Structural controllability: an undirected graph approach

TL;DR

This paper introduces a novel undirected graph-based approach to determine the structural controllability of linear dynamical systems using graph algorithms, providing necessary and sufficient conditions and an efficient algorithm.

Contribution

It presents a new undirected graph method for analyzing structural controllability, differing from traditional approaches, with a focus on bipartite graphs and redundant edges.

Findings

Provides necessary and sufficient conditions for controllability.

Develops an algorithm with analyzed running time.

Applies results to state space systems for new controllability criteria.

Abstract

This paper addresses questions regarding controllability for `generic parameter' dynamical systems, i.e. the question whether a dynamical system is `structurally controllable'. Unlike conventional methods that deal with structural controllability, our approach uses an undirected graph: the behavioral approach to modelling dynamical systems allows this. Given a system of linear, constant coefficient, ordinary differential equations of any order, we formulate necessary and sufficient conditions for controllability in terms of weights of the edges in a suitable bipartite graph constructed from % components with equal bipartite cardinality in the differential-algebraic system. % of equations. A key notion that helps formulate the conditions is that of a `redundant edge'. Removal of all redundant edges makes the inferring of structural controllability a straightforward exercise. We use standard graph algorithms as ingredients to check these conditions and hence obtain an algorithm to check for structural controllability. We provide an analysis of the running time of our algorithm. When our results are applied to the familiar state space description of a system, we obtain a novel necessary and sufficient condition to check structural controllability for this description.