Nonquadratic Lyapunov functions for robust control

Abstract

This paper deals with the problem of controlling a linear continuous-time system with structured time-varying parameter uncertainties and input disturbances with a Lyapunov-function approach. In contrast with most of the previous results in the literature, we do not confine our attention to the class of quadratic Lyapunov functions. Conversely, the basic motivation of this paper is to determine whether there exist other functions that can be conveniently used as candidate Lyapunov functions. This question has a positive answer: the proposed class is that of polyhedral norms or, more generally, of polyhedral Minkowski functionals. We show that the class of these functions is universal in the sense that if the problem of ultimately bounding the state in an assigned convex set via state feedback control can be solved via a Lyapunov function and a continuous state-feedback compensator then it can be solved via a polyhedral Lyapunov function and a (possibly different) continuous control. Moreover, we show that the control can be piecewise linear. A numerical technique for constructing the controller is presented for the case in which the uncertainty constraint sets are polyhedral.