Open access peer-reviewed chapter

Delay-Dependent Generalized H2 Control for Discrete-Time Fuzzy Systems with Infinite-Distributed Delays

By Jun-min Li, Jiang-rong Li and Zhi-le Xia

Submitted: March 14th 2012Reviewed: July 22nd 2012Published: December 5th 2012

DOI: 10.5772/51778

Downloaded: 938

1. Introduction

In recent years, there has been significant interest in the study of stability analysis and controller synthesis for Takagi-Sugeno(T-S) fuzzy systems, which has been used to approximate certain complex nonlinear systems [1]. Hence it is important to study their stability analysis and controller synthesis. A rich body of literature has appeared on the stability analysis and synthesis problems for T-S fuzzy systems [2-6]. However, these results rely on the existence of a common quadratic Lyapunov function (CQLF) for all the local models. In fact, such a CQLF might not exist for many fuzzy systems, especially for highly nonlinear complex systems. Therefore, stability analysis and controller synthesis based on CQLF tend to be more conservative. At the same time, a number of methods based on piecewise quadratic Lyapunov function (PQLF) for T-S fuzzy systems have been proposed in [7-14]. The basic idea of these methods is to design a controller for each local model and to construct a global piecewise controller from closed-loop fuzzy control system is established with a PQLF. The authors in [7,13] considered the information of membership function, a novel piecewise continuous quadratic Lyapunov function method has been proposed for stability analysis of T-S fuzzy systems. It is shown that the PQLF is a much richer class of Lyapunov function candidates than CQLF, it is able to deal with a large class of fuzzy systems and obtained results are less conservative.

On the other hand, it is well known that time delay is a main source of instability and bad performance of the dynamic systems. Recently, a number of important analysis and synthesis results have been derived for T-S fuzzy delay systems [4-7, 11, 13]. However, it should be pointed out that most of the time-delay results for T-S fuzzy systems are constant delay or time-varying delay [4-5, 7, 11, and 13]. In fact, Distributed delay occurs very often in reality and it has been drawing increasing attention. However, almost all existing works on distributed delays have focused on continuous-time systems that are described in the form of either finite or infinite integral and delay-independent. It is well known that the discrete-time system is in a better position to model digitally transmitted signals in a dynamic way than its continuous-time analogue. Generalized H2 control is an important branch of modern control theories, it is useful for handling stochastic aspects such as measurement noise and random disturbances [10]. Therefore, it becomes desirable to study the generalized H2 control problem for the discrete-time systems with distributed delays. The authors in [6] have derived the delay-independent robust H stability criteria for discrete-time T-S fuzzy systems with infinite-distributed delays. Recently, many robust fuzzy control strategies have been proposed a class of nonlinear discrete-time systems with time-varying delay and disturbance [15-33]. These results rely on the existence CLKF for all local models, which lead to be conservative. It is observed, based on the PLKF, the delay-dependent generalized H2 control problem for discrete-time T-S fuzzy systems with infinite-distributed delays has not been addressed yet and remains to be challenging.

Motivated by the above concerns, this paper deals with the generalized H2 control problem for a class of discrete time T-S fuzzy systems with infinite-distributed delays. Based on the proposed Delay-dependent PLKF(DDPLKF), the stabilization condition and controller design method are derived for discrete time T-S fuzzy systems with infinite-distributed delays. It is shown that the control laws can be obtained by solving a set of LMIs. A simulation example is presented to illustrate the effectiveness of the proposed design procedures.

Notation: The superscript “T” stands for matrix transposition, R n denotes the n-dimensional Euclidean space, R n×m is the set of all n×m real matrices, I is an identity matrix, the notation P>0(P≥0) means that P is symmetric and positive(nonnegative) definite, diag{…} stands for a block diagonal matrix. Z - denotes the set of negative integers. For symmetric block matrices, the notation * is used as an ellipsis for the terms that are induced by symmetry. In addition, matrices, if not explicitly stated, are assumed to have compatible dimensions.

2. Problem Formulation

The following discrete-time T-S fuzzy dynamic systems with infinite-distributed delays [6] can be used to represent a class of complex nonlinear time-delay systems with both local analytic linear models and fuzzy inference rules:

Rj:ifs1(t)isFj1ands2(t)isFj2andandsg(t)isFjg,thenx(t+1)=Ajx(t)+Adjd=1μdx(td)+B1ju(t)+Djv(t)z(t)=Cjx(t)+B2ju(t)x(t)=φ(t)tZj=1,2rE1

where R j , jN:={1,2,…, r} denotes the j-th fuzzy inference rule, r the number of the inference rules. F ji (i=1, 2,…, g) are the fuzzy sets, s(t)=[s 1(t), s 2(t),…, s g(t)]∈R s the premise variable vector, x(t)∈R n the state vector, z(t)∈R q the controlled output vector, u(t)∈R m the control input vector, v(t)∈l 2[0 ∞) the disturbance input, φ(t) the initial state, and (A j, A dj , B 1j, D j, C j , B 2j) represent the j-th local model of the fuzzy system (1).

The constants μ d ≥0 (d =1,2, …) satisfy the following convergence conditions:

μ¯:=d=1+μdd=1+dμd+E2

Remark 1. The delay term d=1+μdx(td)in the fuzzy system (1), is the so-called infinitely distributed delay in the discrete-time setting. The description of the discrete-time-distributed delays has been firstly proposed in the [6], and we aim to study the generalized H2 control problem for discrete-time fuzzy systems with such kind of distributed delays in this paper, which is different from one in [6].

Remark 2. In this paper, similar to the convergence restriction on the delay kernels of infinite-distributed delays for continuous-time systems, the constants μ d (d =1,2, …)are assumed to satisfy the convergence condition (2), which can guarantee the convergence of the terms of infinite delays as well as the DDPLKF defined later.

By using a standard fuzzy inference method, that is using a center-average defuzzifiers product fuzzy inference, and singleton fuzzifier, the dynamic fuzzy model (1) can be expressed by the following global model:

x(t+1)=j=1rhj(s(t))[Ajx(t)+Adjd=1μdx(td)+B1ju(t)+Djv(t)]z(t)=j=1rhj(s(t))[Cjx(t)+B2ju(t)]E3

where hj(s(t))=ωj(s(t))j=1rωj(s(t)),ωj(s(t))=i=1gFji(s(t)),with Fji(s(t))being the grade of membership of si(t)inFij,ωj(s(t))0has the following basic property:

ωj(s(t))0,j=1rωj(s(t))0,jNtE4

and therefore

hj(s(t))0,j=1rhj(s(t))=1,jNtE5

In order to facilitate the design of less conservative H2 controller, we partition the premise variable space ΩRsinto m polyhedral regions Ωi by the boundaries [7]

Ωiv={s(t)|hi(s(t))=1,0hi(s(t+δ))0|δ|11,iN}E6

where v is the set of the face indexes of the polyhedral hull with satisfying

Ωi=v(Ωiv)

Based on the boundaries (6), m independent polyhedral regionsΩl,lL={1,2m}can be obtained satisfying

ΩlΩj=Ωiv,lj,l,jLE7

where L denotes the set of polyhedral region indexes.

In each region Ωl, we define the set

M(l):={i|hi(s(t))0,s(t)Ωl,iN},lLE8

Considering (5) and (8), in each region Ωl, we have

iM(l)hi(s(t))=1E9

and then, the fuzzy infinite-distributed delays system (1) can be expressed as follows:

x(t+1)=iM(l)hi(s(t))[Aix(t)+Adid=1μdx(td)+B1iu(t)+Div(t)]z(t)=iM(l)hi(s(t))[Cix(t)+B2iu(t)]s(t)ΩlE10

Remark 3. According to the definition of (8), the polyhedral regions can be divided into two folds: operating and interpolation regions. For an operating region, the set M(l) contains only one element, and then, the system dynamic is governed by the s-th local model of the fuzzy system. For an interpolation region, the system dynamic is governed by a convex combination of several local models.

In this paper, we consider the generalized H2 controller design problem for the fuzzy system (1) or equivalently (10), give the following assumptions.

Assumption 1. When the state of the system transits from the region Ωl to Ωj at the time t, the dynamics of the system is governed by the dynamics of the region model of Ωl at that time t.

For future use, we define a setΘthat represents all possible transitions from one region to itself or another regions, that is

Θ={(l,j)|s(t)Ωl,s(t+1)Ωjl,jL}E11

Here l = j, when the system stays in the same region Ωl, and lj, when the system transits from the region Ωl to another one Ωj.

Considering the fuzzy system (10), choose the following non-fragile piecewise state feedback controller

u(t)=(Kl+ΔKl)x(t)s(t)ΩllLE12

here ΔK l are unknown real matrix functions representing time varying parametric uncertainties, which are assumed to be of the form

ΔKl=ElUl(t)Hl,UlT(t)Ul(t)I,Ul(t)Rl1×l2E13

where E l , H l are known constant matrices, and Ul(t)Rl1×l2are unknown real time varying matrix satisfyingΔUlT(t)ΔUlI.

Then, the closed-loop T-S system is governed by

x(t+1)=A¯clx(t)+Adld=1μdx(td)+Dlv(t)z(t)=C¯clx(t)E14

for s(t)Ωl,lLwhere

A¯cl=iM(l)hiAil,Adl=iM(l)hiAdi,Dl=iM(l)hiDi,C¯cl=iM(l)hiCilAil=AiB1iK¯l,Cil=CiB2iK¯l

Before formulation the problem to be investigated, we first introduce the following concept for the system (14).

Definition 1. [ 10 ] Let a constant γ>0MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4SdCMaeyOpa4JaaGimaaaa@395D@be given. The closed-loop fuzzy system (14) is said to be stable with generalized H2 performance if both of the following conditions are satisfied:

  • The disturbance-free fuzzy system is globally asymptotically stable.

  • Subject to assumption of zero initial conditions, the controlled output satisfies

||z||γ||v||2E15

for all non-zero v ∈ I2.

Now, we introduce the following lemmas that will be used in the development of our main result.

Lemma 1. [ 6 ] Let MRn×nbe a positive semi-definite matrix, xi(t)Rnand constant

ai>0(i=1,2,)MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyamaaBaaaleaacaWGPbaabeaakiabg6da+iaaicdacaGGOaGaamyAaiabg2da9iaaigdacaGGSaGaaGOmaiaacYcacqWIVlctcaGGPaaaaa@41D2@, if the series concerned is convergent, then we have

(i=1aixi)TM(i=1aixi)(i=1ai)i=1aixiMxiE16

Lemma 2. [ 14 ] For the real matrices P1,P2,P3,P4,A,Ad,B,Xj(j=1,,5)and Di(i=1,,10)with compatible dimensions, the inequalities show in (17) and (18) at the following are equivalent, where Uis an extra slack nonsingular matrix.

(a)[He{P1TA}+D1P1TAd+ATP2+D2ATP3+D3ATP4+P1TB+D4X1*He{P2TAd}+D5AdTP3+D6AdTP4+P2TB+D7X2**D8P3TB+D9X3***He{BTP4}+D10X4****X5]0E17
(b)[He{U}P1+UTA2P2+UTAdP3P4+UTB0*D1D2D3D4X1**D5D6D7X2***D8D9X3****D10X4*****X5]0E18

where He{}stands for+T.

3. Main Results

Based on the proposed partition method, the following DDPLKF is proposed to develop the stability condition for the closed-loop system of (14).

V(t)=V1(t)+V2(t)+V3(t)V1(t)=2x(t)TP¯lx(t),V2(t)=d=1μdk=tdt1x(k)TQ¯x(k)V3(t)=d=1μdi=d1l=t+it1η(l)TZ¯η(l)lLE19

whereP¯l=FTPlF,Q¯=FTQF,Z¯=FTZF, andPl,Q,Z>0MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiuamaaBaaaleaacaWGSbaabeaakiaacYcacaWGrbGaaiilaiaadQfacqGH+aGpcaaIWaaaaa@3CC7@,Fis nonsingular matrix, andη(t)=x(t+1)x(t).

Then, we are ready to present the generalized H2 stability condition of (14) in terms of LMIs as follows

Theorem 1. Given a constantγ>0MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4SdCMaeyOpa4JaaGimaaaa@395D@, the closed-loop fuzzy system (14) with infinite distributed delays is stable with generalized H2 performanceγ, if there exists a set of positive definite matricesPl,Q,Z>0MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiuamaaBaaaleaacaWGSbaabeaakiaacYcacaWGrbGaaiilaiaadQfacqGH+aGpcaaIWaaaaa@3CC7@, the nonsingular matrix F and matrices Xli,Yli,lL,i=1,,4satisfying the following LMIs:

CilTCilγ2Pl0iM(l),lLE20
Πill0iM(l),lLE21
Πilj0iM(l),(l,j)ΘE22

where

          Πilj=[He{F}ΛiljYl2+AdiFPj+Yl3Yl4+Di0*Σl1Σl2Σl3Σl4Xl1**Σl5Σl6Σl7Xl2***Σl8Σl9Xl3****Σl10Xl4*****(d=1dμd)1Z]

with

              Λilj=Pj+Yl1+AiFB1iK¯lF,Σl1=μ¯Q2Pl+He{μ¯Xl1Yl1}, Σl2=Xl1+Xl2TYl2, Σl3=Xl3Yl1TYl3,Σl4=Xl4T+Yl4, Σl5=1μ¯QHe{Xl2}, Σl6=Xl3TYl2, Σl7=Xl4,Σl8=d=1μddZHe{Yl3}, Σl9=Yl4T, Σl10=I.

Proof. Taking the forward difference of (19) along the solution of the system (14), we have

ΔV(t)=V(t+1)V(t)=ΔV1+ΔV2+ΔV3

Assuming thats(t)Ωl,s(t+1)Ωj. The difference of Vi(t),i=1,2,3can be calculated, respectively, showing at the following

ΔV1(t)=2[A¯clx(t)+Adld=1μdx(td)+Dlv(t)]TP¯j[η(t)+x(t)]2xT(t)P¯lx(t)E23

ΔV2(t)=d=1μdτ=t+1dtxT(τ)Q¯x(τ)d=1μdτ=tdt1xT(τ)Q¯x(τ)=μ¯xT(t)Q¯x(t)d=1μdxT(td)Q¯x(td)E24

From Lemma1, we have

d=1μdxT(td)Q¯x(td)1μ¯(d=1μdx(td))TQ¯(d=1μdx(td))E25

Substituting (25) into (24), we have

ΔV2(t)μ¯xT(t)Q¯x(t)1μ¯(d=1μdx(td))TQ¯(d=1μdx(td))E26
ΔV3(t)=d=1μddη(t)TZ¯η(t)d=1μdl=tdt1η(l)TZ¯η(l)E27

Observing of the definition of η(t)and system (14), we can get the following equations:

Ξ1=2[xT(t)X¯l1+d=1μdxT(td)X¯l2+ηT(t)X¯l3+vT(t)Xl4U]×[μ¯x(t)d=1μdxT(td)d=1μdl=tdt1η(l)]=0E28
Ξ2=2[xT(t)Y¯l1+d=1μdxT(td)Y¯l2+ηT(t)Y¯l3+vT(t)Yl4U]×[(A¯liI)x(t)+Adi+Div(t)η(t)]=0E29

where X¯li=FTXliF1(i=1,2,3)

Since ±2aTbaTMa+bTM1bholds for compatible vectorsaandb, and any compatible matrixM>0, we have

2[xT(t)X¯l1+d=1μdxT(td)X¯l2+ηT(t)X¯l3+vT(t)Xl4U]×d=1μdl=tdt1η(l)d=1dμdξT(t)[X¯l1X¯l2X¯l3Xl4U]Z¯1[X¯l1X¯l2X¯l3Xl4U]Tξ(t)+d=1μdl=tdt1η(l)Z¯η(l)E30

withξ(t)=[xT(t),d=1μdxT(td),ηT(t),vT(t)]T

Then, from (23-30) and considering (14), we have

ΔV(t)vT(t)v(t)+vT(t)v(t)+Ξ1+Ξ2iM(l)hiξT(t)Ψiljξ(t)+vT(t)v(t)E31

where

Ψilj=[Φilj1Φilj2Φilj3Φilj4*Φilj5Φilj6Φilj7**Φilj8Φilj9***Φilj10]+d=1dμd[X¯l1X¯l2X¯l3Xl4U]Z¯1[X¯l1X¯l2X¯l3Xl4U]TE32

with

           

Then

ΔV(t)vT(t)v(t)<0E33

if

Ψilj0E34

Using lemma 2, (32) is equivalent to (33)

Ξilj=[He{U}P¯i+Y¯l1+UTA¯liY¯l2+UTAdiP¯j+Y¯l3UT(Yl4+Di)0*Σ¯l1Σ¯l2Σ¯l3UΣl4X¯l1**Σ¯l5Σ¯l6UΣl7X¯l1***Σ¯l8UΣl9X¯l1****Σl10Xl4U*****(d=1μdd)1Z¯]E35

whereΣ¯li=FTΣliF1(i=1,2,3,5,6,8,10)

Let U=F1,G=diag(F,F,F,F,I,F), pre- and post multiplying (35) by GT,G

respectively, then Ξiljis equivalent toΠilj.

Thus, if (21) and (22) holds, (32) is satisfied, which implies that

ΔV(t)vT(t)v(t)E36

It is noted that if the disturbance termv(t)=0, it follows from (31) that

ΔV(t)iM(l)hiζT(t)Ωiljζ(t)E37

with ζ(t)=[xT(t),d=1μdxT(td), ηT(t)]T

Ωilj=[Φilj1Φilj2Φilj3*Φilj5Φilj6**Φilj8]+d=1dμd[X¯l1X¯l2X¯l3]Z¯1[X¯l1X¯l2X¯l3]TE38

By Schur’s complement, LMI (32) impliesΩilj<0MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeuyQdC1aaSbaaSqaaiaadMgacaWGSbGaamOAaaqabaGccqGH8aapcaaIWaaaaa@3C44@, thenΔV(t)<0MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeuiLdqKaamOvaiaacIcacaWG0bGaaiykaiabgYda8iaaicdaaaa@3C45@. Therefore, the closed-loop system (14) with v(t) = 0 is globally asymptotically stable.

Now, to establish the generalized H2 performance for the closed-loop system (14), under zero-initial condition, and v(t)≠0, taking summation for the both sides of (36) leads to

V(x(T+1))t=0TvT(t)v(t)E39

It follows from (20) that

zT(t)z(t)=xT(t)C¯clTC¯clx(t)=iM(l)hiλT(t)[CilTCil00000000]λ(t)γ2λT(t)[P000Q000Z]λ(t)=γ2V(t)E40

with

λ(t)=[x(t),d=1μdτ=tdt1x(τ),d=1μdi=d1l=tdt1η(l)]

From (39) and (40), we have

z(t)2γ2v(t)22E41

The proof is completed.

The following theorem shows that the desired controller parameters and considered controller uncertain can be determined based on the results of Theorem 1.This can be easily proved along the lines of Theorem 1, and we, therefore, only keep necessary details in order to avoid unnecessary duplication.

Theorem 2. Consider the uncertain terms (12). Given a constantγ>0MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4SdCMaeyOpa4JaaGimaaaa@395D@, the closed-loop fuzzy system (14) with infinite-distributed delays is stable with generalized H2 performanceγ, if there exists a set of positive definite matricesPl,Q,Z>0MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiuamaaBaaaleaacaWGSbaabeaakiaacYcacaWGrbGaaiilaiaadQfacqGH+aGpcaaIWaaaaa@3CC7@, the nonsingular matrix Fand matrices Xli,Yli,Ml,lL,i=1,2,3,4satisfying the following LMIs:

[PlCiFB2iMlB2iHlF*γ2I+εlElTEl0**εlI]0iM(l),lLE42
ϒill0iM(l),lLE43
ϒilj0iM(l),(l,j)ΘE44

where

           ϒilj=[He{F}ΤiljYl2+AdiFPj+Yl3Yl4+Di00*Σl1Σl2Σl3Σl4Xl1B1iHlF**Σl5Σl6Σl7Xl20***Σl8Σl9Xl30****Σl10Xl40*****Γl0******εlI]MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@AF16@

with

          Τilj=Pj+Yl1+AiFB1iMl,Γl=(d=1dμd)1Z+εlElTEl.

Furthermore, the control law is given by

Kl=MlF1E45

Proof. In (20) and (21), replace Kl¯withKl+ΔKl, and then by S-procedure, we can easily obtain the results of this theorem, and the details are thus omitted.

Remark 4. If the global state space replace the transitionsΘand allPls in Theorem 2 become a commonP, Theorem 2 is regressed to Corollary 1, shown in the following.

Corollary 1. Consider the uncertain terms (12). Given a constantγ>0MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4SdCMaeyOpa4JaaGimaaaa@395D@, the closed-loop fuzzy system (14) with infinite-distributed delays is stable with generalized H2 performanceγ, if there exists a set of positive definite matricesPl,Q,Z>0MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiuamaaBaaaleaacaWGSbaabeaakiaacYcacaWGrbGaaiilaiaadQfacqGH+aGpcaaIWaaaaa@3CC7@, the nonsingular matrix Fand matrices Xli,Yli,Ml,lL,i=1,2,3,4satisfying the following LMIs:

[PCiFB2iMlB2iHlF*γ2I+εlElTEl0**εlI]0iM(l),lLE46
ϒil0iM(l),lLE47

where

                   ϒil=[He{F}ΤilYl2+AdiFPj+Yl3Yl4+Di00*Σl1Σl2Σl3Σl4Xl1B1iHlF**Σl5Σl6Σl7Xl20***Σl8Σl9Xl30****Σl10Xl40*****Γl0******εlI]MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@AD38@

with

                   

4. Numerical Examples

In this section, we will present two simulation examples to illustrate the controller design method developed in this paper.

Example 1. Consider the following modified Henon system with infinite distributed delays and external disturbance

x1(t+1)={cx1(t)+(1c)d=1+μdx1(td)}2+0.1x2(t)0.5d=1+μdx2(td)+u(t)+0.1v(t)x2(t+1)=x2(t)0.5x1(t)z1(t)=(1c)x1(t)+u(t)z2(t)=0.2x2(t)E48

where the constant c[0,1]is the retarded coefficient.

Lets(t)=cx1(t)+(1c)d=1+μdx1(td). Assume thats(t)[1,1]. The nonlinear term s2(t)can be exactly represented as

s2(t)=h1(s(t))(1)s(t)+h2(s(t))(1)s(t)

where theh1(s(t)),h2(s(t))[0,1], andh1(s(t))+h2(s(t))=1. By solving the equations, the membership functions h1(s(t))and h2(s(t))are obtained as

h1(s(t))=12(1s(t)),     h2(s(t))=12(1+s(t))

It can be seen from the aforementioned expressions that h1(s(t))=1and h2(s(t))=0whens(t)=1, and that h1(s(t))=0and h2(s(t))=1whens(t)=1. Then the nonlinear system in (48) can be approximately represented by the following T-S fuzzy model:

             R1:ifs(t)is 1, thenx(t+1)=A1x(t)+Ad1d=1μdx(td)+B11u(t)+D1V(t)      z(t)=C1x(t)+B21u(t)R2:ifs(t)is 1, thenx(t+1)=A2x(t)+Ad2d=1μdx(td)+B12u(t)+D2v(t)      z(t)=C2x(t)+B22u(t)

where

                      A1=[0.90.10.51]A1d=[0.10.500]B11=B12=[10],A2=[0.90.10.51]A2d=[0.10.500]D1=D2=[0.10],C1=C2=[0.1000.2], B21=B22=[10], E1=E2=[0.050], H1=H2=[0.10], e1=10,e2=11,V(t)=0.1cos(t)×exp(-0.05t).

The subspaces can be described by

Ω1={s(t)|1s(t)0}, Ω2={s(t)|0s(t)1}

Choosing the constantsc=0.9,μd=23d,d=10,we easily find thatμ¯=d=1μd=23<d=1dμd=2<+MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqr1ngBPrgifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaadaqdaaqaaiabeY7aTbaacaqG9aWaaabCaeaacqaH8oqBdaWgaaWcbaGaamizaaqabaaabaGaamizaiabg2da9iaaigdaaeaacqGHEisPa0GaeyyeIuoakiabg2da9iaaikdadaahaaWcbeqaaiabgkHiTiaaiodaaaGccqGH8aapdaaeWbqaaiaadsgacqaH8oqBdaWgaaWcbaGaamizaaqabaaabaGaamizaiabg2da9iaaigdaaeaacqGHEisPa0GaeyyeIuoakiabg2da9iaaikdacqGH8aapcqGHRaWkcqGHEisPaaa@5727@, which satisfies the convergence condition (2).

with the H2 performance index γmin=0.11,we solve (42)-(44) and obtain

                    P1=[0.19440.02480.02480.3342] ,  P2=[0.19510.02520.02520.3358] ,  Q=[0.28760.07460.07460.1636]Z=[0.00480.00190.00190.1275] , F=[0.39390.15160.04760.6285],K1=[0.02230.1702],   K2=[ 0.01710.1685]  .      

Simulation results with the above solutions for the H2 controller designs are shown Fig.1 and Fig.2

Figure 1.

The state evolution x1(t) of controlled system.

Figure 2.

The state evolution x 2(t) of controlled systems.

                   

Example 2. Consider a fuzzy discrete time system with the same form as in Example, but with different system matrices given by

                         A1=[0.9860.10.51],  A1d=[0.10.500],  B11=[00.5],  B12=[10],A2=[0.50.60.60.5],  A2d=[0.050.600],   D1D2=[0.10],C1=[0.02000.1], C2=[0.1000.3],   B21=B22=[10],E1=E2=[0.050],  H1=H2=[0.10] ,e1=10,e2=11e3=12,v(t)=0.1cos(t)×exp(-0.05t).                                       

We expanded the state space from [-1,1] to [3,3], the membership functions are given as

                  h1(s(t))={1s(t)[3,1],0.5s(t)+0.5s(t)[1,1].h2(s(t))={0.5s(t)+0.5s(t)[1,1],1s(t)[1,3].                                                                         

The subspaces are given as shown in Fig.3

Figure 3.

Membership functions and partition of subspaces.

Using the Theorem 2 and Corollary 1, respectively, the achievable minimum performance index for the H2 controller can be obtained and is summarized in Table 1.

ApproachPerformance
Common Lyapunov function based generalized H2 performance (Theorem 2)γmin=0,4586
Piecewise Lyapunov function based generalized H2 performance ( Corollary1)γmin=0,3975

Table 1.

Comparison for generalized H2 performance.

        

By using the LMI toolbox, we have

                     P1=[1.53590.57710.57711.4293],   P2=[1.52540.65400.65401.5478],   P3=[1.27540.56340.56341.4983],  Q=[1.81010.15680.15680.5915]Z=[0.03990.02850.02850.4640],  F=[3.10760.71190.86712.5352],K1=[0.00030.2297],  K2=[0.13110.0371],  K3=[0.11250.0005].       

The simulation results with the initial conditions are shown Fig.4 and Fig.5

Figure 4.

Trajectories from two initial conditions

Figure 5.

Trajectories from two initial conditions

5. Conclusions

This paper presents delay-dependent analysis and synthesis method for discrete-time T-S fuzzy systems with infinite-distributed delays. Based on a novel DDPLKF, the proposed stability and stabilization results are less conservative than the existing results based on the CLKF and delay independent method. The non-fragile stated feedback controller law has been developed so that the closed-loop fuzzy system is generalized H2 stable. It is also shown that the controller gains can be determined by solving a set of LMIs. A simulation example was presented to demonstrate the advantages of the proposed approach.

© 2012 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution 3.0 License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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Jun-min Li, Jiang-rong Li and Zhi-le Xia (December 5th 2012). Delay-Dependent Generalized H2 Control for Discrete-Time Fuzzy Systems with Infinite-Distributed Delays, Advances in Discrete Time Systems, Magdi S. Mahmoud, IntechOpen, DOI: 10.5772/51778. Available from:

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