Robust disturbance attenuation for a class of polynomial discrete-time systems with norm-bounded uncertainty: An integrator approach

This paper investigates the problem of designing a nonlinear feedback controller for a class of polynomial discrete-time systems with norm-bounded uncertainties. The objective of a controller design is to achieve robust stabilization and a prescribed level of H
_∞
performance simultaneously. In general, designing a controller for polynomial discrete-time systems is not a trivial problem. This is due to the fact that the relation between the Lyapunov function and the control input is not jointly convex, hence it cannot be solved by the semidefinite programming (SDP). In this paper, to decouple the Lyapunov function from the control input, an integrator is proposed to be incorporated into the controller structures. In doing so, a convex controller design can be rendered, hence the problem can consequently be solved via SDP. Furthermore, based on the sum of squares approach, sufficient conditions for the existence of a controller are given in terms of solvability of polynomial matrix inequalities (PMIs). These PMIs can be solved by the recently developed sum of squares (SOS) solvers. Finally, a numerical example is provided to demonstrate the validity of this integrator approach.