Nominal Stability of the Real-Time Iteration Scheme for Nonlinear Model Predictive Control

We present and investigate a Newton type method for online optimization in nonlinear model predictive control, the so called “real-time iteration scheme”. In this scheme only one Newton type iteration is performed per sampling instant, and the control of the system and the solution of the optimal control problem are performed in parallel. In the resulting combined dynamics of system and optimizer, the actual feedback control in each step is based on the current solution estimate, and the solution estimates are at each sampling instant refined and transferred to the next optimization problem by a specially designed transition. This approach yields an efficient online optimization algorithm that has already been successfully tested in several applications. Due to the close dovetailing of system and optimizer dynamics, however, stability of the closed loop system is not implied by standard nonlinear model predictive control results. In this paper, we give a proof of nominal stability of the scheme which builds on concepts from both, NMPC stability theory and convergence analysis of Newton type methods. The principal result is that – under some reasonable assumptions – the combined system-optimizer dynamics can be guaranteed to converge towards the origin from significantly disturbed system-optimizer states.