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arxiv: 2604.04051 · v1 · submitted 2026-04-05 · 📡 eess.SY · cs.LG· cs.SY· math.DS· nlin.CG

Recognition: 2 theorem links

· Lean Theorem

Extended Hybrid Timed Petri Nets with Semi-Supervised Anomaly Detection for Switched Systems, Modelling and Fault Detection

Fatiha Hamdi , Abdelhafid Zeroual , Fouzi Harrou
Authors on Pith no claims yet

Pith reviewed 2026-05-13 17:07 UTC · model grok-4.3

classification 📡 eess.SY cs.LGcs.SYmath.DSnlin.CG
keywords hybrid systemsfault detectiontimed Petri netssemi-supervised detectionswitched systemsanomaly detectionhybrid observerslinear matrix inequalities
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The pith

An extended timed continuous Petri net with marking-dependent flows generates residuals that semi-supervised detectors use to identify faults in hybrid switched systems.

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

This paper develops a fault detection method for systems that mix continuous flows and discrete switches by building a single model that links both behaviors. Conventional approaches handle the two parts separately and miss how faults interact across them. The new model adds marking-dependent flow functions to a timed Petri net so that changes in one domain automatically affect the other. An observer built on this model produces error signals that anomaly detectors, trained only on fault-free runs, classify into discrete, continuous, or mixed faults. Simulations show the detectors reach high accuracy while keeping false alarms low, especially with one-class SVM and support vector data description.

Core claim

The central claim is that extending hybrid timed continuous Petri nets with marking-dependent flow functions creates an intrinsic coupling between discrete and continuous dynamics, allowing a mode-dependent hybrid observer whose stability is guaranteed by offline LMIs and whose residuals enable semi-supervised anomaly detectors trained solely on normal data to detect discrete, continuous, and hybrid faults with high accuracy and low false alarms.

What carries the argument

Extended Timed Continuous Petri Net (ETCPN) featuring marking-dependent flow functions that intrinsically couple discrete and continuous dynamics in switched systems.

If this is right

  • The observer design ensures stability under arbitrary switching sequences via precomputed gains.
  • Residuals from the model allow detection of three fault classes without requiring labeled fault examples.
  • OC-SVM and SVDD achieve the best balance of detection rate and false alarm rate among tested methods.
  • The approach confines computational load to offline LMI solving, supporting real-time monitoring.
  • Validation on simulated hybrid faults confirms robust performance and fast convergence.

Where Pith is reading between the lines

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

  • This modeling choice could simplify controller design for hybrid systems by providing a unified representation usable beyond detection.
  • If the residuals prove informative across real-world plants, the method might reduce the data requirements for industrial fault monitoring.
  • Extending the framework to include online parameter updates could handle slowly drifting system parameters without retraining the detectors.
  • Neighbouring problems such as hybrid system identification might benefit from the same marking-dependent flow structure.

Load-bearing premise

The hybrid observer remains stable for any switching pattern when its gains come from offline LMIs, and the residuals it produces carry enough information for detectors trained only on normal data to separate all fault types without raising many false alarms.

What would settle it

Running the system under rapid arbitrary mode switches while injecting one of the three fault types and observing whether the observer states diverge or the semi-supervised detectors produce detection rates below 80 percent with elevated false alarms.

Figures

Figures reproduced from arXiv: 2604.04051 by Abdelhafid Zeroual, Fatiha Hamdi, Fouzi Harrou.

Figure 1
Figure 1. Figure 1: Schematic representation of the fault detection process for Hybrid Dynamic Systems (HDS). The system and observer [PITH_FULL_IMAGE:figures/full_fig_p008_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Illustration of subsystem switching logic in a hybrid dynamical system. Transitions between Subsystem 1 and [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Structure of the SDH system. The system integrates Discrete-Event System (DES) dynamics with multiple LTI [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Graphical representation of a DPN showing discrete and continuous places and transitions [PITH_FULL_IMAGE:figures/full_fig_p011_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: TCPN representation of ETCPN submodel (without output case) [PITH_FULL_IMAGE:figures/full_fig_p017_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: TCPN representation of ETCPN submodel (with output case) [PITH_FULL_IMAGE:figures/full_fig_p017_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Graphical representation of upward and downward level crossings for switching conditions [PITH_FULL_IMAGE:figures/full_fig_p018_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Architecture of the ETCPN-based hybrid observer framework [PITH_FULL_IMAGE:figures/full_fig_p020_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Fault detection process using the ETCPN-based residual generator [PITH_FULL_IMAGE:figures/full_fig_p023_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Residual-based fault detection scheme using ETCPN observer and semi-supervised anomaly detection methods [PITH_FULL_IMAGE:figures/full_fig_p026_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: ETCPN representation of the hybrid system with two modes, illustrating discrete-event dynamics, continuous LTI [PITH_FULL_IMAGE:figures/full_fig_p028_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Simulation results of the ETCPN model illustrating (a) evolution of the system input [PITH_FULL_IMAGE:figures/full_fig_p029_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Evolution of the ETCPN switched observer states: estimated trajectories of [PITH_FULL_IMAGE:figures/full_fig_p030_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: Evolution of the ETCPN switched observer modes [PITH_FULL_IMAGE:figures/full_fig_p030_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: Evolution of estimation error dynamics: observer errors rapidly converging to zero, indicating accurate and stable [PITH_FULL_IMAGE:figures/full_fig_p031_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: Evolution of ETCPN switched observer states during discrete-event faults. [PITH_FULL_IMAGE:figures/full_fig_p032_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: Evolution of ETCPN switched observer modes [PITH_FULL_IMAGE:figures/full_fig_p032_17.png] view at source ↗
Figure 18
Figure 18. Figure 18: Estimation error dynamics highlighting fault detection and post-fault recovery. [PITH_FULL_IMAGE:figures/full_fig_p033_18.png] view at source ↗
Figure 19
Figure 19. Figure 19: Injected intermittent output faults affecting continuous dynamics. [PITH_FULL_IMAGE:figures/full_fig_p034_19.png] view at source ↗
Figure 20
Figure 20. Figure 20: Evolution of ETCPN switched observer states under intermittent output faults. [PITH_FULL_IMAGE:figures/full_fig_p035_20.png] view at source ↗
Figure 21
Figure 21. Figure 21: Evolution of ETCPN switched observer modes [PITH_FULL_IMAGE:figures/full_fig_p035_21.png] view at source ↗
Figure 22
Figure 22. Figure 22: Evolution of estimation error dynamics during continuous faults. [PITH_FULL_IMAGE:figures/full_fig_p035_22.png] view at source ↗
Figure 23
Figure 23. Figure 23: Pattern of injected intermittent output sensor faults and discrete mode-blocking events. [PITH_FULL_IMAGE:figures/full_fig_p037_23.png] view at source ↗
Figure 24
Figure 24. Figure 24: Evolution of ETCPN switched observer states under intermittent output faults. [PITH_FULL_IMAGE:figures/full_fig_p038_24.png] view at source ↗
Figure 25
Figure 25. Figure 25: Evolution of ETCPN switched observer modes [PITH_FULL_IMAGE:figures/full_fig_p038_25.png] view at source ↗
Figure 26
Figure 26. Figure 26: Evolution of estimation error dynamics during continuous faults. [PITH_FULL_IMAGE:figures/full_fig_p038_26.png] view at source ↗
read the original abstract

Hybrid physical systems combine continuous and discrete dynamics, which can be simultaneously affected by faults. Conventional fault detection methods often treat these dynamics separately, limiting their ability to capture interacting fault patterns. This paper proposes a unified fault detection framework for hybrid dynamical systems by integrating an Extended Timed Continuous Petri Net (ETCPN) model with semi-supervised anomaly detection. The proposed ETCPN extends existing Petri net formalisms by introducing marking-dependent flow functions, enabling intrinsic coupling between discrete and continuous dynamics. Based on this structure, a mode-dependent hybrid observer is designed, whose stability under arbitrary switching is ensured via Linear Matrix Inequalities (LMIs), solved offline to determine observer gains. The observer generates residuals that reflect discrepancies between the estimated and measured outputs. These residuals are processed using semi-supervised methods, including One-Class SVM (OC-SVM), Support Vector Data Description (SVDD), and Elliptic Envelope (EE), trained exclusively on normal data to avoid reliance on labeled faults. The framework is validated through simulations involving discrete faults, continuous faults, and hybrid faults. Results demonstrate high detection accuracy, fast convergence, and robust performance, with OC-SVM and SVDD providing the best trade-off between detection rate and false alarms. The framework is computationally efficient for real-time deployment, as the main complexity is confined to the offline LMI design phase.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 2 minor

Summary. The paper proposes an Extended Timed Continuous Petri Net (ETCPN) formalism that introduces marking-dependent flow functions to intrinsically couple discrete and continuous dynamics in hybrid switched systems. It designs a mode-dependent hybrid observer whose gains are computed offline via LMIs to guarantee stability under arbitrary switching, generates residuals from the observer, and feeds them to semi-supervised detectors (OC-SVM, SVDD, EE) trained exclusively on nominal data to identify discrete, continuous, and hybrid faults. Simulations are reported to show high detection accuracy, fast convergence, and good trade-offs between detection rate and false alarms, with the main computational burden confined to the offline LMI phase.

Significance. If the stability guarantee and residual informativeness hold without post-hoc tuning, the work would offer a unified modeling and detection framework that avoids treating discrete and continuous faults separately, potentially advancing fault diagnosis for switched hybrid systems by combining Petri-net structure with semi-supervised anomaly detection. The offline LMI design and normal-data-only training are practical strengths for real-time deployment.

major comments (2)
  1. [Observer design and stability] Observer design and stability section: The claim that the mode-dependent hybrid observer remains asymptotically stable under arbitrary switching when gains are obtained from offline per-mode LMIs is load-bearing for the central claim, yet the manuscript does not indicate whether a common quadratic Lyapunov matrix P > 0 is enforced across all modes or whether dwell-time constraints are imposed. For switched linear systems, independent per-mode LMIs without such coupling generally fail to guarantee arbitrary-switching stability, allowing potential divergence under rapid switching; this must be clarified with the explicit LMI formulation and common-P condition if used.
  2. [Simulation and validation] Simulation and validation section: The abstract and results claim high detection accuracy and robust performance for all three fault classes, but without explicit LMI conditions, residual definitions, quantitative tables (detection rates, false-alarm rates, convergence times), or ablation on the semi-supervised detectors, it is impossible to confirm that the reported performance is not the result of post-hoc parameter tuning on the test faults; this undermines the claim that residuals are sufficiently informative when detectors are trained only on normal data.
minor comments (2)
  1. [Modeling] Notation for the marking-dependent flow functions and the ETCPN incidence matrices should be defined more explicitly in the modeling section to avoid ambiguity when deriving the continuous dynamics.
  2. [Introduction] The paper should include a brief comparison table contrasting the proposed ETCPN with standard hybrid Petri nets (e.g., on coupling mechanism and fault modeling capability) to highlight the extension.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive and detailed comments, which help strengthen the manuscript. We address each major point below and agree that clarifications and additions are required for rigor. The revised version will incorporate explicit LMI formulations and quantitative simulation details.

read point-by-point responses

  1. Referee: [Observer design and stability] Observer design and stability section: The claim that the mode-dependent hybrid observer remains asymptotically stable under arbitrary switching when gains are obtained from offline per-mode LMIs is load-bearing for the central claim, yet the manuscript does not indicate whether a common quadratic Lyapunov matrix P > 0 is enforced across all modes or whether dwell-time constraints are imposed. For switched linear systems, independent per-mode LMIs without such coupling generally fail to guarantee arbitrary-switching stability, allowing potential divergence under rapid switching; this must be clarified with the explicit LMI formulation and common-P condition if used.

    Authors: We agree that the common Lyapunov matrix condition was not stated explicitly. The observer design uses a single common quadratic Lyapunov matrix P > 0 shared across all modes, with per-mode gains L_i computed offline via LMIs of the form (A_i - L_i C_i)^T P + P (A_i - L_i C_i) < 0 for each mode i. This common-P formulation guarantees asymptotic stability under arbitrary switching without dwell-time constraints. We will revise the manuscript to present the full set of LMIs, the common-P requirement, and the associated proof sketch. revision: yes

  2. Referee: [Simulation and validation] Simulation and validation section: The abstract and results claim high detection accuracy and robust performance for all three fault classes, but without explicit LMI conditions, residual definitions, quantitative tables (detection rates, false-alarm rates, convergence times), or ablation on the semi-supervised detectors, it is impossible to confirm that the reported performance is not the result of post-hoc parameter tuning on the test faults; this undermines the claim that residuals are sufficiently informative when detectors are trained only on normal data.

    Authors: We acknowledge the need for greater transparency. The revised manuscript will include: (i) the explicit LMI conditions and common P used for observer design, (ii) the precise residual definition r(k) = y(k) - C x̂(k), (iii) quantitative tables reporting detection rates, false-alarm rates, and convergence times for discrete, continuous, and hybrid faults under each detector (OC-SVM, SVDD, EE), and (iv) an ablation study on detector hyperparameters performed exclusively via cross-validation on normal training data. These additions will demonstrate that the reported performance stems from informative residuals rather than test-set tuning. revision: yes

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The paper introduces the ETCPN extension via marking-dependent flow functions, designs a mode-dependent hybrid observer whose gains are obtained by solving LMIs offline, generates residuals from the observer, and feeds them to semi-supervised anomaly detectors (OC-SVM, SVDD, EE) trained exclusively on normal data. None of these steps reduces a reported detection performance metric to a fitted quantity on the same fault test cases, nor does any central claim collapse by definition or self-citation to its own inputs. The LMI-based stability assertion and the anomaly-detection training protocol are independent of the final simulation results; whether the specific LMI formulation actually guarantees arbitrary-switching stability is a separate correctness question outside the scope of circularity analysis.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 1 invented entities

The central claim rests on the new ETCPN formalism, the assumption that LMIs can be solved to guarantee observer stability under arbitrary switching, and the premise that residuals from this observer are informative enough for normal-data-only anomaly detectors.

free parameters (1)
  • Observer gains
    Computed offline via LMI feasibility; treated as design variables rather than fitted to fault data.
axioms (1)
  • domain assumption Existence of feasible LMIs that guarantee stability of the mode-dependent hybrid observer under arbitrary switching
    Invoked to justify offline gain computation without explicit proof in the abstract.
invented entities (1)
  • Extended Timed Continuous Petri Net (ETCPN) no independent evidence
    purpose: Unified model coupling discrete and continuous dynamics via marking-dependent flow functions
    New formalism introduced to enable the hybrid observer; no independent evidence outside the paper is supplied.

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