Economic receding horizon control without terminal constraints

Lars Grüne

doi:10.1016/j.automatica.2012.12.003

Abstract

We consider a receding horizon control scheme without terminal constraints in which the stage cost is defined by economic criteria, i.e., not necessarily linked to a stabilization or tracking problem. We analyze the performance of the resulting receding horizon controller with a particular focus on the case of optimal steady states for the corresponding averaged infinite horizon problem. Using a turnpike property and suitable controllability properties we prove near optimal performance of the controller and convergence of the closed loop solution to a neighborhood of the optimal steady state. Two examples illustrate our findings numerically and show how to verify the imposed assumptions.

Highlights

In this paper we investigate the performance of receding horizon control schemes with general stage costs
In receding horizon control — often called model predictive control (MPC) — a feedback law is synthesized from the first elements of finite horizon optimal control sequences which are iteratively computed along the closed loop solution
We provide sufficient conditions based on certain controllability assumptions and on the turnpike property, which is a classical tool in optimal control [5, Section 4.4], for understanding the optimal dynamics of economic control problems [10]

Summary

Introduction

In this paper we investigate the performance of receding horizon control schemes with general stage costs. Preprint submitted to Automatica orbit for the infinite horizon averaged problem and this solution is used as a terminal constraint for the finite horizon optimal control problem to be solved in each step of the receding horizon scheme In contrast to these references, in this paper we do not impose any terminal constraints. The price we pay for removing the terminal constraints is on the one hand a more involved analysis using stronger assumptions on the underlying finite horizon problems To this end, we provide sufficient conditions based on certain controllability assumptions and on the turnpike property, which is a classical tool in optimal control [5, Section 4.4], for understanding the optimal dynamics of economic control problems [10].

Problem formulation and preliminaries

Motivating examples

Optimal steady states and the turnpike property

Controllability conditions

Trajectory convergence

Conclusions and outlook