# Robust Stability Criteria for T-S Fuzzy Systems with Time-Varying Delays via Nonquadratic Lyapunov-Krasovskii Functional Approach

Authors: Sung Hyun Kim
Journal: Mathematical Problems in Engineering
 (2016)

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## Abstract

This paper tackles the issue of stability analysis for uncertain T-S fuzzy systems with interval time-varying delays, especially based on the nonquadratic Lyapunov-Krasovskii functional (NLKF). To this end, this paper first provides a less conservative relaxation technique and then derives a relaxed robust stability criterion that enhances the interactions among delayed fuzzy subsystems. The effectiveness of our method is verified by two examples.


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## Body

## 1. Introduction

Over the past few decades, Takagi-Sugeno (T-S) fuzzy model has attracted great attention since it can systematically represent nonlinear systems via a kind of interpolation method that connects smoothly some local linear systems based on fuzzy weighting functions [1]. In particular, the T-S fuzzy model has the advantage that it allows the well-established linear system theory to be applied to the analysis and synthesis of nonlinear systems. For this reason, the T-S fuzzy model has been a popular choice not only in consumer products but also in industrial processes (refer to [2] and references therein).As well-known, time-delay phenomena are ubiquitous in practical engineering systems such as aircraft systems, biological systems, and chemical engineering system [3–5]. Recently, thus, the research on nonlinear systems with state delays has been an important issue in the stability analysis of T-S fuzzy systems. In the literature, there are two major research trends to deal with such systems: one focuses on decreasing computational burdens required to solve a set of conditions from the Lyapunov-Krasovskii functional (LKF) approach, and the other focuses on improving the solvability of delay-dependent stability conditions despite significant computational efforts. Strictly speaking, the first trend is mainly based on Jensen’s inequality approach [6–11] and the second one is based on the free-weighing matrix approach [12–16].Recently, it is recognized that the common quadratic Lyapunov function approach leads to overconservative performance for a large number of fuzzy rules [17, 18]. For this reason, it is essential to tackle the issue of stability analysis in the light of the nonquadratic Lyapunov-Krasovskii functional (NLKF) [19–23]. However, to our best knowledge, up to now, little progress has been made toward using NLKFs for the stability analysis. Motivated by the above concern, this paper proposes a relaxed stability criterion for uncertain T-S fuzzy systems with interval time-varying delays, especially obtained by the NLKF approach. To this end, this paper offers a proper relaxation method that can enhance the interactions among delayed fuzzy subsystems. Further, it is worth noticing that Jensen’s inequality, given in [24], is applicable only to the case where the internal matrix is constant, that is, to the case where the common quadratic Lyapunov-Krasovskii functional (CQLKF) is employed. Thus, this paper focuses more on exploring the second trend in the direction of reducing the conservatism that stems from the CQLKF approach, without resorting to any delay-decomposition method. In this sense, this paper provides two examples numerically to show the effectiveness of our method.The rest of the paper is organized as follows. Section2 gives a mathematical description of the system considered here and presents a useful lemma. Section 3 presents the main result of this paper. Furthermore, through numerical examples, Section 4 shows the verification of our results. Finally, Section 5 makes the concluding remarks.Notation. Throughout this paper, standard notions will be adopted. The notations X≥Y and X>Y mean that X-Y is positive semidefinite and positive definite, respectively. In symmetric block matrices, (∗) is used as an ellipsis for terms that are induced by symmetry. For a square matrix X, He(X) denotes X+XT, where XT is the transpose of X. The natation Conv(·) denotes the convex hull; col(v1,v2,…,vn)=v1Tv2T⋯vnTT for any vector vi; diag(A,B) denotes a diagonal matrix with diagonal entries A and B; and Nr+={1,2,…,r}. For any matrix Si or Sij, (1)Sir=S1S2⋯Sr,Sijr×r=S1irTS2irT⋯SrirTT.All matrices, if their dimensions are not explicitly stated, are assumed to be compatible for algebraic operation.


## 2. System Description and Preliminaries

Consider the following uncertain T-S fuzzy system, which represents a class of nonlinear systems: fori∈Nr+,Plant Rule i. IF η1(t) is Fi1 and  ⋯ηs(t) is Fis, THEN(2)x˙t=Aixt+Ad,ixt-dt+Eipt,qt=Gixt+Gd,ixt-dt,xt=ψt,t∈-d2,0,where x(t)∈Rnx and x(t-d(t))∈Rnx denote the state and the delayed state, respectively; the initial condition ψ(t) is a continuously differentiable vector-valued function; Fij denotes a fuzzy set; ηi(t) denotes the ith premise variable; and r denotes the number of IF-THEN rules. In (2), p(t)∈Rnp and q(t)∈Rnq are used to describe the structured feedback uncertainty such that p(t)=Δ(t)q(t) and ΔT(t)Δ(t)≤I∈Rnq×nq. Further, the state delay d(t) is assumed to be unknown and time-varying with known bounds as follows: d1≤d(t)≤d2, where d1 and d2 are constant. Then, the overall T-S fuzzy model is inferred as follows:(3)x˙t=AΘtxt+AdΘtxt-dt+EΘtpt,qt=GΘtxt+GdΘtxt-dt,where A(Θt)=∑i=1r‍θiAi, Ad(Θt)=∑i=1r‍θiAd,i, E(Θt)=∑i=1r‍θiEi, G(Θt)=∑i=1r‍θiGi, and Gd(Θt)=∑i=1r‍θiGd,i in which θi  (=θi(η(t)) denotes the normalized fuzzy weighting function for the ith rule; η(t)=col(η1(x(t)),…,ηs(x(t))) denotes the premise variable vector; and Θt=col(θ1,…,θr) belongs to(4)SΘ=colθ1,…,θr∣∑i=1rθi=1,αi≤θi≤βi,∀i∈Nr+.Assumption 1.
The fuzzy weighting functionsθi are differentiable and Θ˙t=col(θ˙1,…,θ˙r) belongs to(5)SΘ˙=colθ˙1,…,θ˙r∣∑i=1rθ˙i=0,ϱ1,i≤θ˙i≤ϱ2,i,∀i∈Nr+.To simplify the notations, we useθid1=θi(η(t-d1)) and θid2=θi(η(t-d2)). And, for later convenience, we define x¯(t)=col(x(t),x(t-d1),x(t-d(t)),x(t-d2)), η(t)=col(x¯(t),p(t))∈Rnη, and nη=4nx+np. And we use some block entry matrices ei (i=1,2,…,5) such that x(t)=e1η(t), x(t-d1)=e2η(t), x(t-d(t))=e3η(t), x(t-d2)=e4η(t), and p(t)=e5η(t), which implies x¯(t)=e14η(t) by defining e14T=e1T⋯e4T. Then, (3) becomes(6)x˙t=Φtηt,qt=Ψtηt,where Φt=A(Θt)e1+Ad(Θt)e3+E(Θt)e5 and Ψt=G(Θt)e1+Gd(Θt)e3.Lemma 2.
LetΘt∈SΘ be satisfied. Then, the following condition holds:(7)0>M=M0+∑i=1rθiHeMi+∑i=1rθi2Mii+∑i=1r∑j=i+1rθiθjMij+∑j=1i-1θiθjMjiTif there are all decision variables such that(8)
0
>
L
=
L
0
L
i
r
∗
L
i
j
r
r
,
0
<
X
i
+
X
i
T
,
∀
i
∈
N
r
+
,where L0=M0+He(S0-∑i=1r‍αiβiXi), Li=Mi+Si-S0+(αi+βi)Xi, Lii=Mii+He(-Si-Xi), and Lij=Mij-Si-Sj.Lemma 3.
LetΘ˙t∈SΘ˙ be satisfied. Then, the following condition holds:(9)0>Ω+∑i=1rθ˙iPiif there are all decision variables such that(10)0>Ω+∑i=1rϱli,iPi+Q,∀l∈L=l1,…,lr∣li∈N2+,i∈Nr+.Proof.
In view ofΘ˙t∈SΘ˙, we can get(11)θ˙i=∑li=12λlitϱli,i,0=∑i=1rθ˙iN,where coefficients λli are all positive and sum to one and N is a constant slack variable. Then, (9) leads to(12)0>Ω+∑i=1r∑li=12λlitϱli,iPi+N,which holds if (10) holds because ∑li=12‍λlitϱli,iPi+N∈Conv(ϱli,i(Pi+N)), where li denotes the ith element of l∈L.


## 3.Θt-Dependent Stability Criterion

Based on a nonquadratic Lyapunov-Krasovskii functional (NLKF), this section provides a less conservative stability criterion. To this end, we first choose an NLKF of the following form:(13)Vt=V1t+V2t+V3t,V1t=xTtPΘtxt,V2t=∫t-d1txTαQ1Θαxαdα+∫t-d2txTαQ2Θαxαdα,V3t=∫-d10∫t+αtx˙TβR1Θβx˙βdβdα+∫-d2-d1∫t+αtx˙TβR2Θβx˙βdβdα,where P(Θt), Q1(Θα), Q2(Θα), R1(Θβ), and R2(Θβ) are positive definite for all admissible grades. Then, the time derivative of each Vi(t) along the trajectories of (6) is given by(14)V˙1t=ηTtHee1TPΘtΦt+e1TP˙Θte1ηt,V˙2t=ηTte1TQ1Θt+Q2Θte1-e2TQ1Θt-d1e2-e4TQ2Θt-d2e4ηt,V˙3t=ηTtd1ΦtTR1ΘtΦt+δΦtTR2ΘtΦtηt-∫t-d1tx˙TαR1Θαx˙αdα-∫t-d2t-d1x˙TαR2Θαx˙αdα,which leads to(15)V˙t=ηTtΠ1ηt+O1+O2,where(16)Π1=Hee1TPΘtΦt+e1TP˙Θt+Q1Θt+Q2Θte1-e2TQ1Θt-d1e2-e4TQ2Θt-d2e4+d1ΦtTR1ΘtΦt+δΦtTR2ΘtΦt,O1=-∫t-d1tx˙TαR1Θαx˙αdα,δ=d2-d1,O2=-∫t-d2t-dtx˙Tα1R2Θα1x˙α1dα1-∫t-dtt-d1x˙Tα2R2Θα2x˙α2dα2.Remark 4.
Indeed, it is hard to directly use Jensen’s inequality approach to obtain the upper bounds ofO1 and O2 because R1(Θα) and R2(Θα) are set to be dependent on Θα, which motivates the present study.Lemma 5.
Suppose that there exist matricesU0(Θt), U1(Θt), and U2(Θt)∈R4nx×nx and symmetric matrices 0<PΘt,P˙(Θt), 0<Q1(Θt-d1), 0<Q2(Θt-d2), 0<Q1(Θt), 0<Q2(Θt), 0<R1(Θα), 0<R2(Θαp), 0<R1(Θt), 0<R2(Θt)∈Rnx×nx, M0(Θt), M1(Θt), and M2(Θt)∈R4nx×4nx such that(17)0>Π1+Π2+Γp,∀p∈N2+,(18)0≤M0ΘtU0Θt∗R1Θα,0≤MpΘtUpΘt∗R2Θαp,∀p∈N2+,where(19)Π1=Hee1TPΘtΦt+e1TP˙Θt+Q1Θt+Q2Θte1-e2TQ1Θt-d1e2-e4TQ2Θt-d2e4+d1ΦtTR1ΘtΦt+δΦtTR2ΘtΦt,Π2=ΨtTΨt-e5Te5,Γp=Hee14TU0Θte1-e2+e14TU1Θte3-e4+e14TU2Θte2-e3+d1e14TM0Θte14+δe14TMpΘte14.Then, (6) is robustly asymptotically stable for d1≤d(t)≤d2.Proof.
First of all, by incorporating the following equalities into (15),(20)0=x¯TtM0Θtx¯td1-∫t-d1tdα,0=x¯TtM1Θtx¯td2-dt-∫t-d2t-dtdα1,0=x¯TtM2Θtx¯tdt-d1-∫t-dtt-d1dα2,0=2x¯TtU0Θte1-e2ηt-∫t-d1tx˙αdα,0=2x¯TtU1Θte3-e4ηt-∫t-d2t-dtx˙α1dα1,0=2x¯TtU2Θte2-e3ηt-∫t-dtt-d1x˙α2dα2,we can get(21)V˙t=ηTtΠ1+Γ¯pηt+O¯1+O¯2,where(22)Γ¯p=Hee14TU0Θte1-e2+e14TU1Θte3-e4+e14TU2Θte2-e3+d1e14TM0Θte14+δe14T∑p=12λptMpte14,O¯1=-∫t-d1tx¯tx˙αTM0ΘtU0Θt∗R1Θαx¯tx˙αdα,O¯2=-∫t-d2t-dtx¯tx˙α1TM1ΘtU1Θt∗R2Θα1x¯tx˙α1dα1-∫t-dtt-d1x¯tx˙α2TM2ΘtU2Θt∗R2Θα2x¯tx˙α2dα2in which λ1(t)=(d2-d(t))/(d2-d1) and λ2(t)=(d(t)-d1)/(d2-d1). Next, the structured feedback uncertainty, given as 0≤qT(t)q(t)-pT(t)p(t), can be converted into 0≤ηT(t)(ΨtTΨt-e5Te5)η(t), which yields V˙(t)≤ηT(t)(Π1+Π2+Γ¯p)η(t)+O¯1+O¯2. That is, the robust stability for (6) is assured by 0>ηT(t)(Π1+Π2+Γ¯p)η(t)+O¯1+O¯2. Therefore, if (18) holds, then O¯1+O¯2≤0, and hence the robust stability criterion is given by (17) because ∑p=12‍λp(t)Mp(t)∈Conv(Mp(Θt)).In the absence of uncertainties, the T-S fuzzy system becomesx˙(t)=Φtx¯(t), where Φt=A(Θt)e1+Ad(Θt)e3. The following corollary presents the stability criterion for nominal T-S fuzzy systems with time-varying delays.Corollary 6.
Suppose that there exist matricesU0(Θt), U1(Θt), and U2(Θt)∈R4nx×nx and symmetric matrices 0<P(Θt), P˙(Θt), 0<Q1(Θt-d1), 0<Q2(Θt-d2), 0<Q1(Θt), 0<Q2(Θt), 0<R1(Θα), 0<R2(Θαp), 0<R1(Θt), 0<R2(Θt)∈Rnx×nx, M0(Θt),M1(Θt),  and  M2(Θt)∈R4nx×4nx such that(23)0>Π1+Γp,∀p∈N2+,0≤M0ΘtU0Θt∗R1Θα,0≤MpΘtUpΘt∗R2Θαp,∀p∈N2+,where(24)Π1=Hee1TPΘtΦt+e1TP˙Θt+Q1Θt+Q2Θte1-e2TQ1Θt-d1e2-e4TQ2Θt-d2e4+d1ΦtTR1ΘtΦt+δΦtTR2ΘtΦt,Γp=Hee14TU0Θte1-e2+e14TU1Θte3-e4+e14TU2Θte2-e3+d1e14TM0Θte14+δe14TMpΘte14.Then, (6) without uncertainties is asymptotically stable for d1≤d(t)≤d2.Proof.
The proof is omitted since it is analogous to the derivation of Lemma5.


## 4. LMI-Based Stability Criterion

Based on Lemmas2 and 3, to derive a finite number of solvable LMI conditions from (17), this paper simply sets all the decision variables to be of affine dependence on fuzzy-weighting functions:(25)PΘt=∑i=1rθiPi,P˙Θt=∑i=1rθ˙iPi,Q1Θt=∑i=1rθiQ1,i,Q2Θt=∑i=1rθiQ2,i,(26)Q1Θt-d1=∑q=1rθqd1Q1,q,Q2Θt-d2=∑ϕ=1rθϕd2Q2,ϕ,R1Θt=∑i=1rθiR1,i,(27)R2Θt=∑i=1rθiR2,i,R1Θα=∑i=1rθiαR1,i,R2Θαp=∑i=1rθiαpR2,i,(28)U0Θt=∑i=1rθiU0,i,U1Θt=∑i=1rθiU1,i,U2Θt=∑i=1rθiU2,i,M0Θt=∑i=1rθiM0,i,M1Θt=∑i=1rθiM1,i,M2Θt=∑i=1rθiM2,i.Remark 7.
As a way to improve the performance to be considered, we can increase the degree of polynomial dependence on fuzzy-weighting functions, as in [31–33] but this is outside of the intended scope of this paper.Theorem 8.
LetΘ˙t∈SΘ˙ be satisfied. Suppose that there exist matrices U0,i,U1,i,U2,i∈R4nx×nx and S0,Si,Xi∈Rnc×nc(nc=6nx+np+nq), for i∈Nr+, symmetric matrices N,0<Pi, 0<Q1,i, 0<Q2,i, 0<R1,i, and 0<R2,i∈Rnx×nx, for i∈Nr+, and M0,i,M1,i,M2,i∈R4nx×4nx such that, for all q,ϕ∈Nr+, p∈N2+, and l∈L,(29)
0
>
L
l
q
ϕ
,
0
L
p
,
i
r
∗
L
i
j
r
×
r
,
0
<
X
q
+
X
q
T
,
(30)
0
≤
M
0
,
ϕ
U
0
,
ϕ
∗
R
1
,
q
,
0
≤
M
p
,
ϕ
U
p
,
ϕ
∗
R
2
,
q
,where Llqϕ,0=Mlqϕ,0+He(S0-∑i=1r‍αiβiXi), Lp,i=Mp,i+Si-S0+(αi+βi)Xi, Lii=Mii+He(-Si-Xi), and Lij=Mij-Si-Sj in which(31)Mlqϕ,0=diag⁡-I,0,0,4,4lqϕ,0,(32)Mp,i=0001,4i0-12δR2,i0000-12d1R1,i00004,4p,i,(33)Mii=00000002,4ii0003,4ii0∗∗He4,4ii,Mij=00000002,4ij+2,4ji0003,4ij+3,4ji0004,4ij+4,4ji,(34)4,4lqϕ,0=e1T∑i=1rϱli,iPi+Ne1-e2TQ1,qe2-e4TQ2,ϕe4-e5Te5,1,4i=Gie1+Gd,ie3,4,4p,i=e1T12Q1,i+12Q2,ie1+Γp,i,2,4ij=δR2,jAie1+R2,jAd,ie3+R2,jEie5,3,4ij=d1R1,jAie1+R1,jAd,ie3+R1,jEie5,4,4ij=e1TPjAie1+e1TPjAd,ie3+e1PjEie5,Γp,i=e14TU0,ie1-e2+e14TU1,ie3-e4+e14TU2,ie2-e3+12d1e14TM0,ie14+12δe14TMp,ie14.Then, the system in (6) is robustly asymptotically stable for d1≤d(t)≤d2.Proof.
Note thatΘ˙t∈SΘ˙. Thus, in view of Lemma 3, applying the Schur complement to (17) is given by(35)
0
>
-
I
0
0
Ψ
t
0
-
δ
R
2
Θ
t
0
δ
R
2
Θ
t
Φ
t
0
0
-
d
1
R
1
Θ
t
d
1
R
1
Θ
t
Φ
t
∗
∗
∗
Ω
l
+
Γ
p
,
∀
p
∈
N
2
+
,
l
∈
L
,where Ωl=He(e1TP(Θt)Φt)-e2TQ1(Θt-d1)e2-e4TQ2(Θt-d2)e4-e5Te5+e1T(∑i=1r‍ϱli,i(Pi+N)+Q1(Θt)+Q2(Θt))e1. Further, from (26) and (27), (35) and (18) can be converted into(36)0>∑q=1r∑ϕ=1rθqd1θϕd2Mp,lqϕΘt∈Rnc×nc,∀p∈N2+,l∈L,0≤∑ϕ=1r∑q=1rθϕθqαM0,ϕU0,ϕ∗R1,q,0≤∑ϕ=1r∑q=1rθϕθqαpMp,ϕUp,ϕ∗R2,q,where(37)
M
p
,
l
q
ϕ
Θ
t
=
-
I
0
0
Ψ
t
0
-
δ
R
2
Θ
t
0
δ
R
2
Θ
t
Φ
t
0
0
-
d
1
R
1
Θ
t
d
1
R
1
Θ
t
Φ
t
∗
∗
∗
Ω
l
q
ϕ
+
Γ
p
,
Ω
l
q
ϕ
=
H
e
e
1
T
P
Θ
t
Φ
t
-
e
2
T
Q
1
,
q
e
2
-
e
4
T
Q
2
,
ϕ
e
4
-
e
5
T
e
5
+
e
1
T
∑
i
=
1
r
ϱ
l
i
,
i
P
i
+
N
+
Q
1
Θ
t
+
Q
2
Θ
t
e
1
.As a result, from the convexity of fuzzy-weighting functions, (17) and (18) can be assured by (30),(38)0>Mp,lqϕΘt,∀q,ϕ∈Nr+,p∈N2+,l∈L.Further, note that representing (38) in the form of (7) becomes(39)0>Mlqϕ,0+∑i=1rθiHeMp,i+∑i=1rθi2Mii+∑i=1r∑j=i+1rθiθjMij+∑j=1i-1θiθjMjiT,where Mlqϕ,0, Mp,i, Mii, and Mij are defined in (31)–(33). Therefore, from Lemma 2, we can obtain (29) in the sequel without loss of generality.The following corollary presents the LMI-based stability criterion for nominal T-S fuzzy systems with time-varying delays.Corollary 9.
LetΘ˙t∈SΘ˙ be satisfied. Suppose that there exist matrices U0,i,U1,i,U2,i∈R4nx×nx and S0,Si,Xi∈Rnc×nc(nc=6nx), for i∈Nr+, symmetric matrices N,0<Pi, 0<Q1,i, 0<Q2,i, 0<R1,i, and 0<R2,i∈Rnx×nx, for i∈Nr+, and M0,i,M1,i,M2,i∈R4nx×4nx such that, for all q,ϕ∈Nr+, p∈N2+, and l∈L,(40)
0
>
L
l
q
ϕ
,
0
L
p
,
i
r
∗
L
i
j
r
×
r
,
0
<
X
q
+
X
q
T
,
0
≤
M
0
,
ϕ
U
0
,
ϕ
∗
R
1
,
q
,
0
≤
M
p
,
ϕ
U
p
,
ϕ
∗
R
2
,
q
,where Llpϕ,0=Mlpϕ,0+He(S0-∑i=1r‍αiβiXi), Lp,i=Mp,i+Si-S0+(αi+βi)Xi, Lii=Mii+He(-Si-Xi), and Lij=Mij-Si-Sj in which(41)Mlpϕ,0=diag⁡0,0,3,3lpϕ,0,Mp,i=-12δR2,i000-12d1R1,i0003,3p,i,Mii=001,3ii002,3ii∗∗He3,3ii,Mij=001,3ij+1,3ji002,3ij+2,3ji003,3ij+3,3ji,3,3lqϕ,0=e1T∑i=1rϱli,iPi+Ne1-e2TQ1,qe2-e4TQ2,ϕe4,3,3p,i=e1T12Q1,i+12Q2,ie1+Γp,i,1,3ij=δR2,jAie1+R2,jAd,ie3,2,3ij=d1R1,jAie1+R1,jAd,ie3,3,3ij=e1TPjAie1+e1TPjAd,ie3,Γp,i=e14TU0,ie1-e2+e14TU1,ie3-e4+e14TU2,ie2-e3+12d1e14TM0,ie14+12δe14TMp,ie14.Then, (6) without uncertainties is asymptotically stable for d1≤d(t)≤d2.Proof.
The proof is omitted since it is analogous to the derivation of Theorem8.Remark 10.
The number of scalar variables involved in Theorem8 and Corollary 9 is given as follows: #=nc2(2r+1)+0.5nx(nx+1)+nx(38.5nx+8.5)r. Table 1 shows the number for each case of (nx,r). Since the use of slack variables requires more computation cost compared with other methods, there may be the need to balance the tradeoffs between the computational cost and the performance enhancement.Table 1
# involved in Corollary 9 and Theorem 8 (np=1,nq=1).
( n x , r ) (2, 2) (2, 3) (2, 4) (3, 2) (3, 3) (3, 4) (4, 2) (4, 3) (4, 4) Corollary9 1065 1524 1983 2370 3390 4410 4190 5992 7794 Theorem8 1325 1888 2451 2750 3922 5094 4690 6692 8694


## 5. Numerical Examples

To verify the effectiveness of our methods, this paper provides two examples that make some comparisons with other results: one is related to the stability analysis for nominal T-S fuzzy systems and the other is related to the robust stability analysis for T-S fuzzy systems with uncertainties.Example 1.
Consider the following T-S fuzzy system, adopted in [25]:(42)A1=-200-0.9,A2=-10.50-1,Ad,1=-10-1-1,Ad,2=-100.1-1,where θ1=1/(1+exp⁡(-2x1(t))) and θ2=1-θ1. Table 2 shows the maximum allowable upper bound (MAUB) for each d1∈{0.0,0.4,0.8,1.0,1.2}, where m denotes the number of delay segments and (m-1) denotes the degree of delay partitioning. From Table 2, we can see that our method (Corollary 9) provides larger MAUBs in comparison with those of [25, 26]. Hence it can be concluded that the stability criterion in Corollary 9, obtained based on the NLKF, is less conservative than other results. In particular, for d1=1.2 and d2=1.531, Corollary 9 offers the following solutions:(43)P1=10-21.485-0.227-0.2270.368,P2=10-21.078-0.290-0.2900.799,Q1,1=10-33.3150.2530.2530.785,Q1,2=10-33.0820.1630.1630.930,Q2,1=10-35.6070.2540.2543.137,Q2,2=10-35.708-0.001-0.0013.323,R1,1=10-34.661-0.847-0.8471.669,R1,2=10-34.647-0.813-0.8131.671,R2,1=10-37.232-1.319-1.3192.747,R2,2=10-36.538-0.704-0.7042.583.Table 2
Maximum allowable upper bound (MAUB) for eachd1, where m denotes the number of delay segments and (m-1) denotes the degree of delay partitioning.
d 1 0.0 0.4 0.8 1.0 1.2 (m-1) [25] 0.982 1.038 1.158 1.252 1.359 0 [26] 1.221 1.277 1.311 1.358 1.419 1 [26] 1.278 1.303 1.316 1.361 1.425 2 Corollary9 1.302 1.380 1.413 1.462 1.531 0Example 2.
Consider the following T-S fuzzy system:(44)A1=-210.5-1,A2=-200-1,Ad,1=-10-1-1,Ad,2=-1.600-1,E1=0.0300-0.03,E2=0.0300-0.03,G1=1.6000.05,G2=1.600-0.05,Gd1=0.1000.3,Gd2=0.1000.3,where(45)θ1=1-11+exp⁡-6x2-π/4×11+exp⁡-6x2+π/4,θ2=1-θ1.The maximum allowable upper bound (MAUB) for each method is tabulated in Table 3. And, from Table 3, we can see that the proposed method (Theorem 8) achieves larger MAUBs than those of other methods [27–30]. Hence, it can be concluded that the robust stability criterion in Theorem 8, established from the NLKF approach and Lemma 2, is less conservative than those of [27–30].Table 3
Maximum allowable upper bound (MAUB) ford1=0.
Methods [27] [28] [29] [30] Theorem8 d 2 — 0.443 0.499 1.081 1.132


## 6. Concluding Remarks

This paper proposed an NLKF-based method of deriving a less conservative stability criterion for T-S fuzzy systems with time-varying delays. Of course, the proposed method may increase the burden of numerical computation. However, if the computational complexity is out of the practical problem, then our results can be significantly useful.



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