Optimal selection of essential interconnections for structural controllability in heterogeneous subsystems

This paper deals with structural controllability of a linear time-invariant composite system consisting of several heterogeneous subsystems. The interaction links through which the subsystems interact with other subsystems are referred to as interconnections. We assume the composite system to be structurally controllable if all possible interconnections are present. Our objective is to identify a minimum cardinality set of interconnections required to retain the structural controllability of the composite system. We refer to this problem as the optimal essential interconnection selection problem. We approach the problem in a structured framework, where the zero/nonzero structure of the subsystems is used in the analysis instead of the numerical matrices themselves. This analysis applies to an equivalence class of systems with the same sparsity pattern. Firstly, we propose a polynomial time algorithm to solve the optimal essential interconnection selection problem on a structured composite system when each subsystem is irreducible and no subsystem has a perfect matching in its state bipartite graph. Later, we consider the case where one or more subsystems have perfect matching in their state bipartite graphs. For this case, we first prove a lower bound on the number of minimum number of interconnections needed. Subsequently, we provide a polynomial time algorithm based on a minimum weight perfect matching algorithm and a so-called stub-matching algorithm that achieves this bound. We also discuss about how heterogeneity of the subsystems poses different challenges to the homogeneous counterpart and demonstrate the algorithms using illustrative examples.