Wasserstein Distributionally Robust Control and State Estimation for Partially Observable Linear Systems

TL;DR

This work tackles control and state estimation for partially observable linear systems with unknown disturbance and measurement-noise distributions. It introduces a Wasserstein distributionally robust framework (WDR-CE) that jointly addresses disturbances (DRC) and estimation errors (DRSE) by replacing Wasserstein ambiguity with Gelbrich-distance penalties, enabling a scalable SDP-based solution. A separation principle holds, yielding an affine robust controller and a DR Kalman filter that operate with worst-case disturbance statistics, all within an offline-online procedure. The approach provides guaranteed-cost and out-of-sample performance guarantees and demonstrates superior performance and computational efficiency over baselines, including in long-horizon and non-Gaussian scenarios. This framework offers practical robustness against distributional mis-specification in real-world, partially observable control applications.

Abstract

This paper presents a novel Wasserstein distributionally robust control and state estimation algorithm for partially observable linear stochastic systems, where the probability distributions of disturbances and measurement noises are unknown. Our method consists of the control and state estimation phases to handle distributional ambiguities of system disturbances and measurement noises, respectively. Leveraging tools from modern distributionally robust optimization, we consider an approximation of the control problem with an arbitrary nominal distribution and derive its closed-form optimal solution. We show that the separation principle holds, thereby allowing the state estimator to be designed separately. A novel distributionally robust Kalman filter is then proposed as an optimal solution to the state estimation problem with Gaussian nominal distributions. Our key contribution is the combination of distributionally robust control and state estimation into a unified algorithm. This is achieved by formulating a tractable semidefinite programming problem that iteratively determines the worst-case covariance matrices of all uncertainties, leading to a scalable and efficient algorithm. Our method is also shown to enjoy a guaranteed cost property as well as a probabilistic out-of-sample performance guarantee. The results of our numerical experiments demonstrate the performance and computational efficiency of the proposed method.

Figures (7)

• Figure 1: (a) Total costs incurred by LQG, WDRC, DRLQC, and our WDR-CE methods, for nonzero-mean U-Quadaratic distribution with varying $\theta_w$ and $\theta_v$, averaged over 500 simulation runs; (b) Computation times for zero-mean Gaussian distributions averaged over 10 simulation runs. The shaded region represents 25% standard deviation from the mean.
• Figure 2: Total costs incurred by LQG, WDRC, DRLQC, and our WDR-CE methods, for zero-mean (a) Gaussian and (b) U-Quadratic distributions, with varying $\theta_w$ and $\theta_v$, averaged over 500 simulation runs.
• Figure 3: Total costs incurred by WDRC, WDRC+DRKF, WDRC+DRMMSE, and our WDR-CE methods, for (a) Gaussian and (b) U-Quadratic distributions, with varying $\lambda$ and $\theta_v$, averaged over 500 simulation runs.
• Figure 4: Total costs incurred by LQG, WDRC, and our WDR-CE methods for a long horizon $T=200$, for (a) Gaussian and (b) U-Quadratic distributions, with varying $\theta_w$ and $\theta_v$, averaged over 500 simulation runs.
• Figure 5: (a) Out-of-sample performance and (b) reliability of our WDR-CE method estimated over 100 simulation runs with 1000 uncertainty samples. The shaded region represents $25\%$ standard deviation from the mean.

Theorems & Definitions (14)

• Definition 1: Wasserstein Distance
• Definition 2: Gelbrich Bound
• Theorem 1
• Remark 1
• Lemma 1
• Theorem 2: DR Kalman Filter
• Proposition 1
• Remark 2
• Proposition 2
• Theorem 3