Stability analysis of systems with timevarying delay via a novel Lyapunov functional

Abstract

This paper investigates the stability problem for time-varying delay systems. To obtain a larger delay bound, this paper uses the second-order canonical Bessel-Legendre ( BL ) inequality. Secondly, using four couples of integral terms in the augmented Lyapunov-Krasovskii function ( LKF ) to enhance the relationship between integral functionals and other vectors. Furthermore, unlike the construction of the traditional LKF, a novel augmented LKF is constructed with two new delay-product-type terms, which adds more state information and leads to less conservative results. Finally, two numerical examples are provided to demonstrate the effectiveness and the significant improvement of the proposed stability criteria.

Full text

1068 IEEE/CAA JOURNAL OF AUTOMATICA SINICA, VOL. 6, NO. 4, JULY 2019
Stability Analysis of Systems With Time-varying
Delay via a Novel Lyapunov Functional
Yun Chen and Gang Chen
Abstract—This paper investigates the stability problem for
time-varying delay systems. To obtain a larger delay bound,
this paper uses the second-order canonical Bessel-Legendre (B-
L) inequality. Secondly, using four couples of integral terms in
the augmented Lyapunov-Krasovskii function (LKF) to enhance
the relationship between integral functionals and other vectors.
Furthermore, unlike the construction of the traditional LKF,
a novel augmented LKF is constructed with two new delay-
product-type terms, which adds more state information and leads
to less conservative results. Finally, two numerical examples are
provided to demonstrate the effectiveness and the significant
improvement of the proposed stability criteria.
Index Terms—Integral inequality, Lyapunov theorem, stability,
time-delay systems.
I. INTRODUCTION
TIME-DELAYS appear in many engineering systems such
as aircraft, chemical control systems, biology, and power
systems. It is well known that the time delay often causes
the oscillation, deterioration of system performances, and
even instability [1]−[13], accordingly, the main objective of
stability analysis in time delay systems is to find maximum
allowable delay such that systems with time-varying delays
remain stable. Therefore, the stability analysis of time delay
systems has attracted many researchers as a challenging prob-
lem during the past years.
The Lyapunov method is one of the most fruitful fields
in the stability analysis of time delay systems. The first one
is to construct suitable Lyapunov-Krasovskii function (LKF),
recently, in most existing works. Constructing LKF approaches
usually consist of the multiple integral LKF approach and the
augmented LKF approach. It is noteworthy that some results
were reported in [14]−[16] which provided a new direction
to construct LKF. The main contributions of [14]−[16] do not
meet positive definiteness of all matrices in the LKF and delay-
product-type terms are introduced into the LKF. The other
way is to derive more and more tight inequalities, some novel
inequalities have been proposed in the literature such as the
Manuscript received December 11, 2018; accepted January 13, 2019.
This work was supported by the National Natural Science Foundation of
China (61703153) and the Natural Science Foundation of Hunan Province
(2018JJ4075). Recommended by Associate Editor Qinglong Han. (Corre-
sponding author: Gang Chen.)
Citation: Y. Chen and G. Chen, “Stability analysis of systems with time-
varying delay via a novel Lyapunov functional,” IEEE/CAA J. Autom. Sinica,
vol. 6, no. 4, pp. 1068−1073, Jul. 2019.
The authors are both with the School of Electrical and Information
Engineering, Hunan University of Technology, Zhuzhou 412007, and also
with the Key Laboratory for Electric Drive Control and Intelligent Equipment
of Hunan Province, Zhuzhou 412007, China (e-mail: holycy9408@163.com;
drchengang@163.com).
Digital Object Identifier 10.1109/JAS.2019.1911597
Jensen inequality [17], the Wirtinger-based integral inequality
[18], the Bessel-Legendre (B-L) inequality [19] and so on
[20]. These inequalities have been generally used for the topic
in the last several years [21]. However, their applications to
systems with time-varying delays reveal additional difficulties
related to the inverse of the length of integral interval, although
the reciprocally convex inequalities in [22]−[25] are helpful
for solve the problem, the conservative seems to remain for
stability analysis of systems with time-varying delay. Based
on above discussions, naturally the question arises: how less
conservative stability criterion can be derived by construct-
ing suitable LKF, and how to deal with the inverse of the
length of the integral interval without the reciprocal convex
inequality.
The aim of the present work is to answer these questions.
The main contributions of this paper are twofold. On the
one hand, four couples of integral terms (Rs
t−d(t)xT(v)dv,
Rt
sxT(v)dv,Rs
t−hxT(v)dv,Rt−d(t)
sxT(v)dv,Rt
s˙xT(v)dv,
Rs
t−d(t)˙xT(v)dv,Rt−d(t)
s˙xT(v)dv, and Rs
t−h˙xT(v)dv) ap-
pear in the augmented LKF so that the relationship be-
tween other vectors and integral functionals is enhanced.
In addition, a novel augmented LKF is constructed by two
new delay-product-type terms (d(t)ζT
1(t)Q1(t)ζ1(t),(h−
d(t))ζT
2(t)Q2(t)ζ2(t)), which full utilize the information of
time delay derivative. On the other hand, based on the second-
order canonical B-L inequality, less conservative stability
criteria are derived in terms of linear matrix inequalities
(LMIs) and numerical examples are provided to demonstrate
the advantages of the proposed criteria.
The rest of this paper is structured as follows. Section
II gives the problem formulation and necessary preliminary.
The main result for time-varying delay systems, stabilization
analysis is given in Section III. Numerical examples are
presented in Section IV. At last, we draw some conclusions in
Section V.
Throughout this paper, Rndenotes the n-dimensional Eu-
clidean space. Rn×mis the set of all n×mreal matrices.
Sn(respectively, Sn
+) denotes a set of symmetric matrices
(respectively, positive definite matrices). P > 0means that
the matrix Pis symmetric and positive definite. N−1and
NTstand for the inverse and transpose of the matrix N,
respectively. Iand 0represent the identity matrix and a zero
matrix, respectively. The symmetric terms in a symmetric
matrix are denoted by “∗”. diag{·} denotes a block-diagonal
matrix and Sym{X}=X+XT.

CHEN AND CHEN: STABILITY ANALYSIS OF SYSTEMS WITH TIME-VARYING DELAY VIA A NOVEL LYAPUNOV FUNCTIONAL 1069
II. PRELIMINARIES
Consider a linear system with a time-varying delay
(˙x(t) = Ax(t) + Adx(t−d(t))
x(t) = φ(t)t∈[−h, 0] (1)
where φ(t)is the initial condition, and delay d(t)satisfies
0≤d(t)≤h, µ1≤˙
d(t)≤µ2<1.(2)
The following two lemmas are given for deriving the main
results. Firstly, we introduce the inequality that will be the
core of the paper. It corresponds to the inequality shown in
[26], which is also a particular case of [27], [28]. The proof
of this inequality can be found in [27]−[29].
Lemma 1: Given a symmetric positive definite matrix R∈
Rn×n, any scalars a < b, any differentiable function xin [a,
b]→Rn,Φi∈Rn×k(i= 1,2,3), any constant matrices
Hi∈Rn×k(i= 1,2,3) and a vector ξ∈Rk, the following
inequality holds:
−Zb
a
˙xT(s)R˙x(s)ds ≤ξTΘξ(3)
where
Θ = ¯
ΦTH+HT¯
Φ+(b−a)HT¯
R−1H
¯
Φ = [ΦT
1ΦT
2ΦT
3]TH= [HT
1HT
2HT
3]T
Φ1ξ=x(b)−x(a)
Φ2ξ=x(b) + x(a)−2
b−aZb
a
x(s)ds
Φ3ξ=x(b)−x(a)−6
b−aZb
a
x(s)ds
+12
(b−a)2Zb
a
(b−s)x(s)ds
¯
R= diag{R, 3R, 5R}.
Remark 1: The right-hand side of (3) is related to the length
of the integral interval rather than in the form of its inverse.
As pointed out in [30], the change, although trivial, achieves
significant advantages in assessing the stability of time-delay
systems, for the time-varying case, the reciprocally convex
combination inequality is no longer needed, thus resulting in
more relaxed criteria in [31], [32] than the criteria in [18].
Remark 2: It is worth noticing that, compared with Theorem
1 in [33] (N= 2), it can be found that two inequalities are
equivalent when taking H=−X,ξ=ξN, that means the
Lemma 1 is more general. Furthermore, Theorem 1 in [33]
may not have an advantage in constructing the LKF.
Lemma 2 [34]: For a given quadratic function F(s) =
a2s2+a1s+a0,where ai∈R(i= 0,1,2), if following
inequalities holds:
i) F(0) <0; ii) F(h)<0; iii) F(0) −h2a2<0.(4)
Then F(s)<0for ∀s∈[0, h].
III. MAI N RES ULTS
In this section, we shall develop some new stability con-
ditions based on the second-order canonical B-L inequality.
To simplify vector and matrix representation, the following
notations are defined.
δ1(t) = Zt−d(t)
t−h
xT(s)ds
δ2(t) = Zt−d(t)
t−h
(t−d(t)−s)xT(s)
h−d(t)ds
δ3(t) = Zt
t−d(t)
xT(s)ds
δ4(t) = Zt
t−d(t)
(t−s)xT(s)
d(t)ds
η0(t) = £xT(t)xT(t−d(t)) xT(t−h)¤T
η1(t) = £ηT
0(t)xT(s) ˙xT(s)Zs
t−d(t)
xT(v)dv Zt
s
xT(v)dv
Zt
s
˙xT(v)dv Zs
t−d(t)
˙xT(v)dv¤T
η2(t) = £ηT
0(t)xT(s) ˙xT(s)Zs
t−h
xT(v)dv Zt−d(t)
s
xT(v)dv
Zt−d(t)
s
˙xT(v)dv Zs
t−h
˙xT(v)dv¤T
η3(t) = £δT
1(t)
h−d(t)
δT
2(t)
h−d(t)
δT
3(t)
d(t)
δT
4(t)
d(t)¤T
ξ(t) = £ηT
0(t)ηT
3(t) ˙xT(t−d(t)) ˙xT(t−h)¤T
ζ1(t) = £xT(t)xT(t−d(t)) 1
d(t)Zt
t−d(t)
xT(s)ds
1
d2(t)Zt
t−d(t)
(t−s)xT(s)ds¤T
ζ2(t) = £xT(t−d(t)) xT(t−h)1
h−d(t)Zt−d(t)
t−h
xT(s)ds
1
(h−d(t))2Zt−d(t)
t−h
(t−d(t)−s)xT(s)ds¤T
ei=£0n×(i−1)nIn0n×(9−i)n¤, i = 1,2, . . . , 9.
Now, we present our first stability criterion.
Theorem 1: Given scalars h,µ1and µ2, the system (1)
is stable for delay function d(t)satisfying (2) if there exist
matrices S1,S2∈S9n
+,R∈Sn
+,Q11,Q12 ,Q21,Q22 ∈S4n,
and H1,H2∈R3n×9nsuch that the following LMIs are
feasible:
·Ψ(0,˙
d(t)) hHT
1
∗ −he
R¸<0,·Ψ(h, ˙
d(t)) hHT
2
∗ −he
R¸<0(5)
·Ψ(0,˙
d(t)) −h2∆( ˙
d(t)) hHT
1
∗ −he
R¸<0(6)
d(t)Q11 +Q12 >0,(h−d(t))Q21 +Q22 >0(7)
where
Ψ(d(t),˙
d(t)) = Ψ1(d(t),˙
d(t)) + Ψ2(d(t),˙
d(t))

1070 IEEE/CAA JOURNAL OF AUTOMATICA SINICA, VOL. 6, NO. 4, JULY 2019
Ψ1(d(t),˙
d(t)) = Sym{γT
1(˙
d(t))S1[ω30 +d(t)ω31 +d2(t)
×ω32] + γT
2(˙
d(t))S2[ω60 + (h−d(t))ω61
+ (h−d(t))2ω62] + TT
1H1+TT
2H2}
+ [ω11 +d(t)ω12]TS1[ω11 +d(t)ω12 ]
−(1 −˙
d(t))[ω21 +d(t)ω22]TS1[ω21 +d(t)
×ω22] + (1 −˙
d(t))[ω41 + (h−d(t))ω42]T
×S2[ω41 + (h−d(t))ω42]−[ω51
+ (h−d(t))ω52]TS2[ω51 + (h−d(t))ω52 ]
+hωT
0Rω0
Ψ2(d(t),˙
d(t)) = Γ1(˙
d(t))+d(t)Γ2(˙
d(t)) + d2(t)
×Γ3(˙
d(t))
∆( ˙
d(t)) = ∆( ˙
d(t))1+ Γ3(˙
d(t))
∆( ˙
d(t))1= Sym{γT
1(˙
d(t))S1ω32 +γT
2(˙
d(t))S2ω62}
+ωT
12S1ω12 −(1−˙
d(t))ωT
22S1ω22 −ωT
52S2ω52
+ (1 −˙
d(t))ωT
42S2ω42
Γ1(˙
d(t)) = Sym{ΠT
1Q12γ4(˙
d(t)) + ΠT
2(hQ21 +Q22)
×γ6(˙
d(t))}+˙
d(t)ΠT
1Q12Π1
−˙
d(t)ΠT
2(2hQ21 +Q22)Π2
Γ2(˙
d(t)) = Sym{ΠT
1(Q11γ4(˙
d(t)) + Q12γ3(˙
d(t))) + ΠT
2
×(hQ21 +Q22)γ5(˙
d(t)) −ΠT
2Q21γ6(˙
d(t))
+˙
d(t)ΠT
1Q11Π1+˙
d(t)ΠT
2Q21Π2}
Γ3(˙
d(t)) = Sym{ΠT
1Q11γ3(˙
d(t)) −ΠT
2Q21γ5(˙
d(t))}
e
R= diag{R,3R,5R}
ω0=Ae1+Ade2
ω11 =£eT
1eT
2eT
3eT
1ωT
00 0 0 eT
1−eT
2¤T
ω12 =£00000eT
60 0 0¤T
ω21 =£eT
1eT
2eT
3eT
2eT
80 0 eT
1−eT
20¤T
ω22 =£000000eT
60 0¤T
ω30 =£0000eT
1−eT
20 0 0 0¤T
ω31 =£eT
1eT
2eT
3eT
6000eT
1−eT
6eT
6−eT
2¤T
ω32 =£00000eT
7eT
6−eT
70 0¤T
ω41 =£eT
1eT
2eT
3eT
2eT
8000eT
2−eT
3¤T
ω42 =£00000eT
40 0 0¤T
ω51 =£eT
1eT
2eT
3eT
3eT
90 0 eT
2−eT
30¤T
ω52 =£000000eT
40 0¤T
ω60 =£0000eT
2−eT
30 0 0 0¤T
ω61 =£eT
1eT
2eT
3eT
4000eT
2−eT
4eT
4−eT
3¤T
ω62 =£00000eT
5eT
4−eT
50 0¤T
γ1(˙
d(t)) = £ωT
0(1 −˙
d(t))eT
8eT
900(˙
d(t)−1)eT
2
eT
1wT
0−(1 −˙
d(t))eT
8¤T
γ2(˙
d(t)) = £ωT
0(1 −˙
d(t))eT
8eT
90 0 −eT
3
(1 −˙
d(t))eT
2(1 −˙
d(t))eT
8−eT
9¤T
γ3(˙
d(t)) = £ωT
0(1 −˙
d(t))eT
80 0¤T
γ4(˙
d(t)) = £0 0 eT
1−(1 −˙
d(t))eT
2−˙
d(t)eT
6
−(1 −˙
d(t))eT
2+eT
6−2˙
d(t)eT
7¤T
γ5(˙
d(t)) = £−(1 −˙
d(t))eT
8−eT
90 0¤T
γ6(˙
d(t)) = £h(1 −˙
d(t))eT
8heT
9(1 −˙
d(t))eT
2−eT
3+˙
d(t)
×eT
4−eT
3+ (1 −˙
d(t))eT
4+ 2 ˙
d(t)eT
5¤T
Π1=£eT
1eT
2eT
6eT
7¤T
Π2=£eT
2eT
3eT
4eT
5¤T
T1=£eT
2−eT
3eT
2+eT
3−2eT
4eT
2−eT
3−6eT
4+ 12eT
5¤T
T2=£eT
1−eT
2eT
1+eT
2−2eT
6eT
1−eT
2−6eT
6+ 12eT
7¤T.
Proof: Choose the following LKF candidate:
V(xt) = Vz(xt) + Vw(xt) + Vr(xt)(8)
with
Vz(xt) = Zt
t−d(t)
ηT
1(t)S1η1(t)ds +Zt−d(t)
t−h
ηT
2(t)S2η2(t)ds
Vw(xt) = d(t)(ζT
1(t)(d(t)Q11 +Q12)ζ1(t)) + (h−d(t))
×(ζT
2(t)((h−d(t))Q21 +Q22)ζ2(t))
Vr(xt) = Zt
t−hZt
θ
˙xT(s)R˙x(s)dsdθ.
S1>0,S2>0,R > 0,d(t)Q11 +Q12 >0, and
(h−d(t))Q21 +Q22 >0, we can draw a conclusion that
V(xt)is a positive function.
Calculating the derivative of V(xt), we get
˙
Vz(xt) = ξT(t)[ω11 +d(t)ω12 ]TS1[ω11 +d(t)ω12]ξ(t)
−(1 −˙
d(t))ξT(t)[ω21 +d(t)ω22]T
×S1[ω21 +d(t)ω22]ξ(t) + ξT(t)Sym{γT
1(˙
d(t))
×S1[ω30 +d(t)ω31 +d2(t)ω32]}ξ(t)
+ (1 −˙
d(t))ξT(t)[ω41 + (h−d(t))ω42]T
×S2[ω41 + (h−d(t))ω42]ξ(t)
−ξT(t)[ω51 + (h−d(t))ω52]T
×S2[ω51 + (h−d(t))ω52]ξ(t)
+ξT(t)Sym{γT
2(˙
d(t))S2[ω60 + (h−d(t))ω61
+ (h−d(t))2ω62]}ξ(t)
˙
Vw(xt) = ξT(t)Ψ2(d(t),˙
d(t))ξ(t)
where
d(t)˙
ζ1(t) = d(t)γ3(˙
d(t)) + γ4(˙
d(t))
(h−d(t)) ˙
ζ2(t) = d(t)γ5(˙
d(t)) + γ6(˙
d(t))
and
Ψ2(d(t),˙
d(t)) = ˙
d(t)(ΠT
1(2d(t)Q11 +Q12)Π1
−ΠT
2(2(h−d(t))Q21 +Q22)Π2)
+ Sym{ΠT
1(d(t)Q11 +Q12)(d(t)γ3(˙
d(t))
+γ4(˙
d(t))) + ΠT
2((h−d(t))Q21 +Q22)
×(d(t)γ5(˙
d(t)) + γ6(˙
d(t)))}
=d2(t)Γ3(˙
d(t))+d(t)Γ2(˙
d(t)) + Γ1(˙
d(t))

CHEN AND CHEN: STABILITY ANALYSIS OF SYSTEMS WITH TIME-VARYING DELAY VIA A NOVEL LYAPUNOV FUNCTIONAL 1071
˙
Vr(xt) = hωT
0Rω0−Zt
t−d(t)
˙xT(s)R˙x(s)ds
−Zt−d(t)
t−h
˙xT(s)R˙x(s)ds.
By applying Lemma 1, it can be deduced that
−Zt
t−d(t)
˙xT(s)R˙x(s)ds ≤ξT(t)(TT
2H2+HT
2T2
+d(t)HT
2e
R−1H2)ξ(t)
−Zt−d(t)
t−h
˙xT(s)R˙x(s)ds ≤ξT(t)(TT
1H1+HT
1T1
+ (h−d(t))HT
1e
R−1H1)ξ(t)
thus,
˙
V(xt)≤ξT(t)(Ψ(d(t),˙
d(t))
+d(t)HT
2e
R−1H2+ (h−d(t))HT
1e
R−1H1)ξ(t).
According to Lemma 2 and Schur complement, if (5)−(7)
are satisfied, we have ˙
V(xt)<0.¥
Remark 3: Theorem 1 introduces a stability criterion based
on the LKF theory. From the proof, it is clear to see in the
estimation of ˙
V(xt), Lemmas 1 and 2 play an important role
in deriving a larger delay bound. The numerical results in
next section show that Theorem 1 can obtain less conservative
results than some existing stability criteria.
Remark 4: The LKF Vz(xt)is similar to V1(t, xt)in [35],
compared with η1(t)and η2(t)in [35], adding six terms
Rt
sxT(v)dv,Rt−d(t)
sxT(v)dv,Rt
s˙xT(v)dv,Rs
t−d(t)˙xT(v)dv,
Rt−d(t)
s˙xT(v)dv,Rs
t−h˙xT(v)dv, in terms of structure, it is full
of symmetrical beauty. In fact, it can been seen as the relation-
ship between vectors becomes tighter through symmetric and
positive definite matrices S1,S2. From the perspective of the
connection relationship between the second-order canonical
B-L inequality and LKF, a couple of Rs
t−d(t)xT(v)dv and
Rt
sxT(v)dv correspond to δ4(t), a couple of Rs
t−hxT(v)dv and
Rt−d(t)
sxT(v)dv correspond to δ1(t), a couple of Rt
s˙xT(v)dv
and Rs
t−d(t)˙xT(v)dv correspond to x(t)−x(t−d(t)), and
a couple of Rt−d(t)
s˙xT(v)dv and Rs
t−h˙xT(v)dv correspond to
x(t−d(t))−x(t−h). Adding four couples of integral terms in
the augmented LKF can significantly reduce the conservatism
of stability criteria.
Remark 5: In Vw(xt), different from the construction of
the augmented Lyapunov functional and the multiple integral
Lyapunov functional, two new delay product type structure of
the LKF, which using time delay dependent matrices Q1(t)
=d(t)Q11 +Q12,Q2(t)=(h−d(t))Q21 +Q22 instead
of constant matrices. Q11,Q12 ,Q21,Q22 do not need to be
positive definite matrix, which will lead to a better criterion. To
clarify the effect of the proposed Vw(xt)in LKF, the following
corollary obtained.
Corollary 1: Given scalars h,µ1and µ2, the system (1)
is stable for delay function d(t)satisfying (2) if there exist
matrices S1, S2∈S9n
+,R∈Sn
+and H1, H2∈R3n×9nsuch
that LMIs (5) and (6) are satisfied, where
Ψ(d(t),˙
d(t)) = Ψ1(d(t),˙
d(t))
∆( ˙
d(t)) = ∆( ˙
d(t))1
and other terms Ψ1(d(t),˙
d(t)),∆( ˙
d(t))1,e
R,ω0,ω11,ω12 ,
ω21,ω22 ,ω30,ω31 ,ω32,ω41 ,ω42,ω51 ,ω52,ω60 ,ω61,ω62 ,
γ1(˙
d(t)),γ2(˙
d(t)),T1and T2are all defined in Theorem 1.
Proof: The proof is omitted. ¥
Remark 6: Corollary 1 not only shows the innovation and
the less conservativeness of the Vw(xt), but also the superiority
of new three couples of integral terms in Vz(xt)by comparing
with [35]. Furthermore, compared with Theorem 1, Vw(xt)is
removed in V(xt), as a result, the number of decision variables
(NDV) is reduced in Corollary 1.
IV. NUMERICAL EXAMPLES
This section provides two examples to demonstrate the
benefits and the merits of the proposed stability criteria.
Example 1: Consider the system (1) with
A=·−2 0
0−0.9¸, Ad=·−1 0
−1−1¸.
To demonstrate the superiority of our approach, for different
µ, results are compared with the literature and reported in
Table I. Obviously, Theorem 1 and Corollary 1 provides larger
admissible maximum upper bounds than those in [14], [18],
[19], [31], [33], [35]. What is noteworthy is that all the results
by obtained Corollary 1 are larger than those by Proposition 1
in [35], for the reason that added three couples of integral
terms in Corollary 1. Of course at the price of an increase of
the NDV.
TABLE I
THE ACHIEVED MAXIMUM ADMISSIBLE UP PER BO UN DS F OR
VARIO US µ=µ2=−µ1
µ0.1 0.2 0.5 0.8 NDV
[18] 4.703 3.834 2.420 2.137 46
[31] 4.788 4.060 3.055 2.615 282
[14] 4.831 4.142 3.148 2.713 604
[19] (N= 2) 4.93 4.22 3.09 2.66 263
[33] (N= 2) 4.90 4.19 3.16 2.73 276
[35] 4.910 4.216 3.233 2.789 231
Corollary 1 4.937 4.255 3.287 2.867 561
Theorem 1 4.943 4.270 3.322 2.899 705
Example 2: Consider the system (1) with
A=·0 1
−1−2¸, Ad=·0 0
−1 1 ¸.
For this example, the corresponding admissible maximum
upper bounds calculated by Theorem 1, Corollary 1 and the
results reported in [14], [18], [31], [35] are summarized in
Table II. Manifestly, all the results obtained by Theorem 1
and Corollary 1 are larger than others, this also indicates that

1072 IEEE/CAA JOURNAL OF AUTOMATICA SINICA, VOL. 6, NO. 4, JULY 2019
the proposed criteria reduce the conservativeness of stability
analysis. Besides, compared with Corollary 1, Theorem 1 is
more advantageous because Vw(xt)plays a key role in LKF,
especially in the cases of µ= 0.2and µ= 0.5.
TABLE II
UPP ER BOUND O N hFO R VARIOUS µ=µ2=−µ1
µ0.1 0.2 0.5 0.8
[18] 6.590 3.672 1.411 1.275
[31] 7.148 4.466 2.352 1.768
[14] 7.167 4.517 2.415 1.838
[35] 7.230 4.556 2.509 1.940
Corollary 1 7.335 4.704 2.645 2.051
Theorem 1 7.412 4.797 2.735 2.114
V. C ONCLUSION
In this paper, the problem of the stability analysis of
time-varying delay systems has been studied. Four couple
of integral terms and two new delay-product-type terms are
considered in a novel LKF, which fully combines the second-
order canonical B-L inequality. On this basis, less conservative
stability criteria have been derived, the improvements and the
advantages have been verified by given numerical examples.
Future works are to improve and generalize the two new time-
delay-product terms and find appropriate methods to reduce
computational complexity.
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Yun Chen graduated from Shaoyang University,
Shaoyang, China, in 2017. He is currently a mas-
ter student at Hunan University of Technology,
Zhuzhou, China. His research interests include time-
delay systems, event-triggered control, and net-
worked control systems.
Gang Chen received bachelor and Ph.D. degrees
from Central South University in 2001 and 2012, re-
spectively, and received master degree from Guangxi
University in 2006. Now, he is an Associate Profes-
sor at School of Electrical and Information Engi-
neering, Hunan University of Technology, Zhuzhou,
China. His research interests include time-delay sys-
tems, networked control systems, and robust control.