Deterministic and Randomized Actuator Scheduling With Guaranteed Performance Bounds

In this article, 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.

Automated summary

This article investigates actuator selection for linear dynamical systems, developing a framework for designing sparse actuator schedules with guaranteed performance bounds using deterministic and randomized algorithms. Introducing systemic controllability metrics, the study provides a polynomial-time actuator schedule selecting a constant number of actuators at each time step for approximating controllability metrics. The results also apply to sensor selection, demonstrating effectiveness through numerical simulations.