NONQUADRATIC LYAPUNOV FUNCTIONS FOR ROBUST-CONTROL

This paper deals with the problem of controlling a linear continuous-t ime system with structured time-varying parameter uncertainties and in put disturbances with a Lyapunov-function approach. In contrast with m ost of the previous results in the literature, we do not confine our a ttention to the class of quadratic Lyapunov functions. Conversely, the basic motivation of this paper is to determine whether there exist ot her functions that can be conveniently used as candidate Lyapunov func tions. This question has a positive answer: the proposed class is that of polyhedral norms or, more generally, of polyhedral Minkowski funct ionals. We show that the class of these functions is universal in the sense that if the problem of ultimately bounding the state in an assig ned convex set via state feedback control can be solved via a Lyapunov function and a continuous state-feedback compensator then it can be s olved via a polyhedral Lyapunov function and a (possibly different) co ntinuous control. Moreover, we show that the control can be piecewise linear. A numerical technique for constructing the controller is prese nted for the case in which the uncertainty constraint sets are polyhed ral.