Nonsmooth backstepping design for a class of parametric strict-feedback nonlinear systems based on Lipschitz feedback

In order to obtain flexible feedback performance for a class of parametric strict-feedback nonlinear systems, a nonsmooth backstepping design method is developed to construct the nonsmooth feedback controller such that the feedback gain can be designed according to the prespecified regions in the state space. With the nonsmooth controller, the closed-loop system is presented to be discontinuous in the new coordinates. The trajectories of the closed-loop system satisfy a differential inclusion defined by Filippov such that all the trajectories can be shown to cross the boundary of the regions. It implies that the dynamics of the trajectories on the switching condition can be ignored.