# Delay-Dependent Stability Analysis of Uncertain Fuzzy Systems with State and Input Delays under Imperfect Premise Matching

Authors: Zejian Zhang; Xiao-Zhi Gao; Kai Zenger; Xianlin Huang
Journal: Mathematical Problems in Engineering
 (2013)

---

## Abstract

This paper discusses the stability and stabilization problem for uncertain T-S fuzzy systems with time-varying state and input delays. A new augmented Lyapunov function with an additional triple-integral term and different membership functions of the fuzzy models and fuzzy controllers are introduced to derive the stability criterion, which is less conservative than the existing results. Moreover, a new flexibility design method is also provided. Some numerical examples are given to demonstrate the effectiveness and less conservativeness of the proposed method.


---

## Body

## 1. Introduction

Since the Takagi-Sugeno (T-S) [1] fuzzy systems were firstly proposed in 1985, their stability analysis has received considerable research attention [2, 3]. However, most of the existing results are only for the T-S fuzzy systems free of time delay. Actually, time delay often occurs in many practical systems, such as [4, 5]. It has been shown that the existence of time delay usually becomes the source of instability and deteriorated performance of systems. Therefore, study of the time delay is important in both theory and practice [6]. The first stability analysis work on the T-S fuzzy systems with time delay is done in [7, 8] by using the Lyapunov-Razumikhin functional approach. If some uncertainties exist in a T-S fuzzy time delay system, they may also significantly affect the system performances and even cause unstable system outputs. Therefore, the issue of the stability for the uncertain T-S fuzzy time delay systems has been widely explored [9–11]. In the literature, two basic approaches have been utilized, that is, delay-independent approach [12] and delay-dependent approach. The latter makes use of the information on the length of the delays, and it is less conservative than the former one. A lot of stability analysis results have been reported based on the delay-dependent approach [13, 14].During the recent years, some research work for different types of the delays of the T-S fuzzy systems has been published, such as constant delay [15, 16], bounded time delay [17], time varying delay [18], and interval time varying delay [19, 20]. However, all these results are only for the T-S fuzzy systems with state delays. Thus, they may be invalid when applied to the systems with input delays. As we know, input delays extensively exist in industrial processes and can cause instability or serious performance deterioration. In fact, in modern industrial systems, sensors, controllers, and plants are often connected together, and the sampled data and controller signals are transmitted through networks. In view of this, the input delays should be taken into consideration for robust controller design. Intensive results on the stabilization for the T-S fuzzy systems with state and input delays are reported in [21–28]. For example, the work in [21] is based on the linear systems with input delays. In [22, 24], the authors only consider the fuzzy systems with constant input delays, and their results are conservative. In [23], the authors study the fuzzy systems with both the state and input delays. Unfortunately, the results are obtained without any uncertainty, and the state delay is assumed to be equal to the input delay. In [26], the uncertainty has been considered in the analysis, but the state delay is also assumed to be the same as the input delay. Some interesting results for the uncertain fuzzy systems with state and input delays have been obtained in [25, 27, 28], most of which introduce some Lyapunov-Krasovskii functions containing integral terms, for example, ∫t-τtxT(s)Qx(s)ds, and double-integral terms, ∫-τ0∫t+θtx˙T(s)Qx˙(s)dsdθ. Several triple-integral terms are used in the Lyapunov function [29] to yield less conservative results for the fuzzy systems with state delay. A large portion of robust controller design topics have been investigated on the basis of the Parallel Distribution Compensation (PDC) design technique, where the fuzzy controller shares the same premise membership functions as those of the T-S fuzzy time delay model [7–11, 15–20, 22–28]. As a matter of fact, if the membership functions in the premise of the fuzzy rules of the fuzzy controllers are allowed to be designed arbitrarily, we can even achieve better design flexibility. For instance, a fuzzy controller not sharing the same premise rules as those of the T-S fuzzy model referred to as imperfect premise matching is employed to control the nonlinear plants [30, 31], and [12] extends the available results to the T-S fuzzy time-delay systems with only the state delay.In this paper, an augmented Lyapunov-Krasovskii function that contains a triple-integral term is introduced to investigate the stability and stabilization problem for uncertain T-S fuzzy systems with the state and input delays under the imperfect premise matching, in which the fuzzy time delay model and fuzzy controller are with different premise. Some less conservative delay-dependent stability and robust stability conditions are obtained by two integral inequalities and a parameterized model transformation method. Moreover, different from the general PDC design technique, a new design approach of robust stable controllers is proposed. Two simulation examples are further given to illustrate that the proposed design methods are less conservative and more flexible.


## 2. System Description and Preliminaries

Letr be the number of the fuzzy rules describing the time delay nonlinear plant. The ith rule can be represented as follows.Iff1(x(t)) is M1i and … and fp(x(t)) is Mpi(1)x˙(t)=(A1i+ΔA1i(t))x(t)+(A2i+ΔA2i(t))×x(t-d1(t))+(Bi+ΔBi(t))u(t-d2(t)),x(t)=φ(t),t∈[-max(h1,h2),0],
where Mαi is a fuzzy term of rule i corresponding to the function fα(x(t)),α=1,2,…,p;i=1,2,…,r. x(t)∈Rn is the system state vector, and u(t)∈Rm is the input vector. The matrices A1i,A2i, and Bi,  i=1,2,…,r, are of appropriate dimensions. The initial condition φ(t) is a continuous vector-valued function. The delays d1(t) and d2(t) are time varying, and satisfy(2)0≤d1(t)≤h1,d˙1(t)≤μ1,0≤d2(t)≤h2,d˙2(t)≤μ2,
where hi are constants representing the upper bound of the delay. μ is a positive constant. ΔA1i, ΔA2i, and ΔBi denote the uncertainties in the system, and they are the form of
(3)[ΔA1iΔA2iΔBi]=DiKi(t)[E1iE2iEbi],i=1,2,…,r,
where Di, E1i,E2i, and Ebi are known constant matrices of appropriate dimensions, and Ki(t) is unknown matrix function satisfying KiT(t)Ki(t)≤I. I is an appropriately dimensioned identity matrix. Hence, the overall fuzzy model can be formulized as follows:
(4)x˙(t)=∑i=1rwi(x(t))[(A1i+ΔA1i(t))x(t)+(A2i+ΔA2i(t))×x(t-d1(t))+(Bi+ΔBi(t))×u(t-d2(t))],
where
(5)∑i=1rwi(x(t))=1,wi(x(t))≥0,wi(x(t))=μi(x(t))∑i=1rμi(x(t)),μi(x(t))=∏α=1pμMαi(fα(x(t))).wi(x(t)) is the normalized grade of membership function that is a nonlinear function of x(t). μMαi(fα(x(t))) is the grade of the membership corresponding to the fuzzy term of Mαi.Different from the popular PDC design technique, the following fuzzy control law under imperfect premise matching is employed to deal with the problem of stabilization via state feedback. Under the imperfect premise matching, thejth rule of the fuzzy controller is defined as follows.Ifg1(x(t)) is N1i and … and gq(x(t)) is Nqi(6)u(t)=Fjx(t),j=1,2,…,r,
where Nβj denotes the fuzzy set. q is a positive integer, and Fj∈Rm×n is the feedback gain of rule j. The state feedback fuzzy control law is represented by
(7)u(t)=∑j=1rmj(x(t))Fjx(t),
where
(8)∑j=1rmj(x(t))=1,mj(x(t))≥0,mj(x(t))=vj(x(t))∑j=1rvj(x(t)),vj(x(t))=∏β=1qvNβj(gβ(x(t))).mj(x(t)) is the normalized grade of membership function that is a nonlinear function of x(t). vNβj(gβ(x(t))) is the grade of the membership corresponding to the fuzzy term of Nβj.As the time varying delay is included in the control input, we have(9)u(t-d2(t))=∑j=1rmj(x(t-d2(t)))Fjx(t-d2(t)),
where
(10)∑j=1rmj(x(t-d2(t)))=1,mj(x(t-d2(t)))≥0.Lemma 1 (see [32] (Schur complement)).
Given constant matricesΩ1,Ω2, and Ω3, where Ω1=Ω1Tand  Ω2=Ω2T, there is Ω1+Ω3TΩ2-1Ω3<0, if and only if
(11)[Ω1Ω3T*-Ω2]<0or[-Ω2Ω3*-Ω1]<0.Lemma 2 (see [33]).
LetQ=QT,D,E, and K(t) satisfies KT(t)K(t)≤I; the following inequality holds
(12)Q+DK(t)E+ETKT(t)DT<0,
if and only if the following inequality holds for any smaller ε>0:
(13)Q+ε-1DDT+εETE<0.Lemma 3 (see [34]).
For any constant matrixΣ=ΣT>0 and a scalar τ>0 such that the following integrations are well defined, we have
(14)-∫t-τtxT(s)Σx(s)ds≤-1τ(∫t-τtx(s)ds)TΣ(∫t-τtx(s)ds),-∫-τ0∫t+θtxT(s)Σx(s)dsdθ≤-1τ2(∫-τ0∫t+θtx(s)dsdθ)TΣ(∫-τ0∫t+θtx(s)dsdθ).


## 3. Main Results

In this section, some new delay-dependent criteria for the T-S fuzzy systems with state and input delays are proposed by introducing a novel Lyapunov function with an additional triple-integral term under the imperfect premise matching. Moreover, a robust stabilization criterion is also investigated.


### 3.1. Stability of Nominal Fuzzy Systems

Firstly, we consider the control design of a state feedback control law under the imperfect premise matching that stabilizes the following nominal fuzzy time varying delay system:(15)x˙(t)=∑i=1r∑j=1rwi(x(t))mj(x(t-d2(t)))×[A1ix(t)+A2ix(t-d1(t))+BiFjx×(t-d2(t))],x(t)=φ(t),t∈[-max(h1,h2),0].Theorem 4.
Given scalarsh1≥0,h2≥0, μ1,μ2, and ti(i=2,…,8), the closed-loop system (15) is asymptotically stable for any 0≤di(t)≤hi(i=1,2) via the imperfect premise matching controller design technique, if the membership functions of the fuzzy model and fuzzy controller satisfy mj(x(t-d2(t)))-ρjwj(x(t))≥0 for all j, x(t) and x(t-d2(t)), where 0<ρj<1, and there exist  matrices P-11>0,P-22>0, W-11>0,W-22>0, Z-11>0,Z-22>0,N-11>0,N-22>0, Q-i>0, M-i>0, R-i>0(i=1,2), Λi=ΛiT∈R8n×8n>0(i=1,2,…,r),  and
(16)P-=[P-11P-12*P-22]>0,W-=[W-11W-12*W-22]>0,Z-=[Z-11Z-12*Z-22]>0,N-=[N-11N-12*N-22]>0,
and real matrices Yj(j=1,2,…,r), P-12W-12Z-12N-12, and X such that (30)–(32) hold. In addition, the stabilizing control law is given by
(17)u(t)=∑j=1rmj(x(t-d2(t)))YjX-Tx(t).Proof.
Choose a fuzzy weighting-dependent Lyapunov-Krasovskii functional candidate as(18)V(x(t))=V1(x(t))+V2(x(t))+V3(x(t))+V4(x(t))+V5(x(t)),V1(x(t))=ξ1T(t)Pξ1(t)+ξ2T(t)Wξ2(t),V2(x(t))=∫-h10∫t+θtξT(s)Zξ(s)dsdθ+∫-h20∫t+θtξT(s)Nξ(s)dsdθ,V3(x(t))=∫t-d1(t)txT(s)Q1x(s)ds+∫t-d2(t)txT(s)Q2x(s)ds,V4(x(t))=∫t-h1txT(s)M1x(s)ds+∫t-h2txT(s)M2x(s)ds,V5(x(t))=∫-h10∫θ0∫t+λtx˙T(s)R1x˙(s)dsdλdθ+∫-h20∫θ0∫t+λtx˙T(s)R2x˙(s)dsdλdθ,
where
(19)ξ1T(t)=[xT(t)(∫t-h1tx(s)ds)T],ξ2T(t)=[xT(t)(∫t-h2tx(s)ds)T],ξT(t)=[xT(t)x˙T(t)],P=[P11P12*P22]>0,W=[W11W12*W22]>0,Z=[Z11Z12*Z22]>0,N=[N11N12*N22]>0,
and Qi,Mi,andRi(i=1,2) are the positive-definite matrices to be determined.
The derivatives ofV(x(t)) along the trajectories of system (15) are as follows:
(20)V˙1(x(t))=2ξ1T(t)Pξ˙1(t)+2ξ2T(t)Wξ˙2(t)=2[xT(t)(∫t-h1tx(s)ds)T][P11P12*P22]×[x˙(t)x(t)-x(t-h1)]+2[xT(t)(∫t-h2tx(s)ds)T][W11W12*W22]×[x˙(t)x(t)-x(t-h2)],V˙2(x(t))=h1[xT(t)x˙T(t)][Z11Z12*Z22][x(t)x˙(t)]-∫t-h1t[xT(s)x˙T(s)][Z11Z12*Z22][x(s)x˙(s)]ds+h2[xT(t)x˙T(t)][N11N12*N22][x(t)x˙(t)]-∫t-h2t[xT(s)x˙T(s)][N11N12*N22][x(s)x˙(s)]ds,V˙3(x(t))=xT(t)Q1x(t)-(1-d˙1(t))xT(t-d1(t))×Q1x(t-d1(t))+xT(t)Q2x(t)-(1-d˙2(t))xT(t-d2(t))Q2x(t-d2(t))≤xT(t)Q1x(t)+xT(t)Q2x(t)-(1-μ1)×xT(t-d1(t))Q1x(t-d1(t))-(1-μ2)×xT(t-d2(t))Q2x(t-d2(t)),V˙4(x(t))=xT(t)M1x(t)-xT(t-h1)M1x(t-h1)+xT(t)M2x(t)-xT(t-h2)M2x(t-h2),V˙5(x(t))=12h12x˙T(t)R1x˙(t)-∫-h0∫t+θtx˙T(s)×R1x˙(s)dsdθ+12h22x˙T(t)R2x˙(t)-∫-h0∫t+θtx˙T(s)R2x˙(s)dsdθ.
From (15), the following equation can be obtained for any matrices Ti,i=1,2,…,8,
(21)2∑i=1r∑j=1rwi(x(t))mj(x(t-d2(t)))×[(∫t-h2tx(s)ds)TxT(t)T1+xT(t-d1(t))T2+xT(t-h1)T3+(∫t-h1tx(s)ds)TT4+xT(t-d2(t))T5+xT×(t-h2)T6+(∫t-h2tx(s)ds)TT7+x˙T(t)T8]×[A1ix(t)-A2ix(t-d1(t))-BiFjx(t-d2(t))-x˙(t)]=0.
By Lemma 3, we have(22)-∫t-h1t[xt(s)x˙T(s)][Z11Z12*Z22][x(s)x˙(s)]ds≤-1h1[(∫t-h1tx(s)ds)TxT(t)-xT(t-h1)]×[Z11Z12*Z22][∫t-h1tx(s)dsx(t)-x(t-h1)],-∫t-h2t[xT(s)x˙T(s)][N11N12*N22][x(s)x˙(s)]ds≤-1h2[(∫t-h2tx(s)ds)TxT(t)-xT(t-h1)]×[N11N12*N22][∫t-h2tx(s)dsx(t)-x(t-h2)],-∫-hi0∫t+θtx˙T(s)Rix˙(s)dsdθ≤-2hi2(∫-hi0∫t+θtx˙(s)dsdθ)TRi(∫-hi0∫t+θtx˙(s)dsdθ)=-2hi2(hix(t)-∫t-hitx(s)ds)T×Ri(hix(t)-∫t-hitx(s)ds)i=1,2.
Using the above inequalities, and with the zero quantities (21), we can obtain
(23)V˙(xt)≤2[xT(t)(∫t-h1tx(s)ds)T][P11P12*P22]×[x˙(t)x(t)-x(t-h1)]+h1[xT(t)x˙T(t)]×[Z11Z12*Z22][x(t)x˙(t)]+h2[xT(t)x˙T(t)][N11N12*N22]×[x(t)x˙(t)]-1h1[(∫t-h1tx(s)ds)TxT(t)-xT(t-h1)]×[Z11Z12*Z22][∫t-h1tx(s)dsx(t)-x(t-h1)]-1h2[(∫t-h2tx(s)ds)TxT(t)-xT(t-h2)]×[N11N12*N22][∫t-h2tx(s)dsx(t)-x(t-h2)]+xT(t)Q1x(t)+xT(t)Q2x(t)-(1-μ1)xT(t-d1(t))Q1x×(t-d1(t))-(1-μ2)xT(t-d2(t))Q2x×(t-d2(t))+xT(t)M1x(t)-xT(t-h1)M1x×(t-h1)+xT(t)M2x(t)-xT(t-h2)M2x×(t-h2)+12h12x˙T(t)R1x˙(t)-2h12×(h1x(t)-∫t-h1tx(s)ds)TR1(h1x(t)-∫t-h1tx(s)ds)+12h22x˙T(t)R2x˙(t)-2h22(h2x(t)-∫t-h2tx(s)ds)T×R2(h2x(t)-∫t-h2tx(s)ds)+2[xT(t)(∫t-h2tx(s)ds)T]×[W11W12*W22][x˙(t)x(t)-x(t-h2)]+2∑i=1r∑j=1rwi(x(t))mj(x(t-d2(t)))×[(∫t-h2tx(s)ds)TxT(t)T1+xT(t-d1(t))T2+xT(t-h1)T3+(∫t-h1tx(s)ds)TT4+xT(t-d2(t))T5+xT×(t-h2)T6+(∫t-h2tx(s)ds)TT7+x˙T(t)T8]×[A1ix(t)-A2ix(t-d1(t))-BiFjx(t-d2(t))-x˙(t)]=ζT(t)(∑i=1r∑j=1rwi(x(t))mj(x(t-d2(t)))Φij)ζ(t),
where(24)ζT(t)=[xT(t)xT(t-d1(t))xT(t-h1)(∫t-h1tx(s)ds)TxT(t-d2(t))xT(t-h2)(∫t-h2tx(s)ds)Tx˙T(t)].(25)Φij=[Φ11iΦ12iΦ13Φ14Φ15ijΦ16Φ17Φ18*Φ22iA2iTT3TA2iTT4TΦ25ijA2iTT6TA2iTT7TΦ28i**Φ33i-P22T3BiFj00-T3***Φ44T4BiFj00Φ48****Φ55ijFjTBiTT6TFjTBiTT7TΦ58i*****Φ66Φ67-T6******Φ77Φ78*******Φ88],Φ11i=P12+P12T+W12+W12T+h1Z11+h2N11-1h1Z22-1h2N22+Q1+Q2+M1+M2-2R1-2R2+T1A1i+A1iTT1T,Φ12i=T1A2i+A1iTT2T,Φ13=-P12-1h1Z22+A1iTT3T,Φ14=P22-1h2Z12T+1h1R1+A1iTT4T,Φ15ij=T1BiFj+A1iTT5T,Φ16=-W12-1h2N22+A1iTT6T,Φ17=W22-1h2N12T+2h22R2+A1iTT7T,Φ18=P11+W11+h1Z12+h2N12-T1+A1iTT8T,Φ22i=-(1-μ1)Q1+T2A2i+A2iTT2T,Φ25ij=T2BiFj+A2iTT5T,Φ28i=-T2+A2iTT8T,Φ33=-1hZ22-M1,Φ44=-1h1Z11-2h12R1,Φ48=P12T-T4,Φ55i=-(1-μ2)Q2+T5BiFj+FjTBiTT5T,Φ58i=-T5+FjTBiTT8T,Φ66=-1h2N22-M2,Φ67=-1h2N12T-W22,Φ77=-1h2N11-2h22R2,Φ78=W12T-T7,Φ88=h1Z22+h2N22+12h12R1+12h22R2-T8.
From (25), it is obvious that if
(26)∑i=1r∑j=1rwi(x(t))mj(x(t-d2(t)))Φij<0,V˙(x(t))<0.
From (26), we can discover that the feedback gains are predefined. The following proof presents the controller design method. We first pre- and postmultiply both the diag[XXXXXXXX] and transpose to both sides of (25) and  pre- and postmultiply the diag[XX] transpose to both sides of P, W, Z, andN and the X and transpose to both sides of Qi,Mi, and Ri,i=1,2. Let Ti=tiT1(i=2,…,8), and denote new variables X=T1-1, R-1=XR1XT, Q-1=XQ1XT, M-1=XM1XT, R-2=XR2XT, Q-2=XQ2XT,M-2=XM2XT,Λ-i=XΛiXT, Λ-j=XΛjXT,i=1,2,…,r,P-11=XP11XT, P-12=XP12XT, P-22=XP22XT, W-11=XW11XT, W-12=XW12XT, W-22=XW22XT, Z-11=XZ11XT, Z-12=XZ12XT, Z-22=XZ22XT, N-11=XN11XT, N-12=XN12XT, and N-22=XN22XT. With Fi=YiX-T,i=1,2…,r, we next get
(27)∑i=1r∑j=1rwi(x(t))mj(x(t-d2(t)))Φ-ij<0,
where
(28)Φ-ij=[Φ-11iΦ-12iΦ-13Φ-14Φ-15ijΦ-16Φ-17Φ-18*Φ-22it3XA2iTt4XA2iTΦ-25ijt6XA2iTt7XA2iTΦ-28i**Φ-33i-P-22t3BiYj00-t3XT***Φ-44t4BiYj00Φ-48****Φ-55ijt6YjTBiTt7YjTBiTΦ-58i*****Φ-66Φ-67-t6XT******Φ-77Φ-78*******Φ-88],Φ-11i=P-12+P-12T+W-12+W-12T+h1Z-11+h2N-11-1h1Z-22-1h2N-22+Q-1+Q-2+M-1+M-2-2R-1-2R-2+A1iXT+XA1iT,Φ-12i=A2iXT+t2XA1iT,Φ-13=-P-12-1h1Z-22+t3XA1iT,Φ-14=P-22-1h1Z-12T+1h1R-1+t4XA1iT,Φ-15ij=BiYj+t5XA1iT,Φ-16=-W-12-1h2N-22+t6XA1iT,Φ-17=W-22-1h2N-12T+2h22R-2+t7XA1iT,Φ-18=P-11+W-11+h1Z-12+h2N-12-XT+t8XA1iT,Φ-22i=-(1-μ1)Q-1+t2A2iXT+t2XA2iT,Φ-25ij=t2BiYj+t5XA2iT,Φ28i=-t2XT+t8XA2iT,Φ-33=-1h1Z-22-M-1,Φ-44=-1h1Z-11-2h12R-1,Φ-48=P-12T-t4XT,Φ-55i=-(1-μ2)Q-2+t5BiYj+t5YjTBiT,Φ-58i=-t5XT+t8YjTBiT,Φ-66=-1h2N-22-M-2,Φ-67=-1h2N-12T-W-22,Φ-77=-1h2N-11-2h22R-2,Φ-78=W-12T-t7XT,Φ-88=h1Z-22+h2N-22+12h12R-1+12h22R-2-t8XT.
Here,  “*” denotes the transpose elements in the symmetric positions.
If (27) holds, then V˙(x(t))<0. Consider ∑i=1r∑j=1rwi(x(t))(wj(x(t))-mj(x(t-d2(t))))Λi=0, where Λi=ΛiT∈R8n×8n>0,i=1,2,…,r, are arbitrary matrices. These terms are introduced to (27) to alleviate the conservativeness. From (27), we also have
(29)Φ-=∑i=1r∑j=1rwi(x(t))mj(x(t-d2(t)))Φ-ij=∑i=1r∑j=1rwi(x(t))(wj(x(t))-mj(x(t-d2(t)))+ρjwj(x(t))-ρjwj(x(t)))Λi+∑i=1r∑j=1rwi(x(t))mj(x(t-d2(t)))Φ-ij=∑i=1r∑j=1rwi(x(t))wj(x(t))(ρjΦij-ρjΛi+Λi)+∑i=1r∑j=1rwi(x(t))(mj(x(t-d2(t)))-ρjwj(x(t)))×(Φ-ij-Λi)≤∑i=1rwi2(ρiΦ-ii-ρiΛi+Λi)+∑i=1r∑j=1rwi(mj-ρjwj)(Φ-ij-Λi)+∑i=1r∑i<jwiwj(ρjΦ-ij+ρiΦ-ji-ρjΛi-ρiΛj+Λi+Λj)
with mj(x(t-d2(t)))-ρjwj(x(t))≥0 for all j, x(t) and x(t-d2(t)). Let
(30)Φ-ij-Λi<0,(31)ρiΦ-ii-ρiΛi+Λi<0,(32)ρjΦ-ij+ρiΦ-ji-ρjΛi-ρiΛj+Λi+Λj≤0,i<j,
for alli,j=1,2,…,r.
SinceV˙(x(t))<0, the fuzzy control system with the time varying state and input delays (15) is asymptotically stable with the state feedback control law (17).


### 3.2. Robust Stability of Uncertain Fuzzy Systems

We also examine the design of a robust stable controller for the uncertain system (4) under the imperfect premise matching. Consider the following uncertain fuzzy time varying delay control system:
(33)x˙(t)=∑i=1r∑j=1rwi(x(t))mj(x(t-d2(t)))×[(A1i+ΔA1i(t))x(t)+(A2i+ΔA2i(t))×x(t-d1(t))+(Bi+ΔBi(t))×u(t-d2(t))],x(t)=φ(t),t∈[-max(h1,h2),0].
Based on the above results of Theorem 4, a robust stabilization criterion for the T-S fuzzy systems with time varying state and input delays is investigated. The following result can be obtained.Theorem 5.
Given scalarsh1≥0,h2≥0, μ1,μ2, and ti(i=2,…,8), the uncertain fuzzy control systems with the time varying state and input delays (33) is robustly stable for any 0≤di(t)≤hi(i=1,2) via the imperfect premise matching controller design technique, if the membership functions of the fuzzy model and fuzzy controller satisfy mj(x(t-d2(t)))-ρjwj(x(t))≥0 for all j, x(t), and x(t-d2(t)), where 0<ρj<1, and there exist  common matricesP-11>0,P-22>0, W-11>0,W-22>0, Z-11>0,Z-22>0,N-11>0,N-22>0, Q-i>0,M-i>0,R-i>0(i=1,2) and Λi = ΛiT∈R8n×8n > 0(i=1,2,…,r), P->0, W->0,Z->0, N->0 and some matrices Yj(j=1,2,…,r), X and scalars ε1i>0, ε2i>0, εbij>0 satisfy the following LMIs:
(34)Ψij-Λ-i<0,(35)ρiΨii-ρiΛ-i+Λ-i<0,(36)ρjΨij+ρiΨji-ρjΛ-i-ρiΛ-j+Λ-i+Λ-j,≤0i<j,
where Ψij is defined as in (44). The state feedback gains can be constructed as Fi=YiX-T.Proof.
IfA1i,A2i, and Bi in Φ-ij are replaced with A1i+DiKi(t)E1i, A2i+DiKi(t)E2i, and Bi+DiKi(t)Ebi in (28) of Theorem 4, respectively, (27) for system (33) can be rewritten as
(37)∑i=1r∑j=1rwi(x(t))mj(x(t-d(t)))Ωij<0,
where
(38)Ωij=Φ-ij+D^1iTKi(t)E^i+E^iTKiT(t)D^1i+D^2iTKi(t)E^2ij+E^2ijTKiT(t)D^2i+D^biTKi(t)E^bij+E^bijTKiT(t)D^bi,D^1i=[DiT0000000],D^2i=[0DiT000000],D^bi=[0000DiT000],(39)E^1i=[E1iXTt2E1iXTt3E1iXTt4E1iXTt5E1iXTt6E1iXTt7E1iXTt8E1iXT],E^2i=[E2iXTt2E2iXTt3E2iXTt4E2iXTt5E2iXTt6E2iXTt7E2iXTt8E2iXT],E^bij=[EbiYjt2EbiYjt3EbiYjt4EbiYjt5EbiYjt6EbiYjt7EbiYjt8EbiYj].If Ωij<0, (37) holds. According to Lemma 2, it is straightforward to know that Ωij<0 is true, if for each i, j, there exists  scalars ε1i>0, ε2i>0, and εbij>0 such that the following inequality holds:
(40)Φ-ij+ε1iD^1iTD^1i+ε1i-1E^1iTE^1i+ε2iD^2iTD^2i+ε2i-1E^2iTE^2i+εbijD^biTD^bi+εbij-1E^bijTE^bij<0.
Based on Schur complement, (40) is equivalent to the following inequality:
(41)Ψij=[Φ^ijE^1iTE^2iTE^bijT*-ε1iI00**-ε2iI0***-εbijI]<0,
where
(42)Φ^ij=[Φ^11iΦ-12iΦ-13Φ-14Φ-15ijΦ-16Φ-17Φ-18*Φ^22it3XA2iTt4XA2iTΦ-25ijt6XA2iTt7XA2iTΦ-28i**Φ-33i-P-22t3BiYj00-t3XT***Φ-44t4BiYj00Φ-48****Φ^55ijt6YjTBiTt7YjTBiTΦ-58i*****Φ-66Φ-67-t6XT******Φ-77Φ-78*******Φ-88]<0,Φ^11i=Φ-11i+ε1iDiDiT,Φ^22i=Φ-22i+ε2iDiDiT,Φ^55ij=Φ-55ij+εbiDiDiT,
and Φ- are the same as in Theorem 4. Therefore, (41) holds, if and only if the following inequality holds:
(43)∑i=1r∑j=1rwi(x(t))mj(x(t-d(t)))Ψij<0.
To alleviate the conservativeness of our robust stability analysis, similar to the analysis approach of Theorem4, we introduce ∑i=1r∑j=1rwi(wj-mj)Λ-i=0 to (43), and the following proof is the same as in Theorem 4. We can obtain the conditions and (34)–(36). Thus, the fuzzy control system with the time varying state and input delays (34) is robust stable on the basis of the control law Fi=YiX-Tunder these conditions, and (34)–(36) hold.Remark 6.
If we setwi(x(t))=mj(x(t)), d1(t)=d2(t), the closed-loop system in this paper has the same structure as that of paper [26]. On the other hand, let wi(x(t))=mj(x(t)) andd1(t)=d2(t)=d; we can obtain the system in [16]. Therefore, the system studied in this paper is more general. Additionally, the information of the membership functions of the fuzzy time delay models and controllers are all considered in the stability analysis. If the fuzzy models and fuzzy controllers share the same fuzzy membership functions, that is, wi(x(t))=mj(x(t)), the stability analysis can be referred to [25, 27, 28]. In other words, the problem in [25, 27, 28] is actually a special case of the one investigated in this paper.Remark 7.
The augmented Lyapunov function with an additional triple-integral term is introduced in analyzing the stability problem for the T-S fuzzy systems with time varying state and input delays. In addition, two integral inequalities are used to derive Theorem4, and less free-weighting matrices are introduced, which lead to get less conservative results. It can be observed that some existing results are the special cases of this paper. For example, the method in [25] is from (18) with P12=0,P22=0,Z12=Z22=0,N12=N22=0, andR1=R2=0 in Theorem 4. The results in [22] are based on system (33) with d2(t)=0 and (18) with M2=0,Z12=Z22=0,Q1=Q2=0, andR1=R2=0.Remark 8.
Different from the general PDC technique, the design method under the imperfect premise matching is much more flexible, because the membership functions of the fuzzy controllers do not need to be chosen the same as those of the fuzzy time delay models. Instead, they can be designed arbitrarily. Thus, the design flexibility is significantly enhanced. On the other hand, some simple membership functions of the fuzzy controllers might be employed, which can reduce the implementation cost.Remark 9.
In [7–11, 15–20], the authors have considered the robust control for uncertain T-S fuzzy systems with state delays.  Some delay-dependent conditions and fuzzy controllers are obtained with a traditional Lyapunov-Krasovskii functional method and PDC method, which all are the special cases of Theorem 5 in this paper.


## 3.1. Stability of Nominal Fuzzy Systems

Firstly, we consider the control design of a state feedback control law under the imperfect premise matching that stabilizes the following nominal fuzzy time varying delay system:(15)x˙(t)=∑i=1r∑j=1rwi(x(t))mj(x(t-d2(t)))×[A1ix(t)+A2ix(t-d1(t))+BiFjx×(t-d2(t))],x(t)=φ(t),t∈[-max(h1,h2),0].Theorem 4.
Given scalarsh1≥0,h2≥0, μ1,μ2, and ti(i=2,…,8), the closed-loop system (15) is asymptotically stable for any 0≤di(t)≤hi(i=1,2) via the imperfect premise matching controller design technique, if the membership functions of the fuzzy model and fuzzy controller satisfy mj(x(t-d2(t)))-ρjwj(x(t))≥0 for all j, x(t) and x(t-d2(t)), where 0<ρj<1, and there exist  matrices P-11>0,P-22>0, W-11>0,W-22>0, Z-11>0,Z-22>0,N-11>0,N-22>0, Q-i>0, M-i>0, R-i>0(i=1,2), Λi=ΛiT∈R8n×8n>0(i=1,2,…,r),  and
(16)P-=[P-11P-12*P-22]>0,W-=[W-11W-12*W-22]>0,Z-=[Z-11Z-12*Z-22]>0,N-=[N-11N-12*N-22]>0,
and real matrices Yj(j=1,2,…,r), P-12W-12Z-12N-12, and X such that (30)–(32) hold. In addition, the stabilizing control law is given by
(17)u(t)=∑j=1rmj(x(t-d2(t)))YjX-Tx(t).Proof.
Choose a fuzzy weighting-dependent Lyapunov-Krasovskii functional candidate as(18)V(x(t))=V1(x(t))+V2(x(t))+V3(x(t))+V4(x(t))+V5(x(t)),V1(x(t))=ξ1T(t)Pξ1(t)+ξ2T(t)Wξ2(t),V2(x(t))=∫-h10∫t+θtξT(s)Zξ(s)dsdθ+∫-h20∫t+θtξT(s)Nξ(s)dsdθ,V3(x(t))=∫t-d1(t)txT(s)Q1x(s)ds+∫t-d2(t)txT(s)Q2x(s)ds,V4(x(t))=∫t-h1txT(s)M1x(s)ds+∫t-h2txT(s)M2x(s)ds,V5(x(t))=∫-h10∫θ0∫t+λtx˙T(s)R1x˙(s)dsdλdθ+∫-h20∫θ0∫t+λtx˙T(s)R2x˙(s)dsdλdθ,
where
(19)ξ1T(t)=[xT(t)(∫t-h1tx(s)ds)T],ξ2T(t)=[xT(t)(∫t-h2tx(s)ds)T],ξT(t)=[xT(t)x˙T(t)],P=[P11P12*P22]>0,W=[W11W12*W22]>0,Z=[Z11Z12*Z22]>0,N=[N11N12*N22]>0,
and Qi,Mi,andRi(i=1,2) are the positive-definite matrices to be determined.
The derivatives ofV(x(t)) along the trajectories of system (15) are as follows:
(20)V˙1(x(t))=2ξ1T(t)Pξ˙1(t)+2ξ2T(t)Wξ˙2(t)=2[xT(t)(∫t-h1tx(s)ds)T][P11P12*P22]×[x˙(t)x(t)-x(t-h1)]+2[xT(t)(∫t-h2tx(s)ds)T][W11W12*W22]×[x˙(t)x(t)-x(t-h2)],V˙2(x(t))=h1[xT(t)x˙T(t)][Z11Z12*Z22][x(t)x˙(t)]-∫t-h1t[xT(s)x˙T(s)][Z11Z12*Z22][x(s)x˙(s)]ds+h2[xT(t)x˙T(t)][N11N12*N22][x(t)x˙(t)]-∫t-h2t[xT(s)x˙T(s)][N11N12*N22][x(s)x˙(s)]ds,V˙3(x(t))=xT(t)Q1x(t)-(1-d˙1(t))xT(t-d1(t))×Q1x(t-d1(t))+xT(t)Q2x(t)-(1-d˙2(t))xT(t-d2(t))Q2x(t-d2(t))≤xT(t)Q1x(t)+xT(t)Q2x(t)-(1-μ1)×xT(t-d1(t))Q1x(t-d1(t))-(1-μ2)×xT(t-d2(t))Q2x(t-d2(t)),V˙4(x(t))=xT(t)M1x(t)-xT(t-h1)M1x(t-h1)+xT(t)M2x(t)-xT(t-h2)M2x(t-h2),V˙5(x(t))=12h12x˙T(t)R1x˙(t)-∫-h0∫t+θtx˙T(s)×R1x˙(s)dsdθ+12h22x˙T(t)R2x˙(t)-∫-h0∫t+θtx˙T(s)R2x˙(s)dsdθ.
From (15), the following equation can be obtained for any matrices Ti,i=1,2,…,8,
(21)2∑i=1r∑j=1rwi(x(t))mj(x(t-d2(t)))×[(∫t-h2tx(s)ds)TxT(t)T1+xT(t-d1(t))T2+xT(t-h1)T3+(∫t-h1tx(s)ds)TT4+xT(t-d2(t))T5+xT×(t-h2)T6+(∫t-h2tx(s)ds)TT7+x˙T(t)T8]×[A1ix(t)-A2ix(t-d1(t))-BiFjx(t-d2(t))-x˙(t)]=0.
By Lemma 3, we have(22)-∫t-h1t[xt(s)x˙T(s)][Z11Z12*Z22][x(s)x˙(s)]ds≤-1h1[(∫t-h1tx(s)ds)TxT(t)-xT(t-h1)]×[Z11Z12*Z22][∫t-h1tx(s)dsx(t)-x(t-h1)],-∫t-h2t[xT(s)x˙T(s)][N11N12*N22][x(s)x˙(s)]ds≤-1h2[(∫t-h2tx(s)ds)TxT(t)-xT(t-h1)]×[N11N12*N22][∫t-h2tx(s)dsx(t)-x(t-h2)],-∫-hi0∫t+θtx˙T(s)Rix˙(s)dsdθ≤-2hi2(∫-hi0∫t+θtx˙(s)dsdθ)TRi(∫-hi0∫t+θtx˙(s)dsdθ)=-2hi2(hix(t)-∫t-hitx(s)ds)T×Ri(hix(t)-∫t-hitx(s)ds)i=1,2.
Using the above inequalities, and with the zero quantities (21), we can obtain
(23)V˙(xt)≤2[xT(t)(∫t-h1tx(s)ds)T][P11P12*P22]×[x˙(t)x(t)-x(t-h1)]+h1[xT(t)x˙T(t)]×[Z11Z12*Z22][x(t)x˙(t)]+h2[xT(t)x˙T(t)][N11N12*N22]×[x(t)x˙(t)]-1h1[(∫t-h1tx(s)ds)TxT(t)-xT(t-h1)]×[Z11Z12*Z22][∫t-h1tx(s)dsx(t)-x(t-h1)]-1h2[(∫t-h2tx(s)ds)TxT(t)-xT(t-h2)]×[N11N12*N22][∫t-h2tx(s)dsx(t)-x(t-h2)]+xT(t)Q1x(t)+xT(t)Q2x(t)-(1-μ1)xT(t-d1(t))Q1x×(t-d1(t))-(1-μ2)xT(t-d2(t))Q2x×(t-d2(t))+xT(t)M1x(t)-xT(t-h1)M1x×(t-h1)+xT(t)M2x(t)-xT(t-h2)M2x×(t-h2)+12h12x˙T(t)R1x˙(t)-2h12×(h1x(t)-∫t-h1tx(s)ds)TR1(h1x(t)-∫t-h1tx(s)ds)+12h22x˙T(t)R2x˙(t)-2h22(h2x(t)-∫t-h2tx(s)ds)T×R2(h2x(t)-∫t-h2tx(s)ds)+2[xT(t)(∫t-h2tx(s)ds)T]×[W11W12*W22][x˙(t)x(t)-x(t-h2)]+2∑i=1r∑j=1rwi(x(t))mj(x(t-d2(t)))×[(∫t-h2tx(s)ds)TxT(t)T1+xT(t-d1(t))T2+xT(t-h1)T3+(∫t-h1tx(s)ds)TT4+xT(t-d2(t))T5+xT×(t-h2)T6+(∫t-h2tx(s)ds)TT7+x˙T(t)T8]×[A1ix(t)-A2ix(t-d1(t))-BiFjx(t-d2(t))-x˙(t)]=ζT(t)(∑i=1r∑j=1rwi(x(t))mj(x(t-d2(t)))Φij)ζ(t),
where(24)ζT(t)=[xT(t)xT(t-d1(t))xT(t-h1)(∫t-h1tx(s)ds)TxT(t-d2(t))xT(t-h2)(∫t-h2tx(s)ds)Tx˙T(t)].(25)Φij=[Φ11iΦ12iΦ13Φ14Φ15ijΦ16Φ17Φ18*Φ22iA2iTT3TA2iTT4TΦ25ijA2iTT6TA2iTT7TΦ28i**Φ33i-P22T3BiFj00-T3***Φ44T4BiFj00Φ48****Φ55ijFjTBiTT6TFjTBiTT7TΦ58i*****Φ66Φ67-T6******Φ77Φ78*******Φ88],Φ11i=P12+P12T+W12+W12T+h1Z11+h2N11-1h1Z22-1h2N22+Q1+Q2+M1+M2-2R1-2R2+T1A1i+A1iTT1T,Φ12i=T1A2i+A1iTT2T,Φ13=-P12-1h1Z22+A1iTT3T,Φ14=P22-1h2Z12T+1h1R1+A1iTT4T,Φ15ij=T1BiFj+A1iTT5T,Φ16=-W12-1h2N22+A1iTT6T,Φ17=W22-1h2N12T+2h22R2+A1iTT7T,Φ18=P11+W11+h1Z12+h2N12-T1+A1iTT8T,Φ22i=-(1-μ1)Q1+T2A2i+A2iTT2T,Φ25ij=T2BiFj+A2iTT5T,Φ28i=-T2+A2iTT8T,Φ33=-1hZ22-M1,Φ44=-1h1Z11-2h12R1,Φ48=P12T-T4,Φ55i=-(1-μ2)Q2+T5BiFj+FjTBiTT5T,Φ58i=-T5+FjTBiTT8T,Φ66=-1h2N22-M2,Φ67=-1h2N12T-W22,Φ77=-1h2N11-2h22R2,Φ78=W12T-T7,Φ88=h1Z22+h2N22+12h12R1+12h22R2-T8.
From (25), it is obvious that if
(26)∑i=1r∑j=1rwi(x(t))mj(x(t-d2(t)))Φij<0,V˙(x(t))<0.
From (26), we can discover that the feedback gains are predefined. The following proof presents the controller design method. We first pre- and postmultiply both the diag[XXXXXXXX] and transpose to both sides of (25) and  pre- and postmultiply the diag[XX] transpose to both sides of P, W, Z, andN and the X and transpose to both sides of Qi,Mi, and Ri,i=1,2. Let Ti=tiT1(i=2,…,8), and denote new variables X=T1-1, R-1=XR1XT, Q-1=XQ1XT, M-1=XM1XT, R-2=XR2XT, Q-2=XQ2XT,M-2=XM2XT,Λ-i=XΛiXT, Λ-j=XΛjXT,i=1,2,…,r,P-11=XP11XT, P-12=XP12XT, P-22=XP22XT, W-11=XW11XT, W-12=XW12XT, W-22=XW22XT, Z-11=XZ11XT, Z-12=XZ12XT, Z-22=XZ22XT, N-11=XN11XT, N-12=XN12XT, and N-22=XN22XT. With Fi=YiX-T,i=1,2…,r, we next get
(27)∑i=1r∑j=1rwi(x(t))mj(x(t-d2(t)))Φ-ij<0,
where
(28)Φ-ij=[Φ-11iΦ-12iΦ-13Φ-14Φ-15ijΦ-16Φ-17Φ-18*Φ-22it3XA2iTt4XA2iTΦ-25ijt6XA2iTt7XA2iTΦ-28i**Φ-33i-P-22t3BiYj00-t3XT***Φ-44t4BiYj00Φ-48****Φ-55ijt6YjTBiTt7YjTBiTΦ-58i*****Φ-66Φ-67-t6XT******Φ-77Φ-78*******Φ-88],Φ-11i=P-12+P-12T+W-12+W-12T+h1Z-11+h2N-11-1h1Z-22-1h2N-22+Q-1+Q-2+M-1+M-2-2R-1-2R-2+A1iXT+XA1iT,Φ-12i=A2iXT+t2XA1iT,Φ-13=-P-12-1h1Z-22+t3XA1iT,Φ-14=P-22-1h1Z-12T+1h1R-1+t4XA1iT,Φ-15ij=BiYj+t5XA1iT,Φ-16=-W-12-1h2N-22+t6XA1iT,Φ-17=W-22-1h2N-12T+2h22R-2+t7XA1iT,Φ-18=P-11+W-11+h1Z-12+h2N-12-XT+t8XA1iT,Φ-22i=-(1-μ1)Q-1+t2A2iXT+t2XA2iT,Φ-25ij=t2BiYj+t5XA2iT,Φ28i=-t2XT+t8XA2iT,Φ-33=-1h1Z-22-M-1,Φ-44=-1h1Z-11-2h12R-1,Φ-48=P-12T-t4XT,Φ-55i=-(1-μ2)Q-2+t5BiYj+t5YjTBiT,Φ-58i=-t5XT+t8YjTBiT,Φ-66=-1h2N-22-M-2,Φ-67=-1h2N-12T-W-22,Φ-77=-1h2N-11-2h22R-2,Φ-78=W-12T-t7XT,Φ-88=h1Z-22+h2N-22+12h12R-1+12h22R-2-t8XT.
Here,  “*” denotes the transpose elements in the symmetric positions.
If (27) holds, then V˙(x(t))<0. Consider ∑i=1r∑j=1rwi(x(t))(wj(x(t))-mj(x(t-d2(t))))Λi=0, where Λi=ΛiT∈R8n×8n>0,i=1,2,…,r, are arbitrary matrices. These terms are introduced to (27) to alleviate the conservativeness. From (27), we also have
(29)Φ-=∑i=1r∑j=1rwi(x(t))mj(x(t-d2(t)))Φ-ij=∑i=1r∑j=1rwi(x(t))(wj(x(t))-mj(x(t-d2(t)))+ρjwj(x(t))-ρjwj(x(t)))Λi+∑i=1r∑j=1rwi(x(t))mj(x(t-d2(t)))Φ-ij=∑i=1r∑j=1rwi(x(t))wj(x(t))(ρjΦij-ρjΛi+Λi)+∑i=1r∑j=1rwi(x(t))(mj(x(t-d2(t)))-ρjwj(x(t)))×(Φ-ij-Λi)≤∑i=1rwi2(ρiΦ-ii-ρiΛi+Λi)+∑i=1r∑j=1rwi(mj-ρjwj)(Φ-ij-Λi)+∑i=1r∑i<jwiwj(ρjΦ-ij+ρiΦ-ji-ρjΛi-ρiΛj+Λi+Λj)
with mj(x(t-d2(t)))-ρjwj(x(t))≥0 for all j, x(t) and x(t-d2(t)). Let
(30)Φ-ij-Λi<0,(31)ρiΦ-ii-ρiΛi+Λi<0,(32)ρjΦ-ij+ρiΦ-ji-ρjΛi-ρiΛj+Λi+Λj≤0,i<j,
for alli,j=1,2,…,r.
SinceV˙(x(t))<0, the fuzzy control system with the time varying state and input delays (15) is asymptotically stable with the state feedback control law (17).


## 3.2. Robust Stability of Uncertain Fuzzy Systems

We also examine the design of a robust stable controller for the uncertain system (4) under the imperfect premise matching. Consider the following uncertain fuzzy time varying delay control system:
(33)x˙(t)=∑i=1r∑j=1rwi(x(t))mj(x(t-d2(t)))×[(A1i+ΔA1i(t))x(t)+(A2i+ΔA2i(t))×x(t-d1(t))+(Bi+ΔBi(t))×u(t-d2(t))],x(t)=φ(t),t∈[-max(h1,h2),0].
Based on the above results of Theorem 4, a robust stabilization criterion for the T-S fuzzy systems with time varying state and input delays is investigated. The following result can be obtained.Theorem 5.
Given scalarsh1≥0,h2≥0, μ1,μ2, and ti(i=2,…,8), the uncertain fuzzy control systems with the time varying state and input delays (33) is robustly stable for any 0≤di(t)≤hi(i=1,2) via the imperfect premise matching controller design technique, if the membership functions of the fuzzy model and fuzzy controller satisfy mj(x(t-d2(t)))-ρjwj(x(t))≥0 for all j, x(t), and x(t-d2(t)), where 0<ρj<1, and there exist  common matricesP-11>0,P-22>0, W-11>0,W-22>0, Z-11>0,Z-22>0,N-11>0,N-22>0, Q-i>0,M-i>0,R-i>0(i=1,2) and Λi = ΛiT∈R8n×8n > 0(i=1,2,…,r), P->0, W->0,Z->0, N->0 and some matrices Yj(j=1,2,…,r), X and scalars ε1i>0, ε2i>0, εbij>0 satisfy the following LMIs:
(34)Ψij-Λ-i<0,(35)ρiΨii-ρiΛ-i+Λ-i<0,(36)ρjΨij+ρiΨji-ρjΛ-i-ρiΛ-j+Λ-i+Λ-j,≤0i<j,
where Ψij is defined as in (44). The state feedback gains can be constructed as Fi=YiX-T.Proof.
IfA1i,A2i, and Bi in Φ-ij are replaced with A1i+DiKi(t)E1i, A2i+DiKi(t)E2i, and Bi+DiKi(t)Ebi in (28) of Theorem 4, respectively, (27) for system (33) can be rewritten as
(37)∑i=1r∑j=1rwi(x(t))mj(x(t-d(t)))Ωij<0,
where
(38)Ωij=Φ-ij+D^1iTKi(t)E^i+E^iTKiT(t)D^1i+D^2iTKi(t)E^2ij+E^2ijTKiT(t)D^2i+D^biTKi(t)E^bij+E^bijTKiT(t)D^bi,D^1i=[DiT0000000],D^2i=[0DiT000000],D^bi=[0000DiT000],(39)E^1i=[E1iXTt2E1iXTt3E1iXTt4E1iXTt5E1iXTt6E1iXTt7E1iXTt8E1iXT],E^2i=[E2iXTt2E2iXTt3E2iXTt4E2iXTt5E2iXTt6E2iXTt7E2iXTt8E2iXT],E^bij=[EbiYjt2EbiYjt3EbiYjt4EbiYjt5EbiYjt6EbiYjt7EbiYjt8EbiYj].If Ωij<0, (37) holds. According to Lemma 2, it is straightforward to know that Ωij<0 is true, if for each i, j, there exists  scalars ε1i>0, ε2i>0, and εbij>0 such that the following inequality holds:
(40)Φ-ij+ε1iD^1iTD^1i+ε1i-1E^1iTE^1i+ε2iD^2iTD^2i+ε2i-1E^2iTE^2i+εbijD^biTD^bi+εbij-1E^bijTE^bij<0.
Based on Schur complement, (40) is equivalent to the following inequality:
(41)Ψij=[Φ^ijE^1iTE^2iTE^bijT*-ε1iI00**-ε2iI0***-εbijI]<0,
where
(42)Φ^ij=[Φ^11iΦ-12iΦ-13Φ-14Φ-15ijΦ-16Φ-17Φ-18*Φ^22it3XA2iTt4XA2iTΦ-25ijt6XA2iTt7XA2iTΦ-28i**Φ-33i-P-22t3BiYj00-t3XT***Φ-44t4BiYj00Φ-48****Φ^55ijt6YjTBiTt7YjTBiTΦ-58i*****Φ-66Φ-67-t6XT******Φ-77Φ-78*******Φ-88]<0,Φ^11i=Φ-11i+ε1iDiDiT,Φ^22i=Φ-22i+ε2iDiDiT,Φ^55ij=Φ-55ij+εbiDiDiT,
and Φ- are the same as in Theorem 4. Therefore, (41) holds, if and only if the following inequality holds:
(43)∑i=1r∑j=1rwi(x(t))mj(x(t-d(t)))Ψij<0.
To alleviate the conservativeness of our robust stability analysis, similar to the analysis approach of Theorem4, we introduce ∑i=1r∑j=1rwi(wj-mj)Λ-i=0 to (43), and the following proof is the same as in Theorem 4. We can obtain the conditions and (34)–(36). Thus, the fuzzy control system with the time varying state and input delays (34) is robust stable on the basis of the control law Fi=YiX-Tunder these conditions, and (34)–(36) hold.Remark 6.
If we setwi(x(t))=mj(x(t)), d1(t)=d2(t), the closed-loop system in this paper has the same structure as that of paper [26]. On the other hand, let wi(x(t))=mj(x(t)) andd1(t)=d2(t)=d; we can obtain the system in [16]. Therefore, the system studied in this paper is more general. Additionally, the information of the membership functions of the fuzzy time delay models and controllers are all considered in the stability analysis. If the fuzzy models and fuzzy controllers share the same fuzzy membership functions, that is, wi(x(t))=mj(x(t)), the stability analysis can be referred to [25, 27, 28]. In other words, the problem in [25, 27, 28] is actually a special case of the one investigated in this paper.Remark 7.
The augmented Lyapunov function with an additional triple-integral term is introduced in analyzing the stability problem for the T-S fuzzy systems with time varying state and input delays. In addition, two integral inequalities are used to derive Theorem4, and less free-weighting matrices are introduced, which lead to get less conservative results. It can be observed that some existing results are the special cases of this paper. For example, the method in [25] is from (18) with P12=0,P22=0,Z12=Z22=0,N12=N22=0, andR1=R2=0 in Theorem 4. The results in [22] are based on system (33) with d2(t)=0 and (18) with M2=0,Z12=Z22=0,Q1=Q2=0, andR1=R2=0.Remark 8.
Different from the general PDC technique, the design method under the imperfect premise matching is much more flexible, because the membership functions of the fuzzy controllers do not need to be chosen the same as those of the fuzzy time delay models. Instead, they can be designed arbitrarily. Thus, the design flexibility is significantly enhanced. On the other hand, some simple membership functions of the fuzzy controllers might be employed, which can reduce the implementation cost.Remark 9.
In [7–11, 15–20], the authors have considered the robust control for uncertain T-S fuzzy systems with state delays.  Some delay-dependent conditions and fuzzy controllers are obtained with a traditional Lyapunov-Krasovskii functional method and PDC method, which all are the special cases of Theorem 5 in this paper.


## 4. Numerical Examples

In this section, two numerical examples are given to illustrate the conservativeness and effectiveness of the proposed methods. The first example compares our techniques with the existing ones in the literature for stability analysis, which shows that Theorem4 in this paper is less conservative than the other results. The second example is used to illustrate the advantage of the robust stability conditions of Theorem 5 and demonstrate how to design robust fuzzy controllers by using our approach.Example 10.
Consider the following fuzzy system with state and input delays:(44)x˙(t)=∑i=12wi(θ(t))[A1ix(t)+A2ix(t-d1(t))+Biu(t-d2(t))],A11=[00.601],A21=[0.50.902],A12=[1010],A22=[0.9011.6],B1=B2=[11].
The fuzzy membership functions are selected as [31]
(45)w1(x1(t))=(1-c(t)sin(|x1(t)|-4)51+exp-100x1(t)3(1-x1(t)))×cos(x1(t))21+exp-2.5x1(t)(3+(x1(t)/0.42))×w2(x1(t))=1-w1(x1(t)),
where
(46)x1(t)∈[-π2π2],c(t)=sin(x1(t))+140∈[-0.050.05].
If we set d1(t)=d2(t) in system (44), we can obtain the system in [26], which implies that our system is more general. Employing the LMIs in [26] and those in Theorem 4 yields the maximum state and input delays h1, and h2, which guarantee the stability of system (44) and the feedback gains forμ1=0, and μ2=0, when ρ1=0.75,ρ2=0.95,t2=0.1,t3=0.2,t4=0.7,t5=0.1, t6=0.4,t7=0.1, and t8=1.3 as given in Table 1, which clearly shows the superiority of the results derived in this paper over those obtained from [26].Table 1
The maximum allowable time delay and feedback gains (μ1=0,μ2=0).
Paper
h
1
h
2
Feedback gains
[26]
0.3120
0.3120
F
1
=
[
1.0598
-
5.6598
]   F2=[-1.3068-4.1167]
Theorem4
0.33
0.57
F
1
=
[
-
0.0042
-
0.0449
]   F2=[0.00480.0717]Example 11.
Consider the well-known truck-trailer system, which can be described by the following T-S fuzzy system:(47)x˙(t)=∑i=12wi(θ(t))[(A1i+ΔA1i)x(t)+A2ix(t-d1(t))+(Bi+ΔBi)u(t-d2(t))],(48)A11=[-avt-Lt000avt-Lt000av2t-22Lt0vt-t00],A21=[-(1-a)vt-Lt000(1-a)vt-Lt000(1-a)v2t-22Lt000],B1=[vt-lt000],A12=[-avt-Lt000avt-Lt000adv2t-22Lt0dvt-t00],A22=[-(1-a)vt-Lt000(1-a)vt-Lt000(1-a)dv2t-22Lt000],(49)B2=[vt-lt000],
where l=2.8,L=5.5,v=-1.0,t-=2.0,t0=0.5, d=10t0/π, Di=[0.25550.25550.2555]T, E1i=E2i=[0.100]T, Ebi=[0.100]T (i=1,2), and the fuzzy membership functions are selected the same as in Example 10.(1)
Leta=1, system (47) is the same as the fuzzy system of [22], which implies that our system is more general. Table 2 gives the maximum input delay value of h2, for which the stabilization is guaranteed by Theorem 5 as well as the state feedback gains, whenρ1=0.75,ρ2=0.95t2=0.1,t3=0.2,t4=0.7,t5=0.1,t6=0.4,t7=0.8, and t8=2. It is clearly visible that the results derived in this paper are better than those from [22, 28].(2)
In the case ofa≠1, the state and input delays of system (47) are all exist. The approach proposed in [22] can not be used to get the maximum of state delay h1, as the system in [22] only contains the input delay. However, the method in [28] and Theorem 5 can be used. Here, let a=0.7, ρ1=0.75,ρ2=0.95, t2=0.1,t3=0.2,t4=0.7,t5=0.1, t6=0.4,t7=0.8, and t8=3, and Table 3 provides the maximum upper bounds of h1 and h2 and the state feedback gains for which the robust stabilization is guaranteed by Theorem 5.Table 2
The maximum allowable time delay and feedback gains (μ2=0).
Paper
h
2
Feedback gains
[22]
0.75
F
1
=
[
3.4227
-
0.3535
0.0045
]  F2=[3.5215-0.36170.0056]
[28]
0.86
F
1
=
[
3.3219
-
0.2406
0.0025
]   F2=[3.3272-0.24940.0026]
Theorem5
1.2
F
1
=
[
-
0.0212
0.0093
-
0.0024
]   F2=[-0.02500.00440.0014]Table 3
The maximum allowable time delay and feedback gains (μ1=μ2=0).
Paper
h
1
h
2
Feedback gains
[28]
0.1
0.55
F
1
=
[
3.3219
-
0.2406
0.0025
]   F2=[3.3272-0.24940.0026]
Theorem5
0.4
1.7
F
1
=
[
-
0.0128
0.0090
-
0.0017
]   F2=[-0.01940.00120.0009]By Theorem5, we can conclude that the uncertain fuzzy model (47) is robust stable based on the following fuzzy control law:
(50)u(t-h2)=∑j=12mj(x1(t-h2))Fjx(t-h2).
With the PDC design technique, it is required that the fuzzy controller must share the same fuzzy membership functions as those of the fuzzy time delay model. Under such a condition, the implementation cost of the fuzzy controller is high by employing these complex membership functions. However, from Theorem 5, the membership functions do not need to be chosen the same as w1(x1(t)) and w2(x1(t)). Thus, we can select some simple membership functions instead, such as
(51)m1(x1(t-h2))=0.75exp(-x1(t-h2)-0.38)22×0.382+0.05,m2(x1(t-h2))=1-m1(x1(t-h2)).
We also assume that ρ1=0.75, and ρ2=0.95, such that mj(x1(t-h2))-ρjwj(x1(t))>0 for j=1,2 andx1(t-h2). The fuzzy control law u(t-h2) = m1[-0.01280.0090-0.0017]x + m2[-0.01940.00120.0009]x can be employed to stabilize the uncertain fuzzy control systems in (47) with the state and input delays. Figures 1 and 2 illustrate the stable state response and control input under the initial condition of x(0)=[4-12]T.Figure 1
The state responses of system (47) with h1=0.4,h2=1.7.Figure 2
The control input of system (47) with h1=0.4,h2=1.7.Remark 12.
In Example11, our method offers less conservative results in the sense of allowing longer time delay and obtaining smaller feedback control gains. Moreover, compared with [22–28], where the fuzzy controller must have the same fuzzy membership functions as those of the fuzzy time delay model, the above membership functions of our fuzzy controller are much simper, which can considerably lower the implementation cost of the fuzzy controller and enhance the design flexibility.Remark 13.
If the function ofc(t) in w1(x1(t)) is unknown, on the basis of the PDC design technique [7–11, 15–20, 22–28], the fuzzy controller cannot be implemented as c(t) is not available. However, using the proposed design method, we can select some simper and well-known membership functions that are different from those of the fuzzy time delay models, for example, m1(x1(t)) and m2(x1(t)), so as to realize the fuzzy controller. In other words, the proposed design approach can be applied even for the fuzzy time delay models with uncertain grades of membership.


## 5. Conclusions

In this paper, we study the robust stability and stabilization problems for general nonlinear fuzzy systems with time varying state and input delays. A less conservative delay-dependent robust stability criterion has been obtained by introducing a novel augmented Lyapunov function with an additional triple-integral term. Moreover, a new design approach different from the traditional PDC design technique is proposed, which can significantly improve the design flexibility and reduce the implementation cost of the fuzzy controller by arbitrarily selecting simple fuzzy membership functions. Two numerical examples are used to illustrate the conservativeness and effectiveness of our proposed methods.



---

*Source: 324738-2013-04-04.xml*