# Observer-Based Distributed Fault Detection for Heterogeneous Multi-Agent Systems

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. Preliminaries and Problem Formulation

#### 2.1. Graph Theory

**Lemma**

**1**

**.**If the subgraph ${\mathcal{G}}_{s}$ is undirected, and each follower agent has paths to the leader in the graph $\mathcal{G}$, $\mathcal{H}$ is positive definite.

#### 2.2. Unknown Input Observer

**Lemma**

**2**

- (i)
- rank($MY$) = rank(Y)
- (ii)
- $\left[\begin{array}{cc}sI-Q& Y\\ M& 0\end{array}\right]$ is of full column rank for $\forall s\in \mathbb{C}$, $Re(s)\ge 0$, i.e., $(TQ,M)$ is detectable.

**Remark**

**1.**

#### 2.3. Problem Formulation

**Assumption**

**1.**

**Assumption**

**2.**

**Remark**

**2.**

**Assumption**

**3.**

## 3. Results

#### 3.1. State-Feedback-Based Distributed FD

**Remark**

**3.**

**Theorem**

**1.**

**Proof**

**of**

**Theorem**

**1.**

**Remark**

**4.**

**Theorem**

**2.**

**Proof**

**of**

**Theorem**

**2.**

**Remark**

**5.**

**Remark**

**6.**

**Remark**

**7.**

Algorithm 1 Faulty agent location algorithm |

1. In agent i, construct residual generators ${r}_{i{\overline{i}}_{k}}$ and set appropriate thresholds ${\Theta}_{i{\overline{i}}_{k}}>0$, $k=1,\cdots ,|{\overline{\mathcal{N}}}_{i}|$. |

2. Run residual generators ${r}_{i{\overline{i}}_{k}}$, $k=1,\cdots ,|{\overline{\mathcal{N}}}_{i}|$. |

(1) Check ${r}_{i{\overline{i}}_{k}},k=1$. If $\left|\right|{r}_{i{\overline{i}}_{k}}\left|\right|<{\Theta}_{i{\overline{i}}_{k}}$, and $\left|\right|{r}_{ij}\left|\right|\ge {\Theta}_{ij},j\ne {\overline{i}}_{k}$, stop and remove the faulty node. Otherwise, go to step (2). |

⋮ |

($|{\overline{\mathcal{N}}}_{i}|$) Check ${r}_{i{\overline{i}}_{k}},k=\left|{\overline{\mathcal{N}}}_{i}\right|$. If $\left|\right|{r}_{i{\overline{i}}_{k}}\left|\right|<{\Theta}_{i{\overline{i}}_{k}}$, and $\left|\right|{r}_{ij}\left|\right|\ge {\Theta}_{ij},j\ne {\overline{i}}_{k}$, stop and remove the faulty node. |

**Remark**

**8.**

#### 3.2. Output-Feedback-Based Distributed FD

**Assumption**

**4.**

**Remark**

**9.**

**Remark**

**10.**

**Theorem**

**3.**

**Proof**

**of**

**Theorem**

**3.**

**Remark**

**11.**

**Remark**

**12.**

## 4. Simulation Example

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

MAS | Multi-agent systems |

FD | Fault detection |

COR | Cooperative output regulation |

UIO | Unknown input observer |

UAV | Unmanned aerial vehicles |

## References

- Hu, W.F.; Liu, L. Input time delay margin in event-triggered consensus of multiagent systems. IEEE Trans. Cybern.
**2019**, 5, 1849–1858. [Google Scholar] - De la Sen, M.; Alonso-Quesada, S. On finite-time consensus objectives in time-varying interconnected discrete linear dynamic systems under internal and external delays. Adv. Mech. Eng.
**2018**, 7, 1–24. [Google Scholar] [CrossRef] - Su, S.; Huang, J. Cooperative output regulation of linear multi-agent systems. IEEE Trans. Autom. Control
**2012**, 4, 1062–1066. [Google Scholar] - Su, S.; Huang, J. Cooperative output regulation of linear multi-agent systems by output feedback. Syst. Control Lett.
**2012**, 61, 1248–1253. [Google Scholar] [CrossRef] - Li, Z.; Chen, M.Z.Q.; Ding, Z.T. Distributed adaptive controllers for cooperative output regulation of heterogeneous agents over directed graph. Automatica
**2016**, 68, 179–183. [Google Scholar] [CrossRef] - Lv, M.; Liu, L. Cooperative output regulation of linear multi-agent systems by a novel distributed dynamic compensator. IEEE Trans. Autom. Control
**2017**, 62, 6481–6488. [Google Scholar] - Hashikura, K.; Kojima, A.; Suzuki, T.; Yamada, K. Exosystem-observer approach to H
_{2}control with output regulation constraint. IET Control Theory Appl.**2019**, 16, 732–745. [Google Scholar] [CrossRef] - Basu, H.; Yoon, S.Y. Robust cooperative output regulation under exosystem detectability constraint. Int. J. Control
**2020**, 5, 1102–1114. [Google Scholar] [CrossRef] - Deng, H.C.; Yang, G.H. Cooperative adaptive output regulation for linear multi-agent systems with actuator faults. IET Control Theory Appl.
**2017**, 11, 2396–2402. [Google Scholar] [CrossRef] - Deng, H.C.; Yang, G.H. Distributed adaptive fault-tolerant control approach to cooperative output regulation for linear multi-agent systems. Automatica
**2019**, 103, 62–68. [Google Scholar] [CrossRef] - Deng, H.C. Cooperative fault-tolerant output regulation of linear heterogeneous multiagent systems under directed network topology. IEEE Trans. Syst. Man Cybern. Syst.
**2019**, 2019, 2944254. [Google Scholar] [CrossRef] - Meskin, N.; Khorasani, K. Actuator fault detection and isolation for a network of unmanned vehicles. IEEE Trans. Autom. Control
**2009**, 4, 835–840. [Google Scholar] [CrossRef] - Meskin, N.; Khorasani, K.; Rabbath, C.A. A hybrid fault detection and isolation strategy for a network of unmanned vehicles in presence of large environmental disturbances. IEEE Trans. Control Syst. Technol.
**2009**, 6, 1422–1429. [Google Scholar] [CrossRef] - Menon, P.P.; Edwards, C. Robust fault estimation using relative information in linear multi-agent networks. IEEE Trans. Autom. Control
**2014**, 2, 477–482. [Google Scholar] [CrossRef] [Green Version] - Shames, I.; Teixeira, A.M.H.; Sandberg, H.; Johansson, K.H. Distributed fault detection for interconnected second-order systems. Automatica
**2011**, 47, 2757–2764. [Google Scholar] [CrossRef] [Green Version] - Darouach, M.; Zasadzinski, M.; Xu, S.J. Full-order observers for linear systems with unknown inputs. IEEE Trans. Autom. Control
**1994**, 39, 606–609. [Google Scholar] [CrossRef] [Green Version] - Emami, S.A.; Araujo, R.; Asvadi, A. Distributed simultaneous estimation of states and unknown inputs. Syst. Control Lett.
**2020**, 138, 104660. [Google Scholar] [CrossRef] - Hou, M.; Muller, P.C. Fault detection and isolation observers. Int. J. Control
**1994**, 5, 827–846. [Google Scholar] [CrossRef] - Teixeira, A.; Shames, I.; Sandberg, H.; Johansson, K.H. Distributed fault detection and isolation resilient to network model uncertainties. IEEE Trans. Cybern.
**2014**, 11, 2024–2037. [Google Scholar] [CrossRef] [Green Version] - Shi, J.; He, X.; Wang, Z.D.; Zhou, D.H. Distributed fault detection for a class of second-order multi-agent systems: An optimal robust observer approach. IET Control Theory Appl.
**2014**, 8, 1032–1044. [Google Scholar] [CrossRef] - Jia, W.; Wang, J.Z. Partial-Nodes-Based distributed fault detection and isolation for second-order multi-agent systems with exogenous disturbances. IEEE Trans. Cybern.
**2020**. [Google Scholar] [CrossRef] [PubMed] - Liu, X.; Gao, X.; Han, J. Observer-based fault detection for high-order nonlinear multi-agent systems. J. Frankl. Inst.
**2016**, 353, 72–94. [Google Scholar] [CrossRef] - Gao, X.; Liu, X.; Han, J. Reduced order unknown input observer based distributed fault detection for multi-agent systems. J. Frankl. Inst.
**2017**, 354, 1464–1483. [Google Scholar] [CrossRef] - Davoodi, N.M.R.; Khorasani, K.; Talebi, H.A.; Momeni, H.R. Distributed fault detection and isolation filter design for a network of heterogeneous multiagent systems. IEEE Trans. Control Syst. Technol.
**2014**, 3, 1061–1069. [Google Scholar] [CrossRef] - Qin, L.; He, X.; Zhou, D.H. Fault-tolerant cooperative output regulation for multi-vehicle systems with sensor faults. Int. J. Control
**2017**, 10, 2227–2248. [Google Scholar] [CrossRef] - Ni, W.; Cheng, D. Leader-following consensus of multi-agents systems under fixed and switching topologies. Syst. Control Lett.
**2010**, 59, 209–217. [Google Scholar] [CrossRef] - Huang, W.J. Linear output regulation. In Nonlinear Output Regulation: Theory and Applications; King, B., Ed.; SIAM: Philadelphia, PA, USA, 2004; pp. 3–15. [Google Scholar]
- Ding, X. Norm-based residual evaluation and threshold computation. In Model-Based Fault Diagnosis Techniques: Design Schemes, Algorithms and Tools, 2nd ed.; Grimble, M.J., Johnson, M.A., Eds.; Springer: London, UK, 2013; pp. 285–314. [Google Scholar]
- Horn, W.R.A.; Johnson, C.R. Unitary and real orthogonal triangularizations. In Matrix Analysis, 2nd ed.; Cambridge University Press: New York, NY, USA, 2013; pp. 94–101. [Google Scholar]
- Frank, H.P.M.; Ding, S.X. Survey of robust residual generation and evaluation methods in observer-based fault detection systems. J. Process Control
**1997**, 6, 403–424. [Google Scholar] [CrossRef] - Hu, W.; Liu, L. Cooperative output regulation of heterogeneous linear multi-agent systems by event-triggered control. IEEE Trans. Cybern.
**2017**, 1, 105–116. [Google Scholar] [CrossRef]

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2020 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 (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Jia, W.; Wang, J.
Observer-Based Distributed Fault Detection for Heterogeneous Multi-Agent Systems. *Appl. Sci.* **2020**, *10*, 7466.
https://doi.org/10.3390/app10217466

**AMA Style**

Jia W, Wang J.
Observer-Based Distributed Fault Detection for Heterogeneous Multi-Agent Systems. *Applied Sciences*. 2020; 10(21):7466.
https://doi.org/10.3390/app10217466

**Chicago/Turabian Style**

Jia, Wenhao, and Jinzhi Wang.
2020. "Observer-Based Distributed Fault Detection for Heterogeneous Multi-Agent Systems" *Applied Sciences* 10, no. 21: 7466.
https://doi.org/10.3390/app10217466