The Distributionally Robust Infinite-Horizon LQR

TL;DR

The paper addresses robust control under distributional uncertainty for the infinite-horizon LQR by formulating a Wasserstein-2 ambiguity set around a nominal disturbance distribution. It develops a duality-based approach yielding a $\ gamma$-optimal controller that is stabilizing for $ u$ larger than the $H_\infty$ threshold and computes the optimal non-rational controller in frequency domain via a fixed-point method; a convex scheme then produces finite-order rational approximations for practical time-domain implementation. The methodology bridges stochastic and adversarial viewpoints by connecting the worst-case disturbance spectrum to the controller via spectral factors, and it demonstrates an interpolation between $H_2$ and $H_\infty$ as the ambiguity radius varies. Numerical results on aircraft benchmark models show that the infinite-horizon DR-LQR outperforms both standard $H_2$ and $H_\infty$ controllers and that the rational approximations closely match the non-rational optimum, offering scalable, robust performance under distributional uncertainty with efficient computation.

Abstract

We explore the infinite-horizon Distributionally Robust (DR) linear-quadratic control. While the probability distribution of disturbances is unknown and potentially correlated over time, it is confined within a Wasserstein-2 ball of a radius $r$ around a known nominal distribution. Our goal is to devise a control policy that minimizes the worst-case expected Linear-Quadratic Regulator (LQR) cost among all probability distributions of disturbances lying in the Wasserstein ambiguity set. We obtain the optimality conditions for the optimal DR controller and show that it is non-rational. Despite lacking a finite-order state-space representation, we introduce a computationally tractable fixed-point iteration algorithm. Our proposed method computes the optimal controller in the frequency domain to any desired fidelity. Moreover, for any given finite order, we use a convex numerical method to compute the best rational approximation (in $H_\infty$-norm) to the optimal non-rational DR controller. This enables efficient time-domain implementation by finite-order state-space controllers and addresses the computational hurdles associated with the finite-horizon approaches to DR-LQR problems, which typically necessitate solving a Semi-Definite Program (SDP) with a dimension scaling with the time horizon. We provide numerical simulations to showcase the effectiveness of our approach.

Figures (6)

• Figure 1: The power spectrum $N(\mathrm{e}^{{j}\omega})$ of the worst-case disturbance, when $r \in (0.01, 5, 10)$ for system [AC15].
• Figure 2: The worst-case expected LQR cost of the classical controllers $H_2$, $H_\infty$ compared to the DR-LQR, for the system [AC15], for different $r$ values. The DR-LQR minimizes the cost at all $r's$.
• Figure 3: The percentage difference in the worst case LQR cost relative to the DR-LQR (see the legend) for the system [AC15], for different values of $r$. When $r$ is small (large) $r$, the cost of DR-LQR controller closely aligns with that of ${H}_2$ (${ H}_\infty$). For $r=1.5$, the cost of the DR-LQR is less than that of $H_2$ by $22.8\%$, and that of $H_\infty$ by $19.2\%$ .
• Figure 4: The operator norm, $\|T_K^\ast(\mathrm{e}^{{j}\omega})T_K(\mathrm{e}^{{j}\omega})\|$, of different controllers at all frequencies $\omega \in [0,2\pi]$, for system [AC15]. The DR-LQR cost interpolates between $H_2$ and $H_\infty$ based on the value of $r$. When $r$ is small (large), DR closely aligns with $H_2$ ($H_\infty$) across all frequencies.
• Figure 5: Convergence ratio \ref{['eq:ratio']} for different values of $\gamma$. The Fixed-Point algorithm converges at an exponential rate.

Theorems & Definitions (15)

• Definition 1: Worst-case expected LQR cost under $\mathsf{W_2}$-ambiguity
• Theorem 1: Strong Duality
• proof
• Theorem 2: $\gamma$-optimal solution via saddle points
• proof
• Remark 1
• Remark 2
• Corollary 3
• Theorem 4: $\gamma-$optimal solution is a fixed-Point Solution
• proof