NUMERICAL ALGORITHM FOR OPTIMAL MULTI-VARIABLE CONTROL OF AERO ENGINES

Transcription

1 NUMERICL LGORIHM OR OIML MULI-VRILE CONROL O ERO ENGINE ODLansv Vrkn GGKlkov VYrkov Dparmn of oma Conrol sms Ufa a vaon chncal Unvrs KMar r Ufa 45 Rssa Dparmn of omac Conrol an sms Engnrng Unvrs of hffl Mappn r hffl 3JD UK Kwors: aro ngn opmal ml-varabl conrol cross-copl conrol ral-m nmrcal algorhms bsrac hs papr s focs on praccal problms of gas rbn ngn opmal ml-varabl conrol sgn an mplmnaon n algorhm for ral-m rsolon of opmsaon problm s propos for h opmal conrol sgn of h ngn h ampl of h rbo- ngn ml-varabl conrol prsn shows hgh ffcnc of h vlop mho Inrocon Hgh prformanc rqrmns for morn arcraf powr plans rsl n a grar gr of ngra conrol of aro ngns hs rqrs compl conrol ssms whch ar ml-varabl 7 o conrol loops an ml-fnconal Opraon of all conrol loops ms b co-orna an whr possbl copl across all ransns an sasa conons Opraonal rqrmns bcom grar as man ncrass for accrac an qal of ransn prformanc Effcv conrol of powr plans s also connc wh ngraon of ngn conrol wh arcraf conrol hs rqrs nw mhos for analss an sgn of conrol ssms prsn conrol of a gas rbn ngn GE s sall sgn sng frqnc oman mhos of classcal conrol hor Howvr n las ars man rsarch procs hav bn focs on h applcaon of m oman mhos for aro ngn conrol Harfors 997 hs mhos ar sall bas on h varaons of LQ hans 986 Moorhos 994 an H-nfn Garg 993 rrck mhoologs of lnar opmal conrol ro ngn characrscs ffr ovr h ngn fl an can chang rng h ngn s opraon hs maks ncssar o solv h problm of ral-m opmsaon akng no accon nval characrscs of h ngn ch an approach can prov hgh opraonal qals for h arcraf proplson plan or h solon of problms concrnng opmal conrol of arcraf gas rbn ngns whn a mahmacal mol of h plan s obvosl naccra s prfrabl o s approma mhos In ral-m opmal conrol h obcv of opmsaon n h form of ngral fnconals s no fasbl bcas rflcs avrag characrscs of ssm qal Insa local qal crra ar consr for opmsaon g spcfc loss fncons h approma characr of h mahmacal mol of h ngn forcs h sgnr o pass from src opmal analc csons o nmrc ons Morovr h compaonal sp of sng on-boar comprs lms h sarch o opmsaon whn a class of local-opmal conrol ssms h fll rvav of a Lapnov fncon can b s as a local opmal crron whch nsrs asmpoc sabl of h conrol ssm ch conrol along wh garanng sabl also bcoms opmal whn aonal consrans ar mpos o conrol varabls Ml-varabl local-opmal conrol ssms of gas rbn ngns possss h followng avanags: proc nmrcal solons of h opmsaon problm whch ns hr fl of applcaon; h ssms asmpocall convrg o src opmal solons h o no ffr sbsanall from opmal ssms nr small prrbaons; h snhss procr s bas on ssms analss an nabls sgn of ml-fnconal ssms h obcv of h work prsn n hs papr s o vlop an nvsga nmrcal algorhms of mlvarabl GE conrol wh ral-m opmsaon of conrol laws an algorhms along wh acconng for nval characrscs of h ngn roblm formlaon h problm of ml-varabl GE conrol can b formla n rms of h fnon of local opmal conrol ppos ha h conrollr has m npnn conrol acons ops of h conrollr h obcv of h conrol ssm s o manan m conrol laws g mannanc of consan vals of m ngn paramrs

2 h samplng frqnc f of h gal conrol ssm s known h samplng m pro s h followng: / f h ssm of ffrnal qaons scrbs h plan namcs n h followng form: C D whr s h n-mnson sa vcor; s h m- mnson conrol vcor; s h k-mnson obsrvaon op vcor; C an D ar h marcs of h mnsons n n n m k n an k m rspcvl ll lmns of vcors an ar masr b corrsponng snsors h followng consrans ar appl o h frs rvav of h conrol vcor : lm lm lm whr lm an ar movmn sp lms for h acaors h conrol compr sors n s mmor marcs C an D whch pn on h oprang an flgh conons of h ngn h compr opras wh h pro /f ach -h ccl of s opraon crrn vals of lmns of h sa vcor an obsrvaon vcor ar masr an lmns of h conrol vcor ar pass o acaors h vcor z s gvn fnng h program racor h vcor z consss of m lmns of h vcors an whr m s qal o h nmbr of conrol varabls hs mans ha h conrol ssm has o manan m conrol laws smlanosl smpl ampl s mannanc of consan vals of m ngn paramrs Consr a vcor of racor vaon or vcor of rrors : z - z 3 h obcv of conrol s o mak h crrn vcor z rac h man z wh mamm possbl accrac an qal: z z mn 4 whr s h Eclan norm of a vcor long wh lvrng mn lmns of h conrol vcor shol sasf h consrans of Eq In h snhss of ml-varabl GE conrol h ffrnal qaons scrbng h plan ar o b wrn n h form of a ssm of ffrnal mar qaons Dscr qaons ar sall oban from connos ons wll-known mho for gsaon of connos ffrnal qaons consss of sbson of h ffrnaon opraon b a smpl ffrnc hs mho os no prov hgh accrac of ffrnc mar qaons a low qansaon frqncs Hr a possbl mho s consr for obanng ffrnc qaons b gsaon of connos lnar ffrnal qaons of h plan I s a gsaon mho sng h ransonal mar of h plan whch s gnrals o a non-saonar cas an plc pnnc of h obsrvaon vcor on conrol h sorc aa for a scr mol of h plan s h ssm of vcor-mar ffrnal an algbrac qaons rprsn n h normal form: C [ [ [ D n ] k ] m ] whr an ar h sa obsrvaon an conrol vcors rspcvl; an an ar h vcors fnng h man program racor of h plan Marcs C an D hav h mnsons n n n m k n an k m rspcvl hr lmns ar fncons of h ngn sa an amosphrc conons h ssm Eq5 s gvn n phscal co-ornas h solon of h ssm Eq5 wh varabl coffcns can b rprsn n h form: 5 6 whr s h ransonal mar an s h nal m Dgsng m n h ransonal qaon of a vcormar p Eq6 wh h solon of h ssm Eq5 a wo momns of m an can b wrn as: Usng h propr of h ransonal mar an spposng ha h marcs C an D ar consan ovr h samplng nrval on can oban:

3 or hn Eq8 can b rwrn n h followng form: Comparson of Eq an Eq7 gvs 3 h mar s consan ovr h gsaon nrval hnc h mar s ransform o h ponnal mar -: 4 5 Dno ponn of h ransonal mar as follows:! E 6 h conrol vcor changs spws bng consan ovr h samplng pro hrfor h ngral n Eq5 s smplf own o h followng form: 7 whr s h val of conrol a h nrval Dgsng m rsls n 8 whr 3!! E 9 Dgsng h qaon for h obsrvaon h ssm Eq5 can b wrn as follows: D C Combnng Eq8 an Eq gvs scr vcor-mar qaons of h plan Erapola plan moon paramrs ar sma from hr crrn vals: D C C or whr h marcs an can b fn b Eq6 an Eq9 h marcs an ar rmn b D C C 3 h ssm Eq consss of ffrnc vcor-mar qaons wrn n vaons from sa-sa vals rmnng rapola paramrs of h plan from hr crrn vals Consr h ssm Eq for h m momns an an sbrac h scon ssm from h frs on: 4 or n anohr wa: 5 whr 6 orm h vcor z whch scrbs h crrn sa: H H z 7 whr H an H ar vcors of wghng coffcns akng no accon h approma characr of h mahmacal mol of h plan hr vals rflc h lvl of ncran of knowlg concrnng GE namcs an pn on man ngnrng qal crra Consr h Eclan norm as a norm of h vcor n Eq4: [ ] m z z J 8 h prsson for consrans Eq n a scr form s h followng: lm lm 9 or

4 h h lm lm lm lm 3 ach h sp h conrol vcor s chang o prov h mnmm of h obcv fncon Eq8 nr h consrans Eq3 In hs cas opmal conrol rprsns a srs of solons of a qarac programmng problm whn h conrol compr a scr m nsans along wh con of corrsponng conrols hs problm can b solv sng non-lnar programmng mhos applcabl for h conrol compr Dnamc proprs of h plan n a w rang of opraon can b scrb b a s of lnar mols wh varng coffcns hs nabls ngn opmal conrol o b bl as a sqnc of solons of a qarac programmng problm n ral m: m J mn 3 whr h rror vcor ncls lmns of boh vcors an : H H [ ] [ ] wh h followng consrans for conrols: 3 lm 33 3 Ral-m opmsaon algorhm h algorhm propos for h solon of h mahmacal programmng problm Canon 97 n ral-m s bas on analss of GE characrscs as a ml-varabl plan an gnralsaon of rqrmns o gal ml-varabl conrol ssms Consr an algorhm o sarch for h mnmm of h obcv fncon Eq3 h choc of a mho for solon of h problm Eq3 3 an 33 pns on h followng facors: h compaonal powr ncssar for solvng h problm; h rqr accrac of h solon; h man accrac of consrans flflmn; sabl opraon nr shor-m malfncons ransformng Eq an 3 h problm s rprsn n a sanar form: J Q p C whr s sp of chang of h conrol vcor a h h sp of conrol; s a smmrc posv fn m mnson mar; p an ar ranspos colmn vcors h followng solon rprsns h mnmm pon of h obcv fncon Eq34 f h consrans Eq35 ar no consr: C p 36 If h consrans Eq35 ar akn no accon h problm consss n sarchng for h pon of h allow polhron Eq35 lng a h lvl ln Q p C wh h mnmal val of Q: mn Q 37 h sanar problm samn wh h obcv fncon Eq3 wll la o a sarch for h mnmm of h fncon a h bonar of h allow ara hs wll rmn a pon whr rcon of conrol rrors of som paramrs s oban onl b ncrasng conrol rrors n ohrs hrfor n mnmsaon of Eq34 a mchansm shol b vlop o cl possbl ncras n absol val of an componn of h vcor possbl mho for hs nvolvs nrocng no Eq3 aonal wghng coffcns or pnals n h followng form: m J γ mn 38 whr γ ar wghng coffcns calcla sng spcal formla nohr approach consss n vlopng aonal consrans clng ncras n conrol rrors whl sarchng for h mnmm of h obcv fncon Eq3 hs ns h ssm of lmng nqals Eq33 an maks h sarch procr mor complca rasonabl wa of solvng h problm of qarac programmng s h s of a mho acconng for h phscal sns of h obcv fncon Eq3 s s sn from h formla Eq3 h pon lvrng h ncononal mnmm of h obcv fncon s rmn b solvng h ssm of lnar qaons: m 39 h sarch for h mnmm of h obcv fncon Eq3 whn h allow polhron Eq33 sars wh "!# $&%'&-! an hn procs along h bam: λ 4

5 ? owar h pon / nl h mnmm or h bonar of h allow ara h val of λ' whr h bam crosss h bonar of h allow ara s fn as h smalls of λ : λ mn{ λ λ > } 4 whr λ 4 lm If h mnmm of h obcv fncon s ns of h allow ara hn obvosl λ h sp lngh fnng vcor / s slc as λ mn{ λ } 43 h avanag of h sggs mho s ha mnmsaon of a qarac fncon aks onl on raon Calclaon of h fncon gran an nvrs marcs s no ncssar for fnng h ncononal mnmm h sarch for h mnmm of a fncon along a sragh ln conncng h bgnnng of co-ornas wh h pon of h ncononal mnmm provs proporonal rcon of conrol rrors b all op paramrs n as-o-program Gassan algorhm can b s for solvng h ssm Eq39 o fn h co-ornas of h pon of ncononal mnmm gr : racor of sarch for mnmm of obcv fncon gr llsras h sggs mho sng an ampl of mnmsaon of h followng obcv fncon: J W whr W W an h consrans Eq33 gv h allow ara for sarch n h form of a rcangl wh h nal pon X n h cnr h ncononal mnmm of h obcv fncon J #3'45#6&798;:'4-5 < a h nrscon of h sragh lns 5 X W X an 4X W X hs ar shown b hck lns h sarch for W h mnmm procs along h sragh ln conncng h pons an nl h nrscon wh h bonar of h sarch ara pon X mn h fnal pon rmns opmal conrol a h crrn sp h sqnc of calclaons accorng o h consr mho consss of h followng sags: h pon of ncononal mnmm s calcla as a solon of h ssm: m; 45 h paramr λ' s rmn a h nrscon of h bam λ wh h bonar of h allow ara: lm ' λ ; m; 46 paramr λ s rmn accorng o h formla: ' λ λ mn{ } ; 47 h mnmm pon s calcla: λ 48 h vlop algorhm combns hgh sp of calclaon wh low compaonal ns I also provs h man accrac o allow flflmn of consrans n aonal chck of whhr h pon fon s whn h allow ara Eq3 nrocs a gr of robsnss o fals rslng from ransn pss n h compaonal procss hs nabls h algorhm o b rcommn for s n on-boar gal conrol ssms 4 Eampl of rbo ngn conrol h vlop chnq was s for snhss of a wovarabl conrol ssm for a wn-shaf rbo wh a varabl nozzl wo npnn conrol varabls fl flow W f an nozzl ara n allow sgn of ml-varabl conrol ml-varabl ssm was vlop o conrol mamm vals of h followng paramrs: low prssr shaf sp n L hgh prssr shaf sp n H an gas mprar g bhn h low prssr rbn a mamm r hrs opraon h sorc aa for conrol sgn ar mahmacal mols of h plan an rqrmns on namc an sac accrac of conrol p rsponss of h op paramrs shol b aproc who ovrshoo h rs m ms b lss han on scon ac rror of shaf sp conrol s lss han 3 % an of mprar conrol s lss han Invsgaon of h vlop conrol ssm was prform va smlaon of ransns sng mahmacal mols of h plan an conrollr rs an nvsgaon was prform sng lnar mols of h ssm hn wh a prformanc-bas al non-lnar hrmo namc mol of h ngn

6 h program of nvsgaon ncl h followng pons: h s of varos obcv fncons; rspons o prrbaons appl o man vals an conrol varabls; nflnc of pr m la; gr of compnsaon of hrmocopl nra; paramrc prrbaons; posonal rf n hralc ngraor acaors h mahmacal mol scrbs h plan Eq5 acaors h hrmocopl an h vlop conrollr caors of W f an n ar moll wh ngraors wh mamm sps of 3 kg/hr/s an cm /s rspcvl h samplng frqnc for ransfr of conrol ops o h acaors s 4 Hz h samplng pro s 5 s gr shows sp rspons of h clos-loop conrol ssm o a sp of rpm appl o L shaf sp man h n L ransn s monoonc who a pak of "ovr-conrol" an lass abo 3 s h g ransn s fla Racon cas b a sp chang of C D'EG#H&IKJ"LM mprar man s prsn n gr 3 h N g ransonal procss s a spws fncon lasng aron 5 s h L shaf sp has no chang h conrols ar chang wh mamm possbl sp nl h gas mprar achvs s nw val h plos show ha a hgh gr of coplng bwn n L an O g conrol loops s achv hs ampl provs h ffcnc of h mho vlop for sgn of conrol ssms opms b h ngn s rspons sp 5 Conclng rmarks hs papr prsn an approach for opmal ml-varabl conrol of aro ngns Mos gal on-boar ngn conrol ssms oa rprsn mr gal ralsaon of analog conrol laws s n hro-mchancal conrol ssms Howvr sng conrol comprs possss nogh compng powr o b s n opmsaon of ngn prformanc hrogh a nw p of conrol laws an algorhms Implmnaon of opmal ngn conrol las o mor ffcn opraon of h arcraf powr plan va argng global opmsaon crra an b acconng for nval ngn characrscs In h ampl scss a ral-m algorhm was prsn for opmsaon of conrol laws an hr opmal ralsaon I was shown ha h mho vlop allows sgn of ml-varabl conrol of h ngn wh a hgh gr of coplng bwn conrol loops cknowlgmns h ahors wsh o acknowlg h sppor of EU IN programm nr h IN--757 proc gr : p rspons of clos-loop ssm o man of n L rpm gr 3: p rspons of clos-loop ssm o man of g K Rfrnc [] hans M Kapasors Kappos E pang H Lnar-qarac Gassan wh loop-ransfr rcovr mhoolog for h - ngn Jornal of Ganc Conrol an Dnamcs [] Canon MD Cllm CD olak E hor of opmal conrol an mahmacal programmng McGraw-Hll 97 [3] rrck DK Garg bhala rbofan ngn conrol sgn sng robs mlvarabl conrol chnologs IEEE ransacons on Conrol sms chnolog [4] Garg Robs ngra flgh proplson conrol sgn for a OVL arcraf sng H-nfn conrol sgn chnqs omaca [5] Harfors M pplcaon of H-nfn robs conrol o h RM ngn Conrol Engnrng racc [6] Moorhos DJ Crs KD h conrol ssm sgn mhoolog of h OL an manvr chnolog monsraor Inrnaonal Jornal of Conrol