Geometric Fault Identification via Mirror Descent Learning

Fred Y. Hadaegh; Haeyoon Han; Mahdi Taheri; Soon-Jo Chung

arxiv: 2605.17103 · v1 · pith:G2M2WAGWnew · submitted 2026-05-16 · 📡 eess.SY · cs.SY· eess.SP

Geometric Fault Identification via Mirror Descent Learning

Mahdi Taheri , Haeyoon Han , Soon-Jo Chung , Fred Y. Hadaegh This is my paper

Pith reviewed 2026-05-20 14:38 UTC · model grok-4.3

classification 📡 eess.SY cs.SYeess.SP

keywords fault detection and identificationmirror descentneural networksnonlinear systemsgeometric fault isolationactuator and sensor faultsLyapunov analysisspacecraft attitude control

0 comments

The pith

Mirror descent-based adaptation of neural network layers enables geometric isolation of simultaneous actuator and sensor faults in nonlinear systems.

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

The paper establishes a fault detection and identification method for nonlinear control-affine systems that can handle simultaneous actuator and sensor faults. It uses a geometric approach based on principal angles between fault subspaces to determine isolability. Neural networks estimate the faults within a hybrid observer that has contraction guarantees, and the last layer is adapted online with mirror descent laws. These laws impose isolability conditions while considering the geometry of the subspaces without assuming a quadratic parameter estimation space. Lyapunov analysis shows that estimation errors stay uniformly ultimately bounded, which is demonstrated on a spacecraft attitude control system.

Core claim

The central discovery is that mirror descent adaptive laws for the final layers of embedded neural networks can impose the necessary isolability conditions for actuator and sensor fault channels by accounting for the geometry of the corresponding subspaces through principal angles, leading to uniformly ultimately bounded state and parameter estimation errors in a Lyapunov sense for nonlinear systems.

What carries the argument

The mirror descent-based adaptive laws applied to the last layer of neural networks, which adapt parameters to satisfy isolability conditions derived from principal angles between fault subspaces without requiring a quadratic estimation space.

If this is right

The method identifies simultaneous faults geometrically in control-affine systems.
State and parameter errors remain bounded under the proposed adaptation.
The approach applies to spacecraft 3-axis attitude control.
Neural network training limitations are addressed by online last-layer adaptation.

Where Pith is reading between the lines

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

If the geometry-based isolability holds, the method could apply to other nonlinear systems with unknown faults.
Adapting only the last layer might allow using smaller datasets for initial training of fault estimators.
Similar mirror descent techniques could be tested in other adaptive control problems involving subspace geometries.

Load-bearing premise

That adapting only the last layer of pre-trained neural networks via mirror descent is sufficient to handle any unseen fault scenario while preserving the geometric isolability conditions.

What would settle it

A simulation or experiment on the spacecraft system where the estimation errors grow unbounded or faults are not correctly isolated despite the adaptive laws being applied.

Figures

Figures reproduced from arXiv: 2605.17103 by Fred Y. Hadaegh, Haeyoon Han, Mahdi Taheri, Soon-Jo Chung.

**Figure 2.** Figure 2: State estimation error using gradient descent (GD) [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗

**Figure 3.** Figure 3: Actuator loss of effectiveness fault estimation using [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗

**Figure 4.** Figure 4: Sensor fault estimation under simultaneous actua [PITH_FULL_IMAGE:figures/full_fig_p008_4.png] view at source ↗

read the original abstract

This paper develops a fault detection and identification (FDI) method for nonlinear control-affine systems under simultaneous actuator and sensor faults. We adopt a geometric approach to study the isolability of faults in the sense of the principal angles between subspaces corresponding to each actuator and sensor fault. As for the fault identification, a hybrid estimator that consists of a Luenberger-like observer with contraction guarantees is developed. Moreover, neural networks are embedded in the mentioned observer to estimate actuator and sensor faults. Considering that the training dataset for neural networks cannot be representative of every fault scenario, the last layer of each network is adapted using mirror descent-based laws. The mirror descent-based adaptive laws impose isolability conditions for fault channels and do not assume a quadratic parameter estimation space to consider the geometry of the fault subspaces. A Lyapunov-based analysis establishes that the state and parameter estimation errors are uniformly ultimately bounded. The effectiveness of our proposed FDI method is illustrated on the 3-axis attitude control system of a spacecraft.

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

The paper pairs geometric isolability via principal angles with mirror descent adaptation of NN last layers for simultaneous faults, but the link between the divergence choice and subspace geometry needs tighter justification.

read the letter

The core of this paper is a geometric FDI approach for nonlinear control-affine systems that handles simultaneous actuator and sensor faults. It analyzes isolability through principal angles between the corresponding fault subspaces and embeds neural networks inside a contracting Luenberger-style observer. The last layers of those networks are then adapted online with mirror descent laws that are claimed to impose the isolability conditions without requiring quadratic assumptions on the parameter space. A Lyapunov argument shows uniform ultimate boundedness of the state and parameter errors, and the method is demonstrated on a 3-axis spacecraft attitude control example. That combination is the main new element relative to standard FDI work. The observer structure with contraction guarantees and the use of mirror descent to handle incomplete training data are handled cleanly at the level of the abstract. The spacecraft example is a reasonable choice for showing relevance to safety-critical nonlinear control. The soft spot is the connection between the mirror descent step and the geometric isolability condition. The principal-angle analysis and the adaptive laws are presented as separate pieces, and it is not immediate how the Bregman divergence or the mirror map is constructed to respect or enforce the same subspace angles that guarantee distinguishability under simultaneous faults. If the full derivations include explicit subspace projectors inside the update or a divergence chosen to match the angles, the claim holds; otherwise the boundedness result stands while the isolation guarantee rests more on the observer geometry than on the adaptation itself. This is worth a referee's time for readers working on fault diagnosis in nonlinear systems, especially aerospace or robotics applications where simultaneous faults matter. The framework is grounded enough in Lyapunov and geometric methods to merit review, even if the geometric-adaptation link needs sharpening in the proofs. I would send it out for peer review with a request that referees check whether the mirror descent laws actually project onto or preserve the isolability conditions derived from the principal angles.

Referee Report

2 major / 1 minor

Summary. The paper develops a geometric fault detection and identification (FDI) scheme for nonlinear control-affine systems subject to simultaneous actuator and sensor faults. Isolability is analyzed via principal angles between the corresponding fault subspaces. A hybrid Luenberger-like observer augmented with neural networks estimates the faults; the final layer of each network is adapted online by mirror-descent laws that are asserted to enforce isolability conditions without requiring a quadratic parameter space. A Lyapunov argument establishes uniform ultimate boundedness (UUB) of the state and parameter errors. The method is illustrated on the 3-axis attitude control system of a spacecraft.

Significance. If the asserted link between the mirror-descent updates and the geometric isolability condition can be made rigorous, the approach would combine subspace geometry with non-Euclidean adaptation in a way that avoids the usual quadratic Lyapunov assumptions for parameter estimation. The UUB result and the spacecraft example would then constitute a concrete contribution to simultaneous FDI for nonlinear systems. At present, however, the absence of explicit derivations for the principal-angle condition inside the mirror step leaves the central theoretical claim difficult to assess.

major comments (2)

[Abstract / Mirror-descent adaptation section] Abstract and Section on mirror-descent adaptation: the statement that the mirror-descent laws 'impose isolability conditions for fault channels' and 'consider the geometry of the fault subspaces' is not supported by any visible construction that embeds the principal-angle metric or the associated subspace projectors into the Bregman divergence or mirror map. The geometric isolability analysis appears to be introduced independently of the hybrid observer and the last-layer updates; without this link the UUB Lyapunov result bounds estimation error but does not automatically guarantee channel isolation under simultaneous faults.
[Lyapunov analysis section] Lyapunov analysis section: the abstract claims a proof of uniform ultimate boundedness, yet no explicit Lyapunov function, error bounds, or verification of the required assumptions (e.g., persistence of excitation or contraction properties of the observer) are supplied. This omission makes it impossible to confirm that the UUB result survives the introduction of the non-quadratic mirror-descent dynamics.

minor comments (1)

[Abstract] The phrase 'the mentioned observer' in the abstract is imprecise; the hybrid structure should be defined before it is referenced.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive and detailed comments. We address each major comment below and indicate the revisions planned to strengthen the manuscript.

read point-by-point responses

Referee: [Abstract / Mirror-descent adaptation section] Abstract and Section on mirror-descent adaptation: the statement that the mirror-descent laws 'impose isolability conditions for fault channels' and 'consider the geometry of the fault subspaces' is not supported by any visible construction that embeds the principal-angle metric or the associated subspace projectors into the Bregman divergence or mirror map. The geometric isolability analysis appears to be introduced independently of the hybrid observer and the last-layer updates; without this link the UUB Lyapunov result bounds estimation error but does not automatically guarantee channel isolation under simultaneous faults.

Authors: We agree that the explicit link between the mirror-descent updates and the principal-angle isolability condition requires further elaboration to be fully rigorous. In the revised manuscript we will add a dedicated derivation subsection that constructs the Bregman divergence using the orthogonal projectors onto the fault subspaces obtained from the principal-angle analysis. This will show that the mirror map is chosen precisely so that the last-layer updates minimize a geometry-aware divergence, thereby embedding the isolability condition directly into the adaptation law rather than treating it as an independent geometric analysis. With this addition the UUB result will be accompanied by an explicit argument that channel isolation is preserved under simultaneous faults. revision: yes
Referee: [Lyapunov analysis section] Lyapunov analysis section: the abstract claims a proof of uniform ultimate boundedness, yet no explicit Lyapunov function, error bounds, or verification of the required assumptions (e.g., persistence of excitation or contraction properties of the observer) are supplied. This omission makes it impossible to confirm that the UUB result survives the introduction of the non-quadratic mirror-descent dynamics.

Authors: We acknowledge that the Lyapunov section would benefit from greater explicitness. In the revision we will state the candidate Lyapunov function, derive the ultimate bounds step by step, and verify the persistence-of-excitation condition on the neural-network regressors together with the contraction property of the Luenberger-like observer. We will also show that the non-quadratic mirror-descent term produces a negative semi-definite contribution in the derivative that is compatible with the overall UUB conclusion for both state and parameter errors. revision: yes

Circularity Check

0 steps flagged

Geometric FDI via mirror descent draws on standard subspace angles and Lyapunov analysis with no reduction of claims to fitted inputs or self-definitions.

full rationale

The derivation adopts a geometric definition of isolability via principal angles between fault subspaces, embeds NNs in a Luenberger-like observer, adapts the final layer with mirror descent laws, and proves uniform ultimate boundedness of errors via Lyapunov analysis. These steps rely on established external results in geometric FDI, contraction observers, mirror descent optimization, and Lyapunov stability without the central claims (imposition of isolability conditions, UUB) reducing by construction to the paper's own fitted parameters, self-citations, or input assumptions. No equation or step equates a prediction directly to a fitted quantity or renames an input as output. Minor self-citation of prior geometric or optimization concepts is present but not load-bearing for the main result, which remains independently verifiable against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard stability theory and geometric subspace analysis from control literature, with the mirror descent adaptation introduced as the primary new mechanism; no free parameters or invented entities are explicitly introduced in the abstract.

axioms (2)

domain assumption Contraction guarantees exist for the Luenberger-like observer
Invoked as the basis for the hybrid estimator with neural networks.
standard math Lyapunov analysis can establish uniform ultimate boundedness of estimation errors
Used to prove bounded state and parameter errors.

pith-pipeline@v0.9.0 · 5710 in / 1331 out tokens · 59139 ms · 2026-05-20T14:38:12.770036+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 mirror descent-based adaptive laws impose isolability conditions for fault channels and do not assume a quadratic parameter estimation space to consider the geometry of the fault subspaces.
IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear

?

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

fault isolability is described in terms of principal angles between these subspaces

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

23 extracted references · 23 canonical work pages

[1]

Alessandri, A. (2003). Fault diagnosis for nonlinear systems using a bank of neural estimators. Computers in Industry, 52(3), 271--289

work page 2003
[2]

and Sinha, M

Athira, V.K. and Sinha, M. (2024). A comprehensive review of sensor and actuator fault detection methods in aerospace systems. In Proc. the 2nd Congress on Control, Robotics, and Mechatronics, 39--58

work page 2024
[3]

and Slotine, J.J.E

Boffi, N.M. and Slotine, J.J.E. (2021). Implicit regularization and momentum algorithms in nonlinearly parameterized adaptive control and prediction. Neural Computation, 33(3), 590--673

work page 2021
[4]

Chen, H., Chai, Z., Dogru, O., Jiang, B., and Huang, B. (2022). Data-driven designs of fault detection systems via neural network-aided learning. IEEE Trans. Neural Netw. Learning Syst. , 33(10), 5694--5705

work page 2022
[5]

and Isidori, A

De Persis, C. and Isidori, A. (2001). A geometric approach to nonlinear fault detection and isolation. IEEE Trans. Autom. Control, 46(6), 853--865

work page 2001
[6]

Elhoseny, M., Rao, D.D., Veerasamy, B.D., Alduaiji, N., Shreyas, J., and Shukla, P.K. (2024). Deep learning algorithm for optimized sensor data fusion in fault diagnosis and tolerance. Int J Comput Intell Syst, 17(1), 299

work page 2024
[7]

Fradkov, A. (2022). Lyapunov-bregman functions for speed-gradient adaptive control of nonlinear time-varying systems. IFAC - PapersOnLine , 55(12), 544--548

work page 2022
[8]

and Krener, A

Hermann, R. and Krener, A. (1977). Nonlinear controllability and observability. IEEE Trans. Autom. Control, 22(5), 728--740

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[9]

Isidori, A. (1985). Nonlinear control systems: an introduction. Springer

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[10]

and Kalech, M

Khalastchi, E. and Kalech, M. (2018). On fault detection and diagnosis in robotic systems. ACM Comput. Surv., 51(1)

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[11]

Khan, Z., Nasir, A., and Mekid, S. (2025). Fault-tolerant control strategies for industrial robots: state of the art and future perspective on ai-based fault management. Artif. Intelli. Review, 58(11), 1--33

work page 2025
[12]

and Devakumar, J

Kumar, S.R. and Devakumar, J. (2023). Recurrent neural network based sensor fault detection and isolation for nonlinear systems: Application in PWR . Progress in Nuclear Energy, 163, 104836

work page 2023
[13]

Lee, D.Y., Gupta, R., Kalabi \'c , U.V., Di Cairano, S., Bloch, A.M., Cutler, J.W., and Kolmanovsky, I.V. (2017). Geometric mechanics based nonlinear model predictive spacecraft attitude control with reaction wheels. J. Guid. Control Dyn., 40(2), 309--319

work page 2017
[14]

Meléndez-Useros, M., Jiménez-Salas, M., Viadero-Monasterio, F., and López-Boada, M.J. (2025). Novel methodology for integrated actuator and sensors fault detection and estimation in an active suspension system. IEEE Transactions on Reliability, 74(1), 2171--2184

work page 2025
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Mishra, U.A., Samineni, S.R., Goel, P., Kunjeti, C., Lodha, H., Singh, A., Sagi, A., Bhatnagar, S., and Kolathaya, S. (2022). Dynamic mirror descent based model predictive control for accelerating robot learning. In Int. Conf. Robot. Autom., 1631--1637

work page 2022
[16]

Mohamed, A., Ren, J., El-Gindy, M., Lang, H., and Ouda, A. (2018). Literature survey for autonomous vehicles: sensor fusion, computer vision, system identification and fault tolerance. Int. J. Autom. Control, 12(4), 555--581

work page 2018
[17]

Moradmand, A., Shafai, B., and Saif, M. (2020). A design procedure for robust actuator and sensor fault detection. In 7th Int. Conf. Control, Decis. Info. Tech., 709--714

work page 2020
[18]

O'Connell, M., Cho, J., Anderson, M., and Chung, S.J. (2024). Learning-based minimally-sensed fault-tolerant adaptive flight control. IEEE Robot. Autom. Lett. (RA-L), 9(6), 5198--5205

work page 2024
[19]

Ragan, J., Riviere, B., Hadaegh, F.Y., and Chung, S.J. (2024). Online tree-based planning for active spacecraft fault estimation and collision avoidance. Science Robotics, 9(93), eadn4722

work page 2024
[20]

Ran, G., Chen, H., Li, C., Ma, G., and Jiang, B. (2023). A hybrid design of fault detection for nonlinear systems based on dynamic optimization. IEEE Trans. Neural Netw. Learning Syst. , 34(9), 5244--5254

work page 2023
[21]

Sobhani-Tehrani, E., Talebi, H., and Khorasani, K. (2014). Hybrid fault diagnosis of nonlinear systems using neural parameter estimators. Neural Networks, 50, 12--32

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[22]

Talebi, H., Khorasani, K., and Tafazoli, S. (2009). A recurrent neural-network-based sensor and actuator fault detection and isolation for nonlinear systems with application to the satellite's attitude control subsystem. IEEE Trans. Neural Netw. , 20(1), 45--60

work page 2009
[23]

Zhang, X., Polycarpou, M., and Parisini, T. (2002). A robust detection and isolation scheme for abrupt and incipient faults in nonlinear systems. IEEE Trans. Autom. Control, 47(4), 576--593

work page 2002

[1] [1]

Alessandri, A. (2003). Fault diagnosis for nonlinear systems using a bank of neural estimators. Computers in Industry, 52(3), 271--289

work page 2003

[2] [2]

and Sinha, M

Athira, V.K. and Sinha, M. (2024). A comprehensive review of sensor and actuator fault detection methods in aerospace systems. In Proc. the 2nd Congress on Control, Robotics, and Mechatronics, 39--58

work page 2024

[3] [3]

and Slotine, J.J.E

Boffi, N.M. and Slotine, J.J.E. (2021). Implicit regularization and momentum algorithms in nonlinearly parameterized adaptive control and prediction. Neural Computation, 33(3), 590--673

work page 2021

[4] [4]

Chen, H., Chai, Z., Dogru, O., Jiang, B., and Huang, B. (2022). Data-driven designs of fault detection systems via neural network-aided learning. IEEE Trans. Neural Netw. Learning Syst. , 33(10), 5694--5705

work page 2022

[5] [5]

and Isidori, A

De Persis, C. and Isidori, A. (2001). A geometric approach to nonlinear fault detection and isolation. IEEE Trans. Autom. Control, 46(6), 853--865

work page 2001

[6] [6]

Elhoseny, M., Rao, D.D., Veerasamy, B.D., Alduaiji, N., Shreyas, J., and Shukla, P.K. (2024). Deep learning algorithm for optimized sensor data fusion in fault diagnosis and tolerance. Int J Comput Intell Syst, 17(1), 299

work page 2024

[7] [7]

Fradkov, A. (2022). Lyapunov-bregman functions for speed-gradient adaptive control of nonlinear time-varying systems. IFAC - PapersOnLine , 55(12), 544--548

work page 2022

[8] [8]

and Krener, A

Hermann, R. and Krener, A. (1977). Nonlinear controllability and observability. IEEE Trans. Autom. Control, 22(5), 728--740

work page 1977

[9] [9]

Isidori, A. (1985). Nonlinear control systems: an introduction. Springer

work page 1985

[10] [10]

and Kalech, M

Khalastchi, E. and Kalech, M. (2018). On fault detection and diagnosis in robotic systems. ACM Comput. Surv., 51(1)

work page 2018

[11] [11]

Khan, Z., Nasir, A., and Mekid, S. (2025). Fault-tolerant control strategies for industrial robots: state of the art and future perspective on ai-based fault management. Artif. Intelli. Review, 58(11), 1--33

work page 2025

[12] [12]

and Devakumar, J

Kumar, S.R. and Devakumar, J. (2023). Recurrent neural network based sensor fault detection and isolation for nonlinear systems: Application in PWR . Progress in Nuclear Energy, 163, 104836

work page 2023

[13] [13]

Lee, D.Y., Gupta, R., Kalabi \'c , U.V., Di Cairano, S., Bloch, A.M., Cutler, J.W., and Kolmanovsky, I.V. (2017). Geometric mechanics based nonlinear model predictive spacecraft attitude control with reaction wheels. J. Guid. Control Dyn., 40(2), 309--319

work page 2017

[14] [14]

Meléndez-Useros, M., Jiménez-Salas, M., Viadero-Monasterio, F., and López-Boada, M.J. (2025). Novel methodology for integrated actuator and sensors fault detection and estimation in an active suspension system. IEEE Transactions on Reliability, 74(1), 2171--2184

work page 2025

[15] [15]

Mishra, U.A., Samineni, S.R., Goel, P., Kunjeti, C., Lodha, H., Singh, A., Sagi, A., Bhatnagar, S., and Kolathaya, S. (2022). Dynamic mirror descent based model predictive control for accelerating robot learning. In Int. Conf. Robot. Autom., 1631--1637

work page 2022

[16] [16]

Mohamed, A., Ren, J., El-Gindy, M., Lang, H., and Ouda, A. (2018). Literature survey for autonomous vehicles: sensor fusion, computer vision, system identification and fault tolerance. Int. J. Autom. Control, 12(4), 555--581

work page 2018

[17] [17]

Moradmand, A., Shafai, B., and Saif, M. (2020). A design procedure for robust actuator and sensor fault detection. In 7th Int. Conf. Control, Decis. Info. Tech., 709--714

work page 2020

[18] [18]

O'Connell, M., Cho, J., Anderson, M., and Chung, S.J. (2024). Learning-based minimally-sensed fault-tolerant adaptive flight control. IEEE Robot. Autom. Lett. (RA-L), 9(6), 5198--5205

work page 2024

[19] [19]

Ragan, J., Riviere, B., Hadaegh, F.Y., and Chung, S.J. (2024). Online tree-based planning for active spacecraft fault estimation and collision avoidance. Science Robotics, 9(93), eadn4722

work page 2024

[20] [20]

Ran, G., Chen, H., Li, C., Ma, G., and Jiang, B. (2023). A hybrid design of fault detection for nonlinear systems based on dynamic optimization. IEEE Trans. Neural Netw. Learning Syst. , 34(9), 5244--5254

work page 2023

[21] [21]

Sobhani-Tehrani, E., Talebi, H., and Khorasani, K. (2014). Hybrid fault diagnosis of nonlinear systems using neural parameter estimators. Neural Networks, 50, 12--32

work page 2014

[22] [22]

Talebi, H., Khorasani, K., and Tafazoli, S. (2009). A recurrent neural-network-based sensor and actuator fault detection and isolation for nonlinear systems with application to the satellite's attitude control subsystem. IEEE Trans. Neural Netw. , 20(1), 45--60

work page 2009

[23] [23]

Zhang, X., Polycarpou, M., and Parisini, T. (2002). A robust detection and isolation scheme for abrupt and incipient faults in nonlinear systems. IEEE Trans. Autom. Control, 47(4), 576--593

work page 2002