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

53 Fgur. Iplntaton of a gnrc controllr sgn n coornats Evn though th coornat ol.3 os not xplctly pn on th rotor poston, alost all th supplnt functons rur th rotor angl θ asurnt n ral t. hrfor, accurat θ asurnt s crtcal for opraton of th BDC otor an plntaton of controllrs sgn on th coornat ol. Usually, θ s asur by usng transucrs such as optcal ncors or rsolvrs. h asur angular poston can b absolut or rlatv, but car ust b tan whn calculatng th ntal valu of θ. Rcall that th coornat transforatons n. an.0 ar bas on th zro-angl confguraton n whch th an a axs ar algn. Whn plntng thos transforaton calculatons n th controllr, th ntal valu of th 37

54 angl fbac ust b consstnt wth ths zro-angl confguraton. ransucr rsoluton s anothr factor for th plntaton of controllrs u to th ral t coornat transforatons. Only f th transucr rsoluton s hgh nough, th coornat transforatons can b consr al an transparnt to th controllr. For th control sgn purpos, w shall assu that ths supplntal functons for controllr plntaton ar al. Howvr, w n to b awar of th ununss of th BDC otor controllr plntaton coparng to rgular plntaton of a gtal controllr to an analog syst.. oru Control of Snusoal BDC Motors Whn a otor s us as th actuator n a control syst, usually th nput to th otor controllr s a coan toru fro th outr loop. Wth th coornat BDC otor ol, th -axs currnt trns th otor toru shown n.3. Gvn a coan toru c, th sr -axs currnt can b calculat as c c., K In any applcatons, t s a coon practc to forc to as low as possbl n apltu. In anothr wors, th sr -axs currnt shoul b zro,.., c 0. In so applcatons, snc th -axs currnt onats th otor toru, th sr -axs currnt can b us to srv othr purposs such as a rano xctaton sgnal for ntfcaton purpos. hus, th toru control probl s uvalnt to th currnt control probl whr th sr currnts ar trn th outr loop toru coan an othr factors. Varous control tchnus can b utlz to achv th currnt tracng tass. A PI controllr s probably th ost popular canat, spcally n nustral applcatons. 38

55 .. PI controllr h prforanc of a Proportonal an Intgral PI fbac controllr anly pns on ts paratrs, naly th proportonal gan p an th ntgral gan. h gans ar usually sgn on a lnar t nvarant I ol of th plant. If th plant s a nonlnar syst, a PI controllr ay b sgn for th lnarz syst ol aroun so ulbru stats, an th PI gans can b progra as outputs fro looup tabls, whch ar rvn by syst stats. In aton, a fforwar tr s oftn ngag to account for th nonlnarty of th plant syst. o sgn a PI controllr, th BDC otor ol s rstat hr for th control sgn purpos, t t R R n p p n K v, v. h abov uatons ar nonlnar snc th scon tr on th rght han s of ach uaton nvolvs prouct of stats, an. h varabls of ntrsts ar an. narz th abov uatons wth rspct to th an, 0 Δ R t Δ R t Δ n Δ n p p Δ K Δ v v 0 0 Δv 0. Δv, h ulbru pont s 0 R0 np0 0 K 0 v,.34a v R n..34b 0 0 p

56 h PI controllr s sgn wth rspct to Δ t Δ t R Δ RΔ n Δ p p 0 0 n Δ Δv Δv,..35 Gvn sr currnts an, th PI controllr shall stablz th otor currnt stats aroun ths valu. In anothr wors, th sr ulbru pont shall b, c, c,, an th proportonal an ntgral gans shall b sgn to achv crtan ynac prforancs such as ovrshoot, rspons t tc. h fnal control law s thrfor propos as, c, c v R, c np 0, c K 0 p, c, c t,.36a v R, c np 0, c p c, c t..36b An xapl PI controllr s sgn for a otor R 0. 05Ω, H, K N A. h PI gans for th rotor agntc fl rotatng sp at 0 0 ra/s, 0 50 ra/s, 0 00 ra/s, 0 00 ra/s ar sgn usng th MAAB root locus tool. h PI gans can b plnt wth looup tabls that ar rvn by th otor sp, an th PI controllr s oftn call as a gan-schulng controllr Ioannou 996. It provs so aaptaton for prforanc loss caus by syst nonlnarts an changng stats. h PI controllrs ar spl an robust to syst uncrtants. Howvr, t rls on ntgraton of th rror btwn th fbac stats an thr rspctv rfrnc to countract th sturbancs. hs sots ans sacrfc of ynac prforanc. 40

57 abl. PI controllr gans 0 ra/s p Fforwar nvrs ynacs controllr o us a BDC otor as an actuator n a chancal syst such as autootv strng syst, t s srabl to hav th otor controllr b th xact nvrs of th otor lctrc ynacs. hus th otor wll prouc th toru rust fro th outr loop controllr. In anothr wors, th controllr an otor togthr bco a untarygan f-forwar gan n th control loop. In narly all practcal applcatons, th otor rvs a chancal nrta as a part of ts loa. Usually th ynacs of th chancal syst ar orrs of agntu slowr n coparson to th lctrcal ynacs of th otor. hrfor, for th control purpos, th lctrcal ynacs of th otor can b nglct n coparson to th ovrall ynacs of th syst,.. t t R R n p p n K v v 0. 0, Gvn coan currnts an, th nvrs ynacs control law s propos, c, c as 4

58 v v K R c R c, c c, c. c c,.37 Assung al nowlg of th plant, an substtutng.37 bac nto th plant ol, th clos loop syst ynacs bcos t t R R R R, c,, c..38 h clos loop syst ynacs ol.38 conssts of two frst orr low pass fltrs wth th t constants trn by th stator phas paratrs. For th xapl otor nton bfor R 0. 05Ω, H, K N A, th t constants for both th an axs currnt ynacs ar 0.00s. hs s usually fast nough for ost chancal systs. Fgur.3 an.4 copar th stp rspons of th two controllrs n sulaton. h rfrnc currnt has a stp ncras of c A at 0. scons. Zro rotor ovnt was assu for th purpos of currnt ynacs prforanc coparson. In th frst sulaton shown n Fgur.3, t s assu that th controllr paratrs atch th plant paratrs ally. Both controllrs prov uc rspons to th stp nput whn full nowlg of th plant was assu. h PI controllr show slght ovrshoot u to th ntgraton of currnt rror. In Fgur.4, th controllr paratrs has 0% rror n K an 50% rror n R coparng to th plant. Wth paratr rrors prsnt, t too about half scons for th PI controllr to countract th rror an achv zro stay stat rror, whl th fforwar controllr coul not copnsat th stay stator rror caus by th 4

59 paratr rror. 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...3 Controllr slcton h clos loop syst prforanc var or or lss fro on controllr to anothr. h controllr slcton crtra oftn pn on th spcfc applcaton. In th cas of th autootv powr-assst strng syst, t s srabl for th assstng otor to prov a consstnt sns of stffnss to th rvr. Bss, th chattrng ffct of assstng toru shoul b avo, as huan hans ar vry snstv to vbratons. Bas on th sulaton rsults shown n Fgur.3 an.4, th PI controllr ay not b 43

60 abl to prov satsfactory prforanc n both rspcts. Gvn al nowlg of th plant, fn-tunng th gans or ang a rvatv tr ay prov th ovrshoot probl. Howvr, f sgnfcant paratr rror xst btwn th controllr an th plant, th slow copnsaton of th PI controllr shown n Fgur.4 woul a th rvr s flng of th strng whl stffnss vary n th scal of scons. hs s hghly unsrabl. 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. On th othr han, th fforwar otor nvrs controllr provs favorabl 44

61 rsponss for th EPAS applcaton. If thr xst paratr valu rror n th controllr, rvr ay fl that th strng whl s havy but t s consstnt. In aton, f plnt wth so aaptaton tchnus, whr th paratrs nvolv n th control law can b upat wth th actual plant, thn th fforwar controllr provs or srabl prforanc for EPAS. Consrng th ass proucton of th EPAS an ts xpct long srvc lf n such harsh nvronnt, th otor charactrstcs varaton s narly nvtabl across th sa prouct ol an ovr th srvc lf of ach nvual otor. h aaptv fforwar ynacs nvrs controllr appars to b a favorabl canat for th BDC otor n th EPAS applcaton..3 Suary h -coornat ol has bn wly us for th BDC otor control sgn an analyss. On of th an avantags of th -coornat ol s that th rotor angular poston θ s not xplctly nvolv n th otor ynacs uatons. As a rsult, th control an ynacs analyss usng th -coornat ol ar sgnfcantly splr than usng th phas ol. hs chaptr r-xan how th -coornat ol was rv fro th thr-phas prannt agnt synchronous otor ol. h athatcal rvaton rval how θ was cancll through th coornat transforaton fro th thr-phas fra to th -coornats. It also xplan how th an axs ynacs wr obtan an thr rols n th prncpl of th BDC otor opraton. In aton to prov a founaton to th control sgn, th rvaton nlghtn plntaton ssus for th controllr sgn wth th -coornat ol. For xapl, though th rotor angular poston θ s not xplctly ngag n th 45

62 -coornat ol, t s nspnsabl n th coornat transforatons that ust b on n ral t. hrfor th opraton of th BDC otor rurs asurnt of th rotor angular poston θ. Anothr xapl s th zro-angl confguraton that ss trval for th control an analyss n th coornats. Whn plntng a controllr sgn fro th -coornat ol, th zro-angl confguraton s crtcal to nsur th corrct calculaton of th coornat transforaton an synchronzaton of th stator an rotor agntc fls. Wth th -coornat ol, h BDC otor toru control probl can b convrt nto an uvalnt currnt control probl. hr xst nurous controllr canats for th otor currnt control tas. Whl t s a subjctv attr as to th slcton of control structur, th aaptv fforwar nvrs ynacs controllr show srabl prforanc, an thrfor s slct for as th favorabl controllr for th EPAS applcaton. 46

63 Chaptr 3 ADAPIVE PARAMEER ESIMAION Whl sgnng an nvrs otor ynacs controllr such as that of.37, t s assu that th otor paratrs R,, an K ar constant an th paratrs us n th controllr whl xcutng th otor nvrs ol ar th sa as that n th otor. Howvr, agng an changng nvronntal factors, such as tpratur, huty, tc. wll chang th valus of th paratrs R,, an K. hs can gra th ovrall syst prforanc. Oftn, t s ncssary to ta so corrctv acton so that th valu of on or or paratrs n th controllr s as clos as possbl to th actual valu n th otor. In ths chaptr, paratr staton algorths for a fforwar otor nvrs controllr ar propos as a soluton to th abov probl. Fro sngl paratr to ultpl paratrs, th staton an copnsaton schs ar rv an stablty of th staton schs s prov. Iprovnts for th paratr staton ar propos by tghtnng th boun of rror n th schs. 3. Motor Invrs Controllr Mol A BDC otor ynacs nvrs controllr 3. ha bn propos for th EPAS applcaton Klnau t al., 003, V V R, c, c cos δ K, 3.a c V V snδ R, c, c. 3.b whr δ s th phas avanc angl of th control voltag vctor to th axs n th coornat syst, an V s th agntu of control voltag. h phas la δ can b 47

64 st as a functon of th angular vlocty of lctro-agntc fl as δ tan. h R axs coan currnt, c s obtan fro th outr loop toru coan c as c, c. Not that K snδ, 3.a R an R cosδ. 3.b R h control voltag agntu V can b solv by 3.a cosδ -3.b snδ as V R, c RK, c. 3.3 R h axs coan currnt, c can b solv by 3.a cosδ 3.b snδ as, c K. 3.4 R c Euaton 3.3 nvolvs th otor paratrs R,, an K. Usually th nonal valus of ths paratrs ar us for th control sgn. Any varaton n ths paratr valus wll rsult n control voltag offst an thrfor syst prforanc suffrs. Aaptv paratr staton algorths wll b vlop for ths controllr n ths chaptr. t s start wth a splr cas of sngl paratr staton. 48

65 3. Sngl Paratr Estaton In any stuatons, th chang of th otor col rsstanc R s uch hghr f copar wth othr paratrs such as, an K. Assu that R s th sngl paratr that changs an affcts th otor prforanc. Intgraton of th fbac currnt rror sch to stat a otor paratr R n th controllr s nvstgat n ths scton. 3.. Sngl paratr staton In th followng paragraphs, th paratrs us n th controllr ar not by th suffx c. For th sngl paratr staton purpos, w assu that, an K ar nown. R c s th phas rsstanc us n controllr an t ay b ffrnt fro ts countrpart n th otor. Du to saplng lay an analog/gtal convrson, th asur otor vlocty c ay b slghtly ffrnt for th actual rotor vlocty. Wth ths notaton, th control law n 3. can b rstat as V V Rc, c c, c cos δ K, 3.5a c V V snδ Rc, c c, c. 3.5b Applyng th control voltags 3.5 to th otor ynacs ol, w hav t t R R c, c c, c R K c, 3.6a c, c c, c. 3.6b R Dfn th paratr an vlocty rror as Δ R R, 3.7a R c 49

66 Δ, 3.7b c Δ, 3.7c c an th fbac currnt rror as Δ,, 3.7 c Δ,. 3.7 c Assung th syst s n ulbru,.. 0, 0. Substtut 3.7 nto t t 3.6a, nglct th hgh orr rror trs, an rorganz th uaton to sgn, c Δ R RΔ Δ, cδ KΔ, c. 3.8 In noral BDC otor opraton, th currnt s sgnfcant hghr than. h vlocty rror caus by saplng lay s usually lowr than th currnt rror caus by th otor ynacs. hus th frst t on th rght s of 3.8 s th onant factor for Δ R calculaton. hrfor, th paratr rror stator s propos as sgn c ΔRˆ, RΔ. 3.9, c whr Δ Rˆ s th stat rsstanc rror. h controllr valu of th rsstanc can b upat by ntgratng th stat rror as R t t R t0 C t ΔRˆ t t, 3.0 c c t 0 whr 0 C t < s an ntgraton wghtng factor. 50

67 3.. Stablty of th sngl paratr staton sch ogthr wth th paratr staton algorth n 3.9 an 3.0, th fforwar controllr n 3.5 bcos an aaptv controllr bas on th currnt fbac. h stablty of ths clos loop control syst wll b scuss n ths subscton. R Assu th otor col rsstanc R s statonary,.. 0. hus, th rvatv of t Δ R can b foun by ffrntatng 3.6a as ΔR t R Rc t t R t c. 3. h rght s of uaton 3. can b xpan by ffrntatng 3.0. ang th boun zro-an atv nos t of th currnt fbac snsor nto account, th rvatv of Δ R can b obtan as ΔR t, c R t Δ t sgn, c R t Δ Δ K Δ sgn,, c 3. whr C t t. Euaton 3. rprsnts th frst orr nonlnar ffrntal, c uaton that fns th ynacs of th rror n th stat of R. Dffrntatng th currnt rror n 3.7 an 3.7, an substtutng t an wth t 3.6a an 3.6b, th currnt rror ynacs can b obtan as Δ t Δ t R Δ, c K Δ c Δ ΔR Δ, c, 3.3 t, R Δ, c c Δ ΔR Δ, c, 3.4 t, 5

68 Dfn a nw stat vctor n trs of stat x as x [,, R]. h thr stat uatons can b wrttn x& f t, x g t, x 3.5 whr R f t, x sgn, c R t sgn R, c t, c, c 0 x, 3.6 an KΔ, c Δ, c t, c g t, x Δ, c t sgn [ Δ Δ ]. 3.7, c t R t, c K x& f t, x s th nonal syst an g t, x s th prturbaton. Euaton 3.5 rprsnts th syst wth currnt fbac bas otor rsstanc staton. In orr to trn contons for th stablty of th nonal stat-spac ol, th followng thor s us Khall 996. hor t x0 b an ulbru pont for th nonlnar syst x& f t, x, whr n n f : [ 0, D R s contnuously ffrntabl, D { x R x < r}, an th Jacoban atrx f x s boun an pschtz on D, unforly n t. t f A t t, x x x 0. hn th orgn s an xponntally stabl ulbru pont for th 5

69 nonlnar syst f an only f t s an xponntally stabl ulbru pont for th lnar syst x& At x. In ths cas, f t, x n 3.6 s contnuously ffrntabl n x, whch satsfs th contons of th abov thor. [ 0,0,0] x s an ulbru pont for x& f t, x, an R / f x sgn co R t sgn co R / t co co / / Snc th otor sp an th otor toru co ar boun for any otor rvng a non-zro loa, f x s boun an pschtz on any oan fn ovr th opratng rgon of th otor. Applyng ths thor, t can b a sur that x& At x s xponntally stabl as rur by hor by sttng th valu of t as rur. Whn th otor rvs a loa whch s th cas n ost opratons, At s guarant to b Hurwtz f t > 0 s chosn. On rar occasons t s possbl that th otor s bac-rvn by th loa. Unr bac rvng contons, that th followng conton s satsf. t s slct such t R, c sgn, c R, c > hrfor, x [ 0,0,0] s an xponntally stabl ulbru pont for th nonal syst x& f t, x assung t s appropratly schul. If bac rvng s n a rar occurrnc, t can b a clos to zro urng bac-rvng to ffctvly stop paratr staton. hs wll not affct th prforanc of th staton sch. 53

70 Gvn th stablty of ulbru pont [ 0,0,0] x for th nonal syst x& f t, x, th stablty of th rror ynacs 3.5 can b shown by usng thor Khall 996. n hor t D { x R x < r} satsf for all t, x [ 0, D an suppos th followng assuptons ar : f t, x s contnuously ffrntabl an th Jacoban atrx pschtz n x, unforly n t. f x s boun an h orgn x 0 s an xponntally stabl ulbru pont of th nonal syst x & f t, x. 3 h prturbaton tr g t, x s pcws contnuous n t an locally pschtz n x, an satsfs th boun g t, x ρ, t t0 0, x D. t y t an yt not solutons of th nonal syst an th prturb syst, rspctvly. hn, thr xst postv constants β, γ, η, μ, λ an, npnnt of ρ, such that f ρ < η y t < λ, an y t y < μ, thn th solutons y t an, 0 0 t0 yt wll b unforly boun for all t t 0 0 an y ˆ ˆ γ t t t y t 0 y t0 y t 0 βρ. 3.0 In ths cas, g t, x s boun by g t, x ρ ax g t, x. 3. t Snc th sr otor toru, otor sp assung non-zro loa an acclraton ar boun, Δ, Δ,,, c, an, c ar boun n ral systs. In aton, t 54

71 th snsor nos t s also boun. hrfor, ρ ax g t, x s fnt for all t otors, an th contons of hor ar satsf. hus, th solutons of 3.5 an ts nonal syst wll b unforly boun an satsfy 3.0,.. th aaptv control sch for R wll rsult n a boun valu of stat R an th stat vctor x n c 3. s unforly boun Sulaton rsults h control law 3.5 an th aaptaton rul 3.9 an 3.0 wr tst n sulaton for a otor wth followng constant paratrs: R 0. 05Ω, K 0.05V ra / s, 0 4 H. hs otor s a prototyp BDC otor for an EPAS applcaton. It wll b us as a plant for all aaptv controllrs vlop n ths rsarch. Usually th clos loop otor syst nclung th otor an th otor controllr s th actuator for an outr loop syst, an th outr loop syst charactrstcs oftn hav so ffcts on th prforanc of th otor controllr. Howvr, snc our an ntrst s th otor controllr, w shall focus on th otor prforanc an assu that th otor oprats npnntly to th outr loop syst. In practc, ths s slar to a otor bnch tst n whch th otor can b oprat wth arbtrary sp an toru. hrfor, w assu that th otor subjcts to rano coan toru c an npnntly rano otor vlocty n sulatons. Fgur 3. shows a schatc agra of th clos loop otor tsts n sulaton. 55

72 Aaptv Controllr Paratr Estator Paratr, Rano Sgnal Gnrator c Control aw v,v M Motor out Fgur 3. h schatc agra of th otor tst syst n sulaton. h wghtng factor C n 3.0 s st to avo th sngular ponts of 0 an t, c to satsfy th stablty conton 3.5, as n th followng uaton 0.0 > 0, sgn sgn c > 0 C t. 3. 0, sgn sgn c 0 o vrfy th ffctvnss of th paratr staton algorth, th ntal valu of Rc s assu to hav 0% rror to R. Root an suar RMS valu of th stat varabls s us to approxat th boun on th rror as b Δ 6 RMS Δ 3.3 whr Δ rprsnts th rror n th varabls. Sulaton rsults of th sngl paratr staton ar shown n Fgur 3.. Rgarlss of th sgn of ntal rror, Rc convrgs to R wthn 0 scons an thn -4 stays n a boun ax nor of.78 0 Ω 0.34% R. 56

73 Fgur 3. Sngl paratr staton prforanc n sulaton 3.3 Multpl Paratr Estaton In any stuatons, col rsstanc R s usually th onant paratr that s changng n th otor. h sngl paratr staton sch s abl to ntfy ths sngl paratr chang n th otor nvrs ol. Howvr, whn thr xst rrors n or than on paratr, prforanc of th sngl-paratr staton schs wll trorat, an th accuracy of th control syst wll also suffr. In th followng subsctons, a ult-paratr staton sch s propos an th stablty of th sch s prov Multpl paratr staton t s nvstgat th scrt ntgraton of th stat paratr rror n th controllr: R R C t ΔRˆ, 3.4 c c 57

74 K c K C ˆ t ΔK, 3.5 c whr Δ R ˆ an Δ ˆ ar th stat rror, C an C ar K < < ntgraton wghtng functons, an nots th th ntrval. Agan, w not th paratrs an varabls us for coputaton n th controllr by suffx c. h stat rror Δ R ˆ an ΔKˆ wll b rv blow. Rcall th control law V V Rc, c c, c cos δ K, 3.6a c c V V snδ Rc, c c, c. 3.6b whr K an R ay b ffrnt fro thr countrparts n th otor. c c Substtutng th voltags n 3.6 bac nto th otor ynacs ol, w hav t R R c, c c, c K c c K, 3.7a t R c, c c, c. 3.7b R o fn th paratr rror uaton, lt s aopt th sa fnton of rrors n 3.7, Δ R R, 3.8a R c Δ K K K, 3.8b c Δ,, 3.8c c Δ,, 3.8 c Δ, 3.8 c Δ. 3.8f c 58

75 Assung th syst n ulbru: 0, 0, substtutng 3.8 nto t t 3.7a, an nglctng th hgh orr rror trs such as Δ RΔ, w gt Δ R, c ΔK RΔ Δ, cδ KΔ. 3.9 It s rasonabl to assu that Δ Rt an Δ K wll not chang sgnfcantly n on saplng pro of 0.00 scon. In th an t, for ost DC otors, << R an << K ; bcaus of chancal nrta, Δ << Δ. hrfor, th last 3 trs n th rght s of 3.9 can b possbly nglct f copar to th frst tr, w can fn th valu of Δ R an ΔK by solvng followng uatons ΔR ΔK RΔ, 3.30, c ΔR, c ΔK RΔ. 3.3 hrfor, th stat rror Δ R ˆ an ΔKˆ for th staton sch n 3.4 an 3.5 va fbac currnt rror ntgraton was propos as Rc Δ Rc Δ Δ Rˆ, 3.3, c, c ˆ Rc, c Δ Rc, c Δ Δ K. 3.33, c, c 3.3. Proof of stablty Subtractng R fro both ss of 3.4, th rror ynacs n R bcos Δ R ΔR C ΔRˆ o show th stablty of paratr staton n 3.34, w n to fn th ffrnc btwn th stat paratr rror an th actual paratr rror. Rcall 59

76 3.9 that approxats th rlatonshp btwn th actual paratr rror an th currnt rror n ulbru, an nclu th boun zro-an atv nos t of th currnt fbac snsor an th nglct otor lctrcal ynacs, t Δ R, c ΔK RΔ KΔ Δ, cδ t t Dfn a prturbaton gt as g t KΔ Δ, cδ t t Solvng uaton 3.35 at th ntrval an th ntrval, th actual paratr rror Δ R an ΔK can b solv as [ RΔ RΔ ] [ g g ] ΔR, c, c, c, c 3.37 Not that th frst t on th rght s of 3.37 s th sa as ΔR ˆ. In anothr wor, Δ R bcos t [ g g ] Δ R ΔRˆ. 3.38, c, c [ g g ]. 3.39, c, c Substtut th 3.38 an 3.39 nto 3.34, Δ R [ C ] ΔR C By nucton w hav 60

77 ΔR [ C ] ΔR0 E whr, E C j C j j. j 0 0, 3.4 Sttng 0 C < for ach controllr coputaton ntrval [, ], th frst t on th rght of 3.35 wll cras as. In so ntrvals n 3.33 s hgh bcaus of sngularty caus by vry low angular vlocty or coan toru nput. h ntgraton gan C can b st to b zro for ths ntrvals. Othrws, n a physcal syst, Δ,, co, t, an ar all boun,.. < δ for t so postv constant δ. t C ax [ C, C ] ax 3.4 an δ ax[ δ ], whr { C 0, 0,,, } ax. h scon tr E on th rght ss of 3.4 confors th nualty of Cax { C δ } ax ax ax j 0 Cax E δ t ax C b δ C ax ax. Snc 0 < C ax <, b δ ax C as. ax hrfor, th scon t on th rght s of 3.35 wll b boun as. In suary, th staton sch of R n 3.4 an 3.3 wll b boun. h bounnss of K can b prov slarly. h ult-paratr staton algorth n 3.4 an 3.5 s sulat for th sa otor nton n scton 3.. Fgur 3.3 shows th paratr staton prforanc 6

78 n sulatons. h ntal paratr rror wr st as Δ R t 0 ± 0% R an ΔK t 0 ± 6% K. o avo th rror caus by sngularty ponts n 3.3 an 3.33, C an C n 3.4 an 3.5 ar st as followng: 0. t 0.0 C C, t < 0.0 whr t, c, c. In th sulaton, th paratr staton rach th boun of -4 b Δ R % R an b Δ K % K wthn 80 scons. Fgur 3.3 wo-paratr staton prforanc n sulaton 3.4 Iprovng h Dynac Prforanc Of h Aaptv Algorth h sngl an ultpl paratr staton schs vlop n th prvous two sctons ar prov to b stabl an valat to b ffctv n sulatons. Howvr, 6

79 fastr paratr staton convrgnc s always wlco for th ovrall syst prforanc. h convrgnc sp can b prov by tunng th ntgraton gans C an C but wll rsult n nosr paratr staton at stay stat. h xtnt of provnt by gan tunng s lt by th traoff btwn convrgng sp an th boun of stay stat rror. A bttr way, wthout coprosng th ynac prforanc woul b to ruc th prturbaton to th rror ynacs ol. Rcall that both th sngl an ultpl paratr staton schs ar bas on th otor nvrs controllr 3.. h non-ynac otor nvrs ol nglct otor lctrcal ynacs an ntrouc an rror corrsponng to nglct ynacs t an t. If th ynacs ar not nglct to obtan an algorth for th appl voltag V, t s possbl to ruc rrors ntrouc n th arlr staton approach. Furthror, th basc staton sch of 3.9, 3.3 an 3.33 nglct th ffct of Δ an Δ, whch wr caus by th saplng lay. hs trs prsnt n th prturbaton 3.7 an If th otor sp rror s consr n th staton sch, th prforanc of th aaptv algorth can b possbly prov. In th nxt subsctons, ths two possbl provnts for paratr staton wll b xplor Motor lctrcal ynacs An analytcal soluton of th otor lctrcal ynacs of.3 s not avalabl bcaus of th nonlnar ts that nvolv prouct of stats, an. Howvr, snc th otor s rvng th nrta, th lctrcal stats can chang sgnfcantly n uraton of th orr of th lctrcal t constant, whras th chancal stats,.g. 63

80 th otor sp, can harly chang ovr th sa pro u to slow ynacs of th chancal syst. It s possbl to approxat th otor lctrcal ynacs by assung s a constant urng ach saplng ntrval, whch s of th orr of th lctrcal t constant. Wth ths assupton, th otor lctrcal ynacs uaton.3 bcos a fnt nsonal lnar t-nvarant stat uaton urng th saplng ntrval. hs uaton can b solv xactly va th scrt t stat transton atrx Rugh 996. Now consr.3 n atrx for as cosδ K ζ y& y R R u, 3.45 ζ snδ 0 R for t, whr V y,, u, ζ tan δ. R R Euatons 3.45 can b solv wth th stat transton atrx: Φ, t t cos sn [ t ] sn[ t ] [ ] [ t cos t ] 3.46 for t. h soluton s x[ ] Φ[, ] x Φ, θ Guθ, 3.47 whr cosδ G R sn δ R K R 0. h uantts G an u n th abov uaton can b trat as constants urng ach saplng ntrval bcaus of th zro-orr-hol saplng. 64

81 Gvn th coan toru, whch s th xpct output at t, th c valu of th stat varabl s xpct to b. hs valu can b, c obtan fro, c K c, an th unnown V an can b solv as K V, c I [ cos sn ] cosδ snδ I J R R R, 3.48a cosδ snδ K [ sn cos ] J I V J, 3.48b whr R R R I J sn cos, 3.49a cos sn. 3.49b Copar to th non-ynac otor nvrs controllr n 3., th stat transton atrx tho approxats th lctrcal ynacs of th otor. Hraftr t s call ynac otor nvrs controllr. hrfor, whn th otor sp s constant or chang at a lowr fruncy copar to th toru, th algorth wth th stat transton atrx tho can trac th rur toru wth hghr flty f th paratrs n th controllr atch th actual valus n th otor. If thr s scrpancy btwn th paratr valu n otor an n th controllr, th output currnt rror wll rflct ths scrpancy. Consuntly, th prforanc of th staton sch an th 65

82 controllr wll b prov. Fgur 3.4 copars th prforanc of sngl paratr staton aaptv controllr wth non-ynac otor nvrs controllr n 3. an wth approxat ynac otor nvrs controllr h paratr staton algorths n ths two cass ar th sa as n 3.9 an 3.0, an th ntgraton gans n both cass ar st as 0., C t 0, sgn sgn c > sgn sgn 0 c Fgur 3.4 Sngl paratr staton transnt prforancs: non-ynac otor nvrs controllr vs. approxat ynac nvrs controllr. Apparntly, th ynac otor nvrs controllr prov th accuracy of paratr staton an currnt tracng, though th paratr convrgng sp s gnrally th sa. hs also ans that, th ynac otor nvrs controllr allows fastr paratr staton va hghr ntgraton gan gvn th sa rurnt on stay stat paratr staton rror. 66

83 Fgur 3.5 copars th non-ynac otor nvrs controllr an ynac otor nvrs controllr n th cas of th ult-paratr staton. h paratr staton gans ar st as 0. t 0.0 C C t < 0.0 for both controllr. Fgur 3.5 Mult-paratr staton usng non-ynac otor nvrs controllr vs. ynac otor nvrs controllr Wth th ynac otor nvrs controllr provs, th stay stat paratr rror bouns aftr 50s ar b ΔR , b ΔK , whch ar about 7.5% an 9.4% of corrsponng valus n th sulaton wth th non-ynac otor nvrs controllr. Bss, th ynac otor nvrs controllr rsults n lowr ovrshoot of K c 47% of that of th non-ynac nvrs controllr, an th rs t of R c about 0 scons shortr. 67

84 3.4. Motor sp saplng lay Whl calculatng th voltag usng 3.3 or 3.48a, th sr -axs currnt urng th nxt saplng pro s nown but th otor vlocty urng th nxt saplng pro s not. h rror Δ s caus by th us of ths last avalabl sapl otor vlocty. h paratr staton sch of 3.9, 3.3 an 3.33 nglct ths rror caus by th saplng lay. It s possbl to ruc th boun of rror urng staton f Δ s tan nto account. h otor vlocty s not constant urng th saplng pro [, ]., but t contnuously vars fro to. It s rasonabl to approxat th otor vlocty rror by Δ ˆ [ ] urng ths saplng pro. In th an t, th control voltag s calculat bas on varabl valus at th th sapl. If ths valus ar pt n ory for th saplng pro [, ]., th approxat vlocty rror can b us togthr for paratr rror staton. In short, th paratr staton algorth ay ta avantag of, an, c that ar sav n ory for th past a fw sapls. h fforwar controllr, on th othr han, always uss currnt valus of ths varabls for control voltag calculaton. h physcal paratrs of th otor ar statonary, so usng th varabl valus n th past svral sapls wll not ntrouc rror on paratr staton. For th sngl paratr staton, an provnt s thn propos as R t c t ΔRˆ sgn t RΔ Δ ˆ K Δˆ. 3.5, c, c Wth 3.5, f t, x an g t, x n 3.5 bco 68

85 , ˆ ˆ 0 0 0, Δ Δ co co co co co co K t sgn x t sgn t R sgn R R x t f 3.53 an, Δ Δ Δ,,,,,, t R t sgn t t K t c c c c c x g h sturbanc functon n 3.54 has fwr trs that th n 3.7. Mathatcally ts boun fn n 3.9 wll b lowr than or ual to th boun th n 3.7. Physcally, t s ost lly lowr., x t g, x t g, x t g Whn usng th approxat otor sp rror n th ultpl paratr staton, th paratr rror uatons of 3.30~3 bcos ˆ, K R K R c Δ Δ Δ Δ, 3.55 ˆ, Δ Δ Δ Δ K R K R c hus th paratr rror staton sch s chang to ] [ ˆ,, S S R c c Δ Δ Δ, 3.57 ] [ ˆ,,,, S S K c c c c Δ Δ Δ,

86 whr Δ S RΔ K Δ. h prturbaton gt bcos g t Δ, Δ t t c h boun of th prturbaton s thrfor lowr than that of Snc th two ways of prforanc provnt scuss bfor ar appl to th otor nvrs ol an paratr staton algorth rspctvly, t s possbl to cobn th togthr an tghtn th boun of rror, thraftr furthr prov th prcson of th syst. hs provnts ar vrf n sulatons. Fgur 3.6 copars th staton algorth wth provnts an th basln staton algorth for sngl paratr staton. h algorth wth both th approxat otor ynacs an th otor vlocty rror achv th lowst boun of 5 b ΔR.83 0 an b Δ Fgur 3.7 shows th prforanc coparson of th staton sch wth an wthout th approxat Δ copnsaton for ult-paratr staton. h sch wth Δ copnsaton cas 3 has lowr stay stat rror bouns aftr 50s: b ΔR , b ΔK , whch ar about 9.6% an 4.8% of corrsponng valus n th basc staton ol cas. 70

87 Fgur 3.6 Sngl paratr staton prforancs: coparson of th basc staton progra an th prforanc provnts. abl 3. copars th bouns of stat paratr rror aong th four ffrnt cobnatons of otor nvrs ol an paratr staton sch. Fgur 3.8 shows th prforanc coparson of th paratr staton wth th four ffrnt algorths. Apparntly, th aaptv control wth th ynac otor nvrs ol an th Δ copnsator achvs hghst prcson aong th 4 cass. 7

88 Fgur 3.7 Prforanc coparson of staton sch n opn loop sulaton: cas th basc sch; cas 3 wth approxat Δ copnsaton. abl 3. Boun of rrors n opn loop sulaton: two-paratr staton Cas Boun Prcntag Boun Prcntag Motor Estaton of Δ R of b Δ R of ΔK of b Δ K Invrs sch b Δ R Mol copar b Δ K 0-4 copar to Cas # 0-4 to Cas # Non- Dynac Basc sch.39 00%.43 00% Dynac Basc sch % % Nonynac Δ copnsator % % 4 Dynac Δ copnsator % % 7

89 Fgur 3.8 Prforanc coparson of 4 ffrnt staton schs n opn loop sulaton: two-paratr staton 3.5 Multpl Paratr Estaton Usng Gra-Scht Orthonoralzaton In th prvous scton, sngl paratr an ult-paratr staton algorths wr vlop for th control law of 3.. hs algorths wr prov to b stabl an wr valat n sulatons. Howvr, t woul b or srabl f th paratr staton algorths can b fastr whl antanng th boun of th stay stat paratr rror, spcally n th ult-paratr cas. t s r-xan th ult-paratr staton algorths 3.3~33, 3.57~58. In ach of th paratr rror stators, whn th valu of th nonator was clos to zro at any stp, th ntgraton gans n 3.4 an 3.5 wr st to zro. By ong ths, 73

90 th coputaton nos ntrouc by sngularty can b avo. On th othr han, usful nforaton hn n th calculaton was scar. hrfor, f th staton algorth can ta avantag of th nforaton n ths nar sngularty calculatons wthout ang nos, ts prforanc of paratr staton wll b prov. Euatons 3.3 an 3.33 wr obtan by solvng 3.9 wth ata fro two saplng ntrvals. As th actual otor paratrs ar statonary, w can a or ata nto th coputaton. h uaton 3.9 can b n a vctor forat wth th paratr rror as unnown constants. h paratr rrors can thn b solv through th Gra- Scht orthonoralzaton procss. h sngularty probl n 3.3 an 3.33 wll b avo as runant ata ar us for solvng th paratr rror. In aton, toay s gtal controllrs ar capabl of savng a fw stps of ata n ts ory, so th stat valus can b calculat an upat vry svral sapls. Evn though ths s not ltrally ral-t, t s fast nough for statng th physcal otor paratr valus. hs a wll b xplor n ths scton Mult-paratr staton usng Gra-Scht orthonoralzaton Dfn an nnr prouct x t, x on th functon spac f I {all contnuous t functons on th ntrval I t, t } [ 0 0 p t0 p x t, x t x t x t t, 3.60 t 0 whr t ay vary wth t, s a constant, x an x t ar contnuous functons 0 p n f I., c t an t can b orthonoralz wth ths nnr prouct. Consrng t that Δ R an Δ ar both slowly vlopng rrors, t s rasonabl to assu that K Δ R 74

91 an Δ ar constants wthn th ntrval f w choos a short ntrval for th nnr K p prouct n h stat valu of Δ R an Δ K can b coput fro 3.9. As an xapl, w wll fn Δ Rˆ through th followng procss. h projcton of co t onto t, not t s co, t t. 3.6, Subtractng t fro, c t, w gt a vctor orthogonal to t n spac f I, as shown n Fgur 3.9. I I t, c t prp I, c Ω, Ω, Ω Ω Ω t Fgur 3.9 Schatc agra of Gra Scht Orthonoralzaton prp, c, t, c t t. 3.6, ang th nnr prouct of t prp to both ss of 3.9 an nglctng th rlatvly sall agntu of co an Δ, prp,, Δ R ΔK, R Δ K Δ 3.63 c prp c c Not that prp, 0. h stat valu of Δ R s propos as 75

92 Rc prp, Δ Kc prp, Δ Δ Rˆ. 3.64, prp co Slarly, ΔKˆ can b propos as Rc prp, Δ Kc prp, Δ Δ Kˆ. 3.65, prp, c, whr, prp t, c t.,, c, c h abov stat rror Δ Rˆ an ΔKˆ can thn b us to copnsat th paratrs n th controllr as R R C ΔRˆ, 3.66a K c c c K C ˆ ΔK, 3.66b c whr, R c an K c ar th valus of R an n th controllr for t <, C an C ar th wghtng factors an satsfy p p 0 < C <, 0 C. K 3.5. Proof of stablty Subtractng R fro both ss of 3.55a, th rror of foun as Δ R at th ntrval can b Δ R ΔR C ΔRˆ Inclung th boun zro-an atv nos t of th currnt fbac snsor an th nglct otor lctrcal ynacs, uaton 3.9 bcos t 76

93 Δ R, c ΔK RΔ KΔ Δ, cδ t t Dfn a prturbaton gt by g t Δ, cδ t t Applyng th Gra Scht orthonoralzaton procur to 3.68, ΔR prp { prp prp prp,, c. 3.70, Δ R, c, Δ Substtutng 3.64 an 3.69 nto 3.70, Δ t K, Δ t } Rˆ prp Δ Δ. 3.7 R prp, g t,, c Substtut 3.7 nto 3.67, w gt th paratr rror ynacs, g t prp Δ R [ C ] ΔR C. 3.7 prp,, c By nucton w hav ΔR [ C ] ΔR0 D, 3.73 whr, D j prp, g t j C. j prp,, c j Sttng 0 C < for ach ntrval,, th frst t on th rght of [ p p 3.6 wll b crasng as. Snc Δ,,, c, t, an t ar all 77

94 prp, g t boun,.. < δ for so postv constant δ., prp, c [ ] t C ax C, δ ax {,, } ax ax {,, } [ δ ], Cax { Cax ax } δ b D δ ax C j ax Snc 0 Cax <, b δ ax C as. hrfor, th scon t on th ax rght s of 3.73 wll b boun as. In suary, th staton sch of 3.66 s boun. Sulaton rsults of th Gra-Scht staton algorth ar shown n Fgur 3.0. In th sulaton, th orthonoralzaton ntrval was st as 50s, an th controllrsaplng stp was st as s. h ntgraton gans wr st as C C 0.. hans to th orthonoralzaton procss, th algorths of 3.64~65 ha bttr convrgnt sp lss than 5 scons an hghr accuracy than th ult-paratr staton algorths vlop n scton 3.5. Not that th ynacs approxaton an sp lay stator propos n scton 3.4 wr plnt n ths algorth for bttr rsults. 78

95 a Currnt tracng rsults b Paratr staton rsults Fgur 3.0 Paratr staton prforanc wth Gra-Scht orthonoralzaton: th ntal paratr rror 50%R an 5% K 79

96 3.6 Mult-Paratr Estaton Usng an Axs Currnt In scton 3.5, t was shown that th paratr staton prforanc coul b prov by nclung or ata n th stator through th Gra-Scht noralzaton procss. h prforanc provnt was achv at hghr coputaton costs of th nnr prouct Exanng th paratr staton algorths propos n Sctons 3. to 3.5, thy ar all bas on th -axs ynacs of th BDC otor. It s tru that -axs currnt,, has lttl ffct on th toru gnraton, but thr ay b so nforaton about th paratr ffrnc hn n th -axs currnt ynacs. In aton, th -axs currnt,, always accopans th -axs currnt, bcaus thy ar obtan fro th phas currnts though th coornat transforatons scuss n Chaptr. It s possbl to ta avantag of th ynacs for paratr staton. On th othr han, though th bounnss of th propos sngl an ultpl paratr staton algorths ar prov, t s stll ntrstng to s th stochastc charactrstcs of th algorths. In ths scton, w wll scuss th staton prforanc at th prsnc of zro an Gaussan nos Paratr staton t us r-xan th otor ynacs ol R t R t K. out K v, v, 3.75 Notc that th bac EMF constant K was not xplctly nvolv n th -axs currnt 80

97 ynacs uaton. It s possbl to us th -axs currnt ynacs for staton of th phas rsstanc R. h bac EMF constant K can b stat usng th -axs currnt fbac an th stat R. In that cas, t s srabl to hav a -axs coan, c currnt that s npnnt to th -axs coan currnt. h control law of 3. spcf th phas of th voltag vctor, thus s no longr npnnt to, c. hrfor, th control law propos n.37 wll b us n ths scton, whch s rstat hr as, c, c v v R R c, c c, c, c, c K. c, 3.76 Applyng th control voltag 3.76 nto th otor ynacs ol 3.75, an assung zro orr hol saplng, th -axs clos loop otor ynacs at th th sapl bcos [ Rc, c R ] t [ ] [ K K ], c c, 3.77 Assu that th otor chancal ynacs s uch slowr than lctrcal ynacs, an th rotor sp os not chang sgnfcantly,... Substtut th currnt rror an paratr rror fnton 3.8 nto h -axs clos loop otor ynacs can b wrttn as [ R Δ ΔR ] Δ ΔK c t It s rasonabl to assu that th physcal paratrs vary sgnfcantly slowr than th lctrcal ynacs or chancal ynacs. Paratr ntf by usng ata 8

98 hstory svral sapls ago woul not ntrouc sgnfcant rror. h uantty t s not avalabl n ral t, but t can b approxat by xtrapolaton an us n 3.78 for fnng paratr ffrnc. Substtutng th frst orr approxaton t nto 3.78 yls, [ ] [ ] K R R c c Δ Δ Δ Δ Rorganzng 3.79, w hav th paratr rror uaton [ ] R K R c Δ Δ Δ Δ Slarly, th -axs clos loop otor ynacs at th th sapl s [ ] R R t c Δ Δ Δ, 3.8 Substtutng th frst orr approxaton t nto 3.8, anothr paratr rror uaton s obtan as [ R R c Δ Δ Δ ]. 3.8 Onc agan, at stay stat 0 t an 0 t, an uaton 3.80 an 3.8 bco R K R c Δ Δ Δ Δ 3.83 R R c Δ Δ Δ 3.84 Solvng 3.83 an 3.84 for R Δ an K Δ gvs 8

99 ΔR [ R c Δ Δ ], 3.85 ΔK [ R c Δ Δ ΔR] o avo th sngularty an coputaton nos whn, ar clos to zro, thrshols ar st for th rcprocal calculaton of,. In th staton, th asur,, an ar us to calculat th paratr staton Proof of unbas staton Nxt, consr usng th nosy asur sgnals for paratr staton. Snc paratrs vary uch slowr than th otor ynacs, an thrfor can b trat as constants for any spcfc otor. Euaton 3.8 wth th nos trs bcos, [ n ] ΔR R c[ Δ n ] np[ n ] [ Δ n ] nv, 3.87 whr th stat varabls,, an ar trnstc; n, n, n an n v ar snsor nos an actuator sturbanc. h paratr rror ΔR bcos [ n ] { R [ Δ n ] n [ n ] [ Δ n ] n } R Δ. c p 3.88 A constant thrshol ε > 0 s st to avo th sngularty whn th asur -axs currnt s nar zro, th stat currnt rror s calculat as sgn ΔR ax [ n ] [ n ], [ Δ n ] n } v { [ Δ n ] n [ n ] R c p ε v

100 Wth th nos trs n 3.89, w n to show that uantty obtan by 3.89 s an unbas staton of th trnstc paratr rror n Assu th nos trs n 3.89 ar Gaussan an npnnt,.. E [ n ] 0, 3.90a [ n n ] S E, 3. 90b [ n n ] S E, 3. 90c [ n n ] 0 E hus at any nstant, th asur -axs currnt not as x s Gaussan, an ts probablty nsty functon s x ~ f N, S πs xp x S. 3.9 t x x, [ n ] [ n ] ε sgn sgn y, 3.9 ax ε ax, an [ Δ n ] np[ n ] [ Δ n nv y R ] c Euaton 3.89 s th proucton of y an. h varabl y s a suaton of y svral Gaussan varabls, thrfor t s also Gaussan. h varabl y s a functon of th asur currnt x, an t can b r-wrttn as y x ε x x > ε ε 84

101 h probablty nsty functon of y plas s Appnx I for tal rvaton can b foun fro > ε ε ε 0,, y y y f y y f y g y f x x y 3.94 whr, xp x S y y S y y f y π h uaton 3.94 can b us to calculat th an of th frst t n th Unfortunatly, th analytcal soluton of th an an th varanc woul b xtrly coplcat f not nsolvabl. Mont Carlo sulaton s us to fn th an of th abov rano sgnals. In th sulaton, 5000 zro an unt varanc rano nubrs ar gnrat as th nos. A srs of nubrs x μ fro 0 to 0 ar gnrat as th trnstc coponnt. Actual rano varabl, ~ x N x μ wr th su of th nos an th trnstc coponnt. For ach trnstc coponnt nubr, th nvrs of th 5000 rano nubrs ar calculat an th an of nvrs s thn calculat. Fgur 3. shows th strbuton of x y / for four typcal trnstc coponnt valus x μ. Whn x μ s nar zro, th thrshol ε lts a sgnfcant aount of th valus. hus th strbuton curv s havly stort. On th othr han, whn x y / 85

102 th an of x s rlatvly far fro zro, y / x s clos to a noral strbuton wth an of μ x. Fgur 3. h strbuton of y / x gvn x ~ N μ x, for svral valus of μ x. Fgur 3. copars th sulat an of y / x wth th algbrac functon y μ x for μ x [ 0,0]. h sulaton ncats that f th trnstc coponnt s abov ts hghr than th nos varanc, th an of th nvrs rano sgnal s approxatly ual to th nvrs of th trnstc coponnt rgarlss of th valu of th thrshol ε. Howvr, th valu of ε affcts th accuracy whn th μ x s clos to zro. 86

103 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 Slarly, Mont Carlo sulaton s us to vrf th consstncy of th functon, 87

104 sgn x ax [ n ] [ n ], ε [ n ] R c, c In aton to th nvrs of rano sgnal, anothr trnstc nubr s gnrat. to sulat th, c. Fgur 3.3 ncats that th rano sgnal ans ar consstnt to th algbrac functon of th trnstc nubrs whn th sgnal to nos rato SNR s hghr than 4. hs gvs th contons for unbas staton of th. x Fgur 3.3 h an of y / x copars to μ x. Snc, n an n ar npnnt to, th an of th ranng two n v trs on th rght han s of th uaton 3.89 wll b approxatly ual to th 88 n

105 algbrac functon valu of th ans f th nvrs of s consstnt to th algbrac functon valu of ts trnstc coponnt. hs conclus that th Δ R staton algorth s unbas whn th trnstc sgnal s slct to nsur a SNR hghr than 4. In a slar annr, th unbas staton conton can b foun for th ΔK staton hs algorth s na th -solvr bcaus t ngag both an -axs currnt fbac for paratr staton. h sulaton rsult of th -solvr staton algorth s shown n Fgur 3.4. h paratr staton ntgraton gans wr st as 0. C C 0 0., 0., othrws h prforanc of paratr staton was not as goo as that of th algorth usng Gra Scht orthonoralzaton as shown n Fgur 3.3. h an rason was that th rsstanc rror Δ R calculaton only nvolv th tracng rror of, an Δ K calculaton us Δ R rsults. Bcaus of th opratng prncpl of th BDC otor, th agntu of s usually lowr than that of. hs woul a th paratr staton slowr an nosr than that of th Gra Scht orthonoralzaton algorth n prvous scton. 89

106 a Currnt tracng rsults b Paratr staton rsults Fgur 3.4 h -solvr sulaton rsults: 50% R an 5% K 90

107 3.7 Concluson Sngl paratr staton an ultpl paratr staton for a fforwar BDC otor controllr hav bn propos n ths chaptr. Stablty of ths staton schs has bn prov. h stablty proof also ncat that t was possbl to prov th prforanc of staton schs by tghtnng th boun of rrors. hus, th stat transton atrx tho an approxat otor sp rror fbac wr propos to prov th paratr staton prforanc. In aton, t was shown that th staton prforanc coul b prov by ngagng or ata n th paratr rror calculaton. Anothr provnt for ult-paratr staton usng th Gra-Scht orthonoralzaton was propos to arss th ata loss n thos sapls scar for avong sngularty. Stablty of ths tho was prov. h axs currnt an ar always calculat n par. So can b us for paratr staton too. An staton algorth usng both an fbac was propos an unbas staton conton was shown by Mont Carlo sulaton. 9

108 Chaptr 4 RECURSIVE PARAMEER IDENIFICAION In Chaptr 3, svral paratr staton algorths wr vlop for a fforwar BDC control law of uaton 3.. hs algorths wr bas on th sa prncpl of solvng paratr rrors algbracally fro th rfrnc currnt tracng rrors. Aong th svral prforanc provnt thos nvstgat, t was shown that th bttr accuracy an fastr convrgnc coul b achv by ngagng or sapls n th paratr rror calculaton. On of th s ffcts of ngagng or ata was that th paratr-upatng algorth lagg a fw sapls bhn th fforwar controllr. In aton, t also ncras th rano accss ory RAM consupton whn plntng th algorths n a crocontrollr. Obvously, thr xsts a har lt on such xplct hstorc ata usag. o furthr prov th paratr staton prforanc, w ust xplor ffrnt chanss of paratr ntfcaton or ata hstory usag, for nstanc, th rcursv algorths. hs algorths fn so stat varabls such as covaranc atrcs to stor usful nforaton fro hstorc ata, an upat ths stats at vry sapl. hus, whn th outputs ar calculat n ral t fro th stats, all th past ata hstory ar plctly ngag n ths calculatons. h Rcursv ast Suar RS tho jung 00 an th Extn Kalan Fltr EKF Burl 995, Anrws 00 ar two wly us rcursv syst ntfcaton algorths. h RS algorth assus a lnar scrt ol of th subjct syst, an stats th coffcnts n th lnar ol by usng th runant nput an output sgnal asurnt. h EKF assus npnnt Gaussan noss 9

109 an sturbancs n a nonlnar syst, an stats th stats through lnarzaton at vry sapl. So syst paratrs can b stat as augnt stats. In ths chaptr, th Rcursv ast Suar RS an th Extn Kalan Fltr EKF wll b nvstgat for th BDC otor paratr ntfcaton applcaton. 4. Dscrt Mol of th BDC Motor Dynacs In applcatons such as th EPAS, th BDC otor controllrs ar alost xclusvly plnt by a gtal crocontrollr, whch rurs scrt control algorths. In any rfrncs jung 00, Burl 995, Anrws 993, th RS algorth an th EKF ar sgn on th scrt ol of th plant. hrfor, an approxat scrt BDC otor ynacs ol wll b rv n ths scton for th staton algorth sgn purpos. h contnuous t ol of th BDC otor lctrcal ynacs s rstat hr as R t R t out K. K V, V, 4. t n t < n, fn th stat vctor x [, ] an control nput vctor u [ V K, V ]. Introuc th t constant of th stator col R. h stat spac ol of th BDC otor can b obtan as x& y [ K 0] x 0 x. 0 u, 93 4.

110 Mathatcal soluton of th contnuous ynacs 4. s har to obtan u to th fact that th otor vlocty s a functon of t. Howvr, s govrn by th chancal nrta of th otor shaft, thrfor t s rasonabl to assu that changs slowly an can b consr as a constant wthn th typcal saplng ntrval of ~5 llscons. t foun as A, th charactrstc uatons of 4. can b si A s s s s. 4.3 s h stat transton atrx at ach saplng ont can b obtan by Φ whr si A t cos xp sn t sn t t t cos, 4.4 s th nvrs aplac oprator. h stat uaton 4. can b solv as n x[ n ] Φ[ n, n ] x n Φ n, t Gut, 4.5 n whr 0 G. 0 t θ t, an substtut 4.4 nto 4.5, th soluton of 4. bcos x[ n ] xp θ cos xp 0 sn cos cos sn x n sn cos θ sn θ V K θ θ θ V

111 h ntgraton tr n 4.6 ust b solv so as to obtan th scrt ynacs uaton. h ntgraton tr can b rwrttn as, whr I J J I [ ] I 0 cos xp θ θ θ, 4.7a [ ] J 0 sn xp θ θ θ. 4.7b Wth th assupton that s constant wthn th th saplng ntrval, an n 4.7 can b solv through so athatcal anpulaton s Appnx II for tals as I J [ ] [ ] [ ] cos xp sn xp J I. 4.8 t [ ] [ ] [ ] [ H cos sn sn cos ], Substtutng 4.8 bac nto th ntgraton tr n 4.6, th approxat scrt otor ynacs s obtan as V V I J J I K J I H.4.9 h scrt ynacs uaton 4.9 approxats th contnuous ynacs n 4. wth th assupton that th control voltag an otor vlocty hol constant wthn ach saplng ntrval. Snc an accurat scrt soluton of 4. s not avalabl, uaton 4.9 wll b us as th scrt ol of th BDC otor for syst ntfcaton purpos. 95

112 4. Rcursv ast Suar Estaton Gvn th approxat scrt ynacs ol of 4.9, w wll nvstgat f th otor paratrs can b ntf through th rcursv last suar RS tho n ths scton. h ast Suar tho assus that th objct syst confors a lnar scrt rlatonshp btwn th asurnt y an th nput u jung 00 as y a y... ap y p bu... bu ε, 4.0 whr th coffcnts, K, ap, b, K b a, ar syst charactrstcs constants, p an ar postv ntgrs, an ε s th olng rror. Not that both th asurnt y an th nput u can b vctors, an th coffcnts wll b n atrx for n ths cas. Rwrt 4.0 n th for of y x θ ε, 4. whr [ y,, y p, u,, u ],, a p, b, b θ a,. x, [ ] h optal staton of th coffcnt vctor θ nzs th olng rror ε. A typcal way to fn th optal staton of θ s to fn a uaratc cost functon of th olng rror as Q n n [ y θ ] [ x y x θ ] 4. Dnot θˆ as th optal staton of θ that nzs th uaratc cost Q. Snc Q s uaratc, θˆ can b foun by sttng partal rvatv Q to zro as θ Q θ θ ˆ θ n n [ x y x x ˆ θ] 0,

113 whch gvs n n [ x y ] [ ] ˆ θ x x. 4.4 h nvrs calculaton n 4.4 s coputatonally xpnsv an har to plnt n ral t. Fortunatly, thr xsts a rcursv soluton to th atrx nvrs calculaton. n o splfy th notaton, lt s not X n Xn [ x x ] [ Xn Xn P n ] Or. h nvrs atrx at stp n can b xpan as [ X X ] [ x x ] n P n n n x n x n [ ] P n x x n n, an not ts nvrs as P n, 4.5, whr x n n xn x n x. h atrx nvrs n 4.5 can b solv by th Woolbury Matrx Intty Golub an Van oan 996 BCD A A B C DA B DA A. 4.6 n Coparng 4.5 an 4.6, lt A P n, B x, C I, D x, whr n I s th untary atrx. h nvrs atrx at nth stp can b solv rcursvly as [ x P n x ] x P P n P n P n xn n n n n. 4.7 Substtutng 4.7 nto th paratr staton uaton 4.4, th paratr staton at nth stp bcos [ x n P n x n ] [ y n x n ˆ θ n] ˆ θ n θˆ n P n x n

114 t K n P n x n [ x n P n x n ], 4.9 th rcursv last suar RS paratr staton s gvn as θ ˆ n θˆ n K n [ y n x n ˆ θ n]. 4.0 In a BDC otor, th asurnt vctor y conssts of th -axs currnt an th -axs currnt,.. y [, ]. Coparng 4.9 to 4.0, th scrt stat nput vctor s slct as x [,, V, V, ]. Substtutng y an x nto 4.7, 4.9 an 4.0, th RS paratr staton algorth can b plnt for th BDC otor applcaton. h nvrs atrx Pn s th stat atrx 5 5 n th RS staton, an t ps nforaton obtan fro hstorc ata. h stat paratrs θˆ s a 5 atrx, an ts stay stat valu s a last suar staton of th lnar scrt ol btwn y an x. 4.3 RS Sulaton Rsults h RS algorth s plnt for th BDC otor applcaton n Suln. h followng otor paratr valus wr us n th sulaton: R 0. 05Ω, 4 K N A, 0 H, N 3. In th frst sulaton, th otor was runnng p at a constant vlocty 00ra/s. h sulaton ran at a fx stp lngth of s. Fgur 4. show th RS paratr staton rsults. h stay stat valus of th coffcnts wr shown as lgns n ths fgurs. h paratr staton convrg ucly lss than scons. h otor paratr R, K an ar not rctly shown n th staton rsults. But th analytcal valus of th stat paratrs a ~a 5, 98

115 b ~b 5 can b calculat wth th gvn paratr valus of R, K an. Fgur 4. RS staton sulaton rsults of a ~a 5 & b ~b 5 : constant 00ra s. In ach plot th stay stat valu s shown. / Substtut R 0. 05Ω, 4 K N A, 0 H, N 3 an 00ra/s p nto 4.8 an 4.9, th coffcnt atrcs n 4.9 can b calculat as a b a b H , a b 3 3 a4 b4 I J J.0737 I , 99

116 a5 I b5 J K Coparng th stay stat sulaton rsults wth th analytcal solutons, th RS staton approxat th analytcal scrt ol 4.9 closly, but thr wr about 0%~60% rror n th staton rsults. Fgur 4. show th RS staton sulaton rsults wth rano rotor vlocty. It too longr t about 7 scons for th stat paratrs to convrg to thr stay stat valus. Snc th vlocty was not longr constant, th paratr atrcs n 4.9 coul not b solv analytcally. Howvr, th RS algorth convrg to a lnar approxat of th nonlnar otor ynacs. h stay stat valus of a an b pl th currnt asurnts ar ostly corrlat to thr prvous valu. In suary, th RS paratr staton algorth was capabl of fnng a lnar approxat of th otor ynacs. h stat paratrs wr clos to thr analytcal valus f th otor was runnng at sp o.. otor vlocty was constant. Howvr, th RS algorth woul not b vry rctly hlpful f th BDC otor was runnng at varyng vlocty, as th lnar scrt ol assupton was not val. In aton, th otor paratrs of ntrsts R, K an ar not rctly accssbl n th RS staton rsults. hs woul b nconvnnt to us th RS algorth wth th fforwar otor nvrs ynacs controllrs. 00

117 Fgur 4. RS staton sulaton rsults of a ~a 5 & b ~b 5 : varant vlocty. In ach plot th stay stat valu s shown. 4.4 Extn Kalan Fltr h Extn Kalan Fltr EKF, as ncat by ts na, s an xtnson of th lnar Kalan Fltr to nonlnar systs. It s a coonly us algorth for paratr staton probls. Coputatonally, th EKF has slar stps as that of th RS algorth. It stats th covaranc atrx rcursvly n a way slar to how th RS algorth solvs th nvrs atrx n 4.7, an thn t upats th Kalan gans fro th covaranc atrx, slar to th gan calculaton n 4.9. h stats ar upat 0

118 fro th rror btwn th asurnt an th ol prcton bas on th prvous stp stat valus. Howvr, th EKF tas avantag of th xstng nforaton of th nonlnar plant ol, an nvolvs th ol lnarzaton n th covaranc atrx an stat calculaton. As a rsult, th EKF s usually a goo soluton for nonlnar syst staton nclung th paratr ntfcaton probls. Whn th EKF s plnt for syst ntfcaton probls, th paratrs ar oftn trat as nw stat varabls subjct to so stochastc ynacs. h orgnal plant ol s augnt wth th paratrs as nw stat varabls. Usng th sa rcursv procur, th paratr can b stat wth th orgnal stat varabls of th syst. In ths scton th EKF wll b plnt to th BDC otor stat an paratr staton probls. h EKF has a stanar plntaton procur Burl 995, Anrws 00. Frst of all, th plant ol wll b rforulat to nclu Gaussan actuator prturbaton an asurnt nos. Ang th actuator prturbaton w,w to th approxat BDC otor scrt ol 4.9, th currnt stat uatons bco H I I K J J J V I V w. w 4.a Assung paratrs K an ar constants subjct to rano prturbaton w 3 an w 4, th scrt ynacs of th paratrs can b ol as K K w3, 4.b w 4. 4.c 0

119 t th augnt stat b,, K, x. Euatons 4.a, 4.b an 4.c forulat th nonlnar stat ynacs ol of th augnt syst. For splcty, lt us not th nonlnar stat ol as x f x, u, w. 4. v v, bcos h asurnt vctor wth asurnt nos v v y g x, v x v It s usually rasonabl to assu that th plant prturbaton w, w, w w w 3, 4 an asurnt nos v v v, ar Gaussan an npnnt: E[ w w p ] Σwδ p, 4.4a E[ v v p ] Σvδ p, 4.4b E [ w ] 0, 4.4c E [ v ] 0, 4.4 Sw 0 whr Sw Sv 0 Σw an Σ v S w3 0 Sv 0 Sw4 ar th spctral nsty atrx of th actuaton prturbaton an asurnt nos, an δ p s th Drac lta functon, an E [ w w p] s th athatcal xpctaton opraton. 03

120 h nonlnar syst ynacs ol has bn forulat n 4.~4. Wth th nonlnar stat ol, w can fn ts lnar approxaton functons ˆ ˆ x x x f Φ, ˆ ˆ x w x f Γ an ˆ ˆ x x x g C. [] 4 4 ˆ ˆ f J f I H x f x x Φ 4.5 whr V V I J J I K J I H f f, [ ] { } [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] I cos sn cos sn cos sn, [ ] { } [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] J cos sn cos sn cos sn, an s th saplng t. 04

121 0 ˆ f Γ 4.6 w x ˆx 0 ˆ g C. 4.7 x x xˆ Wth th lnar approxaton functons, th a pror covaranc atrx s coput rcursvly fro th a postror covaranc atrx at th prvous stp Φˆ Σ Φˆ Γˆ Σ Γˆ Σ. 4.8 h Kalan gan can b coput as ˆ C C ˆ Σ W [ ˆ C Σ ] G Σ V, 4.9 an a postror covaranc atrx s upat as Σ [ I G C ] Σ ˆ Gvn th stat ynacs ol 4., w can prct th stats at th nxt stp x ˆ f xˆ, u, 4.3 an prct th asurnt as y ˆ g xˆ. 4.3 h EKF stat th stat fro th asurnt by th followng uaton [ y ˆ ] x ˆ xˆ G y h scrt EKF conssts of th uatons 4.8~30 an Bss, th lnarzaton functon Φ ˆ nvolv n uatons 4.8 ust b coput at vry stp. Obvously, th coputaton cost of th lnarzaton functon at vry stp woul b a 05

122 challng for th ral t applcatons. For rsarch purpos, lt s assu that th coputaton powr s not a probl. h EKF can b plnt to th contnuous BDC otor ynacs ol. For tal of th contnuous EKF, plas s Appnx III. 4.5 EKF Sulaton Rsults h EKF staton algorth for th BDC otor applcaton s plnt n SIMUINK for sulaton. In sulatons, th BDC otor plant ol has th followng paratr valus: 4 R 0. 05Ω, K N A, 0 H, N 3. In p so stuatons, th ffct of paratr rror n R an K ay cancl ach othr. For xapl, postv Δ R an ngatv Δ K ay n up wth th sa control voltag v calculat wth zro paratr rrors. It ay b harr for th paratr staton algorths. hrfor, ffrnt cobnatons of ntal paratr rrors n R an K wr sulat to copar th prforanc of paratr staton, as shown n Fgur 4.3~4.7. Du to so stablty ssu, th sulaton saplng ntrval ha to b lss than or ual to 0.5 llscon. In th fgurs, th,as,as wr th asur currnts, whch wr contanat by sulat Gaussan nos; th an ar th otor plant ol calculat currnt, whl th h an h ar th EKF stat currnts, R h an K h ar th EKF stat paratr valus. Rgarlss of ntal paratr rror cobnatons, EKF staton convrgs to th actual paratr valus wthn scon n all sulatons. abl 4. copars th paratr staton accuracy of th EKF at stay stat, gvn ffrnt valu of th Gaussan nos powr nsty Σ v. 06

123 abl 4. EKF paratr staton an an varanc for ffrnt nos lvl. Nos Powr Σ v μ Rˆ varr ˆ μ Kˆ var K ˆ.637x x0-7.54x x x x x x x x x x0-8 Fgur 4.3 EKF staton sulaton: R h 0.5R, K h 00.9K Fgur 4.4 EKF staton sulaton: R h 0.4R, K h 0.K 07

124 Fgur 4.5 EKF staton sulaton: R h 00.8R, K h 0.K Fgur 4.6 EKF staton sulaton: R h 00.8R, K h 0 0.8K 08

125 Fgur 4.7 EKF staton sulaton: R h 0R, K h 0 K h EKF paratr ntfcaton algorth onstrat srabl prforanc n sulatons. h stat paratrs convrg to th rfrnc valu vry ucly an stay n a tght boun at stay stat. Fro th prforanc prspctv, th EKF s supror to th algorths vlop n Chaptr 3. Howvr, th EKF s coputatonal xpnsv u to th lnarzaton coputaton n ral t. It also has stablty ssus n sulaton, an no athatcal proof of stablty s avalabl. 4.6 Suary In th chaptr, two rcursv paratr staton algorths, th RS an th EKF, wr nvstgat for th BDC otor paratr staton. h RS tho, whch s usually us for lnar systs, was shown capabl of fnng a lnar approxat ol for th nonlnar BDC otor ynacs. In sulatons, whn th otor was runnng at constant sp, th stat paratrs wr clos to thr analytcal valus n th plant. But whn th otor was runnng at varyng sp, th RS staton rsults wr not vry anngful for ntfyng th otor ynacs ol. h EKF too avantag of 09

126 th nowlg of th nonlnar plant ol an coput lnarzaton functons for stat prcton an covaranc atrx n ral t. As a rsult, t onstrat supror prforanc n both paratr an stat staton wth fast convrgnc an cnt accuracy n BDC otor applcaton sulatons. Howvr, th EKF was coputatonally xpnsv u to th lnarzaton an th covaranc atrx coputaton n ral t. In aton, th EKF, whch ngag lnarzaton of th nonlnar ynacs, ght hav stablty ssus n practc. In th BDC otor sulatons, th saplng ntrval ha to b st to 0.5 llscons so as to avo nstablty n th sulaton. In suary, th EKF algorth onstrat hghly srabl paratr ntfcaton prforanc for th BDC otor applcaton, but th coputaton cost an th stablty ssu wr barrrs for plntng th EKF n practc. 0

127 Chaptr 5 MODE REFERENCE ADAPIVE CONRO DESIGN W hav scuss svral ffrnt paratr staton thos for th BDC otor applcaton n Chaptr 3 an Chaptr 4. h EKF algorth prov ost srabl prforanc, but ha so ssus wth coputaton cost an stablty. h algorths vlop n Chaptr 3 wr splr than th EKF an stabl, but thr paratr staton prforanc lagg far bhn th EKF. Ially, w woul l to hav a paratr staton algorth wth prforanc of th EKF an splcty an stablty of th algorths n Chaptr 3. t s r-xan th algorths n Chaptr 3 that solv th paratr rror algbracally fro th currnt rrors. Frstly, ths algorths assu stay stat an nglct th currnt rvatvs whn rvng th paratr rror uatons. In aton, n orr to antan stablty, ths algorths scar so ata ponts that ght caus sngularty n th paratr rror calculaton, an thrfor scar usful nforaton hng n ths ata ponts. If w can a full us of th scar nforaton for paratr staton, t s hghly possbl that th prforanc wll b prov. h ol rfrnc aaptv control MRAC s a potntal soluton for th BDC otor control probl wth a goo balanc btwn coputaton coplxty, stablty an prforanc. Gvn a proprly slct rfrnc ynacs ol, th MRAC can ta avantag of th currnt rvatvs n control an paratr staton. Morovr, th MRAC s usually sgn by fnng th non-postv rvatv of a yapunov cost functon. hrfor ts global stablty s guarant, an all ata ponts can b ngag n

128 th aaptaton law. In ths chaptr, w wll xplor th MRAC applcaton to th snusoal BDC otors. 5. Mol Rfrnc Aaptv Control Mol rfrnc aaptv control has bn a wll-vlop approach of th aaptv control Ioannou an Sun 996, ao 003. A typcal MRAC controllr conssts of a rfrnc ol, a control law, an an aaptv chans that upats th controllr paratrs by usng th fbac rror btwn th rfrnc ol an actual plant, as shown n Fgur 5.. Rfrnc θt y t Aaptaton aw - rt Control aw ut Plant yt Fgur 5. A schatc agra of a typcal MRAC controllr o sgn a MRAC, w n to fn a sutabl rfrnc ol for th clos loop syst ynacs, an thn w n to fn a control law for th plant for achv th prary control objctv such as tracng th rfrnc nput rt. h controllr ay not b an al atch for th plant f th paratrs of th plant ar unnown. h aaptaton law wll upat th paratrs n th control law so that th plant output tracs th

129 rfrnc ol output. In th nxt a fw subsctons, w wll follow ths stps to sgn a MRAC controllr for th BDC otor applcaton. 5.. h BDC otor rfrnc ol Rcall that, as w foun n Chaptr, f th BDC otor controllr atchs th plant ally, th BDC otor actuator wll hav a clos loop ynacs as that of.38. For convnnc, th clos loop ynacs s rstat hr as t t R R R R, c,, c. 5. h ynacs ol n 5. was shown to b srabl for th BDC actuator n Chaptr. hrfor, t ay b an al canat rfrnc ol n th MRAC, xcpt that th paratrs ar unnown. t s substtut th paratrs wth thr nonal valu n controllr, an rwrt th rfrnc ol as t t c 0 0 c, c, c, 5. whr c c s th nonal t constant an R c, ar th rfrnc ol stat currnt. h rfrnc ol actually conssts of two npnnt frst orr low pass fltrs, whch ar slar to that of a DC otor. If th clos loop BDC otor ynacs tracs th rfrnc ol, th BDC otor woul hav slar prforanc as that of a DC otor. t s copar th otor ynacs ol 3

130 u u t K t t t t to th rfrnc ol n 5.. h nonlnar tr t an t, an th bac EMF tr K t ar not prsnt n th rfrnc ol. o trac th rfrnc ol, th control law ust b sgn to cancl ths trs. In aton, th paratrs n th otor ynacs ol ay b ffrnt fro thr nonal valus us n th rfrnc ol. h aaptaton law ust b sgn to ntfy th otor paratr valus. 5.. h control law In orr to trac th rfrnc ol, lt s rwrt th otor ynacs ol as c c c c R R K u u t t,,,, If th scon tr on th rght han s of 5.4 wr zro, thn th actual BDC ol woul hav th sa for of th rfrnc ol. As th otor paratrs ar not nown, thr valu n th controllr wll b us nsta. hus, th control law s propos as.,,, c c c c c c c R u R K u 5.5 whr,, an ar th paratr valus n th controllr. c R c c K If th controllr paratrs,, an atch thr rspctv countrparts n th otor plant, th controllr law 5.5 wll b abl to rv th clos loop otor ynacs c R c c K 4

131 to trac th rfrnc ol prfctly. Howvr, snc th paratr valus n th plant ar not nown xactly, w n to fn so way to stat thr valus h aaptaton law t s consr usng th fbac rror btwn th rfrnc ol an th otor plant to stat th plant paratr valus. Dfn th rfrnc currnt rror vctor as. 5.6 W n to fn out how th rfrnc currnt rrors ar rlat to th paratr ffrncs. Substtutng 5.5 nto 5.3, an subtractng th rsultng uatons by th rfrnc ol 5., th rfrnc currnt rror ynacs can b foun as c c c c c c c K K t t 5.7 o splfy th notaton n 5.7, w fn paratr rrors as c ~, 5.8a c c K K ~, 5.8b c ~. 5.8c Substtutng th paratr rrors bac nto 5.7, th rfrnc currnt rror ynacs ol bcos 5

132 0 ~ ~ ~ 0 0 t t t t c c. 5.9 W want th clos loop otor ynacs to trac th rfrnc th rfrnc ol as clos as possbl. In anothr wor, w want th rfrnc currnt rror to b as low as possbl n agntu. In aton, f th paratr valus n controllr atch thr uvalnt n th plant otor, th control law can a th clos loop otor ynacs ol xactly th sa as that of th rfrnc ol. W also want th paratr rrors to b as clos to zro as possbl. o fn th nal currnt rrors an paratr rrors, lt s fn a yapunov canat functon of th rfrnc currnt rrors an paratr rrors as 3 ~ ~ ~ ~, ~, ~,, Q γ γ γ, 5.0 whr γ, γ, an 3 γ ar postv constants. Q s a non-ngatv uaratc functon of th currnt rrors an th paratr rrors. W can gt th nal currnt rrors an paratr rrors by nzng th valu of. h nu of Q can b achv by rvng ts t oan rvatv to b ngatv or zro. Dffrntatng 5. wth rspct to t, w hav Q t Q & & & & & ~ ~ ~ ~ ~ ~ 3 γ γ γ. 5. Substtutng th rfrnc currnt rror ynacs 5.7 nto 5., th rvatv of Q bcos 6

133 [ ] [ ] c c t t t Q & & & ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 3 γ γ γ, 5. h frst two ts on th rght han s of 5. ar non-postv. If w can a th ranng ts of 5. to b zro, thn w can guarant non-postv t oan rvatv of th yapunov cost functonal, whch wll thn approach ts nu asyptotcally. R-organz 5. as. ~ ~ ~ ~ ~ 3 c c t t Q γ γ γ & & & 5.3 Propos th aaptaton laws for th controllr paratrs as t ~ ~ ~ 3 γ γ γ & & &. 5.4 Substtut th aaptaton law 5.4 nto 5.3, th t oan rvatv of th yapunov cost functon bcos non-postv as 0 c c t Q. 5.5 hs guarants that th global asyptotc stablty of th rfrnc currnt tracng rror ynacs ol. h rfrnc ol s globally stabl as t conssts of two frst orr lnar systs wth pols c n th lft half plan HP. hrfor th MRAC 7

134 controllr propos n 5.5 an 5.4 guarants th global asyptotcally stablty of th clos loop BDC otor ynacs. h physcal paratr valus of th otor can b rv fro th aaptaton law 5.4. Assung that R, an K ar constants, an substtutng 5.8 nto 5.4, w hav ~ R Rc Rc & t c t γ, 5.6a c ~ & t K K c c t K c c γ, 5.6b ~ c c γ 3 t &. 5.6c t t h controllr paratr stators can b foun as t R c c γ, 5.7a t K c c γ, 5.7b t c γ 3 t. 5.7c If th nuctanc s assu to b a constant an c, th aaptaton sch bcos R t c K t c c γ, 5.8 γ c

135 5. Sulaton Rsults h MRAC controllr wth th control law of 5.5 an th aaptaton law of 5.8~9 was plnt n Suln an was tst n sulaton for th currnt tracng an paratr ntfcaton prforanc. h otor paratr valus wr as th followng: R 0. 05Ω, 4 K N A, 0 h, n 3. h aaptaton gans n 5.8~9 p wr st as γ γ. h sulaton saplng ntrval was st to s. h controllr was assu to hav 50% rror n R an 0% rror n K ntally. Fgur 5. shows th rfrnc currnt tracng sulaton rsults. h rfrnc currnt tracng rror convrg to zro wthn scons approxatly 000 stps as shown n Fgur 5.. Fgur 5.3 an 5.4 ar sulaton rsults of paratr staton wth two typcal cass of ntal paratr rrors. In both cass, th stat controllr paratrs also convrg to th thr corrsponng valus wthn scons. At stay stat, th an an stanar vaton of th paratr staton rrors wr: 5 K K K μ, μ c % R c R % R, K 0.03 K, c R R c. 9

136 a -axs currnt tracng prforanc b -axs currnt tracng prforanc Fg 5. MRAC rfrnc currnt tracng prforanc n sulaton: R c R, K c 0. K Fg 5.3 MRAC paratr staton prforanc n sulaton, cas : R c R, K c 0. K 0

137 Fg 5.4 MRAC paratr staton prforanc n sulaton, cas : R c R, K c K 5.3 Concluson A ol rfrnc aaptv control algorth has bn rv for th BDC otor currnt control applcaton n ths chaptr. h actuator ynacs ol of uaton.38 was slct as th rfrnc ol for th clos loop BDC otor syst. A fbac control law of uaton 5.7 was propos to achv currnt tracng to th rfrnc ol outputs usng th controllr paratr valus. o ruc tracng rror ntrouc by paratr rror btwn th controllr an th plant, a yapunov cost functon 5.0 was fn on th currnt tracng rrors an th paratr rrors btwn th plant an th controllr. A controllr aaptaton law was sgn by fnng non-postv t-oan rvatv of th uaratc yapunov functon. h aaptaton law guarant th global asyptotcal stablty of th MRAC algorth. Coparng to

138 th algorths vlop n Chaptr 3, th MRAC algorth ngag all fbac ata an rvatv of th otor currnts for paratr aaptaton; thrfor t onstrat supror prforanc n sulatons. Both th paratr staton an th rfrnc ol currnt tracng convrg ucly, an th stay stat currnt tracng rror was vry low. Coparng to th EKF, th MRAC algorth was uch splr an thus fastr an rur lss coputaton rsourc.

139 Chaptr 6 COSED OOP SIMUAION AND CONROER PERFORMANCE EVAUAION Svral ffrnt BDC otor aaptv control algorths hav bn vlop n Chaptr 3~5. All ths algorths wr valat n th otor bnch tst sulatons whr th coan toru c an th otor vlocty wr assu to b two npnnt rano sgnals. In ral applcatons, BDC otors ar usually th actuator of so outr loop systs; thrfor th coan toru c an th otor vlocty ar partly corrlat through th outr loop plant an controllr. In ths chaptr, w wll valuat th aaptv control algorths fro th prvous chaptrs n clos loop sulatons wth outr loop controllr an plant ols. h frst clos loop syst conssts of th EPAS plant ol togthr wth an assstng toru controllr, an th scon s a otor sp control applcaton wth an ant-wnup PI controllr. 6. EPAS clos loop sulaton In th otor bnch tst sulatons, th coan toru c an th otor vlocty, whch wr th nputs to th otor controllr, wr assu to b two npnnt rano sgnals. Howvr, n th applcaton of th clos loop control of a syst, an c cannot b copltly npnnt bcaus th plant output s partly affct by plant nput c. In orr to onstrat th prforanc of th aaptv control algorths n a clos loop sttng, th xapl of an lctrc powr assst strng syst EPAS s consr. 3

140 6.. EPAS clos loop syst ol As shown n Fgur 6., th clos loop autootv EPAS syst conssts of an lctrc control unt ECU, a brushlss DC otor, a toru snsor, a strng han whl, an a st of rac an pnon. h otor toru s transfrr to th strng colun va a wor an wor gar assbly. h toru transucr asurs th toru on th strng colun s. h otor vlocty s asur by a tachotr. hs sgnals along wth otor poston ar collct by th ECU, whch gnrats c accorngly an calculats th voltags to b appl to th otor. Han Whl Input oru snsor s V ECU Motor Rac & Pnon Strng Gar Fgur 6. Schatc agra of a typcal EPAS. h strng syst was ol as a two-ass chancal syst wth vscous frcton Baawy t al It was assu that th rotor shaft of th BDC otor was rgly connct to th strng colun. A Suln ol of th EPAS was us for 4

141 th clos loop sulaton. In ths ol, th rvr han-whl toru nput was ol as a Gaussan rano sgnal pass through an ant-alasng fltr, whch sulat typcal huan rvr fruncy rspons charactrstcs. In aton, two npnnt Gaussan rano sgnals wr plnt n th ol to sulat th roa rsstanc an othr prturbaton toru on th strng colun. In th sulatons, th sa st of rano nputs, nclung th han-whl toru, th roa rsstanc an othr prturbaton toru, wr f nto th EPAS plant ol for ach of th aaptv BDC control algorths n orr to a thr currnt tracng an paratr staton prforanc coparabl. h ECU s b wth th assstng toru controllr an th BDC otor controllr. h forr s rur n th EPAS to achv th srabl assstng toru an strng whl fbac for th rvr. h lattr wll b th aaptv controllr vlop n ths rsarch. h assstng toru controllr vlop by Patanar Patanar 003 was aopt n th clos loop sulatons. W wll copar four typ of aaptv BDC otor controllr n th clos loop sulatons. h xtn Kalan fltr EKF n scton 4.4 was slct as th bnchar paratr staton algorth. h currnt controllr of uaton.37 was chosn as th otor controllr to for a clos loop aaptv controllr usng th EKF, an th controllr paratrs wr upat wth th EKF stat valus. h Gra- Scht orthonoralzaton algorth n scton 3.5 show th bst prforanc aong th algorths that solv th paratr rrors algbracally fro th axs currnt ynacs alon. W not ths class of algorths as th -solvr algorths n th sulatons. h Gra-Scht orthonoralzaton algorth was slct as th 5

142 rprsntatv for th -solvr algorths for th prforanc coparson. h thr aaptv algorth nclu n th valuaton was th on vlop n scton 3.6 that solv th paratr algbracally fro both th an th axs currnt ynacs. It s na as th -solvr hr. h MRAC algorth vlop n Chaptr 5 was th fourth algorth for prforanc coparson n clos loop sulatons. Notc that th EKF ha to run at sulaton stp lngth of 0.5 llscons for stablty n sulaton, whl th sulaton stp lngth was llscons for th othr thr algorths. h EKF was xpct to b approxatly 4 ts fastr than othr algorths n tr of th paratr staton convrgnc sp gvn th sa contons. h BDC otor ol of uaton.38 was us as th otor plant n th clos loop sulaton. h paratr valus n th BDC otor plant ol wr assu to b constant as th followng: 4 R 0. 05Ω, K N A, 0 h, 3. n p 6.. EPAS sulaton rsults an analyss In so stuatons, th ffct of paratr rror n R an K ay cancl ach othr. For xapl, postv Δ R an ngatv Δ K ay n up wth th sa control voltag v as that of zro paratr rrors. In aton, th assstng toru controllr ay hav ffrnt rsponss to postv an ngatv paratr rrors. It ay b harr for th paratr staton algorths n so ntal paratr rrors cobnatons. hrfor, svral ffrnt cobnatons of ntal controllr paratr rrors n R an K wr sulat. For ach st of ntal paratr rrors, th currnt tracng rror an paratr staton rsults fro th four algorths ar ovrlapp n two fgurs rspctvly for coparson of th convrgng sp an stay stat paratr 6

143 staton rror. Fgur 6. shows th sulaton rsults of ΔR 0 50%R an ΔK 0 0% K ΔK 0 5% K ΔK 0 5% K ΔK 0 0% K. Fgur 6.3 shows th sulaton rsults of ΔR 0 50% R an. Fgur 6.4 shows th sulaton rsults of ΔR 0 50% R an. Fgur 6.5 shows th sulaton rsults of ΔR 0 50% R an. abl 6. copars th stay stat an an stanar vaton of th currnt an paratr staton rror. In all sulatons, th paratr staton an currnt tracng ar stabl an convrg to stay stat valus. Postv ntal rror n K ΔK 0 0% K appars to harr than ngatv ntal rror ΔK 0 for both currnt tracng an paratr staton. h rason was that th postv ΔK woul rsult n hghr otor control voltags, an thrfor hghr otor toru than wth ngatv ΔK at hgh otor vlocty. Consuntly, th assstng toru controllr woul rspon or to hgh otor vlocty, an th coan toru ght b slghtly or corrlat to th otor vlocty. Coparng th currnt tracng rror an paratr staton rsults n all sulatons, th EKF onstrat th bst paratr staton an th currnt tracng prforanc aong th four algorths. hs s rasonabl as th EKF s optal f th syst s at an ulbru stat wth Gaussan noss. h Kalan gans ar upat onln wth th otor stats bas on th snstvty of output currnts to th paratrs valus. In aton, th EKF s an obsrvr that tas th control sgnals V, V an otor stats, as nputs. It s rlatvly npnnt to th outr loop controllr. h saplng stp lngth of 0.5 s also contrbuts to th fastr convrgnc of EKF, as th othr thr algorths wr runnng at s. 7

144 h MRAC ha th lowst stanar vaton of th currnt tracng rror at stay stat. hs was u to th fact that th aaptv controllr was sgn to rv th currnt tracng rror to zro. h slghtly hghr an of currnt tracng rror Δ, c n th sulatons was bcaus that th MRAC currnt tracng rrors wr fn as th ffrnc btwn th rfrnc ol outputs an otor currnts. h rfrnc ol ynacs ntrouc so xtra rror nto th coan currnt tracng. Consrng th factor of slowr saplng sp, th MRAC show coparabl convrgnc sp n paratr staton as that of th EKF. h fast convrgnc sp rsult fro th fact that all ata wr us for th paratr staton. Howvr, th MRAC show so stay stat paratr staton rror n R. In ths algorth, th paratr staton was just an ntrat ans for achvng rfrnc currnt tracng. hus th staton coputaton rur rch fruncy contnts of th xctaton sgnals, nclung, c,, c an, to achv al paratr staton. On possbl rason was that th outr loop controllr was sgn to prov so crtan fruncy charactrstcs for rvr han-whl fbac. hus th fruncy contnts of, c an wr rlatvly lt., c ha rch fruncy contnts as t was st as a Gaussan rano sgnal n th sulatons, but th agntu of was sgnfcantly lowr than that of bcaus of th opratng, c prncpl of th BDC otor., c 8

145 a Currnt tracng rsults b Paratr staton rsults 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

146 a Currnt tracng rsults b Paratr staton rsults 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

147 a Currnt tracng rsults b Paratr staton rsults 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

148 a Currnt tracng rsults b Paratr staton rsults 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

149 abl 6. Stay-stat prforancs n EPAS clos loop sulaton Cas Algorth μ ΔK st Δ K μ ΔR st Δ R μ ΔI st Δ I EKF solvr Solvr MRAC h -solvr an th -solvr wr bas on slar prncpl of paratr rror staton. hr paratr staton rsults wr slowr an soothr than that of th EKF an th MRAC. h -solvr an th -solvr wr slar to th MRAC n th sns of usng th corrlaton of currnt rror an currnt for paratr staton. Actually, th -solvr an th -solvr algorths coul b consr as a spcal cas of ol rfrnc control n whch th rfrnc ol woul b a unt gan. Howvr, n both algorths, so ata ponts wr scar urng th paratr rror staton to avo nos fro sngularty spcally atrcs nvrson. hs procur hlp to antan stablty, but rul out usful ata an thrfor ruc paratr staton convrgnc sp. h -solvr ha bttr staton prforanc n K than th -solvr rgarlss of ntal paratr rrors. On th othr han, th -solvr not prfor as wll n th R staton. It show slar stay stat rror n R staton as that of th MRAC. h an rason was that th -solvr stat R anly on th ynacs an K on th ynacs wth both an fbac n vry loop, whl th -solvr stat both paratrs on th ynacs an n asurnt fro two or or loops. 33

150 h -solvr ha or ata ponts than th -solvr, so t prfor bttr n K staton. On th othr han, th agntu of was sgnfcantly lowr than that of, whch rsult n slghtly slowr R staton of th -solvr. Anothr factor for th algorths coparson s th coputaton costs. Aong th four algorths, th EKF ans ost coputng powr for two rasons: t rurs shortr saplng stp lngth to achv stablty of th Extn Kalan Fltr; th lnarzaton calculaton an th sxtn varanc stats n th EKF ns sgnfcantly hghr aount of coputaton n ach stp. h orthonoralzaton n th -solvr rurs storag of so ntrat calculaton rsults onln for a crtan pro 50s n th sulaton. hrfor, th -solvr algorth rurs ost rano accss ory RAM n plntaton. h approxat nvrs otor ynacs an th nnr prouct calculaton n th -solvr consu th scon hghst coputaton powr. h MRAC an th -solvr hav th lowst rurnts on th coputaton rsourc, as thy n last hstorc ata an thr paratr rror calculatons ar rlatvly splr than th EKF an th -solvr. In suary, th EKF prov th ost srabl paratr staton prforanc n th clos loop EPS sulaton at th cost of hgh coputaton powr. Its stablty s not guarant. h -solvr ha th slowst paratr staton convrgnc sp, but th stay stat accuracy was clos to that of th EKF. Its coputaton costs wr scon hghst u to th orthonoralzaton an th nvrs otor ynacs approxaton calculaton. h -solvr rur lss coputaton rsourc, but ts paratr staton prforanc n R was th lowst aong th four algorths. h MRAC ha th scon hghst paratr staton convrgnc sp though th EKF was runnng 34

151 at four ts fastr. It also ha th tghtst stanar vaton n currnt tracng rror. It was conoc on coputaton. Howvr, u to th charactrstcs of th outr loop controllr, th MRAC show stay stat paratr staton rror, as t was an ntrat ans to achv low currnt tracng rror. h outr loop controllr, n th cas th assstng toru controllr, appar to hav so pact on th prforanc of th aaptv BDC otor controllrs. Fro th otor controllr pont of vw, th outr loop controllr an th EPAS plant togthr for a fbac loop fro th otor output toru/vlocty to th coan toru. Obvously, th prforanc of th otor controllr woul subjct to th ovrall syst fbac. h outr loop controllr ha so spcfc fruncy rspons sgn, an narrow th fruncy contnts of th xctaton sgnals to th aaptv controllrs. As a rsult, th prforanc of th aaptv controllrs ruc slghtly whn copar to thos otor bnch tst sulatons. o furthr stuy th outr loop controllr s pact on th prforanc of th aaptv otor controllrs, a clos loop otor sp control applcaton wll b nvstgat n th nxt scton. 6. Clos oop Sulaton Of a Motor Sp Control Applcaton W hav scuss th prforanc of th aaptv BDC otor controllrs n th EPAS clos loop sulatons. h aaptv control algorths suffr slghtly u to th spcfc fruncy rspons of th assstng toru controllr. In ths scton, w wll s how th aaptv algorths prfor n th clos loop sulatons of a typcal sp control applcaton. In ths cas, th outr loop sp controllr has sgnfcantly ffrnt fruncy rspons than that of th EPAS assstng toru controllr. 35

152 In a typcal otor angular sp/poston control applcaton, th outr loop plant can b ol as a scon orr otor shaft ynacs as shown n Fgur 6.6. h loa toru on th otor shaft s assu to b a ban-lt Gaussan rano sgnal. o a sur th sulaton rsults ar coparabl, a rano loa toru t hstory was crat offln an thn was us n th clos loop sulatons of all otor control algorths. Fgur 6.6 h bloc agra of th otor shaft ynacs Suln ol. h outr loop controllr s an ant-wnup PI controllr wth a rfrnc sp nput, as shown n Fgur 6.7. For prforanc coparson purpos, w us th sa BDC otor R 0. 05Ω, 4 K N A, 0 h, n 3 n th sp control p sulatons. h clos loop sulaton rsults for th EPAS syst show that th ntal paratr rrors ha so ffcts on th controllr prforancs. hus, w run th clos loop sulaton wth ffrnt cobnatons of ntal paratr rrors for ach of 36

153 th aaptv control algorths. h clos loop sulaton rsults ar copar n Fgur 6.8 ~ 6., an th corrsponng stay stat currnt tracng an paratr staton rsults ar copar n abl 6.. Fgur 6.7 h bloc agra of th ant-wnup PI controllr Suln ol. abl 6. Stay-stat prforancs n sp control clos loop sulaton Cas Algorth μ ΔK st Δ K μ ΔR st Δ R μ ΔI st Δ I EKF solvr solvr MRAC In gnral, th otor sp control clos loop sulaton show slar rsults as that of th EPAS clos loop sulaton. h EKF ha th fastst paratr staton convrgnc an th fastst currnt tracng. h MRAC was th scon fastst n trs of paratr staton an currnt tracng. h -solvr show slghtly fastr paratr convrgnc than th -solvr, whl th -solvr ha th tghtst stanar 37

154 vaton n paratr staton rrors at stay stat. h -solvr ha th hghst paratr staton rror stanar vaton spcally n R, whch was bcaus of th rlatvly lowr agntu of. Ovrall, th MRAC was th bst cobnaton of paratr staton an rfrnc currnt tracng convrgnc sp, stay stat tracng rror, an coputaton cost. Howvr, th sp control clos loop sulaton rval so ffrnt ynacs n th ntracton btwn th aaptv controllr an th outr loop controllr. Unl n th EPAS clos loop sulatons, non of th aaptv controllr show stay stat paratr staton rror n th otor sp control sulatons. hs was u to th wr fruncy contnts of th ant-wnup PI controllr than that of th assstng toru controllr n th EPAS syst. In aton, th ntal paratr rrors s to hav or ffcts on th paratr staton, spcally for th MRAC. Agan, ths was bcaus that th ant-wnup PI controllr rspon ffrntly to th currnt rror ntrouc by th ntal paratr rror. h rfrnc sp was st to b a constant 50ra/s, thus th was postv n th sulaton. R-xanng th paratr staton uatons 5.8~5.9, th paratr staton woul b anly trn by th currnt tracng rror. Whn th ntal paratr rrors Δ R0 an ΔK 0 ha ffrnt sgn, thy cancll out ach othr to so xtnt whn calculatng th control voltags. h outr loop ant-wnup PI controllr ntgrat th sp rror, thus lay th ffct of currnt rror ntrouc by oppost sgn of Δ R0 an Δ K 0. h n up wth slowr staton of paratr rror, an consuntly th currnt tracng was also slowr whn th Δ R0 an th Δ K 0 ha th sa sgn. Aong th 38

155 four aaptv control algorths, th EKF was th last snstv on to th outr-loop controllr s rspons to th ntal paratr rrors. hs ght b u to ts obsrvr structur an th control voltags wr part of ts nput sgnals to th paratr staton functons. In suary, th otor sp control clos loop sulaton show slar concluson to th EPAS sulaton n trs of th aaptv control algorths coparson. h MRAC algorth appar to b th bst cobnaton of prforanc an coputaton costs. Howvr, th prforanc of th MRAC algorth was slghtly affct by th outr loop controllr s charactrstcs. 6.3 Concluson Four rprsntatv aaptv BDC otor control algorths fro Chaptr 3~5, naly th EKF, th -solvr, th -solvr, an th MRAC, wr copar n clos loop sulaton of an EPAS syst an a sp control applcaton. Gnrally, th sulaton rsults rval a tra-off btwn th prforanc an algorth coplxty or coputaton costs. Bttr prforanc was obtan at th cost of hghr coplxty of th control algorth. Morovr, th sulaton rsults ncat that th outr loop controllr an plant ynacs woul ruc th prforanc of th aaptv controllrs f th outr loop controllr ha rlatvly narrow fruncy rspons or f th outr loop syst ynacs rspon ffrntly to th ffrnt paratr rror cobnatons. 39

156 a Currnt tracng rsults b Paratr staton rsults 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

157 a Currnt tracng rsults b Paratr staton rsults 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

158 a Currnt tracng rsults b Paratr staton rsults 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

159 a Currnt tracng rsults b Paratr staton rsults 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

160 Aong th four algorths, th EKF show th bst paratr staton prforanc n th clos loop sulatons. It was also th last snstv on to th rfrnc sgnals fro th outr loop syst. hs avantags of th EKF wr u to th ral t lnarzaton of th nonlnar ol an th optal Kalan fltrng bas on th lnarz ol. On th othr han, th srabl prforanc of th EKF ca at th hghst coputng cost. Wth th EKF algorth, th controllr n to coput th lnarzaton functon an to upat a 4x4 covaranc atrx for th 4 augnt stats at vry sapl. In aton, th stablty of th EKF was not guarant u to th onln lnarzaton. In th clos loop sulaton, t rur about 4 ts hghr saplng fruncy than othr algorths so as to antan stablty. h -solvr an th -solvr wr bas on th sa prncpl of solvng th paratr rror for th currnt fbac algbracally. h -solvr show th slowst paratr staton convrgnc n th clos loop sulatons. Coputaton of th orthonoralzaton an approxaton of th nvrs otor ynacs consu scon hghst coputaton powr an ory. h -solvr ngag both an for paratr rror calculaton, an thrfor ha fastr staton than th -solvr. Bss, t rur lss coputaton rsourc as t us or ata for th staton. Howvr, u to th opraton prncpl of th BDC otor, th -axs currnt s usually of sgnfcant lowr agntu than th -axs currnt. Consuntly, th -solvr ha th lowst staton prforanc n R aong th four algorths. h MRAC ha scon hghst fast paratr staton convrgnc sp n th clos loop sulatons. It also ha th tghtst varaton of currnt tracng rror. It was conoc n trs of coputaton cost. hs bnfts wr u to that th MRAC too 44

161 avantag of th otor currnt ynacs an us all fbac ata for staton an currnt tracng. Howvr, snc th paratr staton n th MRAC was an ntrat ans to achv low currnt tracng rror, t was or snstv to th outr loop syst ynacs than othr algorths. Fro th usr prspctv, th coan toru currnt tracng s th ost portant crtron for th otor controllr slcton. h stablty of th algorth plays ually portant rol for practcal control plntaton. In aton, th algorth coplxty an coputaton costs ar also portant factors for plntng th control algorth. Consrng ths factors, th MRAC algorth appars to b th ost prosng canat for th BDC otor actuator. 45

162 Chaptr 7 AN OPERAION SIMUAION MODE FOR BDC MOORS In Chaptr, w hav show that th -ol of a snusoal BDC otor s athatcally uvalnt to ts corrsponng stator phas ol. Du to ts splcty an convnnc, th -ol has bn us for th BDC otor controllr vlopnt. W hav also pont out svral supplntal functons ar n to practcally plnt th controllrs sgn on th -ol. hs supplntal functons ay b assu to b al whn sgnng th controllr, but thy hav so pact on th controllr plntaton. Car ust b tan to arss so practcal ssus whn plntng a -coornat controllr, for xapl, th rotor angular poston asurnt an th phas control voltag oulaton. ratonally ths factors can b stu through th tral an rror tho urng prototyp vlopnt n labs. In th EPAS applcaton, u to th hgh toru rurnt an low battry voltag, th BDC otor phas rsstanc s vry low an th pa currnt can b ovr 00 Ap. In ths cas, a nor rror ay rsult n aag of th coponnts such as th nvrtr. hus th tratonal tral an rror tho ay not b an ffcnt way for control prototyp vlopnt. Valaton of control softwar n th sulaton bfor conuctng otor tsts can sgnfcantly ruc th rs of prototyp falur. It s oftn srabl to hav th capablty of sulatng ths practcal plntaton ssus. In ths chaptr, a tal Suln ol of a typcal BDC otor wll b scuss an vlop. W wll scuss two practcal ssus of th controllr ntalzaton an transucr rsoluton n sulaton wth ths uas-physcal BDC ol. 46

163 7. A Quas-Physcal BDC Motor Mol An opratng BDC otor syst conssts of a PMSM otor, an nvrtr, an nvrtr swtchng-logc controllr, a otor controllr, a otor angular splacnt/vlocty transucr, an otor phas currnt transucrs. A typcal BDC otor syst s shown n Fgur 7.. In any cass, th nvrtr swtchng-logc control algorth an th otor controllr ar progra n a crocontrollr. h angular splacnt transucr can b an optcal ncor or a rsolvr. So BDC otors ar upp wth a st of Hall ffct snsors, whch can rplac th otor angular splacnt transucr for th nvrtr swtchng logc control purpos. h nvrtr can b consr as th actuator of th clos loop BDC otor syst. h crocontrollr achvs th otor control by ajustng th swtchng tng an uty cycl of th nvrtr. W hav scuss th PMSM otor phas ol an ts -coornat otor controllr. Howvr, snc ost BDC otors ar thr-phas Y-connct, w wll us th n-to-n otor PMSM otor ol so as to b clos to ral syst. In aton, a coplt sulaton ol of th BDC otor syst ust also nclu othr coponnts such as th nvrtr, th nvrtr swtchng logc controllr, th rotatonal splacnt transucr, th Hall ffct snsors, an th phas currnt snsors. W wll scuss ths coponnts n th followng subsctons. 47

164 Mcrocontrollr Dcor Poston & Sp Angular ransucr PMSM DIO Hall Snsor A/D Phas Currnt Phas Currnt Currnt Snsor Invrtr 3-phas PWM SVPWM DC Powr Fgur 7. A typcal thr-phas BDC otor syst bloc agra. 7.. h n-to-n PMSM ol For a Y-connct PMSM otor, th nutral pont s usually hn n th otor housng an thrfor not accssbl for th controllr. Insta, th nvrtr controls th ln-to-ln voltags across ach par of th thr phass. W wll n a ln-to-ln ol of th otor nsta of th phas ol n uaton.. Rwrt th phas ol. or.9 as t R V E 7. 48

165 whr s th phas currnt vctor, s th phas voltag vctor, ],, [ 3 v v v ],, [ 3 V a a a a a a a a a, an s th phas bac EMF vctor wth ],, [ 3 E 3 sn 3 sn sn 3 3 sn 3 sn sn 3 π θ π θ θ π θ π θ θ p p p p f p p f p p f p n n n K n n n n n n. 7. Dfn ln-to-ln varabl as f f f f f f f f f f f f 7.3 whr can b, or. Multply both ss of 7. by, th ln-toln ol s obtan as f v V R t V R t V R t a a a. 7.4 Wth ln-to-ln control voltag vctor V V V 3 3,, fro th nvrtr, th ln-to-ln currnts wll b solv fro 7.4. Usng th Krchhoff frst law, th phas currnts ar thn calculat fro th ln-to-ln currnts as 3 3,, 49

166 h toru gnrat by th otor can b calculat wth 3 sn 3 sn sn π θ π θ θ t t t 7.6 h ln-to-ln PMSM otor ol s plnt as a stat spac ol n Suln, as shown n Fgur 7.. Fgur 7. h ln-to-ln PMSM ol n Suln. 7.. h H-brg PWM nvrtr ol 50 ypcally, a thr-phas PMSM otor rurs thr-phas AC control voltags. h DC powr supply to th BDC otor cannot rctly gnrat th AC control voltags. It s th functon of th thr-phas H-brg PWM nvrtr to gnrat approprat AC voltags for th PMSM otor. h H-brg conssts of sx or or gtally controll powr transstors such as hxagonal fl ffct transstors HEXFE, as shown n

167 Fgur 7.3. Usually th powr transstor rspons t s n th orr of nano 0-9 scons, whch s sgnfcantly shortr than a cycl of th PWM sgnal usually n 0-5 scons. It s rasonabl to assu that ths trans. hy can b splf as a swtch controll by th PWM puls. If th uppr transstor n a branch of th H-brg s swtch on, th corrsponng phas trnal s connct to th postv trnal of th DC powr supply. If th lowr transstor s swtch on, th corrsponng phas trnal s connct to th ngatv trnal of th DC powr supply. Obvously, th uppr transstor an th lowr transstor n any branch cannot b swtch on sultanously. V c / b Phas G Phas B Phas C -V c / Fgur 7.3 A splf crcut of th BDC otor syst. o calculat th ln-to-ln voltags for th PMSM otor, w fn th groun rfrnc zro potntal pont as th of DC powr supply, as shown n Fgur 7.3. h H-brg sts th otor stator pol to thr V c / or -V c /, whl th ln-to-ln voltag s thr V c or V c. Not that th potntal at th nutral pont of th otor th 5

168 cntr of Y conncton changs wth th ffrnt H-brg swtchng confguraton, thrfor th stator phas voltag ay b ffrnt fro V c /. For ach branch of th H- brg, f th swtch control sgnal s hgh, th corrsponng stator phas potntal s st to V c /; othrws, t s st to -V c /. h phas pol potntal voltags ar thn us to calculat ln-to-ln voltag aong th thr phass. h Suln ol bloc agra of th H-brg s shown n Fgur 7.4. Fgur 7.4 h H-brg nvrtr ol h rotor angular splacnt transucr h uaratur ncrntal ncor s a coonly us rotatonal splacnt transucr n oton control applcatons. A typcal uaratur ncrntal ncor can b ol as θ x 0.5 roun4 cpr 7.7 π 5

169 whr x s th ncor count output, an cpr s th count pr rsoluton. h ncrntal ncor starts countng fro zro whn th functon s nabl. hrfor, an npnnt ntgrator s us to sulat th rlatv nput angl. Fgur 7.5 shows th Suln bloc agra for th ncrntal ncor ol. It s worth ntonng that th ncrntal ncor only asurs a rlatv splacnt fro th ntal poston of th rotor. h ntal rotor angl s ol as a rano nubr unforly strbut on th ntrval of [0, π]. W hav scuss that portanc of th ntal angl n th coornat transforaton calculatons. hrfor, whn usng a rlatv rotor angular splacnt transucr, th otor controllr ns a stratgy to fn th zro-angl confguraton. Fgur 7.5 h BDC otor coponnts bloc agra h Hall ffct snsor rapzoal BDC otors ar usually upp wth a st of thr Hall ffct snsors for th rotor poston fbac. Snusoal BDC otors can b oprat n th sa annr as trapzoal BDC otors f thy ar upp th Hall ffct snsors. hs Hall ffct snsors prov th angular poston of rotor agntc fl an can b us to 53

170 synchronz th stator agntc fl to th rotor prannt agnts. Each of th Hall snsors can b splf as followng functons: H, 0, θ θ < θ π n othrs p whr θ s th rotor angl, np s th nubr of prannt agnt pol pars, an θ s th rlatv angl btwn th rfrnc pont an th angular poston of th hall snsor. θ can b obtan fro th otor anufacturr spcfcaton or fro a otor phas tst SVPWM h Spac Vctor PWM SVPWM s th ost wly us nvrtr swtchng chans for th snusoal BDC otors. It achvs th voltag vctor control by ajustng th tng an uty cycl of th ght swtchng stats of th thr-phas H- brg nvrtr. Assung that stator cols n th thr phass ar ntcal, ach swtchng stat of th H-brg corrspons to a voltag vctor n th thr-phas stator col fra. t s loo at th xapl of th swtchng stat, 0, 0 for th branch a, b, c of th H-brg. h uppr gat of branch a, th lowr gats of branch b an c ar turn on. Rfrrng to Fgur 7.3, th pol of phas A s connct to V c / an th pols of phas B an C ar connct to V c /. h phas voltag vctor wll b V 3 c, V 3 c, V 3 c. Slarly, th ght basc voltag vctors v 0 ~v 7 for a Y- connct otor n th thr-phas fra ar shown n Fgur 7.6 an thr corrsponng swtch stats ar shown n abl 7.. Notc that v 0 an v 7 ar zro vctors. 54

171 Fgur 7.6 h voltag vctors n th spac vctor oulaton. abl 7. h ght basc voltag vctors n th SVPWM Voltag Brg Stat Motor Phas Voltag Vc n to n Voltag Vc Vctors a b c V AN V BN V CN V AB V BC V CA v v 0 0 /3 -/3 -/3 0 - v 0 /3 /3 -/3 0 - v /3 /3 -/3-0 v 4 0 -/3 /3 /3-0 v /3 -/3 /3 0 - v 6 0 /3 -/3 /3-0 v Gvn a voltag vctor n th thr-phas fra, w can fn a lnar cobnaton th ght basc voltag vctors to approxat th vctor by wghng th uty cycl for ach nvrtr stat. For xapl, th vctor of, an th zro vctors, v as v v v0 7 v shown n Fgur 7.6 can b a lnar cobnaton 55

172 v rv rv r r v0 r r v7, 7.8 whr r v sn r r v sn π 3 δ π 3 an r v sn r r ar th uty cycls of th vctors v, v, v sn δ π 3 0 δ < π 3 s th angl btwn th vctor v an th vctor. W can sparat th -D spac n to sx sctors I~VI as shown n Fgur 7.6. Any vctors locat n th 6 sctors can b xprss as a lnar cobnaton of ts narby basc vctors r r r r r v r v rv r r v0 r r v7, 7.9 whr v r r r v sn r v r v sn r v [ π 3 δ π 3 ] sn π 3 δ π 3, sn π 3, π 3 δ < π 3, 0,,5. 0,, K,5 s th sctor nubr corrsponng to th I~VI n Fgur 7.6. h transstors on/off tng can b calculat as 0 r, r z, z r r z 7.0 whr f, an s th carrr PWM sgnal fruncy. h sx-brg transstor z z f z on-off tng calculaton s suarz n abl 7.. Fgur 7.7 shows th sx transstors on-off tng n on PWM cycl whn th voltag vctor n ach sctor. 56

173 abl 7. Suary of th transstor on-off tng calculaton n ach sctor Vctor Hgh Swtchs S, S, S 3 ; ow swtchs S 4 NOS, S 5 NOS, S 6 NOS 3 ; 0 S 0 /, S 0 /, S 3 0 / S 0 /, S 0 /, S 3 0 / S 0 /, S 0 /, S 3 0 / 3 S 0 /, S 0 /, S 3 0 / 4 S 0 /, S 0 /, S 3 0 / 5 S 0 /, S 0 /, S 3 0 / For th sulaton purpos, th physcal swtchng procss s splf as a coparson btwn th noralz PWM cycl t to th uty cycl rato: S t < r tz,,,3 t 0 r tz, whr S s th uppr ar swtch nput, t z s th PWM cycl ntrval, r s th uty cycl rangng fro 0 to. h SVPWM functon tas th sctor nubr an th corrsponng bas voltag vctor tng r, r as nput, an outputs th H-brg branch on tng sgnal S, S, S 3. It s plnt n Suln as shown n Fgur Sulaton confguraton Wth th coponnts ol scuss n th prvous subscton, a coplt ol of a snusoal BDC otor syst s plnt n Suln, as shown n Fgur

174 Fgur 7.7 ransstors on-off tng n th sx sctors 58

175 Fgur 7.8 SVPWM functon plntaton n Suln o nsur goo rsoluton, th SVPWM functon, th nvrtr an th PMSM otor ust b runnng at vry hgh fruncy snc th actual PWM sgnal fruncy s usually 0~0Hz. h saplng stp for ths coponnts was st to μs. h otor controllr, Hall-ffct snsors an ncrntal ncor ol was runnng at s. hrfor, th rat-transton an zro-orr-hol functon hav to b us n orr to coplt th clos loop sulaton. In th sulaton, th sa st of otor paratrs wll b us as n th prvous chaptrs: 4 R 0. 05Ω, K N A, 0 h, 3. n p 7. Stuy of Controllr Iplntaton Issus n Sulaton Wth th BDC otor syst ol, th controllr plntaton ssu can b stu n sulaton. For control algorths vlop on th -coornat ol, th controllr ns to convrt th coan voltag V, V nto th thr-phas voltags. h phas voltags wll b convrt nto a voltag vctor n th stator phas fra, an a sctor nubr an th bas voltag vctor tng r, r wll b calculat fro 59

176 uaton 7.9 as th fnal controllr output. An xapl Suln ol of ths procss s shown n Fg 7.0. Fgur 7.9 h Suln ol of a clos loop BDC otor syst 7.. Intalzaton In aton to th coornat transforaton an SVPWM ncong, th controllr ust al wth th ntal angl probls. If th BDC otor syst s upp wth an ncrntal ncor, th actual rotor angl s unnown whn th crocontrollr s powr up. h controllr ns a sur that th angular poston asurnt s consstnt wth th zro-angl confguraton fn n th coornat transforatons. On soluton s to forc th rotor nto th zro-angl confguraton bfor noral 60

177 opraton. For xapl, th controllr outputs th bas voltag vctor v for long nough t such that th rotor rachs th ulbru stat whr th axs of th rotor agntc fl s algn to th stator phas axs. hs s th xact zro-angl confguraton that was fn for th coornat transforatons n Chaptr. h sttlng t pns on th uty cycl of th controllr output an apng factor of th rotor shafts. It can usually b lt n th orr of 0.s. At th n of th startng sch, th controllr swtchs to th noral opraton ol, an th ncrntal ncor countr s rst to zro. hus, n th noral opraton o, th rotor angl fbac fro th ncor wll b consstnt wth th coornat transforaton calculatons. hs forc-algnnt ntalzaton s plnt n Suln, as shown n Fg 7.. Fgur 7.0 h practcal controllr ol n Suln 6

178 Fgur 7. h startng sch Suln ol h sulaton of th ntalzaton stratgy wth th physcal otor ol valat th ffctvnss of th abov soluton for ral control plntaton. In th sulaton, th controllr coan an algnng voltag vctor n th rcton of th basc voltag vctor v wth a agntu of 0.Vc. h startng o was st to b 0.5 scons long. In th noral opraton o, th coan toru c was st as 0.5N, an a stp loa toru of 0.4N was appl to th rotor shaft at t t s. Fgur 7. shows sulaton rsults of th phas currnts, th actual otor currnts, th rotor vlocty an angular splacnt, th gnrat toru an th ln-to-ln voltags. In th ntalzaton o, th algnng voltag rov th rotor fro a rano ntal angl to th zro-angl confguraton wthn 0. scons. At t0.5s, th controllr swtch to th noral opraton ol, th controllr us th angl fbac fro th ncor for all coornat transforaton calculatons. h otor gnrat th coan toru 6

179 staly, whch ncat that th stator agntc fl was synchronz wth rotor agntc fl. a -coornat coan voltag b phas currnts c -coornat currnt Rotor ynacs Rotor sp an angl n-to-ln voltag PWM Fgur 7. h BDC otor physcal ol clos loop sulaton 63

180 7.. Incrntal ncor rsoluton h uas-physcal BDC otor ol sulaton can b us to stuy so othr practcal ssus, for xapl, th rsoluton of th ncrntal ncor cpr. If th ncrntal ncor s th only angular splacnt transucr, ts rsoluton can b an portant factor to th prforanc of th BDC otor actuator. As shown n th prvous chaptrs, th BDC otor controllr n th -coornat ns otor vlocty to calculat th control voltags. In aton, th coornat transforaton functons an th SVPWM pn on th rotor angular poston fbac to calculat accurat output for th otor. h accuracy of th rotor angl fbac has an pact on alost all th functons n th otor controllr. A prototyp BDC otor for th EPAS applcaton s upp wth a low-rsoluton uaratur ncor 36 cpr. Sulaton rsults wth such an ncor ar shown n Fgur 7.3. In th sulaton, th controllr swtch to noral opraton o at t0.5s. Durng th nxt 0. scons, th otor vlocty calculaton was vry nosy, an th phas currnts wr not n propr snusoal curv shap. h rason was that th lowrsoluton ncor was not abl to asur rotor angular splacnt lss than.5 gr fro on sapl to th nxt. hs ght rsult n about.8ra/s rror n otor vlocty fbac snc th otor controllr saplng t was s. h coornat transforaton calculatons wr naccurat an caus slght loss of synchronzaton btwn th stator an rotor agntc fls. hrfor th phas currnts bca nosy an ll shap. In coparson, Fgur 7.4 shows sulaton rsults wth a hghrsoluton ncor 4096 cpr. Wth th hgh-rsoluton rotor angl fbac, otor vlocty, control voltags an phas currnts wr sooth an n srabl shap for 64

181 propr opraton of th BDC otor. Obvously, th 36-CPR ncor coul not prov nough rsoluton for sooth BDC otor opraton, spcally urng th low sp opraton. a -coornat coan voltag b phas currnts c -coornat currnt Rotor sp an angular splacnt Fgur 7.3 h BDC otor sulaton wth a low-rsoluton ncor cpr t of th Quas-Physcal BDC Motor Mol h uas-physcal BDC otor ol nclus ol of all ncssary coponnts n a BDC otor syst, thrfor t rvals or practcal opraton charactrstcs of th BDC otor. W hav shown th bnft of ths ol n stuy of ntalzaton an ncrntal ncor rsoluton for practcal BDC otor applcatons. Sulaton rsults 65

182 of ths ol can also b us for ucaton purpos. It rvals or tals of th BDC otor opraton than th splr -coornat ol, though th lattr s or convnnt. a -coornat coan voltag b phas currnts c -coornat currnt Rotor sp an angular splacnt Fgur 7.4 h BDC otor sulaton wth a hgh-rsoluton ncor cpr4096 Unfortunatly, sulaton of th uas-physcal BDC otor ol stll cannot rplac ral otor xprnts. On savantag of th sulaton s ts sulaton sp. Snc th SVPWM functon, th nvrtr ol an th PMSM otor ol hav to run at vry short stp μs, th sulaton cannot run at ral t. On a PC upp wth a.8ghz CPU an MAAB 6.5, t too or than 00 scons to coplt a -scon sulaton. In aton, bcaus th PMSM otor ol was solv 500,000 ts ach 66

183 scon, sulaton rsults was rlatvly snstv to th slcton of solvr. In th Suln nvronnt, th o solvr gav th closst sulaton rsults to that of th -coornat ol alon. 7.4 Suary In ths chaptr, a uas-physcal ol was vlop for a typcal thr-phas Y- connct snusoal BDC otor. hs ol nclu th rotor poston/sp transucrs, th nvrtr, th SVPWM algorth an tc. A ln-to-ln PMSM ol rplac th phas ol. Clos loop sulaton of th controllrs an th physcal ol hlp to stuy th plntaton ssus of th -coornat controllrs such as th ntal rotor angular poston an th ncrntal ncor rsoluton. h uas-physcal BDC ol s not a ral-t ol. It was slow n sulaton bcaus vry short saplng stp μs was n to sulat th ntracton btwn th nvrtr an th PMSM otor wth goo rsoluton. For slar rason, th sulaton rsults wr also slghtly snstv to slcton of solvr. Dspt of ths lts, th uas-physcal ol sulaton was prov to b a usful tool to valat th controllr functons that wr not scuss n th -coornat controllr ol, for xapl, th SVPWM an th coornat transforaton functons. Wth th uasphyscal ol sulaton, th rs of upnt aag by th tratonal tral an rror tho can b sgnfcantly ruc. 67

184 Chaptr 8 CONCUSION AND FUURE WORKS 8. Concluson h Elctrcal Powr Strng syst EPAS wll b th an stra of futur autootv powr strng syst for ts avantag of nrgy ffcncy an flxblty. h snusoal brushlss DC otor has bn ntf as on of 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 actuators. Aaptv control s on of th ost sutabl canat tchnus for th paratr varaton probl. h -coornat ol has bn wly us for th BDC otor control an ynacs analyss. W start th aaptv control vlopnt wth rvaton of th -coornat ol for a typcal thr-phas snusoal BDC otor. h athatcal rvaton rval how th rotor angular poston θ was cancll n th coornat transforatons fro th thr-phas fra to th -coornat, but th θ was nspnsabl n th plntaton of th any controllr sgn wth th -coornat ol. h rvaton also xplan th portanc of th zro-angl θ0 fnton. hough t was not shown n th -coornat ol, th zro-angl confguraton trn th phas of alost all snusoal functons n th coornat transforatons. Wth th -coornat ol, svral aaptv algorths wr stu for th snusoal BDC otor applcaton. h frst group of algorths stat th otor paratrs by ntgratng th otor paratr rrors that wr solv algbracally wth 68

185 -axs currnt fbac fro on or or loops. hs algorths wr not as th - solvr algorths. Crtan contons wr st on th solv paratr rrors to avo nos u to sngularty n atrx nvrson opratons. Stablty of th algorths was prov athatcally an provnts wr propos for such algorths by statng th otor ynacs an rotor vlocty saplng lay. In aton, th Gra- Scht orthonoralzaton procss was propos to prov staton prforanc by alng wth th corrlaton btwn th rotor vlocty an coan toru. h scon aaptv algorth was propos to stat th paratr rrors by usng both th -axs an -axs currnt, fbac through th sa procss as that of th -solvr algorths, snc an always xst n par. hs algorth was not as th -solvr. Coparng to th -solvr algorths, th -solvr was splr snc t only rur on loop of currnt fbac to calculat th paratr rror. On th othr han, th -solvr rur arbtrary -axs coan currnt so as to gt srabl staton prforanc. h contons for unbas staton of th paratr rrors wr stu through Mont Carlo sulatons. hs contons turn out to b slar to th contons n th -solvr algorths that wr st on th solv paratr rrors to avo sngularty n atrx nvrson opratons. h thr paratr staton algorth was th xtn Kalan fltr EKF. h EKF too avantag of th nowlg of th nonlnar plant ol. Unl th prvous two algorths that ha a fx gan for ntgratng th paratr rrors, th EKF coput lnarzaton of th nonlnar plant ol an ajust th Kalan gans n ral t. h EKF s clos to optal fltrng gvn stay stat otor opraton. It onstrat supror prforanc n both paratr an stat staton wth fast 69

186 convrgnc an cnt accuracy n sulatons. Howvr, th coputaton cost an th stablty ssu wr barrrs for plntng th EKF n practc. h ol rfrnc aaptv control MRAC was th fourth algorth stu for th BDC otor paratr varaton probl. By tang a rfrnc ol for th clos loop otor ynacs, th MRAC algorth utlz th rvatv of an for th paratr staton an currnt tracng. Coparng to th -solvr an th -solvr, th MRAC prov th ata ffcncy an thrfor th convrgnc sp of paratr staton. h MRAC was sgn to antan non-postv rvatv of a uaratc cost functon of th rfrnc ol tracng rror an th paratr rrors. hrfor, th stablty of ths algorth was guarant. h MRAC was coputatonally splr an fastr than th EKF algorth. For vlopnt purpos, all ths aaptv control algorths wr valat n otor bnch tst sulatons n whch th coan toru an th otor vlocty wr assu to b Gaussan rano sgnals an npnnt to ach othr. In ral applcatons, th coan toru an th otor vlocty ar usually corrlat to so xtnt. o valuat th prforanc of ths algorths, thy wr copar n th clos loop sulaton of an EPAS syst an a otor sp control applcaton. In gnral, th clos loop sulaton rsults rval a tra-off btwn th prforanc an algorth coplxty or coputaton costs. Bttr prforanc was obtan at th cost of hghr coplxty of th control algorth. Morovr, th sulaton rsults ncat that th outr loop controllr an plant ynacs woul ruc th prforanc of th aaptv controllrs f th outr loop controllr ha rlatvly narrow fruncy 70

187 rspons or f th outr loop syst ynacs rspon ffrntly to th ffrnt paratr rror cobnatons. In th clos loop sulatons, th EKF show th bst paratr staton prforanc aong th four algorths. It was also rlatvly nsnstv to th coan sgnals fro th outr loop systs. hs avantags of th EKF wr u to th ral t lnarzaton of th nonlnar ol an th optal Kalan fltrng bas on th lnarz ol. h srabl prforanc of th EKF ca at th hghst coputng cost. Wth th EKF algorth, th controllr n to coput th lnarzaton functon an to upat a 4x4 covaranc atrx for th 4 augnt stats at vry loop. hs rsult n hgh rurnt on coputaton rsourcs. h -solvr wth th Gra-Scht orthonoralzaton show th slowst paratr staton convrgnc, but t ha alost th tghtst stay stat paratr staton rror boun. h orthonoralzaton an approxaton of th nvrs otor ynacs consu scon hghst coputaton rsourc. h -solvr rur lss coputaton rsourc, but ts stay stat accuracy was th lowst aong th four algorths, spcally n R. hs was u to th fact that th -axs currnt was sgnfcantly lowr than th -axs currnt n agntu. h slow paratr staton of ths two algorths was partly bcaus of th nglct currnt ynacs n paratr rror coputaton an th contons for sngularty. h MRAC show th scon hghst fast paratr staton convrgnc sp n th clos loop sulaton. It also ha th tghtst varaton of currnt tracng rror partly bcaus of currnt tracng was ts onant control objctv. Howvr, snc th paratr staton was an ntrat ans to achv rfrnc ol currnt 7

188 tracng, t was rlatvly snstv to th outr loop syst ynacs. Coparng to th EKF an th -solvr wth th Gra-Scht orthonoralzaton, th MRAC algorth was spl an rur lss rsourc. Ovrall, th MRAC appar to b a bst cobnaton of accuracy an coputng coplxty aong th four algorths. h clos loop sulatons also rval that th outr loop syst ynacs appar to hav pact on th prforanc of th aaptv BDC otor controllrs. Fro th otor controllr s pont of vw, th outr loop controllr an th EPAS plant togthr for a fbac loop fro th otor output toru/vlocty to th coan toru. h outr loop syst ha so spcfc fruncy rspons u to ts sgn. hs usually narrow th fruncy contnts of th coan toru, whch was on of th xctaton sgnals to th aaptv controllrs. As a rsult, th prforanc of th aaptv controllrs ruc slghtly whn copar to th otor bnch tst sulatons whr th coan toru an vlocty wr assu to b rano an npnnt. h EKF, wth ts obsrvr structur, appar to b lss snstv to th outr loop syst ynacs. hs was bcaus that th EKF too th otor control voltags an currnt asurnts as ts nput sgnals, an th coan toru was not rctly nvolv n ts staton procss. Fro th usr prspctv, th coan toru or currnt tracng s th ost portant crtron for th otor controllr slcton. h stablty of th algorth plays ually portant rol for practcal control plntaton. In aton, th algorth coplxty an coputaton costs ar also portant factors for plntng th control algorth. Consrng ths factors, th MRAC algorth appars to b th ost prosng control canat for th BDC otor actuator n th EPAS applcaton. 7

189 On of th unu charactrstcs of th BDC otor control probl s that th control algorths vlop n th -coornat cannot b rctly us for ral otor control applcaton. Coornat transforatons ar nspnsabl btwn th -coornat controllr an physcal otor syst. In aton, th control voltag ust b nco n a sutabl nvrtr actuaton sch such as th spac vctor puls wth oulaton. o facltat th BDC otor control plntaton, a uas-physcal BDC otor ol was vlop. So practcal control plntaton ssus wr stu n sulaton, nclung unnown ntal angl, an rotor angular splacnt fbac rsoluton. Evn though th sulaton coul not run ral-t, an th sulaton rsults wr slghtly snstv to solvr, th sulaton rsults prov that t was a usful tool to valat th controllr functons such as th SVPWM an th coornat transforatons. 8. Futur Wors Str-by-Wr s on of th potntal canat tchnologs for futur autootv strng systs. Coparng to th tratonal hyraulc or chancal strng syst, th Str-by-Wr syst rplac th harwar ln btwn th strng control strng whl an th roa whls wth lctroncally transtt coan sgnal. On th on han, ths tchnology crats gratr flxblty of harwar nstallaton an softwar rconfguraton that nabls ntllgnt rvr assstanc an vhcl stablty control. On th othr han, th lctronc sgnal transsson s on of th an ssus for th Str-by-Wr n trs of fal safty an th robustnss of th lctronc control syst. h snusoal BDC otor s an al canat actuator for th Str-by-Wr systs. Fault tolranc wll b a ncssty for th BDC otor control n such as syst. 73

190 h BDC otor opraton nvolvs th rotor poston/sp transucr, PWM nvrtr that conssts of thr-phas rctfr an powr transstors, an th PMSM. ypcal fault nclus poston transucr falur, powr transstor falur, an otor stator col opn crcut, short to groun, an tc. h paratr staton algorth an th EKF can b plnt for th fault agnoss. Statstcal stuy has to b carr out so as to obtan th fault tcton thrshols. Snsorlss control has bn a hot rsarch an vlopnt rcton for BDC otors. Snsorlss ans that th PMSM wll b oprat wthout a rotor angular poston snsor. Usually so crtan typ of obsrvr s ploy for asyptotcal staton of th otor sp an angular poston. For th trapzoal BDC otors, BEMF zro crossng tcton s also propos to rplac th phas Hall snsors. It s a goo canat tchnology for th fal-saf opraton of th BDC otor n cas of poston snsor fault. h PWM nvrtr provs not only a ffctv way for th PMSM opraton wth a DC powr supply, t also gv so flxblty for th BDC otor falsav opraton n cas of stator wnng fault such as opn crcut an nvrtr brg coponnt falur. 74

191 Appnx I HE PROBABIIY DENSIY FUNCION DERIVAION xp, ~ x x x x x x N x σ μ π σ σ μ A. > ε ε ε x x x y A. y F y an ar th cuulatv strbuton functon of y an x. F x x If ε 0 < y, thn ε x ε 0, < y y F y x F y F x x y A.3 If ε > y, thn 0 > > x ε [ ] ε ε ε, 0 0 > > > y F F x F y F x x x y A.4 If ε < y, thn ε > > x 0 [ ] ε ε ε, 0 0 < > > y F F x F y F x x x y A.5 If ε 0 > y, thn ε x ε 0, > y y F y x F y F x x y A.6 75

192 > > < ε ε ε ε ε, 0, 0, y F F y y F y y F y F x x x x y A.7 x x g y, x x g, y y g A.8 > ε ε ε 0,, y y y f y y f y g y f x x y A.9 whr xp x x x x x y y y y f y σ μ μ πσ. A.0 76

193 Appnx II SOUION OF HE INEGRAION ERMS IN EQUAION 4.6 t [ ] I 0 cos θ θ θ, A.a [ ] J 0 sn θ θ θ. A.b hs two trs can b solv by th followng anpulaton: [ ] I 0 sn θ θ [ ] [ ] 0 0 sn sn θ θ θ θ [ ] [ ] 0 sn sn θ θ θ [ ] [ ] 0 sn sn θ θ θ [ ] J sn, or [ J I sn ]. A. Slarly, [ ] J 0 cos θ θ 77

194 [ ] [ ] 0 0 cos cos θ θ θ θ [ ] [ ] 0 cos cos θ θ θ [ ] I cos [ ] cos I J A.3 Wrt A. an A.3 n atrx for as [ ] [ ] cos sn J I. A.4 h soluton of can b obtan as J I [ ] [ ] [ ] cos xp sn xp J I. A.5 78

195 Appnx III CONINUOUS EXENDED KAMAN FIER FOR BDC MOORS A contnuous vrson of th xtn Kalan fltr for th BDC otor paratr ntfcaton probl s rv blow. h BDC otor ynacs ol can b rforulat wth th actuator prturbaton : w,w t w t w t V t t K t V t t t t t t, A3.a Assung paratrs an K ar constants subjct to rano prturbaton an, th ynacs of th paratrs can b ol as w 3 w 4 w 3 t t A3.b w 4 t t K A3.c t th stat, thn th augnt stat uaton bcos K,,, x ,, 4 3 t w t w t w t w t V t t K t V t t t t t t t t x w u x f x& 79

196 A3. an th asurnt wth asurnt nos v v, v bcos v t y t x t. A v t It s usually rasonabl to assu that th plant prturbaton w, w, w w w 3, 4 an asurnt nos v v v, ar Gaussan an npnnt: E [ w tw t θ ] Σwδ θ, A3.4a E [ v tv t θ ] Σvδ θ, A3.4b E [ wt ] 0, A3.4c E [ vt ] 0, A3.4 Sw 0 whr Sw Sv 0 Σw an Σ v S w3 0 Sv 0 Sw4 ar th spctral nsty atrx of th actuaton prturbaton an asurnt nos, an δ θ s th Drac lta functon, an E[ w tw t θ ] s th athatcal xpctaton opraton. h lnarz ol can b obtan as ˆ f F t x t t t t t 0 t, A3.5 t

197 ˆ u f t u B A ˆ ˆ x x x g C, A3.7 Wth th lnar approxaton functons, th a pror covaranc atrx s coput rcursvly fro th a postror covaranc atrx at prvous stp t Σ C Σ C t Σ Σ t F t Σ t FtΣ t Σ v W ˆ ˆ & A3.8 h Kalan gan can b coput as v Σ C t Σ G t A3.9 An th stat can b stat wth th asurnt as [ ] ˆ ˆ ˆ ˆ ˆ t t t V t t K t V t t t t x C G t B x F x u &. A3.0 h contnuous Extn Kalan Fltr conssts of th uatons A3.8, A3.9 an A3.0. 8

198 REFERENCES Acaran,., t al, 00, A robust controllr sgn for rv by wr hyraulc powr strng syst, Proc. 00 A. Control Conf., Anchorag, AK, pp Anwar, S., 004, A Paratrc Mol of an Ey Currnt Elctrc Machn for Autootv Brang Applcatons, IEEE rans. Contr. Sys. ch., Vol., No. 3. Batzl,. D., K. Y., 005, Elctrc Propulson Wth Snsorlss Prannt Magnt SynchroNous Motor: Iplntaton an Prforanc, IEEE RANSACIONS ON ENERGY CONVERSION, Vol. 0, No. 3. Bolognan, S., t al, 00, EKF-Bas Snsorlss IPM SynchroNous Motor Drv For Flux-Wanng-Applcatons, IEEE. Burton A. W., 00, Control Objctvs an Systs Analyss: InNovaton Drvrs for Elctrc Powr-Assst Strng, Proc. Arcan Control Confrnc, pp Cao, C., t al, 003, Paratr Convrgnc n Nonlnarly Paratrz Systs, IEEE rans. Autoat. Contr., Vol. 48, No. 3. Caut, S., t al, 00, Robust Control an Stablty Analyss of narz Syst wth Paratr Varaton: Applcaton to Inucton Motors, Procngs of th 40th IEEE Confrnc on Dcson an Control, pp Ctn, A., E., t al, 005, Coplant Control of Elctrc PowrAssst Strng Systs, SAE chncal Papr

199 Chn, H. C., t al, 000, Robust Currnt Control for Brushlss DC Motors, IEE Proc. Elctr. Powr Appl., Vol. 47, No. 6, pp Chn, H., t al, 006, PMSM Srvo Drv Syst for Elctrc Powr Strng Bas on wo-dgr-of-fro oru Control, Intrnatonal Confrnc of Inustral chnology, pp Chn, J., ang P. C., 999, A slng o currnt control sch for PWM brushlss DC otor rvs, IEEE ransactons On Powr Elctroncs, Vol. 4, No. 3. Chang, w., Zhu,., Patanar R., 007, Man Valu Engn Molng an Valaton for a 4-stro Sngl Cylnr Gasoln Engn, rns n Appl Scncs Rsarch, Vol., No.. Cho, C., t al, 007, Mult-oan olng of Elctrc Powr Strng wth PMSM Drv Syst, IEMDC '07, pp Cho, H., t al, 005, On-Cntr Hanlng Charactrstcs of Motor Drvn Powr Strng Syst, Procngs of th 005 IEEE Intrnatonal Confrnc on Mchatroncs, July. Duby, G. K., 00, Funantals of Elctrcal Drvs, Alpha Scnc Intrnatonal t., Pangbourn. Fan, J. C., Kobayash., 998, A Spl Aaptv PI Controllr for nar Systs wth Constant Dsturbancs, IEEE rans. Autoat. Contr., Vol. 43, No. 5. Forra, A., t al, 00, sgn of robust controllrs wth constrants on th control sgnal applcaton for Burshlss D C rvs, IECON 0, pp

200 G, S. S., Hong, F.,,. H., 003, Aaptv Nural Ntwor Control of Nonlnar Systs Wth UnNown Dlays, IEEE rans. Autoat. Contr., Vol. 48, No.. Gobbo, D. D., t al, 00, Exprntal Applcaton of Extn Kalan Fltrng for Snsor Valaton, IEEE rans. Contr. Sys. ch., Vol. 9, No.. Golub, G. H., Van oan, C. F., 996, Matrx Coputatons, 3r ton, Johns Hopns Unvrsty Prss, pp.50. Grwal, M. S., Anrws, A. P., 00, Kalan Fltrng: hory an Practc Usng MAAB, n ton, John Wly & Sons. Gun, R. A., yn, C., 000, A Novl Applcaton Of h Fast Sulat Dffuson Algorth For Dynacal Paratr Intfcaton Of Brushlss Motor Drv Systs, ISCAS 000-IEEE Intrnatonal Syposu on Crcuts an Systs, Gnva. Hau, M. E., t al, 003, A Snsorlss Intal Rotor Poston Estaton Sch for a Drct oru Controll Intror Prannt Magnt Synchronous Motor Drv, IEEE RANSACIONS ON POWER EECRONICS, Vol. 8, No. 6. Hartr, W., t al, 000, Futur Elctrcal Strng Systs: Ralzatons wth Safty Rurnts, SAE chncal Papr Hoang., t al, 994, Analyss an plntaton of a ral-t prctv currnt controllr for prannt-agnt synchronous srvo rvs, IEEE RANSACIONS ON INDUSRIA EECRONICS, Vol. 4., No.. 84

201 Hossan, S., Dshpan, U., 003, Molng Of Practcal Nonalts In A Prannt Magnt SynchroNous Motor Drv, IEEE Inustry Applcatons Confrnc, Vol. 3, pp Hotzl, R., Karsnt,., 998, Aaptv racng Stratgy for a Class of Nonlnar Systs, IEEE rans. Autoat. Contr., Vol. 43, No. 9. Huang, B., 00, Dtcton of Abrupt Changs of otal ast Suars Mols an Applcaton n Fault Dtcton, IEEE rans. Contr. Sys. ch., Vol. 9, No.. Huang, J., 00, An Aaptv Copnsator for a Class of narly Paratrz Systs, IEEE rans. Autoat. Contr., Vol. 47, No. 3. Ils-Klupnr, D., t al, 005, Elctrc Actuaton chnologs for Autootv Strng Systs, SAE chncal Papr IoanNou, P. A., Sun, J., 996, Robust Aaptv Control, Prntc-Hall, Nw Jrsy. Isrann, R., achann, K. H., Mato, D., 99, Aaptv Control Systs, Prntc Hall, Hl Hpsta. Jang, Z.-P., Hll, D. J., 999, A Robust Aaptv Bacstppng Sch for Nonlnar Systs wth Unol Dynacs, IEEE rans. Autoat. Contr., Vol. 44, No. 9. Johnson, J. P., t al, 999, Rvw of Snsorlss Mthos for Brushlss DC, IEEE. Khall, H. K., 996, Nonlnar Systs, n ton, Prntc-Hall, Nw Jrsy. Khorra, F., Krshnaurthy, P., Mlot, H., 003, Molng an Aaptv Nonlnar Control of Elctrc Motors., Sprngr, Nw Yor. K, K. H., t al, 994, Paratr staton an control for prannt agnt synchronous otor rv usng ol rfrnc aaptv tchnu, IEEE. 85

202 K, K., t al, 999, A Currnt Control for a Prannt Magnt SynchroNous Motor wth a Spl Dsturbanc Estaton Sch, IEEE rans. Contr. Sys. ch., Vol. 7, No. 5. Klnau, J., Patanar, R. P., Collr, S., Chany, A., 003, Fbac paratr staton for lctrc achns, US app #. 0/0393, Pub. # US 003/ A. Klugr, M., Harrs, J., 007, Ful EcoNoy Bnfts of Elctrc an Hyraulc Off Engn Accssors, SAE chncal Papr Ko, J. S., 998, Asyptotcally stabl aaptv loa toru obsrvr for prcson poston control of BDC otor, IEE Proc. Elctr. Powr Appl., Vol. 45, No. 4., pp Kosatopoulos, E. B., IoanNou, P. A., 00, Robust Swtchng Aaptv Control of Mult-Input Nonlnar Systs, IEEE rans. Autoat. Contr., Vol. 47, No. 4. Kovuhulrungsr,., Kos,., 006, Prcs Sp Estaton Fro a ow- Rsoluton Encor by Dual-Saplng-Rat Obsrvr, IEEE/ASME RANSACIONS ON MECHARONICS, Vol., No. 6. Kraus, P. C., 00, Analyss of Elctrc Machnry an Drv Systs n ton, Wly-IEEE Prss, Nw Yor. Krstc, M., Kanllaopoulos, I., Kootovc, P.V., 995, Nonlnar an Aaptv Control Dsgn, Wly, Nw Yor., C. K., Kwo, N. M., 995, Varabl Structur Controllr wth Aaptv Swtchng Surfacs, Procngs of th Arcan Control Confrnc, Vol., pp

203 , S., t al, 004, Iprov Vlocty Estaton for ow-sp an ransnt Rgs Usng ow-rsoluton Encors, IEEE/ASME RANSACIONS ON MECHARONICS, Vol. 9, No. 3. ao,., Chn,., 000, An Exponntally Stabl Aaptv Frcton Copnsator, IEEE rans. Autoat. Contr., Vol. 45, No.5. anon, S., saals, K. S., 000, Mol Rfrnc Aaptv an Nonaaptv Control of nar -Varyng Plants, IEEE rans. Autoat. Contr., Vol. 45, No. 7. n, F. J., n, Y. S., 999, A Robust PM SynchroNous Motor Drv wth Aaptv Uncrtanty Obsrvr, IEEE ransactons on Enrgy Convrson, Vol. 4, No. 4. n, W., Qan, C., 00, Aaptv Control of Nonlnarly Paratrz Systs: A Nonsooth Fbac Frawor, IEEE rans. Autoat. Contr., Vol. 47, No. 5. n, W., Qan, C., 00, Aaptv Control of Nonlnarly Paratrz Systs: h Sooth Fbac Cas, IEEE rans. Autoat. Contr., Vol. 47, No. 8. u, Y.,, X. Y., 003, Robust Aaptv Control of Nonlnar Systs Rprsnt by Input Output Mols, IEEE rans. Autoat. Contr., Vol. 48, No. 6, oh, A. P., Qu, C. Y., Fong, K. F., 003, Aaptv Control of Dscrt Systs Wth Concav/Convx Paratrzatons, IEEE rans. Autoat. Contr., Vol. 48, No. 6. ow. S., t al, 996, Motor ntty-a otor ol for toru analyss an control, IEEE RANSACIONS ON INDUSRIA EECRONICS, Vol.43, No., pp

204 MarNo, R., Prsaa, S., an o, P., 999, Global Aaptv Output Fbac Control of Inucton Motors wth uncrtan Rotor Rsstanc, IEEE rans. Autoat. Contr., Vol. 44, No. 5. MarNo, R., o, P., 999, An Aaptv Output Fbac Control for a Class of Nonlnar Systs wth -Varyng Paratrs, IEEE rans. Autoat. Contr., Vol. 44, No.. Matsu, N., Ohash, H., 99, DSP-bas aaptv control of a brushlss otor, IEEE transactons on nustry applcatons, Vol. 8, No.. McCann, R., t al, 998, Influnc of Motor Drv Paratrs on th Robust Stablty of Elctrc Powr strng Systs, Powr Elctroncs n ransportaton, pp Mllr, D. E., 003, A Nw Approach to Mol Rfrnc Aaptv Control, IEEE rans. Autoat. Contr., Vol. 48, No. 5. Mlan, R., Bortoff, B. A., 999, Obsrvr-Bas Aaptv Control of a Varabl Rluctanc Motor: Exprntal Rsults, IEEE rans. Contr. Sys. ch., Vol. 7, No. 5. Mr, S., Elbulu, M. E., Zngr, D. S., 998, PI an Fuzzy Estators for unng th Stator Rsstanc n Drct oru Control of Inucton Machns, IEEE ransactons on Powr Elctroncs, Vol. 3, No., pp Mr, S., Husan, I., Elbulu, M. E., 998, Swtch Rluctanc Motor Molng wth On-n Paratr Intfcaton, IEEE ransactons on Inustry Applcatons, Vol. 34, No. 4, pp

205 Moha, O. A., t al, 005, Physcal Molng of PM SynchroNous Motors for Intgrat Couplng Wth Machn Drvs, IEEE RANSACIONS ON MAGNEICS, Vol. 4, No. 5. Moslr, O., t al, 999, Mol-Bas Fault DagNoss of an Actuator Syst Drvn by th Brushlss DC Motor, Procngs of Arcan Control confrnc. Moslr, O., Isrann, R., 000, Applcaton of Mol-Bas Fault Dtcton to a Brushlss DC Motor, IEEE ransactons on Inustral Elctroncs, Vol. 47, No. 5, pp Narnra, K. S., Balarshnan, J., 997, Aaptv Control Usng Multpl Mols, IEEE rans. Autoat. Contr., Vol. 4, No.. Narnra, K. S., Xang, C., 000, Aaptv Control of Dscrt- Systs Usng Multpl Mols, IEEE rans. Autoat. Contr., Vol. 45, No. 9. Ojo, O., t al, 00, Mols For h Control An Sulaton Of SynchroNous yp Machn Drvs Unr Varous Fault Contons, Inustry Applcatons Confrnc, Vol.3, pp: Ong, C., 998, Dynac Sulaton of Elctrc Machnry Usng Matlab /Suln, Prntc Hall, Nw Jrsy. Par,. J., Han, C. S.,, S. H., 005, Dvlopnt of th lctronc control unt for th rac-actuatng str-by-wr usng th harwar-n-th-loop sulaton syst, Mchatroncs, Vol. 5, pp Parar, M., Hung J. Y., 004, A Snsorlss Optal Control Syst for an Autootv Elctrc Powr Assst Strng Syst, IEEE ransactons On Inustral Elctroncs, Vol. 5, No.. 89

206 Patanar R., Zhu,., 004, Ral-t Multpl Paratr Estaton for Voltag Controll Brushlss DC Motor Actuators, Proc. Arcan Control Confrnc. Patanar, R., Zhu,., 007, Brushlss DC otor actuator halth ontorng an graaton copnsaton va ral-t ultpl paratr staton, Intrnatonal Journal of Autoaton an Control, Vol., No., pp Pat, Y., Krshnaprasa, P., 994, Ratonal wavlts n ol ructon an syst ntfcaton, Proc. IEEE Conf. on Dcson an Control, a Buna Vsta, F. PatNo, H. D., u D., 000, Nural Ntwor-Bas Mol Rfrnc Aaptv Control Syst, IEEE rans. Syst. Man an Cybrn., Vol. 30, No.. Plczws, P. M., Kunz, U. H., 990, h optal control of a constran rv syst wth brushlss DC otor, IEEE ransactons on Inustral Elctroncs, Vol. 37, No. 5. Ptrovc, V., t al, 000, Dsgn an Iplntaton of an Aaptv Controllr for oru Rppl Mnzaton n PM SynchroNous Motors, IEEE ransactons On Powr Elctroncs, Vol. 5, No. 5, pp Pllay, P, Krshnan, R., 989, Molng Sulaton An Analyss Of Prannt- Magnt Motor Drvs. Part I. h Prannt-Magnt SynchroNous Motor Drv, IEEE RANSACIONS ON INDUSRY APPICAIONS, Vol. 5, No.. Pllay, P, Krshnan, R., 989, Molng Sulaton an Analyss of Prannt- Magnt Motor Drvs Part II: h Brushlss DC Motor Drv, IEEE RANSACIONS ON INDUSRY APPICAIONS, Vol. 5, No.. 90

207 Qu, Z., 00, Robust Control of Nonlnar Systs by Estatng Varant Uncrtants, IEEE rans. Autoat. Contr., Vol. 47, No.. Rahan, M. A., t al, 003, Nonlnar Control of Intror Prannt-Magnt SynchroNous Motor, IEEE RANSACIONS ON INDUSRY APPICAIONS, Vol. 39, No.. Rahan, M. A., Zhou P., 996, Analyss of brushlss prannt agnt synchronous otors, IEEE RANSACIONS ON INDUSRIA EECRONICS, Vol.43, No.. Rahan, M. F., t al, 003, A Drct oru-controll Intror Prannt-Magnt SynchroNous Motor Drv Wthout a Sp Snsor, IEEE RANSACIONS ON ENERGY CONVERSION, Vol. 8, No.. Ravanth, R., Myn, S. P., 999, Bouns on Achvabl Prforanc n th Intfcaton an Aaptv Control of -Varyng Systs, IEEE rans. Autoat. Contr., Vol. 44, No. 4. Rovthas, G. A., 999, Robustfyng Nonlnar Systs Usng Hgh-Orr Nural Ntwor Controllrs, IEEE rans. Autoat. Contr., Vol. 44, No.. Rubaa, A., Kotaru, R., 000, Onln Intfcaton an Control of a DC Motor Usng arnng Aaptaton of Nural Ntwors, IEEE ransactons On Inustry Applcatons, Vol. 36, No. 3, pp Rugh, W. J., 996, nar Syst hory n ton, Prntc-Hall, Nw Jrsy. Say, A. H., 00, A Frawor for Stat-Spac Estaton wth Uncrtan Mols, IEEE rans. Autoat. Contr., Vol. 46, No. 7. 9

208 Shyu, K. K., Shh, H. J., Fu, S. S., 998, Mol Rfrnc Aaptv Sp Control for Inucton Motor Dv Usng Nural Ntwors, IEEE rans. In. Elctron., Vol. 39, No., pp80-8. Song, J., t al, 004, Mol vlopnt an control thoology of a nw lctrc powr strng syst, Proc. Inst. Mch. Eng. D. J. Autoobl Eng., Vol. 8, No. 9. Sozr, Y., t al, 997, Drct ol rfrnc aaptv control of prannt agnt brushlss DC otors, Procngs of th 997 IEEE Intrnatonal Confrnc on Control Applcatons. Sozr, Y., Kaufan, H., orry, D. A., 997, Drct Mol Rfrnc Aaptv Control of Prannt Magnt Brushlss DC Motors, Proc. h 997 IEEE Intl. Conf. On. Contr. Appl., pp Sugtan, N., t al, 997, Elctrc powr strng wth H-n?nty control sgn to obtan roa nforaton, Proc. 997 Ar. Control Conf., Albuuru, NM, pp Surshbabu, N., Farrll J. A., 999, Wavlt-Bas Syst Intfcaton for Nonlnar Control, IEEE rans. Autoat. Contr., Vol. 44, No.. a, M., Hngw, P., ozua, M., 004, Molng an control of strng syst of havy vhcls for autoat hghway systs, IEEE/ASME rans. Mchatron., Vol. 9, No. 4, pp ashta,., Matsu N., 994, Snsorlss Brushlss DC Motor Drv wth EMF constant Intfr, IEEE. 9

209 anaa, H., t al, 007, Dvlopnt of oru Controll Actv Strng wth Iprovng th Vhcl Stablty for Brushlss EPS, SAE chncal Papr ao, G., 003, Aaptv Control Dsgn an Analyss, John Wly & Sons, Nw Jrsy. ao, G., Josh, S. M., Ma, X., 00, Aaptv Stat Fbac an racng Control of Systs wth Actuator Falurs, IEEE rans. Autoat. Contr., Vol. 46, No.. lfor, D., t al, 005, Onln Intfcaton of Inucton Machn Elctrcal Paratrs for Vctor Control oop unng, IEEE RANSACIONS ON INDUSRIA EECRONICS, Vol. 50, No.. rzc, B., Jarc, M., 00, Dsgn an Iplntaton of th Extn Kalan Fltr for th Sp an Rotor Poston Estaton of Brushlss DC Motor, IEEE ransactons On Inustral Elctroncs, Vol. 48, No. 6. an, D., Yn, G., X, G., 004, Mol an H-nf robust control sgn for lctrcpowr strng syst, Proc. 004 Int. Conf. Intll. Mchatron. Auto., Chngu, P. R. Chna, pp Urasa, N., t al, 000, An Accurat Molng For Prannt Magnt SynchroNous Motor Drvs, IEEE 000. Wang, J.B., aw, C. M., 997, Control of nucton otor rv for provng opratng charactrstcs an ynac rspons, ASME/IEEE Mchatroncs, Vol. 7, No. 7, pp Xu, H., IoanNou, P. A., 003, Robust Aaptv Control for a Class of MIMO Nonlnar Systs Wth Guarant Error Bouns, IEEE rans. Autoat. Contr., Vol. 48, No

210 YANG Gu-j, t al, 003, Mol of Prannt Magnt Inuctor yp SynchroNous Motor, IEEE. Zarba, A.., ubaa, M. K., Stuntz, R. M., 998, Control an strng fl ssus n th sgn of an lctrc powr strng syst, Proc. 998 A. Control Conf., Phlalpha, PA, pp Zn I., orn., Forgz C., 000, An Extn Kalan Fltr an an Approprat Mol for th Ral-t Estaton of th Inucton Motor Varabls an Paratrs, url: 00/ IASED_MECO_Zn.pf. Zhang, Q., 00, Aaptv Obsrvr for Multpl-Input Multpl-Output MIMO nar -Varyng Systs, IEEE rans. Autoat. Contr., Vol. 47, No. 3. Zhang, Q., 994, Usng wavlts n Nonparatrc staton, Proc. 33r IEEE Conf. Dcson an Control, a Buna Vsta, F. Zhang, Q., Bnvnst, A., 99, Wavlt ntwors, IEEE rans. Nural Ntwors, Vol. 3. Zhang,., t al, 000, Aaptv Control of Frst-Orr Systs wth Nonlnar Paratrzaton, IEEE rans. Autoat. Contr., Vol. 45, No. 8. Zhang, Y., Fan, B., IoanNou, P. A., 003, Bacstppng Control of nar - Varyng Systs Wth KNown an UnNown Paratrs, IEEE rans. Autoat. Contr., Vol. 48, No.. Zhang, Y., IoanNou P. A., 000, A Nw nar Aaptv Controllr: Dsgn Analyss an Prforanc, IEEE rans. Autoat. Contr., Vol. 45, No

211 Zhao, J., Kanllaopoulos, I., 00, Actv Intfcaton for Dscrt- Nonlnar Control Part I: Output-Fbac Systs, IEEE rans. Autoat. Contr., Vol. 47, No.. Zhao, J., Kanllaopoulos, I., 00, Actv Intfcaton for Dscrt- Nonlnar Control Part II: Strct-Fbac Systs, IEEE rans. Autoat. Contr., Vol. 47, No.. Zhu,., Patanar R., 004, Aaptv Estaton for Brushlss DC Motor Actuators n Elctrc Powr Assst Strng, SAE tchncal papr Zhu,., Patanar, R., 006, Multpl Paratr Estaton Usng Gra-Scht OrthoNoralzaton for Brushlss DC Motor Actuators, Intrnatonal Journal of Vhcl Autonoous Systs, Vol. 4, No., pp