Mathematical Problems in Engineering

Volume 2015, Article ID 783275, 8 pages

http://dx.doi.org/10.1155/2015/783275

## Extremum Seeking Based Fault-Tolerant Cooperative Control for Multiagent Systems

School of Information Science and Engineering, Central South University, Changsha 410075, China

Received 18 July 2014; Revised 17 October 2014; Accepted 27 October 2014

Academic Editor: Fanglai Zhu

Copyright © 2015 Fu Jiang 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

We propose a novel fault-tolerant cooperative control strategy for multiagent systems. A set of unknown input observers for each agent are constructed for fault detection. Then a real-time adaptive extremum seeking algorithm is utilized for adaptive approximation of fault parameter. We prove that the consensus can be still reached by regulating the interconnection weights and changing the connection topology of the fault agent. A numerical simulation example is given to illustrate the feasibility and effectiveness of the proposed method.

#### 1. Introduction

Recent years have seen a growing interest in the cooperative control of multiagent systems [1, 2]. Cooperative multiagent system refers to the concept that multiple agents work together to complete a task or achieve a target state according to the cooperative control law [3]. With the rapid development of embedded systems, complex algorithms can be effectively implemented in multiagent systems.

In multiagent systems, a fault occurring in any agent may have an impact on other agents, which is different from the traditional faults occurring in isolated systems [4]. Moreover, when faults occur to agents, the topology of the multiagent system may change [5]. Therefore, fault detection should perform fast detecting to avoid affecting other agents, and fault-tolerant control should make the system has endurance to the failures while keeping the topology structure.

Fault detection of multiagent systems has to be completed before fault tolerance. In the last decade, scholars proposed different methods for fault detection, such as observer-based methods [6, 7], parity equation [8], and the identification-based method [9]. Shames et al. derived sufficient conditions for the existence of unknown input observers for second-order linear time invariant systems [6], which constituted the basis of the current study. Then, they extended these conditions to imprecise models [7]. For parity space method, residual errors are obtained by collecting system input and output. Chan et al. developed a parity space-based estimator, which is sensitive to specific faults [8]. The literature [9] overviewed the problem of identifying. The identification-based method means that residuals for output variables are generated with adaptive nonparametric or parametric models. However, all these methods need a great amount of computations and long computing time when the system has large numbers of agents, which are not acceptable for practice.

Once a faulty agent is detected, fault-tolerant control is taken for handling faults. In this paper, fault-tolerant control is divided into two steps, namely, fault parameter approximation and adjusting some interconnection weights. Adaptive fault parameter approximation is developed on the basis of parameter estimation. Generally, fault parameters are estimated using the nonlinear neural network [10, 11]. We transform the fault parameter approximation problem into the optimization problem by using extremum seeking. Compared with the classical neural network method, the advantages of the proposed approach are that the approximation is real-time and online without any offline training.

Moreover, the extremum seeking based parameter approximation is significantly simplified. The design process of extremum seeking does not call for the understanding of the input and output characteristics of the system [12, 13].

In step two, the faulty system is recovered by adjusting some weights of the cooperative protocol. There have been several studies in recovering faulty multiagent systems. Semsar-Kazerooni and Khorasani [14] and Azizi and Khorasani [15] used fault-tolerant control algorithms to recover an actuator fault detected by FDI. Furthermore, Azizi and Khorasani put forward a two-level architecture which contains partial recovery and cooperative recovery [15]. Yang et al. proposed a cooperative protocol to adjust fault parameters for a target aggregation problem of nonlinear multiagent systems [5]. However, in these studies, the cooperative fault-tolerant control was used to adjust interconnection weights without isolating out the faulty agent, which leads to a lot of calculation when faulty agent has a number of neighbors.

The rest of the paper is organized as follows. Section 2 provides some preliminary knowledge and formulates the problems. Section 3 focuses on fault detection. In Section 4, an adaptive fault parameter approximation algorithm using extremum seeking is proposed. In Section 5, the cooperative fault-tolerant control of multiagent systems is discussed. In Section 6, an example of a multiagent team is given to demonstrate the effectiveness of the proposed scheme. In Section 7, conclusion is drawn.

#### 2. Preliminary Knowledge

Agents and their link topology are mapped based on the graph theory [16]. We consider a system constituted by agents; is an undirected graph with vertex set and edge set , where represents agent . The edge denotes a connection between agent and agent , and is the weight of the interconnection. The set represents all the neighboring agents that are interconnected with . Agent is supposed to have a double-integrator dynamics:where and are the position and velocity of agent and is the controlled input based on the following formula:Formula (2) achieves the position and velocity consensus. The term characterizes the time jump function of an actuator fault, denotes faulty time of agent ,* if *, ,* else *. The variable is the fault parameter of agent . The system dynamics in the presence of a fault are written as follows:where , , , , , and . We designed ( means an identity matrix with the dimension ) to observe all the states of the multiagent system. The following text gave some details of for the velocity consensus and position consensus problems:where ; is the Laplacian matrix of the graph, where ,* if* (, (otherwise, ), and . By this definition, every row sum of the matrix is zero, so the Laplacian matrix always has a zero eigenvalue, right eigenvector , and .

#### 3. Fault Detection and Isolation

In this paper, a set of state observers are constructed for the second-order system using the unknown input observer (UIO) method. UIO refers to a robust fault diagnosis scheme for multiagent systems. Once an actuator failure is detected, residual errors are used to locate the faulty agent. In order to reduce the amount of calculation, only neighbors are observed for each agent. We rewrite (3) aswhere is with the th column deleted, is the th column of , is with the th component deleted, and is the th component of . Suppose graph is interconnected and the topology of the system is fixed.

A full-order observer for system (5) is described byChoosing the matrixes , , , and satisfies the following conditions:

Then, there exists a UIO [7] for agent as follows:where and are the estimated state and the observer’s state for agent and , , , and are unknown matrices of appropriate dimension, which must be designed such that will asymptotically converge to . The unknown input observer is constructed to achieve the decoupling from , by designing matrixes , , , and . Matrix is a stability matrix; that is, it has all its eigenvalues in the left-hand side of the complex plane.

Thus, we can obtain the observer error and residual dynamics aswhere is the observer error and is the corresponding residual, which is a fault indicator function that satisfies

The detection and isolation condition for fault are given as follows:where and are isolation thresholds. If the above condition is satisfied, we can conclude that there is a fault affecting the system’s th component.

The proposed approach in this section is feasible if a single additive fault exists. In order to isolate multiple faults, one can repeat the abovementioned fault detection procedure for each of the potential fault combinations. We can derive similar observers for all faults and then adopt the detection and isolation condition to isolate each of them.

#### 4. Extremum Seeking for Approximating Fault Parameters

##### 4.1. Single Faulty Agent Case

The fault detection scheme makes use of observers called unknown input observer, as described in the previous section. Then residuals and their thresholds are designed to generate false alarms, which is used for fast network fault location. Under the assumption of only one faulty agent in the network (suppose the th agent is faulty), the proposed extremum seeking framework is shown in Figure 1.