A Universal Feedback Controller for Discontinuous Dynamical Systems Using Nonsmooth Control Lyapunov Functions

Abstract

The consideration of nonsmooth Lyapunov functions for proving stability of feedback discontinuous systems is an important extension to classical stability theory since there exist nonsmooth dynamical systems whose equilibria cannot be proved to be stable using standard continuously differentiable Lyapunov function theory. For dynamical systems with continuously differentiable flows, the concept of smooth control Lyapunov functions was developed by Artstein to show the existence of a feedback stabilizing controller. A constructive feedback control law based on a universal construction of smooth control Lyapunov functions was given by Sontag. Even though a stabilizing continuous feedback controller guarantees the existence of a smooth control Lyapunov function, many systems that possess smooth control Lyapunov functions do not necessarily admit a continuous stabilizing feedback controller. However, the existence of a control Lyapunov function allows for the design of a stabilizing feedback controller that admits Filippov and Krasovskii closed-loop system solutions. In this paper, we develop a constructive feedback control law for discontinuous dynamical systems based on the existence of a nonsmooth control Lyapunov function defined in the sense of generalized Clarke gradients and set-valued Lie derivatives.The notation used in this paper is fairly standard. Specifically, R denotes the set of real numbers, R n denotes the set of n  1 real column vectors, Z þ denotes the set of non-negative integers, and (Á) T denotes transpose. We write @S and S to denote the boundary and the closure of the subset S & R n , respectively. Furthermore, we write k Á k for the Euclidean vector norm on R n , B e ðaÞ; a 2 R n ; e > 0, for the open ball centered at a with radius e, distðp; MÞ for the distance from a point p to the set M, that is, distðp; MÞ ¼ D inf x2M k p À x k, and xðtÞ ! M as t ! 1 to 1 Downloaded From: http://dynamicsystems.asmedigitalcollection.asme.org/ on 05/25/2015 Terms of Use: http://asme.org/terms