A Stability Condition for Online Feedback Optimization without Timescale Separation

TL;DR

This paper proves that Online Feedback Optimization can be stable without requiring the controller to operate slower than the plant, using a Lyapunov-based analysis that is independent of plant time constants.

Contribution

It introduces a new stability condition for OFO that does not depend on timescale separation, enhancing responsiveness and scalability.

Findings

Stability of OFO without timescale separation is theoretically proven.

The proposed condition is independent of plant time constants.

Numerical examples validate the theoretical results.

Abstract

Online Feedback Optimization (OFO) is a control approach to drive a dynamical plant to an optimal steady state. By interconnecting optimization algorithms with real-time plant measurements, OFO provides all the benefits of feedback control, yet without requiring exact knowledge of plant dynamics for computing a setpoint. On the downside, existing stability guarantees for OFO require the controller to evolve on a sufficiently slower timescale than the plant, possibly affecting transient performance and responsiveness to disturbances. In this paper, we prove that, under suitable conditions, OFO ensures stability without any timescale separation. In particular, the condition we propose is independent of the time constant of the plant, hence it is scaling-invariant. Our analysis leverages a composite Lyapunov function, which is the $\max$ of plant-related and controller-related components. We corroborate our theoretical results with numerical examples.