Mathematical Problems in Engineering

Volume 2015 (2015), Article ID 436935, 7 pages

http://dx.doi.org/10.1155/2015/436935

## Robust Fault Diagnosis Design for Linear Multiagent Systems with Incipient Faults

^{1}College of Automation Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China^{2}College of Aerospace Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China

Received 30 July 2014; Accepted 24 September 2014

Academic Editor: Peng Shi

Copyright © 2015 Jingping Xia et al. 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

The design of a robust fault estimation observer is studied for linear multiagent systems subject to incipient faults. By considering the fact that incipient faults are in low-frequency domain, the fault estimation of such faults is proposed for discrete-time multiagent systems based on finite-frequency technique. Moreover, using the decomposition design, an equivalent conclusion is given. Simulation results of a numerical example are presented to demonstrate the effectiveness of the proposed techniques.

#### 1. Introduction

In the past two decades, the study of multiagent systems has been a very hot topic and attracted considerable attention [1–7]. Multiagent systems appear in various fields, such as cooperative control of unmanned air vehicles and satellite formation flying. However, fault diagnosis is an important problem in automatic control systems and has been an active research area during the past three decades. Fruitful results related to this topic can be found in several excellent works [8–10] and references therein. However, there are very few results about fault diagnosis of multiagents.

In general, according to the varying rate of faults, the faults can be classified into abrupt and incipient in practical systems. Abrupt faults denote system parameter changes faster than the fault-free cases, whose property is able to detect this type of fault quickly to avoid serious consequences, while incipient faults are represented by drift-type changes and are more important in slowly developing problems. Compared with abrupt faults, early fault detection is the key objective for the class of incipient faults. Therefore, the development of effective fault diagnosis schemes for incipient faults is very important. In [11, 12], an adaptive online approximation method was proposed to detect incipient faults. And a system decomposition based method was proposed in [13, 14] to achieve incipient fault diagnosis, while, in this paper, we consider the problem of fault diagnosis from the perspective of frequency property of incipient faults.

As we know the performance is the maximum singular value of the transfer function of the studied system. However, if the possible occurring faults are classified into a finite-frequency range, the nominal design for the entire range will bring much conservatism. For example, incipient faults belong to low-frequency range. Reference [15] not only considered properties in different frequency domains, but also provided exact linear matrix inequalities (LMIs) characterization in finite-frequency ranges, which can be viewed a useful tool to handle incipient fault diagnosis. Therefore, the issue of incipient fault estimation of discrete-time multiagent systems is a meaningful research and motivates our study.

In this paper, on the basis of existing results [16–18], main contributions of this paper are as follows. According to the frequency range of incipient faults, a fault estimation observer is proposed to achieve incipient fault estimation for multiagent systems using the finite-frequency design. Moreover, an equivalent conclusion of the obtained results is derived by the decomposition technique.

The rest of this paper is organized as follows. Preliminaries and the system description of multiagents have been presented in Section 2. In Section 3, the fault estimation observer of multiagents with incipient faults is presented. Simulation results are presented in Section 4 to show the effectiveness of the proposed approach, followed by some conclusions in Section 5.

#### 2. Preliminaries and Problem Statement

##### 2.1. Graph Theory

Consider a directed graph with a nonempty finite set of nodes , a set of edges or arcs , and the associated adjacency matrix . In this note, the graph is assumed to be time-invariant; that is, is constant. An edge rooted at node and ended at node is denoted by , which means information can flow from node to node . is the weight of edge and if ; otherwise . We assume there are no repeated edges and no self-loops, that is, , with . Node is called a neighbor of node if . The set of neighbors of node is denoted by . Define the in-degree matrix as with and the Laplacian matrix as . In this paper, we consider the case of undirected graph.

##### 2.2. Problem Statement

Consider the following multiagent systems with nodes and a communication graph : where is the state vector, is the input vector, is the output vector, is the disturbance and noise which belong to , and represents the actuator fault. , , , , , and are constant real matrices of appropriate dimensions. It is supposed that matrices and are of full rank. It is supposed that the pair is observable.

The dynamics of the leader or control node, labeled 0, is given by where is the state vector and is the output vector. It can be considered as a command generator, which generates the desired target trajectory. The leader node can be observed from a small subset of nodes in graph . If node observes the leader, an edge is said to exist with weighting gain . We refer to node with as a pinned or controlled node. Denote the pinning matrix as .

Lemma 1 (see [15]). *For the following linear discrete-time system**its transfer function can be written as . Given a symmetric matrix , the following two statements are equivalent.*(i)*The finite-frequency inequality *(ii)*There exist Hermitian matrices and (where ), and wherefor low-frequency range , and for high-frequency range .*

*3. Main Results*

*For the dynamics (1), the fault estimation observer of the th node is constructed:where is the observer state, is the observer output, is an estimate of , and , are observer gain matrices of node . is the neighborhood output estimation error of the th node, that is, *

*Remark 2. *Here, it is assumed that the state of leader mode is measured or available, so the observed output obtained from the observer is equal to the measured output, that is, .

For the th node, let one gets where .

*Based on (11), the following augmented system can be obtained: *

*Furthermore, denote new vectors and matrices: then for the th node, it follows that*

*Denote the global variable then the global error dynamics can be expressed aswhere denotes Kronecker product [19].*

*Theorem 3. Given scalars . The global error dynamics (16) satisfy the performances and if there exist symmetric positive definite matrices , , symmetric matrices , , and matrices , such that the following conditions hold:where and ; then the fault estimation observer gain matrix is given by .*

*Remark 4. *In general, the noises are in high-frequency domain and incipient faults are in low-frequency one. So according to [17] and Lemma 1, the proof of Theorem 3 can be deduced by choosing symmetric positive definite matrices , , symmetric matrices , , and a matrix , . For brevity, this proof is not provided here.

*Remark 5. *From the process of fault estimation observer design, we can see that there is no parameter value of leader node 0 because of the assumption that the state of leader node is measurable, as shown in Remark 2. Since undirected graph is considered in this paper, matrix is symmetric such that one of the eigenvalues of is zero, and the others are positive. Therefore, the presented leader node can be viewed as a virtual node, whose function is to make matrix nonsingular.

*Remark 6. *From [20], we can get the matrix is nonsingular. Furthermore, based on the decomposition design in [21], an equivalent conclusion of Theorem 3 is obtained, that is, Theorem 7.

*Theorem 7. Given scalars . The global error dynamics (16) satisfy the performances and if there exist symmetric positive definite matrices , , symmetric matrices , , and matrices , such that the following conditions hold:where () are the eigenvalues of and the fault estimation observer gain matrix is given by .*

*Proof. *Since matrix is symmetric positive definite, there always exists a nonsingular matrix such that and , where and are positive. Under the coordinate transformation And let then it gets Since is a diagonal matrix, the transfer function where is orthogonal. It follows that Moreover, by choosing the same matrix variables , , , , , shown in Theorem 3, one gets the two conditions of Theorems 3 and 7 are equivalent.

*Remark 8. *Theorem 7 is an equivalent conclusion of Theorem 3 by using decomposition technique. And note that the eigenvalues of matrix are positive; that is, , .

*4. Simulation Results*

*In this section, the following example is presented to illustrate the effectiveness of the proposed method. It is assumed that there are four agents of multiagent systems, that is, , and each node is The disturbance matrices are and . Here, actuator faults are considered, that is, the fault distribution matrix .*

*For these multiagent systems subject to four nodes, we assume that each node only contacts with its nearest neighbors, as shown in Figure 1. From Figure 1, one gets Laplacian matrix And only the first node links the leader node, so we have It is readily verified that is nonsingular and its eigenvalues are positive, that is, 0.1864, 2.0000, 2.4707, and 4.3429.*