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Removing Time-Scale Separation in Feedback-Based Optimization via Estimators

Niloufar Yousefi, John W. Simpson-Porco

TL;DR

The paper tackles the instability and limited performance of traditional feedback-based optimization (FBO) caused by the need for time-scale separation between the plant and controller. It introduces estimator-based FBO (EE-FBO), which embeds a dynamic plant-model estimator to predict steady-state outputs and disturbances, feeding these predictions into the gradient-based controller. The authors prove unconditional stability for both a full-model estimator and a reduced-model estimator in two contexts: a standard LTI plant and a singularly perturbed two-timescale plant, with convergence rates limited by the open-loop dynamics rather than controller gain. A power-system frequency control case with inverter-based resources demonstrates significant transient-performance gains, with the reduced-model EE-FBO achieving performance close to the full-model version. Collectively, the work provides a principled spectrum of EE-FBO designs that trade model information for improved closed-loop performance and broadens the practical applicability of FBO in fast, disturbance-rejecting settings.

Abstract

Feedback-based optimization (FBO) provides a simple control framework for regulating a stable dynamical system to the solution of a constrained optimization problem in the presence of exogenous disturbances, and does so without full knowledge of the plant dynamics. However, closed-loop stability requires the controller to operate on a sufficiently slower timescale than the plant, significantly constraining achievable closed-loop performance. Motivated by this trade-off, we propose an estimator-based modification of FBO which leverages dynamic plant model information to eliminate the time-scale separation requirement of traditional FBO. Under this design, the convergence rate of the closed-loop system is limited only by the dominant eigenvalue of the open-loop system. We extend the approach to the case of design based on only an approximate plant model when the original system is singularly perturbed. The results are illustrated via an application to fast power system frequency control using inverter-based resources.

Removing Time-Scale Separation in Feedback-Based Optimization via Estimators

TL;DR

The paper tackles the instability and limited performance of traditional feedback-based optimization (FBO) caused by the need for time-scale separation between the plant and controller. It introduces estimator-based FBO (EE-FBO), which embeds a dynamic plant-model estimator to predict steady-state outputs and disturbances, feeding these predictions into the gradient-based controller. The authors prove unconditional stability for both a full-model estimator and a reduced-model estimator in two contexts: a standard LTI plant and a singularly perturbed two-timescale plant, with convergence rates limited by the open-loop dynamics rather than controller gain. A power-system frequency control case with inverter-based resources demonstrates significant transient-performance gains, with the reduced-model EE-FBO achieving performance close to the full-model version. Collectively, the work provides a principled spectrum of EE-FBO designs that trade model information for improved closed-loop performance and broadens the practical applicability of FBO in fast, disturbance-rejecting settings.

Abstract

Feedback-based optimization (FBO) provides a simple control framework for regulating a stable dynamical system to the solution of a constrained optimization problem in the presence of exogenous disturbances, and does so without full knowledge of the plant dynamics. However, closed-loop stability requires the controller to operate on a sufficiently slower timescale than the plant, significantly constraining achievable closed-loop performance. Motivated by this trade-off, we propose an estimator-based modification of FBO which leverages dynamic plant model information to eliminate the time-scale separation requirement of traditional FBO. Under this design, the convergence rate of the closed-loop system is limited only by the dominant eigenvalue of the open-loop system. We extend the approach to the case of design based on only an approximate plant model when the original system is singularly perturbed. The results are illustrated via an application to fast power system frequency control using inverter-based resources.

Paper Structure

This paper contains 12 sections, 4 theorems, 43 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Background: Feedback-Based Optimization via Gradient Flow
  3. Estimator-Based Performance Enhancement of FBO
  4. Estimator Design Based on Full Plant Model
  5. Stability Analysis
  6. Model-Information vs. Performance Trade-Offs in FBO
  7. Estimator-Based FBO for Two-Timescale Systems
  8. Estimator Design Based on Reduced Plant Model
  9. Stability Analysis
  10. Application to Fast Frequency Control of Power Systems using IBRs
  11. Conclusion
  12. Proofs

Key Result

Theorem 2.1

Assume that $A$ is Hurwitz, that Assumption Ass:Cost holds, and select $\eta \in (0,2\mu_{f}/(\ell_f + \ell_g \|\Pi_u\|_{2}^2)^2)$. Then (i) for each $w \in \mathbb{R}^{q}$, the closed-loop system Eq:LTI, Eq:GradientController possesses a unique equilibrium point $(\bar{x},\bar{u}^{\star})$ with cor

Figures (4)

  • Figure 1: Feedback-based optimization with an estimator. The estimator predicts output and its steady state, feeding these into the controller.
  • Figure 2: Block diagram of closed-loop system in error coordinates, where $\mathcal{H}_{w}(u,\tilde{v}) = P_{\mathcal{U}}(u - \eta F_w(u,\Tilde{v})) - u$.
  • Figure 3: Block diagram of the interconnected estimator, controller, and plant, where $\mathcal{H}_{w}(u,\tilde{v},e_2) = P_{\mathcal{U}}(u - \eta F_{w}(u,\tilde{v},e_2)) - u$.
  • Figure 4: Response power system under $40$MW load change under FBO and EE-FBO methods.

Theorems & Definitions (4)

  • Theorem 2.1: Low-Gain Stability of FBO
  • Proposition 3.1: Exponential ISS of a Projection Algorithm
  • Theorem 3.2: Unconditional Stability of EE-FBO
  • Theorem 4.1: Unconditional Stability of EE-FBO Based on Reduced Model