pith. sign in

arxiv: 2504.07847 · v2 · pith:Y3T5EGZFnew · submitted 2025-04-10 · 🧮 math.OC

An update-resilient Kalman filtering approach

Shenglun Yi , Mattia Zorzi This is my paper

Pith reviewed 2026-05-25 07:52 UTC · model grok-4.3

classification 🧮 math.OC
keywords robust Kalman filterupdate-resilient filterambiguity setobservation uncertaintyfilter stabilityleast favorable modelminimax filtering
0
0 comments X

The pith

Update-resilient Kalman filter derived when uncertainty affects only the observations

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper proposes a robust filtering paradigm in which model uncertainty, expressed via an ambiguity set, is confined to the observations. It derives the corresponding estimator, called the update-resilient Kalman filter, and presents it as novel relative to existing minimax game-based methods. The work also characterizes the least favorable state space model and establishes stability properties of the filter. Numerical examples are provided to illustrate performance.

Core claim

When model uncertainty is restricted to the observations through an ambiguity set, the corresponding robust estimator is the update-resilient Kalman filter. This estimator is derived, shown to differ from prior minimax approaches, accompanied by an explicit characterization of the least favorable state space model, and supported by a stability analysis.

What carries the argument

The update-resilient Kalman filter, the robust estimator obtained by restricting the ambiguity set to the observation model

If this is right

  • The filter is distinct from existing minimax game-based filtering approaches.
  • The associated least favorable state space model can be explicitly characterized.
  • Stability of the filter holds under the stated conditions on the observation uncertainty.
  • Numerical examples confirm effectiveness of the estimator in practice.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Limiting uncertainty to observations may reduce computational burden relative to full-model ambiguity sets.
  • The construction could be tested on systems with intermittent sensor errors to check practical resilience.
  • Similar resilient estimators might be derived for other linear estimators by isolating uncertainty to one equation block.

Load-bearing premise

Model uncertainty described through an ambiguity set is present only in the observations.

What would settle it

A concrete counter-example in which the derived filter loses its claimed robustness or stability when uncertainty is also introduced into the state dynamics would falsify the scope of the guarantees.

Figures

Figures reproduced from arXiv: 2504.07847 by Mattia Zorzi, Shenglun Yi.

Figure 1
Figure 1. Figure 1: Minimum eigenvalue of Rk as a function of ϕk with k = 10. The largest value of ϕk such that Rk is positive definite is approximately equal to 0.095. 5 10 15 20 25 30 35 40 45 50 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 KF P-RKF U-RKF [PITH_FULL_IMAGE:figures/full_fig_p021_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Variance of the estimation error when KF (black line), P-RKF (red line) and [PITH_FULL_IMAGE:figures/full_fig_p021_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Variance of the estimation error when KF (black line), P-RKF (red line) and [PITH_FULL_IMAGE:figures/full_fig_p021_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Risk sensitivity parameter θt of P-RKF (red line) and U-RKF (blue line) when c = 5 · 10−2 . m k c p [PITH_FULL_IMAGE:figures/full_fig_p022_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Mass-spring-damper system. that it considers model uncertainties also in the process equation (1). Finally, [PITH_FULL_IMAGE:figures/full_fig_p022_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Average variance of the displacement for the different filters in the presence [PITH_FULL_IMAGE:figures/full_fig_p024_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Average variance of the displacement for the different filters in the case of [PITH_FULL_IMAGE:figures/full_fig_p024_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Average variance of the displacement for the different filters in the presence [PITH_FULL_IMAGE:figures/full_fig_p025_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Average variance of the displacement for the different filters in the case of [PITH_FULL_IMAGE:figures/full_fig_p025_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Variance of the estimation error when the standard KF (black line), P-RSF [PITH_FULL_IMAGE:figures/full_fig_p032_10.png] view at source ↗
read the original abstract

We propose a new robust filtering paradigm considering the situation in which model uncertainty, described through an ambiguity set, is present only in the observations. We derive the corresponding robust estimator, referred to as update-resilient Kalman filter, which appears to be novel compared to existing minimax game-based filtering approaches. Moreover, we characterize the corresponding least favorable state space model and analyze the filter stability. Finally, some numerical examples show the effectiveness of the proposed estimator.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper proposes a robust Kalman filtering paradigm in which model uncertainty (via an ambiguity set) is confined exclusively to the observation equation. It derives a corresponding minimax estimator termed the update-resilient Kalman filter, claims novelty relative to existing game-based approaches, characterizes the associated least-favorable state-space model, proves stability of the resulting filter, and illustrates performance via numerical examples.

Significance. If the derivation and stability analysis hold under the stated restriction, the work supplies a targeted robust estimator with an explicit least-favorable model and Riccati-based stability guarantees. This is a modest but concrete contribution within the minimax filtering literature, provided the restriction to observation uncertainty is maintained.

major comments (2)
  1. [Problem formulation and stability analysis] The derivation of the update-resilient filter and the subsequent stability argument both rely on the explicit restriction that the ambiguity set acts only on the observation equation (as stated in the abstract and used to obtain the saddle-point structure). The manuscript should state this modeling choice as an assumption in the problem formulation section and verify that the closed-form recursion and Riccati bounds do not extend without modification if process-model uncertainty is later introduced.
  2. [Introduction / related-work discussion] The claim that the filter is novel compared with existing minimax approaches requires a precise contrast (e.g., difference in the resulting Riccati equation or in the least-favorable dynamics) rather than a general statement; without that comparison the novelty assertion remains difficult to assess.
minor comments (2)
  1. [Numerical examples] Numerical examples are referenced but not described in the abstract; ensure that the full manuscript supplies sufficient detail on the simulated systems, ambiguity-set parameters, and performance metrics so that the effectiveness claims can be reproduced.
  2. [Throughout] Notation for the ambiguity set and the least-favorable model should be introduced once and used consistently; avoid re-defining symbols across sections.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments. We address each major point below and will revise the manuscript accordingly.

read point-by-point responses

  1. Referee: [Problem formulation and stability analysis] The derivation of the update-resilient filter and the subsequent stability argument both rely on the explicit restriction that the ambiguity set acts only on the observation equation (as stated in the abstract and used to obtain the saddle-point structure). The manuscript should state this modeling choice as an assumption in the problem formulation section and verify that the closed-form recursion and Riccati bounds do not extend without modification if process-model uncertainty is later introduced.

    Authors: We agree that the observation-only restriction is essential to the saddle-point structure and resulting recursions. In the revised manuscript we will add this explicitly as Assumption 1 in the problem formulation section. We will also insert a short remark noting that the closed-form filter and Riccati stability bounds rely on this restriction and would require a different minimax formulation (and modified analysis) if process-model uncertainty were included. revision: yes

  2. Referee: [Introduction / related-work discussion] The claim that the filter is novel compared with existing minimax approaches requires a precise contrast (e.g., difference in the resulting Riccati equation or in the least-favorable dynamics) rather than a general statement; without that comparison the novelty assertion remains difficult to assess.

    Authors: We will strengthen the novelty discussion. In the revised introduction we will add a direct comparison: unlike typical game-theoretic filters that place ambiguity in both process and observation models (yielding coupled Riccati equations and joint least-favorable dynamics), our observation-only ambiguity set produces a distinct Riccati recursion whose solution depends only on modified observation-noise terms, together with a least-favorable model whose process dynamics remain nominal. This explicit contrast will be placed in both the introduction and the related-work subsection. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation follows from minimax setup under stated assumption

full rationale

The paper derives the update-resilient Kalman filter from a minimax game under the explicit modeling choice that ambiguity is confined to the observation equation. The least-favorable state-space model and stability bounds are obtained directly from the resulting saddle-point problem and Riccati recursion; neither step reduces to a fitted parameter renamed as a prediction nor relies on a self-citation chain whose content is itself unverified. The restriction to observation-only uncertainty is an assumption, not a self-definitional loop, and the central claims remain independent of the paper's own outputs.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

No free parameters, axioms, or invented entities can be identified from the abstract alone; the ambiguity set is referenced but not specified.

pith-pipeline@v0.9.0 · 5585 in / 1108 out tokens · 46406 ms · 2026-05-25T07:52:29.677905+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

28 extracted references · 28 canonical work pages

  1. [1]

    Abadeh, V

    S. Abadeh, V. Nguyen, D. Kuhn, and P. Esfahani. Wasserstein distributionally robust Kalman filtering. In Adv. Neural Inf. Process. Syst. , pages 8474–8483, 2018

    work page 2018

  2. [2]

    Aubin and I

    J. Aubin and I. Ekeland. Applied nonlinear analysis . Courier Corporation, 2006

    work page 2006

  3. [3]

    F. S. Cattivelli and A. H. Sayed. Diffusion strategies for distributed Kalman filtering and smoothing. IEEE Trans. Autom. Control, 55(9):2069–2084, 2010

    work page 2069

  4. [4]

    Cover and J

    T. Cover and J. Thomas. Information Theory. Wiley, New York, 1991

    work page 1991

  5. [5]

    El Ghaoui and G

    L. El Ghaoui and G. Calafiore. Robust filtering for discrete-time systems with bounded noise and parametric uncertainty. IEEE Trans. Autom. Control, 46(7):1084–1089, 2001. 32

    work page 2001

  6. [6]

    Hansen and T

    L. Hansen and T. Sargent. Robustness. Princeton University Press, Princeton, NJ, 2008

    work page 2008

  7. [7]

    Hassibi, A

    B. Hassibi, A. Sayed, and T. Kailath. Indefinite-Quadratic Estimation and Control- A Unified Approach to H2 and H ∞ Theories. SIAM, Philadelphia, 1999

    work page 1999

  8. [8]

    Huang, D

    J. Huang, D. Shi, and T. Chen. Distributed robust state estimation for sensor networks: A risk-sensitive approach. In 2018 IEEE Conference on Decision and Control (CDC), pages 6378–6383, 2018

    work page 2018

  9. [9]

    Huang, Y

    Y. Huang, Y. Zhang, N. Li, and J. Chambers. A robust Gaussian approximate fixed-interval smoother for nonlinear systems with heavy-tailed process and measurement noises. IEEE Signal Process. Lett., 23(4):468–472, 2016

    work page 2016

  10. [10]

    Huang, Y

    Y. Huang, Y. Zhang, Y. Zhao, L. Mihaylova, and J. Chambers. Robust Rauch– Tung–Striebel smoothing framework for heavy-tailed and/or skew noises. IEEE Trans. Aerosp. Electron. Syst., 56(1):415–441, 2019

    work page 2019

  11. [11]

    S. Kim, V. M. Deshpande, and R. Bhattacharya. Robust kalman filtering with probabilistic uncertainty in system parameters. IEEE Control Syst. Lett. , 5(1):295–300, 2021

    work page 2021

  12. [12]

    Levy and R

    B. Levy and R. Nikoukhah. Robust least-squares estimation with a relative entropy constraint. IEEE Trans. Informat. Theory , 50(1):89–104, 2004

    work page 2004

  13. [13]

    Levy and R

    B. Levy and R. Nikoukhah. Robust state-space filtering under incremental model perturbations subject to a relative entropy tolerance. IEEE Trans. Autom. Control, 58:682–695, Mar. 2013

    work page 2013

  14. [14]

    Levy and M

    B. Levy and M. Zorzi. A contraction analysis of the convergence of risk-sensitive filters. SIAM J Control Optim , 54(4):2154–2173, 2016

    work page 2016

  15. [15]

    Longhini, M

    A. Longhini, M. Perbellini, S. Gottardi, S. Yi, H. Liu, and M. Zorzi. Learning the tuned liquid damper dynamics by means of a robust EKF. In American Control Conference (ACC), pages 60–65. IEEE, 2021

    work page 2021

  16. [16]

    V. A. Nguyen, S. Shafieezadeh-Abadeh, D. Kuhn, and P. Mohajerin Esfahani. Bridging bayesian and minimax mean square error estimation via Wasserstein distributionally robust optimization. Math. Oper. Res., 48(1):1–37, 2023

    work page 2023

  17. [17]

    K. D.T. Rocha and M. H. Terra. Robust Kalman filter for systems subject to parametric uncertainties. Syst. Control Lett., 157:105034, 2021

    work page 2021

  18. [18]

    Speyer and W

    J. Speyer and W. Chung. Stochastic Processes, Estimation, and Control . Advances in Design and Control. Soc. Indust. Appl. Math., Philadelphia, 2008

    work page 2008

  19. [19]

    Y. Xu, W. Xue, C. Shang, and H. Fang. On globalized robust kalman filter under model uncertainty. IEEE Trans. Autom. Control, 70(2):1147–1160, 2025

    work page 2025

  20. [20]

    S. Yi, T. Su, and Z. Tang. Robust adaptive kalman filter for structural performance assessment. Int. J. Robust Nonlinear Control , 34(9):5966–5982, 2024. 33

    work page 2024

  21. [21]

    Yi and M

    S. Yi and M. Zorzi. Low-rank Kalman filtering under model uncertainty. In 59th IEEE Conference on Decision and Control (CDC), pages 2930–2935, 2020

    work page 2020

  22. [22]

    Yi and M

    S. Yi and M. Zorzi. Robust Kalman filtering under model uncertainty: The case of degenerate densities. IEEE Trans. Autom. Control, 67(7):3458–3471, 2021

    work page 2021

  23. [23]

    M. Yoon, V. Ugrinovskii, and I. Petersen. Robust finite horizon minimax filtering for discrete-time stochastic uncertain systems. Syst. Control Lett. , 52(2):99–112, 2004

    work page 2004

  24. [24]

    Zenere and M

    A. Zenere and M. Zorzi. On the coupling of model predictive control and robust Kalman filtering. IET Control. Theory Appl. , 12(13):1873–1881, 2018

    work page 2018

  25. [25]

    F. Zhu, Y. Huang, C. Xue, L. Mihaylova, and J. Chambers. A sliding window variational outlier-robust kalman filter based on student’s t-noise modeling. IEEE Trans. Aerosp. Electron. Syst., 58(5):4835–4849, 2022

    work page 2022

  26. [26]

    M. Zorzi. Robust Kalman filtering under model perturbations. IEEE Trans. Autom. Control, 62(6):2902–2907, 2016

    work page 2016

  27. [27]

    M. Zorzi. Convergence analysis of a family of robust Kalman filters based on the contraction principle. SIAM J Control Optim , 55(5):3116–3131, 2017

    work page 2017

  28. [28]

    Zorzi and B

    M. Zorzi and B. Levy. Robust Kalman filtering: Asymptotic analysis of the least favorable model. In 57th IEEE Conference on Decision and Control (CDC) , pages 7124–7129, Dec 2018. 34

    work page 2018