Table of Contents Author Guidelines Submit a Manuscript
Mathematical Problems in Engineering
Volume 2013, Article ID 465671, 7 pages
http://dx.doi.org/10.1155/2013/465671
Research Article

Fault Tolerant Consensus of Multi-Agent Systems with Linear Dynamics

School of Electronics and Information, Jiangsu University of Science and Technology, Zhenjiang, Jiangsu 212003, China

Received 14 November 2013; Accepted 7 December 2013

Academic Editor: Hao Shen

Copyright © 2013 Jianzhen Li. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

This paper deals with the consensus problem of linear multi-agent systems with actuator faults. A fault estimator based consensus protocol is provided, together with a convergence analysis. It is shown that the consensus errors of all agents will converge to a small set around the origin, if parameters in the consensus protocol are properly chosen. A numerical example is given to illustrate the effectiveness of the proposed protocol.

1. Introduction

Recently, distributed consensus problems for multi-agent systems have become a hot research area in the control theory community [112]. This is partly because of their widespread applications in areas such as robots, flocking, unmanned air vehicles, sensor fusion, and microgrids (see [1318] and the references therein). For multi-agent systems, consensus means the group of agents asymptotically agree on certain quantities of interest that depends on the states of all agents [19]. In the research of multi-agent systems, the main challenge is how to design simple control rule for simple agents to achieve a prescribed group behavior.

Fault detection and fault accommodation are very important problems for control systems [2022]. The fault detection and the fault tolerant consensus problems of multi-agent systems have attracted attention of researchers in the last few years. In [23], the fault detection problem was considered for discrete-time multi-agent systems with first-order dynamics, while the continuous-time second-order multi-agent systems were considered in [24]. The fault tolerant consensus problem for first-order multi-agent systems was investigated in [25], under the assumption that the faults are detected in time. The resilient consensus problem was considered in [26], for multi-agent systems with adversary agents. A consensus protocol is provided; under which consensus can be achieved, if the number of adversary agents satisfies a certain condition related to the degree of the communication graph.

The fault tolerant consensus problem for high-order linear multi-agent systems has not been considered in the literature, which motivated the work in this paper. In this paper, consensus problems will be considered for linear multi-agent systems with actuator faults. A fault estimator is provided, based on which a consensus protocol is derived. It is proved that consensus error can converge to a small set around the origin, if parameters in the fault estimator and the consensus protocol are properly chosen. The rest of the paper is organized as follows. Section 2 formulates the fault tolerant consensus problem of the multi-agent systems with linear dynamics. The main results are presented in Section 3. A numerical example is given in Section 4 to illustrate the proposed results and the paper is concluded in Section 5.

Notations. Throughout this paper, matrix means that is symmetric positive definite. Consider . Similarly, .

2. Problem Formulation

Consider a team of agents with the following linear dynamics: where is the state of agent , is the control input, and is the output. denotes the actuator fault of agent . If , then no actuator fault occurs at agent . In this paper, we have the following assumptions.

Assumption 1. Assume that , , for .

Assumption 2. is observable, and is full rank.

Remark 3. Assumption 2 means that the state of the agent can be constructed from the output, and there is no subsystem decoupled from the faults in the linear description [20]. Under Assumption 2, there exist a symmetric positive definite and matrices and such that
The communication topology among the agents can be represented by an undirected graph , where is the node set and is the edge set. An edge if agent and agent can access information from each other. An undirected path is a sequence of undirected edges of the form , where . An undirected graph is connected if for any there exists a path between them. The neighbor set of agent is defined as . The adjacency matrix is defined as if and otherwise. The Laplacian matrix is defined as , where . It is well known that if the communication topology is connected, then has a simple zero eigenvalue and nonnegative eigenvalues .

Definition 4. We say algorithm asymptotically solves the consensus problem if as , for any .
The objective of this paper is to derive a consensus protocol under which consensus can be achieved even if some agents are subject to actuator faults.

3. Main Results

This section gives the fault tolerant consensus protocol for multi-agent systems described in the last section. Before giving the consensus protocol, we give the fault estimator first.

3.1. Fault Estimator

Consider the following fault estimator, which is motivated by the fault estimator in [20]: where and are estimations of , , and , respectively; and ; matrices and are chosen according to (3); matrix and constant scalar are chosen such that .

Lemma 5. Define . Under Assumptions 1 and 2, estimator (4) guarantees that converge exponentially to the following set: where .

Proof. The proof is similar to the proof of Theorem 1 in [20] and hence is omitted here.

3.2. Fault Tolerant Consensus

Assumption 6. is controllable, and .
Under Assumption 6, there exists a matrix such that ; that is, . Based on estimator (4), we have the following consensus protocol: where is a parameter matrix to be designed. With (6), system (1) becomes Let . It is easy to see that consensus is achieved if , for . From (1) and (6), we have Notice that , , and , and (8) can be rewritten as For undirected graphs, we have . It is easy to see that which, together with (9), leads to Define and . Equation (11) can be written in a compact form as where
It can be seen that , , , , and . Before giving the main results of this paper, the following lemma is needed.

Lemma 7 (see [10]). Let and be matrices previously defined; the following statements hold.(1)The eigenvalues of are with multiplicity and with multiplicity . The vectors and are the left and right eigenvectors of associated with zero eigenvalue, respectively.(2)There exists an orthogonal matrix with last column , such that
Next, we give the main results of this paper.

Theorem 8. Suppose the undirected graph is connected, and the nonzero eigenvalues of are . Using protocol (6), with the fault estimator (4), consensus errors will converge to a small ball around the origin if there exists a symmetric positive definite matrix such that the following LMI holds: and is chosen as .

Proof. Suppose that the communication graph is connected; the nonzero eigenvalues of are and there exists a symmetric positive definite matrix such that (15) holds. Let ; we have
Define where . From (1), (2), and (4), we have that Equation (18) can be rewritten in a compact form as
Consider the following Lyapunov function: where is defined in Lemma 5. It follows from (12) that From (19) we have Notice that and it follows that Let ; we have and , where is defined in Lemma 7. Define By the definition of we know that , where . It follows that From (21) we have Similarly, from (24) we have It then follows that where .
Notice that and are bounded, for . It is easy to see that is bounded, and the bound is determined by and . We assume that . Equation (29) can be rewritten as where and It is easy to see that . From the assumption of the theorem, we know that . By the definition of , we know that . It is easy to see that if is small enough. From (30) we know that if which implies that will converge into , yielding that will converge to a small ball around the origin. The proof is completed.

Remark 9. Theorem 8 shows that can converge to a small ball surrounding the origin. From the proof, we know that this set is determined by and . Since and can be selected freely, this ball can be chosen arbitrarily small. However, if is chosen too small, may converge very slowly. To overcome this problem, dynamically changing and can be used in the practice. This is out of the scope of this paper and will be considered in our future research.

4. A Numerical Example

Consider a multi-agent system consisting of agents with , for , and for . The communication topology is given in Figure 1, with the following Laplacian matrix: The eigenvalues of are , , , and . According to Theorem 8, can be chosen as and parameters in the estimator (4) can be chosen as , , and Figures 2 and 3 show, respectively, the position and velocity responses of nodes 1–4. It can be seen that consensus can be achieved in this case.

465671.fig.001
Figure 1: The network topology associated with agents to .
465671.fig.002
Figure 2: Position responses of agents 1–4.
465671.fig.003
Figure 3: Velocity responses of agents 1–4.

5. Conclusions and Future Work

The fault-tolerant consensus problem for multi-agent systems with actuator faults was considered. A fault estimator based consensus protocol is provided, together with a sufficient condition under which the consensus can be achieved. It is proved that consensus errors of all agents can converge to a small set around the origin. The numerical example confirmed the proposed theoretical results. In practice, many systems have stochastic Markovian jumping dynamics [2732]. Future research efforts will be devoted to the fault tolerant consensus problem of multi-agent systems with stochastic Markovian jumping dynamics.

Acknowledgments

This study was supported by National Natural Science Foundation of China under Grants 61203024, 61100116, 61374063, and 61304249, Natural Science Fundamental Research Project of Jiangsu Colleges and Universities under Grant 12KJB120001, and the Natural Science Foundation of Jiangsu Province of China under Grant BK2011492.

References

  1. R. Olfati-Saber and R. M. Murray, “Consensus problems in networks of agents with switching topology and time-delays,” IEEE Transactions on Automatic Control, vol. 49, no. 9, pp. 1520–1533, 2004. View at Publisher · View at Google Scholar · View at Scopus
  2. W. Ren and R. W. Beard, “Consensus seeking in multiagent systems under dynamically changing interaction topologies,” IEEE Transactions on Automatic Control, vol. 50, no. 5, pp. 655–661, 2005. View at Publisher · View at Google Scholar · View at Scopus
  3. W. Ren, “Multi-vehicle consensus with a time-varying reference state,” Systems and Control Letters, vol. 56, no. 7-8, pp. 474–483, 2007. View at Publisher · View at Google Scholar · View at Scopus
  4. W. Yu, G. Chen, and M. Cao, “Some necessary and sufficient conditions for second-order consensus in multi-agent dynamical systems,” Automatica, vol. 46, no. 6, pp. 1089–1095, 2010. View at Publisher · View at Google Scholar · View at Scopus
  5. H. Zhao, S. Xu, D. Yuan, J. Lu, and Y. Zou, “Minimum communication cost consensus in multi-agent systems with Markov chain patterns,” IET Control Theory and Applications, vol. 5, no. 1, pp. 63–68, 2011. View at Publisher · View at Google Scholar · View at Scopus
  6. H. Zhao, W. Ren, D. Yuan, and J. Chen, “Distributed discrete-time coordinated tracking with Markovian switching topologies,” Systems & Control Letters, vol. 61, pp. 766–772, 2012. View at Google Scholar
  7. D. Yuan, S. Xu, H. Zhao, and Y. Chu, “Accelerating distributed average consensus by exploring the information of second-order neighbors,” Physics Letters A, vol. 374, no. 24, pp. 2438–2445, 2010. View at Publisher · View at Google Scholar · View at Scopus
  8. D. Yuan, S. Xu, H. Zhao, and Y. Chu, “Distributed average consensus via gossip algorithm with real-valued and quantized data for 0<q<1,” Systems and Control Letters, vol. 59, no. 9, pp. 536–542, 2010. View at Publisher · View at Google Scholar · View at Scopus
  9. S. Liu, T. Li, and L. Xie, “Distributed consensus for multiagent systems with communication delays and limited data rate,” SIAM Journal on Control and Optimization, vol. 49, no. 6, pp. 2239–2262, 2011. View at Publisher · View at Google Scholar · View at Scopus
  10. P. Lin, Y. Jia, and L. Li, “Distributed robust H consensus control in directed networks of agents with time-delay,” Systems and Control Letters, vol. 57, no. 8, pp. 643–653, 2008. View at Publisher · View at Google Scholar · View at Scopus
  11. Z. Li, Z. Duan, G. Chen, and L. Huang, “Consensus of multi-agent systems and synchronization of complex networks: a unified viewpoint,” IEEE Transactions on Circuits and Systems I, vol. 57, pp. 213–224, 2010. View at Google Scholar
  12. G. Wen, G. Hu, W. Yu, J. Cao, and G. Chen, “Consensus tracking for higher-order multi-agent systems with switching directed topologies and occasionally missing control inputs,” Systems & Control Letters, vol. 62, pp. 1151–1158, 2013. View at Google Scholar
  13. R. W. Beard, J. Lawton, and F. Y. Hadaegh, “A coordination architecture for spacecraft formation control,” IEEE Transactions on Control Systems Technology, vol. 9, no. 6, pp. 777–790, 2001. View at Publisher · View at Google Scholar · View at Scopus
  14. B. Jiang, M. Staroswiecki, and V. Cocquempot, “Cooperative target tracking control of multiple robots,” IEEE Transactions on Industrial Electronics, vol. 59, no. 8, pp. 3232–3240, 2012. View at Google Scholar · View at Scopus
  15. J. A. Fax and R. M. Murray, “Information flow and cooperative control of vehicle formations,” IEEE Transactions on Automatic Control, vol. 49, no. 9, pp. 1465–1476, 2004. View at Publisher · View at Google Scholar · View at Scopus
  16. R. Olfati-Saber and J. S. Shamma, “Consensus filters for sensor networks and distributed sensor fusion,” in Proceedings of the 44th IEEE Conference on Decision and Control, and the European Control Conference (CDC-ECC '05), pp. 6698–6703, Seville, Spain, December 2005. View at Publisher · View at Google Scholar · View at Scopus
  17. H. Su, X. Wang, and Z. Lin, “Flocking of multi-agents with a virtual leader,” IEEE Transactions on Automatic Control, vol. 54, no. 2, pp. 293–307, 2009. View at Publisher · View at Google Scholar · View at Scopus
  18. A. Bidram, A. Davoudi, F. L. Lewis, and Z. Qu, “Secondary control of microgrids based on distributed cooperative control of multi-agent systems,” IET Control Theory and Applications, vol. 7, pp. 822–831, 2013. View at Google Scholar
  19. R. Olfati-Saber, J. A. Fax, and R. M. Murray, “Consensus and cooperation in networked multi-agent systems,” Proceedings of the IEEE, vol. 95, no. 1, pp. 215–233, 2007. View at Publisher · View at Google Scholar · View at Scopus
  20. B. Jiang, M. Staroswiecki, and V. Cocquempot, “Fault accommodation for nonlinear dynamic systems,” IEEE Transactions on Automatic Control, vol. 51, no. 9, pp. 1578–1583, 2006. View at Publisher · View at Google Scholar · View at Scopus
  21. S. X. Ding, Model-Based Fault Diagnosis Techniques: Design Schemes, Springer, Berlin, Germany, 2008.
  22. H. Shen, X. Song, and Z. Wang, “Robust fault tolerant control of uncertain fractional-order systems against actuator faults,” IET Control Theory & Applications, vol. 7, pp. 1233–1241, 2013. View at Google Scholar
  23. F. Pasqualetti, A. Bicchi, and F. Bullo, “Consensus computation in unreliable networks: a system theoretic approach,” IEEE Transactions on Automatic Control, vol. 57, no. 1, pp. 90–104, 2012. View at Publisher · View at Google Scholar · View at Scopus
  24. I. Shames, A. M. H. Teixeira, H. Sandberg, and K. H. Johansson, “Distributed fault detection for interconnected second-order systems,” Automatica, vol. 47, no. 12, pp. 2757–2764, 2011. View at Publisher · View at Google Scholar · View at Scopus
  25. H. Yang, M. Staroswiecki, B. Jiang, and J. Liu, “Fault tolerant cooperative control for a class of nonlinear multi-agent systems,” Systems and Control Letters, vol. 60, no. 4, pp. 271–277, 2011. View at Publisher · View at Google Scholar · View at Scopus
  26. H. LeBlanc, H. Zhang, X. Koutsoukos, and S. Sundaram, “Resilient asymptotic consensus in robust networks,” IEEE Journal on Selected Areas in Communications, vol. 31, pp. 766–781, 2013. View at Google Scholar
  27. H. Shen, S. Xu, J. Lu, and J. Zhou, “Passivity-based control for uncertain stochastic jumping systems with mode-dependent round-trip time delays,” Journal of the Franklin Institute, vol. 349, no. 5, pp. 1665–1680, 2012. View at Publisher · View at Google Scholar · View at Scopus
  28. S. He and F. Liu, “Adaptive observer-based fault estimation for stochastic Markovian jumping systems,” Abstract and Applied Analysis, vol. 2013, Article ID 176419, 11 pages, 2013. View at Publisher · View at Google Scholar
  29. S. He and F. Liu, “Robust stabilization of stochastic Markovian jumping systems via proportional-integral control,” Signal Processing, vol. 91, no. 11, pp. 2478–2486, 2011. View at Publisher · View at Google Scholar · View at Scopus
  30. Z. Wu, P. Shi, H. Su, and J. Chu, “Passivity analysis for discrete-time stochastic markovian jump neural networks with mixed time delays,” IEEE Transactions on Neural Networks, vol. 22, no. 10, pp. 1566–1575, 2011. View at Publisher · View at Google Scholar · View at Scopus
  31. Z. Wu, P. Shi, H. Su, and J. Chu, “Asynchronous l2-l1 filtering for discrete-time stochastic markov jump systems with randomly occurred sensor nonlinearities,” Automatica. In press.
  32. B. Zhang, W. Zheng, and S. Xu, “Filtering of Markovian jump delay systems based on a new performance index,” IEEE Transactions on Circuits and Systems I, vol. 60, pp. 1250–1263, 2013. View at Google Scholar