# New Approach on Robust and Reliable DecentralizedH∞ Tracking Control for Fuzzy Interconnected Systems with Time-Varying Delay

Authors: Xinrui Liu; Qiuye Sun; Xinming Hou
Journal: ISRN Applied Mathematics
 (2014)

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## Abstract

This paper investigates the robust and reliable
decentralizedH∞ tracking control issue for the fuzzy large-scale
interconnected systems with time-varying delay, which are composed
of a number of T-S fuzzy subsystems with interconnections.
Firstly, the ordinary fuzzy interconnected systems are equivalently transformed to the fuzzy descriptor systems; then, according to
the Lyapunov direct method and the decentralized control theory
of large-scale interconnected systems, the new linear matrix
inequalities- (LMIs-) based conditions with some free variables
are derived to guarantee the H∞ tracking performance not only
when all control components are operating well, but also in the
presence of some possible actuator failures. Moreover, there is no
need for the precise failure parameters of the actuators, rather than
the lower and upper bound. Finally, two simulation examples are
provided to illustrate the effectiveness of the proposed method.


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## Body

## 1. Introduction

Large-scale interconnected systems, such as electrical power systems, computer communication systems, economic systems, and process control systems, have attracted great interests from many researchers in recent years. Takagi-Sugeno (T-S) fuzzy model has become a popular and effective approach to control complex systems, and a lot of significant results on stabilization andH∞ control via linear matrix inequality (LMI) approach have been reported; see [1–4]. Compared with the centralized control, the decentralized scheme is preferred in the control design issue of the large-scale interconnected systems [5]. Recently, there are some works about stability and stabilization of fuzzy large-scale systems [6–9]. It is well known that delays appear in many dynamic systems, which are potential causes of system instability [10, 11]. The tasks of stabilization and tracking are two typical control problems. In general, tracking problems are more difficult than stabilization problems especially for nonlinear systems [12]. Reference [13] has given decentralized H∞ fuzzy model reference tracking control design, and the stable conditions in the sense of Lyapunov are given. The technology of descriptor model transformation is used in [14, 15]. A T-S fuzzy descriptor tracking control design for nonlinear systems with a guaranteed H∞ model reference tracking performance is discussed [16]. However, in practical situations, failure of actuators often occurs. Thus, an important requirement is to design a reliable controller such that the stability and performance of the closed-loop system can tolerate actuator failures [17–20].In this paper, we investigate the robust and reliable decentralizedH∞ tracking control issue for the fuzzy interconnected systems with time delay, which are composed of a number of T-S fuzzy subsystems with interconnections. Firstly, the ordinary fuzzy interconnected systems are equivalently transformed to the fuzzy descriptor systems; then, according to the Lyapunov direct method and the decentralized control theory of large-scale interconnected systems, the new LMIs-based conditions with some free variables are derived to guarantee the H∞ tracking performance not only when all control components are operating well, but also in the presence of some possible actuator failures. Finally, two simulation examples are provided to illustrate the effectiveness of the proposed method.The innovation of this paper can be summarized as follows:(1) the more practical H∞ performance index is used, which considers not only the effect of tracking error e(t) but also the effect of control u(t); (2) utilizing the descriptor model transformation, the new LMIs-based reliable H∞ performance conditions with some free variables are derived; (3) there is no need for the precise failure parameters of the actuators, rather than the lower and upper bound of failure parameters.In the following sections, the identity matrices and zero matrices are denoted byI and 0, respectively. XT denotes the transpose of matrix X. ℝn denotes the n-dimensional Euclidean space. The standard notation >(<) is used to denote the positive (negative-) definite ordering of matrices. Inequality X>Y shows that the matrix X-Y is positively definite. The symbol of * denotes the transposed element in the symmetric position.


## 2. Systems Description

Suppose there are the interconnected systems consisting ofJ interconnected subsystems Si, i=1,…,J. Each rule of the subsystem Si is represented by a T-S fuzzy model as follows:
(1)Sil:{Ifξi1isMi1land⋯andξigiisMigil,Thenx˙i(t)=Ailxi(t)+B1ilwi(t)+B2ilui(t)+∑j=1,j≠iJCjilxj(t-τij(t))xi(t)=ϕi(t),t∈[-τ¯,0]i=1,…,J,l=1,…,ri,
where xi(t)∈Rni, ui(t)∈Rmi, and wi(t) denote the state, input, and disturbance vector, respectively. Ail,B1il,B2il denote system matrices and Cjil denote the interconnection matrices between the ith and jth subsystem of the lth rule. τij(t) are the time-varying delays, τij(t)≤τij0, τ˙ij(t)≤H~ij, τ¯=maxi,jτij0, and ri is the number of IF-THEN rules of the subsystem Si. The initial conditions ϕi(t) are the differentiable functions for t∈[-τ¯,0]. Here, we denote XtT=[x1T(t),…,xJT(t)], ΦT(t)=[ϕ1T(t),…,ϕJT(t)].If we utilize the singleton fuzzifier, product fuzzy inference, and central-average defuzzifier, (1) can be inferred as
(2)x˙i(t)=∑l=1riμil(ξiq(t))(∑j=1,j≠iJCjilxj(t-τij(t))Ailxi(t)+B1ilwi(t)+B2ilui(t)+∑j=1,j≠iJCjilxj(t-τij(t))),
where
(3)λil(ξiq(t))=∏q=1giMiql(ξiq(t)),μil(ξiq(t))=λil(ξiq(t))∑l=1riλil(ξiq(t)),
where ξi1,…,ξigi are the premise variables and Miql(ξiq(t)) is the grade of membership of ξiq(t) in Miql. It can be seen that λil(ξiq(t))≥0, μil(ξiq(t))≥0 and ∑l=1riμil(ξiq(t))=1, i=1,…,J, l=1,…,ri.Consider the reference model for theith subsystem as follows:
(4)x˙ri(t)=Arixri(t)+Brivi(t),
where xri(t) denote the reference states, Ari denote the specific asymptotically stable matrices, Bri denote the system matrices with appropriate dimensions, and vi(t) denotes the bounded reference input.Lemma 1 (see [10]).
For any constant symmetric matrixM∈ℝn×n, M=MT>0, scalar α>0, vector function ζ:[0,α]→ℝn, such that the integrations in the following are well defined; then
(5)α∫0αζT(β)Mζ(β)dβ≥(∫0αζ(β)dβ)TM∫0αζ(β)dβ.Lemma 2 (see [10]).
For any matrixQ∈ℝl×n, any constant ɛ>0, and any positive definite matrix T, all ζ∈ℝl, η∈ℝn, the following result holds:
(6)2ζTQη≤εζTQT-1QTζ+1εηTTη.Lemma 3 (see [2]).
For any real matrixXl,Yl,1≤l≤n, and S>0 with appropriate dimensions, one has
(7)2∑l=1n∑m=1nμlμmXlTSYm≤∑l=1nμl(XlTSXl+YlTSYl),
where μl(1≤l≤n) are defined as μl(M(k))≥0, ∑l=1nμl(M(k))=1.


## 3.H∞ Tracking Control Design

According to the conventional parallel distributed compensation (PDC) concept, the fuzzy controllers corresponding toSi are used as follows:
(8)Fil:{Ifξi1isMi1land⋯andξigiisMigil,Thenui(t)=Kil(xi(t)-xri(t)),
where Kil are the controllers gains of the lth rule for subsystem Si.And the final output of the fuzzy controllers for each subsystemSi is
(9)ui(t)=∑l=1riμilKil(xi(t)-xri(t)).Instead of actuator outage, a more general actuator failure model is adopted in this paper. LetuiF(t) be the control input vector after failures have occurred. The following actuator failure model is adopted:
(10)uiF(t)=αiui(t)=αi∑l=1riμilKil(xi(t)-xri(t)),
where αi=diag(αi1,…,αimi) with 0≤α_ij≤αij≤α¯ij≤1, i=1,…,J, j=1,…,mi.Note that the parametersα_ij and α¯ij characterize the admissible failures of the jth actuator in the ith subsystem. Obviously, when α_ij=α¯ij=0, the failure model (10) corresponds to the case of the jth actuator outage. When 0≤α_ij≤α¯ij≤1, it corresponds to the case of partial failure of the jth actuator. When α_ij=α¯ij=1, it implies that there is no failure in the jth actuator. Denote the matrix set Sηi={λLi:λLi=diag(λLi1,…,λLimi)}, where λLij=α_ijorα¯ij, i=1,…,J, j=1,…,mi, L=1,…,2mi-li, li≤mi is the number of no failure actuators for subsystem Si.Lemma 4 (representation theorem [21]).
S
=
{
x
∣
A
x
=
b
,
x
≥
0
} is nonempty and bounded multiple-surface set, and there exist finite limit points x(1),…,x(k); then, x∈S if and only if x=∑j=1kζjx(j), ∑j=1kζj=1, ζj≥0, j=1,…,k.
Here, letSi={xi∣Aixi=bi,xi≥0}, and Ai=I; that is, Si={xi∣xi=bi,xi≥0}, where bi=[αi1⋯αi(mi-li)]T. Then, there exist finite limit points xi(1)=(α_i1⋯α_i(mi-li)),…,xi(2mi-li)=(α¯i1⋯α¯i(mi-li)) for Si.
Given any two pointsXi=(xij)∈Si, Yi=(yij)∈Si, α_ij≤xij, yij≤α¯ij, j=1,…,mi-li, for any real number λ∈[0,1], then α_ij≤λxij+(1-λ)yij≤α¯ij; that is λXi+(1-λ)Yi∈Si. Therefore, Si is nonempty and bounded multiple-surface set.
Let theli no failure actuators are the last li actuators of ith subsystem, according to Lemma 4, one has
(11)αi=diag(αi1⋯αi(mi-li)1⋯1)=ξi1diag(α_i1⋯α_i(mi-li)1⋯1)+⋯+ξi2mi-lidiag(α¯i1⋯αi(mi-li)1⋯1),
where ζiL≥0, ∑L=12mi-liζiL=1.
Therefore,αi∈Sαi′, where Sαi′={αi:αi=∑L=12mi-liζiLλLi,ζiL≥0,∑L=12mi-liζiL=1}.
Denote the tracking error by(12)ei(t)=xi(t)-xri(t).
Then, the whole closed-loop fuzzy interconnected systems become(13)x-˙i(t)=∑j=1,j≠iJ∑l=1ri∑m=1riμilμim(A-ilmx-i(t)+C-jilx-j(t-τij(t))(t-τij(t))+B-ilw-i(t)),
where
(14)x-i(t)=[xi(t)ei(t)],w-i(t)[wi(t)vi(t)],A-ilm=∑L=12mi-li[AilB2ilλLiKimAil-AriAri+B2ilλLiKim],C-jil=[Cjil0Cjil0],B-il=[B1il0B1il-Bri].
Equation (13) can be transformed to be the descriptor system form as follows:
(15)Eξ˙i=∑j=1,j≠iJ∑l=1ri∑m=1riμilμim(A~ilmξi+C~jilx-j(t-τij(t))+B~ilw-i),
where
(16)ξi=[x-ix-˙i],E=[I000],A~ilm=[0IA-ilm-I],C~jil=[0C-jil],B~il=[0B-il].Definition 5.
TheH∞ tracking control problem for the interconnected systems (13) is to design the controllers to minimize the prescribed level of disturbance attenuation ∑i=1Jγi>0, i=1,…,J, if the following two conditions are satisfied.(1)
Whenw-i=0, the whole fuzzy interconnected systems (13) are asymptotically stable.(2)
For the zero initially condition(ΦT(t)=0,t∈[-τ-,0]), ∑i=1J∫0∞(eiT(t)Qiei(t)+uiT(t)Riui(t))dt≤∑i=1Jγi2∫0∞w-iT(t)w-i(t)dt.Remark 6.
From Definition5, we can see that, compared with the condition ∑i=1J∫0∞eiT(t)Qiei(t)dt≤∑i=1Jγi2∫0∞w-iT(t)w-i(t)dt commonly used for the H∞ tracking control problem, the practical H∞ performance index considers not only the effect of e(t) but also the effect of u(t). That is to say, the condition that the reducing of the tracking error e(t) needs to cost the much bigger gain of the controller can be avoided effectively.Theorem 7.
For the given constantδi, ηi, if there exist matrices Y2i1, Y2i2, Y2i3, Y2i4, Pi2, Ti12, Sji12, Sii12, Ei1pq, Ei2pq, Eij3pq, Eij4pq, Ei5pq, p,q=1,2, and positive definite matrices W2im, Pi1, Pi3, Ti11, Ti22, Sji11, Sji22, Sii11, Sii22, satisfying following eigenvalue problem (17), then the whole interconnected nonlinear systems are asymptotically stable with the H∞ tracking performance index γi, i,j=1,…,J, j≠i, l,m=1,…,ri, L=1,…,2mi-li(17)∑i=1Jγiminsubjectto∑i=1Jγi(18)and(19)(18)[Π~11Π~12Π~13Π~14Π~15Π~16Π~17Π~18δiB1il0Y2i3(W2il)TEi111Ei112*Π~22Π~23Π~24Π~25Π~26Π~27Π~28δiB1il-δiBriηiY2i3ηi(W2il)TEi121Ei122**Π~33Π~34Π~35Π~36Π~37Π~38B1il000Ei211Ei212***Π~44Π~45Π~46Π~47Π~48B1il-Bri00Ei221Ei122****Π~55Π~56Π~57Π~58Π~59(Ei521)T00Eij311Eij312*****Π~66Π~67Π~68Π~69(Ei522)T00Eij321Eij322******Π~77Π~78Π~79-(Ei521)T00Eij411Eij412*******Π~88Π~89-(Ei522)T00Eij421Eij422********-γi2I000Ei511Ei512*********-γi2I00Ei521Ei522**********-Qi-1000***********-Ri-100************-Ti11τji0-Ti12τji0*************-Ti22τji0]<0(19)[Ti11Ti12*Ti22]>0,[Pi1Pi2*Pi3]>0,[Sji11Sji12*Sji22]>0,[Sii11Sii12*Sii22]>0,
where
(20)Π~11=δiAilY2i1+δi(AilY2i1)T+δiB2ilλLiW2im+δi(B2ilλLiW2im)T+Sii11,Π~12=δiAilY2i2+δiηiB2ilλLiW2im+δi((Ail-Ari)Y2i1)T+δi(AriY2i3+B2ilλLiW2im)T+Sii12,Π~22=δi(Ail-Ari)Y2i2+δi((Ail-Ari)Y2i2)T+δiηi(AriY2i3+B2ilλLiW2im)+δiηi(AriY2i3+B2ilλLiW2im)T+Sii22,Π~13=(AilY2i1)T+(B2ilλLiW2im)T+Pi1-δiY2i1,Π~14=((Ail-Ari)Y2i1)T+(AriY2i3+B2ilλLiW2im)T+Pi2-δiY2i2,Π~23=(AilY2i2)T+ηi(B2ilλLiW2im)T+(Pi2)T-δiY2i3,Π~24=((Ail-Ari)Y2i2)T+ηi(AriY2i3+B2ilλLiW2im)T+Pi3-δiηiY2i3,Π~33=-(Y2i1)T-Y2i1+τji0Ti11,Π~34=-(Y2i3)T-Y2i2+τji0Ti12,Π~44=-ηi(Y2i3)T-ηiY2i3+τji0Ti22,Π~15=Ei111,Π~ijlm16=Ei112,Π~25=Ei121,Π~ijlm26=Ei122,Π~35=Ei211,Π~ijlm36=Ei212,Π~45=Ei221,Π~ijlm46=Ei222,Π~55=(Eij311)T+Eij311,Π~56=(Eij321)T+Eij312,Π~66=(Eij322)T+Eij322,Π~17=-Ei111+δiCjilY2i1,Π~27=-Ei121+δiCjilY2i1,Π~18=-Ei112+δiCjilY2i2,Π~28=-Ei122+δiCjilY2i2,Π~37=-Ei211+CjilY2i1,Π~47=-Ei221+CjilY2i1,Π~38=-Ei212+CjilY2i2,Π~48=-Ei222+CjilY2i2,Π~57=-Eij311+(Eij411)T,Π~58=-Eij312+(Eij412)T,Π~67=-Eij321+(Eij421)T,Π~68=-Eij322+(Eij422)T,Π~77=-Eij411-(Eij411)T-(1-H~ij)Sji11,Π~78=-Eij412-(Eij412)T-(1-H~ij)Sji12,Π~88=-Eij422-(Eij422)T-(1-H~ij)Sji22,Π~59=(Ei511)T,Π~69=(Ei512)T,Π~79=-(Ei511)T,Π~89=-(Ei512)T.Proof.
We choose the following Lyapunov function for the whole interconnected system (13):
(21)V(Xt)=V1(Xt)+V2(Xt)+V3(Xt),V1(Xt)=∑i=1JξiT(t)ETP~iξi(t),V2(Xt)=∑i=1J∑j=1,j≠iJ∫t-τji(t)tx-i(s)TSix-i(s)ds,V3(Xt)=∑i=1J∑j=1,j≠iJ∫-τji00∫t+θtx-˙i(s)TTix-˙i(s)dsdθ,
where Si, Ti are positive definite matrices, ETP~i=P~iTE≥0, and denote
(22)P~i=[P0i0P1iP2i],P0i=P0iT.
Computing the time derivative ofV(Xt), we have
(23)V˙1(Xt)=∑i=1J∑l=1ri∑m=1riμilμim((∑j=1,j≠iJC~jilx¯j(t-τij(t)))ξiT(A~ilmTP~i+P~iTA~ilm)ξi+ξiTP~iT(∑j=1,j≠iJC~jilx¯j(t-τij(t)))+(∑j=1,j≠iJx¯jT(t-τij(t))C~jilT)P~iξi+ξiTP~iTB~ilw¯i+w¯iTB~ilTP~iξi(∑j=1,j≠iJC~jilx¯j(t-τij(t))))V˙1(Xt)=∑i=1J∑j=1,j≠iJ∑l=1ri∑m=1riμilμim[ξix¯j(t-τij(t))w¯i]T[A¯ilmTP1i+P1iTA¯ilmA¯ilmTP2i+P0iT-P1iTP1iTC¯jilP1iTB¯il*-P2i-P2iTP2iTC¯jilP2iTB¯il**00***0]×[ξix¯j(t-τij(t))w¯i],V˙2(Xt)≤∑i=1J∑j=1,j≠iJ(x¯i(t)TSix¯i(t)-(1-H~ij)x¯j(t-τij(t))TSjx¯j(t-τij(t))),V˙3(Xt)=∑i=1J∑j=1,j≠iJ(τji0x-˙iTTix-˙i-∫t-τji0tx-˙iT(s)Tix-˙i(s)ds)V˙3(Xt)≤∑i=1J∑j=1,j≠iJ(τji0x-˙iTTix-˙i-∫t-τij(t)tx-˙jT(s)Tjx-˙j(s)ds).
From the Leibniz-Newton formula, the following equation is considered:(24)2(x¯iTEi1+x-˙iTEi2+x¯jTEij3+x¯jT(t-τij(t))Eij4+w¯iTEi5)×(x¯j-x¯j(t-τij(t))-∫t-τij(t)tx-˙j(s)ds)=0.
From (24), we have
(25)2(x¯iTEi1+x-˙iTEi2+x¯jTEij3+x¯jT(t-τij(t))Eij4(t-τij(t))+w¯iTEi5x¯iTEi1+x-˙iTEi2+x¯jTEij3+x¯jT(t-τij(t)))(x¯j-x¯j(t-τij(t)))=Γ¯ijTΦij1Γ¯ij,
where
(26)Γij=[ξix¯jx¯j(t-τij)],Γ¯ij=[ξix¯jx¯j(t-τij)w¯i],Φij1=[00Ei1-Ei10*0Ei2-Ei20**Eij3+Eij3T-Eij3+Eij4TEi5T***-Eij4-Eij4T-Ei5T****0],-2(x¯iTEi1+x-˙iTEi2+x¯jTEij3+x¯jT(t-τij(t))Eij4(t-τij(t))+w¯iTEi5x¯iTEi1+x-˙iTEi2+x¯jTEij3+x¯jT(t-τij(t))Eij4)∫t-τij(t)tx-˙j(s)ds≤τij0Γ¯ijTE¯iTj-1E¯iTΓ¯ij+∫t-τij(t)tx-˙jT(s)Tjx-˙j(s)ds,
where E¯i=[Ei1Ei2Eij3Eij4Ei5]T.
According to Lemma3, we have
(27)eiT(t)Qiei(t)+uiT(t)Riui(t)=∑l=1ri∑m=1riμilμimeiT(t)(Qi+(Kil)TRiKim)ei(t)≤∑l=1riμileiT(t)(Qi+(Kil)TRiKil)ei(t).
LettingP1i=δiP2i and using (15), we have
(28)V˙+∑i=1JeiT(t)Qiei(t)+∑i=1JuiT(t)Riui(t)V˙-∑i=1Jγi2w¯iTw¯iV˙+∑i=1J∑j=1,j≠iJ2(x¯iTEi1+x-˙iTEi2+x¯jTEij3(t-τij(t))+x¯jT(t-τij(t))Eij4+w¯iTEi5x¯iTEi1+x-˙iTEi2+x¯jTEij30)V˙×(x¯j-x¯j(t-τij(t))-∫t-τij(t)tx-˙j(s)ds)≤∑i=1J∑j=1,j≠iJ∑l=1ri∑m=1riμilμimΓ¯ijTΠijlmΓ¯ij,
where
(29)Πijlm=[Πijlm11Πijlm12Πijlm13Πijlm14Πijlm15*Πijlm22Πijlm23Πijlm24Πijlm25**Πijlm33Πijlm34Πijlm35***Πijlm44Πijlm45]+τij0E¯iTj-1E¯iT.
From (29), we can see that, if Πijlm<0, then Πijlm′<0, where
(30)Πijlm11=δiP2iTA¯ilm+δiA¯ilmTP2i+Q¯i+Si,Πijlm12=A¯ilmTP2i+P0iT-δiP2iT,Πijlm22=-P2iT-P2i+τji0Ti,Πijlm13=Ei1,Πijlm23=Ei2,Πijlm33=Eij3+Eij3T,Πijlm14=-Ei1+δiP2iTC¯jil,Πijlm24=-Ei2+P2iTC¯jil,Πijlm34=-Eij3+Eij4T,Πijlm15=δiP2iTB¯il,Πijlm44=-Eij4-Eij4T-(1-H~ij)Sj,Πijlm25=P2iTB¯il,Πijlm35=Ei5T,Πijlm45=-Ei5T,Πijlm55=-γi2I,Q¯i=diag[0Qi+(Kil)TRiKil],Πijlm′=[Πijlm11Πijlm12Πijlm13Πijlm14Ei1*Πijlm22Πijlm23Πilm24Ei2**Πijlm33Πijlm34Eij3***Πijlm44Eij4****-Tjτij0].
Ifw¯i=0,
(31)V˙≤∑i=1J∑j=1,j≠iJ∑l=1ri∑m=1riμilμimΓijTΠijlm′Γij<0,
then the whole nonlinear interconnected systems are asymptotically stable.
Ifw¯i≠0,
(32)V˙+∑i=1JeiT(t)Qiei(t)+∑i=1JuiT(t)Riui(t)-∑i=1Jγi2w¯iT(t)w¯i(t)<0.
With zero initial condition (Φ(t)=0, t∈[-τ¯,0]), hence V(Φ(t))=0, t∈[-τ¯,0], and V(X∞)≥0, integrating both sides of (32) from 0 to ∞, we have
(33)V(X∞)-V(Φ(0))≤∑i=1Jγi2∫0∞w¯iT(t)w¯i(t)dt≤-∑i=1J∫0∞(eiT(t)Qiei(t)+uiT(t)Riui(t))dt
implying that ∑i=1J∫0∞(eiT(t)Qiei(t)+uiT(t)Riui(t))dt≤∑i=1Jγi2∫0∞w¯iT(t)w¯i(t)dt.
LetY2i=P2i-1, E~i1=Y2iTEi1Y2i, E~i2=Y2iTEi2Y2i, E~ij3=Y2iTEij3Y2i, E~ij4=Y2iTEij4Y2i, E~i5=Ei5Y2i. By Schur complement and premultiplying and postmultiplying to (29) by positive-define matrix diag[P2i-TP2i-TP2i-TP2i-TIP2i-T], diag[P2i-1P2i-1P2i-1P2i-1IP2i-1], respectively, we have
(34)Π^ijlm=[Π^ijlm11Π^ijlm12Π^ijlm13Π^ijlm14Π^ijlm15E~i1*Π^ijlm22Π^ijlm23Π^ijlm24Π^ijlm25E~i2**Π^ijlm33Π^ijlm34Π^ijlm35E~ij3***Π^ijlm44Π^ijlm45E~ij4****Π^ijlm55E~i5*****-T~iτji0],
where
(35)Π^ijlm11=δiA¯ilmY2i+δiY2iTA¯ilmT+Y2iTQ¯iY2i+S~ii,Π^ijlm12=Y2iTA¯ilmT+Pi-δiY2i,Π^ijlm22=-Y2iT-Y2i+τji0T~i,Π^ijlm13=E~i1,Π^ijlm23=E~i2,Π^ijlm33=E~ij3+E~ij3T,Π^ijlm14=-E~i1+δiC¯jilY2i,Π^ijlm24=-E~i2+C¯jilY2i,Π^ijlm34=-E~ij3+E~ij4T,Π^ijlm15=δiB¯il,Π^ijlm25=B¯il,Π^ijlm44=-E~ij4-E~ij4T-(1-H~ij)S~ji,Π^ijlm35=E~i5T,Π^ijlm45=-E~i5T,Π^ijlm55=-γi2I,Q¯i=diag[0Qi+(Kil)TRiKil],T~i=Y2iTTiY2i,S~ji=Y2iTSjY2i.
DenotingW2im=KimY2i3, Y2i4=ηiY2i3, E~i1=[Ei111Ei112Ei121Ei122], E~i2=[Ei211Ei212Ei221Ei222], E~ij3=[Eij311Eij312Eij321Eij322], E~ij4=[Eij411Eij412Eij421Eij422], E~i5=[Ei511Ei512Ei521Ei522], Pi=[Pi1Pi2*Pi3], T~i=[Ti11Ti12*Ti22], S~ji=[Sji11Sji12*Sji22], S~ii=[Sii11Sii12*Sii22], Y2i=[Y2i1Y2i2Y2i3Y2i4], using (13), and by Schur complement, the proof is completed.Remark 8.
Firstly, utilizing the descriptor model transformation, LMIs-based conditions with some free variables are derived. Secondly, whenw¯i(t)=0, compared with the stable conditions in the sense of Lyapunov in [13], we obtain the asymptotically stable conditions. Finally, compared with [16], the time-varying delay fuzzy large systems are considered, and a more practical H∞ performance index is considered. Therefore, obtained results are new and less conservative.
IfJ=1, then the T-S model-based interconnected systems (13) can be transformed into a common T-S system as follows:
(36)x-˙(t)=∑l=1r∑m=1rμlμm(A¯lmx¯(t)+B¯lw¯(t)),
where
(37)x¯(t)=[x(t)e(t)],w¯(t)=[w(t)v(t)],B¯l=[B1l0B1l-Br],A¯lm=∑L=12m1-l1[AlB2lλLKmAl-ArAr+B2lλLKm], and l1 denotes the number of no failure actuators.Corollary 9.
For the given constantδ, η, if there exist matrices Y1, Y2, Y3, Y4, P2 and positive definite matrices W2m, P1, P3, satisfy (38), then the T-S fuzzy system (36) is asymptotically stable with the H∞ tracking performance index γ, and the gain of controllers Km=W2mY3-1, l,m=1,…,r, L=1,…,2mi-l1. 							(38)minsubjecttoγ(39)and(40),(39)[P1P2*P3]>0,(40)[Π¯11Π¯12Π¯13Π¯14δB1l0Y3Π¯18*Π¯22Π¯23Π¯24δB1l-δBrηiY3Π¯28**Π¯33Π¯34B1l000***Π¯44B1l-Br00****-γ2I000*****-γ2I00******-Q-10*******Π¯88]<0,
where
(41)Π¯11=δAlY1+δ(AlY1)T+δB2lλLW2m+δ(B2lλLW2m)T,Π¯12=δAlY2+δηB2lλLW2m+δ((Al-Ar)Y1)T+δ(ArY3+B2lλLiW2m)T,Π¯22=δ(Al-Ar)Y2+δ((Al-Ar)Y2)T+δη(ArY3+B2lλLW2m)+δη(ArY3+B2lλLW2m)T,Π¯13=(AlY1)T+(B2lλLW2m)T+P1-δY1,Π¯14=((Al-Ar)Y1)T+(ArY3+B2lλLW2m)T+P2-δY2,Π¯23=(AlY2)T+η(B2lλLW2m)T+P2T-δY3,Π¯24=((Al-Ar)Y2)T+η(ArY3+B2lλLW2m)T+P3-δηY3,Π¯33=-Y1T-Y1,Π¯34=-Y3T-Y2,Π¯44=-ηY3T-ηY3,Π¯18=(W2l)T,Π¯28=η(W2l)T,Π¯88=-R-1.


## 4. Illustrative Examples

Example 1.
Consider the two-machine interconnected systems which are composed of two subsystems as follows [13]:
(42)x˙i1(t)=xi2(t),x˙i2(t)=-DdiM1ixi2(t)+1M1iui(t)+∑j=1,j≠i2E1iE1jYijMi×(cos(δij0-θij)(t-τij(t))-cos(xi1(t)-xj1(t-τij(t))+δij0-θij))+B1iwi2,i=1,2.
We assume the two-machine interconnected systems’ parameters as follows:(43)E11=1.017,E12=1.005,M11=1.03,M12=1.25,Dd1=0.8,Dd2=1.2,Y12=Y21=1.98,θ12=-θ21=1.5,δ120=-δ210=1.2,B11=B12=diag[1,1],τ12(t)=τ21(t)=0.6sin2(t)+1.2.
The systems are approximated by the following nine-rule fuzzy model which is the same as [13], and the details of the rules are omitted here.
Other parameters are shown as follows:(44)Ari=[01-100-101],Bri=[1001],v1(t)=[0100cos(0.5t)],v2(t)=[0100sin(0.5t)].
Here, we assume that there exists the actuator failure only in the actuator of the first subsystem. Letα_1=0.2, α¯1=1, α_2=α¯2=α2=1; that is λ11=0.2, λ21=1, λ12=1. The actuator failure parameter α1 is shown in Figure 1.
Letδ1=δ2=1, η1=η2=1, Q1=Q2=diag(1,1), R1=R2=0.02, according to the conditions proposed in Theorem 7; we have K1l=[-42.6186-18.7306], K2l=[-13.7741-8.6137], l=1,…,9. For the initial condition x1(0)=[-13]T, x2(0)=[1-2]T and the disturbances w12(t)=sin(2πt)e-0.1t, w22(t)=cos(2πt)e-0.1t, the tracking trajectories are shown in Figures 2 and 3, in which xrij,i=1,2, j=1,2, denote the states of the reference model, xij,i=1,2, j=1,2, denote the states when there are no failure actuators, and xfij,i=1,2, j=1,2, denote the states when there exist partial failure actuators for t∈[0.2,5]. It can be seen that there are nearly no differences between xfij and xij. Therefore, when there exist the actuator failures, the performances of the system are not influenced. Compared with the result K1l≈[-140-67], K2l≈[-183-85], l=1,…,9 in [13], the gains of controllers are much smaller, and the control performances are nearly the same.Figure 1
The actuator failure parameter.Figure 2
The states of subsystem 1.Figure 3
The states of subsystem 2.Example 2.
Consider the following tracking system [16]:
(45)x˙(t)=[-1.410-1010-1-2]x(t)+[010]u(t)+[111]w(t),x˙r(t)=[-110.6-0.5-10-1-1-3]xr(t)+[08sin(t)0].
Letα_=0.2, α¯=1, the actuator failure parameter α is the same as Example 1, δ=1, η=0.8, Q=diag(1,1,1), R=0.01, according to the conditions proposed in Corollary 9; we have γ2=1.1291, K=[-2.9191-12.6548-3.3239]. For the initial condition x(0)=[22]T, xr(0)=[00]T, and the disturbance w(t)=0.1cos(t), the tracking trajectories are shown in Figures 4, 5, and 6. Compared with the result γ2=1.0751, K=103*[-4.1688-9.1600-1.3698] in [16], the gain of controller is much smaller, and the control performances are nearly the same. For the different δ, η, the trajectories of γ2 are shown in Figure 7.Figure 4
The trajectories of the statex1, reference state xr1, the state xf1 using [16], and the disturbance w.Figure 5
The trajectories of the statex2, reference state xr2, the state xf2 using [16], and the disturbance w.Figure 6
The trajectories of the statex3, reference state xr3, the state xf3 using [16], and the disturbance w.Figure 7
The trajectories ofγ2 about δ and η.


## 5. Conclusion

This paper investigates the robust and reliable decentralizedH∞ fuzzy tracking control issue for the fuzzy interconnected systems with time-varying delay, which consist of a number of T-S fuzzy subsystems with interconnections. Firstly, the ordinary fuzzy interconnected systems are equivalently transformed to the fuzzy descriptor systems; then the new LMIs-based conditions with some free variables are derived to guarantee the H∞ tracking performance not only when all control components are operating well, but also in the presence of some possible actuator failures. Finally, two simulation examples are provided to illustrate the effectiveness of the proposed method.



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*Source: 705609-2014-02-05.xml*