Team-Optimal Solution of Finite Number of Mean-Field Coupled LQG Subsystems

TL;DR

This paper derives a unique, linear, and identical optimal control law for finite mean-field coupled LQG subsystems, with gains computed via Riccati equations independent of the number of subsystems, achieving centralized performance in a decentralized setting.

Contribution

It provides a novel solution for finite mean-field coupled LQG systems, showing the optimal control law is identical across subsystems and can be computed efficiently.

Findings

Optimal control law is unique, linear, and identical for all subsystems.

Optimal gains are obtained from Riccati equations independent of subsystem count.

Decentralized control achieves the same performance as centralized control.

Abstract

A decentralized control system with linear dynamics, quadratic cost, and Gaussian disturbances is considered. The system consists of a finite number of subsystems whose dynamics and per-step cost function are coupled through their mean-field (empirical average). The system has mean-field sharing information structure, i.e., each controller observes the state of its local subsystem (either perfectly or with noise) and the mean-field. It is shown that the optimal control law is unique, linear, and identical across all subsystems. Moreover, the optimal gains are computed by solving two decoupled Riccati equations in the full observation model and by solving an additional filter Riccati equation in the noisy observation model. These Riccati equations do not depend on the number of subsystems. It is also shown that the optimal decentralized performance is the same as the optimal centralized performance. An example, motivated by smart grids, is presented to illustrate the result.