MALIKA Sader, WANG Fuyong, LIU Zhongxin,*, and CHEN Zengqiang

1.College of Artificial Intelligence, Nankai University, Tianjin 300350, China;

2.Tianjin Key Laboratory of Intelligent Robotics, Nankai University, Tianjin 300350, China

Abstract: In this paper, the distributed fuzzy fault-tolerant tracking consensus problem of leader-follower multi-agent systems(MASs) is studied.The objective system includes actuator faults,mismatched parameter uncertainties, nonlinear functions, and exogenous disturbances under switching communication topologies.To solve this problem, a distributed fuzzy fault-tolerant controller is proposed for each follower by adaptive mechanisms to track the state of the leader.Furthermore, the fuzzy logic system is utilized to approximate the unknown nonlinear dynamics.An error estimator is introduced between the mismatched parameter matrix and the input matrix.Then, a selective adaptive law with relative state information is adopted and applied.When calculating the Lyapunov function’s derivative,the coupling terms related to consensus error and mismatched parameter uncertainties can be eliminated.Finally, a numerical simulation is given to validate the effectiveness of the proposed protocol.

Keywords: distributed fuzzy fault-tolerant control (FTC), tracking consensus problem, leader-follower multi-agent system, mismatched parameter uncertainty.

1.Introduction

In the past few decades, the dynamics of multi-agent systems (MASs) have been investigated in various science fields [1−3].Among them, the study of the consensus in MASs can be traced back long ago.Based on a distributed model proposed by Reynolds in 1987, a simple model is primarily developed by Vicsek in [4] to study the emergence of self-ordered motion in the system.

Subsequently, Jadbabaie et al.[5] studied the consensus of Vicsek’s model.In addition, Fax et al.[6] applied the graph and the matrix theory to solve the consensus problem via a coordinated position controller based on cooperative frameworks.Recently, the consensus problem has become a hot research topic because of its application in some special fields [7−11].

While investigating the consensus of MASs, people find out that the agents in MASs are usually under complex environments with various disturbances, such as channel noises, source noises, and sink noises.For these reasons, the stability of MASs with external disturbances has been studied in recent years [12−14].

In general, due to the existence and unavoidability of complex nonlinear functions in most real MASs [15],more and more consideration should be given to solve the nonlinear function.There are several different ways of tackling the nonlinear function, such as the neural network [16] and the fuzzy logic system (FLS) [17].By the finite-time passivity, in [18], the finite-time synchronization of nonlinear MASs was investigated.In [19], the adaptive fuzzy containment control for nonlinear MASs with input delay was studied.Recently, a distributed consensus control method has been proposed by adaptive mechanisms in [20].The mentioned method is fully distributed.

In addition, in MASs, the parameter uncertainties phenomena could not be ignored.The uncertainties parameters can be roughly categorized into matched parameter uncertainties and mismatched parameter uncertainties.In[21], the containment problem was investigated for a class of MASs in the presence of time-varying uncertainties.In [22], the consensus of fractional-order singular uncertain MASs is studied.The problem of fault diagnosis for uncertain MASs was considered in [23].Note that the above works only consider the matched parameter uncertainties.Recently, there are several papers focused on the mismatched parameter uncertainties.In [24,25], the consensus of MASs with mismatched parameter uncertainties is considered.

In a real application, actuator faults are also considered as a significant problem.Actuator faults are normally caused by improper operation or component aging[26], and they are classified into four types: stuck, outage,bias, and loss of effectiveness (LOE), respectively.These failures have become a research topic due to the severe related degradation of global behavior caused by them[27−29].For this shortcoming, fault-tolerant control(FTC) methods of various types, such as the hierarchical control scheme [30], the decentralized output slidingmode controller [31], and the distributed learning control approach [13], are reported in previous kinds of literature.Besides, the distributed controller has drawn wide attention as a promising method to realize the consensus of MASs with four types of actuator faults.Besides, most works, which focus on the FTC, commonly assume the communication network topology is fixed.

However, this is mostly because while the communication network is switching, the Lyapunov function is invalid.Hence, the switching communication network (SCT)is more complex.In [32], a distributed FTC method is developed for SCT in MASs considering unknown uncertainties and external disturbance.This method removes the common assumption for the fixed communication topologies and compensates for the shortcoming of actuator faults, including the outage and stuck faults.Nevertheless, in real world, the nonlinear function satisfies the Lipschitz condition under some certain conditions.

In this paper, the tracking consensus of MASs with external disturbance, nonlinear function, and mismatched parameter uncertainties subject to actuator faults under switching communication topologies is studied.New ideas and innovative points of this article lie in the followings:

(i) Different from the existing papers dealing with actuator faults, such as [29−32], in this paper, the considered system with mismatched parameter uncertainties,nonlinear function, and actuator faults under switching communication topologies are more general.For the considered system, a new distributed fuzzy fault-tolerant controller is proposed via adaptive mechanisms.

(ii) The proposed controller’s adaptive module is employed to estimate the weight vector norm in FLS rather than in traditional Lipschitz conditions.

(iii) For the case of matched uncertainties, the design of the distributed adaptive protocol is relatively simple.However, for the case of mismatched uncertainties under switching communication topologies, the existing controllers [21−25] cannot solve the issue in this work.Here,a norm error estimation is introduced between the mismatched parameter matrix and the input matrix.On this basis, a selective adaptive law with relative state information is adopted and applied.When calculating the derivative of the Lyapunov function, the coupling terms related to consensus error and mismatched parameter uncertainties can be eliminated.

This article is organized as follows.In Section 2, the preliminaries are introduced.In Section 3, the main results are shown.In Section 4, a simulation example based on the aircraft model is shown.Ultimately, conclusions are summarized in Section 5.

2.Preliminaries and problem statement

2.1 Graph theory

Consider a graph Gσ(t)=(V,Eσ(t),Aσ(t)) in the MAS,which is defined by several parameters, including a set ofNnodes (a nonempty finite) V={v1,···,vN}, a set of edges Eσ(t)=V×V, an associated weighted adjacency matrix Aσ(t)=[aij(t)]∈RN×N, and a switching signal σ(t):[0,+∞)→P, where P is an indices set for total graphs.The (vi,vj)∈Eσ(t)is an edge which indicates the information exchange among agentsiandj.aij(t) is a symbol of the weight of edge (vi,vj), which satisfiesaij(t)>0 if (vi,vj)∈Eσ(t)(i≠j) , otherwiseaij(t)=0.An uniformly bounded non-overlapping time interval [td,td+1)(t0=0,d=1,2,···) exists with an infinite sequence, in which the interaction graph is time invariant in each time interval, meanwhile, a dwell time τd>0 exists which satisfiestd+1−td≥τd.Ni(t) ={j|(vj,vi) ∈Eσ(t)} is the neighbor set of nodei.In one graph, a sequence of connected edges indicates a path.An undirected graph will be obtained, ifaij(t)=aji(t).An undirected graph is considered to be connected when the condition that a path exists in each pair of nodes is met.The Laplacian matrix can be represented as Lσ(t)=[lij(t)] withlij(t)= −aij(t).Denote the leader adjacency matrix asDσ(t)= diag(d1(t),···,dN(t)) withdi(t)>0 if and only if the nodevican access the leader information.

2.2 Problem formulation

Consider a team of nonlinear MAS links withN+1 agents.In this system, the dynamic of the leader is illustrated by

wherex0∈Rn,u0∈Rm, and ΔA(t) indicate the state of leader, input, and mismatched parameter uncertainties, respectively.It is assumed here that (A,B) is stabilizable.For the mismatched parameter uncertainties, according to[24] and [25], it is assumed that ΔA(t)=DN(t) whereDis a known real constant matrix, andN(t) satisfies||N(t)||≤θ*with the unknown constant parameter θ*.

Theith follower is illustrated by

wherei=1,···,N;, andfi(xi)∈Rmrepresent the state, control input, time-varying exogenous disturbance, and nonlinear function, respectively.

Remark 1The system models (1) and (2) indicate the mismatched parameter uncertainties MASs.They are widely considered in early papers to achieve consensus of leader-follower MASs (see [24,25] for details).

Inspired by [25], for agenti, actuator faults in modem(m=1,···,M) are described by

wherek=1,2,···,h.represents the actuator of agenti.is an unknown time-varying efficiency factor.denotes the stuck value.There is≤1.andare constants representing the lower bound and the upper bound ofrespectively.

Remark 2Whenandare different selected values, the fault model (3) describes different actuator faults.Table 1 illustrates the fault model.

Table 1 Actuator fault model

To simplify the presentation, (3) can be rewritten as follows:

where

Remark 3Before starting, the following lemmas and assumptions, more explanation for actuator faults will be given here.Abnormal operations or components aging,commonly appearing in the physical layer, are the main reasons of actuator faults.Thus, an appropriate controller will be proposed to compensate for actuator faults for they are urgent problems to be solved.

Assumption 1The undirected graphsare connected and fixed across each interval [td,td+1).

Remark 4Assumption 1 is an essential condition in MASs under switching communication topologies.

Lemma 1[15] If Assumption 1 holds, then Hp=Lp+Dpis positive definite.

Assumption 2[32] The actuator bias fault is bounded,i.e.,.

Remark 5Assumption 2 indicates that the actuator bias fault is bounded.It is widely used in the robust FTC of MASs, and many practical systems satisfy this assumption (see [32] for more detail).

Assumption 3rank[Bρi]=rank[B].

Remark 6Assumption 3 is a common assumption that can be used to solve the FTC problem of MASs.

Lemma 2[26] If Assumption 3 holds, there existsµi>0 such thatBρiBT≥µiBTB.

Lemma 3[19] Let the domain of continuous functionfi(xi) be a compact set Ω.Then, for any πi(t)>0, there exists a fuzzy logic system such that

Using Lemma 3, it has

where φi(xi)=diagindicates the fuzzy basis function with φiq∈Rl.ϑi=[ϑi1,···,ϑim]Tindicates the unknown parameter with πiq∈Rl(q=1,···,m).Moreover,there is an unknown constant π0>0 such that||πi(t)||≤π0.

Remark 7[30] The basis function φi(·) in Lemma 3 satisfies≤lI.

Define consensus error as δi=xi−x0, then

The relative information from neighborsis written by

wherexi−xjdenotes the relative information form agentjto agenti.

where

The aim of this paper is to find a controller such that all followers asymptotically converge to the state of leader.

3.Main results

3.1 Distributed fuzzy fault-tolerant consensus protocol design

Before starting this subsection, we introduce the following notations for agenti:

wherec0>1/(2λ0) with

andyi=[yi,1,···,yi,h]Tare defined by

An appropriate time-varying distributed controller is designed for agentiby

where

P>0 withthe uniform continuous function is written as χi(t) with+∞.

whereri,1is a positive constant.

timation ofli,2which is adjusted by

whereri,2is a positive constant, ςibelongs to any neighbor ofis determined by

with ζi0being a positive scalar.

Remark 8The norm error estimatoris introduced between the mismatched parameter matrix and the input matrix.On this basis, a selective adaptive law with relative state information is adopted and applied.When calculating the derivative of the Lyapunov function, the coupling terms related to consensus error and mismatched parameter uncertainties can be eliminated.

whereri,3is a positive constant.

whereri,4is a positive constant.

In (13), Υi=diag(Υi,k(Ξk,1), where Υi,k(Ξi,k)=1,ifΞi,k≥0,thenΥi,k(Ξi,k)=0;ifΞi,k<0 with Ξi,k=andare adjusted by

3.2 Stability analysis

In this subsection, first, for convenience, we denote the notations:

then, a theorem is proposed to characterize the sufficient condition for the control objective.

Theorem 1Consider systems (1) and (2) under Assumptions 1−3.Then the controller (8) ensures that the tracking consensus problem is solvable if the topology dwell time satisfies, wherefor allandb0=minp∈P(bp).

ProofNow, a Lyapunov function is considered by

Based on the control objective, the proof process has been divided into Step 1 and Step 2.

Step 1Fort∈[td,td+1), let σ(t)=p,p∈P.Herein,from (14), we have

where

Let I be defined as follows:

For alli∈I, there exist constantsk∈Σi,s∉Σisuch thatFrom (7), it follows that

Substituting (16) into (15) yields

From (17), we have

Step 2Fort∈[td,td+1), we define=maxp∈P{Φp}.From the definition of Hp, there exist>0 and>0 such that

which implies

ForT>0, there exists anh≥0 such thatth

Remark 9For the matched parameter uncertainties,according to the reference [21], ΔA(t)=BN(t), whereBis a known real constant matrix (1),N(t) satisfiesN(t)TN(t)≤I.Theui,2is designed as follows:

Remark 10For the case ΔA(t)=0, theui,2withis removed from the controller (8).

4.Numerical example

In this section, an example based on a reduced-order aircraft model is presented.Consider leader-follower MASs(1) and (2) under the switching communication topologies with four followers labeled as 1, 2, 3, 4 and one leader labeled as 0.The communication switching topologies of the system is shown in Fig.1.The topology switching signal of the example is given in Fig.2.The matrices of systemA,B,N(t) , andDare selected as follows：

Fig.1 Network topologies with four followers

Fig.2 Switching signal of communication graphs

The nonlinear function is described asgi=[−0.13xi2xi3,0,0,0,0]T(i=1,2,3,4).

The disturbance iswi=[1sint]T.The input is

wherem=3,4,si=[1,sint]T,

The initial values and control parameters of this example are shown below:

Moreover, in this example, for agent 2, the first actuator ha outage fault.For agent 3, the second actuator has bias fault, which is described by ψ3,2=5+0.1sint,t≥30s.For agent 1, the third actuator has stuck fault that ψ1,3=4+0.2sint,t≥48s.For agent 4, the second actuator has LOE faults.

The system in [33] fails to consider switching communication topologies, actuator faults, and mismatched parameter uncertainties.The state trajectories by using method (8) in this paper are shown in Fig.3, Fig.5, and Fig.7.The state trajectories by using the method in [33] are shown in Fig.4, Fig.6, and Fig.8.From these figures, it is easy to see that in the presence of the above actuator faults and mismatched parameter uncertainties under the switching communication topologies, the agents converge to zero by using the developed controller (8), while they are divergent by using the controller in [33].The estimations of adaptive parameters are given in Fig.9−Fig.13.Furthermore, these figures are provided to demonstrate the validity and applicability of the proposed control scheme.

Fig.3 Response curves of xi1(t)(i=0,···,4) using controller (8)

Fig.4 Response curves of xi1(t)(i=0,···,4) using controller in [33]

Fig.5 Response curves of xi2(t)(i=0,···,4) using controller (8)

Fig.6 Response curves of xi2(t)(i=0,···,4) using controller in [33]

Fig.7 Response curves of xi3(t)(i=0,···,4) using controller (8)

Fig.8 Response curves of xi3(t)(i=0,···,4) using controller in [33]

Fig.9 Response curves of (i=1,···,4)

Fig.10 Response curves of (i=1,···,4)

Fig.11 Response curves of (i=1,···,4)

Fig.12 Response curves of (i=1,···,4)

Fig.13 Response curves of and(i=1,···,4)

5.Conclusions

In this paper, the tracking consensus problem for a class of nonlinear leader-follower MAS with external disturbance, mismatched parameter uncertainties, and actuator faults under switching communication topologies is studied.A new distributed fuzzy FTC is designed under the case of switching communication topologies and actuator faults.The effectiveness of the developed approach is shown by a simulation example.

It is worth pointing out that this paper only considers the distributed FTC problem under the undirected network topology case.Therefore, the extension of the current result to a more general switching directed topology is a challenging task, and this problem will be further investigated in the future work.Besides, it is challenging to achieve the consensus of MASs by the event-triggered scheme.Even though the convergence does not affect the established result in theorem, it is still an interesting future work for us.