Hindawi Publishing Corporation
Discrete Dynamics in Nature and Society
Volume 2013, Article ID 929725, 10 pages
http://dx.doi.org/10.1155/2013/929725
Research Article
New Delay-Dependent Robust Stability Criterion for LPD
Discrete-Time Systems with Interval Time-Varying Delays
Narongsak Yotha1 and Kanit Mukdasai2,3
1
Department of Mathematics, Faculty of Science, Chiang Mai University, Chiang Mai 50200, Thailand
Department of Mathematics, Faculty of Science, Khon Kaen University, Khon Kaen 40002, Thailand
3
Centre of Excellence in Mathematics, CHE, Si Ayutthaya Road, Bangkok 10400, Thailand
2
Correspondence should be addressed to Kanit Mukdasai; kanit@kku.ac.th
Received 18 October 2012; Accepted 13 January 2013
Academic Editor: Xiaohui Liu
Copyright © 2013 N. Yotha 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.
This paper investigates the problem of robust stability for linear parameter-dependent (LPD) discrete-time systems with interval
time-varying delays. Based on the combination of model transformation, utilization of zero equation, and parameter-dependent
Lyapunov-Krasovskii functional, new delay-dependent robust stability conditions are obtained and formulated in terms of linear
matrix inequalities (LMIs). Numerical examples are given to demonstrate the effectiveness and less conservativeness of the proposed
methods.
1. Introduction
Systems with time delay exist in many fields such as electric
systems, chemical processes systems, networked control systems, telecommunication systems, and economical systems.
Over the past decades, the problem of robust stability analysis
for uncertain systems with time delay has been widely investigated by many researchers. Commonly, stability criteria for
uncertain systems with time delay are generally divided into
two classes: a delay-independent one and a delay-dependent
one. The delay-independent stability criteria tends to be more
conservative, especially for a small size delay; such criteria do
not give any information on the size of delay. On the other
hand, delay-dependent stability criteria are concerned with
the size of delay and usually provide a maximal delay size.
Discrete-time systems with state delay have strong background in engineering applications, among which networkbased control has been well recognized to be a typical example. If the delay is constant in discrete systems, one can transform a delayed system into a delay-free one by using state augmentation techniques. However, when the delay is large, the
augmented system will become much complex and thus difficult to analyze and synthesize [1]. In recent years, robust stability analysis of continuous-time and discrete-time systems
subject to time-invariant parametric uncertainty has received
considerable attention. An important class of linear timeinvariant parametric uncertain system is a linear parameterdependent (LPD) system in which the uncertain state matrices are in the polytope consisting of all convex combination
of known matrices. To address this problem, several results
have been obtained in terms of sufficient (or necessary and
sufficient) conditions, see [1–24] and references cited therein.
Most of these conditions have been obtained via the Lyapunov theory approaches in which the parameter-dependent
Lyapunov functions have been employed. These conditions
are always expressed in terms of LMIs which can be solved
numerically by using available tools such as the LMI Toolbox in MATLAB. Recently, delay-dependent robust stability
criteria for LPD continuous-time systems with time delay
have been taken into consideration. Sufficient conditions for
robust stability of time-delay systems have been presented via
Lyapunov approaches [8, 16, 21]. However, much attention has
been focused on the problem of robust stability analysis for
LPD discrete-time systems with time delay [10, 13, 22].
In this paper, we focus on the delay-dependent robust
stability criterion for LPD discrete-time systems with interval
time-varying delays. Based on the combination of model
2
Discrete Dynamics in Nature and Society
transformation, utilization of zero equation, and parameterdependent Lyapunov functional, new delay-dependent
robust stability conditions are obtained and formulated in
terms of linear matrix inequalities (LMIs). Finally, numerical
examples are given to illustrate that the resulting criterion
outperforms the existing stability condition.
Definition 2 (see [19]). The system (1) when 𝐴(𝛼) = 𝐴, 𝐵(𝛼) =
𝐵, and 𝐴, 𝐵 ∈ 𝑀𝑛×𝑛 is said to be asymptotically stable if there
exists a positive definite function 𝑉(𝑘, 𝑥(𝑘)) : Z+ × R𝑛 → R
such that
2. Problem Formulation and Preliminaries
along any trajectory of the solution of the system (1) when
𝐴(𝛼) = 𝐴, 𝐵(𝛼) = 𝐵.
We introduce some notations, definitions, and propositions
that will be used throughout the paper. Z+ denotes the set of
nonnegative integer numbers; R𝑛 denotes the 𝑛-dimensional
space with the vector norm ‖ ⋅ ‖; ‖𝑥‖ denotes the Euclidean
vector norm of 𝑥 ∈ R𝑛 , that is, ‖𝑥‖2 = 𝑥𝑇 𝑥; 𝑀𝑛×𝑟 denotes the
space of all real matrices of (𝑛 × 𝑟)-dimensions; 𝐴𝑇 denotes
the transpose of the matrix 𝐴; 𝐴 is symmetric if 𝐴 = 𝐴𝑇 ; 𝐼
denotes the identity matrix; 𝜆(𝐴) denotes the set of all eigenvalues of 𝐴; 𝜆 max (𝐴) = max{Re 𝜆 : 𝜆 ∈ 𝜆(𝐴)}; 𝜆 min (𝐴) =
min{Re𝜆 : 𝜆 ∈ 𝜆(𝐴)}; 𝜆 max (𝐴(𝛼)) = max{𝜆 max (𝐴 𝑖 ) : 𝑖 =
1, 2, . . . , 𝑁}; 𝜆 min (𝐴(𝛼)) = min{𝜆 min (𝐴 𝑖 ) : 𝑖 = 1, 2, . . . , 𝑁};
matrix 𝐴 is called a semipositive definite (𝐴 ≥ 0) if 𝑥𝑇 𝐴𝑥 ≥ 0,
for all 𝑥 ∈ R𝑛 ; 𝐴 is a positive definite (𝐴 > 0) if 𝑥𝑇 𝐴𝑥 > 0 for
all 𝑥 ≠ 0; matrix 𝐵 is called a seminegative definite (𝐵 ≤ 0) if
𝑥𝑇 𝐵𝑥 ≤ 0, for all 𝑥 ∈ R𝑛 ; 𝐵 is a 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 delays of the form
𝑥 (𝑘 + 1) = 𝐴 (𝛼) 𝑥 (𝑘) + 𝐵 (𝛼) 𝑥 (𝑘 − ℎ (𝑘)) ,
𝑥 (𝑠) = 𝜙 (𝑠) ,
+
𝑠 ∈ {−ℎ2 , −ℎ2 + 1, . . . , −1, 0, } ,
(1)
(2)
Δ𝑉 (𝑘, 𝑥 (𝑘)) = 𝑉 (𝑘 + 1, 𝑥 (𝑘 + 1)) − 𝑉 (𝑘, 𝑥 (𝑘)) < 0, (6)
Proposition 3 ([7, the Schur complement lemma]). Given
constant symmetric matrices 𝑋, 𝑌, and 𝑍 of appropriate
dimensions with 𝑌 > 0, then 𝑋 + 𝑍𝑇 𝑌−1 𝑍 < 0 if and only
if
𝑋 𝑍𝑇
(
) < 0 or
𝑍 −𝑌
−𝑌 𝑍
( 𝑇 ) < 0.
𝑍 𝑋
Proposition 4 (see [9]). For any constant matrix 𝑊 ∈
𝑀𝑛×𝑛 , 𝑊 = 𝑊𝑇 > 0, two integers 𝑟𝑀 and 𝑟𝑚 satisfying
𝑟𝑀 ≥ 𝑟𝑚 , and vector function 𝑥 : [𝑟𝑚 , 𝑟𝑀] → R𝑛 , the following
inequality holds:
𝑟𝑀
𝑇
𝑟𝑀
𝑟𝑀
𝑖=𝑟𝑚
𝑖=𝑟𝑚
( ∑ 𝑥 (𝑖)) 𝑊 ( ∑ 𝑥 (𝑖)) ≤ (𝑟𝑀 − 𝑟𝑚 +1) ∑ 𝑥𝑇 (𝑖) 𝑊𝑥 (𝑖) .
𝑖=𝑟𝑚
(8)
Rewrite the system (1) in the following system:
𝑥 (𝑘 + 1) = 𝑥 (𝑘) + 𝑦 (𝑘) ,
𝑛
where 𝑘 ∈ Z , 𝑥(𝑘) ∈ R is the system state and 𝜙(𝑠) is an
initial value at 𝑠. 𝐴(𝛼), 𝐵(𝛼) ∈ 𝑀𝑛×𝑛 are uncertain matrices
belonging to the polytope of the form
𝑁
𝐴 (𝛼) = ∑𝛼𝑖 𝐴 𝑖 ,
𝑖=1
𝑁
𝑁
(3)
∑𝛼𝑖 = 1, 𝛼𝑖 ≥ 0, 𝐴 𝑖 , 𝐵𝑖 ∈ 𝑀𝑛×𝑛 , 𝑖 = 1, . . . , 𝑁.
𝑖=1
In addition, we assume that the time-varying delay ℎ(𝑘) is
upper and lower bounded. It satisfies the following assumption of the form
0 < ℎ1 ≤ ℎ (𝑘) ≤ ℎ2 ,
(4)
where ℎ1 and ℎ2 are known positive integers.
Definition 1 (see [19]). The system (1) is said to be robustly
stable if there exists a positive definite function 𝑉(𝑘, 𝑥(𝑘)) :
Z+ × R𝑛 → R such that
Δ𝑉 (𝑘, 𝑥 (𝑘)) = 𝑉 (𝑘 + 1, 𝑥 (𝑘 + 1)) − 𝑉 (𝑘, 𝑥 (𝑘)) < 0, (5)
along any trajectory of the solution of the system (1).
𝑘−1
𝑦 (𝑘) = [𝐴 (𝛼) + 𝐵 (𝛼) − 𝐼] 𝑥 (𝑘) − 𝐵 (𝛼) ∑ 𝑦 (𝑖) .
(9)
𝑖=𝑘−ℎ(𝑘)
3. Robust Stability Conditions
𝐵 (𝛼) = ∑𝛼𝑖 𝐵𝑖 ,
𝑖=1
(7)
In this section, we study the robust stability criteria for the
system (1) by using the combination of model transformation, the linear matrix inequality (LMI) technique, and the
Lyapunov method. We introduce the following notations for
later use:
𝑁
𝑗
𝐶𝑗 (𝛼) = ∑𝛼𝑖 𝐶𝑖 ,
𝑖=1
𝑁
𝑗
𝐸𝑗 (𝛼) = ∑𝛼𝑖 𝐸𝑖 ,
𝑖=1
𝑁
𝑗
𝐿 𝑗 (𝛼) = ∑𝛼𝑖 𝐿 𝑖 ,
𝑖=1
𝑁
𝑄 (𝛼) = ∑𝛼𝑖 𝑄𝑖 ,
𝑖=1
𝑁
𝑗
𝐷𝑗 (𝛼) = ∑𝛼𝑖 𝐷𝑖 ,
𝑖=1
𝑁
𝑗
𝐺𝑗 (𝛼) = ∑𝛼𝑖 𝐺𝑖 ,
𝑖=1
𝑁
𝑃 (𝛼) = ∑𝛼𝑖 𝑃𝑖 ,
𝑖=1
𝑁
𝑅 (𝛼) = ∑𝛼𝑖 𝑅𝑖 ,
𝑖=1
Discrete Dynamics in Nature and Society
𝑁
3
𝑇
𝑁
𝑆 (𝛼) = ∑𝛼𝑖 𝑆𝑖 ,
𝑖=1
𝑖=1
𝑁
𝑈 (𝛼) = ∑𝛼𝑖 𝑈𝑖 ,
𝑇
𝑁
2
1
Σ34
𝑖 = 𝐺𝑖 − 𝐺𝑖 ,
𝑖=1
3
1
Σ39
𝑖 = 𝐺𝑖 −𝐺𝑖 ,
𝑉 (𝛼) = ∑𝛼𝑖 𝑉𝑖 ,
𝑖=1
𝑁
𝑇
𝑖=1
𝑁
𝑌 (𝛼) = ∑𝛼𝑖 𝑌𝑖 ,
𝑇
𝑖=1
3
3
Σ66
𝑖 = − 𝑈𝑖 − 𝐶𝑖 − 𝐶𝑖 ,
𝑇
𝑖=1
𝑇
𝑉𝑖 , 𝑊𝑖 , 𝑋𝑖 , 𝑌𝑖 , 𝑍𝑖 ∈ 𝑀𝑛×𝑛 ,
(11)
𝑗 = 1, 2, 3, 𝑖 = 1, 2, . . . , 𝑁, 𝜌 = ℎ2 − ℎ1 .
Theorem 5. The system (1)-(2) is robustly stable if there
exist positive definite symmetric matrices 𝑃𝑖 , 𝑄𝑖 , 𝑅𝑖 , 𝑆𝑖 , 𝑇𝑖 , 𝑈𝑖 ,
𝑉𝑖 , 𝑊𝑖 , 𝑋𝑖 , 𝑌𝑖 , and 𝑍𝑖 , any appropriate dimensional matrices
𝑗
𝑗
𝑗
𝑗
𝑗
𝑀𝑖 , 𝑁𝑖 ,𝐾𝑖 , 𝐿 𝑖 , 𝐶𝑖 , 𝐷𝑖 , 𝐸𝑖 , 𝐺𝑖 , and 𝐿 𝑖 , 𝑗 = 1, 2, 3, 𝑖 = 1, 2, . . . , 𝑁
satisfying the following LMIs:
Σ11
𝑖,𝑗
[
[∗
[
[∗
[
[∗
[
[
∏=[∗
[
𝑖,𝑗
[∗
[
[∗
[
[∗
[
∗
[
Σ12
𝑖,𝑗
Σ22
𝑖,𝑗
∗
∗
∗
∗
∗
∗
∗
Σ13
𝑖,𝑗
0
Σ33
𝑖
∗
∗
∗
∗
∗
∗
Σ14
𝑖,𝑗
0
Σ34
𝑖
Σ44
𝑖
∗
∗
∗
∗
∗
Σ15
𝑖,𝑗
0
0
0
Σ55
𝑖
∗
∗
∗
∗
Σ16
𝑖
0
Σ36
𝑖,𝑗
0
0
Σ66
𝑖
∗
∗
∗
Σ17
𝑖,𝑗
0
0
Σ47
𝑖,𝑗
0
0
Σ77
𝑖,𝑗
∗
∗
Σ18
0
𝑖,𝑗
]
Σ28
0 ]
𝑖,𝑗
]
]
0 Σ18
𝑖,𝑗 ]
18 ]
0 Σ𝑖,𝑗 ]
]
Σ18
0 ],
𝑖,𝑗
]
0 0 ]
]
0 0 ]
]
0 ]
Σ88
]
𝑖,𝑗
∗ Σ99
𝑖,𝑗 ]
(10)
∏ < −𝐼,
2
𝐼,
𝑁−1
∏+∏<
𝑖,𝑗
𝑗,𝑖
= 𝑄𝑖 + 𝑅𝑖 + 𝑇𝑖 + ℎ1 𝑀𝑖 + 𝑁𝑖 +
𝑇
𝑇
+ ℎ2 𝐾𝑖 + 𝐿 𝑖 +
𝑇
𝑇
+ 𝐶𝑖1 + 𝐶𝑖1 + 𝐷𝑖1 + 𝐷𝑖1 + 𝐸𝑖1 + 𝐸𝑖1 + 𝐿2𝑖 𝐴 𝑗
𝑇
𝑖 = 1, . . . , 𝑁 − 1, 𝑗 = 𝑖 + 1, . . . , 𝑁,
[
𝑀𝑖 𝑁𝑖
] ≥ 0,
𝑁𝑖𝑇 𝑌𝑖
𝑖 = 1, 2, . . . , 𝑁,
(14)
𝐾𝑖 𝐿 𝑖
] ≥ 0,
𝐿𝑇𝑖 𝑍𝑖
𝑖 = 1, 2, . . . , 𝑁.
(15)
𝐿𝑇𝑖
Proof. Consider the following parameter-dependent
Lyapunov-Krasovskii function for the system (9) of the form
𝑇
+ 𝐴𝑇𝑖 𝐿2𝑗 + 𝐿2𝑖 𝐵𝑗 + 𝐵𝑖𝑇 𝐿2𝑗 − 𝐿2𝑖 − 𝐿2𝑖 ,
𝐴𝑇𝑖 𝐿1𝑗
+
𝐵𝑖𝑇 𝐿1𝑗
−
𝐿1𝑖
−
𝑇
Σ15
𝑖 = −
+ 𝐸𝑖2 ,
Σ17
𝑖 = −
𝑇
𝐷𝑖1
+ 𝐷𝑖3 ,
Σ18
𝑖,𝑗
𝑇
𝐸𝑖1
𝐸𝑖3
−
+
ℎ22 𝑉𝑖
Σ22
𝑖
= 𝑃𝑖 +
+
ℎ12 𝑈𝑖
𝑇
𝐿2𝑖
5
𝑉 (𝑘) = ∑𝑉𝑖 (𝑘) ,
+ 𝑃𝑖 ,
(16)
𝑖=1
𝑇
2
1
Σ13
𝑖 = 𝐶𝑖 − 𝑁𝑖 − 𝐶𝑖 ,
𝑇
𝐸𝑖1
(12)
(13)
[
𝑁𝑖𝑇
𝑖 = 1, 2, . . . , 𝑁,
𝑖,𝑖
where
= −
𝑇
3
3
Σ99
𝑖,𝑗 = − 𝑋𝑖 − 𝐺𝑖 − 𝐺𝑖 .
Moreover,
=
𝑇
3
3
Σ77
𝑖 = −𝑉𝑖 − 𝐷𝑖 − 𝐷𝑖 ,
3
3
3
𝑇 3
Σ88
𝑖,𝑗 = − 𝑊𝑖 − 𝐸𝑖 − 𝐸𝑖 − 𝐿 𝑖 𝐵𝑗 − 𝐵𝑖 𝐿 𝑗 ,
𝑗
𝑗
𝑗
𝑗 𝑗
𝐶𝑖 , 𝐷𝑖 , 𝐸𝑖 , 𝐺𝑖 , 𝐿 𝑖 , 𝑃𝑖 , 𝑄𝑖 , 𝑅𝑖 , 𝑆𝑖 , 𝑇𝑖 , 𝑈𝑖 ,
Σ12
𝑖,𝑗
𝑇
3
2
Σ58
𝑖 = −𝐸𝑖 − 𝐸𝑖 ,
𝑇
∑𝛼𝑖 = 1, 𝛼𝑖 ≥ 0,
Σ11
𝑖,𝑗
𝑇
2
2
Σ55
𝑖 = − 𝑇𝑖 − 𝐸𝑖 − 𝐸𝑖 ,
𝑁
𝑇
3
2
Σ49
𝑖 = −𝐺𝑖 − 𝐺𝑖 ,
𝑁
𝑍 (𝛼) = ∑𝛼𝑖 𝑍𝑖 ,
𝑖=1
𝑇
2
2
2
2
Σ44
𝑖 = −𝑅𝑖 −𝑆𝑖 −𝐷𝑖 −𝐷𝑖 −𝐺𝑖 −𝐺𝑖 ,
3
2
Σ47
𝑖 = − 𝐷𝑖 − 𝐷𝑖 ,
𝑋 (𝛼) = ∑𝛼𝑖 𝑋𝑖 ,
𝑖=1
𝑇
3
2
Σ36
𝑖 = −𝐶𝑖 − 𝐶𝑖 ,
𝑇
𝑁
𝑊 (𝛼) = ∑𝛼𝑖 𝑊𝑖 ,
𝑇
2
2
1
1
Σ33
𝑖 = − 𝑄𝑖 + 𝑆𝑖 − 𝐶𝑖 − 𝐶𝑖 + 𝐺𝑖 + 𝐺𝑖 ,
𝑇 (𝛼) = ∑𝛼𝑖 𝑇𝑖 ,
2
1
Σ14
𝑖 = 𝐷𝑖 − 𝐿 𝑖 − 𝐷𝑖 ,
Σ16
𝑖 =
𝑇
𝐿2𝑖 𝐵𝑗
+
𝑇
+
𝐴𝑇𝑖 𝐿3𝑗
ℎ22 𝑊𝑖
+ ℎ2 𝑍𝑖 − 𝐿1𝑖 − 𝐿1𝑖 ,
𝑇
−𝐶𝑖1
+ 𝐶𝑖3 ,
+
𝐵𝑖𝑇 𝐿3𝑗
𝑉1 (𝑘) = 𝑥𝑇 (𝑘) 𝑃 (𝛼) 𝑥 (𝑘) ,
−
𝐿3𝑖 ,
2
+ 𝜌 𝑋𝑖 + ℎ1 𝑌𝑖
𝑇
where
1
3
Σ28
𝑖,𝑗 = −𝐿 𝑖 𝐵𝑗 − 𝐿 𝑖 ,
𝑘−1
𝑘−1
𝑖=𝑘−ℎ1
𝑖=𝑘−ℎ2
𝑉2 (𝑘) = ∑ 𝑥𝑇 (𝑖) 𝑄 (𝛼) 𝑥 (𝑖)+ ∑ 𝑥𝑇 (𝑖) 𝑅 (𝛼) 𝑥 (𝑖)
𝑘−ℎ1 −1
𝑘−1
𝑖=𝑘−ℎ2
𝑖=𝑘−ℎ(𝑘)
+ ∑ 𝑥𝑇 (𝑖) 𝑆 (𝛼) 𝑥 (𝑖)+ ∑ 𝑥𝑇 (𝑖) 𝑇 (𝛼) 𝑥 (𝑖) ,
4
Discrete Dynamics in Nature and Society
𝑘−1
−1
𝑘−1
𝑉3 (𝑘) = ℎ1 ∑ ∑ 𝑦𝑇 (𝑖) 𝑈 (𝛼) 𝑦 (𝑖)
+2 ∑ 𝑦(𝑖)𝑇 𝐿𝑇3 (𝛼) [−𝑦 (𝑘)+[𝐴 (𝛼)+𝐵 (𝛼)−𝐼]
𝑗=−ℎ1 𝑖=𝑘+𝑗
𝑖=𝑘−ℎ(𝑘)
𝑘−1
𝑘−1
−1
+ ℎ2 ∑ ∑ 𝑦𝑇 (𝑖) 𝑉 (𝛼) 𝑦 (𝑖)
×𝑥 (𝑘)−𝐵 (𝛼) ∑ 𝑦 (𝑖)] ,
𝑖=𝑘−ℎ(𝑘)
𝑗=−ℎ2 𝑖=𝑘+𝑗
Δ𝑉2 (𝑘) = 𝑉2 (𝑘 + 1) − 𝑉2 (𝑘)
𝑘−1
−1
+ ℎ2 ∑ ∑ 𝑦𝑇 (𝑖) 𝑊 (𝛼) 𝑦 (𝑖) ,
= 𝑥𝑇 (𝑘) 𝑄 (𝛼) 𝑥 (𝑘) − 𝑥𝑇 (𝑘 − ℎ1 ) 𝑄 (𝛼) 𝑥 (𝑘 − ℎ1 )
𝑗=−ℎ2 𝑖=𝑘+𝑗
−ℎ1 −1 𝑘−1
+ 𝑥𝑇 (𝑘) 𝑅 (𝛼) 𝑥 (𝑘) − 𝑥𝑇 (𝑘 − ℎ2 ) 𝑅 (𝛼) 𝑥 (𝑘 − ℎ2 )
𝑗=−ℎ2 𝑖=𝑘+𝑗
+ 𝑥𝑇 (𝑘 − ℎ1 ) 𝑆 (𝛼) 𝑥 (𝑘 − ℎ1 ) − 𝑥𝑇 (𝑘 − ℎ2 )
𝑉4 (𝑘) = 𝜌 ∑ ∑ 𝑦𝑇 (𝑖) 𝑋 (𝛼) 𝑦 (𝑖) ,
𝑉5 (𝑘) =
0
𝑘−1
× 𝑆 (𝛼) 𝑥 (𝑘 − ℎ2 ) + 𝑥𝑇 (𝑘) 𝑇 (𝛼) 𝑥 (𝑘)
∑ 𝑦𝑇 (𝑖) 𝑌 (𝛼) 𝑦 (𝑖)
∑
𝑗=−ℎ1 +1 𝑖=𝑘−1+𝑗
+
0
∑
− 𝑥𝑇 (𝑘 − ℎ (𝑘)) 𝑇 (𝛼) 𝑥 (𝑘 − ℎ (𝑘)) .
𝑘−1
(20)
∑ 𝑦𝑇 (𝑖) 𝑍 (𝛼) 𝑦 (𝑖) .
𝑗=−ℎ2 +1 𝑖=𝑘−1+𝑗
(17)
Evaluating the forward deference of 𝑉(𝑘), it is defined as
From Proposition 4, we have
Δ𝑉3 (𝑘) = 𝑉3 (𝑘 + 1) − 𝑉3 (𝑘)
𝑘−1
Δ𝑉 (𝑘) = 𝑉 (𝑘 + 1) − 𝑉 (𝑘) .
(18)
= ℎ12 𝑦𝑇 (𝑘) 𝑈 (𝛼) 𝑦 (𝑘) − ℎ1 ∑ 𝑦𝑇 (𝑖) 𝑈 (𝛼) 𝑦 (𝑖)
𝑖=𝑘−ℎ1
𝑘−1
+ ℎ22 𝑦𝑇 (𝑘) 𝑉 (𝛼) 𝑦 (𝑘) − ℎ2 ∑ 𝑦𝑇 (𝑖) 𝑉 (𝛼) 𝑦 (𝑖)
Let us define, for 𝑖 = 1, 2, . . . , 5,
𝑖=𝑘−ℎ2
𝑘−1
Δ𝑉𝑖 (𝑘) = 𝑉𝑖 (𝑘 + 1) − 𝑉𝑖 (𝑘) .
+ ℎ22 𝑦𝑇 (𝑘) 𝑊 (𝛼) 𝑦 (𝑘) − ℎ2 ∑ 𝑦𝑇 (𝑖) 𝑊 (𝛼) 𝑦 (𝑖)
(19)
Then along the solution of the system (9), we obtain
𝑖=𝑘−ℎ2
≤
ℎ12 𝑦𝑇 (𝑘) 𝑈 (𝛼) 𝑦 (𝑘)
𝑇
𝑘−1
− ( ∑ 𝑦 (𝑖))
𝑖=𝑘−ℎ1
𝑘−1
Δ𝑉1 (𝑘) = 𝑉1 (𝑘 + 1) − 𝑉1 (𝑘)
= 𝑥𝑇 (𝑘 + 1) 𝑃 (𝛼) 𝑥 (𝑘 + 1) − 𝑥𝑇 (𝑘) 𝑃 (𝛼) 𝑥 (𝑘)
= 2𝑥𝑇 (𝑘) 𝑃 (𝛼) 𝑦 (𝑘) + 𝑦𝑇 (𝑘) 𝑃 (𝛼) 𝑦 (𝑘)
+ 2𝑥𝑇 (𝑘) 𝐿𝑇1 (𝛼) [ − 𝑦 (𝑘) + [𝐴 (𝛼) + 𝐵 (𝛼) − 𝐼]
𝑘−1
× 𝑥 (𝑘) −𝐵 (𝛼) ∑ 𝑦 (𝑖)]
𝑖=𝑘−ℎ(𝑘)
+ 2𝑦𝑇 (𝑘) 𝐿𝑇2 (𝛼) [ − 𝑦 (𝑘) + [𝐴 (𝛼) + 𝐵 (𝛼) − 𝐼]
𝑘−1
×𝑥 (𝑘) − 𝐵 (𝛼) ∑ 𝑦 (𝑖)]
𝑖=𝑘−ℎ(𝑘)
× 𝑈 (𝛼) ( ∑ 𝑦 (𝑖)) + ℎ22 𝑦𝑇 (𝑘) 𝑉 (𝛼) 𝑦 (𝑘)
𝑖=𝑘−ℎ1
𝑇
𝑘−1
𝑘−1
− ( ∑ 𝑦 (𝑖)) 𝑉 (𝛼) ( ∑ 𝑦 (𝑖))
𝑖=𝑘−ℎ2
𝑖=𝑘−ℎ2
𝑘−1
𝑇
+ ℎ22 𝑦𝑇 (𝑘) 𝑊 (𝛼) 𝑦 (𝑘) − ( ∑ 𝑦 (𝑖))
𝑖=𝑘−ℎ(𝑘)
𝑘−1
× 𝑊 (𝛼) ( ∑ 𝑦 (𝑖)) ,
𝑖=𝑘−ℎ(𝑘)
Δ𝑉4 (𝑘) = 𝑉4 (𝑘 + 1) − 𝑉4 (𝑘)
𝑘−ℎ1 −1
= 𝜌2 𝑦𝑇 (𝑘) 𝑋 (𝛼) 𝑦 (𝑘) − 𝜌 ∑ 𝑦𝑇 (𝑖) 𝑋 (𝛼) 𝑦 (𝑖)
𝑖=𝑘−ℎ2
Discrete Dynamics in Nature and Society
5
𝑇
𝑘−ℎ1 −1
2 𝑇
0
× ∑ 𝑦 (𝑘 − 1 + 𝑖)
≤ 𝜌 𝑦 (𝑘) 𝑋 (𝛼) 𝑦 (𝑘) − ( ∑ 𝑦 (𝑖)) 𝑋 (𝛼)
𝑖=−ℎ1 +1
𝑖=𝑘−ℎ2
= ℎ1 𝑥𝑇 (𝑘) 𝑀 (𝛼) 𝑥 (𝑘) + 2𝑥𝑇 (𝑘) 𝑁 (𝛼) 𝑥 (𝑘)
𝑘−ℎ1 −1
× ( ∑ 𝑦 (𝑖)) ,
− 2𝑥𝑇 (𝑘) 𝑁 (𝛼) 𝑥 (𝑘 − ℎ1 ) ,
𝑖=𝑘−ℎ2
(24)
Δ𝑉5 (𝑘) = 𝑉5 (𝑘 + 1) − 𝑉5 (𝑘)
0
= ℎ1 𝑦𝑇 (𝑘) 𝑌 (𝛼) 𝑦 (𝑘) − ∑ 𝑦𝑇 (𝑘 − 1 + 𝑖) 𝑌 (𝛼)
𝑖=−ℎ1 +1
and we conclude that
0
− ∑ 𝑦𝑇 (𝑘 − 1 + 𝑖) 𝑍 (𝛼) 𝑦 (𝑘 − 1 + 𝑖)
× 𝑦 (𝑘 − 1 + 𝑖)
𝑖=−ℎ2 +1
0
≤ ℎ2 𝑥𝑇 (𝑘) 𝐾 (𝛼) 𝑥 (𝑘) + 2𝑥𝑇 (𝑘) 𝐿 (𝛼)
+ℎ2 𝑦𝑇 (𝑘) 𝑍 (𝛼) 𝑦 (𝑘)− ∑ 𝑦𝑇 (𝑘−1+𝑖) 𝑍 (𝛼)
𝑖=−ℎ2 +1
0
× 𝑦 (𝑘 − 1 + 𝑖) .
(21)
We can show that
𝑖=−ℎ2 +1
= ℎ2 𝑥𝑇 (𝑘) 𝐾 (𝛼) 𝑥 (𝑘) + 2𝑥𝑇 (𝑘) 𝐿 (𝛼) 𝑥 (𝑘)
− 2𝑥𝑇 (𝑘) 𝐿 (𝛼) 𝑥 (𝑘 − ℎ2 ) .
0
𝑇
(25)
× ∑ 𝑦 (𝑘 − 1 + 𝑖)
2𝑥 (𝑘) 𝑁 (𝛼) ∑ 𝑦 (𝑘 − 1 + 𝑖)
𝑖=−ℎ1 +1
It is obvious that
0
+ ∑ 𝑦𝑇 (𝑘−1+𝑖) 𝑌 (𝛼) 𝑦 (𝑘−1+𝑖)+ℎ1 𝑥𝑇 (𝑘) 𝑀 (𝛼) 𝑥 (𝑘)
𝑖=−ℎ1 +1
𝑘−1
Υ ≡ 𝑥 (𝑘) − 𝑥 (𝑘 − ℎ1 ) − ∑ 𝑦 (𝑖) = 0,
𝑖=𝑘−ℎ1
𝑇
𝑀 (𝛼) 𝑁 (𝛼)
𝑥 (𝑘)
] [ 𝑇
]
= ∑ [
𝑦 (𝑘 − 1 + 𝑖)
𝑁
(𝛼) 𝑌 (𝛼)
𝑖=−ℎ +1
0
𝑘−1
Φ ≡ 𝑥 (𝑘) − 𝑥 (𝑘 − ℎ2 ) − ∑ 𝑦 (𝑖) = 0,
1
𝑖=𝑘−ℎ2
𝑥 (𝑘)
×[
] ≥ 0.
𝑦 (𝑘 − 1 + 𝑖)
(22)
Ψ ≡ 𝑥 (𝑘) − 𝑥 (𝑘 − ℎ (𝑘)) −
(26)
𝑘−1
∑ 𝑦 (𝑖) = 0,
𝑖=𝑘−ℎ(𝑘)
It is easy to see that
𝑘−ℎ1 −1
Ω ≡ 𝑥 (𝑘 − ℎ1 ) − 𝑥 (𝑘 − ℎ2 ) − ∑ 𝑦 (𝑖) = 0.
0
2𝑥𝑇 (𝑘) 𝐿 (𝛼) ∑ 𝑦 (𝑘 − 1 + 𝑖)
𝑖=𝑘−ℎ2
𝑖=−ℎ1 +1
0
+ ∑ 𝑦𝑇 (𝑘−1+𝑖) 𝑍 (𝛼) 𝑦 (𝑘−1+𝑖)+ℎ2 𝑥𝑇 (𝑘) 𝐾 (𝛼) 𝑥 (𝑘)
The following equations are true for any polytopic matrices
with appropriate dimensions:
𝑖=−ℎ2 +1
=
0
∑ [
𝑖=−ℎ2 +1
𝑇
𝐾 (𝛼) 𝐿 (𝛼)
𝑥 (𝑘)
]
] [ 𝑇
𝑦 (𝑘 − 1 + 𝑖)
𝐿 (𝛼) 𝑍 (𝛼)
×[
[2𝑥𝑇 (𝑘) 𝐶𝑇 (𝛼) + 2𝑥𝑇 (𝑘 − ℎ1 ) 𝐶𝑇 (𝛼)
[2𝑥𝑇 (𝑘) 𝐷𝑇 (𝛼) + 2𝑥𝑇 (𝑘 − ℎ2 ) 𝐷𝑇 (𝛼)
𝑇
− ∑ 𝑦 (𝑘 − 1 + 𝑖) 𝑌 (𝛼) 𝑦 (𝑘 − 1 + 𝑖)
𝑖=−ℎ1 +1
𝑇
(27)
+2( ∑ 𝑦 (𝑖)) 𝐶3𝑇 (𝛼)] × Υ = 0,
𝑖=𝑘−ℎ1
]
(23)
By (22) and (23), we can obtain
2
𝑇
𝑘−1
𝑥 (𝑘)
] ≥ 0.
𝑦 (𝑘 − 1 + 𝑖)
0
1
[
𝑇
≤ ℎ1 𝑥 (𝑘) 𝑀 (𝛼) 𝑥 (𝑘) + 2𝑥 (𝑘) 𝑁 (𝛼)
1
[
𝑘−1
2
𝑇
+2( ∑ 𝑦 (𝑖)) 𝐷3𝑇 (𝛼)] × Φ = 0,
𝑖=𝑘−ℎ2
]
(28)
6
Discrete Dynamics in Nature and Society
[2𝑥𝑇 (𝑘) 𝐸𝑇 (𝛼) + 2𝑥𝑇 (𝑘 − ℎ (𝑘)) 𝐸𝑇 (𝛼)
1
[
2
(29)
𝑇
𝑘−1
+2( ∑ 𝑦 (𝑖)) 𝐸3𝑇 (𝛼)] × Ψ = 0,
𝑖=𝑘−ℎ(𝑘)
]
[2𝑥𝑇 (𝑘 − ℎ1 ) 𝐺𝑇 (𝛼) + 2𝑥𝑇 (𝑘 − ℎ2 ) 𝐺𝑇 (𝛼)
1
[
2
(30)
𝑇
𝑘−ℎ1 −1
+2( ∑ 𝑦 (𝑖)) 𝐺3𝑇 (𝛼)] × Ω = 0.
𝑖=𝑘−ℎ2
]
Δ𝑉 (𝑘) ≤ 𝜉𝑇 (𝑘) ∑ ∑𝛼𝑖 𝛼𝑗 ∏𝜉 (𝑘) ,
𝑖=1 𝑗=1
(31)
𝑖,𝑗
where 𝜉𝑇 (𝑘) = [𝑥(𝑘)𝑇 𝑦(𝑘)𝑇 𝑥(𝑘−ℎ1 )𝑇 𝑥(𝑘−ℎ2 )𝑇 𝑥(𝑘−ℎ(𝑘))𝑇
𝑘−ℎ1 −1 𝑇
𝑘−1
𝑘−1
𝑇
𝑇
𝑇
𝑦 (𝑖)]
∑𝑘−1
∑𝑖=𝑘−ℎ
𝑖=𝑘−ℎ1 𝑦 (𝑖) ∑𝑖=𝑘−ℎ2 𝑦 (𝑖) ∑𝑖=𝑘−ℎ(𝑘) 𝑦(𝑖)
and ∏𝑖,𝑗 is defined in (10). Due to the fact that
we obtain the following identities:
𝑁−1
∑𝑁
𝑖=1
2
𝛼𝑖 = 1,
𝑁−1
𝑖=1
𝑖=1 𝑗=𝑖+1
𝑁
Σ13
0
Σ33
∗
∗
∗
∗
∗
∗
Σ14
0
Σ34
Σ44
∗
∗
∗
∗
∗
Σ15
0
0
0
Σ55
∗
∗
∗
∗
Σ16
0
Σ36
0
0
Σ66
∗
∗
∗
Σ17
0
0
Σ47
0
0
Σ77
∗
∗
Σ18 0
Σ28 0 ]
]
0 Σ18 ]
]
0 Σ18 ]
]
]
Σ18 0 ] ,
]
0 0 ]
]
0 0 ]
]
Σ88 0 ]
∗ Σ99 ]
(37)
𝑇
𝑇
𝑁−1
𝑁
2
(𝑁 − 1) ∑𝛼𝑖2 − 2 ∑ ∑ 𝛼𝑖 𝛼𝑗 = ∑ ∑ [𝛼𝑖 − 𝛼𝑗 ] ≥ 0.
𝑖=1 𝑗=𝑖+1
(33)
𝑇
𝑇
+ 𝐶1 + 𝐶1 + 𝐷1 + 𝐷1 + 𝐸1 + 𝐸1 + 𝐿2 𝐴
𝑇
𝑇
+ 𝐴𝑇 𝐿2 + 𝐿2 𝐵 + 𝐵𝑇 𝐿2 − 𝐿2 − 𝐿2 ,
𝑇
Σ12 = 𝐴𝑇 𝐿1 + 𝐵𝑇 𝐿1 − 𝐿1 − 𝐿2 + 𝑃,
𝑇
𝑇
Σ13 = 𝐶2 − 𝑁 − 𝐶1 ,
Σ14 = 𝐷2 − 𝐿 − 𝐷1 ,
𝑇
𝑁
∑ ∑ 𝛼𝑖 𝛼𝑗 ∏ = ∑𝛼2 ∏ + ∑ ∑ 𝛼𝑖 𝛼𝑖 [∏ + ∏] , (32)
𝑖=1 𝑗=1
𝑖,𝑗
𝑖=1
𝑖,𝑖
𝑖=1 𝑗=𝑖+1
𝑗,𝑖
[ 𝑖,𝑗
]
𝑁
Σ12
Σ22
∗
∗
∗
∗
∗
∗
∗
Σ11 = 𝑄 + 𝑅 + 𝑇 + ℎ1 𝑀 + 𝑁 + 𝑁𝑇 + ℎ2 𝐾 + 𝐿 + 𝐿𝑇
𝑁 𝑁
𝑁
Σ11
[∗
[
[∗
[
[∗
[
[
∏=[∗
[
[∗
[
[∗
[
[∗
[∗
where
It follows from (18)–(30) that
𝑁 𝑁
According to Theorem 5, we have Corollary 6 for the delaydependent asymptotically stability criterion of the system
(35)-(36). We introduce the following notations for later use:
𝑇
Σ15 = − 𝐸1 + 𝐸2 ,
Σ16 = −𝐶1 + 𝐶3 ,
𝑇
Σ17 = − 𝐷1 + 𝐷3 ,
𝑇
𝑇
Σ18 = − 𝐸1 + 𝐸3 − 𝐿2 𝐵 + 𝐴𝑇 𝐿3 + 𝐵𝑇 𝐿3 − 𝐿3 ,
Σ22 = 𝑃 + ℎ12 𝑈 + ℎ22 𝑉 + ℎ22 𝑊 + 𝜌2 𝑋
𝑇
+ ℎ1 𝑌 + ℎ2 𝑍 − 𝐿1 − 𝐿1 ,
By (31)–(33), if the conditions (12)–(15) are true, then
𝑇
Δ𝑉 (𝑘) < −𝜔‖𝑥‖2 ,
(34)
where 𝜔 > 0. This means that the system (1)-(2) is robustly
stable. The proof of the theorem is complete.
If 𝐴(𝛼) = 𝐴 and 𝐵(𝛼) = 𝐵 when 𝐴, 𝐵 ∈ 𝑀𝑛×𝑛 then the
system (1)-(2) reduces to the following system:
𝑥 (𝑘 + 1) = 𝐴𝑥 (𝑘) + 𝐵𝑥 (𝑘 − ℎ (𝑘)) ,
(35)
𝑠 ∈ {−ℎ2 , . . . , −1, 0} .
(36)
𝑥 (𝑠) = 𝜙 (𝑠) ,
Take the Lyapunov-Krasovskii functional as (16), where
𝑃(𝛼) = 𝑃, 𝑄(𝛼) = 𝑄, 𝑅(𝛼) = 𝑅, 𝑆(𝛼) = 𝑆, 𝑇(𝛼) = 𝑇, 𝑈(𝛼) =
𝑈, 𝑉(𝛼) = 𝑉, 𝑊(𝛼) = 𝑊, 𝑋(𝛼) = 𝑋, 𝑌(𝛼) = 𝑌, and 𝑍(𝛼) = 𝑍
when 𝑃, 𝑄, 𝑅, 𝑆, 𝑇, 𝑈, 𝑉, 𝑊, 𝑋, 𝑌, 𝑍 ∈ 𝑀𝑛×𝑛 . Moreover, let us
set polytopic matrices with appropriate dimensions of the
forms 𝐶𝑗 (𝛼) = 𝐶𝑗 , 𝐷𝑗 (𝛼) = 𝐷𝑗 , 𝐸𝑗 (𝛼) = 𝐸𝑗 , 𝐺𝑗 (𝛼) = 𝐺𝑗 ,
and 𝐿 𝑗 (𝛼) = 𝐿𝑗 when 𝐶𝑗 , 𝐷𝑗 , 𝐸𝑗 , 𝐺𝑗 , 𝐿𝑗 ∈ 𝑀𝑛×𝑛 , 𝑗 = 1, 2, 3.
Σ28 = − 𝐿1 𝐵 − 𝐿3 ,
𝑇
𝑇
𝑇
Σ33 = − 𝑄 + 𝑆 − 𝐶2 − 𝐶2 + 𝐺1 + 𝐺1 ,
𝑇
Σ34 = 𝐺2 −𝐺1 ,
𝑇
Σ36 = − 𝐶3 − 𝐶2 ,
Σ39 = 𝐺3 − 𝐺1 ,
𝑇
𝑇
Σ44 = − 𝑅 − 𝑆 − 𝐷2 − 𝐷2 − 𝐺2 − 𝐺2 ,
𝑇
Σ47 = − 𝐷3 − 𝐷2 ,
𝑇
𝑇
Σ49 = − 𝐺3 − 𝐺2 ,
Σ55 = −𝑇 − 𝐸2 − 𝐸2 ,
𝑇
Σ58 = − 𝐸3 − 𝐸2 ,
𝑇
Σ66 = − 𝑈 − 𝐶3 − 𝐶3 ,
𝑇
𝑇
Σ77 = −𝑉 − 𝐷3 − 𝐷3 ,
𝑇
Σ88 = − 𝑊 − 𝐸3 − 𝐸3 − 𝐿3 𝐵 − 𝐵𝑇 𝐿3 ,
𝑇
Σ99 = − 𝑋 − 𝐺3 − 𝐺3 .
(38)
𝑥(𝑘)
Discrete Dynamics in Nature and Society
10
8
6
4
2
0
−2
−4
−6
7
system is robustly stable for discrete delay time satisfying
ℎ1 = 2, ℎ2 = 16, and
4.3970 −0.4104
𝑃1 = 106 × [
],
−0.4104 0.3531
0
5
10
15
20
25
30
𝑘
5.2274 −0.0613
𝑃2 = 105 × [
],
−0.0613 0.3488
4.9438 −1.3988
],
𝑄1 = 104 × [
−1.3988 1.3372
𝑥 1 (𝑘)
𝑥2 (𝑘)
Figure 1: The Simulation solution of the states 𝑥1 (𝑘) and 𝑥2 (𝑘)
in Example 7 for LPD discrete-time delayed system with initial
conditions 𝑥1 (𝑘) = −5.5 and 𝑥2 (𝑘) = 8.5, 𝑘 = −16, . . . , −2, −1, 0
and 𝛼1 = 1/3, 𝛼2 = 2/3 by using method of Runge-Kutta order 4
(ℎ = 0.01) with Matlab.
Corollary 6. The system (35)-(36) is asymptotically stable if
there exist positive definite symmetric matrices 𝑃, 𝑄, 𝑅, 𝑆, 𝑇,
𝑈, 𝑉, 𝑊, 𝑋, 𝑌, and 𝑍, any appropriate dimensional matrices
𝑀, 𝑁, 𝐾, 𝐿, 𝐶𝑗 , 𝐷𝑗 , 𝐸𝑗 , 𝐺𝑗 , and 𝐿𝑗 , 𝑗 = 1, 2, 3, satisfying the
following LMIs:
∏ < 0,
𝑀 𝑁
[ 𝑇 ] ≥ 0,
𝑁 𝑌
(39)
𝐾 𝐿
[ 𝑇 ] ≥ 0.
𝐿 𝑍
1.6042 −0.3173
𝑄2 = 104 × [
],
−0.3173 0.1431
𝑅1 = 104 × [
1.8144 −0.5737
],
−0.5737 0.4286
𝑅2 = 103 × [
4.2669 −0.8605
],
−0.8605 0.3167
𝑆1 = 104 × [
2.3161 −0.7409
],
−0.7409 0.5638
𝑆2 = 103 × [
5.5557 −1.1238
],
−1.1238 0.4007
𝑇1 = 105 × [
2.2862 −0.3474
],
−0.3474 0.2764
𝑇2 = 104 × [
5.0867 0.3198
],
0.3198 0.2709
𝑈1 = 104 × [
9.6266 −0.3736
],
−0.3736 2.6793
𝑈2 = 104 × [
6.7104 −1.2896
],
−1.2896 0.6493
4. Numerical Examples
𝑉1 = 103 × [
2.6354 −0.1810
],
−0.1810 0.2286
Example 7. Consider the following LPD discrete-time system
with interval time-varying delays (1)-(2) with
𝑉2 = 103 × [
1.1098 −0.2116
],
−0.2116 0.0795
0.80 0
],
𝐴1 = [
0.01 0.60
𝐵1 = [
0.10 0
],
0.20 0.10
1.2581 0.0031
],
𝑊1 = 104 × [
0.0031 0.0279
0.90 0
𝐴2 = [
],
0.05 0.90
𝐵2 = [
−0.10 0
],
−0.20 −0.10
(40)
9.8860 0.9947
],
𝑊2 = 103 × [
0.9947 0.4654
𝑋1 = 103 × [
3.6535 −0.2316
],
−0.2316 0.3050
ℎ(𝑘) = 2 + 14cos2 (𝑘𝜋/2) with initial condition 𝜙(𝑘) = [ −5.5
8.5 ],
𝑘 ∈ [−16, 0]. The numerical solutions 𝑥1 (𝑘) and 𝑥2 (𝑘) of (1)(2) with (40) are plotted in Figure 1.
𝑋2 = 103 × [
1.5049 −0.2858
],
−0.2858 0.1068
𝑌1 = 105 × [
1.0677 −0.0668
],
−0.0668 0.3783
Solution. By using the LMI Toolbox in MATLAB (with
accuracy 0.01) and conditions (12)–(15) of Theorem 5, this
𝑌2 = 104 × [
8.4989 −1.5451
],
−1.5451 1.0863
8
Discrete Dynamics in Nature and Society
𝑍1 = 104 × [
4.8154 −0.3746
],
−0.3746 0.6475
𝐺21 = 104 × [
−1.6145 −0.0032
],
−0.0032 −1.6397
𝑍2 = 104 × [
2.5697 −0.4901
],
−0.4901 0.2043
𝐺12 = 104 × [
0.6849 0.2738
],
0.2738 1.3873
−1.5983 −0.1755
],
−0.1755 −1.6231
𝐺22 = 104 × [
1.3546 0.0545
],
0.0545 1.6203
𝐶21
−2.1448 0.1127
= 10 × [
],
0.1127 −1.6904
𝐺13 = 104 × [
1.8742 −0.0945
],
−0.0945 1.7071
𝐶12
1.2837 0.1710
= 10 × [
],
0.1710 1.5002
𝐺23 = 104 × [
1.7014 −0.0146
],
−0.0146 1.6454
𝐶11 = 104 × [
4
4
𝐶22 = 104 × [
𝐶13 = 103 × [
1.6947 −0.0260
],
−0.0260 1.6512
−7.3049 0.2491
],
0.2491 9.7027
−0.1374 0.3389
],
𝐶23 = 104 × [
0.3389 1.4742
−0.9350 −0.1962
𝐷11 = 104 × [
],
−0.1962 −1.4853
−1.7099 0.0229
],
𝐷21 = 104 × [
0.0229 −1.6489
0.5714 0.2950
𝐷12 = 104 × [
],
0.2950 1.3754
1.3928 0.0393
𝐷22 = 104 × [
],
0.0393 1.6269
1.9066 −0.0891
],
𝐷13 = 104 × [
−0.0891 1.7055
1.7454 −0.0290
𝐷23 = 104 × [
],
−0.0290 1.6519
−4.2910 −1.6701
],
𝐸11 = 104 × [
−1.6701 −1.5104
−3.7600 −0.2382
],
𝐸21 = 104 × [
−0.2382 −1.7532
−7.0237 0.5763
𝐸12 = 104 × [
],
0.5763 0.8408
−0.7534 −0.1269
],
𝐸22 = 104 × [
−0.1269 1.4957
1.0553 −0.0488
𝐸13 = 105 × [
],
−0.0488 0.2403
3.8446 0.2721
],
𝐸23 = 104 × [
0.2721 1.7426
𝐺11 = 104 × [
−1.1174 −0.1492
],
−0.1492 −1.4929
𝐿11 = 106 × [
5.3779 −0.2824
],
−0.2824 0.4796
𝐿12 = 106 × [
2.3117 0.0294
],
0.0294 0.3636
𝐿21 = 106 × [
4.0552 −0.3696
],
−0.3696 0.2311
𝐿22 = 104 × [
9.4267 0.9110
],
0.9110 7.4877
𝐿31 = 105 × [
−2.9313 −0.2209
],
−0.2209 −0.4362
𝐿32 = 105 × [
2.1784 −0.0837
],
−0.0837 −0.1597
𝐾1 = [
897.9190 −266.0787
],
−266.0787 187.1557
𝐾2 = [
236.7231 −47.9750
],
−47.9750 19.2914
𝑀1 = 103 × [
6.1473 −1.9404
],
−1.9404 1.4158
𝑀2 = 103 × [
1.6184 −0.3310
],
−0.3310 0.1305
𝑁1 = 103 × [
−5.4598 0.9628
],
0.9628 −2.2399
𝑁2 = 103 × [
−3.9963 0.7762
],
0.7762 −0.4562
𝐿 1 = 103 × [
−2.4982 0.3102
],
0.3102 −0.3738
𝐿 2 = 103 × [
−1.1912 0.2324
].
0.2324 −0.0945
(41)
For the given ℎ1 , Table 1 lists the comparison of the upper
bounds delay ℎ2 for robust stability of the system (1)-(2) with
(40) by the different method. By a conditions in Theorem 5,
we can see from Table 1 that our result is superior to those in
[22, Theorem 1].
Discrete Dynamics in Nature and Society
9
Table 1: Comparison of the maximum allowed time delay ℎ2 for
different conditions.
ℎ1
Liu et al. [10] (2006)
Zhang et al. [22] (2010)
Theorem 5
2
2
12
>15
4
4
13
>16
5
5
14
>17
6
6
15
>18
Table 2: Comparison of the maximum allowed time delay ℎ2 for
different conditions.
ℎ1
Fridman and Shaked [6] (2005)
Gao and Chen [1](2007)
Zhang et al. [22] (2010)
Zhang et al. [23] (2011)
Corollary 6
7
13
14
15
15
>18
10
15
15
17
17
>19
15
19
18
21
21
>24
20
23
22
25
25
>27
Example 8. Consider the system (35) with
𝐴=[
0.80 0
],
0.05 0.90
𝐵=[
−0.10 0
].
−0.20 −0.10
(42)
For the given ℎ1 , we calculate the allowable maximum value
of ℎ2 that guarantees the asymptotic stability of the system
(35) with (42). By using different methods, the calculated
results are presented in Table 2. From the table, we can see
that Corollary 6 in this paper provides the less conservative
results.
Acknowledgments
This work is supported by National Research Council of
Thailand and Khon Kaen University, Thailand (Grant no. kku
fmis 121740).
References
[1] H. Gao and T. Chen, “New results on stability of discrete-time
systems with time-varying state delay,” Institute of Electrical and
Electronics Engineers, vol. 52, no. 2, pp. 328–334, 2007.
[2] T. Botmart and P. Niamsup, “Robust exponential stability
and stabilizability of linear parameter dependent systems with
delays,” Applied Mathematics and Computation, vol. 217, no. 6,
pp. 2551–2566, 2010.
[3] B. T. Cui and M. G. Hua, “Robust passive control for uncertain
discrete-time systems with time-varying delays,” Chaos, Solitons
and Fractals, vol. 29, no. 2, pp. 331–341, 2006.
[4] M. Fang, “Delay-dependent stability analysis for discrete singular systems with time-varying delays,” Acta Automatica Sinica,
vol. 36, no. 5, pp. 751–755, 2010.
[5] E. Fridman and U. Shaked, “Parameter dependent stability
and stabilization of uncertain time-delay systems,” Institute of
Electrical and Electronics Engineers, vol. 48, no. 5, pp. 861–866,
2003.
[6] E. Fridman and U. Shaked, “Stability and guaranteed cost control of uncertain discrete delay systems,” International Journal
of Control, vol. 78, no. 4, pp. 235–246, 2005.
[7] K. Gu, V. L. Kharitonov, and J. Chen, Stability of Time-Delay
Systems, Birkhauser, Berlin, Germany, 2003.
[8] Y. He, M. Wu, J.-H. She, and G.-P. Liu, “Parameter-dependent
Lyapunov functional for stability of time-delay systems with
polytopic-type uncertainties,” Institute of Electrical and Electronics Engineers, vol. 49, no. 5, pp. 828–832, 2004.
[9] X. Jiang, Q. L. Han, and X. Yu, “Stability criteria for linear
discrete-time systems with interval-like time-varying delay,” in
Proceedings of the American Control Conference (ACC ’05), pp.
2817–2822, June 2005.
[10] X. G. Liu, R. R. Martin, M. Wu, and M. L. Tang, “Delaydependent robust stabilisation of discrete-time systems with
time-varying delay,” IEE Proceedings: Control Theory and Applications, vol. 153, no. 6, pp. 689–702, 2006.
[11] W.-J. Mao, “An LMI approach to 𝐷-stability and 𝐷-stabilization
of linear discrete singular system with state delay,” Applied
Mathematics and Computation, vol. 218, no. 5, pp. 1694–1704,
2011.
[12] W.-J. Mao and J. Chu, “𝐷-stability and 𝐷-stabilization of
linear discrete time-delay systems with polytopic uncertainties,”
Automatica, vol. 45, no. 3, pp. 842–846, 2009.
[13] K. Mukdasai, “Robust exponential stability for LPD discretetime system with interval time-varying delay,” Journal of Applied
Mathematics, vol. 2012, Article ID 237430, 13 pages, 2012.
[14] K. Mukdasai and P. Niamsup, “Robust stability of discrete-time
linear parameter dependent system with delay,” Thai Journal of
Mathematics (Annual Meeting in Mathematics), vol. 8, no. 4,
Special issue, pp. 11–20, 2010.
[15] P. T. Nam, H. M. Hien, and V. N. Phat, “Asymptotic stability
of linear state-delayed neutral systems with polytope type
uncertainties,” Dynamic Systems and Applications, vol. 19, no. 1,
pp. 63–72, 2010.
[16] P. Niamsup and V. N. Phat, “𝐻∞ control for nonlinear timevarying delay systems with convex polytopic uncertainties,”
Nonlinear Analysis: Theory, Methods & Applications, vol. 72, no.
11, pp. 4254–4263, 2010.
[17] V. N. Phat and P. T. Nam, “Exponential stability and stabilization of uncertain linear time-varying systems using parameter
dependent Lyapunov function,” International Journal of Control,
vol. 80, no. 8, pp. 1333–1341, 2007.
[18] M. De la Sen, “Robust stabilization of a class of polytopic
linear time-varying continuous systems under point delays and
saturating controls,” Applied Mathematics and Computation,
vol. 181, no. 1, pp. 73–83, 2006.
[19] S. Udpin and P. Niamsup, “Robust stability of discrete-time LPD
neural networks with time-varying delay,” Communications in
Nonlinear Science and Numerical Simulation, vol. 14, no. 11, pp.
3914–3924, 2009.
[20] Z. G. Wu, P. Shi, H. Su, and J. Chu, “Delay-dependent exponential stability analysis for discrete-time switched neural networks
with time-varying delay,” Neurocomputing, vol. 74, no. 10, pp.
1626–1631, 2011.
[21] Y. Xia and Y. Jia, “Robust stability functionals of state delayed
systems with polytopic type uncertainties via parameterdependent Lyapunov functions,” International Journal of Control, vol. 75, no. 16-17, pp. 1427–1434, 2002.
[22] W. Zhang, Q. Y. Xie, X. S. Cai, and Z. Z. Han, “New stability criteria for discrete-time systems with interval time-varying delay
and polytopic uncertainty,” Latin American Applied Research,
vol. 40, no. 2, pp. 119–124, 2010.
10
[23] W. Zhang, J. Wang, Y. Liang, and Z. Z. Han, “Improved delayrange-dependent stability criterion for discrete-time systems
with interval time-varying delay,” Journal of Information and
Computational Science, vol. 14, no. 8, pp. 3321–3328, 2011.
[24] W.-A. Zhang and L. Yu, “Stability analysis for discrete-time
switched time-delay systems,” Automatica, vol. 45, no. 10, pp.
2265–2271, 2009.
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