Economic Linear Quadratic MPC With Non-Unique Optimal Solutions

TL;DR

This paper investigates the stability of economic linear quadratic MPC when the rotated cost is only positive semi-definite, connecting it to dissipativity and Riccati equation properties, and extends stability results without terminal constraints.

Contribution

It bridges the gap between semi-definite rotated costs and stability in economic MPC, linking dissipativity, Riccati equations, and optimal control problem properties.

Findings

Established connection between semi-definite rotated cost and stability.

Analyzed properties of the constrained generalized Riccati equation.

Extended stability results to cases without terminal constraints.

Abstract

Asymptotic stability in economic receding horizon control can be obtained under a strict dissipativity assumption, related to positive-definiteness of a so-called rotated cost, and through the use of suitable terminal cost and constraints. In the linear-quadratic case, a common assumption is that the rotated cost is positive definite. The positive semi-definite case has received surprisingly little attention, and the connection to the standard dissipativity assumption has not been investigated. In this paper, we fill this gap by connecting existing results in economic model predictive control with the stability results for the semi-definite case, the properties of the constrained generalized discrete algebraic Riccati equation, and of two optimal control problems. Moreover, we extend recent results relating exponential stability to the choice of terminal cost in the absence of terminal constraints.