Optimal control of network-coupled subsystems: Spectral decomposition and low-dimensional solutions

TL;DR

This paper presents a spectral decomposition-based method for optimal control of large-scale network-coupled systems, reducing complexity by focusing on eigenvalues of the coupling matrix.

Contribution

The paper introduces a novel spectral decomposition approach that simplifies optimal control synthesis for network-coupled systems by decoupling dynamics based on eigenvalues.

Findings

Decouples large network systems into smaller subsystems using spectral decomposition.

Reduces computational complexity by depending on eigenvalue count rather than network size.

Provides a scalable framework for optimal control in large-scale networks.

Abstract

In this paper, we investigate optimal control of network-coupled subsystems where the dynamics and the cost couplings depend on an underlying undirected weighted graph. The graph coupling matrix in the dynamics may be the adjacency matrix, the Laplacian matrix, or any other symmetric matrix corresponding to the underlying graph. The cost couplings can be any polynomial function of the underlying coupling matrix. We use the spectral decomposition of the graph coupling matrix to decompose the overall system into (L+1) systems with decoupled dynamics and cost, where L is the rank of the coupling matrix. Furthermore, the optimal control input at each subsystem can be computed by solving (Ldist + 1) decoupled Riccati equations where Ldist (Ldist \leq L) is the number of distinct non-zero eigenvalues of the coupling matrix. A salient feature of the result is that the solution complexity depends on the number of distinct eigenvalues of the coupling matrix rather than the size of the network. Therefore, the proposed solution framework provides a scalable method for synthesizing and implementing optimal control laws for large-scale network-coupled subsystems.