On construction of smooth Lyapunov functions for non-smooth systems

The construction of smooth Lyapunov functions for non-smooth systems is reported. Although non-smooth Lyapunov functions are believed to be natural for non-smooth dynamic systems (Shevitz and Paden 1994), the determination of their generalized derivatives on the discontinuity surface 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 construction is facilitated by keeping the inner product of the discontinuous part of the rate of the state vector and the vector related to the gradient of a Lyapunov function with the limit value as zero when the solution trajectory approaches the discontinuity surface, and a zero value on the discontinuity surface. Four examples, including systems with stick-slip friction and sliding-mode control systems, are used to demonstrate the applicability of the method.