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

Fatiha Hamdi, Abdelhafid Zeroual, Fouzi Harrou

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.

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

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.

Paper Structure

This paper contains 20 sections, 2 theorems, 46 equations, 26 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Related Works
  3. Problem statement and materials
  4. Hybrid Dynamical System Class
  5. Petri Net Definitions and Preliminaries
  6. Proposed Extended Timed Continuous Petri Net (ETCPN)
  7. Marking Evolution of the ETCPN
  8. Hybrid Discrete-Time ETCPN Model representation
  9. Hybrid Evolution of the ETCPN Dynamics Model
  10. ETCPN-Based Residual Generator Framework
  11. Discrete Event Observer Design
  12. Discrete-Time LTI Observer Design
  13. Residual Generation and Fault Detection Methodology
  14. Anomaly Detection Techniques
  15. Detection performance evaluation metrics
  16. ...and 5 more sections

Key Result

Theorem 1

Given the ETCPN definitions and the assumptions presented above, the incidence matrix $W_q^C$ can be formulated as follows: $\blacktriangleleft$$\blacktriangleleft$

Figures (26)

  • 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)
  • ...and 21 more figures

Theorems & Definitions (10)

  • Definition 3.1
  • Definition 3.2
  • Definition 3.3
  • Definition 4.1
  • Remark
  • Definition 4.2: Enabling Conditions in ETCPN
  • Theorem 1
  • Theorem 2
  • proof
  • proof