Distributionally Robust LQG with Kullback-Leibler Ambiguity Sets

TL;DR

This paper develops a distributionally robust LQG controller that accounts for model uncertainties using Kullback-Leibler divergence, maintaining linearity and providing a convergent computational scheme.

Contribution

It introduces a novel robustification of LQG with ambiguity sets based on relative entropy, extending to decision-dependent uncertainties with an approximation method.

Findings

Optimal control remains linear under distributional robustness.

The proposed iterative scheme converges to saddle points.

The approach effectively handles model misspecification in stochastic systems.

Abstract

The Linear Quadratic Gaussian (LQG) controller is known to be inherently fragile to model misspecifications common in real-world situations. We consider discrete-time partially observable stochastic linear systems and provide a robustification of the standard LQG against distributional uncertainties on the process and measurement noise. Our distributionally robust formulation specifies the admissible perturbations by defining a relative entropy based ambiguity set individually for each time step along a finite-horizon trajectory, and minimizes the worst-case cost across all admissible distributions. We prove that the optimal control policy is still linear, as in standard LQG, and derive a computational scheme grounded on iterative best response that provably converges to the set of saddle points. Finally, we consider the case of endogenous uncertainty captured via decision-dependent ambiguity sets and we propose an approximation scheme based on dynamic programming.