Extension of local linear controllers to global piecewise affine controllers for uncertain non-linear systems

A two-step controller synthesis method is proposed in this paper for a class of uncertain nonlinear systems described by piecewise affine differential inclusions. In the first step, a robust linear controller is designed for the linear differential inclusion that describes the dynamics of the nonlinear system close to the equilibrium point. In the second step, a stabilizing piecewise affine controller is designed that coincides with the linear controller in a region around the equilibrium point. The proposed method has two objectives: global stability and local performance. It thus enables to use well known techniques in linear control design for local stability and performance while delivering a global piecewise affine controller that is guaranteed to stabilize the nonlinear system. To construct the required theoretical framework, a stability theorem for nonsmooth Lyapunov functions is presented and proved. The new method will be applied to two examples.