# Actuator Placement for Optimizing Network Performance under Controllability Constraints

## TL;DR

This paper addresses the challenge of selecting actuators in large-scale networks to minimize energy consumption while maintaining controllability, using a matroid optimization framework and greedy algorithms with performance guarantees.

## Contribution

It introduces a novel approach that reformulates actuator placement as a matroid optimization problem with a weakly submodular objective, providing new performance guarantees.

## Key findings

- Greedy algorithm achieves near-optimal solutions with performance bounds.
- Matroid constraints can be checked via maximum matching in auxiliary bipartite graphs.
- The approach effectively balances energy efficiency and controllability in network control.

## Abstract

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, greedy algorithm can be used to obtain an approximate solution. To provide a performance guarantee for this approach, we first define the submodularity ratio for the metric under consideration and then reformulate the structural controllability constraint as a matroid constraint. This shows that the problem under study can be characterized by a matroid optimization involving a weakly submodular objective function. Then, we derive a novel performance guarantee for the greedy algorithm applied to this class of optimization problems. 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.