Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2011, Article ID 463603, 16 pages
doi:10.1155/2011/463603
Research Article
New Robust Exponential Stability Criterion
for Uncertain Neutral Systems with
Discrete and Distributed Time-Varying
Delays and Nonlinear Perturbations
Sirada Pinjai and Kanit Mukdasai
Department of Mathematics, Khon Kaen University, Khon Kaen 40002, Thailand
Correspondence should be addressed to Kanit Mukdasai, kanit@kku.ac.th
Received 26 April 2011; Accepted 29 June 2011
Academic Editor: Martin D. Schechter
Copyright q 2011 S. Pinjai and K. Mukdasai. This is an open access article distributed under
the Creative Commons Attribution License, which permits unrestricted use, distribution, and
reproduction in any medium, provided the original work is properly cited.
We investigate the problem of robust exponential stability for uncertain neutral systems
with discrete and distributed time-varying delays and nonlinear perturbations. Based on the
combination of descriptor model transformation, decomposition technique of coefficient matrix,
and utilization of zero equation and new Lyapunov functional, sufficient conditions for robust
exponential stability are obtained and formulated in terms of linear matrix inequalities LMIs.
The new stability conditions are less conservative and more general than some existing results.
1. Introduction
In the past decades, the problem of stability for neutral differential systems, which have
delays in both its state and the derivatives of its states, has been widely investigated by many
researchers. Such systems are often encountered in engineering, biology, and economics.
The existence of time delay is frequently a source of instability or poor performances in
the systems. Recently, some stability criteria for neutral system with time delay have been
given 1–13. The problem of delay-dependent stability for uncertain neutral systems with
time-varying delays was considered 6. Reference 13 investigated the problem of delaydependent robust stability for delay neutral type control system with time-varying structured
uncertainties and time-varying delays. The problem of a new delay-dependent robust
stability criterion for neutral systems was investigated. The considered system has timevarying structured uncertainties and interval time-varying delays in 7. However, many
researchers have studied the problem of stability for systems with discrete and distributed
delays such as 12 which presented some stability conditions for uncertain neutral systems
with discrete and distributed delays. The robust stability of uncertain linear neutral systems
2
Abstract and Applied Analysis
with discrete and distributed delays has been studied in 3. In 14, 15, they studied
the problem of stability for linear switching system with discrete and distributed delays.
Moreover, a descriptor model transformation and a corresponding Lyapunov-Krasovskii
functional have been introduced for stability analysis of systems with delays in 10, 16.
In this paper, we will study the problems of robust exponential stability for
uncertain neutral systems with discrete and distributed time-varying delays and nonlinear
perturbations. Based on a combination of descriptor model transformation, decomposition
technique of coefficient matrix, and utilization of zero equation and new Lyapunov functional
which improved delay-dependent stability criteria for the considered systems, sufficient
conditions for the robust exponential stability are obtained and formulated in terms of linear
matrix inequalities LMIs. The new stability conditions will be less conservative and more
general than some existing results.
2. Problem Formulation and Preliminaries
We introduce some notations and definitions that will be used throughout the paper. R
denotes the set of all real nonnegative numbers; Rn denotes the n-dimensional space with
the vector norm · ; x denotes the Euclidean vector norm of x ∈ Rn ; Rn×r denotes
the set of n × r real matrices; AT denotes the transpose of the matrix A; A is symmetric
if A AT ; I denotes the identity matrix; λA denotes the set of all eigenvalues of A;
λmax A max{Re λ : λ ∈ λA}; λmin A min{Re λ : λ ∈ λA}; matrix A is called
semipositive definite A ≥ 0 if xT Ax ≥ 0, for all x ∈ Rn ; A is positive definite A > 0 if
0; matrix B is called seminegative definite B ≤ 0 if xT Bx ≤ 0, for all
xT Ax > 0 for all x /
n
0; A > B means A − B > 0; A ≥ B
x ∈ R ; B is negative definite B < 0 if xT Bx < 0 for all x /
means A − B ≥ 0; C−h, 0, Rn denotes the space of all continuous vector functions mapping
−h, 0 into Rn ; ∗ represents the elements below the main diagonal of a symmetric matrix.
Consider the uncertain neutral system with discrete and distributed time-varying
delays and nonlinear perturbations of the form
ẋt − Cẋt − rt Atxt Btxt − ht f1 t, xt
f2 t, xt − ht Dt
t
xsds,
t ≥ 0,
t−gt
xt0 θ φθ,
ẋt0 θ ϕθ,
At A ΔAt,
∀θ ∈ − max r, h, g , 0 ,
Bt B ΔBt,
2.1
Dt D ΔDt,
where xt ∈ Rn is the state, A,B,C, D ∈ Rn×n are known real constant matrices. rt, ht, and
gt are neutral, discrete, and distributed time-varying delays, respectively,
0 ≤ rt ≤ r,
0 ≤ ht ≤ h,
ṙt ≤ δ < 1,
ḣt ≤ μ < ∞,
0 ≤ gt ≤ g,
2.2
Abstract and Applied Analysis
3
where h, r, g, μ, and δ are given positive constants. The initial condition functions φt,
ϕt ∈ C− max{r, h, g}, 0, Rn denote continuous vector-valued initial function of t ∈
− max{r, h, g}, 0 with the norm φ sups∈− max{r,h,g},0 φs, ϕ sups∈− max{r,h,g},0 ϕs.
The uncertain matrices ΔAt, ΔBt, and ΔDt are norm bounded and can be described as
ΔAt EFtN1 ,
ΔBt EFtN2 ,
ΔDt EFtN3 ,
2.3
where E, Ni , i 1, 2, 3, are known constant matrices with appropriate dimensions. The
uncertain matrix Ft satisfies
FtT Ft ≤ I.
2.4
The uncertainties f1 · and f2 · represent the nonlinear parameter perturbations with respect
to the current state xt and the delayed state xt − ht, respectively, and are bounded in
magnitude:
f1T t, xtf1 t, xt ≤ β1 xT txt,
2.5
f2T t, xt
− htf2 t, xt − ht ≤ β2 x t − htxt − ht,
T
where η, ρ are given positive constants. First, we consider nominal system 2.1 which is
defined to be
ẋt − Cẋt − rt Axt Bxt − ht f1 t, xt f2 t, xt − ht
t
D
xsds, t ≥ 0,
2.6
t−gt
xt0 θ φθ,
ẋt0 θ ϕθ,
∀θ ∈ − max r, h, g , 0 .
Rewrite the system 2.6 in the following descriptor system:
ẋt yt,
yt Axt Bxt − ht f1 t, xt f2 t, xt − ht
t
Cyt − rt D
xsds.
t−gt
2.7
4
Abstract and Applied Analysis
In order to improve the bound of the discrete delay ht in 2.6, let us decompose the constant
matrix B as
B B1 B2 ,
2.8
where B1 , B2 ∈ Rn×n are constant matrices. Utilize the following zero equation:
0 Gxt − Gxt − ht − G
t
ẋsds,
2.9
t−ht
where G ∈ Rn×n will be chosen to guarantee the exponential stability of the system 2.6. By
2.8 and 2.9, the system 2.7 can be represented in the form of a descriptor system with
discrete and distributed delays and nonlinear perturbations:
ẋt yt Gxt − Gxt − ht − G
t
ysds,
t−ht
yt A1 xt B2 xt − ht f1 t, xt f2 t, xt − ht
Cyt − rt − B1
t
ysds D
t−ht
2.10
t
xsds,
t−gt
where A1 A B1 .
Definition 2.1. The system 2.1 is robustly exponentially stable if there exist real scalars α > 0,
M > 0 such that, for each φt ∈ C− max{r, h, g}, 0, Rn , the solution xt, φ, ϕ of the system
2.1 satisfies
x t, φ, ϕ ≤ M max φ, ϕ e−αt ,
∀t ∈ R .
2.11
Lemma 2.2 Jensen’s inequality 17. For any constant symmetric matrix R > 0, scalar h > 0,
and vector function ẋt : −h, 0 → Rn such that the following integral is well defined, one has
⎤
⎤T ⎡
⎤⎡
−R R
xt
⎦.
⎦ ⎣
⎦⎣
−h
ẋ sRẋsds ≤ ⎣
t−h
xt − h
R −R xt − h
t
⎡
T
xt
2.12
3. Exponential Stability Conditions
In this section, we first study the exponential stability criteria for the nominal system 2.1 by
using the combination of linear matrix inequality LMI technique and Lyapunov method.
Abstract and Applied Analysis
5
Assumption 1. All the eigenvalues of matrix C are inside the unit circle.
Theorem 3.1. Under Assumption 1 and given positive constants h, r, g, μ, δ, α, β1 , and β2 , the
nominal system 2.1 is exponentially stable if there exist positive definite matrices Pi , i 1, 2, . . . , 6,
any appropriate dimensional matrices W1 , Q1 , Q2 , Mi , i 1, 2, . . . , 8, and positive constants 1 , 2
satisfying the following LMI:
⎡
Ω11 Ω12 Ω13
⎢
⎢ ∗ Ω22 Ω23
⎢
⎢
⎢ ∗
∗ Ω33
⎢
⎢
⎢ ∗
∗
∗
⎢
Ω⎢
⎢
⎢ ∗
∗
∗
⎢
⎢
⎢ ∗
∗
∗
⎢
⎢
⎢ ∗
∗
∗
⎣
∗
∗
∗
Ω14 Ω15
Ω16
Ω24
Q2T
Ω34
Ω44
∗
∗
∗
Ω17
Ω18
⎤
⎥
Q2T C Q2T D⎥
⎥
⎥
−M5 −M6 −M7 −M8 ⎥
⎥
⎥
−M5 −M6 −M7 −M8 ⎥
⎥
⎥ < 0,
⎥
−1 I
0
0
0 ⎥
⎥
⎥
∗
−2 I
0
0 ⎥
⎥
⎥
∗
∗
Ω77
0 ⎥
⎦
∗
Q2T
∗
∗
∗
3.1
Ω88
where Ω11 W1 W1T Q1T A1 AT1 Q1 P2 − e−2αh P4 g 2 P6 M1T M1 1 β12 I 2αP1 , Ω22 −Q2 −Q2T h2 P3 h2 P4 P5 , Ω23 Q2T B2 −M2T , Ω24 Q2T B1 −M2T , Ω33 −e−2αh P2 μP2 −e−2αh P4 −
M3T − M3 2 β22 I, Ω34 −M3T − M4 , Ω44 −e−2αh P3 − M4 − M4T , Ω77 −1 − δe−2αr P5 , Ω88 −e−2αg P6 , Ω12 P1 − Q1T AT1 Q2 M2 , Ω13 −W1 Q1T B2 e−2αh P4 − M1T M3 , Ω14 −W1 − Q1T B1 − M1T M4 , Ω15 Q1T M5 , Ω16 Q1T M6 , Ω17 Q1T C M7 , Ω18 Q1T D M8 .
Moreover, the solution xt, φ, ϕ satisfies the inequality
x t, φ, ϕ ≤
N
max φ, ϕ e−αt ,
λmin P1 t ∈ R ,
3.2
where
N λmax P1 hλmax P2 h3 λmax P3 h3 λmax P4 rλmax P5 g 3 λmax P6 .
3.3
Proof. Construct a Lyapunov functional candidate for the nominal system 2.1 of the form
V t 6
i1
Vi t,
3.4
6
Abstract and Applied Analysis
where
V1 t x tP1 xt T
V2 t t
xt
T I 0
0 0
yt
0
P1
Q1 Q2
xt
,
yt
e2αs−t xT sP2 xsds,
t−ht
0 t
V3 t h
−h
3.5
0 t
V4 t h
V5 t e2αs−t yT θP3 yθdθ ds,
ts
t
−h
e
y θP4 yθdθ ds,
2αs−t T
ts
e2αs−t yT sP5 ysds,
t−rt
V6 t g
0 t
−g
e2αs−t xT θP6 xθdθ ds.
ts
The derivative of V · along the trajectories of nominal system 2.1 is given by
V̇ t 6
3.6
V̇i t,
i1
where
V̇1 t 2
xt
T yt
P1 Q1T
0 Q2T
ẋt
0
2x tP1 yt Gxt − Gxt − ht − G
T
2x
ysds
t−ht
T
t
− yt A1 xt B2 xt − ht f1 t, xt
tQ1T
f2 t, xt − ht Cyt − rt − B1
2y
tQ2T
ysds D
t−ht
T
t
t
xsds
t−gt
− yt A1 xt B2 xt − ht f1 t, xt f2 t, xt − ht
Cyt − rt − B1
t
t−ht
ysds D
t
xsds
t−gt
2αxT tP1 xt − 2αV1 t,
V̇2 t xT tP2 xt − 1 − ḣt e−2αht xT t − htP2 xt − ht − 2αV2 t
≤ xT tP2 xt − e−2αh xT t − htP2 xt − ht μxT t − htP2 xt − ht − 2αV2 t.
3.7
Abstract and Applied Analysis
7
Obviously, for any scalar s ∈ t − h, t, we get e−2αh ≤ e−2αs−t ≤ 1. Together with Lemma 2.2
Jensen’s inequality, we obtain
V̇3 t h y tP3 yt − h
2 T
≤ h y tP3 yt − h
2 T
0
e2αs yT t sP3 yt sds − 2αV3 t
−h
t
e2αs−t yT sP3 ysds − 2αV3 t
t−h
≤ h2 yT tP3 yt − he−2αh
t
yT sP3 ysds − 2αV3 t
t−h
≤ h2 yT tP3 yt − e−2αh
t
yT sdsP3
t
t−h
≤ h y tP3 yt − e
2 T
−2αh
t
y sdsP3
T
t
t−ht
V̇4 t ≤ h2 yT tP4 yt − e−2αh
ysds − 2αV3 t
t−h
t
ysds − 2αV3 t,
3.8
t−ht
yT sdsP4
t
t−ht
ysds − 2αV4 t
t−ht
h2 yT tP4 yt − e−2αh xT t − xT t − ht P4 xt − xt − ht − 2αV4 t,
V̇5 t xT tP5 xt − 1 − ṙte−2αrt xT t − rtP2 xt − rt − 2αV5 t
≤ xT tP5 xt − 1 − δe−2αr xT t − rtP5 xt − rt − 2αV5 t,
V̇6 t ≤ g x tP6 xt − e
2 T
−2αg
t
x sdsP6
T
t
t−gt
xsds − 2αV6 t.
t−gt
From the Leibniz-Newton formula, the following equation is true for any real matrices
Mi , i 1, 2, . . . , 8, with appropriate dimensions
2 x
T
tM1T
y
T
tM2T
f1T t, xtM5
x t −
T
f2T t, xt
× xt − xt − ht −
−
htM3T
y t −
T
ysds
t−ht
yT sdsM4
t−ht
htM6T
t
t
0.
rtM7T
t
t−gt
x sdsM8
T
3.9
8
Abstract and Applied Analysis
From 2.5, we obtain for any scalars 1 , 2 > 0
0 ≤ 1 β1 xT txt − 1 f1T t, xtf1 t, xt,
0 ≤ 2 β2 x t − htxt − ht −
T
2 f2T t, xt
3.10
− htf2 t, xt − ht.
Let us define W1 P1 G. From 3.6, 3.9, and 3.10, we obtain
V̇ t 2αV t ≤ ξT tΩξt,
where ξT t xT t, yT t, xT t − ht,
t
rt, t−gt xT sds and
⎡
Ω11 Ω12 Ω13
⎢
⎢
⎢ ∗ Ω22 Ω23
⎢
⎢
⎢ ∗
∗ Ω33
⎢
⎢
⎢ ∗
∗
∗
⎢
Ω⎢
⎢
⎢ ∗
∗
∗
⎢
⎢
⎢ ∗
∗
∗
⎢
⎢
⎢
∗
∗
⎢ ∗
⎣
∗
∗
∗
t
t−ht
yT sds, f T t, xt, g T t, xt − ht, yT t −
Ω14 Ω15
Ω16
Ω24
Q2T
Ω34
Ω44
∗
∗
∗
∗
3.11
Ω17
Ω18
⎤
⎥
⎥
Q2T C Q2T D⎥
⎥
⎥
−M5 −M6 −M7 −M8 ⎥
⎥
⎥
−M5 −M6 −M7 −M8 ⎥
⎥
⎥.
⎥
−1 I
0
0
0 ⎥
⎥
⎥
0
0 ⎥
∗
−2 I
⎥
⎥
⎥
0 ⎥
∗
∗
Ω77
⎦
∗
∗
∗
Ω88
Q2T
3.12
It is a fact that, if Ω < 0, then
V̇ t 2αV t ≤ 0,
∀t ∈ R ,
3.13
V t ≤ V 0e−2αt ,
∀t ∈ R .
3.14
which gives
From 3.14, it is easy to see that
λmin P1 xt2 ≤ V t ≤ V 0e−2αt ,
V 0 6
Vi 0,
i1
3.15
Abstract and Applied Analysis
9
where
V1 0 xT 0P1 x0,
V2 0 0
−h0
V3 0 h
V4 0 h
V5 0 0 0
−h
−h
−r0
3.16
e2αs yT θP4 yθdθ ds,
s
e2αs yT sP5 ysds,
0 0
−g
e2αs yT θP3 yθdθ ds,
s
0 0
0
V6 0 g
e2αs xT sP2 xsds,
e2αs xT θP6 xθdθ ds.
s
Therefore, we get
2
λmin P1 xt2 ≤ V 0e−2αt ≤ N max φ, ϕ e−2αt ,
3.17
where N λmax P1 hλmax P2 h3 λmax P3 h3 λmax P4 rλmax P5 g 3 λmax P6 . From
3.17, we get
x t, φ, ϕ ≤
N
max φ, ϕ e−αt ,
λmin P1 ∀t ∈ R .
3.18
This means that the system 2.1 is exponentially stable. The proof of the theorem is complete.
If C ΔAt ΔBt f1 t, xt f2 t, xt − ht Dt 0, then system 2.1
reduces to the following system:
ẋt Axt Bxt − ht,
t ≥ 0,
3.19
xt0 θ φθ,
ẋt0 θ ϕθ,
∀θ ∈ −h, 0.
10
Abstract and Applied Analysis
Take the Lyapunov functional as
V̇ t 4
3.20
V̇i t,
i1
where V1 t, V2 t, V3 t, and V4 t are defined in Theorem 3.1. According to Theorem 3.1, we
have the following corollary for the delay-dependent exponential stability criterion of system
3.19.
Corollary 3.2. For given positive constants h, μ, and α, the system 3.19 is exponentially stable
if there exist positive definite matrices Pi , i 1, 2, . . . , 4, and any appropriate dimensional matrices
W1 , Q1 , Q2 , Mi , i 1, 2, . . . , 4, satisfying the following LMI:
⎡
⎤
Φ11 Φ12 Φ13 Φ14
⎢
⎥
⎢ ∗ Φ22 Φ23 Φ24 ⎥
⎢
⎥
Φ⎢
⎥ < 0,
⎢ ∗
⎥
∗
Φ
Φ
33
34 ⎦
⎣
∗
∗
∗ Φ44
3.21
where Φ11 W1 W1T Q1T A1 AT1 Q1 P2 − e−2αh P4 M1T M1 2αP1 , Φ22 −Q2 − Q2T h2 P3 h2 P4 , Φ23 Q2T B2 − M2T , Φ24 Q2T B1 − M2T , Φ33 −e−2αh P2 μP2 − e−2αh P4 − M3T − M3 , Ω34 −M3T − M4 , Φ44 −e−2αh P3 − M4 − M4T , Φ12 P1 − Q1T AT1 Q2 M2 , Φ13 −W1 Q1T B2 e−2αh P4 − M1T M3 , Φ14 −W1 − Q1T B1 − M1T M4 .
Moreover, the solution xt, φ, ϕ satisfies the inequality
x t, φ, ϕ ≤
K
max φ, ϕ e−αt ,
λmin P1 t ∈ R ,
3.22
where
K λmax P1 hλmax P2 h3 λmax P3 h3 λmax P4 .
3.23
If ΔAt ΔBt Dt 0, then system 2.1 reduces to the following system:
ẋt − Cẋt − rt Axt Bxt − ht f1 t, xt f2 t, xt − ht,
t ≥ 0,
3.24
xt0 θ φθ,
ẋt0 θ ϕθ,
∀θ ∈ − max{r, h}, 0.
Abstract and Applied Analysis
11
Take the Lyapunov functional as
V̇ t 5
3.25
V̇i t,
i1
where V1 t, V2 t, V3 t, V4 t, and V5 t are defined in Theorem 3.1. According to
Theorem 3.1, we have the following corollary for the delay-dependent exponential stability
criterion of system 3.24.
Corollary 3.3. For given positive constants h, r, μ, δ, α, β1 , and β2 , the system 3.24 is exponentially
stable if there exist positive definite matrices Pi , i 1, 2, . . . , 5, any appropriate dimensional matrices
W1 , Q1 , Q2 , Mi , i 1, 2, . . . , 7, and positive constants 1 , 2 satisfying the following LMI:
⎡
Υ11 Υ12 Υ13
⎢
⎢ ∗ Υ22 Υ23
⎢
⎢
⎢∗
∗ Υ33
⎢
⎢
⎢
Υ⎢ ∗
∗
∗
⎢
⎢∗
∗
∗
⎢
⎢
⎢∗
∗
∗
⎣
∗
∗
∗
Υ14
Υ24
Υ34
Υ44
∗
∗
∗
Υ15
Υ16
Q2T
Q2T
Υ17
⎤
⎥
Q2T C ⎥
⎥
⎥
−M5 −M6 −M7 ⎥
⎥
⎥
−M5 −M6 −M7 ⎥
⎥ < 0,
⎥
−1 I
0
0 ⎥
⎥
⎥
∗
−2 I
0 ⎥
⎦
∗
∗
Υ77
3.26
where Υ11 W1 W1T Q1T A1 AT1 Q1 P2 − e−2αh P4 M1T M1 1 β12 I 2αP1 , Υ22 −Q2 − Q2T h2 P3 h2 P4 P5 , Υ23 Q2T B2 − M2T , Υ24 Q2T B1 − M2T , Υ33 −e−2αh P2 μP2 − e−2αh P4 − M3T −
M3 2 β22 I, Υ34 −M3T − M4 , Υ44 −e−2αh P3 − M4 − M4T , Υ12 P1 − Q1T AT1 Q2 M2 , Υ13 −W1 Q1T B2 e−2αh P4 − M1T M3 , Υ14 −W1 − Q1T B1 − M1T M4 , Υ15 Q1T M5 , Υ16 Q1T M6 , Υ17 Q1T C M7 , Υ77 −1 − δe−2αr P5 .
Moreover, the solution xt, φ, ϕ satisfies the inequality
x t, φ, ϕ ≤
K
max φ, ϕ e−αt ,
λmin P1 t ∈ R ,
3.27
where
K λmax P1 hλmax P2 h3 λmax P3 h3 λmax P4 .
3.28
4. Robust Exponential Stability Conditions
Theorem 4.1. Under Assumption 1 and given positive constants h, r, g, μ, δ, α, β1 , and β2 , the
system 2.1 is exponentially stable if there exist positive definite matrices Pi , i 1, 2, . . . , 6, any
12
Abstract and Applied Analysis
appropriate dimensional matrices W1 , Q1 , Q2 , Mi , i 1, 2, . . . , 8, and positive constants 1 , 2 , and ε
satisfying the following LMI:
⎡
Ω11 Ω12 Ω13
⎢
⎢ ∗ Ω22 Ω23
⎢
⎢
⎢ ∗
∗ Ω33
⎢
⎢
⎢ ∗
∗
∗
⎢
⎢
⎢ ∗
∗
∗
⎢
Ω
⎢
⎢ ∗
∗
∗
⎢
⎢
⎢ ∗
∗
∗
⎢
⎢
⎢ ∗
∗
∗
⎢
⎢
⎢ ∗
∗
∗
⎣
∗
∗
∗
Ω14 Ω15
Ω16
Ω17
Ω24
Q2T
Q2T C Q2T D Q1T E
Q2T
Ω18 Q1T E εN1T
Ω34 −M5 −M6 −M7 −M8
Ω44 −M5 −M6 −M7 −M8
∗
−1 I
0
0
0
∗
∗
−2 I
0
0
∗
∗
∗
Ω77
0
∗
∗
∗
∗
Ω88
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
⎤
⎥
0 ⎥
⎥
⎥
0 εN2T ⎥
⎥
⎥
0
0 ⎥
⎥
⎥
0
0 ⎥
⎥
⎥ < 0,
0
0 ⎥
⎥
⎥
0
0 ⎥
⎥
⎥
T⎥
0 εN3 ⎥
⎥
−εI
0 ⎥
⎦
∗
4.1
−εI
where Ω11 W1 W1T Q1T A1 AT1 Q1 P2 − e−2αh P4 g 2 P6 M1T M1 1 β12 I 2αP1 , Ω22 −Q2 −Q2T h2 P3 h2 P4 P5 , Ω23 Q2T B2 −M2T , Ω24 Q2T B1 −M2T , Ω33 −e−2αh P2 μP2 −e−2αh P4 −
M3T − M3 2 β22 I, Ω34 −M3T − M4 , Ω44 −e−2αh P3 − M4 − M4T , Ω77 −1 − δe−2αr P5 , Ω88 −e−2αg P6 , Ω12 P1 − Q1T AT1 Q2 M2 , Ω13 −W1 Q1T B2 e−2αh P4 − M1T M3 , Ω14 −W1 − Q1T B1 − M1T M4 , Ω15 Q1T M5 , Ω16 Q1T M6 , Ω17 Q1T C M7 , Ω18 Q1T D M8 .
Moreover, the solution xt, φ, ϕ satisfies the inequality
x t, φ, ϕ ≤
N
max φ, ϕ e−αt ,
λmin P1 t ∈ R ,
4.2
where
N λmax P1 hλmax P2 h3 λmax P3 h3 λmax P4 rλmax P5 g 3 λmax P6 .
4.3
Proof. Replacing A1 , B2 , and D in 3.26 with A1 t A B1 EFtN1 , B2 t B2 EFtN2 ,
and Dt D EFtN3 , respectively, we find that condition 3.26 for system 2.1 is
equivalent to the following condition:
Ω ΓFtΠ ΠT FtT ΓT < 0,
4.4
where ΓT ET Q1 , ET Q2 , 0, 0, 0, 0, 0, 0 and Π N1 , 0, N2 , 0, 0, 0, 0, N3 . Using Lemma 1 in
9, we have that there exists a positive number ε > 0. We can find that 4.4 is equivalent to
the LMI as follows:
Ω ε−1 ΓΓT εΠT Π < 0.
4.5
Abstract and Applied Analysis
13
Applying Lemma 1 Schur complement lemma in 12, we obtain
⎡
Ω
Γ
εΠT
⎤
⎢ T
⎥
⎢ Γ −εI 0 ⎥ < 0.
⎣
⎦
εΠ 0 −εI
4.6
Hence, we conclude that 4.6 is equivalent to 4.1. The proof of the theorem is complete.
If f1 t, xt f2 t, xt − ht Dt 0, then system 2.1 reduces to the following
system:
ẋt − Cẋt − rr A EFtN1 xt B EFtN2 xt − ht,
xt0 θ φθ,
ẋt0 θ ϕθ,
∀θ ∈ − max{r, h}, 0.
t ≥ 0,
4.7
Take the Lyapunov functional as
V̇ t 5
4.8
V̇i t,
i1
where V1 t, V2 t, V3 t, V4 t, and V5 t are defined in Theorem 3.1. According to Theorems
3.1 and 4.1, we have the following corollary for the delay-dependent robust exponential
stability of system 4.7.
Corollary 4.2. Under Assumption 1 and given positive constants h, r, μ, δ, and α, the system 4.7 is
robustly exponentially stable if there exist positive definite matrices Pi , i 1, 2, . . . , 5, any appropriate
dimensional matrices W1 , Q1 , Q2 , M7 , Mi , i 1, 2, . . . , 4, and positive constant ε satisfying the
following LMI:
⎡
Ω11 Ω12 Ω13
⎢
⎢ ∗ Ω22 Ω23
⎢
⎢
⎢ ∗
∗ Ω33
⎢
⎢
⎢ ∗
∗
∗
Ω
⎢
⎢
⎢ ∗
∗
∗
⎢
⎢
⎢ ∗
∗
∗
⎣
∗
∗
∗
Ω14 Ω17 Q1T E εN1T
⎤
⎥
0 ⎥
⎥
⎥
T⎥
0 εN2 ⎥
⎥
0
0 ⎥
⎥ < 0,
⎥
0
0 ⎥
⎥
⎥
−εI
0 ⎥
⎦
Ω24 Q2T C Q1T E
Ω34 −M7
Ω44 −M7
∗
Ω77
∗
∗
∗
∗
∗
4.9
−εI
where Ω11 W1 W1T Q1T A1 AT1 Q1 P2 − e−2αh P4 M1T M1 2αP1 , Ω22 −Q2 − Q2T h2 P3 h2 P4 P5 , Ω23 Q2T B2 −M2T , Ω24 Q2T B1 −M2T , Ω33 −e−2αh P2 μP2 −e−2αh P4 −M3T −M3 , Ω34 −M3T −M4 , Ω44 −e−2αh P3 −M4 −M4T , Ω77 −1−δe−2αr P5 , Ω12 P1 −Q1T AT1 Q2 M2 , Ω13 −W1 Q1T B2 e−2αh P4 − M1T M3 , Ω14 −W1 − Q1T B1 − M1T M4 , Ω17 Q1T C M7 .
14
Abstract and Applied Analysis
Table 1: Comparison of the maximum allowed time delay hmax .
hmax δ 0.9
1.18
1.37
1.43
2.00
Methods
Fridman and Shaked 16
He et al. 4
Zhu and Yang 18
Our results
hmax δ 1
0.99
1.34
1.39
1.51
hmax δ 1.2
0.99
1.34
1.36
1.51
hmax δ 1.5
0.99
1.34
1.35
1.51
Moreover, the solution xt, φ, ϕ satisfies the inequality
x t, φ, ϕ ≤
Z
max φ, ϕ e−αt ,
λmin P1 t ∈ R ,
4.10
where
Z λmax P1 hλmax P2 h3 λmax P3 h3 λmax P4 rλmax P5 .
4.11
5. Numerical Examples
In order to show the effectiveness of the approaches presented in Sections 3 and 4, three
numerical examples are provided.
Example 5.1. Consider the asymptotic stability α 0 of the linear system 3.19 with timevarying delays when
−2 0
−1 0
A
,
B
.
5.1
0 −0.9
−1 −1
Decompose matrix B as follows: B B1 B2 , where
−0.83
0
B1 ,
−1 −0.83
−0.17
0
B2 .
0
−0.17
5.2
Table 1 lists the comparison of the upper bounds delays for asymptotic stability α 0
of system 5.1 by different methods. We can see from Table 1 that our results Corollary 3.2
are superior to those in Lemma 4 16, Lemma 1 4, and Theorem 1 18.
Example 5.2. Consider the exponential stability of the neutral system 3.24 with time-varying
delays and nonlinear perturbations with
A
−2
0
0 −0.9
,
β1 0.05,
B
−1 0
−1 −1
β2 0.1,
,
0.1 0
C
,
0 0.1
ht rt.
Decompose matrix B as follows which is the same as the decomposition in Example 5.1.
5.3
Abstract and Applied Analysis
15
Table 2: Comparison of the maximum allowed time delay hmax .
hmax δ 0
1.29
1.62
0.69
0.75
0.57
0.61
Methods
Chen et al. 1 α 0.1
Our results α 0.1
Chen et al. 1 α 0.5
Our results α 0.5
Chen et al. 1 α 0.7
Our results α 0.7
hmax δ 0.5
0.94
1.10
0.60
0.61
0.53
0.54
hmax δ 0.9
0.54
0.62
0.46
0.47
0.43
0.43
Table 3: Comparison of the maximum allowed time delay hmax .
Methods
Han 2
He et al. 5
Nian et al. 9
Our results
hmax c 0.1
1.48
1.75
1.86
1.96
hmax c 0.15
1.33
1.49
1.63
1.87
hmax c 0.2
1.16
0.27
1.42
1.75
hmax c 0.3
0.79
0.91
1.06
1.50
hmax c 0.4
0.37
0.63
0.76
1.23
Table 2 lists the comparison of the upper bounds delay for exponential stability of
5.3 by different methods. It is clear that our results Corollary 3.3 are superior to those in
Theorem 1 1.
Example 5.3. Consider the asymptotic stability α 0 of the uncertain neutral system 4.7
with time-varying delays with
A
−2
0
0 −0.9
E I,
,
−1 0
B
,
−1 −1
N1 N2 0.2I,
C
c 0
0 c
,
0 ≤ c < 1,
5.4
ht rt.
Decompose matrix B as follows which is the same as the decomposition in Example 5.1.
Table 3 shows the comparison of the upper bounds delay allowed obtained from
Corollary 4.2 for asymptotic stability α 0 of system 4.7 with 5.4 by other methods.
It can be found from Table 3 that our results are significantly better than those in 2, 5, 9.
6. Conclusions
The problem of robust exponential stability for uncertain neutral systems with discrete
and distributed time-varying delays and nonlinear perturbations was studied. Based on
the combination of descriptor model transformation, decomposition technique of coefficient
matrix, and utilization of zero equation and new Lyapunov functional, sufficient conditions
for robust exponential stability were obtained and formulated in terms of linear matrix
inequalities LMIs. Numerical examples have shown significant improvements over some
existing results.
16
Abstract and Applied Analysis
Acknowledgment
This work was supported by the Higher Education Research Promotion and National
Research University Project of Thailand, Office of the Higher Education Commission, through
the Cluster of Research to Enhance the Quality of Basic Education.
References
1 Y. Chen, A. Xue, R. Lu, and S. Zhou, “On robustly exponential stability of uncertain neutral systems
with time-varying delays and nonlinear perturbations,” Nonlinear Analysis, vol. 68, no. 8, pp. 2464–
2470, 2008.
2 Q.-L. Han, “Robust stability of uncertain delay-differential systems of neutral type,” Automatica, vol.
38, no. 4, pp. 719–723, 2002.
3 Q.-L. Han, “A descriptor system approach to robust stability of uncertain neutral systems with
discrete and distributed delays,” Automatica, vol. 40, no. 10, p. 1791–1796 2005, 2004.
4 Y. He, Q.-G. Wang, C. Lin, and M. Wu, “Delay-range-dependent stability for systems with timevarying delay,” Automatica, vol. 43, no. 2, pp. 371–376, 2007.
5 Y. He, M. Wu, J.-H. She, and G.-P. Liu, “Delay-dependent robust stability criteria for uncertain neutral
systems with mixed delays,” Systems & Control Letters, vol. 51, no. 1, pp. 57–65, 2004.
6 O. M. Kwon, J. H. Park, and S. M. Lee, “Augmented Lyapunov functional approach to stability of
uncertain neutral systems with time-varying delays,” Applied Mathematics and Computation, vol. 207,
no. 1, pp. 202–212, 2009.
7 O. M. Kwon, J. H. Park, and S. M. Lee, “On delay-dependent robust stability of uncertain neutral
systems with interval time-varying delays,” Applied Mathematics and Computation, vol. 203, no. 2, pp.
843–853, 2008.
8 J. H. Park, “Novel robust stability criterion for a class of neutral systems with mixed delays and
nonlinear perturbations,” Applied Mathematics and Computation, vol. 161, no. 2, pp. 413–421, 2005.
9 X. Nian, H. Pan, W. Gui, and H. Wang, “New stability analysis for linear neutral system via state
matrix decomposition,” Applied Mathematics and Computation, vol. 215, no. 5, pp. 1830–1837, 2009.
10 J. Tian, L. Xiong, J. Liu, and X. Xie, “Novel delay-dependent robust stability criteria for uncertain
neutral systems with time-varying delay,” Chaos, Solitons and Fractals, vol. 40, no. 4, pp. 1858–1866,
2009.
11 B. Wang, X. Liu, and S. Zhong, “New stability analysis for uncertain neutral system with time-varying
delay,” Applied Mathematics and Computation, vol. 197, no. 1, pp. 457–465, 2008.
12 L. Xiong, S. Zhong, and J. Tian, “New robust stability condition for uncertain neutral systems with
discrete and distributed delays,” Chaos, Solitons and Fractals, vol. 42, no. 2, pp. 1073–1079, 2009.
13 Z. Zhao, W. Wang, and B. Yang, “Delay and its time-derivative dependent robust stability of neutral
control system,” Applied Mathematics and Computation, vol. 187, no. 2, pp. 1326–1332, 2007.
14 F. Gao, S. Zhong, and X. Gao, “Delay-dependent stability of a type of linear switching systems with
discrete and distributed time delays,” Applied Mathematics and Computation, vol. 196, no. 1, pp. 24–39,
2008.
15 P. Li, S.-M. Zhong, and J.-Z. Cui, “Stability analysis of linear switching systems with time delays,”
Chaos, Solitons and Fractals, vol. 40, no. 1, pp. 474–480, 2009.
16 E. Fridman and U. Shaked, “Delay-dependent stability and control: constant and time-varying
delays,” International Journal of Control, vol. 76, no. 1, pp. 48–60, 2003.
17 K. Gu, V. L. Kharitonov, and J. Chen, Stability of Time-Delay Systems,, Birkhäuser, Berlin, Germany,
2003..
18 X. Zhu and G. Yang, “Delay-dependent stability criteria for systems with differentiable time delays,”
Acta Automatica Sinica, vol. 34, no. 7, pp. 765–771, 2008.
Advances in
Operations Research
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Advances in
Decision Sciences
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Mathematical Problems
in Engineering
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Journal of
Algebra
Hindawi Publishing Corporation
http://www.hindawi.com
Probability and Statistics
Volume 2014
The Scientific
World Journal
Hindawi Publishing Corporation
http://www.hindawi.com
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
International Journal of
Differential Equations
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Volume 2014
Submit your manuscripts at
http://www.hindawi.com
International Journal of
Advances in
Combinatorics
Hindawi Publishing Corporation
http://www.hindawi.com
Mathematical Physics
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Journal of
Complex Analysis
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
International
Journal of
Mathematics and
Mathematical
Sciences
Journal of
Hindawi Publishing Corporation
http://www.hindawi.com
Stochastic Analysis
Abstract and
Applied Analysis
Hindawi Publishing Corporation
http://www.hindawi.com
Hindawi Publishing Corporation
http://www.hindawi.com
International Journal of
Mathematics
Volume 2014
Volume 2014
Discrete Dynamics in
Nature and Society
Volume 2014
Volume 2014
Journal of
Journal of
Discrete Mathematics
Journal of
Volume 2014
Hindawi Publishing Corporation
http://www.hindawi.com
Applied Mathematics
Journal of
Function Spaces
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Optimization
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014