Nonlinear robust state feedback control of uncertain polynomial discrete-time systems: An integral action approach

This paper examines the problem of designing a nonlinear robust state feedback controller for uncertain polynomial discrete-time systems. In general, this is a challenging controller design problem due to the fact that the relation between Lyapunov function and the control input is not jointly convex, hence, this problem cannot be solved by a semidefinite programming (SDP). In this paper, a novel approach is proposed, where an integral action is incorporated into the controller design so that a convex solution to the problem can be rendered. Based on the sum of squares (SOS) approach, sufficient conditions for the existence of a nonlinear state feedback controller for polynomial discrete-time systems are given in terms of solvability of polynomial matrix inequalities (PMIs), which can be solved by the recently developed SOS solver. Numerical examples are provided to demonstrate the validity of this integral action approach.