Sparsity in Partially Controllable Linear Systems

TL;DR

This paper investigates partially controllable linear systems with sparsity patterns, providing structural conditions and finite-sample guarantees for learning control, and demonstrating improved performance over classical methods through simulations.

Contribution

It introduces a novel analysis of partially controllable systems with sparsity, offering structural insights and finite-sample guarantees using high-dimensional statistical techniques.

Findings

Structural conditions for irrelevant variables in control.

Finite-sample guarantees for learning control in sparse systems.

Simulation results show improvements over certainty-equivalence methods.

Abstract

A fundamental concept in control theory is that of controllability, where any system state can be reached through an appropriate choice of control inputs. Indeed, a large body of classical and modern approaches are designed for controllable linear dynamical systems. However, in practice, we often encounter systems in which a large set of state variables evolve exogenously and independently of the control inputs; such systems are only partially controllable. The focus of this work is on a large class of partially controllable linear dynamical systems, specified by an underlying sparsity pattern. Our main results establish structural conditions and finite-sample guarantees for learning to control such systems. In particular, our structural results characterize those state variables which are irrelevant for optimal control, an analysis which departs from classical control techniques. Our algorithmic results adapt techniques from high-dimensional statistics -- specifically soft-thresholding and semiparametric least-squares -- to exploit the underlying sparsity pattern in order to obtain finite-sample guarantees that significantly improve over those based on certainty-equivalence. We also corroborate these theoretical improvements over certainty-equivalent control through a simulation study.