Beyond Quadratic Costs: A Bregman Divergence Approach to H$_\infty$ Control

TL;DR

This paper extends $H_$ control to include strictly convex penalties using Bregman divergences, resulting in a closed-form, nonlinear controller with robust performance guarantees for linear systems.

Contribution

It introduces a novel Bregman divergence-based approach to $H_$ control, enabling the use of non-quadratic convex costs while maintaining tractability and robustness.

Findings

Derives a Riccati-like identity for the new control framework.

Provides necessary and sufficient conditions for optimality.

Enables nonlinear controllers with safety and sparsity features.

Abstract

In the past couple of decades, non-quadratic convex penalties have reshaped signal processing and machine learning; in robust control, however, general convex costs break the Riccati and storage function structure that make the design tractable. Practitioners thus default to approximations, heuristics or robust model predictive control that are solved online for short horizons. We close this gap by extending $H_\infty$ control of discrete-time linear systems to strictly convex penalties on state, input, and disturbance, recasting the objective with Bregman divergences that admit a completion-of-squares decomposition. The result is a closed-form, time-invariant, full-information stabilizing controller that minimizes a worst-case performance ratio over the infinite horizon. Necessary and sufficient existence/optimality conditions are given by a Riccati-like identity together with a concavity requirement; with quadratic costs, these collapse to the classical $H_\infty$ algebraic Riccati equation and the associated negative-semidefinite condition, recovering the linear central controller. Otherwise, the optimal controller is nonlinear and can enable safety envelopes, sparse actuation, and bang-bang policies with rigorous $H_\infty$ guarantees.