Minimal Actuator Selection

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.

💡 Research Summary

This paper addresses the fundamental problem of selecting the smallest possible subset of actuators from a given admissible set so that a linear time‑invariant system becomes controllable. Unlike much of the existing literature, which either focuses on performance‑oriented actuator placement or assumes that the input matrix can be freely designed, the authors consider the realistic scenario where the input matrix (B) is fixed and only its columns (i.e., the available actuators) may be chosen.

The authors begin by formulating the controllability requirement using the Popov‑Belevitch‑Hautus (PBH) test: for every eigenvalue (\lambda) of the state matrix (A), the matrix (