[Paper Review] Robust Kalman Filtering Under Model Uncertainty: the Case of Degenerate Densities

Shenglun Yi, Mattia Zorzi|arXiv (Cornell University)|Aug 25, 2021

Target Tracking and Data Fusion in Sensor Networks33 references4 citations

TL;DR

This paper proposes a robust Kalman filtering framework for state estimation under model uncertainty, specifically addressing degenerate (low-rank) transition densities. By formulating a minimax game over an ambiguity set defined via relative entropy, the method derives a low-rank, risk-sensitive Riccati-like iteration that maintains numerical stability and outperforms the standard Kalman filter in high-dimensional or degenerate settings.

ABSTRACT

We consider a robust state space filtering problem in the case that the transition probability density is unknown and possibly degenerate. The resulting robust filter has a Kalman-like structure and solves a minimax game: the nature selects the least favorable model in a prescribed ambiguity set which also contains non-Gaussian probability densities, while the other player designs the optimum filter for the least favorable model. It turns out that the resulting robust filter is characterized by a Riccati-like iteration evolving on the cone of the positive semidefinite matrices. Moreover, we study the convergence of such iteration in the case that the nominal model is with constant parameters on the basis of the contraction analysis in the same spirit of Bougerol. Finally, some numerical examples show that the proposed filter outperforms the standard Kalman filter.

Motivation & Objective

• Address the failure of standard Kalman filters in high-dimensional or degenerate systems where covariance matrices become indefinite.
• Extend robust Kalman filtering to cases where the transition density is possibly degenerate, enabling application in weather forecasting and oceanography.
• Develop a minimax filtering framework that accounts for model uncertainty via a relative entropy-based ambiguity set containing non-Gaussian and degenerate densities.
• Ensure numerical stability and convergence of the filter in the case of constant system parameters using contraction analysis.
• Derive the least favorable model and a robust filter structure that generalizes risk-sensitive filtering to low-rank settings.

Proposed method

• Formulate the robust filtering problem as a minimax game: nature selects the least favorable model from an ambiguity set, while the filter designer optimizes for that model.
• Define the ambiguity set as a Kullback-Leibler ball around the nominal Gaussian model, allowing for degenerate densities via a relative entropy constraint.
• Derive the least favorable model analytically, showing that only the state covariance matrix is perturbed, preserving the mean and cross-covariance.
• Propose a low-rank, risk-sensitive Riccati iteration that evolves on the cone of positive semidefinite matrices, with the iteration structure adapted to handle degeneracy.
• Use contraction analysis in the spirit of Bougerol to establish convergence of the robust filter under stabilizability, observability, and small ambiguity set conditions.
• Apply matrix decomposition (e.g., SVD) to handle low-rank structures, ensuring the filter remains computationally feasible in high-dimensional systems.

Experimental results

Research questions

• RQ1 How can robust Kalman filtering be extended to systems with degenerate (low-rank) transition densities?
• RQ2 What is the structure of the least favorable model in the presence of model uncertainty and degenerate densities?
• RQ3 Can a robust filter be designed that maintains stability and convergence when the nominal model has constant parameters and the ambiguity set is small?
• RQ4 How does the proposed filter compare numerically to the standard Kalman filter in terms of estimation accuracy under model mismatch?
• RQ5 What conditions ensure the convergence of the robust Riccati iteration in the degenerate case?

Key findings

• The proposed robust filter has a Kalman-like structure and is characterized by a Riccati-like iteration that evolves on the cone of positive semidefinite matrices, even when the transition density is degenerate.
• The least favorable model is shown to perturb only the state covariance matrix, preserving the mean and cross-covariance, which simplifies the robust design.
• Convergence of the robust filter is guaranteed under the conditions of stabilizability, observability of the reachable subspace, and a sufficiently small ambiguity set.
• The filter is numerically stable and outperforms the standard Kalman filter in numerical examples involving high-dimensional or degenerate systems.
• The robust filter corresponds to a low-rank risk-sensitive Riccati iteration, which is new in the literature and handles degeneracy via pseudo-inverses and subspace projections.
• The relative entropy constraint ensures that the ambiguity set includes non-Gaussian and degenerate densities, making the framework applicable to real-world problems like weather forecasting.