Adaptive control of sinusoidal brushless DC motor actuators

Transcription

1 Mchgan chnologcal Unvrsty Dgtal Mchgan ch Dssrtatons, Mastr's hss an Mastr's Rports - Opn Dssrtatons, Mastr's hss an Mastr's Rports 008 Aaptv control of snusoal brushlss DC otor actuators angtao Zhu Mchgan chnologcal Unvrsty Copyrght 008 angtao Zhu Rcon Ctaton Zhu, angtao, "Aaptv control of snusoal brushlss DC otor actuators", Dssrtaton, Mchgan chnologcal Unvrsty, Follow ths an atonal wors at: Part of th Mchancal Engnrng Coons

2 Aaptv control of Snusoal Brushlss DC Motor actuators BY IANGAO ZHU A DISSERAION Subtt n partal fulfllnt of th rurnts for th gr of DOCOR OF PHIOSOPHY Mchancal Engnrng-Engnrng Mchancs MICHIGAN ECHNOOGICA UNIVERSIY 008 Copyrght angtao Zhu 008

3 hs ssrtaton, "Aaptv Control of Snusoal Brushlss DC Motor Actuators," s hrby approv n partal fulfllnt of th rurnts for th gr of DOCOR OF PHIOSOPHY n th fl of Mchancal Engnrng- Engnrng Mchancs. DEPARMEN: Mchancal Engnrng-Engnrng Mchancs Sgnaturs: Dssrtaton Avsor ypwrttn Na: Goron G. Parr Dat Dpartnt Char ypwrttn Na: Wlla W. Prbon Dat

4 ACKNOWEDGEMEN I woul l to xprss y grattu to all thos who gav th possblty to coplt ths thss. I a ply nbt to y avsors Prof. Goron Parr an Dr. Ravnra Patanar, whos hlp, stulatng suggstons an ncouragnt hlp n all th t of rsarch for an wrtng of ths thss. h brs of y ssrtaton cott, Dr. Roshan D Souza, Prof. Charls Van Karsn, an Prof. Clar Gvns, hav gnrously gvn thr t an xprts to bttr y wor. I than th for thr contrbuton an support. Espcally, I woul l to gv y spcal thans to y wf Naa whos patnt lov nabl to coplt ths wor. I woul also l to than y o Yufn for hr unwavrng fath n.

5 HIS DISSERAION IS DEDICAED O MY WIFE NADIA AND MY DAUGHER KIMBERIE v

6 ABE OF CONENS abl of Contnts...v st of Fgurs... x st of abls... x Abstract... xv Chaptr Introucton.... Elctrcal Powr-Assst Strng Syst.... Elctrc Motor Actuator For EPAS BDC Motors an Control Ovrvw tratur Rvw on BDC Motor Control....5 Aaptv Control for Snusoal BDC Motors Publcatons....7 Suary...3 Chaptr Snusoal Brushlss DC Motor Molng...5. Prannt Magnt Synchronous Motor Molng h thr-phas ol h rct uaratur coornat ol Iplntaton of coornat controllrs oru Control of Snusoal BDC Motors PI controllr Fforwar nvrs ynacs controllr Controllr slcton...43 v

7 .3 Suary...45 Chaptr 3 Aaptv Paratr Estaton Motor Invrs Controllr Mol Sngl Paratr Estaton Sngl paratr staton Stablty of th sngl paratr staton sch Sulaton rsults Multpl Paratr Estaton Multpl paratr staton Proof of stablty Iprovng h Dynac Prforanc Of h Aaptv Algorth Motor lctrcal ynacs Motor sp saplng lay Multpl Paratr Estaton Usng Gra-Scht Orthonoralzaton Mult-paratr staton usng Gra-Scht orthonoralzaton Proof of stablty Mult-Paratr Estaton Usng an Axs Currnt Paratr staton Proof of unbas staton Concluson...9 Chaptr 4 Rcursv Paratr Intfcaton Dscrt Mol of th BDC Motor Dynacs...93 v

8 4. Rcursv ast Suar Estaton RS Sulaton Rsults Extn Kalan Fltr EKF Sulaton Rsults Suary...09 Chaptr 5 Mol Rfrnc Aaptv Control Dsgn Mol Rfrnc Aaptv Control h BDC otor rfrnc ol h control law h aaptaton law Sulaton Rsults Concluson... Chaptr 6 Clos loop Sulaton an Controllr Prforanc Evaluaton EPAS clos loop sulaton EPAS clos loop syst ol EPAS sulaton rsults an analyss Clos oop Sulaton Of a Motor Sp Control Applcaton Concluson...39 Chaptr 7 An Opraton Sulaton Mol for BDC Motors A Quas-Physcal BDC Motor Mol h n-to-n PMSM ol h H-brg PWM nvrtr ol h rotor angular splacnt transucr...5 v

9 7..4 h Hall ffct snsor SVPWM Sulaton confguraton Stuy of Controllr Iplntaton Issus n Sulaton Intalzaton Incrntal ncor rsoluton t of th Quas-Physcal BDC Motor Mol Suary...67 Chaptr 8 Concluson an Futur Wors Concluson Futur Wors...73 Appnx I h probablty nsty functon rvaton...75 Appnx II Soluton of th Intgraton rs n Euaton Appnx III Contnuous Extn Kalan Fltr for BDC Motors...79 Rfrncs...8 v

10 IS OF FIGURES Fgur. Schatc agra of a thr-phas BDC otor wth on par of rotor prannt agnt pols....5 Fgur. ypcal trapzoal an snusoal bac EMF wavfor...6 Fgur.3 Schatc agra of a typcal BDC otor control syst...7 Fgur.4 BDC otor toru wavfor whn oprat wth bac EMF zro crossng snsng chans...9 Fgur.5 ypcal aaptv controllr bloc agra...7 Fgur. h thr stator phas, th coornats an th ab coornats...9 Fgur. Iplntaton of a gnrc controllr sgn n coornats...37 Fgur.3 Stp rspons of th BDC otor wth th PI controllr an th fforwar controllr at 0 0, assung full nowlg of th plant paratrs...43 Fgur.4 Stp rspons of th BDC otor wth th PI controllr an th fforwar controllr at 0 0, wth 0% rror n K an 50% rror n R...44 Fgur 3. h schatc agra of th otor tst syst n sulaton Fgur 3. Sngl paratr staton prforanc n sulaton...57 Fgur 3.3 wo-paratr staton prforanc n sulaton...6 Fgur 3.4 Sngl paratr staton transnt prforancs: non-ynac otor nvrs controllr vs. approxat ynac nvrs controllr...66 x

11 Fgur 3.5 Mult-paratr staton usng non-ynac otor nvrs controllr vs. ynac otor nvrs controllr...67 Fgur 3.6 Sngl paratr staton prforancs: coparson of th basc staton progra an th prforanc provnts...7 Fgur 3.7 Prforanc coparson of staton sch n opn loop sulaton: cas th basc sch; cas 3 wth approxat Δ copnsaton....7 Fgur 3.8 Prforanc coparson of 4 ffrnt staton schs n opn loop sulaton: two-paratr staton...73 Fgur 3.9 Schatc agra of Gra Scht Orthonoralzaton...75 Fgur 3.0 Paratr staton prforanc wth Gra-Scht orthonoralzaton: th ntal paratr rror 50% R an 5% K...79 Fgur 3. h strbuton of y / x gvn x ~ N μ x, for svral valus of μ x...86 Fgur 3. h an of y xr x x gvn x ~ N μ x, copars to th algbrac functon y xr μ x μ x...87 Fgur 3.3 h an of y / x copars to μ x Fgur 3.4 h -solvr sulaton rsults: 50% R an 5% K...90 Fgur 4. RS staton sulaton rsults of a ~a 5 & b ~b 5 : constant 00ra s / Fgur 4. RS staton sulaton rsults of a ~a 5 & b ~b 5 : varant vlocty...0 Fgur 4.3 EKF staton sulaton: R h 0.5R, K h 00.9K...07 Fgur 4.4 EKF staton sulaton: R h 0.4R, K h 0.K...07 x

12 Fgur 4.5 EKF staton sulaton: R h 00.8R, K h 0.K...08 Fgur 4.6 EKF staton sulaton: R h 00.8R, K h 0 0.8K...08 Fgur 4.7 EKF staton sulaton: R h 0R, K h 0 K...09 Fgur 5. A schatc agra of a typcal MRAC controllr... Fg 5. MRAC rfrnc currnt tracng prforanc n sulaton: R c R, K c 0. K...0 Fg 5.3 MRAC paratr staton prforanc n sulaton, cas : R c R, K c 0. K...0 Fg 5.4 MRAC paratr staton prforanc n sulaton, cas : R c R, K c K... Fgur 6. Schatc agra of a typcal EPAS...4 Fgur 6. Coparson of th aaptv BDC otor controllrs n EPAS clos loop sulaton: ΔR 0 50%R an Δ K 0 0% K...9 Fgur 6.3 Coparson of th aaptv BDC otor controllrs n EPAS clos loop sulaton: ΔR 0 50%R an Δ K 0 5% K...30 Fgur 6.4 Coparson of th aaptv BDC otor controllrs n EPAS clos loop sulaton: ΔR 0 50%R an Δ K 0 5% K...3 Fgur 6.5 Coparson of th aaptv BDC otor controllrs n EPAS clos loop sulaton: ΔR 0 50%R an Δ K 0 0% K...3 Fgur 6.6 h bloc agra of th otor shaft ynacs Suln ol Fgur 6.7 h bloc agra of th ant-wnup PI controllr Suln ol...37 x

13 Fgur 6.8 Coparson of th aaptv BDC otor controllrs n th otor sp control clos loop sulaton: Δ R 0 50% R an Δ K 0 0% K...40 Fgur 6.9 Coparson of th aaptv BDC otor controllrs n th otor sp control clos loop sulaton: Δ R 0 50% R an Δ K 0 5% K...4 Fgur 6.0 Coparson of th aaptv BDC otor controllrs n th otor sp control clos loop sulaton: Δ R 0 50% R an Δ K 0 5% K...4 Fgur 6. Coparson of th aaptv BDC otor controllrs n th otor sp control clos loop sulaton: Δ R 0 50% R an Δ K 0 0% K...43 Fgur 7. A typcal thr-phas BDC otor syst bloc agra...48 Fgur 7. h ln-to-ln PMSM ol n Suln...50 Fgur 7.3 A splf crcut of th BDC otor syst....5 Fgur 7.4 h H-brg nvrtr ol...5 Fgur 7.5 h BDC otor coponnts bloc agra Fgur 7.6 h voltag vctors n th spac vctor oulaton Fgur 7.7 ransstors on-off tng n th sx sctors...58 Fgur 7.8 SVPWM functon plntaton n Suln...59 Fgur 7.9 h Suln ol of a clos loop BDC otor syst...60 Fgur 7.0 h practcal controllr ol n Suln...6 Fgur 7. h startng sch Suln ol...6 Fgur 7. h BDC otor physcal ol clos loop sulaton...63 Fgur 7.3 h BDC otor sulaton wth a low-rsoluton ncor cpr Fgur 7.4 h BDC otor sulaton wth a hgh-rsoluton ncor cpr x

14 IS OF ABES abl. PI controllr gans...4 abl 3. Boun of rrors n opn loop sulaton: two-paratr staton...7 abl 4. EKF paratr staton an an varanc for ffrnt nos lvl...07 abl 6. Stay-stat prforancs n EPAS clos loop sulaton...33 abl 6. Stay-stat prforancs n sp control clos loop sulaton...37 abl 7. h ght basc voltag vctors n th SVPWM...55 abl 7. Suary of th transstor on-off tng calculaton n ach sctor...57 x

15 ABSRAC Elctrcal Powr Assst Strng syst EPAS wll lly b us on futur autootv powr strng systs. h snusoal brushlss DC BDC otor has bn ntf as on of th ost sutabl actuators for th EPAS applcaton. Motor charactrstc varatons, whch can b ncat by varatons of th otor paratrs such as th col rsstanc an th toru constant, rctly part naccuracs n th control sch bas on th nonal valus of paratrs an thus th whol syst prforanc suffrs. h otor controllr ust arss th t-varyng otor charactrstcs probl an antan th prforanc n ts long srvc lf. In ths ssrtaton, four aaptv control algorths for brushlss DC BDC otors ar xplor. h frst algorth ngags a splf nvrs -coornat ynacs controllr an solvs for th paratr rrors wth th -axs currnt fbac fro svral past saplng stps. h controllr paratr valus ar upat by slow ntgraton of th paratr rrors. Iprovnt such as ynac approxaton, sp approxaton an Gra-Scht orthonoralzaton ar scuss for bttr staton prforanc. h scon algorth s propos to us both th -axs currnt an th -axs currnt fbac for paratr staton snc always accopans. Stochastc contons for unbas staton ar shown through Mont Carlo sulatons. Stuy of th frst two aaptv algorths ncats that th paratr staton prforanc can b achv by usng or hstory ata. h Extn Kalan Fltr EKF, a rprsntatv rcursv staton algorth, s thn nvstgat for th BDC otor applcaton. xv

16 Sulaton rsults valat th supror staton prforanc wth th EKF. Howvr, th coputaton coplxty an stablty ay b barrrs for practcal plntaton of th EKF. h fourth algorth s a ol rfrnc aaptv control MRAC that utlzs th sr otor charactrstcs as a rfrnc ol. Its stablty s guarant by yapunov s rct tho. Sulaton shows supror prforanc n trs of th convrgnc sp an currnt tracng. hs algorths ar copar n clos loop sulaton wth an EPAS ol an a otor sp control applcaton. h MRAC s ntf as th ost prosng canat controllr bcaus of ts cobnaton of supror prforanc an low coputatonal coplxty. A BDC otor controllr vlop wth th -coornat ol cannot b plnt wthout svral supplntal functons such as th coornat transforaton an a DC-to-AC currnt ncong sch. A uas-physcal BDC otor ol s vlop to stuy th practcal plntaton ssus of th coornat control stratgy, such as th ntalzaton an rotor angl transucr rsoluton. hs ol can also b bnfcal urng frst stag vlopnt n autootv BDC otor applcatons. xv

17 Chaptr INRODUCION Elctrcal powr assst strng syst EPAS wll lly b us for futur powr strng systs an thus thy ar on ntns targt of tchnology nnovaton. Brushlss DC BDC otors ar oftn slct as th actuator for EPAS n any publcatons. h charactrstcs of nvual BDC otors vary wth nvronntal factors, srvc lf, an anufacturng procss. A BDC otor controllr that antans th actuator prforanc rgarlss of ths varatons s hghly sr n EPAS applcatons an s th focus of ths rsarch.. Elctrcal Powr-Assst Strng Syst Autootv powr-assst strng systs hav tratonally bn onat by hyraulc pups. h always-on hyraulc pup consus powr fro th ngn whnvr th ngn s runnng. hs rsults n unwant ful consupton f th rvr s not ovng th strng whl. In contrast, lctrc otors only consu powr whn th rvr ans assstng forc. hs has bn rcognz as an ffctv way to prov ful conoy. Klugr an Harrs Klugr an Harrs 007 copar ngn bra spcfc ful consupton BSFC ffcts of off-ngn accssors on a havy-uty truc an a lght uty n van. hrough varous rvng cycl sulatons, t was foun that confgurng vhcls wth off-ngn accssors s a tho that provs ful savngs rangng fro 3% to 5%. For havy-uty trucs th provnts rang fro 3-%, an 8-5% for

18 nvans. Powr strng s on of th ajor contrbutors for ts low opratonal cycl 0%. An EPAS offrs about 80% lowr nrgy consupton than th hyraulc pup, an th osson of th hyraulc flu crass th nvronntal pact Hartr 000. Slarly, Burton Burton 00 show an avrag 3.0% ructon of ful consupton on a.6 car n cobn rvng cycl, an up to 3.5% ructon n th cty rvng cycl. In toay s autootv art, th avantag n ful conoy alon can a th EPAS an attractv opton ovr th tratonal hyraulc powr strng syst. In aton, th EPAS s or copact an thrfor asr for assbly an confguraton. It can b aapt to a w varty of vhcl applcatons wth sall changs to harwar. Furthror, th EPAS s or flxbl n trs of functon. Its softwar control stratgs can b of to prov a w rang of hanlng charactrstcs wthout changng harwar. For xapl, t s srabl that strng systs gv hgh assstanc to th rvr at low vhcl sps whl low assstanc at hgh sps. Ctn t al. Ctn 005 show that ths coul not b achv by convntonal HPAS systs n whch a torson bar wth constant chancal proprts nrta, stffnss an apng was us to trn th assstanc aount. A coplant controllr for th EPAS was propos n ths wor an t was shown that ths coul b achv by changng th vrtual syst ynac paratrs an th control gans. h flxblty of th EPAS also provs or opportunty for bttr vhcl hanlng prforanc. anaa t al. anaa 007 vlop an actv strng control algorth for an EPAS. hs algorth ncras th strng rturn toru n a rgon whr th algnnt toru was saturat u to th rvr's xcssv strng anuvr

19 on a slppry roa. Kubota t al. Kubota 007 propos an algorth by usng th EPAS to countract strng-pull caus by latral sturbancs such as roa contour, suspnson algnnt rror an tr proprts. h sturbanc was stat by fnng th an assstng toru n straght ln rvng an cancl by th EPAS. Du to ths avantags an progrss n lctrc otor tchnology, battry an crocontrollrs, EPAS wll bco th an stra of th futur powr strng syst an an ntns targt of tchnology nnovaton.. Elctrc Motor Actuator For EPAS h lctrc otor s th hart of th EPAS. Elctrc otors ar th ost coonly us actuators n chatronc systs. o na a fw avantags, thy hav hghr ffcncy an hghr powr nsty powr/ass copar to thr chancal or hyraulc countrparts. ratonally, brush DC otors an asynchronous or synchronous AC otors ar th ost wly us u to thr low costs. Howvr, th rlablty an antnanc cost ntrouc by th brush has always bn a ltng factor for brush DC otors applcaton to th EPAS. ratonal AC otors ar rarly us n autootv applcatons bcaus of unavalablty of th AC powr supply an thr unsrabl low sp charactrstcs. In th past two cas, thr has bn sgnfcant progrss on rar-arth prannt agnt synchronous PMS otors. hs otors ar upp wth prannt agnt rotors. Swtchng vcs, such as brushs, ar no longr n for thr opraton. ogthr wth an nvrtr, a swtchng logc controllr an so rotor poston fbac chans, a PMS otor can b oprat n slf-controll o on a DC currnt sourc. Such a otor s oftn call a brushlss DC BDC otor. Du to th absnc of 3

20 brushs an a coutator, BDC otors hav a nubr of avantags n antnanc, rlablty an ffcncy copar to convntonal DC or AC otors. Ils-Klupnr Ils-Klupnr 005 rvw lctrc actuator canats for th lctrcal powr strng applcaton, nclung brush an brushlss rv systs bas on prannt agnt brush DC DC, nucton IM, prannt agnt trapzoal an snusoal synchronous, swtch-rluctanc SR, an rluctanc synchronous RS achns. A w rang of factors wr consr n valuaton, nclung but not lt to toru nsty, pa to contnuous toru capablty, varabl sp control, toru pulsatons, tpratur snstvty, acoustc nos, powr convrtr rurnts, anufacturng, rlablty, custor accptanc, cost. h snusoal an trapzoal BDC otors wr ntf as th ost sutabl actuators for th EPAS applcaton. h trapzoal BDC otors hav bn plnt n proucton EPAS by Dlph Corporaton snc 004. hough th BDC otors hav bn accpt as th ost sutabl canat actuators for th EPAS applcaton, t s a fact that th BDC otors rur or sophstcat control than brush DC otors, spcally snusoal BDC otors. hs s th bacgroun an otvaton of ths rsarch..3 BDC Motors an Control Ovrvw A BDC otor s an AC synchronous otor wth prannt agnts on th rotor an wnngs on th stator. Most BDC otors hav thr phas stator wnngs, whl thr rotors can hav svral pars of rotor agnt pols. Fgur. s a schatc agra of a thr-phas BDC otor wth on par of rotor agnt pols. h nrgz stator wnngs crat an lctroagntc fl, an th rotor s attract to algn wth th stator 4

21 fl. Whn currnt s suppl to th stators n an approprat sunc, th stator lctroagntc fl rotats an rvs th rotor agnts. h stator lctroagntc fl an th rotor usually rotat at th sa sp, an th phas la btwn th stator fl an th rotor ns to b antan to gnrat constant toru. Masurnt of th rotor poston s n for a BDC otor s opraton to proprly sunc th stator currnt. b V co rotor N a co S Stator wnngs co c Fgur. Schatc agra of a thr-phas BDC otor wth on par of rotor prannt agnt pols. co ncat coon lns; a, b, an c ncat stator trnals. BDC otors can b catgorz as trapzoal or snusoal accorng to th wavfor of thr bac lctrootv forc EMF, as shown n Fgur.. Structurally, ths two typs of BDC otors ar ffrnt n th way that thr stator slots an cols strbut long th stator nnr prphry, whch crats ffrnt wavfors of bac EMF n stator cols as th agnts rotatng wth th rotor. Both typs of BDC otors 5

22 rur th currnt an bac EMF n ach stator phas to b synchronz so as to gnrat constant toru. h gnrat toru s proportonal to th phas currnt valu on ach phas, an th total rotor shaft toru output s th suaton of toru gnrat on all phass. Rotor Angl Rotor Angl Phas A-B Phas A-B Phas B-C Phas B-C Phas C-A Phas C-A a rapzoal bac EMF b Snusoal bac EMF Fgur. ypcal trapzoal an snusoal bac EMF wavfor Fgur.3 s a schatc agra of a typcal thr-phas BDC otor control syst. It usually conssts of an nvrtr, a cro-controllr that controls th swtchng logc, a rotor poston fbac chans an a prannt agnt synchronous otor. In th outr loop control syst, th BDC otor control syst plays th rol of ts actuator, 6

23 so th objctv of BDC otor control s to gnrat approprat sr toru for th outr loop syst. Fgur.3 Schatc agra of a typcal BDC otor control syst. PMSM s prannt agnt synchronous otor. Slar to brush DC otors, th gnrat toru fro a BDC otor s gnrally proportonal to th phas currnts, so th toru control s bascally th phas currnt control. Howvr, for th BDC otor, currnt control conssts of two sub-tass: stator an rotor flux synchronzaton, an control of th phas currnt valus. h forr tas nsurs consstnt toru gnraton, whl th lattr trns th agntu of th gnrat toru. Both tass ar accoplsh through th thr-phas nvrtr of Fgur.3. o antan th synchronzaton, th controllr ynacally cs that a crtan st of gats s to b turn on an th ranng gats to b off bas on th rotor poston. h phas currnt valu control s usually achv by ajustng th tng of thos gats 7

24 to b turn on. ypcally ths s on by thr hystrss currnt control or puls wth oulaton PWM control. In th hystrss currnt control, th phas branch gats ar swtch on f th fbac currnt s outs a prst ban of thr corrsponng rfrnc valus, an vc vrsa. Apparntly, th gat swtchng fruncy vars as th currnt rror vars. In th PWM control o, th gats ar swtch at a fx fruncy, an th currnt valu s controll through ajustng th PWM uty cycl. h control prncpl for th trapzoal BDC otors s that currnt shoul flow n only two of th thr phass at a t xas Instrunts Europ, 997. hr shoul b no toru proucton urng th rgon of bac EMF zro crossng for ach nvual stator phas. rapzoal BDC otors ar oftn upp wth transucrs to tct th bac EMF zro-crossng rgons. h nvrtr gat swtchng logc can b obtan through a truth tabl bas on th status of a st of Hall ffct snsor outputs. hortcally, constant toru can b gnrat wth th rotor poston fbac, as th bac EMF s constant whn th phass ar swtch on. Howvr, u to th phas nuctanc, th stator phas currnt cannot b stablsh nstantanously, thus toru rppl s nvtabl at vry phas coutaton as shown n Fgur.4a. Snusoal BDC otor can also oprat n ths way, but th toru rppl wll b n snusoal shap u to th snusoal bac EMF an phas coutaton, as shown n Fgur.4b. In ost trapzoal BDC otors, u to th fact that only two phass ar on at a t an a sngl currnt flow through th, t s possbl to control th currnt wth on currnt snsor on th nvrtr nput ln fro th DC powr supply. Snc th swtchng logc s fx for ach nvual otor an s oftn progra n harwar l a truth 8

25 tabl, th currnt controllr only ns to control th currnt valu. In ths aspct, t s ut slar to th brush DC otors. Rotor Angl, θ a oru gnrat by a rapzoal BDC otor Rotor Angl, θ b oru gnrat by a Snusoal BDC otor Fgur.4 BDC otor toru wavfor whn oprat wth bac EMF zro crossng snsng chans. Snusoal BDC otors ar capabl of gnratng constant toru f th phas currnts ar controll to b snusoal an n-phas wth thr corrsponng bac EMF. hs usually rurs hgh rsoluton of rotor poston fbac, spcally n th applcatons whr otor sp an loa vary sgnfcantly. In aton, th nvrtr swtchng schs ar or sophstcat as all stator phass ar on urng th opraton of th otor. On of th ost wly us thos for constant toru gnraton n snusoal BDC otors s th Spac Vctor Puls Wth Moulaton SVPWM or Fl Ornt Control. In th SVPWM sch, svral bas voltag vctors ar fn n th stator agntc fl coornats that ar trn by th nvrtr gat on/off stats. In ach of th PWM pros th controllr sns a st of gat on/off coans that corrspon to spcfc bas voltag vctors. hrough ajustng th uty cycls of 9

26 ach of ths bas voltag vctors, any voltag vctor n ths coornat can b approxat by a lnar cobnaton of th bas vctors. hr-phas snusoal BDC otors, whn controll to achv constant toru gnraton, n all thr-phas currnts to b snusoal an n phas wth thr corrsponng bac EMF. Unl th sngl currnt control n trapzoal BDC otors, all phas currnts of th snusoal BDC otor contrbut to th total toru output at all ts. hs usually rurs two or thr currnt snsors for fbac control to achv rfrnc toru tracng. h rct-uaratur ol., whch s obtan fro th thr-phas ol through two coornat transforatons, s oftn plnt for th otor ynacs analyss an control vlopnt. R t R t K n p p n K v, v,. whr subscrpts an ncat th rct an uaratur coornat varabls, s th currnt, v s th control voltag, R s th col rsstanc, s th col nuctanc, s th rotor vlocty, an n p s th nubr of rotor prannt agnt pols pars. Wth th ol, th thr phas currnts can b transfor nto two npnnt vrtual currnts, n th rct an uaratur coornats. If th otor s oprat unr ts rat sp, th gnrat toru s approxatly proportonal to th uaratur currnt coponnt. hrfor, gvn a rfrnc toru fro th outr loop controllr, th rfrnc currnt can b calculat. Snc th rct axs currnt os not contrbut to th toru gnraton, t s oftn srabl to p t as clos to zro 0

27 as possbl to prov nrgy ffcncy. Control tchnus such as proportonal ntgral ffrntal PID control can b plnt to achv rfrnc currnt tracng..4 tratur Rvw on BDC Motor Control hough ts applcaton to th EPAS s rlatvly nw, BDC otor control has bn an actv subjct n tchncal publcatons snc th 980s. Control tchnus foun n publcatons arss sp/currnt-tracng prforanc, robustnss to paratr varaton, toru rppl, saturaton an othr ssus rlat to BDC otors. Blow s an ovrvw of rfrncs ost closly rlat to ths wor. Pllay an Krshnan Pllay 989 prsnt a coornat ol for an nustral snusoal PMSM rv an a thr orr stat spac ol for ts sp control applcaton. Sulaton rsults wr shown for a puls wth oulaton PWM currnt control an hystrss currnt control. Rfrnc phas currnts wr transfor fro th sr currnts n coornats, whch n turn wr trn by th sp rror an toru coan. Plczws an Kunz Plczws 990 sgn an optal controllr to arss th voltag saturaton of BDC otors. Insta of usng a coornat ol, a 4 th orr stat spac ol was us for th BDC otor ynacs nclung rotor chancal ynacs an stator col lctrcal ynacs. h stator currnt ynacs wr splf as on of th stat uatons. Matsu an Ohash Matsu 99 vlop a gtal sgnal procssor DSP bas aaptv controllr for a BDC otor. Spac Vctor Puls Wth Moulaton SVPWM was shown to b supror to th Hall-snsor bas vctor slcton PWM n trs of currnt control prforanc. h DSP controllr was runnng at 00μs. h rotor

28 poston fbac was fro a rsolvr upat at vry 800μs an ntrpolat vry 00μs. In ths wor, th PWM fruncy was st to 0Hz. Hoang t al. Hoang 994 rctly us th nvrs of th coornat ol of th BDC otor to calculat coan voltag. h rvatv was approxat by bacwars-fnt ffrnc ol. In aton, an ntgral of fbac currnt rror was plnt for control corrcton. K t al. K 995 propos an aaptv currnt controllr for th PMSM. It was shown that paratr staton was robust to othr un-stat paratr rror, an ths controllr coul b us to stat otor sp f snsorlss control was sr. ow t al. ow 996 fn a otor ntty bas on th BDC currnt fruncy contnt. It was shown that an optal rv currnt coul b trn n ths way, whch gav sooth an axal toru. Sozr t al. Sozr 997 copar rct ol rfrnc aaptv control DMRAC wth nrct ol rfrnc aaptv control IMRAC. DMRAC gav goo rsults on sturbanc rjcton n loa, st pont an paratrs. IMRAC ha ffcults whn paratrs chang fast. Chn t al. Chn 006 sgn a two-gr-of-fro controllr for th BDC otor currnt tracng. hs controllr show bttr sturbanc rjcton than proportonal ntgral controllr. Howvr, snc th axs ynacs an bac EMF wr fn as sturbancs, th copnsaton through fltrng th fbac woul b latr than rct staton of th bac EMF an axs currnt. hrfor, t suffr th sa probl as th PI controllr, though to a lowr xtnt.

29 Rahan t al. Rahan 003 propos an aaptv bacstppng control for a PMS otor, an show t was globally asyptotcally stabl usng yapunov s rct tho. Caut t al. Caut 00 rv a robust controllr for a class of nonlnar syst wth paratr varaton bas on th lnar atrx nualty MI approach an polytopc ol. Stablty was analyz usng a paratr-pnnt yapunov functon an th global stablty was prov n th prsnc of th nonlnarty that was gnor urng th lnarzaton procss. hs controllr was plnt on an nucton otor an sulaton show asyptotc tracng of th sp trajctory. Forra t al. Forra 00 stu robust control for BDC otors n th prsnc of control voltag saturaton. A BDC otor ol was ntf as a n orr transfr functon fro rfrnc currnt to sp output by usng th Auto-Rgrssv Xognous ARX tho. Gan schulng, bas on th sp rror, was us for th control stratgy. Chn t al. Chn 000 propos a cobnaton of a PI controllr, fforwar controllr an a robust controllr bas on rct sturbanc cancllaton for th BDC otor phas currnt control. hs controllr yl fast phas currnt rspons for both trapzoal an snusoal BDC otors. Rubaa an Kotaru Rubaa 000 stu a thr-layr fforwar artfcal nural ntwor FANN an a ynac bac propagaton DBP nural ntwor controllr for a BDC otor applcaton. An aaptv onln tranng stratgy was propos for th FANN. It convrg uch fastr than th DBP larnng algorth wth a constant larnng rat. h stablty of th nural ntwor controllrs was not scuss. 3

30 Ptrovc t al. Ptrovc 000 stu th 6 th an th orr haroncs of th prannt agnt flux n BDC otors an thr ffct on th toru rppl. Snc th coffcnts of ths haronc coponnts wr changng wth t, th authors propos a yapunov stabl aaptv staton algorth. Sulaton show al cancllaton of th toru rppl. Howvr, thy pont out th practcal prforanc woul b lt by th saplng of th controllr nputs an outputs. A lowr sp lt occurr wth ncras lay n th sp asurnt. Othr harwar ssus, l a t an swtch voltag rop copnsaton, ha to b solv for succssful controllr plntaton. hr controllr was runnng on a MS30C3 floatng pont DSP at a saplng ntrval of 500μs. Chn an ang Chn 999 propos a slng-o controllr for BDC otors. hs controllr was plnt as harwar logc crcut, an thrfor ha ssntally no saplng lay probls. st ata show fast currnt stp rspons wth controllr plnt on a FPGA. n an n n 999 propos a robust controllr by cobnng an ntgral proportonal tracng controllr an an aaptv uncrtanty obsrvr. A lup uncrtanty was fn to captur th paratrc an nonparatrc ol uncrtanty. A yapunov functon of th tracng rror was us to rv th aaptaton law. Snsorlss control s on of th latst trns n BDC otor control publcatons ashta 994, Rahan 003, Hau 004, 004, Bolognan 00, K 003, Johnson 999 tc.. Hr th wor snsorlss ans that no rotor poston snsors ar us. h rotor poston an sp ar stat usng th bac EMF asurnt or othr nrct thos. Howvr, ost of ths publcatons ncat unrlabl sp 4

31 an angl staton at low rotor sp, whch s an nhrnt probl of snsorlss staton algorths. hs s unsrabl for autootv strng syst applcatons, consrng th stuaton that assstng toru s rur whl th rvr hols th strng whl at so crtan angl. Many of th BDC otor control rfrncs abov wr bas on th coornat ol. h coornat ol has bn wll stablsh for AC otors nclung nucton, synchronous, an PMS otors Kraus 986, Pllay an Krshnan 989, Rahan an Zhou 994, Yang t al. 003, an tc. hs AC otors usually ploy a snusoal AC powr supply at a fx fruncy that s uvalnt to th rotor sp. h coornat transforatons fro th phas ol to th coornat ol cancl th rotor angl nvolv n th ynac uatons., thus th coplxty of analyss s ruc. BDC otors o not hav such -ol-frnly powr supply. hus, control vloprs ust progra th nvrtr to approxat th AC powr supply usng th rotor angl asurnt. hs usually rurs nowlg of th nvrtr an powr lctroncs. In so stuatons, such as n th applcaton to EPAS, t s oftn srabl to hav a full ol of th BDC otor nclung th nvrtr an th PMS otor so that th control progra can b tst n sulaton. Unfortunatly, ths typ of ol only appars n vry lt publcatons. For xapl, Hossan an Dshpan Hossan an Dshpan 003 vlop a tal BDC otor ol n Suln wth thral graaton, coggng toru, frcton loss an gtal controllr uantzaton phnona. Sulaton rsults ar shown for 000rp an N loa toru. h sulaton wth non-alts atch wll wth tst bnch asurnts. Urasa t al. Urasa 000 5

32 analyz powr loss factors n a BDC otor, an ntf th corrsponng toru loss uatons. In suary, any control tchnus hav bn provn ffctv n BDC otor control applcatons. It s not uncoon that ultpl solutons xst for th sa probl. Howvr, t s notworthy that aaptv control has bn a popular choc for BDC otors, spcally snusoal BDC otors..5 Aaptv Control for Snusoal BDC Motors Snusoal BDC otors ar on of th ost sutabl actuators for th EPAS applcatons bcaus of thr hgh rlablty, low antnanc cost an clos-to-dcotor ynac prforanc. any othr chancal or lctrcal systs, lctrc otor charactrstcs vary aong nvuals n th sa ol an fro th sa anufacturng procss. Charactrstcs ay also chang wth srvc lf an nvronntal factors such as tpratur. hs s a srous potntal probl for th EPAS applcaton consrng th autootv ass proucton, th xpct long srvc lf, an th harsh worng nvronnt. hrfor, t s crtcal for th EPAS to b upp wth a controllr that s abl to achv th assstng toru gnraton tas an b robust to otor paratr varaton. h otor paratrs ost lly to vary nclu col rsstanc, col nuctanc, toru constant. h varatons rctly part naccuracs to th ol-bas control sch u to ts us of th nonal paratr valus. In any practcal applcatons of otor control, t s oftn th cas that cost an sgn consratons prohbt th us of snsors plac rctly on th otor wnngs or th agnts to ontor paratr varatons. o nsur auat toru control an accptabl fruncy oan 6

33 prforanc, t s srabl to copnsat th controllr for varatons n otor paratrs. Usually, th paratrs chang orrs of agntu slowr than th otor lctrcal ynacs. Aaptv control appars to b a favorabl choc for th snusoal BDC otor applcaton n EPAS. In gnral, an aaptv controllr conssts of a control law that s sgn usng nonal plant paratrs, an an aaptaton law that stats an upats th paratrs usng plant stats an/or output fbac, as shown n Fgur.5. h aaptaton ps th control law upat for th varyng charactrstcs of th plant, an antans srabl clos loop prforanc vn whn th plant s changng. Aaptv control s a wll-vlop branch of control thory. hr xst a larg nubr of rfrncs covrng th topcs of paratr staton an aaptv control Ioannou an Sun 996, ao 004, jung 998, tc. t spac n ths ssrtaton woul not allow a survy of all ths aaptv control tchnus. Howvr, so basc concpts wll b scuss, an th focus wll b on so spcfc aaptv tchnus that ar potntal canats for th snusoal BDC otor control n EPAS applcatons. θt Aaptv aw rt Controllr ut Plant yt Fgur.5 ypcal aaptv controllr bloc agra 7

34 Dpnng on how th control law an aaptaton law ar sgn, aaptv control can b classf as rct or nrct. In th nrct aaptv control approach, th plant paratrs ar us xplctly to calculat th controllr paratrs. h aaptaton law stats th plant paratrs, an thrfor nrctly upats th controllr. In contrast, th rct aaptv control ngags th plant ol n th controllr paratrs plctly. h aaptaton ajusts th control law paratrs wthout calculatng th plant paratrs. In autootv control applcatons, t s oftn srabl to hav on-ln plant ol nforaton for control an agnostcs purposs. h nrct aaptv control obvously fts ths rurnt bttr. Mol rfrnc aaptv control s on of th an approachs to aaptv control. A rfrnc ol s sgn for al prforanc of th clos loop syst wth consraton of th plant ynacs. h controllr rvs th plant output to trac th output fro th rfrnc ol. h aaptaton law uss th controllr coan, plant output an th rfrnc ol trajctory tracng rror to upat th controllr paratrs. In th EPAS applcaton, th al cas s that th BDC otor prfors l a brush DC otor. hs ncats that th MRAC s a lly a canat for th BDC otor control, an th rfrnc ol s that of a brush DC otor. h snusoal BDC otor s a ult-nput ult-output MIMO nonlnar syst an t varyng as shown n ts coornat ol.. Aong th aaptv tchnus, yapunov s Drct tho was oftn us to rv th aaptaton law for nonlnar systs an to prov th stablty of th clos loop syst. h Extn Kalan Fltr EKF has bn an ffctv way for both stat an paratr staton n nonlnar systs. It calculats th optal Kalan gans on-ln for th lnarzaton of 8

35 th syst, an s rlatvly optal at stay stat. hs two thos ar canats for th aaptaton law for th snusoal BDC otor aaptv control. In rcnt yars, nural ntwors wr prsnt n nurous publcatons for thr applcaton n aaptv control Rovthas 999, Patno an u 000. Varous typs of nural ntwors wr ngag to approxat th unnown plant ynacs an th ntwor wght factors wr aaptvly upat on-ln. Howvr, th snusoal BDC otor ol. has bn wly us an prov ffctv for olng ts ynacs. It s not ncssary to us such a blac box ol, thrfor ths tho s not consr n ths rsarch. In th followng paragraphs, a fw rfrncs publsh n th past ca wll b rvw, whch ngag th aaptv tchnu canats for th snusoal BDC otor applcaton. Zhang t al. Zhang t al. 000 stu a class of frst-orr nonlnarly paratrz systs. By utlzng a spcal proprty of th systs consr, an ntgral-typ yapunov functon was ntrouc to construct a yapunov-bas controllr an paratr upatng laws. It was shown that globally asyptotc tracng coul b achv, an xplct transnt bouns on th tracng rror wr prov for ffrnt chocs of yapunov functons. Hotzl an Karsnt Hotzl an Karsnt 998 prsnt an aaptv fbac tracng stratgy for a class of uncrtan sngl-nput/sngl-output systs n strct paratrc fbac for wth nonlnar t-varyng paratrzaton. h tracng sch was bas on a bacstppng sgn. A local stablty rsult was obtan va yapunov argunts. 9

36 An aaptv control sch was propos Marno an o 999 to solv th asyptotc tracng output fbac probl for a class of obsrvabl, nu phas, nonlnar systs wth output pnnt nonlnarts ultplyng t-varyng paratrs. Proof of asyptotcal stablty was achv by showng yapunov stablty of th syst. oh t al. oh t al. 003 propos an aaptv controllr that nvolv two tunng functons that wr trn by a n-ax optzaton approach. h propos algorth was shown to b yapunov stabl an capabl of achvng zro tracng rror n stay stat. Zhang t al. Zhang t al. 003 propos a bacstppng controllr for lnar t varyng V systs wth nown an unnown paratrs. h controllr was rv by a srs of yapunov canat functons, global stablty was guarant by choosng crtan sgn paratrs proprly. Zhang an Ioannou Zhang an Ioannou 000 prsnt a nw crtanty uvalnc bas aaptv controllr by a cobnng bacstppng bas control law wth a noralz aaptv law. h nw aaptv controllr guarant stablty an prforanc, as wll as paratrc robustnss for th non-aaptv controllr wthout th us of hghr orr nonlnarts. ao an Chn ao an Chn 000 prsnt an xponntally stabl aaptv copnsaton for Coulob frcton n a spl srvo control syst. Stablty was prov usng th yapunov stablty thor. h propos sch prov xponntal convrgnc for th Coulob frcton coffcnt staton an stat tracng rrors vn wthout prsstncy of xctaton. 0

37 Kosatopoulos an Ioannou Kosatopoulos an Ioannou 00 propos a swtchng aaptv controllr for ult-nput nonlnar systs whos ynacs wr nonlnarly affct by xtrnal nput sturbancs. By ang us of th noton of robust control yapunov functons an a of vrson of th swtchng aaptv controllr t was shown that th propos controllr guarant boun clos loop sgnals an convrgnc of th stat to a rsual st. anon an saals anon an saals 000 arss th ol rfrnc aaptv control probl of lnar t-varyng plants. A grant-bas aaptv law wth projcton an noralzaton was rv to stat th unnown controllr paratrs. It was shown that, for a class of possbly fast t-varyng plants, bounnss of th clos loop sgnals an sall tracng rrors n th an-suar noralz sns coul b achv, prov that only th unstructur part of th sr controllr was slowly t-varyng. Qu Qu 00 propos that, spt of thr nonlnarty an t varanc, uncrtants or thr bounng functons coul b stat as long as thy wr gnrat by xosysts whos ols wr thr nown or partally nown. hs was ralz by fnng a control algorth that satsf functon, Q. Q & < 0 of th yapunov canat Mlan an Bortoff Mlan an Bortoff 999 prsnt an obsrvr bas aaptv control through bacstppng control approach, whch nsur asyptotcal stablty. Exprntal coparson wth a full stat fbac controllr show bttr transnt prforanc an sallr stay stat rror of ths obsrvr-bas controllr.

38 Jang an Hll Jang an Hll 999 prsnt a constructv robust aaptv nonlnar control sch that coul b classf as a robustfcaton of th aaptv bacstppng algorth. Sulatons of a spl pnulu wth unnown paratrs an wthout vlocty asurnt llustrat th prforanc of th controllr. Gobbo t al. Gobbo t al. 00 propos a snsor falur tcton an ntfcaton sch by usng an EKF to stat th fault-rlat paratrs, whch wr procss by a cson algorth to tct possbl falurs. Exprntal rsults, by applyng ffrnt typs of falurs on th snsors of th nvrt pnulu, valat th ffctvnss of th approach. Zn t al. Zn t al. prsnt an ffcnt scrt-t scon-orr ol of an nucton otor for th rotor flux an ral-t paratr staton usng an EKF. Exprntal rsults show grat accuracy an fast convrgnc of th stat paratrs. In concluson, yapunov s Drct tho s stll on of th ost coonly us ways for on-ln paratr staton n varous latst aaptv control applcatons. h Extn Kalan Fltr appars to b an opton for rvng nrct aaptaton law for any nonlnar systs. MRAC has bn on of th an approachs to aaptv control. hs thos wll b valuat for th nrct aaptv control applcaton on th snusoal BDC otor..6 Publcatons hr journal paprs an two confrnc paprs hav bn publsh urng th author s octoral stuy. h sngl paratr staton algorth usng th -axs currnt ynacs was prsnt n a SAE confrnc papr Zhu an Patanar 004. h ult-paratr staton algorth usng th -axs currnt ynacs procss was

39 prsnt n Arcan Control Confrnc 004 Patanar an Zhu 004. h ultparatr staton algorth wth th Gra-Scht procss was publsh n th Intrnatonal Journal of Vhcl Autoaton Systs Zhu an Patanar 006. h paratr staton algorths can also b plnt for agnostcs an actuator halth ontorng. A papr on ths topc was publsh n th Intrnatonal Journal of Autoaton an Control Patanar an Zhu 007. Anothr papr about olng an sulaton of a sngl cylnr ntrnal cobuston ngn Chang, Zhu an Patanar 007 was publsh on th rns n Appl Scncs Rsarch Journal..7 Suary Elctrcal Powr-Assst Strng systs wll lly b us for th futur powr strng systs bcaus of ts avantag of nrgy ffcncy, flxblty an rlablty. h snusoal brushlss DC otor has bn ntf as th ost sutabl canat actuator for th EPAS. h long srvc lf, harsh worng nvronnt an ass proucton pos otor paratr varaton probl for th EPAS actuator controllr. Aaptv control s an al tchnu to arss ths probl whl achvng th control goals. Spcfcally, th nrct aaptv control, th ol rfrnc aaptv control, th yapunov tho, an th Extn Kalan Fltr ar consr as canat aaptv tchnus for th snusoal BDC otor applcaton. hy wll b xplor n tal n latr chaptrs. h ssrtaton s organz as followng: Chaptr wll scuss th -ol of th snusoal BDC otor n tal. Bas on th -ol, Chaptr 3 wll prsnt svral nrct aaptv algorths that paratr ar stat by solvng algbrac uatons forulat by svral loops currnt fbac. In Chaptr 4, rcursv last suar 3

40 algorth an xtn Kalan fltr wll b rv for th paratr staton probl. Chaptr 5 wll prsnt a ol rfrnc aaptv controllr for th BDC otor applcaton. In Chaptr 6, th algorths vlop n Chaptr 3 to Chaptr 5 ar copar n clos loop sulaton of an EPAS ol an a sp control applcaton. Chaptr 7 wll scuss so practcal control plntaton ssus though sulaton of a uas-physcal ol of th BDC otor syst, nclung coponnts such as th nvrtr, th spac vctor puls wth oulaton SVPWM, an tc. Chaptr 8 wll conclu th ssrtaton an scuss so futur rsarch rctons. 4

41 Chaptr SINUSOIDA BRUSHESS DC MOOR MODEING A BDC otor s th cobnaton of a prannt agnt synchronous otor PMSM, an H-brg DC-AC nvrtr, a rotor poston fbac chans, an a gtal controllr. Fro a usrs prspctv, th otor only ns DC powr an os not hav coutaton vcs such as brushs, thus t s call a Brushlss DC otor. h controllr s usually sgn n slf-controll o. ogthr wth th H-brg nvrtr, t gnrats AC currnt n ach phas of th prannt agnt synchronous otor wth a DC powr supply. Assung al opraton of th nvrtr, a BDC otor s actually a PMSM otor. h rct uaratur ol, whch s a wll-stablsh ol for AC nucton otor an synchronous otor, can b us for th BDC otor ynacs analyss an control sgn. Snc ths rsarch anly concntrats on th snusoal BDC otor, an consrng th fact that ost snusoal BDC otors ar thr-phas Y-connct, a -axs ol for such otors wll b rv n ths chaptr, an toru/currnt control algorths bas on ths ol wll b scuss.. Prannt Magnt Synchronous Motor Molng h actual otor n a snusoal BDC Motor s a prannt agnt synchronous otor, whch conssts of a rotor wth prannt agnts an svral phass of snusoally strbut stator wnngs. A 3-phas PMSM ol s prsnt n ths scton an a -coornat ol s rv fro th 3-phas ol. 5

42 .. h thr-phas ol h lctrcal ynacs of a thr-phas PMSM otor can b ol as Khorra 003, Kraus 00: Φ3 V 3 R3. t V [ ] [, ], an [ Φ, Φ Φ ] whr, V V, 3 V, 3 3, 3 Φ ar th vctors of 3, th phas voltags, currnts, an fluxs rspctvly. R s th phas rsstanc. h flux 3 lnag vctor Φ 3 s gvn as Φ f f,. whr s th nuctanc atrx of th stator cols an f 3f f.4 3 f ar th uvalnt nuctanc of th prannt agnt on th rotor. h uantty f s th fcttous currnt u to th prannt agnt. h nuctanc trs n.3 an.4 can b calculat as cosn, a g p θ π a g cosnpθ, 3 π 33 a g cosnpθ, 3 6

43 a π g cosnpθ, 3 a π 3 3 g cosnpθ,.5 3 a 3 3 g cosn p θ, cos n, f 0 p θ π f 0 cos npθ, 3 π 3 f 0 cos npθ. 3 whr,,, an ar postv constants, n s th nubr of agnt pol a g 0 pars, an θ s th rotor poston. h frst trs of ths nuctanc trs stan for th nuctanc wthn th loop for by th stator ron an ar gap btwn th stator an th rotor, an thy ar nvarant to rotor poston. h scon trs rprsnt th nuctanc n th loop for by th stator ron an th prannt agnts. hy ar snusoal functons of th rotor angl bcaus th prannt agnts ar rotatng wth th rotor. h lctroagntc toru gnrat by th otor can b calculat as p θ 3f 3f.6 whr an s th nuctanc atrx 3 3 f.7 3 f ff an ff s a postv constant assocat to prannt agnts. It s notworthy that th nuctanc atrx n.3 an.7 ar sytrcal, an th nuctanc trs contan snusoal functons of th rotor angl wth a phas shft of 7 π 3

44 to ach othr. hs s u to th fact that th stator phass ar sytrcally strbut n th stator cylnrcal nnr surfac, an th stator phass ar ntcal n trs of thr agntc an lctrcal charactrstcs. Usng.5,.7 can b xpan as. 3 sn 3 sn sn 3 sn sn 3 sn 3 sn 3 sn sn π θ π θ θ π θ θ π θ π θ π θ θ p p p f p p p p g p p p p g p n n n n n n n n n n n n.8 h uatons. an.8 can b rctly us for AC synchronous otors n whch th phas currnts an voltags ar snusoal at a fx fruncy that s uvalnt to th rotor agntc fl sp. Howvr, thy ar not convnnt for varyng sp applcatons such as that of BDC otors, as th rotor angl s xplctly nvolv n ths uatons. hrfor, t s srabl to splfy ths uatons. h ost coonly us tho s to transfor th fro th stator phas coornats nto th rct uaratur coornats... h rct uaratur coornat ol Bfor prforng coornat transforatons, lt us s how th coornats ar fn. t us us a thr phas sngl agnt PMSM otor for llustraton, as shown n Fg.. h coornats stan for th rct uaratur coornats, an ths coornat syst s fx on th rotor agnt. h rct axs s algn wth th agnt north pol axs, an th uaratur axs s 90 gr countr-clocws to th axs. Slarly, a coornat syst, not as th ab coornats, s fn as th 8

45 fx fra on th stator. h a axs s algn to th phas axs, an th b axs s 90 gr countr-clocws to th a axs. h orgns n both coornats ar th cntr of th rotor. In ths xapl, th rotor angl θ s th angular splacnt btwn th coornats an th ab coornats. If th rotor s upp wth or than on par of prannt agnts, t can b ol wth an uvalnt sngl par agnt rotor wth rotor angls of n p θ, whr n p s th nubr of agnt pars. b θ a 3 Fgur. h thr stator phas, th coornats an th ab coornats In th confguraton shown n Fgur., th zro angl θ0 s th rotor poston whr th axs s algn to th a axs. Howvr, th zro angl rotor poston s not unu an can b fn n othr confguratons. For xapl, t coul b fn whn th an a axs ar algn. h zro-angl poston ay s to b trval for th 9

46 control an analyss n th coornats. Howvr, t s ssntal for th synchronzaton btwn th stator th rotor agntc fls. It trns th phas of alost all th snusoal functons nvolv n th - ab coornat transforaton an ts nvrs transforaton. h fnton of th zro-angl poston ust b consstnt n th coornat transforatons spcally whn plntng th controllr sgn n th coornats. In th followng sctons, th a- algnnt confguraton show n Fgur. s aopt. o splfy th phas ol., lt us xpan th flux trs by substtutng.3~.5 nto., t..9 θ θ 3 3 3f 3 3 f V3 R3 Notc that [ ] [ ] [ ].0a an slarly 3f θ [ ] 0..0b In aton, snc th nutral pont s not accssbl n alost all Y-connct BDC otors, accorng to Krchhoff frst law, th su of phas currnts ust b zro: [ ] 0..0c 3 Apparntly, th suaton of all thr uatons n.9 gvs a trval uaton wth zro on both ss. hs ncats that only two of th thr phas uatons ar npnnt, an t s possbl to transfor th nto splr fors., 30

47 h currnts, voltags an fluxs n th thr-stator phass can b consr as vctors n th stator fx ab coornats. hy can b projct to th a an b axs wth a transforaton f,. 0ab c f 3 whr stanng for,, or f v Φ. [ ] f f a f b s th vctor of transfor 0ab f 0 uantts. h transforaton atrx c s gvn by c Notc that th transforaton atrx satsfs c c I. h coffcnt nsurs 3 that th transforaton antans nrgy consrvaton. hrough th transforaton., th nuctanc atrx 3 bcos ab c c 0 a g cosn pθ g snn pθ..3 0 g snn pθ a g cosn pθ 3 Also, abf c 3f cos npθ..4 3 sn npθ h gnrat toru can b wrttn n th nw coornats as 3

48 [ ] cos sn 6 cos 3 sn 3 θ θ θ θ θ θ θ θ p b p a f p p b a g p b a p g p f f c c c c c c n n n n n n n 0abf 0ab 0ab 0ab 0ab 3f h lctrcal ynacs can b xprss n th coornats as 0ab 0ab 0ab 0abf 0ab 0ab 0ab 0ab V R t f θ θ..6 h frst uaton n th 3 vctor uaton.6 s algbrac snc th lnts of th frst row of ar zro, an, as shown n.0c, 0ab hrfor, th frst uaton can b gnor n th analyss of th ynacs an n th control sgn. h scon an thr uatons prov ffrntal uatons that govrn th ynacs of an : a b sn cos 6 sn cos cos sn 3 θ θ θ θ θ θ p p f p p b p a p b p a g p b a b a b a n n n n n n n n R v v & & ab.7 wth cos sn sn cos 3 θ θ θ θ p g a p g p g p g a n n n n ab..8 h orgnal thr phas ol. an.8 ar transfor nto th splr fors of.5 an.7 wth only two ynac uatons. Howvr, th rotor angular splacnt θ stll xplctly prsnts n th uatons. hs uatons ar stll nconvnnt for analyss an control sgn. 3

49 Rcall that th coornat syst s fx on th rotor, an th projcton fro th ab coornats to th coornats contan th snusoal functon of θ. It s possbl that th poston pnnc of th toru xprsson.5 an th lctrcal ynacs.7 can b lnat through th ab- projcton, whch s gvn by a transforaton f..9 0 pf 0ab whr 0 0 p 0 cos n pθ sn n pθ,.0 0 sn n pθ cos n pθ an slar to.. Cobn. an.9, h coornat varabls can b obtan fro th orgnal thr phas varabls by f f wth π π pc cos npθ cos npθ cos n 3 pθ π π sn npθ sn npθ sn n θ 3 p 3 Notc that I, whch ans ts nvrs s ual to ts transpos. Usng th transforaton., th nuctanc atrx n th coornats can b obtan as a g 0, a g an th nuctanc trs for th rotor prannt agnt n th coornats bco 33

50 f 3f Notc that both an o not xplctly pn on th rotor poston, an thy 0 0f ar uch splr than thr countrparts n th phas ol as shown n.3 an.4. o obtan th ynac uatons n th coornats, car ust b tan snc th transforaton atrx conssts of functons of th rotor angl. Not that Φ 3 3 3f f f f,.5 t t t an rcall that f s a constant, th lctrcal ynacs n th coornats can b rv fro. as 0 0 t 00 0f f V0 R0..6 t Notc that np t 0 0 Agan, th frst uaton n th 3 vctor uatons.6 s algbrac, an thrfor can b gnor. h ynac uatons n th coornats can b foun as t t R R n p p n K v, v..8 Slarly, th toru xprsson n.5 can b transfor nto th coornats as 34

51 6 3 n p g n p f..9 Introucng th notaton K, 3 a 3 a K 3n,, p g 3 n p g g f,.30 th lctrcal ynac ol of th BDC otor transfor nto th coornats as R np t R np t K K. K v, v In th toru uaton, th frst tr, K.3 ncats th contrbuton of th stator nucton flux, whl th scon tr K s th contrbuton of th prannt agnt flux. In th noral otor sp rang, th prannt agnts play th onant rol n th flux lnag. Hnc, n any applcatons th tr K s uch lowr than th K, an thrfor can b nglct. h toru uaton bcos K..3 h coornat ol of.3 s poston npnnt an thrfor or convnnt than th 3-phas ol of. for analyss an control purpos. It has bn us n any rfrncs for BDC otor control sgn. Slarly, all control algorths vlop n latr chaptrs wll b bas on ths -coornat ol. 35

52 ..3 Iplntaton of coornat controllrs Bfor ovng onto th control sgn topcs, th practcal plntaton of a controllr sgn on th coornat ol srvs a lttl bt or attnton. h coornat ol n.3 s poston npnnt, thus t s frnly for control vlopnt. Howvr, all physcally accssbl varabls fro a otor, such as currnt, voltag an bac EMF, ar n th stator phas oan. o practcally plnt a controllr sgn on th coornat ol, th coornat transforatons n. an.0 ust b on n ral t. hus, an actual BDC otor control syst ns a fw or functons than th coornat controllr alon, as shown n Fgur.. h asur phas currnts ust b transfor nto currnts n coornats, through two coornat transforatons. hn th controllr output voltags v, v ust b transfor bac to phas voltag for th ral otor. h phas voltags ar snusoal functons of th rotor angular poston. Usually so spcal oulaton tho such as th spac vctor puls wth oulaton SVPWM s nvolv to plnt ths altrnatv voltags fro th DC powr supply. h oulaton rurs wll-coornat softwar logc an powr lctroncs harwar opraton. 36