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Optimization of Linear Multi-Agent Dynamical Systems via Feedback Distributed Gradient Descent Methods

Amir Mehrnoosh, Gianluca Bianchin

TL;DR

Under convexity and smoothness assumptions, convergence to a critical optimization point is established, and under restricted strong convexity, it is proved linear convergence to a neighborhood of the optimum, with its size dependent on the stepsize.

Abstract

Feedback optimization is an increasingly popular control paradigm to optimize dynamical systems, accounting for control objectives that concern the system operation at steady-state. Existing feedback optimization techniques heavily rely on centralized systems and controller architectures, and thus suffer from scalability and privacy issues when systems become large-scale. In this paper, we propose a distributed architecture for feedback optimization inspired by distributed gradient descent, whereby each agent updates its local control variable by combining the average of its neighbors with a local negative gradient step. Under convexity and smoothness assumptions for the cost, we establish convergence of the control method to a critical optimization point. By reinforcing the assumptions to restricted strong convexity, we show that our algorithm converges linearly to a neighborhood of the optimal point, where the size of the neighborhood depends on the choice of the stepsize. Simulations corroborate the theoretical results.

Optimization of Linear Multi-Agent Dynamical Systems via Feedback Distributed Gradient Descent Methods

TL;DR

Under convexity and smoothness assumptions, convergence to a critical optimization point is established, and under restricted strong convexity, it is proved linear convergence to a neighborhood of the optimum, with its size dependent on the stepsize.

Abstract

Feedback optimization is an increasingly popular control paradigm to optimize dynamical systems, accounting for control objectives that concern the system operation at steady-state. Existing feedback optimization techniques heavily rely on centralized systems and controller architectures, and thus suffer from scalability and privacy issues when systems become large-scale. In this paper, we propose a distributed architecture for feedback optimization inspired by distributed gradient descent, whereby each agent updates its local control variable by combining the average of its neighbors with a local negative gradient step. Under convexity and smoothness assumptions for the cost, we establish convergence of the control method to a critical optimization point. By reinforcing the assumptions to restricted strong convexity, we show that our algorithm converges linearly to a neighborhood of the optimal point, where the size of the neighborhood depends on the choice of the stepsize. Simulations corroborate the theoretical results.
Paper Structure (11 sections, 4 theorems, 48 equations, 2 figures)

This paper contains 11 sections, 4 theorems, 48 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Problem Setting
  3. System to control
  4. Distributed structure of the controller
  5. Control objectives as an optimization problem
  6. Design of the F-DGD Method
  7. Convergence Analysis and Error Bounds
  8. Asymptotic convergence
  9. Control error bounds
  10. Simulation Results
  11. Conclusions

Key Result

Theorem IV.1

(Convergence of the state sequences) Let Assumptions ass:stabilityPlant-ass:convexityLocalCost hold, $W$ be such that $\beta<1,$ and $\eta \leq \bar{\eta} := \min \{ \eta_1, \eta_2, \eta_3\},$ with with $\mu$ an arbitrary constant, $0< \mu \leq 1 - \frac{{(1- \lambda_N(W)) + \eta {L_{\Phi}}}}{2}$, and $L_h = \Vert (I-A)^{\hbox{[}.9]{$ - $} 1} B S \Vert$. Then, the sequences $x^k,$$u^{k}_{(i)}$ ge

Figures (2)

  • Figure 1: Distributed system architecture considered in this work (cf. \ref{['eq:plantModel_distributed']}). Each local controller actuates the corresponding subsystem (green lines), by using global feedback information (red lines), see \ref{['eq:distributed_control_algorithm']}.
  • Figure 2: Error $\bar{e}^k = \frac{1}{N} \sum_{i=1}^{N} \Vert u_{(i)}^k - u^{*k} \Vert$, inputs, and outputs of the proposed decentralized algorithm with different stepsizes, where $u^{*k} = \text{Proj}_\mathcal{A^*}(\frac{1}{N} \sum_{i=1}^{N} u_{(i)}^k)$.

Theorems & Definitions (12)

  • Remark 1
  • Remark 2
  • Remark 3
  • Theorem IV.1
  • proof
  • Proposition IV.2
  • proof
  • Lemma IV.3
  • Remark 4
  • Theorem IV.4
  • ...and 2 more