Local Stability and Convergence of Unconstrained Model Predictive Control

TL;DR

This paper analyzes the local stability and convergence properties of unconstrained nonlinear MPC using a linear plant model, deriving explicit error estimates and demonstrating exponential convergence under certain conditions.

Contribution

It provides explicit error bounds and convergence rates for unconstrained nonlinear MPC based on a linear model, including effects of prediction and control horizons.

Findings

MPC controls converge exponentially to infinite-horizon solutions when T - τ approaches infinity.

Local stability is achieved when the plant model closely matches the nonlinear dynamics near the origin.

Numerical simulations validate the theoretical convergence rates.

Abstract

The local stability and convergence for Model Predictive Control (MPC) of unconstrained nonlinear dynamics based on a linear time-invariant plant model is studied. Based on the long-time behavior of the solution of the Riccati Differential Equation (RDE), explicit error estimates are derived that clearly demonstrate the influence of the two critical parameters in MPC: the prediction horizon $T$ and the control horizon $\tau$. In particular, if the MPC-controller has access to an exact (linear) plant model, the MPC-controls and the corresponding optimal state trajectories converge exponentially to the solution of an infinite-horizon optimal control problem when $T-\tau \rightarrow \infty$. When the difference between the linear model and the nonlinear plant is sufficiently small in a neighborhood of the origin, the MPC strategy is locally stabilizing and the influence of modeling errors can be reduced by choosing the control horizon $\tau$ smaller. The obtained convergence rates are validated in numerical simulations.