Finite-Time Vibration Control of Earthquake Excited Linear Structures with Input Time- Delay and Saturation

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1 JOURNAL OF LOW FREQUENCY NOISE VIBRAION AND ACIVE CONROL Pages 7 Fiie-ime Vibraio Corol of Earhquake Excied Liear Srucures wih Ipu ime- Delay ad Sauraio Falu Weg * Yuachu Dig Ji Ge Limig Liag ad Guoliag Yag Faculy of Elecrical Egieerig ad Auomaio Jiagxi Uiversiy of Sciece ad echology Gazhou Jiagxi Chia. School of Resources ad Eviromeal Egieerig Jiagxi Uiversiy of Sciece ad echology Gazhou Jiagxi Chia. Deparme of Mechaical ad Idusrial Egieerig Uiversiy of oroo oroo ON Caada MS G8. Firs submied: 8 February /Revised: 7 Jue /Acceped: Jue ABSRAC he problem of fiie-ime vibraio corol of earhquake excied liear srucures wih ipu ime-delay ad sauraio is cocered i his paper. he objecive of desigig corollers is o guaraee he fiie-ime sabiliy of closed-loop sysems while aeuaig earhquake-iduced vibraio of he srucures. Firs based o marix rasformaio he srucural sysem is described as a sae-space model which coais acuaor sauraio ad ipu ime-delay. he based o a Lyapuov fucioal ad fiie-ime sabiliy aalysis mehod some sufficie codiios for he exisece of sauraioolera fiie-ime vibraio-aeuaio corollers are obaied. By solvig hese codiios he desired corollers ca be obaied for he closed-loop sysem o be fiie-ime sable wih a prescribed level of disurbace aeuaio. I is show by he simulaio resuls ha compared wih some Lyapuov asympoic sabiliy resuls fiie-ime sabiliy corol ca resul i beer sae resposes. Furhermore sauraio-olera coroller ca resul i a much lower coroller gai ha he oes wihou cosiderig acuaor sauraios. Keywords: Fiie-ime sabiliy; sauraio; vibraio corol; liear srucure; ime-delay.. INRODUCION Due o he limiaio of lad more ad more high-risig buildigs are buil up i rece years. Such kids of buildigs o oly help save he source of lad bu also prese a beer scee o people. However some srog earhquakes ad wid happe frequely i pas several years. Such as he 8 We Chua Earhquake he Yu Shu Earhquake ad he Japaese suami ec. have resuled i collapse of may buildigs ad deah of may people. As srucural buildigs become higher ad higher heir sabiliy ad solidiy are challeged ad cao be guaraeed oly by hose passive ad semi-acive corol mehods. hus he saus of acive corol [] for srucural buildigs becomes more ad more sigifica ad may achievemes have bee reached by he scholars durig he las decades such as oupu-feedback corol [-] classical H corol [-] eergy-o-peak corol [6-8] robus sampled-daa corol [9] slidig mode corol [-] adapive Correspodig auhor. el.: ; Fax: ; 9@zju.edu.c Suppor by Naioal Naural Sciece Foudaios (Nos. 668 ad 67) Jiagxi Provicial Naural Sciece Foudaios (Nos. GJJ ad GJJ8) ad he Naural Sciece Foudaio of Jiagxi Uiversiy of Sciece ad echology (o. NSFJ-K6) of Chia. Vol. No.

2 Fiie-ime Vibraio Corol of Earhquake Excied Liear Srucures wih Ipu ime-delay ad Sauraio corol [] fuzzy corol [] eural eworks [6] opimal corol [7-8] ec. have bee applied o he vibraio aeuaio for buildigs srucures. Furhermore some acive corol devices also were desiged for applyig hose corol algorihms. For example mageorheological dampers [9-] acive mass damper (AMD) [-] acive brace sysem (ABS) [-] ec. have bee used for vibraio aeuaio. O he oher had mos of hose acuaio devices are subjec o ampliude sauraio ha for he physical ipus such as force orque hrus sroke volage curre ad flow rae of all coceivable applicaios of curre echology are ulimaely limied ad uexpeced large ampliude disurbaces ca also push a sysem s acuaors io sauraio hus forcig he sysem o operae i a oliear mode for which i was o desiged ad i which i may be usable [6]. hus coroller desig for buildigs srucural sysems which ivolves acuaor sauraio is also eeded. Foruaely some exisig achievemes already cosidered his problem (see [7-8] ad hose refereces herei). Furhermore ime delay or rasporaio lag is commoly ecouered whe he corol forces are applied o he pracical sysems. hus aoher impora issue of srucural corol is he ime delay problem whe he corol forces are applied o he srucures [9-]. Recely he sudy of srucural sysems wih corol ipu ime-delay has received icreasig aeio for example i erms of he feasibiliy of cerai delay-depede liear marix iequaliies (LMIs) he robus H disurbace aeuaio problem for ucerai srucural sysems wih corol ipu ime-delay was researched by [8]. [] addressed a covex opimizaio approach o he problem of sae-feedback H corol desig for vibraio reducio of base-isolaed buildig srucures wih delayed measuremes. By combiig he radom search of geeic algorihms ad he solvabiliy of LMIs [] ivesigaed he H coroller desig approach for vibraio aeuaio of seismic-excied buildig srucures wih ime delay i corol ipu chael. Based o LMI echique he problem of robus acive vibraio corol for a class of elecro-hydraulic acuaed srucural sysems wih ime delay i he corol ipu chael ad parameer uceraiies appearig i all he mass dampig ad siffess marices was ivesigaed i []. By cosiderig he acuaor sauraio ad corol ipu ime-delay he acive vibraio corol for a class of earhquake-excied srucural sysems was preseed i [7 8]. For more resuls abou he sauraio ad ime-delay he readers ca refer o [8-9 6] ad hose refereces herei. O he oher had i is worh o poi ou ha mos of he exisig resuls are obaied based o Lyapuov sabiliy aalysis mehod which cares abou he asympoic covergece of he sysem while. However i is ofe he peak values of displacemes or acceleraios make he buildigs damaged. hus a beer performace ca be expeced if fiie-ime sabiliy aalysis mehod is ake io cosideraio. he cocep of fiie-ime sabiliy was firs iroduced i he Russia lieraure []. A sysem is said o be fiie-ime sable if give a boud o he iiial codiio is sae does o exceed a cerai domai durig a specified ime ierval. Recely he problem of fiie-ime sabiliy of regular or sigular sysems has received cosiderable aeio. For example based o Lyapuov fucioal [] discussed he fiie-ime sabiliy of a class of sochasic oliear sysem ad some sufficie sabiliy codiios were obaied. I erms of LMI echique some sufficie codiios were give i [6] for he ucerai discree sigular sysems o be fiie-ime sable ad sabilizable. For more resuls abou he fiie-ime sabiliy he readers ca refer o [7-] ad he refereces herei. his paper cocers he problem of fiie-ime vibraio corol of earhquake excied liear srucures wih ipu ime-delay ad sauraio. he mai coribuio of his paper cosiss i hree aspecs. Firs based o he marix rasform he sae-space model of srucural sysems which coai parameer uceraiies ipu ime-delay ad acuaor sauraio is obaied. he he sigal ipu ime-delay ad acuaor sauraio are cosidered i he sabilizaio coroller desig for srucural sysems. Secod he cocep of fiie-ime sabiliy is expaded o he sabiliy aalysis of srucural sysems wih ipu imedelay ad sauraio ad he LMIs-based codiios are esablished for he 6 JOURNAL OF LOW FREQUENCY NOISE VIBRAION AND ACIVE CONROL

3 Falu Weg Yuachu Dig Ji Ge Limig Liag ad Guoliag Yag srucural sysems o be fiie-ime sabilizable. By solvig hese LMIs he desired acuaor sauraio-olera coroller ca be obaied for he closed-loop sysem wih ipu ime-delay o be fiie-ime sable wih he performace z <γ ω. hird a PNP Lyapuov fucioal is uilized o obai he fiie-ime sabilizaio codiios ad he less coservaiveess of PNP Lyapuov fucioal has bee show i []. he orgaizaio of his paper is as follows. Secio formulaes he problem ad preses he dyamic models. he mai resuls are give i secio. he illusraive examples are give i secio o show he applicabiliy ad improveme of he preseed approaches. Fially he paper is cocluded i secio. Noaio: hroughou his paper for real marices X ad Y he oaio X Y (respecively X > Y) meas ha he marix X Y is semi-posiive defiie (respecively posiive defiie). I is he ideiy marix wih appropriae dimesio ad a superscrip represes raspose. x expresses he -orm of x. We defie sym{ M } M + M. For a symmeric marix * deoes he symmeric erms. he symbol R sads for he -dimesioal Euclidea space ad R m is he se of m real marices.. PROBLEM FORMULAION AND DYNAMIC MODELS Cosider a degree-of-freedom srucural sysem. he sysem uder cosideraio is depiced i Fig.. he liear srucural model equaio ca be wrie wih [ ] Mx () + Cx () + Kx () = H u( τ ) + H ω x () () m m m g where xm() xm() xm() xm() x m () is he iersorey relaive drif of he h floor; u( τ) is he corol force ipu ad τ is he corol forces ipu imedelay; x () is he ipu disurbace belogs o L [ ) H R m gives he g locaios of hese corollers H ω R is a vecor deoig he ifluece of disurbace exciaio ad M C K R are he mass dampig ad siffess marices of he sysem respecively. From Fig. we ca obai = [ ] M = m m m m m m c c = c C c c u m x m () c k u m x m () u m c k x m () x g c k Figure. degree-of-freedom srucural sysem. Vol. No. 7

4 Fiie-ime Vibraio Corol of Earhquake Excied Liear Srucures wih Ipu ime-delay ad Sauraio K = k k k k k H ω m m =. m Defiig he sae variables as = x() x m () xm () wrie i he followig sae-space form: he equaio () ca be Ex () = Ax() + Bu( τ) + B ω() z () = Cx () z [ ] x () = Φ() τ ω () where C z is real cosa marix wih appropriae dimesios Φ() is he iiial codiio o he segme [ τ ] ω() = x saisfies g () ω () ω() d d d ad = I E I =. = Bω A = B M K C H H ω Whe cosiderig possible acuaor sauraio we iroduce a sae-feedback coroller i he form of u () () = σ ( Fx) () where F is he acuaor sauraio-olera coroller gai o be desig laer. he fucio σ (.): R m R m is a sadard sauraio fucio wih he limi of u limi for he ih acuaor i.e. σ (Fx()) = [σ (u ()) σ (u ()) σ (u m ())] where σ (u i ()) = sig (u i ())mi{ u i () u limi }. he we ca use he rasform σ (Fx()) = ψ()fx() [-6] where ψ() = diag{ψ () ψ () ψ m ()} ψ i ()Δ = σ (u i ())/u i () wih ψ i () = if u i () =. o obai he high gai coroller as ha i [] he commad o he ih acuaor is allowed o be ε i u limi for a arbirary scalar ε i >. herefore he resulig ψ i () will be bouded by ε i ad ha is ε i ψ i () i = m. By defiig ψ i = (ε i + )/ Δψ i = δ i ()ψ i δ i δ i () δ i = ( ε i )/( + ε i ) i = m we ca depic he coroller () as u() = (ψ + Δψ)Fx() where ψ = m diag{ψ ψ ψ m } Δψ = diag{δψ Δψ Δψ m } = δψ i ei ( δi () δi) fi. i= e i ad f i are all colum vecors wih he ih iems o be ad ohers o be. Obviously here has δ i ()/δ i. I pracice he mass dampig ad siffess are usually subjeced o possible perurbaios such as measureme error he chages i eviromeal emperaure ad plasic deformaio ec.. By assumig ha he ucerai m j mj mj k c c c j = where m k c ( m k c j kj kj j j j j j j j j j) are he lower (upper) bouds of he mass siffess ad dampig respecively ad deoig mˆ j = ( mj + mj) Δ mˆ = θ mˆ θj θj < θ j = ( mj mj) ( mj + m j j j j) j = k = k + k Δ kˆ = θ kˆ j j j θj θj < j = j j j k k k + k j = θ 8 JOURNAL OF LOW FREQUENCY NOISE VIBRAION AND ACIVE CONROL

5 Falu Weg Yuachu Dig Ji Ge Limig Liag ad Guoliag Yag cˆ Δ cˆ = θ ( + ) cˆ θ θ < = c c c+ c j = cj + cj j j j ( j) ( j) θ ( j) j = we ca describe he ucerai sysem by sae space equaio of he form: E( θ ) x () = A( θ) x() + Bu( τ) + Bω ( θ) ω() u () = ( ψ +Δψ) Fx () z () = Cx z () x () = Φ() [ τ ] () where ucerai marices E(θ ) A(θ ) B ω (θ )saisfyig E B = j j j j E( θ ) = E + θ E A θ = A + θ A B θ = B + θ B ω ω j ωj j= j= j= I B E = mˆ ω = mˆ mˆ mˆ j jej fj Mˆ = mˆ e f ω j j j j Mˆ = Cˆ = mˆ mˆ mˆ mˆ mˆ mˆ cˆ cˆ cˆ cˆ cˆ kˆ kˆ Kˆ kˆ = kˆ kˆ I A = Kˆ Cˆ A = kˆ e f j j j j A cˆ e f j =. + j= j ( + j) ( + j) e j R (j = ) f j R (j = ) e j R (j = ) f j R (j = ) f j R (j = ) are all colum vecors. Defiiio. he sysem is said o be fiie-ime H sabilizable wih respec o (c c c R γ d) if here exiss a coroller gai F such ha he closed-loop sysem has x () sup { Φ () s RΦ() s } c sup { Φ () s RΦ () s } c Rx () c ad s [ τ ] s [ τ ] z < γω for ay [ Τ] ω () ω() d d where < c < c c > R > > γ > d. Defiiio. he sysem () is said o be robusly fiie-ime H sabilizable wih respec o (c c c R γ d) if he sysem is fiie-ime H sabilizable for all admissible uceraiies. Lemma [7]: Give ay marices X V ad U wih appropriae dimesios such ha U >. he we have XUX XV + VX + VU V. () Vol. No. 9

6 Fiie-ime Vibraio Corol of Earhquake Excied Liear Srucures wih Ipu ime-delay ad Sauraio Lemma [7]: Le ϖ() be a oegaive fucio such ha ϖ() a + b ϖ(s) ds for some cosas a b > he we have ϖ() a exp(b). Lemma [8]: give marices χ μ ad ν wih appropriae dimesios ad wih χ symmerical he χ + μf()ν + ν Τ F() Τ μ Τ < holds for ay F() saisfyig F() Τ F() I if ad oly if here exiss a scalar λ > such ha χ + λμμ Τ + λ ν Τ ν <.. MAIN RESULS heorem : he sysem () wihou uceraiies is fiie-ime H sabilizable wih respec o (c c c R γ d) for cosa ime-delay τ ad cosa α if here exis posiive defiie symmeric marices P Q Q Q Q symmeric marices Z Z marices Z Z Z G Y i (i = 6) H i (i = 6) osigular marix S posiive scalars r r r m ε ε ε ε ε ε 6 ad scalars β β β β saisfyig he followig LMIs Π = Ξ τy τ Q SC z Ξ 9 I Ξ 99 < (6) Π = P+ τz + τz + τz + Q + τq + τq Z Z + τz τz Q τq Q + Q + Q + Q + Z Z Z + τ τ τ Z > (7) P+ τz + τz + τ Z + Q + τq + τ Q > ε R (8) P ε R < (9) < Q < ε R () < Q < ε R () < Q < ε R () < Q < ε 6 R () τ ε τ ετ ε τ ε τ + + ε τ γ ε () < α c c c c c c 6 c d ce where Ξ = Ξ + m r Γ Γ i i i i= Ξ = Ξ Ξ Ξ Ξ Ξ B + H ω 6 Ξ Ξ Ξ Ξ β B H ω 6 Ξ Ξ Ξ β B Ξ Ξ β B + Y Ξ β B H γ I ω ω 6 ω 6 P Z Z P = Z Z Z R R = R R JOURNAL OF LOW FREQUENCY NOISE VIBRAION AND ACIVE CONROL

7 Falu Weg Yuachu Dig Ji Ge Limig Liag ad Guoliag Yag { } ( ) Ξ = Q + τq τ Q Q + Sym H + Z Z + τz τz + A S + Bψ G ατz α P+ τ Z + Q + τq + τ Q Ξ = Z + Z τz + τz H + H + Q + β SA + β G ψ B Ξ = P+ τz + τz + τ Z + H E S + β SA + β G ψ B τ Ξ = τ Q + Q β E S β SE Ξ = Z Z Z + Z + τq + H + Y + β SA + β G ψ B Ξ = Q H H Q Ξ = H β E S Ξ = H + H Bψ G+ β SA + β G ψ B Ξ = Z + Z + Z Z H + Y Ξ = Z Z + τz τz + Y β SE Ξ = Q + Y + Y Ξ = H H β Bψ G Ξ = H β Bψ G β SE Ξ = H + Y β Bψ G Ξ = H H β G ψ B β Bψ G Γ = δe ψ B βδie iψ B β δie iψ B β δie iψ B β δie iψ B i i i Y = Y Y Y Y Y Y 6 Ξ = [ Λ Λ Λ ] 9 m Λ i = fig fi G Ξ = diag{ r r r }. 99 m Furhermore a sae-feedback coroller is described as F = GS. Proof: See he Appedix Remark. I is worh o poi ou ha he codiios i heorem are acually LMIs whe he real scalars β β β ad β are priori gives. Furhermore he variables β β β ad β supply a addiioal degree of freedom for he feasibiliy of LMIs (6)-(). hus he codiios i heorem ca be solved direcly by he powerful LMI oolbox i Malab. Moreover he corollers obaied by heorem are wih fixed gais which are easy o be used for calculaig he corol forces ad saisfy he real-ime requireme of pracical srucural sysems. Remark. he closed-loop sysems which are composed of he fiie-ime H sabilizaio corollers obaied by heorem have o oly he H performace z < γ ω for [ ) bu also he sae cosrai x() Rx() κ for [ ). I is well kow ha mos earhquakes especially heir peak exciaios las a very shor ime. hus whe he peak exciaio comes durig he ime ierval [ ] he sae cosrai ca make sure ha he sae resposes saisfy x() Rx() κ. he as he exciaios become weak he H performace ca sill esure ha he sae resposes saisfy z < γ ω. Furhermore if we choose α = LMI (6) reduces o a H sae feedback coroller desig codiio. ha is o say H sabiliy is oly a special case of fiie-ime sabiliy hus he less coservaiveess of heorem is obvious. Based o heorem we ca obai he followig Corollary ad which guaraee he H sabiliy of he closed-loop sysems wih ad wihou sauraio. Corollary : he sysem () wihou uceraiies is H sabilizable for cosa ime-delay τ if here exis posiive defiie symmeric marices P Q Q Q Q symmeric marices Z Z marices Z Z Z G Y i (i = 6) H i (i = 6) osigular marix S posiive scalars r r r m ad scalars β β β β saisfyig (7) ad he followig LMI Vol. No.

8 Fiie-ime Vibraio Corol of Earhquake Excied Liear Srucures wih Ipu ime-delay ad Sauraio m Ξ + riγiγi τy SCz Ξ9 i= Π = τ Q I Ξ99 < () where Ξ = Ξ Ξ Ξ Ξ Ξ B + H ω 6 Ξ Ξ Ξ Ξ β B H ω 6 Ξ Ξ Ξ β B Ξ Ξ β B + Y Ξ β B H γ I ω ω 6 ω 6 { } Ξ = Q + τq Q τ Q + Sym H + Z Z + τz τz + A S + Bψ G Furhermore a sae-feedback coroller is described as F = GS. Corollary : he sysem () wihou uceraiies ad sauraio is H sabilizable for cosa ime-delay τ if here exis posiive defiie symmeric marices P Q Q Q Q symmeric marices Z Z marices Z Z Z G Y i (i = 6) H i (i = 6) osigular marix S posiive scalars r r r m ad scalars β β β β saisfyig ad he followig LMI. Π = Ξ τy SCz τ Q. (6) < I Furhermore a sae-feedback coroller is described as F = GS. heorem : he sysem () is robusly fiie-ime H sabilizable wih respec o (c c c R γ d) for cosa ime-delay τ ad cosa α if here exis posiive defiie symmeric marices P Q Q Q Q symmeric marices Z Z marices Z Z Z G Y i (i = 6) H i (i = 6) osigular marix S posiive scalars r r r m r r r r r r () ε ε ε ε ε ε 6 ad scalars β β β β saisfyig equaio (7)-() ad he followig LMI Γ M * M < (7) where ( ) Γ = Π + θ Γ Γ rmˆ + rkˆ θ Γ Γ + r cˆ θ Γ Γ j j j j j j j j j j + j j + j + j + j j= Γ = e β e β e β e β e j j j j j j m Γ = e β e β e β e β e j j j j j j m Λ = Μ = Λ Λ Λ Λ Λ Λ f S f ( ) j j j m JOURNAL OF LOW FREQUENCY NOISE VIBRAION AND ACIVE CONROL

9 Falu Weg Yuachu Dig Ji Ge Limig Liag ad Guoliag Yag Λ = f S M = diag{ r r r r r r }. j j m Furhermore a sae-feedback coroller is described as F = GS. + θ Proof: Replacig E A ad B ω wih E E A A ad B + θ B ω j= j ω j j= j respecively equaio (6) ca be expressed as j + θ j= j j ( ( + j) j ( + j) ( + j) j= Π + θ mˆ Γ Λ + θ kˆ Γ Λ + θ cˆ j j j j j j j j Γ Λ + θ mˆ Λ Γ + θ kˆ Λ Γ + θ cˆ Λ Γ <. j j j j j j j j ( + j) j ( + j) ( + j) ) (8) By Lemma equaio (8) holds if ad oly if here exis posiive scalars r r r r r r () such ha ( Π + r θ mˆ Γ Γ + r Λ Λ + r θ kˆ j j j j j j j j j j j ΓjΓ j + r Λ Λ j= + r θ cˆ Γ Γ + r Λ Λ ) <. + j + j j + j + j + j + j + j j j j (9) Applyig he Schur compleme LMI (9) is equivale o LMI (7). his complees he proof. Remark : i is worh o poi ou based o replacig E A ad B ω i Corollary wih E ad Bω + θ B + θ E A + θ A j= j ω j j= j j j= j j respecively i is easy o obai he robus H sabilizaio codiios for he sysem wih ad wihou sauraio ad he operaio is similar o heorem. For breviy hey are omied here.. ILLUSRAIVE EXAMPLE Cosider he srucural sysem wih =. he srucural parameers are mˆ i = kg kˆi = 98kN/m ad ĉ =.7kNs/m(i = ) [8]. he he sae space equaio () has he followig parameers: H = diag{ } x x x x x x x E( θ )= E + θ mˆ e f = [ ] B ( θ )= B + θ mˆ e f ω ω j= j j j j j= j j j j A( θ )= A + θ kˆ e f + θ ( + ) cˆ j j j j e ( + ) f j j j ( + j ) j= where A = Kˆ I Cˆ = I E Mˆ Mˆ = mˆ mˆ mˆ mˆ mˆ mˆ Kˆ = kˆ kˆ kˆ kˆ kˆ Cˆ = cˆ cˆ cˆ cˆ cˆ e = = f e = Vol. No.

10 Fiie-ime Vibraio Corol of Earhquake Excied Liear Srucures wih Ipu ime-delay ad Sauraio f = e = f = e = e = f = f = e = = e f = f = f = 6 e = = e6 f = B = ω mˆ mˆ mˆ f = f = f =.. Assume ha he displacemes ad velociies of he hree soreys are all measurable for feedback i his case. he corolled oupu is chose o be he iersorey relaive drifs ha is z() = [x m () x m () x m ()]. Cosider he maximum acuaor oupu force limi u limi = N ad suppose ha ε i = where i =. he we ca ge he permissible maximum corol sigal before sauraio u beflimi = ε i u limi = = Ν ha is whe he corol sigals before sauraio u befi saisfy ubef i N he desiged corollers should have he desired performaces. I order o verify he dyamics of he closed-loop sysem a ime hisory of acceleraio (see Fig. ) from EI Cero 9 earhquake exciaio is applied o his sysem ad his exciaio saisfies ω () ω() d = 8.7. hus we ca choose d = 9. Firs cosider he sysem wihou uceraiies ha is θ i = (i = ) θ i = (i = 6). By choosig τ = ms β = β = β = β =. γ =. c = c =. c =. α =. d = 9 R = I = s we solve he LMIs (6)-() ad obai S = ω() (m/s ) ime (s) Figure. he ime hisory of acceleraio from EI Cero 9 earhquake exciaio. JOURNAL OF LOW FREQUENCY NOISE VIBRAION AND ACIVE CONROL

11 Falu Weg Yuachu Dig Ji Ge Limig Liag ad Guoliag Yag ad a sae feedback coroller which has he followig gai marix F = () For descripio i breviy we deoe his desiged coroller as coroller I hereafer. O he oher had By choosig τ = ms β = β = β = β =. γ =. we solve he Corollary ad obai a sauraio-olera H sae feedback coroller which has he followig gai marix F = () ad his desiged coroller is deoed as coroller II hereafer. Furhermore by choosig τ = ms β = β =. β = β = γ =. ψ = I we solve he Corollary ad obai a H sae feedback coroller which does o cosider he ipu sauraio ad has he followig gai marix F = () his desiged coroller is deoed as coroller III hereafer. I order o faciliae he compariso we obai aoher eergy-o-peak sae feedback coroller which does o ivolve he fiie-ime sabiliy by solvig heorem i [9] wih γ =. ad his coroller has he followig gai F = which is deoed as coroller IV hereafer. he we choose parial sae resposes of he ope-loop sysem which is excied by he earhquake show i fig. as he iiial codiio ad i is show i Fig.. Afer doig some calculaios we obai he - iiial codiios saisfyig sup Φ s RΦ s =.99 < c = { } [ ] s Φ Φ sup s R s =.68 < c =.. [ ] s ms ms { } () he displacemes ad acceleraios resposes of he ope-loop ad closedloop sysems which are composed wih he coroller I II III ad IV are compared i Figs. ad ad he maximum displacemes acceleraios ad sup x Rx of he ope-loop ad closed-loop sysems are show i able. [ ] s { () ()} Magiude Figure. he iiial codiios of he srucural sysem x.8 x.6 x x. x. x 6 ime (s) Vol. No.

12 Fiie-ime Vibraio Corol of Earhquake Excied Liear Srucures wih Ipu ime-delay ad Sauraio X (m) Ope-loop Coroller III Coroller I Coroller IV.... ime (s) Coroller II X (m)... ime (s). X (m). ime (s) Figure. Displacemes resposes of he ope-loop ad closed-loop sysems which are composed wih corollers I o IV respecively (τ = ms). Ope-loop Coroller III Coroller I Coroller IV Coroller II X (m/s ) X (m/s ) ime (s) ime (s) X (m/s ) ime (s) Figure. Acceleraios resposes of he ope-loop ad closed-loop sysems which are composed wih corollers I o IV respecively (τ = ms). Obviously he corollers are all effecive i aeuaig he srucural vibraios. However i ca be foud from able ha he maximum resposes obaied by coroller I is less ha hose obaied by he ohers corollers. Furhermore sup x Rx obaied by coroller I is. which is less ha he [ ] s { () ()} 6 JOURNAL OF LOW FREQUENCY NOISE VIBRAION AND ACIVE CONROL

13 Falu Weg Yuachu Dig Ji Ge Limig Liag ad Guoliag Yag he maximum displacemes acceleraios ad able. sup x sysems which are composed wih corollers I o IV respecively of he ope-loop ad closed-loop Ope-loop Coroller I Coroller II Coroller III Coroller IV x max (cm) x max (cm) x max (cm) ẍ max (m/s ) ẍ max (m/s ) ẍ max (m/s ) sup [ ] s { x () Rx() } [ ] s { () Rx() } u ime (s) u ime (s) u ime (s) Figure 6. Corol forces of he srucural sysem which is composed wih coroller I (τ = ms). permissible value c =. ad sup [ ] s { x () Rx() } obaied by coroller II III ad IV are..6. respecively which are all higher ha he permissible value c =.. ha is o say oly he fiie-ime sabiliy coroller I saisfies he fiie-ime sabiliy codiio ad he effeciveess of heorem is obvious. he le s come o see he corol forces which are ploed i Figs. 6 o 9 where meas sigals before sauraio ad meas sigals afer sauraio. I ca be foud ha sauraios happe i all closed-loop sysems. he maximum corol sigals before sauraio of hose acuaors are show i able. I ca be obaied from able ha he maximum corol sigals before sauraio obaied by coroller I ad II are 67.N ad 8.6N respecively which are all Vol. No. 7

14 Fiie-ime Vibraio Corol of Earhquake Excied Liear Srucures wih Ipu ime-delay ad Sauraio u ime (s) u ime (s) u ime (s) Figure 7. Corol forces of he srucural sysem which is composed wih coroller II (τ = ms). u ime (s) u ime (s) u ime (s) Figure 8. Corol forces of he srucural sysem which is composed wih coroller III (τ = ms). less ha he permissible limiaio N. However he maximum corol sigals before sauraio obaied by coroller III ad IV are 76. N ad.9 6 N respecively which are all higher ha he permissible limiaio N. 8 JOURNAL OF LOW FREQUENCY NOISE VIBRAION AND ACIVE CONROL

15 Falu Weg Yuachu Dig Ji Ge Limig Liag ad Guoliag Yag x 6 u ime (s) x 6 u ime (s) x 6 u ime (s) Figure 9. Corol forces of he srucural sysem which is composed wih coroller IV (τ = ms). able. he maximum corol sigals before sauraio of he closed-loop sysems which are composed wih corollers I o IV respecively Coroller I Coroller II Coroller III Coroller IV u u u Now le s come o see he ucerai case ad he maximum acuaor oupu force limi u limi = 8N ad suppose ha ε i = where i =. he we ca ge he permissible maximum corol sigal before sauraio u beflimi = ε i u limi = 8 = 8N ha is whe he corol sigals before sauraio u befi saisfy u befi 8N he desiged corollers should have he desired performaces. Furhermore he uceraiies are applied o he mass siffess ad dampig coefficies of he firs sorey ad he parameer uceraiies saisfy θ. θ. θ.. By choosig τ = ms β =. β = β =. β = γ =. c = c α =. d = 9 R = I = s we solve he LMIs = c = 6 (7)-() ad (7) ad obai Vol. No. 9

16 Fiie-ime Vibraio Corol of Earhquake Excied Liear Srucures wih Ipu ime-delay ad Sauraio S = ad a robus sae feedback coroller which has he followig gai marix F = () For descripio i breviy we deoe his desiged coroller as coroller V hereafer. For breviy we cosider he omial case (θ i =(i = ) θ i =(i = 6) correspods o case ) ad four-verex cases where he mass siffess ad dampig coefficies are give as heir verex values respecively. Case correspods o mˆ =. kg kˆ =. 98kN/m ad ĉ =..7kNs/m; Case correspods o mˆ =. kg kˆ =. 98kN/m ad ĉ =.8.7kNs/m; Case correspods o mˆ =.8 kg kˆ =. 98kN/m ad ĉ =..7kNs/m; Case correspods o mˆ =.8 kg kˆ =. 98kN/m ad ĉ =.8.7kNs/m. he parameers of soreys ad are he same as hose of he omial sysem. he sysem has he same iiial codiios as hose show i Fig.. Afer some simple calculaios i is foud ha { () ()} sup Φ s RΦ s =.998 < c s [ ] ms { } sup Φ ( sr ) Φ ( s) =.8 < c. s [ ] ms Uder he same exciaio meioed above he displacemes ad acceleraios resposes of he ope-loop ad closed-loop sysems i Case are ploed i Figs. ad respecively which show he effeciveess of coroller V i aeuaig he vibraio of he srucural sysem. he maximum displacemes acceleraios ad sup x Rx of he ope-loop ad closed-loop sysems i he five cases are [ ] s { () ()} show i able where Ope meas Ope-loop sysem ad Closed meas Closedloop sysem. we ca obai from able ha beer maximum resposes are reached for all closed-loop cases o maer he parameer uceraiies exis or o. he maximum displacemes acceleraios ad able. [ ] s { x () Rx() } closed-loop sysems i five cases (τ = ms) of he ope-loop ad Case Case Case Case Case Cases Ope Closed Ope Closed Ope Closed Ope Closed Ope Closed x max (cm) x max (cm) x max (cm) xmax( m s ) sup xmax( m s ) xmax( m s ) sup [ ] s { x () Rx() } JOURNAL OF LOW FREQUENCY NOISE VIBRAION AND ACIVE CONROL

17 Falu Weg Yuachu Dig Ji Ge Limig Liag ad Guoliag Yag. Ope-loop Coroller V x (m). ime (s). Ope-loop Coroller V x (m). ime (s). Ope-loop Coroller V x (m). ime (s) Figure. Displacemes resposes of he ope-loop ad closed-loop sysems i case (τ = ms). Furhermore he values of { () ()} sup [ ] s { x () Rx ()} obaied i he five cases all saisfy sup x Rx c. hus i is validaed ha he desiged fiie-ime sabiliy [ ] s coroller V is robus o parameer uceraiies. he correspodig ipu forces are ploed i Fig. which shows he sauraios happe i he closed-loop sysems. he maximum corol sigals before sauraio of hose acuaors are show i able. We ca obai from able ha he maximum corol sigal before sauraio is 6.9N which is less ha he permissible limiaio 8N.. CONCLUSION he fiie-ime vibraio corol of earhquake excied liear srucures wih ipu imedelay ad sauraio has bee ivesigaed i his paper. Firs by iroducig a rasform marix ψ() he liear srucural sysem is described as a sae-space model which coais acuaor sauraio ad ipu sigals ime-delay. Secodly based o fiie-ime sabiliy aalysis mehod some sufficie codiios for he exisece of sauraio-olera fiie-ime vibraio-aeuaio corollers are obaied. If he feasibiliy problem of hese codiios is solvable he desired coroller ca be able. he maximum corol sigals before sauraio of he closed-loop sysems i five cases Case Case Case Case Case u u u Vol. No. 6

18 Fiie-ime Vibraio Corol of Earhquake Excied Liear Srucures wih Ipu ime-delay ad Sauraio Ope-loop Coroller V ẍ (m) ime (s) Ope-loop Coroller V ẍ (m) ime (s) Ope-loop Coroller V ẍ (m) ime (s) Figure. Acceleraios resposes of he ope-loop ad closed-loop sysems i case (τ = ms). u ime (s) u ime (s) u ime (s) Figure. Corol forces of he closed-loop sysem i case (τ = ms). 6 JOURNAL OF LOW FREQUENCY NOISE VIBRAION AND ACIVE CONROL

19 Falu Weg Yuachu Dig Ji Ge Limig Liag ad Guoliag Yag obaied for he closed-loop sysem o be fiie-ime sable wih a prescribed level of disurbace aeuaio. he codiio is also exeded o he ucerai case. Fially he simulaio resuls show he effeciveess of he desiged corollers. For he vibraio corol cosidered i his aricle he sysemaic sae is chose as he cosraied variables. However acceleraio is also a impora facor ifluecig he safey of srucures. hus obaiig he corollers wih acceleraio cosrai cosidered is a ieresig challege for fuure research. Furhermore i should be highlighed ha he proposed mehodology ca be of grea ieres o a wide variey of egieerig areas where he sae cosrai ad vibraio aeuaio are ecouered. APPENDIX Proof of heorem : We firs cosider he sysem () wihou uceraiies ha is θ i =(i = ) θ i =(i = ). By subsiuig he corol law u() = (ψ +Δψ)Fx() io he sysem () ad accordig o x () x ( τ ) = x () s ds τ we ca obai he followig closed-loop sysem ( ) Ex () = A + B ψ +Δψ F x() B ψ + Δ ψ F x s ds+ Bω ω() τ () z () = Cx (). z Choose a Lyapuov-Krasovskii fucioal cadidae as V() = V () + V () (6) where V() = x() Px() + x () s Q x() s ds+ x( s) Q x( s) dsdε τ τ + ε + τ x () s Qx () s dsdε + τ x ( s) Q x ( s) dsdεdθ τ + ε τ θ + ε ( s) τ τ τ θ s s s τ + ( s) ( s) ( s) V () = x dsz x ds+ x Z x ds+ x Z x dsdθ + x dsz x dsdθ + x dsdθz x dsdθ ( s) τ τ + θ τ + θ τ + θ P = S PS > Q i = S Q i S >(i = ) Z i = S Z i S (i = ) are ay marices wih appropriae dimesios ad wih Z Z symmerical. Accordig o equaio (7) ad (6) we have V () x ds Q+ Q x ds ( s) τ τ ( s) τ τ τ ( τ ) ( τ x() x() s ds Q x() τ x() s ds τ ) x () ( P Q ) x () ( τ s ) ( τ τ s τ ) x x ds Q x x ds + V ( () () () () S S ζ() Π ζ() > S S (7) where ζ () = () x x ds τ ( s) soluio of sysem () is give by. he he derivaive of V() alog he Vol. No. 6

20 Fiie-ime Vibraio Corol of Earhquake Excied Liear Srucures wih Ipu ime-delay ad Sauraio V () = x () Px() + x () Q x() Furhermore we have x ( τ) Q x( τ) + τx Q x x Q x ds ( ) ( s) ( s) + τ + τ x () Qx () τ x ( ε) Qx ( ε) dε τ + τ x () Q x () τ x ( s) Q x ( s) dsdθ τ + θ ( ( τ ) s τ x() Z ( x() x( τ )) x () Z ( τ x() x() s ds τ ) x() Z ( τ x ( x() x( τ ))) ( x() x( τ )) Z τ x() x() s ds τ x( s) dsz τ x ( x x( τ )) τ ( τ τ ) τ V () = x x Z x ds + x Z x ds ( s) τ x () + β x ( τ) + β x () + β x () s ds + β x () s ds S τ A + B ψ +Δψ F x() B ψ + Δψ F x s ds+ B ω() E x () = (( ( ) ) ( ) τ () ω τ ) + ) ( τ ) + x x + x Z x x ds. () ( ) () () s (8) (9) () where S = S. For ay marices H i = S H i S (i = ) H 6 = H 6 S here holds x () H + τ + x ( ) H x () H + x () s dsh + + ω x () s dsh () H6 τ τ ( x () x ( τ) x ( ε) dε) =. τ () Y Accordig o lemma for ay marices Y =[Y Y Y Y Y Y 6 ] = S YS i i (i = ) Y = Y S here holds 6 6 Furhermore we have () s () () x Q x ds ξ Y x ds + τξ YQ Y ξ τ ( s) ( s) τ. () τ x () ε Q x () ε dε x () ε dεq x () ε dε τ τ τ x () = x ( τ) Q Q Q x (). x ( τ) () 6 JOURNAL OF LOW FREQUENCY NOISE VIBRAION AND ACIVE CONROL

21 Falu Weg Yuachu Dig Ji Ge Limig Liag ad Guoliag Yag ( s) ( s) τ + θ ( τ s τ ). ( () () s τ ) τ x Q x dsdθ τx x ds Q x x ds () he by oig (8)-() we ca obai where ξ () = τ () () () x ω x x x s ds x s ds () () τ τ ad Ξ Ξ Ξ Ξ Ξ SBω + H 6 Ξ Ξ Ξ Ξ βsbω H 6 Ξ Ξ Ξ β Ξ = SBω Ξ Ξ βsbω + Y6 Ξ βsbω H 6 Ξ = Q + τq Q τ Q + Sym H + Z Z + τz τz + SA + SB ψ +Δψ F { ( ) } Ξ = Z + Z τz + τz H + H + Q + β A S + β F ψ +Δψ B S ξ V () Ξ + τyq Y ξ () () Ξ = Q H H Q () Ξ = P+ τz + τz + τ Z + H SE + β AS + β F ψ +Δψ BS Ξ = H β SE τ Ξ = τ Q + Q βse βes Ξ = Z Z Z + Z + τq + H + Y + β AS + β F ψ +Δψ BS Ξ = H + H SB ψ + Δ ψ F + β A S + β F ψ +Δψ B S Ξ = Z + Z + Z Z H + Y Ξ = Z + Z + Z Z H + Y Ξ = Z Z + τz τz + Y β E S Ξ = Q + Y + Y Ξ = H H β SB ψ + Δψ F Ξ = H β SB ψ + Δψ F β E S Ξ = H + Y β SB ψ + Δψ F Ξ = H H β F ψ + Δψ B S β SB ψ + Δψ F. Vol. No. 6

22 Fiie-ime Vibraio Corol of Earhquake Excied Liear Srucures wih Ipu ime-delay ad Sauraio } By pre ad pos-muliplyig (6) wih diag{s S S S S I S I I m+ ad is raspose cosiderig G = FS ad accordig o he Schur Complime we ca obai Ξ + τyq Y { } + diag C C α P+ τz + τz + τ Z + Q + τq + τ Q γ <. z z (6) From () ad (6) i is easy o obai () () V () < αv () + ω γ ω z()() z (7) Iegraig boh sides of (7) from o wih [ Τ] i follows By lemma i has Accordig o (7) we have () () α ω γ V () < V() + Vsds () + s ω sds. (8) < α α V () V() e + e ω () sγ ω() sds. (9) V () x ( P+ τz+ τz + τ Z+ Q+ τq+ τ Q ) x () () ( ( ) ) () () λ R P+ τz + τz + τ Z + Q + τq + τ Q R x Rx mi () where R = S RS > he x () Rx() ( ( ) ) λ R P+ τz + τz + τ Z + Q + τq + τ Q R mi τ + λ + τ + λ τ + V() V() max R PR c c c max R Q R c V ()() τ τ + λ λ τ λ () max ( R Q R ) c + max ( R QR ) c + max ( R Q R ) c P Z Z where ξ = θ P = Z Z x x ds x dsd () ( s) ( s) τ τ θ. Z Furhermore i holds ω () s γ ω() s ds < γ d. () I view of (9)-() i yieds 66 JOURNAL OF LOW FREQUENCY NOISE VIBRAION AND ACIVE CONROL

23 Falu Weg Yuachu Dig Ji Ge Limig Liag ad Guoliag Yag x () Rx() α ( ( ) ) λ R P+ τz + τz + τ Z + Q + τq + τ Q R mi e τ τ λmax ( R PR ) c + τ c + c + λmax ( R QR ) τc + λmax ( R QR ) c τ + λmax ( R QR ) c + τ λmax ( R QR ) c + γ d. () By cosiderig he codiios (8)-() we ca obai x () Rx() c. Nex we will esablish he z() < γ ω () performace of he sysem uder zero iiial codiio ha is Φ() = [ τ ] ad V() = =. Iegraig boh side of (7) from o wih [ ) by Lemma i yields Obviously equaio () implies ( γω () ω() () () ) α V() < e s s ds z s z s ds () () () () () z s z s ds < γ ω s ω s ds. From Defiiio we kow he sysem is fiie-ime H sabilizable wih respec o (c c c R γ d ). his complees he proof. REFERENCES [] Nor K.A.M. Muhalif A.G.A. ad Wahid A.N. A coloy Opimizaio for Coroller ad Sesor-Acuaor Locaio i Acive Vibraio Corol Joural of Low Frequecy Noise Vibraio ad Acive Corol () 9 8. [] Palacios-Quiñoeroa F. Rubió -Massegœa J. Rossella J.M. ad Karimi H.R. Feasibiliy Issues i Saic Oupu-Feedback Coroller Desig wih Applicaio o Srucural Vibraio Corol Joural of he Frakli Isiue () 9. [] Rubió -Massegúa J. Rossella J.M. Karimib H.R. ad Palacios-Quiñoeroa F. Saic Oupu-Feedback Corol uder Iformaio Srucure Cosrais Auomaica 9 () 6. [] Zhag B. ad ag G. Acive Vibraio H Corol of Offshore Seel Jacke Plaforms Usig Delayed Feedback Joural of Soud ad Vibraio () [] Palacios-Quiñoeroa F. Rubió-Massegúa J. Rossella J.M. Karimi H.R. Opimal Passive-Dampig Desig Usig a Deceralized Velociy-Feedback H Approach. Modelig Ideificaio ad Corol () [6] Zhag W. Che Y. ad Gao H. Eergy-o-Peak Corol for Seismic-Excied Buildigs wih Acuaor Fauls ad Parameer Uceraiies Joural of Soud ad Vibraio () 8 6. [7] Palacios-Quiñoeroa F. Rubió-Massegúa J. Rossella J.M. ad Karimi H.R. Vibraio Corol for Adjace Srucures Usig Local Sae Iformaio Mecharoics () 6. [8] Du H. ad Zhag N. Eergy-o-Peak Corol of Seismic-Excied Buildigs wih Ipu Delay Srucural Corol ad Healh Moiorig 7 (7) [9] Du H. Zhag N. Samali B. ad Naghdy F. Robus Sampled-Daa Corol of Srucures Subjec o Parameer Uceraiies ad Acuaor Sauraio Egieerig Srucures [] A. Oveisi ad M. Gudarzi Adapive Slidig Mode Vibraio Corol of a Vol. No. 67

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