On the robustness of optimal regulators for nonlinear discrete-time systems

In this paper the robustness of nonlinear discrete-time systems is analyzed. The nominal plant is supposed to be controlled by means of a feedback control law which is optimal with respect to some given criterion. The robustness of the closed-loop system is studied for two different classes of perturbations in the control law, which are called gain and additive nonlinear perturbations. The results are entirely based on the existence of a stationary solution of the dynamic programming equation (DPE), which provides directly a Lyapunov function associated to the closed-loop system. The convexity of that solution and the use of the Taylor formula appear to be the key to establish the robustness properties of the nominal plant. Two examples are solved in order to show an interesting fact: the existence of a compromise between the robustness of the system subjected to the two different classes of perturbations.