Rethinking Strict Dissipativity for Economic MPC

TL;DR

This paper introduces the concept of two-storage strict dissipativity for economic MPC, providing a more verifiable condition for stability that relates to value functions and includes new terminal cost strategies.

Contribution

It proposes a novel two-storage strict dissipativity condition, linking it to optimal control and stability, and discusses new terminal cost designs for finite-horizon stability guarantees.

Findings

Two-storage strict dissipativity is necessary and sufficient for asymptotic stability.

The new condition relates directly to value functions and is easier to verify.

New terminal cost strategies are proposed for finite-horizon stability.

Abstract

Stability of economic model predictive control can be proven under the assumption that a strict dissipativity condition holds. This assumption has a clear interpretation in terms of the so-called rotated stage cost, which must have its minimum at the optimal steady state. However, contrary to dissipativity, for strict dissipativity the storage function cannot be immediately related to the value function of an optimal control problem formulated with the economic stage cost. We propose the novel concept of two-storage strict dissipativity, which requires two storage functions to satisfy dissipativity and be separated by a positive definite function. This new condition can be immediately related to optimal control by means of value functions and might be easier to verify than strict dissipativity. Furthermore, we prove that two-storage strict dissipativity is sufficient and necessary for asymptotic stability, it is related to strict dissipativity, and also to alternative approaches relying on the so-called cost-to-travel. Finally, we discuss commonly used and new terminal cost designs that guarantee asymptotic stability in the finite-horizon case.