Minimal Actuator Selection

TL;DR

This paper addresses the minimal actuator selection problem in control systems, providing a precise characterization and efficient algorithms for choosing the fewest actuators to ensure controllability, even under robustness constraints.

Contribution

It generalizes previous approaches by formulating the problem as an integer linear program and a set multicover problem, enabling the use of existing algorithms for optimal actuator selection.

Findings

The problem can be cast as an integer linear program.

When actuation channels are independent, it reduces to a set multicover problem.

The approach applies even for robust actuator selection against faults.

Abstract

Selecting a few available actuators to ensure the controllability of a linear system is a fundamental problem in control theory. Previous works either focus on optimal performance, simplifying the controllability issue, or make the system controllable under structural assumptions, such as in graphs or when the input matrix is a design parameter. We generalize these approaches to offer a precise characterization of the general minimal actuator selection problem where a set of actuators is given, described by a fixed input matrix, and goal is to choose the fewest actuators that make the system controllable. We show that this problem can be equivalently cast as an integer linear program and, if actuation channels are sufficiently independent, as a set multicover problem under multiplicity constraints. The latter equivalence is always true if the state matrix has all distinct eigenvalues, in which case it simplifies to the set cover problem. Such characterizations hold even when a robust selection that tolerates a given number of faulty actuators is desired. Our established connection legitimates a designer to use algorithms from the rich literature on the set multicover problem to select the smallest subset of actuators, including exact solutions that do not require brute-force search.