Hindawi Publishing Corporation
Journal of Applied Mathematics
Volume 2013, Article ID 367457, 16 pages
http://dx.doi.org/10.1155/2013/367457
Research Article
Delay-Dependent Synchronization for
Complex Dynamical Networks with Interval Time-Varying
and Switched Coupling Delays
T. Botmart1,2 and P. Niamsup2,3,4
1
Department of Mathematics, Faculty of Science, Srinakharinwirot University, Bangkok 10110, Thailand
Center 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
4
Materials Science Research Center, Faculty of Science, Chiang Mai University, Chiang Mai 50200, Thailand
2
Correspondence should be addressed to P. Niamsup; piyapong.n@cmu.ac.th
Received 29 December 2012; Accepted 31 January 2013
Academic Editor: Xinzhi Liu
Copyright © 2013 T. Botmart and P. Niamsup. 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 local exponential synchronization for complex dynamical networks with interval time-varying delays in the
dynamical nodes and the switched coupling term simultaneously. The constraint on the derivative of the time-varying delay
is not required which allows the time delay to be a fast time-varying function. By using common unitary matrix for different
subnetworks, the problem of synchronization is transformed into the stability analysis of some linear switched delay systems. Then,
when subnetworks are synchronizable and nonsynchronizable, a delay-dependent sufficient condition is derived and formulated
in the form of linear matrix inequalities (LMIs) by average dwell time approach and piecewise Lyapunov-Krasovskii functionals
which are constructed based on the descriptor model of the system and the method of decomposition. The new stability condition
is less conservative and is more general than some existing results. A numerical example is also given to illustrate the effectiveness
of the proposed method.
1. Introduction
Complex dynamical network, as an interesting subject, has
been thoroughly investigated for decades. These networks
show very complicated behavior and can be used to model
and explain many complex systems in nature such as computer networks [1], the World Wide Web [2], food webs
[3], cellular and metabolic networks [4], social networks
[5], and electrical power grids [6]. In general, a complex
network is a large set of interconnected nodes, in which
a node is a fundamental unit with specific contents. As
an implicit assumption, these networks are described by
the mathematical term graph. In such graphs, each vertex
represents an individual element in the system, while edges
represent the relationship between them. Two nodes are
joined by an edge if and only if they interact.
In the last decade, the synchronization of complex
dynamical networks has attracted much attention by several
researchers in this field [7–12]. Synchronization of complex
dynamical networks is one of the most important dynamical mechanisms for creating order in complex dynamical
networks. Wang and Chen introduced a uniform dynamical
network model and also investigated its synchronization [10,
11, 13]. They have shown that the synchronizability of a scalefree dynamical network is robust against random removal
of nodes and yet is fragile to specific removal of the most
highly connected nodes [11]. Li and Chen [7] considered the
synchronization of complex dynamical network models with
coupling delays (continuous and discrete-time) and derived
delay-independent and delay-dependent synchronization
conditions. By utilizing Lyapunov functional method, Yue
and Li [12] studied the synchronization of continuous
2
and discrete complex dynamical networks with interval
time-varying delays in the dynamical nodes and the coupling
term simultaneously in which delay-dependent synchronization conditions are derived in the form of linear matrix
inequalities (LMIs). In [8], Li et al. studied the synchronization problem of general complex dynamical networks
with time-varying delays in the network couplings and in
the dynamical nodes, respectively. However, the time-varying
delays are required to be differentiable.
It is well known that the existence of time delay in
a system may cause instability and oscillations. Examples
of time-delay systems are chemical engineering systems,
biological modeling, electrical networks, physical networks,
and many others, [14–20]. The stability criteria for systems
with time delays can be classified into two categories: delayindependent and delay-dependent. Delay-independent criteria do not employ any information on the size of the
delay, while delay-dependent criteria make use of such
information at different levels. Delay-dependent stability
conditions are generally less conservative than the delayindependent ones especially when the delay is small [20].
Recently, delay-dependent stability for interval time-varying
delay was investigated in [12, 15–17, 19]. Interval time-varying
delay is a time delay that varies in an interval in which the
lower bound is not restricted to be 0. Jiang and Han [17]
considered the problem of robust 𝐻∞ control for uncertain
linear systems with interval time-varying delay based on
the Lyapunov functional approach in which restriction on
the differentiability of the interval time-varying delay was
removed. Shao [19] presented a new delay-dependent stability
criterion for linear systems with interval time-varying delay
without introducing any free-weighting matrices. In order
to reduce further the conservatism introduced by descriptor
model transformation and bounding techniques, a freeweighting matrices method is proposed in [15, 21–24]. An
advantage of this transformation is to transform the original
system to an equivalent descriptor representation without
introducing additional dynamics in the sense defined in
[21]. In [22], based on the descriptor model transformation
and decomposition technique of coefficient matrix, some
new robust stability criteria are derived. In [12], the synchronization problem has been investigated for continuous/discrete complex dynamical networks with interval timevarying delays. Based on piecewise analysis method and the
Lyapunov functional method, some new delay-dependent
synchronization criteria are derived in the form of LMIs by
introducing free-weighting matrices. It will be pointed out
later that some existing results require more free-weighting
matrix variables than our result.
Switched systems are special class of hybrid systems
which consist of a family of continuous-time or discretetime subsystems and a switching law that orchestrates the
switching between them. Such systems have drawn considerable attention in control and computer communities in
the last decade [25–31]. In [32], some stability properties of
switched linear systems consisting of both stable and unstable
subsystems have been derived by using an average dwell
time approach and piecewise Lyapunov functions. It was
shown that when the average dwell time is sufficiently large
Journal of Applied Mathematics
and the total activation time of the unstable subsystems is
relatively small compared with that of stable subsystems, then
global exponential stability is guaranteed. Liu and Zhao [9]
investigated synchronization of complex delayed networks
with switching topology via switched system stability theory.
In [9], when all subnetworks are synchronizable, a delaydependent synchronization condition is given in terms of
LMIs which guarantees the solvability of the synchronization problem under an average dwell time scheme. Liu
et al. [33] studied the local and global exponential synchronization of a complex dynamical network with switching topology and time-varying coupling delays. However,
the time-varying delays in the dynamical nodes were not
considered.
In this paper, we will investigate the local exponential synchronization of complex dynamical networks with
interval time-varying delays in both dynamical nodes and
in switched coupling terms simultaneously. The restriction
on the differentiability of interval time-varying delays is
removed, which means that a fast time-varying delay is
allowed. We use a common unitary matrix for different
subnetworks to transform the synchronization problem of
the network into the stability analysis of 𝑁 − 1 pieces of
linear switched systems. Then by using the average dwell
time approach, the descriptor model transformation, and the
decomposition technique of coefficient matrix, a new class of
piecewise Lyapunov-Krasovskii functionals are constructed
in order to get improved delay-dependent synchronization
criteria which are derived in the form of LMIs. Finally,
numerical examples are given to demonstrate that the derived
conditions are less conservative than some existing results
given in the literature.
The rest of the paper is organized as follows. In Section 2,
a complex dynamical network model with interval timevarying and switched coupling delays as well as some useful
lemmas is given. In Section 3, common unitary matrix and
descriptor system transformation strategy are introduced
to reformulate the network model. Then delay-dependent
synchronization criteria are presented based on this novel
model. Numerical examples that illustrated the obtained
results are given in Section 4. The paper ends with conclusions in Section 5 and cited references.
2. Network Model and
Mathematic Preliminaries
The following notation will be used in this paper: 𝑅𝑛
denotes the 𝑛-dimensional Euclidean space; 𝜆 min (⋅) and
𝜆 max (⋅) denote the minimum and maximum eigenvalue of
a real symmetric matrix, respectively; ‖ ⋅ ‖ denotes the
Euclidean vector norm of 𝑥; 𝐴𝑇 denotes the transpose of the
vector/matrix 𝐴; 𝐴 is symmetric if 𝐴 = 𝐴𝑇 ; 𝐼 denotes the
̇ + 𝜃)‖}.
identity matrix; ‖ ⋅ ‖cl = sup−ℎ𝑀 ≤𝜃≤0 {‖𝑥(𝑡 + 𝜃)‖, ‖𝑥(𝑡
The notation diag{. . .} stands for a block-diagonal matrix.
Journal of Applied Mathematics
3
Consider a complex dynamical switched network consisting of 𝑁 identical coupled nodes, with each node being an
𝑛-dimensional dynamical system as
𝑥̇ 𝑖 (𝑡) = 𝑓 (𝑥𝑖 (𝑡) , 𝑥𝑖 (𝑡 − ℎ (𝑡)))
𝑁
+ 𝑐𝜎(𝑡) ∑ 𝑔𝑖𝑗𝜎(𝑡) 𝐴 𝜎(𝑡) 𝑥𝑗 (𝑡 − ℎ (𝑡)) ,
(1)
𝑗=1
𝑥 (𝑡) = 𝜙 (𝑡) ,
where 𝑥𝑖 (𝑡) = (𝑥𝑖1 (𝑡), 𝑥𝑖2 (𝑡), . . . , 𝑥𝑖𝑛 (𝑡))𝑇 ∈ 𝑅𝑛 is the state
vector of 𝑖th node, for all 𝑖 ∈ {1, 2, . . . , 𝑁} are the state of node
𝑖th; 𝑓(⋅) ∈ 𝑅𝑛 is the continuously differentiable function;
𝜎(𝑡) : [0, +∞) → 𝑀 = {1, 2, . . . , 𝑚} is piecewise constant
and left continuous and is called a switching signal; the
constant 𝑐𝜎(𝑡) > 0 is the coupling strength; 𝐴 𝜎(𝑡) = (𝑎𝑖𝑗 )𝜎(𝑡)
𝑛×𝑛 ∈
𝑅𝑛×𝑛 is a constant inner-coupling matrix, if some pairs (𝑖, 𝑗),
1 ≤ 𝑖, 𝑗 ≤ 𝑛, with 𝑎𝑖𝑗𝜎(𝑡) ≠ 0 which means two coupled nodes
are linked through their 𝑖th and 𝑗th state variables, otherwise
𝑁×𝑁
is the outer-coupling
𝑎𝑖𝑗𝜎(𝑡) = 0; 𝐺𝜎(𝑡) = (𝑔𝑖𝑗 )𝜎(𝑡)
𝑁×𝑁 ∈ 𝑅
matrix of the network, in which 𝑔𝑖𝑗𝜎(𝑡) is defined as follows: if
there is a connection between node 𝑖 and node 𝑗 (𝑗 ≠ 𝑖), then
𝑔𝑖𝑗𝜎(𝑡) = 𝑔𝑗𝑖𝜎(𝑡) = 1; otherwise, 𝑔𝑖𝑗𝜎(𝑡) = 𝑔𝑗𝑖𝜎(𝑡) = 0 (𝑗 ≠ 𝑖), and the
diagonal elements of matrix 𝐺 are defined by
𝑁
𝑁
𝑗=1,𝑖 ≠ 𝑗
𝑗=1,𝑖 ≠ 𝑗
𝑖 = 1, 2, . . . , 𝑁.
Definition 3 (see [12]). The delayed dynamical switched
network (1) is said to achieve asymptotic synchronization if
𝑥1 (𝑡) = 𝑥2 (𝑡) = ⋅ ⋅ ⋅ = 𝑠 (𝑡)
𝑡 ∈ [𝑡0 − ℎ𝑀, 𝑡0 ] , ℎ𝑀 ≥ 0,
𝑔𝑖𝑖𝜎(𝑡) = − ∑ 𝑔𝑖𝑗𝜎(𝑡) = − ∑ 𝑔𝑗𝑖𝜎(𝑡) ,
Definition 2 (see [26]). For any 𝑇2 > 𝑇1 ≥ 0, let 𝑁𝜎(𝑡) (𝑇1 , 𝑇2 )
denote the number of switching of 𝜎(𝑡) over (𝑇1 , 𝑇2 ). If
𝑁𝜎(𝑡) (𝑇1 , 𝑇2 ) ≤ 𝑁0 + ((𝑇2 − 𝑇1 )/𝑇𝑎 ) holds for 𝑇𝑎 > 0, 𝑁0 ≥ 0,
then 𝑇𝑎 is called average dwell time and 𝑁0 is called a chatter
bound.
(2)
It is assumed that switched network (1) is connected in the
sense that there are no isolated clusters; that is namely, 𝐺𝜎(𝑡)
is an irreducible matrix. The time delay ℎ(𝑡) is a time-varying
continuous function satisfying
(3)
The initial condition function 𝜙(𝑡) is a continuous vectorvalued function for 𝑡 ∈ [𝑡0 − ℎ𝑀, 𝑡0 ].
Corresponding to the switching signal 𝜎(𝑡), we have the
switching sequence Σ = {𝑥𝑡0 ; (𝑖0 , 𝑡0 ), (𝑖1 , 𝑡1 ), . . ., (𝑖𝑘 , 𝑡𝑘 ), . . . , |
𝑖𝑘 ∈ 𝑀}, and when 𝑡 ∈ [𝑡𝑘 , 𝑡𝑘+1 ), the 𝑖𝑘 -th subnetwork is
activated. Assume that there are no jumps in the state at the
switching instants, and every network in (1) is connected; that
is, 𝑘 ∈ 𝑀, 𝐺𝑘 is irreducible.
Remark 1. For fixed 𝜎(𝑡), the system model (1) turns into the
delayed complex dynamical network proposed by Yue and Li
[12] as
𝑁
𝑥̇ 𝑖 (𝑡) = 𝑓 (𝑥𝑖 (𝑡) , 𝑥𝑖 (𝑡 − ℎ (𝑡))) + 𝑐 ∑𝑔𝑖𝑗 𝐴𝑥𝑗 (𝑡 − ℎ (𝑡)) ,
𝑗=1
(4)
𝑖 = 1, 2, . . . , 𝑁.
Therefore, (1) is a general complex networks model, with (4)
as a special case.
(5)
where 𝑠(𝑡) is a solution of an isolated node satisfying
𝑠̇ (𝑡) = 𝑓 (𝑠 (𝑡) , 𝑠 (𝑡 − ℎ (𝑡))) .
(6)
Definition 4 (see [34]). If the matrix 𝐴 ∈ 𝑀𝑛×𝑛 is similar to a
diagonal matrix, then 𝐴 is said to be diagonalizable.
In this paper, we assume that 𝑠(𝑡) is an orbitally stable
solution of the above system. Clearly, the stability of the
synchronized states (5) of network (1) is determined by the
dynamics of the isolated node, the coupling strength 𝑐𝜎(𝑡) , the
inner-coupling matrix 𝐴 𝜎(𝑡) , the outer-coupling matrix 𝐺𝜎(𝑡) ,
and the fast time-varying delays ℎ(𝑡).
The following lemmas will be used in the proof of the
main result.
Lemma 5 (see [34]). Let G be a family of diagonalizable
matrices. Then G is a commuting family (under multiplication)
if and only if it is a simultaneously diagonalizable family.
Lemma 6 (see [14]). For any constant symmetric matrix 𝑀 ∈
𝑅𝑛×𝑛 , 𝑀 = 𝑀𝑇 > 0, 0 ≤ ℎ𝑚 ≤ ℎ(𝑡) ≤ ℎ𝑀, 𝑡 ≥ 0, and any
differentiable vector function 𝑥(𝑡) ∈ 𝑅𝑛 , we have
𝑡
(a) [∫
𝑡−ℎ𝑚
0 ≤ ℎ𝑚 ≤ ℎ (𝑡) ≤ ℎ𝑀.
as 𝑡 󳨀→ ∞,
𝑇
𝑥̇ (𝑠) 𝑑𝑠] 𝑀 [∫
𝑡−ℎ𝑚
≤ ℎ𝑚 ∫
𝑡
𝑡−ℎ𝑚
(b) [∫
𝑡−ℎ𝑚
𝑡−ℎ(𝑡)
𝑡
𝑥̇ (𝑠) 𝑑𝑠]
𝑥̇ 𝑇 (𝑠) 𝑀𝑥̇ (𝑠) 𝑑𝑠,
𝑇
𝑥̇ (𝑠) 𝑑𝑠] 𝑀 [∫
𝑡−ℎ𝑚
𝑡−ℎ(𝑡)
≤ (ℎ (𝑡) − ℎ𝑚 ) ∫
𝑡−ℎ𝑚
𝑡−ℎ(𝑡)
≤ (ℎ𝑀 − ℎ𝑚 ) ∫
𝑡−ℎ𝑚
𝑡−ℎ(𝑡)
𝑥̇ (𝑠) 𝑑𝑠]
(7)
𝑥̇ 𝑇 (𝑠) 𝑀𝑥̇ (𝑠) 𝑑𝑠
𝑥̇ 𝑇 (𝑠) 𝑀𝑥̇ (𝑠) 𝑑𝑠.
3. Synchronization of Switched Network
In this section, we will obtain some delay-dependent synchronization criteria for general complex dynamical network
with interval time-varying delay and switched coupling
̃ 𝑖𝜎(𝑡) =
̃𝑖 = 𝐽(𝑡), 𝐵
delays (1) by strict LMI approaches. Let 𝐴
󸀠
𝐽ℎ (𝑡 − ℎ(𝑡)) + 𝑐𝜎(𝑡) 𝜆 𝑖𝜎(𝑡) 𝐴 𝜎(𝑡) , 𝐽(𝑡) = 𝑓 (𝑠(𝑡), and 𝑠(𝑡 −
ℎ(𝑡))) ∈ 𝑅𝑛×𝑛 is the Jacobian of 𝑓(𝑥(𝑡), 𝑥(𝑡 − ℎ(𝑡))) at 𝑠(𝑡)
with the derivative of 𝑓(𝑥(𝑡), 𝑥(𝑡 − ℎ(𝑡))) with respect to 𝑥(𝑡).
4
Journal of Applied Mathematics
𝐽ℎ (𝑡 − ℎ(𝑡)) = 𝑓󸀠 (𝑠(𝑡), 𝑠(𝑡 − ℎ(𝑡))) ∈ 𝑅𝑛×𝑛 is the Jacobian
of 𝑓(𝑥(𝑡), 𝑥(𝑡 − ℎ(𝑡))) at 𝑠(𝑡 − ℎ(𝑡)) with the derivative of
𝑓(𝑥(𝑡), 𝑥(𝑡 − ℎ(𝑡))) with respect to 𝑥(𝑡 − ℎ(𝑡)).
Lemma 7. Consider the switched coupling delays dynamical
network described by (1). Let 0 = 𝜆 1𝜎(𝑡) > 𝜆 2𝜎(𝑡) ≥ 𝜆 3𝜎(𝑡) ≥
⋅ ⋅ ⋅ ≥ 𝜆 𝑁𝜎(𝑡) be the eigenvalues of the outer-coupling matrix
𝐺𝜎(𝑡) . If the 𝑛-dimensional linear switched system
̃𝑖 𝑧𝑖 (𝑡) + 𝐵
̃ 𝑖𝜎(𝑡) 𝑧𝑖 (𝑡 − ℎ (𝑡)) ,
𝑧̇ 𝑖 (𝑡) = 𝐴
Proof. To investigate the stability of the synchronized states
(5), set
𝑖 = 1, 2, . . . , 𝑁.
(9)
Substituting (9) into (1), for 1 ≤ 𝑖 ≤ 𝑁, we have
𝑒𝑖̇ (𝑡) = 𝑥̇ 𝑖 (𝑡) − 𝑠̇ (𝑡)
= 𝑓 (𝑥𝑖 (𝑡) , 𝑥𝑖 (𝑡 − ℎ (𝑡))) − 𝑓 (𝑠 (𝑡) , 𝑠 (𝑡 − ℎ (𝑡)))
𝑁
(10)
+ 𝑐𝜎(𝑡) ∑ 𝑔𝑖𝑗𝜎(𝑡) 𝐴 𝜎(𝑡) 𝑒𝑗 (𝑡 − ℎ (𝑡)) .
𝑗=1
Since 𝑓(⋅) is continuous differentiable, it is easy to show that
the origin of the nonlinear system (1) is an asymptotically
stable equilibrium point if it is an asymptotically stable
equilibrium point of the following linear switched timevarying delays systems:
𝑒𝑖̇ (𝑡) = 𝐽 (𝑡) 𝑒𝑖 (𝑡) + 𝐽ℎ (𝑡) 𝑒𝑖 (𝑡 − ℎ (𝑡))
(11)
+ 𝑐𝜎(𝑡) 𝐴 𝜎(𝑡) (𝑒1 (𝑡 − ℎ (𝑡)) , . . . , 𝑒𝑁 (𝑡 − ℎ (𝑡)))
𝑛×𝑛
where 𝐽(𝑡) = 𝑓 (𝑠(𝑡), 𝑠(𝑡 − ℎ(𝑡))) ∈ 𝑅
is the Jacobian of
𝑓(𝑥(𝑡), 𝑥(𝑡 − ℎ(𝑡))) at 𝑠(𝑡) and 𝐽ℎ (𝑡 − ℎ(𝑡)) = 𝑓󸀠 (𝑠(𝑡), 𝑠(𝑡 −
ℎ(𝑡))) ∈ 𝑅𝑛×𝑛 is the Jacobian of 𝑓(𝑥(𝑡), 𝑥(𝑡−ℎ(𝑡))) at 𝑠(𝑡−ℎ(𝑡)).
Letting 𝑒(𝑡) = (𝑒1 (𝑡), . . . , 𝑒𝑁(𝑡)) ∈ 𝑅𝑛×𝑁 and 𝑒(𝑡 − ℎ(𝑡)) =
(𝑒1 (𝑡 − ℎ(𝑡)), . . . , 𝑒𝑁(𝑡 − ℎ(𝑡))) ∈ 𝑅𝑛×𝑁, we have
𝑇
+ 𝑐𝜎(𝑡) 𝐴 𝜎(𝑡) 𝑒 (𝑡 − ℎ (𝑡)) 𝐺𝜎(𝑡)
.
𝑖 = 1, 2, . . . , 𝑁, (15)
̃ 𝑖𝜎(𝑡) = 𝐽ℎ (𝑡 − ℎ(𝑡)) + 𝑐𝜎(𝑡) 𝜆 𝑖𝜎(𝑡) 𝐴 𝜎(𝑡) .
̃𝑖 = 𝐽(𝑡) and 𝐵
where 𝐴
Thus, we have transformed the stability problem of the
synchronized state (5) to the stability problem of the 𝑁
pieces of 𝑛-dimensional linear switched time-varying delays
differential equations (15). Note that 𝜆 1𝑘 = 0 corresponding
to the synchronization of the system states (5), where the
state 𝑠(𝑡) is an orbitally stable solution of the isolated node
as assumed above in (5). If the following 𝑁 − 1 pieces of 𝑛dimensional linear switched time-varying delays systems
̃𝑖 𝑧𝑖 (𝑡) + 𝐵
̃ 𝑖𝜎(𝑡) 𝑧𝑖 (𝑡 − ℎ (𝑡)) ,
𝑧̇ 𝑖 (𝑡) = 𝐴
𝑖 = 2, 3, . . . , 𝑁, (16)
are asymptotically (or exponentially) stable, then 𝑒(𝑡) will
tend to the origin asymptotically (or exponentially), which
is equivalent to the fact that the synchronized states (5) are
asymptotically (or exponentially) stable. This completes the
proof.
𝑖 = 2, 3, . . . , 𝑁.
(12)
̃𝑖 𝑧𝑖 (𝑡) + 𝐵
̃ 𝑖 𝑧𝑖 (𝑡 − ℎ (𝑡)) ,
𝑦𝑖 (𝑡) = 𝐴
(18)
𝑖 = 2, 3, . . . , 𝑁.
(19)
To derive delay-dependent stability conditions, which include
the information of the time-varying delay ℎ(𝑡), one usually
uses the fact
𝑡
𝑧 (𝑡 − ℎ (𝑡)) = 𝑧 (𝑡) − ∫
𝑡
= 𝑧 (𝑡) − ∫
(13)
(17)
We rewrite system (17) in the following equivalent descriptor
system form:
𝑡−ℎ(𝑡)
Obviously, 𝐺𝑘 is diagonalizable for fixed 𝜎(𝑡) = 𝑘 ∈ 𝑀. If 𝐺𝑘 ,
𝐺𝑙 , for all 𝑘, and 𝑙 ∈ 𝑀 commute pairwise;, that is, 𝐺𝑘 𝐺𝑙 =
𝐺𝑙 𝐺𝑘 , then based on Lemma 5, one can get a common unitary
matrix 𝑈 ∈ 𝑅𝑁×𝑁 with 𝑢𝑖 ∈ 𝑅𝑁 such that
𝑈𝑇 𝐺𝑘 𝑈 = Γ𝑘 ,
̃𝑖 𝑧𝑖 (𝑡) + 𝐵
̃ 𝑖𝜎(𝑡) 𝑧𝑖 (𝑡 − ℎ (𝑡)) ,
𝑧̇ 𝑖 (𝑡) = 𝐴
𝑧̇ 𝑖 (𝑡) = 𝑦𝑖 (𝑡)
𝑇
𝜎(𝑡)
𝜎(𝑡)
× (𝑔𝑖1
, . . . , 𝑔𝑖𝑁
) ,
𝑒̇ (𝑡) = 𝐽 (𝑡) 𝑒 (𝑡) + 𝐽ℎ (𝑡 − ℎ (𝑡)) 𝑒 (𝑡 − ℎ (𝑡))
(14)
namely,
̃𝑖 𝑧𝑖 (𝑡) + 𝐵
̃ 𝑖 𝑧𝑖 (𝑡 − ℎ (𝑡)) ,
𝑧̇ 𝑖 (𝑡) = 𝐴
𝑗=1
= 𝐽 (𝑡) 𝑒𝑖 (𝑡) + 𝐽ℎ (𝑡) 𝑒𝑖 (𝑡 − ℎ (𝑡))
+ 𝑐𝜎(𝑡) 𝐴 𝜎(𝑡) 𝑧 (𝑡 − ℎ (𝑡)) Γ𝜎(𝑡) ,
Next, we consider the nonswitched system
𝑁
+ 𝑐𝜎(𝑡) ∑𝑔𝑖𝑗𝜎(𝑡) 𝐴 𝜎(𝑡) 𝑒𝑗 (𝑡 − ℎ (𝑡))
󸀠
𝑧̇ (𝑡) = 𝐽 (𝑡) 𝑧 (𝑡) + 𝐽ℎ (𝑡 − ℎ (𝑡)) 𝑧 (𝑡 − ℎ (𝑡))
𝑖 = 2, 3, . . . , 𝑁 (8)
is asymptotically (or exponentially) stable, then the synchronized state (5) is asymptotically (or exponentially) stable.
𝑒𝑖 (𝑡) = 𝑥𝑖 (𝑡) − 𝑠 (𝑡) ,
where 𝑈𝑇 𝑈 = 𝐼, Γ𝑘 = diag{𝜆 1𝑘 , . . . , 𝜆 𝑁𝑘 }, 𝜆 𝑖𝑘 , 𝑖 = 1, 2, . . . , 𝑁
are eigenvalues of 𝐺𝑘 . In addition, with (2) and the irreducible
feature of 𝐺𝑘 , we may choose 𝑢1 = (1/√𝑁)(1, 1, . . . , 1)𝑇 such
that 𝜆 1𝑘 = 0, for all 𝑘 ∈ 𝑀.
From (12), by using the nonsingular transform 𝑒(𝑡)𝑈 =
𝑧(𝑡) = (𝑧1 (𝑡), . . . , 𝑧𝑁(𝑡)) ∈ 𝑅𝑁×𝑁, we have the following
matrix equation:
𝑡−ℎ(𝑡)
𝑧̇ (𝑠) 𝑑𝑠
(20)
𝑦 (𝑠) 𝑑𝑠,
to transform the original system to a system with distributed
delays. In order to improve the bound of time-varying delay
̃ 𝑖1 + 𝐵
̃ 𝑖2 , where 𝐵
̃ 𝑖1
̃𝑖 = 𝐵
ℎ(𝑡), let us decompose the matrix 𝐵
̃
are 𝐵𝑖2 are appropriate constant matrices to be determined
later. Then the original system (17) can be represented in the
Journal of Applied Mathematics
5
form of a descriptor system with time-varying and distributed
delays as
𝑧̇ 𝑖 (𝑡) = 𝑦𝑖 (𝑡)
̃ 𝑖1 ) 𝑧𝑖 (𝑡) + 𝐵
̃𝑖 + 𝐵
̃ 𝑖2 𝑧𝑖 (𝑡 − ℎ (𝑡))
0 = − 𝑦𝑖 (𝑡) + (𝐴
̃ 𝑖1 ∫
−𝐵
𝑡
𝑡−ℎ(𝑡)
𝑦𝑖 (𝑠) 𝑑𝑠,
(21)
𝑖 = 2, 3, . . . , 𝑁.
𝑖 = 2, . . . , 𝑁, where
Σ𝑖11 Σ𝑖12 Σ𝑖13
∗ Σ𝑖22 0
∗
∗ Σ𝑖33
∗
∗
∗
∗
∗
∗
∗
∗
∗
[
[
[
[
Σ𝑖 = [
[
[
[
Σ𝑖14 0
Σ𝑖24 0
Σ𝑖34 0
Σ𝑖44 Σ𝑖45
∗ Σ𝑖55
∗
∗
Σ𝑖16
Σ𝑖26 ]
]
0 ]
],
Σ𝑖46 ]
]
0 ]
Σ𝑖66 ]
̃𝑖 + 𝐵
̃ 𝑖1 ) + (𝐴
̃𝑖 + 𝐵
̃ 𝑖1 )𝑇 𝑃𝑖2 + 𝑄𝑖
Σ𝑖11 = 𝑃𝑖2𝑇 (𝐴
Let 𝛿 = ℎ𝑀 − ℎ𝑚 . Let 𝑉𝑖 (𝑡) be defined by
𝑉𝑖 (𝑡) = 𝑉𝑖1 (𝑡) + 𝑉𝑖2 (𝑡) + 𝑉𝑖3 (𝑡) + 𝑉𝑖4 (𝑡) + 𝑉𝑖5 (𝑡) + 𝑉𝑖6 (𝑡) ,
(22)
where
+ 𝑅𝑖 − 𝑒−𝛼ℎ𝑚 𝑈𝑖 + 𝛼𝑃𝑖1 ,
̃𝑖 + 𝐵
̃ 𝑖1 )𝑇 𝑃𝑖3 ,
Σ𝑖12 = 𝑃𝑖1𝑇 − 𝑃𝑖2𝑇 + (𝐴
Σ𝑖13 = 𝑒−𝛼ℎ𝑀 𝑈𝑖 ,
𝑉𝑖1 (𝑡) = 𝜉𝑖𝑇 (𝑡) 𝐸𝑃𝑖 𝜉𝑖 (𝑡) = 𝑧𝑖𝑇 (𝑡) 𝑃𝑖1 𝑧𝑖 (𝑡) ,
𝑡
𝑡−ℎ𝑚
𝑡
𝑉𝑖3 (𝑡) = ∫
𝑡−ℎ𝑀
0
0
𝑉𝑖5 (𝑡) = ℎ𝑀 ∫
−ℎ𝑀
−ℎ𝑚
−ℎ𝑀
∫
𝑡
𝑡+𝜃
∫
𝑡
𝑡+𝜃
𝑡
∫
𝑡+𝜃
𝑦𝑖𝑇 (𝑠) 𝑆𝑖 𝑒𝛼(𝑠−𝑡) 𝑦𝑖
𝑡−ℎ(𝑡)
𝑇
𝐼 0 0
] ,
0 0 0
̃ 𝑖1 − 𝑃𝑖4 ,
Σ𝑖26 = − 𝑃𝑖3𝑇 𝐵
(𝑠) 𝑑𝑠 𝑑𝜃,
(23)
Σ𝑖44 = − 2𝑒−𝛼𝛿 𝑊𝑖 ,
Σ𝑖45 = 𝑒−𝛼𝛿 𝑊𝑖 ,
𝑇
𝑦𝑖𝑇 (𝑠) 𝑑𝑠] ,
𝑃𝑖 = [
̃ 𝑖2 𝑃𝑇 ,
Σ𝑖46 = 𝐵
𝑖4
Σ𝑖55 = − 𝑒−𝛼ℎ𝑀 𝑅𝑖 − 𝑒−𝛼𝛿 𝑊𝑖 ,
𝑃𝑖1 0 0
].
𝑃𝑖2 𝑃𝑖3 𝑃𝑖4
For given 𝛼 > 0, the following lemma provides an estimation
of 𝑉𝑖 (𝑡).
Lemma 8. Assume that the time-varying delay ℎ(𝑡) satisfies
0 ≤ ℎ𝑚 ≤ ℎ(𝑡) ≤ ℎ𝑀 and let 𝛼 > 0. If there exist symmetric
positive definite matrices 𝑃𝑖1 > 0, 𝑄𝑖 > 0, 𝑅𝑖 > 0, 𝑆𝑖 > 0,
𝑈𝑖 > 0, and 𝑊𝑖 > 0 and appropriately dimensioned matrices
𝑃𝑖2 , 𝑃𝑖3 , and 𝑃𝑖4 for 𝑖 = 2, 3, . . . , 𝑁, such that the following LMIs
hold:
Σ𝑖1 = Σ𝑖 − [0 0 𝐼 −𝐼 0 0]
𝑇
× 𝑒−𝛼𝛿 𝑊𝑖 [0 0 𝐼 −𝐼 0 0] < 0,
Σ𝑖2 = Σ𝑖 − [0 0 0 𝐼 −𝐼 0]
−𝛼𝛿
×𝑒
Σ𝑖33 = − 𝑒−𝛼ℎ𝑚 𝑄𝑖 − 𝑒−𝛼ℎ𝑚 𝑈𝑖 − 𝑊𝑖 ,
Σ𝑖34 = 𝑒−𝛼𝛿 𝑊𝑖 ,
𝑧̇ 𝑇𝑖 (𝑠) 𝑊𝑖 𝑒𝛼(𝑠−𝑡) 𝑧̇ 𝑖 (𝑠) 𝑑𝑠 𝑑𝜃,
𝑡
(26)
̃ 𝑖2 ,
Σ𝑖24 = 𝑃𝑖3𝑇 𝐵
𝑧̇ 𝑇𝑖 (𝑠) 𝑈𝑖 𝑒𝛼(𝑠−𝑡) 𝑧̇ 𝑖 (𝑠) 𝑑𝑠 𝑑𝜃,
𝜉𝑖 (𝑡) = [𝑧𝑖𝑇 (𝑡) 𝑦𝑖𝑇 (𝑡) ∫
𝐸=[
2
2
𝑈𝑖 + ℎ𝑀
𝑆𝑖 + 𝛿2 𝑊𝑖 − 𝑃𝑖3𝑇 − 𝑃𝑖3 ,
Σ𝑖22 = ℎ𝑚
𝑧𝑖𝑇 (𝑠) 𝑅𝑖 𝑒𝛼(𝑠−𝑡) 𝑧𝑖 (𝑠) 𝑑𝑠,
−ℎ𝑚
𝑉𝑖6 (𝑡) = 𝛿 ∫
̃ 𝑖1 − (𝐴
̃𝑖 + 𝐵
̃ 𝑖1 )𝑇 𝑃𝑖4 ,
Σ𝑖16 = − 𝑃𝑖2𝑇 𝐵
𝑧𝑖𝑇 (𝑠) 𝑄𝑖 𝑒𝛼(𝑠−𝑡) 𝑧𝑖 (𝑠) 𝑑𝑠,
𝑉𝑖2 (𝑡) = ∫
𝑉𝑖4 (𝑡) = ℎ𝑚 ∫
̃ 𝑖2 ,
Σ𝑖14 = 𝑃𝑖2𝑇 𝐵
(24)
̃ 𝑖1 − 𝐵
̃ 𝑇 𝑃𝑖4 .
Σ𝑖66 = − 𝑒−𝛼ℎ𝑀 𝑆𝑖 − 𝑃𝑖4𝑇 𝐵
𝑖1
Then, along the trajectory of system (17), we have
𝑉𝑖 (𝑡) ≤ 𝑒−𝛼(𝑡−𝑡0 ) 𝑉𝑖 (𝑡0 ) ,
𝑡 ≥ 𝑡0 ≥ 0.
(27)
Proof. By taking the derivative of Lyapunov-Krasovskii functional candidate (22) along the trajectory of the system (21),
we obtain
𝑇
𝑊𝑖 [0 0 0 𝐼 −𝐼 0] < 0,
(25)
𝑉̇ 𝑖 (𝑡) = 𝑉̇ 𝑖1 (𝑡) + 𝑉̇ 𝑖2 (𝑡) + 𝑉̇ 𝑖3 (𝑡) + 𝑉̇ 𝑖4 (𝑡) + 𝑉̇ 𝑖5 (𝑡) + 𝑉̇ 𝑖6 (𝑡) ,
(28)
6
Journal of Applied Mathematics
where
On the other hand, we have
𝑉̇ 𝑖1 (𝑡) = 2𝑧𝑖𝑇 (𝑡) 𝑃𝑖1 𝑧̇ 𝑖 (𝑡)
= 2𝜉𝑖𝑇 (𝑡) 𝑃𝑖𝑇 [
− 𝛿∫
𝑡−ℎ𝑚
𝑡−ℎ𝑀
𝑧̇ 𝑖 (𝑡)
],
0
𝑧̇ 𝑇𝑖 (𝑠) 𝑊𝑖 𝑧̇ 𝑖 (𝑠) 𝑑𝑠
= −𝛿 ∫
𝑡−ℎ(𝑡)
𝑉̇ 𝑖2 (𝑡) = −𝛼𝑉𝑖2 (𝑡) + 𝑧𝑖𝑇 (𝑡) 𝑄𝑖 𝑧𝑖 (𝑡)
− 𝛿∫
− 𝑧𝑖𝑇 (𝑡 − ℎ𝑚 ) 𝑄𝑖 𝑒−𝛼ℎ𝑚 𝑧𝑖 (𝑡 − ℎ𝑚 ) ,
(29)
− ℎ𝑀𝑒
∫
𝑡
𝑦𝑖𝑇 (𝑠) 𝑆𝑖 𝑦𝑖
𝑡−ℎ𝑀
− 𝛿𝑒−𝛼𝛿 ∫
𝑡−ℎ𝑀
− (ℎ (𝑡) − ℎ𝑚 ) ∫
𝑡−ℎ(𝑡)
− (ℎ (𝑡) − ℎ𝑚 ) ∫
(𝑠) 𝑑𝑠,
− (ℎ (𝑡) − ℎ𝑚 ) ∫
𝑡−ℎ(𝑡)
𝑡−ℎ𝑀
𝑧̇ 𝑇𝑖 (𝑠) 𝑊𝑖 𝑧̇ 𝑖 (𝑠) 𝑑𝑠.
= −𝛾 ∫
≤ −𝛾 ∫
− ℎ𝑚 ∫
𝑡−ℎ𝑚
𝑡−ℎ(𝑡)
(𝑠) 𝑑𝑠
𝑡−ℎ(𝑡)
𝑡−ℎ𝑚
𝑡
𝑡−ℎ𝑚
𝑧̇ 𝑇𝑖 (𝑠) 𝑑𝑠]
= −[𝑧𝑖 (𝑡) − 𝑧𝑖 (𝑡 − ℎ𝑚 )] 𝑈𝑖 [𝑧𝑖 (𝑡) − 𝑧𝑖 (𝑡 − ℎ𝑚 )]
−𝑧𝑖𝑇 (𝑡) 𝑈𝑖 𝑧𝑖
(𝑡) +
2𝑧𝑖𝑇 (𝑡) 𝑈𝑖 𝑧𝑖
𝑡
𝑡−ℎ𝑀
𝑡
𝑡−ℎ(𝑡)
𝑡
≤ −[∫
𝑡−ℎ(𝑡)
𝑡−ℎ(𝑡)
𝑡−ℎ(𝑡)
(𝑡 − ℎ𝑚 )
− 𝛿∫
𝑡−ℎ𝑚
𝑡−ℎ𝑀
𝑇
𝑆𝑖 [∫
𝑡
𝑡−ℎ(𝑡)
𝛿𝑧̇ 𝑇𝑖 (𝑠) 𝑊𝑖 𝑧̇ 𝑖 (𝑠) 𝑑𝑠
(ℎ (𝑡) − ℎ𝑚 ) 𝑧̇ 𝑇𝑖 (𝑠) 𝑊𝑖 𝑧̇ 𝑖 (𝑠) 𝑑𝑠.
From Lemma 6, we have
𝑦𝑖𝑇 (𝑠) 𝑆𝑖 𝑦𝑖 (𝑠) 𝑑𝑠
𝑦𝑖𝑇 (𝑠) 𝑑𝑠]
= − (1 − 𝛾) ∫
𝑡−ℎ𝑚
𝑦𝑖𝑇 (𝑠) 𝑆𝑖 𝑦𝑖 (𝑠) 𝑑𝑠
≤ −ℎ (𝑡) ∫
𝑧̇ 𝑇𝑖 (𝑠) 𝑊𝑖 𝑧̇ 𝑖 (𝑠) 𝑑𝑠
≤ − (1 − 𝛾) ∫
− 𝑧𝑖𝑇 (𝑡 − ℎ𝑚 ) 𝑈𝑖 𝑧𝑖 (𝑡 − ℎ𝑚 ) ,
− ℎ𝑀 ∫
(33)
𝑡−ℎ𝑚
(30)
𝑇
𝑧̇ 𝑇𝑖 (𝑠) 𝑊𝑖 𝑧̇ 𝑖 (𝑠) 𝑑𝑠
(ℎ𝑀 − ℎ (𝑡)) 𝑧̇ 𝑇𝑖 (𝑠) 𝑊𝑖 𝑧̇ 𝑖 (𝑠) 𝑑𝑠,
− (ℎ𝑀 − ℎ (𝑡)) ∫
𝑧̇ 𝑇𝑖 (𝑠) 𝑑𝑠] 𝑈𝑖 [∫
𝑧̇ 𝑇𝑖 (𝑠) 𝑊𝑖 𝑧̇ 𝑖 (𝑠) 𝑑𝑠.
𝛿𝑧̇ 𝑇𝑖 (𝑠) 𝑊𝑖 𝑧̇ 𝑖 (𝑠) 𝑑𝑠
𝑡−ℎ𝑚
𝑇
𝑡
=
𝑡−ℎ(𝑡)
𝑡−ℎ𝑀
≤ −[∫
𝑧̇ 𝑇𝑖 (𝑠) 𝑊𝑖 𝑧̇ 𝑖 (𝑠) 𝑑𝑠
Let 𝛾 = (ℎ(𝑡) − ℎ𝑚 )/𝛿. Then, we obtain
Using Lemma 6, we obtain
𝑧̇ 𝑇𝑖 (𝑠) 𝑈𝑖 𝑧̇ 𝑖
𝑡−ℎ(𝑡)
(31)
𝑧̇ 𝑇𝑖 (𝑠) 𝑊𝑖 𝑧̇ 𝑖 (𝑠) 𝑑𝑠
≤ −[𝑧𝑖 (𝑡 − ℎ𝑚 ) − 𝑧𝑖 (𝑡 − ℎ (𝑡))]
𝑇
× 𝑊𝑖 [𝑧𝑖 (𝑡 − ℎ𝑚 ) − 𝑧𝑖 (𝑡 − ℎ (𝑡))]
𝑦𝑖𝑇 (𝑠) 𝑑𝑠] .
(32)
𝑧̇ 𝑇𝑖 (𝑠) 𝑊𝑖 𝑧̇ 𝑖 (𝑠) 𝑑𝑠
𝑡−ℎ(𝑡)
𝑡−ℎ𝑀
𝑡−ℎ𝑀
𝑡
𝑡−ℎ𝑚
𝑧̇ 𝑇𝑖 (𝑠) 𝑊𝑖 𝑧̇ 𝑖 (𝑠) 𝑑𝑠
𝑡−ℎ𝑀
𝑉̇ 𝑖6 (𝑡) ≤ −𝛼𝑉𝑖6 (𝑡) + 𝑦𝑖𝑇 (𝑡) 𝛿2 𝑊𝑖 𝑦𝑖 (𝑡)
𝑡−ℎ𝑚
𝑡−ℎ𝑚
− (ℎ𝑀 − ℎ (𝑡)) ∫
𝑧̇ 𝑇𝑖 (𝑠) 𝑈𝑖 𝑧̇ 𝑖 (𝑠) 𝑑𝑠,
2
𝑆𝑖 𝑦𝑖 (𝑡)
𝑉̇ 𝑖5 (𝑡) ≤ −𝛼𝑉𝑖5 (𝑡) + 𝑦𝑖𝑇 (𝑡) ℎ𝑀
−𝛼ℎ𝑀
𝑧̇ 𝑇𝑖 (𝑠) 𝑊𝑖 𝑧̇ 𝑖 (𝑠) 𝑑𝑠
𝑡−ℎ(𝑡)
2
𝑉̇ 𝑖4 (𝑡) ≤ −𝛼𝑉𝑖4 (𝑡) + 𝑦𝑖𝑇 (𝑡) ℎ𝑚
𝑈𝑖 𝑦𝑖 (𝑡)
𝑡−ℎ𝑚
𝑡−ℎ(𝑡)
= − (ℎ𝑀 − ℎ (𝑡)) ∫
− 𝑧𝑖𝑇 (𝑡 − ℎ𝑀) 𝑅𝑖 𝑒−𝛼ℎ𝑀 𝑧𝑖 (𝑡 − ℎ𝑀) ,
𝑡
𝑧̇ 𝑇𝑖 (𝑠) 𝑊𝑖 𝑧̇ 𝑖 (𝑠) 𝑑𝑠
𝑡−ℎ𝑀
𝑉̇ 𝑖3 (𝑡) = −𝛼𝑉𝑖3 (𝑡) + 𝑧𝑖𝑇 (𝑡) 𝑅𝑖 𝑧𝑖 (𝑡)
− ℎ𝑚 𝑒−𝛼ℎ𝑚 ∫
𝑡−ℎ𝑚
− [𝑧𝑖 (𝑡 − ℎ (𝑡)) − 𝑧𝑖 (𝑡 − ℎ𝑀)]
𝑇
Journal of Applied Mathematics
7
× 𝑊𝑖 [𝑧𝑖 (𝑡 − ℎ (𝑡)) − 𝑧𝑖 (𝑡 − ℎ𝑀)]
Next, we define 𝑉(𝑡) as follows:
𝑇
− 𝛾[𝑧𝑖 (𝑡 − ℎ (𝑡)) − 𝑧𝑖 (𝑡 − ℎ𝑀)]
𝑉𝑖 (𝑡) = 𝑉𝑖1 (𝑡) + 𝑉𝑖2 (𝑡) + 𝑉𝑖3 (𝑡) + 𝑉𝑖4 (𝑡) + 𝑉𝑖5 (𝑡) + 𝑉𝑖6 (𝑡) ,
(38)
× 𝑊𝑖 [𝑧𝑖 (𝑡 − ℎ (𝑡)) − 𝑧𝑖 (𝑡 − ℎ𝑀)]
− (1 − 𝛾) [𝑧𝑖 (𝑡 − ℎ𝑚 ) − 𝑧𝑖 (𝑡 − ℎ (𝑡))]
𝑇
× 𝑊𝑖 [𝑧𝑖 (𝑡 − ℎ𝑚 ) − 𝑧𝑖 (𝑡 − ℎ (𝑡))] .
(34)
where
𝑉𝑖1 (𝑡) = 𝜉𝑖𝑇 (𝑡) 𝐸𝑃𝑖 𝜉𝑖 (𝑡) = 𝑧𝑖𝑇 (𝑡) 𝑃𝑖1 𝑧𝑖 (𝑡) ,
Therefore, from (21) and (30)–(34), it follows that
𝑉𝑖2 (𝑡) = ∫
𝑉̇ 𝑖 (𝑡) + 𝛼𝑉𝑖 (𝑡) ≤ 𝜉1𝑇 (𝑡) Σ𝑖 𝜉1 (𝑡)
𝑇
𝑉𝑖3 (𝑡) = ∫
𝑇
− (1 − 𝛾) [𝑧𝑖 (𝑡 − ℎ𝑚 ) − 𝑧𝑖 (𝑡 − ℎ (𝑡))]
× 𝑒−𝛼𝛿 𝑊𝑖 [𝑧𝑖 (𝑡 − ℎ𝑚 ) − 𝑧𝑖 (𝑡 − ℎ (𝑡))]
= 𝜉1𝑇 (𝑡) [𝛾Σ𝑖1 + (1 − 𝛾) Σ𝑖2 ] 𝜉1 (𝑡) ,
(35)
where Σ𝑖1 , Σ𝑖2 , and Σ𝑖 are defined in (24), (25), and (26),
respectively, and
𝑉𝑖4 (𝑡) = ℎ𝑚 ∫
𝜉1 (𝑡) = [ 𝑧𝑖𝑇 (𝑡) 𝑦𝑖𝑇 (𝑡) 𝑧𝑖𝑇 (𝑡 − ℎ𝑚 ) 𝑧𝑖𝑇 (𝑡 − ℎ (𝑡))
𝑡−ℎ(𝑡)
𝑉𝑖5 (𝑡) = ℎ𝑀 ∫
0
−ℎ𝑀
𝑉𝑖6 (𝑡) = 𝛿 ∫
−ℎ𝑚
−ℎ𝑀
∫
𝑡
𝑡+𝜃
∫
𝑧̇ 𝑇𝑖 (𝑠) 𝑈𝑖 𝑒𝛽(𝑡−𝑠) 𝑧̇ 𝑖 (𝑠) 𝑑𝑠 𝑑𝜃,
𝑡
𝑡+𝜃
∫
𝑡
𝑡+𝜃
𝑦𝑖𝑇 (𝑠) 𝑆𝑖 𝑒𝛽(𝑡−𝑠) 𝑦𝑖 (𝑠) 𝑑𝑠 𝑑𝜃,
𝑧̇ 𝑇𝑖 (𝑠) 𝑊𝑖 𝑒𝛽(𝑡−𝑠) 𝑧̇ 𝑖 (𝑠) 𝑑𝑠 𝑑𝜃,
𝜉𝑖 (𝑡) = [𝑧𝑖𝑇 (𝑡) 𝑦𝑖𝑇 (𝑡) ∫
𝑡
𝑡−ℎ(𝑡)
𝑇
𝐼 0 0
] ,
0 0 0
(39)
𝑇
𝑦𝑖𝑇 (𝑠)𝑑𝑠] ,
𝑃 0 0
𝑃𝑖 = [ 𝑖1
].
𝑃𝑖2 𝑃𝑖3 𝑃𝑖4
For given 𝛼 > 0, the following lemma provides an estimation
of 𝑉𝑖 (𝑡).
Since 0 ≤ 𝛾 ≤ 1, 𝛾Σ𝑖1 + (1 − 𝛾)Σ𝑖2 is a convex combination of
Σ𝑖1 and Σ𝑖2 . Therefore, 𝛾Σ𝑖1 + (1 − 𝛾)Σ𝑖1 < 0 is equivalent to
Σ𝑖1 < 0 and Σ𝑖2 < 0. Thus, it follows from (24), (25), and (35)
that
𝑡 ≥ 𝑡0 ≥ 0.
0
(36)
𝑦𝑖𝑇 (𝑠) 𝑑𝑠 ] .
𝑉̇ 𝑖 (𝑡) + 𝛼𝑉𝑖 (𝑡) ≤ 0,
𝑧𝑖𝑇 (𝑠) 𝑅𝑖 𝑒𝛽(𝑡−𝑠) 𝑧𝑖 (𝑠) 𝑑𝑠,
−ℎ𝑚
𝐸=[
𝑇
𝑡
𝑡−ℎ𝑀
× 𝑒−𝛼𝛿 𝑊𝑖 [𝑧𝑖 (𝑡 − ℎ (𝑡)) − 𝑧𝑖 (𝑡 − ℎ𝑀)]
𝑧𝑖𝑇 (𝑡 − ℎ𝑀) ∫
𝑧𝑖𝑇 (𝑠) 𝑄𝑖 𝑒𝛽(𝑡−𝑠) 𝑧𝑖 (𝑠) 𝑑𝑠,
𝑡−ℎ𝑚
− 𝛾[𝑧𝑖 (𝑡 − ℎ (𝑡)) − 𝑧𝑖 (𝑡 − ℎ𝑀)]
𝑡
𝑡
Lemma 10. Assume that the time-varying delay ℎ(𝑡) satisfies
0 ≤ ℎ𝑚 ≤ ℎ(𝑡) ≤ ℎ𝑀 and let 𝛽 > 0. If there exist symmetric
positive definite matrices 𝑃𝑖1 > 0, 𝑄𝑖 > 0, 𝑅𝑖 > 0, 𝑆𝑖 > 0, 𝑈𝑖 > 0,
and 𝑊𝑖 > 0 and appropriately dimensioned matrices 𝑃𝑖2 , 𝑃𝑖3 ,
and 𝑃𝑖4 for 𝑖 = 2, 3, . . . , 𝑁, such that the following symmetric
LMIs hold:
(37)
Σ𝑖1 = Σ𝑖 − [0 0 𝐼 −𝐼 0 0]
By integrating this inequality from 𝑡0 to 𝑡, we obtain (27).
Remark 9. If 𝛼 = 0 in (24) and (25), then it follows from
Lemma 8 that the network system (17) is asymptotically
stable.
𝑇
× 𝑊𝑖 [0 0 𝐼 −𝐼 0 0] < 0,
Σ𝑖2 = Σ𝑖 − [0 0 0 𝐼 −𝐼 0]
𝑇
× 𝑊𝑖 [0 0 0 𝐼 −𝐼 0] < 0,
(40)
8
Journal of Applied Mathematics
𝑖 = 2, . . . , 𝑁, where
We rewrite system (44) in the following equivalent descriptor
system form:
Σ𝑖11 Σ𝑖12 Σ𝑖13
∗ Σ𝑖22 0
∗
∗ Σ𝑖33
∗
∗
∗
∗
∗
∗
∗
∗
[∗
Σ𝑖14 0
Σ𝑖24 0
Σ𝑖34 0
Σ𝑖44 Σ𝑖45
∗ Σ𝑖55
∗
∗
[
[
[
Σ𝑖 = [
[
[
[
Σ𝑖11 =
̃𝑖
𝑃𝑖2𝑇 (𝐴
Σ𝑖16
Σ𝑖26 ]
]
0 ]
],
Σ𝑖46 ]
]
0 ]
Σ𝑖66 ]
𝑧̇ 𝑖 (𝑡) = 𝑦𝑖 (𝑡) ,
̃ 𝑖𝜎(𝑡)1 ) 𝑧𝑖 (𝑡)
̃𝑖 + 𝐵
0 = − 𝑦𝑖 (𝑡) + (𝐴
𝑇
̃ 𝑖1 ) + (𝐴
̃𝑖 + 𝐵
̃ 𝑖1 ) 𝑃𝑖2
+𝐵
(45)
̃ 𝑖𝜎(𝑡)2 𝑧𝑖 (𝑡 − ℎ (𝑡)) − 𝐵
̃ 𝑖𝜎(𝑡)1 ∫
+𝐵
+ 𝑄𝑖 + 𝑅𝑖 − 𝑈𝑖 − 𝛽𝑃𝑖1 ,
𝑡
𝑡−ℎ(𝑡)
𝑦𝑖 (𝑠) 𝑑𝑠.
𝑇
̃𝑖 + 𝐵
̃ 𝑖1 ) 𝑃𝑖3 ,
Σ𝑖12 = 𝑃𝑖1𝑇 − 𝑃𝑖2𝑇 + (𝐴
Σ𝑖13 = 𝑈𝑖 ,
̃ 𝑖2 ,
Σ𝑖14 = 𝑃𝑖2𝑇 𝐵
𝑇
̃ 𝑖1 − (𝐴
̃𝑖 + 𝐵
̃ 𝑖1 ) 𝑃𝑖4 ,
Σ𝑖16 = − 𝑃𝑖2𝑇 𝐵
Σ𝑖22 =
2
ℎ𝑚
𝑈𝑖
+
2
ℎ𝑀
𝑆𝑖
2
+ 𝛿 𝑊𝑖 −
𝑃𝑖3𝑇
(41)
− 𝑃𝑖3 ,
̃ 𝑖2 ,
Σ𝑖24 = 𝑃𝑖3𝑇 𝐵
We consider the case when synchronizable and nonsynchronizable subnetworks coexist. Without loss of generality, we suppose that each subnetwork 𝑘 ∈ 𝑆𝑠 = {1, 2, . . . , 𝑟}
is synchronizable, where 1 ≤ 𝑟 < 𝑚, and each subnetwork
𝑙 ∈ 𝑆𝑛 = {𝑟 + 1, 𝑟 + 2, . . . , 𝑚} is non-synchronizable.
For each subnetwork of the system (45) which satisfies
Lemma 8, the Lyapunov-Krasovskii functional candidate can
be chosen as
̃ 𝑖1 − 𝑃𝑖4 ,
Σ𝑖26 = − 𝑃𝑖3𝑇 𝐵
Σ𝑖33 = − 𝑄𝑖 − 𝑈𝑖 − 𝑊𝑖 ,
𝑉𝑖𝑘 (𝑡) = 𝑧𝑖𝑇 (𝑡) 𝐸𝑃𝑖𝑘1 𝑧𝑖 (𝑡) + ∫
Σ𝑖34 = 𝑊𝑖 ,
𝑡
𝑡−ℎ𝑚
Σ𝑖44 = − 2𝑊𝑖 ,
𝑡
+∫
Σ𝑖45 = 𝑊𝑖 ,
𝑡−ℎ𝑀
̃ 𝑖2 𝑃𝑇 ,
Σ𝑖46 = 𝐵
𝑖4
+ ℎ𝑚 ∫
Σ𝑖55 = − 𝑅𝑖 − 𝑊𝑖 ,
𝑇
𝑉𝑖 (𝑡) ≤ 𝑒
0
+ ℎ𝑀 ∫
∫
𝑉𝑖 (𝑡0 ) ,
0
𝑡 ≥ 𝑡0 ≥ 0.
−ℎ𝑚
+ 𝛿∫
(42)
−ℎ𝑀
𝑡
𝑡+𝜃
∫
−ℎ𝑀
Then, along the trajectory of system (17), we have
𝛽(𝑡−𝑡0 )
𝑧𝑖𝑇 (𝑠) 𝑅𝑖𝑘 𝑒𝛼(𝑠−𝑡) 𝑧𝑖 (𝑠) 𝑑𝑠
−ℎ𝑚
̃ 𝑖1 − 𝐵
̃ 𝑃𝑖4 .
Σ𝑖66 = − 𝑆𝑖 − 𝑃𝑖4𝑇 𝐵
𝑖1
𝑧𝑖𝑇 (𝑠) 𝑄𝑖𝑘 𝑒𝛼(𝑠−𝑡) 𝑧𝑖 (𝑠) 𝑑𝑠
𝑧̇ 𝑇𝑖 (𝑠) 𝑈𝑖𝑘 𝑒𝛼(𝑠−𝑡) 𝑧̇ 𝑖 (𝑠) 𝑑𝑠𝑑𝜃
𝑡
𝑡+𝜃
∫
𝑡
𝑡+𝜃
𝑦𝑖𝑇 (𝑠) 𝑆𝑖𝑘 𝑒𝛼(𝑠−𝑡) 𝑦𝑖 (𝑠) 𝑑𝑠𝑑𝜃
𝑧̇ 𝑇𝑖 (𝑠) 𝑊𝑖𝑘 𝑒𝛼(𝑠−𝑡) 𝑧̇ 𝑖 (𝑠) 𝑑𝑠𝑑𝜃,
(46)
Proof. With the same argument as in Lemma 8, by taking the
derivative of Lyapunov-Krasovskii functional candidate (38)
along the trajectory of the system (21), we obtain
𝑉̇ 𝑖 (𝑡) − 𝛽𝑉𝑖 (𝑡) ≤ 0,
𝑡 ≥ 𝑡0 ≥ 0.
(43)
where 𝑖 = 2, 3, . . . , 𝑁, 𝑘 ∈ 𝑆𝑠 , 𝑃𝑖𝑘1 , 𝑄𝑖𝑘 , 𝑅𝑖𝑘 , 𝑈𝑖𝑘 , 𝑆𝑖𝑘 , and 𝑊𝑖𝑘
are positive definite matrices and
Integrating this inequality from 𝑡0 to 𝑡, we get (42).
We are now ready to derive some new sufficient conditions for exponential synchronization of the switched
network
̃𝑖 𝑧𝑖 (𝑡) + 𝐵
̃ 𝑖𝜎(𝑡) 𝑧𝑖 (𝑡 − ℎ (𝑡)) ,
𝑧̇ 𝑖 (𝑡) = 𝐴
𝑖 = 2, 3, . . . , 𝑁. (44)
𝜉𝑖 (𝑡) = [𝑧𝑖𝑇 (𝑡) 𝑦𝑖𝑇 (𝑡) ∫
𝑡
𝑡−ℎ(𝑡)
𝑇
𝐼 0 0
𝐸=[
] ,
0 0 0
𝑇
𝑦𝑖𝑇 (𝑠) 𝑑𝑠] ,
𝑃𝑖𝑘 = [
(47)
0
𝑃𝑖𝑘1 0
].
𝑃𝑖𝑘2 𝑃𝑖𝑘3 𝑃𝑖𝑘4
Journal of Applied Mathematics
9
Similarly, for each subnetwork of the system (45) which
satisfies Lemma 10, the Lyapunov-Krasovskii functional candidate can be chosen as
𝑡
𝑉𝑖𝑙 (𝑡) = 𝑧𝑖𝑇 (𝑡) 𝐸𝑃𝑖𝑙1 𝑧𝑖 (𝑡) + ∫
𝑡−ℎ𝑚
𝑡
+∫
𝑡−ℎ𝑀
+ ℎ𝑚 ∫
𝑧𝑖𝑇 (𝑠) 𝑅𝑖𝑙 𝑒𝛽(𝑡−𝑠) 𝑧𝑖 (𝑠) 𝑑𝑠
0
−ℎ𝑚
+ ℎ𝑀 ∫
∫
0
−ℎ𝑚
−ℎ𝑀
𝑡
𝑡+𝜃
∫
−ℎ𝑀
+ 𝛿∫
𝑧𝑖𝑇 (𝑠) 𝑄𝑖𝑙 𝑒𝛽(𝑡−𝑠) 𝑧𝑖 (𝑠) 𝑑𝑠
𝑡+𝜃
∫
𝑡
𝑡+𝜃
(𝑠) 𝑑𝑠 𝑑𝜃,
(49)
From Lemmas 8 and 10, the following properties of the
Lyapunov-Krasovskii functional candidate (49) are obtained.
(i) There exist 𝑎, 𝑏 > 0 such that
󵄩2
󵄩2
󵄩
󵄩
𝑎󵄩󵄩󵄩𝑧𝑖 (𝑡)󵄩󵄩󵄩 ≤ 𝑉𝑖𝑘,𝑖𝑙 (𝑡) ≤ 𝑏󵄩󵄩󵄩𝑧𝑖 (𝑡0 )󵄩󵄩󵄩cl ,
𝑖 = 2, 3, . . . , 𝑁, 𝑘, 𝑙 ∈ 𝑀.
𝑖 = 2, 3, . . . , 𝑁, 𝑘, 𝑙 ∈ 𝑀.
(50)
(51)
(52)
Now, for any piecewise constant switching signal 𝜎(𝑡) and
any 0 ≤ 𝑡0 < 𝑡, we let 𝑇− (𝑡0 , 𝑡) (𝑇+ (𝑡0 , 𝑡) resp.) denote the
total activation time of the synchronizable subnetworks (the
ones of non-synchronizable subnetworks resp.) during (𝑡0 , 𝑡).
Then, we choose a scalar 𝛼∗ ∈ (0, 𝛼) arbitrarily to propose the
following switching law.
𝑐0 = 𝑒𝑁0 ln 𝜇 ,
𝑇𝑎 ≥
𝑇𝑎∗
𝜆=
∀𝑘∈𝑀
𝑏𝑖 = max (𝜆 max (𝑃𝑖𝑘1 )) + ℎ𝑚 max (𝜆 max (𝑄𝑖𝑘 ))
∀𝑘∈𝑀
∀𝑘∈𝑀
+ ℎ𝑀 max (𝜆 max (𝑅𝑖𝑘 )) +
(57)
2
ℎ𝑚
max (𝜆 (𝑈 ))
2 ∀𝑘∈𝑀 max 𝑖𝑘
2
ℎ𝑀
𝛿2
max (𝜆 max (𝑆𝑖𝑘 )) + max (𝜆 max (𝑊𝑖𝑘 )) .
2 ∀𝑘∈𝑀
2 ∀𝑘∈𝑀
+
−𝛼(𝑡−𝑡𝑘 )
𝑉𝑖𝑘 (𝑡𝑘 ) , if 𝜎 (𝑡) = 𝑘 ∈ 𝑆𝑠 ,
{𝑒
𝑉𝑖 (𝑡) ≤ { 𝛽(𝑡−𝑡 )
𝑘
𝑉𝑖𝑙 (𝑡𝑘 ) ,
if 𝜎 (𝑡) = 𝑙 ∈ 𝑆𝑛 .
𝑒
{
(58)
Since 𝑉𝑖𝜎(𝑡𝑘 ) (𝑡𝑘 ) ≤ 𝜇𝑉𝑖𝜎(𝑡𝑘− ) (𝑡𝑘− ) is true from (51) at the
switching point 𝑡𝑘 , where 𝑡𝑘− = lim𝑡 → 𝑡𝑘 𝑡 and from 𝑘 =
𝑁𝜎 (𝑡0 , 𝑡) ≤ 𝑁0 + ((𝑡 − 𝑡0 )/𝑇𝑎 ), we obtain
+
(𝑡𝑘 ,𝑡)−𝛼𝑇− (𝑡𝑘 ,𝑡)
𝑉𝑖𝜎(𝑡𝑘 ) (𝑡𝑘 )
+
(𝑡𝑘 ,𝑡)−𝛼𝑇− (𝑡𝑘 ,𝑡)
𝜇𝑉𝑖𝜎(𝑡𝑘− ) (𝑡𝑘− )
+
(𝑡𝑘 ,𝑡)−𝛼𝑇− (𝑡𝑘 ,𝑡)
𝜇𝑒𝛽𝑇
𝑉𝑖 (𝑡) ≤ 𝑒𝛽𝑇
≤ 𝑒𝛽𝑇
≤ 𝑒𝛽𝑇
+
(53)
ln 𝜇
= ∗ .
𝛼
1 ∗ ln 𝜇
),
(𝛼 −
2
𝑇𝑎
+
(𝑡𝑘−1 ,𝑡𝑘 )−𝛼𝑇− (𝑡𝑘−1 ,𝑡𝑘 )
× 𝑉𝑖𝜎(𝑡𝑘−1 ) (𝑡𝑘−1 )
holds on time interval (𝑡0 , 𝑡). Meanwhile, we choose 𝛼∗ < 𝛼
as the average dwell time scheme: for any 𝑡 > 𝑡0 ,
𝑡 − 𝑡0
𝑁𝜎 (𝑡0 , 𝑡) ≤ 𝑁0 +
,
𝑇𝑎
𝑖 = 2, 3, . . . , 𝑁, (56)
𝑎𝑖 = min (𝜆 min (𝑃𝑖𝑘1 )) ,
(S1): Determine the switching signal 𝜎(𝑡) so that
𝑇− (𝑡 , 𝑡) 𝛽 + 𝛼∗
inf + 0
≥
𝑡≥𝑡0 𝑇 (𝑡 , 𝑡)
𝛼 − 𝛼∗
0
(55)
Proof. Suppose that 𝑡 ∈ [𝑡𝑘 , 𝑡𝑘+1 ). For piecewise LyapunovKrasovskii functional candidate (49), along trajectory of
network system (45), we have
(iii) The Lyapunov functional candidate (49) satisfies
−𝛼(𝑡−𝑡0 )
𝑉𝑖𝑘 (𝑡0 ) , if 𝜎 (𝑡) = 𝑘 ∈ 𝑆𝑠 ,
{𝑒
𝑉𝑖 (𝑡) ≤ { 𝛽(𝑡−𝑡 )
0
𝑉𝑖𝑙 (𝑡0 ) ,
if 𝜎 (𝑡) = 𝑙 ∈ 𝑆𝑛 .
𝑒
{
𝑊𝑖𝑝 ≤ 𝜇𝑊𝑖𝑞 ,
where
∀𝑘∈𝑀
(ii) There exists a constant 𝜇 ≥ 1 such that
𝑉𝑖𝑘 (𝑡) ≤ 𝜇𝑉𝑖𝑙 (𝑡) ,
𝑆𝑖𝑝 ≤ 𝜇𝑆𝑖𝑞 ,
𝑖
(48)
𝑖 = 2, 3, . . . , 𝑁, 𝜎 (𝑡) ∈ 𝑀.
𝑈𝑖𝑝 ≤ 𝜇𝑈𝑖𝑞 ,
𝑅𝑖𝑝 ≤ 𝜇𝑅𝑖𝑞 ,
𝑐 𝑏 −𝜆(𝑡−𝑡0 ) 󵄩󵄩
󵄩
󵄩
󵄩󵄩
󵄩󵄩𝑧𝑖 (𝑡0 )󵄩󵄩󵄩cl ,
󵄩󵄩𝑧𝑖 (𝑡)󵄩󵄩󵄩 ≤ √ 0 𝑖 𝑒
𝑎
where 𝑖 = 2, 3, . . . , 𝑁, 𝑙 ∈ 𝑆𝑛 , 𝑃𝑖𝑙1 , 𝑄𝑖𝑙 , 𝑅𝑖𝑙 , 𝑈𝑖𝑙 , 𝑆𝑖𝑙 , and 𝑊𝑖𝑙 are
positive definite matrices.
Consider the following piecewise Lyapunov-Krasovskii
functional candidate:
𝑉𝑖 (𝑡) = 𝑉𝑖𝜎(𝑡) ,
𝑄𝑖𝑝 ≤ 𝜇𝑄𝑖𝑞 ,
Then the synchronous manifold (5) of network (1) is locally
exponentially stable for switching signal satisfying (53) and
(54), and the state decay estimate is given by
𝑦𝑖𝑇 (𝑠) 𝑆𝑖𝑙 𝑒𝛽(𝑡−𝑠) 𝑦𝑖 (𝑠) 𝑑𝑠 𝑑𝜃
𝑧̇ 𝑇𝑖 (𝑠) 𝑊𝑖𝑙 𝑒𝛽(𝑡−𝑠) 𝑧̇ 𝑖
𝑃𝑖𝑝1 ≤ 𝜇𝑃𝑖𝑞1 ,
∀𝑝, 𝑞 ∈ 𝑀.
𝑧̇ 𝑇𝑖 (𝑠) 𝑈𝑖𝑙 𝑒𝛽(𝑡−𝑠) 𝑧̇ 𝑖 (𝑠) 𝑑𝑠 𝑑𝜃
𝑡
Theorem 11. For a given constant 𝛼 > 0, 𝛽 > 0 and timevarying delay satisfying (3), suppose that the subnetwork (1 ≤
𝑘 ≤ 𝑟) of switched network (1) satisfies the conditions of
Lemma 8, and the others satisfy the conditions of Lemma 10.
Assume that there exists 𝜇 ≥ 1 such that
(54)
≤ 𝜇𝑒𝛽𝑇
(𝑡𝑘−1 ,𝑡)−𝛼𝑇− (𝑡𝑘−1 ,𝑡)
𝑉𝑖𝜎(𝑡𝑘−1 ) (𝑡𝑘−1 )
..
.
+
≤ 𝜇𝑘 𝑒𝛽𝑇
+
≤ 𝑒𝛽𝑇
(𝑡0 ,𝑡)−𝛼𝑇− (𝑡0 ,𝑡)
𝑉𝑖𝜎(𝑡0 ) (𝑡0 )
(𝑡0 ,𝑡)−𝛼𝑇− (𝑡0 ,𝑡)+(𝑁0 +((𝑡−𝑡0 )/𝑇𝑎 )) ln 𝜇
𝑉𝑖𝜎(𝑡0 ) (𝑡0 ) .
(59)
10
Journal of Applied Mathematics
Table 1: Comparison of the maximum value ℎ𝑀 for the asymptotic stability of system (65) by different methods.
𝑐
0.3
0.960
1.345
1.8894
1.354
1.9158
1.389
2.1331
Li et al. 2008 [8]
Yue and Li 2010 [12]
Ours
Yue and Li 2010 [12]
Ours
Yue and Li 2010 [12]
Ours
ℎ𝑚 = 0
ℎ𝑚 = 0.1
ℎ𝑚 = 0.5
Under the switching law (𝑆1) for any 𝑡0 , 𝑡, we have
𝛽𝑇+ (𝑡0 , 𝑡) − 𝛼𝑇− (𝑡0 , 𝑡) ≤ − 𝛼∗ (𝑇+ (𝑡0 , 𝑡) + 𝑇− (𝑡0 , 𝑡))
= − 𝛼∗ (𝑡 − 𝑡0 ) .
(60)
Thus,
𝑉𝑖 (𝑡) ≤ 𝑒𝑁0 ln 𝜇 𝑒−(𝛼
∗
−((ln 𝜇)/𝑇𝑎 ))(𝑡−𝑡0 )
𝑉𝑖𝜎(𝑡0 ) (𝑡0 )
≤ 𝑐0 𝑒−2𝜆(𝑡−𝑡0 ) 𝑉𝑖𝜎(𝑡0 ) (𝑡0 ) ,
(61)
󵄩
󵄩2
𝑉𝑖𝜎(𝑡0 ) (𝑡0 ) ≤ 𝑏𝑖 󵄩󵄩󵄩𝑧𝑖 (𝑡0 )󵄩󵄩󵄩cl .
−2
[1
[
𝐺=[
[0
[0
[1
1
−3
1
1
0
0
1
−2
1
0
0
1
1
−3
1
1
0]
]
0]
].
1]
−2]
(66)
(63)
(64)
0.39𝑐𝜆 𝑖
0
0
̃ 𝑖1 = [ 0
0 ],
0.39𝑐𝜆 𝑖
𝐵
0
0.39𝑐𝜆 𝑖 ]
[ 0
which means that synchronous solution to (1) is locally
exponentially stable. The proof is completed.
̃ 𝑖2 = [
𝐵
[
0.61𝑐𝜆 𝑖
0
0
0 ],
0
0.61𝑐𝜆 𝑖
0
0
0.61𝑐𝜆 𝑖 ]
(67)
𝑖 = 2, . . . , 5.
4. Numerical Examples
In this section, we give examples to show the effectiveness of
theoretical results obtained in Lemma 8 and Theorem 11.
Example 12. Consider a nonswitched network model with 5
nodes, where each node is a three-dimensional stable linear
system described by
𝑥̇ 𝑖1 (𝑡) = − 𝑥𝑖1 (𝑡)
𝑥̇ 𝑖2 (𝑡) = − 2𝑥𝑖2 (𝑡)
𝐽ℎ (𝑡−ℎ(𝑡)) = 0. Assume that the inner-coupling matrix is 𝐴 =
diag{1, 1, 1}, and the outer-coupling matrix is given by the
following irreducible symmetric matrix satisfying condition
(2):
̃𝑖 =
̃𝑖 = diag{−1, −2, −3}, 𝐵
In this example, we have 𝐴
diag{𝑐𝜆 𝑖 , 𝑐𝜆 𝑖 , 𝑐𝜆 𝑖 }, and 𝑖 = 2, . . . , 5. Let us decompose the
̃𝑖 = 𝐵
̃ 𝑖1 + 𝐵
̃ 𝑖2 , where
matrix 𝐵
Therefore,
𝑐 𝑏 −𝜆(𝑡−𝑡0 ) 󵄩󵄩
󵄩󵄩
󵄩
󵄩
󵄩󵄩𝑧𝑖 (𝑡)󵄩󵄩󵄩 ≤ √ 0 𝑖 𝑒
󵄩󵄩𝑧𝑖 (𝑡0 )󵄩󵄩󵄩cl .
𝑎𝑖
0.6
0.464
0.587
0.9049
0.587
0.9276
0.605
1.1145
(62)
Combining (61) and (62) leads to
𝑏 −2𝜆(𝑡−𝑡0 ) 󵄩󵄩
󵄩2 1
󵄩󵄩
󵄩2
󵄩󵄩𝑧𝑖 (𝑡)󵄩󵄩󵄩 ≤ 𝑉𝑖 (𝑡) ≤ 𝑖 𝑐0 𝑒
󵄩󵄩𝑧𝑖 (𝑡0 )󵄩󵄩󵄩cl .
𝑎𝑖
𝑎𝑖
0.5
0.562
0.731
1.0983
0.731
1.1219
0.740
1.3127
The eigenvalues of 𝐺 are 𝜆 𝑖 = {0, −1.382, −2.382, −3.618, −
4.618}, 𝑖 = 1, 2, . . . , 5. Therefore, if the delayed subsystem in
(17) is asymptotically stable, then the synchronized state 𝑠(𝑡)
is asymptotically stable.
where 𝑐0 = 𝑒𝑁0 ln 𝜇 , 𝜆 = (1/2)(𝛼∗ − ((ln 𝜇)/𝑇𝑎 )). According to
(49), we have
󵄩2
󵄩
𝑎𝑖 󵄩󵄩󵄩𝑧𝑖 (𝑡)󵄩󵄩󵄩 ≤ 𝑉𝑖 (𝑡) ,
0.4
0.710
0.950
1.3923
0.951
1.4171
0.967
1.6165
(65)
𝑥̇ 𝑖3 (𝑡) = − 3𝑥𝑖3 (𝑡) ,
which is asymptotically stable at the equilibrium point 𝑠(𝑡) =
0, and its Jacobin matrices are 𝐽(𝑡) = diag{−1, −2, −3},
By using Lemma 8, we will give the maximum upper
bounds ℎ𝑀 of the time-varying delay for different lower
bounds ℎ𝑚 and coupling strength 𝑐. Applying Lemma 8 with
𝛼 = 0, we obtain the maximum upper bound of delays
ℎ𝑀 as shown in Table 1. We see that, Lemma 8 provides a
less conservative result than those obtained via the methods
of [8, 12]. When ℎ𝑚 ≠ 0 especially, the result in [8] is not
discussed while Lemma 8 in this paper also considers the case
ℎ𝑚 ≠ 0. Note that we use MATLAB LMI Control Toolbox in
order to solve the LMI in (24) and (25).
The numerical simulations are carried out using the
explicit Runge-Kutta-like method (dde45), interpolation
and extrapolation by spline of the third order. Figure 1
shows the synchronization state curves 𝑥𝑖1 (𝑡), 𝑥𝑖2 (𝑡), 𝑥𝑖3 (𝑡),
Journal of Applied Mathematics
11
strengths are 𝑐1 = 0.3 and 𝑐2 = 0.6, and the outer-coupling
matrices are
𝑥𝑖1 (𝑡)
0.5
0
−0.5
0
5
10
15
20
25
Time (s)
30
35
40
(a)
𝑥𝑖2 (𝑡)
1
0
−1
0
5
10
15
20
25
Time (s)
30
35
40
−2
[1
[
𝐺1 = [
[0
[0
[1
1
−3
1
1
0
0
1
−2
1
0
0
1
1
−3
1
1
0]
]
0]
],
1]
−2]
−4
[1
[
𝐺2 = [
[1
[1
[1
1
−4
1
1
1
1
1
−4
1
1
1
1
1
−4
1
1
1]
]
1]
].
1]
−4]
(71)
(b)
𝑥𝑖3 (𝑡)
1
0
−1
0
5
10
15
20
25
Time (s)
30
35
40
(c)
Figure 1: Synchronization state curves for the dynamic nodes
delayed network (65) with 𝑐 = 0.5 and ℎ(𝑡) = 0.5 + 0.715| cos 𝑡|.
𝑖 = 1, 2, . . . , 5 with 𝑐1 = 0.5, ℎ(𝑡) = 0.5 + 0.715| cos 𝑡| and
the initial function
𝜙1 (𝑡) = [−0.2 cos 𝑡, −0.5 cos 𝑡, 1.1 cos 𝑡] ,
𝜙2 (𝑡) = [0.5 cos 𝑡, 0.7 cos 𝑡, −0.7 cos 𝑡] ,
𝜙3 (𝑡) = [0.3 cos 𝑡, 1.2 cos 𝑡, 0.4 cos 𝑡] ,
(68)
𝜙4 (𝑡) = [0.35 cos 𝑡, −0.5 cos 𝑡, −1.25 cos 𝑡] ,
𝜙5 (𝑡) = [−0.4 cos 𝑡, −1.1 cos 𝑡, 0.8 cos 𝑡] .
Example 13. Consider a linear switched network model with
5 nodes where each node is described by the following
delayed dynamical system:
𝑥̇ 𝑖1 (𝑡) = − 1𝑥𝑖1 (𝑡) − 𝑥𝑖1 (𝑡 − ℎ (𝑡))
𝑥̇ 𝑖2 (𝑡) = − 2𝑥𝑖2 + 𝑥𝑖1 (𝑡 − ℎ (𝑡)) − 𝑥𝑖2 (𝑡 − ℎ (𝑡))
(69)
𝑥̇ 𝑖3 (𝑡) = − 1.5𝑥𝑖3 − 𝑥𝑖3 (𝑡 − ℎ (𝑡)) ,
which is asymptotically stable at the equilibrium point 𝑠(𝑡) =
0, 𝑠(𝑡 − ℎ(𝑡)) = 0. Jacobian matrices 𝐽(𝑡) and 𝐽ℎ (𝑡 − ℎ(𝑡)) are
−1 0
0
𝐽 (𝑡) = [ 0 −2 0 ] ,
[ 0 0 −1.5]
−1 0 0
𝐽ℎ (𝑡 − ℎ (𝑡)) = [ 1 −1 0 ] .
[ 0 0 −1]
(70)
=
Assume that the inner-coupling matrices are 𝐴 1
diag{0.5, 0.5, 0.5}, 𝐴 2 = diag{−1, −1, −1} and the coupling
The eigenvalues of 𝐺1 and 𝐺2 are 𝜆 𝑖1 = {0, −1.382, −2.382, −
3.618, −4.618}, 𝜆 𝑖2 = {0, −5, −5, −5, −5}, and 𝑖 = 1, 2, . . . , 5,
respectively. Obviously, there exists a unitary matrix 𝑈 which
diagonalizes 𝐺1 and 𝐺2 simultaneously. The subnetwork associated with {𝑐1 , 𝐴 1 , 𝐺1 } is synchronizable and the subnetwork
associated with {𝑐2 , 𝐴 2 , 𝐺2 } is non-synchronizable.
From the above conditions, we obtain matrices
−1 0
0
̃𝑖 = [ 0 −2 0 ] ,
𝐴
[ 0 0 −1.5]
𝑘
−1 + 𝑐𝑘 𝑎11
𝜆 𝑖𝑘
0
0
[
]
𝑘
̃ 𝑖𝑘 = [
𝐵
𝜆 𝑖𝑘
0
1
−1 + 𝑐𝑘 𝑎22
],
𝑘
𝑎
𝜆
0
0
−1
+
𝑐
𝑘 33 𝑖𝑘 ]
[
(72)
̃ 𝑖𝑘 =
𝑖 = 2, . . . , 5, and 𝑘 = 1, 2. Let us decompose the matrix 𝐵
̃
̃
𝐵𝑖𝑘1 + 𝐵𝑖𝑘2 , where
̃ 𝑖11 = 0.93𝐵
̃ 𝑖1 ,
𝐵
̃ 𝑖12 = 0.07𝐵
̃ 𝑖1 ,
𝐵
̃ 𝑖21 = 0.86𝐵
̃ 𝑖2 ,
𝐵
̃ 𝑖22 = 0.14𝐵
̃ 𝑖2 ,
𝐵
𝑖 = 2, . . . , 5.
(73)
Given ℎ1 = 0.3, ℎ2 = 0.68, 𝛼 = 1.25, and 𝛽 = 0.6,
it is found that LMIs (24), (25), (42), and (42) have feasible
solutions. We may choose 𝜇 = 1.35. Let 𝛼∗ = 0.55 <
𝛼. Then, from the switching law (𝑆1), it is required that
inf 𝑡≥𝑡0 (𝑇− (𝑡0 , 𝑡)/𝑇+ (𝑡0 , 𝑡)) ≥ (𝛽 + 𝛼∗ )/(𝛼 − 𝛼∗ ) = 1.6429 and
the average dwell time is computed as 𝑇𝑎 ≥ 𝑇𝑎∗ = (ln 𝜇)/𝛼∗ =
0.5456.
Let ℎ(𝑡) = 0.3 + 0.38| sin 𝑡| and the initial function
𝜙1 (𝑡) = [−0.4 cos 𝑡, −0.5 cos 𝑡, 1.1 cos 𝑡] ,
𝜙2 (𝑡) = [0.5 cos 𝑡, 0.7 cos 𝑡, −0.7 cos 𝑡] ,
𝜙3 (𝑡) = [0.89 cos 𝑡, 1.2 cos 𝑡, 0.4 cos 𝑡] ,
𝜙4 (𝑡) = [1.35 cos 𝑡, −1.5 cos 𝑡, −1.25 cos 𝑡] ,
𝜙5 (𝑡) = [−0.8 cos 𝑡, −1.1 cos 𝑡, 0.8 cos 𝑡] .
(74)
12
Journal of Applied Mathematics
1
1
5
2
4
5
2
3
4
3
𝐺2
𝐺1
(a)
(b)
Figure 2: The structure of complex networks with 𝑁 = 5.
200
1
𝑥𝑖1 (𝑡)
𝑥𝑖1 (𝑡)
2
0
−1
0
1
2
3
4
5
6
7
8
9
0
−200
−400
10
0
1
2
3
4
Time (s)
200
100
0
−100
−200
8
9
10
7
8
9
10
7
8
9
10
𝑥𝑖2 (𝑡)
1
𝑥𝑖2 (𝑡)
7
(a)
(a)
0
−1
0
1
2
3
4
5
6
Time (s)
7
8
9
10
0
1
2
3
𝑥𝑖3 (𝑡)
1
0
−1
0
1
2
3
4
5
6
Time (s)
4
6
5
Time (s)
(b)
(b)
𝑥𝑖3 (𝑡)
6
5
Time (s)
7
8
9
10
10
0
−10
−20
−30
0
1
2
3
4
6
5
Time (s)
(c)
(c)
Figure 3: Synchronization state curves for the synchronizable
subnetwork (69).
Figure 4: Synchronization state curves for the nonsynchronizable
subnetwork (69).
Figure 2 shows the structure of complex networks. Figures 3
and 4 show the synchronization state curves of the synchronizable subnetwork and non-synchronizable subnetwork,
respectively. Figure 5 shows the synchronization state curves
𝑥𝑖1 (𝑡), 𝑥𝑖2 (𝑡), 𝑥𝑖3 (𝑡), 𝑖 = 1, 2, . . . , 5 of switched network with
𝜎(𝑡) satisfying (𝑆1). Figure 6 shows the switching 𝜎(𝑡) of the
switched delay network with average dwell time. We see that
the synchronization state converges to zero under the above
conditions.
Example 14. We consider nonlinear switched network model
with 5 nodes in which each node is a Lorenz chaotic system
with time-varying delay described by [35] as
𝑥̇ 𝑖1 (𝑡) = 𝑎 (𝑥𝑖2 (𝑡) − 𝑥𝑖1 (𝑡)) ,
𝑥̇ 𝑖2 (𝑡) = 𝑐𝑥𝑖1 − 𝑥𝑖2 (𝑡) − 𝑥𝑖1 (𝑡) 𝑥𝑖3 (𝑡 − ℎ (𝑡)) ,
𝑥̇ 𝑖3 (𝑡) = 𝑥𝑖1 (𝑡) 𝑥𝑖2 (𝑡 − ℎ (𝑡)) − 𝑏𝑥𝑖3 (𝑡 − ℎ (𝑡)) ,
(75)
𝑥𝑖1 (𝑡)
Journal of Applied Mathematics
2
1
0
−1
−2
0
1
2
3
4
5
6
Time (s)
13
7
8
9
10
𝑥𝑖2 (𝑡)
(a)
2
1
0
−1
−2
−10 10 0
̃𝑖 = [−28 −1 0] ,
𝐴
[ 0 0 0]
0
1
2
3
4
5
6
Time (s)
7
8
9
10
𝑥𝑖3 (𝑡)
(b)
2
1
0
−1
−2
0
1
2
3
4
5
6
Time (s)
7
8
9
10
̃ 𝑖11 = 0.8𝐵
̃ 𝑖1 ,
𝐵
̃ 𝑖21 = 0.86𝐵
̃ 𝑖2 ,
𝐵
Figure 5: Synchronization state curves for the switched delay
network (69).
3
2.5
2
𝜎(𝑡)
𝑐 𝑎𝑘 𝜆
0
0
[ 𝑘 11 𝑖𝑘
]
𝑘
̃
𝐵𝑖𝑘 = [ 0
𝜆 𝑖𝑘
−1
𝑐𝑘 𝑎22
],
𝑘
𝑎
𝜆
0
0
−1.3
+
𝑐
𝑘 33 𝑖𝑘 ]
[
(77)
̃ 𝑖𝑘 =
𝑖 = 2, . . . , 5, and 𝑘 = 1, 2. Let us decompose the matrix 𝐵
̃
̃
𝐵𝑖𝑘1 + 𝐵𝑖𝑘2 , where
(c)
̃ 𝑖12 = 0.2𝐵
̃ 𝑖1 ,
𝐵
̃ 𝑖22 = 0.14𝐵
̃ 𝑖2 ,
𝐵
𝑖 = 2, . . . , 5.
(78)
Given ℎ1 = 0.2, ℎ2 = 0.55, 𝛼 = 0.9, and 𝛽 = 0.3,
it is found that LMIs (24), (25), (42), and (42) have feasible
solutions. We may choose 𝜇 = 1.253. Let 𝛼∗ = 0.45 <
𝛼. Then, from the switching law (𝑆1), it is required that
inf 𝑡≥𝑡0 (𝑇− (𝑡0 , 𝑡)/𝑇+ (𝑡0 , 𝑡)) ≥ (𝛽 + 𝛼∗ )/(𝛼 − 𝛼∗ ) = 1.6667 and
the average dwell time is computed as 𝑇𝑎 ≥ 𝑇𝑎∗ = (ln 𝜇)/𝛼∗ =
0.5012.
Let ℎ(𝑡) = 0.2 + 0.35| cos 𝑡| and the initial function
𝜙1 (𝑡) = [15 cos 𝑡, 20 cos 𝑡, 5 cos 𝑡] ,
1.5
𝜙2 (𝑡) = [5 cos 𝑡, 15 cos 𝑡, −2 cos 𝑡] ,
𝜙3 (𝑡) = [10 cos 𝑡, 5 cos 𝑡, 3 cos 𝑡] ,
1
(79)
𝜙4 (𝑡) = [20 cos 𝑡, 10 cos 𝑡, −1 cos 𝑡] ,
0.5
0
Assume that the inner-coupling matrices are 𝐴 1 =
diag{0.4, 0.4, 0.4} and 𝐴 2 = diag{−0.2, −0.2, −0.2}, the coupling strengths 𝑐1 = 0.5, and 𝑐2 = 0.3, and the outercoupling matrices 𝐺1 , and 𝐺2 are the same as in Example 13.
The subnetwork associated with {𝑐1 , 𝐴 1 , 𝐺1 } is synchronizable
and the subnetwork associated with {𝑐2 , 𝐴 2 , 𝐺2 } is nonsynchronizable.
From the above conditions, we obtain
𝜙5 (𝑡) = [15 cos 𝑡, 20 cos 𝑡, 8 cos 𝑡] .
0
1
2
3
4
5
6
Time (s)
7
8
9
10
Figure 6: The switching signal 𝜎(𝑡) of the switched delay network
(69) with average dwell time.
where 𝑎 = 10, 𝑏 = 1.3, and 𝑐 = −28. It is asymptotically
stable at the equilibrium point 𝑠(𝑡) = 0, 𝑠(𝑡 − ℎ(𝑡)) = 0 and its
Jacobian matrices are
−10 10 0
𝐽 (𝑡) = [−28 −1 0] ,
[ 0 0 0]
0 0 0
𝐽ℎ (𝑡 − ℎ (𝑡)) = [0 0 −1 ] .
[0 0 −1.3]
(76)
Figures 7 and 8 show the synchronization errors between the
states of node 𝑖 and node 𝑖+1, 𝑒𝑖𝑗 (𝑡) = 𝑥𝑖𝑗 −𝑥(𝑖+1)𝑗 , 𝑖 = 1, . . . , 4,
and 𝑗 = 1, . . . , 3 of the synchronizable subnetwork and nonsynchronizable subnetwork, respectively. Figure 9 shows the
synchronization errors between the states of node 𝑖 and node
𝑖 + 1 of switched network with 𝜎(𝑡) satisfying (𝑆1). Figure 10
shows the switching 𝜎(𝑡) of the switched delay network with
average dwell time. We see that the synchronization state
converges to zero under the above conditions.
Remark 15. The advantage of Examples 13 and 14 is the lower
bound of the delay ℎ𝑚 ≠ 0. Moreover, in these examples we
still investigate interval time-varying delays in the dynamical
nodes and the switched coupling term simultaneously, hence
the synchronization conditions derived in [33] cannot be
applied to these examples.
4
2
0
−2
−4
𝑒𝑖1 (𝑡)
Journal of Applied Mathematics
𝑒𝑖1 (𝑡)
14
0
0.5
1
1.5
2
2.5
3
Time (s)
3.5
4
4.5
5
10
5
0
−5
−10
0
1
2
3
4
2
0
−2
−4
0
0.5
1
1.5
2
2.5
3
Time (s)
3.5
4
4.5
5
20
10
0
−10
−20
0
1
2
3
(b)
𝑒𝑖3 (𝑡)
𝑒𝑖3 (𝑡)
0
−2
0
0.5
1
1.5
2
2.5
3
Time (s)
7
8
9
4
5
Time (s)
6
7
8
9
6
7
8
9
(b)
2
−4
6
(a)
𝑒𝑖2 (𝑡)
𝑒𝑖2 (𝑡)
(a)
4
5
Time (s)
3.5
4
4.5
5
40
20
0
−20
−40
0
1
2
3
4
5
Time (s)
(c)
(c)
Figure 7: Synchronization errors between the states of node 𝑖 and
node 𝑖 + 1 for the synchronizable subnetwork (75).
Figure 9: Synchronization errors between the states of node 𝑖 and
node 𝑖 + 1 for the switched delay network (75).
3
𝑒𝑖1 (𝑡)
2
2.5
0
2
0
0.5
1
1.5
2
2.5
Time (s)
3
3.5
4
(a)
𝑒𝑖2 (𝑡)
1.5
1
2
0.5
0
−2
0
0.5
1
1.5
2
2.5
Time (s)
3
3.5
4
200
100
0
−100
−200
0
0
1
2
3
4
5
Time (s)
6
7
8
9
Figure 10: The switching signal 𝜎(𝑡) of the switched delay network
(75) with average dwell time.
(b)
𝑒𝑖3 (𝑡)
𝜎(𝑡)
−2
5. Conclusion
0
0.5
1
1.5
2
2.5
Time (s)
3
3.5
4
(c)
Figure 8: Synchronization errors between the states of node 𝑖 and
node 𝑖 + 1 for the non-synchronizable subnetwork (75).
In this paper, the synchronization problem has been investigated for complex dynamical networks with interval timevarying and switched coupling delays. Both interval timevarying delays in dynamical nodes and interval time-varying
delays in switched couplings have been considered. We
transformed the synchronization problem of the switched
Journal of Applied Mathematics
network into the stability analysis of linear switched systems. By using the average dwell time approach and piecewise Lyapunov-Krasovskii functionals which are constructed
based on the descriptor model transformation and decomposition technique of coefficient matrix, new delay-dependent
synchronization criteria was derived in terms of linear matrix
inequalities. Numerical examples were given to illustrate that
the derived criteria are less conservative than some existing
results.
Acknowledgments
The first author is supported by the Center of Excellence in
Mathematics, Thailand, and Commission for Higher Education. The second author is supported by the Center of
Excellence in Mathematics, Thailand and Commission for
Higher Education, Thailand. The authors also wish to thank
the National Research University Project under Thailand’s
Office of the Higher Education Commission for financial
support. They would like to thank Professor Xinzhi Liu for
the valuable comments and suggestions.
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