Hindawi Publishing Corporation
Journal of Applied Mathematics
Volume 2012, Article ID 237430, 13 pages
doi:10.1155/2012/237430
Research Article
Robust Exponential Stability for
LPD Discrete-Time System with Interval
Time-Varying Delay
Kanit Mukdasai1, 2
1
2
Department of Mathematics, Faculty of Science, Khon Kaen University, Khon Kaen 40002, Thailand
Centre of Excellence in Mathematics, CHE, Si Ayuthaya Road, Bangkok 10400, Thailand
Correspondence should be addressed to Kanit Mukdasai, kanit@kku.ac.th
Received 5 April 2012; Accepted 11 July 2012
Academic Editor: B. V. Rathish Kumar
Copyright q 2012 Kanit 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.
This paper investigates the problem of robust exponential stability for uncertain linear-parameter
dependent LPD discrete-time system with delay. The delay is of an interval type, which means
that both lower and upper bounds for the time-varying delay are available. The uncertainty under
consideration is norm-bounded uncertainty. Based on combination of the linear matrix inequality
LMI technique and the use of suitable Lyapunov-Krasovskii functional, new sufficient conditions
for the robust exponential stability are obtained in terms of LMI. Numerical examples are given to
demonstrate the effectiveness and less conservativeness of the proposed methods.
1. Introduction
Over the past decades, the problem of stability analysis of delay discrete-time systems
has been widely investigated by many researchers. Because the existence of time delay is
frequent, a source of oscillation instability performances degradation of systems. Stability
criteria for discrete-time systems with time delay is generally divided into two classes: delayindependent ones and delay-dependent ones. Delay-independent stability criteria tend 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 stability criteria is concerned
with the size of the delay and usually provide a maximal delay size. Moreover, robust stability
of linear continuous-time and discrete-time systems subject to time-invariant parametric
uncertainty has 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
2
Journal of Applied Mathematics
matrices. To address this problem, several results have been obtained in terms of sufficient
or necessary and sufficient conditions; see 1–15 and references cited therein. Most of these
conditions have been obtained via the Lyapunov theory approaches in which parameter
dependent Lyapunov functions have been employed. These conditions are always expressed
in terms of linear matrix inequalities LMIs which can be solved numerically by using
available tools such as LMI toolbox in MATLAB. The results have been obtained for robust
stability for LPD systems in which time delay occur in state variable such as 6, 11, 14
present sufficient conditions for robust stability of LPD continuous-time system with delays.
However, a few results have been obtained for robust stability for LPD discrete-time systems
with delay.
In this paper, we deal with the problem of robust exponential stability for uncertain
LPD discrete-time system with interval time-varying delay. Combined with the linear matrix
inequality technique and the use of suitable Lyapunov-Krasovskii functional, new sufficient
conditions for the robust exponential stability are obtained in terms of LMI. Finally, numerical
examples have demonstrated the effectiveness of the criteria.
2. Problem Formulation and Preliminaries
We introduce some notations and definitions that will be used throughout the paper. Z
denotes the set of non negative integer numbers; Rn denotes the n-dimensional space with
the vector norm · ; x denotes the Euclidean vector norm of x ∈ Rn ; that is, x2 xT x;
Mn×r denotes the space of all matrices of n × r-dimensions; AT denotes 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{Reλ : λ ∈ λA}; Matrix A is called semi positive definite
A ≥ 0 if xT Ax ≥ 0, for all x ∈ Rn ; A is positive definite A > 0 if xT Ax > 0 for all x / 0;
Matrix B is called semi-negative definite B ≤ 0 if xT Bx ≤ 0, for all x ∈ Rn ; B is negative
definite B < 0 if xT Bx < 0 for all x / 0; A > B means A − B > 0; A ≥ B means A − B ≥ 0; ∗
represents the elements below the main diagonal of a symmetric matrix.
Consider the following uncertain LPD discrete-time system with interval time-varying
delay in the state
xk 1
Aα ΔAkxk Bα ΔBkxk − hk,
xs
φs,
s
−h2 , . . . , −1, 0,
2.1
where k ∈ Z , xk ∈ Rn is the system state and φs is a initial value at s. Aα, Bα ∈ Mn×n
are uncertain matrices belonging to the polytope of the form
Aα
N
αi Ai ,
Bα
N
i 1
N
αi
i 1
1,
αi ≥ 0,
αi Bi ,
i 1
Ai , Bi ∈ Mn×n ,
2.2
i
1, . . . , N.
Journal of Applied Mathematics
3
ΔAk and ΔBk are unknown matrices representing time-varying parameter uncertainties,
we assumed to be of the form
ΔAk
A1 α
KαΔkA1 α,
N
i 1
N
αi
αi A1i ,
1,
ΔBk
N
αi Bi1 ,
B1 α
KαΔkB1 α,
N
αi Ki ,
Kα
i 1
αi ≥ 0,
i 1
A1i , Bi1 ∈ Mn×n ,
i 1
i
2.3
1, . . . , N.
The class of parametric uncertainties Δk, which satisfies
Δk
FkI − JFk−1 ,
2.4
is said to be admissible where J is a known matrix satisfying
I − JJ T > 0,
2.5
FkT Fk ≤ I.
2.6
and Fk is uncertain matrix satisfying
In addition, we assume that the time-varying delay hk is upper and lower bounded. It
satisfies the following assumption of the form
h1 ≤ hk ≤ h2 ,
2.7
where h1 and h2 are known positive integers.
Definition 2.1. The uncertain LPD discrete-time-delayed system in 2.1 is said to be robustly
exponentially stable if there exist constant scalars 0 < a < 1 and b > 0 such that
2
xk2 ≤ bak sup φl ,
−h2 ≤l≤0
2.8
for all admissible uncertainties.
Lemma 2.2 see 5 Schur complement lemma. Given constant matrices X, Y, Z of appropriate
dimensions with Y > 0. Then X ZT Y −1 Z < 0 if and only if
X ZT
Z −Y
<0
or
−Y Z
< 0.
ZT X
2.9
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Journal of Applied Mathematics
Lemma 2.3 see 2. Given constant matrices M1 , M2 , and M3 of appropriate dimensions with
M1 M1T . Then,
M1 M2 ΔkM3 M3T ΔkT M2T < 0,
where Δk
2.10
FkI − JFk−1 , FkT Fk ≤ I, for all k ∈ Z if and only if
M1 −1 M3T M2
I −J
−J T I
−1
−1 M3T M2
T
< 0,
2.11
for some scalar > 0.
3. Main Results
In this section, we present our main results on the robust exponential stability criteria for
uncertain LPD discrete-time system with interval time-varying delays. We introduce the
following notation for later use:
Aα
Aα ΔAk,
Bα
Bα ΔBk,
h
h2 − h1 1.
3.1
1, . . . , N,
3.2
Lemma 3.1. For any Aα, Bα, h in 3.1, P α and Qα given by
P α
N
i 1
αi Pi ,
Qα
N
αi Qi ,
N
i 1
i 1
αi
1, αi ≥ 0, i
are parameter-dependent positive definite Lyapunov matrices such that
AT αP αBα
AT αP αAα − P α hQα
< 0,
BT αP αAα
BT αP αBα − Qα
3.3
⎡
⎤
−P α hQα
0
AαT P α −1 A1 αT
0
⎢
⎥
∗
−Qα BαT P α −1 B1 αT
0
⎢
⎥
⎢
⎥
⎢
∗
∗
−P α
0
P αKα⎥ < 0.
⎢
⎥
⎣
⎦
∗
∗
∗
−I
J
∗
∗
∗
∗
−I
3.4
if and only if
Journal of Applied Mathematics
5
Proof. Consider
AT αP αBα
AT αP αAα − P α hQα
BT αP αAα
BT αP αBα − Qα
AT αP αAα AT αP αBα
−P α hQα
0
0
−Qα
BT αP αAα BT αP αBα
3.5
AT α
−P α hQα
0
P α Aα Bα .
T
0
−Qα
B α
We assume that
A α
−P α hQα
0
P α Aα Bα < 0.
T
0
−Qα
B α
T
3.6
Using Lemma 2.2, we obtain
⎡
⎤
−P α hQα
0
AαT KαΔkA1 αT
⎢
⎥
∗
−Qα BαT KαΔkB1 αT ⎦ < 0.
⎣
−1
∗
∗
−P α
3.7
We rewrite the latter inequality as
⎡
⎤ ⎡
⎤
−P α hQα
0
AαT
0
⎢
T ⎥
∗
−Qα Bα ⎦ ⎣ 0 ⎦Δk A1 α B1 α 0
⎣
Kα
∗
∗
−P α−1
⎡
⎤T
3.8
0
T
A1 α B1 α 0 ΔkT ⎣ 0 ⎦ < 0.
Kα
Using Lemma 2.3, inequality 3.8 holds if and only if there exists > 0 such that
⎡
⎤
−P α hQα
0
AαT
⎢
⎥
∗
−Qα BαT ⎦
⎣
∗
∗
−P α−1
⎤
⎡
0
−1 A1 αT
⎥ I −J
⎢ −1
T
⎣ B1 α
0 ⎦
−J I
0
Kα
−1
⎤T
⎡
0
−1 A1 αT
⎥
⎢ −1
0 ⎦ < 0.
⎣ B1 αT
0
Kα
3.9
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Journal of Applied Mathematics
If we apply to 3.9, then we obtain
⎡
⎤
−P α hQα
0
AαT −1 A1 αT
0
⎢
∗
−Qα BαT −1 B1 αT
0 ⎥
⎢
⎥
⎢
⎥
−1
⎢
⎥ < 0.
∗
∗
−P
0
Kα
α
⎢
⎥
⎣
∗
∗
∗
−I
J ⎦
∗
∗
∗
∗
−I
3.10
Premultiplying 3.10 by diag{I, I, P α, I, I} and postmultiplying by diag{I, I, P α, I, I}, we
get that 3.4 and the lemma is proved.
Lemma 3.2. If there exist positive definite symmetric matrices Pi , Qi , i
real numbers , ζ such that
1, 2, . . . , N, and positive
⎡
⎤
T
−Pi hQi 0 ATi Pi −1 A1i
0
⎢
⎥
T
⎢
0 ⎥
∗
−Qi BiT Pi −1 Bi1
⎢
⎥
< −ζI, i 1, 2, . . . , N,
⎢
0
Pi Ki ⎥
∗
∗
−Pi
⎢
⎥
⎣
⎦
∗
∗
∗
−I
J
∗
∗
∗
∗
−I
⎤
⎡
⎤ ⎡
T
T
−Pj hQj 0 ATj Pi −1 A1j
0
−Pi hQi 0 ATi Pj −1 A1i
0
⎥
⎢
⎥ ⎢
T
T
⎢
∗
−Qj BjT Pi −1 Bj1
0 ⎥ ⎢
0 ⎥
∗
−Qi BiT Pj −1 Bi1
⎥
2ζI
⎢
⎥ ⎢
⎥
⎢
⎢
0
Pi Kj ⎥
∗
∗
−Pi
∗
∗
−Pj
0
Pj Ki ⎥ < N − 1 ,
⎢
⎥ ⎢
⎢
⎥
⎣
∗
∗
∗
−I
J ⎦ ⎣
∗
∗
∗
−I
J ⎦
∗
∗
∗
∗
−I
∗
∗
∗
∗
−I
i
1, . . . , N − 1,
j
i 1, . . . , N,
3.11
then, for any Aα, A1 α, Bα, B1 α, Kα, h in 3.1, P α and Qα are parameter-dependent
positive definite Lyapunov matrices in Lemma 3.1 such that 3.4 holds.
Proof. Consider
⎤
⎡
−P α hQα
0
AαT P α −1 A1 αT
0
⎥
⎢
∗
−Qα BαT P α −1 B1 αT
0
⎥
⎢
⎥
⎢
⎢
∗
∗
−P α
0
P αKα⎥
⎥
⎢
⎦
⎣
∗
∗
∗
−I
J
∗
∗
∗
∗
−I
⎡
⎤
T
−Pi hQi 0 ATi Pj −1 A1i
0
⎢
⎥
T
N
N ⎢
0 ⎥
∗
−Qi BiT Pj −1 Bi1
⎢
⎥
.
αi αj ⎢
0
Pi Kj ⎥
∗
∗
−Pi
⎢
⎥
i 1j 1
⎣
⎦
∗
∗
∗
−I
J
∗
∗
∗
∗
−I
3.12
Journal of Applied Mathematics
Using the fact that
N
i 1
αi
7
1, we obtain the following identities:
N
N αi αj Ai Bj
N
N
N−1
α2 Ai Bi αi αj Ai Bj Aj Bi ,
i 1j 1
i 1
i 1 j i1
N
N
N−1
αi αj ζ
N − 1 α2i ζ − 2
i 1
N
N−1
i 1 j i1
2
αi − αj ζ ≥ 0.
3.13
i 1 j i1
Then, it follows from 3.11, 3.12, and 3.13 that 3.4 holds. The proof of the lemma is
complete.
Theorem 3.3. The system 2.1 is robustly exponentially stable if the LMI conditions 3.11 are
feasible.
Proof. Consider the following Lyapunov-Krasovskii function for system 2.1 of the form
V xk
V1 xk V2 xk V3 xk,
3.14
where
V1 xk
xT kP αxk,
V2 xk
k−1
xT iQαxi,
i k−hk
V3 xk
−h
1 1
k−1
3.15
T
x lQαxl.
j −h2 2 l kj−1
A Lyapunov-Krasovskii difference for the system 2.1 is defined as
ΔV xk
ΔV1 xk ΔV2 xk ΔV3 xk.
3.16
Taking the difference of V1 xk and V2 xk, the increments of V1 xk and V2 xk are
ΔV1 xk
V1 xk 1 − V1 xk
xT k 1P αxk 1 − xT kP αxk
xT kAT αP αAαxk xT k − hkBT αP αAαxk
xT k − hkBT αP αBαxk − hk
xT kAT αP αBαxk − hk − xT kP αxk,
3.17
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Journal of Applied Mathematics
ΔV2 xk
V2 xk 1 − V2 xk
k
k−1
xT iQαxi −
i k1−hk1
xT iQαxi
i k−hk
xT kQαxk − xT k − hkQαxk − hk
k−h
1
xT iQαxi −
i k1−hk1
k−1
k−1
3.18
xT iQαxi
i k1−hk1
xT iQαxi.
i k1−h1
Form hk ≥ h1 , the two last terms of the right-hand side of the latter equality yield
k−1
k−1
xT iQαxi −
i k1−h1
xT iQαxi ≤ 0.
3.19
i k1−hk1
Thus, we obtain
ΔV2 xk ≤ xT kQαxk − xT k − hkQαxk − hk
k−h
1
3.20
xT iQαxi.
i k1−hk1
The increment of V3 xk is easily computed as
ΔV3 xk
V3 xk 1 − V3 xk
⎤
⎡
−h
k−1
k−1
1 1
⎣xT kQαxk xT lQαxl −
xT lQαxl⎦
j −h2 2
l kj−1
h2 − h1 xT kQαxk −
k−h
1
l kj−1
3.21
xT iQαxi.
i k1−h2
It is easy to see that
T
Qαxk−h
ΔV2 xk ΔV3 xk ≤ h2 − h1 1xT kQαxk − xk−h
k−h
1
xT iQαxi −
i k1−hk1
k−h
1
i k1−h2
xT iQαxi,
3.22
Journal of Applied Mathematics
for simplicity, we let xk − hk
k−h
1
9
xk−h . Since, hk ≤ h2 , we obtain that
xT iQαxi −
i k1−hk1
k−h
1
xT iQαxi ≤ 0.
3.23
i k1−h2
Therefore, we conclude that
T
ΔV xk ≤ xT kAT αP αAαxk xk−h
BT αP αAαxk
T
xT kAT αP αBαxk−h xk−h
BT αP αBαxk−h
− xT kP αxk h2 − h1 1xT kQαxk
3.24
T
− xk−h
Qαxk−h .
It follows form 3.24 that
ΔV xk ≤ Y T
Δ11 α
AT αP αBα
Y,
BT αP αAα BT αP αBα − Qα
3.25
where Δ11 α
AT αP αAα − P α hQα and Y T
3.25, and Lemma 3.1, and 3.2, we obtain
xkT xk − hkT . By 3.11,
ΔV xk < −ωx2 ,
3.26
where ω > 0. By 3.14, it is easy to see that
V xk ≤ β1 x2 β1 h
k−1
xi2 ,
3.27
1, 2, . . . , N}.
3.28
i k−h2
where
β1
max {λmax Pi , λmax Qi ; i
It can be shown that there always exists a scalar θ > 1 satisfying
θ − 1β1 − λθ h2 θh2 θ − 1β1 h
0.
3.29
10
Journal of Applied Mathematics
For any scalar θ > 1, it follows from 3.26 and 3.27 that
θk1 V xk 1− θk1 V xk
θk1 V xk 1 − V xk θk θ − 1V xk
k−1
< α1 θθk xk2 α2 θθk
3.30
xi2 ,
i k−h2
where
α1 θ
θ − 1β1 − λθ,
3.31
θ − 1β1 h.
α2 θ
Therefore, for any integer T ≥ h2 1, summing up both sides of 3.30 from 0 to T − 1 gives
θT V xT − V x0 ≤ α1 θ
T −1
θi xi2 α2 θ
i 0
T −1 i−1
θi xl2 .
3.32
i 0 l i−h2
For h2 ≥ 1,
T −1 i−1
i 0 l i−h2
θi xl2 ≤
−1 lh
2
θi xl2 T −1−h
2 lh
2
l −h2 i 0
l 0 i l1
T −1 T −1
θi xl2
θi xl2
l T −h2 i l1
−1
≤ h2
θlh2 xl2 h2
l −h2
h2
T −1
T −1−h
2
θlh2 xl2
3.33
l 0
θlh2 xl2
l T −h2
T −1
2
≤ h2 h2 1θh2 sup φl h2 θh2 θl xl2 .
−h2 ≤l≤0
l 1
From 3.32 and 3.33, we obtain
2
θT V xT ≤ V x0 h2 h2 1θh2 α2 θ sup φl
−h2 ≤l≤0
α1 θ α2 θh2 θ
h2
T −1
l 0
3.34
l
2
θ xl .
Journal of Applied Mathematics
11
Observe
2
V x0 ≤ β1 β1 hh2 sup φl ,
V xT ≥ γxT 2 ,
γ
−h2 ≤l≤0
min{λmin Pi ; i
3.35
1, 2, . . . , N}.
Then, it follows from 3.29, 3.33, and 3.35 that
xT 2 ≤
h2 h2 1θh2 α2 θ β1 β1 hh2 1 T
γ
θ
2
sup φl .
3.36
−h2 ≤l≤0
By Definition 2.1, this means that the system 2.1 is robustly exponentially stable. The proof
of the theorem is complete.
4. Numerical Example
Example 4.1. Consider the following uncertain LPD discrete-time system with time-varying
delays 2.1 where hk 2 coskπ/2, that is,. h1 1, h2 3 and
A1
B2
−0.6 0.02
,
0.02 −0.6
A2
−0.7 0.03
,
0.03 −0.7
B1
−0.6 0.02
,
0.02 −0.08
0.006 0.0002
,
0.0002 0.006
−0.8 0.03
0.005 0.0001
,
A11
,
A12
0.03 −0.09
0.0001 0.005
−0.007 0.0005
−0.004 0.0002
B11
,
B21
,
0.0005 −0.007
0.0002 −0.004
0.01 0.003
0.02 0.001
K1
,
K2
,
0.003 0.01
0.001 0.02
4.1
0.001 0 and J
0 0.001 . By using LMI Toolbox in MATLAB, we use condition 3.11 in
Theorem
3.3
for
The solutions
of LMI verify as follows of the form 1,
31.3635 1.2365this
37.6354
example.
0.2543 , Q
9.4325 0.5587 , and Q
10.8564 1.3856
,
P
P1
2
1
2
1.2365 29.4763
0.2543 41.3745
0.5587 11.4534
1.3856 11.9781 see
Figure 1.
Example 4.2. Consider the following the LPD discrete-time system with time-varying delays
2.1 where, ΔAk ΔBk 0 with
A1
B1
0.60 0
,
0.01 0.60
0.10 0
,
0.20 0.10
A2
B2
0.80 0
,
0.05 0.70
−0.10
0
.
−0.20 −0.10
4.2
12
Journal of Applied Mathematics
5
The state x(k)
4
3
x1 (k)
2
1
0
x2 (k)
−1
−2
0
2
4
6
8
10
12
14
16
Time (s)
Figure 1: The simulation solution of the states x1 k and x2 k in Example 4.1 for uncertain LPD discretetime delayed system with initial conditions x1 k 2 and x2 k 4, k −3, −2, −1, 0, and α1 α2 1/2 by
using the method of Runge-Kutta order 4h 0.01 with Matlab.
Table 1: Comparison of the maximum allowed time delay h2 .
Methods
Liuet al. 7 2006
Our results
h2 h1
2
4
2
h2 h1
4
6
4
h2 h1
5
7
5
h2 h1
7
9
7
Table 1 lists the comparison of the upper-bound delay for asymptotic stability of system 2.1
where ΔAk ΔBk 0 by different method. We apply Theorem 3.3 and see from Table 1
that our result is superior to those in 7, Theorem 3.2.
Acknowledgments
This work was supported by the Thailand Research Fund TRF, the Office of the
Higher Education Commission OHEC, and the Khon Kaen University Grant number
MRG5580006, and the author would like to thank his advisor Associate Professor Dr.
Piyapong Niamsup from the Department of Mathematics, Faculty of Science, Chiang Mai
University, Chiang Mai, Thailand.
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