# Chaotic Tolerant Synchronization Analysis with Propagation Delay and Actuator Faults

Authors: Zhang Qunli
Journal: Mathematical Problems in Engineering
 (2015)

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## Abstract

The criteria for tolerant synchronization with a constant propagation delay and actuator faults are presented by using matrix analysis techniques. A new algorithm, which constructs the extended error systems in order to make the conservation of the stability lower, is proposed. Based on proper Lyapunov-Krasovskii functional, the novel delay-dependent fault tolerant synchronization analyses are derived. Finally, numerical examples show the effectiveness of the proposed method.


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## Body

## 1. Introduction

The realization of OGY (Ott-Grebogi-Yorker) chaos-control method [1] and PC (Pecora and Carroll) synchronization method [2] has been attracting researchers’ attention since the 1990s. Chaos synchronization [3–13] is of great practical significance and has aroused great interest in recent years.They all focused on the design of synchronization under normal operating conditions in the above works described. But, in practical chaotic secure communication systems, sensors, actuators, and inner components may inevitably fail, which can lead to sharp performance decline of chaotic secure. For the reason, fault tolerant synchronization and control [14–20] of chaotic systems have been the hot topic of intensive researches recently.More recent works studied the fault tolerant synchronization and control, but they were limited to construct Lyapunov-Krasovskii functional(1)Vt=V1et+V2f~t,which separates e(t) and f~(t), where e(t) is the synchronization error and f~(t) is the error of fault function. Motivating the limitation and the extended transformation [21, 22], the effective new method, which does the extended transformation for the error system, to consider Lyapunov-Krasovskii functional(2)Vt=VeTt,f~TtT,which do not separate e(t) and f~(t), for the chaotic fault tolerant synchronization with a constant propagation delay and actuator faults, makes the conservation descent. Finally, numerical examples are given to verify the above method.Notations. In this paper, R,Rn, and Rn×m denote, respectively, the real number, the real n-vectors, and the real n×m matrices. The superscript “T” stands for the transpose of a matrix. The symbol  X>Y(X≥Y), where X and Y are symmetric matrices, means that X-Y is positive definite (positive semidefinite). I is the identity matrix of appropriate dimensions. “∗” denotes the matrix entries implied by symmetry.


## 2. Preliminaries and Systems Description

Consider a chaotic master system with the actuator faults itemf(t) in the following form:(3)x˙t=Axt+gxt+hxt-θ+Eft,yt=Cxt+Dft,where x(t)∈Rn is the measurable state vector. y(t)∈Rp is the output vector, and A,C,D, and E are proper dimension constant matrices. g(x(t)), h(x(t-θ)) are known continuous nonlinear functions. θ>0 is the delay.Assumption 1.
There exist the matricesU1,U2,M1,M2,W1,W2,V1,V2∈Rn×n, and the nonlinear functions g(·), and h(·) satisfy(4)U1gx-gy-U2x-yTM1gx-gy-M2x-y≤0,(5)W1hx-hy-W2x-yTV1hx-hy-V2x-y≤0for all x,y∈Rn.
The slave system linked with the chaotic master system (3) is described by(6)x^˙t=Ax^t+gx^t+hx^t-θ+Ef^-Ly^t-τ-yt-τ,y^t=Cx^t+Df^t,where x^(t)∈Rn is the measurable state vector and τ>0 is a constant propagation delay.
Lete(t)=x^(t)-x(t), f~(t)=f^(t)-f(t), and φ(t)=y^(t)-y(t)=Ce(t)+Df~(t), and then(7)e˙t=Aet+gx^t-gxt+hx^t-θ-hxt-θ+Ef~t-LCet-τ-LDf~t-τ,φt=Cet+Df~t.Lemma 2 (see [23, 24]).
For any constant matrixW∈Rn×n, W>0, scalar 0<h(t)<h, and vector function w(t):[0,h]→Rn such that the integrations concerned are well defined, and then(8)∫0htwsdsTW∫0htwsds≤ht∫0htwTsWwsds.Lemma 3 (see [25]).
LetQ>0,  H,F(t), and E be real matrices of appropriate dimensions, with F(t) satisfying FT(t)F(t)≤I. Then, the following inequalities are equivalent:(1)
Q+HF(t)E+ETFT(t)HT<0;(2)
there exists a scalarε>0 such that Q+ε-1HHT+εETE<0.Lemma 4 (see [26–28]).
LetA,L,E, and F(t) be real matrices of appropriate dimensions, with F(t) satisfying FT(t)F(t)≤I. Then, one has the following:(1)
for any scalarε>0,(9)LFE+ETFTLT<ε-1LLT+εETE;(2)
for any matrixP>0 and scalar ε>0, such that εI-EFET>0,(10)A+LFETPA+LFE≤ATPA+ATPEεI-ETPE-1ETPA+εLTL.


## 3. Fault Tolerant Synchronization Analysis

### 3.1. Fault Tolerant Synchronization Analysis Whenf(t) and f^(t) Are Derivable on t

Letf~˙(t)=-Gε(t)=-GCe(t)-GDf~(t), and we have(11)e˙t=Aet+gx^t-gxt+hx^t-θ-hxt-θ+Ef~t-LCet+LCet-LCet-τ-LDf~t+LDf~t-LDf~t-τ=A-LCet+gx^t-gxt+hx^t-θ-hxt-θ+E-LDf~t+LC∫t-τte˙sds+LD∫t-τtf~˙sds.Suppose η(t)=e(t)f~(t), and then(12)
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,where(13)B=A-LCE-LD-GC-GD,pt=gx^t-gxt,qt=hx^t-θ-hxt-θ,R=LCLD00.From the assumption, we have(14)gx-gyTU1TM1gx-gy-gx-gyTU1TM2x-y-x-yTU2TM1gx-gy+x-yTU2TM2x-y≤0.So, we can get from assumption that(15)ptTU1TM1pt-ptTU1TM2et-etTU2TM1pt+etTU2TM2et≤0,eTtf~Tt-U2TM2000etf~t+eTtf~TtU2TM1R10R2pt0+pTt0U1TM20R3R4etf~t+pTt0-U1TM1R5R6R7pt0≥0,in which the proper dimension matrices Ri∈Rn×n, i=1,2,…,7, are arbitrary.That is,(16)ηTt-U2TM2000ηt+ηTtU2TM1R10R2pt0+pt0TU1TM20R3R4ηt+pt0T-U1TM1R5R6R7pt0≥0,ηTt12S1+S1Tηt+ηTt12S2+S3Tpt0+pt0T12S3+S2Tηt+pt0T12S4+S4Tpt0≥0,where(17)S1=-εU2TM2000,S2=εU2TM1εR10εR2,S3=εU1TM20εR3εR4,S4=-εU1TM1εR5εR6εR7,ε>0is any constant. Consider(18)ηTtS1+S1Tηt+ηTtS2+S3Tpt0+pt0TS3+S2Tηt+pt0TS4+S4Tpt0≥0,ξTtΛ1ξt≥0,where(19)ξt=ηTtη˙TtpTt0qTt0∫t-τtη˙Tsds∫t-θtη˙TsdsT,Λ1=S1+S1T00S2+S3T00∗00000∗∗S4+S4T000∗∗∗000∗∗∗∗00∗∗∗∗∗0.Imitating the above inference, we obtain from inequality (5) the following:(20)ξTtΛ2ξt≥0,where(21)Λ2=δY1TΔY100Y1TΔY30Y1TΔY2∗00000∗∗0000∗∗∗Y3TΔY30Y3TΔY2∗∗∗∗00∗∗∗∗∗Y2TΔY2,δ>0,Y1=I000,Y2=-I000,Y3=00IR8,Δ=-W2TV2-V2TW2W2TV1+V2TW1∗-W1TV1-V1TW1.We choose Lyapunov-Krasovskii functional(22)Vt=ηTtPηt+∫-τ0∫t+rtη˙TsQη˙sdsdr+∫-θ0∫t+rtη˙TsΘη˙sdsdr.Differentiating V(t) with respect to t, the following result is yielded:(23)V˙t=2ηTtPη˙t+τη˙TtQη˙t-∫t-τtη˙TsQη˙sds=2ηTtPBηt+2ηTtPpt0+2ηTtPR∫t-τtη˙sds+τη˙TtQη˙t-∫t-τtη˙TsQη˙sds+θη˙TtΘη˙t-∫t-θtη˙TsΘη˙sds.From Lemma 2, we have(24)V˙t≤ηTtPB+BTPηt+2ηTtPpt0+2ηTtPqt0+2ηTtPR∫t-τtη˙sds+τη˙TtQη˙t-1τ∫t-τtη˙sdsTQ∫t-τtη˙sds+θη˙TtΘη˙t-1θ∫t-τtη˙sdsTΘ∫t-τtη˙sds=ξTtΠξt≥0,where(25)Π=PB+BTP0PPPR0∗τQ0000∗∗0000∗∗∗000∗∗∗∗-1τQ0∗∗∗∗∗-1θΘ.From model (12), we have(26)0=2Q1ηt+Q2η˙tT-η˙t+Bηt+pt0+R∫t-τtη˙sds=ξTtΞξt,where(27)Ξ=Q1TB+BTQ1-Q1T+BTQ2TQ1TQ1TQ1TR0∗-Q2T-Q2Q2TQ2TQ2TR0∗∗0000∗∗∗000∗∗∗∗00∗∗∗∗∗0.From formulas (18), (20), (24), and (26), we get(28)V˙t≤ξTtΛ1+Λ2+Π+Ξξt=ξTtΩξt,where Ω=Λ+Π+Ξ.Based on the above derivation, we have the following result.Theorem 5.
The fault tolerant synchronization (3) and (6) is achieved if there exist constants τ>0, θ>0, δ>0, and ε>0, the positive definite matrices P=PT>0, Q=QT>0, and Θ>0, and the matrices Qj,Mj,Uj, j=1,2, L,C,D,Ri, i=1,2,…,8, such that the matrix Ω<0.Remark 6.
After the extended transformatione(t)→η(t), system (7) is turned into system (12). Based on the Lyapunov functional in [15–21], we take Lyapunov functional(29)Vt=ηTtPηt+∫-τ0∫t+rtη˙TsQη˙sdsdr+∫-θ0∫t+rtη˙TsΘη˙sdsdr,and we get matrix Ω. The conservation of stability of error system (7) can be decreased by choosing the matrices Ri, i=1,2,…,8, and the constant ε>0.Corollary 7.
WhenD=0 in system (3), result similar to Theorem 5 can be obtained.Whenh(x(t-θ))=0, inequality (18) is transformed to(30)σTtΛσt≥0,where(31)σt=ηTtη˙Ttpt0T∫t-τtη˙sdsTT,Λ=S1+S1T0S2+S3T0∗000∗∗S4+S4T0∗∗∗0,S1=-εU2TM2000,S2=εU2TM1εR10εR2,S3=εU1TM20εR3εR4,S4=-εU1TM1εR5εR6εR7,ε>0is any constant. The proper dimension matrices Ri∈Rn×n, i=1,2,…,7, are arbitrary.We choose Lyapunov functional(32)Vt=ηTtPηt+∫-τ0∫t+θtη˙TsQη˙sdsdθ,and differentiating V(t) with respect to t and using Lemma 2 yield(33)V˙t≤σTtΠσt,where(34)Π=PB+BTP0PPR∗τQ00∗∗00∗∗∗-1τQ.From model (12), we have(35)0=2Q1ηt+Q2η˙tT-η˙t+Bηt+pt0+R∫t-τtη˙sds=σTtΞσt,where(36)Ξ=Q1TB+BTQ1-Q1T+BTQ2TQ1TQ1TR∗-Q2T-Q2Q2TQ2TR∗∗00∗∗∗0.Based on the above derivation, we have the following result.Corollary 8.
The fault tolerant synchronization (3) and (5) is achieved if there exist constants τ,ε>0, the positive definite matrices P=PT>0, Q=QT>0, and the matrices Qj,Mj,Uj, j=1,2, Qj,Mj,Uj, j=1,2,  L,C,D,Ri, i=1,2,…,7, such that matrix Ω<0.


### 3.2. Fault Tolerant Synchronization Analysis Whenf(t) and f^(t) Are Derivable on x(t)

Letdf(t)/dx(t)=ħ(x(t))=H·F(x(t))·N and df^(t)/dx(t)=ħ(x^(t))=H·F(x^(t))·N,  H,N be proper dimension matrices, with satisfying FT(x(t))F(x(t))≤I, where H,N are proper dimension matrices and I is identity matrix. Then,(37)e˙t=Aet+gx^t-gxt+hx^t-θ-hxt-θ+Ef~t-LCet+LCet-LCet-τ-LDf~t+LDf~t-LDf~t-τ=A-LCet+gx^t-gxt+E-LDf~t+LC∫t-τte˙sds+LD∫t-τtf~˙sds.Suppose η(t)=e(t)f~(t); we have(38)
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.According to df(t)/dx(t)=ħ(x(t))=H·F(x(t))·N, df^(t)/dx(t)=ħ(x^(t))=H·F(x^(t))·N, we get(39)f~t=∫01ħ1-λxt+λx^tetdλ=∫01H·F1-λxt+λx^t·N·etdλ,-δf~t+δ∫01H·F1-λxt+λx^t·N·etdλ=0,δ∫01H·F1-λxt+λx^t·Ndλ,-δIηt=0,where δ>0. Consider(40)I000η˙t=A-LCE-LDδ∫01H·F1-λxt+λx^t·Ndλ-δIηt+gx^t-gxt0+LCLD00∫t-τtη˙sds,E1η˙t=Bηt+pt0+qt0+R∫t-τtη˙sds,where(41)E1=I000,B=A-LCE-LDδ∫01H·F1-λxt+λx^t·Ndλ-δI,pt=gx^t-gxt,R=LCLD00.From the assumption, we have(42)ςTtΛ1ςt≥0,where(43)ςt=ηTte˙TtpTt0qTt0∫t-τtη˙Tsds∫t-θte˙TsdsT,Λ1=S1+S1T00S2+S3T00∗00000∗∗S4+S4T000∗∗∗000∗∗∗∗00∗∗∗∗∗0,S1=-εU2TM2000,S2=εU2TM1εR10εR2,S3=εU1TM20εR3εR4,S4=-εU1TM1εR5εR6εR7,ε>0is any constant. The proper dimension matrices Ri∈Rn×n, i=1,2,…,7, are arbitrary(44)ςTtΛ2ςt≥0,where(45)Λ2=δY1TΔY100Y1TΔY30Y1TΔY2∗00000∗∗0000∗∗∗Y3TΔY30Y3TΔY2∗∗∗∗00∗∗∗∗∗-W2TV2-V2TW2,δ>0,Y1=I000,Y2=-I000,Y3=00IR8,Δ=-W2TV2-V2TW2W2TV1+V2TW1∗-W1TV1-V1TW1.We choose Lyapunov function(46)Vt=eTtP1et+∫-τ0∫t+rte˙TsQe˙sdsdr+∫-θ0∫t+rte˙TsΘe˙sdsdr=ηTtPE1ηt+∫-τ0∫t+θte˙TsQe˙sdsdθ+∫-θ0∫t+rte˙TsΘe˙sdsdr,where P=P1P20P3, and it is easy to know that PE1=E1TPT.The derivation ofV(t) on t is(47)V˙t=2ηTtPE1η˙t+τe˙TtQe˙t-∫t-τte˙TsQe˙sds+θe˙TtΘe˙t-∫t-θte˙TsΘe˙sds=2ηTtPBηt+2ηTtPpt0+2ηTtPR∫t-τtη˙sds+τe˙TtQe˙t-∫t-τte˙TsQe˙sds+θe˙TtΘe˙t-∫t-θte˙TsΘe˙sds.From Lemma 2, we have(48)V˙t≤ηTtPB+BTPTηt+2ηTtPpt0+2ηTtPqt0+2ηTtPR∫t-τtη˙sds+τe˙TtQe˙t-1τ∫t-τte˙TsdsTQ∫t-τte˙Tsds+θe˙TtΘe˙t-1θ∫t-θte˙TsdsTΘ∫t-θte˙sds=ςTtΠςt,where(49)Π=PB+BTPT0PPPR0∗τQ0000∗∗0000∗∗∗000∗∗∗∗-1τQ0∗∗∗∗∗-1θΘ.From model (3), we have(50)0=2Q1et+Q2e˙t+Q3∫t-τtη˙sdsT-e˙t+A-LC,E-LDηt+pt+qt+LC,LD∫t-τtη˙sds=ςTtΞςt,where(51)Ξ=Ξ11Ξ12Ξ13Ξ13Ξ140∗Ξ22Ξ23Ξ23Ξ240∗∗Ξ330Ξ340∗∗∗Ξ33Ξ340∗∗∗∗Ξ440∗∗∗∗∗0,Ξ11=Q1T0A-LC,E-LD+Q1T0A-LC,E-LDT,Ξ12=-Q1T0+Q2TA-LC,E-LDT,Ξ13=Q1T0I,R8,Ξ14=Q1T0LC,LD+Q3TA-LC,E-LDT,Ξ22=-Q2+Q2T,Ξ23=Q2TI,R9,Ξ24=-Q3+Q2TLC,LDT,Ξ33=0,Ξ34=Q3TI,R9,Ξ44=Q3TLC,LD+Q3TLC,LDT,where the proper dimension matrices R9∈Rn×n is arbitrary.From formulas (42), (44), (48), and (50), we get(52)V˙t≤ςTtΛ+Π+Ξςt=ςTtΩςt,where(53)
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,where Z=P1(A-LC)P1(E-LD)-δP20-δP3.From Lemma3, we obtain that S1+S1T+Ξ11+PB+BTPT<0 is equivalent to(54)S1+S1T+Ξ11+Z+ZT+ε-1δ2P2HHTP2TP2HHTP3T∗P3HHTP3T+εNTN000<0.Based on the above derivation, we have the following result.Theorem 9.
The fault tolerant synchronization (3) and (6) is achieved if there exist constants τ,ε>0, δ>0, the positive definite matrices P1=P1T>0, Q=QT>0, and the matrices Qj,Mj,Uj, j=1,2, Qj,Mj,Uj, j=1,2, L,C,D,Ri, i=1,2,…,9, such that matrix Σ<0, where(55)Σ=Σ11Ω12∗Ω22,Θ11=S1+S1T+Ξ11+Z+ZT+ε-1δ2P2HHTP2TP2HHTP3T∗P3HHTP3T+εNTN000.Remark 10.
After extending system (37) to singular system (40) by the transformation e(t)→η(t), taking proper matrices Ri, i=1,2,…,9, and the constant ε>0 can decrease the conservation of stability of error system (37) by constructing the Lyapunov functional(56)Vt=eTtP1et+∫-τ0∫t+rte˙TsQe˙sdsdr+∫-θ0∫t+rte˙TsΘe˙sdsdr=ηTtPE1ηt+∫-τ0∫t+θte˙TsQe˙sdsdθ+∫-θ0∫t+rte˙TsΘe˙sdsdr.When h(x(t-θ))=0, inequality (26) is transformed to(57)ωTtΛωt≥0,where(58)ωt=ηTte˙Ttpt0T∫t-τte˙sdsTT,Λ=S1+S1T0S2+S3T0∗000∗∗S4+S4T0∗∗∗0,S1=-εU2TM2000,S2=εU2TM1εR10εR2,S3=εU1TM20εR3εR4,S4=-εU1TM1εR5εR6εR7,ε>0is any constant. The proper dimension matrices Ri∈Rn×n, i=1,2,…,7, are arbitrary.
We choose Lyapunov function(59)Vt=eTtP1et+∫-τ0∫t+θte˙TsQe˙sdsdθ=ηTtPE1ηt+∫-τ0∫t+θte˙TsQe˙sdsdθ,where P=P1P20P3, and it is easy to know that PE1=E1TPT, and differentiating V(t) with respect to t and using Lemma 2 yield(60)V˙t≤ωTtΠωt,where(61)Π=PB+BTPT0PPR∗τQ00∗∗00∗∗∗-1τQ.From model (12), we have(62)0=2Q1et+Q2e˙t+Q3∫t-τtη˙sdsT-e˙t+A-LC,E-LDηt+pt+LC,LD∫t-τtη˙sds=σTtΞσt,where(63)Ξ=Ξ11Ξ12Ξ13Ξ14∗Ξ22Ξ23Ξ24∗∗Ξ33Ξ34∗∗∗Ξ44,Ξ11=Q1T0A-LC,E-LD+Q1T0A-LC,E-LDT,Ξ12=-Q1T0+Q2TA-LC,E-LDT,Ξ13=Q1T0I,R8,Ξ14=Q1T0LC,LD+Q3TA-LC,E-LDT,Ξ22=-Q2+Q2T,Ξ23=Q2TI,R8,Ξ24=-Q3+Q2TLC,LDT,Ξ33=0,Ξ34=Q3TI,R8,Ξ44=Q3TLC,LD+Q3TLC,LDT,where the proper dimension matrices R8∈Rn×n is arbitrary.Based on the above derivation, we have the following result.Corollary 11.
The fault tolerant synchronization (3) and (5) is achieved if there exist constants τ,ε>0, δ>0, the positive definite matrices P1=P1T>0, Q=QT>0, and the matrices Qj,Mj,Uj, j=1,2, Qj,Mj,Uj, j=1,2, L,C,D,Ri, i=1,2,…,8, such that matrix Θ<0, where(64)Θ=Θ11Ω12∗Ω22,Θ11=S1+S1T+Ξ11+Z+ZT+ε-1δ2P2HHTP2TP2HHTP3T∗P3HHTP3T+εNTN000.


## 3.1. Fault Tolerant Synchronization Analysis Whenf(t) and f^(t) Are Derivable on t

Letf~˙(t)=-Gε(t)=-GCe(t)-GDf~(t), and we have(11)e˙t=Aet+gx^t-gxt+hx^t-θ-hxt-θ+Ef~t-LCet+LCet-LCet-τ-LDf~t+LDf~t-LDf~t-τ=A-LCet+gx^t-gxt+hx^t-θ-hxt-θ+E-LDf~t+LC∫t-τte˙sds+LD∫t-τtf~˙sds.Suppose η(t)=e(t)f~(t), and then(12)
e
˙
t
=
A
-
L
C
,
E
-
L
D
e
t
f
~
t
+
g
x
^
t
-
g
x
t
+
h
x
^
t
-
θ
-
h
x
t
-
θ
+
L
C
,
L
D
∫
t
-
τ
t
e
˙
s
d
s
∫
t
-
τ
t
f
~
˙
s
d
s
=
A
-
L
C
,
E
-
L
D
η
t
+
g
x
^
t
-
g
x
t
+
L
C
,
L
D
∫
t
-
τ
t
η
˙
s
d
s
,
f
~
˙
t
=
-
G
C
,
-
G
D
e
t
f
~
t
=
-
G
C
,
-
G
D
η
t
,
η
˙
t
=
A
-
L
C
E
-
L
D
-
G
C
-
G
D
η
t
+
g
x
^
t
-
g
x
t
0
+
h
x
^
t
-
θ
-
h
x
t
-
θ
0
+
L
C
L
D
0
0
∫
t
-
τ
t
η
˙
s
d
s
,
η
˙
t
=
B
η
t
+
p
t
0
+
q
t
0
+
R
∫
t
-
τ
t
η
˙
s
d
s
,where(13)B=A-LCE-LD-GC-GD,pt=gx^t-gxt,qt=hx^t-θ-hxt-θ,R=LCLD00.From the assumption, we have(14)gx-gyTU1TM1gx-gy-gx-gyTU1TM2x-y-x-yTU2TM1gx-gy+x-yTU2TM2x-y≤0.So, we can get from assumption that(15)ptTU1TM1pt-ptTU1TM2et-etTU2TM1pt+etTU2TM2et≤0,eTtf~Tt-U2TM2000etf~t+eTtf~TtU2TM1R10R2pt0+pTt0U1TM20R3R4etf~t+pTt0-U1TM1R5R6R7pt0≥0,in which the proper dimension matrices Ri∈Rn×n, i=1,2,…,7, are arbitrary.That is,(16)ηTt-U2TM2000ηt+ηTtU2TM1R10R2pt0+pt0TU1TM20R3R4ηt+pt0T-U1TM1R5R6R7pt0≥0,ηTt12S1+S1Tηt+ηTt12S2+S3Tpt0+pt0T12S3+S2Tηt+pt0T12S4+S4Tpt0≥0,where(17)S1=-εU2TM2000,S2=εU2TM1εR10εR2,S3=εU1TM20εR3εR4,S4=-εU1TM1εR5εR6εR7,ε>0is any constant. Consider(18)ηTtS1+S1Tηt+ηTtS2+S3Tpt0+pt0TS3+S2Tηt+pt0TS4+S4Tpt0≥0,ξTtΛ1ξt≥0,where(19)ξt=ηTtη˙TtpTt0qTt0∫t-τtη˙Tsds∫t-θtη˙TsdsT,Λ1=S1+S1T00S2+S3T00∗00000∗∗S4+S4T000∗∗∗000∗∗∗∗00∗∗∗∗∗0.Imitating the above inference, we obtain from inequality (5) the following:(20)ξTtΛ2ξt≥0,where(21)Λ2=δY1TΔY100Y1TΔY30Y1TΔY2∗00000∗∗0000∗∗∗Y3TΔY30Y3TΔY2∗∗∗∗00∗∗∗∗∗Y2TΔY2,δ>0,Y1=I000,Y2=-I000,Y3=00IR8,Δ=-W2TV2-V2TW2W2TV1+V2TW1∗-W1TV1-V1TW1.We choose Lyapunov-Krasovskii functional(22)Vt=ηTtPηt+∫-τ0∫t+rtη˙TsQη˙sdsdr+∫-θ0∫t+rtη˙TsΘη˙sdsdr.Differentiating V(t) with respect to t, the following result is yielded:(23)V˙t=2ηTtPη˙t+τη˙TtQη˙t-∫t-τtη˙TsQη˙sds=2ηTtPBηt+2ηTtPpt0+2ηTtPR∫t-τtη˙sds+τη˙TtQη˙t-∫t-τtη˙TsQη˙sds+θη˙TtΘη˙t-∫t-θtη˙TsΘη˙sds.From Lemma 2, we have(24)V˙t≤ηTtPB+BTPηt+2ηTtPpt0+2ηTtPqt0+2ηTtPR∫t-τtη˙sds+τη˙TtQη˙t-1τ∫t-τtη˙sdsTQ∫t-τtη˙sds+θη˙TtΘη˙t-1θ∫t-τtη˙sdsTΘ∫t-τtη˙sds=ξTtΠξt≥0,where(25)Π=PB+BTP0PPPR0∗τQ0000∗∗0000∗∗∗000∗∗∗∗-1τQ0∗∗∗∗∗-1θΘ.From model (12), we have(26)0=2Q1ηt+Q2η˙tT-η˙t+Bηt+pt0+R∫t-τtη˙sds=ξTtΞξt,where(27)Ξ=Q1TB+BTQ1-Q1T+BTQ2TQ1TQ1TQ1TR0∗-Q2T-Q2Q2TQ2TQ2TR0∗∗0000∗∗∗000∗∗∗∗00∗∗∗∗∗0.From formulas (18), (20), (24), and (26), we get(28)V˙t≤ξTtΛ1+Λ2+Π+Ξξt=ξTtΩξt,where Ω=Λ+Π+Ξ.Based on the above derivation, we have the following result.Theorem 5.
The fault tolerant synchronization (3) and (6) is achieved if there exist constants τ>0, θ>0, δ>0, and ε>0, the positive definite matrices P=PT>0, Q=QT>0, and Θ>0, and the matrices Qj,Mj,Uj, j=1,2, L,C,D,Ri, i=1,2,…,8, such that the matrix Ω<0.Remark 6.
After the extended transformatione(t)→η(t), system (7) is turned into system (12). Based on the Lyapunov functional in [15–21], we take Lyapunov functional(29)Vt=ηTtPηt+∫-τ0∫t+rtη˙TsQη˙sdsdr+∫-θ0∫t+rtη˙TsΘη˙sdsdr,and we get matrix Ω. The conservation of stability of error system (7) can be decreased by choosing the matrices Ri, i=1,2,…,8, and the constant ε>0.Corollary 7.
WhenD=0 in system (3), result similar to Theorem 5 can be obtained.Whenh(x(t-θ))=0, inequality (18) is transformed to(30)σTtΛσt≥0,where(31)σt=ηTtη˙Ttpt0T∫t-τtη˙sdsTT,Λ=S1+S1T0S2+S3T0∗000∗∗S4+S4T0∗∗∗0,S1=-εU2TM2000,S2=εU2TM1εR10εR2,S3=εU1TM20εR3εR4,S4=-εU1TM1εR5εR6εR7,ε>0is any constant. The proper dimension matrices Ri∈Rn×n, i=1,2,…,7, are arbitrary.We choose Lyapunov functional(32)Vt=ηTtPηt+∫-τ0∫t+θtη˙TsQη˙sdsdθ,and differentiating V(t) with respect to t and using Lemma 2 yield(33)V˙t≤σTtΠσt,where(34)Π=PB+BTP0PPR∗τQ00∗∗00∗∗∗-1τQ.From model (12), we have(35)0=2Q1ηt+Q2η˙tT-η˙t+Bηt+pt0+R∫t-τtη˙sds=σTtΞσt,where(36)Ξ=Q1TB+BTQ1-Q1T+BTQ2TQ1TQ1TR∗-Q2T-Q2Q2TQ2TR∗∗00∗∗∗0.Based on the above derivation, we have the following result.Corollary 8.
The fault tolerant synchronization (3) and (5) is achieved if there exist constants τ,ε>0, the positive definite matrices P=PT>0, Q=QT>0, and the matrices Qj,Mj,Uj, j=1,2, Qj,Mj,Uj, j=1,2,  L,C,D,Ri, i=1,2,…,7, such that matrix Ω<0.


## 3.2. Fault Tolerant Synchronization Analysis Whenf(t) and f^(t) Are Derivable on x(t)

Letdf(t)/dx(t)=ħ(x(t))=H·F(x(t))·N and df^(t)/dx(t)=ħ(x^(t))=H·F(x^(t))·N,  H,N be proper dimension matrices, with satisfying FT(x(t))F(x(t))≤I, where H,N are proper dimension matrices and I is identity matrix. Then,(37)e˙t=Aet+gx^t-gxt+hx^t-θ-hxt-θ+Ef~t-LCet+LCet-LCet-τ-LDf~t+LDf~t-LDf~t-τ=A-LCet+gx^t-gxt+E-LDf~t+LC∫t-τte˙sds+LD∫t-τtf~˙sds.Suppose η(t)=e(t)f~(t); we have(38)
e
˙
t
=
A
-
L
C
,
E
-
L
D
e
t
f
~
t
+
g
x
^
t
-
g
x
t
+
h
x
^
t
-
θ
-
h
x
t
-
θ
+
L
C
,
L
D
∫
t
-
τ
t
e
˙
s
d
s
∫
t
-
τ
t
f
~
˙
s
d
s
=
A
-
L
C
,
E
-
L
D
η
t
+
g
x
^
t
-
g
x
t
+
h
x
^
t
-
θ
-
h
x
t
-
θ
+
L
C
,
L
D
∫
t
-
τ
t
η
˙
s
d
s
.According to df(t)/dx(t)=ħ(x(t))=H·F(x(t))·N, df^(t)/dx(t)=ħ(x^(t))=H·F(x^(t))·N, we get(39)f~t=∫01ħ1-λxt+λx^tetdλ=∫01H·F1-λxt+λx^t·N·etdλ,-δf~t+δ∫01H·F1-λxt+λx^t·N·etdλ=0,δ∫01H·F1-λxt+λx^t·Ndλ,-δIηt=0,where δ>0. Consider(40)I000η˙t=A-LCE-LDδ∫01H·F1-λxt+λx^t·Ndλ-δIηt+gx^t-gxt0+LCLD00∫t-τtη˙sds,E1η˙t=Bηt+pt0+qt0+R∫t-τtη˙sds,where(41)E1=I000,B=A-LCE-LDδ∫01H·F1-λxt+λx^t·Ndλ-δI,pt=gx^t-gxt,R=LCLD00.From the assumption, we have(42)ςTtΛ1ςt≥0,where(43)ςt=ηTte˙TtpTt0qTt0∫t-τtη˙Tsds∫t-θte˙TsdsT,Λ1=S1+S1T00S2+S3T00∗00000∗∗S4+S4T000∗∗∗000∗∗∗∗00∗∗∗∗∗0,S1=-εU2TM2000,S2=εU2TM1εR10εR2,S3=εU1TM20εR3εR4,S4=-εU1TM1εR5εR6εR7,ε>0is any constant. The proper dimension matrices Ri∈Rn×n, i=1,2,…,7, are arbitrary(44)ςTtΛ2ςt≥0,where(45)Λ2=δY1TΔY100Y1TΔY30Y1TΔY2∗00000∗∗0000∗∗∗Y3TΔY30Y3TΔY2∗∗∗∗00∗∗∗∗∗-W2TV2-V2TW2,δ>0,Y1=I000,Y2=-I000,Y3=00IR8,Δ=-W2TV2-V2TW2W2TV1+V2TW1∗-W1TV1-V1TW1.We choose Lyapunov function(46)Vt=eTtP1et+∫-τ0∫t+rte˙TsQe˙sdsdr+∫-θ0∫t+rte˙TsΘe˙sdsdr=ηTtPE1ηt+∫-τ0∫t+θte˙TsQe˙sdsdθ+∫-θ0∫t+rte˙TsΘe˙sdsdr,where P=P1P20P3, and it is easy to know that PE1=E1TPT.The derivation ofV(t) on t is(47)V˙t=2ηTtPE1η˙t+τe˙TtQe˙t-∫t-τte˙TsQe˙sds+θe˙TtΘe˙t-∫t-θte˙TsΘe˙sds=2ηTtPBηt+2ηTtPpt0+2ηTtPR∫t-τtη˙sds+τe˙TtQe˙t-∫t-τte˙TsQe˙sds+θe˙TtΘe˙t-∫t-θte˙TsΘe˙sds.From Lemma 2, we have(48)V˙t≤ηTtPB+BTPTηt+2ηTtPpt0+2ηTtPqt0+2ηTtPR∫t-τtη˙sds+τe˙TtQe˙t-1τ∫t-τte˙TsdsTQ∫t-τte˙Tsds+θe˙TtΘe˙t-1θ∫t-θte˙TsdsTΘ∫t-θte˙sds=ςTtΠςt,where(49)Π=PB+BTPT0PPPR0∗τQ0000∗∗0000∗∗∗000∗∗∗∗-1τQ0∗∗∗∗∗-1θΘ.From model (3), we have(50)0=2Q1et+Q2e˙t+Q3∫t-τtη˙sdsT-e˙t+A-LC,E-LDηt+pt+qt+LC,LD∫t-τtη˙sds=ςTtΞςt,where(51)Ξ=Ξ11Ξ12Ξ13Ξ13Ξ140∗Ξ22Ξ23Ξ23Ξ240∗∗Ξ330Ξ340∗∗∗Ξ33Ξ340∗∗∗∗Ξ440∗∗∗∗∗0,Ξ11=Q1T0A-LC,E-LD+Q1T0A-LC,E-LDT,Ξ12=-Q1T0+Q2TA-LC,E-LDT,Ξ13=Q1T0I,R8,Ξ14=Q1T0LC,LD+Q3TA-LC,E-LDT,Ξ22=-Q2+Q2T,Ξ23=Q2TI,R9,Ξ24=-Q3+Q2TLC,LDT,Ξ33=0,Ξ34=Q3TI,R9,Ξ44=Q3TLC,LD+Q3TLC,LDT,where the proper dimension matrices R9∈Rn×n is arbitrary.From formulas (42), (44), (48), and (50), we get(52)V˙t≤ςTtΛ+Π+Ξςt=ςTtΩςt,where(53)
Ω
=
Λ
+
Π
+
Ξ
=
Ω
11
Ω
12
∗
Ω
22
,
Ω
11
=
S
1
+
S
1
T
+
Ξ
11
+
Y
1
T
Δ
Y
1
+
P
B
+
B
T
P
T
Ξ
12
∗
τ
Q
+
Ξ
22
,
Ω
12
=
P
+
Ξ
13
S
2
+
S
3
T
+
Ξ
13
+
P
+
Y
1
T
Δ
Y
3
P
R
+
Ξ
14
Y
1
T
Δ
Y
2
Ξ
23
Ξ
23
Ξ
24
0
,
Ω
22
=
S
4
+
S
4
T
0
Ξ
34
0
∗
Y
3
T
Δ
Y
3
Ξ
34
Y
3
T
Δ
Y
2
∗
∗
-
τ
-
1
Q
+
Ξ
44
0
∗
∗
∗
-
W
2
T
V
2
-
V
2
T
W
2
,
P
B
+
B
T
P
T
=
P
1
A
-
L
C
P
1
E
-
L
D
-
δ
P
2
0
-
δ
P
3
+
P
1
A
-
L
C
P
1
E
-
L
D
-
δ
P
2
0
-
δ
P
3
T
+
δ
P
2
H
∫
0
1
F
1
-
λ
x
t
+
λ
x
^
t
d
λ
N
0
P
3
H
∫
0
1
F
1
-
λ
x
t
+
λ
x
^
t
d
λ
N
0
+
δ
P
2
H
∫
0
1
F
1
-
λ
x
t
+
λ
x
^
t
d
λ
N
0
P
3
H
∫
0
1
F
1
-
λ
x
t
+
λ
x
^
t
d
λ
N
0
T
=
Z
+
Z
T
+
δ
P
2
H
P
3
H
∫
0
1
F
1
-
λ
x
t
+
λ
x
^
t
d
λ
N
0
+
δ
P
2
H
P
3
H
∫
0
1
F
1
-
λ
x
t
+
λ
x
^
t
d
λ
N
0
T
,where Z=P1(A-LC)P1(E-LD)-δP20-δP3.From Lemma3, we obtain that S1+S1T+Ξ11+PB+BTPT<0 is equivalent to(54)S1+S1T+Ξ11+Z+ZT+ε-1δ2P2HHTP2TP2HHTP3T∗P3HHTP3T+εNTN000<0.Based on the above derivation, we have the following result.Theorem 9.
The fault tolerant synchronization (3) and (6) is achieved if there exist constants τ,ε>0, δ>0, the positive definite matrices P1=P1T>0, Q=QT>0, and the matrices Qj,Mj,Uj, j=1,2, Qj,Mj,Uj, j=1,2, L,C,D,Ri, i=1,2,…,9, such that matrix Σ<0, where(55)Σ=Σ11Ω12∗Ω22,Θ11=S1+S1T+Ξ11+Z+ZT+ε-1δ2P2HHTP2TP2HHTP3T∗P3HHTP3T+εNTN000.Remark 10.
After extending system (37) to singular system (40) by the transformation e(t)→η(t), taking proper matrices Ri, i=1,2,…,9, and the constant ε>0 can decrease the conservation of stability of error system (37) by constructing the Lyapunov functional(56)Vt=eTtP1et+∫-τ0∫t+rte˙TsQe˙sdsdr+∫-θ0∫t+rte˙TsΘe˙sdsdr=ηTtPE1ηt+∫-τ0∫t+θte˙TsQe˙sdsdθ+∫-θ0∫t+rte˙TsΘe˙sdsdr.When h(x(t-θ))=0, inequality (26) is transformed to(57)ωTtΛωt≥0,where(58)ωt=ηTte˙Ttpt0T∫t-τte˙sdsTT,Λ=S1+S1T0S2+S3T0∗000∗∗S4+S4T0∗∗∗0,S1=-εU2TM2000,S2=εU2TM1εR10εR2,S3=εU1TM20εR3εR4,S4=-εU1TM1εR5εR6εR7,ε>0is any constant. The proper dimension matrices Ri∈Rn×n, i=1,2,…,7, are arbitrary.
We choose Lyapunov function(59)Vt=eTtP1et+∫-τ0∫t+θte˙TsQe˙sdsdθ=ηTtPE1ηt+∫-τ0∫t+θte˙TsQe˙sdsdθ,where P=P1P20P3, and it is easy to know that PE1=E1TPT, and differentiating V(t) with respect to t and using Lemma 2 yield(60)V˙t≤ωTtΠωt,where(61)Π=PB+BTPT0PPR∗τQ00∗∗00∗∗∗-1τQ.From model (12), we have(62)0=2Q1et+Q2e˙t+Q3∫t-τtη˙sdsT-e˙t+A-LC,E-LDηt+pt+LC,LD∫t-τtη˙sds=σTtΞσt,where(63)Ξ=Ξ11Ξ12Ξ13Ξ14∗Ξ22Ξ23Ξ24∗∗Ξ33Ξ34∗∗∗Ξ44,Ξ11=Q1T0A-LC,E-LD+Q1T0A-LC,E-LDT,Ξ12=-Q1T0+Q2TA-LC,E-LDT,Ξ13=Q1T0I,R8,Ξ14=Q1T0LC,LD+Q3TA-LC,E-LDT,Ξ22=-Q2+Q2T,Ξ23=Q2TI,R8,Ξ24=-Q3+Q2TLC,LDT,Ξ33=0,Ξ34=Q3TI,R8,Ξ44=Q3TLC,LD+Q3TLC,LDT,where the proper dimension matrices R8∈Rn×n is arbitrary.Based on the above derivation, we have the following result.Corollary 11.
The fault tolerant synchronization (3) and (5) is achieved if there exist constants τ,ε>0, δ>0, the positive definite matrices P1=P1T>0, Q=QT>0, and the matrices Qj,Mj,Uj, j=1,2, Qj,Mj,Uj, j=1,2, L,C,D,Ri, i=1,2,…,8, such that matrix Θ<0, where(64)Θ=Θ11Ω12∗Ω22,Θ11=S1+S1T+Ξ11+Z+ZT+ε-1δ2P2HHTP2TP2HHTP3T∗P3HHTP3T+εNTN000.


## 4. Numerical Examples

Example 1.
Consider a typical delayed Hopfield neural networks [29–32] with two neurons(65)A=-I,gxt=2.0-0.1-5.03.0tanh⁡x1ttanh⁡x2t,hxt-θ=0.15-0.1-0.2-2.5tanh⁡x1t-θtanh⁡x2t-θ,τ=0.01,θ=1,LC=LD=-38I,E=2I,xt=x1x2.When f(t) and f^(t) are derivable on t, we take(66)ft=0,0T,0≤t≤20,2sin⁡tT,t≥2,f~˙t=0.2000x1t-x3tx2t-x4t-2f~t.Simulation results are shown in Figure 1.
Whenf(t) and f^(t) are derivable on x(t), we take(67)ft=0,0T,0≤t≤100.2sin⁡x1t,0.4sin⁡x2tT,t≥10.Simulation results are shown in Figure 2.Figure 1
The state responses ofx(t) and x^(t) when f(t) and f^(t) are derivable on t.Figure 2
The state responses ofx(t) and x^(t) when f(t) and f^(t) are derivable on x(t).Example 2.
Consider chaotic Lü system [31, 32]; that is,(68)A=-36360020000-3,gxt=0-x1x3x1x2,where x(t)=x1x2x3. We choose(69)LC=0-3800-380000,LD=0000-380000,E=200020002,τ=0.01.When f(t) and f^(t) are derivable on t, we take(70)ft=0,0,0T,0≤t≤20,2sin⁡t,0T,t≥2,(71)f~˙t=0.2000x1t-x3tx2t-x4t-2f~t.Simulation results are shown in Figure 3.
Whenf(t) and f^(t) are derivable on x(t), we take(72)ft=0,0,0T,0≤t≤50,0.4sin⁡x1t,0.4sin⁡x2tT,t≥5.Simulation results are shown in Figure 4.Figure 3
The state responses ofx(t) and x^(t) when f(t) and f^(t) are derivable on t.Figure 4
The state responses ofx(t) and x^(t) when f(t) and f^(t) are derivable on x(t).


## 5. Conclusions

In this paper, taking into account the constant propagation delay and actuator faults and the extended transformation of the error systems, the problem of the chaotic fault tolerant synchronization has been addressed. The conservation of stability of the error systems is lower than that of other existing literatures.



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*Source: 785861-2015-10-05.xml*