FINITE-TIME BOUNDEDNESS AND STABILIZATION OF SWITCHED LINEAR SYSTEMS

Transcription

1 K Y BERNETIKA VOLUM E 46 21, NUMBER 5, P AGES FINITE-TIME BOUNDEDNESS AND STABILIZATION OF SWITCHED LINEAR SYSTEMS Habo Du, Xangze Ln and Shhua L In ths paper, fnte-tme boundedness and stablzaton problems for a class of swtched lnear systems wth tme-varyng exogenous dsturbances are studed Frstly, the concepts of fnte-tme stablty and fnte-tme boundedness are extended to swtched lnear systems Then, based on matrx nequaltes, some suffcent condtons under whch the swtched lnear systems are fnte-tme bounded and unformly fnte-tme bounded are gven Moreover, to solve the fnte-tme stablzaton problem, stablzng controllers and a class of swtchng sgnals are desgned The man results are proven by usng the multple Lyapunov-lke functons method, the sngle Lyapunov-lke functon method and the common Lyapunov-lke functon method, respectvely Fnally, three examples are employed to verfy the effcency of the proposed methods Keywords: swtched lnear systems, fnte-tme boundedness, multple Lyapunov-lke functons, sngle Lyapunov-lke functon, common Lyapunov-lke functon Classfcaton: 93A14, 93C1,93D15, 93D21 1 INTRODUCTION A swtched system belongs to a specal class of hybrd systems It conssts of a famly of subsystems descrbed by dfferental or dfference equatons and a swtchng law that orchestrates swtchng between these subsystems [18] Swtched systems arse as models for phenomena whch can not be descrbed by exclusvely contnuous or exclusvely dscrete processes Examples nclude manufacturng control [2], traffc control [28], chemcal processng [1] and communcaton networks [25], etc Due to ther success n practcal applcatons and mportance n theory development, swtched systems have been attractng consderable attenton durng the last decades The basc research topcs for swtched systems are the ssues of stablty and stablzablty whch have attracted most of the attenton [14, 8, 16, 7, 12, 21, 23, 24, 33, 15, 32, 22] Due to the hybrd nature of swtched systems operaton, the stablty analyss of swtched systems s more dffcult to deal wth than that of contnuous systems or dscrete systems, and so becomes a challengng ssue In ths respect, Lyapunov stablty theory and ts varatons or generalzatons stll play a domnant role Analyss method can be roughly dvded nto the common Lyapunov functon method, the sngle Lyapunov functon method and the multple Lyapunov

2 Fnte-tme Boundedness and Stablzaton of Swtched Lnear Systems 871 functons method For the former, a common Lyapunov functon for all subsystems guarantees stablty under an arbtrary swtchng sgnal However, many swtched systems, whch may not possess a common Lyapunov functon, yet stll are stable under some properly chosen swtchng sgnals The multple Lyapunov functons and the sngle Lyapunov functon methods have been proven to be powerful and effectve tools for fndng such a swtchng sgnal e g, tme-dependent swtchng sgnals and state-dependent swtchng sgnals [14, 7, 12, 21, 23, 24] For more analyss and synthess results of swtched systems, the readers are referred to the lterature [8, 16, 33, 15, 32, 22] and the references theren Up to now, most of the exstng lterature related to stablty of swtched systems focuses on Lyapunov asymptotc stablty, whch s defned over an nfnte tme nterval However, n many practcal applcatons, the man concern s the behavor of the system over a fxed fnte tme nterval For nstance, the problem of sendng a rocket from a neghborhood of a pont A to a neghborhood of a pont B over a fxed tme nterval; the problem, n a chemcal process, of keepng the temperature, pressure or some other parameters wthn specfed bounds n a prescrbed tme nterval In these cases, fnte-tme stablty could be used, whch focuses ts attenton on the transent behavor over a fnte tme nterval rather than on the asymptotc behavor of a system response More specfcally, a system s sad to be fnte-tme stable f, gven a bound on the ntal condton, ts state remans wthn prescrbed bounds n the fxed tme nterval It should be noted that fnte-tme stablty and Lyapunov asymptotc stablty are ndependent concepts: a system could be fntetme stable but not Lyapunov asymptotcally stable, and vce versa Some early results on fnte-tme stablty problems can be found n [13, 9, 3] Recently, based on the lnear matrx nequalty theory, many valuable results have been obtaned for ths type of stablty [3, 2, 34, 1, 35, 4, 5], to name just a few In [3, 2], the authors have extended the concept fnte-tme stablty to fnte-tme boundedness In [34, 1], the defnton of fnte-tme stablty have been extended to the systems wth mpulsve effects In [35], fnte-tme control problem has been dscussed for the systems subject to tme-varyng exogenous dsturbance More analyss and synthess results of fnte control problem can be found n [4, 5] and the references theren In addton, t should be ponted out that the authors of [6, 19, 17, 29] have presented some results of fnte-tme stablty for dfferent systems, but the fnte-tme stablty whch conssts of Lyapunov stablty and fnte-tme convergence s dfferent from that n ths paper and [13, 9, 3, 3, 2, 34, 1, 35, 4, 5] So far, the stablty analyss for swtched systems and the fnte-tme stablty for dfferent systems have been extensvely studed by many researchers However, lttle work has been done for the fnte-tme stablty of swtched systems It s well-known that a nonlnear tme-varyng system or a lnear tme-varyng system can often be approxmated by usng several lnear tme-nvarant systems around the equlbrum pont These lnear systems beng only vald around the equlbrum pont, t s mportant to be able to avod large excursons of the states Consderng such reason and the wde applcaton of swtched systems n practce, t motvates us to nvestgate the fnte-tme stablty and stablzaton problems for a class of swtched lnear systems In [31], the authors have studed the fnte-tme stablty and

3 872 H DU, X LIN AND S LI practcal stablty for a class of swtched systems However, the suffcent condtons are dffcult to test and the problem of controller desgn s not consdered To the best of the authors knowledge, the proposed work n ths paper on fnte-tme boundedness and stablzaton problems for swtched lnear systems has not been studed n the prevous lterature The contrbuton of ths paper s twofold Frst, the defnton of fnte-tme boundedness s extended to swtched lnear systems and some suffcent condtons are gven n terms of matrx nequaltes whch are easy to test Second, the problem of control synthess ncludng both swtchng sgnals and feedback swtchng controllers s dscussed These swtchng sgnals contan tme-dependent swtchng sgnals and state-dependent sgnals, whch are sutable for dfferent cases The rest of the paper s organzed as follows In Secton 2, some notatons and problem formulatons are presented Secton 3 provdes the man results of ths paper Some suffcent condtons whch guarantee a class of swtched lnear systems fnte-tme bounded and unformly fnte-tme bounded are gven In Secton 4, the fnte-tme stablzaton problem s nvestgated Fnally, three numercal examples are presented to llustrate the effcency of the proposed methods n Secton 5 Concludng remarks are gven n Secton 6 2 PRELIMINARIES AND PROBLEM FORMULATION In ths paper, let P >, <, denote a symmetrc postve defnte postvesemdefnte, negatve defnte, negatve-semdefnte matrx P For any symmetrc matrx P, λ max P and λ mn P denote the maxmum and mnmum egenvalues of matrx P, respectvely The dentty matrx of order n s denoted as I n or, smply, I f no confuson arses Consder a class of swtched lnear control systems of the form ẋt = A σt xt + B σt ut + G σt ωt, x = x, 1 where xt R n s the state, ut R p s the control nput and ωt R r s the exogenous dsturbance σt: [, M = {1, 2,, m} s the swtchng sgnal whch s a pecewse constant functon dependng on tme t or state xt A, B and G are constant real matrces for M Assumpton 21 The exogenous dsturbance ωt s tme-varyng and satsfes the constrant ω T tωtdt d, d It should be ponted out that the assumpton about the dsturbance ωt n ths paper s dfferent from that of [3, 34], where the external dsturbance s constant Assumpton 22 [14] The state of the swtched lnear system does not jump at the swtchng nstants, e, the trajectory xt s everywhere contnuous The swtchng sgnal σt has fnte number of swtchng on any fnte nterval tme Correspondng to the swtchng sgnal σt, we have the followng swtchng sequence {x ;, t,, k, t k,, k M, k =, 1, },

4 Fnte-tme Boundedness and Stablzaton of Swtched Lnear Systems 873 whch means that k th subsystem s actvated when t [t k, t k+1 The am of ths paper s to fnd a class of swtchng sgnals under whch system 1 wth zero nput s fnte-tme bounded and deal wth the fnte-tme stablzaton problem for system 1 3 FINITE-TIME BOUNDEDNESS ANALYSIS In ths secton, some fnte-tme boundedness crtera for swtched lnear control system 1 wth nput ut = are presented Frstly, let us extend the defntons of fnte-tme stablty and fnte-tme boundedness n [3] to swtched lnear systems Defnton 31 Gven three postve constants c 1, c 2, T f, wth c 1 < c 2, a postve defnte matrx R, and a swtchng sgnal σt, the swtched lnear system ẋt = A σt xt, x = x 2 s sad to be fnte-tme stable wth respect to c 1, c 2, T f, R, σ, f x T Rx c 1 xt T Rxt < c 2, t [, T f ] Next, consder the case when the state s subject to some external dsturbance sgnals Ths leads us to gve the defnton of fnte-tme boundedness, whch recovers Defnton 31 as a specal case Defnton 32 Gven four postve constants c 1, c 2, d, T f, wth c 1 < c 2, a postve defnte matrx R, and a swtchng sgnal σt, the swtched lnear system ẋt = A σt xt + G σt ωt, x = x 3 s sad to be fnte-tme bounded wth respect to c 1, c 2, d, T f, R, σ, f x T Rx c 1 xt T Rxt < c 2, t [, T f ], ωt : T f ω T tωtdt d Remark 33 It should be remarked that the concepts of fnte-tme stablty and fnte-tme boundedness are dfferent from the concept of reachable set Reachable set s defned as the set of states that a system attans under gven some bounded nputs and startng from some gven ntal condtons In most analyss about reachable set, t s assumed that the system s asymptotcally stable [11] However, a system s fnte-tme stable or bounded f, gven a bound on the ntal state and bounded constant dsturbances, the state remans wthn the prescrbed bound n the fxed tme nterval In the analyss of fnte-tme stablty or boundedness, the assumpton of system asymptotc stablty s unnecessary For more detaled dscussons about the dfference between two approaches can be found n Remark 4 of [3] In the sequel, based on the multple Lyapunov-lke functons and the sngle Lyapunov-lke functon methods, some suffcent condtons whch guarantee system 3 fnte-tme bounded are gven, respectvely

5 874 H DU, X LIN AND S LI 31 Suffcent condtons for fnte-tme boundedness based on multple Lyapunov-lke functons In ths subsecton, some suffcent condtons whch guarantee system 3 fnte-tme bounded are presented Before gvng the man result, let us revew the defnton of average dwell-tme whch wll be useful n the subsequent analyss Defnton 34 [12] For any T t, let N σ t, T denote the number of swtchng of σt over t, T If N σ t, T N + T t/τ a holds for τ a >, N s a nonnegatve nteger, then τ a s called average dwell-tme Wthout loss of generalty, n ths paper we choose N =, as [24] Now, based on the multple Lyapunov-lke functons method and an average dwelltme technque, the followng theorem provdes suffcent condtons for fnte-tme boundedness of system 3 Theorem 35 For any M, let P = R /2 P R /2 and suppose that there exst matrces P >, Q > and constants α, γ > such that A P + P A T α P G Q Q G T <, 4a γ Q c 1 λ 1 + γd λ 3 < c 2 λ 2 e αt f If the average dwell-tme of the swtchng sgnal σ satsfes τ a τ a = T f lnµ lnc 2 /λ 2 lnc 1 /λ 1 + γd/λ 3 αt f, 4b 4c then system 3 s fnte-tme bounded wth respect to c 1, c 2, d, T f, R, σ, where λ 1 = mn M λ mn P, λ 2 = max M λ max P, λ 3 = mn M λ mn Q, α = max M α, γ = max M γ, µ = λ 2 /λ 1 P r o o f Choose a pecewse Lyapunov-lke functon as follows V t = V σt t = x T t P σt xt Step 1: When t [t k, t k+1, the dervatve of V t wth respect to t along the trajectory of system 3 yelds V t = x T t P σt k A σt k + A T σt k P σt k xt + xt t P σt k G σt k ωt + ω T tg T σt k P σt k xt = x T t, ω T t P σt k A σt k + A T σt k G T σt k P σt k P σt k G σt k P σt k xt ωt 5

6 Fnte-tme Boundedness and Stablzaton of Swtched Lnear Systems 875 Assumng condton 4a s satsfed, then pre- and post-multplyng 4a by the postve symmetrc matrx P 1 Q C A, we obtan the equvalent condton P A + A T Combnng 5 and 6 leads to P G T P α P P G γ Q < 6 V t < x T t, ω T t α σtk P σt k xt γ σtk Q ωt σt k 7 = α σtk V σtk t + γ σtk ω T tq σt k ωt By calculaton, we have d e α σtk t V σtk t < e α σt t k γ σtk ω T tq dt σt k ωt 8 Integratng 8 from t k to t gves V t < e α σt k t t k V σtk t k + γ σtk t e ασtkt s ω T sq σt k ωsds 9 t k Then, due to the defntons of λ 1 and λ 2, for, j M and x R n, we have x T P x λ max P x T x = 1/λ mn P x T x 1/λ 1 x T x, x T P j x λ mn P j x T x = 1/λ max P j x T x 1/λ 2 x T x Snce λ 2 /λ 1 = µ, then x T P x 1/λ 1 x T x λ 2 /λ 1 x T P j x = µx T P j x It follows that x T R 1/2 P R 1/2 x µx T R 1/2 P j R 1/2 x, e, x T P x µx T P j x Wthout loss of generalty, at the swtchng nstant t k, assume σt k =, σt k = j, where σt k = lm v σt k + v Notcng that xt k = xt k from Assumpton 22, we obtan V σtk t k µv σt k t k, 1 where xt k = lm v xt k + v By 9 and 1, and notcng that α = max M α, γ = max M γ, we have t V t < e αt tk µv σt k t k + γ e αt s ω T sq σt k ωsds 11 t k Step 2: For any t, T f, let N be the number of swtchng of σt over, T f, whch mples that N σ, t N Notcng that µ 1 snce λ 2 λ 1, and usng the

7 876 H DU, X LIN AND S LI teratve method n step 1, we have V t < e αt µ N V σ + γµ N t1 + γµ N t2 =e αt µ N V σ + γ e αt s ω T sq σ ωsds t e αt s ω T sq σt ωsds + + γ 1 e αt s ω T sq σt k ωsds t 1 t k t e αt s µ Nσs,t ω T sq σs ωsds e αt f µ N V σ + γe αt f µ N λ max Q σs Tf e αt f µ N V σ + γe αt f µ N d/λ 3, ω T sωsds where λ max Q σs = 1/λ mnq σs = 1/λ 3 The relaton N T f /τ a leads to On the other hand, V t < e αt f µ T f τa V t λ mn P σt xt trxt = 12 Vσ + γd/λ λ max P σt xt trxt 1 λ 2 x T trxt, 14 V σ λ max P σ xt Rx = [1/λ mn P σ ]x T Rx c 1 /λ 1 15 Puttng together 13, 14 and 15, we get x T trxt λ 2 V t < λ 2 e αt f µ T f τa c 1 /λ 1 + γd/λ 3 16 Assumng condton 4b s satsfed, then the followng proof can be dvded nto two cases Case 1: µ = 1, whch s a trval case, from 4b, we have λ 2 e αt f c 1 /λ 1 + γd/λ 3 < c 2 17 Substtutng 17 nto 16 leads to x T trxt < c 2 Case 2: µ > 1, from 4b, we obtan lnc 2 /λ 2 lnc 1 /λ 1 + γd/λ 3 αt f > Then, assumng condton 4c s satsfed, we have Ths leads to ln c 2 ln c 1 + γd αt T f f λ 2 λ 1 λ 3 τ a lnµ µ T f τa Therefore, t follows from 16 and 18 that c 2 ln λ = 2 c 1 /λ 1 + γd/λ 3 e αt f lnµ c 2 λ 2 c 1 /λ 1 + γd/λ 3 e αt f 18 x T trxt < c 2 19 Notcng that the trajectory of system 3 remans contnuous at nstant T f, we conclude that 19 holds for all t [, T f ]

8 Fnte-tme Boundedness and Stablzaton of Swtched Lnear Systems 877 Remark 36 Let N =, whch mples that there s no swtchng over [, T f ], then the system 3 degenerates nto an ordnary lnear system and Theorem 35 contans Lemma 6 of [3] as a specal case Remark 37 The functon V t n the proof of Theorem 35 belongs to multple Lyapunov-lke functons Unlke the classcal quadratc Lyapunov functon for swtched lnear systems n the case of asymptotcal stablty, there s no requrement of negatve defnteness or negatve semdefnteness on V t Ths s the specfcal dfference between these two knds of Lyapunov functons Actually, f the exogenous dsturbance ωt = and we lmt the constants α < M, then V t wll be a negatve defnte functon In ths case, we can obtan that the system 3 s asymptotcally stable on the nfnte nterval [, + f the average dwell-tme τ a lnµ/α The detaled proof can be found n [14] In the case of fnte-tme stablty for system 2, t s easy to obtan the suffcent condtons from Theorem 35, e, the case of d = Corollary 38 For any M, let P = R /2 P R /2 and suppose that there exst matrces P > and constants α such that A P + P A T α P <, µ < c 2 c 1 e αt f If the average dwell-tme of the swtchng sgnal σ satsfes τ a τ a = T f lnµ lnc 2 /c 1 lnµ αt f, 2a 2b 2c then system 2 s fnte-tme stable wth respect to c 1, c 2, T f, R, σ, where λ 1 = mn M λ mn P, λ 2 = max M λ max P, α = max M α, µ = λ 2 /λ 1 The advantage of multple Lyapunov-lke functons les n ther flexblty, snce dfferent Lyapunov-lke functons are constructed for each subsystem From Remark 36, we know that condtons 4a, 4b mply that each subsystem s fntetme bounded wth respect to c 1, c 2, d, T f, R That s to say, f the swtchng s slow enough satsfyng condton 4c, then the whole swtched systems s fntetme bounded However, n some practcal cases, all subsystems are not fnte-tme bounded, then the condton 4a can not be guaranteed In ths case, the whole swtched systems may stll be fnte-tme bounded by properly choosng the swtchng sgnal In the next subsecton, we wll nvestgate ths case 32 Suffcent condtons for fnte-tme boundedness based on a sngle Lyapunov-lke functon In ths subsecton, some suffcent condtons for fnte-tme boundedness of system 3 are derved by applyng a sngle Lyapunov-lke functon method Assume each subsystem of system 3 s not fnte-tme bounded Ths assumpton s due to that

9 878 H DU, X LIN AND S LI f at least one of the ndvdual subsystems s fnte-tme bounded, ths problem s trval just keep σt = p, where p s the ndex of ths fnte-tme bounded subsystem In the sequel, a class of state-dependent swtchng sgnals are desgned such that system 3 s fnte-tme bounded Theorem 39 For any M, let P = R /2 PR /2 and suppose that there exst matrces P >, Q > and constants α, β, γ >, m =1 β = 1, such that β A P + T PA α P β G Q =1 =1 β QG T <, 21a γq =1 If the swtchng sgnal σt s desgned as c 1 λ mn P + γd λ mn Q < c 2 λ max P e αt f 21b σt = {, f y T tω yt < and σt = ; argmn{y T tω j yt, j M}, f y T tω yt and σt =, 21c then system 3 s fnte-tme bounded wth respect to c 1, c 2, d, T f, R, σ, where argmn stands for the ndex whch attans the mnmum among M, σ = argmn{y T Ω j y, j M}, yt = xt ωt, Ω = P A + A T P α P P G G T P γq P r o o f Choose a sngle Lyapunov-lke functon as follows V t = x T t P xt Accordng to 21c, we assume that the th subsystem s actve when t [t k, t k+1 wthout loss of generalty, whch means that σt = σt k =, when t [t k, t k+1 Then the dervatve of V t wth respect to t along the trajectory of system 3 yelds V t = x T t P A σtk + A T σt k P xt + x T t P G σtk ωt + ω T tg T σt k P xt = x T t, ω T t P A σtk + A T σt k P P G σtk G T σt k P xt ωt 22 In what follows, we wll show that x T t, ω T t P A σtk + A T σt k P P G σtk G T σt k P < x T t, ω T t α P γq xt ωt xt, ωt 23

10 Fnte-tme Boundedness and Stablzaton of Swtched Lnear Systems 879 holds for any t [t k, t k+1 If 23 s not true, then there exsts a tme t [t k, t k+1 and a non-zero state x T t, ω T t T, such that x T t, ω T t P A σtk + A T P σt k α P P G σtk P γq G T σt k xt ωt 24 By the defntons of yt and Ω, 24 mples that y T tω σtk yt Then accordng to the swtchng law 21c, we have y T tω j yt, j M Snce β, m =1 β = 1, we obtan m =1 β y T tω yt, e, y T t =1 β P A + A T P α P β G T P γq =1 =1 β P G yt 25 Assumng condton 21a s satsfed, then pre- and post-multplyng 21a by the P postve symmetrc matrx Q, we obtan the equvalent condton =1 β P A + A T P α P β G T P γq =1 =1 β P G < 26 Obvously, combnng 25 and 26 leads a contradcton snce yt Thus, 23 holds for any t [t k, t k+1 By 22 and 23, we have for any t [t k, t k+1, V t < x T t, ω T t α P γq By calculaton, we have xt ωt = αv t + γω T tq ωt 27 d e αt V t < e αt γω T tq ωt 28 dt Note that V t remans contnuous at any swtchng nstant due to the defnton of V t and 28 holds for any subsystem f the subsystem s actve Integratng 28 from to t gves V t < e αt V + γ t e αt f V + γe αt f e αt s ω T sq ωsds Tf ω T sq ωsds e αt f γd V + λ mn Q 29

11 88 H DU, X LIN AND S LI On the other hand, V t = x T t P xt + ω T Q ω λ mn P x T trxt = 1 λ max P xt trxt, 3 V = x T P x λ max P x T Rx = [1/λ mn P]x T Rx 31 Puttng together 29-31, we get x T trxt λ max PV t < λ max Pe αt f V + γd/λ mn Q λ max Pe αt f c 1 /λ mn P + γd/λ mn Q Assumng condton 21b s satsfed, then we obtan x T trxt < c 2 32 Remark 31 Note that even though each subsystem of system 3 s not fntetme bounded wth respect to c 1, c 2, d, T f, R, the swtched system 3 may stll be fnte-tme bounded wth respect to c 1, c 2, T f, R, σ by properly choosng the swtchng sgnal σ In the case of asymptotcal stablty of swtched systems, under the assumpton that all the subsystems are not asymptotcal stable, how to desgn the swtchng sgnals s an nterestng topc Here, n the case of fnte-tme boundedness of swtched systems, we have dscussed the smlar topc n Theorem 39 Smlarly, n the case of fnte-tme stablty for system 2, we have the followng corollary Corollary 311 For any M, let P = R /2 PR /2 and suppose that there exst a matrx P > and constants α, β, m =1 β = 1, such that β A P + T PA α P <, =1 c 1 λ mn P + d λ mn Q < c 2 λ max P e αt f 33a 33b If the swtchng sgnal σt s desgned as {, f x σt = T tω xt < and σt = ; argmn{x T tω j xt, j M}, f x T tω xt and σt =, 33c then system 2 s fnte-tme stable wth respect to c 1, c 2, T f, R, σ, where σ = argmn{x T Ω j x, j M}, Ω = P A + A T P α P The multple Lyapunov-lke functons and the sngle Lyapunov-lke functon methods have been employed to desgn dfferent swtchng sgnals n dfferent cases However, by the condton 4c, 21c, there are some constrants on the swtchng sgnals Hence, Theorems 35 and 39 may not be sutable for the cases of fast swtchng or stochastc swtchng In the followng, we wll gve some condtons whch guarantee system 3 fnte-tme bounded under an arbtrary swtchng

12 Fnte-tme Boundedness and Stablzaton of Swtched Lnear Systems Suffcent condtons for unform fnte-tme boundedness In the above two subsectons, suffcent condtons for fnte-tme boundedness of swtched lnear system 3 are presented To solve the case of arbtrary swtchng, the common Lyapunov-lke functon method wll be used In the sequel, some suffcent condtons are presented for fnte-tme boundedness of system 3 under un arbtrary swtchng, whch s called unform fnte-tme boundedness n ths paper Defnton 312 Gven four postve constants c 1, c 2, d, T f, wth c 1 < c 2, and a postve defnte matrx R, the swtched lnear system 3 s sad to be unformly fnte-tme bounded wth respect to c 1, c 2, d, T f, R, f x T Rx c 1 xt T Rxt < c 2, t, T f ], ωt : T f ω T tωtdt d holds for any swtchng sgnal σt Remark 313 The meanng of unformty n Defnton 312 and n [14] are dentcal, whch s wth respect to the swtchng sgnal, rather than tme Theorem 314 For any M, let P = R /2 PR /2 and suppose that there exst matrces P >, Q > and constants α, γ > such that A P + T PA α P G Q <, 34a γ Q QG T c 1 λ mn P + γd λ mn Q < c 2 λ max P e αt f, 34b then system 3 s unformly fnte-tme bounded wth respect to c 1, c 2, d, T f, R, where α = max M α, γ = max M γ P roof Choose a common Lyapunov-lke functon as follows V xt = x T t P xt Substtutng P σt, Q σt wth P, Q nto the proof of Theorem 35, t s easy to get the concluson 4 FINITE-TIME STABILIZATION Havng gven the fnte-tme boundedness analyss for the swtched lnear systems 3, n the followng, let us nvestgate the fnte-tme stablzaton ssue Here, n ths paper, the followng swtchng state feedback controller ut = K σt xt 35 s desgned to stablze system 1 Substtutng 35 nto system 1, we get the closed-loop system 36 as follows ẋt = A σt + B σt K σt xt + G σt ωt, x = x 36 In the sequel, some suffcent condtons for fnte-tme stablzaton and unform fnte-tme stablzaton of system 1 are presented, respectvely

13 882 H DU, X LIN AND S LI Theorem 41 For any M, let P = R /2 P R /2 and suppose that there exst matrces P >, Q >, M and constants α, γ > such that A P + P A T + B M + M TBT α P G Q Q G T <, 37a γ Q c 1 λ 1 + γd λ 3 < c 2 λ 2 e αt f 37b Then, under the feedback controller ut = K xt = M P xt and any swtchng sgnal σ wth average dwell-tme satsfyng τ a τ a = T f lnµ lnc 2 /λ 2 lnc 1 /λ 1 + γd/λ 3 αt f, 37c system 1 s fnte-tme bounded wth respect to c 1, c 2, d, T f, R, σ, where λ 1 = mn M λ mn P, λ 2 = max M λ max P, λ 3 = mn M λ mn Q, α = = max M α, γ = max M γ, µ = λ 2 /λ 1 P r o o f Applyng Theorem 35 to the closed-loop system 36 and changng varables as M σt = Kσt Pσt, t s easy to obtan the result Actually, n Theorem 41, t s easy to fnd that each subsystem of system 1 can be fnte-tme stablzed by lnear state feedback controller just keep σt =, M However, n some cases, although each subsystem can not be fntetme stablzed by any lnear state feedback, the whole swtched system may stll be fnte-tme stablzed under a sutable swtchng sgnal Theorem 42 For any M, let P = R /2 PR /2 and suppose that there exst matrces P >, Q >, M, and constants α, β, γ >, m =1 β = 1, such that β A P + T PA + B M + M T BT α P β G Q =1 =1 β QG T <, 38a γq =1 c 1 λ mn P + γd λ mn Q < c 2 λ max P e αt f 38b Then, f the feedback controller s chosen as ut = K xt = M P xt and the swtchng sgnal σt s desgned as {, f y σt = T tω yt < and σt = ; argmn{y T tω j yt, j M}, f y T tω yt and σt 38c =, system 1 s fnte-tme bounded wth respect to c 1, c 2, d, T f, R, σ, where σ = argmn{y T Ω j y, j M}, y T t = x T t, ω T t, P Ω = A + B M P + A + B M P T P α P P G P γq G T

14 Fnte-tme Boundedness and Stablzaton of Swtched Lnear Systems 883 P r o o f Applyng Theorem 39 to the closed-loop system 36 and changng varables as M σt = Kσt P, t s easy to obtan the result Consequently, the unform fnte-tme stablzaton problem for swtched lnear control system 1 can be easly solved by Theorem 314 Theorem 43 For any M, let P = R /2 PR /2 and suppose that there exst matrces P >, Q >, M and constants α, γ > such that A P + T PA + B M + M TBT α P G Q <, 39a γ Q QG T c 1 λ mn P + γd λ mn Q < c 2 λ max P e αt f 39b Then, under the feedback controller ut = K xt = M P xt, system 1 s unformly fnte-tme bounded wth respect to c 1, c 2, d, T f, R, where α = max M α, γ = max M γ P r o o f Applyng Theorem 314 to the closed-loop system 36 and changng varables as M σt = Kσt P, we obtan the result Remark 44 From a vewpont of computaton, two remarks are gven Frst, t should be noted that the condtons n Theorem 35 to Theorem 43, e, condtons 4a, 21a, 34a, 37a, 38a and 39a, are not standard lnear matrx nequaltes LMIs condtons Actually, these matrx nequaltes are blnear matrx nequaltes BMIs [26, 27] due to the product of unknown scalars and matrces As suggested n [15, 21], one possble way to compute the BMI problem s to grd up the unknown scalars, and then solve a set of LMIs for fxed values of these parameters That s to say, once some values are fxed for α, β, γ, these condtons can be translated nto LMIs condtons and thus solved nvolvng Matlab s LMI control toolbox Second, the condtons 4b, 21b, 34b, 37b, 38b and 39b, are not lnear matrx nequaltes LMIs condtons For convenence of computaton, as n [3], these condtons can be guaranteed by the followng LMI condtons, M, θ 1 I < P < I, θ 2 I < Q, c 2e αt f c1 γd c1 θ 1 γd θ2 >, 4a 4b 4c for some postve numbers θ 1 and θ 2 It should be ponted out that the condton 4 s just a suffcent condton for the condtons 4b, 21b, 34b, 37b, 38b and 39b To some extent, ths suffcent condton s conservatve

15 884 H DU, X LIN AND S LI Remark 45 Fnte-tme boundedness and stablzaton problems have been dscussed based on three knds of Lyapunov-lke functons In practcal applcatons, the choosng order s gven as follows: Frst, we choose the common Lyapunov-lke functon method If there exsts a common Lyapunov-lke functon for all subsystems, then the fnte-tme boundedness can be guaranteed under an arbtrary swtchng However, n many cases, t s dffcult to fnd a common Lyapunov-lke functon Second, we choose the multple Lyapunov-lke functons method If there exst dfferent Lyapunov-lke functons for each subsystem, then the fnte-tme boundedness can be guaranteed f the swtchng s slow enough However, f there are some subsystems whch are not fnte-tme bounded, then the common Lyapunov-lke functon and the multple Lyapunov-lke functons methods can not be appled Thrd, we need to employ the sngle Lyapunov-lke functon method Even all the subsystems are not fnte-tme bounded, the whole system may stll be fnte-tme bounded under a sutable swtchng sgnal 5 NUMERICAL EXAMPLES AND SIMULATIONS Example 51 Consder the swtched lnear system gven by ẋt = A σt xt + G σt ω wth A 1 =, A 2 2 =, G 1 1 = G 2 =, x = 1 1 T, ωt = 2 sn2t + 2 1sn3t + cos5t T Note that each subsystem s not asymptotcally stable because A 1 and A 2 are not Hurwtz Then, we consder the fnte-tme boundedness The parameters are gven as c 1 = 1, c 2 = 2, d = 34, T f = 1 and matrx R = I We frst apply Theorem 314, e, the common Lyapunov-lke functon method We can not fnd any feasble soluton Then, we apply Theorem 35 and solve correspondng matrx nequaltes Solvng 4a and 4 for α = γ = 1 M leads to feasble solutons P 1 = Q 1 = , P 2 = , Q 2 = 24 whch satsfy the condton 4b Therefore, accordng to 4c, for any swtchng sgnal σ 1 t wth average dwell-tme τ a τa = 4245, system 41 s fnte-tme bounded wth respect to 1, 2, 34, 1, I, σ 1 Fg 1a shows the state trajectory over 1s under a perodc swtchng sgnal σ 1 wth nterval tme T = 45s from the ntal state x From Fg 1b, t s easy to see that system 41 s fnte-tme bounded wth respect to 1, 2, 34, 1, I, σ 1 If the swtchng s too frequent, t s possble that the system s not fnte-tme bounded Fg 2a shows the state trajectory of system 41 over 1s under a perodc swtchng sgnal σ 2 wth nterval tme T = 135s from the ntal state x From Fg 1b, we can fnd that system 41 s not fnte-tme bounded wth respect to 1, 2, 34, 1, I, σ 2,,

16 Fnte-tme Boundedness and Stablzaton of Swtched Lnear Systems 885 x x T trxt c 2 = x 1 5 c 1 = Tmes a Phase plot of state xt b Tme hstory of xt T Rxt Fg 1 The response of system 41 under the swtchng sgnal σ c 2 =2 x 2 x T trxt x 1 5 c 1 = Tmes a Phase plot of state xt b Tme hstory of xt T Rxt Fg 2 The response of system 41 under the swtchng sgnal σ 2 In the next example, we consder the case that each subsystem s not fnte-tme bounded However, by properly choosng the swtchng sgnal, the swtched system s stll fnte-tme bounded Example 52 Consder the swtched lnear system gven by wth A 1 = ẋt = A σt xt + G σt ω 42, A 2 = , G 1 = G 2 = I, ωt = 1 snt, 1 cost, 6snt + 2 cost T, x = 1,, T

17 886 H DU, X LIN AND S LI Fg 3 The state trajectores for swtched system 42 and each subsystem The parameters are gven as c 1 = 1, c 2 = 2, d = 37, T f = 1 and matrx R = I Frstly, we apply Theorems 35 and 314, we can not get any feasble soluton Actually, t s shown that both the subsystems are not fnte-tme founded wth respect to 1, 2, 37, 1, I by smulatons Fg 3 shows the state trajectores of all subsystems over 1s from the ntal state x Then, we apply Theorem 39 and solve correspondng matrx nequaltes Solvng 21a and 4 for α = 5, β 1 = 65, β 2 = 35, γ = 1 leads to feasble solutons P = , Q = Therefore, accordng to Theorem 39, f the swtchng sgnal σ 3 t s desgned as 21c, system 42 s fnte-tme bounded wth respect to 1, 2, 37, 1, I, σ 3 Fg 3 shows the state trajectory of swtched system 42 over 1s under the swtchng sgnal σ 3 t from the ntal state x It s easy to see that system 42 s fntetme bounded wth respect to 1, 2, 37, 1, I, σ 3 from Fg 4a, whch presents the state trajectory of system 42 under the swtchng sgnal σ 3 t Moreover, the swtchng sgnal σ 3 t s shown n Fg 4b Example 53 Consder the fnte-tme stablzaton problem for swtched system ẋt = A σt xt + B σt ut + G σt ωt 43 The correspondng parameters are specfed as follows

18 Fnte-tme Boundedness and Stablzaton of Swtched Lnear Systems 887 Trajectory 2 1 State trajectory x1 State trajectory x2 State trajectory x3 Swtchng sgnal Tmes Tmes a The state trajectory of swtched system 42 b The swtchng sgnal σ 3 Fg 4 The state trajectory of swtched system 42 and the swtchng sgnal σ A 1 =, B 25 1 =, A 2 =, B 3 2 =, G 5 1 = 4 cos1t + 3 G 2 = I, ωt =, c 2 sn3t 1 = 1, c 2 = 1, d = 1, T f = 1, R = I By Theorem 43, solvng 39a and 4, we obtan the matrx solutons for α = γ = 6 M as follows P =, Q =, M 1 = , M 2 = Thus, accordng to Theorem 43, under the followng state feedback controllers u 1 t = xt, u 2 t = xt, system 43 s unformly fnte-tme bounded wth respect to 1, 1, 1, 1, I 6 CONCLUSION In ths paper, fnte-tme boundedness and stablzaton problems have been nvestgated for a class of swtched lnear systems Some suffcent condtons have been provded for fnte-tme boundedness of swtched lnear systems In addton, the fnte-tme stablzaton problem has also been studed A challengng and nterestng future research topc s how to extend the results n ths paper to swtched nonlnear systems ACKNOWLEDGEMENT Ths work was supported by Natural Scence Foundaton of Chna , and Specalzed Research Fund for the Doctoral Program of Hgher Educaton of Chna Receved Aprl 19, 21

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