Risk-sensitive Affine Control Synthesis for Stationary LTI Systems

TL;DR

This paper introduces a risk-sensitive control synthesis method for stationary LTI systems that minimizes risk measures over the stationary distribution, extending W-CVaR optimization to handle nonzero-mean noise and affine controllers with a convergent BMI-based algorithm.

Contribution

It extends W-CVaR optimization to affine controllers with nonzero-mean noise, reformulates the problem as a BMI, and proposes a convergent alternating optimization algorithm.

Findings

The proposed method effectively synthesizes risk-sensitive controllers.

The approach outperforms naive LQR in numerical experiments.

The BMI formulation enables robust control design.

Abstract

To address deviations from expected performance in stochastic systems, we propose a risk-sensitive control synthesis method to minimize certain risk measures over the limiting stationary distribution. Specifically, we extend Worst-case Conditional Value-at-Risk (W-CVaR) optimization for Linear Time-invariant (LTI) systems to handle nonzero-mean noise and affine controllers, using only the first and second moments of noise, which enhances robustness against model uncertainty. Highlighting the strong coupling between the linear and bias terms of the controller, we reformulate the synthesis problem as a Bilinear Matrix Inequality (BMI), and propose an alternating optimization algorithm with guaranteed convergence. Finally, we demonstrate the numerical performance of our approach in two representative settings, which shows that the proposed algorithm successfully synthesizes risk-sensitive controllers that outperform the na\"ive LQR baseline.