Intelligent Control and Automation, 2011, 2, 203213 doi:10.4236/ica.2011.23025 Published Online August 2011 (http://www.SciRP.org/journal/ica) Copyright © 2011 SciRes. ICA H∞ FiniteTime Control for Switched Linear Systems with TimeVarying Delay Hao Liu, Yi Shen School of Astronautics, Harbin Institute of Technology, Harbin, China Email: lh_hit_1985@163.com Received May 6, 2011; revised May 23, 2011; accepted May 30, 2011 Abstract Finitetime boundedness and H∞ finitetime boundedness of switched linear systems with timevarying delay and exogenous disturbances are addressed. Based on average dwell time (ADT) and freeweight matrix technologies, sufficient conditions which can ensure finitetime boundedness and H∞ finitetime bounded ness are given. And then in virtue of the results on finitetime boundedness, the state memory feedback con troller is designed to H∞ finitetime stabilize a timedelay switched system. These conditions are given in terms of LMIs and are delaydependent. An example is given to illustrate the efficiency of the proposed method. Keywords: Switched System, TimeDelay, H∞ FiniteTime Boundedness, ADT 1. Introduction A switched system is a special kind of hybrid system, which is composed of a family of subsystems and a switching sequence orchestrating the switching between the subsystems. Recently, switched systems have re ceived a great deal of attention, and commonly been found in automotive engine control systems, network control, process control, traffic control, etc. Many im portant progress and remarkable results have been made on basic problems concerning stability and design of switched systems [110]. For recent progress, readers can refer to survey papers [1113] and the references therein. Many Lyapunov function techniques are effective tools dealing with switched systems [1417]. Average dwell time and dwell time (DT) approaches were employed to study the stability and stabilization of timedependent switched systems [1820]. Timedelay, which is a common phenomenon en countered in many engineering process, is known to be great sources of poor performance and instability. For switched systems, because of the complicated behavior caused by the interaction between the continuous dy namics and discrete switching, the problem of time de lays is more difficult to study [21]. The current methods of stabilization for timedelay systems can be classified into two categories: delayindependent and de laydependent stabilization [2224]. In [25], by using free weighting matrix scheme and average dwell time method incorporated with a piecewise Lyapunov functional, ex ponentially stability and L2gain were analyzed for a class of switched systems with timevarying delays and disturbance input. In [26], the robust stability, robust stabilization and H∞ control problems for timedelay discrete switched singular systems with parameter un certainties are discussed. Up to now, most of existing literature related to stabil ity of switched systems investigates Lyapunov asymp totic stability, which is defined over an infinite time in terval. However, in practice, one is interested in not only system stability (usually in the sense of Lyapunov) but also a bound of system trajectories over a fixed short time [27]. The finitetime stability is a different stability concept which admits the state does not exceed a certain bound during a fixed finitetime interval. Some early results on finitetime stability can be found in [2830]. Finitetime stability and stabilization for discrete linear system were investigated in [31]. In [32], finitetime sta bilization of linear timevarying systems has been dis cussed. It should be pointed out that a finitetime stable system may not be Lyapunov asymptotical stable, and a Lyapunov asymptotical stable system may not be fi nitetime stable since the transient of a system response may exceed the bound [33]. So far, however, compared with numerous research results about Lyapunov stability, few results on finitetime stability have been given in
H. LIU ET AL. 204 literature about the finitetime boundedness switched systems with timedelay. This motivates us to study in this area. In [27], finitetime boundedness and finitetime weighted L2gain for a class of switched delay systems with timevarying exogenous disturbances is investigated. In [33], the problems of finitetime stability analysis and stabilization for switched nonlinear discretetime systems are addressed, and then the results are extended to H∞ finitetime boundedness of switched nonlinear dis cretetime systems. In [34], finitetime stability and sta bilization problems for a class of switched linear systems were studied, and the state feedback controllers and a class of switching signals with average dwelltime have been designed to stabilize the switched linear control systems. However, to the best of authors’ knowledge, there is no result available yet on finitetime stability of switched systems with timevarying delay. Thus, it is necessary to investigate finitetime stability and finitetime bounded ness for a class of switched linear systems with timevarying delay, which is an important property for switched system. Our contributions are given as follows: 1) Definitions of finitetime boundedness and H∞ fi nitetime are extended to switched linear systems with timevarying delay. 2) Sufficient conditions for fi nitetime boundedness and H ∞ finitetime boundedness of switched linear systems with timevarying delay are given. 3) A set of memory state feedback controllers are designed to guarantee the closedloop switched system with timevarying delay H∞ finitetime bounded. The paper is organized as follows. In Section 2, some definitions and problem formulations are presented. In Section 3, based on ADT technology and LMIs, suffi cient conditions which ensure finitetime stability of switched linear systems with timevarying delay are given. In Section 4, sufficient conditions which guaran tee the switched system has H∞ finitetime are presented. In Section 5, a set of memory state feedback controllers are designed, which can guarantee the closedloop switched system H∞ finitetime bounded. Finally, an example is presented to illustrate the efficiency of the proposed method in Section 6. Conclusions are given in Section 7. Notations: The notations used in this paper are stan dard. The notation P > 0 means that P is a real symmetric and positive definite; the symbol ‘*’ within a matrix represents the symmetric term of the matrix; the super script ‘T’ stands for matrix transposition; Rn denotes the ndimensional Euclidean space; I and 0 represent the identity matrix and a zero matrix, respectively; stands for a blockdiagonal matrix. diag{} max P and denote the maximum and minimum ei genvalues of matrix P, respectively; Notations ‘sup’ and ‘inf’ denote the supremum and infimum, respectively. P min 2. Preliminaries and Problem Formulation In this paper, a switched linear system with timevarying delay is described as follows: ,0 tdtt t ttt tAxtAxtdt ButGt ztCxtDut Et xt tt (1) where n tR is the state, is the control input, m ut R m zt R is the measurement output, t , dt , t,()t,()t,()t and ()t are real known constant matrices with appropriate dimensions, A ()t B G C D E is the continuous vector valued function specifying the initial state of the system, ()t is the timevarying exogenous noise signal and satisfies Assumption 1, N:0tI 1,2,, is the switching signal, corresponding to it, the switching sequence 00011 ;, ,,,,,,,,0,1, kk k xitititi Ik k i means that the th subsystem is activated when 1 , kk ttt . dt denotes the timedelay satisfying Assumption 2. Assumption 1. The exogenous noise signal is time varying and satisfies 0d, 0 Ttttd d . (2) Assumption 2. The timevarying delay satisfies 0d, d()1.tth (3) Remark 1. It should be pointed out that the Assump tion 2 about timevarying delay d(t) in this paper is dif ferent from that of [27], where the timedelay is constant. In [33], the concept of finitetime boundedness and H∞ finitetime boundedness for discrete switched system were proposed. In this paper, we extend the definitions to continuous switched linear system with timevarying delay. First, the following three lemmas are presented, which play important roles in our further derivation. Lemma 1 [35]. The linear matrix inequality 11 12 21 22 SS S0 SS 22 , where and are equivalent to 11 11 T SS22 22 T SS S0 , 1 1222 12 SSSS T 11 0 . Lemma 2 [36]. For any, let 0Tt ,NtT de note the switching number of t over . If ,tT 0 ,a NtT NTt (4) holds for and an integer , then is called an average dwelltime. Copyright © 2011 SciRes. ICA
H. LIU ET AL.205 0 Lemma 3 [37]. For given symmetrical matrix X, 11 1 0 * PXQ R and are satisfied 22 2 0 * PXQ R simultaneously, if and only if the following inequality holds 12 1 2 1 2 *0 ** PPQQ R R (5) Definition 1. (Finitetime stability) Switched system (1) with and is said to be fi nitetime stability with respect to 0ut 0t ,,,,, f TdR , where 0 and, R is positive definite matrix and is a switching signal. If 0d t T xtRxt , 0, t T , whenever xRx 0 T sup . If the above condition holds for any switching signal t , system (1) is said to be uniformly finitetime stability with respect to. ,, ,, f TdR Remark 2. As can be seen from Definition 1, the con cept of finitetime stability and Lyapunov asymptotic stability are different. A Lyapunov asymptotically stable switched system may not be finitetime stable if its states exceed the prescribed bounds. Remark 3. The meaning of “uniformity” in Definition 1 is with respect to the switching signal, rather than the time, which is identical to that of [11]. Next, the definitions of finitetime boundedness and H∞ finitetime boundedness for switched system with timevarying delay are introduced. Definition 2. (Finitetime boundedness) Switched sys tem (1) with is said to be finitetime bound edness with respect to 0ut ,,,,, f TdR , where 0 and , R is positive definite matrix and 0d t is a switching signal. If T xtRxt , 0, tT 0 T sup , , whenever 0 : f TT tt xRx dttd . Definition 3. (H∞ finitetime boundedness) Switched system (1) with is said to be H∞ finitetime boundedness with respect to 0ut (,,,,, ) f TdR , where 0 ,, 0d0 , R is positive definite matrix and ()t is a switching signal, following conditions should be satisfied: 1) Switched system (1) is finitetime bounded. 2) Under zeroinitial condition, 0t ,0t , the output z(t) satisfies 2 00 dd ff TT TT ztzttt tt . (6) In this paper, the main purpose is to find sufficient conditions, which can ensure the finitetime boundedness and H∞ finitetime boundedness, and apply these condi tions to design H∞ finitetime stabilizing controller. Remark 4. Definition 3 means that once a switching signal is given, a switched system is H∞ finitetime boundedness if, given a bound on initial state and a H∞ gain , the state remains within the prescribed bound in the fixed finitetime interval. 3. FiniteTime Stability and Bounded Analysis In this section, we focus on finitetime boundedness of switched timedelay system (1) with , that is 0ut ()()() ,0 ,0 tdt t xtAxtAxt dtGtt xtt t (7) Now, let us discuss the finitetime boundedness of switched timedelay system (6). For a symmetric positive definite matrix nn RR , it is easy to verify that R can be factorized according to , where 1/21/2 T RR R 12 R is also a symmetric positive definite matrix. Theorem 1. For any i, let I1212 ii PRPR, 12 12 ii QRQR, . Suppose that there exist matrices , , , , , , 1/ 2 i SR 0 i Q 2, , 0 ii i 1/ 2 i SR 0 i S 0 i P 11, 1 22 XX X 0 i W1,i N2,i N * i X and constants 0 i , 0 such that 11 1213 22 23 33 * ** 0 (8) 11, 12,1, 22, 2, *0 ** i ii i ii i XX N XN eS (9) 234max1 sup if ii T i iI eedWe (10) where 111, 1,11, 121, 2,12, 13 222, 2,22, 23 33 . , , (1 ), , .. i TT iiiiiiiiii iii TT idiiidiiii T iii ii TT idiidiii i T di ii T iii i T PPAPQASAN NX PAAS ANNX PGAS G he QASANNX ASG GSG W If the average dwell time of the switching signal satis fies Copyright © 2011 SciRes. ICA
H. LIU ET AL. 206 * 123 4 ln ln lnii aa f T ee (11) then the switched systems is finitetime boundedness with respect to (,,,,,) f TdR , where 1 , max sup iI dW 0ln if TN ij PP , , ij QQ , ij SS ,ij I, , iI i max , 1mi infiI i P n 2 supiI maxi P , 3max supiI i Q , 4max supiI i S . Proof. Choose a Lyapunovlike function as follows 1, 2,3,iiii VtVtVtVt Vt (12) where 1, () 2, () 0() 3, , d, dd. i i T ii tts T ii tdt tts T ii t Vt xtPxt Vte xsQxss Vte xsSxss When , taking the derivative of V(t) with respect to t along the trajectory of switched system (7), we have 1 , kk ttt 1, TT ii TT ii ii TT di i T idi TT ii T ii VtxtPxtxtPxt xtAPPAxt xtdtAPxt xtPAxtdt tGPxt xtPGt i (13) 2, 2, () 2, 1 1 i i T iii i dt T i T ii i T i VtVt xtQxt dt extdt Qxt dt Vt xtQxt he xtdt Qxtdt (14) 3, 3, 0 3, () d d i i T iiii T i T ii i tst T i t VtVt xtSxt ext Sxt Vt xtSxt exsSxss (15) From the LeibnizNewton formula, the following equation is true for any matrices , , 1, i N2,i N iI with appropriate dimensions 1, 2, () 2d d0 i TT i t tdt N xsxtsN xtxtdtxs s (16) For any matrices , with appropriate dimensions, we have 0 i X iI 1111 () d0 t TT ii tdt tXtsXs s (17) where 1 T TT txtxtdt . Then, it follows from (13)(17) that 1, 2,3, 11 1213 22 23 33 () 1, 2, () 11 () 2 * ** 2 d d i iiiii T tstT i t TT ii t tdt tTT ii tdt Vt VtVtVtVt Vt xt xt xt dtxt dt tt exsSxsds xtN xtdtN xs s sXs stWt 2 33 () ,,d T tTT i tdt tt tsts stWt (18) Assuming conditions (8) and (9) are satisfied, we ob tain T i VtVttW t i (19) By calculation, we have () () () () d tk k k tk k k tt tk tts T t t Vt eVt esW ss (20) Since 1 , ij PP , ij QQ , i SS j and 12 PR12 ii PR , 12 12 ii QRQR, 12 i SR j 12 i SR , then , , , , iji ij PPQQ SS ijI (21) Assume that k t i and at switching instant . According to (19), we obtain k t j k t kk k tt Vt Vt k (22) For any 0, tT, let N be the switching number of σ(t) over (0, Tf ). Using the iterative method, we have Copyright © 2011 SciRes. ICA
H. LIU ET AL.207 1 2 1 1 (0) () (0) 0 1 () () 0 (,) () 0 0 () 0 max 0 (0) d d d 0 d 0 d 0sup k k f f f tN t NtsT tts NT t t tts T t t tN ttsNst T s TN tTNT s TN i iI Vt eV esWss esWss esWss eV esW eV esWss eV dW ss (23) where maxiI i . Noticing that 0fa NNT , then 0 max 00sup ffa TNT iI i Vt eVdW (24) On the other hand, 1212 min 1 inf TT ii TT i iI VtxtPxtx tRPRxt PxtRxt xtRxt (25) 0 (0 00 00 (0) max 0 max 00 max 00 23 4 0(0) 0 dd 00 sup sup TsT sT T T T VxPxexsQx exsS xss Px Rx eQ xRx eS xRx ee dss (26) Taking (24)(26) into account, we obtain 0 23 4max 1 sup fa T iTNT iI xtRxt eedW e (27) 1) When 1 , from (10), ff TT T xtRxte e (28) 2) When 1 , from (11), 123 4 ln ln lnii f a T ee (29) Substituting (29) into (27) yields T xtRxt (30) According to definition 2, we can conclude that the switched timedelay system (6) is finitetime bounded with respect to ( ,,,,,) f TdR . The proof is com pleted. Remark 5. In the proof of Theorem 1, there is no re quirement of negative definitiveness on , which is different from the classical Lyapunov function for switched systems in the case of asymptotical stability. In order to reduce the conservatism of the theorem condi tions, freeweighing matrix method is introduced. When Vt 1 , one obtains τa, in other words, there is no restric tion on the average dwell time for switching signal. When the timevarying exogenous noise signal 0t , the results about finitetime stability can be obtained and given in the following corollary. Corollary 1. Assume that the switched timedelay system (6) satisfies 0ut and . For any 0t iI , let 12 ii PRP12 R , 12 i QR12 i QR , 12 12 i SR 0 i Q i SR . Suppose that there exist matrices , , , 0 i P 0 i S 11,12, 22, 0 * ii i i XX XX 0 , , and constants 1, i N2,i N i , 0 such that 11 12 22 0 * (31) 11, 12,1, 22, 2, *0 ** i iii ii i XXN XN eS (32) 23 41 if ii T eee (33) where 11 1, 1,11, 121,2,12, 222, 2,22, , , 1. i TT i iiiiiiiii T ii i TT idiiidiiii TT idiidiii APPAP QASA NN X PAAS ANNX he QASANNX i If the ADT of the switching signal satisfies * 123 4 ln ln lnii aa f T ee (34) then the switched system is finitetime stability with re spect to ,,,, f TR , where 0ln f TN , 1 , ij PP , ij QQ , ij SS , ,ij I , iIi max 1 inf , , mini P iI Copyright © 2011 SciRes. ICA
H. LIU ET AL. 208 2max sup iI i P 4max sup iI i S , , . 3max sup iI i Q ut Remark 6. It is easy to find that some differences be tween Lyapunov asymptotical stability and finitetime stability. Conditions (33) and (34) must be satisfied for finitetime stability, which is not necessary for asymp totical stability. Thus, the two concepts are independent. However, in previous research, there are few results on finitetime stability, which needs our full investigation. 4. H∞ FiniteTime Boundedness Analysis In this section, we discuss H∞ finitetime boundedness of switched timedelay system (1) with . First, consider the following switched timedelay system 0 () () () [,0] tdt tt t () () tAxt ztC xt xtt t Axtdt G E t I t (35) Theorem 2. For any i , let 1212 ii PRPR , 1212 ii QRQR, 12 12 i 0 N i SR 0 i S 1, i2, N SR 0 i P i Q 11, 12, 22, 0 * ii i i XX XX . Suppose that there exist matrices , , , , , and constants i0 i and 0 11 12 22 ** ii CC 11, i such that 13 23 2 *0 TT ii TT iiii i CE IGSGEE (36) 12, 1, 22, 2, *0 ** i ii i ii i XXN XN eS (37) 2 41 if T ede (38) If the ADT of the switching signal satisfies * ln aa 2 10 ln ln ln f f T dTN (0,, (39) then the switched systems is H∞ finitetime boundedness with respect to , ,,) f TdR , where 1 , ij PP ij QQ, , ij SS , , ,ijI maxiI i 2max sup iI i P 4max sup iI i S , , , , . mini P iI 1 infiI 3 sup maxi Q Proof. Assuming condition (36) is satisfied, then we obtain 11 1213 22 23 2 * ** 0 *000 ** T iii TT ii ii T ii GSG CC CE EE (40) Since 0 *0000 ** T TT iiii i ii T ii i CCCEC CE EE E 0 (41) which implies that 11 1213 22 23 2 *0 ** T iii IGSG (42) From Theorem 1, conditions (37)(39) can ensure that the switched timedelay system (35) is finitetime bounded with respect to (0, ,,,,) f TdR . Next, we will prove condition (6) is satisfied under zero initial condition. Choose the following Lyapunov function 1, 2, 3,iiii VtVtVtVt Vt , where 1, () 2, () 0 3, , d, dd . i i T ii tts T ii tdt ttsT ii t Vt xtPxt Vte xsQxss Vte xsSxss When 1 , kk ttt , by virtue of (36), we can obtain () 2 () (()()()())d ()d k tk k tk k k tt kk kk tt k t tts TT t t tt ts k tt Vt eVt esszsz eVte s ss s (43) Since 1 , ij PP , ij QQ , i SS j and 12 PR12 ii PR , 1212 ii QRQR, 12 12 ii SRSR ,jI , then ,,, jij QS Si iji PPQ . In what follows, assume that k ti and at switching instant . We have k tj k t kk k tt Vt Vt k (44) Since max i iI , then it follows from (43) and (44) that Copyright © 2011 SciRes. ICA
H. LIU ET AL.209 s ss ∞ d k k k tt k t tts t Vt eVt es (45) When (0, Tf ), let N be the switching number of σ(t) over (0, Tf ). Using the iterative method, we have t 1 2 1 (0) 0 1 0 (,) 0 0 0 0d ()d d 0 d 0 d k f f tts tN N tts N t tts t tN tts Nst TN tTN VteVes s ess ess eV ess eV ess (46) Under zero initial condition, (46) implies 0 0d f tTN Vtes s (47) that is , 0 , 2 0 d d f f tTNst T tTNst T ezszss es (48) Setting t = Tf, we obtain 2 00 dd ff TT TT zszsss ss (49) Therefore, according to Definition 3, the proof is com pleted. 5. FiniteTime Stabilization In this section, the static state feedback controllers are designed. Based on the results in the previous section, the closedloop system Hfinitetime bounded with respect to 0, ,,,, f TdR can be ensured by memory state feedback controllers 1, 2,ii utK xtKxtdt. Applying the memory state feedback controllers into switched timedelay system (1), we can obtain the closedloop switched system as follows () () [,0] tdt t tt t tAxtAxtdt Gt ztCxtDxt dtEt xtt t (50) where 1,ttt AABK t , 2,dt dttt AABK , 1,ttt CCDK t , 2,tt DDK t . From condition (36), we have 11 12 22 2 11 *0 0 ** 0 ** *0 ** ** TT iii i TT di i TT ii i i PGA C AD IEE G S I (51) where 111, 1,11, 121,2,12, 222, 2,22, 1. i TT iiiiiiiii T idi iii T iii i APPAPQNNX PA NNX he QNNX , , i According to Lemma 3, (37) and (51) are equivalent to the following inequality 11 121, 22 2, 2 11 *0 **0 00 ***00 ** **0 ** *** i TT iiii i TT dii i TT ii i i i PGAN C AND IEE G S eS I (52) where 111, 1, 121,2, 222, 2, , , 1. i TT iiiiiiiii T idi ii T iii PPAPQNN PA NN he QNN For matrix Inequality (52), let , 1, 2, 0 i TT i ii P MNN idi A A I , then 11 12 22 * 0 01 i TT ii ii iii i AMMA QP he Q (53) Let 1 1 1, 2, 0 i i ii P MLL and 11 ag,, ,, ii TMIIS di T T I. Premultiplying Equation (52) by and postmultiplying Equation (52) by T, we ave h Copyright © 2011 SciRes. ICA
H. LIU ET AL. Copyright © 2011 SciRes. ICA 210 11 121314 11 22 2 11 1 0 *0 **000 ** *00 ** **0 ** *** i i TT ii diiii TT ii i i i G QAI QD IEE G S eS I (54) where 11 111112 11 11 11 12 121 22 11 13 11 14 1 1 21 i i i i i TT iiii diiiidiiiiiiii iiiidiiiii ii i TT ii iidi TT ii iii PAQAAPAQPQPPhe Q PQ AQheQ QheQ PA QA PC QD 1 where , 1 1,iii LQ 1 2,iii Q , . Denote , , 0 ii i R 1 ii PP , 1 ii SS , 1 ii Q 1 1, 1,iii YKPQ, , . By Schur complement (Lemma 1), we can obtain the following Theorem. 1 2,ii Q 2,i YK Theorem 3. For given 0 , iR , 0iR . Suppose that there exist matrices0 i P, 0 i Q, 0 i S, , and constants 1, i Y2,i Y0 i , 0 and such that the following conditions are satisfied iI 11 121314 2223 2, 2 1 0 *0 0 **00 0 0 ***00 0 ****0 0 ** ** *0 ** ** ** i ii TT iii TT ii i i i i GP IYD IEE G S eS I Q (55) 2 41 if iT ede (56) where 111, 1, 2 2, 2, 12 2, 131, 2, 141,2, 1, 1, , i i i TTTT iiiiiidi idiiiiii TT iiiiiiiii iiiidiiiiiii i TTTTTT ii iidi ii iii TTT iii iii PAAPQ AAQYBBY YBBYPhe Q PQ AQBYheQ PAQAYBYB PCY DY 2 22 23 2, , 21, . i i di TT i ii i TTT iii ii D QheQ QAY B If the ADT of the switching signal satisfies * 2 10 ln ln lnln f aa f T dTN (57) then the memory state feedback gains 1 1, 1,iii YP and 1 2, 2,iii YQ ensure closedloop switched timedelay system (50) H finitetime bounded with respect to ∞ 0, ,,,, f TdR . Remark 7. In Theorem 3, i and i are adjustable parameters. By virtue of the method in [38], these pa rameters can be obtained. Remark 8. It should be pointed out that the conditions in Theorems 1, 2, 3 and Corollary 1 are not standard LMIs conditions. However, once some values are fixed for i , these conditions, i.e., (10) and (38) can be trans lated into LMIs conditions. As in [27], (10) and (38) can be rewritten in the following forms 1) The condition (10) can be guaranteed by the fol lowing LMI condition, that is, for any, there exists some positive numbers iI 1 , , , and 2 3 4 5 such that 1i2 PI (58) 3 0i Q I (59)
H. LIU ET AL.211 I I 4 0i S (60) 5 0i W (61) 23 451 if ii T eede (62) 2) The condition (38) can be guaranteed by the fol lowing LMI condition, that is, for any, there exists some positive numbers , , and satisfying (58)(60) such that iI 1 2 3 4 23 41 if ii T eee . (63) 6. Numerical Simulation and Results In this section, for given and , an example is em ployed to verify the method proposed above. Consider a switched linear system with timevarying delay as fol lows () ()()tdt t tAxtAxtdt Gt (64) with , , , , , , 1 1.7 1.70 1.31 0.7 0.7 10.6 A 1.51.70.1 1.3 10.3 0.7 10.6 1 G 0.7 0 0 2 10 1.3 0.1 1.5 0.1 d A 2 110 0.7 00.6 1.7 01.7 A 1 1 1 0.1 0.6 1.8 1d A ()xt ,0th, , 21 GG 0.2 0.02h, . The values of , T, and are selected as follows: d R 0.5 , , , , 10 f T0.01dRI0.05 i , 0.01 . When 2 and 30 , by virtue of Theorem 1, one obtains a. For any switching signal * 2.4659 t with average dwell time a * a 2s , switched linear system with timedelay is finitetime bounded with re spect to (0.5, 30, 10, 0.01, I, σ). The state trajectory over 0~10 s under a periodic switching signal with interval time is shown in Figure 1. It is obvious that switched linear system (64) is finitetime bounded. The state trajectory over 0 ~ 10 s under a periodic switching signal with interval time is shown in Figure 2. As can be seen from figure 2, switched linear system (64) is not finitetime bounded any more. 2.5T s T 7. Conclusions In this paper, unlike most existing research results fo 0123 45678910 2 1. 5 1 0. 5 0 0.5 1 1.5 Time /s Trajectories x 1 x 2 x 3 Figure 1. The histories of the state trajectory of switched system under a periodic switching signal with interval time ΔT = 2.5s. 0 123 4 5 67 8910 8 6 4 2 0 2 4 6 Time/s Traject ories x 1 x 2 x 3 Figure 2. The histories of the state trajectory of switched system under a periodic switching signal with interval time ΔT = 2s. cusing on Lyapunov stability property of switched timevarying delay system, we mainly discuss finitetime boundedness and H∞ finitetime boundedness of switched linear systems with timevarying delay. As the main contribution of this paper, sufficient conditions which can guarantee finitetime boundedness and H∞ finitetime boundedness of switched linear systems with timevarying delay are proposed. And then based on the results on finitetime boundedness, the memory state feedback controller is designed to H∞ finitetime stabilize a switched linear system with timevarying delay. An important and challenging further investigation is how to extend the results in this paper to uncertain switched systems and switched nonlinear systems. 8. Acknowledgements The authors would like to thank the EditorinChief, the Associate Editor, and the reviewers for their insightful and constructive comments, which help to enrich the content and improve the presentation of this paper. 9. References [1] J. Liu, X. Z. Liu and W. C. Xie, “DelayDependent Ro bust Control for Uncertain Switched Systems with TimeDelay,” Nonlinear Analysis: Hybrid Systems, Vol. 2, Copyright © 2011 SciRes. ICA
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