A revisit to MPC of discrete-time nonlinear systems

In this paper, model predictive control (MPC) of discrete-time nonlinear systems with guaranteed nominal stability is revisited. In general the optimal cost function of the optimization problem might be discontinuous in the state of the systems, though it is not valuable to be chosen as a candidate Lyapunov function. Compared with the existing results, in this paper the optimal cost function is only used to show the convergence of the system trajectory to its equilibrium. Instead stability is proven in terms of a candidate Lyapunov function which is locally twice continuously differentiable in a vicinity of the equilibrium. Furthermore, asymptotic stability is achieved by the stability of the considered systems together with the convergence of the system trajectory to its equilibrium. In the end, locally robustly asymptotic stability of model predictive control is proven based on the locally continuous Lyapunov funciton. That is, locally inherent robustness of MPC of nonlinear systems with respect to input constraints, state constraints and terminal constraints is proven.