# Fault Diagnosis and Fault-Tolerant Control for Leader-Follower Multi-Agent Systems with Time Delay

Authors: Chang Zhang; Yawei Wu; Lina Yao
Journal: Journal of Control Science and Engineering
 (2022)

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## Abstract

Under the guidance of the directed graph, this paper studies the fault diagnosis (FD) and fault-tolerant control (FTC) of the leader-follower multi-agent systems (MASs) with time delay. Actuator faults and disturbances are considered in this study, which make the work of this paper more challenging. Firstly, a robust distributed observer on the basis of the relative estimation error is put forward to obtain the fault estimation information. Then, after getting the fault information obtained by the observer, a dynamic compensation FTC protocol on the basis of the relative output tracking error is developed to make the output of followers can track the leader's output. Finally, the proposed algorithm is verified by an example and satisfactory results are obtained.


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## Body

## 1. Introduction

In recent years, with the large-scale application of MAS such as UAV (unmanned aerial vehicle) formation flying, multi-unmanned vehicles, and so on, the control of MAS has gradually become a research hotspot. For instance, in [1], a pinning control strategy is proposed for multi-agent systems with exogenous disturbances. In [2], heterogeneous multi-agent system with input constraints composed by second-order linear and nonlinear agents is discussed. A point-to-point tracking control based on iterative learning is proposed in [3] for a class of nonlinear multi-agent systems. A new collaborative output regulation control method is proposed in [4] for the problem of resilient practical cooperative output regulation of heterogeneous multi-agent systems subject to denial-of-service (DoS) attacks. The problem of event-triggered consistent control of nonlinear uncertain systems in the presence of unknown parameters and external disturbances is studied in [5]. Also, there are many other research results on multi-agent control methods such as [6–10]. Compared with a single agent, MAS can obviously accomplish some more challenging tasks, but at the same time, the topology of the multi-agent system becomes more complex. With the topology of MASs becoming more complex, how to ensure the safety and reliability of MASs has also attracted much attention. In order to improve the reliability of MAS, it is necessary to study FD and FTC of MAS.In the design of FTC algorithm, the prior knowledge of fault estimation information is usually required. So, FD is essential to obtain the information. Nowadays, the problem of FD in MAS has received widespread attention. Adaptive observer, sliding mode observer (SMO), and unknown input observer (UIO) are commonly used in FD of MAS. In [11], for a linear MAS which has actuator faults, an adaptive distributed fault estimation observer on the basis of the relative output estimation error is proposed. In literature [12], a robust distributed fault estimation algorithm on the basis of SMO is developed for linear MAS. In [13], an UIO is constructed for linear MAS with actuator fault only according to the relative output information. In addition, some other algorithms have been presented in [14–20] for the problem of fault diagnosis in multi-agent systems.It is worth noting that most of the aforementioned studies do not consider the occurrence of time delay in MAS. In practical systems, communication delay will inevitably occur, so it is of great practical importance to consider the possible time delay when studying multi-agent systems. FD and FTC problems for stochastic distribution control systems with time delay have been studied in [21, 22], but it does not take into account the perturbation of the system. The FD and FTC method for systems with fast time-varying delay was proposed in the literature [23]. But it is based on the classical observer error to construct the fault diagnosis algorithm. The FD and FTC problems for nonlinear systems with time delay and linear time-delaying systems containing different parameters have been studied in the literature [24, 25]. In [26], a distributed fault estimation algorithm on the basis of relative output estimation error is developed for interconnected SDC system with time delay. However, most of the aforementioned studies dealing with the time delay problem are for single systems. Few studies have been conducted for fault diagnosis of multi-agent systems with time delay, which is one of the motivations for the research in this paper.After obtaining the fault estimation information, the FTC strategy can be designed based on the obtained information so that the output of the follower system tracks that of the leader’s system even after a failure occurs. Fault-tolerant control of multi-agent systems has attracted a lot of attention from researchers in recent years in order to reduce the damage caused by the system fault. An adaptive backstepping sliding mode FTC strategy is studied for a second-order MAS with actuator fault in [27]. The problem of consensus tracking for MAS with actuator fault types of interruption and partial fault is studied in [28]. But time delay is not considered in this literature. In literature [29], a distributed adaptive FTC protocol on the basis of the network local information is introduced for a kind of MAS, and it also does not contain time delay of the system. In addition, a fault-tolerant feedback controller according to the relative output of neighboring agents is often used to control MAS [16, 30–33].In this research, a FD and FTC strategy is devised for MAS with time delay. The major contributions of this research cover the following. (1) Time delay and disturbance are considered in MAS, which increases the complexity of FD and FTC. (2) A novel FD observer according to the relative estimation error is developed and the parameters may be gained by addressing linear matrix inequality (LMI). (3) A dynamic compensation FTC protocol on the basis of the relative output tracking error is studied, and the corresponding parameters can be gained according to LMI.The structure of this paper is arranged as follows. In Section2, the system model of leader-follower MAS is introduced and some necessary assumptions are established. In Section 3, this paper proposes a robust observer on the basis of relative estimate error to get the fault estimation information. In Section 4, a dynamic compensation controller according to the relative output tracking error is designed to control the leader-follower MAS. In Section 5, a MAS with five agents is taken as an example under a directed graph to prove the validity of the introduced scheme. At last, the conclusions are drawn in Section 6.Notation. The matrix of n×n is expressed as Rn×n. The inverse and transpose of matrix A are expressed as A−1,AT, respectively. A+ is the pseudo-inverse of matrix A. The n×n identity matrix can be expressed as In. Kronecker product is represented by the symbol ⊗. ⋅∞ and ⋅ represent the infinite norm and 2-norm of the vector or matrix, respectively.The Basics of Algebraic Graph Theory. In algebraic graph theory, each agent can be considered as a node in the graph. The information exchange of agents can be represented by G=V,E,S, where V=V1,V2,…,Vn represents the set containing all the nodes in the graph, E⊂V×V represents the set containing all the edges, and Eji=Vj,Vi represents that agent i can transmit the information to agent j. The adjacency matrix of algebraic graph theory can be expressed as S=Sij∈Rn×n, where sii=0 and if Eij∈E holds, sij=1; otherwise, sij=0. Furthermore, for a directed graph, the in degree and out degree of node i are defined as ∑j=1Naij∑j=1Naji, respectively. In this study, undirected graphs are considered, where ∑j=1Naij∑j=1Naji. The Laplace matrix of graph G can be expressed as L=Lij∈Rn×n=D−A, where D≜diag∑j∈V1a1j∑j∈V2a2j,…,∑j∈Vnanj. The matrix G=diaggi∈RN×N can represent the communication between the leader system and the follower system, where the i-th agent can gain the information of the leader, gi=1, or else, gi=0.


## 2. Model Description

Considering the MAS withN followers and 1 leader, the state equation of the i-th follower system is shown as follows:(1)x˙it=Axit+Adxit−d+Buit+Efit+Fdit,yit=Cxiti=1,…,N,where xi∈Rn denotes the state of the follower system, yi∈Rm denotes the output, d represents a constant delay, ui∈Rp is the control input of the system, fit∈Rs stands for actuator fault and it is bounded, and dit∈Rl represents the external disturbance or system uncertainty. In this article, A, AdB, E, F, C are constant real matrices of appropriate dimensions. It is supposed that matrices C and E are of full rank and the pair (A, C) is observable.The leader’s system state space equation is shown below:(2)x˙lt=Axlt+Adxlt−d+Brt,ylt=Cxlt,where xl∈Rn represents the state vector of the leader system, yl∈Rm is the output vector, and r∈Rp is the known given reference input trajectory.Assumption 1.
It is assumed that system parametersB,E satisfy the following condition: rankB,E=rankB.
It is worth noting that when Assumption1 is satisfied, there is a matrix B∗ that satisfies the following condition:(3)I−BB∗E=0.


## 3. Fault Diagnosis

FD is required to get an estimation of the fault. Design the following robust observer as(4)x^˙i=Ax^i+Adx^it−d+Buit+Ef^i,y^i=Cx^ii=1,⋯N,where x^i∈Rn denotes the state estimation vector, y^i∈Rp denotes the output estimation vector, and f^it indicates the estimation vector of the fault, which can be obtained by the dynamic compensator gx.Different from the traditional residual form, relative estimation errors are often used in MAS, and it can be shown as follows:(5)εi=∑j∈Niaijyit−y^i−yjt−y^jt+giyit−y^it.The dynamic compensatorgx can be indicated as(6)x˙kit=Akxkit+Bkεit,f^it=Ckxkit+Dkεit,where xki∈Rni is the state vector of the dynamic compensator.Denote(7)exi=xi−x^i,efi=fi−f^i.Then, the observation error system is formulated as follows:(8)e˙xit=Aexit+Adexit−d+Efit−f^it+Fdit.Denotezi=exiTefiTT,wi=fiTdiTT, and the following formula can be further gained as(9)e˙xit=Aexit+Adexit−d+C1wi−Ef^i,zi=A1exi+C2wi−D1f^i,where(10)C1=EF,A1=I0,C2=00I0,D1=0I.Denote(11)ex=ex1Tex2T⋯exNTT,u=u1Tu2T⋯uNTT,d=d1Td2T⋯dNTT,ε=ε1Tε2T⋯εNTT,f=f1Tf2T⋯fNTT,f^=f^1Tf^2T⋯f^NTT,xk=xk1Txk2T⋯xkNTT,w=wk1Twk2T⋯wkNTT.Combining (5) and (9), it can be deduced that(12)e˙xt=IN⊗Aext+IN⊗Adext−d+IN⊗C1wt−IN⊗Ef^t,zt=IN⊗A1ext+IN⊗C2wt−IN⊗D1f^t,εt=L+G⊗Cext.The global system of the dynamic compensatorgx can be expressed as follows:(13)x˙gt=IN⊗Agxgt+IN⊗Bgεt,f^t=IN⊗Cgxgt+IN⊗Dgεt.Denoteζt=exTtxgTtT, and combine equations (12) and (13), and it can be gained that(14)ζ˙t=A¯ζt+Ad¯ξt−d+B¯wt,z=Hξt+C¯wt,where(15)A¯=IN⊗A−L+G⊗EDgC−IN⊗ECgL+G⊗BgCIN⊗Ag,Ad¯=IN⊗Ad000,B¯=IN⊗C10,H=IN⊗A1−L+G⊗D1DgC−IN⊗D1Cg,C¯=IN⊗C2.It can be further found that(16)A¯=A1¯+L¯K¯C1¯,H=H1+D¯K¯C1¯,where(17)A1¯=I⊗A000,L¯=−I⊗E00I⊗I,K¯=I⊗DgI⊗CgI⊗BgI⊗Ag,C1¯=L+G⊗C00IN⊗I,D¯=−I⊗D10,H1=I⊗A10.Lemma 1.
Given a symmetric matrixM=M11M12∗M22, where M11 and M22 are symmetric matrices, the following inequality condition can be obtained: 1M<0,2M22<0,M11−M12M22−1M12T<0,3M11<0,M22−M12TM11−1M12<0.Lemma 2.
For the parameterβ>0, if there are positive definite symmetric matrices (PDSM) P,Q meeting the LMI(18)Ψ=ATP+PA+QPAdPBCT∗−Q00∗∗−βIDT∗∗∗−βI<0,then the following linear system:(19)x˙t=Axt+Adxt−d+Bvt,yt=Cxt+Dvt,should meet the H∞ performance index yt2≤β2vt2.Proof.
Select the Lyapunov function shown below:(20)Vt=xTtPxt+∫t−dtxTtQxt.
It can be further obtained that(21)V˙t=xTtPA+ATP+Qxt++2xTtPAdxt−d+2xTtPBvt−xTt−dQxt−d.
When linear system (19) meets the H∞ performance index J=1/βyTtyt−βvTtvt<0, the inequality Ξ=V˙+J<0 holds. It can be found that(22)Ξ=xtxt−dvtTPA+ATP+QPAdPB∗−Q0∗∗−βI+1βCT0DTC0Dxtxt−dvt<0.
Combined with Lemma1, it can be obtained that Ξ<0 is equivalent to Ψ<0. The proof is completed.Theorem 1.
For scalarsδ1>0,δ2>0, if there is a PDSM P¯=IN⊗P1100IN⊗P22 and matrix Y=L+G⊗Y11IN⊗Y12L+G⊗Y21IN⊗Y22 satisfying the following LMI:(23)π=π11π12I⊗Ad0I⊗C1π16I⊗P110∗π22000π260I⊗P22∗∗−δ2I00000∗∗∗−δ2I0000∗∗∗∗−δ1II⊗C2T00∗∗∗∗∗−δ1I00∗∗∗∗∗∗−δ2−1I0∗∗∗∗∗∗∗−δ2−1I<0,where Y=K¯C1¯P¯, π11=IN⊗P11A+IN⊗ATP11−L+G⊗EY11−L+GT⊗Y11TET, π12=−IN⊗EY12−L+GT⊗Y21T, π22=IN⊗Y22+IN⊗Y22T, π16=IN⊗P11A1T−L+GT⊗Y11TD1T, π26=−IN⊗Y12TD1Tthen system (14) is stable and satisfies the H∞ performance index zt2≤δ12vt2.Proof.
From Lemma2, if the H∞ performance index zt2≤δ12vt2 is satisfied in system (14), then there are P,Q to make the following inequality hold:(24)A¯TP+PA¯+QPA¯dPB¯HT∗−Q00∗∗−δ1IC¯T∗∗∗−δ1I<0.
By combining equation (16), it can be further found that(25)A¯1TP+PA¯1+PL¯K¯C¯1+C¯1TK¯TA¯TPT+QPA¯dPB¯H1T+C¯1TK¯TD¯T∗−Q00∗∗−δ1IC¯T∗∗∗−δ1I<0.
Multiplying the left side and the right side of inequality (25) by diagP−1III, it can be obtained that(26)ΦA¯dB¯P−1C¯1TK¯TD¯T+P−1H1T∗−Q00∗∗−δ1IC¯T∗∗∗−δ1I<0,where Φ=P−1A¯1T+A¯1P−1+L¯K¯C¯1P−1+P−1C1T¯K¯TL¯T+P−1QP−1T.
Combining Lemma1 and defining Y=K¯C¯1P−1,P−1=P¯, it can be found that(27)P¯A¯1T+A¯1P¯+L¯Y¯+YL¯TA¯dB¯YTD¯T+XH1TP¯∗−Q000∗∗−δ1IC¯T0∗∗∗−δ1I0∗∗∗∗−Q−1<0.
For convenience, chooseQ=δ2I and bring A¯1,L¯,A¯d,D¯,H1,C¯,P¯,Y into the inequality (27), and it can found that π<0. The proof is completed. After P¯,Y is gained by solving LMI π<0, the parameter matrix K¯ of the dynamic compensator gx can be obtained by K¯=YC¯1P¯+.


## 4. Fault-Tolerant Control

After getting the fault estimation information, the FTC strategy based on the relative output estimation is devised to make the output of the follower system still track the leader’s output after a failure occurs.The state tracking error between thei-th follower and leader is indicated as follows:(28)eit=xit−x0t.Then, it can be further found that(29)e˙it=Aeit+Adeit−d+Buit+Efit+Fdit−Brt.Denote(30)et=e1Tte2Tt⋯eNTtT.Then, the global system may be indicated that(31)e˙it=IN⊗Aet+IN⊗Adet−d+IN⊗But+IN⊗Eft−IN⊗Brt+IN⊗Fdt.Define the relative output tracking error of thei-th agent as shown below:(32)μit=∑j∈NIaijyit−yjt+giyit−y0t=L+G⊗Cet.A dynamic compensation FTC strategy according to the relative output tracking error is developed as follows:(33)ψ˙it=Ak1ψit+Bk1μit,uit=Ck1ψit+Dk1μit−B∗Ef^i+r,where ψit∈Rq is the state vector of the dynamic compensation fault-tolerant controller.Denote(34)ψt=ψ1Ttψ2Tt⋯ψNTtT,μt=μ1Ttμ2Tt⋯μNTtT.Then, the global system of the dynamic compensation FTC strategy (33) can be indicated as follows:(35)ψ˙t=IN⊗Ak1ψt+IN⊗Bk1μt,ut=IN⊗Ck1ψt+IN⊗Dk1μt−IN⊗B∗Ef^+IN⊗Ir.DefineSt=eTtψTtT, d¯=efTdTT and substitute equations (35) and (32) into (31), and it can be obtained that(36)S˙t=AcSt+AdcSt−d+E¯d¯,et=CcSt,where(37)Ac=IN⊗A+L+G⊗BDk1CI⊗BCk1L+G⊗Bk1CI⊗Ak1,Adc=I⊗Ad000,E¯=I⊗EI⊗F00,Cc=I0.Then, it can be further found that(38)Ac=Ac1+Lc1K1¯Cc¯,where(39)Ac1=IN⊗A000,Lc1=IN⊗B00IN⊗I,Cc¯=L+G⊗C00IN⊗I,K1¯=IN⊗Dk1IN⊗Ck1IN⊗Bk1IN⊗Ak1.Theorem 2.
For scalarsγ1>0,γ2>0, if there is a PDSM P1=IN⊗P111IN⊗P112IN⊗P112TIN⊗P122 and a matrix Yc=IN⊗Yc11IN⊗Yc12IN⊗Yc21IN⊗Yc22 satisfying the following LMI:(40)Π=Π11Π12I⊗P111Ad0I⊗P111EI⊗P111FI00∗Π22I⊗P112TAd0I⊗P112TEI⊗P112TF0I0∗∗−γ2I000000∗∗∗−γ2I00000∗∗∗∗−γ1I0000∗∗∗∗∗−γ1I000∗∗∗∗∗∗−γ1I00∗∗∗∗∗∗∗−γ1I0∗∗∗∗∗∗∗∗−γ1I<0,where Yc=P1Lc1K1¯, Π11=I⊗ATP111+I⊗P111A+L+G⊗Yc11C+L+GT⊗CTYc11T+γ2I, Π12=I⊗ATP112+I⊗Yc12+L+GT⊗CTYc21T, Π22=I⊗Yc22+I⊗Yc22T+γ2I then system (36) is stable and meets H∞ performance index et2≤γ12d¯2.Proof.
It is known from Lemma2 that when system (36) meets the H∞ performance index et2≤γ12d¯2, there are positive definite matrices P1,Q1 satisfying the following inequality:(41)AcTP1+P1Ac+Q1P1AdcPE¯Cc∗−Q100∗∗−γ1I0∗∗∗−γ1I<0.
Substituting equation (38) into (41), it can be found that(42)Ac1TP1+P1Ac1+P1Lc1K1¯Cc¯+CcT¯K1T¯Lc1TP1+Q1P1AdcPE¯Cc∗−Q100∗∗−γ1I0∗∗∗−γ1I<0.
DefineYc=P1Lc1K1¯, and it can be further obtained that(43)Ac1TP1+P1Ac1+YcCc¯+CcT¯Yc+Q1P1AdcPE¯Cc∗−Q100∗∗−γ1I0∗∗∗−γ1I<0.
For convenience, chooseQ1=γ2I and substitute P1,Yc,Ac,Adc,E¯,Cc into (43), and it can obtained that Π<0. The proof is completed. After P1,Yc is gained by solving LMI Π<0, the parameter matrix K1¯ of the dynamic compensation FTC strategy can be gained by K1¯=P1Lc1+Yc.


## 5. Simulation Example

In the simulation section, we choose a MAS consisting of five kinds of aircraft to illustrate the effectiveness of the proposed algorithm in the paper; in addition, the state delay caused by wireless transmission is considered in the aircraft model. The given parameters of the aircraft are shown as follows:(44)A=−0.02271−0.0002−17.1−0.178−12.200−6.67,Ad=−0.5000.5−0.50.500−3.35,B=006.67,E=006.67,F=0.20.20.2,C=010001,d=2,dit=0.01sint,rt=2sin0.7t.The topology structure of the system is given in Figure1. The code of 0 acts as the leader, and other agents represent the followers. The Laplacian matrix L and the leader adjacency matrix G are shown below:(45)L=100−1−11000−11000−11,G=1000000000100000.Figure 1
The topology of the MAS.The parameters of the dynamic compensatorgx are designed as follows:(46)δ1=2.5,δ2=1,Ak=−98.4895−0.0112−0.0977−0.0109−98.01000.2348−0.09760.2383−98.2363,Bk=−2.31212.269728.5611−25.7432−69.857897.7782,Ck=53−74.5154.7,Dk=−86.8803−114.7595.The parameters of the dynamic compensation FTC strategy based on the relative output tracking error are shown as follows:(47)γ1=2,γ2=2,B∗E=1,Ak1=−0.31390.02940.00080.0297−0.5474−0.00150.00270.000−0.5770,Bk1=0.0032−0.0054−0.04400.07070.23290.3992,Ck1=−0.28820.27562.5922,Dk1=−33.2715−56.1697.In order to verify the effectiveness of the proposed FD algorithm, two different fault forms, time-varying fault and constant fault, are selected here for the experiment. The specific forms of the two faults are described as follows.Case 1.
Only agents 1 and 4 have constant faults, while agents 2 and 3 have no faults.(48)f1=0t<101t≥10,f4=0t<102t≥10,f2=f3=0.Case 2.
Only agents 1 and 4 have time-varying faults, while agents 2 and 3 have no faults.(49)f1=0t<10sin0.5t−10t≥10,f4=0t<102cos0.5t−10t≥10,f2=f3=0.
Based on the proposed fault diagnosis algorithm, the observer is designed to estimate the constant and time-varying fault, respectively, and the results are shown in Figures2 and 3. From the simulation results, it can be seen that the designed observer has good estimation results for both constant and time-varying faults. The output trajectory of the multi-agent system in which the fault occurred is shown in Figures 4 and 5. It can be seen from the figures that the output of the followers no longer tracks the leader’s trajectory due to the presence of the fault. After fault-tolerant control is carried out on the system, its output trajectory when failure occurs is shown in Figures 6 and 7. By comparison, it can be seen that the fault-tolerant control algorithm proposed in this paper can make the MAS have good tracking performance even after the fault occurs.Figure 2
Constant fault and fault estimation.Figure 3
Time-varying fault and fault estimation.Figure 4
Profiles of output trajectoriesyi1 without fault-tolerant controller.Figure 5
Profiles of output trajectoriesyi2 without fault-tolerant controller.Figure 6
Profiles of output trajectoriesyi1 with fault-tolerant controller.Figure 7
Profiles of output trajectoriesyi2 with fault-tolerant controller.


## 6. Conclusions

A novel fault diagnosis and fault-tolerant control (FTC) protocol is proposed for the leader-follower multi-agent systems (MASs) with time delay in this paper. At first, design an observer on the basis of the relative estimation error information to estimate the fault of the original MAS. It is worth noting that the fault estimation information is obtained by the dynamic compensator proposed in this paper. Then, after obtaining fault information, a novel dynamic compensation FTC protocol based on relative output tracking error is used in the system to allow the follower's output to still track the leader's output when a fault occurs. In the end, an example of five-aircraft MAS is given to prove the validity of the developed algorithm. However, there are some limitations in this article. The time delay of the system is time invariant and the leader and follower have the same system structure. So, the research on heterogeneous agent systems is interesting which will be focused in our future study.



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