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Minimal Actuator Selection

Luca Ballotta, Geethu Joseph

TL;DR

The paper addresses the problem of selecting the smallest subset of actuators from a fixed set to ensure controllability of a linear system, using the PBH test and Jordan decomposition to derive a binary ILP. Under a technical independence assumption, the problem is shown to be equivalent to a set multicover problem (reducing to set cover when all eigenvalues are simple), and it extends naturally to robust, fault-tolerant scenarios by incorporating additional actuator redundancy. The authors prove NP-hardness and provide pathways to exact or approximate solutions using ILP solvers and well-established multicover algorithms, enabling practical offline design and reconfiguration. This work bridges actuator allocation and combinatorial optimization, and suggests future directions like joint sensor-actuator selection and time-varying actuator schedules for scalable systems.

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.

Minimal Actuator Selection

TL;DR

The paper addresses the problem of selecting the smallest subset of actuators from a fixed set to ensure controllability of a linear system, using the PBH test and Jordan decomposition to derive a binary ILP. Under a technical independence assumption, the problem is shown to be equivalent to a set multicover problem (reducing to set cover when all eigenvalues are simple), and it extends naturally to robust, fault-tolerant scenarios by incorporating additional actuator redundancy. The authors prove NP-hardness and provide pathways to exact or approximate solutions using ILP solvers and well-established multicover algorithms, enabling practical offline design and reconfiguration. This work bridges actuator allocation and combinatorial optimization, and suggests future directions like joint sensor-actuator selection and time-varying actuator schedules for scalable systems.

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.
Paper Structure (10 sections, 6 theorems, 20 equations, 2 tables, 1 algorithm)

This paper contains 10 sections, 6 theorems, 20 equations, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Minimal Actuator Selection
  3. Robust Minimal Actuator Selection
  4. Algorithms for Actuator Selection
  5. Conclusion
  6. Proof of \ref{['thm:reformulation']}
  7. Proof of \ref{['prop:setmulticover']}
  8. Proof of \ref{['cor:setcover']}
  9. Proof of \ref{['thm:robust-reformulation']}
  10. Proof of \ref{['prop:robust_setmulticover']}

Key Result

Theorem 1

Consider the discrete-time linear dynamical system eq:sysmodel. Let $\boldsymbol{W}^{(i)}\in\mathbb{R}^{\alpha_i\times m}$, $i\in[p]$, be obtained from alg:BILP. Then, the minimal actuator selection problem eq:actuator_selection is equivalent to where $\mathcal{S}^*=\operatorname{supp}\left\{\boldsymbol{y}^*\right\}$ and $\odot$ is the Hadamard product.

Theorems & Definitions (15)

  • Theorem 1
  • proof
  • Definition 1: Full spark frame alexeev2012full
  • Theorem 2
  • proof
  • Corollary 3
  • proof
  • Theorem 4
  • proof
  • Corollary 5
  • ...and 5 more