Regret of $H_\infty$ Preview Controllers

TL;DR

This paper analyzes how increasing the preview length in $H_ fty$ and regret-optimal controllers allows them to asymptotically match the performance of an ideal non-causal controller, with convergence results supported by a numerical example.

Contribution

It establishes the convergence of $H_ fty$ preview controllers' performance to the non-causal optimal and shows regret minimization with increasing preview length.

Findings

$H_ fty$ performance converges to non-causal optimal as preview length increases.

Optimal regret of preview controllers approaches zero with longer preview.

Numerical example confirms theoretical convergence results.

Abstract

This paper studies preview control in both the $H_\infty$ and regret-optimal settings. The plant is modeled as a discrete-time, linear time-invariant system subject to external disturbances. The performance baseline is the optimal non-causal controller that has full knowledge of the disturbance sequence. We first review the construction of the $H_\infty$ preview controller with $p$-steps of disturbance preview. We then show that the closed-loop $H_\infty$ performance of this preview controller converges as $p\to \infty$ to the performance of the optimal non-causal controller. Furthermore, we prove that the optimal regret of the preview controller converges to zero. These results demonstrate that increasing preview length allows controllers to asymptotically achieve non-causal performance in both the $H_\infty$ and regret frameworks. A numerical example illustrates the theoretical results.