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With the rising importance of large-scale network control, the problem of actuator placement has received increasing attention. Our goal in this paper is to find a set of actuators minimizing the metric that measures the average energy consumption of the control inputs while ensuring structural controllability of the network. As this problem is intractable, the greedy algorithm can be used to obtain an approximate solution. To provide a performance guarantee for this approach, we first define a new notion of submodularity ratio and show that the metric under consideration enjoys the notion of weak submodularity corresponding to this ratio. We then reformulate the structural controllability constraint as a matroid constraint. This shows that the problem under study can be characterized by the optimization of a weakly submodular function under a matroid constraint. For the greedy algorithm applied to this class of optimization problems, we derive a novel performance guarantee. Finally, we show that the matroid feasibility check for the greedy algorithm can be cast as a maximum matching problem in a certain auxiliary bipartite graph related to the network graph.

Actuator Placement for Optimizing Network Performance under Controllability Constraints