Deterministic and Randomized Actuator Scheduling With Guaranteed Performance Bounds

TL;DR

This paper presents deterministic and randomized algorithms for selecting a sparse set of actuators in large-scale linear systems, ensuring performance bounds on controllability and observability metrics with efficiency independent of system size.

Contribution

It introduces a new framework with systemic controllability metrics and provides polynomial-time algorithms for actuator scheduling with guaranteed performance bounds.

Findings

Constant number of actuators selected on average per time step

Guaranteed approximation of controllability and observability metrics

Algorithms are efficient and scalable to large systems

Abstract

In this paper, we investigate the problem of actuator selection for linear dynamical systems. We develop a framework to design a sparse actuator schedule for a given large-scale linear system with guaranteed performance bounds using deterministic polynomial-time and randomized approximately linear-time algorithms. First, we introduce systemic controllability metrics for linear dynamical systems that are monotone and homogeneous with respect to the controllability Gramian. We show that several popular and widely used optimization criteria in the literature belong to this class of controllability metrics. Our main result is to provide a polynomial-time actuator schedule that on average selects only a constant number of actuators at each time step, independent of the dimension, to furnish a guaranteed approximation of the controllability metrics in comparison to when all actuators are in use. Our results naturally apply to the dual problem of sensor selection, in which we provide a guaranteed approximation to the observability Gramian. We illustrate the effectiveness of our theoretical findings via several numerical simulations using benchmark examples.