Input Selection for Performance and Controllability of Structured Linear Descriptor Systems

A common approach to controlling complex networks is to directly control a subset of input nodes, which then controls the remaining nodes via network interactions. While techniques have been proposed for selecting input nodes based on either performance metrics or controllability, a unifying approach based on joint consideration of performance and controllability is an open problem. In this paper, we develop a submodular optimization framework for selecting input nodes based on joint performance and controllability in structured linear descriptor systems. We develop our framework for arbitrary linear descriptor systems. In developing our framework, we first prove that selecting a minimum-size set of input nodes for controllability is a matroid intersection problem that can be solved in polynomial-time in the network size. We then prove that input selection to maximize a performance metric with controllability as a constraint is equivalent to maximizing a monotone submodular function with two matroid basis constraints, and derive efficient approximation algorithms with provable optimality bounds for input selection. Finally, we present a graph controllability index metric, which characterizes the largest controllable subgraph of a given complex network, and prove its submodular structure, leading to input selection algorithms that trade-off performance and controllability. We provide improved optimality guarantees for known systems such as strongly connected networks, consensus networks, networks of double integrators, and networks where all system parameters (e.g., link weights) are chosen independently and at random.