Optimal Network Topology Design in Composite Systems with Constrained Neighbors for Structural Controllability

Composite systems are large complex systems consisting of interconnected agents (subsystems). Our focus is on controllability of linear time-invariant composite systems. In a composite system, often only a few of the agents called as leaders receive input. In such a case, the agents share/communicate their private state information with pre-specified neighboring agents so as to achieve controllability. Our objective in this paper is to identify an optimal network topology, optimal in the sense of minimum cardinality information transfer between agents to guarantee the controllability of the composite system when the possible neighbor set of each agent is pre-specified. We focus on graph-theoretic analysis referred to as structural controllability as numerical entries of system matrices in complex systems are mostly unknown. We first prove that given a set of agents and the possible set of neighbors, finding a minimum cardinality set of information (interconnections) that must be shared to accomplish structural controllability of the composite system is NP-hard. We obtain the NP-hardness result using reduction from a degree constrained spanning tree problem. Then we present a polynomial-time algorithm that finds a 2-optimal solution to this NP-hard problem.