Stability analysis of model predictive controllers using Mixed Integer Linear Programming

It is a well known fact that finite time optimal controllers, such as MPC do not necessarily result in closed loop stable systems. Within the MPC community it is common practice to add a final state constraint and/or a final state penalty in order to obtain guaranteed stability. However, for more advanced controller structures it can be difficult to show stability using these techniques. Additionally in some cases the final state constraint set consists of so many inequalities that the complexity of the MPC problem is too big for use in certain fast and time critical applications. In this paper we instead focus on deriving a tool for a-postiori analysis of the closed loop stability for linear systems controlled with MPC controllers. We formulate an optimisation problem that gives a sufficient condition for stability of the closed loop system and we show that the problem can be written as a Mixed Integer Linear Programming Problem (MILP).

I. Introduction

A linear Model Predictive Controller (MPC) solves, online in each sample instant, a finite time horizon optimal control problem of the form \begin{align*} V_{k}^{\ast}=\underset{{x_{k+i}, u_{k+i}}}{\min.}\quad & \sum_{i=0}^{N-1}\ell(x_{k+i}, u_{k+i})+\Psi(x_{k+N}) \tag{1a}\\ \mathrm{s}.\mathrm{t}.\quad &x_{k+i+1}=Ax_{k+i}+Bu_{k+i} \tag{1b}\\ &Ex_{k+i}\leq f\quad i=1, \ldots, N-1 \tag{1c}\\ &Tx_{k+N}\leq t \tag{1d}\\ &Gu_{k+i}\leq h\quad i=1, \ldots, N-1 \tag{1e} \end{align*} and implements the optimal solution, , as the input to the system in a receding horizon fashion.