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: qlzhang@mail.neu.edu.cn).

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: lijinna_0904@126.com).

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

= = = =

1434

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)

1435

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= −

1436

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.

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