# Cyberphysical System Modeled with Complex Networks and Hybrid Automata to Diagnose Multiple and Concurrent Faults in Manufacturing Systems

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Background

#### 2.1. Cyber–Physical Systems

#### 2.2. Hybrid Systems

#### 2.3. Hybrid Automata

$Q=(q1,q2,\dots )$ is a finite set of discrete states;

$X\subseteq {R}^{n}$ represents the state space of the continuous state variables;

$f:QxX\to {R}^{n}$ assigns to each discrete state $q\in Q$ an analytic vector field $f(q,\xb7);$

$Init\subseteq Q\phantom{\rule{0.277778em}{0ex}}x\phantom{\rule{0.277778em}{0ex}}X$ is the set of initial states;

$Inv:Q\to {2}^{x}$ assigns to each state $q\in Q$ a set $Inv\left(q\right)\subseteq X$ called the invariant set;

$E\subseteq QxQ$ is the set of discrete transitions;

$G:E\to {2}^{x}$ assigns to each discrete transition $(q,{q}^{\prime})\in E$ a guard set $G(q,{q}^{\prime})\subset X$;

$R:ExX\to {2}^{x}$ is a reset map.

#### 2.4. Complex Networks

## 3. Matherial and Methods

#### 3.1. Cyber–Physical System Components

#### 3.2. Cyber–Physical System Implementation

#### 3.3. Physical Hybrid Model of Warehouse Operation

#### 3.4. Digital Hybrid Warehouse Operation Model

#### 3.5. Complex Network of Automated Warehouse

#### Construction of Adjacency Matrix

#### 3.6. Data Models and Equations

#### 3.6.1. Digital Data Model

#### 3.6.2. Analog Data Model

#### 3.7. Development of the System

#### 3.7.1. Physical System Development

Algorithm 1: Development of the digital and analog physical system: |

#### 3.7.2. Virtual System Development

Algorithm 2: Development of the digital and analog virtual system: |

#### 3.7.3. Digital Control Operation and Fault Diagnosis

Algorithm 3: Development of the digital control |

## 4. Results and Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

Virtual System Variables: | |

$D{I}_{V{S}_{k}}$ | Virtual Binary Inputs Array |

$DI{d}_{V{S}_{k}}$ | Virtual Binary Inputs Difference |

$DI{m}_{V{S}_{k}}$ | Virtual Binary Inputs Memory |

$D{O}_{V{S}_{k}}$ | Virtual Binary Outputs Array |

$DO{d}_{V{S}_{k}}$ | Virtual Binary Outputs Difference |

$DO{m}_{V{S}_{k}}$ | Virtual Binary Outputs Memory |

$S{t}_{V{S}_{k}}$ | State virtual system |

Physical System Variables: | |

$D{I}_{P{S}_{k}}$ | Physical Binary Inputs Array |

$DI{d}_{P{S}_{k}}$ | Physical Binary Inputs Difference |

$DI{m}_{P{S}_{k}}$ | Physical Binary Inputs Memory |

$D{O}_{P{S}_{k}}$ | Physical Binary Outputs Array |

$DO{d}_{P{S}_{k}}$ | Physical Binary Outputs Difference |

$DO{m}_{P{S}_{k}}$ | Physical Binary Outputs Memory |

$S{t}_{P{S}_{k}}$ | State physical system |

## Appendix A

#### Appendix Adjacency Matrix Construction

## References

- TcaciucGherasim, S.A. A Solution for an Industrial Automation and SCADA System. In Proceedings of the 2022 International Conference and Exposition on Electrical And Power Engineering (EPE), Iaşi, Romania, 20–22 October 2022; IEEE: Piscataway, NJ, USA, 2022; pp. 297–300. [Google Scholar]
- Khadra, A.; Rammal, R. SCADA System for Solar Backup Power System Automation. In Proceedings of the 2022 International Conference on Smart Systems and Power Management (IC2SPM), Beirut, Lebanon, 10–12 November 2022; IEEE: Piscataway, NJ, USA, 2022; pp. 75–79. [Google Scholar]
- Nuhel, A.K.; Sazid, M.M.; Ahmed, K.; Bhuiyan, M.N.M.; Hassan, M.Y.B. A PI Controller-based Water Supplying and Priority Based SCADA System for Industrial Automation using PLC-HMI Scheme. In Proceedings of the 2022 IEEE International Conference on Artificial Intelligence in Engineering and Technology (IICAIET), Kinabalu, Malaysia, 13–15 September 2022; IEEE: Piscataway, NJ, USA, 2022; pp. 1–6. [Google Scholar]
- Hamzah, M.; Islam, M.M.; Hassan, S.; Akhtar, M.N.; Ferdous, M.J.; Jasser, M.B.; Mohamed, A.W. Distributed Control of Cyber Physical System on Various Domains: A Critical Review. Systems
**2023**, 11, 208. [Google Scholar] [CrossRef] - Zhang, K.; Shi, Y.; Karnouskos, S.; Sauter, T.; Fang, H.; Colombo, A.W. Advancements in industrial cyber-physical systems: An overview and perspectives. IEEE Trans. Ind. Inform.
**2022**, 19, 716–729. [Google Scholar] [CrossRef] - Villalonga, A.; Beruvides, G.; Castano, F.; Haber, R.E. Cloud-based industrial cyber–physical system for data-driven reasoning: A review and use case on an industry 4.0 pilot line. IEEE Trans. Ind. Inform.
**2020**, 16, 5975–5984. [Google Scholar] [CrossRef] - Alshalalfah, A.L.; Mohamed, O.A.; Ouchani, S. A framework for modeling and analyzing cyber-physical systems using statistical model checking. Internet Things
**2023**, 22, 100732. [Google Scholar] [CrossRef] - Li, P.; Zhang, F.; Yang, Y.; Ma, X.; Yao, S.; Yang, P.; Zhao, Z.; Lai, C.S.; Lai, L.L. The integrated modeling of microgrid cyber physical system based on hybrid automaton. Front. Energy Res.
**2022**, 10, 748828. [Google Scholar] [CrossRef] - Staroletov, S. Automatic proving of stability of the cyber-physical systems in the sense of Lyapunov with KeYmaera. In Proceedings of the 2021 28th Conference of Open Innovations Association (FRUCT), Moscow, Russia, 25–29 January 2021; IEEE: Piscataway, NJ, USA, 2021; pp. 431–438. [Google Scholar]
- Ravasio, D.; Tuissi, L.; Spinelli, S.; Ballarino, A. A Compressed Air Network Energy-Efficient Hierarchical Unit Commitment and Control. In Proceedings of the 2023 15th International Conference on Computer and Automation Engineering (ICCAE), Sydney, Australia, 3–5 March 2023; IEEE: Piscataway, NJ, USA, 2023; pp. 469–473. [Google Scholar]
- Guo, Z.; Zhang, Y.; Zhao, X.; Song, X. CPS-based self-adaptive collaborative control for smart production-logistics systems. IEEE Trans. Cybern.
**2020**, 51, 188–198. [Google Scholar] [CrossRef] [PubMed] - Sun, T.; Xia, W.; Zhao, X.; Sun, X.M. A Novel Mathematical Characterization for Switched Linear Systems Based on Automata and Its Stabilizability Analysis. IEEE Trans. Control. Netw. Syst.
**2023**. [Google Scholar] [CrossRef] - Alonso, M.; Turanzas, J.; Amaris, H.; Ledo, A.T. Cyber-physical vulnerability assessment in smart grids based on multilayer complex networks. Sensors
**2021**, 21, 5826. [Google Scholar] [CrossRef] [PubMed] - Zhao, Z.; Xu, Y.; Li, Y.; Zhao, Y.; Wang, B.; Wen, G. Sparse actuator attack detection and identification: A data-driven approach. IEEE Trans. Cybern.
**2023**, 53, 4054–4064. [Google Scholar] [CrossRef] [PubMed] - Pósfai, M.; Barabasi, A.L. Network Science; Citeseer: London, UK, 2016. [Google Scholar]
- Giudici, R. Introducción a la Teoría de Grafos; Equinoccio: Baruta, Miranda, Venezuela, 1997. [Google Scholar]
- Mu, D.; Yue, X.; Ren, H. Robustness of Cyber-Physical Supply Networks in Cascading Failures. Entropy
**2021**, 23, 769. [Google Scholar] [CrossRef] [PubMed] - Platzer, A. Logical Foundations of Cyber-Physical Systems; Springer: Cham, Switzerland, 2018; Volume 662. [Google Scholar]
- Lin, H.; Antsaklis, P.J. Hybrid Dynamical Systems; Foundations and Trends® in Systems and Control: Delft, The Netherlands, 2015. [Google Scholar]
- Meyn, S. Control Techniques for Complex Networks; Cambridge University Press: Cambridge, UK, 2008. [Google Scholar]

**Figure 6.**(

**A**) Sensors and actuators in conveyor belt download. (

**B**) Sensors and actuators in conveyor belt load. (

**C**) Sensors and actuators in elevator base.

**Figure 9.**(

**a**) Physical System Controller; (

**b**) Virtual System Controller; (

**c**) Supervisory Controller.

State | Output | State | Output | State | Output | State | Output |
---|---|---|---|---|---|---|---|

${q}_{P0}$ | Leds ^{1} | ${q}_{P5}$ | Elevator motors (QW30) | ${q}_{P10}$ | Leds ^{1} | ${q}_{P15}$ | Elevator motors |

${q}_{P1}$ | Conveyor Motors (Q10.6, Q10.7) | ${q}_{P6}$ | Motor forks (Q11.0 Right) | ${q}_{P11}$ | Elevator motors | ${q}_{P16}$ | Motor forks |

${q}_{P2}$ | Motor forks (Q11.1 Left) | ${q}_{P7}$ | Motor forks (Q11.2 Up) | ${q}_{P12}$ | Motor forks | ${q}_{P17}$ | Motor forks |

${q}_{P3}$ | Motor forks (Q11.2 Up) | ${q}_{P8}$ | Motor forks (Q11.0 Right) | ${q}_{P13}$ | Motor forks | ${q}_{P18}$ | Motor forks |

${q}_{P4}$ | Motor forks (Q11.1 LeftR) | ${q}_{P9}$ | Elevator motors (QW30) | ${q}_{P14}$ | Motor forks | ${q}_{P19}$ | Conveyor motor |

^{1}Led Outputs = Q10.0, Q10.1, Q11.3, Q10.2, Q10.3, Q11.4, Q10.3.

Input | Denomination | Output | Denomination | Input | Denomination | Output | Denomination |
---|---|---|---|---|---|---|---|

I10.0 | Start button L | Q10.0 | Start (Light) | I11.0 | Sensor mov X | Q11.0 | Forks right |

I10.1 | Stop Button L | Q10.1 | Start L (Light) | I11.1 | Sensor mov Z | Q11.1 | Forks left |

I10.2 | Reset button L | Q10.2 | Stop L (Light) | I11.2 | Emergency stop | Q11.2 | Forks Up |

I10.3 | Sensor conveyor 1L | Q10.3 | Stop (Light) | I11.3 | Start button D | Q11.3 | Start D (Light) |

I10.4 | Sensor conveyor 2L | Q10.4 | Reset L (Light) | I11.4 | Stop button D | Q11.4 | Stop D (Light) |

I10.5 | Sensor forks right | Q10.5 | Reset (Light) | I11.5 | Reset button D | Q11.5 | Reset D (Light) |

I10.6 | Sensor forks middle | Q10.6 | Conveyor 1L | I11.6 | Sensor conveyor 1D | Q11.6 | Conveyor 1D |

I10.7 | Sensor forks left | Q10.7 | Conveyor 2L | I11.7 | Sensor conveyor 2D | Q11.7 | Conveyor 2D |

I12.0 | Sensor current | QW30 | Target position |

No. Simulation | State | Input/Output Induce Fault | Input Physical Vector | Input Virtual Vector | Output Physical Vector | Output Virtual Vector | Fault Detection |
---|---|---|---|---|---|---|---|

1 | ${q}_{P3}$ | I10.4, Q11.4 | I10.4 = 0 | I10.4 = 1 | Q11.4 = 0 | Q11.4 = 1 | I10.4, Q11.4—Correct |

2 | ${q}_{P9}$ | I10.5, Q11.0 | I10.5 = 0 | I10.5 = 1 | Q11.0 = 0 | Q11.0 = 1 | I10.5, Q11.0—Correct |

3 | ${q}_{P14}$ | I11.0, I11.1, Q11.4 | I11.0 = 0, I11.1 = 0 | I11.0 = 1, I11.1 = 1 | Q11.4 = 0 | Q11.4 = 1 | I11.0, I11.1, Q11.4—Correct |

4 | ${q}_{P19}$ | I10.5, Q11.6, Q11.7 | I10.5 = 0 | I10.5 = 1 | Q11.6 = 0, Q11.7 = 0 | Q11.6 = 1, Q11.7 = 1 | I10.5, Q11.6, Q11.7—Correct |

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |

© 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Velazquez, A.; Martell, F.; Sanchez, I.Y.; Paredes, C.A.
Cyberphysical System Modeled with Complex Networks and Hybrid Automata to Diagnose Multiple and Concurrent Faults in Manufacturing Systems. *Appl. Sci.* **2023**, *13*, 10603.
https://doi.org/10.3390/app131910603

**AMA Style**

Velazquez A, Martell F, Sanchez IY, Paredes CA.
Cyberphysical System Modeled with Complex Networks and Hybrid Automata to Diagnose Multiple and Concurrent Faults in Manufacturing Systems. *Applied Sciences*. 2023; 13(19):10603.
https://doi.org/10.3390/app131910603

**Chicago/Turabian Style**

Velazquez, Alejandro, Fernando Martell, Irma Y. Sanchez, and Carlos A. Paredes.
2023. "Cyberphysical System Modeled with Complex Networks and Hybrid Automata to Diagnose Multiple and Concurrent Faults in Manufacturing Systems" *Applied Sciences* 13, no. 19: 10603.
https://doi.org/10.3390/app131910603