Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Networked Control and Mean Field Problems Under Diagonal Dominance: Decentralized and Social Optimality

Vivek Khatana, Duo Wang, Petros Voulgaris, Nicola Elia, Naira Hovakimyan

TL;DR

The paper tackles scalable decentralized control for networks of heterogeneous agents coupled through a mean-field-type social cost. It develops an input-output framework with a Youla-Kucera parametrization and a diagonal-dominance condition on the collective dynamics, proving that selfish, decentralized controllers become asymptotically optimal for both $H_\infty$ and $H_2$ norms as the number of agents $n$ grows, provided the dominance is $o(\sqrt{n})$. It extends results from homogeneous to heterogeneous agents and validates the theory via a case study showing sharp bounds: decentralized controllers approach the centralized optimum, while violating the dominance condition degrades performance. The work offers a scalable, practically implementable approach for large-scale networked control in domains like power systems, traffic, and multi-robot swarms by leveraging local maps and the dominance of local dynamics over aggregate couplings.

Abstract

In this article, we employ an input-output approach to expand the study of cooperative multi-agent control and optimization problems characterized by mean-field interactions that admit decentralized and selfish solutions. The setting involves $n$ independent agents that interact solely through a shared cost function, which penalizes deviations of each agent from the group's average collective behavior. Building on our earlier results established for homogeneous agents, we extend the framework to nonidentical agents and show that, under a diagonal dominant interaction of the collective dynamics, with bounded local open-loop dynamics, the optimal controller for $H_\infty$ and $H_2$ norm minimization remains decentralized and selfish in the limit as the number of agents $n$ grows to infinity.

Networked Control and Mean Field Problems Under Diagonal Dominance: Decentralized and Social Optimality

TL;DR

The paper tackles scalable decentralized control for networks of heterogeneous agents coupled through a mean-field-type social cost. It develops an input-output framework with a Youla-Kucera parametrization and a diagonal-dominance condition on the collective dynamics, proving that selfish, decentralized controllers become asymptotically optimal for both and norms as the number of agents grows, provided the dominance is . It extends results from homogeneous to heterogeneous agents and validates the theory via a case study showing sharp bounds: decentralized controllers approach the centralized optimum, while violating the dominance condition degrades performance. The work offers a scalable, practically implementable approach for large-scale networked control in domains like power systems, traffic, and multi-robot swarms by leveraging local maps and the dominance of local dynamics over aggregate couplings.

Abstract

In this article, we employ an input-output approach to expand the study of cooperative multi-agent control and optimization problems characterized by mean-field interactions that admit decentralized and selfish solutions. The setting involves independent agents that interact solely through a shared cost function, which penalizes deviations of each agent from the group's average collective behavior. Building on our earlier results established for homogeneous agents, we extend the framework to nonidentical agents and show that, under a diagonal dominant interaction of the collective dynamics, with bounded local open-loop dynamics, the optimal controller for and norm minimization remains decentralized and selfish in the limit as the number of agents grows to infinity.

Paper Structure

This paper contains 9 sections, 5 theorems, 48 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Notations And Definitions
  3. Problem Setup
  4. (Scaled) $\mathcal{H}_2$ Case
  5. 2-block problems
  6. ${\cal H}_\infty$ case
  7. ${\cal H}_2$ case
  8. Case Study
  9. Concluding Remarks

Key Result

Lemma 1

Let Assumption assmp:bound_on_system_matrices holds and let $\alpha = o(\sqrt{n})$, then under the constraint $Q \in \mathcal{Q}^\alpha$ it holds that

Figures (1)

  • Figure 1: Agents' response to noisy inputs with a common sinusoidal signal. Experiment with $n=60,120$ agents. For visualization, the majority of agent responses are depicted in gray, and a randomly selected subset is emphasized in color.

Theorems & Definitions (15)

  • Definition 1
  • Definition 2
  • Lemma 1
  • proof
  • Theorem 1
  • proof
  • Lemma 2
  • proof
  • Theorem 2
  • proof
  • ...and 5 more