Input-to-state stability: an unifying framework for robust model predictive control

This paper deals with the robustness of Model Predictive Controllers for constrained uncertain nonlinear systems. The uncertainty is assumed to be modeled by a state and input dependent signal and a disturbance signal. The framework used for the analysis of the robust stability of the systems controlled by MPC is the wellknown Input-to-State Stability. It is shown how this notion is suitable in spite of the presence of constraints on the system and of the possible discontinuity of the control law. For the case of nominal MPC controllers, existing results on robust stability are extended to the ISS property, and some novel results are presented. Afterwards, stability property of robustMPC is analyzed. The proposed robust predictive controller uses a semi-feedback formulation and the notion of sequence of reachable sets (or tubes) for the robust constraint satisfaction. Under mild assumptions, input-to-state stability of the predictive controller based on nominal predicted cost is proved. Finally, stability of min-max predictive controllers is analyzed and sufficient conditions for the closed-loop system exhibits input-to-state practical stability property are stated. It is also shown how using a modified stage cost can lead to the ISS property. It is remarkable that ISS of predictive controllers is preserved in case of suboptimal solution of the minimization problem.