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Online Feedback Optimization for Monotone Systems without Timescale Separation

Mattia Bianchi, Florian Dörfler

TL;DR

Online Feedback Optimization (OFO) steers a dynamical plant toward a cost-minimizing steady state using real-time measurements and input-output sensitivities. The paper removes the traditional timescale-separation requirement by exploiting monotone plant structure and a small-gain theorem, proving global convergence to the optimal operating point for any controller gain $\alpha$, under conditions depending only on steady-state maps $k_x$ and $k_y$. It provides verifiable sufficient conditions and regularization guidance to ensure the required convexity properties, including in stable LTI systems with quadratic costs. Numerical examples on LTI and gene-expression models illustrate robust convergence and trade-offs between convergence speed and transient overshoot, supporting the practicality of OFO without explicit time-scale separation.

Abstract

Online Feedback Optimization (OFO) steers a dynamical plant to a cost-efficient steady-state, only relying on input-output sensitivity information, rather than on a full plant model. Unlike traditional feedforward approaches, OFO leverages real-time measurements from the plant, thereby inheriting the robustness and adaptability of feedback control. Unfortunately, existing theoretical guarantees for OFO assume that the controller operates on a slower timescale than the plant, which can affect responsiveness and transient performance. In this paper, we focus on relaxing this ``timescale separation'' assumption. Specifically, we consider the class of monotone systems, and we prove that OFO can achieve an optimal operating point, regardless of the time constants of controller and plant. By leveraging a small gain theorem for monotone systems, we derive several sufficient conditions for global convergence. Notably, these conditions depend only on the steady-state behavior of the plant, and they are entirely independent of the transient dynamics.

Online Feedback Optimization for Monotone Systems without Timescale Separation

TL;DR

Online Feedback Optimization (OFO) steers a dynamical plant toward a cost-minimizing steady state using real-time measurements and input-output sensitivities. The paper removes the traditional timescale-separation requirement by exploiting monotone plant structure and a small-gain theorem, proving global convergence to the optimal operating point for any controller gain , under conditions depending only on steady-state maps and . It provides verifiable sufficient conditions and regularization guidance to ensure the required convexity properties, including in stable LTI systems with quadratic costs. Numerical examples on LTI and gene-expression models illustrate robust convergence and trade-offs between convergence speed and transient overshoot, supporting the practicality of OFO without explicit time-scale separation.

Abstract

Online Feedback Optimization (OFO) steers a dynamical plant to a cost-efficient steady-state, only relying on input-output sensitivity information, rather than on a full plant model. Unlike traditional feedforward approaches, OFO leverages real-time measurements from the plant, thereby inheriting the robustness and adaptability of feedback control. Unfortunately, existing theoretical guarantees for OFO assume that the controller operates on a slower timescale than the plant, which can affect responsiveness and transient performance. In this paper, we focus on relaxing this ``timescale separation'' assumption. Specifically, we consider the class of monotone systems, and we prove that OFO can achieve an optimal operating point, regardless of the time constants of controller and plant. By leveraging a small gain theorem for monotone systems, we derive several sufficient conditions for global convergence. Notably, these conditions depend only on the steady-state behavior of the plant, and they are entirely independent of the transient dynamics.

Paper Structure

This paper contains 17 sections, 9 theorems, 38 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Notation and background
  3. Problem statement
  4. OFO for monotone systems
  5. Sufficient conditions
  6. Numerical examples
  7. LTI system
  8. Control of gene expression
  9. Research directions
  10. Proof of Lemma \ref{['lem:SGT']}
  11. Proof of Proposition \ref{['prop:wellposedness']}
  12. Proof of Theorem \ref{['th:main']}
  13. Preliminary results for the proof of Lemma \ref{['lem:asmi']}
  14. Proof of Lemma \ref{['lem:asmi']}
  15. Proof of Lemma \ref{['lem:asmii']}
  16. ...and 2 more sections

Key Result

Lemma 1

Let the following conditions hold: Then, all bounded solutions of the closed-loop system converge to $x^\star = k_x (u^\star)$.

Figures (2)

  • Figure 1: Performance of the controller \ref{['eq:OFO']} for the linear plant in Section \ref{['sec:LTI']} and different values of the control gain $\alpha$. Input constraints (dotted lines) are always satisfied.
  • Figure 2: Gene expression control for the plant in Section \ref{['sec:NL']} in closed-loop with \ref{['eq:OFO']}. Input constraints are shown by the dotted line.

Theorems & Definitions (15)

  • Lemma 1: Small-gain theorem
  • Proposition 1
  • Theorem 1
  • Remark 1
  • Remark 2
  • Remark 3
  • Lemma 2
  • Lemma 3
  • Lemma 4
  • Corollary 1
  • ...and 5 more