Structural controllability: an undirected graph approach

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 the differential-algebraic system. 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.