LQR Control with Sparse Adversarial Disturbances

TL;DR

This paper analyzes the impact of sparse adversarial disturbances on LQR control, deriving policies and bounds that characterize the suboptimality of standard controllers under such disturbances.

Contribution

It introduces a disturbance-aware policy with a recurrence structure and provides regret bounds comparing it to optimal offline control.

Findings

The disturbance-aware policy converges to the blind online policy with sublinear disturbance growth.

Finite-horizon regret between blind online and optimal offline policies is quadratic in disturbances.

The analysis characterizes the suboptimality of LQR under sparse adversarial disturbances.

Abstract

Recent developments in cyber-physical systems and event-triggered control have led to an increased interest in the impact of sparse disturbances on dynamical processes. We study Linear Quadratic Regulator (LQR) control under sparse disturbances by analyzing three distinct policies: the blind online policy, the disturbance-aware policy, and the optimal offline policy. We derive the two-dimensional recurrence structure of the optimal disturbance-aware policy, under the assumption that the controller has information about future disturbance values with only a probabilistic model of their locations in time. Under mild conditions, we show that the disturbance-aware policy converges to the blind online policy if the number of disturbances grows sublinearly in the time horizon. Finally, we provide a finite-horizon regret bound between the blind online policy and optimal offline policy, which is proven to be quadratic in the number of disturbances and in their magnitude. This provides a useful characterization of the suboptimality of a standard LQR controller when confronted with unexpected sparse perturbations.