[ieee 2009 ieee international conference on control and automation (icca) - christchurch, new...
TRANSCRIPT
Abstract—Exponential stability and controller design problems for networked control systems (NCSs) with multiple- packet transmission are studied. Various state feedback NCSs, such as having time-delay and dropped out packets, having time-delay and no dropped out packet, having dropped out packets and no time-delay, having no time-delay and no dropped out packet in sensor-to-controller and controller-to- actuator, are modeled as asynchronous dynamical systems constrained by event rates. Based on the theory of asynchronous dynamical system, Lyapunov stability theory and linear matrix inequality method, the negative semi-definite matrix conditions of exponential stability for the state feedback NCSs with multiple-packet transmission and controller design method are presented. Numerical examples show that the negative semi- definite matrix conditions of exponential stability and the controller design method are feasible.
I. INTRODUCTION T is well known that there is transmission time-delay,
data packet dropout and multiple-packet transmission caused by the network in networked control system (NCS). They have certain impact on the stability of NCS, which is an important feature that reflects the normal operation state of the system. The papers [1]-[3] present exponential stability definition and fundamental theorem of asynchronous dynamical system (ADS), and study the exponential stability of NCS; the papers [4]-[6] study the asymptotic stability and robust stability of NCS with time-varying delay; the papers [7]-[9] focus on the global exponential stability of model- based NCS; the papers [3] and [10]-[15] focus on the exponential stability and the asymptotic stability of multi- packet transmission NCS, but they have not referred to the packet dropout case. This paper focuses on exponential stability and controller design for the NCSs with multiple- packet transmission and state feedback. Considering the case
Manuscript received March 27, 2009. This work was supported by
National Nature Science Foundation under Grant 60574011. X. D. Dang is with the School of Information Science and Engineering
Technology, Shenyang Ligong University, Shenyang, Liaoning, CO 110168, P.R. China, and with the Institute of Systems Science, Northeastern University, Shenyang, Liaoning, CO 110004, P.R. China (phone: 024- 83685236; fax: 024-82175127; e-mail: xddang2005@126. com).
Q. L. Zhang is with the Institute of Systems Science, Key Laboratory of Integrated Automation of Process Industry, Ministry of Education, Northeastern University, Shenyang, Liaoning, CO 110004, P.R. China (e-mail: [email protected]).
J. N. Li is with the Department of Mathematics and Physics, Shenyang Institute of Chemical Technology, Shenyang, Liaoning, CO 110142,P.R. China(e-mail: [email protected]).
with time-delay and packet dropout, the case with time-delay and no packet dropout, the case with packet dropout and no time-delay and the case with no packet dropout and no time-delay, we construct the models of state feedback NCSs with multiple-packet, present the negative semi-definite matrix conditions of exponential stability and controller design.
II. DESCRIPTIONS OF THE PROBLEM For convenience of investigation, we make the following
rational assumptions: 1) Sensors are time driven. All clocks of nodes in the NCS
are synchronized and sampling period is h. Based on a static scheduling algorithm, only one packet can be transmitted at a time.
2) Both controllers and actuators are event driven. Contro- llers receive signal in sensors transmission sequence. Actuators accept signal in controller transmission sequence.
3) The loss rate is constant if there is data packet dropout in NCS.
4) All signals are transmitted without scheduling reordering. 5) sc
kτ and cakτ denote sensor-to-controller transmission delay
and controller-to-actuator transmission delay, respect- tively. Total time-delay sc ca
k k k hτ τ τ= + ≤ is time invariable. [10]-[12] The time-delay which all devices process data is kτ . kτ = 0 denotes no time-delay.
Thus, the structure of the multiple-packet transmission NCS is illustrated as Fig. 1 and Fig. 2.
Taking ( ) ( )k kh= for short (omitting the sample period
mark h ). Notation 1 Taking vector
T T T T1 2( ) ( ) ( ) ( )mx k x k x k x k= T T T T
1 2( ) ( ( ) ( ) ( ))mx k x k x k x k= T T T T1 2( ) ( ( ) ( ) ( ))nv k v k v k v k= T T T T1 2( ) ( ( ) ( ) ( ))nu k u k u k u k=
In Fig. 1 and Fig. 2, ( )x k and ( )u k are the controlled plant state and input, respectively. Both 1K and 2K are the switches of the network. is ( js ) ( 1,2, , ; 1,2, ,i m j n= = ) denotes
event containing data packet ( )ix k ( ( )jv k ). iss ( jts ) ( 1,2s = ,
Stability of State Feedback NCSs with Multiple-Packet Ttransmission
Xiangdong Dang, Qingling Zhang, Jinna Li
I
2009 IEEE International Conference on Control and AutomationChristchurch, New Zealand, December 9-11, 2009
ThMPo5.8
978-1-4244-4707-7/09/$25.00 ©2009 IEEE 1433
Fig. 1. Structure of NCS with multiple-packet transmission and data packet dropout.
Fig. 2. Structure of NCS with multiple-packet transmission and no data packet dropout.
1,2t = , Fig. 1; 1s = , 1t = , Fig. 2)are the state of 1K ( 2K ). If the state of 1K ( 2K ) is 1is ( 1js ), which denotes the data packet
( )ix k ( ( )jv k ) of is ( js ) is unloosed, ( ) ( )i ix k x k=
( ( ) ( ))j ju k v k= . If the state of 1K ( 2K ) is 2is ( 2js ) (Fig. 1),
which denotes the data packet ( )ix k ( ( )jv k ) of is ( js ) is lost,
( ) ( 1)( ( ) ( 1))i i j jx k x k u k u k= − = − . If the state of 1K ( 2K ) is
neither 1is ( 1js ) nor 2is ( 2js ) (Fig. 1) and if the state of
1K ( 2K ) is not 1is ( 1js ) (Fig. 2), which denote the data packet
( )ix k ( ( )jv k ) of is ( js ) is pending state, ( ) ( 1)i ix k x k= −
( ( ) ( 1))j ju k u k= − . So, ( )ix k and ( )ju k can be expressed as in Fig. 1:
1
1 2
1 2
( ) if the state of is ;( ) ( 1) if the state of is ;
( 1) if the state of is neither nor
i i
i i i
i i i
x k K sx k x k K s
x k K s s= −
−
2 1
2 2
2 1 2
( ) if the state of is ;
( ) ( 1) if the state of is ;
( 1) if the state of is neither nor
j j
j j j
j j j
v k K s
u k u k K s
u k K s s
= −
−
in Fig. 2:
1
1 1
( ) if the state of is ;( )
( 1) if the state of is not ;i i
ii i
x k K sx k
x k K s=
−
2 1
2 1
( ) if the state of is ;( )
( 1) if the state of is not ;j j
jj j
v k K su k
u k K s=
−
Thus, ( )x k and ( )u k can be described as
ˆ ˆ( ) ( ) ( ) ( 1) (Fig.1)is i is i ix k x k x k= Φ Φ + Φ Φ + Φ − 12s ,= (1) ˆ ˆ( ) ( ) ( ) ( 1) (Fig.1)jt j jt j ju k v k u kΨ Ψ Ψ Ψ Ψ= + + − 12t ,= (2)
ˆ( ) ( ) ( 1) (Fig.2)i ix k x k x k= Φ + Φ − (3)
ˆ( ) ( ) ( 1) (Fig.2)j ju k v k u kΨ Ψ= + − (4) where
1 1 1 1
2 2 1 2
1 1 2 1
2 2 2 2
ˆ, 0( 1), if the state of is ;ˆ0, ( 2), if the state of is ;ˆ, 0( 1), if the state of is ;ˆ0, ( 2), if the state of is .
i i i
i i i
j j j
j j j
I s K s
I s K s
I t K s
I t K s
Ψ Ψ
Ψ Ψ
Φ = Φ = =
Φ = Φ = =
= = =
= = =
(0, ,0, ,0, ,0)i iidiag ϕΦ = , 1iiϕ = , ˆ ;i iIΦ = − Φ
(0, ,0, ,0, ,0)j jjdiagΨ ψ= , ˆ1, .jj j jIψ Ψ Ψ= = −
III. MATHEMATICAL MODEL Consider the continuous model of controlled plant
( ) ( ) ( )kx t Ax t Bu t τ= + − (5) where ( )x t and ( )u t are the controlled plant state and input, respectively. A and B are constant matrices with appropriate dimensions. The discrete model of system (5) is given by
1 0 1( 1) ( ) ( )+ ( 1)x k A x k B u k B u k+ = + − (6) where for 0kτ ≠ (with time-delay)
1 0 10, , k
k
h hAh As As
hA e B e ds B B e ds B
τ
τ
−
−= = ⋅ = ⋅ (7)
for 0kτ ≡ (with no time-delay)
1 0 10, , 0
hAh AsA e B e ds B B= = ⋅ = (8)
The discrete controller model is ( ) ( )v k Kx k= − (9)
where ( )x k , ( )v k and K are the controller input, output and gain. Define the vector
T T T TZ( ) [ ( ) ( ) ( 1)]k x k x k u k= − (10) If the states of 1K and 2K are iss and jts , the multiple-packet transmission NCS model can be obtained by (1), (2) ((3), (4)), (6), (9) and (10).
( ) ( )1 lZ k Z k+ = Ω (11) where
1 1 2
3 1 4 5
6 7
0 l isjt
A KA
K
π ππ π π
π πΩ = Ω = (12)
Notation 2 For having time-delay ( 0kτ ≠ ) and packet dropout
1 0 2 0 7 1 3 4 8 9
5 3 2 6 7 8
9 3 1
, + , = , + ,ˆ ˆ ˆ ˆ, , , = + ,
. 1, 2, , ; 1, 2; 1, 2, , ; 1,2.
jt j is i
jt j jt j j is i i
B B
i m s j n t
π Ψ Ψ π π Β π π π Κ
π = π π π Ψ Ψ π Ψ Ψ Ψπ = π π
= − = Φ Φ =
= − = + Φ Φ Φ
= = = =
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Notation 3 For having time-delay ( 0kτ ≠ ) and no packet dropout
1 0 2 0 7 1 3 4 8 9
5 3 2 6 7 8 9 3 1
, + , = , + ,ˆ ˆ, , , = , .
1, 2, , ; 1; 1,2, , ; 1.
j i
j j i
B B
i m s j n t
π Ψ π π Β π π π Κ
π = π π π Ψ π Ψ π = π π
= − = Φ =
= − = Φ
= = = =
Notation 4 For having packet dropout and no time-delay ( 0kτ ≡ )
2 0 7 1 3, ,Bπ π= - 9π and , , ,i s j t are same as Notation 2. Notation 5 For having no time-delay ( 0kτ ≡ ) and no
packet dropout 2 0 7 1 3, ,Bπ π= - 9π and , , ,i s j t are same as Notation 3.
Notation 6 The incidence rate of event ( )is jts s is ( )is jtr r ,
1,2; 1,2;s t= = and 2 1 2 11 ( 1 )i i j jr r r r= − = − ; the incidence
rate of event ( )i js s is ( )i jr r , and1
1m
ii
r=
= (1
1n
jj
r=
= ); the
incidence rate of event l isjtΩ = Ω is ˆ ( )l is i jt ir r r r r= ,and
1
ˆ 1N
ll
r=
= , 4N mn= (with packet dropout), N mn= (with no
packet dropout).
IV. EX PONENTIAL STABILITY AND CONTROLLER DESIGN
Lemma 1[1]-[3] For the ADS ( ) ( )( )1 sx k f x k+ = , s=1,2,
, N , the incidence rates of events s=1,2, , N are rs, and
1
1N
ss
r=
= . If there exists a Lyapunov function ( ( ))V x k :
R Rn+→ and scalars as>0 satisfying
11s
N rss
a=
>∏ (13)
( )( ) ( )( ) ( ) ( )( )21 1sV x k V x k a V x k−+ − ≤ − (14)
Then the ADS is exponentially stable.
Lemma 2[16] For symmetric matrix 11 12
21 22
s sS s s= , if s11 is
reversible, and s11<0, then 122 21 11 120 0S s s s s−≤ ⇔ − ≤ .
Theorem 1 For system (11), the incidence rate l̂r of event
( )l isjtΩ = Ω , and 1
ˆ 1N
ll
r=
= , if there exist symmetric positive
definite matrices P,Q,R and scalars ( ) 0l isjta a= > (l=1,2, ,N) satisfying
ˆ1
1lN r
lla
=>∏ (15)
1 1 2
3 1 4 5
6 7-2
-2
0 0 * 0 * * 0
ˆ * * * 0 0
* * * * 0
* * * * *
ll
l
P A P KQ RQ A P Q R
R KQ R
a P
a Q
π ππ π π
π πΩ =
-
-
-
-
--2
0
la R
≤
-
(16)
where the notation “∗” denotes symmetric element, the following is the same. l=1,2, ,N, N=4mn(with packet dropout); N=mn (with no packet dropout). Then
1) system (11) is exponentially stable. 2) in (16), let W=KQ, if there exist feasible solutions P, Q,
R, W, we can obtain control law ( ) ( )1v k WQ x k−= − . Proof The inequality (15) is precisely (13) in Lemma 1.
Supposing ( )P diag P Q R= , choosing a Lyapunov function T 1( ( )) ( ) ( )V Z k Z k P Z k−= , then
2
T 1 2 T 1
T T 1 2 1
( ( 1)) ( ( ))
( 1) ( 1) ( ) ( )
( )( ) ( ) 0
l
l
l l l
V Z k a V Z k
Z k P Z k a Z k P Z k
Z k P a P Z k
−
− − −
− − −
+ −
= + + −
= Ω Ω − ≤
It is equivalent to[2]-[4],[6] T 1 2 1 0l l lP a P− − −Ω Ω − ≤
By Lemma 2, the above inequality is equivalent to
T 2 1
0
l
ll l
P
a P− −
− ΩΩ = ≤
Ω −
The above inequality is equivalent to
T 2
( ) ( )
0
l
l
l l
diag I P diag I P
P P
P a P−
Ω
− Ω= ≤
Ω −
Substituting for (12) and ( )P diag P Q R= in the above inequality, we have
1 1 2
3 1 4 5
6 7-2
-2
-2
0 0 * 0 * * 0
* * * 0 0
* * * * 0
* * * * *
l
l
l
P A P KQ RQ A P Q R
R KQ R
a P
a Q
a R
π ππ π π
π π
−−
−
−
−
−
0≤
That is (16). From Lemma 1, conclusion 1) holds. In (16), supposing W=KQ, we can obtain the equivalent
inequality of (16) 1 1 2
3 1 8 9 5
6 7-2
-2
0 0 * 0 * * 0
* * * 0 0
* * * * 0
* * *
ll
l
P A P W RQ A P Q W R
R W R
a P
a Q
π ππ π π π
π π
−− +
−Ω =
−
−-2
0
* * la R
≤
−
(17)
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If (17) have the feasible solutions P , Q , R ,W , the conclu-
sion 2) can be obtained by substituting for 1K WQ−= in (9). The proof is completed. Remark 1 Theorem 1 includes the following 4 cases: 1) Using (1), (2), (7) and Notation 2, we have the case with
time-delay and packet dropout; 2) Using (3), (4), (7) and Notation 3, we have the case with
time-delay and no packet dropout; 3) Using (1), (2), (8) and Notation 4, we have the case with
packet dropout and no time-delay; 4) Using (3), (4), (8) and Notation 5, we have the case with
no time-delay and no packet dropout; Remark 2 For m+n data packets of events ( 1,2, , )is i m=
and ( 1, 2, , )js j n= , if there exists a data loss rate which is not zero at least, then the system has packet dropout; if all data loss rates are zero forever, then the system has no packet dropout.
V. NUMERICAL EXAMPLES Consider the continuous plant model
( ) ( ) ( )kx t Ax t Bu t τ= + − (18)
where 1
2
( )( )
( )x t
x tx t
= ,0.2 0 1
, 0 0.1 1
A B= = , sampling
period 0.7h s= . The discrete equation of (18) is 1 0 1( 1) ( ) ( )+ ( 1)x k A x k B u k B u k+ = + − (19)
T1 2 1 2Z( ) [ ( ) ( ) ( ) ( ) ( 1)]k x k x k x k x k u k= − (20)
A. Case with Time-Delay and Packet Dropout
0.01k s=τ , 1 01 1503 0 0 7398, 0 1 0725 0 7144. .
A B. .
= = ,
10 01160 0107.
B.
= .The incidence rate 11 21 0.9r r= = ,
12 22 0.1r r= = , 11 0.9r = , 12 0.1r = . The incidence rates of event 1s , 2s and 1s are 1 2 0.5r r= = , 1 1r = . The incidence rates of event 1 1111Ω = Ω , 2 1112Ω = Ω , 3 1211Ω = Ω , 4 1212Ω = Ω ,
5 2111Ω = Ω , 6 2112Ω = Ω , 7 2211Ω = Ω and 8 2212Ω = Ω are
1̂ 0.405,r = 2̂ 0.045,r = 3̂ 0.045,r = 4̂ 0.005,r = 5 1ˆ ˆr r= ,
6 7 2ˆ ˆ ˆr r r= = and 8 4ˆ ˆr r= . Choose scalars 1 5 21.1361,a a a= = = ,
3 6 7 0.7288a a a= = = , 4 8a a= = 0.6421 ,then 1 2ˆ ˆ1 2r ra a
8̂8 1.04287665 1ra > > . 1 2
ˆΦ =Φ =1 00 0
, 1 2Φ̂ = Φ =
0 00 1
, 2m = , 1n = . For (16) and Remark 1 1) in Theorem
1, using the feasible solver feasp in MATLAB LMI toolbox, we get the feasible solutions as follows
13 0.1240 0.015610
0.0156 0.7284P −= × ,
8 0.0000 0.000010
0.0000 5.3745Q
−= ×
−,
14=1.1661 10R −× , 1310 [0.0321 0.4377]W −= × . From Theorem 1(Remark 1 1)), we know that the system is
exponentially stable with the control law ( ) [0.0334 0.0000] ( )v k x k= − −
B. Case with Time-Delay and no Packet Dropout
0.02k s=τ , 1 01 1503 0 0 7284, 0 1 0725 0 7036. .
A B. .
= = ,
10 02300 0215.B.
= .Here, the incidence rate 11 21 11 1r r r= = = ,
1 2 0.5r r= = , 1 1r = . Choose scalars 1 2 1.1361, a a= = then 1 2ˆ ˆ
1 2 1.1361 1r ra a = > . 1Φ , 2Φ , 1Φ̂ and 2Φ̂ are same as Case A. For (16) and Remark 1 2) in Theorem 1, using the feasible solver feasp in MATLAB LMI toolbox, we have the feasible solutions as follows
10 0.0848 0.018110
0.0181 0.3786P −= × ,
8
0.0000 0.01260.0126 7.8414 10
Q =×
,
12=6.6406 10R −× , 1010 [0.0416 0.3386]W −= × . From Theorem 1(Remark 1 2)), we know that the system is
exponentially stable with the control law ( ) [0.0704 0.0000] ( )v k x k= − −
C. Case with Packet Dropout and no Time-Delay
Here, 1 01 1503 0 0 7514, 0 1 0725 0 7251. .
A B. .
= = , 1B 0= . The
others are same as Case A. For (16) and Remark 1 3) in Theorem 1, using the feasible solver feasp in MATLAB LMI toolbox, we get the following feasible solutions
10 0.3486 0.008510
0.0085 0.4205P −= × ,
9 0 1467 0 0002100 0002 0 1466
. .Q
. .− −= ×
−,
11=3.3175 10R −× , 1210 [0.0241 0.2184]W −= × . From Theorem 1(Remark 1 3)), we know that the system is
exponentially stable with the control law ( ) [0.0002 0.0015] ( )v k x k= −
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D. Case with no Time-Delay and no Packet Dropout
Here, 1 01 1503 0 0 7514, 0 1 0725 0 7251. .
A B. .
= = , 1B 0= . 1Ω ,
2Ω and the others are same as Case B. For (16) and Remark 1 4) in Theorem 1, using the feasible solver feasp in MATLAB LMI toolbox, we obtain the following feasible solutions
13 0.2829 0.000410
0.0004 0.3523P − −
= ×−
,
12 0.1448 0.001710
0.0017 0.1464Q − −
= ×−
,
8=6.3050 10R × , 1410 [0.2504 0.2169]W −= × . From Theorem 1(Remark 1 4)), we know that the system is
exponentially stable with the control law ( ) [0.0174 0.0150] ( )v k x k= −
VI. CONCLUSIONS In this paper, the state feedback NCSs with multiple-packet
transmission in sensor-to-controller and controller-to- actuator are modeled as ADS constrained by event rates. Using the theory of ADS, Lyapunov stability theory and linear matrix inequality method, we present the negative semi-definite matrix conditions of exponential stability and the controller design method for the multiple-packet transmission state feedback NCS with time-delay and packet dropout, with time-delay and no packet dropout, with packet dropout and no time-delay and with no time-delay and no packet dropout. Illustrative examples show that the feasibility of the negative semi-definite matrix conditions of exponential stability. The negative semi-definite matrix conditions include the negative definite matrix conditions of exponential stability.
ACKNOWLEDGMENT This work is supported by the National Natural Science
Foundation of China under Grant No. 60574011.
REFERENCES [1] A. Hassibi, S. P. Boyd, and J. P. How, “Control of asynchronous
dynamical systems with rate constraints on events” in Proceeding of the 38th Conference on Decision and Control, Phoenix, USA, 1999, pp. 1345-1351.
[2] A. Rabello, and A. Bhaya, “Stability of asynchronous dynamical systems with rate constraints and applications,” IEE Proceedings Control Theory Application, vol. 150, no. 5, pp. 546-550, Sep. 2003.
[3] W. Zhang, M. S. Branicky, and S. M. Phillips, “Stability of networked control systems,” IEEE Control Systems Magazine, vol. 21, no. 1, pp. 84-99, February 2001.
[4] G. M. Xie and L. Wang “Stabilization of networked control systems with time-varying network-induced delay,” in Proceedings of the 43rd IEEE Conference on Decision and Control, Nassau, Bahamas, 2004, pp. 3551-3556.
[5] S. B. Li, Z. Wang and Y. X. Sun, “Delay-dependent controller design for networked control systems with long time delays: an iterative LMI
method,” in Proceedings of the 5th World Congress on Intelligent Control and Automation, Hangzhou, China, 2004, pp. 1338-1342.
[6] Z. Z. Qiu, Q. L. Zhang, X. F. Zhang and L. Yang, “Robust stability for a class of networked control systems based on state observer,” International Journal of Information and systems sciences, vol.3, no.4, pp. 594-603, December 2007.
[7] S. M. Mu, T. G. Chu, and L.Wang, “An improved model-based control scheme for networked systems,” in IEEE International Conference on Systems, Man and Cybernetics, Hague, Netherlands, 2004, pp. 6131-6136.
[8] L. A. Montestruque and P. J. Antsaklis, “State and output feedback control in model-based networked control systems,” in Proceedings of the 41st IEEE Conference on Decision and Control, Las Vegas, USA, 2002, pp. 1620-1625.
[9] L. A. Montestruque and P. J. Antsaklis, “On the model-based control of networked systems,” Automatica, vol. 39, no. 10, pp. 1837-1843, October 2003.
[10] Z. G. Sun, L. Xiao and D. S. Zhu, “Analysis of networked control systems with multiple-packet transmission,” in Proceeding of the 5th World Congress on Intelligent Control and Automation, Hangzhou, China, 2004, pp.1357-1360.
[11] L. Tian, Y. Ge and Z. A. Liu, “Analysis of networked control systems with communication constraints,” in Proceeding of the 6th World Congress on Intelligent Control and Automation, Dalian, China, 2006, pp. 4535-4538.
[12] M. Zheng, Q. L. Zhang, M. Song and C. Jiang, “Stability analysis of multi-packet transmission networked control systems with short delay,” Information and Control, vol.36, no.3, pp.293-301, June 2007.
[13] Y. Ge, Z. A. Liu, Q. G. Chen and M. Jiang, “Stability of networked control systems with communication constraints,” Journal of University of Science and Technology of China, vol. 38, no. 3, pp 266-271, Mar. 2008.
[14] G. M. Xie and L.Wang, “Stabilization of NCSs: Asynchronous Partial Transfer Approach,” in 2005 American Control Conference, Portland, OR, USA, 2005, pp. 1226-1230.
[15] M. Yu and W. Tan, “Robust stabilization of networked control systems: an LMI approach,” in Proceedings of the 26th Chinese Control Conference, Zhangjiajie, Hunan, China, 2007, pp. 59-63.
[16] S. G. Wang, M. X. Wu and Z. Z. Jia, Matrix Inequality. Beijing: Science Press, 2006.
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