FiniteTime Nonfragile Synchronization of Stochastic Complex Dynamical Networks with SemiMarkov Switching Outer Coupling
World Journal
FiniteTime Nonfragile Synchronization of Stochastic Complex Dynamical Networks with SemiMarkov Switching Outer Coupling
Rathinasamy Sakthivel 0 1
Ramalingam Sakthivel 2
Boomipalagan Kaviarasan 2
Chao Wang 3
YongKi Ma 4
Hiroki Sayama
0 Department of Mathematics, Bharathiar University , Coimbatore 641046 , India
1 Department of Mathematics, Sungkyunkwan University , Suwon 440746 , Republic of Korea
2 Department of Mathematics, Anna University Regional Campus , Coimbatore 641046 , India
3 Department of Mathematics, Yunnan University , Kunming, Yunnan 650091 , China
4 Department of Applied Mathematics, Kongju National University , Chungcheongnamdo 32588 , Republic of Korea
The problem of robust nonfragile synchronization is investigated in this paper for a class of complex dynamical networks subject to semiMarkov jumping outer coupling, timevarying coupling delay, randomly occurring gain variation, and stochastic noise over a desired finitetime interval. In particular, the network topology is assumed to follow a semiMarkov process such that it may switch from one to another at different instants. In this paper, the random gain variation is represented by a stochastic variable that is assumed to satisfy the Bernoulli distribution with white sequences. Based on these hypotheses and the LyapunovKrasovskii stability theory, a new finitetime stochastic synchronization criterion is established for the considered network in terms of linear matrix inequalities. Moreover, the control design parameters that guarantee the required criterion are computed by solving a set of linear matrix inequality constraints. An illustrative example is finally given to show the effectiveness and advantages of the developed analytical results.

1. Introduction
During the past twenty years, the investigation of complex
dynamical networks (CDNs) that consist of a huge number
of interacting dynamical nodes has received a great deal of
attention from various science and engineering areas, such as
social networks, ecological preypredator networks, protein
networks, power grids, and ecosystems [1, 2]. It should be
mentioned that the analysis of dynamical behaviors of CDNs
has become a hot research topic in recent years. Among
many dynamical behaviors, synchronization phenomenon
is the most important behavior and several interesting and
efficient methodologies have been developed in the literature
to solve the synchronization problem of various kinds of
CDNs; for instance, see [3?7]. In [4], the problem of inner
synchronization of a CDN has been investigated by
considering two different types of guaranteed cost dynamic feedback
controllers, where the control gains corresponding to two
feedback controllers have different dimensions subject to the
topological structure of the CDN. In [5], the problem of
outer synchronization between two hybridcoupled delayed
dynamical networks has been discussed by using the
aperiodically adaptive intermittent pinning control, where a simple
and elegant pinnednode selection scheme is proposed to
achieve the required result. It is worth mentioning that the
CDN representing realtime systems is generally affected by
external noise factors or stochastic disturbances [8]. Thus,
the consideration of external noise factors in the study of
synchronization of CDNs is of great importance in the
viewpoints of both theoretical and practical. By taking this
fact into account, in recent years, research communities
have eagerly investigated the problems of synchronization of
CDNs with external stochastic disturbances, see for
example [9?12]. Sakthivel et al. [10] presented some sufficient
conditions that ensure the synchronization and solve the
state estimation problem of discretetime stochastic complex
networks in the presence of uncertain inner couplings, where
the interval matrix approach is employed to characterize the
uncertainties encountered in the inner coupling terms. Li
et al. [11] developed a new synchronization criterion for a
class of discretetime stochastic complex networks subject to
partial mixed impulsive effects, where by using the Lyapunov
stability theory and the variation of parameters formula, the
required criterion is obtained.
It should be pointed out that the interconnection
topology among the nodes of CDNs plays a significant role in the
study of synchronization problem. In the existing literature,
there have been two reported kinds of interconnection
topologies, which are constant/fixed topology [13] and
timevarying topology [14]. In [14], it has been illustrated that the
timevarying interconnection topology is more general than
the fixed one. Nevertheless, in most of the real networks, the
connectivity of the network topology might be unfixed or
randomly changing due to new creation or link failures. In
order to tackle these types of issues, it is more appropriate
to model the CDNs with randomly switching topologies that
are governed by a Markov process. Based on this scenario
and following the seminal works reported in [15, 16], some
interesting and significant results about synchronization of
CDNs with Markov jump topologies have been discussed;
see [17?19]. Specifically, in [18], the problem of
nonfragile synchronization for a class of discretetime complex
networks subject to Markov jumping switching topology
has been investigated in a unified framework that includes
the nonfragile H? synchronization control and nonfragile
l2?l? synchronization control problem as its special cases.
However, it is worth pointing out that the aforementioned
papers have considered constant transition rates in the
Markov process. It should be mentioned that the Markov
process might consist of timevarying transition rates when
modeling practical systems. Such kind of process is known
as semiMarkov process and few interesting results regarding
semiMarkov jump systems have been addressed in recent
literature [20?23]. Interestingly, in [21], by employing the
supplementary variable technique and plant transformation,
the state estimation and sliding mode control problems
have been investigated for semiMarkovian jump systems
in the presence of mismatched uncertainties. Apparently,
semiMarkov jump systems are comparatively more general
than the traditional Markov jump systems [24]. Following
the aforementioned seminal works, the concept of
semiMarkov process has further been employed in the network
topology to obtain the synchronization criteria for CDNs (see
[25, 26] and the references therein). However, only very few
results about the synchronization of CDNs with semiMarkov
jump topology have been reported in the literature, which
stimulates us to do this present work.
It is worth mentioning that most of the available
controller design approaches have a predominant assumption
that the designed controller can be implemented accurately.
But in some real situations, such an assumption is not always
true as the controllers are often very sensitive or fragile to
their parameters? variations. Furthermore, it should be noted
that a small perturbation in controllers may lead to
undesirable oscillatory behavior or even instability [27]. Hence, it is
desirable as well as necessary to ensure the insensitivity of the
controller to certain parameter perturbations without loss of
the robust stability and thus, the investigation of nonfragile or
resilient controller design that has been capable of tolerating
some level of controller parameter gain variations has been
enormously increased in recent years [28?31]. To mention a
few, in [28], a robust resilient control problem of
discretetime Markov jump nonlinear systems has been solved by
employing the linear matrix inequality and stochastic analysis
techniques; in [29], based on the dissipative theory and
the eventtriggered sampling scheme, the nonfragile control
design problem for a class of networkbased singular systems
with input timevarying delay and external disturbances has
been addressed. Therefore, it is reasonable to consider the
nonfragile control design in the study of synchronization
of CDNs. It is noteworthy that only few research papers
regarding the nonfragile control design for achieving the
synchronization of CDNs have been published; see [32, 33].
On the other hand, it is worth mentioning that most of the
existing results based on the classical control theory dealt
with the asymptotic property of control system trajectories
over an infinitetime interval and did not possess any
restriction to the system states. But in many practical problems, it
is required that the described system state does not exceed
a certain bound during a fixed finitetime interval [34].
According to this fact, a great number of interesting results
on finitetime control design have been proposed for the
synchronization of various CDNs; for instance, see [35?38].
To the best of our knowledge, however, the problem of robust
nonfragile synchronization has not been fully investigated for
a class of CDNs over a prescribed f initetime interval.
Motivated by the above analysis, in this paper, we focus
on the finitetime nonfragile synchronization problem is
investigated for a class of CDNs subject to semiMarkov jump
topology and stochastic noises. More precisely, a new
delaydependent sufficient condition under which the considered
CDNs are synchronized to the target network within a given
finitetime interval is developed in terms of linear matrix
inequalities by utilizing the Lyapunov stability theory and
the stochastic analysis techniques. Subsequently, based on
the developed condition, a design algorithm of the proposed
nonfragile state feedback controller that can ensure the
finitetime stochastic synchronization of the addressed network
is presented. Eventually, a numerical example is shown to
illustrate the effectiveness of the proposed theoretical results.
The rest of this paper is organized as follows: in Section 2,
the problem formulation of the network model under study
and the preliminaries required to obtain the main results
are given. T he f initetime stochastic synchronization criterion
for the considered network model is presented in Section 3.
A numerical example and its simulations are provided in
Section 4. Conclusion of this paper is given in Section 5.
2. Problem Formulation and Preliminaries
In this paper, we consider a class of complex dynamical
networks (CDNs) with semiMarkov jump outer coupling
and stochastic noise, which consists of identical nodes
and is defined over the Wiener process probability space
(?,F, P), where ? is the sample space, F is the algebra of
events and P is the probability measure defined on F. Such a
network model can be described in the following form:
where () ? R denotes the state vector of the th node;
is a known real constant matrix with suitable dimension;
(?, ?) ? R represents a nonlinear vectorvalued function;
the constant > 0 is the coupling strength of network;
(()) are the elements of the outer coupling matrix (())
which describes the network topological structure and is
assumed to follow a semiMarkov process () which to be
defined later. In particular, (()) is defined as follows:
if there exists a connection between node and node ,
then (()) > 0 ; otherwise, (()) = 0. Further, the
diagonal elements of the outer coupling matrix are given
as follows: (()) = ? ?=1, =? (()); ? represents the
inner coupling matrix and is a positive diagonal matrix with
appropriate dimension; () ? R is the control input of the
th node which to be defined later; the function (?, ?, ?) : R ?
R ? R ? R is the noise intensity vectorvalued function;
() is a 1dimensional Brownian motion defined on the
probability space (?,F, P) with E{()} = 0, E{ 2()} = 1
and E{()()} = 0 for =? , where E is the mathematical
expectation; () is the timevarying delay function satisfying
0 ? 1 ? () ? 2 < ? and (? ) ? < 1 , where 1, 2, and
are known scalars; and () denotes the initial value of the th
node?s state and is assumed to be a continuous vectorvalued
function.
Now, let us define the semiMarkov jump process of
the outer coupling matrix. The process {(), ? 0} is a
continuoustime homogeneous semiMarkov process with
right continuous trajectories and takes values in a finite set
S = {1, 2, . . . N,}. More precisely, () is associated with the
transition probability matrix ? = [ (?)]N?N which is given
by the following transition rates:
Prob { ( + ? ) = 
( ) = }
{
= {
{1 +
(?)? + (?),
if ?= ,
(?)? + (?), if = ,
where ? > 0 is the sojourn time, lim??0((?)/?) = 0 and
(?) ? 0for ?= is the transition rate from mode at time
to mode at time + ? and (?) = ? ?=N1, ? = (?). For
notational simplicity, we hereafter denote the semiMarkov
process parameter () by . For example, (()) is denoted
by .
To synchronize all the identical nodes in the network
(
1
) to a common value, let us define the synchronization error
vector as () = () ? (), where () ? R is the state
vector of the unforced isolated node that can be expressed as
() = [() + (, ())] and is assumed to be noisefree,
that is, (, (), ( ? ())) = 0. Based on this error vector,
we now choose a robust state feedback controller to achieve
the synchronization of network (
1
), which is insensitive to the
uncertain perturbations or gain fluctuations and of the form:
( ) = (
+
( )?
( )) ( ),
= 1, 2, . . . , ,
(
3
)
where is the feedback controller gain matrix that is to
be determined in the forthcoming section, ? () is a
timevarying matrix representing the controller gain fluctuations,
and () is a stochastic variable describing the randomly
occurring controller gain fluctuations. It is here assumed
that ? () takes the form ? () = ? () , where
and are known real constant matrices and ?p() is
an unknown timevarying matrix satisfying ? ()? () ? .
Further, it is assumed that the stochastic variable () obeys
the Bernoulli distribution with the following probability
rules: (i) Prob{() = 1} = E{()} = and (ii) Prob {() =
0} = 1 ? E{()} = 1 ? , where ? [0, 1].
Then, by using (
1
) and (
3
), the closedloop form of the
error system can be obtained as follows:
+
( )?
( )) ( )+ ( ,
( ))
( ) = [( +
(
4
)
? S, = 1, 2, . . . , ,
where (, ()) = (, ()) ? (, ()) and (?, (), ( ?
())) = (, (), ( ? ())) ? (, (), ( ? ())) . By
using the Kronecker product properties and mathematical
manipulations, the error system (
4
) can be written in the
following compact form:
( ) = [(( ? ) + (
+ (
?
? ( )
?
)
)
+ ( ( )? )(
?
? ( )
)) ( )+
(, ( )) (
5
)
(
2
)
+ (
? ?) ( ?
( ))] + ?(, ( ), (
? ( )))
( ),
where () = [ 1 (), 2 ( ), . . . , ()] , (, ()) = [ (,
1()), (, 2()), . . . , (, ())] , and (?, () , (?
())) = [(?, 1(), 1( ? ())), (?, 2(), 2( ? ())), . . .,
(?, (), ( ? ()))] .
In order to develop the main results, the following
assumptions and definition are required.
Assumption 1. For the nonlinear function (?, ?), there exists
a known real constant matrix G such that ?(, ())? ?
G? ()? for any () ? R .
Assumption 2. The noise intensity function (?, ?, ?) : R+ ?
R ? R ? R is uniformly Lipschitz continuous in
terms of the following inequality of trace inner
product: trace{ (, (), ( ? ()))(, (), ( ? ()))} ?
() () + ( ? ()) ( ? ()), where and ( =
1, 2, . . . , ) are known nonnegative constants.
Assumption 3. For each ? S, all the real parts of
eigenvalues of (()) are negative except an eigenvalue 0
with multiplicity 1, which means that the reverse of the graph
generated by the matrix (()) contains a rooted spanning
directed tree for every ? S.
Definition 4 (see [34]). The considered network (
1
) is said
to be stochastically synchronized in finitetime with respect
to ( 1, 2, 2, ?, ) if there exist positive definite matrix
( ? S)and positive constants ?, 1, 2 with 2 > 1 such
that the following condition holds:
E {
( 0) (
Based on the LyapunovKrasovskii stability theory, this
section aims to develop a new set of delaydependent sufficient
conditions that can guarantee the stochastic synchronization
of the considered network model (
1
) over a finitetime
interval. Moreover, based on these conditions, a design of the
robust nonfragile state feedback control (
3
) for the network
model under consideration is provided in terms of linear
matrix inequalities (LMIs).
Theorem 5. Consider the network model (
1
) with Assumptions
1?3. For given positive scalars 1, 2, ?, , , , ? [0, 1 ],
1, 2, symmetric matrix ( ? S), and diagonal matrices
3, 4, the considered network (
1
) is stochastically synchronized
in finitetime under the nonfragile controller (
3
), if there exist
symmetric matrices > 0 ( ? S), > 0 ( = 1, 2, 3,)
V > 0 (V = 1, 2)and positive scalars ( ? S), 1 such that
the following matrix inequalities hold:
?
< 0,
]
]] < 0,
]
where
4 = max { max (
? ?3 ) , ? S} ,
5 = max { max (
( )(
3
( , , ) = ?
=1
( )(
?
) ( ),
( )(
+ ?
+ ?
?()
= 1 ?
0
?
? 1 +
(
, , )
= m ( , , )
and the rest of elements of ?
(?)are zero.
Proof. To develop the finitetime stochastic synchronization
criterion for the network model (
1
), it is enough to establish
the finitetime stochastic stability criterion for the
closedloop error system (
5
). For this purpose, we select the
LyapunovKrasovskii functional as follows:
where
( , , ) ,
(
11
)
( )(
where m( , , ) = m 1( , , ) + m 2( , , ) + m 3( , , )
and ( , , ) = ( , , )/ .
Now, by calculating the time derivative of ( , , ) along
the solution trajectories of the error system (
5
), we can get
m 1 ( , , ) = 2
( )(
?
) [( ? ) + (
?
? (
? ( ))+
) + ( )(
?
? ( )
) + ( ( )? )
?
? ( )
) ( )+ (
? )
? ( )+ trace { (, ( ), ( ?
( )))(
?
)
? (, ( ), ( ?
( )))},
m 2 ( , , ) ?
( )((
? 1) + (
( )(
? 1)
( )(
? 2) ( ).
Further, by applying Jensen?s single integral inequality [6]
to the integral terms in (
16
), we can get the following
inequalities:
( )(
( )(
trace { (, ( ), ( ?
( )))(
?
)
? (, ( ), ( ?
( )))} ?
? trace { (, ( ), ( ?
( )))
? (, ( ), ( ?
( )))} ?
(
( )(
? 3)
? ( )+
( ?
( ))(
? 4) ( ?
( ))) ,
(
14
)
(
15
)
(
16
)
(
17
)
(
18
)
where ( ? S) are positive scalars and 3, 4 are known
constant matrices.
Moreover, according to Assumption 1, we can obtain the
following inequality:
( )(
and the elements of ? (?), , and are defined in the
theorem statement. Moreover, based on Lemma 2 in [6],
for any positive scalar 1, the righthand side of (
20
) can
equivalently be written as
.
?1/2
=
?1/2, ?
Based on the Schur complement, it is noted that (
22
) is
equivalent to the lef thand side of (
8
). T hus, it can be observed
that E{m( , , )} < 0 if the LMIs (
7
) and (
8
) hold.
Furthermore, if there exists a constant > 0, it yields
that E{m( , , )} < E{( , , )} . From which, it can
be obtained that E{? ( , , )} < E{( 0, 0, 0)}, where
0 = (0). Next, define the following new parameters: ? =
?1/2 ?1/2, ?1 ?1/2 1 ?1/2, ?2 = ?1/2 ?1/2,
?3
?1/2
?
= 3 1 = 1 2 =
2 ?1/2. Then, it follows from condition (
11
) and 0 ?
? that
?1/2
2
?1/2, ?
Then, by combining (
13
)?(
19
) and taking mathematical
expectation, it can be obtained that
, , )}
where
? E {m ( , , )} ? E { ( )
2
+ 1 ?
0
( )(
(
20
)
(
21
)
(
23
)
=
(max { max (
? ? )}
?
+ 1 max { max (
+ 2 max { max (
+ 2 max { max (
+ 13 max { max (
+ 132 max { max (
? 1
?
? 2
? ?1 )}
? ?2 )}
? ?3 )}
? ?1 )}
? ?2 )})
?
? E { ( )(
?
) ( )} ? 1
It is clear to see that the inequality (
25
) is the same as
that in (
9
) which is the desired condition. Hence, it can be
concluded that the closedloop error system (
5
) is
finitetime stochastically stable which means that the considered
network model (
1
) is stochastically synchronized within
a prescribed finitetime interval. Thus, the proof of this
theorem is completed.
Theorem 6. Consider the network model (
1
) with Assumptions
1?3. For given positive scalars 1, 2, ?, , , , ? [0, 1], 1, 2,
symmetric matrix ( ? S)and diagonal matrices 3, 4, the
considered network (
1
) is stochastically synchronized in
finitetime under the nonfragile state feedback controller (
3
), if there
exist symmetric matrices > 0 ( ? S), ? > 0 ( = 1, 2, 3),
?V > 0 (V = 1, 2), any matrices ( ? S) with appropriate
dimensions and positive scalars ? , ( ? S), , ( = 1, 2, 3 ),
V (V = 1, 2), 1 such that the following matrix inequalities
hold:
? ?
> 0,
? S,
?
? (?) = [??
?1
<
<
?1
,
0 < ?
<
0 < ?V < V
?1
?1
,
,
? ) (
?
) + (
?
) (
+ (
?
) + (
?
) + (
? ?1 )
? )
+ (
+
??13 = (
??15 = (
??18 = (
??19 = (
??110 = (
??111 = (
(?)(
?
?
?
?
?
) ,
?
and the remaining parameters of ?? (?)are zero. Moreover, if
the obtained LMIs are feasible, then the desired state feedback
controller gain matrices in (
3
) are computed by = ?1.
Proof. Let
=
?1 and pre and postmultiply the matrix
(
?
? (?) by diag{??(????????????)?,?.??.?.?,??(????????????)??,?(????????????? ), (
4
), ( ? ), ( ? ), ( ?
the following new variables:
)}. Now, we introduce
= ? ( = 1, 2, 3),
?
V = ?V (V = 1, 2), and = .
Moreover, from Theorem 5, it is noticed that 7 < (
?1/2 ?1/2) < 1, 0 < ( ? ?1/2 1 ?1/2) <
?1/2 ?1/2 ?1/2 ?1/2
?
2, 0 <
2
) < 3, 0 < (
?
( ? ?1/2 1 ?1/2) < 5 and 0 < ( ? 2 ) < 6.
According to the congruence transformation, these relations
can be changed into ?11( ? ?1) < ( ? ) < ?71( ?
0 < (
?1), 0 < (
? ? ) <
?2
? ?V ) < 7
V+4(
?2
7
?
+1 ( ?
?1) (V = 1, 2). Now, if we set
?1) ( = 1, 2, 3) and
7 = 1, ?11 = , +1 ? ( = 1, 2, 3)and V+4 ? V (V = 1, 2),
then the constraints in (
28
) can easily be deduced. Moreover,
the conditions in (
26
), (
27
), and (
29
) can be obtained from (
7
),
(
8
), and (
9
), respectively, which are the desired conditions.
Hence, the proof is completed.
3
?1/2
) < 4, 0 <
?1/2
Remark 7. It should be mentioned that the constraints
in (
27
) are cannot be solved directly via MATLAB LMI
control toolbox due to the existence of the timevarying
terms ?=N1 (?). To overcome this difficulty, the transition
rates (?) are assumed to be bounded and satisfy
? ? (?) ? + , since they are partially measurable in
practice which is mentioned in [26]. Moreover, in this case,
the following assumptions are made as in [26]:
,
N
?
=1
, ,
= 1,
Now, we are able to present the suf f icient conditions
guaranteeing the stochastic synchronization of the
considered network (
1
) over a finitetime interval in terms of LMIs
in which all the elements are either constants or constant
matrices according to Remark 7. Thus, we have the following
theorem.
Theorem 8. For given positive scalars 1, 2, ?, , , , ?
[0, 1], 1, 2, symmetric matrix ( ? S) and diagonal
matrices 3, 4, the considered network (
1
) with Assumptions
1?3 is stochastically synchronized in finitetime under the
nonfragile controller (
3
), if there exist symmetric matrices >
0 ( ? S), ? > 0 ( = 1, 2, 3,) ?V > 0 (V = 1, 2), any
matrices ( ? S)with appropriate dimensions and positive
scalars ? ( ? S), , ( = 1, 2, 3), V(V = 1, 2), 1 such that
the following matrix inequality, (
26
), (
28
), and (
29
) hold:
? ) (
+ (
?
) + (
and the remaining elements of ??, are the same as those
defined in Theorem 6. Further, the nonfragile state feedback
controller gain matrices in (
3
) are calculated by = ?1.
Proof. Based on Remark 7, the timevarying element (?)
may take values in the interval [ ? , + ]. T hen, by using (
31
)
and following the similar lines in the proof of Theorem 6, it is
easy to obtain the inequality (
32
) which completes the proof.
Remark 9. It should be noted that, so far in the literature,
several control approaches have been proposed for the
synchronization problem of several CDNs [3?6], wherein the
interconnection topology among the nodes are assumed to
be fixed. However, in practice, this assumption is practically
dif f icult or even impossible. However, yet now, there were
no results reported in the existing literature for the
synchronization analysis of stochastic CDNs with switching topology.
According to this fact, in this paper, finitetime
synchronization problem of stochastic CDNs with switching topology is
investigated. Furthermore, due to random behavior in the
dynamics of stochastic CDNs, it is very difficult to determine
the exact fixed control value. Therefore, in this paper, the
feedback control gain is considered with uncertain terms,
which is more significant to reflect the realistic scenarios.
4. An Illustrative Example
This section provides an illustrative example to verify the
developed theoretical results in the previous section. For the
sake of simplicity, consider a class of CDNs in the form of
(
1
) with five identical nodes and the state vector of each node
being threedimensional, that is, = 5 and = 3 .
Let us select the network matrix and the nonlinear
function as
1
?1.4
0
It is clear to see that (, ()) satisfies Assumption 1 with
G = diag {0.5, 0.5, 0.5}. In this example, we consider the
semiMarkov jump topology with two modes, whose connectivity
graph is shown in Figure 1. The inner coupling matrix is
assumed to be ? = diag {0.2, 0.5, 0.7}and the coupling
strength is chosen as = 0.5 . The timevarying delay is taken
as () = 0.5 + 0.5 sin() from which it can be obtained that
1 = 0, 2 = 1 and = 0.5 .
Based on Figure 1, the jumping coupling configuration
matrices (()) for () = 1, 2 can be expressed as
(
34
)
(
35
)
Moreover, the elements 12(?)and 21(?)of transition rate
matrix are assumed to lie in the intervals [0.1, 2]and [0.8, 1.7],
respectively. So, in light of (
31
), the transition rates 12(?)
2
and 21(?)can be represented as 12(?) = ?=1 12, and
21(?) = ?2=1 21, , respectively, with 12,1 = 0.1, 12,2 =
2, 21,1 = 0.8, and 21,2 = 1.7. T he stochastic variable
representing the controller gain fluctuations is chosen as
() = 0.25 + 0.25 sin(). Furthermore, the uncertain
parameter matrices in the control gain are taken as
1
4
(a)
and ? () = sin (). The rest of parameters involved in the
simulation are set to be 1 = 1, 2 = 2, = 0.01, and ? = 1.
Then, by solving the LMIs (
26
), (
28
), and (
29
) in Theorem 6
along with (
32
) in Theorem 8 with the aid of MATLAB LMI
control toolbox, we can get a set of feasible solutions from
which the nonfragile state feedback control gain matrices can
be obtained as follows:
Here, our aim is to design the nonfragile state feedback
controller such that the considered network (
1
) is robustly
2
3
2
4
0.6
Time (sec)
synchronized with the target network within a desired
finitetime interval. For the simulation purposes, we set the initial
conditions for the states of the nodes and the isolated node as
follows: 1(0) = [10 11 10] , 2(0) = [9 11 10] , 3(0) =
and (0) = [9 8 9] . The noise intensity function is taken
as (, (), ( ? ())) = [0.1 sin( 1 ()) 0.1sin( 2 ( ?
())) 0.1 sin( 3 ( ? ()))] .
Based on these values, simulations are drawn in Figures
2?12. Specifically, the state responses of the first, second,
and third nodes together with the isolated node are plotted
in Figures 2, 3, and 4, respectively, wherein the dotted line
represents the isolated node and the dashed lines denote the
five identical nodes. It can easily be observed from these
figures that the states of the nodes are exactly synchronized
with the states of the isolated node within short period
which shows the efficiency of the proposed nonfragile control
strategy. Moreover, the corresponding error state responses
and the control response curves are given in Figures 5?7
and Figures 8?10, respectively. Further, Figure 11 shows the
jumping mode of the semiMarkov switching topology. In
10.5
10
9.5
8.5
9
8
7.5
0
0
0.2
s2(t)
x12(t)
x22(t)
0.2
s3(t)
x13(t)
x23(t)
0.6
Time (sec)
0.6
Time (sec)
0.6
Time (sec)
0.6
Time (sec)
0.8
1
addition, to realize the finitetime synchronization, the time
evolution of () () ( = 1, 2, 3, 4, 5), ( ? S)is depicted
in Figure 12. It can be seen from Figure 12 that the states of the
error system do not exceed the prescribed threshold 2 = 2,
which means that the synchronization of considered network
(
1
) is achieved within a given f initetime interval. T hus, it
can be concluded from the simulations that the designed
nonfragile control algorithm effectively works even in the
presence of stochastic noise and timevarying coupling delay.
5. Conclusion
In this paper, we have studied the robust finitetime
nonfragile synchronization problem for a class of CDNs with
semiMarkov jump outer coupling, timevarying coupling
delay, randomly occurring gain variation and stochastic
noise. In particular, we have considered the semiMarkov
switching topology to obtain the synchronization criterion.
Moreover, we have introduced a stochastic variable
satisfying the Bernoulli distribution to represent the random
gain variations in the controller design. By employing the
LyapunovKrasovskii stability theory and some stochastic
analysis techniques, we then have developed a new
finitetime stochastic synchronization criterion for the considered
network in terms of linear matrix inequalities and have
presented a design algorithm for the proposed nonfragile
state feedback controller to a solution of the obtained set
0
0.2
e13(t)
e23(t)
e33(t)
0.4
0.6
Time (sec)
0.8
1
0.5
1
0
1.5
3
2
1
0.5
0
0
0
0.2
0.6
Time (sec)
0.6
Time (sec)
0.6
Time (sec)
of linear matrix inequalities. At last, we have provided a
numerical example to verify the obtained theoretical results.
In addition, it should be pointed out that one of the future
research topics would be to investigate the problem of
finitetime mixed ? and passivity synchronization of stochastic
singular CDNs with semiMarkov switching outer coupling
delay and actuator saturation.
Conflicts of Interest
The authors declare that there are no conflicts of interest
regarding the publication of this paper.
0.4
0.6
Time (sec)
0.8
1
1
ei2(t) (i = 1, 2, 3, 4, 5)
c1 = 1
c2 = 2
e1(t)
e2(t)
2
?2
e3(t)
e4(t)
e5(t)
4
2
ei3(t) (i = 1, 2, 3, 4, 5)
0
Acknowledgments
The work of YongKi Ma was supported by the National
Research Foundation of Korea (NRF) grant funded by the
Korean Government (MSIP) (no. 2015R1C1A1A01054663).
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