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Distributionally Robust Kalman Filtering over Finite and Infinite Horizon

Taylan Kargin, Joudi Hajar, Vikrant Malik, Babak Hassibi

TL;DR

This work develops a distributionally robust Kalman filtering framework under a Wasserstein-2 ambiguity set, accommodating arbitrary temporal correlations in disturbances and measurement noise. It characterizes finite-horizon robustness via an SDP and infinite-horizon robustness via a dual Toeplitz formulation, proving linear estimators are optimal for Gaussian nominal disturbances and that the steady-state error remains bounded. A key innovation is a frequency-domain Frank-Wolfe algorithm that computes the non-rational optimal filter and a method to approximate it with finite-order rational transfer functions for practical state-space realization. The approach yields filters that interpolate between the nominal Kalman and robust $ ext{H}_\infty$ filters, with demonstrated accuracy and scalability in frequency- and time-domain experiments, and shows competitiveness against prior DR estimators. This enables robust, real-time filtering in scenarios with model misspecification and correlated disturbances, extending Kalman filtering to more realistic, uncertain environments.

Abstract

This paper investigates the distributionally robust filtering of signals generated by state-space models driven by exogenous disturbances with noisy observations in finite and infinite horizon scenarios. The exact joint probability distribution of the disturbances and noise is unknown but assumed to reside within a Wasserstein-2 ambiguity ball centered around a given nominal distribution. We aim to derive a causal estimator that minimizes the worst-case mean squared estimation error among all possible distributions within this ambiguity set. We remove the iid restriction in prior works by permitting arbitrarily time-correlated disturbances and noises. In the finite horizon setting, we reduce this problem to a semi-definite program (SDP), with computational complexity scaling with the time horizon. For infinite horizon settings, we characterize the optimal estimator using Karush-Kuhn-Tucker (KKT) conditions. Although the optimal estimator lacks a rational form, i.e., a finite-dimensional state-space realization, it can be fully described by a finite-dimensional parameter. {Leveraging this parametrization, we propose efficient algorithms that compute the optimal estimator with arbitrary fidelity in the frequency domain.} Moreover, given any finite degree, we provide an efficient convex optimization algorithm that finds the finite-dimensional state-space estimator that best approximates the optimal non-rational filter in ${\cal H}_\infty$ norm. This facilitates the practical implementation of the infinite horizon filter without having to grapple with the ill-scaled SDP from finite time. Finally, numerical simulations demonstrate the effectiveness of our approach in practical scenarios.

Distributionally Robust Kalman Filtering over Finite and Infinite Horizon

TL;DR

This work develops a distributionally robust Kalman filtering framework under a Wasserstein-2 ambiguity set, accommodating arbitrary temporal correlations in disturbances and measurement noise. It characterizes finite-horizon robustness via an SDP and infinite-horizon robustness via a dual Toeplitz formulation, proving linear estimators are optimal for Gaussian nominal disturbances and that the steady-state error remains bounded. A key innovation is a frequency-domain Frank-Wolfe algorithm that computes the non-rational optimal filter and a method to approximate it with finite-order rational transfer functions for practical state-space realization. The approach yields filters that interpolate between the nominal Kalman and robust filters, with demonstrated accuracy and scalability in frequency- and time-domain experiments, and shows competitiveness against prior DR estimators. This enables robust, real-time filtering in scenarios with model misspecification and correlated disturbances, extending Kalman filtering to more realistic, uncertain environments.

Abstract

This paper investigates the distributionally robust filtering of signals generated by state-space models driven by exogenous disturbances with noisy observations in finite and infinite horizon scenarios. The exact joint probability distribution of the disturbances and noise is unknown but assumed to reside within a Wasserstein-2 ambiguity ball centered around a given nominal distribution. We aim to derive a causal estimator that minimizes the worst-case mean squared estimation error among all possible distributions within this ambiguity set. We remove the iid restriction in prior works by permitting arbitrarily time-correlated disturbances and noises. In the finite horizon setting, we reduce this problem to a semi-definite program (SDP), with computational complexity scaling with the time horizon. For infinite horizon settings, we characterize the optimal estimator using Karush-Kuhn-Tucker (KKT) conditions. Although the optimal estimator lacks a rational form, i.e., a finite-dimensional state-space realization, it can be fully described by a finite-dimensional parameter. {Leveraging this parametrization, we propose efficient algorithms that compute the optimal estimator with arbitrary fidelity in the frequency domain.} Moreover, given any finite degree, we provide an efficient convex optimization algorithm that finds the finite-dimensional state-space estimator that best approximates the optimal non-rational filter in norm. This facilitates the practical implementation of the infinite horizon filter without having to grapple with the ill-scaled SDP from finite time. Finally, numerical simulations demonstrate the effectiveness of our approach in practical scenarios.
Paper Structure (46 sections, 21 theorems, 123 equations, 5 figures, 2 tables, 4 algorithms)

This paper contains 46 sections, 21 theorems, 123 equations, 5 figures, 2 tables, 4 algorithms.

Table of Contents

  1. Introduction
  2. Related Works
  3. Contributions
  4. Problem Setup
  5. The Finite-Horizon Distributionally Robust Filtering
  6. The Infinite-Horizon Distributionally Robust Filtering
  7. Tractable Convex Formulations
  8. An SDP for the Finite-Horizon Filtering
  9. A Concave-Convex Optimization for the Infinite-Horizon Filtering
  10. An Efficient Algorithm
  11. Iterative Optimization Methods in the Frequency-Domain
  12. Rational Approximation using H inf-norm
  13. Numerical Experiments
  14. Frequency Domain Evaluations
  15. Time Domain Evaluations
  16. ...and 31 more sections

Key Result

Theorem 3.1

Let $T\!>\!0$ be a fixed horizon and $\Pi_T$ be the class of non-linear causal estimators. Suppose that the nominal $\operatorname{\mathbb{P}}^\circ_T$ is Gaussian. Then, the following holds: Moreover, eq:minimax admits a saddle point $(\pi_T^\star, \operatorname{\mathbb{P}}_T^\star)$ such that the worst-case distribution $\operatorname{\mathbb{P}}_T^\star$ is Gaussian and the optimal causal filt

Figures (5)

  • Figure 1: DR-KF versus the $\mathcal{H}_2, \mathcal{H}_{\infty}$ filters and the variation of the expected MSE with $r$.
  • Figure 2: The average MSE of the different filters for the tracking problem, under (a) white noise, (b) correlated Gaussian noise, and (c) worst-case noise for the finite horizon DR KF for the system in \ref{['subsec:sim_freq']}. While the $H_2$ filter (KF) performs best in (a), it behaves poorly in (b), (c). The DRKF achieves the lowest error in (b) and (c), and the finite and infinite horizon achieve similar average MSE at the end of the horizon.
  • Figure 3: Average MSE for the KF, our DRKF, and the DRKF from shafieezadeh_2018, for system in section \ref{['sec:epfl']}.
  • Figure 4: The frequency response of different filters ($\mathcal{H}_2, \mathcal{H}_{\infty}$ and DRKF) for the tracking problem in section \ref{['ap:sim']}. The worst-case expected MSE is 3.99 for $H_\infty$ , 3.77 for $H_2$ and 3.47 (lowest) for DRKF.
  • Figure 5: The average MSE of the different filters horizon under different disturbances for the tracking problem in section \ref{['ap:sim']}. (a) is white noise, while (b) is the worst-case noise for the finite horizon DR KF (SDP). While the KF performs best under gaussian noise, the DRKF achieves the lowest error in most of other scenarios (including the more realistic case of correlated noise), and the finite and infinite horizon achieve similar avergae MSE at the end of the horizon.

Theorems & Definitions (34)

  • Remark 2.2
  • Remark 2.4
  • Theorem 3.1: Minimax duality
  • Lemma 3.3
  • Theorem 3.4: An SDP formulation for finite-horizon $\mathsf{W_2}$-DR-KF
  • Remark 3.5
  • Corollary 3.6
  • Lemma 3.7
  • Theorem 3.8: Convex formulation of infinite-horizon $\mathsf{W_2}$-DR-KF
  • Corollary 3.9
  • ...and 24 more