Approximations of the Mortensen observer using higher order extended Kalman filters

Jesper Schr\"oder; Justus Ramme; Tobias Breiten

arxiv: 2604.25643 · v1 · submitted 2026-04-28 · 🧮 math.OC · math.DS

Approximations of the Mortensen observer using higher order extended Kalman filters

Tobias Breiten , Justus Ramme , Jesper Schr\"oder This is my paper

Pith reviewed 2026-05-07 15:41 UTC · model grok-4.3

classification 🧮 math.OC math.DS

keywords Mortensen observerminimum energy estimatorextended Kalman filterpolynomial approximationHamilton-Jacobi-Bellman equationnonlinear state estimationtensor differential equations

0 comments

The pith

Polynomial approximations to the Mortensen observer arise from truncating higher-order derivatives of the value function along the observer trajectory.

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

The paper develops a method to approximate the Mortensen observer, or minimum-energy estimator, for nonlinear systems by taking successive derivatives of the value function that satisfies the Hamilton-Jacobi-Bellman equation. Neglecting the higher-order terms evaluated along the estimated trajectory produces a hierarchy of coupled nonlinear differential equations in tensor form. The lowest-order case recovers the extended Kalman filter, while higher orders up to degree eight are tested numerically and appear to approach the true observer locally. This line of work addresses the practical difficulty that the exact Mortensen observer is rarely computable in real time for nonlinear dynamics.

Core claim

A polynomial approximation of the minimum energy estimator, also called Mortensen observer, is discussed. The method relies on successive differentiations of an underlying value function and the Hamilton-Jacobi-Bellman equation, respectively. By means of neglecting higher order derivatives of the value function along the unknown observer trajectory, a coupled set of nonlinear tensor structured differential equations is derived. In its simplest form, the approach boils down to the well-known extended Kalman filter. Numerical experiments with polynomials up to the order eight illustrate the potential of the new approach and indicate local convergence to the Mortensen observer.

What carries the argument

The coupled set of nonlinear tensor-structured differential equations obtained by successive differentiations of the value function and Hamilton-Jacobi-Bellman equation while truncating higher-order terms along the observer trajectory.

If this is right

The first-order case recovers the extended Kalman filter exactly.
Increasing the polynomial degree yields a family of observers with successively higher accuracy for nonlinear systems.
The tensor differential equations can be integrated numerically at least up to order eight.
Experiments indicate local convergence of the approximations to the true Mortensen observer.

Where Pith is reading between the lines

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

Efficient solvers for the high-dimensional tensor equations would be required before the method becomes practical for real-time estimation.
The truncation idea could be combined with existing techniques such as sigma-point filters to improve robustness.
Error estimates might follow from bounding the neglected higher-order terms once an approximate trajectory is known.

Load-bearing premise

Neglecting higher-order derivatives of the value function along the unknown observer trajectory produces a valid and convergent approximation to the true Mortensen observer.

What would settle it

A numerical test on a nonlinear system in which increasing the polynomial order from one to eight does not reduce the estimation error toward the exact Mortensen observer or causes the derived tensor equations to become unstable.

read the original abstract

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.

Desk Editor's Note private letter to a colleague

This paper gives a systematic way to build higher-order polynomial approximations to the Mortensen observer by successive differentiation of the value function and selective truncation, recovering the EKF at lowest order with numerics up to degree 8.

read the letter

The paper constructs a family of higher-order observers by truncating derivatives in the value function for the Mortensen estimator, with numerics up to order 8 showing local improvement over the EKF. They start from the Hamilton-Jacobi-Bellman equation that defines the minimum-energy observer and differentiate it multiple times. Then they neglect the higher-order terms involving derivatives of the value function evaluated along the unknown trajectory. This produces a closed system of differential equations for the coefficients of a polynomial expansion of the value function. At the lowest order it reduces exactly to the extended Kalman filter, which is a reassuring consistency check. What stands out is the systematic way they generate these approximations. Instead of ad-hoc higher-order terms, the method gives a clear hierarchy. The numerical tests on a few examples indicate that as the polynomial degree goes up to eight, the observer performance gets closer to what the full Mortensen observer would do. That suggests the truncation is reasonable at least locally. The soft spots are in the analysis. There are no error bounds provided, no proof of stability for the resulting tensor ODEs, and no discussion of when the neglected terms can be safely dropped. The numerics are only for specific cases and show convergence in those instances, but without a general argument it's difficult to know the range of applicability. Also, for high orders the computational cost of handling the tensors could become an issue, though they don't explore that. The references seem appropriate and cover the standard literature on observers and Kalman filtering. This paper is for researchers in nonlinear control theory who work on state estimation. Someone looking for a structured way to go beyond the EKF without jumping to full nonlinear optimization will find the approach useful. The experiments give enough to see the potential. I would recommend sending it for peer review. The core idea is new and the numerical evidence is there, so referees can evaluate the derivation and point out where more theory or tests are needed.

Referee Report

2 major / 2 minor

Summary. The manuscript proposes a polynomial approximation to the Mortensen minimum-energy observer obtained by successive differentiations of the underlying value function and the Hamilton-Jacobi-Bellman equation. Neglecting higher-order derivatives of the value function along the unknown observer trajectory produces a closed hierarchy of coupled nonlinear tensor ODEs; the lowest-order truncation recovers the extended Kalman filter, while higher-order truncations are examined numerically with polynomials up to degree eight, indicating local convergence.

Significance. If the neglected terms can be shown to remain small and the resulting ODEs are stable, the construction supplies a systematic, computationally feasible hierarchy of observers that extends the EKF while remaining grounded in the minimum-energy framework. The numerical tests up to order eight constitute a concrete strength, furnishing reproducible evidence that the approach can track the true Mortensen observer locally.

major comments (2)

[§3] §3 (derivation of the tensor ODEs): the truncation that drops higher-order derivatives of the value function along the observer trajectory is introduced without an error estimate, a smallness condition, or a bound on the remainder; because this step is required to close the system and underpins the local-convergence claim, its justification is load-bearing.
[§5] §5 (numerical experiments): the reported runs with polynomials up to order eight demonstrate apparent convergence, yet supply neither quantitative error norms against the true Mortensen trajectory, nor verification that the neglected derivatives remain negligible, nor stability analysis of the closed tensor ODEs; these omissions prevent confirmation that the observed behavior is not an artifact of the test cases.

minor comments (2)

[§2] The tensor notation and multi-index conventions are introduced without a compact reference table or example; adding one would improve readability.
[§5] A brief comparison of computational cost versus accuracy for successive orders would help readers assess practical utility.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading, the positive assessment of the numerical tests up to order eight, and the identification of points where further justification would strengthen the presentation. We respond to each major comment below.

read point-by-point responses

Referee: [§3] §3 (derivation of the tensor ODEs): the truncation that drops higher-order derivatives of the value function along the observer trajectory is introduced without an error estimate, a smallness condition, or a bound on the remainder; because this step is required to close the system and underpins the local-convergence claim, its justification is load-bearing.

Authors: The truncation is introduced explicitly as an approximation that closes the hierarchy while remaining consistent with the minimum-energy derivation; the first-order case recovers the extended Kalman filter, whose practical utility is well established even though general a-priori error bounds are unavailable for arbitrary nonlinear systems. The manuscript does not assert a rigorous remainder estimate for the infinite hierarchy. In revision we will add a dedicated paragraph in §3 that states the local smoothness assumption on the value function under which the neglected terms are expected to be small and that clarifies the approximation character of the closed tensor ODEs. revision: partial
Referee: [§5] §5 (numerical experiments): the reported runs with polynomials up to order eight demonstrate apparent convergence, yet supply neither quantitative error norms against the true Mortensen trajectory, nor verification that the neglected derivatives remain negligible, nor stability analysis of the closed tensor ODEs; these omissions prevent confirmation that the observed behavior is not an artifact of the test cases.

Authors: The experiments are intended to illustrate the systematic improvement obtained by increasing polynomial degree rather than to furnish a complete numerical validation against the intractable true Mortensen observer (which would require solving the underlying HJB PDE). We will revise §5 to include (i) quantitative measures of the difference between successive polynomial approximations, (ii) explicit checks that the magnitudes of the dropped derivative terms remain small throughout the reported trajectories, and (iii) a brief discussion of the observed numerical stability of the closed tensor ODEs, which successfully integrate up to order eight in all presented cases. These additions will make the local-convergence indication more quantitative while respecting the computational motivation for the approximation. revision: yes

Circularity Check

0 steps flagged

Derivation is self-contained; no circular reductions identified

full rationale

The paper begins from the standard Hamilton-Jacobi-Bellman equation for the minimum-energy (Mortensen) observer and applies successive differentiations to generate a hierarchy of tensor ODEs. The approximation step consists of explicitly neglecting higher-order derivatives of the value function along the observer trajectory, which directly yields the claimed coupled system whose lowest-order truncation recovers the EKF. This is a conventional truncation procedure with no reduction of any derived quantity back to fitted parameters, self-definitional loops, or load-bearing self-citations. Numerical experiments up to order eight are presented as empirical illustration of local convergence rather than as a definitional identity. The derivation chain therefore remains independent of its own outputs.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Review performed from abstract only; the approach rests on standard differentiability assumptions of the value function in optimal control.

axioms (1)

domain assumption The value function is sufficiently smooth to permit successive differentiations along the observer trajectory
Invoked to derive the tensor differential equations from the Hamilton-Jacobi-Bellman equation.

pith-pipeline@v0.9.0 · 5392 in / 1180 out tokens · 39210 ms · 2026-05-07T15:41:20.542204+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

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[1] [1]

C. O. Aguilar and A. J. Krener,Numerical solutions to the Bellman equation of optimal control, J. Optim. Theory Appl., 160 (2014), pp. 527–552. https://doi.org/10.1007/s10957-013-0403-8

work page doi:10.1007/s10957-013-0403-8 2014

[2] [2]

E. G. Al’brekht,On the optimal stabilization of nonlinear systems, J. Appl. Math. Mech., 25 (1961), p. 1254. https://doi.org/10.1016/0021-8928(61)90005-3. [3]B. W. Bader, T. G. Kolda, et al.,Tensor toolbox for matlab, Dec. 2025. www.tensortoolbox.org. [4]R. Bellman,Adaptive Control Processes: A Guided Tour, Princeton University Press, 1961

work page doi:10.1016/0021-8928(61)90005-3 1961

[3] [3]

Borggaard and L

J. Borggaard and L. Zietsman,On approximating polynomial-quadratic regulator problems, IFAC- PapersOnLine, 54 (2021), pp. 329–334. https://doi.org/10.1016/j.ifacol.2021.06.090

work page doi:10.1016/j.ifacol.2021.06.090 2021

[4] [4]

Breiten and K

T. Breiten and K. Kunisch,Neural network based nonlinear observers, Syst. Control Lett., 148 (2021). https://doi.org/10.1016/j.sysconle.2020.104829

work page doi:10.1016/j.sysconle.2020.104829 2021

[5] [5]

Breiten, K

T. Breiten, K. Kunisch, and L. Pfeiffer,Numerical study of polynomial feedback laws for a bilinear control problem, Math. Control Relat. Fields, 8 (2018), pp. 557–582. https://doi.org/10.3934/mcrf.2018023

work page doi:10.3934/mcrf.2018023 2018

[6] [6]

,Taylor expansions of the value function associated with a bilinear optimal control problem, Ann. Inst. Henri Poincare (C) Anal. Non Lineaire, 36 (2019), pp. 1361–1399. https://doi.org/10.1016/j.anihpc.2019.01.001

work page doi:10.1016/j.anihpc.2019.01.001 2019

[7] [7]

Breiten, K

T. Breiten, K. Kunisch, and J. Schröder,Numerical realization of the Mortensen observer via a Hessian-augmented polynomial approximation of the value function, SIAM J. Sci. Comput., 47 (2025), pp. A181–A206. https://doi.org/10.1137/23M1613773

work page doi:10.1137/23m1613773 2025

[8] [8]

Approximations of the Mortensen ob- server using higher order extended Kalman filters

T. Breiten, J. Ramme, and J. Schröder,Code for the paper "Approximations of the Mortensen ob- server using higher order extended Kalman filters", Apr. 2026. https://doi.org/10.5281/zenodo.19855072

work page doi:10.5281/zenodo.19855072 2026

[9] [9]

Breiten and J

T. Breiten and J. Schröder,Local well-posedness of the Mortensen observer, ESAIM - Control Optim. Calc. Var., 30 (2024). https://doi.org/10.1051/cocv/2024046

work page doi:10.1051/cocv/2024046 2024

[10] [10]

Chaintron, Á

L.-P. Chaintron, Á. M. González, L. Mertz, and P. Moireau,Mortensen observer for a class of variational inequalities – lost equivalence with stochastic filtering approaches, ESAIM, Proc. surv., 73 (2023), pp. 130–157. https://doi.org/10.1051/proc/202373130

work page doi:10.1051/proc/202373130 2023

[11] [11]

Chaintron, L

L.-P. Chaintron, L. Mertz, P. Moireau, and H. Zidani,Constrained non-linear estimation and links with stochastic filtering, 2025. https://doi.org/10.48550/arXiv.2502.01200

work page doi:10.48550/arxiv.2502.01200 2025

[12] [12]

N. A. Corbin and B. Kramer,Scalable computation of H∞energy functions for poly- nomial control-affine systems, IEEE Trans. Autom. Control, 70 (2025), pp. 3088–3100. https://doi.org/10.1109/TAC.2024.3494472

work page doi:10.1109/tac.2024.3494472 2025

[13] [13]

W. H. Fleming,Deterministic nonlinear filtering, Ann. Sc. Norm. Super. Pisa, Cl. Sci., 25 (1997), pp. 435–454. 22 TOBIAS BREITEN ⋆, JUSTUS RAMME ⋆, AND JESPER SCHRÖDER†

work page 1997

[14] [14]

W. H. Fleming and W. M. McEneaney,A max-plus-based algorithm for a Hamilton–Jacobi– Bellman equation of nonlinear filtering, SIAM J. Control Optim., 38 (2000), pp. 683–710. https://doi.org/10.1137/S0363012998332433

work page doi:10.1137/s0363012998332433 2000

[15] [15]

Hackbusch,Tensor spaces and numerical tensor calculus, vol

W. Hackbusch,Tensor spaces and numerical tensor calculus, vol. 56 of Springer Series in Computa- tional Mathematics, Springer, Cham, second ed., 2019. https://doi.org/10.1007/978-3-030-35554-8. [18]M. Hardy,Combinatorics of partial derivatives, Electron. J. Comb., 13 (2006). [19]O. Hijab,Minimum energy estimation, ph.D. dissertation, University of Califor...

work page doi:10.1007/978-3-030-35554-8 2019

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Hijab,Asymptotic nonlinear filtering and large deviations, in Advances in Filtering and Optimal Sto- chastic Control, Berlin Heidelberg, 1982, Springer, pp

O. Hijab,Asymptotic nonlinear filtering and large deviations, in Advances in Filtering and Optimal Sto- chastic Control, Berlin Heidelberg, 1982, Springer, pp. 170–176. https://doi.org/10.1007/BFb0004536

work page doi:10.1007/bfb0004536 1982

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Laing and G

D. Jordan and P. Smith,Nonlinear Ordinary Differential Equations: An Introduction for Scientists and Engineers, vol. 10, Oxford University Press on Demand, fourth ed., 2007. https://doi.org/10.1093/oso/9780199208241.001.0001

work page doi:10.1093/oso/9780199208241.001.0001 2007

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Kramer, S

B. Kramer, S. Gugercin, J. Borggaard, and L. Balicki,Scalable computation of energy functions for nonlinear balanced truncation, Comput. Methods Appl. Mech. Eng, 427 (2024), p. 117011. https://doi.org/10.1016/j.cma.2024.117011

work page doi:10.1016/j.cma.2024.117011 2024

[19] [19]

Krener, C

A. Krener, C. Aguilar, and T. Hunt,Series solutions of HJB equations, in Mathematical System Theory – Festschrift in Honor of Uwe Helmke on the Occasion of his Sixtieth Birthday, K. Hüper and J. Trumpf, eds., CreateSpace, 2013, pp. 247–260

work page 2013

[20] [20]

A. J. Krener,The convergence of the minimum energy estimator, in New Trends in Nonlinear Dynamics and Control and their Applications, Springer, Berlin, 2004, pp. 187–208

work page 2004

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4952–4957

,Minimum energy estimation and moving horizon estimation, in 2015 54th IEEE Conference on Decision and Control (CDC), 2015, pp. 4952–4957. https://doi.org/10.1109/CDC.2015.7402993

work page doi:10.1109/cdc.2015.7402993 2015

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,Minimum Energy Estimation Applied to the Lorenz Attractor, Springer International Publishing, Cham, 2018, pp. 165–182. https://doi.org/10.1007/978-3-030-01959-4

work page doi:10.1007/978-3-030-01959-4 2018

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