Stability of Finite Horizon Optimisation based Control without Terminal Weight

TL;DR

This paper introduces a new stability analysis method for finite horizon MPC without terminal weight, using an auxiliary value function called OSVF, and proposes a stabilising algorithm based on this approach.

Contribution

It defines the OSVF as a control Lyapunov function and develops a new stabilising MPC algorithm (CMPC) that guarantees stability without terminal weight.

Findings

OSVF can serve as a control Lyapunov function for stability.

The proposed CMPC algorithm is recursively feasible and stabilising.

Numerical examples validate the effectiveness of the stability condition.

Abstract

This paper presents a stability analysis tool for model predictive control (MPC) where control action is generated by optimising a cost function over a finite horizon. Stability analysis of MPC with a limited horizon but without terminal weight is a well known challenging problem. We define a new value function based on an auxiliary one-step optimisation related to stage cost, namely optimal one-step value function (OSVF). It is shown that a finite horizon MPC can be made to be asymptotically stable if OSVF is a (local) control Lyapunov function (CLF). More specifically, by exploiting the CLF property of OSFV to construct a contractive terminal set, a new stabilising MPC algorithm (CMPC) is proposed. We show that CMPC is recursively feasible and guarantees stability under the condition that OSVF is a CLF. Checking this condition and estimation of the maximal terminal set are discussed. Numerical examples are presented to demonstrate the effectiveness of the proposed stability condition and corresponding CMPC algorithm.