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