On Lyapunov's stability analysis of non-smooth systems with applications to control engineering

The extension of Lyapunov's stability theory to nun-smooth systems by Shevitz and Paden (Trans. Automat. Control 39 (1994) 1910) is modified with the goal of simplifying the procedure for construction of non-smooth Lyapunov functions. Shevitz and Paden's extension is built upon Filippov's solution theory and Clarke's generalized gradient. One important step in using their extension is to determine the generalized derivative of a nun-smooth Lyapunov function on a discontinuity surface, which involves the estimation uf an intersection of a number of convex sets. Such a determination is complicated and can become unmanageable for many systems. We propose to estimate the derivative of a nun-smooth Lyapunov function using the extreme points of Clarke's generalized gradient as opposed to the whole set. Such a modification not only simplifies the form. but also reduces the number of the convex sets involved in the estimation of the generalized derivative. This makes the stability analysis For some non-smooth systems practically easier. Three examples, including a mathematical system, a system with stick-slip friction compensator and an actuator having interaction with the environment, art: used For demonstration. (C) 2001 Elsevier Science Ltd. All rights reserved.