Stochastic Optimal Control of Structural Systems

Transcription

1 he Open Aomaon and Conol Syem Jonal, 8,, -9 Sochac Opmal Conol o Scal Syem Open Acce Z.G. Yng Depamen o Mechanc, Zheang nvey, angzho 37, P. R. Chna Abac: he ochac opmal conol an mpoan eeach bec n cal engneeng. Recenly, a ochac opmal nonlnea conol mehod ha been popoed baed on he ochac dynamcal pogammng pncple and ochac aveagng mehod. he acve and em-acve ochac opmal conol mehod have been he developed o cal yem. he conol aaon o bond, paal ae obevaon, ec. have been aken no accon by he ochac opmal conol. he peen pape vey hee eeach developmen. INRODCION he ong nonlnea ochac vbaon o engneeng ce ch a all bldng and lage bdge eqenly caed by evee loadng ch a eahqake and om. he ochac opmal conol an mpoan eeach bec n cal engneeng []. he mahemac heoy o ochac opmal conol ha been developed [- ]. oweve, he lnea-qadac-ga (LQG conol mehod ha moly ed o cal vbaon conol [5]. Recenly, ome ochac opmal nonlnea conol mehod [6, 7] have been popoed, n pacla, he mehod baed on he ochac dynamcal pogammng pncple and ochac aveagng mehod [8-7]. h conol mehod can acheve bee conol eecvene and ecency han he LQG conol mehod. hee eeach developmen ae veyed a ollow: ( ochac opmal conol law; ( opmal acve conol; (3 opmal em-acve conol; ( developmen o ochac opmal conol mehod. SOCASIC OPIMAL CONROL LAW [8, 9] A conolled and ochacally exced nonlnea cal yem wh ml-degee-o-eedom can be expeed a X M X ( + C( X, X X V + ( X, X W + B X whee X he n-dmenonal cal dplacemen veco, V (X he cal poenal, W( he ochac excaon veco, he conol oce veco. Rewe Eq. ( n he ollowng om o qa amlonan eqaon: Q P P c + kwk + Q ( (a (b whee he amlonan ncon, Q and P ae epecvely dplacemen and momenm, W k ( he ochac excaon, amed a Gaan whe noe, he conol. Apply he ochac aveagng mehod o yem ( yeld aveaged Iô ochac deenal eqaon. In he negable and non-eonan cae, o nance, he Iô eqaon d [ m ( + < > ]d + ( db whee he negal o amlonan yem, m and σ ae epecvely he aveaged d and don coecen, B ( he n Wene poce, < > he aveage opeao. he obecve o ochac opmal conol o degn a conol whch mnmze he peomance ndex o yem ( o (3. o ne me-neval epone conol, he peomance ndex J E[ L(, d + ( ( ] and o nne me-neval egodc conol, J lm # L(, d (3 (a (b whee L he peomance ncon, Ψ he emnal co. Eq. (3 and ( cone a ochac opmal conol poblem. By ng he ochac aveagng mehod, he dmenon o yem eqaon can been edced, he dynamcal pogammng eqaon become non-degeneae and he epone conol conveed no he enegy conol. Accodng o he ochac dynamcal pogammng pncple, he dynamcal pogammng eqaon can been bl, o yem (3 and ndex (b a Adde coepondence o h aho a he Depamen o Mechanc, Zheang nvey, angzho 37, P.R. Chna; E-mal: yngzg@z.ed.cn # # V # V mn{ L(, + [ m + < > ] + k k } # P # # # (5 87-3/8 8 Benham Open

2 Sochac Opmal Conol o Scal Syem he Open Aomaon and Conol Syem Jonal, 8, Volme 5 whee V he vale ncon. hen he opmal conol law can be obaned by mnmzng he le de o Eq. (5. Le ncon L be L (, g( + < R > he opmal conol R P V whee / P a lnea ncon o velocy geneally o ha a qa-lnea dampng oce. he vale ncon V obaned by olvng he ollowng eqaon: $ + k$ k m R + g( # < he ochac opmal conol mehod above ha been appled o a ngle-degee-o-eedom yem wh cbc nonlneay and wo-degee-o-eedom nonlnea yem nde Gaan whe noe and emc excaon. Nmecal el how ha he conol eecvene and ecency by ng he popoed mehod ae bee han by ng he LQG mehod n em o oo-mean-qae epone (g. and. K (% g. (. Conol eecvene (K-pecenage edcon n oomean-qae dplacemen Popoed opmal conol LQG conol Excaon neny Popoed opmal LQG conol Excaon neny g. (. Conol ecency (µ-ao o conol eecvene o nomalzed oo-mean-qae conol. OPIMAL ACIVE CONROL [, ] he opmal eedback conol (7 can be mplemened by acve conol devce, o nance, acve ma dampe (AMD. he opmal acve conol mehod ha been appled > (6 (7 (8 o he vbaon conol o a hyeec colmn yem and copled adacen ce. Opmal Acve Conol o yeec Colmn A conolled and paamecally/exenally exced hyeec colmn yem can be expeed a X + % X + [# k k $ ] X + ( # Z + n Z AX # X Z X Z Z n (9a (9b whee X he dmenonle hozonal dplacemen, Z he hyeec oce ayng BW hyee Eq. (9b, η( and ξ( ae epecvely he vecal paamec excaon and hozonal exenal excaon, amed a Gaan whe noe, he conol. ly, he hyeec oce Z epaaed eqvalenly no qa-lnea dampng oce and nonlnea elac eong oce o yeld eqvalen nonhyeec yem X + [ $ + $ ( ] X + # + k X + X ( whee he oal yem enegy. hen applyng he ochac aveagng mehod yeld aveaged Iô ochac deenal eqaon # d [ m( + < > ]d + ( d # X ( Accodng o he ochac dynamcal pogammng pncple, he dynamcal pogammng eqaon o yem ( and ndex (b obaned a # # V # V mn{ L(, + ( m+ < > + } # P # # ( o he ncon L qadac n conol, he opmal acve conol V V X R X R (3 he epone ac o he conolled yem can be evalaed by bng he opmal conol no Eq. (9a and ng he ochac aveagng mehod. Nmecal el ae hown n g. (3. and (% (/ Excaon neny g. (3. Conol eecvene (Κ-pecenage edcon n oomean-qae dplacemen and ecency (µ-ao o conol eecvene o nomalzed oo-mean-qae conol o he hyeec colmn.

3 6 he Open Aomaon and Conol Syem Jonal, 8, Volme Z.G. Yng Opmal Acve Conol o Copled Adacen Sce A conolled and copled ml-oey ce yem nde emc gond moon excaon can be expeed a M X + C X + KX x g ME + P M X + C X + K X x g M E + P (a (b whee X and X ae epecvely he dplacemen veco o ce and, x g ( he emc excaon wh he K powe pecal deny, he coplng conol oce veco. By he cal mode anomng, edcng and ochac aveagng, he Iô ochac deenal eqaon obaned d [ m ( + < v > ]d + ó ( db( Q (5 whee he modal enegy veco o he copled ce, Q he modal dplacemen veco, v he modal conol oce veco, m and ae epecvely he d coecen veco and don coecen max. o yem (5 and ndex (b, he dynamcal pogammng eqaon obaned a mn {L(,+ (m+ < Q v > V + ( V } # (6 o he ncon L qadac n conol, he opmal acve conol R p (P # #V # Q + P # #V # # Q # (7 he epone ac o he conolled yem can be evalaed by bng he opmal conol no Eq. ( and ng he ochac aveagng mehod. Nmecal el ae hown n g. (a and (b. OPIMAL SEMI-ACIVE CONROL [-] he em-acve conol depend on a malle powe pply and moe gncan. he dynamc chaacec o em-acve conol devce need o be aken no accon by conol. he MR dampe a clacal em-acve conol devce, whch can been decbed by he Bngham model o Boc-Wen model [8, 9]. he MR-LCD anohe emacve conol devce. Sem-Acve Conol Law Baed on he Bngham Model o MR Dampe By applyng he ochac aveagng mehod and ochac dynamcal pogammng pncple o yem ( o (, he opmal acve conol (7 can be deemned. oweve, he em-acve MR dampe canno alway mplemen he opmal acve conol law. Accodng o he Bngham model, he conol podced by an MR dampe cq gn( Q (8 he pa o conol (8 a pave conol componen whch can be ncopoaed n yem, and he econd pa o conol (8 a em-acve conol componen whch ha he amplde adable by he appled volage and n he oppoe decon o velocy. I he em-acve conol (8 dagee wh he opmal conol, e o be zeo. heeoe, he opmal em-acve conol #%, R gn( Q, P V $ < gn( Q (9a (9b he ecacy o em-acve conol (9 geneally lowe han ha o he coepondng acve conol (7. oweve, nde cean condon, he elaon alway hold o ha he em-acve conol (9 agee wh he acve conol (7. h he em-acve MR dampe can peom he opmal acve conol law. Sem-Acve Conol Law Baed on he Boc-Wen Model o MR Dampe he Boc-Wen model can qe decbe he dynamc chaacec o MR dampe ch a hyee. Accodng o h model, he conol podced by an MR dampe c Q z z A Q # Q z loo loo dplacemen Κ dplacemen µ d Κ d µ g. (. Conol eecvene (K and ecency (µ o he copled adacen ce. (a -oey ce Κ and µ dplacemen Κ dplacemen µ d Κ d µ Κ and µ (b -oey ce n n Q z z (a (b

4 Sochac Opmal Conol o Scal Syem he Open Aomaon and Conol Syem Jonal, 8, Volme 7 he coecen c and α can be epaaed no wo pa, epecvely. he pa c p and α p ae conan, and he econd pa ae popoonal o he appled volage. ha c c + c p + p (a,b Sbng em-acve conol ( and ( no yem (, conveng no eqvalen non-hyeec yem and ncopoang he pave conol componen n he yem yeld Q P V P c + kwk ( c + Q P Q (a (b By ng he ochac aveagng mehod and ochac dynamcal pogammng, he opmal em-acve conol volage obaned a c ( Q R (3 I he gh-hand de o Eq. (3 negave, volage e o be zeo. nde cean condon, he gh-hand de o Eq. (3 can be alway non-negave. g. (5 how he conol eecvene and ecency o a ngle-degeeo-eedom yem. K and g. (5. Sem-acve conol eecvene (K and ecency (µ baed on he Boc-Wen model. Sem-Acve Conol Law o MR-LCD MR-LCD a new em-acve conol devce combnng magneo-heologcal ld and ned lqd colmn dampe, whch can be nalled a he op loo o bldng o vbaon conol. A conolled bldng ce wh an MR-LCD nde wnd loadng can be expeed a M X + CX + KX + w E D m D y + ( y + k D y md E Excaon neny X (a (b Eq. (a and (b ae epecvely he bldng and MR-LCD eqaon o moon. D he neacon oce. w ( he wnd excaon wh he Davenpo pecm. he conol oce podced by he MR ld, whch can be epaaed no pave pa p and em-acve pa. Makng he modal anomaon o (a, combnng wh (b and lneazng acally yeld Z AZ + ( + (5 Accodng o he ochac dynamcal pogammng pncple, he opmal em-acve conol obaned #, R gn( y, B PZ gn( y $ < (6a (6b By bng he em-acve conol no Eq. ( and ng he acal lneazaon mehod, he epone o he conolled yem can be evalaed. Nmecal el ae hown n g. (6. egh (m g. (6. Conol eecvene o he 5-oey bldng wh MR-LCD. DEVELOPMENS O SOCASIC OPIMAL CONROL MEOD [5-7] In ac, an opmal eedback conol aeced by acao aaon, paal ae obevaon and conol me delay. he conol mehod need o ake hee aco no accon. Sochac Opmal Conol Law o Acao Saaon he ymmec bonded conol conan can be expeed a b (7 nde h condon, he opmal bonded conol deved om (7 $, # b gn(, R Dplacemen Ineoy d P V < b % b Acceleaon Pecenage edcon K (% (8a (8b Nmecal el how ha conol (8 ha hghe ecency han he coepondng bang-bang conol (g. 7 and can aenae he chaeng o bang-bang conol (g. 8.

5 8 he Open Aomaon and Conol Syem Jonal, 8, Volme Z.G. Yng Conol ecency g. (7. Conol ecency o popoed ochac opmal bonded conol and bang-bang conol. PSD g. (8. Powe pecal deny (PSD o acceleaon epone o bang-bang conol (BBC and popoed ochac opmal bonded conol (OBC. Sochac Opmal Conol Law o Paally Obevable Syem Acal conol yem ae paally obevable de o nevable ae meang and emang eo. he ochac opmal conol poblem o paally obevable yem can be conveed no ha o compleely obevable yem by he epaaon heoem. he ochac opmal conol poblem o a paally obevable yem can be expeed a dx [ AX + G( X ]d + Bd + CdB( d 3 Y [ DX + E( X ]d + d + C db + C db ( (9a (9b J E{ L(X,d + (X( } (9c Eq. (9a, (9b and (9c ae he yem eqaon o moon, obevaon eqaon and conol peomance ndex, epecvely. I Eq. (9a and (9b ae lnea, he paally obevable conol poblem (9 can be conveed no he ollowng compleely obevable conol poblem: dxˆ ( AXˆ + Bd + ( R D C + C d ˆ C C B( (3a J E{ L( ˆX, d + ( ˆX( } (3b Popoed opmal conol Bang-bang conol Conol oce bond (z OBC (black BBC (gey whee Xˆ he opmal emaon o X. Emaon eo ~ X X Xˆ he Gaan poce wh covaance R C ayng R + C ARC + RC A ( RC D + C C C ( DRC + C C CC (3 he opmal conol o compleely obevable yem (3 can be deemned by ng he mehod above. I Eq. (9a and (9b ae nonlnea, epaae conol no wo pa,.e., + and elec ch ha G+B and E+ ae poenal ncon, and hen conve he paally obevable conol poblem (9 no he ollowng compleely obevable conol poblem: dxˆ ( AXˆ + B d + ( R D C + C C C d V (3a dv dy DXˆd (3b J E{ L ( ˆX, d + ( ˆX( } (3c whee V I he nnovaon poce. he opmal conol o compleely obevable yem (3 can be deemned mlaly. Nmecal el ae hown n g. (9. g. (9. Conol eecvene o he paally obevable nonlnea yem. Sochac Opmal me-delayed Conol Law a he opmal me-delay conol poblem can be expeed Q P # # P $ $ c + k # Q # P J lm k + ( Q, P (33a (33b # L(Q, P, d (33c whee τ he me delay n conol. Baed on he ochac aveagng mehod o me-delayed yem, he ae anomaon obaned a Q Eecvene (K /R /R Rao o excaon neny o obevaon noe (e/e P Q ( # Q co # n /R (3a

6 Sochac Opmal Conol o Scal Syem he Open Aomaon and Conol Syem Jonal, 8, Volme 9 P P ( # Q n + P co and he Iô ochac deenal eqaon (3b d [m (+ < P >]d + k (db k ( (35a he coepondng peomance ndex o nne meneval egodc conol J lm # L(, d (35b Eq. (35 a non-me-delay opmal conol poblem and opmal conol can be deemned by ng he mehod above. he nvee anomaon o (3 P Q Q Q co + n P P # Q n + P co ng (36 yeld he opmal me-delayed conol # # R ( (36a (36b (37 whee ( τ epeen he me-delayed ae ncon. hee conol mehod have epecvely aken no accon he eec o acao aaon, paal ae obevaon and conol me delay, and howeve, need o mpove he. ACKNOWLEDGEMEN h dy wa ppoed by he Zheang Povncal Naal Scence ondaon o Chna nde gan no. Y6787. REERENCES [] G.W. one, L.A. Begman,.K. Caghey, A.G. Chaako, R.O. Cla, S.. Ma, R.E. Skelon,.. Soong, B.. Spence and J..P Yao, Scal conol: pa, peen, and e, ASCE Inenaonal Jonal o Engneeng Scence, 3(9: , 997. [] W.. lemng and R.W. Rhel, Deemnc and Sochac Opmal Conol, Beln: Spnge-Velag, 975. [3] R.. Sengel, Sochac Opmal Conol: heoy and applcaon, New Yok: John Wley Son, 986. [] M.. Dmenbeg, D.V. Iochenko and A.S. Ba, Opmal bonded conol o eady-ae andom vbaon, Pobablc Engneeng Mechanc, 5: 38-6,. [5].. Soong, Acve Scal Conol: heoy and pacce, New Yok: John Wley Son, 99. [6] J.N. Yang, A.K. Agawal and S. Chen, Opmal polynomal conol o emcally exced non-lnea and hyeec ce, Eahqake Engneeng and Scal Dynamc, 5: -3, 996. [7] L.G. Cepo and J.Q. Sn, Sochac Opmal Conol o Nonlnea Syem va Sho-me Gaan appoxmaon and Cell Mappng, Nonlnea Dynamc, 8: 33-,. [8] W.Q. Zh and Z.G. Yng, Opmal Nonlnea eedback Conol o Qa-amlonan Syem, Scence n Chna, See A, : 3-9, 999. [9] W.Q. Zh, Z.G. Yng and.. Soong, An Opmal Nonlnea eedback Conol Saegy o Randomly Exced Scal Syem, Nonlnea Dynamc, : 3-5,. [] W.Q. Zh, Z.G. Yng, Y.Q. N and J.M. Ko, Opmal nonlnea ochac conol o hyeec yem, ASCE Inenaonal Jonal o Engneeng Scence, 6: 7-3,. [] Z.G. Yng, Y.Q. N and J.M. Ko, Non-lnea Sochac Opmal Conol o Copled-ce Syem o Ml-degee-oeedom, Jonal o Sond and Vbaon, 7: 83-6,. [] Z.G. Yng, W.Q. Zh and.. Soong, A Sochac Opmal em-acve Conol Saegy o ER/MR Dampe, Jonal o Sond and Vbaon, 59: 5-6, 3. [3] Z.G. Yng, Y.Q. N and J.M. Ko, A New Sochac Opmal Conol Saegy o yeec MR Dampe, Aca Mechancal Solda Snca, 7: 3-9,. [] Z.G. Yng, Y.Q. N and J.M. Ko, Sem-acve Opmal Conol o Lneazed Syem wh Ml-degee o eedom and Applcaon, Jonal o Sond and Vbaon, 79: , 5. [5] Z.G. Yng and W.Q. Zh, A Sochacally Aveaged Opmal Conol Saegy o Qa amlonan Syem wh Acao Saaon, Aomaca, : 577-8, 6. [6] W.Q. Zh and Z.G. Yng, Nonlnea Sochac Opmal Conol o Paally Obevable Lnea Sce, Scal Engneeng, : 333-,. [7] Z.G. Yng and W.Q. Zh, A Sochac Opmal Conol Saegy o Paally Obevable Nonlnea Qa-amlonan Syem, Jonal o Sond and Vbaon, 3: 8-96, 8. [8] S.J. Dyke, B.. Spence, M.K. San and J.D.Calon, Modelng and Conol o Magneoheologcal Dampe o Semc Repone Redcon, Sma Maeal and Sce, 5: , 996. [9] M.D. Syman and M.C. Conannon, Sem-acve Conol Syem o Semc Poecon o Sce: a Sae-o-he-a Revew, Scal Engneeng, : 69-87, 999. Receved: Jne 9, 8 Reved: Jne, 8 Acceped: Jne 6, 8 Z.G. Yng; Lcenee Benham Open. h an open acce acle lcened nde he em o he Ceave Common Abon Non-Commecal Lcene (hp://ceavecommon.og/lcene/by-nc/3./ whch pem neced, non-commecal e, dbon and epodcon n any medm, povded he wok popely ced.