Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2012, Article ID 348418, 20 pages
doi:10.1155/2012/348418
Research Article
Robust Exponential Stability Criteria of
LPD Systems with Mixed Time-Varying Delays and
Nonlinear Perturbations
Kanit Mukdasai,1, 2 Akkharaphong Wongphat,1
and Piyapong Niamsup2, 3
1
Department of Mathematics, Faculty of Science, Khon Kaen University, Khon Kaen 40002, Thailand
Centre of Excellence in Mathematics, CHE, Si Ayutthaya Road, Bangkok 10400, Thailand
3
Department of Mathematics, Faculty of Science, Chiang Mai University, Chiang Mai 50200, Thailand
2
Correspondence should be addressed to Piyapong Niamsup, piyapong.n@cmu.ac.th
Received 15 October 2012; Accepted 15 November 2012
Academic Editor: Elena Braverman
Copyright q 2012 Kanit Mukdasai et al. 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.
This paper investigates the problem of robust exponential stability for linear parameter-dependent
LPD systems with discrete and distributed time-varying delays and nonlinear perturbations.
Parameter dependent Lyapunov-Krasovskii functional, Leibniz-Newton formula, and linear
matrix inequality are proposed to analyze the stability. On the basis of the estimation and
by utilizing free-weighting matrices, new delay-dependent exponential stability criteria are
established in terms of linear matrix inequalities LMIs. Numerical examples are given to
demonstrate the effectiveness and less conservativeness of the proposed methods.
1. Introduction
Over the past decades, dynamical systems with state delays have attracted much interest in
the literature over the half century, especially in the last decade. Since time delay is frequently
a source of instability or poor performances in various systems such as electric, chemical
processes, and long transmission line in pneumatic systems 1. The problems of stability
and stabilization for dynamical systems with or without state delays have been intensively
studied in the past years by many researchers in mathematics and control communities
2, 3. Stability criteria for dynamical systems with time delay is generally divided into
two classes: delay-independent one and delay-dependent one. Delay-independent stability
criteria tends to be more conservative, especially for small size delay, such criteria do
not give any information on the size of the delay. On the other hand, delay-dependent
2
Abstract and Applied Analysis
stability criteria concerned with the size of the delay and usually provide a maximal delay
size. Various stability of linear continuous-time and discrete-time systems subject to timeinvariant parametric uncertainty have received considerable attention. An important class
of linear time-invariant parametric uncertain system is linear parameter-dependent LPD
system in which the uncertain state matrices are in the polytope consisting of all convex
combination of known matrices. Most of sufficient or necessary and sufficient conditions
have been obtained via Lyapunov-Krasovskii theory approaches in which parameterdependent Lyapunov-Krasovskii functional has been employed. These conditions are always
expressed in terms of linear matrix inequalities LMIs. The results have been obtained for
robust stability for LPD systems in which time delay occurs in state variable such as 4–6
which present sufficient conditions for robust stability of LPD continuous-time system with
delays.
Recently, many researchers have studied the problem of stability for time-delay
systems with nonlinear perturbations such as 7 which considers the robust stability for
a class of linear systems with interval time-varying delay and nonlinear perturbations. In 8,
exponential stability of time-delay systems with nonlinear uncertainties is studied. Based
on the Lyapunov theory approach and the approaches of decomposing the matrix, a new
exponential stability criterion is derived in terms of LMI. In 9, they propose a new delaydependent stability criterion in terms of linear matrix inequality for dynamic systems with
time-varying delays and nonlinear perturbations by using Lyapunov theory. However, many
researchers have studied the problem of stability for systems with discrete and distributed
delays such as 10 which presented some stability conditions for uncertain neutral systems
with discrete and distributed delays. The robust stability of uncertain linear neutral systems
with discrete and distributed delays has been studied in 11. In 12, 13, 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
functionals have been introduced for stability analysis of systems with delays in 14, 15.
In this paper, we will investigate the problems of robust exponential stability for
LPD system with mixed time-varying delays and nonlinear perturbations. Based on the
combination of Leibniz-Newton formula and linear matrix inequality, the use of suitable
Lyapunov-Krasovskii functional, new delay-dependent exponential stability criteria will be
obtained in terms of LMIs. Finally, numerical examples will be given to show the effectiveness
of the obtained results.
2. Problem Formulation and Preliminaries
We introduce some notations, definition, and lemmas 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 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}; λmax Aα max{λmax Ai : i 1, 2, . . . , N};
λmin Aα min{λmin Ai : i 1, 2, . . . , N}; matrix A is called a semipositive definite A ≥
0;
0 if xT Ax ≥ 0, for all x ∈ Rn ; A is a positive definite A > 0 if xT Ax > 0 for all x /
matrix B is called a seminegative definite B ≤ 0 if xT Bx ≤ 0, for all x ∈ Rn ; B is a negative
0; A > B means A − B > 0; A ≥ B means A − B ≥ 0;
definite B < 0 if xT Bx < 0 for all x /
C−h, 0, Rn denotes the space of all continuous vector functions mapping −h, 0 into Rn
Abstract and Applied Analysis
3
where h max{h, g}; ∗ represents the elements below the main diagonal of a symmetric
matrix.
Consider the system described by the following state equation of the form
ẋt Aαxt Bαxt − ht ft, xt gt, xt − ht
t
Cα
xsds, t > 0;
t−gt
xt φt,
2.1
t ∈ −h, 0 ,
ẋt ψt,
where xt ∈ Rn is the state variable, Aα, Bα, Cα ∈ Rn×n are uncertain matrices belonging
to the polytope
Aα N
αi Ai ,
i1
N
αi 1,
Bα N
αi Bi ,
Cα i1
N
αi Ci ,
i1
αi ≥ 0, Ai , Bi , Ci ∈ R
n×n
2.2
, i 1, . . . , N.
i1
ht and gt are discrete and distributed time-varying delays, respectively, satisfying
0 ≤ ht ≤ h,
ḣt ≤ hd ,
0 ≤ gt ≤ g,
2.3
where h, hd , g are given positive real constants. Consider the initial functions φt, ψt ∈
C−h, 0, Rn with the norm φ supt∈−h,0 φt and ψ supt∈−h,0 ψt. The
uncertainties f·, g· represent the nonlinear parameter perturbations with respect to the
current state xt and the delayed state xt−ht, respectively, and are bounded in magnitude
of the form
f T t, xtft, xt ≤ η2 xT txt,
g T t, xt − htgt, xt − ht ≤ ρ2 xT t − htxt − ht,
2.4
where η, ρ are given real constants.
Definition 2.1. The system 2.1 is robustly exponentially stable, if there exist positive real
numbers β and M such that for each φt, ψt ∈ C−h, 0, Rn , the solution xt, φ, ψ of the
system 2.1 satisfies
x t, φ, ψ ≤ M max φ, ψ e−βt ,
∀t ∈ R .
2.5
4
Abstract and Applied Analysis
Lemma 2.2 Schur complement lemma, see 9. Given constant symmetric matrices X, Y, Z
where Y > 0. Then X ZT Y−1 Z < 0 if and only if
X ZT
Z −Y
<0
or
−Y Z
< 0.
ZT X
2.6
Lemma 2.3 Jensen’s inequality, see 1. For any constant matrix Q ∈ Rn×n , Q QT > 0, scalar
h > 0, vector function ẋ : 0, h → Rn such that the integrations concerned are well defined, then
−h
0
−h
ẋ s tQẋs tds ≤ −
T
Rearranging the term
−h
0
−h
0
−h
0
−h
T 0
ẋs tds Q
ẋs tds .
2.7
−h
ẋs tds with xt − xt − h, one can yield the following inequality:
ẋT s tQẋs tds ≤
xt
xt − h
T −Q Q
xt
.
Q −Q xt − h
2.8
Lemma 2.4 see 16. Let xt ∈ Rn be a vector-valued function with first-order continuousderivative entries. Then, the following integral inequality holds for any matrices X, Mi ∈ Rn×n ,
i 1, 2, . . . , 5 and a scalar function h : ht ≥ 0:
t
xt
ẋT sX ẋsds ≤
−
xt
− h
t−h
T
⎡
⎤
M1T M1 −M1T M2 xt ⎣
⎦
−M1 M2T −M2T − M2 xt − h
T xt
M3 M4
xt
h
,
M4T M5 xt − h
xt − h
2.9
where
⎤
X M1 M2
⎢MT M M ⎥
3
4 ⎥ ≥ 0.
⎢ 1
⎦
⎣
T
T
M2 M4 M5
⎡
2.10
Abstract and Applied Analysis
5
3. Main Results
In this section, we first study the robust exponential stability criteria for the system 2.1
by using the combination of linear matrix inequality LMI technique and Lyapunov theory
method. We introduce the following notations for later use:
Pj α N
j
αi Pi ,
Wj α i1
N
j
αi Wi ,
i1
Ml α N
αi Mil ,
j
j
N
j
αi Ni ,
Qj α i1
Rs α i1
j
Nj α N
αi Rsi ,
N
i1
i1
N
j
αi Qi ,
i1
αi 1,
3.1
αi ≥ 0,
j
Pi , Wi , Ni , Qi , Mil , Rsi ∈ Rn×n , j 1, 2, . . . , 6, l 1, 2, . . . , 5, s 1, 2, 3, i 1, 2, . . . , N;
⎤
⎡ 11
15
14
Σi,j,k Σ12
Σ13
Σ16
Σ17
i,j Σi,j Σi,j
i
i,j,k
i,j,k
⎥
⎢
⎢ ∗ Σ22 Σ23 Σ24 Σ25 Σ26 Σ27 ⎥
⎢
i,j
i,j
i,j
i
i,j,k
i,j,k ⎥
⎥
⎢
⎥
⎢
⎥
⎢
T
33
34
35
3
37
⎥
⎢ ∗
∗
Σ
Σ
Σ
−N
Σ
i
i
i
i
i,j ⎥
⎢
⎥
⎢
⎥
⎢
45
44
4T
47 ⎥
⎢ ∗
∗
∗
Σ
Σ
−N
Σ
i
i
i,j ⎥,
i
3.2
⎢
⎥
⎢
i,j,k
⎥
⎢
⎥
⎢ ∗
∗
∗
∗ Σ55
Σ56
Σ57
i
i
i,j ⎥
⎢
⎥
⎢
⎥
⎢
66
⎥
⎢ ∗
∗
∗
∗
∗
Σ
0
i
⎥
⎢
⎥
⎢
⎣
77 ⎦
∗
∗
∗
∗
∗
∗
Σi,j,k
where
T
T
1
1
T 1
2
2 T 5
−2βh 5
Σ11
Pi Qi1 Qi1 Qi4 Aj ATi Qj4
i,j,k 2βPi Pi Aj Ai Pj Pi h Ai Pj Ak − e
T
T
T
Ni1 Ni1 Wi1 Aj ATi Wj1 hMi1 hMi1 h2 Mi3 1 η2 I g 2 Pi6 ,
T
T
T
1
2
1
T 5
4
2 T 5
−2βh 5
Σ12
Pi hR2i − hMi1
i,j,k Pi Bj Qi − Qi Ai Qj Qi Bj h Ai Pj Bk e
T
T
T
− Ni1 Ni2 Wi1 Bj Wi2 Aj hMi2 h2 Mi4 ,
T
T
T
T
T
T
3
1
T
3
1
4
2 T 5
Σ13
i,j Ni Wi Ai Wj Pi Qi h Ai Pj ,
4
1
T
4
1
4
2 T 5
Σ14
i,j Ni Wi Ai Wj Pi Qi h Ai Pj ,
5
1
T
5
3
4
T 6
Σ15
i,j Ni − Wi Ai Wj Qi − Qi Ai Qj ,
T
6
Abstract and Applied Analysis
T
T
1
1
6
Σ16
i −Qi − Ni Ni ,
T
T
2 T 5
1
T
6
1
4
Σ17
i,j,k h Ai Pj Ck Wi Cj Ai Wj Pi Cj Qi Cj ,
T
T
2
2
5
T 5
−2βh 2
Σ22
Pi hd Pi2 h2 BiT Pj5 Bk − e−2βh Pi5 h2 R1i
i,j,k −Qi − Qi Qi Bj Bi Qj − e
T
T
T
T
− Ni2 − Ni2 Wi2 Bj BiT Wj2 − hR2i − hR2i − hMi2 − hMi2 h2 Mi5 2 ρ2 I,
T
T
T
T
2
3
T
3
5
2 T 5
Σ23
i,j Wi − Ni Bi Wj Qi h Bi Pj ,
2
4
T
4
5
2 T 5
Σ24
i,j Wi − Ni Bi Wj Qi h Bi Pj ,
T
T
2
5
T
5
T 6
3
5
Σ25
i,j −Wi − Ni Bi Wj Bi Qj − Qi − Qi ,
T
T
2
2
6
Σ26
i −Qi − Ni − Ni ,
T
T
2 T 5
2
T
6
5
Σ27
i,j,k h Bi Pj Ck Wi Cj Bi Wj Qi Cj ,
T
3
3
2 5
Σ33
i Wi Wi h Pi − 1 I,
T
3
4
2 5
Σ34
i W i W i h Pi ,
T
3
5
6
Σ35
i −Wi Wi Qi ,
T
2 5
3
6
Σ37
i,j h Pi Cj Wi Cj Wi ,
T
4
4
2 5
Σ44
i Wi Wi h Pi − 2 I,
T
4
5
6
Σ45
i −Wi Wi Qi ,
T
2 5
4
6
Σ47
i,j h Pi Cj Wi Cj Wi ,
T
T
5
5
6
6
2 3
2 4
Σ55
i −Wi − Wi − Qi − Qi h Pi h Pi ,
T
T
3
5
Σ56
i −Qi − Ni ,
T
T
5
6
6
Σ57
i,j Wi Cj − Wi Qi Cj ,
T
6
6
−2βh 4
Σ66
Pi ,
i −Ni − Ni − e
T
2βg 6
Pi h2 CiT Pj5 Ck CiT Wj6 Wi6 Cj .
Σ77
i,j,k −e
3.3
Theorem 3.1. For given positive real constants h, hd , g, η and ρ, system 2.1 is robustly
exponentially stable with a decay rate β, if there exist positive definite symmetric matrices Pis , any
Abstract and Applied Analysis
7
appropriate dimensional matrices Wis , Qis , Nis , Mir , Rti , s 1, 2, . . . , 6, r 1, 2, . . . , 5, t 1, 2, 3,
i 1, 2, . . . , N and positive real constants 1 and 2 satisfying the following LMIs:
⎡ 1 2⎤
Ri Ri
⎦ > 0,
⎣
∗ R3i
i 1, 2, . . . , N,
⎡ −2βh 3
⎤
e
Pi − R3i Mi1 Mi2
⎢
⎥
⎢
∗
Mi3 Mi4 ⎥
⎢
⎥ ≥ 0,
⎣
⎦
∗
∗ Mi5
< −I,
3.4
i 1, 2, . . . , N,
i 1, 2, . . . , N,
3.6
i,i,i
i,i,j
i,j,i
i,j,k
<
j,i,i
i,k,j
1
N − 12
j,i,k
j,k,i
i 1, 2, . . . , N − 2,
I,
i 1, 2, . . . , N, i /
j, j 1, 2, . . . , N,
k,i,j
<
k,j,i
6
N − 12
3.5
3.7
I,
3.8
j i 1, . . . , N − 1, k j 1, . . . , N.
Moreover, the solution xt, φ, ψ satisfies the inequality
x t, φ, ψ ≤
N
max φ, ψ e−βt ,
λmin P1 α
∀t ∈ R ,
3.9
3
3
3
where N λmax P
1 α hλmax P2 α h λmax P3 α h λmax P4 α h λmax P5 α R1 α R2 α
3
h λmax RT α R3 α .
2
Proof. Choose a parameter-dependent Lyapunov-Krasovskii functional candidate for the
system 2.1 of the form
V t 7
Vi t,
i1
where
V1 t xT tP1 αxt
⎡
⎤
⎤T ⎡
⎤⎡
⎤⎡
xt
I 0 0
0
0
xt
P1 α
⎣xt − ht⎦ ⎣0 0 0⎦⎣Q1 α Q2 α Q3 α⎦⎣xt − ht⎦,
ẋt
ẋt
0 0 0 Q4 α Q5 α Q6 α
3.10
8
Abstract and Applied Analysis
V2 t t
e2βs−t xT sP2 αxsds,
t−ht
V3 t h
V4 t h
V5 t h
V6 t h
V7 t g
0 t
e2βs−t ẋT sP3 αẋsds dθ,
−h
tθ
−h
tθ
0 t
e2βs−t ẋT sP4 αẋsds dθ,
0 t
−h
e2βs−t ẋT sP5 αẋsds dθ,
tθ
⎤
⎡
T R1 α R2 α xθ
−
hθ
⎦ xθ − hθ ds dθ,
⎣ T
e2βθ−t
ẋs
ẋs
R2 α R3 α
θ−hθ
t θ
−h
0 t
−g
e2βs−t xT sP6 αxsdsdθ.
tθ
3.11
Calculating the time derivatives of Vi t, i 1, 2, 3, . . . , 6, along the trajectory of 2.1 yields
⎡
⎤
T
T
⎤
⎤T P1 α Q1 α Q4 α ⎡
xt
⎢
⎥ ẋt
T
T
⎢
Q2 α Q5 α⎥
V̇1 t 2⎣xt − ht⎦ ⎢ 0
⎥⎣ 0 ⎦
⎣
⎦
0
ẋt
0
Q3T α Q6T α
⎡
3.12
⎡
⎤
T
T
⎡
⎤T P1 α Q1 α Q4 α ⎡
⎤
⎢
⎥ ω11
xt
T
T
⎢ 0
⎥
Q
Q
α
α
2
5
⎥⎣ω21 ⎦,
2⎣xt − ht⎦ ⎢
⎢
⎥
⎣
⎦
T
T
ẋt
0
Q3 α Q6 α ω31
where
ω11 Aαxt Bαxt − ht ft, xt gt, xt − ht Cα
t
xsds,
t−gt
ω21 xt − xt − ht −
t
ẋsds,
t−ht
ω31 Aαxt Bαxt − ht ft, xt gt, xt − ht Cα
t
xsds − ẋt.
t−gt
3.13
Abstract and Applied Analysis
9
Taking the time-derivative of V2 t leads to
V̇2 t xT tP2 αxt − 1 − ḣt e−2βht xT t − htP2 αxt − ht − 2βV2 t
≤ xT tP2 αxt − e−2βh xT t − htP2 αxt − ht hd xT t − htP2 αxt − ht
− 2βV2 t.
3.14
Obviously, for any scalar s ∈ t − h, t, we get e−2βh ≤ e−2βs−t ≤ 1. Together with Lemma 2.3
Jensen’s inequality, we obtain
V̇3 t h ẋ tP3 αẋt − h
2 T
≤ h2 ẋT tP3 αẋt − h
0
−h
e2βs ẋT t sP3 αẋt sds − 2βV3 t
t
≤ h ẋ tP3 αẋt − he
2 T
3.15
e2βs−t ẋT sP3 αẋsds − 2βV3 t
t−h
−2βh
t
ẋT sP3 αẋsds − 2βV3 t.
t−h
Following the estimation of V̇3 t, we have
V̇4 t ≤ h2 ẋT tP4 αẋt − he−2βh
≤ h ẋ tP4 αẋt − e
2 T
−2βh
t
t
ẋT sP4 αẋsds − 2βV4 t
t−h
ẋ sdsP4 α
T
t−h
≤ h2 ẋT tP4 αẋt − e−2βh
t
t
ẋsds − 2βV4 t
3.16
t−h
ẋT sdsP4 α
t
t−ht
ẋsds − 2βV4 t.
t−ht
From 3.16, it follows that
V̇5 t ≤ h ẋ tP5 αẋt − e
2 T
−2βh
t
t−ht
ẋ sdsP5 α
T
t
ẋsds − 2βV5 t
t−ht
h2 ẋT tP5 αẋt − e−2βh xT t − xT t − ht P5 αxt − xt − ht − 2βV5 t
h Aαxt Bαxt − ht ft, xt gt, xt − ht Cα
2
t
T
xsds
t−gt
10
Abstract and Applied Analysis
× P5 α Aαxt Bαxt − ht ft, xtgt, xt − ht
Cα
t
xsds − e−2βh xT t − xT t − ht P5 αxt − xt − ht
t−gt
− 2βV5 t.
3.17
Taking the time derivative of V6 t and V7 t, we obtain
V̇6 t hhtxT t − htR1 αxt − ht 2hxT t − htR2 αxt
t
− 2hxT t − htR2 αxt − ht h
ẋT sR3 αẋsds − 2βV6 t
t−h
≤ h2 xT t − htR1 αxt − ht 2hxT t − htR2 αxt
t
T
− 2hx t − htR2 αxt − ht h
ẋT sR3 αẋsds − 2βV6 t;
V̇7 t ≤ g 2 xT tP6 αxt − e−2βg
t−h
t
xT sdsP6 α
t−gt
t
3.18
xsds − 2βV7 t.
t−gt
From the Leibinz-Newton formula, the following equation is true for any real matrices Ni α,
i 1, 2, . . . , 6 with appropriate dimensions
2 xT tN1T α xT t − htN2T α f T t, xtN3T α g T t, xt − htN4T α
ẋ
T
tN5T α
t
ẋ
t−ht
T
sdsN6T α
× xt − xt − ht −
t
3.19
ẋsds 0.
t−ht
From the utilization of zero equation, the following equation is true for any real matrices Wi ,
i 1, 2, . . . , 5 with appropriate dimensions
2 xT tW1T α xT t − htW2T α f T t, xtW3T α g T t, xt − htW4T α
ẋ
T
tW5T α
t
x
t−gt
T
sdsW6T α
× Aαxt Bαxt − ht ft, xt gt, xt − ht Cα
t
xsds − ẋt 0.
t−gt
3.20
Abstract and Applied Analysis
11
From 2.4, we obtain for any positive real constants 1 and 2 ,
0 ≤ 1 η2 xT txt − 1 f T t, xtft, xt,
0 ≤ 2 ρ2 xT t − htxt − ht − 2 g T t, xt − htgt, xt − ht.
3.21
By 3.5, Lemma 2.4 and the integral term of the right-hand side of V̇3 t and V̇6 t, we obtain
−h
t
ẋT s e−2βh P3 α − R3 α ẋsds
t−h
≤h
⎡
⎤
T MT α M1 α −MT α M2 α 1
1
xt
xt
⎣
⎦
xt − ht
−M1 α M2T α −M2T α − M2 α xt − ht
3.22
T xt
xt
M3 α M4 α
.
M4T α M5 α xt − ht
xt − ht
h2
According to 3.12–3.22, it is straightforward to see that
V̇ t ≤ ζT t
N N
N αi αj αk
ζt − 2βV t,
i1 j1 k1
3.23
i,j,k
t
where ζT t xT t, xT t − ht, f T t, xt, g T t, xt − ht, ẋT t, t−ht ẋT sds,
t
xT sds and i,j,k is defined in 3.2. The facts that N
i1 αi 1, we obtain the following
t−gt
identities:
⎡
⎤
N N
N N N
N
⎦
αi αj αk
α3i
α2i αj ⎣
i1 j1 k1
i1
i,j,k
i,i,i
i1 i /
j,j1
i,i,j
i,j,i
j,i,i
⎡
⎤
N
N−2
N−1
⎦.
αi αj αl ⎣
i1 ji1kj1
i,j,k
i,k,j
j,i,k
j,k,i
k,i,j
3.24
k,j,i
We define Φ and Λ as
Φ≡
N N
N N
N
2
αi αi − αj N − 1 α3i −
α2i αj ≥ 0,
i1 j1
Λ≡
N N−1
N
i1
i1 j /
i;j1
N
N N−1
N−2
N−1
2
αi αj − αk N − 2
α2i αj − 6
αi αj αl ≥ 0.
i1 j /
i;j1 k /
i;k2
i1 j /
i;j1
i1 ji1 kj1
3.25
12
Abstract and Applied Analysis
From N − 1Φ Λ ≥ 0, we obtain
N
α3i −
i1
N N
1
N − 12
N−2
N−1
6
α2i αj −
N − 12
i1 i / j,j1
N
αi αj αl ≥ 0.
3.26
i1 ji1kj1
By 3.23–3.26, if the conditions 3.6–3.8 are true, then
V̇ t 2βV t ≤ 0,
∀t ∈ R ,
3.27
V t ≤ V 0e−2βt ,
∀t ∈ R .
3.28
which gives
From 3.28, it is easy to see that
λmin P1 αxt2 ≤ V t ≤ V 0e−2βt ,
V 0 3.29
6
Vi 0,
i1
where
V1 0 xT 0P1 αx0,
t
e2βs xT sP2 αxsds,
V2 0 −h0
V3 0 h
V4 0 h
V5 0 h
V6 0 h
0 0
−h
e
θ−hθ
2βθ
0 0
−h
θ
−h
θ
−h
θ
0 0
0 0
e2βs ẋT sP3 αẋsds dθ,
e2βs ẋT sP4 αẋsds dθ,
e2βs ẋT sP5 αẋsds dθ,
xθ − hθ
ẋs
V7 0 g
0 0
−g
3.30
T R1 α R2 α
RT2 α R3 α
xθ − hθ
ds dθ,
ẋs
e2βs xT sP6 αxsds dθ.
θ
Therefore, we get
2
λmin P1 αxt2 ≤ V 0e−2βt ≤ N max φ, ϕ e−2βt ,
3.31
Abstract and Applied Analysis
13
3
3
3
where N
λmax P
1 α hλmax P2 α h λmax P3 α h λmax P4 α h λmax P5 α R
α
R
α
1
2
h3 λmax RT α R3 α . From 3.31, we get
2
x t, φ, ϕ ≤
N
max φ, ϕ e−βt ,
λmin P1 α
∀t ∈ R .
3.32
This means that system 2.1 is robustly exponentially stable. The proof of the theorem is
complete.
If Aα A, Bα B and Cα 0 when A and B are appropriate dimensional
constant matrices, then system 2.1 reduces to the following system:
ẋt Axt Bxt − ht ft, xt gt, xt − ht,
xt φt,
ẋt ψt,
t > 0;
t ∈ −h, 0.
3.33
Take the Lyapunov-Krasovskii functional as
V t 6
Vi t,
3.34
i1
where
⎤
⎤⎡
⎤⎡
⎤T ⎡
I 0 0
P1 0 0
xt
xt
V1 t xT tP1 xt ⎣xt − ht⎦ ⎣0 0 0⎦⎣Q1 Q2 Q3 ⎦⎣xt − ht⎦,
ẋt
0 0 0 Q4 Q5 Q6
ẋt
t
V2 t e2βs−t xT sP2 xsds,
⎡
t−ht
V3 t h
V4 t h
V5 t h
V6 t h
t θ
−h
θ−hθ
0 t
−h
tθ
−h
tθ
−h
tθ
0 t
0 t
e2βθ−t
e2βs−t ẋT sP3 ẋsds dθ,
3.35
e2βs−t ẋT sP4 ẋsds dθ,
e2βs−t ẋT sP5 ẋsds dθ,
xθ − hθ
ẋs
T R1 R2
RT2 R3
xθ − hθ
ds dθ.
ẋs
According to Theorem 3.1, we have the following Corollary 3.2 for the delay-dependent
exponential stability criteria of system 3.33.
Corollary 3.2. For given positive real constants h, hd , η and ρ, system 3.33 is exponentially stable
with a decay rate β, if there exist positive definite symmetric matrices Pi , i 1, 2, . . . , 5, any appropriate
14
Abstract and Applied Analysis
dimensional matrices Qi , Ni , i 1, 2, . . . , 6, Wi , Mi , i 1, 2, . . . , 5, Ri , i 1, 2, 3 and positive real
constants 1 and 2 satisfying the following LMIs:
R1 R2
> 0,
∗ R3
⎤
⎡ −2βh
P3 − R3 M1 M2
e
⎣
∗
M3 M4 ⎦ ≥ 0,
∗
∗ M5
⎡
Σ11 Σ12 Σ13
⎢ ∗ Σ22 Σ23
⎢
⎢∗
∗ Σ33
⎢
⎢
⎢∗
∗
∗
⎢
⎢
⎣∗
∗
∗
∗
∗
∗
⎤
Σ14 Σ15 Σ16
Σ24 Σ25 Σ26 ⎥
⎥
Σ34 Σ35 −N3T ⎥
⎥
⎥ < 0,
Σ44 Σ45 −N4T ⎥
⎥
⎥
∗ Σ55 Σ56 ⎦
∗
∗ Σ66
where
Σ11 2βP1 P1 A AT P1 P2 h2 AT P5 A − e−2βh P5 Q1 Q1T Q4T A AT Q4
N1T N1 W1T A AT W1 hM1T hM1 h2 M3 1 η2 I,
Σ12 P1 B Q2 − Q1T AT Q5 Q4T B h2 AT P5 B e−2βh P5 hRT2 − hM1T
− N1T N2 W1T B W2T A hM2 h2 M4 ,
Σ13 N3 W1T AT W3 P1 Q4T h2 AT P5 ,
Σ14 N4 W1T AT W4 P1 Q4T h2 AT P5 ,
Σ15 N5 − W1T AT W5 Q3 − Q4T AT Q6 ,
Σ16 − Q1T − N1T N6 ,
Σ22 Q2T − Q2 Q5T B BT Q5 − e−2βh P2 hd P2 h2 BT P5 B − e−2βh P5 h2 R1
− N2T − N2 W2T B BT W2 − hRT2 − hR2 − hM2T − hM2 h2 M5 2 ρ2 I,
Σ23 W2T − N3 BT W3 Q5T h2 BT P5 ,
Σ24 W2T − N4 BT W4 Q5T h2 BT P5 ,
Σ25 − W2T − N5 BT W5 BT Q6 − Q3 − Q5T ,
Σ26 − Q2T − N2T − N6 ,
Σ33 W3T W3 h2 P5 − 1 I,
3.36
Abstract and Applied Analysis
15
Σ34 W3T W4 h2 P5 ,
Σ35 − W3T W5 Q6 ,
Σ44 W4T W4 h2 P5 − 2 I,
Σ45 − W4T W5 Q6 ,
Σ55 − W5T − W5 − Q6T − Q6 h2 P3 h2 P4 ,
Σ56 − Q3T − N5T ,
Σ66 − N6T − N6 − e−2βh P4 .
3.37
Moreover, the solution xt, φ, ψ satisfies the inequality
x t, φ, ψ ≤
N
max φ, ψ e−βt ,
λmin P1 ∀t ∈ R ,
where N λmax P1 hλmax P2 h3 λmax P3 h3 λmax P4 h3 λmax P5 h3 λmax 3.38
R
1
RT2
R2
R3
.
4. Numerical Examples
In order to show the effectiveness of the approaches presented in Section 3, four numerical
examples are provided.
Example 4.1. Consider the LPD time-delay system 2.1 with the following parameters N 3:
−2 0
−3 1
−3 0
,
A2 ,
A3 ,
1 −3
0 −4
0 −2
1 0
−1 1
−1 0
B1 ,
B2 ,
B3 ,
1 −1
0 −1
0 −1
0.2 0.1
−0.3 0.2
−0.4 0.1
C1 ,
C2 ,
C3 ,
0.1 −0.3
0.1 0.2
0.1 0.5
0.3 sin tx1 t − ht
0.2 sin tx1 t
,
gt, xt − ht ,
ft, xt 0.2 cos tx2 t
0.3 cos tx2 t − ht
0.3t
−7
2
2
,
gt 0.4 cos t,
φt ht 0.2134 sin
, t ∈ −0.4, 0.
5
0.4268
A1 4.1
It is easy to see that hd 0.3, η 0.2, ρ 0.3, and g 0.4. Find the discrete delay time h to
guarantee system 2.1 with the above parameters to be robustly exponentially stable with a
decay rate β 0.15.
16
Abstract and Applied Analysis
Solution 1. By using the LMI Toolbox in Matlab with accuracy 0.01 and conditions 3.4–
3.8 of Theorem 3.1, this system is robustly exponentially stable for discrete delay satisfying
h 0.2134 and
225.4987 −143.6565
316.6633 0.3122
361.5534 −0.4974
,
P21 ,
P31 ,
−143.6565 300.6876
0.3122 426.8816
−0.4974 306.5360
153.3987 −105.3621
284.5598 −17.5849
283.8792 −0.3896
,
P22 ,
P32 ,
P12 −105.3621 242.2669
−17.5849 306.8985
−0.3896 254.6003
484.7945 31.1598
380.0951 9.9856
391.5353 −0.0741
P13 ,
P23 ,
P33 ,
31.1598 398.0539
9.9856 387.3373
−0.0741 392.5712
285.1892 −5.6549
229.7296 −0.3010
239.9236 0.0430
P14 ,
P24 ,
P34 ,
−5.6549 230.3520
−0.3010 239.7532
0.0430 251.9889
285.3879 −8.7124
231.9963 −0.7154
239.9146 −0.0007
P15 ,
P25 ,
P35 ,
−8.7124 227.7653
−0.7154 239.6643
−0.0007 251.9987
195.2662 −48.4444
285.1103 −2.8941
291.3897 2.4418
P16 , P26 ,
P36 ,
−48.4444 261.3741
−2.8941 286.4945
2.4418 303.1851
153.8322 −6.2058
1, 057.7 119.5
−442.4
2.6
1
1
1
Q1 ,
Q2 ,
Q3 ,
−6.2058 511.1849
119.5 1, 113.8
2.6 −1, 942.6
−1, 438.7 472.6
−3, 162.4 −435.4
1, 383.2 −0.1
Q12 ,
Q22 ,
Q32 ,
472.6 503.4
−435.4 −107.8
−0.1 1, 874.5
−1, 217.0 −623.5
−9.6
−1, 232.0
564.8910 −1.6044
3
3
3
Q1 ,
Q2 ,
Q3 ,
−623.5 −135.3
−1, 232.0
91.0
−1.6044 −90.0137
−61.6 −1, 470.3
−4, 191.2
13.6
3, 768.2 −0.5
,
Q24 ,
Q34 ,
Q14 −1, 470.3 −4, 166.8
13.6
−4, 402.9
−0.5 −89.7
214.0
−15, 185.0
7, 593.6 7, 529.0
3, 542.1
0.0
5
5
5
Q1 ,
Q2 ,
Q3 ,
−15, 185.0 15, 214.0
7, 529.0 1, 325.3
0.0
−3, 023.0
−40.4375 9.9389
54.7453 −6.5632
53.4114 −0.2701
Q16 ,
Q26 ,
Q36 ,
9.9389 28.8514
−6.5632 45.3682
−0.2701 42.9778
−123.7682 9.3117
−1, 007.0 −119.6
483.1 −2.9
1
1
1
N1 ,
N2 ,
N3 ,
9.3117 −461.4212
−119.6 −1, 067.4
−2.9 1, 962.9
1, 415.9 −464.4
3, 122.6 434.1
−1, 429.4
0.1
N12 ,
N22 ,
N32 ,
−464.4 −534.9
434.1 64.8
0.1
−1, 922.0
38.2260 −6.0579
−13.4412 6.7579
−9.1322 −0.0352
3
3
3
N1 ,
N2 ,
N3 ,
−6.0579 −13.3891
6.7579 9.3172
−0.0352 −24.2347
31.0135 −6.3425
−14.1576 6.0573
−9.6531 0.0282
N14 ,
N24 ,
N34 ,
−6.3425 −14.6572
6.0573 7.3776
0.0282 −23.7517
1, 234.7 657.0
−18.2 1, 241.1
−604.6748 1.6013
N15 ,
N25 ,
N35 ,
657.0 110.7
1, 241.1 −134.2
1.6013 49.2281
P11 The state x(t)
Abstract and Applied Analysis
6
4
2
0
−2
−4
−6
−8
0
0.5
17
1
1.5
2
2.5
3
3.5
4
4.5
5
Time (s)
x1 (t)
x2 (t)
Figure 1: State trajectories x1 t and x2 t of LPD time-delay system 2.1 with 4.1, α1 α2 α3 1/3 by
using program dde45lin with Matlab.
14.9282 −0.7772
36.9436 0.4195
,
N26 ,
−0.7772 33.2457
0.4195 33.4742
222.2533 104.5561
2.1314 0.0476
M11 ,
M21 ,
104.5561 54.1789
0.0476 3.1486
−11.9251 −13.1913
1.4344 −2.6047
M12 ,
M22 ,
−13.1913 −5.6759
−2.6047 3.3445
393.2821 56.3348
269.8280 −0.1703
3
3
M1 ,
M2 ,
56.3348 295.3780
−0.1703 270.2562
−63.2823 −34.4679
−0.8017 −0.1312
M14 ,
M24 ,
−34.4679 −20.2534
−0.1312 −0.8706
208.1212 −29.9392
268.6888 0.6640
5
5
M1 ,
M2 ,
−29.9392 252.5413
0.6640 268.0517
196.3135 −36.7812
268.6863 0.6631
R11 ,
R12 ,
−36.7812 248.7288
0.6631 268.0488
28.5196 7.4817
1.4407 −2.6031
2
2
R1 ,
R2 ,
7.4817 4.9413
−2.6031 3.3546
158.8158 −17.7934
178.2539 4.6824
R31 ,
R32 ,
−17.7934 171.3160
4.6824 181.6405
N16 33.2443 −0.0821
,
−0.0821 25.6059
2.0463 −0.0108
M31 ,
−0.0108 1.4850
1.0670 −0.0153
M32 ,
−0.0153 0.0826
270.0284 −0.0192
3
M3 ,
−0.0192 268.5208
−0.6584 −0.0064
M34 ,
−0.0064 −1.2408
268.9342 −0.0012
5
M3 ,
−0.0012 268.6650
268.9326 −0.0013
R13 ,
−0.0013 268.6593
1.0720 −0.0153
2
R3 ,
−0.0153 0.0894
183.6198 −0.0347
R33 ,
−0.0347 184.1092
N36 4.2
and 1 414.9151 and 2 381.9944. It is known that the maximum value of h for the stability
of this system is h 0.6246. The stability is also assured for h < 0.6246. The numerical solution
x1 t and x2 t of 2.1 with 4.1 are plotted in Figure 1.
Example 4.2. Consider the following linear systems, which are considered in 17:
18
Abstract and Applied Analysis
Table 1: Upper bounds of time delays in Example 4.2 for various conditions.
Cao and Lam 18 2000
Zuo and Wang 3 2006
Chen et al. 17 2008
Corollary 3.2
η 0, ρ 0.1
hd 0.5
0.5467
1.1424
1.1425
2.0925
η 0, ρ 0.1
hd ≥ 1
—
—
0.7355
0.8412
η 0.1, ρ 0.1
hd 0.5
0.4950
1.0097
1.0097
1.8235
η 0.1, ρ 0.1
hd ≥ 1
—
—
0.7147
0.8406
Table 2: Upper bounds of time delays in Example 4.3 for various conditions.
Kwon and Park 19 2008
Corollary 3.2
η 0.05, ρ 0.1
β0
β 0.1
3.40
1.36
2.87
1.53
β 0.3
0.76
0.97
β 0.5
0.55
0.75
Table 3: Comparison of convergence rate obtained for Corollary 3.2 and from 8, 20 in Example 4.4.
Method
Mondié and Kharitonov 20
Nam 8
Corollary 3.2
ẋt Year
2005
2009
2012
Convergence rate β
0.470
1.153
1.410
−1.2 0.1
−0.6 0.7
xt xt − ht ft, xt gt, xt − ht,
−0.1 −1
−1 −0.8
4.3
where ft, xt ≤ ηxt, gt, xt − ht ≤ ρxt − ht.
By Corollary 3.2 to the system 4.3, we can obtain the maximum upper bounds of the
time delay under different values of η, ρ, and hd as shown in Table 1. From Table 1, we see
that Corollary 3.2 gives larger delay bounds than some of the recent results in literatures.
Example 4.3. Consider the following linear systems, which are considered in 19:
ẋt −2 0
−1 0
xt xt − h ft, xt gt, xt − h,
0 −1
−1 −1
4.4
where ft, xt ≤ ηxt, gt, xt − ht ≤ ρxt − ht. By using Corollary 3.2 to the
system 4.4, we obtain the maximum upper bounds of the time delay for different values of
η, ρ, and hd as shown in Table 2. From Table 2, it can be seen that Corollary 3.2 gives larger
delay bounds than the recent results in 19.
Example 4.4. Consider the following linear systems, which is considered in 8:
ẋt −4 1
0.1 0
xt xt − h ft, xt gt, xt − h,
0 −4
4 0.1
4.5
Abstract and Applied Analysis
19
where ft, xt ≤ 0.2xt, gt, xt − h ≤ 0.2xt − h. The maximum value of
convergence rate is 1.410 by using Corollary 3.2 for system 4.5. From Table 3, we can see
that Corollary 3.2 gives larger convergence rate than the results in 8, 20.
5. Conclusions
The problem of robust exponential stability for LPD systems with time-varying delays and
nonlinear perturbations was studied. Based on the combination of Leibniz-Newton formula
and linear matrix inequality, the use of suitable Lyapunov-Krasovskii functional, new delaydependent exponential stability criteria are formulated in terms of LMIs. Numerical examples
have shown significant improvements over some existing results.
Acknowledgments
This work is supported by the Thailand Research Fund TRF, the Office of the Higher
Education Commission OHEC, Khon Kaen University grant number MRG5580006,
Faculty of Science, Khon Kaen University and the Centre of Excellence in Mathematics, the
Commission on Higher Education, Thailand.
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Journal of
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International
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Stochastic Analysis
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Discrete Dynamics in
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Optimization
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