Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Residual-Aware Distributionally Robust EKF: Absorbing Linearization Mismatch via Wasserstein Ambiguity

Minhyuk Jang, Jungjin Lee, Astghik Hakobyan, Naira Hovakimyan, Insoon Yang

Abstract

The extended Kalman filter (EKF) is a cornerstone of nonlinear state estimation, yet its performance is fundamentally limited by noise-model mismatch and linearization errors. We develop a residual-aware distributionally robust EKF that addresses both challenges within a unified Wasserstein distributionally robust state estimation framework. The key idea is to treat linearization residuals as uncertainty and absorb them into an effective uncertainty model captured by a stage-wise ambiguity set, enabling noise-model mismatch and approximation errors to be handled within a single formulation. This approach yields a computable effective radius along with deterministic upper bounds on the prior and posterior mean-squared errors of the true nonlinear estimation error. The resulting filter admits a tractable semidefinite programming reformulation while preserving the recursive structure of the classical EKF. Simulations on coordinated-turn target tracking and uncertainty-aware robot navigation demonstrate improved estimation accuracy and safety compared to standard EKF baselines under model mismatch and nonlinear effects.

Residual-Aware Distributionally Robust EKF: Absorbing Linearization Mismatch via Wasserstein Ambiguity

Abstract

The extended Kalman filter (EKF) is a cornerstone of nonlinear state estimation, yet its performance is fundamentally limited by noise-model mismatch and linearization errors. We develop a residual-aware distributionally robust EKF that addresses both challenges within a unified Wasserstein distributionally robust state estimation framework. The key idea is to treat linearization residuals as uncertainty and absorb them into an effective uncertainty model captured by a stage-wise ambiguity set, enabling noise-model mismatch and approximation errors to be handled within a single formulation. This approach yields a computable effective radius along with deterministic upper bounds on the prior and posterior mean-squared errors of the true nonlinear estimation error. The resulting filter admits a tractable semidefinite programming reformulation while preserving the recursive structure of the classical EKF. Simulations on coordinated-turn target tracking and uncertainty-aware robot navigation demonstrate improved estimation accuracy and safety compared to standard EKF baselines under model mismatch and nonlinear effects.

Paper Structure

This paper contains 26 sections, 6 theorems, 91 equations, 3 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notation
  4. Problem Setup
  5. Distributionally Robust State Estimation
  6. Wasserstein Ambiguity Sets
  7. Residual-Aware Approach for Distributionally Robust EKF
  8. Exact Nonlinear Error Dynamics and Effective Uncertainty Model
  9. Stage-wise SDP Reformulation Under Local Linear-Gaussian Surrogate
  10. Bounding the Linearization Residuals
  11. Computable Effective Radius and Recursive Distributionally Robust Certificate
  12. Oracle effective radius
  13. Surrogate posterior MSE envelope
  14. Computable prior bound and effective radius
  15. Certificate recursion and main result
  16. ...and 11 more sections

Key Result

Proposition 1

Fix $t \ge 1$, and let $\Sigma_{x,t-1}$ be the posterior covariance carried from stage $t-1$. Under the local linear-Gaussian approximation, the stage-wise surrogate of DR-MMSE estimation problem eq:stagewise-minimax is equivalent to the following finite-dimensional SDP: where Moreover, the SDP eq:sdp_t attains its optimum. If $\Sigma_{x,t}^*$ and $\Sigma_{\epsilon,t}^*$ are optimal for eq:sdp_t

Figures (3)

  • Figure 1: Coordinated-turn experiment under nominal covariance misspecification. (a) EKF with nominal statistics and (b) DR-EKF trajectories for the Monte Carlo run whose EKF MSE is the median over 100 runs. (c) Average MSE versus the initial turn rate $\omega_0$, where increasing $|\omega_0|$ corresponds to stronger nonlinearity; shaded regions denote $0.1$ times the standard deviation over 100 runs.
  • Figure 2: Closed-loop navigation under uncertainty-aware MPC. (a) Representative trajectories for EKF with nominal statistics, EKF with true noise statistics, and the proposed DR-EKF; crosses indicate collisions, and five of the 100 Monte Carlo runs are shown for clarity. (b) Time evolution of the safety margin $\delta_t$; shaded bands denote $\pm1$ empirical standard deviation over 100 runs.
  • Figure 3: Closed-loop navigation under uncertainty-aware MPC with pedestrian prediction on the ZARA02 scene from the ETH/UCY dataset. (a) Representative trajectories for EKF with nominal statistics, EKF with true noise statistics, and the proposed DR-EKF; crosses mark collisions. (b) Time evolution of the safety margin $\delta_t$; shaded bands indicate $\pm1$ empirical standard deviation across 50 runs.

Theorems & Definitions (16)

  • Remark 1
  • Proposition 1: Stage-wise SDP reformulation
  • Remark 2: Initial stage $t=0$
  • Lemma 1: Quadratic remainder bound
  • Lemma 2: Oracle residual radii
  • Lemma 3: Surrogate posterior MSE envelope
  • Lemma 4: Computable one-step radius bound
  • Theorem 1: Distributionally robust true MSE certificate
  • Remark 3
  • Remark 4
  • ...and 6 more