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A Hybrid Systems Model of Feedback Optimization for Linear Systems: Convergence and Robustness

Oscar Jed Chuy, Matthew Hale, Ricardo Sanfelice

TL;DR

This work develops a hybrid modeling framework that couples continuous-time linear plant dynamics with discrete-time gradient-based optimization to perform feedback optimization in real time. It establishes well-posedness, completeness, and Zeno-free behavior of the hybrid model, and proves exponential convergence to a neighborhood of a goal state with robustness to disturbances in measurements, plant matrices, and input-timing. Theoretical guarantees are complemented by simulations showing disturbance rejection in the closed loop. The approach provides a robust, practical bridge between continuous-time physical dynamics and discrete-time computational updates for feedback optimization in linear systems.

Abstract

Feedback optimization algorithms compute inputs to a system using real-time output measurements, which helps mitigate the effects of disturbances. However, existing work often models both system dynamics and computations in either discrete or continuous time, which may not accurately model some applications. In this work, we model linear system dynamics in continuous time, and we model the computations of inputs in discrete time. Therefore, we present a novel hybrid systems model of feedback optimization. We first establish the well-posedness of this hybrid model and establish completeness of solutions while ruling out Zeno behavior. Then we show the state of the system converges exponentially fast to a ball of known radius about a desired goal state. Next we analytically show that this system is robust to perturbations in (i) the values of measured outputs, (ii) the matrices that model the linear time-invariant system, and (iii) the times at which inputs are applied to the system. Simulation results confirm that this approach successfully mitigates the effects of disturbances.

A Hybrid Systems Model of Feedback Optimization for Linear Systems: Convergence and Robustness

TL;DR

This work develops a hybrid modeling framework that couples continuous-time linear plant dynamics with discrete-time gradient-based optimization to perform feedback optimization in real time. It establishes well-posedness, completeness, and Zeno-free behavior of the hybrid model, and proves exponential convergence to a neighborhood of a goal state with robustness to disturbances in measurements, plant matrices, and input-timing. Theoretical guarantees are complemented by simulations showing disturbance rejection in the closed loop. The approach provides a robust, practical bridge between continuous-time physical dynamics and discrete-time computational updates for feedback optimization in linear systems.

Abstract

Feedback optimization algorithms compute inputs to a system using real-time output measurements, which helps mitigate the effects of disturbances. However, existing work often models both system dynamics and computations in either discrete or continuous time, which may not accurately model some applications. In this work, we model linear system dynamics in continuous time, and we model the computations of inputs in discrete time. Therefore, we present a novel hybrid systems model of feedback optimization. We first establish the well-posedness of this hybrid model and establish completeness of solutions while ruling out Zeno behavior. Then we show the state of the system converges exponentially fast to a ball of known radius about a desired goal state. Next we analytically show that this system is robust to perturbations in (i) the values of measured outputs, (ii) the matrices that model the linear time-invariant system, and (iii) the times at which inputs are applied to the system. Simulation results confirm that this approach successfully mitigates the effects of disturbances.

Paper Structure

This paper contains 19 sections, 4 theorems, 21 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notation
  4. Feedback Optimization Background and Setup
  5. Background on Hybrid Systems
  6. A Hybrid Model for Feedback Optimization
  7. Overview of Hybrid Feedback Optimization
  8. Optimization Problem Setup
  9. Hybrid Modeling and Flow and Jump Sets
  10. Flow Map Definition
  11. Jump Map Definition
  12. Properties of Hybrid Feedback Optimization
  13. Well-Posedness and Existence of Solutions
  14. Algorithm Framework
  15. Convergence Analysis
  16. ...and 4 more sections

Key Result

Lemma 1

The hybrid feedback optimization model $\mathcal{H}_{FO}$ in eq:hybridFO is well-posed in the sense that it satisfies Definition def:hybridcond.

Figures (1)

  • Figure 1: A visual representation of the input evolution of $\mathcal{H}_{FO}$. There are $\alpha(p)$ iterations of gradient descent when computing the $(p+1)^{\textnormal{th}}$ input. The input is changed at hybrid times of the form $(t_{\textcolor{black}{\bar{\alpha}\left(p\right)}+p},\textcolor{black}{\bar{\alpha}\left(p\right)}+p)$.

Theorems & Definitions (11)

  • Remark 1
  • Remark 1
  • Remark 2
  • Remark 3
  • Definition 1: Hybrid Basic Conditions Hybridbook
  • Lemma 1
  • proof
  • Proposition 1: Completeness of Maximal Solutions
  • proof
  • Lemma 2: hybridfeedcntrl
  • ...and 1 more