# Game-Theoretic Mixed $H_2/H_{\infty}$ Control with Sparsity Constraint for Multi-agent Networked Control Systems

## TL;DR

This paper develops a game-theoretic approach with sparsity constraints for robust multi-agent control, improving communication efficiency and robustness against model uncertainties in networked systems.

## Contribution

It introduces a novel PALM-based algorithm for solving sparsity-constrained mixed $H_2/H_{ extinfty}$ control problems and analyzes a noncooperative game for distributed control.

## Key findings

- The proposed algorithm converges to an approximate GNE.
- The method outperforms previous sparsity-constrained controllers.
- A partially-distributed solution is achieved for identical $H_2$ objectives.

## Abstract

Multi-agent networked control systems (NCSs) are often subject to model uncertainty and are limited by large communication cost, associated with feedback of data between the system nodes. To provide robustness against model uncertainty and to reduce the communication cost, this paper investigates the mixed $H_2/H_{\infty}$ control problem for NCS under the sparsity constraint. First, proximal alternating linearized minimization (PALM) is employed to solve the centralized social optimization where the agents have the same optimization objective. Next, we investigate a sparsity-constrained noncooperative game, which accommodates different control-performance criteria of different agents, and propose a best-response dynamics algorithm based on PALM that converges to an approximate Generalized Nash Equilibrium (GNE) of this game. A special case of this game, where the agents have the same $H_2$ objective, produces a partially-distributed social optimization solution. We validate the proposed algorithms using a network with unstable node dynamics and demonstrate the superiority of the proposed PALM-based method to a previously investigated sparsity-constrained mixed $H_2/H_{\infty}$ controller.