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REVIEW 4 major objections 5 minor 22 references

A robust approach to sigma point Kalman filtering

T0 review · 4 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read A minimax game with a least-favorable covariance makes sigma-point filters robust to model uncertainty.

desk verdict The robust sigma-point filtering idea is new and the update-resilient variant is sound, but the prediction-resilient theorem's Gaussian center is mis-specified, leaving the main optimality claim unsupported. read the letter →

arxiv 2506.04815 v2 pith:OVYR6MQV submitted 2025-06-05 math.OC

Shenglun Yi , Mattia Zorzi This is my paper
classification math.OC MSC 93E1193E2062F3565C05
keywords robustnonlinearstateestimationminimaxgamesigma-pointKalmanfilterKullback-LeiblerambiguitysetleastfavorablemodelMetropolis-Hastingssimulatorunscenteduncertainty
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper proposes two robust filters, prediction-resilient and update-resilient, for nonlinear state-space models whose true dynamics differ from the nominal model. It sets up a dynamic minimax game in which the estimator minimizes mean-squared error and an adversarial player chooses the least favorable model inside a Kullback-Leibler ball around the nominal model. The main result is a closed-form robust predictor: generate $\sigma$ points from an inflated covariance, propagate them as in standard $\sigma$-point filtering, and update the least favorable covariance as $(P^{-1} - \theta I)^{-1}$ with $\theta$ chosen so that the worst-case KL divergence equals the tolerance $c_t$. If the construction holds, the filter attains the optimal worst-case mean-squared error over the ambiguity set and reduces exactly to the standard $\sigma$-point filter when $c_t = 0$.

What carries the argument

The load-bearing object is $\bar p_t(z_t|Y_{t-1}) = N(\bar m_t, \bar K_t)$, the $\sigma$-point approximation of the pseudo-nominal conditional density of $(x_{t+1}, y_t)$, with blocks (14)-(18) from Proposition 1. This approximation turns the intractable minimax problem over non-Gaussian densities into the Gaussian robust least-squares setting of [13, Theorem 1], which supplies the minimizer (the conditional mean) and the maximizer (the least favorable Gaussian covariance). The second mechanism is covariance inflation: replace a covariance $P$ by $(P^{-1} - \theta I)^{-1}$, where $\theta$ solves $\gamma(P, \theta) = c_t$ and $\gamma(P,\theta) := \frac12(\log\det(I-\theta P) + \mathrm{tr}((I-\theta P)^{-1}-I))$. The filter alternates this inflation with ordinary $\sigma$-point propagation: $\sigma$ points from $\tilde P_t$, update with the Kalman gain, propagate through $f$, reform $P_{t+1}$, and inflate again.

What would settle it

Compute whether the density defined by (2), (14)-(18) integrates to one over $z_t$ for a fixed nonlinear model and a fixed observation $y_t$; if the integral over $x_{t+1}$ is not 1, or the implied conditional density of $x_{t+1}$ given $y_t$ is not Gaussian, the KL ball in (29) has no well-defined center and the claimed worst-case optimality does not follow.

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Extended reading notes

Core claim

The paper's central claim is Theorem 1: the estimator solving the approximate minimax game (28), in which the adversarial player maximizes prediction error over the KL ball centered at the $\sigma$-point approximation, is exactly the $\sigma$-point predictor (30)-(35), provided the $\sigma$ points are generated from the least favorable a priori covariance $\tilde P_t$. The nominal prediction covariance $P_{t+1}$ is formed by ordinary $\sigma$-point propagation, and the least favorable prediction covariance is $\tilde P_{t+1} = (P_{t+1}^{-1} - \theta_t I)^{-1}$, where $\theta_t > 0$ is the unique solution of $\gamma(P_{t+1}, \theta_t) = c_t$. The companion update-resilient filter applies the same covariance inflation to the filtering covariance $P_{t|t}$ instead. The paper further claims that the least favorable model, though generally nonlinear and non-Gaussian, is available in explicit approximate form, and it provides an MCMC simulator for generating data from that model; the simulations show the filter matched to the least favorable model performs best and that all robust variants beat standard $\sigma$-point filters, particle filters, and a maximum-correntropy UKF.

Load-bearing premise

The derivation assumes the approximate joint density of next state and current observation built from (14)-(18) is a genuine, well-normalized Gaussian density, even though the next-state mean in (14) depends on the observation, which a joint density over both variables cannot do.

Editorial extensions

If this is right

  • If Theorem 1 is correct, no estimator using the same observations can have smaller worst-case mean-squared error over the KL ambiguity set than the prediction-resilient filter.
  • Setting $c_t = 0$ gives $\theta_t = 0$ and makes both new filters identical to the standard sigma-point Kalman filter, so the robust filters are strict generalizations.
  • The explicit least favorable model and its MCMC simulator allow a user to generate adversarial trajectories consistent with the chosen tolerance, making the filter's uncertainty assessment testable before deployment.
  • The simulations indicate that the update-resilient filter is preferable when measurement uncertainty dominates and the prediction-resilient filter when uncertainty is balanced between process and measurement.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If the covariance-inflation construction holds, the same recipe should extend to any sigma-point rule (unscented, cubature, Gauss-Hermite, or custom), because the proof of Theorem 1 never uses the specific weights beyond the mean and covariance approximations.
  • The MCMC simulator turns the robust filter into a stress-testing tool: one can draw worst-case data for a chosen tolerance $c_t$ and empirically check whether the estimator's coverage of the true state matches the promised level.
  • A concrete repair path suggested by the proof structure is to replace (14)-(18) with a proper joint Gaussian density whose mean does not depend on $y_t$; with such a well-defined center, the KL-ball argument and the worst-case claim would apply directly.
  • The numerical results suggest KL-robustness is the right notion for parametric model mismatch, while heavy-tail methods target a different failure mode, so a combined approach could cover both parametric drift and impulsive outliers.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 5 minor

Summary. The paper proposes two robust extensions of sigma-point Kalman filters: a prediction-resilient filter (Section III) and an update-resilient filter (Section V), both formulated as solutions of a dynamic minimax game with a KL-divergence ambiguity set. The center of the ambiguity set is approximated via sigma-point transformations, leading to closed-form estimators that reduce to the standard UKF/CKF when the tolerance c_t is zero. The authors also design an MCMC-based simulator to sample from the approximate least favorable model, and report numerical experiments on a synthetic worst-case scenario and on a mass-spring system with parametric uncertainty.

Significance. If the main theorem were valid, the paper would provide a principled, closed-form robustification of sigma-point filters with a worst-case optimality guarantee, which is a valuable contribution to nonlinear state estimation. The paper contains several strengths: Algorithms 1 and 3 are concrete and reduce exactly to UKF/CKF at c_t=0; the approximation framework is clearly motivated; and the MCMC simulator with a convergence argument is a useful practical tool. The numerical study is extensive and shows consistent improvement over standard filters. However, the central worst-case optimality claim for the prediction-resilient filter rests on a flawed construction: the proposed center of the ambiguity ball is not a well-defined joint density, and the proof of Theorem 1 is therefore invalid. Since this is the paper's main theoretical contribution, the result cannot be accepted as stated.

major comments (4)
  1. [Section III.A-B, Proposition 1 and Eq. (14)] The object \bar p_t(z_t|Y_{t-1}) defined through (14)-(18) is not a genuine joint Gaussian density over z_t = (x_{t+1}, y_t). The block mean \bar m_{x_{t+1}} in (14) depends on the observation y_t through the term (y_t - \bar m_{y_t}) appearing in both the numerator and the denominator. A density of z_t conditional on Y_{t-1} cannot have a mean depending on its own y_t component; its normalization over z_t is then not well defined, and in scalar linear cases the denominator in (14) can vanish. Consequently, the ambiguity ball \bar B_t in (29) is not a set of densities, D_KL(\tilde p_t, \bar p_t) is undefined, and the external result [13, Theorem 1] cannot be invoked. This invalidates the proof of Theorem 1 and the claimed worst-case optimality of the prediction-resilient filter.
  2. [Section III.B, Theorem 1 and Eq. (40)] The proof of Theorem 1 is circular. The center \bar m_{x_{t+1}} in (40) is defined using \hat x_{t+1} = \sum_i W_m^i \hat X_{t+1}^i, which is precisely the estimator (30) whose optimality is to be proven. Proposition 1 was constructed so that the conditional mean under (2) equals the sigma-point predictor. Thus the statement that (30) solves the approximate minimax problem (28) is a restatement of the construction of \bar p_t, not an independent consequence of robust least-squares theory. The theorem therefore does not establish that the filter attains the optimal worst-case MSE against the KL ball.
  3. [Section VI.A, Fig. 4 and Eq. (43)] The worst-case numerical evaluation is self-referential by design. The target density (43) is built from \theta_t and \hat x_{t+1} produced by the same filter being tested, and the simulator draws from that filter-specific target. Figure 4 then shows each filter winning on data generated from its own least favorable model, which is a consistency check of the MCMC sampler rather than an independent validation of filter optimality. The conclusion that 'for each dataset generated from a specific least favorable model, the optimal filter is the one designed for that particular model' is an artifact of the construction and should be reframed.
  4. [Section V, Eq. (62)] The update-resilient filter does not suffer from the density defect of the prediction-resilient filter: its center (62) is a genuine Gaussian over (x_t, y_t) with mean independent of y_t. However, the paper's central claim regarding the prediction-resilient filter remains unsupported. The authors should either provide a corrected construction of the prediction-resilient center that yields a well-defined density, or explicitly restrict the mathematical claims to the update-resilient case while presenting the prediction-resilient algorithm as a heuristic with empirical support.
minor comments (5)
  1. [Section III.A, Eq. (14)] The expression for \bar m_{x_{t+1}} is very difficult to parse because \xi appears both as a symbol and as a denominator. Please rewrite the equation with clearly indexed intermediate quantities, and state the domain of validity (e.g., when the denominator is nonzero).
  2. [Section III.B, after Eq. (26)] The approximation \tilde p_t(x_t|Y_{t-1}) \simeq N(\hat x_t, \tilde P_t) is introduced after Proposition 1, but the notation \tilde P_t is used in Theorem 1 before its definition. Please define \tilde P_t explicitly when it is first used.
  3. [Section IV.B, Eq. (52)] The Monte Carlo approximation of M_t is described, but the choice of r and the claimed 'cumulative relative error over the time interval [0,50] approximately equal to 10%' in Section VI.A are not derived. Please provide the error-propagation calculation or state it as an empirical observation.
  4. [Section VI.B, Table I] The layout of Table I is confusing: the columns labeled '\sigma', 'MC-UKF', 'N_p', and 'PF' appear to mix parameter values and MSE results, and the cells for the standard filters (UKF, CKF) are not clearly separated from the rows for c. Please restructure the table so that each estimator's MSE is in its own column with a clear header.
  5. [References and notation] The paper uses [9] and [10] as 'unpublished' or 'submitted' references; please ensure all external results cited in the proofs (especially [13, Theorem 1]) are stated precisely, as the correctness of those statements is load-bearing for the theoretical claims.

Circularity Check

2 steps flagged · score 8.0 of 10

The prediction-resilient filter's minimax optimality is self-definitional: the center of the KL ball in (14) is engineered so its conditional mean is the sigma-point predictor, so Theorem 1 returns that predictor by construction; the Section VI.A worst-case simulator is likewise built from the filter's own outputs.

  1. self definitional [Section III.A Proposition 1 (Eq. 14) and Section III.B Theorem 1 (proof, Eqs. 40 and 30)]
    "¯mxt+1 + ¯Kxt+1yt ¯K−1 yt (yt − ¯myt) ... = ξ = Pp i=1 W i mδi. The latter coincides with the sigma point Kalman predictor ˆxt+1 in (12) ... The density ¯pt(zt|Yt−1) = N( ¯mt, ¯Kt) is defined as in Proposition 1 with µ = ˆxt and P = ˜Pt ... by [13, Theorem 1] if follows that: i) the minimizer of (28) is ¯mxt+1 + ¯Kxt+1yt ¯K−1 yt (yt − ¯myt) which coincides with ˆxt+1 defined in (30)."

    Equation (14) is solved so that the conditional mean of the Gaussian center ¯pt equals ξ, the sigma-point predictor. In Theorem 1 the same construction is reused with ξ = ˆxt+1, the very robust estimator whose optimality is being proved. Applying [13, Theorem 1] then 'derives' as the minimizer the same conditional mean that was inserted into the center by definition. The ambiguity-set center is thus built from the estimator it is supposed to justify, making the claim that (30) solves (28) a tautology rather than an independent minimax consequence. Separately, the y_t-dependence of ¯mxt+1 in (14) means ¯pt is not a normalizable density over z_t, so the KL ball (29) and the appeal to [13, Theorem 1] are not well defined; this is a correctness failure that compounds the circularity.

  2. other [Section IV (Eqs. 42-43) and Section VI.A (Fig. 4)]
    "˜ϕ0 t (zt|xt) = 1 Mt exp( θt 2 ∥xt+1 − ˆxt+1∥2 ) ϕt(zt|xt) where ˆxt+1 and θt are the state predictor and the risk sensitivity parameter obtained by the prediction resilient filter ... Fig. 4(a)-(b) ... The results highlight that in each case, the best filter is the one constructed with the corresponding least favorable model."

    The approximate least favorable model (43) used by the MH simulator is centered at the filter's own output ˆxt+1 with the filter's own risk parameter θt. Trajectories drawn from this target density are therefore biased toward states near the filter's predictions. The subsequent MSE comparison (Fig. 4) shows that the filter used to build the target wins on its own generated data. This is a self-consistency check, not independent evidence of worst-case optimality; it 'predicts' a result that is forced by how the data were generated.

full rationale

The central circular step is in Theorem 1: the ambiguity-set center ¯pt is defined, through (14) and Proposition 1, so that its conditional mean equals the sigma-point predictor. Theorem 1 then invokes the external robust least-squares result [13] to conclude that the minimizer of the approximate game is that same sigma-point predictor. The estimator is inserted into the center before the minimax problem is solved, so the robust-optimality claim reduces to a definition. The separate defect that ¯mxt+1 depends on y_t, making ¯pt not a genuine Gaussian density, further invalidates the appeal to [13, Theorem 1], but the score reflects the self-definitional structure rather than the measure-theoretic flaw. Section VI.A compounds this by generating 'least favorable' data from a density built from the filter's own ˆxt+1 and θt, so the reported wins in Fig. 4 are expected by construction. There is real independent content: the update-resilient filter (Section V) uses a center (62) whose mean is independent of y_t and is not engineered to match the robust estimator (65), so it avoids this circularity; the mass-spring benchmark with mismatched parameters (Section VI.B) provides an external, non-self-referential evaluation; and the MCMC proposal machinery can stand on its own. No load-bearing self-citation chain is needed for the main result, since [13] is an external theorem and [10] is used only for the simulator proposal in the update-resilient case. However, the paper's headline claim for the prediction-resilient filter is forced by definition, so the circularity score is high.

Assumptions & free parameters 3 free parameters · 6 assumptions · 0 invented entities

The central claims rest on six premises: the stated regularity of f and h; the asserted convexity and compactness for the minimax theorem; the unquantified sigma point substitution for the true pseudo-nominal density; the Gaussian closing of the least favorable a priori density; the treatment of the Proposition 1 object as a genuine joint Gaussian center in the prediction-resilient case; and the external theorem [13]. The main free knobs are the tolerance c_t, the UKF tuning constants, and the MCMC accuracy settings. No new physical entities are postulated: the least favorable model is a constructed statistical object generated by the simulator.

free parameters (3)
  • tolerance c_t = c in {0.0001, 0.001, 0.01, 0.03, 0.05, 0.1} in Table I; c=1e-3 in Fig. 4
    Radius of the KL ambiguity ball, chosen by the user. Performance peaks at an intermediate c and degrades for large c (over-conservatism, noted by the authors). No a priori selection rule is given; the favorable rows of Table I are grid-selected.
  • UKF tuning (a, b, kappa) = a=0.5, b=2, kappa=1 (experiments); kappa=2 (Example 1)
    Standard choices from [17] for the unscented transformation. They fix the sigma point weights and therefore the computed P_{t+1} and theta_t.
  • MCMC accuracy settings (r_0, tau*, r_max) = r_0=100, tau*=2e-3, r_max=4000
    Control the Monte Carlo estimate of the normalizing constant M_t in the simulator. They affect the generated least favorable data but not the filter recursions themselves.
assumptions (6)
  • domain assumption f and h are bounded on any compact set in R^n (Section II).
    Stated as 'mild assumption'; used to claim the prediction resilient filter is well defined on the finite horizon and that the MH chain converges (Section IV, convergence paragraph).
  • domain assumption The sets B_t and G_t are convex and compact, so von Neumann's minimax theorem yields a saddle point.
    Stated in Section III.B. Compactness of the estimator class G_t (finite second moment estimators) is asserted, not proved.
  • ad hoc to paper Replacing the true pseudo-nominal density p_t by the sigma point approximation pbar_t preserves the solution of the game (27) up to acceptable error.
    Section III.B introduces the approximate game (28) and Theorem 1 solves it; the approximation error between (27) and (28) is never quantified.
  • ad hoc to paper The least favorable a priori density at the next step is Gaussian, N(xhat_{t+1}, Ptilde_{t+1}) (eq. (39)) and N(xhat_{t+1}, P_{t+1}) (eq. (66)).
    This Gaussian closing step lets the recursion continue with sigma points; the paper acknowledges it is an approximation.
  • ad hoc to paper The density pbar_t(z_t|Y_{t-1}) = N(mbar_t, Kbar_t) defined by (14)-(18) is treated as a valid joint Gaussian despite mbar_{x_{t+1}} in (14) depending on y_t.
    Theorem 1 applies [13, Theorem 1], which requires a genuine Gaussian center for the KL ball. The y_t-dependent mean makes (2) not a proper Gaussian density over z_t; this is the main technical fragility of the prediction-resilient derivation.
  • standard math [13, Theorem 1] (Levy and Nikoukhah, robust least-squares estimation with a relative entropy constraint).
    External theorem used as the engine for Theorems 1 and 2: minimizer is the conditional mean, maximizer is the exponentially tilted Gaussian N(m, Ktilde).

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Pith. "Pith review of A robust approach to sigma point Kalman filtering." pith.science (2026). https://pith.science/paper/OVYR6MQV

@misc{pith2026250604815,
  author       = {Pith},
  title        = {Pith review of: A robust approach to sigma point Kalman filtering},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/OVYR6MQV}},
  note         = {Machine review of arXiv:2506.04815}
}
read the original abstract

We propose a robust estimator for nonlinear state-space models and provide a clear interpretation of it as the minimizer of a minimax game. The corresponding maximizer searches for the least favorable model over an ambiguity set whose center is obtained by approximating the nominal model through a sigma-point transformation. Moreover, we develop a Markov Chain Monte Carlo (MCMC) scheme for generating adversarial data from it, thereby allowing the assessment of the resulting uncertainty.

Figures

Figures reproduced from arXiv: 2506.04815 by the authors.

Figure 1
Figure 1. State prediction using the standard UKF (red line) and UTF (blue line). [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 1
Figure 1. As we can see, these estimators are different and thus this approximate conditional [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗
Figure 2
Figure 2. Boxplot of the value of r needed in the MH algorithm and corresponding to the data simulator for P-UKF, P-CKF, U-UKF and U-CKF. 0 200 400 600 800 1000 1200 1400 1600 0 0.2 0.4 0.6 P-UKF 0 200 400 600 800 1000 1200 1400 1600 0 0.2 0.4 0.6 P-CKF 0 200 400 600 800 1000 1200 1400 1600 0 0.2 0.4 0.6 U-UKF 0 200 400 600 800 1000 1200 1400 1600 0 0.2 0.4 0.6 U-CKF [PITH_FULL_IMAGE:figures/full_fig_p030_2.png] view at source ↗
Figures from the paper (5 more)
Figure 3
Figure 3. Figure 3: Acceptance rate in the MH algorithm corresponding to the data simulator for P-UKF, P-CKF, U-UKF and U-CKF. [PITH_FULL_IMAGE:figures/full_fig_p030_3.png]
Figure 4
Figure 4. Figure 4: Mean squared error of the filters when the least favorable data are generated by: (a) P-UKF simulator; (b) P-CKF [PITH_FULL_IMAGE:figures/full_fig_p031_4.png]
Figure 5
Figure 5. Figure 5: Spring-mass system. B. State estimation for mass-spring system We consider a mass-spring system, as shown in [PITH_FULL_IMAGE:figures/full_fig_p032_5.png]
Figure 6
Figure 6. Figure 6: Mean squared error of the state for the sigma point Kalman filters with [PITH_FULL_IMAGE:figures/full_fig_p035_6.png]
Figure 7
Figure 7. Figure 7: One realization of the displacement and the measurement trajectory. [PITH_FULL_IMAGE:figures/full_fig_p035_7.png]

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Works this paper leans on

22 extracted references · 21 canonical work pages

  1. [1]

    Stochastic stability of the discrete-time extended Kalman filter,

    K. Reif, S. Gunther, E. Yaz, and R. Unbehauen, “Stochastic stability of the discrete-time extended Kalman filter,”IEEE Transactions on Automatic Control, vol. 44, no. 4, pp. 714–728, 1999

    work page 1999

  2. [2]

    S ¨arkk¨a and L

    S. S ¨arkk¨a and L. Svensson,Bayesian filtering and smoothing, vol. 17. Cambridge university press, 2023

    2023

  3. [3]

    A new method for the nonlinear transformation of means and covariances in filters and estimators,

    S. Julier, J. Uhlmann, and H. F. Durrant-Whyte, “A new method for the nonlinear transformation of means and covariances in filters and estimators,”IEEE Transactions on Automatic Control, vol. 45, no. 3, pp. 477–482, 2000

    work page 2000

  4. [4]

    Cubature Kalman filters,

    I. Arasaratnam and S. Haykin, “Cubature Kalman filters,”IEEE Transactions on Automatic Control, vol. 54, no. 6, pp. 1254– 1269, 2009

    work page 2009

  5. [5]

    Improved calibration of numerical integration error in sigma-point filters,

    J. Pr ¨uher, T. Karvonen, C. J. Oates, O. Straka, and S. S ¨arkk¨a, “Improved calibration of numerical integration error in sigma-point filters,”IEEE Transactions on Automatic Control, vol. 66, no. 3, pp. 1286–1292, 2020

    work page 2020

  6. [6]

    Discrete-time nonlinear filtering algorithms using Gauss–Hermite quadrature,

    I. Arasaratnam, S. Haykin, and R. J. Elliott, “Discrete-time nonlinear filtering algorithms using Gauss–Hermite quadrature,” Proceedings of the IEEE, vol. 95, no. 5, pp. 953–977, 2007

    work page 2007

  7. [7]

    Nonlinear filtering method using a switching error model for outlier-contaminated observations,

    A. Nakabayashi and G. Ueno, “Nonlinear filtering method using a switching error model for outlier-contaminated observations,”IEEE Transactions on Automatic Control, vol. 65, no. 7, pp. 3150–3156, 2019

    work page 2019

  8. [8]

    Robust stable iterated unscented Kalman filter based on maximum correntropy criterion,

    H. Zhao, B. Tian, and B. Chen, “Robust stable iterated unscented Kalman filter based on maximum correntropy criterion,” Automatica, vol. 142, p. 110410, 2022

    work page 2022

Show all 22 references
  1. [9]

    Robust state-space filtering under incremental model perturbations subject to a relative entropy tolerance,

    B. Levy and R. Nikoukhah, “Robust state-space filtering under incremental model perturbations subject to a relative entropy tolerance,”IEEE Transactions on Automatic Control, vol. 58, pp. 682–695, Mar. 2013

    2013

  2. [10]

    An update-resilient Kalman filtering approach,

    S. Yi and M. Zorzi, “An update-resilient Kalman filtering approach,”Submitted, 2024

    2024

  3. [11]

    Robust Kalman filtering under model perturbations,

    M. Zorzi, “Robust Kalman filtering under model perturbations,”IEEE Transactions on Automatic Control, vol. 62, no. 6, pp. 2902–2907, 2016

    2016

  4. [12]

    Robust Kalman filtering with probabilistic uncertainty in system parameters,

    S. Kim, V . Deshpande, and R. Bhattacharya, “Robust Kalman filtering with probabilistic uncertainty in system parameters,” IEEE Control Systems Letters, vol. 5, no. 1, pp. 295–300, 2020

    2020

  5. [13]

    Robust least-squares estimation with a relative entropy constraint,

    B. Levy and R. Nikoukhah, “Robust least-squares estimation with a relative entropy constraint,”IEEE Transactions on Information Theory, vol. 50, no. 1, pp. 89–104, 2004

    2004

  6. [14]

    Robust Kalman filtering under model uncertainty: The case of degenerate densities,

    S. Yi and M. Zorzi, “Robust Kalman filtering under model uncertainty: The case of degenerate densities,”IEEE Transactions on Automatic Control, vol. 67, no. 7, pp. 3458–3471, 2021

    2021

  7. [15]

    On globalized robust Kalman filter under model uncertainty,

    Y . Xu, W. Xue, C. Shang, and H. Fang, “On globalized robust Kalman filter under model uncertainty,”IEEE Transactions on Automatic Control, vol. 70, no. 2, pp. 1147–1160, 2025

    2025

  8. [16]

    Learning the tuned liquid damper dynamics by means of a robust EKF,

    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,” in2021 American Control Conference (ACC), pp. 60–65, IEEE, 2021

    2021

  9. [17]

    The unscented kalman filter,

    E. A. Wan and R. Van Der Merwe, “The unscented kalman filter,”Chapter 7 of: Kalman filtering and neural networks, pp. 221–280, 2001

    2001

  10. [18]

    Cover and J

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

    1991

  11. [19]

    On the coupling of model predictive control and robust Kalman filtering,

    A. Zenere and M. Zorzi, “On the coupling of model predictive control and robust Kalman filtering,”IET Control Theory & Applications, vol. 12, no. 13, pp. 1873–1881, 2018

    2018

  12. [20]

    Robust Kalman filtering: Asymptotic analysis of the least favorable model,

    M. Zorzi and B. C. Levy, “Robust Kalman filtering: Asymptotic analysis of the least favorable model,” inIEEE Conference on Decision and Control (CDC), pp. 7124–7129, 2018

    2018

  13. [21]

    Stationarity and convergence of the Metropolis-Hastings algorithm: Insights into theoretical aspects,

    S. D. Hill and J. C. Spall, “Stationarity and convergence of the Metropolis-Hastings algorithm: Insights into theoretical aspects,”IEEE Control Systems Magazine, vol. 39, no. 1, pp. 56–67, 2019

    2019

  14. [22]

    H. K. Khalil and J. W. Grizzle,Nonlinear systems, vol. 3. Prentice hall Upper Saddle River, NJ, 2002. September 9, 2025 DRAFT

    2002

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