IEEE TRANSACTIONS ON NEURAL NETWORKS AND LEARNING SYSTEMS, VOL. 23, NO.

5, MAY 2012

725

H State Estimation for Discrete-Time Complex

Networks With Randomly Occurring Sensor
Saturations and Randomly Varying Sensor Delays
Derui Ding, Zidong Wang, Senior Member, IEEE, Bo Shen, and Huisheng Shu

Abstract In this paper, the state estimation problem is investigated for a class of discrete time-delay nonlinear complex
networks with randomly occurring phenomena from sensor
measurements. The randomly occurring phenomena include
randomly occurring sensor saturations (ROSSs) and randomly
varying sensor delays (RVSDs) that result typically from networked environments. A novel sensor model is proposed to
describe the ROSSs and the RVSDs within a unified framework
via two sets of Bernoulli-distributed white sequences with known
conditional probabilities. Rather than employing the commonly
used Lipschitz-type function, a more general sector-like nonlinear
function is used to describe the nonlinearities existing in the
network. The purpose of the addressed problem is to design a
state estimator to estimate the network states through available
output measurements such that, for all probabilistic sensor
saturations and sensor delays, the dynamics of the estimation
error is guaranteed to be exponentially mean-square stable and
the effect from the exogenous disturbances to the estimation
accuracy is attenuated at a given level by means of an H -norm.
In terms of a novel LyapunovKrasovskii functional and the
Kronecker product, sufficient conditions are established under
which the addressed state estimation problem is recast as solving
a convex optimization problem via the semidefinite programming
method. A simulation example is provided to show the usefulness
of the proposed state estimation conditions.
Index Terms Complex networks, randomly occurring sensor
saturations, randomly varying sensor delays, state estimation.

I. I NTRODUCTION

OMPLEX networks are everywhere. Many phenomena

in nature can be modeled as coupled networks, such
as brain structures, proteinprotein interactions, social interactions, the Internet, and the World Wide Web. All such

Manuscript received July 25, 2011; revised December 19, 2011; accepted
January 30, 2012. Date of publication March 1, 2012; date of current version
May 2, 2012. This work was supported in part by the Engineering and
Physical Sciences Research Council of the U.K. under Grant GR/S27658/01,
the Royal Society of the U.K., the National Natural Science Foundation
of China under Grant 61028008, Grant 61134009, Grant 61104125, and
Grant 60974030, the Natural Science Foundation of Universities in Anhui
Province of China under Grant KJ2011B030, and the Alexander von Humboldt
Foundation, Germany.
D. Ding, B. Shen, and H. Shu are with the School of Information
Science and Technology, Donghua University, Shanghai 200051, China
(e-mail:
deruiding2010@gmail.com;
shenbodh@gmail.com;
hsshu@dhu.edu.cn).
Z. Wang is with the Department of Information Systems and Computing,
Brunel University, Uxbridge UB8 3PH, U.K., and also with the School of
Information Science and Technology, Donghua University, Shanghai 200051,
China (e-mail: zidong.wang@brunel.ac.uk).
Color versions of one or more of the figures in this paper are available
online at http://ieeexplore.ieee.org.
Digital Object Identifier 10.1109/TNNLS.2012.2187926

networks can be represented in terms of nodes, edges, and

coupling strengths, indicating complex connections between
the nodes. Typical complex networks that have been thoroughly investigated include communication networks, social
networks, electrical power grids, cellular and metabolic networks, and the Internet. Random graphs are known to be able
to describe the large-scale networks with no explicit design
principles, and so the early study on complex networks has
been the territory of graph theory since the seminal work in
[1] and [2]. Recently, due to the discovery of the smallworld and scale-free properties [3], [4], the dynamical
behaviors of complex networks have attracted ever-increasing
research interest from a variety of communities such as
mathematicians, statisticians, computer scientists, and control
engineers. As a result, a number of dynamic analysis issues
have been extensively investigated for complex networks, such
as the stability and stabilization, synchronization, pinning
control, and spread mechanism, see [5][23] and the references
therein.
In the past decade, special attention has been paid to the
stability and synchronization problems of various complex networks. Generally speaking, there are mainly two approaches
that shed insightful light on the stability and synchronization
phenomena in various real-world complex networks. The first
one is the matrix eigenvalue analysis method that has been
widely applied (see [11], [14], [15], [18], [19], [22]) for
pinning-control or impulsive-control problems. For example,
in [11], the important concept of virtual control has been
proposed to show, in a nice way, that the pinned nodes
virtually control other dynamical nodes through coupling
and eventually lead to the synchronization of the whole
network. The other approach is the linear matrix inequality
(LMI) technique which has recently been adopted (see [8],
[9], [12], [23][26]) for complex networks with or without
time delays. For instance, in [24], the relationship between
the stability of the whole network and the stability of its
corresponding subsystems has been discussed, and a special
decentralized control strategy has been employed to derive
some necessary and sufficient conditions for the stability
and stabilizability for linear networks. Furthermore, in [26],
an attempt was made to address the synchronization problem for stochastic discrete-time complex networks with time
delays.
The vast literature on the stability and synchronization
problems of complex networks has implicitly assumed that

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IEEE TRANSACTIONS ON NEURAL NETWORKS AND LEARNING SYSTEMS, VOL. 23, NO. 5, MAY 2012

the states of the complex networks under investigation are

fully accessible. This is, unfortunately, not always the case in
practice. For example, as a typical example of complex networks, wireless sensor networks exhibit complicated coupling
between the sensor nodes and network-induced phenomena
such as random packet dropouts, random sensor saturations,
and random sensor communication delays. These phenomena,
together with the large scale of the networks, often give rise
to the unavailability of part of the sensors, i.e., only partial
information from the sensor measurements is available. In
this case, it becomes necessary to estimate the states of the
nodes through partial but available measurements, and then
use the estimated node states to carry out specified tasks
such as dynamics analysis and synchronization control. The
state estimation problem for various complex networks has
recently drawn particular research attention, see [26][28].
For instance, in [26], by using a novel LyapunovKrasovskii
functional and the Kronecker product, the state estimation
problem has been studied for an array of discrete-time complex
networks with the simultaneous presence of both discrete and
distributed time delays. Very recently, the state estimation
problem over a finite horizon has been investigated in [28]
for a class of time-varying complex networks in terms of a
new concept called H -synchronization, and the estimator
gain has been obtained by utilizing recursive linear matrix
inequalities.
For the purpose of estimating the network states in reality,
the available measurement outputs are collected from all
sensors, which are then processed to minimize the effects from
the possible noise and various kinds of incomplete information such as the missing measurement and communication
delays. It is now well known that sensors cannot provide
signals of unlimited amplitude due primarily to physical or
technological constraints. This phenomenon is referred to as
sensor saturation. Saturation brings in nonlinear characteristics which can severely restrict the application of traditional estimator design schemes. Specifically, this kind of
characteristics not only degrades the estimation performance
that can be achieved without saturation but may also lead to
undesirable oscillatory or even unstable behavior. Because of
the practical importance of sensor saturations, much attention
has been given to the filtering and control problems for
systems with sensor saturation [29], [30]. In most existing
literature, the saturation is actually assumed to occur definitely. Such an assumption is, however, not always true. For
example, in a network environment, the sensor saturations
may occur in a probabilistic way and the saturation level
may be randomly changeable as well. This is mainly due to
the random occurrence of network-induced phenomena such
as random sensor failures leading to intermittent saturation,
sensor aging resulting in changeable saturation level, sudden
environment changes, etc. Such a phenomenon of sensor saturation, namely, randomly occurring sensor saturation (ROSS),
has been largely overlooked in the area probably because of the
difficulty in their mathematical analysis. The main motivation
of the present paper is, therefore, to investigate how ROSS
influences the performance of state estimation for complex
networks.

In addition to the appearance of ROSSs, the sensor measurement delay results in another common phenomenon that
occurs in a random way especially when the sensors are
connected via communication networks. Sensor delays may
be induced by a variety of reasons such as an asynchronous time-division-multiplexed network, intermittent sensor
failures, random congestion of packet transmissions, etc. Such
phenomena are customarily referred to as randomly varying
sensor delays (RVSDs), see [31] and [32] for more details.
In many cases, the RVSDs are a source of instability and
performance deterioration of a complex network equipped
with a large number of sensors. One of the most popular
ways to describe the RVSDs is to use a Bernoulli-distributed
(binary switching) white sequence specified by a conditional
probability distribution in the sensor output. This approach
was first proposed in [33] to deal with the optimal recursive
estimation problem. Recently, it was used in [32] for filtering
problems and in [34] for control designs. Obviously, to reflect
the network reality, it makes practical sense to consider both
the ROSSs and RVSDs where their occurrence probabilities
can be estimated via statistical tests. To date, to the best of
our knowledge, the estimation problem for complex networks
with both ROSSs and RVSDs remains an open yet challenging
issue, and the main purpose of this paper to narrow such a
gap. It is worth pointing out that the main difficulty lies in
how to establish a unified framework to account for the two
phenomena of ROSSs and RVSDs.
Summarizing the above discussions, the focus of this paper
is on the state estimation problem for a class of discrete timedelay complex networks with randomly occurring phenomena including ROSSs and RVSDs that result typically from
networked environments. Two sets of Bernoulli-distributed
white sequences with known conditional probabilities are
introduced to describe the ROSSs and the RVSDs within a
unified framework. A general sector-like nonlinear function
is employed to describe the inherently nonlinear nature of
the complex networks. By employing the Lyapunov stability
theory combined with the stochastic analysis approach, a
delay-dependent criterion is established that guarantees the
existence of the desired estimator gains, and then the explicit
expression of such estimator gains is characterized in terms
of the solution to a convex optimization problem via the
semidefinite programming method. Moreover, a simulation
example is provided to show the effectiveness of the proposed
estimator design scheme. The main contribution of this paper
is twofold: 1) novel sensor model is established to account
for both the ROSSs and RVSDs in a unified framework; and
2) based on this sensor model, the estimator design approach
is proposed to ensure that the error dynamics is exponentially
mean-square stable and the H performance constraint is
satisfied.
The rest of this paper is organized as follows. In Section II,
a class of discrete time-delayed complex networks with both
the ROSSs and RVSDs is presented. In Section III, by employing the Lyapunov stability theory, some sufficient conditions
are established in the form of LMI and then the explicit
expression of the estimator gains is given. In Section IV,
an example is presented to demonstrate the effectiveness

DING et al.: H STATE ESTIMATION FOR DISCRETE-TIME COMPLEX NETWORKS

of the results obtained. Finally, conclusions are drawn in

Section V.
Notation: The notation used here is fairly standard except
where otherwise stated. Rn and Rnm denote, respectively,
the n-dimensional Euclidean space and the set of all n m
real matrices. The set of all positive integers is denoted by N.
l2 ([0, ); Rn ) is the space of square-summable n-dimensional
vector functions over [0, ). I denotes the identity matrix
of compatible dimension. The notation X Y (respectively,
X > Y ), where X and Y are symmetric matrices, means
that X Y is positive semidefinite (respectively, positive
definite). M T represents the transpose of M. Sym{A} denotes
the symmetric matrix A + A T . For a matrix A Rnn ,
max (A) and min (A) denote the maximum and minimum
eigenvalue of A, respectively. E{x} stands for the expectation
of stochastic variable x. ||x|| describes the Euclidean norm of
a vector x. The shorthand diag{M1 , M2 , . . . , Mn } denotes a
block diagonal matrix with diagonal blocks being the matrices
M1 , ..., Mn . The symbol denotes the Kronecker product. In
symmetric block matrices, the symbol is used as an ellipsis
for terms induced by symmetry.
II. P ROBLEM F ORMULATION AND P RELIMINARIES
Consider the following array of discrete time-delayed complex networks consisting of N coupled nodes:

x i (k + 1) = f (x i (k)) + g(x i (k (k)))

+
wi j x j (k) + L i v 1 (k),
(1)
j
=1

z i (k) =M x i (k),

x i (s) =i (s), s [M , 0], i = 1, 2, . . . , N

where x i (k) Rn is the state vector of the i th node,
z i (k) Rr is the output of the i th node, and v 1 (k) is
the disturbance input belonging to l2 ([0, ); Rq ).  =
diag{r1 , r2 , . . . , rn } is the matrix linking the j th state variable
if r j = 0, and W = (wi j ) NN is the coupled configuration matrix of the network with wi j 0 (i = j ) but
not all zero. L i and M are constant matrices with appropriate dimensions, and i (s) is a given initial condition
sequence.
The positive integer (k) describes the known time-varying
delay satisfying 0 < m (k) M , where m and M
are known positive integers representing the minimum and
maximum delays, respectively. The nonlinear vector-valued
functions f and g : Rn
Rn are assumed to be continuous
and satisfy f (0) = 0, g(0) = 0 and the sector-bounded
conditions
f

[ f (x) f (y) 1 (x y)]T

f

[ f (x) f (y) 2 (x y)] 0,

g

[g(x) g(y) 1 (x y)]T

g

[g(x) g(y) 2 (x y)] 0

f

(2)

for all x, y Rn , where 1 , 2 , 1 and 2 are real matrices

of appropriate dimensions.

727

In this paper, the N sensors with both saturations and delays

are modeled by
yi (k) = i (k)[i (k) (C x i (k)) + (1 i (k))C x i (k)]
+(1 i (k))[i (k) (C x i (k d))
+(1 i (k))C x i (k d)] + G i v 2 (k),
i = 1, 2, . . . , N

(3)

where yi (k) Rm is the measurement output of the node i ,

v 2 (k) is the disturbance input which belongs to l2 ([0, ); R p ),
the sensor delay d is a scalar satisfying 0 < d M , and
G i , C are known matrices with appropriate dimensions. The
saturation function : Rm
Rm is defined as
T

(x) = (x 1 ) (x 2 ) (x m )
(4)
where x i is the i th element of the vector x and (x i ) =
sign(x i )min{1, |x i |}. Here, the notation of sign denotes the
signum function. Later, we will slightly abuse the notation by
using to denote both the scalar-valued and vector-valued
saturation functions. Note that, without loss of generality,
the saturation level is taken as unity. The variables i (k)
and i (k) (i = 1, 2, . . . , N) are Bernoulli-distributed white
sequences taking values on 0 and 1 with the following
probabilities:


Prob{ki = 1} = i
Prob{ki = 1} = i
and
i
Prob{k = 0} = 1 i
Prob{ki = 0} = 1 i
where i , i [0, 1] are known constants. Throughout the
paper, the stochastic variables i (k) and i (k) are mutually
independent in all i .
Remark 1: As is well known, in an abstract model for
complex networks, the nodes in the same cluster usually
possess the same attributes or properties. For example, in a
homogeneous sensor network, the sensor nodes are typically
identical in terms of battery energy and hardware complexity.
An interesting topic for complex networks is to examine how
the nodes interact with each other to form a rich dynamics
through links according to a given topology dynamics. Therefore, it is reasonable to assume that the nodes information
is collected by means of the same type of measurements.
Moreover, without loss of generality, the disturbances in
measurements are set to be same. In the case of different
kinds of disturbances, similar results can be obtained readily
by using an augmented method.
Remark 2: Note that the zero-row-sum property of the
configuration matrix W is quite important for many traditional methods to deal with the dynamics analysis issues of
complex networks. By assuming the zero-row-sum property,
eigenvalue-based matrix analysis methods could be employed
to construct the difference of signals, see [11], [14][16],
[18], [19], [22], [26], [28] for more details. These methods,
however, are no longer valid for the problem addressed in
this paper because of the sensor saturation phenomenon. One
of the main contributions of this paper is the development
of a new methodology to deal with the phenomenon of
both ROSSs and RVSDs without requiring the zero-row-sum
property.

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IEEE TRANSACTIONS ON NEURAL NETWORKS AND LEARNING SYSTEMS, VOL. 23, NO. 5, MAY 2012

Remark 3: The measurement model proposed in (3) provides a novel unified framework to account for the phenomenon of both ROSSs and RVSDs. The stochastic variable i (k)
characterizes the random nature of sensor saturation, whereas
the stochastic variable i (k) is used to describe the phenomenon of the probabilistic sensor delay. By combining these
two stochastic variables, model (3) represents the following
four different phenomena: 1) when i (k) = 1 and i (k) = 0,
sensor i works normally; 2) when i (k) = 1 and i (k) = 1,
model (3) is reduced to yi (k) = (C x i (k)), which means that
the measurements received by sensor i are saturated; 3) when
i (k) = 0 and i (k) = 0, it can be seen from model (3) that
the measurements at previous d time instant are employed
by the estimator i instead of the one at current time instant;
and 4) when i (k) = 0 and i (k) = 1, model (3) implies
that the measurements are not only delayed but also saturated
before they enter into the estimator i . In addition, it is easy to
observe that the time delay in the measurements takes random
values as 0 when i (k) = 1 and d when i (k) = 0. Such
kind of phenomenon is referred to as the randomly varying
delays.
Based on the measurement yi (k), we construct the following
state estimator for node i :

xi (k + 1) = f (xi (k)) + g(xi (k (k)))

+ K i (yi (k) C xi (k)),

z i (k) =M xi (k),

(5)

xi (s) =0, s [M , 0], i = 1, 2, . . . , N

where xi (k) Rn is the estimate of the state x i (k), z i (k) Rr

is the estimate of the output z i (k), and K i Rnm is the
estimator gain matrix to be designed.
For simplicity, we introduce the following notations:
x k = [ x 1T (k) x 2T (k) x NT (k) ]T ,
x k = [ x1T (k) x2T (k) x NT (k) ]T ,
z k = [ z 1T (k) z 2T (k) z TN (k) ]T ,
z k = [ z 1T (k) z 2T (k) z TN (k) ]T , z k = z k z k ,
v k = [ v 1T (k) v 2T (k) ]T , k = (k), ek = x k xk ,
f (x k ) = [ f T (x 1 (k)) f T (x 2 (k)) f T (x N (k)) ]T ,
g(x k ) = [ g T (x 1 (k)) g T (x 2 (k)) g T (x N (k)) ]T ,
 (C x k ) = [ T (C x 1 (k)) T (C x 2 (k)) T (C x N (k)) ]T ,
fk = f (x k ) f (xk ), g k = g(x k ) g(xk ),
K = diag{K 1 , K 2 , . . . , K N }, C = I C,
L = [ L T L T L T ]T , L = [ L K G ],
G=

1
[ G 1T

2
G 2T

N
G TN ]T ,

M = I M,

E i = diag{0, . . . , 0, I, 0, . . . , 0}.



i1

Ni

By using the Kronecker product, the error dynamics of the

state estimation can be obtained from (1), (3), and (5) as

follows:

k + (W  + K C)x
k

ek+1 = fk + g kk K Ce

N

kK

+ Lv
i (k)i (k)E i  (C x k )

i=1

N


+ (1 i (k))i (k)E i C x k

+
+

i=1
N


i=1
N


i (k)(1 i (k))E i  (C x kd )

(6)


(1 i (k))(1 i (k))E i C x kd ,

i=1

k.
z k = Me

Then, by setting k = [ x kT ekT ]T , we have the following

augmented system:

k+1 = W 1 k + W2 kd + fk + gkk

kd )

+H  (CSk ) + H   (CS

N

i
k)

+ i=1 ( k i )Gi  (CS

N

k
+ i=1
(ki i )Gi CS

N
kd )
(7)
+ i=1
( ki i )Gi  (CS

N

kd + Lv k ,
+ i=1 ( k i )Gi CS

=
M
,

k
k

i = [1T (i ), 2T (i ), . . . , NT (i ), 1T (i ), 2T (i ),

. . . , NT (i )]T , i [M , 0]
where
fk = [ f T (x k ) fkT ]T , gk = [ g T (x k ) g kT ]T ,
ki = i (k)i (k), ki = i (k)(1 i (k)),
ki = (1 i (k))i (k), ki = (1 i (k))(1 i (k)),

i = i i ,  = diag{ 1 I, 2 I, . . . , N I },
i = i (1 i ),  = diag{1 I, 2 I, . . . , N I },

i = i (1 i ),  = diag{ 1 I, 2 I, . . . , N I },
i = (1 i )(1 i ),  = diag{ 1 I, 2 I, . . . , N I },
H = [ 0 K T ]T , Gi = [ 0 E iT K T ]T ,
M = [ 0 M ], S = [ I 0 ],


W 
0
W1 =
,
W  + K (I  )C K C




0
0
L
0
W2 =
,
L
=
.
L K G.
K  C 0
As analyzed in [30] and [35], the saturation function
k ) satisfies
 (CS
T

[
(CS)
CS]
0
[
(CS)
(I )CS]

(8)

where  = diag{1 , 2 , . . . , m } and 0  I . Also, it

follows from (2) that
f
f
[ fk (I 1 )k ]T [ fk (I 2 )k ] 0,
g

[ gk (I 1 )k ]T [ gk (I 2 )k ] 0.

(9)

Definition 1 [36]: The augmented system (7) with v k = 0

is said to be exponentially mean-square stable if there exist
constants > 0 and 0 < h < 1 such that
E{||k ||2 } h k

max

i[M , 0]

E{||i ||2 }, k N.

DING et al.: H STATE ESTIMATION FOR DISCRETE-TIME COMPLEX NETWORKS

The purpose of this paper is to design a set of state

estimators of the form (5) for the complex networks (1) with
the sensor model (3) containing both ROSSs and RVSDs. More
specifically, we are interested in looking for the parameters
K i (i = 1, 2, . . . , N) such that the following requirements are
met simultaneously:
1) the zero-solution of the augmented system (7) with
v k = 0 is exponentially mean-square stable;
2) under the zero initial condition, for a given disturbance
attenuation level > 0 and all nonzero v k , the output
error z k satisfies


1 
E{||z k ||2 } 2
||v k ||2 .
N
k=0

(10)

k=0

Remark 4: In terms of (6) and (10), it can be seen that

the value of E{||z k ||2 } would become larger as the number of
the nodes increases. Theoretically, the disturbance attenuation
level for the overall network should account for the average
disturbance rejection performance which is insensitive to the
change of the number of the nodes in the estimator design.
For this purpose, the term of 1/N is used to accommodate the
average H index over the complex network so that the scalar
reflects the practical significance of the H disturbance
rejection level.
III. M AIN R ESULTS
In this section, the stability and the H performance
are analyzed for the augmented system (7). A sufficient
condition is given to guarantee that the augmented system
(7) is exponentially mean-square stable and the H performance is achieved for all probabilistic sensor saturations and
sensor delays. Then, the explicit expression of the desired
estimator gains is proposed in terms of the solution to certain matrix inequalities derived according to the obtained
condition.
Theorem 1: Let the estimator parameters K i (i =
1, 2, . . . , N) and the diagonal matrix  be given. The zero
solution of the augmented system (7) with v k = 0 is exponentially mean-square stable if there exist positive definite
matrices Q i (i = 1, 2, . . . , 6) and positive scalars j ( j =
1, 2, . . . , 4) satisfying

11 W1T P1 W2 0 14

22
0 W2T P1

33 0

1 =

44

(11)
T
T
16
W1 P1 H 
W1 P1

W2T P1 W2T P1 H 
27

gT

2 2
0
0



<0
P1 H
P1 H
P1



55
P1 H
P1 H

66
67

77

729

where
i = i (1 i )[1 i (1 i )],
i = (1 i )(1 i )[1 (1 i )(1 i )],

i = i (1 i )[1 i (1 i )], i = i i (1 i i ),

i = i

i
i

=
=

= i i (1 i )(1 i ),

i i2 (1 i ),
i2 i (1 i ),

= i (1 i )2 (1 i ),

i = i (1 i )(1 i )2 ,

1 fT f
f
f
f
f
1 = I Sym{ 1 2 }, 2 = I (1 + 2 )/2,
2
1 gT g
g
g
g
g
1 = I Sym{ 1 2 }, 2 = I (1 + 2 )/2,
2
g
P1 = diag{I Q 1 , I Q 2 }, 33 = P3 2 1 ,
P2 = diag{I Q 3 , I Q 4 }, 44 = P1 1 I,
P3 = diag{I Q 5 , I Q 6 }, 55 = P1 2 I,
11 = W1T P1 W1 P1 + P2 + (M m + 1)P3
f

1  3 S T C T (I )CS
1

N


+
(i + i + i + i )S T C T GiT P1 Gi CS,
i=1

14 = W1T P1 + 1 2 , 67 = T HT P1 H  ,

16 = W1T P1 H  + 3 S T C T (I  + I )/2,
fT

22 = W2T P1 W2 P2 4 S T C T (I )CS

+
27 =
66 =

77 =

N


( i + i + i + i )S T C T GiT P1 Gi CS,

i=1
W2T P1 H  + 4 S T C T (I  + I )/2,
T HT P1 H  3 I
N


+
( i + i + i + i )GiT P1 Gi ,
i=1
T T
H P1 H  4 I
N


+
( i + i + i + i )GiT P1 Gi .
i=1

Proof: Construct the following Lyapunov function for (7):

V (k) = V1 (k) + V2 (k) + V3 (k)

(12)

where
V1 (k) = k P1 k +

k1


i P2 i ,

i=kd

V2 (k) =

k1

i=kk

i P3 i , V3 (k) =

k
m


k1


i P3 i .

j =kM +1 i= j

Calculating the difference of V1 (k) along the trajectory of

(7) with v k = 0 and taking the mathematical expectation, we
have
E{V1 (k)} = E{V1 (k + 1) V1 (k)}
= E{k+1 P1 k+1 k P1 k + k P2 k kd P2 kd }

T
= E kT W1T P1 W1 k + kd
W2T P1 W2 kd

730

IEEE TRANSACTIONS ON NEURAL NETWORKS AND LEARNING SYSTEMS, VOL. 23, NO. 5, MAY 2012

T
+ fkT P1 fk + gk
P g
k 1 kk
T
T T
k)
+
(CSk ) H P1 H   (CS
kd ) T HT P1 H   (CS
kd )
+
T (CS

+
+
+
+

N

i=1
N

i=1
N

i=1
N


= E k P3 k kk P3 kk +

k )GiT P1 Gi  (CS
k)
i  T (CS

k
m


i P3 i

i=kk+1 +1

k1


k1


i P3 i

i=kk +1

i P3 i

i=km +1

k
m


k
i kT S T C T GiT P1 Gi CS

E k P3 k kk P3 kk +

kd )GiT P1 Gi  (CS
kd )
i  T (CS

E{V
3 (k)} = E{V3 (k + 1) V3 (k)}

m +1 
k
m
k
k1
k



E
i P3 i
i P3 i

j =kM +2 i= j
j =kM +1 i= j

m
k
m
k
k1
k





E
i P3 i
i P3 i

j =kM +1 i= j +1
j =kM +1 i= j

k

(k P3 k j P3 j )
E

j =kM +1

k
m


(15)
i P3 i .
E (M m )k P3 k

i P3 i

i=kM +1

(14)

T
kd
i kd
S T C T GiT P1 Gi CS

i=1

+2kT W1T P1 W2 kd + 2kT W1T P1 fk

k)
+2kT W1T P1 gkk + 2kT W1T P1 H   (CS
T
T

T
kd ) + 2kd W2T P1 fk
+2k W1 P1 H  (CS

T
T
k)
+2kd
W2T P1 gkk + 2kd
W2T P1 H   (CS
T
kd ) + 2 fkT P1 gkk
+2kd
W2T P1 H   (CS

k ) + 2 fkT P1 H   (CS
kd )
+2 fkT P1 H   (CS
T
k)
+2 gk
P1 H   (CS

i=kM +1

T
kd )
+2gk
P H   (CS
k 1
k ) T HT P1 H   (CS
kd )
+2
T (CS
N


k
k )GiT P1 Gi CS
i  T (CS
2
i=1
N

k )GiT P1 Gi  (CS
kd )
2
i  T (CS
i=1
N


kd
k )GiT P1 Gi CS
2
i  T (CS
i=1
N


kd )
2
i kT S T C T GiT P1 Gi  (CS
i=1
N


kd
2
i kT S T C T GiT P1 Gi CS
i=1
N


kd
kd )GiT P1 Gi CS
2
i  T (CS
i=1

k P1 k + k P2 k kd P2 kd } .

Similarly, we can derive

E{V
2 (k)} = E{V2 (k + 1) V2 (k)}
k
k1



=E
i P3 i
i P3 i

i=kk+1 +1
i=kk

= E k P3 k kk P3 kk

k1
k1



+
i P3 i
i P3 i

i=kk+1 +1

i=kk +1

From the elementary inequality 2a T b a T a + b T b, it is

straightforward to see that
k )GiT P1 Gi CS
k
2
T (CS
k )GiT P1 Gi  (CS
k)
 T (CS

k
+kT S T C T GiT P1 Gi CS
T
k )Gi P1 Gi  (CS
kd )
2
(CS

(16)

k )GiT P1 Gi  (CS
k)
 T (CS
T
T
kd )
+
(CSkd )Gi P1 Gi  (CS

(17)

kd
k )GiT P1 Gi CS
2
(CS
k )GiT P1 Gi  (CS
k)
 T (CS
T

T
kd
+kd
S T C T GiT P1 Gi CS
T T T T
kd )
2k S C Gi P1 Gi  (CS

k
kT S T C T GiT P1 Gi CS
T
T
kd )
+
(CSkd )Gi P1 Gi  (CS

(13)

kd
2kT S T C T GiT P1 Gi CS
T T T T
k
k S C Gi P1 Gi CS
T
T T T
kd
+kd S C Gi P1 Gi CS
T
T
kd
2
(CSkd )Gi P1 Gi CS
T
T
kd )
 (CSkd )Gi P1 Gi  (CS
T
kd .
+kd
S T C T GiT P1 Gi CS

(18)

(19)

(20)

(21)

Furthermore, in terms of (13)(21), we can obtain

E{V (k)} =E{V (k + 1) V (k)}
=

3

i=1

1 k }
E{Vi (k)} E{kT 

(22)

DING et al.: H STATE ESTIMATION FOR DISCRETE-TIME COMPLEX NETWORKS

On the other hand, according to the definition of V (k), one

derives

where


T
T
k = kT kd
fkT
k
k
k )  T (CS
kd )
gkk  T (CS

T

E{V (k)} 1 E{||k || } + 2

(i + i + i + i )S T C T GiT P1 Gi CS,

i=1
N


( i + i + i

E{||i ||2 }

i=kM

22 = W2T P1 W2 P2

+

k1


11 = W1T P1 W1 P1 + P2 + (M m + 1)P3

N


731

+ i )S T C T GiT P1 Gi CS,

+3

k1


E{||i ||2 }

(26)

i=kM

where 1 = max (P1 ), 2 = max (P2 ), and 3 = (M m +

1)max (P3 ).
For any scalar > 1, together with (12), the above
inequality implies that

i=1

k+1 E{V (k + 1)} k E{V (k)}

= k+1 E{V (k)} + k ( 1)E{V (k)}

66 = T HT P1 H 

N


+
( i + i + i + i )GiT P1 Gi ,
77 =


[( 1)1 0 ]k E{||k ||2 }

k1

k E{||i ||2 }.
+( 1)(2 + 3 )

i=1
T T

H P1 H 
N


+
( i + i + i + i )GiT P1 Gi ,

1 =


Then, along similar lines as the proof of Theorem 1 in [19],

we can achieve

i=1

11 W T

1

P1 W2 0

22
0

P3

W1T
W2T

P1
P1

k E{V (k)} E{V (0)} + (1 () + 2 ())

0
P1

+2 ()

E{||i || }

E{V (0)} (2M + 1) max E{||i ||2 }.

M i0

2 [gkk (I 1 )kk ]T [gkk (I 2 )kk ]
k ) (I )CS
k ]T [
k ) CS
k]
(CS
(CS
3 [
kd ) (I )CS
kd ]T
4 [
(CS
kd ) CS
kd ]
[
(CS
(23)

Since 1 < 0, there must exist a sufficiently small scalar

0 > 0 such that
(24)

Then, it is easy to see from (23) and (24) that the following
inequality holds:
E{V (k)} 0 E{||k ||2 }.

i=0
2

(28)

Meanwhile, it follows easily from (26) that

E{V (k)}

1 k 1 [ fk (I f )k ]T [ fk (I f )k ]
E kT 
1
2

1 + 0 diag{I, 0} < 0.

i E{||i ||2 }

where 1 () = ( 1)1 0 , 2 () = M M ( 1)
(2 + 3 ).
Let 0 = min (P1 ) and = max{1 , 2 , 3 }. It is obvious
from (12) that
(29)
E{V (k)} 0 E{||k ||2 }.

Subsequently, from (8) and (9), it follows that:

E{kT 1 k }.

k1


M i0

W1T P1 W1T P1 H  W1T P1 H 

W2T P1 W2T P1 H  W2T P1 H 

0
0
0

.
P1
P1 H 
P1 H 

P1
P1 H
P1 H

T
T


66
H P1 H
77

(27)

i=kM

(25)

(30)

In addition, it can be verified that there exists a scalar 0 > 1

such that
(31)
1 (0 ) + 2 (0 ) = 0.
Therefore, it is not difficult to see from (28), (30), and (31)
that
k0 E{V (k)} (2M + 1) max E{||i ||2 }
M i0

E{||i ||2 }.
+2 (0 )

(32)

M i0

And then, it is obvious from (29) and (32) that

E{||k ||2 }
 k
(33)
(2M + 1) + M 2 (0 )
1
max E{||i ||2 }.

M i0
0
0
According to Definition 1, the augmented system (7) with
v k = 0 is exponentially mean-square stable, which completes
the proof.

732

IEEE TRANSACTIONS ON NEURAL NETWORKS AND LEARNING SYSTEMS, VOL. 23, NO. 5, MAY 2012

Now, let us consider the H performance of the overall

estimation process. In the following theorem, a sufficient condition is obtained that guarantees both the exponential meansquare stability and the H performance for the augmented
system (7).
Theorem 2: Let the disturbance attenuation level > 0,
the estimator parameters K i (i = 1, 2, . . . , N), and the
diagonal matrix  be given. Then the augmented system (7) is
exponentially stable in the mean square sense for v k = 0 and,
under the zero initial condition, satisfies the H performance
constraint (10) for all nonzero v k , if there exist positive definite
matrices Q i > 0 (i = 1, 2, . . . , 6) and positive scalars j
( j = 1, 2, . . . , 4) satisfying

11 W1T P1 W2 0 14 W1T P1

22
0 W2T P1 W2T P1

gT

33 0 2 2

44
P1
2 =

55

(34)

T
T

W1 P1 H
W1 P1 L
16
W2T P1 H 
27
W2T P1 L

0
0
0

P1 H 
P1 L
P1 H 
<0

P1 H 
P1 L
P1 H 


T
67
H P1 L
66

 HT P1 L

77

LT P1 L 2 I
where
11 =W1T P1 W1 P1 + P2 + (M m + 1)P3
f
+ 1 MT M
1 1 3 S T C T (I )CS
N
N


+
(i + i + i + i )S T C T GiT P1 Gi CS
i=1

and other parameters are defined as in Theorem 1.

Proof: According to Theorem 1, it is easily seen that the
zero solution of (7) with v k = 0 is exponentially stable in the
mean square since the inequality (11) is implied by (34). It
remains to show that, under zero initial condition, the output
error z k satisfies the H performance constraint (10).
Choosing the Lyapunov function similar to one in the proof
of Theorem 1, we can calculate
E{V (k)}
E{kT 1 k + 2v kT LT P1 W1 k
+2v kT LT P1 W2 kd + 2v kT LT P1 fk
k)
+2v kT LT P1 gkk + 2v kT LT P1 H   (CS
T T

T
T
kd ) + v k L P1 Lv k }
+2v k L P1 H  (CS

where
L
=
[ LT P1 W1 LT P1 W2 0 LT P1 LT P1
T

T

L P1 H L P1 H ].
In order to analyze the H performance of (7), we introduce
!
s

1
J (s) = E
(37)
||z k ||2 2 ||v k ||2
N
k=0

where s is nonnegative integer.

Under the zero initial condition, one has
s

1
||z k ||2
J (s) = E
N
k=0

2 ||v k ||2 + V (k) E{V (s + 1)}

!
s

1
E
||z k ||2 2 ||v k ||2 + V (k)
N
k=0

s 

E
kT 2 k < 0.

Letting s , it follows from the above inequality that


1 
E{||z k ||2 } 2
||v k ||2 .
N
k=0

where k and 1 have been defined previously.

Setting k = [ kT v kT ]T , the inequality (35) can be rewritten
as

 !
L T
T 1

E{V (k)} E k
(36)
k
LT P1 L

k=0

The proof is now complete.

Up to now, the analysis problem of the estimator performance has been solved. Finally, we are in a position to
consider the H estimator design problem for the complex
network (1). The following result can be easily accessible from
Theorem 2, and the proof is therefore omitted.
Theorem 3: Let the disturbance attenuation level > 0
and the diagonal matrix  be given. For the discrete timedelayed complex networks (1) with the sensor model (3)
containing both ROSSs and RVSDs, the augmented system
(7) is exponentially stable in the mean-square sense for v k =
0 and satisfies the H performance constraint (10) under
the zero initial condition for all nonzero v k , if there exist
positive definite matrices Q i > 0 (i = 1, 2, . . . , 6), matrices
Yi (i = 1, 2, . . . , N), and positive scalars j ( j = 1, 2, . . . , 4)
satisfying

RT F T F T F T F T

1
2
3
4
Q2 0
0
0
0

Q2 0
0
0
<0

(39)

Q
0
0
2

Q2 0

Q2
where

(35)

(38)

k=0

=

11

12
22

13
0
33

0
0
0
44

0
0
0
0
55

DING et al.: H STATE ESTIMATION FOR DISCRETE-TIME COMPLEX NETWORKS

16
26
36
0
0
66

17
27
37
0
57
67
77

18
0
0
0
0
68
78
88

19
0
39
0
0
69
79
0
99

1,10
2,10

3,10

0
,
6,10

7,10

8,10

9,10
10,10

39 = 4 I C T ( + I )/2,
3,10 = [ (I C)T  Y T L 0 ], 44 = I Q 4 ,
g

Z2 =
Z3 =

2
(2)
2 E 2 Y T
T
(3)
2 E2Y
(4)
2 E 2 Y T

66 = diag{I Q 1 , I Q 2 } 1 I,
67 = diag{I Q 1 , I Q 2 },
68 = 78 = [ 0  Y T ]T ,

(2)
T
 N E N Y T ] ,
 T ]T ,
(3)
N EN Y

(4)
 N E N Y T ]T ,
F1 = [ Z 1 (I C) 0 0 0 0 0 0 0 0 0 ],
F2 = [ 0 0 Z 2 (I C) 0 0 0 0 0 0 0 ],
F3 = [ 0 0 0 0 0 0 0 Z 3 0 0 ],
F4 = [ 0 0 0 0 0 0 0 0 Z 4 0 ],
R = [ Y (I  )(I C) Y (I C) Y  (I C)
0 0 0 0 Y  Y  0 Y G ],
"



(1)
i =
+ i + i + i ,
" i

i + i + i + i ,
(2)
i =
"

=
i + i + i + i ,
(3)
i
"

(4)
i =
i + i + i + i ,

Z4 =

11 = (W T W ) ((Q 1 + Q 2 )) 3 I (C T C)

+Sym{(W )T Y (I  )(I C)}
f

12
13
16

I (Q 1 Q 3 (M m + 1)Q 5 ) 1 1 ,
= (W )T Y (I C),
= (W )T Y  (I C),
= [ W T ( Q 1 ) + 1 2f T
W T ( Q 2 ) + (I C)T (I  )Y T ],

17 = [ W T ( Q 1 )
18

W T ( Q 2 ) + (I C)T (I  )Y T ],
= (W T )Y T  + 3 I C T ( + I )/2,
= (W T )Y T  ,

19
1,10 = [ W T ((Q 1 + Q 2 ))L + (I C)T (I  )Y T L
(W )T Y G ],
22 = I (Q 2 Q 4 (M m + 1)Q 6 )
1
f
1 1 + (I (M T M)),
N
26 = [ 0 (I C)T Y T + 1 2f T ],
27 = [ 0 (I C)T Y T ],
2,10 = [ (I C)T Y T L 0 ],
33 = I Q 3 4 I (C T C),
36 = 37 = [ 0 (I C)T  Y T ],

gT

55 = diag{I Q 5 , I Q 6 } 2 1 , 57 = 2 2 ,

Y = diag{Y1 , Y2 , . . . , Y N }, Q2 = I Q 2 ,
Z 1 = [ (1) E 1 Y T (1) E 2 Y T (1) E N Y T ]T ,
1
T
[ (2)
1 E1 Y
T
[ (3)
1 E1 Y
T
[ (4)
1 E1 Y

733

69 = 79 = [ 0  Y T ]T ,
77 = diag{I Q 1 , I Q 2 } 2 I,
88 = 3 I, 8,10 = [  Y T L 0 ],
99 = 4 I, 9,10 = [  Y T L 0 ],


(I Q 2 )L 0
6,10 = 7,10 =
,
(I Q 2 )L Y G
 T

L (I (Q 1 + Q 2 ))L 2 I L T Y G
10,10 =

2 I
and other parameters are defined as in Theorem 1. Moreover,
if the above inequality is feasible, the desired state estimator
gains can be determined by
K i = Q 1
2 Yi .

(40)

Remark 5: At present, a large number of results on complex

networks are available in the literature that require the symmetry and the zero-row-sum properties for the configuration
matrix W . However, these results cannot be applied to the
state estimation problem with sensor saturations considered in
this paper. Our main results in Theorems 13 are applicable
to a wide class of complex networks including both directed
and undirected networks, and the measurement model in
this paper is quite comprehensive and encompasses networkinduced phenomena including both the ROSSs and the RVSDs.
Comparing with existing literature, our results are obtained on
a new state estimation problem for a more general model using
in-depth stochastic analysis tools.
Remark 6: A general form of sector-like nonlinear function,
instead of the commonly used Lipschitz-type function, is
employed to describe the nonlinearities existing in the network. The main results established contain all the information
of the complex networks including the physical parameters,
lower and upper bounds of the network state delay, the sensor
delays, the H disturbance rejection attenuation level, as well
as the occurrence probabilities of sensor saturations and sensor
delays. Notice that the system governing the error dynamics
involves both the fixed and varying time delays. To reduce the
possible design conservatism, a novel LyapunovKrasovskii
functional is proposed in which the first component V1 (k) is
used to account for the fixed time delay and other components
are constructed to cater for the varying time delays. In the
next section, a simulation example is provided to show the
usefulness of the proposed design procedure for the desired
state estimators.
Remark 7: Note that, for the standard LMI system, the
algorithm has a polynomial-time complexity. That is, the
number N () of flops needed to compute an -accurate
solution is bounded by O(MN 3 log(V/)), where M is the
total row size of the LMI system, N is the total number

IEEE TRANSACTIONS ON NEURAL NETWORKS AND LEARNING SYSTEMS, VOL. 23, NO. 5, MAY 2012
1.2

0.8
0.6
0.4
0.2
0
0.2
0.4
0.6

Fig. 1.

10

15

20
Time (k)

25

30

35

40

Measurements from sensor 1.

1

Actual measurements y21

Ideal measurements y21
Actual measurements y22
Ideal measurements y22

0.5

IV. N UMERICAL E XAMPLES

0.5

Consider a discrete time-delayed complex network (1) with

three nodes. The coupling configuration matrix is assumed to
be W = (wi j ) MM , with

0.6 0.6
0
W = 0.6 1.1 0.5
0
0.5 0.5
and the inner-coupling matrix is given as  = diag{0.1, 0.1}.
The disturbance matrices and the output matrix are

T

T
L 1 = 0.04 0.03 , L 2 = 0.02 0.04 ,

T
L 3 = 0.02 0.03 , M = [0.70 0.65].

Actual measurements y11

Ideal measurements y11
Actual measurements y12
Ideal measurements y12

Sensor 1

of scalar decision variables, V is a data-dependent scaling

factor, and is the relative accuracy set for the algorithm.
Let us now look at the H state estimation problem for the
complex network (1) with the measurements (3), where the
network size is N and the variable dimensions can be seen
from x i (k) Rn , yi (k) Rm , z i (k) Rr , xi (k) Rn
(i = 1, 2, . . . , N), v 1 (k) Rq , and v 2 (k) R p . From
Theorem 3, we have M = 8n N + 2m N +r N and N = 3n 2 +
3n +m 2 N +4. Therefore, the computational complexity of the
LMIs-based H state estimation algorithm can be represented
as O(n 7 N + m 7 N 4 ). Similarly, the computational complexity
of the proposed condition in Theorem 1 is also O(n 7 N).
Obviously, the computational complexity of the LMI-based
algorithms depends polynomially on the network size and
the variable dimensions. In order to reduce the computation
burden, a possible way is to obtain the estimator gains node
by node and then representing the computational complexity
as O(n 7 N). Fortunately, research on LMI optimization is a
very active area in the applied mathematics, optimization, and
the operations research community, and substantial speedups
can be expected in the future.

Sensor 2

734

1.5

Fig. 2.

10

15

20
Time (k)

25

30

35

40

Measurements from sensor 2.

In this example, the probabilities are taken as 1 =

0.88, 2 = 0.85, 3 = 0.87, and 1 = 0.91, 2 =
0.92, 3 = 0.9, the delay parameters are chosen as m = 1,

The nonlinear vector-valued functions f (x i (k)) and M = 3, the disturbance attenuation level is = 0.92, and the
diagonal matrix is  = diag{0.7, 0.7}. By using the MATLAB
g(x i (k)) are chosen as
(with YALMIP 3.0 and SeDuMi 1.1), we solve LMI (39) and


0.6x i1 (k) + 0.3x i2 (k) + tanh(0.3x i1 (k))
obtain a set of feasible solutions as follows:
,
f (x i (k)) =


0.6x i2 (k) tanh(0.2x i2 (k))


6.6408 2.5172
1 = 14.6065, Q 1 =
,
0.02x i1 (k) + 0.06x i2 (k)
2.5172 1.3932
g(x i (k)) =
.


0.03x i1 (k) + 0.02x i2 (k) + tanh(0.01x i1 (k))
5.4508 1.9621
,
2 = 53.3398, Q 2 =
Then, it is easy to see that the constraint (2) can be met
1.9621 3.0666


with
0.2706 0.0581




=
0.3624,
Q
=
,

3
3
0.6 0.3
0.3 0.3
f
f
0.0581 0.1037
, 2 =
,
1 =


0 0.4
0 0.6
0.1516 0.1421




,
4 = 2.1997, Q 4 =
0.02 0.06
0.02 0.06
g
g
0.1421
0.3701
, 2 =
.
1 =


0.03 0.02
0.02 0.02.
0.5182 0.1538
,
Q5 =
0.1538 0.0527
Consider the sensors with both the ROSSs and RVSDs


described by (3) with the following parameters:
0.4129 0.1257
Q6 =
,




0.1257
0.0975
0.03
0.02


G1 =
, G2 =
,
0.0023 0.3538
0.02
0.04
,
Y1 =




0.3023 0.0403
0.06
0.8 0.6


, C=
.
G3 =
0.0467 0.3267
0.02
0.9 0.4
,
Y2 =
0.3426 0.0630

DING et al.: H STATE ESTIMATION FOR DISCRETE-TIME COMPLEX NETWORKS

1.5

V. C ONCLUSION

Actual measurements y31

Ideal measurements y31
Actual measurements y32
Ideal measurements y32

Sensor 3

0.5

0.5

Fig. 3.

10

15

20
Time (k)

25

30

35

40

Measurements from sensor 3.

1.5

735

Output estimation error z1 of estimator 1

Output estimation error z2 of estimator 2
Output estimation error z3 of estimator 3

In this paper, we have investigated the H state estimation

problem for a class of complex networks with time-varying
delay and incomplete information. The considered incomplete
information includes both the ROSSs and RVSDs. In order to
take both ROSSs and RVSDs into account in a unified way,
a novel sensor model has been proposed by using two sets
of Bernoulli-distributed white sequences with known conditional probabilities. Then, some estimators have been designed
such that the augmented system is exponentially mean-square
stable and the estimation error satisfies the specified H
performance requirement. Finally, the developed estimation
approach has been demonstrated by a numerical simulation
example. Further research topics include the extension of our
results to more general complex networks with varying time
delays in measurements and also to network control systems
with both ROSSs and RVSDs.

Output estimation errors

R EFERENCES
0.5

0.5

Fig. 4.

10

15

20
Time (k)

25

30

35

40

Estimator errors z i (k) (i = 1, 2, 3).


Y3 =


0.0162 0.3243
.
0.2813 0.0882.

Then, according to (40), the desired estimator parameters can

be designed as


0.0466 0.0782
,
K1 =
0.1284 0.0369


0.0411 0.0682
,
K2 =
0.1380 0.0231


0.0468 0.0638
.
K3 =
0.1216 0.0121
In the simulation, the exogenous disturbance inputs are
selected as
6 cos(0.8k)
.
v 1 (k) = 5 exp(0.2k) sin(k), v 2 (k) =
k +1
The discrete time-varying delay (k) satisfies (k) = 2 +
sin(k/2), and the constant delay is selected as d = 1. The
initial values i (k) (i = 1, 2, 3; k = 3, 2, 1, 0) are
generated that obey uniform distribution over [1.4, 1.4].
Simulation results are shown in Figs. 14, where Figs. 13
plot the actual measurements and ideal measurements for
sensors 13, respectively, and Fig. 4 depicts the output errors.
The simulation results confirm that the designed H estimator
performs very well.

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Derui Ding received the B.Sc. degree in industrial engineering and the M.Sc. degree in detection
technology and automation equipments from Anhui
Polytechnic University, Wuhu, China, in 2004 and
2007, respectively. He is currently pursuing the
Ph.D. degree in control science and engineering with
Donghua University, Shanghai, China.
He is a Lecturer with Anhui Polytechnic University. He is an active reviewer for many international
journals. His current research interests include nonlinear stochastic control and filtering, as well as
complex networks and sensor networks.

Zidong Wang (SM03) was born in Jiangsu, China,

in 1966. He received the B.Sc. degree in mathematics from Suzhou University, Suzhou, China,
in 1986, the M.Sc. degree in applied mathematics
and the Ph.D. degree in electrical engineering from
the Nanjing University of Science and Technology,
Nanjing, China, in 1990 and 1994, respectively.
He is currently a Professor of dynamical systems
and computing with the Department of Information Systems and Computing, Brunel University,
Uxbridge, U.K. From 1990 to 2002, he held teaching
and research appointments in various universities in China, Germany, and U.K.
He has published more than 100 papers in refereed international journals.
His current research interests include dynamical systems, signal processing,
bioinformatics, and control theory and applications.
Prof. Wang is a fellow of the Royal Statistical Society and has been a
member of program committee for many international conferences. He has
been a recipient of the Alexander von Humboldt Research Fellowship, Germany, the Japan Society for the Promotion of Science Research Fellowship,
Japan, the William Mong Visiting Research Fellowship, Hong Kong. He
serves as an Associate Editor for 11 international journals, including the IEEE
T RANSACTIONS ON AUTOMATIC C ONTROL, the IEEE T RANSACTIONS ON
C ONTROL S YSTEMS T ECHNOLOGY, the IEEE T RANSACTIONS ON N EURAL
N ETWORKS , the IEEE T RANSACTIONS ON S IGNAL P ROCESSING, and the
IEEE T RANSACTIONS ON S YSTEMS , M AN , AND C YBERNETICS PART C.

Bo Shen received the B.Sc. degree in mathematics

from Northwestern Polytechnical University, Xian,
China, in 2003, and the Ph.D. degree in control
theory and control engineering from Donghua University, Shanghai, China, in 2011.
He is currently with the School of Information
Science and Technology, Donghua University. From
2009 to 2010, he was a Research Assistant with the
Department of Electrical and Electronic Engineering, University of Hong Kong, Hong Kong. From
2010 to 2011, he was a Visiting Ph.D. Student with
the Department of Information Systems and Computing, Brunel University,
Uxbridge, U.K. He is an active reviewer for many international journals. His
current research interests include nonlinear control and filtering, stochastic
control and filtering, as well as complex networks and genetic regulatory
networks.

Huisheng Shu received the B.Sc. degree in mathematics from Anhui Normal University, Wuhu, China,
in 1984, the M.Sc. degree in applied mathematics
and the Ph.D. degree in control theory and control
engineering from Donghua University, Shanghai,
China, in 1990 and 2005, respectively.
He is currently a Professor with Donghua University. He has published 16 papers in refereed
international journals. His current research interests
include mathematical theory of stochastic systems,
robust controls, and robust filtering.