Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Decentralized Feedback Optimization via Sensitivity Decoupling: Stability and Sub-optimality

Wenbin Wang, Zhiyu He, Giuseppe Belgioioso, Saverio Bolognani, Florian Dörfler

TL;DR

This work presents a fully decentralized feedback optimization controller for net-worked systems to lift the communication burden and improve scalability and proves that the proposed decentralized controller yields solutions that correspond to the Nash equilibria of a non-cooperative game.

Abstract

Online feedback optimization is a controller design paradigm for optimizing the steady-state behavior of a dynamical system. It employs an optimization algorithm as a dynamic feedback controller and utilizes real-time measurements to bypass knowing exact plant dynamics and disturbances. Different from existing centralized settings, we present a fully decentralized feedback optimization controller for networked systems to lift the communication burden and improve scalability. We approximate the overall input-output sensitivity matrix through its diagonal elements, which capture local model information. For the closed-loop behavior, we characterize the stability and bound the sub-optimality due to decentralization. We prove that the proposed decentralized controller yields solutions that correspond to the Nash equilibria of a non-cooperative game.

Decentralized Feedback Optimization via Sensitivity Decoupling: Stability and Sub-optimality

TL;DR

This work presents a fully decentralized feedback optimization controller for net-worked systems to lift the communication burden and improve scalability and proves that the proposed decentralized controller yields solutions that correspond to the Nash equilibria of a non-cooperative game.

Abstract

Online feedback optimization is a controller design paradigm for optimizing the steady-state behavior of a dynamical system. It employs an optimization algorithm as a dynamic feedback controller and utilizes real-time measurements to bypass knowing exact plant dynamics and disturbances. Different from existing centralized settings, we present a fully decentralized feedback optimization controller for networked systems to lift the communication burden and improve scalability. We approximate the overall input-output sensitivity matrix through its diagonal elements, which capture local model information. For the closed-loop behavior, we characterize the stability and bound the sub-optimality due to decentralization. We prove that the proposed decentralized controller yields solutions that correspond to the Nash equilibria of a non-cooperative game.
Paper Structure (16 sections, 6 theorems, 32 equations, 4 figures)

This paper contains 16 sections, 6 theorems, 32 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Preliminaries and Problem Formulation
  3. Notation
  4. Problem Formulation
  5. Design of the Decentralized Controller
  6. Decentralized Controller
  7. A Game-Theoretic Interpretation
  8. Performance Analysis
  9. Interconnection with an Algebraic Plant
  10. Interconnection with an LTI Plant
  11. Numerical Results
  12. Conclusion
  13. Proof of Lemma \ref{['lemma: unique stationary point']}
  14. Proof of Theorem \ref{['theorem: no plant dynamic strongly convex']}
  15. Proof of Theorem \ref{['theorem: no plant dynamic strongly convex vanilla']}
  16. ...and 1 more sections

Key Result

Theorem III.1

Given Assumption assumption: chapter 2, the stationary points of the controller eq: decentralized algorithm system, if they exist, equal to the Nash equilibria of the following convex game:

Figures (4)

  • Figure 1: Example of a system with three agents. The black line indicates the coupling dynamics between agents. The red dashed line represents the decentralized update using local inputs and outputs.
  • Figure 2: An 8-node DC power system.
  • Figure 3: Relative distance to the globally optimal point (G=1).
  • Figure 4: Relative distance to the globally optimal point with different values of $G$ (i.e., different degrees of diagonal dominance).

Theorems & Definitions (13)

  • Theorem III.1
  • Proof III.1
  • Lemma IV.1
  • Proof IV.1
  • Theorem IV.1
  • Proof IV.2
  • Theorem IV.2
  • Proof IV.3
  • Remark IV.1
  • Theorem IV.3
  • ...and 3 more