Controllability and observability of grid graphs via reduction and symmetries

TL;DR

This paper analyzes the controllability and observability of grid graph-based systems by leveraging graph symmetries and eigenvector structures, providing criteria to simplify analysis in network control and related fields.

Contribution

It introduces a novel approach using graph decompositions and symmetry analysis to characterize controllability and observability of grid-structured systems.

Findings

Provides necessary and sufficient conditions for controllability and observability.

Shows how symmetry reduces analysis complexity.

Applies criteria to example systems for validation.

Abstract

In this paper we investigate the controllability and observability properties of a family of linear dynamical systems, whose structure is induced by the Laplacian of a grid graph. This analysis is motivated by several applications in network control and estimation, quantum computation and discretization of partial differential equations. Specifically, we characterize the structure of the grid eigenvectors by means of suitable decompositions of the graph. For each eigenvalue, based on its multiplicity and on suitable symmetries of the corresponding eigenvectors, we provide necessary and sufficient conditions to characterize all and only the nodes from which the induced dynamical system is controllable (observable). We discuss the proposed criteria and show, through suitable examples, how such criteria reduce the complexity of the controllability (respectively observability) analysis of the grid.