Nonlinear feedback laws for practical stabilization of systems with input and state constraints

This paper presents a method for designing nonlinear state feedback laws for systems with input and state constraints. The objective is to achieve practical stabilization with large stability region and strong disturbance rejection. Two invariant sets will be constructed within the state constraints: the outer one for stabilization and the inner one for asymptotic disturbance rejection. The nonlinear feedback law is designed such that all trajectories starting from the outer invariant set will enter the inner invariant set and stay there. Both invariant sets will be constructed by using the convex hull function, a recently introduced non-quadratic Lyapunov function. Since the invariant sets are convex hull of ellipsoids, they are able to incorporate the input and state constraints more effectively than simple ellipsoids, thus promising larger stability region within state constraint and stronger disturbance rejection capability. Since the convex hull functions are constructed from quadratic functions, the optimization problems can be treated with LMI-based method. Numerical examples demonstrate the effectiveness of the design methods.