Nonlinear State Feedback Control for A Class of Polynomial Discrete-time Systems with Norm-Bounded Uncertainties: An Integrator Approach

This paper investigates the problem of designing a nonlinear state feedback controller for a class of uncertain polynomial discrete-time systems using rational Lyapunov functions. The uncertainty that under consideration is modelled as a norm-bounded uncertainty. In general, the problem of designing a controller for polynomial discrete-time systems cannot be formulated as a convex problem. This is due to the fact that the Lyapunov function and the control input is not jointly convex, hence it cannot be solved by a semidefinite programming (SDP). In this paper, we propose a novel approach where an integrator is introduced to convexify the nonconvex controller design problem, so that it can be solved easily by SDP. Furthermore, based on the sum of squares approach, sufficient conditions for the existence of a rational polynomial state feedback controller for a polynomial discrete-time systems are given in terms of solvability of polynomial matrix inequalities. These inequalities are then solved by the recently developed sum of squares (SOS) solvers. Finally, numerical example is provided to demonstrate the validity of this integrator approach.