Minimal Actuator Placement with Optimal Control Constraints

TL;DR

This paper addresses the challenge of selecting the minimal set of actuators in a linear control system to ensure controllability and meet control effort constraints, introducing a supermodular structure and an efficient approximation algorithm.

Contribution

It proves the NP-hardness of the problem, reveals its supermodular structure, and provides a near-optimal polynomial-time approximation algorithm with proven bounds.

Findings

The algorithm achieves an O(log n) approximation factor.

The problem is NP-hard, indicating computational complexity.

Algorithm performs efficiently on large random networks.

Abstract

We introduce the problem of minimal actuator placement in a linear control system so that a bound on the minimum control effort for a given state transfer is satisfied while controllability is ensured. We first show that this is an NP-hard problem following the recent work of Olshevsky. Next, we prove that this problem has a supermodular structure. Afterwards, we provide an efficient algorithm that approximates up to a multiplicative factor of O(logn), where n is the size of the multi-agent network, any optimal actuator set that meets the specified energy criterion. Moreover, we show that this is the best approximation factor one can achieve in polynomial-time for the worst case. Finally, we test this algorithm over large Erdos-Renyi random networks to further demonstrate its efficiency.