# Fault Tolerant Consensus of Multi-Agent Systems with Linear Dynamics

Authors: Jianzhen Li
Journal: Mathematical Problems in Engineering
 (2013)

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## 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.


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## Body

## 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. Section2 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 A>0 means that A is symmetric positive definite. Consider 1n=[1,1,…,1]T∈Rn×1. Similarly, 0n=[0,0,…,0]T∈Rn×1.


## 2. Problem Formulation

Consider a team ofn agents with the following linear dynamics:
(1)x˙i(t)=Axi(t)+Bui(t)+Efi(t),(2)yi(t)=Cxi(t),
where xi(t)∈ℝm is the state of agent i, ui(t)∈ℝp is the control input, and yi(t)∈ℝl is the output. fi(t)∈ℝm denotes the actuator fault of agent i. If fi(t)≡0, then no actuator fault occurs at agent i. In this paper, we have the following assumptions.Assumption 1.
Assume that∥fi(t)∥≤f0, ∥f˙i(t)∥≤f1, for i=1,…,n.Assumption 2.
(
A
,
C
) is observable, and CE is full rank.Remark 3.
Assumption2 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 P and matrices F and R such that
(3)PE=CTR,(A-FC)TP+P(A-FC)<0.
The communication topology among then agents can be represented by an undirected graph 𝒢=(𝒱,ℰ), where 𝒱={1,…,n} is the node set and ℰ∈𝒱×𝒱 is the edge set. An edge (j,i)∈ℰ if agent i and agent j can access information from each other. An undirected path is a sequence of undirected edges of the form (i1,i2),(i2,i3),…, where ij∈𝒱. An undirected graph is connected if for any i,j∈𝒱 there exists a path between them. The neighbor set 𝒩i of agent i is defined as 𝒩i≜{j∣(j,i)∈ℰ}. The adjacency matrix 𝒜≜[aij]∈ℝn×n is defined as aij=aji>0 if (j,i)∈ℰ and aij=0 otherwise. The Laplacian matrix ℒ is defined as ℒ≜D-A, where D=diag{∑j=1n‍a1j,…,∑j=1n‍anj}. It is well known that if the communication topology is connected, then ℒ has a simple zero eigenvalue and n-1 nonnegative eigenvalues λ2≤λ3≤⋯≤λn.Definition 4.
We say algorithmui(t) asymptotically solves the consensus problem if xi(t)-(∑j=1Nxj(t))/n→0 as t→∞, for any i=1,…,n.
The objective of this paper is to derive a consensus protocolui(t) 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]:
(4)x^˙i(t)=Ax^i(t)+Bui(t)+Ef^i(t)+Feyi(t),f^˙i(t)=ΓRTeyi(t)-σΓf^i(t),eyi(t)=Cexi(t),
where x^i(t) and f^i(t) are estimations of xi(t), yi(t), and fi(t), respectively; exi(t)=xi(t)-x^i(t) and eyi(t)=yi(t)-y^i(t); matrices F and R are chosen according to (3); matrix Γ>0 and constant scalar σ>0 are chosen such that σ-λmax(Γ-1)>0.Lemma 5.
Defineefi(t)≜fi(t)-f^i(t). Under Assumptions 1 and 2, estimator (4) guarantees that (exi,efi) converge exponentially to the following set:
(5)S={(exi,efi)∣exiTΠ1exi-[σ-λmax(Γ-1)]∥efi∥2≤λmax(Γ-1)f12+σf02[σ-λmax(Γ-1)]∥efi∥2},
where Π1=P(A-FC)+(A-FC)TP.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.
(
A
,
B
) is controllable, and rank(B,E)=rank(B).
Under Assumption6, there exists a matrix B such that (I-BB*)E=0; that is, BB*E=E. Based on estimator (4), we have the following consensus protocol:
(6)ui(t)=K∑j=1naij[xi(t)-xj(t)]-B*Ef^i(t),
where K is a parameter matrix to be designed. With (6), system (1) becomes
(7)x˙i(t)=Axi(t)+BK∑j=1naij[xi(t)-xj(t)]-BB*Ef^i(t)+Efi(t).
Let ei(t)≜xi(t)-(1/n)∑j=1n‍xj(t). It is easy to see that consensus is achieved if ei(t)=0, for i=1,…,n. From (1) and (6), we have
(8)e˙i(t)=Axi(t)+BK∑j=1naij[xi(t)-xj(t)]-BB*Ef^i(t)+Efi(t)-1n∑j=1n{Axj(t)+BK∑l=1najl[xj(t)-xl(t)]oooooooooo-BB*Ef^j(t)+Efj(t)∑l=1najl[xj(t)-xl(t)]}.
Notice that -BB*Ef^i(t)+Efi(t)=Eefi(t), xi(t)-xj(t)=ei(t)-ej(t), and Axi(t)-(1/n)∑j=1n‍Axj(t)=Aei(t), and (8) can be rewritten as
(9)e˙i(t)=Aei(t)+BK∑j=1naij[ei(t)-ej(t)]+Eefi(t)-1n∑j=1n{BK∑l=1najl[xj(t)-xl(t)]+Eefj(t)}.
For undirected graphs, we have aij=aji. It is easy to see that
(10)∑j=1nBK∑l=1najl[xj(t)-xj(t)]=∑j=1nBK∑l=1najlxj(t)-∑j=1nBK∑l=1najlxl(t)=∑j=1nBK∑l=1najlxj(t)-∑l=1nBK∑j=1naljxj(t)=0
which, together with (9), leads to
(11)e˙i(t)=Aei(t)+BK∑j=1naij[ei(t)-ej(t)]+E{efi(t)-1n∑j=1nefj(t)}.
Define e(t)≜[e1T(t)⋯enT(t)]T and ef(t)≜[ef1T(t)⋯efnT(t)]T. Equation (11) can be written in a compact form as
(12)e˙(t)=(IN⊗A+ℒ⊗BK)e(t)+(H⊗E)ef(t),
where
(13)H=[n-1n-1n⋯-1n-1nn-1n⋯-1n⋮⋮⋱⋮-1n-1n⋯n-1n].
It can be seen thatH=In-1n1T/n, H2=H, 1nTH=0nT, H1n=0n, and Hℒ=ℒH=ℒ. Before giving the main results of this paper, the following lemma is needed.Lemma 7 (see [10]).
LetH and ℒ be matrices previously defined; the following statements hold.(1)
The eigenvalues ofH are 1 with multiplicity n-1 and 0 with multiplicity 1. The vectors 1nT and 1n are the left and right eigenvectors of H associated with zero eigenvalue, respectively.(2)
There exists an orthogonal matrixU∈ℝn×n with last column 1n/n, such that
(14)UTHU=[In-10n-1*0],UTℒU=[Δ0n-1*0].
Next, we give the main results of this paper.Theorem 8.
Suppose the undirected graph𝒢 is connected, and the nonzero eigenvalues of ℒ are λ2≤⋯≤λn. Using protocol (6), with the fault estimator (4), consensus errors {ei(t):i=1,…,n} will converge to a small ball around the origin if there exists a symmetric positive definite matrix X such that the following LMI holds:
(15)AX+XAT-λ2BBT<0,
and K is chosen as K=-BTX-1.Proof.
Suppose that the communication graph is connected; the nonzero eigenvalues ofℒ are λ2≤⋯≤λn and there exists a symmetric positive definite matrix X such that (15) holds. Let Q=X-1; we have
(16)QA+ATQ-λ2QBBTQ<0.
Define(17)ex(t)≜[ex1T(t)⋯exnT(t)]T,f(t)≜[f1T(t)⋯fnT(t)]T,f˙(t)≜[f˙1T(t)⋯f˙nT(t)]T,
where exi(t)=xi(t)-x^i(t). From (1), (2), and (4), we have that
(18)e˙xi(t)=(A-KC)exi(t)+Eefi(t),e˙fi(t)=f˙i(t)-ΓRCexi(t)+σΓefi(t)-σΓfi(t).
Equation (18) can be rewritten in a compact form as
(19)e˙x(t)=[In⊗(A-KC)]ex(t)+(In⊗E)ef(t),e˙f(t)=f˙(t)-(In⊗ΓRC)ex(t)+(In⊗σΓ)ef(t)-(In⊗σΓ)f(t).
Consider the following Lyapunov function:(20)V(t)=V1(t)+γV2(t)+γV3(t),V1(t)=eT(t)(In⊗Q)e(t),V2(t)=exT(t)(H⊗P)ex(t),V3(t)=efT(t)(H⊗Γ-1)ef(t),
where P is defined in Lemma 5. It follows from (12) that
(21)V˙1(t)=eT(t){In⊗(QA+ATQ)+ℒoooooooo⊗(QBK+KTBTQ)}e(t)+2eT(t)(H⊗QE)ef(t).
From (19) we have
(22)V˙2(t)=exT(t){H⊗Π1}ex(t)+2exT(t)(H⊗PE)ef(t),V˙3(t)=-2efT(t)[(H⊗RC)ex(t)+(H⊗σIm)ef(t)oooooooooo+(H⊗Γ-1)f˙(t)-(H⊗σIm)f(t)].
Notice that PE=CR and
(23)-2efT(t)(H⊗Γ-1)f˙(t)≤efT(t)[H⊗λmax(Γ-1)Im]ef(t)+f˙T(t)[H⊗λmax(Γ-1)Im]×f˙(t),-2efT(t)(H⊗σIm)f(t)≤efT(t)[H⊗σIm]ef(t)+fT(t)[H⊗σIm]f(t);
it follows that
(24)V˙2(t)+V˙3(t)≤exT(t){H⊗Π1}ex(t)-efT(t)[H⊗(σIm-Γ-1)]ef(t)+f˙T(t)[H⊗λmax(Γ-1)Im]f˙(t)+fT(t)[H⊗σIm]f(t).
Let U=[U11n/n]; we have U1THU1=IN-1 and U1TℒU1=Δ, where U is defined in Lemma 7. Define
(25)e~(t)≜(UT⊗Im)e(t)≜[e~1T(t)⋯e~nT(t)]T,e~x(t)≜(UT⊗Im)ex(t)≜[e~x1T(t)⋯e~xnT(t)]T,e~f(t)≜(UT⊗Im)ef(t)≜[e~f1T(t)⋯e~fnT(t)]T,f~(t)≜(UT⊗Im)f(t)≜[f~1T(t)⋯f~nT(t)]T,f˙~(t)≜(UT⊗Im)f˙(t)≜[f˙~1T(t)⋯f˙~nT(t)]T.
By the definition of e(t) we know that e(t)=(H⊗Im)x(t), where x(t)=[x1T(t)⋯xnT(t)]T. It follows that
(26)e~n(t)=(1nTn⊗Im)(H⊗Im)x(t)=0nm.
From (21) we have
(27)V˙1(t)=e~T(t)(UT⊗Im){In⊗(QA+ATQ)oooooooooooooooo+ℒ⊗(QBK+KTBTQ)}×(U⊗Im)e~(t)+e~T(t)(UT⊗Im)(H⊗QE)(U⊗Im)e~f(t)=e~T(t){[Δ0n-1*0]In⊗(QA+ATQ)oooooooo+[Δ0n-1*0]⊗(QBK+KTBTQ)}e~(t)+e~T(t)([In-10n-1*0]⊗Q)e~f(t)≤∑i=1n-1[e~iT(t)(QA+ATQooooooooooo-λmin(Δ)QBBTQ)e~i(t)+e~iT(t)QEe~fi(t)].
Similarly, from (24) we have
(28)V˙2(t)+V˙3(t)≤∑i=1n-1[e~xiT(t)Π1e~xi(t)-e~fiT(t)(σIm-Γ-1)e~fi(t)+f~˙iT(t)λmax(Γ-1)f~˙i(t)+σImf~iT(t)f~i(t)].
It then follows that
(29)V˙(t)≤∑i=1n-1{f~˙iT(t)[e~iT(t)Π2e~i(t)+2e~iT(t)QEe~fi(t)]+γ[f~˙iT(t)λmax(Γ-1)f~˙i(t)+σImf~iT(t)f~i(t)e~xiT(t)Π1e~xi(t)-e~fiT(t)(σIm-Γ-1)e~fi(t)+f~˙iT(t)λmax(Γ-1)f~˙i(t)+f~˙iT(t)λmax(Γ-1)f~˙i(t)+σImf~iT(t)f~i(t)]},
where Π2=QA+ATQ-λmin(Δ)QBBTQ.
Notice thatfi(t) and f˙i(t) are bounded, for i=1,…,n. It is easy to see that f~˙iT(t)λmax(Γ-1)f~˙i(t)+σf~iT(t)f~i(t) is bounded, and the bound is determined by σ and Γ. We assume that |f~˙iT(t)λmax(Γ-1)f~˙i(t)+σf~iT(t)f~i(t)|<β(σ,Γ). Equation (29) can be rewritten as
(30)V˙(t)≤∑i=1n-1{[e~iT(t)Π1e~i(t)+2e~iT(t)QEe~fi(t)]+γ[e~xiT(t)Π2e~xi(t)-e~fiT(t)(σIm-Γ-1)e~fi(t)+γβ(σ,Γ)e~iT(t)Π1e~i(t)]}=∑i=1n-1[ξiT(t)Ωξi(t)+γβ(σ,Γ)],
where ξi(t)≜[e~iT(t)e~xiT(t)e~fiT(t)]T and
(31)Ω=[Π20QE0γΠ10ETQ0γ(Γ-1-σIm)].
It is easy to see that λmin(Δ)=λ2. From the assumption of the theorem, we know that Π2<0. By the definition of Π1, we know that Π1<0. It is easy to see that Ω<0 if γ is small enough. From (30) we know that V˙(t)≤0 if
(32)ξiT(t)Ωξi(t)+γβ(σ,Γ)≤0
which implies that ξi(t) will converge into S1={ξi:|ξiT(t)Ωξi(t)|≤γβ(σ,Γ)}, yielding that {ei(t):i=1,…,n} will converge to a small ball around the origin. The proof is completed.Remark 9.
Theorem8 shows that ei(t) can converge to a small ball surrounding the origin. From the proof, we know that this set is determined by σ and Γ. Since σ and Γ can be selected freely, this ball can be chosen arbitrarily small. However, if σ is chosen too small, ei(t) may converge very slowly. To overcome this problem, dynamically changing σ and Γ can be used in the practice. This is out of the scope of this paper and will be considered in our future research.


## 3.1. Fault Estimator

Consider the following fault estimator, which is motivated by the fault estimator in [20]:
(4)x^˙i(t)=Ax^i(t)+Bui(t)+Ef^i(t)+Feyi(t),f^˙i(t)=ΓRTeyi(t)-σΓf^i(t),eyi(t)=Cexi(t),
where x^i(t) and f^i(t) are estimations of xi(t), yi(t), and fi(t), respectively; exi(t)=xi(t)-x^i(t) and eyi(t)=yi(t)-y^i(t); matrices F and R are chosen according to (3); matrix Γ>0 and constant scalar σ>0 are chosen such that σ-λmax(Γ-1)>0.Lemma 5.
Defineefi(t)≜fi(t)-f^i(t). Under Assumptions 1 and 2, estimator (4) guarantees that (exi,efi) converge exponentially to the following set:
(5)S={(exi,efi)∣exiTΠ1exi-[σ-λmax(Γ-1)]∥efi∥2≤λmax(Γ-1)f12+σf02[σ-λmax(Γ-1)]∥efi∥2},
where Π1=P(A-FC)+(A-FC)TP.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.
(
A
,
B
) is controllable, and rank(B,E)=rank(B).
Under Assumption6, there exists a matrix B such that (I-BB*)E=0; that is, BB*E=E. Based on estimator (4), we have the following consensus protocol:
(6)ui(t)=K∑j=1naij[xi(t)-xj(t)]-B*Ef^i(t),
where K is a parameter matrix to be designed. With (6), system (1) becomes
(7)x˙i(t)=Axi(t)+BK∑j=1naij[xi(t)-xj(t)]-BB*Ef^i(t)+Efi(t).
Let ei(t)≜xi(t)-(1/n)∑j=1n‍xj(t). It is easy to see that consensus is achieved if ei(t)=0, for i=1,…,n. From (1) and (6), we have
(8)e˙i(t)=Axi(t)+BK∑j=1naij[xi(t)-xj(t)]-BB*Ef^i(t)+Efi(t)-1n∑j=1n{Axj(t)+BK∑l=1najl[xj(t)-xl(t)]oooooooooo-BB*Ef^j(t)+Efj(t)∑l=1najl[xj(t)-xl(t)]}.
Notice that -BB*Ef^i(t)+Efi(t)=Eefi(t), xi(t)-xj(t)=ei(t)-ej(t), and Axi(t)-(1/n)∑j=1n‍Axj(t)=Aei(t), and (8) can be rewritten as
(9)e˙i(t)=Aei(t)+BK∑j=1naij[ei(t)-ej(t)]+Eefi(t)-1n∑j=1n{BK∑l=1najl[xj(t)-xl(t)]+Eefj(t)}.
For undirected graphs, we have aij=aji. It is easy to see that
(10)∑j=1nBK∑l=1najl[xj(t)-xj(t)]=∑j=1nBK∑l=1najlxj(t)-∑j=1nBK∑l=1najlxl(t)=∑j=1nBK∑l=1najlxj(t)-∑l=1nBK∑j=1naljxj(t)=0
which, together with (9), leads to
(11)e˙i(t)=Aei(t)+BK∑j=1naij[ei(t)-ej(t)]+E{efi(t)-1n∑j=1nefj(t)}.
Define e(t)≜[e1T(t)⋯enT(t)]T and ef(t)≜[ef1T(t)⋯efnT(t)]T. Equation (11) can be written in a compact form as
(12)e˙(t)=(IN⊗A+ℒ⊗BK)e(t)+(H⊗E)ef(t),
where
(13)H=[n-1n-1n⋯-1n-1nn-1n⋯-1n⋮⋮⋱⋮-1n-1n⋯n-1n].
It can be seen thatH=In-1n1T/n, H2=H, 1nTH=0nT, H1n=0n, and Hℒ=ℒH=ℒ. Before giving the main results of this paper, the following lemma is needed.Lemma 7 (see [10]).
LetH and ℒ be matrices previously defined; the following statements hold.(1)
The eigenvalues ofH are 1 with multiplicity n-1 and 0 with multiplicity 1. The vectors 1nT and 1n are the left and right eigenvectors of H associated with zero eigenvalue, respectively.(2)
There exists an orthogonal matrixU∈ℝn×n with last column 1n/n, such that
(14)UTHU=[In-10n-1*0],UTℒU=[Δ0n-1*0].
Next, we give the main results of this paper.Theorem 8.
Suppose the undirected graph𝒢 is connected, and the nonzero eigenvalues of ℒ are λ2≤⋯≤λn. Using protocol (6), with the fault estimator (4), consensus errors {ei(t):i=1,…,n} will converge to a small ball around the origin if there exists a symmetric positive definite matrix X such that the following LMI holds:
(15)AX+XAT-λ2BBT<0,
and K is chosen as K=-BTX-1.Proof.
Suppose that the communication graph is connected; the nonzero eigenvalues ofℒ are λ2≤⋯≤λn and there exists a symmetric positive definite matrix X such that (15) holds. Let Q=X-1; we have
(16)QA+ATQ-λ2QBBTQ<0.
Define(17)ex(t)≜[ex1T(t)⋯exnT(t)]T,f(t)≜[f1T(t)⋯fnT(t)]T,f˙(t)≜[f˙1T(t)⋯f˙nT(t)]T,
where exi(t)=xi(t)-x^i(t). From (1), (2), and (4), we have that
(18)e˙xi(t)=(A-KC)exi(t)+Eefi(t),e˙fi(t)=f˙i(t)-ΓRCexi(t)+σΓefi(t)-σΓfi(t).
Equation (18) can be rewritten in a compact form as
(19)e˙x(t)=[In⊗(A-KC)]ex(t)+(In⊗E)ef(t),e˙f(t)=f˙(t)-(In⊗ΓRC)ex(t)+(In⊗σΓ)ef(t)-(In⊗σΓ)f(t).
Consider the following Lyapunov function:(20)V(t)=V1(t)+γV2(t)+γV3(t),V1(t)=eT(t)(In⊗Q)e(t),V2(t)=exT(t)(H⊗P)ex(t),V3(t)=efT(t)(H⊗Γ-1)ef(t),
where P is defined in Lemma 5. It follows from (12) that
(21)V˙1(t)=eT(t){In⊗(QA+ATQ)+ℒoooooooo⊗(QBK+KTBTQ)}e(t)+2eT(t)(H⊗QE)ef(t).
From (19) we have
(22)V˙2(t)=exT(t){H⊗Π1}ex(t)+2exT(t)(H⊗PE)ef(t),V˙3(t)=-2efT(t)[(H⊗RC)ex(t)+(H⊗σIm)ef(t)oooooooooo+(H⊗Γ-1)f˙(t)-(H⊗σIm)f(t)].
Notice that PE=CR and
(23)-2efT(t)(H⊗Γ-1)f˙(t)≤efT(t)[H⊗λmax(Γ-1)Im]ef(t)+f˙T(t)[H⊗λmax(Γ-1)Im]×f˙(t),-2efT(t)(H⊗σIm)f(t)≤efT(t)[H⊗σIm]ef(t)+fT(t)[H⊗σIm]f(t);
it follows that
(24)V˙2(t)+V˙3(t)≤exT(t){H⊗Π1}ex(t)-efT(t)[H⊗(σIm-Γ-1)]ef(t)+f˙T(t)[H⊗λmax(Γ-1)Im]f˙(t)+fT(t)[H⊗σIm]f(t).
Let U=[U11n/n]; we have U1THU1=IN-1 and U1TℒU1=Δ, where U is defined in Lemma 7. Define
(25)e~(t)≜(UT⊗Im)e(t)≜[e~1T(t)⋯e~nT(t)]T,e~x(t)≜(UT⊗Im)ex(t)≜[e~x1T(t)⋯e~xnT(t)]T,e~f(t)≜(UT⊗Im)ef(t)≜[e~f1T(t)⋯e~fnT(t)]T,f~(t)≜(UT⊗Im)f(t)≜[f~1T(t)⋯f~nT(t)]T,f˙~(t)≜(UT⊗Im)f˙(t)≜[f˙~1T(t)⋯f˙~nT(t)]T.
By the definition of e(t) we know that e(t)=(H⊗Im)x(t), where x(t)=[x1T(t)⋯xnT(t)]T. It follows that
(26)e~n(t)=(1nTn⊗Im)(H⊗Im)x(t)=0nm.
From (21) we have
(27)V˙1(t)=e~T(t)(UT⊗Im){In⊗(QA+ATQ)oooooooooooooooo+ℒ⊗(QBK+KTBTQ)}×(U⊗Im)e~(t)+e~T(t)(UT⊗Im)(H⊗QE)(U⊗Im)e~f(t)=e~T(t){[Δ0n-1*0]In⊗(QA+ATQ)oooooooo+[Δ0n-1*0]⊗(QBK+KTBTQ)}e~(t)+e~T(t)([In-10n-1*0]⊗Q)e~f(t)≤∑i=1n-1[e~iT(t)(QA+ATQooooooooooo-λmin(Δ)QBBTQ)e~i(t)+e~iT(t)QEe~fi(t)].
Similarly, from (24) we have
(28)V˙2(t)+V˙3(t)≤∑i=1n-1[e~xiT(t)Π1e~xi(t)-e~fiT(t)(σIm-Γ-1)e~fi(t)+f~˙iT(t)λmax(Γ-1)f~˙i(t)+σImf~iT(t)f~i(t)].
It then follows that
(29)V˙(t)≤∑i=1n-1{f~˙iT(t)[e~iT(t)Π2e~i(t)+2e~iT(t)QEe~fi(t)]+γ[f~˙iT(t)λmax(Γ-1)f~˙i(t)+σImf~iT(t)f~i(t)e~xiT(t)Π1e~xi(t)-e~fiT(t)(σIm-Γ-1)e~fi(t)+f~˙iT(t)λmax(Γ-1)f~˙i(t)+f~˙iT(t)λmax(Γ-1)f~˙i(t)+σImf~iT(t)f~i(t)]},
where Π2=QA+ATQ-λmin(Δ)QBBTQ.
Notice thatfi(t) and f˙i(t) are bounded, for i=1,…,n. It is easy to see that f~˙iT(t)λmax(Γ-1)f~˙i(t)+σf~iT(t)f~i(t) is bounded, and the bound is determined by σ and Γ. We assume that |f~˙iT(t)λmax(Γ-1)f~˙i(t)+σf~iT(t)f~i(t)|<β(σ,Γ). Equation (29) can be rewritten as
(30)V˙(t)≤∑i=1n-1{[e~iT(t)Π1e~i(t)+2e~iT(t)QEe~fi(t)]+γ[e~xiT(t)Π2e~xi(t)-e~fiT(t)(σIm-Γ-1)e~fi(t)+γβ(σ,Γ)e~iT(t)Π1e~i(t)]}=∑i=1n-1[ξiT(t)Ωξi(t)+γβ(σ,Γ)],
where ξi(t)≜[e~iT(t)e~xiT(t)e~fiT(t)]T and
(31)Ω=[Π20QE0γΠ10ETQ0γ(Γ-1-σIm)].
It is easy to see that λmin(Δ)=λ2. From the assumption of the theorem, we know that Π2<0. By the definition of Π1, we know that Π1<0. It is easy to see that Ω<0 if γ is small enough. From (30) we know that V˙(t)≤0 if
(32)ξiT(t)Ωξi(t)+γβ(σ,Γ)≤0
which implies that ξi(t) will converge into S1={ξi:|ξiT(t)Ωξi(t)|≤γβ(σ,Γ)}, yielding that {ei(t):i=1,…,n} will converge to a small ball around the origin. The proof is completed.Remark 9.
Theorem8 shows that ei(t) can converge to a small ball surrounding the origin. From the proof, we know that this set is determined by σ and Γ. Since σ and Γ can be selected freely, this ball can be chosen arbitrarily small. However, if σ is chosen too small, ei(t) may converge very slowly. To overcome this problem, dynamically changing σ and Γ 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 of4 agents with
(33)A=[0100],B=E=[01],C=[1001];f1(t)=f2(t)=f3(t)≡0, f4(t)=0 for t<1s, and f4(t)=0.2 for t≥1s. The communication topology is given in Figure 1, with the following Laplacian matrix:
(34)ℒ=[1-100-12-100-12-1001-1].
The eigenvalues of ℒ are 0, 0.5858, 2, and 3.4142. According to Theorem 8, K can be chosen as
(35)K=[-0.4699-2.5507],
and parameters in the estimator (4) can be chosen as σ=1, Γ=1, and
(36)R=[0-1.3094].
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.Figure 1
The network topology associated with agents1 to 4.Figure 2
Position responses of agents 1–4.Figure 3
Velocity responses of agents 1–4.


## 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.



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*Source: 465671-2013-12-29.xml*