Robust Regret Control with Uncertainty-Dependent Baseline

TL;DR

This paper introduces a robust regret control framework for uncertain linear systems, where the baseline adapts to system uncertainty, and the controller design is formulated as a $$-synthesis problem.

Contribution

It develops a method to achieve robust regret bounds by linking regret minimization to $H_$ performance, using linear approximation for nonlinear uncertainty dependence.

Findings

Controller achieves robust additive regret relative to the baseline.

Recasting regret conditions as a $$-synthesis problem simplifies design.

Numerical example demonstrates effectiveness of the approach.

Abstract

This paper proposes a robust regret control framework in which the performance baseline adapts to the realization of system uncertainty. The plant is modeled as a discrete-time, uncertain linear time-invariant system with real-parametric uncertainty. The performance baseline is the optimal non-causal controller constructed with full knowledge of the disturbance and the specific realization of the uncertain plant. We show that a controller achieves robust additive regret relative to this baseline if and only if it satisfies a related, robust $H_\infty$ performance condition on a modified plant. One technical issue is that the modified plant can, in general, have a complicated nonlinear dependence on the uncertainty. We use a linear approximation step so that the robust additive regret condition can be recast as a standard $\mu$-synthesis problem. A numerical example is used to demonstrate the proposed approach.