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Dissipativity-based state estimation of delayed static neural networks
Accepted Manuscript
Dissipativity-based State Estimation of Delayed Static Neural Networks Yanchai Liu , Ting Wang , Mengshen Chen , Hao Shen , Yueying Wang , Dengping Duan PII: DOI: Reference:
S0925-2312(17)30592-1 10.1016/j.neucom.2017.03.059 NEUCOM 18292
To appear in:
Neurocomputing
Received date: Revised date: Accepted date:
15 July 2016 14 February 2017 22 March 2017
Please cite this article as: Yanchai Liu , Ting Wang , Mengshen Chen , Hao Shen , Yueying Wang , Dengping Duan , Dissipativity-based State Estimation of Delayed Static Neural Networks, Neurocomputing (2017), doi: 10.1016/j.neucom.2017.03.059
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ACCEPTED MANUSCRIPT
Dissipativity-based State Estimation of Delayed Static Neural Networks
Yanchai Liua, Ting Wangb, Mengshen Chen b, Hao Shenb, Yueying Wanga,*, Dengping Duana a
b
School of Aeronautics and Astronautics, Shanghai Jiao Tong University, Shanghai 200240, China School of Electrical Engineering and Information, Anhui University of Technology, Ma’anshan 243002, China
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Abstract
This paper proposes a dissipativity-based state estimation methodology for static neural networks with time-varying delay. An Arcak-type observer is used to construct the
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estimation error system. To reduce the conservatism of observer design, a LyapunovKrasovskii functional (LKF) is adopted to fully exploit the available characteristics about activation function. In addition, a relaxed constraint condition is put forward to keep the whole LKF positive without requiring parts of involved matrices to be positive.
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By adopting the LKF and constraint condition, estimation conditions with a strict dissipative performance are obtained, which ensures the asymptotic stability of
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estimation error system. The computation of gain matrices about observer can be
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transformed into a convex optimization problem. Two examples are given to illustrate
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the validity and advantage of provided methodology.
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Keywords: static neural networks, state estimation, dissipativity , time-varying delay
*Correspondence author at: School of Aeronautics and Astronautics, Shanghai Jiao Tong University, Shanghai 200240, China. Tel:+86-21-34207170 Email addresses: [email protected](Yanchai Liu), [email protected] (Ting Wang), [email protected] (mengshen chen), [email protected](Hao Shen), , [email protected] ( Yueying Wang*). [email protected](Dengping Duan)
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1. Introduction According to the basic variables used when modeling [1], roughly speaking, recurrent neural networks are generally divided into two categories: local field neural networks
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and static neural networks. The former use local field states of neurons as basic variables, while the static neural networks adopt neuron states as basic variables. It has been clearly pointed out in [2] that, only under certain strict conditions, the two classes of neural networks can be equivalent. During the past years, static neural networks have
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been extensively applied in various fields, including associative memory and combinational optimization. Recently, the research on delayed recurrent neural networks has gradually become a hot topic (see e.g. [3]-[13]) because time delays are unavoidable in implementing a recurrent neural network because of the finite
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conduction velocity and switching speed about amplifier, which can lead to
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performance degradation or even instability of the neural networks. Meanwhile, a recurrent neural network with large scale often contains an amount of
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neuronal connections to deal with complex nonlinearities. As a consequence, the comple acquisition of the information about neuronal states may be unrealistic, while it
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is necessary in system modeling. So, how to efficiently estimate the states of neural
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networks gradually becomes an important research topic (see e.g. [14]-[23]). It is worth mentioning that aforementioned results are confined to local field neural networks, thus the obtained estimation methodologies are not entirely suitable for static neural networks. In addition, the behaviors about static neural networks are usually complicated escpecially when they suffer from various stochastic disturbances. It is of vital importance to suppress the infuluence from the external disturbaces when
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estimating their states. For this reason, considerable attention has been devoted to estimation problem for delayed static neural networks [24]-[30] with H / H 2 performance. To mention a few, the state estimation problem for time-delay static neural networks was investigated in [25], where estimation conditions with H / H 2
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performance were provided. Further improved results were given in [26], [27], where reciprocally convex combination approach was adopted. In [28], a suitable LKF was constructed, and less conservative results than [26], [27] were further achieved. The design problem of Arcak-type observer with H 2 performance for delayed static neural
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networks was studied in [29]. Based on the same estimator, the state estimation problem with guaranteed H performance was further studied in [30], where much better performance has been achieved than the results in [26] and [27].
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On the other hand, dissipativity theory [31], [32] originates from electrical networks, and, by using an input-output description, presents a tool for analysis and synthesis of
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control systems [33]. Dissipativity is characterized by storage functions and supply
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rates, which represent the energy stored inside the system and energy supplied from outside the system, respectively. It can be viewed as a more general performance index
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containing some well-known ones such as H performance constraint, the positive real
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performance, the mixed performance, and the sector bounded performance [34], [35]. Thus, from practical application point of view, it is more desirable to expect that the neural networks have the dissipativity property to achieve effective noise attenuation. Although some research efforts have been devoted to dissipativity analysis for neural networks (see, e.g., [36]-[38]), there has been very little literature reported on state estimation with guaranteed dissipative performance for static neural networks, which is the motivation about this paper.
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In this paper, we focus on state estimation with guaranteed dissipative performance for delayed static neural networks. The main contributions of the paper can be summarized as follows. 1) Dissipativity performance is introduced into the state estimation problem for delayed static neural networks, which is more general than
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previous H / H 2 performance. 2) An improved LKF is adopted to fully emploit the characteristics about the activation function. Moreover, a more relaxed condition is derived to keep the whole LKF positive without requiring some involved matrices to be positive. 3) An observer design criterion is offered for delayed static neural networks,
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which guarantees the asymptotic stability about estimation error system with a strict dissipative performance. Simulation results show that less conservativeness is achieved by our design method than the existing ones.
Notation. The notations used in this paper are standard. The set about m n real mn
. The block diagonal matrix and identity matrix are,
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matrices is represented by
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respectively, represented by diag
and
I . The space of square-integrable vector
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functions defined on 0, is denoted by L2 0, . The transpose of the matrix is denoted by superscript “ T ” , the inverse is represented by “ 1 ”, and the induced
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symmetric term isdenoted by “ ”.
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2. Problems formulation and preliminaries Choose a static neural networks with time-varying delay and external disturbance as
x ( t ) x ( t ) g x t d (t ) y (t ) x (t ) x t d (t ) 2 (t ) z (t ) x (t ) x (t ) (t ), t d ,0
4
(t )
1
(1)
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where x(t )
n
is the neuronal state; y (t )
m
is the measurable output; (t )
the external disturbance belonging to L2 0, ; z(t )
(t ) is the initial function;
1
the connection weight matrix;
1
,
2
,
nn
,
n
p
q
is
is output of neural network;
T is the exogenous input;
0 is a diagonal matrix;
,D,
1
,
nn
2
is
and
satisfies: 0 d (t ) d , d ( t )
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are appropriate dimensioned matrices; The time delay d (t ) is the time-varying and
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with d and being positive scalars. g x(t ) g1 x1 (t ) , g2 x2 (t ) ,
(2)
, gn xn (t )
T
denotes the continuous activation function and satisfies
0
gi ( c ) g i ( d ) li , gi (0) 0 , c d cd
, i 1,2,
,n
(3)
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To reduce design conservatism, the Arcak-type full-order observer is adopted to
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estimate the states of static neural network (1), which is described by
1
y (t ) yˆ (t )
2
y (t ) yˆ (t ) (
and
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with
CE
PT
xˆ (t ) xˆ (t ) g xˆ t d (t ) yˆ (t ) xˆ (t ) xˆ t d (t ) zˆ(t ) xˆ (t ) xˆ (t ) xˆ (0), t d ,0
1
2
4)
being the gain matrices of the observer to be computed.
Define the estimation errors about state and output as (t ) x(t ) xˆ (t )
and z (t ) z(t ) zˆ(t ) , respectively. Thus, based on (1) and (3), we obtain the estimation error system as
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(t ) z (t ) (t )
(t )
2
2
t d (t ) g t , d (t ), x(t ), (t )
1
2
2
( t ) (5)
where
For
estimation
s( , z ) :
q
p
x t d (t )
error
system
g
xˆ t d (t )
(5),
we
1
y(t ) yˆ (t )
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g t, d (t ), x(t ), (t ) g
introduce
a
function
satisfying s(0,0) 0 , which is called a supply rate if for all
s ( ), z ( ) d , t2
t1
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input-output pairs ( , z ) satisfying (5), s( , y ) satisfies
t2 t1 0 . System (5) is dissipative under the supply rate s( , z ) , if under zero initial condition, the following dissipation inequality t*
s ( ), z ( ) d 0 , t
*
0
(6)
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0
holds for any nonzero input (t ) . It can be found that if there exists a nonnegative
satisfying V (0) 0 and
ED
n
definite function V ( x ) :
V x(t ) V x(0) s ( ), z ( ) d , t * 0
PT
*
t*
(7)
0
CE
then system (5) is dissipative under the supply rate s( , z ) . Accordingly, the function
V ( x ) is called a storage function for system (5). In this paper, we consider a quadratic
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supply rate s( , z ) , which is proposed in the following definition Definition 1: The estimation error system (5) is called strictly (
,
,
) - -dissipative
for any t p 0 and some scalar 0 , if under zero initial condition, the following inequality is satisfied:
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T
tp
0
where
p p
z (t ) (t ) *
,
pq
tp z (t ) T (t ) dt 0 (t ) (t )dt
and
qq
are real matrices, with
(8)
and
being
symmetric matrices.
0 and
I , thus the performance
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0,
Remark 1. Here, it is assumed that
defined in Definition 1 includes H , positive realness, and passivity as special cases. Specially, when
0 , and
I ,
H performance index. When
0,
( 2 )I , inequality (7) degenerates to an I , and
2 I , inequality (7) degenerates
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to strict passivity or strictly positive realness.
Lemma 1[39]. Given time delay 0 d (t ) d , and and any appropriate dimensioned
Z 0 , the following condition is satisfied: R
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R matrices R and Z such that *
ED
t R d T ( ) R ( )d T (t ) t d *
where
Z ( t ) R
PT
(t ) 1T (t ) 2T (t ) 3T (t ) 4T (t ) 1 (t ) t d (t ) t d
CE
2 (t ) t d (t ) t d
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4 (t ) (t ) t d (t )
T
t d ( t ) 2 ( )d 3 (t ) (t ) t d (t ) d d (t ) t d
R 0 2 t ( )d , R d (t ) t d ( t ) 0 3R
The main aim about this paper is to design an Arcak-type observer (4) to make the
estimation error system (5) with (t ) 0 be asymptotically stable, and be strictly (
,
,
) - -dissipative under zero initial condition.
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3. Main results
In this section, we shall investigate the state estimation problem for delayed static neural networks (1) with dissipativity performance. To make full use of the information
into the following two suitable parts:
g e t , d (t ), x(t ), (t ) g g
xˆ t d (t )
x t d (t )
g
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of the activation function, the nonlinear function ge t, d (t ), x(t ), (t ) can be divided
xˆ t d (t )
g xˆ t d (t ) y (t ) yˆ (t ) (t ), t d (t ) , (t ) 1
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g e1 t d (t ) g e 2
(9)
Before proceeding further, we introduce the following useful proposition based on (3). Proposition 1. For any matrices Si diag si1, si 2 ,
(t ) 2 gT1 (t ) S1g 1 (t ) 0
(10)
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2 gT1 (t ) S1
, sin 0 , i 1,2,3
2 gT1 t d (t ) S2
ED
t d (t ) 2 gT1 t d (t ) S2 g 1 t d (t ) 0
2 gT 2 (t ), t d (t ) , (t ) S3
1
(t )
2
PT
K1 (t ) 1 t d (t ) 2 gT2 (t ), t d (t ) , (t ) S3 g 2 (t ), t d (t ) , (t ) 0
(11)
(12)
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Remark 2. As illustrated in [30], the slope constants about the activation function has
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an important impact on design conservatism of observer. However, it should be pointed out that the slope information in (5) cannot be used directly due to the existence of the term
1
y(t ) yˆ (t ) in
g t, d (t ), x(t ), (t ) . To facillatate the adoption of information,
the nonlinear part of the estimation error system is divided into two parts. Theorem 1. For given scalars d 0 , , and 0 , if there exist matrices 0
p p
,
pq
and
qq
with
8
and
being symmetric, symmetric
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P13 Q Q12 ˆ P23 , Q 11 0 , Q , R 0 , U 0 , diagonal matrices * Q 22 P33
P12 P22 *
Si 0 , i 1,2,3 , 1 diag 11 , 12 ,
, 1n 0 , 2 diag 21, 22 ,
Z appropriately dimensioned matrices Z 11 Z 21
, 2 n 0 and any
Z12 , M , G1 , G2 such that the Z 22
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P11 matrices P * *
following conditions are feasible for d (t ) 0, d , d (t ) 0, : Z12 Z 22 0 0 3R
(13)
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R 0 Z11 * 3R Z 21 R * * * * *
1,4 0 3,4 4,4 * * * * * *
1,5 2,5 3,5 4,5 5,5 * * * * *
PT
CE
AC
*
R 0 0 0 hQˆ R
1,6 1,7 d (t ) P13 2,7 3,6 0 4,6 0 4 Z 22 0 6,6 0 * 7,7 * * * * * *
ED
1,3 1,1 1,2 * G2 2,2 * 3,3 * * * * * * * * * * * * * * * * * * * * * *
P13 P23 P33
M
P11 R P12 * P22 * * * *
M M 3,8 0 0 0 0 8,8 * *
(14)
1,9 M T G1T 0 0 0 0 0 9,9 *
1,10 2,10 0 0 0 0 0 0 0 G1 2 I (15)
where
1,1 P12 P12T Q11 Qˆ dU
1,2 P11
T
MT
T
T
G2T M
M T
1
9
T
M T G2
T
G2T 4R
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1,3 G2
T T Z11T Z12T Z 21 Z 22 2R 1 d (t ) ( P12 P13 )
T T 1,4 Z11T Z12T Z21 Z22 P12 T T 1,5 2Z 21 2Z 22 d d (t ) P23T
1,7 Q12
T
1,9 M
T
1,10
M
T
T
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1,6 6R d (t ) P33T
S1T
G1T 1
G2
2
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2,2 M M T d 2 R 2,5 d d (t ) P12 T
1T
2,10 M
1
T
G2
T2
M
2,7
2
ED
T T 3,3 1 d (t ) Q11 8R Z11T Z11 Z12T Z12 Z 21 Z 21 Z 22 Z 22
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T T 3,4 2 R Z11T Z12T Z21 Z22
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T T 3,5 6R 2Z 21 2Z 22 1 d (t ) d d (t ) ( P22T P23T )
3,6 6R 2Z12 2Z 22 1 d (t ) d (t )( P23 P33T )
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3,8 1 d (t ) Q12
T
LT S2T
4,4 Qˆ 4 R 5,5 12 R d d (t ) U
4,5 6R d d (t ) P22T 4,6 2Z12 2Z22 d (t ) P23
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6,6 12 R d (t )U
7,7 S1 S1T Q22
8,8 S2 S2T 1 d (t ) Q22
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9,9 S3 S3T Then, the estimation error system (5) is globally asymptotically stable and strictly (
,
) - -dissipative. Moreover, the gain matrices
,
can be computed as S3 L G1 , 1
2
M 1G2
and
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1
1
2
of observer (4)
(16)
Proof. For estimation error system (5), we choose a LKF as 6
V (t ) Vi (t ) i 1
T
P11 * *
(t ) P13 t d ( t ) P23 ( )d t d P33 t ( )d t d ( t )
ED
(t ) t d ( t ) V1 (t ) ( )d t d t ( )d t d ( t )
M
where
PT
n
wiT
i 1
0
n
wiT
1i li g 1i ( ) d 2 0
CE
V2 (t ) 2
P12 P22 *
i 1
2i g 1i ( ) d
( ) Q11 Q12 ( ) V3 (t ) d t d ( t ) g 1i ( ) * Q22 g 1i ( ) T
AC
t
t
V4 (t ) T ( )Qˆ ( )d t d
V5 (t ) d
0
d
t
t
T ( ) R ( )d d
11
(17)
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V6 (t )
0
d
t
t
T ( )U ( )d d
First, we need to prove V (t ) 0 . It can be verified that T
T
P11 R * * *
P12 P22 *
P13 P23 P33
*
*
(18)
(t ) R t d ( t ) 0 t d ( )d d 0 t ( )d t d ( t ) dQˆ R ( )
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(t ) t d ( t ) ( ) d 1 t t d t d t d ( ) d t d ( t ) ( )
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(t ) (t ) P11 P12 P13 t d ( t ) 1 t t d ( t ) V1 (t ) V4 (t ) V5 (t ) ( )d * P22 P23 ( )d d t d d t d t d t * * P33 t ( )d ( )d t d ( t ) t d ( t ) t t 1 T T ( )Qˆ ( )d (t ) ( ) R (t ) ( ) d t d t d d
From (14), we obtain that V1 (t ) V4 (t ) V5 (t ) 0 . Moreover, from assumption, we
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known that V2 (t ) and V3 (t ) are also positive definite. So, the whole LKF in (17) is
ED
positive definite. Taking the time derivative of the LKF and introducing dissipativity
AC
CE
PT
performance index result in
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z (t ) 2 zT (t ) (t ) wT (t )(
V (t ) zT (t )
I ) (t )
T
(t ) (t ) P11 P12 P13 t d ( t ) 2 ( )d * P22 P23 1 d (t ) t d (t ) (t d ) t d t * * P33 (t ) 1 d (t ) t d (t ) ( )d t d ( t ) 2 T (t ) T L1 (t ) 2 gT1 (t ) 1 (t ) 2 gT1 e(t ) 2 (t )
(t ) Q11 Q12 (t ) g 1i (t ) * Q22 g 1i (t )
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T
t d (t ) Q11 Q12 t d (t ) 1 d (t ) g 1i t d (t ) * Q22 g 1i t d (t )
T
t
T (t )Qˆ (t ) T (t d )Qˆ (t d ) d 2 T (t ) R (t ) d T ( ) R ( )d d T (t )U (t )
t d
T (t )
T ( )U ( )d
t
t d ( t )
(t ) 2 T (t )
T
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t d
t d ( t )
T
T ( )U ( )d
(t ) T (t )(
(19)
I ) (t )
By using Lemma 1, the following inequality holds:
M
t R d T ( ) R ( )d T (t ) t d *
Z ( t ) R
(20)
ED
where R and (t ) are defined in Lemma 1.
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For any scalar and real matrix M with appropriate dimension, the following condition holds:
CE
0 T (t ) M T (t ) M
2
(t )
2
t d (t ) g t, d (t ), x(t ), (t )
1
2
2
(t )
AC
(t )
(21
)
Thus, adding the left parts of inequalities (10)-(12) into (19), and in the light of (20) and (21), we can obtain that for all nonzero (t ) L2 0, V (t ) zT (t )
z (t ) 2 zT (t ) (t ) T (t )(
13
I ) (t ) T (t ) (t) 0
(22)
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where
(t ) T T T T (t ) (t ) t d (t ) (t d )
g eT1 t d (t ) g eT2 (t ), t d (t ) , (t ) T (t )
1 t T ( )d t d ( t ) d (t ) T
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g eT1 (t )
t d ( t ) 1 T ( )d t d d d (t )
Under zero conditions, we obtain V (0) 0 and V () 0 . Integrating both sides of (22) over the time period 0 to t p results in (t ) V (t p ) 0 , which ensures (8), that is,
,
,
) - -dissipative. Following the similar procedures used
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system (5) is strictly (
above, we can obtain the estimation error system (5) with (t ) 0 is asymptotically stable. The proof is completed.
Remark 3. By adopting the LKF in (17), a dissipativity-based state estimation criterion
M
is presented delayed static neural networks. To reduce conservativeness, a relaxed
ED
condition (14) is provided to keep the whole LKF positive without requiring the involved matrices P and Qˆ to be positive, which is another advantage of proposed
PT
design method.
Remark 4. The number about decision variables involved in Theorem is
CE
13n 2 (2m 9)n , while the numbers in [28] and [30] are, respectively,
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9.5n 2 (m 5.5)n and 9n2 (2m q 4)n , which shows more computational burden will be consumed by our method. However, it also means that the solution of obtained results in our paper will be searched in a bigger set. Remark 5. It is worth mentioning the achieved estimation performance can be further enhanced by other ways. For instance, an efficient way is to adopt the latest inequality techniques [40], [41] to estimate the upper bound on derivatives for LKF (17).
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As a consequence, more computational time will be consumed. Moreover, it is noted that only asymptotic stability is ensured. How to adopt the finite-time control/filtering techniques [42]-[46] to investigate the finite-time state estimation problem for static neural networks is a topic theoretical significance.
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Remark 6. The computation about optimal dissipative performance index max can be transformed into the following convex optimization algorithm:
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Algorithm 1: max , subject to the conditions in (13) - (15).
4. Numerical examples
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In this subsection, two examples are provided to illustrate the validity of achieved estimation methodology.
ED
Example 1. Choose the system parameters of delayed static neural network (1) as
PT
0 1.06 0 0 1.42 0 , 0 0.88 0 0.2 0.2 0.2 ,
CE
1
T
2
0.32 0.85 1.36 1.10 0.41 0.50 , 0.42 0.82 0.95
0.4 0.3 , T
1 0 0.5 1 0 1 , 0 1 1
0 1 0.5 , 0 0.5 0.6
0 1 0.2 . 0 0 0.5
AC
We first illustrate the less conservativeness of proposed approach. In order to
facilitate comparison, we set
I3 ,
0 , and
2 . In this situation, the
disspativity-based stability criterion proposed in Theorem 1 degenerates to a stability criterion for error estimation system (5) with guaranteed H performance index. Let
0.5 , d 0.8 and 0.6 , Table I lists the comparison results about index min under
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various constants
, which shows that the estimation performance achieved by our
methodology is better than ones in the most recent works [28] and [30]. Moreover, the allowable maximum slope constant by using our method is
1.65 with minimum
index min 155.5177 , while the allowable ones in [28] and [30] are 1.42I and 1.49I ,
Under slope constant
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respectively.
diag 1.2,1.3,1.4 and 0.5 , Table II gives the
comparison results about optimal H performance min under different delay bounds d , which further confirm less conservativeness of our method over the most recent works
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[28] and [30].
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Table I: Optimal H performance index min via different slope constants Methods 1.2I 1.3I 1.4I 1.5I [28] 1.0918 2.2097 11.9007 infeasible [30] 0.6536 0.9805 2.2387 infeasible Theorem 1 2.0434e-06 0.6461 1.0098 1.8932
CE
PT
ED
Table II: Optimal H performance index min via different upper bounds d Methods 0.7 0.8 0.9 1.0 [28] 1.3675 2.4425 7.1801 infeasible [30] 0.5953 1.0052 32.0824 infeasible Theorem 1 2.6353e-07 0.5535 0.9600 2.2973 For simulation, we choose activation function f ( x) 1.4 tanh( x) ,
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1.5 1.5 1.5 , and T
1.5I 3 ,
4.0 , time delay d (t ) 0.4 0.4sin(1.5t ) , which implies
1.4 , d 0.8 and 0.6 . By using Theorem 1, the optimal dissipativity
performance is computed as max 1.8143 , and the corresponding gain matrices of observer as
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-0.0380 0.0083 0.0894 , 1 (1.0e 06) * 0.0014 -0.0090 -0.1703
2
-0.6131 -0.1763 -0.6389 0.2901 -0.7613 -0.9732
By choosing the initial condition as (0) 2 2 3 , Fig. 1 dipicts the responses T
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of estimation errors under disturbance signal (t ) sin(t )e5t . The simulation results confirm the validity of achieved method to design the dissipativity-based observer for delayed static neural networks.
3
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2.5 2 1.5
0.5
M
e(t)
1
0
ED
-0.5 -1
PT
-1.5 0
2
4
6
8
10
t/sec
Fig. 1. Responses of state estimation error (t )
AC
CE
-2
e(1) e(2) e(3)
Example 2. Choose the parameters of delayed static neural network as
0 7.0214 , 7.4367 0
2
0.1 ,
1 0 ,
6.4993 12.0257 , 0.6867 5.6614
1 1 , 0 1
1
0.2 , 0.2
2 0 .
Table III: Optimal H performance index min via different slope constants Methods 0.4I 0.5I 0.6I 0.7I 0.8I
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[28] [30] Theorem 1
0.1829 0.0910 0.0805
0.2636 0.1179 0.0937
0.6121 0.1755 0.1153
infeasible 0.3783 0.1546
infeasible infeasible 0.2459
Denote 0.2 , d 0.4 and 0.7 , the comparison results about optimal H performance min via different slope constants
are depicted in Table III. It is
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observed from Table III that the performance achieved by our approach is better than ones in the most recent works [28] and [30], which further confirm the advantage of our methodology. Moreover, the maximum slope constants computed in [28] and [30] are
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0.65I and 0.77I , respectively, while applying our method, the maximum one
0.94 , with corresponding H performance index min 14.0796 . By making the
first gain matrix in Arcak-type observer (4) be zero, the observer reduces to a Luenberger-type one. In this case, the obtained maximum slope constant reduces to
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0.92 , which verifies the advantage of Arcak-type observer
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5. Conclusion
In this paper, the issue about dissipativity-based state estimation has been addressed for
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delayed static neural networks. By employing the LKF, an observer design procedure with strict dissipative performance have been presented. The computation about
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maximum dissipative performance and corresponding gain matrices of the observer can
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be transformed into a convex optimization problem. Two examples have been given to illustrate the less conservativeness of provided design method than most recent results. In future, we will use the proposed methodogy to estimate the neuronal states of projection neural networks [47], which is a typical static neural network.
Acknowledgement
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This work was supported by the China Postdoctoral Science Foundation [grant number 2016M590360]; the National Natural Science Foundation of China [grant
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numbers 11272205 and 51304066].
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YanchaiLiu received the B.Eng. degree in electronic engineeringform the Southeast University, Nanjing, China, the M. Eng. degree in communication engineering from the Shanghai Jiao Tong University, Shanghai, China.She is now working towardthe Ph.D degreewith Shanghai Jiao Tong University. Her research interests include digital signal processing, image processing, and artificial intelligence.
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Ting Wang is now a M.S. candidate at the School of Electrical and Information Engineering, Anhui University of Technology, China. Her current research interests include power systems, stochastic switched systems, complex networks, robust control and filtering.
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Mengshen Chen is now a M.S. candidate at the School of Electrical and Information Engineering, Anhui University of Technology, China. His current research interests include complex networks, Markov jump systems, switched systems, robust control and filtering
Hao Shen received the Ph.D. degree in control theory and control engineering from the Nanjing University of Science and Technology, Nanjing, China, in 2011.
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He was a Post-Doctoral Fellow with the Department of Electrical Engineering, Yeungnam University, Gyeongsan, Korea, from 2013 to 2014. Since 2011, he has been with the Anhui University of Technology, Ma’anshan, China, where he is currently an Associate Professor with the School of Electrical and Information Engineering. His current research interests include stochastic hybrid systems, network control systems, nonlinear systems, and their application.
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Yueying Wang received the B.Eng. degree in mechanical engineering and automation form the Beijing Institute of Technology, Beijing, China, in 2006, the M. Eng. degree in navigation, guidance, and control, and Ph.D. degree incontrol science and engineeringfrom the Shanghai Jiao Tong University, Shanghai, China, in 2010 and 2015, respectively. He is currently a Post-Doctoral Fellow at the School of Aeronautics and Astronautics, Shanghai Jiao Tong University. His current research interests include sliding mode control, sampleddata control, robust filtering, and their applications.
Dengping Duan received the Ph.D. degree from Harbin Institute of Technology, China. He is now a Professor and Ph.D. supervisor in the School of Aeronautics and Astronautics, Shanghai Jiao Tong University, China. His research interests include system theory and flight control.
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Dissipativity-based State Estimation of Delayed Static Neural Networks Yanchai Liu , Ting Wang , Mengshen Chen , Hao Shen , Yueying Wang , Dengping Duan PII: DOI: Reference:
S0925-2312(17)30592-1 10.1016/j.neucom.2017.03.059 NEUCOM 18292
To appear in:
Neurocomputing
Received date: Revised date: Accepted date:
15 July 2016 14 February 2017 22 March 2017
Please cite this article as: Yanchai Liu , Ting Wang , Mengshen Chen , Hao Shen , Yueying Wang , Dengping Duan , Dissipativity-based State Estimation of Delayed Static Neural Networks, Neurocomputing (2017), doi: 10.1016/j.neucom.2017.03.059
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Dissipativity-based State Estimation of Delayed Static Neural Networks
Yanchai Liua, Ting Wangb, Mengshen Chen b, Hao Shenb, Yueying Wanga,*, Dengping Duana a
b
School of Aeronautics and Astronautics, Shanghai Jiao Tong University, Shanghai 200240, China School of Electrical Engineering and Information, Anhui University of Technology, Ma’anshan 243002, China
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Abstract
This paper proposes a dissipativity-based state estimation methodology for static neural networks with time-varying delay. An Arcak-type observer is used to construct the
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estimation error system. To reduce the conservatism of observer design, a LyapunovKrasovskii functional (LKF) is adopted to fully exploit the available characteristics about activation function. In addition, a relaxed constraint condition is put forward to keep the whole LKF positive without requiring parts of involved matrices to be positive.
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By adopting the LKF and constraint condition, estimation conditions with a strict dissipative performance are obtained, which ensures the asymptotic stability of
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estimation error system. The computation of gain matrices about observer can be
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transformed into a convex optimization problem. Two examples are given to illustrate
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the validity and advantage of provided methodology.
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Keywords: static neural networks, state estimation, dissipativity , time-varying delay
*Correspondence author at: School of Aeronautics and Astronautics, Shanghai Jiao Tong University, Shanghai 200240, China. Tel:+86-21-34207170 Email addresses: [email protected](Yanchai Liu), [email protected] (Ting Wang), [email protected] (mengshen chen), [email protected](Hao Shen), , [email protected] ( Yueying Wang*). [email protected](Dengping Duan)
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1. Introduction According to the basic variables used when modeling [1], roughly speaking, recurrent neural networks are generally divided into two categories: local field neural networks
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and static neural networks. The former use local field states of neurons as basic variables, while the static neural networks adopt neuron states as basic variables. It has been clearly pointed out in [2] that, only under certain strict conditions, the two classes of neural networks can be equivalent. During the past years, static neural networks have
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been extensively applied in various fields, including associative memory and combinational optimization. Recently, the research on delayed recurrent neural networks has gradually become a hot topic (see e.g. [3]-[13]) because time delays are unavoidable in implementing a recurrent neural network because of the finite
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conduction velocity and switching speed about amplifier, which can lead to
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performance degradation or even instability of the neural networks. Meanwhile, a recurrent neural network with large scale often contains an amount of
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neuronal connections to deal with complex nonlinearities. As a consequence, the comple acquisition of the information about neuronal states may be unrealistic, while it
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is necessary in system modeling. So, how to efficiently estimate the states of neural
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networks gradually becomes an important research topic (see e.g. [14]-[23]). It is worth mentioning that aforementioned results are confined to local field neural networks, thus the obtained estimation methodologies are not entirely suitable for static neural networks. In addition, the behaviors about static neural networks are usually complicated escpecially when they suffer from various stochastic disturbances. It is of vital importance to suppress the infuluence from the external disturbaces when
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estimating their states. For this reason, considerable attention has been devoted to estimation problem for delayed static neural networks [24]-[30] with H / H 2 performance. To mention a few, the state estimation problem for time-delay static neural networks was investigated in [25], where estimation conditions with H / H 2
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performance were provided. Further improved results were given in [26], [27], where reciprocally convex combination approach was adopted. In [28], a suitable LKF was constructed, and less conservative results than [26], [27] were further achieved. The design problem of Arcak-type observer with H 2 performance for delayed static neural
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networks was studied in [29]. Based on the same estimator, the state estimation problem with guaranteed H performance was further studied in [30], where much better performance has been achieved than the results in [26] and [27].
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On the other hand, dissipativity theory [31], [32] originates from electrical networks, and, by using an input-output description, presents a tool for analysis and synthesis of
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control systems [33]. Dissipativity is characterized by storage functions and supply
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rates, which represent the energy stored inside the system and energy supplied from outside the system, respectively. It can be viewed as a more general performance index
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containing some well-known ones such as H performance constraint, the positive real
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performance, the mixed performance, and the sector bounded performance [34], [35]. Thus, from practical application point of view, it is more desirable to expect that the neural networks have the dissipativity property to achieve effective noise attenuation. Although some research efforts have been devoted to dissipativity analysis for neural networks (see, e.g., [36]-[38]), there has been very little literature reported on state estimation with guaranteed dissipative performance for static neural networks, which is the motivation about this paper.
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In this paper, we focus on state estimation with guaranteed dissipative performance for delayed static neural networks. The main contributions of the paper can be summarized as follows. 1) Dissipativity performance is introduced into the state estimation problem for delayed static neural networks, which is more general than
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previous H / H 2 performance. 2) An improved LKF is adopted to fully emploit the characteristics about the activation function. Moreover, a more relaxed condition is derived to keep the whole LKF positive without requiring some involved matrices to be positive. 3) An observer design criterion is offered for delayed static neural networks,
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which guarantees the asymptotic stability about estimation error system with a strict dissipative performance. Simulation results show that less conservativeness is achieved by our design method than the existing ones.
Notation. The notations used in this paper are standard. The set about m n real mn
. The block diagonal matrix and identity matrix are,
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matrices is represented by
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respectively, represented by diag
and
I . The space of square-integrable vector
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functions defined on 0, is denoted by L2 0, . The transpose of the matrix is denoted by superscript “ T ” , the inverse is represented by “ 1 ”, and the induced
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symmetric term isdenoted by “ ”.
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2. Problems formulation and preliminaries Choose a static neural networks with time-varying delay and external disturbance as
x ( t ) x ( t ) g x t d (t ) y (t ) x (t ) x t d (t ) 2 (t ) z (t ) x (t ) x (t ) (t ), t d ,0
4
(t )
1
(1)
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where x(t )
n
is the neuronal state; y (t )
m
is the measurable output; (t )
the external disturbance belonging to L2 0, ; z(t )
(t ) is the initial function;
1
the connection weight matrix;
1
,
2
,
nn
,
n
p
q
is
is output of neural network;
T is the exogenous input;
0 is a diagonal matrix;
,D,
1
,
nn
2
is
and
satisfies: 0 d (t ) d , d ( t )
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are appropriate dimensioned matrices; The time delay d (t ) is the time-varying and
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with d and being positive scalars. g x(t ) g1 x1 (t ) , g2 x2 (t ) ,
(2)
, gn xn (t )
T
denotes the continuous activation function and satisfies
0
gi ( c ) g i ( d ) li , gi (0) 0 , c d cd
, i 1,2,
,n
(3)
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To reduce design conservatism, the Arcak-type full-order observer is adopted to
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estimate the states of static neural network (1), which is described by
1
y (t ) yˆ (t )
2
y (t ) yˆ (t ) (
and
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with
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xˆ (t ) xˆ (t ) g xˆ t d (t ) yˆ (t ) xˆ (t ) xˆ t d (t ) zˆ(t ) xˆ (t ) xˆ (t ) xˆ (0), t d ,0
1
2
4)
being the gain matrices of the observer to be computed.
Define the estimation errors about state and output as (t ) x(t ) xˆ (t )
and z (t ) z(t ) zˆ(t ) , respectively. Thus, based on (1) and (3), we obtain the estimation error system as
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(t ) z (t ) (t )
(t )
2
2
t d (t ) g t , d (t ), x(t ), (t )
1
2
2
( t ) (5)
where
For
estimation
s( , z ) :
q
p
x t d (t )
error
system
g
xˆ t d (t )
(5),
we
1
y(t ) yˆ (t )
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g t, d (t ), x(t ), (t ) g
introduce
a
function
satisfying s(0,0) 0 , which is called a supply rate if for all
s ( ), z ( ) d , t2
t1
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input-output pairs ( , z ) satisfying (5), s( , y ) satisfies
t2 t1 0 . System (5) is dissipative under the supply rate s( , z ) , if under zero initial condition, the following dissipation inequality t*
s ( ), z ( ) d 0 , t
*
0
(6)
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0
holds for any nonzero input (t ) . It can be found that if there exists a nonnegative
satisfying V (0) 0 and
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n
definite function V ( x ) :
V x(t ) V x(0) s ( ), z ( ) d , t * 0
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*
t*
(7)
0
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then system (5) is dissipative under the supply rate s( , z ) . Accordingly, the function
V ( x ) is called a storage function for system (5). In this paper, we consider a quadratic
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supply rate s( , z ) , which is proposed in the following definition Definition 1: The estimation error system (5) is called strictly (
,
,
) - -dissipative
for any t p 0 and some scalar 0 , if under zero initial condition, the following inequality is satisfied:
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T
tp
0
where
p p
z (t ) (t ) *
,
pq
tp z (t ) T (t ) dt 0 (t ) (t )dt
and
qq
are real matrices, with
(8)
and
being
symmetric matrices.
0 and
I , thus the performance
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0,
Remark 1. Here, it is assumed that
defined in Definition 1 includes H , positive realness, and passivity as special cases. Specially, when
0 , and
I ,
H performance index. When
0,
( 2 )I , inequality (7) degenerates to an I , and
2 I , inequality (7) degenerates
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to strict passivity or strictly positive realness.
Lemma 1[39]. Given time delay 0 d (t ) d , and and any appropriate dimensioned
Z 0 , the following condition is satisfied: R
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R matrices R and Z such that *
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t R d T ( ) R ( )d T (t ) t d *
where
Z ( t ) R
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(t ) 1T (t ) 2T (t ) 3T (t ) 4T (t ) 1 (t ) t d (t ) t d
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2 (t ) t d (t ) t d
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4 (t ) (t ) t d (t )
T
t d ( t ) 2 ( )d 3 (t ) (t ) t d (t ) d d (t ) t d
R 0 2 t ( )d , R d (t ) t d ( t ) 0 3R
The main aim about this paper is to design an Arcak-type observer (4) to make the
estimation error system (5) with (t ) 0 be asymptotically stable, and be strictly (
,
,
) - -dissipative under zero initial condition.
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3. Main results
In this section, we shall investigate the state estimation problem for delayed static neural networks (1) with dissipativity performance. To make full use of the information
into the following two suitable parts:
g e t , d (t ), x(t ), (t ) g g
xˆ t d (t )
x t d (t )
g
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of the activation function, the nonlinear function ge t, d (t ), x(t ), (t ) can be divided
xˆ t d (t )
g xˆ t d (t ) y (t ) yˆ (t ) (t ), t d (t ) , (t ) 1
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g e1 t d (t ) g e 2
(9)
Before proceeding further, we introduce the following useful proposition based on (3). Proposition 1. For any matrices Si diag si1, si 2 ,
(t ) 2 gT1 (t ) S1g 1 (t ) 0
(10)
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2 gT1 (t ) S1
, sin 0 , i 1,2,3
2 gT1 t d (t ) S2
ED
t d (t ) 2 gT1 t d (t ) S2 g 1 t d (t ) 0
2 gT 2 (t ), t d (t ) , (t ) S3
1
(t )
2
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K1 (t ) 1 t d (t ) 2 gT2 (t ), t d (t ) , (t ) S3 g 2 (t ), t d (t ) , (t ) 0
(11)
(12)
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Remark 2. As illustrated in [30], the slope constants about the activation function has
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an important impact on design conservatism of observer. However, it should be pointed out that the slope information in (5) cannot be used directly due to the existence of the term
1
y(t ) yˆ (t ) in
g t, d (t ), x(t ), (t ) . To facillatate the adoption of information,
the nonlinear part of the estimation error system is divided into two parts. Theorem 1. For given scalars d 0 , , and 0 , if there exist matrices 0
p p
,
pq
and
qq
with
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and
being symmetric, symmetric
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P13 Q Q12 ˆ P23 , Q 11 0 , Q , R 0 , U 0 , diagonal matrices * Q 22 P33
P12 P22 *
Si 0 , i 1,2,3 , 1 diag 11 , 12 ,
, 1n 0 , 2 diag 21, 22 ,
Z appropriately dimensioned matrices Z 11 Z 21
, 2 n 0 and any
Z12 , M , G1 , G2 such that the Z 22
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P11 matrices P * *
following conditions are feasible for d (t ) 0, d , d (t ) 0, : Z12 Z 22 0 0 3R
(13)
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R 0 Z11 * 3R Z 21 R * * * * *
1,4 0 3,4 4,4 * * * * * *
1,5 2,5 3,5 4,5 5,5 * * * * *
PT
CE
AC
*
R 0 0 0 hQˆ R
1,6 1,7 d (t ) P13 2,7 3,6 0 4,6 0 4 Z 22 0 6,6 0 * 7,7 * * * * * *
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1,3 1,1 1,2 * G2 2,2 * 3,3 * * * * * * * * * * * * * * * * * * * * * *
P13 P23 P33
M
P11 R P12 * P22 * * * *
M M 3,8 0 0 0 0 8,8 * *
(14)
1,9 M T G1T 0 0 0 0 0 9,9 *
1,10 2,10 0 0 0 0 0 0 0 G1 2 I (15)
where
1,1 P12 P12T Q11 Qˆ dU
1,2 P11
T
MT
T
T
G2T M
M T
1
9
T
M T G2
T
G2T 4R
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1,3 G2
T T Z11T Z12T Z 21 Z 22 2R 1 d (t ) ( P12 P13 )
T T 1,4 Z11T Z12T Z21 Z22 P12 T T 1,5 2Z 21 2Z 22 d d (t ) P23T
1,7 Q12
T
1,9 M
T
1,10
M
T
T
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1,6 6R d (t ) P33T
S1T
G1T 1
G2
2
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2,2 M M T d 2 R 2,5 d d (t ) P12 T
1T
2,10 M
1
T
G2
T2
M
2,7
2
ED
T T 3,3 1 d (t ) Q11 8R Z11T Z11 Z12T Z12 Z 21 Z 21 Z 22 Z 22
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T T 3,4 2 R Z11T Z12T Z21 Z22
CE
T T 3,5 6R 2Z 21 2Z 22 1 d (t ) d d (t ) ( P22T P23T )
3,6 6R 2Z12 2Z 22 1 d (t ) d (t )( P23 P33T )
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3,8 1 d (t ) Q12
T
LT S2T
4,4 Qˆ 4 R 5,5 12 R d d (t ) U
4,5 6R d d (t ) P22T 4,6 2Z12 2Z22 d (t ) P23
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6,6 12 R d (t )U
7,7 S1 S1T Q22
8,8 S2 S2T 1 d (t ) Q22
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9,9 S3 S3T Then, the estimation error system (5) is globally asymptotically stable and strictly (
,
) - -dissipative. Moreover, the gain matrices
,
can be computed as S3 L G1 , 1
2
M 1G2
and
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1
1
2
of observer (4)
(16)
Proof. For estimation error system (5), we choose a LKF as 6
V (t ) Vi (t ) i 1
T
P11 * *
(t ) P13 t d ( t ) P23 ( )d t d P33 t ( )d t d ( t )
ED
(t ) t d ( t ) V1 (t ) ( )d t d t ( )d t d ( t )
M
where
PT
n
wiT
i 1
0
n
wiT
1i li g 1i ( ) d 2 0
CE
V2 (t ) 2
P12 P22 *
i 1
2i g 1i ( ) d
( ) Q11 Q12 ( ) V3 (t ) d t d ( t ) g 1i ( ) * Q22 g 1i ( ) T
AC
t
t
V4 (t ) T ( )Qˆ ( )d t d
V5 (t ) d
0
d
t
t
T ( ) R ( )d d
11
(17)
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V6 (t )
0
d
t
t
T ( )U ( )d d
First, we need to prove V (t ) 0 . It can be verified that T
T
P11 R * * *
P12 P22 *
P13 P23 P33
*
*
(18)
(t ) R t d ( t ) 0 t d ( )d d 0 t ( )d t d ( t ) dQˆ R ( )
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(t ) t d ( t ) ( ) d 1 t t d t d t d ( ) d t d ( t ) ( )
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(t ) (t ) P11 P12 P13 t d ( t ) 1 t t d ( t ) V1 (t ) V4 (t ) V5 (t ) ( )d * P22 P23 ( )d d t d d t d t d t * * P33 t ( )d ( )d t d ( t ) t d ( t ) t t 1 T T ( )Qˆ ( )d (t ) ( ) R (t ) ( ) d t d t d d
From (14), we obtain that V1 (t ) V4 (t ) V5 (t ) 0 . Moreover, from assumption, we
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known that V2 (t ) and V3 (t ) are also positive definite. So, the whole LKF in (17) is
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positive definite. Taking the time derivative of the LKF and introducing dissipativity
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CE
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performance index result in
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z (t ) 2 zT (t ) (t ) wT (t )(
V (t ) zT (t )
I ) (t )
T
(t ) (t ) P11 P12 P13 t d ( t ) 2 ( )d * P22 P23 1 d (t ) t d (t ) (t d ) t d t * * P33 (t ) 1 d (t ) t d (t ) ( )d t d ( t ) 2 T (t ) T L1 (t ) 2 gT1 (t ) 1 (t ) 2 gT1 e(t ) 2 (t )
(t ) Q11 Q12 (t ) g 1i (t ) * Q22 g 1i (t )
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T
t d (t ) Q11 Q12 t d (t ) 1 d (t ) g 1i t d (t ) * Q22 g 1i t d (t )
T
t
T (t )Qˆ (t ) T (t d )Qˆ (t d ) d 2 T (t ) R (t ) d T ( ) R ( )d d T (t )U (t )
t d
T (t )
T ( )U ( )d
t
t d ( t )
(t ) 2 T (t )
T
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t d
t d ( t )
T
T ( )U ( )d
(t ) T (t )(
(19)
I ) (t )
By using Lemma 1, the following inequality holds:
M
t R d T ( ) R ( )d T (t ) t d *
Z ( t ) R
(20)
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where R and (t ) are defined in Lemma 1.
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For any scalar and real matrix M with appropriate dimension, the following condition holds:
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0 T (t ) M T (t ) M
2
(t )
2
t d (t ) g t, d (t ), x(t ), (t )
1
2
2
(t )
AC
(t )
(21
)
Thus, adding the left parts of inequalities (10)-(12) into (19), and in the light of (20) and (21), we can obtain that for all nonzero (t ) L2 0, V (t ) zT (t )
z (t ) 2 zT (t ) (t ) T (t )(
13
I ) (t ) T (t ) (t) 0
(22)
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where
(t ) T T T T (t ) (t ) t d (t ) (t d )
g eT1 t d (t ) g eT2 (t ), t d (t ) , (t ) T (t )
1 t T ( )d t d ( t ) d (t ) T
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g eT1 (t )
t d ( t ) 1 T ( )d t d d d (t )
Under zero conditions, we obtain V (0) 0 and V () 0 . Integrating both sides of (22) over the time period 0 to t p results in (t ) V (t p ) 0 , which ensures (8), that is,
,
,
) - -dissipative. Following the similar procedures used
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system (5) is strictly (
above, we can obtain the estimation error system (5) with (t ) 0 is asymptotically stable. The proof is completed.
Remark 3. By adopting the LKF in (17), a dissipativity-based state estimation criterion
M
is presented delayed static neural networks. To reduce conservativeness, a relaxed
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condition (14) is provided to keep the whole LKF positive without requiring the involved matrices P and Qˆ to be positive, which is another advantage of proposed
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design method.
Remark 4. The number about decision variables involved in Theorem is
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13n 2 (2m 9)n , while the numbers in [28] and [30] are, respectively,
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9.5n 2 (m 5.5)n and 9n2 (2m q 4)n , which shows more computational burden will be consumed by our method. However, it also means that the solution of obtained results in our paper will be searched in a bigger set. Remark 5. It is worth mentioning the achieved estimation performance can be further enhanced by other ways. For instance, an efficient way is to adopt the latest inequality techniques [40], [41] to estimate the upper bound on derivatives for LKF (17).
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As a consequence, more computational time will be consumed. Moreover, it is noted that only asymptotic stability is ensured. How to adopt the finite-time control/filtering techniques [42]-[46] to investigate the finite-time state estimation problem for static neural networks is a topic theoretical significance.
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Remark 6. The computation about optimal dissipative performance index max can be transformed into the following convex optimization algorithm:
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Algorithm 1: max , subject to the conditions in (13) - (15).
4. Numerical examples
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In this subsection, two examples are provided to illustrate the validity of achieved estimation methodology.
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Example 1. Choose the system parameters of delayed static neural network (1) as
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0 1.06 0 0 1.42 0 , 0 0.88 0 0.2 0.2 0.2 ,
CE
1
T
2
0.32 0.85 1.36 1.10 0.41 0.50 , 0.42 0.82 0.95
0.4 0.3 , T
1 0 0.5 1 0 1 , 0 1 1
0 1 0.5 , 0 0.5 0.6
0 1 0.2 . 0 0 0.5
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We first illustrate the less conservativeness of proposed approach. In order to
facilitate comparison, we set
I3 ,
0 , and
2 . In this situation, the
disspativity-based stability criterion proposed in Theorem 1 degenerates to a stability criterion for error estimation system (5) with guaranteed H performance index. Let
0.5 , d 0.8 and 0.6 , Table I lists the comparison results about index min under
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various constants
, which shows that the estimation performance achieved by our
methodology is better than ones in the most recent works [28] and [30]. Moreover, the allowable maximum slope constant by using our method is
1.65 with minimum
index min 155.5177 , while the allowable ones in [28] and [30] are 1.42I and 1.49I ,
Under slope constant
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respectively.
diag 1.2,1.3,1.4 and 0.5 , Table II gives the
comparison results about optimal H performance min under different delay bounds d , which further confirm less conservativeness of our method over the most recent works
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[28] and [30].
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Table I: Optimal H performance index min via different slope constants Methods 1.2I 1.3I 1.4I 1.5I [28] 1.0918 2.2097 11.9007 infeasible [30] 0.6536 0.9805 2.2387 infeasible Theorem 1 2.0434e-06 0.6461 1.0098 1.8932
CE
PT
ED
Table II: Optimal H performance index min via different upper bounds d Methods 0.7 0.8 0.9 1.0 [28] 1.3675 2.4425 7.1801 infeasible [30] 0.5953 1.0052 32.0824 infeasible Theorem 1 2.6353e-07 0.5535 0.9600 2.2973 For simulation, we choose activation function f ( x) 1.4 tanh( x) ,
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1.5 1.5 1.5 , and T
1.5I 3 ,
4.0 , time delay d (t ) 0.4 0.4sin(1.5t ) , which implies
1.4 , d 0.8 and 0.6 . By using Theorem 1, the optimal dissipativity
performance is computed as max 1.8143 , and the corresponding gain matrices of observer as
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-0.0380 0.0083 0.0894 , 1 (1.0e 06) * 0.0014 -0.0090 -0.1703
2
-0.6131 -0.1763 -0.6389 0.2901 -0.7613 -0.9732
By choosing the initial condition as (0) 2 2 3 , Fig. 1 dipicts the responses T
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of estimation errors under disturbance signal (t ) sin(t )e5t . The simulation results confirm the validity of achieved method to design the dissipativity-based observer for delayed static neural networks.
3
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2.5 2 1.5
0.5
M
e(t)
1
0
ED
-0.5 -1
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-1.5 0
2
4
6
8
10
t/sec
Fig. 1. Responses of state estimation error (t )
AC
CE
-2
e(1) e(2) e(3)
Example 2. Choose the parameters of delayed static neural network as
0 7.0214 , 7.4367 0
2
0.1 ,
1 0 ,
6.4993 12.0257 , 0.6867 5.6614
1 1 , 0 1
1
0.2 , 0.2
2 0 .
Table III: Optimal H performance index min via different slope constants Methods 0.4I 0.5I 0.6I 0.7I 0.8I
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[28] [30] Theorem 1
0.1829 0.0910 0.0805
0.2636 0.1179 0.0937
0.6121 0.1755 0.1153
infeasible 0.3783 0.1546
infeasible infeasible 0.2459
Denote 0.2 , d 0.4 and 0.7 , the comparison results about optimal H performance min via different slope constants
are depicted in Table III. It is
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observed from Table III that the performance achieved by our approach is better than ones in the most recent works [28] and [30], which further confirm the advantage of our methodology. Moreover, the maximum slope constants computed in [28] and [30] are
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0.65I and 0.77I , respectively, while applying our method, the maximum one
0.94 , with corresponding H performance index min 14.0796 . By making the
first gain matrix in Arcak-type observer (4) be zero, the observer reduces to a Luenberger-type one. In this case, the obtained maximum slope constant reduces to
M
0.92 , which verifies the advantage of Arcak-type observer
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5. Conclusion
In this paper, the issue about dissipativity-based state estimation has been addressed for
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delayed static neural networks. By employing the LKF, an observer design procedure with strict dissipative performance have been presented. The computation about
CE
maximum dissipative performance and corresponding gain matrices of the observer can
AC
be transformed into a convex optimization problem. Two examples have been given to illustrate the less conservativeness of provided design method than most recent results. In future, we will use the proposed methodogy to estimate the neuronal states of projection neural networks [47], which is a typical static neural network.
Acknowledgement
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This work was supported by the China Postdoctoral Science Foundation [grant number 2016M590360]; the National Natural Science Foundation of China [grant
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numbers 11272205 and 51304066].
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YanchaiLiu received the B.Eng. degree in electronic engineeringform the Southeast University, Nanjing, China, the M. Eng. degree in communication engineering from the Shanghai Jiao Tong University, Shanghai, China.She is now working towardthe Ph.D degreewith Shanghai Jiao Tong University. Her research interests include digital signal processing, image processing, and artificial intelligence.
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Ting Wang is now a M.S. candidate at the School of Electrical and Information Engineering, Anhui University of Technology, China. Her current research interests include power systems, stochastic switched systems, complex networks, robust control and filtering.
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CE
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Mengshen Chen is now a M.S. candidate at the School of Electrical and Information Engineering, Anhui University of Technology, China. His current research interests include complex networks, Markov jump systems, switched systems, robust control and filtering
Hao Shen received the Ph.D. degree in control theory and control engineering from the Nanjing University of Science and Technology, Nanjing, China, in 2011.
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He was a Post-Doctoral Fellow with the Department of Electrical Engineering, Yeungnam University, Gyeongsan, Korea, from 2013 to 2014. Since 2011, he has been with the Anhui University of Technology, Ma’anshan, China, where he is currently an Associate Professor with the School of Electrical and Information Engineering. His current research interests include stochastic hybrid systems, network control systems, nonlinear systems, and their application.
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CE
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Yueying Wang received the B.Eng. degree in mechanical engineering and automation form the Beijing Institute of Technology, Beijing, China, in 2006, the M. Eng. degree in navigation, guidance, and control, and Ph.D. degree incontrol science and engineeringfrom the Shanghai Jiao Tong University, Shanghai, China, in 2010 and 2015, respectively. He is currently a Post-Doctoral Fellow at the School of Aeronautics and Astronautics, Shanghai Jiao Tong University. His current research interests include sliding mode control, sampleddata control, robust filtering, and their applications.
Dengping Duan received the Ph.D. degree from Harbin Institute of Technology, China. He is now a Professor and Ph.D. supervisor in the School of Aeronautics and Astronautics, Shanghai Jiao Tong University, China. His research interests include system theory and flight control.
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