中国博士论文经典文献

A novel approach on stabilization for linear systems with time-varying input delay qYun Liu a ,Li-Sheng Hu a ,⇑,Peng Shi b ,c ,daDept.of Automation,Shanghai Jiao Tong University,200240Shanghai,ChinabFaculty of Advanced Technology,University of Glamorgan,Pontypridd,CF371DL,United Kingdom cSchool of Engineering and Science,Victoria University,Melbourne,Vic 8001,Australia dSchool of Mathematics and Statistics,University of South Australia,Mawson Lakes 5095,Australiaa r t i c l e i n f o Keywords:Linear systemsStability and stabilization Time-delaysa b s t r a c tThis paper presents new results on delay-dependent stability and stabilization for linear systems with interval time-varying delays.Some less conservative delay-dependent crite-ria for determining the stability of the time-delay systems are obtained in this paper.Based on the stability conditions,we propose a new state transformation technology to facilitate controller designing efficiently and computationally.The method is also applicable to the existing stability conditions reported by now,while the existing technologies may fail to derive computational control procedures from the stability conditions.Finally,some numerical examples well illustrate the effectiveness of the proposed method.Ó2011Elsevier Inc.All rights reserved.1.IntroductionDuring the last few decades,there has been a growing research interest on the analysis and synthesis of time-delay systems,which widely exist in various physical and engineering systems such as chemical processes,neural networks and long transmission lines in pneumatic systems [4,5,8,16,23].In the literature,Lyapunov–Razumikhin functional and Lyapunov–Krasovskii functional are widely used approaches for time-delay systems to obtain a delay-independent or a delay-dependent stability condition [3,6,19,21,22].Basically,delay-independent conditions are more conservative than the delay-dependent conditions especially when the delay is small.Therefore,more attention is paid to the study of delay-dependent conditions [7,14,25,26].Currently,the results of delay-dependent stability mainly focus on time-varying delay with range zero to an upper bound.In practice,the range of delay may vary in a range for which the lower bound is not restricted to be zero.For this reason,the stability of the systems with such interval time-varying delays has attracted considerable attention [10,12,18].For example,in [9],a discretized Lyapunov functional approach is employed to obtain stability criteria for linear uncertain systems with interval time-varying delays.By using free-weighting matrix,He et al.[10]present some less conservative stability condi-tions.This result is improved by Shao [17]where a new Lyapunov functional with fewer matrix variables is constructed and the convex analysis method is applied.Recently,these results are further extended in [20]by proposing a new type of augmented Lyapunov functional containing some triple-integral terms.Introducing some integral terms,such as R tt Àd 1x ðs Þds ,in the augmented vector,less conservative results can be obtained by using the Lyapunov functional with 0096-3003/$-see front matter Ó2011Elsevier Inc.All rights reserved.doi:10.1016/j.amc.2011.11.056qThis work is partially supported by Natural Science Foundation of China (NSFC)under Grants 60474020,60774006,61034008,61025016,and Engineering and Physical Science Research Council,UK under Grant EP/F029195.⇑Corresponding author.E-mail addresses:lshu@ (L.-S.Hu),pshi@ (P.Shi).5938Y.Liu et al./Applied Mathematics and Computation218(2012)5937–5947triple-integral terms.It has been shown in[1,15,20]that these new types of Lyapunov functionals can be applied efficiently to derive less conservative stability results for time-delay systems.Nevertheless,there still exists some room for further improvement,which is one of the motivation of this paper.On the other hand,control problems for time-delay systems are important issues.In[27],an integral-inequality meth-od is proposed for the delay-dependent stabilization problem of linear systems with time-varying state and input delays. For constant but unknown time-delay,by introducing a state-transformation to describe the delay-dependence dynamics, some control design schemes based on quadratic H2performance,H1criteria and simultaneous H2=H1synthesis are established in[13].For a stabilizable and detectable linear system with an arbitrarily large delay in the input channel, by explicit construction of stabilizing feedback laws,Lin and Fang[11]show that the system can be asymptotically sta-bilized by either linear state or output feedback as long as all the open-loop poles are on the closed left-half plane.How-ever,to the best of the authors’knowledge,few results are reported in the existing literature on the control designing problem for linear system with interval time-varying input delay.It is not easy to apply the results in[9,10,17,20]to ob-tain control procedures because of the cross terms of matrix variables involved[24].This is the other motivation for the study of this paper.In the paper,we propose new results on delay-dependent stability and stabilization for time-delay ing a new Lyapunov functional,some less conservative delay-dependent stability conditions are derived for the linear systems with time-varying delay in a range.As pointed above,the existing stability results may be difficultly to obtain computa-tional controller design procedure.For linear system with time-varying input delay,by carefully selecting the matrix vari-ables,and choosing a nonsingular transformation matrix T which transforms the state x of origin system to a new auxiliary state x¼Tx,the cross terms coming from the stability condition can be removed.Assisted by this technology,a less con-servative designing procedure is obtained from the stability conditions.The technology is also applicable to the existing stability conditions reported.The main results are presented in terms of linear matrix inequalities(LMIs)which can be easily tested by using the Matlab’s LMI Tool-box[2].Finally,some numerical examples well demonstrate the effectiveness of the proposed method.Notation:Throughout the paper,R n denotes the n dimensional Euclidean space with vector norm kÁk,R nÂm is the set of all nÂm real matrices,I is the identity matrix with appropriate dimensions,and the superscripts‘‘À1’’and‘‘T’’stand for the inverse and transpose of a matrix,respectively.The notation X>0(respectively,X P0),for X2R nÂn means that the matrix X is symmetric and positive definite(respectively,positive semi-definite).2.Problem formulationConsider the following time-delay systems_xðtÞ¼AxðtÞþBuðtÀdðtÞÞ;ð1ÞxðhÞ¼/ðhÞ;h2½Àd2;0where x(t)2R n denotes the state vector,u(t)2R m denotes the control input.A and B are constant real matrices and the pair (A,B)is assumed to be stabilizable./(h)is a continuous vector-valued initial function on[Àd2,0].d(t)denotes the time-vary-ing delay and satisfiesd16dðtÞ6d2;ð2Þ_dðtÞ6l;ð3Þwhere06d1<d2and06l are constants.The controller takes following formuðtÞ¼FxðtÞ;ð4Þwhere F is controller gain matrix with appropriate dimensions.Substituting(4)into(1),we obtain the following closed-loop system_xðtÞ¼AxðtÞþAxðtÀdðtÞÞ;dð5ÞxðhÞ¼/ðhÞ;h2½Àd2;0 ;where A d=BF.Then the problems considered in this paper can be formulated as:(i)to set up a delay-dependent stability for the time-delay system(5)with the given A and A d;(ii)to design state feedback controller(4)such that system(1)asymp-totically stable.At the end of this section,we introduce the following lemma which is useful in the derivation of our results.Lemma1[5].For any constant matrix W>0,scalars a<b and vector function x(s):[a,b]?R n such that the following integrations are well defined,thenðbÀaÞZ ba x TðsÞW xðsÞds PZ bax TðsÞdsWZ baxðsÞds;ðbÀaÞ22Z baZ bhx TðsÞW xðsÞdsd h PZ baZ bhx TðsÞdsd h WZ baZ bhxðsÞdsd h;ðbÀaÞ22Z baZ hax TðsÞW xðsÞdsd h PZ baZ hax TðsÞds d h WZ baZ haxðsÞds d h:3.Stability analysisIn this section,we consider asymptotic stability of time-delay system(5).Using a new Lyapunov functional,some delay-dependent stability criteria are obtained with the consideration of range for the time-varying delay.Theorem1.For the given scalars06d1<d2and l,system(5)with d(t)satisfying(2)and(3)is asymptotically stable if there exist matrices P>0,Q1>0,Q2>0,R i>0,S i>0,Z i>0,i=1,...,4,and Y j,j=1,2,3with appropriate dimensions such that S1þQ1ÀQ2>0;ð6ÞN0¼C0PÇTþÇP C TþW0þKþY e A eþA T e Y T e<0;ð7ÞN1¼C1PÇTþÇP C T1þW1þKþY e A eþA T e Y T e<0;ð8Þwhere d12=d2Àd1,C0¼e1e4e6d1e8d12e10½ ;C1¼e1e4e6d1e8d12e9½ ;Ǽe2e5e7e1Àe4e4Àe6½ ;W0¼Àð1ÀlÞd212e3Q1e T3Àd212e10S1e T10Àðe4Àe3Þð2S2þZ1Þðe T4Àe T3ÞÀðe3Àe6ÞS2ðe T3Àe T6Þ;W1¼Àð1ÀlÞd212e3Q2e T3Àd212e9ðS1þQ1ÀQ2Þe T9Àðe4Àe3ÞS2ðe T4Àe T3ÞÀðe3Àe6Þð2S2þZ2Þðe T3Àe T6Þ;K¼diag R3þd21S3;R4þd21ð2S4þZ3þZ4Þ;0;R1ÀR3þd212ðQ1þS1Þ;R2ÀR4þd212ð2S2þZ1þZ2Þ;ÀR1;ÀR2;Àd21S3;0;0()Àðe1Àe4ÞS4e T1Àe T4ÀÁÀ2ðe1Àe8ÞZ3e T1Àe T8ÀÁÀ2ðe8Àe4ÞZ4e T8Àe T4ÀÁÀ2ðe4Àe9ÞZ1e T4Àe T9ÀÁÀ2ðe3Àe10ÞZ1ðe T3Àe T10ÞÀ2ðe9Àe3ÞZ2e T9Àe T3ÀÁÀ2ðe10Àe6ÞZ2e T10Àe T6ÀÁY e¼e1Y1þe2Y2þe3Y3;A e¼Ae T1Àe T2þA d e T3and e i2R10nÂn,i=1,...,10are block entry matrices,for example,e T3¼½00I0000000 . Proof.Construct a Lyapunov functional candidate asVðtÞ¼X3i¼0V iðtÞ;where V0(t)=f T(t)P f(t)with fðtÞ¼col xðtÞ;xðtÀd1Þ;xðtÀd2Þ;R ttÀd1xðsÞds;R tÀd1tÀd2xðsÞdsn o,V1ðtÞ¼d212ZÀd1ÀdðtÞx TðtþsÞQðsÞxðtþsÞdsþZ tÀd1tÀd2x TðsÞR1xðsÞdsþZ tÀd1tÀd2_x TðsÞR2_xðsÞdsþZ ttÀd1x TðsÞR3xðsÞdsþZ ttÀd1_x TðsÞR4_xðsÞds;V2ðtÞ¼d12ZÀd1Àd2Z tÀd1tþhx TðsÞS1xðsÞds d hþd12ZÀd1Àd2Z tÀd1tþh_x TðsÞS2_xðsÞdsd hþd1Z0Àd1Z ttþhx TðsÞS3xðsÞds d hþd1Z0Àd1ÂZ ttþh_x TðsÞS4_xðsÞdsd h;Y.Liu et al./Applied Mathematics and Computation218(2012)5937–59475939V3ðtÞ¼ZÀd1Àd2ZÀd1gZ tÀd1tþh_x TðsÞZ1_xðsÞdsd h d gþZÀd1Àd2Z gÀd2Z tÀd1tþh_x TðsÞZ2_xðsÞdsd h d gþZ0Àd1Z0gZ ttþh_x TðsÞZ3_xðsÞds d h d gþZ0Àd1Z gÀd1Z ttþh_x TðsÞZ4_xðsÞds d h d gandQðsÞ¼sþd212Q1Àsþd112Q2for s2[Àd2,Àd1].Doing the time derivative of V(t)along the trajectory of system(5),we have_VðtÞ¼2f TðtÞP_fðtÞÀð1À_dðtÞÞd212x TðtÀdðtÞÞQðÀdðtÞÞxðtÀdðtÞÞþx TðtÞðR3þd21S3ÞxðtÞþ_x TðtÞR4þd212ð2S4þZ3þZ4Þ!_x TðtÞþx TðtÀd1ÞR1ÀR3þd212ðQ1þS1ÞxðtÀd1Þþ_x TðtÀd1ÞR2ÀR4þd212ð2S2þZ1þZ2Þ!_xðtÀd1ÞÀx TðtÀd2ÞR1xðtÀd2ÞÀ_x TðtÀd2ÞR2_xðtÀd2ÞÀd12Z tÀd1tÀdðtÞx TðsÞðQ1ÀQ2ÞxðsÞdsÀd12Z tÀd1tÀd2ðx TðsÞS1xðsÞþ_x TðsÞS2_xðsÞÞdsÀd1Z ttÀd1ðx TðsÞS3xðsÞdsþ_x TðsÞS4_xðsÞÞdsÀZÀd1Àd2Z tÀd1tþh_x TðsÞZ1_xðsÞdsd hÀZÀd1Àd2Z tþhtÀd2_x TðsÞZ2_xðsÞds d hÀZ0Àd1Z ttþh_x TðsÞZ3_xðsÞdsd hÀZ0Àd1Z tþhtÀd1_x TðsÞZ4_xðsÞdsd h:ð9ÞSetting nðtÞ¼col xðtÞ;_xðtÞ;xðtÀdðtÞÞ;xðtÀd1Þ;_xðtÀd1Þ;xðtÀd2Þ;_xðtÀd2Þ;11R ttÀd1xðsÞds;11R tÀd1tÀdðtÞxðsÞds;12R tÀdðtÞtÀd2xðsÞdsn o and a¼dðtÞÀd112,we have Q(Àd(t))=(1Àa)Q1+a Q2,andf TðtÞ¼n TðtÞCðaÞ;_f TðtÞ¼n TðtÞÇ;where C(a)=[e1e4e6d1e8a d12e9+(1Àa)d12e10].According to the expression of system(5),we note that for any matrices Y i,i=1,2,3with appropriate dimensions,the following equality holds2ðx TðtÞY1þ_x TðtÞY2þx TðtÀdðtÞÞY3ÞðAxðtÞÀ_xðtÞþA d xðtÀdðtÞÞÞ¼0:Furthermore,it is equivalent ton TðtÞðY e A eþA Te Y TeÞnðtÞ¼0:ð10ÞUsing Lemma1,one can obtaind1Z ttÀd1x TðsÞS3xðsÞds P d21n TðtÞe8S3e T8nðtÞ;ð11Þd1Z ttÀd1_x TðsÞS4_xðsÞds P n TðtÞðe1Àe4ÞS4ðe T1Àe T4ÞnðtÞ;ð12ÞZ0Àd1Z ttþh_x TðsÞZ3_xðsÞdsd h P2n TðtÞðe1Àe8ÞZ3ðe T1Àe T8ÞnðtÞ;ð13ÞZ0Àd1Z tþhtÀd1_x TðsÞZ4_xðsÞds d h P2n TðtÞðe8Àe4ÞZ4e T8Àe T4ÀÁnðtÞ:ð14ÞWith the condition(6),we haveZ tÀd1 tÀd2x TðsÞS1xðsÞdsþZ tÀd1tÀdðtÞx TðsÞðQ1ÀQ2ÞxðsÞds¼Z tÀd1tÀdðtÞx TðsÞðS1þQ1ÀQ2ÞxðsÞdsþZ tÀdðtÞtÀd2x TðsÞS1xðsÞds P a d12n TðtÞe9ðS1þQ1ÀQ2Þe T9nðtÞþð1ÀaÞd12n TðtÞe10S1e T10nðtÞ:ð15Þ5940Y.Liu et al./Applied Mathematics and Computation218(2012)5937–5947Noticing thatd12dðtÞÀd1Pd2ÀdðtÞd12þ1¼2Àa;d12 2PdðtÞÀd112þ1¼1þa;we obtaind12Z tÀd1tÀd2_x TðsÞS2_xðsÞds¼d12Z tÀd1tÀdðtÞ_x TðsÞS2_xðsÞdsþd12Z tÀdðtÞtÀd2_x TðsÞS2_xðsÞdsPð2ÀaÞn TðtÞðe4Àe3ÞS2ðe T4Àe T3ÞnðtÞþð1þaÞn TðtÞðe3Àe6ÞS2ðe T3Àe T6ÞnðtÞ:ð16ÞSimilarly,ZÀd1Àd2Z tÀd1tþh_x TðsÞZ1_xðsÞdsd h¼ZÀd1ÀdðtÞZ tÀd1tþh_x TðsÞZ1_xðsÞdsd hþZÀdðtÞÀd2Z tÀdðtÞtþh_x TðsÞZ1_xðsÞds d hþðd2ÀdðtÞÞÂZ tÀd1tÀdðtÞ_x TðsÞZ1_xðsÞds P2n TðtÞðe4Àe9ÞZ1ðe T4Àe T9ÞnðtÞþ2n TðtÞðe3Àe10ÞZ1ðe T3Àe T10ÞnðtÞþð1ÀaÞn TðtÞðe4Àe3ÞZ1ðe T4Àe T3ÞnðtÞð17ÞandZÀd1Àd2Z tþhtÀd2_x TðsÞZ2_xðsÞdsd h¼ZÀd1ÀdðtÞZ tþhtÀdðtÞ_x TðsÞZ2_xðsÞds d hþZÀdðtÞÀd2Z tþhtÀd2_x TðsÞZ2_xðsÞds d hþðdðtÞÀd1ÞÂZ tÀdðtÞtÀd2_x TðsÞZ2_xðsÞds P2n TðtÞðe9Àe3ÞZ2ðe T9Àe T3ÞnðtÞþ2n TðtÞðe10Àe6ÞZ2ðe T10Àe T6ÞnðtÞþa n TðtÞðe3Àe6ÞZ2ðe T3Àe T6ÞnðtÞ:ð18ÞApplying(3)and adding the left-hand side of(10)into the right-hand side of(9),from(11)–(18)we obtain _VðtÞ6n TðtÞNðaÞnðtÞ;where N(a)=(1Àa)N0+a N1.We note that N(a)is convex in a2[0,1],thus,it is negative definite only if its vertices are,i.e. N0<0and N1<0.One can see that if(6)–(8)are satisfied,then_VðtÞ6À k xðtÞk2for a sufficiently small >0,from which we conclude that system(5)is asymptotically stable according to Lyapunov stability theory[8].This ends the proof.h When d1=0,Theorem1reduces to the following delay-dependent stability criterion.Corollary1.Given scalars d2>0,d1=0and l,system(5)with time-varying delay d(t)satisfying(2)and(3)is asymptotically stable if there exist matrices e P>0;Q i>0;R i>0;S i>0;Z i>0;i¼1;2,and Y j,j=1,2,3such that S1þQ1ÀQ2>0;e N0¼e C0e P eÇTþeÇe P e C Tþe W0þe KþY~e A~eþA T~e Y T~e<0;e N1¼e C1e P eÇTþeÇe P e C T1þe W1þe KþY~e A~eþA T~e Y T~e<0;wheree C0¼~e1~e4d2~e7½ ;e C1¼~e1~e4d2~e6½ ;eǼ~e2~e5~e1À~e4½ ;e W0¼Àd22~e3Q1~e T3Àd22~e7S1~e T7Àð~e1À~e3Þð2S2þZ1Þ~e T1À~e T3ÀÁÀð~e3À~e4ÞS2~e T3À~e T4ÀÁ;e W1¼Àd22~e3Q2~e T3Àd22~e6ðS1þQ1ÀQ2Þ~e T6Àð~e1À~e3ÞS2~e T1À~e T3ÀÁÀð~e3À~e4Þð2S2þZ2Þ~e T3À~e T4ÀÁ;e K¼diag R1þd22ðQ1þS1Þ;R2þd22ð2S2þZ1þZ2Þ;0;ÀR1;ÀR2;0;0()À2ð~e1À~e6ÞZ1~e T1À~e T6ÀÁÀ2ð~e3À~e7ÞZ1~e T3À~e T7ÀÁÀ2ð~e6À~e3ÞZ2~e T6À~e T3ÀÁÀ2ð~e7À~e4ÞZ2~e T7À~e T4ÀÁ;Y.Liu et al./Applied Mathematics and Computation218(2012)5937–59475941Y~e¼~e1Y1þ~e2Y2þ~e3Y3;A~e¼A~e T1À~e T2þA d~e T3and~e i2R7nÂn;i¼1;...;7are block entry matrices,for example,~e T3¼½00I0000 .Remark1.Theorem1and Corollary1give stability criteria of system(5)with d(t)satisfying(2)and(3)for d1P0and d1=0, respectively.They can be applied to both slow and fast time-varying delays only if l is known.However,the information of delay rate may not be known in many cases,or d(t)even not be differentiable,then Theorem1and Corollary1fail to work. Regarding these circumstance,rate-independent criterions for d(t)only satisfying(2)with d1P0or d1=0can be derived by choosing Q(s)=0(i.e.,Q1=0and Q2=0)in Theorem1or Corollary1,respectively.Therefore,we have the following results.For the case when d(t)satisfies(2)and l is unknown.The following result can be obtained from Theorem1by setting Q1=Q2=0.Corollary2.For the given scalars06d1<d2,system(5)with d(t)satisfying(2)is asymptotically stable if there exist matrices P>0,R i>0,S i>0,Z i>0,i=1,...,4,and Y j,j=1,2,3with appropriate dimensions such thatC0PÇTþÇP C TþW l;0þK lþY e A eþA T e Y T e<0;C1PÇTþÇP C T1þW l;1þK lþY e A eþA T e Y T e<0;whereW l;0¼Àd212e10S1e T10Àðe4Àe3Þð2S2þZ1Þðe T4Àe T3ÞÀðe3Àe6ÞS2ðe T3Àe T6Þ;W l;1¼Àd212e9S1e T9Àðe4Àe3ÞS2ðe T4Àe T3ÞÀðe3Àe6Þð2S2þZ2Þðe T3Àe T6Þ;K l¼diag R3þd21S3;R4þd21ð2S4þZ3þZ4Þ;0;R1ÀR3þd212S1;R2ÀR4þd212ð2S2þZ1þZ2Þ;ÀR1;ÀR2;Àd21S3;0;0()Àðe1Àe4ÞS4e T1Àe T4ÀÁÀ2ðe1Àe8ÞZ3ðe T1Àe T8ÞÀ2ðe8Àe4ÞZ4e T8Àe T4ÀÁÀ2ðe4Àe9ÞZ1e T4Àe T9ÀÁÀ2ðe3Àe10ÞZ1e T3Àe T10ÀÁÀ2ðe9Àe3ÞZ2e T9Àe T3ÀÁÀ2ðe10Àe6ÞZ2e T10Àe T6ÀÁand C0,C1,!,Y e,A e are the same as those defined in Theorem1.For d1=0and unknown l,setting Q1=Q2=0in Corollary1,we have the following result.Corollary3.Given scalars d2>0and d1=0,system(5)with time-varying delay d(t)satisfying(2)is asymptotically stable if there exist matrices e P>0;R i>0;S i>0;Z i>0;i¼1;2,and Y j,j=1,2,3such thate C0e P eÇTþeÇe P e C Tþe W l;0þe K lþY~e A~eþA T~e Y T~e<0;e C1e P eÇTþeÇe P e C T1þe W l;1þe K lþY~e A~eþA T~e Y T~e<0;wheree W l;0¼Àd22~e7S1~e T7Àð~e1À~e3Þð2S2þZ1Þ~e T1À~e T3ÀÁÀð~e3À~e4ÞS2~e T3À~e T4ÀÁ;e W l;1¼Àd22~e6S1~e T6Àð~e1À~e3ÞS2~e T1À~e T3ÀÁÀð~e3À~e4Þð2S2þZ2Þ~e T3À~e T4ÀÁ;e K l¼diag R1þd22S1;R2þd222ð2S2þZ1þZ2Þ;0;ÀR1;ÀR2;0;0()À2ð~e1À~e6ÞZ1~e T1À~e T6ÀÁÀ2ð~e3À~e7ÞZ1~e T3À~e T7ÀÁÀ2ð~e6À~e3ÞZ2~e T6À~e T3ÀÁÀ2ð~e7À~e4ÞZ2~e T7À~e T4ÀÁand e C0;e C1;eÇ;Y~e;A~e are the same as those defined in Corollary1.4.Matrix transformation for stabilizationIn this section,we present a computational procedure for the gain matrix F of the controller(4)such that the system(1) stable.Noticing A d=BF in Theorem1,the controller gain F appear in the term Y e A e and its symmetric one,that is,the inequal-ities(7)and(8)involve nonlinear terms Y i BF in unknown matrix variables Y i,for i=1,2,3,and F,which makes the inequal-ities listed in Theorem1be not computational for F.For the sequel development,introduce a matrix transformation technique to present a computational procedure for the gain F.Suppose that the rank of the matrix B is r,i.e.rank(B)=r,satisfying16r6m.Then there exists an invertible trans-formation matrix T2R nÂn satisfying5942Y.Liu et al./Applied Mathematics and Computation218(2012)5937–5947TB ¼0B T 0ÂÃT;ð19Þwhere 02R r Âm and rank ð0Þ¼r .With the help of such matrix T ,introducing a new state x ðt Þ; x ðt Þ:¼Tx ðt Þfor the closed-loop system (5),then (5)is equiv-alent to_ xðt Þ¼A x ðt ÞþBF x ðt Àd ðt ÞÞ; x ðh Þ¼ /ðh Þ;h 2½Àd 2;0 ;ð20Þwhere A ¼TAT À1;¼TB ;¼FT À1; /ðh Þ¼T /ðh Þ.Then we have the following computational result for F .Theorem 2.For given scalars 06d 1<d 2and 06l ,if there exist matrices P >0,Q 1>0,Q 2>0,R i >0,S i >0,Z i >0,i =1,...,4,andmatrices b F ¼0b F T 0h i T with b F 02R r Âm ,and b Y j ¼b Y j ;110b Y j ;21b Y 0"#with b Y 02R r Âr ;j ¼1;2;3such that the following LMIs hold:S 1þQ 1ÀQ 2>0;ð21ÞX 0¼C 0P ÇT þÇP C T 0þW 0þK þb Y e b A e þb A T e b Y T e þðe 1þe 2þe 3Þb Fe T 3þe 3b F T ðe 1þe 2þe 3ÞT <0;ð22ÞX 1¼C 1P ÇT þÇP C T 1þW 1þK þb Y e b A e þb A T e b Y T e þðe 1þe 2þe 3Þb Fe T 3þe 3b F T ðe 1þe 2þe 3ÞT <0;ð23Þwhere b Y e ¼e 1b Y 1þe 2b Y 2þe 3b Y 3and b A e ¼Ae T 1Àe T 2,then system (1)is asymptotically stabilized by controller (4)with F ¼FT ;ð24Þwhere is the matrix satisfyingB 0F ¼b Y À10b F 0:ð25ÞProof.Similar to the proof of Theorem 1,we choose the same Lyapunov functional candidate with state x ðt Þfor system (20).Considering the system (20)and the definition of b Y i ,we haveb Yi ¼0ðb Y 0B 0F ÞT h i T;i ¼1;2;3:Setting b F 0¼b Y 0B 0F ,from the definition of b F ,we find thatb F ¼b Yi BF ;i ¼1;2;3;which impliesb Y e BFe T 3þe 3F T B T b Y T e ¼ðe 1þe 2þe 3Þb Fe T 3þe 3b F T ðe 1þe 2þe 3ÞT :Following a similar proof procedure of Theorem 1,we know that system (20)with F satisfying (25)is asymptotically stable,ifthe inequalities (21)–(23)hold.Then equivalently,system (1)is asymptotically stabilized by (4)with F given by (24).This ends the proof.hFor d 1=0,similar as the proof of Theorem 2,we have the following result.Corollary 4.Given scalars d 2>0,d 1=0and l ,if there exist matrices e P>0;Q i >0;R i >0;S i >0;Z i >0;i ¼1;2,and matrices b F ¼0b F 0h i with bF 02R r Âm ,and b Y j ¼bY j ;110b Y j ;21b Y 0"#with b Y 02R r Âr ;j ¼1;2;3such that the following LMIs hold:S 1þQ 1ÀQ 2>0;e X 0¼e C e P e ÇT þe Çe P e C T þe W 0þe K þb Y ~e b A ~e þb A T ~e b Y T ~e þð~e 1þ~e 2þ~e 3Þb F ~e T 3þ~e 3b F T ð~e 1þ~e 2þ~e 3ÞT <0;e X 1¼e C e P e ÇT þe Çe P e C T þe W 1þe K þb Y ~e b A ~e þb A T ~e b Y T ~e þð~e 1þ~e 2þ~e 3Þb F ~e T 3þ~e 3b F T ð~e 1þ~e 2þ~e 3ÞT <0;where b Y ~e ¼~e 1b Y 1þ~e 2b Y 2þ~e 3b Y 3and b A ~e ¼~T 1À~e T 2,then the system (1)is asymptotically stabilized by controller (4)with F ¼FT,where F is the matrix satisfying B 0F ¼b Y À10b F 0.When l is unknown and 06d 16d(t)6d 2,by setting Q 1=Q 2=0in Theorem 2,we have the following result.Y.Liu et al./Applied Mathematics and Computation 218(2012)5937–59475943。