Limitations and Tradeoffs in Minimum Input Selection Problems

In this paper, the problem of the actuator selection for linear dynamical networks is investigated. We develop a framework to design a sparse actuator schedule for a given large-scale linear system with guaranteed performance bounds using a polynomial-time algorithm. We first introduce a notion of systemic controllability metrics for linear dynamical networks that are monotone, convex, and homogeneous with respect to the controllability Gramian matrix of the network. It is shown that several popular and widely used optimization criteria in the literature belong to this new class of controllability metrics. By leveraging recent advances in sparsification literature and the famous Kadison-Singer conjecture, we then prove that there exists a sparse actuator schedule that chooses on average a constant number of actuators at each time, to approximate all systemic controllability metrics.