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

Fatiha Hamdi; Abdelhafid Zeroual; Fouzi Harrou

eess.SYApr 5, 2026

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

Authors: Fatiha Hamdi, Abdelhafid Zeroual, Fouzi Harrou

Organizations: Department of Electronics, Faculty of technology, LAAAS Laboratory, Batna 2 University, Batna, Algeria. · Department of Electrical Engineering, Faculty of Science and Applied Sciences, Larbi Ben M’hidi University, Oum El Bouaghi, Algeria. · Computer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division, King Abdullah University of Science and Technology (KAUST), Thuwal, 23955-6900, Saudi Arabia.

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.

Figures & tables

Figure 1: Schematic representation of the fault detection process for Hybrid Dynamic Systems (HDS). The system and observer plans generate outputs

(y,\psi)

and

(\hat{y},\hat{\psi})

, respectively. Residuals

r^{C}

and

r^{D}

are computed and compared against detection thresholds to identify faults affecting continuous (

f^{C}

) and discrete (

f^{D}

) dynamics, producing fault indicators

\Phi^{C}

and

\Phi^{D}

Figure 2: Illustration of subsystem switching logic in a hybrid dynamical system. Transitions between Subsystem 1 and Subsystem 2 are governed by the switching signal

S_{q}

and conditions

g_{1}

and

g_{2}

Figure 3: Structure of the SDH system. The system integrates Discrete-Event System (DES) dynamics with multiple LTI subsystems. Transitions between LTI subsystems are governed by the discrete-event controller.

Figure 4: Graphical representation of a DPN showing discrete and continuous places and transitions

Figure 5: TCPN representation of ETCPN submodel (without output case)

Figure 6: TCPN representation of ETCPN submodel (with output case)

Figure 7: Graphical representation of upward and downward level crossings for switching conditions

Figure 8: Architecture of the ETCPN-based hybrid observer framework

Figure 9: Fault detection process using the ETCPN-based residual generator

Figure 10: Residual-based fault detection scheme using ETCPN observer and semi-supervised anomaly detection methods

Figure 11: ETCPN representation of the hybrid system with two modes, illustrating discrete-event dynamics, continuous LTI subsystems, and switching logic within a unified modelling framework.

Panel of Figure 12 — Figure 12: Simulation results of the ETCPN model illustrating (a) evolution of the system input $U(k)$ , (b) continuous model states $x_{1}(k)$ and $x_{2}(k)$ , and (c) switching behaviour through discrete modes $q(k)$

Figure 13: Evolution of the ETCPN switched observer states: estimated trajectories of

x_{1}(k)

and

x_{2}(k)

tracking the actual system states.

Figure 14: Evolution of the ETCPN switched observer modes

q(k)

: estimated discrete modes accurately capturing the system’s switching behavior.

Figure 15: Evolution of estimation error dynamics: observer errors rapidly converging to zero, indicating accurate and stable hybrid state estimation.

Figure 16: Evolution of ETCPN switched observer states during discrete-event faults.

Figure 17: Evolution of ETCPN switched observer modes

q(k)

illustrating mode blocking faults.

Figure 18: Estimation error dynamics highlighting fault detection and post-fault recovery.

Method	Accuracy	Recall	FPR	F1 Score
OC-SVM	0.911	0.800	0.057	0.80
EE	0.844	0.700	0.114	0.67
SVDD	0.889	0.900	0.114	0.78

Table 1: Fault detection results using residual-based methods for discrete-event faults (Case 1).

Figure 19: Injected intermittent output faults affecting continuous dynamics.

Figure 20: Evolution of ETCPN switched observer states under intermittent output faults.

Figure 21: Evolution of ETCPN switched observer modes

q(k)

during fault occurrences.

Figure 22: Evolution of estimation error dynamics during continuous faults.

Method	Accuracy	Recall	FPR	F1 Score
One-Class SVM	0.943	1.000	0.057	0.83
EE	0.911	1.000	0.121	0.86
SVDD	0.889	1.000	0.152	0.83

Table 2: Fault detection results for Case 2 (continuous-time faults) using residual-based methods.

Figure 23: Pattern of injected intermittent output sensor faults and discrete mode-blocking events.

Figure 24: Evolution of ETCPN switched observer states under intermittent output faults.

Figure 25: Evolution of ETCPN switched observer modes

q(k)

during fault occurrences.

Figure 26: Evolution of estimation error dynamics during continuous faults.

Method	Accuracy	Recall	FPR	F1 Score
OCSVM	0.900	0.750	0.029	0.828
EE	0.860	0.562	0.000	0.720
SVDD	0.880	0.688	0.029	0.786

Table 3: Detection performance across the three methods on Case 3.

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Rapid and accurate fault detection in high-voltage transmission networks is essential for grid reliability and equipment protection. Transmission fault datasets are frequently imbalanced, and certain fault types produce electrical signatures that fall within the normal operating envelope, causing single-model classifiers to fail on safety-critical cases. This paper proposes a hybrid two-stage machine learning pipeline that decouples detection from classification. Stage 1 combines an Isolation Forest anomaly detector with an optional supervised binary detector through an OR-fusion rule; the supervised branch is allocated automatically during training for any fault class the anomaly detector cannot resolve, and is omitted when no such class exists. Stage 2 applies a Random Forest multiclass classifier only to samples flagged by Stage 1. Feature engineering is expressed as a per-measurement-point operator mapping six raw channels to eighteen features, including zero-sequence symmetrical components derived from Fortescue's theorem, yielding 18L features for L measurement points. On the TLFaultDataset, the pipeline raises Line-fault end-to-end accuracy from 31.3% to 95.8%. On an independent single-point dataset, the same framework attains 97.25% end-to-end accuracy across all classes including normal operation, exceeding the TLFed federated benchmark of 94.84% without GPU or federated infrastructure, at 0.05 ms per sample on CPU. Ablation on both datasets shows zero-sequence features resolving the three-phase versus three-phase-to-ground ambiguity, raising the F1-score of that class pair from 0.39 to 0.997. The direction of the zero-sequence signature is found to be system-dependent, motivating a learned decision boundary in place of a fixed relay threshold.

Abstract

Semi-Supervised Anomaly Detection

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