ON CONSTRUCTION OF SMOOTH LYAPUNOV FUNCTIONS FOR NONSMOOTH SYSTEMS

The construction of smooth Lyapunov functions for non-smooth systems i s reported. Although non-smooth Lyapunov functions are believed to be natural for non-smooth dynamic systems (Shevitz and Paden 1994), the d etermination of their generalized derivatives on the discontinuity sur face can be extremely difficult, especially when solution trajectories approach the intersection of discontinuity surfaces. The construction of smooth Lyapunov functions avoids such a difficulty. Such a constru ction is facilitated by keeping the inner product of the discontinuous part of the rate of the state vector and the vector related to the gr adient of a Lyapunov function with the limit value as zero when the so lution trajectory approaches the discontinuity surface, and a zero val ue on the discontinuity surface. Four examples, including systems with stick-slip friction and sliding-mode control systems, are used to dem onstrate the applicability of the method.