Absolute stability and stabilization for Lurie networked control

INTERNATIONAL JOURNAL OF ROBUST AND NONLINEAR CONTROL

Int.J.Robust.Nonlinear Control2011;21:1667–1676

Published online20September2010in Wiley Online Library(https://www.360docs.net/doc/c017187852.html,).DOI:10.1002/rnc.1658

Absolute stability and stabilization for Lurie networked control

systems

Hong-Bing Zeng1,2,Yong He1,Min Wu1,?,?and Shen-Ping Xiao2

1School of Information Science and Engineering,Central South University,

Changsha410083,People’s Republic of China

2School of Electrical and Information Engineering,Hunan University of Technology,Zhuzhou412008,

People’s Republic of China

SUMMARY

This paper investigates the problem of absolute stability and stabilization for networked control systems (NCSs)with the controlled plant being Lurie systems(Lurie NCSs),in which the network-induced delays are assumed to be time-varying and bounded.By considering the relationship between the network-induced delay and its upper bound,an improved stability criterion for networked control system is proposed. Furthermore,the resulting condition is extended to design a state feedback controller by employing an improved cone complementary linearization(ICCL)algorithm.A numerical example is worked out to illustrate the effectiveness and the bene?ts of the proposed method.Copyright?2010John Wiley& Sons,Ltd.

Received21February2010;Revised7May2010;Accepted10August2010

KEY WORDS:networked control systems(NCSs);stability and stabilization;network-induced delay;

cone complementary linearization(CCL)algorithm

1.INTRODUCTION

The so-called networked control systems(NCSs)are spatially distributed systems in which the communication between the sensors,the controllers,and the actuators occurs through a shared digital communication https://www.360docs.net/doc/c017187852.html,pared with the traditional point-to-point feedback control systems,NCSs bring many great advantages,such as low cost,reduced weight and power require-ments,?exibility of operation,simple diagnosis and maintenance,and high reliability,etc.There-fore,NCSs become more and more popular in many practical applications in the recent years. In an NCS,network-induced delays including sensor-to-controller and controller-to-actuator with variable lengths occur while exchanging data among devices connected to the shared network medium,which will inevitably degrade the performance of control systems and even destabilize the systems[1–9].It is indispensable to design a valid controller to guarantee the networked-based control systems to be stable within a maximum allowable delay bound(MADB)[10].

Recently,delay-dependent stability for non-networked control systems with delay have attracted much attention and numerous results have been derived[11–18].Among them,a free-weighting matrix approach[18]is reported to cover the results using Moon et al.’s inequality[19]and the descriptor system approach[12]becomes one of the most effective methods in dealing with the delay-dependent problem,see e.g.[20,21].As for NCSs,the work in[6]established some stability

?Correspondence to:Min Wu,School of Information Science and Engineering,Central South University,Changsha 410083,People’s Republic of China.

?E-mail:min@https://www.360docs.net/doc/c017187852.html,

1668H.-B.ZENG ET AL.

conditions under an assumption that the networked-induced delay is less than the sampling period for continuous-time systems based on sample-date method.In[22],a new model for networked-induced delay being larger than the sampling period was presented,where some new criteria have been derived to stabilize an NCS by the free-weighting matrix approach.The similar idea combining the augmented Lyapunov functional in[23]is also used to study the H∞control problem for NCSs in[24].In addition,by considering the relationship between the network-induced delay and its upper bound,an improved result for networked control systems is proposed in[25].On the other hand,the focus has been on NCSs with the controlled plant being linear systems in all the above literatures while the nonlinear phenomenon is essential and universal.Thus,it is more signi?cant to study nonlinear NCSs.Recently,a special NCS,Lurie NCS,has been investigated in[26].However,as pointed out in[25],some useful terms are ignored in the derivative of the Lyapunov–Krasovskii functional,which may lead to considerable conservativeness.Therefore, there is room for further investigation.

On the other hand,in order to get a networked state feedback controller gain,an inequality approach is employed in[26].Without iteration and tuning parameters,the inequality approach is simple and convenient,but the result obtained is considerably conservative.In addition,a method that tunes some parameters is presented in[22].However,it is very dif?cult to choose the parameters.Moon et al.modi?ed the cone complementary linearization(CCL)algorithm[27]and proposed an LMI-based iterative algorithm to solve the problem of delay-dependent state feedback stabilization controller design[19].However,the stopping conditions for iteration are very strict and the gain matrix and other Lyapunov matrices derived in the previous step of iteration must satisfy one or more matrix inequalities in[19].With a new stopping condition,an improved cone complementary linearization(ICCL)algorithm has been proposed in[25].

In this paper,a new absolute stability condition for Lurie NCSs is?rst established without ignoring any terms in the derivative of Lyapunov–Krasovskii functional,in which the relationship among network-induced delay,its upper bound and the difference between them is considered. Then,based on the resulting condition,a state feedback controller design method is presented by employing the ICCL algorithm.A numerical example is worked out to illustrate the effectiveness and the merits of the proposed method.

Notation:Throughout this paper,superscripts‘?1’and‘T’stand for the inverse and transpose of a matrix,respectively; n denotes the n-dimensional Euclidean space; n×m is the set of all n×m real matrices;P>0means that the matrix P is positive de?nite;I is an appropriately dimensioned identity matrix;diag{...}denotes a block-diagonal matrix;and the symmetric terms in a symmetric

matrix are denoted by ,e.g.[X Y

Z ]=[X

Y T

Y

Z

],where X and Z are symmetric matrices.

2.PROBLEM STATEMENT Consider the following Lurie control system:

???

??˙x(t)=Ax(t)+Bu(t)+Dw(t) z(t)=Cx(t)

w(t)=? (t,z(t))

(1)

where x(t)∈ n is the state vector,u(t)∈ m is the controlled input vector.z(t)∈ p is the measured output.A,B,C and D are constant matrices with appropriate dimensions. (t,z(t))is a memoryless, nonlinear function that is piecewise continuous in t,globally Lipschitz in z(t), (t,0)=0,and satis?es the following sector condition?t 0and?z(t)∈ p

T(t,z(t))[ (t,z(t))? z(t)] 0(2) where is a real diagonal matrix.The set of all functions satisfying the above sector condition is denoted by (·)∈F[0, ].

ABSOLUTE STABILITY AND STABILIZATION

1669

Remark 1

(·)∈F [ 1, 2]can be transformed into (·)∈F [0, ]by applying the loop transformation suggested in [28].Therefore,we only consider the case (·)∈F [0, ]in this paper.For convenience of investigation,the following de?nition and assumptions are introduced:De?nition 1

The Lurie system (1)is said to be absolutely stable in the sector F [0, ]if the system is globally uniformly asymptotically stable for any nonlinear function (·)satisfying (2).Assumption 1

(a)The considered NCS consists of a time-driven sensor,event-driven controller and event-driven actuator,which are all connected through a control network.The calculated time-delay is viewed as a part of the network-induced delay of controller-to-actuator.

(b)The controller always uses the most recent data and discards the old data.When old data

arrive at the controller,it is treated as packet loss.

(c)The real input realized through a zeroth-order holding in (1)is a piecewise constant function.From the above assumptions,by applying a similar modeling technique employed in [22,24],the closed-loop system with memoryless state feedback controller can be formulated as

˙x (t )=Ax (t )+Bu (t )+Dw (t ),t ∈[i k h + k ,i k +1h + k +1)

u (t +

)=Kx (t ? k ),

t ∈{i k h + k ,k =1,2,...}

(3)

where h is the sampling period,k =1,2,3,...are the sequence numbers of the most recent data

available on the controller and assumed not to change their value until new data arrive at the controller,i k is an integer denoting the sequence number of the sensor’s sampling time points and {i 1,i 2,i 3,...}?{1,2,3,...}, k denotes the time delay from the instant i k h when the sensor nodes sample the sensor data from a plant to the instant when the actuators transfer the data to the plant.

Following Assumption 1,it always holds that i k +1>i k .When {i 1,i 2,i 3,...}={1,2,3,...},it means that no packet dropout occurs in the transmission.On the other hand,if =i k +1?i k >1,then there are ?1consecutive data packets dropped or lost.Therefore,(3)represents the NCS with the effects of the network-induced delay and data packet dropout being simultaneously considered.Without loss of generality,we assume that u (t )=0before the ?rst control signal reaches the plant and a constant >0exists such that

(i k +1?i k )h + k +1 ,

k =1,2,...

(4)Then,it can be seen that system (3)can be rewritten in the following equivalent form:

˙x (t )=Ax (t )+BKx (i k h )+Dw (t ),

t ∈[i k h + k ,i k +1h + k +1),

k =1,2,...

(5)

Similar to [8,26,29],we de?ne

k (t )=t ?i k h ,

t ∈[i k h + k ,i k +1h + k +1),

k =1,2,...

(6)

From the de?nition in (6),it is easy to obtain k k (t ) (i k +1?i k )h + k +1 in every internal

[i k h + k ,i k +1h + k +1).Clearly, ∞k =1[i k h + k ,i k +1h + k +1)=[t 0,∞).Then,Equation (5)can be described by the following system:

˙x (t )=Ax (t )+BKx (t ? k (t ))+Dw (t )x (t )= (t ),

t ∈(t 0? ,t 0)

(7)

where (t )is the initial condition function of the systems,which is a continuous and differentiable

vector-valued one.

1670

H.-B.ZENG ET AL.

Remark 2

A natural assumption is that the network-induced delay and the data packet dropout are bounded,that is, k and i k +1?i k have upper bounds,and this guarantees the existence of ;otherwise,the NCS must not be stabilized.Then, k (t )is a bounded piecewise continuous function.Therefore,system (7)is equivalent to a Lurie system with a bounded time-varying delay.

3.MAIN RESULTS

In this section,we shall ?rst establish an asymptotic stability criterion for the Lurie NCS (7)with a given gain K by using the Lyapunov–Krasovskii method.

Theorem 1

Given a scalar >0and a controller gain matrix K ,closed-loop system (7)with nonlinear function (·)∈F [0, ]is absolutely stable if there exist matrices P =P T >0,Q =Q T >0,Z =Z T >0,N =[N T 1N T 2]T ,and M =[M T 1M T 2

]T ,such that the following matrix inequalities hold:? 1=??

???????

??????

11 12?M 1P D ?C T T

? N 1 A T Z

22?M 20? N 2 K T

B T

Z ?Q 000 ?2I 0 D T

Z ? Z 0

? Z ?

?

?

?

???????

????

<0(8)

? 2=?????????

??????

11 12?M 1P D ?C T T

? M 1 A T Z

22?M 20? M 2 K T

B T

Z ?Q 000 ?2I 0 D T

Z ? Z 0

? Z

?

??

?

???????

????

<0(9)

where

11=PA +A T P +Q +N 1+N T 1

12=PBK ?N 1+N T 2+M 1 22=?N 2?N T 2+M 2+M T 2

.Proof

Choose a Lyapunov–Krasovskii functional candidate to be

V (x t )=x T

(t )Px (t )+

t

t ?

x T

(s )Qx (s )d s +

?

t

t +

˙x T (s )Z ˙x (s )d s d

(10)

where P =P T >0,Q =Q T >0,and Z =Z T >0are to be determined.

ABSOLUTE STABILITY AND STABILIZATION

1671

From the Leibniz–Newton formula,the following equations are true for any matrices N =[N T 1N T 2]T ,and M =[M T 1M T 2

]T with appropriate dimensions 0=2 T (t )N x (t )?x (t ? k (t ))? t

t ? k (t )

˙x (s )d s (11)

0=2 T (t )M

x (t ? k (t ))?x (t ? )?

t ? k (t )

t ?

˙x (s )ds

(12)

where (t )=[x T (t )x T (t ? k (t ))]T .

On the other hand,from (2),we have

0 ?2w T (t )w (t )?2w T (t ) Cx (t )

(13)

Calculating the derivative of V (x t )along the solutions of system (7)and adding the right sides

of (11)–(13)into it yield

˙V (x t )=2x T (t )P ˙x (t )+x T (t )Qx (t )?x T (t ? )Qx (t ? )

+ ˙x T (t )Z ˙x

(t )? t

t ?

˙x T (s )Z ˙x (s )d s 2x T (t )P ˙x (t )+x T (t )Qx (t )?x T (t ? )Qx (t ? )

+ ˙x T (t )Z ˙x (t )? t t ? k (t )

˙x T (s )Z ˙x (s )d s ? t ? k (t )

t ?

˙x T (s )Z ˙x (s )d s +2 T (t )N

x (t )?x (t ? k (t ))?

t

t ? k (t )

˙x (s )d s

+2 T (t )M x (t ? k (t ))?x (t ? )?

t ? k (t )

t ?

˙x (s )d s

?2w T (t )w (t )?2w T (t ) Cx (t )

=1 t t ? k (t )

T (t ,s )[ 1+ T 3Z 3] (t ,s )d s +1

t ? k (t )t ?

T (t ,s )[ 2+ T 3Z 3] (t ,s )d s (14)

where

1=?

???????? 11+ 12

?M 1P D ?C T T

? N 1

22?M 20? N 2

?Q 00

?2I 0

? Z ?

??

?????? 2=?

?????? 11+ 12?M 1

P D ?C T T

? M 1 22?M 20? M 2

?Q 00

?2I 0

? Z

???

??

??

1672

H.-B.ZENG ET AL.

3=[A BK 0D 0]

(t ,s )=[x T (t )x T (t ? k (t ))x T (t ? )w T (t )˙x

T (s )]T .Thus,if 1+ T 3

Z 3<0and 2+ T

3Z 3<0,which are equivalent to (8)and (9)by Schur complements [30],respectively,˙V

(x t )0.Therefore,system (7)is uniformly asymptotically stable.Furthermore,it is noted that V (x t ) min (P ) x (t ) 2and lim x (t ) →∞ min (P ) x (t ) 2=∞.Then,in view of Lyapunov–Krasovskii stability theory [14],system (7)is globally uniformly asymptotically stable for all (·)∈F (0, ).Therefore,according to De?nition 1,system (7)is absolutely stable.This completes the proof.

Remark 3In the proof of Theorem 1in [26],the term ? t t ? ˙x T ( )R ˙x ( )d is increased to ? t t ? k (t ))˙x T ( )R ˙x ( )d ,which leads to considerable conservativeness.In contrast,in the above proof of Theorem 1in this paper,? t t ? ˙x T ( )Z ˙x ( )d is taken to be ? t t ? k (t )˙x T (s )Z ˙x (s )d s ? t ? k (t )t ?h

˙x T (s )Z ˙x (s )d s .On the other hand,a new term, t

t ? x T (s )Qx (s )d s ,is introduced in the Lyapunov–Krasovskii functional (10).Furthermore,the relationships among k (t ), ? k (t ),and have been taken into account without ignoring useful terms in the derivative of the Lyapunov–Krasovskii functional.Thus,the stability criterion in Theorem 1may be less conservative.Remark 4

In (8)and (9)of Theorem 1,setting M 1=0,M 2=0,N 1=?R ,N 2=R ,Z = R ,and Q =εI with εbeing a suf?ciently small scalar,we can prove that (8)and (9)in this paper are equivalent to (8)in [26]by using Schur complements.Thus,the criterion in [26]is a special case of the theorem.In the following,Theorem 1is extended to design a stabilization controller gain K for system (7).Theorem 2

Given a scalar >0,closed-loop system (7)with nonlinear function (·)∈F [0, ]is abso-lutely stable if there exist matrices L =L T >0,W =W T >0,R =R T >0,S =[S T 1S T 2

]T ,and T =[T T 1T T 2]T ,and V ,such that the following matrix inequalities hold:

1=?

?

?

??

?

?

??

???

?

?

??

?

11 12?T 1D ?LC T T ? S 1 L A T

22?T 20? S 2 V T B T ?W 000 ?2I 0 D T ? LR ?1L 0 ? R

?????????????????<0(15) 2=?????????

??????

11 12?T 1D ?LC T T

? T 1 L A T

22?T 20? T 2 V T

B T ?W 000 ?2I 0 D T ? L R ?1L

? R

?

?

?

?

????

???

????

<0(16)

ABSOLUTE STABILITY AND STABILIZATION

1673

where

11=AL +LA T +W +S 1+S T 1

12=BV ?S 1+S T 2+T 1 22=?S 2?S T 2+T 2+T T 2.

Moreover,a stabilizing controller gain is given by K =VL ?1.

Proof

Pre-and post-multiply ?

i ,i =1,2in (8)and (9)by diag {P ?1,P ?1,P ?1,I ,P ?1,Z ?1}and diag {P ?1,P ?1,P ?1,I ,P ?1,Z ?1},respectively,and make the following changes to the variables:

L :=P ?1,R :=Z ?1,

V :=KL ,

W :=LQL ,

S i :=LN i L ,

T i :=L M i L ,

i =1,2

Matrix inequalities (15),(16)are derived.This completes the proof.

It is noted that the conditions in Theorem 2cannot be implemented by using a numerical software due to the existence of the nonlinear term LR ?1L in (15)and (16).In order to get a networked state feedback controller gain,an inequality approach is employed in [26].Without iteration and tuning any parameters,the inequality approach is simple and convenient,but the result obtained is considerably conservative.In [22]and [24],the networked state feedback controller gain is obtained by tuning some parameters,while there are no valid methods to get these parameters.However,an ICCL algorithm proposed in [25]can be employed to solve this non-convex problem.De?ne a new variable U such that U LR ?1L ,and replace conditions (15)and (16)with

??????????

??? 11 12?T 1D ?LC T T ? S 1 LA T 22?T 20? S 2 V T B T ?W 000 ?2I 0 D T ? U 0 ? R ??

??

???

??????<0

(17)????

??????

??? 11 12?T 1D ?LC T T ? T 1 L A T 22?T 20? T 2 V T B T ?W 000 ?2I 0 D T ? U 0 ? R ??

?????

??????<0

(18)and

U LR ?1L

(19)

(19)is equivalent to U ?1?L ?1RL ?1 0,which can be expressed as

U ?1L ?1

L ?1R

?1 0

(20)

1674H.-B.ZENG ET AL.

using Schur complements.Thus,by introducing new variables P,H,Z,the original conditions (15)and(16)are represented as(17)and(18)and

H P P Z

0,H=U?1,P=L?1,Z=R?1(21)

Then,this non-convex problem is converted into the following LMI-based nonlinear minimization problem:

Minimize tr{LP+UH+RZ}

subject to(17),(18)and

H P P Z

0,

L I

I P

0,

U I

I H

0,

R I

I Z

0.(22)

Then,the following algorithm is given to obtain the maximum max.

Algorithm1

Step1.Choose a suf?ciently small initial >0such that there exists a feasible solution to(17),

(18),and(22).Set max= .

Step2.Find a feasible set(P0,L0,W,S,T,Z0,R0,U0,H0,V)satisfying(17),(18)and

(22).Set k=0.

Step3.Solve the following LMI problem for the variables(P,L,W,S,T,Z,R,U,H,V)

Minimize tr{LP k+L k P+UH k+U k H+RZ k+R k Z}

subject to(17),(18),and(22).

Set P k+1=P,L k+1=L,U k+1=U,H k+1=H,R k+1=R,Z k+1=Z.

Step4.If LMIs(8),and(9)are feasible with a given K derived in Step3for the variables (P,Q,Z,N,M),then set max= and return to Step2after increasing to some extent.

If LMIs(8),and(9)are infeasible within a speci?ed number of iterations,then exit.

Otherwise,set k=k+1and go to Step3.

4.NUMERICAL EXAMPLE

In the following,a numerical example is given to illustrate the merits of the proposed method. Example1

Consider system(1)with

A=

01

1?2

,B=

1

,C=[1?0.5],D=

1

, (·)∈F[0,1].

When the controller gain matrix K is given as[?0.7539?0.3319],the derived max,which ensures the stability of the system(7)is1.2562in[26].However,the corresponding value obtained by Theorem1is1.2890.On the other hand,if the controller gain K is not assumed,it is reported in[26]that closed-loop system(7)is stable for =1.2841with K=[?0.5324?0.2419]. Using Algorithm1,it is found that the closed-loop system(7)is stable for =1.4584with K=[?0.6130?0.2687]after1iteration and =1.5250with K=[?0.5347?0.2469]after84 iterations.It is clear that our result is better than those in[26].

ABSOLUTE STABILITY AND STABILIZATION1675

5.CONCLUSION

The problem of absolute stability and stabilization for Lurie NCSs has been studied in this paper. An improved absolute stability condition for networked closed-loop system with a given controller gain matrix has been?rst established by considering the relationship between the network-induced delay and its upper bound.Furthermore,based on the resulting condition,a networked state feedback controller gain has been obtained by employing an ICCL algorithm.Finally,a numerical example has been given to illustrate the effectiveness and the bene?ts of the proposed method.

ACKNOWLEDGEMENTS

This work was supported in part by the National Natural Science Foundation of China under Grant Nos. 60425310and60974026,the Doctor Subject Foundation of China under Grant No.200805330004,the Scienti?c Research Fund of Hunan Provincial Education Department under Grant No.10C0628and the National Science Fund for Distinguished Youth Scholars of Human Province under Grant No.08JJ1010, Publishing Arts Research Council under Grant No.98-1846389.

REFERENCES

1.Lee KC,Lee K,Lee MH.Worst case communication delay of real-time industrial switched ethernet with multiple

levels.IEEE Transactions on Industrial Electronics2006;53:1669–1676.

2.Tang B,Liu GP,Gui WH.Improvement of state feedback controller design for networked control systems.IEEE

Transactions on Circuits and Systems II:Express Briefs2008;55(5):464–468.

3.Walsh G,Ye H,Bushnell L.Stability analysis of networked control systems.IEEE Transactions on Control

Systems Technology2002;10(3):438–446.

4.Yang FW,Wang ZD,Hung YS,Gani M.H∞control for networked systems with random communication delays.

IEEE Transactions on Automatic Control2006;51(3):511–518.

5.Yang https://www.360docs.net/doc/c017187852.html,worked control systems:a brief survey.IEE Proceedings of Control Theory and Application2006;

153(4):403–412.

6.Zhang W,Branicky MS,Phillips SM.Stability of networked control systems.IEEE Control Systems Magazine

2001;21:84–99.

7.Wang Y.H∞control of networked control systems via LMI approach.International Journal of Innovative

Computing,Information and Control2007;3:343–352.

8.Peng C,Tian YC,Tad′e MO.State feedback controller design of networked control systems with interval time-

varying delay and nonlinearity.International Journal of Robust and Nonlinear Control2008;18(12):1285–1301.

9.Gao H,Chen T,Lam J.A new delay system approach to network-based control.Automatica2008;44(1):39–52.

10.Kim D,Lee Y,Kwon W,Park H.Maximum allowable delay bounds of networked control systems.Control

Engineering Practice2003;11(11):1301–1313.

11.Boukas EK,Al-Muthairi NF.Delay-dependent stabilization of singular linear systems with delays.International

Journal of Innovative Computing,Information and Control2006;2:283–291.

12.Fridman E,Shaked U.Delay-dependent stability and H∞control:constant and time-varying delays.International

Journal of Control2003;76:48–60.

13.Gao H,Lam J,Wang C,Wang Y.Delay-dependent output-feedback stabilisation of discrete-time systems with

time-varying state delay.IEE Proceedings of Control Theory and Application2004;151(6):691–698.

14.Gu K,Kharitonov VL,Chen J.Stability of Time-Delay Systems.Birkhuser:Boston,2003.

15.Jiang X,Han QL.On H∞control for linear systems with interval time-varying delay.Automatica2005;

41(12):2099–2106.

16.Lin C,Wang QG,Lee TH.A less conservative robust stability test for linear uncertain time-delay systems.IEEE

Transactions on Automatic Control2006;51(1):87–91.

17.Zhang XM,Wu M,She JH,He Y.Delay-dependent stabilization of linear systems with time-varying state and

input delays.Automatica2005;41(8):1405–1412.

18.He Y,Wu M,She JH,Liu GP.Delay-dependent robust stability criteria for uncertain neutral systems with mixed

delays.Systems and Control Letters2004;51(1):57–65.

19.Moon YS,Park P,Kwon WH,Lee YS.Delay-dependent robust stabilization of uncertain state-delayed systems.

International Journal of Control2001;74(14):1447–1455.

20.He Y,Wu M,She JH,Liu GP.Parameter-dependent Lyapunov functional for stability of time-delay systems with

polytopic-type uncertainties.IEEE Transactions on Automatic Control2004;49(5):828–832.

21.Wu M,He Y,She JH,Liu GP.Delay-dependent criteria for robust stability of time-varying delay systems.

Automatica2004;40(8):1435–1439.

22.Yue D,Han QL,Peng C.State feedback controller design of networked control systems.IEEE Transactions on

Circuits and Systems,Part II2004;51(11):640–644.

1676H.-B.ZENG ET AL.

23.Wu M,He Y,She JH.New delay-dependent stability criteria and stabilizing method for neutral systems.IEEE

Transactions on Automatic Control2004;49(12):2266–2271.

24.Yue D,Han QL,Lam https://www.360docs.net/doc/c017187852.html,work-based robust H∞control of systems with uncertainty.Automatica2005;

41(6):999–1007.

25.He Y,Liu GP,Rees D,Wu M.Improved stabilisation method for networked control systems.IET Control Theory

and Applications2007;1(6):1580–1585.

26.Hao F,Zhao X.Absolute stability of Lurie networked control systems.International Journal of Robust and

Nonlinear Control2010;20(12):1326–1337.

27.Ghaoui LE,Oustry F,AitRami M.A cone complementarity linearization algorithm for static output-feedback

and related problems.IEEE Transactions on Automatic Control1997;42:1171–1176.

28.Khalil HK.Nonlinear Systems.Prentice-Hall:Upper Saddle River,NJ,1996.

29.Jiang X,Han QL.New stability criteria for linear systems with interval time-varying delay.Automatica2008;

44(10):2680–2685.

30.Boyd S,Ghaoui LE,Feron E,Balakrishnan V.Linear Matrix Inequality in System and Control Theory.SIAM:

Philadelphia,1994.