On input-to-state stability of min-max nonlinear model predictive control

In this paper we consider discrete-time nonlinear systems that are affected, possibly simultaneously, by parametric uncertainties and disturbance inputs. The min-max model predictive control (MPC) methodology is employed to obtain a controller that robustly steers the state of the system towards a desired equilibrium. The aim is to provide a priori sufficient conditions for robust stability of the resulting closed-loop system using the input-to-state stability framework. First, we show that only input-to-state practical stability can be ensured in general for perturbed nonlinear systems in closed-loop with min-max MPC schemes and we provide explicit bounds on the evolution of the closed-loop system state. Then, we derive new conditions that guarantee input-to-state stability of the min-max MPC closed-loop system, using a dual-mode approach. An example illustrates the presented theory.