Stability of finite horizon model predictive control with incremental input constraints

Model predictive control of discrete-time nonlinear systems with incremental input constraints is proposed in this paper. Firstly, the existence of the terminal set and terminal penalty is proven on the assumption that the considered system is twice continuously differentiable. Secondly, properties of the optimal cost function are exploited. It shows that the optimal cost function is positive semi-definite, continuous at the equilibrium and monotonically decreasing along the predicted trajectory. The systems under control converge to the equilibrium since the optimal cost function is monotonically decreasing. Thirdly, stability of nonlinear systems is proven in terms of the classical Lyapunov Theorem, where an upper bound of the optimal cost function in the terminal set is chosen as a candidate Lyapuonv function. Finally, the system is asymptotically stable since the system state converges to the equilibrium and the system is stable.