Generalised Regret Optimal Controller Synthesis for Constrained Systems

TL;DR

This paper introduces a synthesis method for generalized dynamic regret in constrained systems, enabling adaptive control strategies that outperform classical methods under various disturbance conditions.

Contribution

It extends dynamic regret synthesis to include pointwise disturbance bounds and system constraints, providing a unified adaptive control framework.

Findings

Optimal solutions are no worse than bounded energy solutions.

The method guarantees performance bounds based on disturbance weights.

Numerical comparisons show advantages over traditional $ ext{H}_2$ and $ ext{H}_ extinfty$ controllers.

Abstract

This paper presents a synthesis method for the generalised dynamic regret problem, comparing the performance of a strictly causal controller to the optimal non-causal controller under a weighted disturbance. This framework encompasses both the dynamic regret problem, considering the difference of the incurred costs, as well as the competitive ratio, which considers their ratio, and which have both been proposed as inherently adaptive alternatives to classical control methods. Furthermore, we extend the synthesis to the case of pointwise-in-time bounds on the disturbance and show that the optimal solution is no worse than the bounded energy optimal solution and is lower bounded by a constant factor, which is only dependent on the disturbance weight. The proposed optimisation-based synthesis allows considering systems subject to state and input constraints. Finally, we provide a numerical example which compares the synthesised controller performance to $\mathcal{H}_2$- and $\mathcal{H}_\infty$-controllers.