Mathematical Problems in Engineering

Volume 2013 (2013), Article ID 465671, 7 pages

http://dx.doi.org/10.1155/2013/465671

## 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 [1–12]. This is partly because of their widespread applications in areas such as robots, flocking, unmanned air vehicles, sensor fusion, and microgrids (see [13–18] 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 [20–22]. 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.

#### 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 [27–32]. 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.

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