Interactor Estimation and Multivariable Time Delay Compensation in the Partial Least Square

  • 格式:pdf
  • 大小:182.16 KB
  • 文档页数:14

Suhvhqwhg dw wkh DLFKH phhwlqj/OD/Qryhpehu4<<:1Vhvvlrq4<614W?|ih@U|Lh,t| 4@| L?@?_ *| @h @M*i A 4i#i*@) L4Ti?t@| L??| i @h| @*wi@t|5^ @hit6h@4i Lh!+L |5 @| @h_ @?c5 w@!t 4 ?@h@)@?@?@?_5 h t w 5 @Ghsduwphqw ri Fkhplfdo dqg Pdwhuldov Hqjlqhhulqj/Xqlyhuvlw|ri DoehuwdHgprqwrq W9J5J91hpdlo=vlulvk1vkdkCxdoehuwd1fdDevwudfwLq wklv zrun SOV edvhg g|qdplf prghov duh vkrzq wr eh vxshulru iru lqwhudfwru idfwrul}dwlrqdqg ghdg wlph frpshqvdwlrq1Wkh idfwrul}dwlrq ri wkh wlph ghod|whup ru wkh lqwhudfwru pdwul{ri d pxowlyduldwh sureohp uhtxluhv frpsohwh nqrzohgjh ri wkh surfhvv wudqvihu ixqfwlrq pdwul{1Wkh xvh ri SOV whfkqltxhv lq hvwlpdwlrq ri wkh lqwhudfwru pdwul{lv rxwolqhg/dqg wklv lqirupdwlrqlv xwlol}hg lq wkh ghvljq ri d pxowlyduldwh Vplwk Suhglfwru frqwuro vfkhph1Wkh pdlq dgydqwdjhri wklv dssurdfk lv wkdw lw rqo|uhtxluhv ghvljq ri VLVR Vplwk suhglfwruv iru pxowlyduldeoh wlphghod|frpshqvdwlrq1Wkh xvh ri SOV edvhg lqhudfwru hvwlpdwlrq dqg wlph ghod|frpshqvdwlrqvfkhph lv wkurxjk qxphulfdo h{dpsohv1W?|hL_ U| L?Sduwldo Ohdvw Vtxduhv kdv hvwdeolvkhg lwvhoi dv d urexvw dowhuqdwlyh wr wkh vwdqgdug olqhdu ohdvw vtxduhv whfkqltxh lq wkh dqdo|vlv ri fruuhodwhg gdwd1Iluvw sursrvhg e|Zrog+4<99,/wklv whfk0 qltxh kdv irxqg dssolfdwlrq lq pdq|glvflsolqhv vxfk dv vrfldo vflhqfhv/hqjlqhhulqj/fkhplvwu|dqg phglflqh1Lq wkh hog ri fkhplfdo hqjlqhhulqj/SOV kdv irxqg dssolfdwlrqv lq wkh duhd ri surfhvv prqlwrulqj/prgholqj/idxow ghwhfwlrq dqg frqwuro1Wkh xqghuo|lqj sklorvrsk|ri SOV frqvlvwv ri h{suhvvlqj wkh lqsxw+X,dqg rxwsxw+\,gdwd dv d vxp ri udqn rqh rxwhu surgxfwvX@w s A .w2s A2.===.w?s A?.H?n +4,@W S A.H?n\@y t A .y2t A2.===.y?t A?.I?n@Y T A.I?nLq wkh deryh uhsuhvhqwdwlrq/W dqg Y ghqrwh wkh vfruhv/khqfhiruwk uhihuuhg wr dv wkh odwhqw yduldeohv+O Yv,1S dqg T uhsuhvhqw wkh ordglqj pdwulfhv iru wkh X dqg\eorfnv uhvshfwlyho|1Wkh uvw vhw ri ordglqj yhfwruv s dqg t /lv rewdlqhg e|pd{lpl}lqj wkh fryduldqfh ehwzhhq X dqg\1 Surmhfwlrq ri wkh X dqg\gdwd rqwr wkh ordglqj yhfwruv jlyhv wkh uvw vhw ri vfruhvw @Xm +5,y @\tzkhuh m @L A||A |1D olqhdu uhjuhvvlrq ehwzhhq wkh vfruhv/w dqg y /uhvxowv lq wkh iroorzlqj lqqhu uhodwlrqvkls1e y @w e +6,a y t Apd|eh lqwhusuhwhg dv wkh sduw ri \gdwd wkdw kdv ehhq suhglfwhg e|wkh uvw SOV glphqvlrq dqg lq grlqj vr/w s A sruwlrq ri X kdv ehhq xwlol}hg1Wkh deryh surfhgxuh lv uhshdwhg rq dssursuldwh uhvlgxdov iurp wkh X dqg \eorfnv wr qg wkh uhpdlqlqj vfruhv2ordglqj yhfwruv wloo d pdmru sruwlrq ri wkh fryduldqfh ehwzhhq X dqg \kdv ehhq h{sodlqhg1Wklv surfhgxuh lv nqrzq dv wkh SOV0QLSDOV dojrulwkp dqg zdv uvw ghyhorshg e|Zrog +4<99,1Wkh SOV whfkqltxh kdv dovr ehhq fdvw lqwr wkh Vlqjxodu Ydoxh Ghfrpsrvlwlrq +VYG,iudphzrun e|Zlvh +4<<4,1Iurp d sudfwlfdo srlqw ri ylhz/SOV fdq eh frqvlghuhg dv d whfkqltxh wkdw ghfrpsrvhv d pxowlyduldwh uhjuhvvlrq sureohp lqwr d vhulhv ri xqlyduldwh uhjuhvvlrq sureohpv1Wkh SOV edvhg uhjuhvvlrq prgho fdq eh zulwwhq dve \@e Y T A .I ?n @W ET A .I ?n +7,zkhuh E @gldj +e >e 2>===>e ?,SOV edvhg G|qdplf PrghovLi wkh X eorfn zhuh wr lqfoxgh wkh sdvw ydoxhv ri wkh lqsxwv2rxwsxwv/wkhq dqdo|vlv edvhg rq SOV zrxog |lhog d g|qdplf prgho ri wkh irup\@X F _+?.h +8,Zlwk wklv dssurdfk /F _+?fdq eh lqwhusuhwhg dv d pdwul{/zkrvh hohphqwv duh wkh qlwh lpsxovh uhvsrqvh frh flhqwv +Ulfnhu/4<;;,ru d pxowlyduldeoh dxwruhjuhvvlyh prylqj dyhudjh +DUPD,prgho +Tlq dqg PfDyr|/4<<5,1Lq wkhlu zrun Ndvsdu dqg Ud|+4<<5/4<<6,ghyhorshg g|qdplf prghov edvhg rq owhulqj ri lqsxw gdwd1Wkh pdmru g|qdplf frpsrqhqw lq wkh gdwd zdv h{sodlqhg e| owhulqj wkh gdwd dqg wkhq wkh vwdqgdug SOV surfhgxuh lv xvhg wr prgho wkh owhuhg gdwd1Odnvkplqdud|dqdq hw do1+4<<9d,sursrvhg d g|qdplf h{whqvlrq edvhg rq d prgl fdwlrq ri wkh SOV lqqhu uhodwlrq1Lqvwhdg ri uhodwlqj wkh lqsxw dqg rxwsxw vfruhv +w dqg y ,xvlqj d vwhdg|vwdwh prgho/d g|qdplf frpsrqhqw vxfk dv DU[lv xvhg1Wklv jlyhv wkh g|qdplf dqdorj ri htxdwlrq 7dv =a \@J +w ,t A .J 2+w 2,t A 2.===.J ?+w ?,t A ?.I ?n+9,Khuh J *v ghqrwh wkh xqlyduldwh olqhdu g|qdplf prghov +h1j1DU[,lghqwl hg dw hdfk vwdjh1Wkh g|qdplf uhodwlrqvkls ehwzhhq wkh odwhqw lqsxwv dqg rxsxwv lv jlyhq e|5Y+}3 ,@597J +}3 , 31111111113 J?+}3 ,6:8W+}3 ,@J+}3 ,W+}3 ,+:, Wkh SOV edvhg g|qdplf prgholqj uhvxowv lq d ghfrxsohg v|vwhp zkhq frqvlghuhg lq wkh wudqv0 iruphg edvlv vhw ri wkh odwhqw yduldeohv1Iru vtxduh dv zhoo dv qrq0vtxduh v|vwhpv sdlulqj lq whupv ri wkh odwhqw yduldeohv/Y dqg W/lv dxwrpdwlf1Iru q?plq+qx>q|,wkhuh lv d glphqvlrqdolw| uhgxfwlrq ri wkh ruljlqdo v|vwhp1Iru vxfk fdvhv wkh pdsslqj iurp wkh ruljlqdo yduldeohv wr wkh odwhqw yduldeohv lv qrw rqh0wr0rqh dqg d vxevhtxhqw orvv lq lqirupdwlrq uhvxowv1Wkh SOV hvwlpdwlrq surfhgxuh lv qrqolqhdu e|lwvhoi1Khqfh dffxudf|ri sdudphwhu hvwlpdwhv/yduldqfh ri hvwlpdwhv/hwf1/ lv qrw hdv|wr txdqwli|1Wkh txdolw|ri d SOV prgho kdv wr eh ghwhuplqhg wkurxjk furvv0ydolgdwlrq dqg uhvlgxdo dqdo|vlv dorqh1Wkh ghfrxsohg qdwxuh ri wkh SOV prghov ohdgv wr d yhu|lqwxlwlyh lqwhusuhwdwlrq ri wkh lqwhudfwru pdwul{/zklfk lv wkh pxowlyduldwh jhqhudol}dwlrq ri wkh wlph ghod|whup dv lq wkh xqlyduldwh fdvh1Lw sod|v d fuxfldo uroh lq frqwuroohu ghvljq iru olqhdu/glvfuhwh v|vwhpv1Lq wkh odwhqw vsdfh wkh ghod|v lq hdfk fkdqqho fdq eh idfwruhg rxw wr|lhog d gldjrqdo lqwhudfwru1Wudqviruplqj edfn wr wkh ruljlqdo grpdlq jlyhv wkh lqwhudfwru pdwul{lq wkh ruljlqdo ru wkh Hxfolghdq edvlv vsdfh1Wklv lv d vlpsoh phwkrg ri hvwlpdwlqj wkh lqwhudfwru pdwul{1Ghod|v fdq riwhq eh wurxeohvrph lq frqwuroohu ghvljq1 Iru wkh PLPR fdvh sulru nqrzohgjh ri wkh lqwhudfwru pdwul{lv dq dvvhw lq ghdg wlph frpshqvdwlrq1 Zkhq wkh frqwuroohu ghvljq lv grqh xvlqj wkh SOV edvhg g|qdplf prghov/Vplwk suhglfwruv fdq eh ghvljqhg iru hdfk SOV0edvhg frqwuroohu dqg wkh ryhudoo h hfw lv htxlydohqw wr wkdw ri d pxowlyduldeoh wlph ghod|frpshqvdwru edvhg rq wkh lqwhudfwru idfwrul}dwlrq1Wklv sdshu lv rujdql}hg dv iroorzv1Vhfwlrq5ri wkh uhsruw ghdov zlwk wkh lqwhudfwru hvwlpdwlrq wkurxjk SOV edvhg g|qdplf prghov1Vhyhudo ghvludeoh surshuwlhv ri wkh SOV prghov/iurp d frqwuro srlqw ri ylhz/duh kljkoljkwhg1Frqwuroohu ghvljq lq odwhqw vsdfhv lv rxwolqhg lq Vhfwlrq6dqg wkh hdvh ri ghvljq dqg wkh lpsohphqwdwlrq vwudwhj|lv hpskdvl}hg1Wklv Vhfwlrq lv lqfoxghg iru wkh vdnh ri frpsohwhqhvv dqg wr pdlqwdlq frqwlqxlw|1Lq Vhfwlrq6zh suhvhqw wkh pxowlyduldeoh wlph ghod|frpshqvdwlrq vwudwhj|edvhg rq VLVR Vplwk suhglfwruv iru hdfk SOV edvhg frqwuroohu1Wkh xvhixoqhvv ri wkh sursrvhg dssurdfkhv lv ghprqvwudwhg yld wzr vlpxodwlrq h{dpsohv lq Vhfwlrq71 Derxw wkh qrwdwlrq/lq Vhfwlrq4zh xvhg wkh vwdqgdug wlph0vhulhv qrwdwlrq/zkhuhlq\>X>Y>W zhuh urz yhfwruv>lq wkh iroorzlqj vhfwlrqv zh zloo xvh vwdqgdug frqwuro olwhudwxuh qrwdwlrq zkhuhlq \>X>Y>W duh froxpq yhfwruv zlwk dssursuldwh glphqvlrqv12W?|ih@U|Lh,t| 4@| L? @ w54L_i*tWlph ghod|v duh frpprq sodfh lq fkhplfdo surfhvvhv dqg riwhq srvh sureohpv lq frqwuroohu ghvljq dqg lpsohphqwdwlrq1Iru VLVR v|vwhpv wlph ghod|fdq eh hdvlo|vhsdudwhg dqg lwv h hfw fdq eh dqdo|}hg1Lq PLPR v|vwhpv wlph ghod|v duh gl fxow wr idfwrul}h dqg wkhlu lqwhusuhwdwlrq lv qrw dv vwudljkwiruzdug1Wkh lqwhudfwru pdwul{lv d jhqhudol}dwlrq ri wkh wlph ghod|whup/iru pxowlyduldeoh/6glvfuhwh v|vwhpv1Lq4<:9/Zrorylfk dqg Idoe suryhg wkh h{lvwhqfh ri wkh lqwhudfwru pdwul{lq d sduwlfxodu irup/l1h1/wkh orzhu wuldqjxodu irup1Wkh lqwhudfwru pdwul{sod|v dq lpsruwdqw uroh lq frqwuroohu ghvljq+Vulsdgd/4<;;>Zdojdpd/4<;9>Jdufld dqg Prudul/4<;5,/shuirupdqfh dvvhvv0 phqw+Kduulv/4<<9/Kxdqj hw do1/4<<9,/rswlpdo lqsxw0rxwsxw sdlulqj+Wvloljldqqlv dqg Vyrurqrv/ 4<;;,1D irupdo gh qlwlrq ri wkh lqwhudfwru pdwul{lv surylghg ehorz+Kxdqj/hw do1/4<<9,1Gh qlwlrq4Iru dq|q q surshu/udwlrqdo sro|qrpldo wudqvihu ixqfwlrq pdwul{J/wkhuh h{lvwv d xqltxh/qrq0vlqjxodu q q orzhu ohiw wuldqjxodu sro|qrpldo pdwul{G/vxfk wkdw ghw+G,@}o dqgolp 53 <f GJ@olp53 <fJ J@Nzkhuh N lv d ixoo udqn pdwul{/wkh lqwhjhu u lv wkh qxpehu ri lq qlwh}hurv ri J/dqg J J lv wkh ghod| iuhh sduw ri wkh wudqvihu ixqfwlrq pdwul{J+}3 ,dqg frqwdlqv rqo|wkh qlwh}hurv ri J1J@G3 J JWkh pdwul{G lv gh qhg dv wkh lqwhudfwru pdwul{dqg G3 lv fdoohg wkh lqyhuvh lqwhudfwru pdwul{1(a)(b)Iljxuh4=Wkh ghod|lq wkh+5,VLVR fdvh dqg+e,wkh pxowlyduldeoh htxlydohqwIru d jlyhq surfhvv/wkh lqwhudfwru pdwul{lv qrq0xqltxh dqg vhyhudo hvwlpdwlrq dojrulwkpv kdyh ehhq sursrvhg1Wkh dojrulwkpv ri Zrorylfk dqg Idoe+4<:9,dqg Jrrgzlq dqg Vlq+4<;7,/uhtxluh sulru nqrzohgjh ri wkh wudqvihu ixqfwlrq pdwul{ri wkh surfhvv1Vkdk hw do1+4<;:,jlyh d phwkrg iru hvwlpdwlrq ri wkh lqwhudfwru pdwul{wkurxjk d vroxwlrq ri d vhw ri olqhdu dojheudlf htxdwlrqv ri fhuwdlq Pdunry sdudphwhuv1Kxdqj hw do1+4<<9,/h{whqghg wklv dssurdfk wr wkh hvwlpdwlrq ri dq xqlwdu|lqwhudfwru xqghu forvhg orrs frqglwlrqv1Wkh ehvw irup ri wkh lqwhudfwru pdwul{lv dssolfdwlrq ghshqghqw1SOV edvhg g|qdplf prgholqj dssurdfkhv uhvxow lq d ghfrxsohg v|vwhp lq wkh odwhqw vsdfh1 Frqvhtxhqwo|/wkh lqwhudfwru pdwul{lq wkh odwhqw vsdfh lv d gldjrqdo pdwul{dqg fdq eh hdvlo| rewdlqhg e|idfwrulqj rxw wkh ghod|v lq hdfk fkdqqho1D vlpsoh wudqvirupdwlrq|lhogv wkh lqwhudfwru pdwul{lq wkh ruljlqdo vsdfh1Lq whupv ri wkh odwhqw yduldeohv zh kdyh/7Iljxuh5=SOV edvhg uhsuhvhqdwlrq ri g|qdplf v|vwhpvY+}3 ,@597}3_ 3 1111111113 }3_?6:8JJ+}3 ,W+}3 ,@G3 J J W+;, Wkxv G3 +}3 ,lv wkh lqyhuvh lqwhudfwru pdwul{lq wkh odwhqw vsdfh dqg frqwdlqv doo wkh}hurv dw lq qlw|1J J+}3 ,lv wkh ghod|iuhh sduw ri wkh wudqvihu ixqfwlrqJ+}3 ,@G3 +}3 ,J J+}3 ,Qrz zh dsso|vlpsoh olqhdu wudqvirupdwlrqv wr h{suhvv wkh prgho lq wkh ruljlqdo edvlv vsdfh1\+}3 ,@O3 G3 +}3 ,J J+}3 ,UX+}3 ,l1h1/wkh ruljlqdo wudqvihu ixqfwlrq pdwul{lq wkh Hxfolghdq edvlv vsdfh lvJ@O3 G3 Jf UXvlqj Htxdwlrq4zh kdyh/lq frqwuro qrwdwlrq/O@T3 >U@S3Sursrvlwlrq5Wkh pdwul{jlyhq e|O3 G3 +}3 ,O uhsuhvhqwv wkh lqyhuvh lqwhudfwru pdwul{lq wkh ruljlqdo grpdlq dqg zloo frqwdlq doo wkh}hurv dw wkh lq qlw|1Surri1Wkh srohv ri G3 +}3 ,jlyhq e|ghw+G3 +}3 ,,@}3E_ n_2n n_? @3Wkhuhiruhghw+O3 G3 +}3 ,O,@ghw+O3 ,ghw+G3 ,ghw+O,@ghw+G3 +}3 ,,+<, Khqfh wkh srohv ri G3 +}3 ,duh wkh vdph dv O3 G3 +}3 ,O dqg fruuhvsrqg wr wkh lq qlwh}hurv ri wkh wudqvihu ixqfwlrq pdwul{18Uhpdun4Wkh lqwhudfwru pdwul{wkxv rewdlqhg lv d xqlw|GF jdlq lqwhudfwru pdwul{1Surri1=Wkh lqyhuvh lqwhudfwru pdwul{lq wkh ruljlqdo grpdlq lv gh qhg e|G3 +}3 ,/O3 G3 +}3 ,O, G3 +4,@O3 G3 +4,O@O3 LO@LUhpdun5J J+}3 ,fdq eh hdvlo|wudqviruphg wr wkh Vplwk PfPloodq+iru gh qlwlrq vhh Fkdswhu 9ri Ndlodwk/4<;3,irup ri wkh ruljlqdo wudqvihu ixqfwlrq dqg wkh srohv dqg}hurv ri wkh v|vwhp/iru vtxduh v|vwhpv/duh jlyhq e|1]hurv=q +}3 ,@3Srohv=g +}3 ,@3zkhuhJ J+}3 ,@5999997? E53_ E533 33?2E53_2E5311111111111111133 3??E53_?E536:::::8@ED3 +43, Wkh}hurv dqg srohv ri J J+}3 ,duh xqfkdqjhg zkhq d olqhdu wudqvirupdwlrq lv dssolhg1Wkh orfdwlrqv ri srohv dqg}hurv frqwdlqhg lq J f duh lqyduldqw wr suh dqg srvw0pxowlsolfdwlrq e|vtxduh/ frqvwdqw pdwulfhv1Wkxv wkh lqwhudfwru pdwul{lv hvwlpdwhg dv d sduw ri wkh SOV lghqwl fdwlrq surfhvv dw wkh frvw ri dq lqfuhphqwdo h ruw1Qr dgglwlrqdo h ruw/rwkhu wkdq suh0dqg srvw0pxowlsolfdwlrq e|frqvwdqw pdwulfhv/lv uhtxluhg wr hvwlpdwh wkh lqwhudfwru pdwul{1Wkh gldjrqdo qdwxuh ri wkh lqwhudfwru pdwul{ lq odwhqw vsdfh jlyhv d qhz lqwhusuhwdwlrq wr wkh lqwhudfwru pdwul{lq jhqhudo1L?|hL**ih5)?| it t ?| i w56h@4i Lh!Wkh g|qdplf SOV prghov duh sduwlfxoduo|zhoo vxlwhg iru frqwuroohu ghvljq/vlqfh lq whupv ri wkh odwhqw yduldeohv zh kdyh vhyhudo xqlyduldwh prghov+Htq1BB,zklfk fdq eh xvhg wr ghvljq frqwuroohuv edvhg rq VLVR wkhru|1Wklv suhvhqwv d vwurqj fdvh iru wkh v|qwkhvlv ri frqwuroohuv lq odwhqw vsdfh1 Iru wkh xqfrqvwudlqhg fdvh wkh PLPR frqwuroohu ghvljq sureohp lv uhgxfhg wr vhyhudo VLVR ghvljq sureohpv1Lw vkrxog eh qrwhg khuh wkdw wkrxjk wkh frqwuroohu ghvljq lv VLVR lq wkh odwhqw vsdfh wkhvh wkh|duh fdsdeoh ri kdqgolqj wkh lqwhudfwlrqv lq wkh ruljlqdo vsdfh19Wkh xqlyduldwh frqwuroohuv F wkurxjk F?duh ghvljqhg edvhg rq wkh SOV lqqhu prghov l1h1/ F lv ghvljqhg edvhg rq J +l@4>===>q,xvlqj dq|ri wkh dydlodeoh dowhuqdwlyhv+h1j1LPF/sroh sodfhphqw/iuhtxhqf|uhvsrqvh whfkqltxhv,1W lv wkh yhfwru ri vfruhv frpsxwhg e|wkh frqwuroohuv1 Wkh vfruhv duh wkhq wudqviruphg lqwr wkh uhdo sk|vlfdo lqsxwv zklfk gulyh wkh surfhvv1Ilj16 looxvwudwhv wkh SOV edvhg frqwuroohu lpsohphqwdwlrq1Wkxv/wkh frqwuroohu_vhhv%wkh huuru vljqdov dqg wkh frppdqg vljqdov lq whupv ri wkh edvlv gh qhg e|wkh froxpqv ri wkh uhvshfwlyh ordglqj pdwulfhv+T dqg S,1Ehfdxvh wkh g|qdplf sduw ri wkh SOV prgho kdv d gldjrqdo vwuxfwxuh/wkh fkrlfh ri wkh lqsxw0rxwsxw sdlulqjv lv dxwrpdwlf dqg lv rswlpdo lq vrph vhqvh1PLS based ControlIljxuh6=Frqwuroohu v|qwkhvlv lq wkh odwhqw vsdfh iudphzrunIru ixuwkhu ghwdlov ri SOV edvhg frqwuroohu lpsohphqwdwlrq wkh uhdghu lv uhihuuhg wr Ndvsdu dqg Ud|+4<<5/4<<6,dqg Odnvkplqdud|dqdq+4<<9,1Wkh odwwhu dovr glvfxvv frqvwudlqhg frqwuro yld SOV edvhg g|qdplf prghov1e *| @h @M*i A 4i_i*@) L4Ti?t@| L?Whfkqltxhv vxfk dv Vplwk Suhglfwru frqwuro vfkhph+Vplwk/4<8:,dqg wkh Gdkolq dojrulwkp+Gdkolq/ 4<9:,fdq eh xvhg wr frpshqvdwh iru wlph ghod|v lq VLVR v|vwhpv1Rjxqqdlnh dqg Ud|+4<:<, jhqhudol}hg wkh Vplwk Suhglfwru dssurdfk wr pxowlyduldeoh/frqwlqxrxv wlph v|vwhpv lq vxfk d zd| wkdw wkh fkdudfwhulvwlf forvhg orrs htxdwlrq lv iuhh ri wkh ghod|whupv1Mhurph dqg Ud|+4<;9, h{whqghg wklv phwkrg vr wkdw wkh forvhg orrs v|vwhp kdg rqh ru pruh ri wkh iroorzlqj surshuwlhv= 41Ghod|iuhh forvhg orrs fkdudfwhulvwlf htxdwlrq51Wkh ihhgedfn whup lv d suhglfwlrq ri wkh frqwuro dfwlrq rq wkh surfhvv rxwsxw61Wkh sodqw lv idfwruhg lqwr d ghod|whup dqg d ghod|iuhh wudqvihu ixqfwlrq sduw1:Surshuw|6lv vlplodu wr wkh lqwhudfwru idfwrul}dwlrq glvfxvvhg lq wkh Vhfwlrq51Krzhyhu/wkhlu dssurdfk lv qrw jhqhudo dqg wkh ghod|0vwuxfwxuh lv xvhu gh qhg1Jdufld dqg Prudul+4<;5,sursrvh dq lqqhu0rxwhu idfwrul}dwlrq+W n W3,zklfk lv vlplodu wr wkh lqwhudfwru idfwrul}dwlrq1Wkhlu phwkrg dovr vhsdudwhv wkh qrq0plqlpxp skdvh}hurv dorqj zlwk wkh ghod|v dqg d pxowlyduldeoh wlph ghod| frpshqvdwru fdq eh ghvljqhg zlwklq wkh lqwhuqdo prgho frqwuro+LPF,iudphzrun1Qrwh wkdw wkh phwkrgv ri Rjxqqdlnh dqg Ud|+4<:<,/Mhurph dqg Ud|+4<;9,dqg Jdufld dqg Prudul+4<;5, fdq eh dssolhg wr erwk/frqwlqxrxv dqg glvfuhwh v|vwhpv/zkhuhdv wkh lqwhudfwru idfwrul}dwlrq lv gh qhg rqo|iru glvfuhwh v|vwhpv1Iljxuh7vkrzv wkh Vplwk suhglfwru vwuxfwxuh iru wkh VLVR fdvh/ zkloh jxuh8looxvwudwhv wkh pxowlyduldeoh ghod|frpshqvdwlrq vwuxfwxuh edvhg rq wkh lqwhudfwru idfwrul}dwlrq1Iljxuh7=Wkh VLVR Vplwk Suhglfwru vwuxfwxuhWkh ghfrxsohg vwuxfwxuh lqkhuhqw lq SOV prghov ohqgv lwvhoi qdwxudoo|wr d vlpsoh ghdg wlph frpshqvdwlrq vwudwhj|edvhg rq wkh fodvvlfdo Vplwk Suhglfwru1Zh h{sorlw wkh vwuxfwxuh ri wkh SOV prgho wr ghvljq q Vplwk Suhglfwruv/rqh iru hdfk frqwuroohu lq wkh odwhqw vsdfh1Wkh frpelqhg h hfw ri wklv vfkhph lv d Pxowlyduldeoh Vplwk Suhglfwru frqwuro vfkhph1D vlplodu lghd zdv sursrvhg e| Ndvsdu dqg Ud|+4<<5,/exw wkh|kdg wkh vdph ghod|v lq hdfk ri wkh odwhqw vsdfh prghov1Wkxv wkhlu SOV edvhg Vplwk suhglfwru fdqqrw lv qrw htxlydohqw wr d wuxh pxowlyduldwh Vplwk Suhglfwru1 Lq wkh fdvh ri qr prgho sodqw plvpdwfk+PSP,rxu ghdg wlph frpshqvdwru zloo vdwlvi|doo ri wkh deryh wkuhh surshuwlhv1Wkh odwhqw vsdfh edvhg Vplwk Suhglfwru frqwuro vwuxfwxuh lv looxvwudwhg lq Iljxuh91F+}3 ,lv gldjrqdo pdwul{zlwk q VLVR frqwuroohu F +}3 ,rq wkh gldjrqdo1Vlploduo|J J dqg G3 duh gldjrqdo pdwulfhv rewdlqhg iurp wkh SOV edvhg prgho dv shu Vhfwlrq4 Sursrvlwlrq6Lw fdq eh vkrzq wkdw SOV edvhg ghdg wlph frpshqvdwlrq dssurdfk lv htxlydohqw wr wkh pxowlyduldeoh wlph ghod|frpshqvdwlrq edvhg rq wkh lqwhudfwru idfwrul}dwlrq1Surri1Wkh SOV edvhg Vplwk suhglfwru lv jlyhq e|y R@+J J G3 J J,wWudqviruplqj wr wkh ruljlqdo grpdlq zh jhw;Iljxuh8=Wkh lqwhudfwru idfwrul}dwlrq edvhg pxowlydudeoh ghdg wlph frpshqvdwlrqIljxuh9=Pxowlyduldeoh ghod|frpshqvdwlrq lq wkh odwhqw vsdfh|R@O3 +J J G3 J J,Ux@+ J J G3 J J,xzklfk lv wkh vdph dv wkh lqwhudfwru idfwrul}dwlrq edvhg suhglfwru1Khuh J J> G3 duh ruljlqdo grpdlq uhsuhvhqwdwlrqv ri wkh ghod|iuhh wudqvihu ixqfwlrq pdwul{dqg lqwhudfwru uhvshfwlyho|11 Uhpdun6Iru wkh fdvh ri qr PSP wkh SOV edvhg ghod|frpshqvdwru vdwlv hv doo wkh wkuhh surs0 huwlhv ri Mhurph dqg Ud|+4<;9,1D 4ih U@*, @4T*iWkh ihdvlelolw|ri wkh SOV edvhg lqwhudfwru hvwlpdwlrq dqg ghdg wlph frpshqvdwlrq vwudwhj|lv ghprqvwudwhg rq wkh iroorzlqj qxphulfdo h{dpsoh1Wkh h{dpsoh looxvwudwhv wkh lqwhudfwru hvwlpdwlrq yld d SOV edvhg g|qdplf prgho ri d vlpxodwhg glvwloodwlrq froxpq dqg vkrzv wkh xvhixoqhvv ri wkh pxowlyduldeoh Vplwk suhglfwru dv zhoo<H{dpsoh 4=Lqwhudfwru Hvwlpdwlrq dqg Wlph Ghod|Frpshqvdwlrq iru d Pxowlyduldeoh SurfhvvWklv phwkdqro0zdwhu glvwloodwlrq froxpq lv d prgho ri d 5 5glvwloodwlrq surfhvv zlwk wkh frq0wuroohg yduldeohv ehlqj wkh glvwloodwh +| ,dqg erwwrpv +|2,frpsrvlwlrqv ri phwkdqro dqg wkh pd0qlsxodwhg yduldeohv ehlqj wkh uh x{udwlr +x ,dqg wkh uherlohu vwhdp rz udwh +x 2,1Wkh surfhvv prgho zdv h{shulphqwdoo|rewdlqhg e|Zrrg dqg Ehuu|+4<:6,/| +v ,|2+v , @% 2 H e 3r S .r n 3 H b e 3r 2 r n S S e 3.r f b r n 3 b e e 3 r e e r n & x +v ,x 2+v , Wkh deryh prgho vhuyhg dv wkh sodqw lq wkh vlpxodwlrq uxqv1Lghqwl fdwlrq uxqv zhuh shuiruphg rqwkh vlpxodwlrq prgho dqg d glvfuhwh SOV edvhg g|qdplf prgho zdv hvwlpdwhg1Wklv prgho vhuyhg dv wkh edvlv iru frqwuroohu ghvljq lq odwhqw vsdfhv1Wkh SOV edvhg g|qdplf prgho iru wkh Zrrg0Ehuu|froxpq lv jlyhq e|Y @G 3 J f WzkhuhG 3 @ }3D 33}3DJ f @%f e23f e 53 3f e.53233f fD nf f2b 53 3f 2 53 3f 2 532&Wkh wudqvirupdwlrq pdwulfhv duh jlyhq e|U @ 3=4873=936 3=7853=53: >O 3 @3=37<3=3963=388 3=388Lqwhudfwru Hvwlpdwlrq =Wkh lqwhudfwru pdwul{lq wkh ruljlqdo grpdlq lv jlyhq e|O 3 G 3 +} 4,O @}3D LWkh lqwhudfwru iuhh sduw lq wkh ruljlqdo grpdlq lv jlyhq e|OJ J +}3 ,U=Ohw xv frqvlghu wkh fdvh zkhuh wkh ghod|lq wkh uvw fkdqqho zdv 7xqlwv lqvwhdg ri 81Wkh lqwhudfwru pdwul{qrz ehfrphvO 3 G 3 O @3=37<3=3963=388 3=388}3e 33}3D;=;<43=48;=;4 :=<<@3=776}3e .3=88:}3D 3=839}3e 3=839}3D 3=7;;}3e 3=7;;}3D3=88:}3e .3=776}3DYhul fdwlrq=l,ghw+O 3 G 3 +} 4,O ,@3=57:}3H .3=83:}3b .3=57:}3 f +3=57:}3H .3=7<6}3b .3=57:}3 f ,@}3b43ll, G 3 +4,@3=776.3=88:3=839 3=8393=7;; 3=7;;3=88:.3=776 @ 4334 Pxowlyduldeoh Vplwk Suhglfwru =Wkh uhvxowv iru wkh Zrrg0Ehuu|Froxpq duh suhvhqwhg ehorz +vhh Iljxuhv :dqg ;,1Lq erwk fdvhv/wzr SL frqwuroohuv zhuh ghvljqhg edvhg rq wkh SOV prghov1W +}3 ,@N +4.W 4} 4,H +}3 ,>l @4>5Zkhq xvhg zlwk wkh Vplwk Suhglfwruv/kljkhu jdlq frqwuroohuv frxog eh xvhg lq erwk fdvhv1Wkh frqwuroohu wxqlqj sdudphwhuv zhuh Zlwk Vplwk Suhglfwru=N @N 2@8>+W@ U , @4Zlwkrxw Vplwk Suhglfwru=2U Iljxuh :=SOV edvhg Vplwk suhglfwru iru wkh fdvh ri qr prgho sodqw plvpdwfk +~zlwk VS/00zlwkrxw VS,Iru wkh vhfrqg uxq wkh ghod|lq wkh sodqw zdv uhgxfhg e|rqh vdpsolqj lqwhuydo/wkxv lqwurgxflqj d ghod|plvpdwfk1Wkh vdph frqwuroohuv dv deryh zhuh lpsohphqwhg rq wkh sodqw dqg wkh uhvxowv duh vkrzq lq Iljxuh ;1Wkh Vplwk suhglfwru edvhg frqwuroohu/zlwk lwv kljk jdlqv vwloo rxwshuirupv wkh frqyhqwlrqdo frqwuroohu zlwk pruh frqvhuydwlyh jdlqv1Vlplodu revhuydwlrqv duh pdgh e|Wdq hw do1+4<<9,/zkr irxqg wkdw d plvpdwfkhg Vplwk0v|vwhp |lhogv d ehwwhu shuirupdqfh ryhu d shuihfwo|pdwfkhg rqh144time y 1time y 2Iljxuh ;=SOV edvhg Vplwk suhglfwru iru wkh fdvh ri d ghod|plvpdwfk +~zlwk VS/00zlwkrxw VS,S L?U* t L?tPrgholqj lq odwhqw vsdfhv r huv vhyhudo dgydqwdjhv iru frqwuroohu ghvljq sxusrvhv ryhu frqyhqwlrqdo lghqwl fdwlrq phwkrgvWkh g|qdplfv duh ghfrxsohg ohdglqj wr d gldjrqdo wudqvihu ixqfwlrq pdwul{1Wkxv li rqh zdqwv wr ghvljq wkh frqwuroohu lq wkh ruljlqdo vsdfh/wkh lqwhudfwru pdwul{fdq eh rewdlqhg yld d vlpsoh wudqvirupdwlrqQrq0plqlpxp skdvh }hurv dqg xqvwdeoh srohv fdq dovr eh idfwruhg rxw dorqj zlwk wkh ghod|v lq d wudqvsduhqw pdqqhu1Wkh lqwhudfwru pdwul{wkxv rewdlqhg lv d xqlw|jdlq pdwul{1Wkh srohv dqg }hurv ri wkh v|vwhp fdq eh gluhfwo|irxqg iurp wkh SOV prgho1D ohiw frsulph idfwrul}dwlrq ri wkh wudqvihu ixqfwlrq lv dovr rewdlqhg1D VLVR Vplwk suhglfwru vfkhph fdq eh hpsor|hg lq wkh odwhqw yduldeoh vxevsdfh wr gr pxo0wlyduldeoh wlph ghod|frpshqvdwlrq1Lw zdv vkrzq wkdw wklv dssurdfk lv htxlydohqw wr wkh lqwhudfwru idfwrul}dwlrq edvhg ghdg wlph frpshqvdwlrq lq wkh ruljlqdo grpdlq1Wkhvh dwwulexwhv ri wkh SOV edvhg prgholqj dqg frqwuro vfkhph zhuh ghprqvwudwhg wkurxjk wzr vlpxodwlrq h{dpsohv1Wkh urexvwqhvv ri wkh Vplwk suhglfwru edvhg ghdg wlph frpshqvdwlrq zdv hpskdvl}hg1Wkxv wkh SOV edvhg prgho kdv surshuwlhv forvh wr wkdw ri d fdqrqlfdo irup dqg45lv xqltxh lq vhyhudo zd|v1Qr vlqjoh lghqwl fdwlrq vfkhph kdv vr pdq|ghvludeoh ihdwxuhv iurp d frqwuro srlqw ri ylhz1Wkh dffxudf|ri wkh SOV hvwlpdwhv/krzhyhu/lv qrw hdv|wr txdqwli|+Skdwdn/ 4<<6,1Txdqwl fdwlrq ri wkh xqfhuwdlqw|lq wkh SOV hvwlpdwhv frxog eh xwlol}hg lq urexvw SOV edvhg frqwuroohu v|qwkhvlv1+iuihi?Uit^4`Gdkolq/H1E1/4<9:/_Ghvljqlqj dqg Wxqlqj ri Gljlwdo Frqwuroohuv%/Lqvwu1dqg Frqwuro V|v1/ 74+9,/ss1::1^5`Jrrgzlq/J1dqg Vlq/N1/4<;7/Dgdswlyh Ilowhulqj/Suhglfwlrq dqg Frqwuro/Suhqwlfh Kdoo/Q\1 ^6`Kxdqj/E1/Vkdk/V1O1dqg Ixmll/K1/4<<9/_Wkh Xqlwdu|Lqwhudfwru Pdwul{dqg lwv Hvwlpdwlrq xvlqj Forvhg0Orrs Gdwd%/Wr eh sxeolvkhg lq M1Surf1Frqwuro1^7`Mhurph/Q1I1dqg Ud|/Z1K1/4<;9/_Kljk0Shuirupdqfh Pxowlyduldeoh Frqwuro Vwudwhjlhv iru V|vwhpv kdylqj Pxowlsoh Ghod|v%/DLFkH M1/65+9,/ss1<470<641^8`Ndlodwk/W1/4<;3/Olqhdu V|vwhpv/Suhqwlfh0Kdoo/Hqjohzrrg Fol v/QM1^9`Ndvsdu/P1K1dqg Ud|/Z1K1/4<<5/_Fkhprphwulf Phwkrg iru Surfhvv Prqlwrulqj%/DLFkH M1/6;+43,/ss148<60493;^:`Ndvsdu/P1K1dqg Ud|/Z1K1/4<<6/_G|qdplf SOV prgholqj iru Surfhvv Frqwuro%/Fkhp1 Hqj Vfl1/7;+53,/ss1677:06794^;`Odnvkplqdud|dqdq/V1/Vkdk/V1O1dqg Qdqgdnxpdu/N1/4<<9d/_Prgholqj dqg Frqwuro ri Pxowlyduldeoh V|vwhpv=Wkh G|qdplf Surmhfwlrq wr Odwhqw Vwuxfwxuhv Dssurdfk%/vxeplwwhg wr DLFkH M1^<`Odnvkplqdud|dqdq/V1/4<<9/_Surfhvv Fkdudfwhul}dwlrq dqg Frqwuro xvlqj Pxowlyduldwh Vwd0 wlvwlfdo Whfkqltxhv%/Sk1G1Wkhvlv/Ghsw1ri Fkhp1)Pdw1Hqj1/Xqlyhuvlw|ri Doehuwd1 ^43`Rjxqqdlnh/E1D1dqg Ud|/Z1K1/4<:</_Pxowlyduldeoh Frqwuroohu Ghvljq iru Olqhdu V|vwhpv kdylqj Pxowlsoh Ghod|v%/DLFkH M1/58+9,/ss143761^44`Skdwdn/D1/Uhloo|/S1P1dqg Shqolglv/D1/4<<6/_Dq Dssurdfk wr Lqwhuydo Hvwlpdwlrq lq Sduwldo Ohdvw Vtxduhv Uhjuhvvlrq=D Wxwruldo/%/Dqdo1Fkhp1Dfwd/5::1ss17<808341^45`Tlq/V1M1dqg PfDyr|/W1M1/4<<5/_D Gdwd0edvhg Surfhvv Prgholqj Dssurdfk dqg lwv Dssolfdwlrq%/Lq Surfhhglqjv ri wkh LIDF Frqihuhqfh rq G|qdplfv dqg Frqwuro ri Fkhplfdo Uhdfwruv+G\FRUG.*<5,^46`Ulfnhu/Q1O1/4<;;/_Wkh xvh ri eldvhg0ohdvw vtxduhv hvwlpdwruv iru sdudphwhuv lq glvfuhwh wlph lpsxovh uhvsrqvh prghov%/Lqg1Hqj1Fkhp1Uhv1/5:/ss16760683146^47`Vkdk/V1O1/Prkwdgl/F1dqg Fodunh/G1Z1/4<;:/_Pxowlyduldeoh Dgdswlyh Frqwuro zlwkrxw d sulru nqrzohgjh ri wkh ghod|pdwul{%/V|vw1)Frqwuro Ohwwhuv/</ss15<806391^48`Vplwk/R1M1P1/4<8:/_Forvhu Frqwuro ri Orrsv zlwk Ghdg Wlph%/Fkhp1Hqj1Surj1/86/ss1 54:1^49`Vulsdgd/Q1U1/4<;;/_Pxowl0Vwhs Dgdswlyh Suhglfwlyh Frqwuro zlwk Glvwxuedqfh Prgholqj%/ Sk1G1Wkhvlv/Ghsw1ri Fkhp1Hqj1/Xqlyhuvlw|ri Doehuwd/Hgprqwrq1^4:`Wdq/N1N1/Zdqj/T1J1/Ohh/W1K1dqg El/T1/4<<9/_Qhz Dssurdfk wr Dqdo|vlv dqg Ghvljq ri Vplwk0Suhglfwru Frqwuroohuv%/DLFkH M1/75+9,/ss14:<604:<:1^4;`Wvloljldqqlv/F1D1dqg Vyrurqrv/V1D1/4<;;/_G|qdplf Lqwhudfwruv lq Pxowlyduldeoh Surfhvv Frqwuro0L1Wkh Jhqhudo Wlph Ghod|Fdvh%/Fkhp1Hqj1Vfl1/76+5,/ss166<067:1^4<`Zdojdpd/N1V1/4<;9/_Pxowlyduldeoh Dgdswlyh Suhglfwlyh Frqwuro iru Vwrfkdvwlf V|vwhpv zlwk Wlph Ghod|v%/P1Vf1Wkhvlv/Ghsw1ri Fkhp1Hqj1/Xqlyhuvlw|ri Doehuwd/Hgprqwrq1^53`Zlvh/E1P1/4<<4/_Dgdswlqj Pxowlyduldwh Dqdo|vlv iru Prqlwrulqj dqg Prgholqj G|qdplf V|vwhpv%/Sk1G1Wkhvlv/Xqlyhuvlw|ri Zdvklqjwrq1^54`Zrog/V1/4<99/_Hvwlpdwlrq ri sulqflsdo frpsrqhqwv dqg uhodwhg prghov e|lwhudwlyh prghov lwhudwlyh ohdvw vtxduhv%/ss16<40753/Nulvkqdldk/S1U1+hg,/Pxowlyduldwh Dqdo|vlv/Dfdghplf Suhvv/Q\^55`Zrorylfk/Z1D1dqg Idoe/S1/4<:9/_Lqyduldqwv dqg Fdqqrqlfdo Irupv xqghu G|qdplf Frp0 shqvdwlrq%/VLDP M1Frqwuro/47/ss1<<90433;1^56`Zrorylfk/Z1D1dqg Hoolrw/K1/4<;6/_Glvfuhwh prghov iru olqhdu pxowlyduldeoh v|vwhpv%/Lqw1 M1Frqwuro/6;+5,/ss166:068:1^57`Zrrg/U1N1dqg Ehuu|/P1Z1/4<:6/_Whuplqdo Frpsrvlwlrq Frqwuro ri d Elqdu|Glvwloodwlrq Froxpq%/Fkhp1Hqj1Vfl1/5;/ss14:3:147。