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 article for a class of uncertain non-linear 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 non-linear system close to the equilibrium point. In the second step, a stabilising 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 us 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 stabilise the non-linear system. To construct the required theoretical framework, a stability theorem for non-smooth Lyapunov functions is presented and proved. The new method will be applied to two examples.