable system parameter dependence from linear to 'speed-convex'. Other nonlinear adaptive controllers based on the Lyapunov approach were proposed by Praly (1991) and Pomet and Praly (1992).
Recently, renewed interest in passivity has produced new results not only in linear adaptive control (Krstic et al., 1993;