Actuator Placement for Symmetric Structural Controllability With Heterogeneous Costs

In this letter, we consider the problem of optimal actuator placement for networked linear time-invariant systems with underlying undirected topologies. We aim to leverage the undirected topology for the design of an actuator deployment strategy that guarantees symmetric structural controllability, i.e., generic controllability of the system subject to symmetric weights. Naturally, the objective is to make this design as cost effective and computationally efficient as possible. Therefore, we focus on two related (yet different) problems: 1) to determine the subset of state nodes that need to be actuated to ensure symmetric structural controllability at the lowest cost incurred by the actuators and 2) given a symmetrically structurally controllable system (whose input nodes may actuate on more than one state node), to determine a subset of its input nodes that still ensure symmetric structural controllability, at the lowest possible cost. We show that both problems can be solved in polynomial time by implicitly providing an efficient algorithm that determines their solutions. Several numerical experiments are provided to illustrate our main results.