Adaptive Neural Backstepping Control Design for A Class of Nonsmooth Nonlinear Systems

This paper investigates the problems of backstepping control for a class of nonsmooth nonlinear systems. With the help of set-valued functions (or maps) and set-valued derivatives, a new Lyapunov criterion is developed for nonsmooth systems. The concept of semi-globally uniformly ultimately bounded (SGUUB) stability that is widely used for smooth nonlinear systems in lower triangular form is, for the first time, applied to the nonsmooth case, and such a concept provides a theory foundation for the subsequent backstepping control design. Then, two types of continuous controllers are designed based on the backstepping technique and adaptive neural network (NN) algorithm. In the light of Cellina approximate selection theorem and smooth approximation theorem for Lipschitz functions, the system under investigation is first transformed into an equivalent model. Then, the first type of controller can be efficiently designed by using the traditional backstepping technique. On the other hand, when the unknown uncertainty is taken into account in the equivalent systems, another type of controller is designed based on adaptive NN control strategy. It is proved that in both cases, all the closed-loop signals are SGUUB by using our proposed controllers. Finally, a numerical example with two types of controllers is supplied to show the availability of the provided control schemes.