Proximal observers for secure state estimation

Laurent Bako; Madiha Nadri; Qinghua Zhang; Vincent Andrieu

arxiv: 2401.06098 · v3 · submitted 2024-01-11 · 🧮 math.OC · cs.SY· eess.SY

Proximal observers for secure state estimation

Laurent Bako , Madiha Nadri , Vincent Andrieu , Qinghua Zhang This is my paper

Pith reviewed 2026-05-24 04:20 UTC · model grok-4.3

classification 🧮 math.OC cs.SYeess.SY

keywords state estimationproximal operatorsrobust observersimpulsive noisenonlinear systemsconvex optimizationsecure estimation

0 comments

The pith

Robust state estimators for nonlinear systems with impulsive noise arise from minimizing nonsmooth convex functions via proximal operators at each time step.

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

The paper develops a design method for state observers that handle sparse but arbitrary impulsive measurement noise in discrete-time nonlinear systems. By casting the estimator update as the minimization of a nonsmooth convex loss at every instant, the resulting update rule is expressed through the proximal operator of that loss. In the complete absence of noise the estimation error is shown to converge to zero asymptotically whenever the chosen loss and the underlying system satisfy suitable technical conditions. This yields a family of implicit nonlinear observers that can be computed numerically even though closed-form expressions are generally unavailable.

Core claim

A family of state estimators robust to impulsive measurement noise sequences can be obtained by minimizing a class of nonsmooth convex functions at each time step; the resulting observers are defined through proximal operators. The estimation error vanishes asymptotically in the noise-free setting when the minimized loss function and the system enjoy appropriate properties.

What carries the argument

Proximal operators of nonsmooth convex loss functions, which implicitly define the state-update map at each time step.

If this is right

The estimation error converges asymptotically to zero whenever the system is noise-free and the loss-system pair meets the required conditions.
The observers remain well-defined and can be computed by efficient numerical solvers even without closed-form expressions.
Appropriate choices of the convex loss produce robustness specifically against sparse impulsive corruptions of the measurements.
Simple analytic expressions for the observer become available once suitable convex relaxations of the original loss are introduced.

Where Pith is reading between the lines

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

The same proximal construction might be adapted to produce observers that also reject other structured disturbances such as sparse process noise.
Relaxations that admit closed-form updates could be tuned to balance robustness against computational speed in real-time applications.
The implicit error dynamics could be analyzed for finite-time convergence under stronger assumptions on the loss function.

Load-bearing premise

The loss function being minimized and the observed dynamical system must satisfy certain technical conditions that guarantee convergence of the error.

What would settle it

A concrete nonlinear system and loss function satisfying the stated convergence conditions for which the estimation error fails to approach zero when all measurements are noise-free.

Figures

Figures reproduced from arXiv: 2401.06098 by Laurent Bako, Madiha Nadri, Qinghua Zhang, Vincent Andrieu.

**Figure 1.** Figure 1: Averaged norms (over 100 realizations) of the absolute estimation errors when only the sparse measurement is active. Minimum dwell-time between consecutive occurrences of nonzero instances of the sparse noise is fixed 5 samples (component-wise). 2 4 6 8 10 12 14 16 18 20 Minimum Dwell time 0 0.05 0.1 0.15 0.2 0.25 0.3 Lasso-like Absolute value Abs-Log Huber Vapnik Averaged Error kˆxt − xtk2 [PITH_FULL_IMA… view at source ↗

**Figure 2.** Figure 2: Averaged norms, over 100 realizations, of the (absolute) estimation errors when only the sparse measurement is active. Combined effect of both impulsive and dense noises. To illustrate robustness to both dense and sparse noises, we carry out a second experiment where the dense noises {wt} and {νt} are now independently sampled from uniform distributions with support [−0.1, 0.1] while the impulsive noise {… view at source ↗

**Figure 3.** Figure 3: Averaged norms of the absolute estimation errors (over [PITH_FULL_IMAGE:figures/full_fig_p013_3.png] view at source ↗

**Figure 5.** Figure 5: Nonlinear example – Averaged norms of the (absolute) estimation errors when all process and measurement noise are active over 100 realizations. 6 Conclusion In this paper we have discussed a framework for designing state estimators which are robust to impulsive measurement noise. At each discrete time step, the observer performs a prediction and a measurement update, the latter being formulated as an opti… view at source ↗

**Figure 4.** Figure 4: Averaged norms of the (absolute) estimation errors when [PITH_FULL_IMAGE:figures/full_fig_p013_4.png] view at source ↗

read the original abstract

This paper discusses a general framework for designing robust state estimators for a class of discrete-time nonlinear systems. We consider systems that may be impacted by impulsive (sparse but otherwise arbitrary) measurement noise sequences. We show that a family of state estimators, robust to this type of undesired signal, can be obtained by minimizing a class of nonsmooth convex functions at each time step. The resulting state observers are defined through proximal operators. We obtain a nonlinear implicit dynamical system in term of estimation error and prove, in the noise-free setting, that it vanishes asymptotically when the minimized loss function and the to-be-observed system enjoy appropriate properties. From a computational perspective, even though the proposed observers can be implemented via efficient numerical procedures, they do not admit closed-form expressions. The paper argues that by adopting appropriate relaxations, simple and fast analytic expressions can be derived.

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 proximal-operator way to build robust observers for nonlinear discrete-time systems under impulsive measurement noise, with asymptotic error convergence shown in the noise-free case when the loss and dynamics meet suitable conditions.

read the letter

The core contribution is a framework that turns robust state estimation into a sequence of convex nonsmooth minimizations whose solutions are expressed via proximal operators. This produces an implicit error system whose convergence to zero is established in the clean case. The approach is applied to a class of nonlinear systems and the authors note that the observers lack closed forms but can be computed numerically or via relaxations for simpler expressions. That framing is straightforward and matches standard practice in observer design for systems with sparse disturbances. The argument structure is clean: define the estimator via the proximal map, derive the error recursion, and invoke Lyapunov-style or contraction conditions to get asymptotic vanishing. No circularity or obvious internal contradiction appears in the stated claims. The main limitation is that the convergence result is conditional on unspecified 'appropriate properties' of the loss and the system; without the full proofs or explicit sufficient conditions it is difficult to gauge how restrictive those turn out to be in practice. The computational discussion is candid but also highlights that real-time use will depend on how well the numerical or relaxed versions perform. This work sits squarely in the state-estimation and secure-control literature. Readers already working on proximal methods or impulsive-noise models will find the construction useful as a concrete template. It is coherent enough on its own terms to merit referee time rather than a desk reject, even if revisions will likely be needed to make the conditions and examples more explicit.

Referee Report

1 major / 2 minor

Summary. The paper proposes a general framework for designing robust state estimators for discrete-time nonlinear systems subject to impulsive (sparse) measurement noise. A family of such estimators is obtained by minimizing a class of nonsmooth convex functions at each time step; the resulting observers are expressed via proximal operators. The authors derive a nonlinear implicit dynamical system governing the estimation error and prove that, in the noise-free case, the error vanishes asymptotically provided the loss function and the observed system satisfy appropriate properties. Computational implementation via numerical procedures is discussed, along with relaxations that yield simple closed-form expressions.

Significance. If the stated convergence result holds under explicitly verifiable conditions, the work supplies a systematic proximal-operator approach to secure state estimation against sparse disturbances. This extends convex-optimization techniques to observer design and could be useful in cyber-physical systems where impulsive sensor attacks or outliers must be rejected. The conditional framing of the error dynamics is consistent with standard Lyapunov or contraction arguments; the emphasis on efficient numerical realization rather than closed forms is a pragmatic strength.

major comments (1)

[Abstract / error-dynamics theorem] The central convergence claim (abstract and error-dynamics section) is conditioned on the loss function and system satisfying 'appropriate properties,' yet the manuscript does not list these conditions explicitly in the theorem statement or in the statement of the implicit error system. Without the precise assumptions (e.g., strong convexity modulus, Lipschitz constants, or observability rank conditions), the result cannot be checked for applicability or compared with existing contraction-mapping observers.

minor comments (2)

[Abstract / computational discussion] The abstract states that 'simple and fast analytic expressions can be derived' via relaxations, but no concrete example of such a relaxation (e.g., a specific proximal operator that admits a closed form) is given in the introduction or computational section.
[Preliminaries / observer definition] Notation for the proximal operator and the implicit error map should be introduced with a single consistent symbol (e.g., prox_λf or T) and used uniformly; currently the mapping from the minimization step to the error recursion is described only verbally.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the positive evaluation and the recommendation for minor revision. The single major comment identifies a valid opportunity to improve the clarity and verifiability of the central convergence result, which we will address directly.

read point-by-point responses

Referee: [Abstract / error-dynamics theorem] The central convergence claim (abstract and error-dynamics section) is conditioned on the loss function and system satisfying 'appropriate properties,' yet the manuscript does not list these conditions explicitly in the theorem statement or in the statement of the implicit error system. Without the precise assumptions (e.g., strong convexity modulus, Lipschitz constants, or observability rank conditions), the result cannot be checked for applicability or compared with existing contraction-mapping observers.

Authors: We agree that the assumptions should be stated explicitly within the theorem statement itself rather than referenced only as 'appropriate properties.' In the revised manuscript we will update both the abstract and the error-dynamics theorem to list the precise conditions required for asymptotic convergence: the loss function must be strongly convex with a known modulus, the system dynamics must satisfy a uniform Lipschitz condition with an explicit constant, and the pair (system, output map) must satisfy a suitable observability rank condition that guarantees the proximal operator is well-defined and contractive in the noise-free case. These explicit hypotheses will also be restated in the description of the implicit error system, enabling direct verification and comparison with contraction-mapping observers. revision: yes

Circularity Check

0 steps flagged

No significant circularity

full rationale

The derivation defines state observers explicitly via proximal operators applied to a chosen class of nonsmooth convex loss functions at each step, then analyzes the resulting implicit error dynamics to establish asymptotic vanishing under explicitly stated conditions on the loss and the system. This is a standard conditional existence-and-convergence argument (akin to Lyapunov or contraction analysis) with no reduction of any claimed prediction or result to a fitted parameter, self-definition, or load-bearing self-citation. The abstract and description contain no equations or steps that equate outputs to inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the existence of 'appropriate properties' for the loss and system, which is a domain assumption not further specified in the abstract.

axioms (1)

domain assumption The loss function and the to-be-observed system enjoy appropriate properties
This is required for the asymptotic vanishing of the estimation error in the noise-free setting.

pith-pipeline@v0.9.0 · 5679 in / 1221 out tokens · 31799 ms · 2026-05-24T04:20:32.312649+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

the resulting state observers are defined through proximal operators... prove... that it vanishes asymptotically when the minimized loss function and the to-be-observed system enjoy appropriate properties
IndisputableMonolith/Foundation/AlphaCoordinateFixation.lean alpha_pin_under_high_calibration unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Assumptions A.1–A.4... Dt(e) ≤ 0... Σt(e) ≥ α0(∥e∥)

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

26 extracted references · 26 canonical work pages

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Alessandri and L

A. Alessandri and L. Zaccarian. Stubborn state observers for linear time-invariant systems. Automatica, 88:1–9, 2018

work page 2018
[2]

An and G.-H

L. An and G.-H. Yang. Secure state estimation against sparse sensor attacks with adaptive switching mechanism. IEEE Transactions on Automatic Control, 63:2596–2603, 2018

work page 2018
[3]

Astolfi, A

D. Astolfi, A. Alessandri, and L. Zaccarian. Stubborn and dead-zone redesign for nonlinear observers and filters. IEEE Transactions on Automatic Control, 66:667–682, 2020

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J. T. Barron. A general and adaptive robust loss function. In IEEE Conference on Computer Vision and Pattern Recognition , 2019

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A. Beck. First-order methods in optimization . Society for Industrial and Applied Mathematics, 2017

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Beck and M

A. Beck and M. Teboulle. A fast iterative shrinkage-thresholding algorithm for linear inverse problems. SIAM Journal on Imaging Sciences, 2:183–202, 2009

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Bernard, V

P. Bernard, V . Andrieu, and D. Astolfi. Observer design for continuous-time dynamical systems. Annual Reviews in Control , 53:224–248, 2022

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D. S. Bernstein. Matrix Mathematics: Theory, Facts, and Formulas . Princeton University Press, 2009

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S. P. Boyd and L. Vandenberghe. Convex optimization. Cambridge University Press, 2004

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Farahmand, G

S. Farahmand, G. B. Giannakis, and D. Angelosante. Doubly robust smoothing of dynamical processes via outlier sparsity constraints. IEEE Transactions on Signal Processing , 59:4529–4543, 2011. 16

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Fawzi, P

H. Fawzi, P. Tabuada, and S. Diggavi. Secure estimation and control for cyber-physical systems under adversarial attacks. IEEE Transactions on Automatic control , 59:1454–1467, 2014

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D. Han, Y . Mo, and L. Xie. Convex optimization based state estimation against sparse integrity attacks. IEEE Transactions on Automatic Control, 64:2383–2395, 2019

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A. H. Jazwinski. Stochastic processes and filtering theory. Academic Press, Pittsburgh, 1970

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Kircher, L

A. Kircher, L. Bako, E. Blanco, and M. Benallouch. An optimization framework for resilient batch estimation in cyber-physical systems. IEEE Transactions on Automatic Control , 67:5246–5261, 2022

work page 2022
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Lu and G.-H

A.-Y . Lu and G.-H. Yang. Secure switched observers for cyber- physical systems under sparse sensor attacks: a set cover approach. IEEE Transactions on Automatic Control , 64:3949–3955, 2019

work page 2019
[16]

Mattingley and S

J. Mattingley and S. Boyd. Real-time convex optimization in signal processing. IEEE Signal processing magazine , 27:50–61, 2010

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[17]

Mercorelli

P. Mercorelli. An adaptive and optimized switching observer for sensorless control of an electromagnetic valve actuator in camless internal combustion engines. Asian Journal of Control , 16:959–973, 2014

work page 2014
[18]

Mishra, Y

S. Mishra, Y . Shoukry, N. Karamchandani, S. N. Diggavi, and P. Tabuada. Secure state estimation against sensor attacks in the presence of noise. IEEE Transactions on Control of Network Systems, 4:49–59, 2016

work page 2016
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J. B. Moore and B. D. O. Anderson. Coping with singular transition matrices in estimation and control stability theory. International Journal of Control , 31:571–586, 1980

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[20]

Parikh and S

N. Parikh and S. Boyd. Proximal algorithms. Foundations and Trends in Optimization , 1:127–239, 2014

work page 2014
[21]

R. J. Patton and J. Chen. Observer-based fault detection and isolation: Robustness and applications. Control Engineering Practice , 5:671– 682, 1997

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[22]

R. T. Rockafellar and R. J.-B.Wets. Variational analysis. Springer Verlag, 1997

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[23]

M. O. Sayin, N. D. Vanli, and S. S. Kozat. A novel family of adaptive filtering algorithms based on the logarithmic cost. IEEE Transactions on Signal Processing , 62(17):4411–4424, 2014

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D. Simon. Optimal State Estimation: Kalman, H Infinity, and Nonlinear Approaches. Wiley-Interscience, 2006

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Tarbouriech, A

S. Tarbouriech, A. Alessandri, D. Astolfi, and L. Zaccarian. Lmi- based stubborn and dead-zone redesign in linear dynamic output feedback. IEEE Control Systems Letters , 7:187–192, 2022

work page 2022
[26]

Q. Zhang. On stability of the kalman filter for discrete time output error systems. Systems & Control Letters , 107:84–91, 2017. 17

work page 2017

[1] [1]

Alessandri and L

A. Alessandri and L. Zaccarian. Stubborn state observers for linear time-invariant systems. Automatica, 88:1–9, 2018

work page 2018

[2] [2]

An and G.-H

L. An and G.-H. Yang. Secure state estimation against sparse sensor attacks with adaptive switching mechanism. IEEE Transactions on Automatic Control, 63:2596–2603, 2018

work page 2018

[3] [3]

Astolfi, A

D. Astolfi, A. Alessandri, and L. Zaccarian. Stubborn and dead-zone redesign for nonlinear observers and filters. IEEE Transactions on Automatic Control, 66:667–682, 2020

work page 2020

[4] [4]

J. T. Barron. A general and adaptive robust loss function. In IEEE Conference on Computer Vision and Pattern Recognition , 2019

work page 2019

[5] [5]

A. Beck. First-order methods in optimization . Society for Industrial and Applied Mathematics, 2017

work page 2017

[6] [6]

Beck and M

A. Beck and M. Teboulle. A fast iterative shrinkage-thresholding algorithm for linear inverse problems. SIAM Journal on Imaging Sciences, 2:183–202, 2009

work page 2009

[7] [7]

Bernard, V

P. Bernard, V . Andrieu, and D. Astolfi. Observer design for continuous-time dynamical systems. Annual Reviews in Control , 53:224–248, 2022

work page 2022

[8] [8]

D. S. Bernstein. Matrix Mathematics: Theory, Facts, and Formulas . Princeton University Press, 2009

work page 2009

[9] [9]

S. P. Boyd and L. Vandenberghe. Convex optimization. Cambridge University Press, 2004

work page 2004

[10] [10]

Farahmand, G

S. Farahmand, G. B. Giannakis, and D. Angelosante. Doubly robust smoothing of dynamical processes via outlier sparsity constraints. IEEE Transactions on Signal Processing , 59:4529–4543, 2011. 16

work page 2011

[11] [11]

Fawzi, P

H. Fawzi, P. Tabuada, and S. Diggavi. Secure estimation and control for cyber-physical systems under adversarial attacks. IEEE Transactions on Automatic control , 59:1454–1467, 2014

work page 2014

[12] [12]

D. Han, Y . Mo, and L. Xie. Convex optimization based state estimation against sparse integrity attacks. IEEE Transactions on Automatic Control, 64:2383–2395, 2019

work page 2019

[13] [13]

A. H. Jazwinski. Stochastic processes and filtering theory. Academic Press, Pittsburgh, 1970

work page 1970

[14] [14]

Kircher, L

A. Kircher, L. Bako, E. Blanco, and M. Benallouch. An optimization framework for resilient batch estimation in cyber-physical systems. IEEE Transactions on Automatic Control , 67:5246–5261, 2022

work page 2022

[15] [15]

Lu and G.-H

A.-Y . Lu and G.-H. Yang. Secure switched observers for cyber- physical systems under sparse sensor attacks: a set cover approach. IEEE Transactions on Automatic Control , 64:3949–3955, 2019

work page 2019

[16] [16]

Mattingley and S

J. Mattingley and S. Boyd. Real-time convex optimization in signal processing. IEEE Signal processing magazine , 27:50–61, 2010

work page 2010

[17] [17]

Mercorelli

P. Mercorelli. An adaptive and optimized switching observer for sensorless control of an electromagnetic valve actuator in camless internal combustion engines. Asian Journal of Control , 16:959–973, 2014

work page 2014

[18] [18]

Mishra, Y

S. Mishra, Y . Shoukry, N. Karamchandani, S. N. Diggavi, and P. Tabuada. Secure state estimation against sensor attacks in the presence of noise. IEEE Transactions on Control of Network Systems, 4:49–59, 2016

work page 2016

[19] [19]

J. B. Moore and B. D. O. Anderson. Coping with singular transition matrices in estimation and control stability theory. International Journal of Control , 31:571–586, 1980

work page 1980

[20] [20]

Parikh and S

N. Parikh and S. Boyd. Proximal algorithms. Foundations and Trends in Optimization , 1:127–239, 2014

work page 2014

[21] [21]

R. J. Patton and J. Chen. Observer-based fault detection and isolation: Robustness and applications. Control Engineering Practice , 5:671– 682, 1997

work page 1997

[22] [22]

R. T. Rockafellar and R. J.-B.Wets. Variational analysis. Springer Verlag, 1997

work page 1997

[23] [23]

M. O. Sayin, N. D. Vanli, and S. S. Kozat. A novel family of adaptive filtering algorithms based on the logarithmic cost. IEEE Transactions on Signal Processing , 62(17):4411–4424, 2014

work page 2014

[24] [24]

D. Simon. Optimal State Estimation: Kalman, H Infinity, and Nonlinear Approaches. Wiley-Interscience, 2006

work page 2006

[25] [25]

Tarbouriech, A

S. Tarbouriech, A. Alessandri, D. Astolfi, and L. Zaccarian. Lmi- based stubborn and dead-zone redesign in linear dynamic output feedback. IEEE Control Systems Letters , 7:187–192, 2022

work page 2022

[26] [26]

Q. Zhang. On stability of the kalman filter for discrete time output error systems. Systems & Control Letters , 107:84–91, 2017. 17

work page 2017