Active Model Based Predictive Control for Unmanned Helicopter in Full Flight Envelope

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1 he 2 IEEE/RSJ Inernonl Conference on Inellgen Robos nd Sysems Ocober 8-22, 2, pe, wn Acve Model Bsed Predcve Conrol for Unmnned Helcoper n Full Flgh Envelope Dle Song, Junong Q, Jnd Hn, nd Gungjun Lu Absrc- For he conrol of unmnned helcopers n full flgh envelope, n cve model bsed conrol scheme s developed n hs pper. An dpve se-membershp fler (ASMF) s used o onlne esme boh he model error due o flgh mode chnge nd s boundry, kng dvnge of ASMF, so h he model error cn be ssumed unknown bu bounded (UBB). he proposed pproch s prccl becuse he model error depends on boh helcoper dynmcs nd flgh ses, nd my no be ssumed s whe nose. An cve modelng bsed sonry ncremen predcve conrol (AMSIPC) s lso proposed bsed on he esmed model error nd s boundry o opmlly compense he model error, s well s he erodynmcs me dely. he proposed mehod hs been mplemened on he ServoHel-2 unmnned helcoper plform nd expermenlly esed, nd he resuls hve demonsred s effecveness. I. INRODUCION Unmnned helcopers re ncresngly populr plforms for unmnned erl vehcles (UAVs). Wh he bles such s hoverng, kng off nd lndng verclly, unmnned helcopers exend he poenl pplcons of UAVs. However, due o he complex mechnsm nd complced ero-flow durng flgh, s lmos mpossble o ccurely model he dynmcs of n unmnned helcoper n full flgh envelope, nd he sgnfcn model uncernes ssoced wh nomnl model my degrde he performnce nd even sbly of n onbord conroller. Due o he dffculy n obnng hgh fdely full envelope model, he mul-mode modelng echnque hs been proposed for helcopers such s l-roor rcrf XV- 5 [], helcoper BO-5 [2], UH-6 [3], R-5 [4] nd X- Cell [5]. he mode-dependen model, whch s denfed nd smplfed ccordng o specfc flgh mode such s hoverng, crusng, kng off nd lndng, cn be used for conrol desgn for he correspondng flgh mode. However, he mode-dependen conrol suffers from les wo problems: one s he dffculy n ccommodng he mode rnson dynmcs [6], nd he oher s he compenson of he model drf due o flgh dynmcs chnge whn one hs pper s suppored by Nonl Scence Foundon of Chn (67528). Dle Song s wh Se Key Lborory of Robocs, Shenyng Insue of Auomon, Chnese Acdemy of Scences, nd he Grdue School of Chnese Acdemy of Scences, Bejng, Chn (Correspondng Auhor, phone: , e-ml: dlesong@s.cn). Junong Q s wh Se Key Lborory of Robocs, Shenyng Insue of Auomon, Chnese Acdemy of Scences, Shenyng, Chn (e-ml: qj@s.cn). Jnd Hn s wh Se Key Lborory of Robocs, Shenyng Insue of Auomon, Chnese Acdemy of Scences, Shenyng, Chn (e-ml: jdhn@s.cn). Gungjun Lu s wh Deprmen of Aerospce Engneerng, Ryerson Unversy, orono, Onro M5B2K3, Cnd (e-ml: gjlu@ryerson.c) prculr mode. Up o now, for he purpose of prccl mplemenon, he mode rnson problem cn be prlly del wh by lmng he mode swchng condons, e.g., mode chnge s mde hrough hoverng mode. Robus nd dpve conrol echnques [7-8], on he oher hnd, hve been used o del wh he model shf whn flgh mode. However, such conrol schemes normlly need o know he boundry of nernl nd exernl uncernes nd relve nose dsrbuon, whch re dffcul o denfy ccurely for helcoper n full flgh envelope. In recen yers, he encourgng chevemen n sequenl esmon mkes n mporn drecon for onlne modelng nd model-reference conrol [9]. Among sochsc esmons, he mos populr one s he Klmn-ype flers (KFs) [,, 2]. Alhough wdely used, he KFs suffer from sensvy o bs nd dvergence n he esmes, relyng on ssumpons on ssc dsrbuon such s whe nose nd known men or covrnce for opml esmon. In mny cses, s more prccl o ssume h he noses or uncernes re unknown bu bounded (UBB). In vew of hs, he se-membershp fler (SMF), whch compues compc fesble se n whch he rue se or prmeer les only under he UBB nose ssumpon, provdes n rcve lernve [3-4]. On he conrol ssue, model predcve conrol (MPC) cn compense for he erodynmcs dely nd does no requre hgh ccurcy reference nonlner model [5]. Among hese mehods, lner generlzed predcve conrol (GPC) hs become one of he mos populr MPC mehods n ndusry nd cdem. However, he norml GPC s sensve o process nose nd model errors, whch re unknown bu bounded for helcopers n full flgh envelope. In hs pper, for relzng he couplng conrol of unmnned helcopers n full flgh envelope, n cve modelng bsed conroller s developed bsed on modfed generlzed predcve conrol nd dpve se-membershp fler esmon (ASMF). he me vryng model error nd s boundry re esmed by he dpve se-member fler Bsed on hs cve esmon nd he modfed GPC conroller s developed. Flgh expermens hve been conduced o es he performnce of he proposed conroller on our UAV plform, nd expermenl resuls hve demonsred he effecveness of he proposed mehod. II. ACIVE MODEL BASED CONROL SCHEME AND REFERENCE MODEL OF A HELICOPER Fg. llusres he cve model bsed conrol scheme. he error beween he reference model nd he cul dynmcs of he conrolled pln s esmed by n on-lne modelng sregy. he conrol, whch s desgned ccordng //$25. 2 IEEE 66

2 o he reference model, should be ble o compense he esmed model error nd n rel me. In he followngs of hs pper, we use he ASMF s he cve modelng lgorhm nd he modfed GPC s he conrol. Reference Inpu Conroller Model Errors Reference Model Acve Modelng Pln Pln Oupu Fgure : he scheme of cve model bsed conrol A reference model s ypclly obned by lnerzng he nonlner dynmcs of helcoper one flyng mode. he model errors from lnerzon, exernl dsurbnce, smplfcon, nd un-modeled dynmcs cn be consdered s ddonl process nose [6]. hus, lnerzed sespce model for helcoper dynmcs n full flgh envelope cn be formuled s X AX BU Bf f( X, XW, ) () Y CX 3 where X R s he se, ncludng 3-xs velocy, pch nd roll ngle, 3-xs gyro vlues, flppng ngles of mn roor nd sblzer br nd he feedbck of yw gyro. 8 Y R s he oupu, ncludng 3-xs velocy, pch nd roll ngle nd 3-xs gyro vlues, A nd B conn prmeers h cn be denfed n dfferen flgh modes, nd we use hem o descrbe he prmeers n hoverng 4 mode. U R s he conrol npu vecor. he del of buldng he nomnl model nd physcl menng of prmeers s explned n Ref.[7]. III. ASMF BASED ACIVE MODEL ERROR ESIMAION As llusred n Fg., dopng he cve modelng process o ge he model error f nd sysem se X s he bss for elmnon of he model error. Conroller cn only work bsed on nomnl model nd feedbck of se nd model error from cve modelng process. In hs secon, he cve modelng process s bul bsed on n dpve semembershp fler (ASMF) [4] snce he UBB process nose. Frs, we mus obn he reference equon for esmon. Compred wh he smplng frequency (ofen >5Hz for flgh conrol) of he conrol sysem, he model error f ( X, X, W) cn be consdered s slow-vryng vecor, whch mens f f h, where f s he smplng vlue of f ( X, X, W) smplng me, nd h s he ssumed unknown bu bounded (UBB) process nose. Le he exended smplng se X X f, nd hen we cn obn he dscree equon from Eq. () s X AdX BdU W Y CdX V (2) where Ad Bf Bd Ad 3 3 I, Bd , d C C d 83, W W h, Bf I3 3 nd f s 3 vecor for model errors. Here, s he smplng me, Im m s he m m un mrx nd m n s he m n zero mrx. { Ad, Bd, Cd} s he dscree expresson of sysem { A, B, C }. Boh he model error nd he process nose W re vehcle dynmcs nd flgh ses dependen, nd do no necessrly hve norml dsrbuon. hus, he Klmn ype fler cnno be ppled, nd dpve se-membershp fler, whch s developed for UUB process nose nd cn ge he uncern boundres of he ses, s consdered o esme he ses nd model errors here. In hs secon we only presen he resul of ASMF nd plese refer o [4] for he dels bou ASMF. Wh respec o Eq. (2), we cn buld he dpve se-membershp fler s Eq. (3), r m rm pm C d P Cd R W P Cd W K ˆ ( ˆ Y Cd X ) W ( Y Cd X ) ˆ ˆ ˆ X X K( Y Cd X ) P P P P ( Cd W Cd ) ˆ X ˆ Ad X BdU r( Q ) r( Q ) r( Ad P Ad ) A d P Ad Q P (3) where Q R 88 nd R R re he nl ellpcl boundry of process nose W nd mesuremen nose V, respecvely, rm s he mxmum egenvlue of R smplng me, p m s he mxmum egenvlue of CdP Cd smplng me, operonl symbol r() s he rce of mrx, nd re he dpve prmeers 67

3 ˆ of he fler, X s he esmon of he exended se X smplng me, ˆ X s he esmon of ˆ X smplng me, P s he esmon of ellpcl boundry P smplng me, nd P s he esmon of ellpcl boundry P smplng me. We cn lso obn he boundry of he h elemen X of exended se X s X ˆ, ˆ P X P P s he h dgonl elemen of mrx P. smplng me, where IV. MODIFIED GPC FOR UNMANNED HELICOPERS We descrbe he norml GPC n Pr A, nd hen, he modfed scheme s proposed n Pr B&C o elmne he negve nfluence of model errors n rel pplcons. A. Prelmnry work for Generlzed Predcve Conrol Generlly, for lner sysem wh cuor me dely lke, X AdX Bduk W (4) y CdX n where R s he sysem se vecor smplng me, X l y R m s he oupu vecor, u R s he conrol npu vecor, k s he cuors me-dely, nd W s process nose; rdonl Generlzed Predcve Conrol (GPC) cn be desgned s [8]. However, wh pplcon o he unmnned helcopers, hs knd of GPC lgorhm hs he followng hree dsdvnges: ) I cnno rejec he nfluence of workng mode chnges,.e., f X x x ( x) (5) where x s he curren operon pon, ( x ) s he vld rnge for model lnerzon nd x s he bsolue se me, he bsed predcon wll brng sedy errors for velocy rckng. 2) Norml GPC s sensve o msmch of he nomnl model, whch mens slow chnge n prmeers ( Ad, Bd) my resul n predcon error nd unsble conrol. 3) he rnsen model errors of he nomnl model from exernl dsurbnce, esmed by ASMF, cnno be elmned, nd hs wll lso resul n he nsbly of he close loop. B. Sonry Incremen Predcve Conrol o rejec he nfluence of workng mode chnge nd sensvy o nomnl prmeers chnge n rel pplcon, we ssume h he process nosew s ncremen n Eq. (4) s sonry rndom process, whch mens W W W W (6) s norml dsrbuon. Where q s he dfference operor; q s one-sep dely fcor. hus, Eq. (4) cn be rewren s follows, X AdX Bduk W (7) o vod he sep sgnl reference rckng, whch s dngerous for unmnned helcoper sysem, we use low pss fler o clcule he se-pon npus of he oupu n he fuure -h sep, =,, p. l Le SP R be he se-pon npu me, hen we hve rk SP ( rk SP), p (8) where s he cu-off frequency of he fler, he nl vlue r k yˆ k, r k s he -h se-pon npu, nd yˆ k s he esme of oupu me +k. hus, he se-pon problem s solved nd he oupu predcon cn be mplned bsed on ncremen model (7) s follows: Le Xˆ ˆ ˆ X X, hen Xˆ k Xˆ ˆ k AdXk Bdu ˆ X A ˆ X k d k m { nd A B d } u m m n m ˆ X { n k AdBd} um, p m n Hence, he bove problem ), whch comes from workng mode chnge, s solved becuse x dsppers n predcve equon (9). We cn obn he followng predcon mrx for he oupu from Eq. (9) : ˆ Y yˆk yˆk2... yˆkp ˆ ˆ ˆ Cd Xk Cd Xk2... Cd Xkp G u u... up Y GU (9) () where Y s he known pr of p seps predcon, whch cnno be nfluenced by curren conrol npu, nd mrx G hs he followng form: CdBd... CdBd CdAdBd CdBd... G () p p2 Cd AdBd Cd AdBd... CdB d Compred wh he norml GPC, he predcon of SIPC hs beer chrcerscs h cn be descrbed by he followng heorem, whch solves he bove problem 2) n Pr A. heorem: for nomnl model (7), he se predcon obned by Eqs. (9) mnns unbsed even when he nomnl model prmeers ( Ad, Bd) chnge onlne. he chrcersc s lso mnned n norml GPC condons, where W s norml dsrbuon. 68

4 In order o reduce he compuonl burden of Eq. (), we propose here sep pln echnque, u u (2) where s m m dgonl mrx presenng he lengh of one sep. hen, we cn smplfy Eq. () by only clculng he unknown conrol, ˆ p Y Y G Imm... u (3) Y G u 2 where Im ms m m un mrx. he cos funcon of he sonry ncremen predcve conrol s desgned s: ˆ ( ) ( ˆ J R Y W R Y ) u u (4) lp lp k k2... kp, W R mm R where R r r r s he wegh mrx for rckng error, nd s he wegh mrx of he conrol ncremen. By mnmzng he cos funcon of Eq. (4), we cn clcule he conrol vecor s follows: f ( ) u ( G WG ) G W( R Y ) K R Y (5) where K f ( G2WG2 ) G2W cn be compleed offlne. Consequenly, he sonry ncremen predcve conroller (SIPC) cn be desgned s followngs. Sep I: Mke ncremen predcon Bsed on he curren nd hsory mesure vlue, use Eqs. (9-) o obn he predcon for fuure oupu Yˆ nd nl pln pon rk yˆ k. Sep II: Pln for he se-pon npu Use Eq. (8) o pln he fuure se-pons, nd obn k k kp R r r 2... r Sep III: Recedng horzon opmzon Clcule he conrol ncremen u, bsed on Eq. (5). Sep IV: Conrol mplemenon Curren conrol npu u u u, whch s used s he conrol o he rel pln. Afer h, go bck o sep I he nex me nsn. C. Opml sregy for model error compenson In order o compense he model error n Eq. (), he conrol vecor hs o mch he followng equon, whch cn be drecly obned from Eq. (): BU d Bf f BU d (6) where U s he conrol vecor need o be clculed by he predcve conroller n secon IV.B, desgned bsed on he orgnl model () whou he model error f. hus, we nroduce he followng cos funcon wh qudrc form o solve he bove problem ). U rgmn J( U) U J( U) BU d Bf f BU d H BU d Bf f BU d (7) where H s wegh mrx, whch cn be seleced. Acully, he convergence of ASMF lgorhm s lso nfluenced by he conrol con U. hs s becuse he sbly of he ASMF cn be represened by he fler prmeer, whle n Eq. (3) cn be rewren s follows, ˆ ˆ ( Y CdX ) W ( Y Cd X ) ( ˆ Y Cd( AdX BdU)) (8) ( ( ˆ W Y C A X B U ))] d d d In [4], hs been shown he sbly of he ASMF cn be represened by he fler prmeer,.e., he ASMF s sble when. Frsly, defne ˆ J ( U, Y) ( Y Cd( AdX BdU)) (9) ( ˆ W Y Cd( AdX BdU))] hen, J ( U, Y ) mx J ( U, Y ) ˆ X mx{ ˆ Y Cd Ad X BdU W Xˆ ˆ Y Cd AdX BdU } We should selec n U o mke J ( U, Y ) smll s fr s possble, h s, J ( Y ) mn J ( U, Y ) (2) U (2) We nroduce he followng cos funcon J( U ) wh consderon of boh (7) nd (2) he sme me: U rgmn J( U) U (22) J( U) J( U) J ( U, Y ) where R re he posve defne wegh mrx. o mnmze J( U ), consderng J( U), he conrol cn J ( ) be obned U,.e., U J( U) 2( MU N) (23) U where M Bd HBd Bd Cd W CdBd N Bd H( Bf f BdU ) Bd Cd W Y hus, we cn obn he opml conrol h mnmzes J ( U ) s: 69

5 U( Y ) M N ( BdHBd Bd Cd W CdBd) Bd Cd W Y BdH( Bf f BdU ) (24) For he unknown mesuremen me + n Eq. (24), we consder h he conrol sysem s sble, so, Y ( Y). Here, ( Y ) s he ellpcl domn of Y. hus, we frs defne rry S o nclude he esme of he -h elemen s wo boundry endpons s ˆ { ( ) h ( { } )} S Y Y Mx CdCol j (25) jl { pll, l,...,3} where Y s he -h elemen n he vecor Y, Yˆ s he correspondng oupu Y s endpons esmon. For se S, {, 2,...,8} nd h s or for every, s he bsolue vlue of he -h elemen n vecor, nd he funcon Col{} j s defned s follows: 3 Col{} j j... j (26) hen, we defne se S o descrbe ll possble endpon vecor of hey s ˆ EP EP 3 S Yˆ S... S (27) where Y s he possble endpon (EP) for oupu Y nex smplng me +. hus, he proposed cve modelng bsed predcve conroller cn be mplemened by usng he followng seps: Sep I: Mke ncremen predcon Bsed on he curren esmed se ˆ X, use he sonry ncremen predcve conroller, s n secon IV.B, o obn he nomnl conrol npu U ; Sep II: Model error esmon nd elmnon Bsed onu, compue he opml conrol npu U : ) Esme he vlues nd boundres of se X nd model error f, usng ASMF n (3); b) Clcule he correspondng ˆ EP U ( Y ) for every ˆ EP Y n se S by Eq. (24); c) For every ˆ EP U( Y ) n sep ), use Eq. (9) o obn he mxmum of funcon ˆEP ( ( ˆEP J U Y ), Y ), nd ge he ˆ EP Y o le ˆEP ˆEP ˆEP Y rg Mx J { U( Y ), Y} ; ˆ EP Y S d) he correspondng ˆ EP U( Y ) s he opml conrol U me,.e. ˆ EP U U( Y ). Sep III: Recedng horzon sregy Go bck o sep I he nex me nsn. V. FLIGH EXPERIMEN All flgh ess re conduced on he Servohel-2 seup, more dels of hs expermenl plform cn be found n [9]. Fgure 2: SERVOHELI-2 smll-sze helcoper plform Generlzed predcve conrol (GPC), sonry ncremen predcve conrol (SIPC) nd cve model bsed sonry ncremen predcve conrol (AMSIPC) re ll esed n he sme flgh condons, nd he comprson resuls re shown n Fgs We use he denfed prmeers n Ref.[7] s nomnl model. I cn be seen h, when he helcoper ncreses s longudnl velocy nd chnges flgh mode from hoverng o crusng, GPC (brown lne) hs sedy velocy error (>2%) nd ncresng poson error becuse of he model errors. SIPC (blue lne) hs smller velocy error becuse uses ncremen model o rejec he nfluence of he chngng operon pon nd dynmcs slow chnge durng he flgh. However, he ncremen model my enlrge he model errors due o he uncern prmeers nd sensor/process noses, resulng n he oscllons (>3%) n he consn velocy perod (clerly seen n Fg. 3 nd 4). Whle for AMSIPC (green lne), becuse he model error hs been onlne esmed by he ASMF, he proposed AMSIPC successfully reduces velocy oscllons (<5%) nd rckng errors (<5%). VI. CONCLUSIONS An cve model bsed predcve conrol scheme ws proposed n hs pper o compense model error due o flgh mode chnge nd model uncernes, nd relze full flgh envelope conrol whou mul-mode models nd mode-dependen conrols. he ASMF ws doped s n cve modelng echnque o onlne esme he error beween reference model nd rel dynmcs. he proposed conrol scheme ws mplemened on our developed ServoHel-2 unmnned helcoper. Expermenl resuls hve demonsred cler mprovemens over he norml GPC whou cve modelng enhncemen. I should be noed h, presen, we hve only esed he conrol scheme wh respec o he flgh mode chnge from hoverng o crusng, nd vce vers. Furher mode chnge condons, ncludng ground effec, coordned urnng nd ec, wll be flgh-esed n ner fuure. REFERENCE [] schler M.B., Frequency-domn Idenfcon of XV-5 l-roor Arcrf Dynmcs n Hoverng Flgh, Journl of he Amercn Helcoper Socey, Vol. 3 (2), pp.38-48, 985. [2] schler M. B. nd Cuffmn M. G., Frequency-Response Mehod for Roorcrf Sysem Idenfcon: Flgh Applcon o BO-I5 Coupled Roor/Fuselge Dynmcs, Journl of he Amercn Helcoper Socey, Vol. 37 (3), pp.3-7,

6 [3] Flecher J. W., Idenfcon of UH-6 Sbly Dervve Models n Hover from Flgh es D, Journl of he Amercn Helcoper Socey, Vol. 4 (), pp.8-2, 995. [4] Meler B., schler M. B. nd Knde., Sysem Idenfcon of Smll-Sze Unmnned Helcoper Dynmcs, Amercn Helcoper Socey 55h Annul Forum Proceedngs, Vol. 2, pp.76-77, Monrel, Quebec, Cnd, My 25-27, 999. [5] Gvrles V., Meler B. nd Feron E., Nonlner Model for Smllscle Acrobc Helcoper, Proceedngs of he Amercn Insue of Aeronucs Gudnce, Nvgon, nd Conrol Conference, pp.8, Monrel, Quebec, Cnd, Augus 6-9, 2. [6] Mssmlno M. nd Vlero S., A Full Envelope Smll Commercl Arcrf Flgh Conrol Desgn Usng Mulvrble Proporonl-Inegrl Conrol, IEEE rnscons on Conrol Sysems echnology, Vol. 6 (), pp.69-76, Jnury, 28. [7] Voorslujs M. nd Mulder A., "Prmeer-dependen robus conrol for roorcrf UAV," AIAA Gudnce, Nvgon, nd Conrol Conference nd Exhb, pp.-, Sn Frncsco, Clforn, USA, Augus 5-8, 25. [8] Bjnens B., Chu Q.P. nd Voorslujs M., "Adpve feedbck lnerzon flgh conrol for helcoper UAV," AIAA Gudnce, Nvgon, nd Conrol Conference nd Exhb, pp.-, Sn Frncsco, Clforn, USA, Augus 5-8, 25. [9] Hykn S. nd De Fres N., Specl Issue on Sequenl Se Esmon, Proceedngs of he IEEE, Vol. 92(3), pp , 24. [] Lerro D. nd Br-Shlom Y. K., rckng wh Debsed Conssen Convered Mesuremens vs. EKF, IEEE rnscons on Aerosp. Elecron.Sys.,AES-29, pp.5-22, 993. [] Juler S. nd Uhlmnn J., Unscened flerng nd nonlner esmon, Proceedngs of he IEEE, Vol. 92(3), pp , 24. [2] Song Q., Jng Z., nd Hn J. D., UKF-Bsed Acve Model nd Adpve Inverse Dynmcs Conrol for Moble Robo, IEEE Inernonl Conference on Robocs nd Auomon, 27. [3] Shmm J. S. nd u K. Y., Approxme se-vlued observers for nonlner sysems, IEEE rnscons on Auomc Conrol, Vol. 42(5), pp , 997. [4] Zhou B., Hn J.D. nd Lu G., A UD fcorzon-bsed nonlner dpve se-membershp fler for ellpsodl esmon, Inernonl Journl of Robus nd Nonlner Conrol, Vol 8 (6), pp.53-53, November, 27. [5] Dng B. C., X Y. G., A Synhess Approch of On-lne Consrned Robus Model Predcve Conrol. Auomc. Vol. 4, No., 24, pp [6] Crssds J. L., Robus Conrol of Nonlner Sysems Usng Model- Error Conrol Synhess, Journl of gudnce, conrol nd dynmcs, Vol. 22 (4), pp.595-6, 999. [7] Song D.L., Q J.., D. L., Hn J.D. nd Lu G., Modelng Smllsze Unmnned Helcoper Usng Opml Esmon n he Frequency Domn, Inernonl Conference on Mechroncs nd Mchne Vson n Prcce, pp.97-2, Aucklnd, New Zelnd, December 2-4, 28. [8] Gregor K. nd Igor S., rckng-error Model-bsed Predcve Conrol for Moble Robos n rel me. Robocs nd Auonomous Sysems. Vol. 55, No. 7, 27, pp [9] Q J.., Song D.L., D. L., Hn J.D., he ServoHel-2 Roorcrf UAV Projec, Inernonl Conference on Mechroncs nd Mchne Vson n Prcce, pp.92-96, Aucklnd, New Zelnd, December2-4, 28. Fgure 3: Longudnl rckng resuls: () velocy; (b) poson error (<5s hoverng, >5s crusng) Fgure 4: Lerl rckng resuls: () velocy; (b) poson error (25s~8s crusng, ohers hoverng) Fgure 5: Vercl rckng resuls: () velocy; (b) poson error (<5s hoverng; >5s crusng) 62

Advanced Electromechanical Systems (ELE 847)

Advanced Electromechanical Systems (ELE 847) (ELE 847) Dr. Smr ouro-rener Topc 1.4: DC moor speed conrol Torono, 2009 Moor Speed Conrol (open loop conrol) Consder he followng crcu dgrm n V n V bn T1 T 5 T3 V dc r L AA e r f L FF f o V f V cn T 4