# Mathematical Modelling and Predictive Control of Permanent Magnet Synchronous Motor Drives

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1 ransactons on Elctrcal Engnrng, Vol. (), o. 4 4 athatcal ollng an Prctv ontrol o Prannt agnt ynchronos otor Drvs Květoslav la Dpt. o aptv ysts, Insttt o Inoraton hory an toaton o th R Po Voárnso věží 4, 8 8 Prag 8, zch Rpblc, -al: bstract h papr als wth a athatcal ollng o th thr-phas Prannt agnt ynchronos otors (P) an thr ol-bas control. hs otors ar s n rvs o robots an achn tools. h constrcton o thr athatcal ol s scss hr wth rspct to a ol-bas control sgn. h ol s copos va athatcal-physcal analyss. h analyss s otln n th an thortcal ponts. s a prosng ol-bas approach, th prctv control s xplan. It rprsnts jst a prosng altrnatv to th stanar solton bas on th vctor casca control. Kywors Prannt agnt synchronos otor, athatcal ollng, scrt prctv control, ltstp xplct control law, sqar-root optzaton. I. IRODUIO ynchronos otors wth a thr-phas stator wnng an a rotor wth prannt agnts (ltrnat rrnt otors) blong to th latst gnraton o otors. hy ar appl as rvs to achn tools an robots. Unl Drct rrnt (D) /brshs/ otors an Elctrcally ot (E) /D brshlss/ otors, th Prannt agnt ynchronos otors (P) (Fg. ) ay b congr as lnar otors, whch nowaays co n s n robotc applcatons as wll. h otors wor on th prncpl o sltanos control o aplt an rqncy o all thr trnal haronc crrnts wth th Pls-Wth-olaton (PW). h stator o a thr-phas otor rprsnts thr snsoally strbt wnngs wth axs splac by. Whn th wnngs ar xct by balanc thr-phas snsoal crrnts, th cobn ct s qvalnt to a sngl snsoally strbt wnng xct by a constant crrnt an rotatng at th stator rqncy. h rotor agntc l s sppl by prannt agnts nsta o lctroagnts [6]. In ths papr, th athatcal ollng o th P rvs wll b xplan rspctng a spcc ol-bas control sgn. onstrcton o th ol wll ars ro th athatcal physcal analyss an wll b shown n stanar coponnt ors an coplx plan or spac n rnt coornat systs spl control sgn. Fro th control pont o vw, thr ar thr an tass: poston control, sp control an crrnt (torq) control. h tass ar closly rlat to a control congraton or control loops. n otr loop s th poston loop, a l loop s th sp loop an an ntrnal loop s th crrnt loop. hs papr wll ocs on th sp control tas, whch wll b st n th llstratv xapls. onsqntly, th sp an crrnt loops wll b nvstgat. h tas wll b scss or th convntonal control approach bas on th vctor control wth a casca o PI controllrs an or an avanc control approach bas on th Gnralz Prctv ontrol (GP) [], [4], [6]. h casca congraton ans st o atonoos PI controllrs, whr tal rlatons ar xtrnal strbancs. h sttng o PI controllrs s lt only on svral statc constants. hr x congraton os not gv any spac or so possbl provnts or.g. ocatons solvng rthr control rqrnts. On th othr han, th GP s nvstgat as a gnral, spl lxbl altrnatv, whch can solv both sp an crrnt loops togthr wth th spac or solton o atonal rqrnts on th control. h papr s organz as ollows. h scton II als wth a stabl athatcal-physcal ol or th control sgn. h scton III scsss th ol ocaton an rlat assptons or th prctv control sgn. h scton IV brly scrbs th convntonal loop scha o th vctor control. h scton V concrns wth th an ponts o th GP sgn. In th scton, thr s a rvaton o qatons o th prctons an xplanaton o th sqar-root nzng procr o th qaratc crtron. h gnraton o control actons as a rslt o th nzaton s scss. h scton VI onstrats th bhavor o th convntonal vctor control an th ol prctv control by a coparatv xapl. ' ' ' ' ' rotor wth prannt agnts ' ' ' ' stator wnngs Fg.. chatc cross scton o P ynchronos otor wth pol par nbr p an pol nbr pp 6 ( p)

2 ransactons on Elctrcal Engnrng, Vol. (), o. 4 5 II. ORO-ORIEED ODE OF P DRIVE athatcal-physcal ol o th P rvs s portant both or th otln o th convntonal vctor control [], [6] an anly or th ol-bas control approachs n gnral. h ol srvs as a slaton ol or rap prototypng o th controllrs. h ol o prannt agnt synchronos otors arss ro svral natral laws an rlatons. ot, that th ocs s gvn on th stator part o th otor, whr th lctrc wnng (cols) ar blt n. Fro th rotor pont o vw, only nowlg o agntc proprts o prannt agnts s ncssary.. Us otaton h ol covrs th rlatons o th crrnt an voltag qlbr an approprat rlatons o th voltag strbton or nval phass o th thr-phas syst. h ol contans a nbr o paratrs. hr notaton an approprat nts ar gvn as ollows: R - stator rsstanc [Ω, Oh] - stator nctanc (srac P) [H, Hnry] - rotor agntc lx [Wb, Wbr] p - nbr o pol pars, pp p - pol nbr - vscos cocnt o th loa [g s - ] J - ont o loa nrta [g ] I U - spply crrnt [] - spply voltag [V],, - crrnts o nval phass,, [],, - voltags o nval phass,, [V] β, - crrnts n th β syst [], - voltags n th β syst [V] β, q - crrnts n th q syst [], q - voltags n th q syst [V] n, - chancal sp [rp], rqncy [Hz; s - ] n, - lctrcal sp [rp ], rqncy [Hz ; s - ] - chancal anglar sp [ra s - ] - lctrcal anglar sp [ra s - ] ϑ - chancal angl poston [ra] ϑ - lctrcal angl poston [ra ] τ - otor rvng torq [] τ - loa torq []. Intal Physcal Dscrptron t th syst o th qatons scrbng th physcal bass o th P bgn by an qaton o stator crrnt qlbr: () an analogosly by an qaton o stator voltag qlbr: () Frthr crcal rlaton s th stator voltag strbton xprss by a st o th ollowng qatons: R R ( ) () t t R R ( ) (4) t t R R ( ) (5) t t whr ach ln blongs to th approprat nval phas. h qatons () - (5) xprss th lctro-agntc proprts o th stator col wnng (Fg. ). ' O ' Fg.. Pol prannt agnt l wnngs or 6 pols h athatcal ol n th two-nsonal (D) spac o th thr-phas -- syst s coplt by th rlaton o lctro-chancal proprts xprss by th qaton o th torq qlbr: J && ϑ τ J & ' τ τ J & pτ pτ whr τ s a otor (rvng) torq gvn by (6) p τ R{ U I } R I (7) s a chancal loss an τ s a loa torq. ll ths qantts ollow ro th law o th nrgy consrvaton: P R{ U l. powr npt I } P ch. loa losss n ron ( ag.) τ R I P P col losss P F P ch. losss (8)

3 ransactons on Elctrcal Engnrng, Vol. (), o. 4 6 r β β r r q r β β q r r ϑ Fg.. D -- an -β coornat systs. plng ransoratons h qatons () - (6) consttt th ntal ol rprsntaton n th x D thr-phas syst or nval,, phass. hat ol can b spl both or th slaton an control sgn by two spcc transoratons. h rst s orwar lar transoraton (Fg. ): β, onsrng th crrnt qlbr (), thn th transoraton can b rc as ollows β, (9) () hs transoraton convrts () - (6) ro th D -- phas syst nto th D - β syst. h ncat transorng procr s val or both crrnt, voltag an lx coponnts consrng approprat physcal qantts rspctvly. It rprsnts rcton o thr phass or thr approprat phas axs n only two - β axs. h axs ar x to th stator coornat syst.. to th ntal -- phas syst. h transor qatons ar xprss as ollows: β R sn( ϑ ) & ϑ () t R β β cos( ϑ & ) ϑ () t J & ϑ p ( cosϑ snϑ ) pτ β () h scon transoraton s th orwar Par transoraton shown n Fg. 4: q cosϑ snϑ snϑ cosϑ β (4) hat transoraton convrts th D - β syst () - () nto th D - q syst. h - q syst nl th two x - β axs syst s consttt by two rotatng - q axs. Fg. 4. D - β an - q coornat systs h axs ar connct to th rotatng lctroagntc l o th stator col wnng or rotatng rotor wth prannt agnts. h P s a synchronos otor as t s nton rctly n ts labl. hs, th sp o th lctroagntc rotatng l s qal th sp o th rotor an proportonally synchronos wth th npt crrnt rqncy. h qatons () - () applyng (4) gt th ors: R q (5) t R (6) q q q t J & ϑ p pτ q (7) D. Drvaton o th ransoratons n oplx pac h ncat transoratons n th prvos sbscton can b rv also n a or copact or n th coplx spac. I th ntal qatons () - (5) ar consr, thn th lar transoraton s n by ans o coplx varabl as ollows: j π j π 4 j sng rprsntaton o coplx varabl as thn β (8) ϕ j cos ϕ j snϕ (9) R ( ) j( Rβ ( β β )) 44 t t 4444 wth β β cos ϑ, sn ϑ () las ntcally to th qatons () an (). nalogcally, th sa staton s at th Par transoraton. t th rvaton start ro th qatons () - (5) agan. hn, ro th gotrcal pont o vw, th qatons nalz by th Par transoraton wth natral nclson o th lar transoraton ar th ollowng: wth j jϑ j j j ϑ π π ϑ 4 jϑ j j q β jϑ j ϑ cos ϑ j snϑ jϑ () ()

4 ransactons on Elctrcal Engnrng, Vol. (), o. 4 7 hn, th ncat xprsson las to th or () jϑ R q 4444 t 4444 () j ( R ) q q t whch gvs ntcal qatons or an q n by th qatons (5) an (6). ot, th sybol s n th xplanaton abov rprsnts th rsltant ncssary npt stator voltag sppl by a powr spply. E. Rsltng athatcal ol o P rv h rsltant athatcal ol conssts o two rst orr rntal qatons n th crrnt pont o vw n th rotatng rrnc ra an on scon orr rntal qaton n th rotaton angl (anglar poston o th rotatng rrnc ra) pont o vw: t R q (4) t q R q q (5) J && ϑ p q q pτ (6) h - q ol (4) - (6) can b xprss jst n th approprat stat-spac l or (7): t R q τ q R p J J q p J τ q (7) hs ol or rprsnts as spl as possbl th athatcal-physcal scrpton stabl or slaton an spl bass or th ol-bas control sgn. h ol (7) contans nonlnar lnts. hy wll b scss n scton III. ccorng to th ncat ol ors an corrsponng transoratons n ths scton, th sal nstral control,.. th casca PI control, s strctr as wll. h br scrpton o th casca control wll b gvn n th scton IV. Fnally, or rthr xplanaton, th ll stat vctor [, q,, τ ] s ass to b nown ro asr varabls ([ (),, τ ] ) nclng also th anglar poston ϑ. h anglar poston ϑ s not ncl nto th stat vctor to rct rlaton to th anglar sp: t ϑ (8) III. ODE ODIFIIO D UPIO FOR ODE-ED ORO DEIG s was nton, th stabl ol or th olbas control sgn s a ol n th - q coornat syst (7). In spt o ts splcty, t contans two nonlnar trs. hs, or th ol bas control, th ol (7) has to b lnarz, so that th prctv control, a ltstp approach, can b ralz. h nonlnar trs ay b lnarz as ollows: q q τ th rrnc stat varabls ar slct to b zros r q r (9), r, r, τ () h lnarzaton or lnarzng coposton (9) arss ro th ollowng a [8] an spcc rrnc stat: ( x, y, ( x, ( xr, y, ( xr,.( x xr) y, yr, ( xr,.( z zr) ( x xr) yr, zr ) ( z zr) ( xr, y, ( xr,.( y yr) yr, ( y yr),, z ) () ( xr yr r hn, th rsltng lnarz or s: t R q τ R p J q J q p J τ () hs ol or rprsnts alray th sal stat-spac ol, bt wth t-varant trs: x( t) ( t) x( t) ( t) () t (t) s a t-varant stat-spac atrx, s a constant npt atrx. h varancs o (t) ar gvn by th varabl lnts,.. ( t) ( ( t)). h ol (), as aganst (7), can b alray scrtz by th stanar xponntal scrtzaton procr to th or: x x (4) y x (5)

5 ransactons on Elctrcal Engnrng, Vol. (), o. 4 8 r p ( ) ontrollr r orq ( q ) ontrollr Flx ( ) ontrollr q Invrs Par ransor (-q»-β) β nwav Gnrator P sn cos ϑ ϑ & nsng q Forwar Par ransor (-q «-β) β Forwar lar ransor (-β «) Fg. 5. p control o P by vctor control (two-stp casca control) IV. UU DE PI ORO s was nton, th sal nstral control,.. th casca PI control, ollows th rctly scrb way n scton II. tr asrnt o nval phas crrnts an asrnt or staton rotor poston an rotor sp th crrnts ar transor stpws by th orwar lar transoraton an by th orwar Par transoraton nto th - q coornat syst. In t, th an control opraton s xct. h sgn control actons ( - q voltags) ar convrt va th nvrs Par transoraton bac to th - β syst ( - β voltags). h control actons n th - β syst ar l to th nwav gnrator, whch gnrats approprat nval voltag agnts or nval -- phass. It s llstrat n Fg. 5. hat scha o th P sp control conssts o two ntrconnct loops. h an (astr) loop s a sp loop. h sbsary (slav) loop s a crrnt loop ralz as two paralll lgs corrsponng to th torq an lx control rspctvly. Each loop or lg contans an solat PI controllr. Fro th control thory pont o vw, ths arrangnt rprsnts at last sx control paratrs (gans, t constants), whch ar sally prcally or by spl ato-tnng algorth st p [9]. w [ r, r ] Gnralz Prctv ontrollr q In spcc cass, th PI control s spplnt by a l wanng to rach th hgh sp rgon, to ncrasng th Elctro agntc Fl voltag an nt spply voltag []. h l wanng s on by th crrnt - coponnt, whch procs a agntc lx oppost to th prannt agnt lx, s Fg. 6. ot that th otpt o th crrnt controllr (crrnt coponnt n th q axs) st b lt accorng to th rsng crrnt coponnt n th axs wth rspct to ax allow val o th crrnt agnt. U ax I ax w loop o l wanng ax w q /t arctg Par ransoraton cn w q w qw w lar ransoraton ϑ PW, a b Fg. 6. p control o P wth l wanng loop. Invrs Par ransor (-q»-β) β nwav Gnrator P c P sn cos ϑ ϑ & nsng q Forwar Par ransor (-q «-β) β Forwar lar ransor (-β «) Fg. 7. p control o P by Gnralz Prctv ontrol

6 ransactons on Elctrcal Engnrng, Vol. (), o. 4 9 V. PREDIIVE ORO h Prctv ontrol s a lxbl an powrl control approach []. Its llstratv scha or an applcaton to th sp control o th P rvs s n Fg. 7. h bass o th Prctv control s a nzaton o a qaratc crtron (6), n whch th tr syst otpts ar sbsttt by thr prctons (7) xprss by th ol gvn by (4) an (5) [], []: n J n J J n[ Qy (ˆ y w) Q ] (6) yˆ G, x, G O (7) whr ŷ, w an ar vctors o th prctons (tr prct syst otpts), rrncs an control actons (syst npts) or a gvn prcton horzon : yˆ [ yˆ,, yˆ ], w [ w,, w ], [,, ] an Q y an Q ar th wghtng control paratrs: otpt an npt atrx pnalzatons. h prctons yˆ,, yˆ n approprat t nstants o th prcton horzon can b xprss rcrrntly by th ol qatons (4) an (5) accorng to th orla (7). h ors o th qaratc cost ncton as wll as qatons o th prctons pn on control rqrnts gvn by sr or consr applcaton [4], [], []. h nzaton o th crtron (6) can b prov by svral ways. h powrl on s a way va a solton bas on th last sqars [7] appl to th algbrac qaton syst: Q y J ) y w Q y G Q y ( w ) Q Q b J n b b Q Q b / Q (8) (9) whr Q s an orthogonal atrx, whch rarrang th atrx nto th ppr rght trangl atrx R or R rspctvly as t s ncat: R c (4) b R c c z (4) h vctor c z s a lost vctor, whos Eclan nor c z s qal val o th sqar root J (.. J c z c z ). o obtan nnown control actons, only th ppr part o th syst (4) s s or nal control trnaton as ollows. R c ( R ) c (4) nc a atrx R s ppr trangl, thn th control s gvn rctly by th bac-rn procr. h ncat way rprsnts a pr solton, whch can b rach on-ln. Drnt, th ost rlat optzaton way at th GP s a qaratc prograng,.. optzaton o th objctv ncton (4) by algorths o th qaratc prograng [4]: nf( ) n{ ( G Q G Q ) ( w) G } y H g n{ H g }, b (4) h qaratc prograng can solv qalty an nqalty constrans as t s ncat n (4). Howvr, or th P rvs, t s a qt t-consng way apart ro th pr-copt oln plntatons []. Fnally, th splst way, possbly talor or ast ynac systs as th P rv ar, s a rct sarch o local n o th qaratc cost ncton (qaratc crtron). hs way can la to xplct ors o control laws, whch can b or th P rvs pr-copt o-ln. hn, rng th ral-t (on-ln) control, control actons ar trn by slcton o an approprat control law corrsponng to th topcal stat o th syst. In cas o th P rv control, th slct paratr or stat varabl s anglar vlocty (s ol ()). h scrb way las to th ollowng coptaton or ( G Q yw G Q ) G Q yw ( w ) (44) an corrsponng xplct control law o th constant vlocty-pnnt gans: w x (45) w VI. OPRIVE EXPE In ths scton, thr s a br scrpton o on coparatv xapl o th ata ro a ral xprnt an ata obtan by slaton. h ral xprnt was ralz on th ns P rv wth th typ sgnaton: FK7-5K-G [9]. In Fg. 8, thr s t hstory o th ral asr ata ro th ral xprnt. In Fg. 9, thr s t hstory o th slaton ata. h coparatv slaton s prov by th athatcal ol ro scton II. h ol paratrs o th P rv wr tan ro a anal [9] or th otor nton abov. h grs show slar corss o th corrsponng t hstors o physcal qantts: chancal sp, phas voltags () an phas crrnts (). h obvos soothnss o th slaton s cas by consrng th otor as al syst wthot any strbanc. h both xprnts rn or a tranglar prol o th sr rotatonal sp vals wthn th ntrval ±rp. h conton on zro (n) crrnts was ncl both n th ral xprnt an slaton. x

7 ransactons on Elctrcal Engnrng, Vol. (), o. 4 [rp] t [s] 5 [V] t [s]. [] t [s] Fg. 8. p control o P by two-stp casca PI control t hstors o ral xprnt, saplng pro s.5s [rp] t [s] 5 [V] t [s]. [] t [s] Fg. 9. p control o P by Gnralz Prctv ontrol t hstors o slaton; horzon 8, saplng pro s.5s VII. OUIO h papr als wth a sty o th Prctv ontrol sgn or th P rvs. hr athatcal ol was xplan an s n th ol-bas control sgn. h nstral casca PI control was brly xplan as wll. h coparatv xapl onstrats th slarty o th nstral ralzaton an ol-bas sgn. h Prctv ontrol s a prosng way to optz th rv control wth a possblty to consr othr rqrnts or rv constrants, whch cannot b sply solv by convntonal control systs. REFEREE []. Orys, D. lar. stat - spac scrpton or GP controllrs. Int. J. ysts I., Vol. 4, o. 9, 99, pp [] K. la, J. öh, P. Píša, oncpts o ol-as ontrol an rajctory Plannng or Paralll Robots. Proc. o th IED Int. on. on Robotcs an pplcatons. Grany. 7, pp. 5-. []. Proop, P. Grasbl, -Phas P Vctror ontrol: Dsgn o otor ontrol pplcaton, Frrscal conctor, 5. [4] K. la, D. Voš, p ontrol o P Drvs by Gnralz Pr. lgorths. IEO Proc.,, pp. -7. [5] E. antana, E., W. aral, Prctv lg. or ontrollng p an Rotor Flx o Incton otor. IEEE rans. on nstral lctroncs, Vol. 55, o., 8, pp [6]. olognan,. Prtt,. Zglotto, Dsgn an Iplntaton o P or ED. IEEE rans. on IE, 56/6, 9, pp [7] h. awson, R. Hanson, olvng ast qar Probls, Prntc-Hall, Inc., w Yor, 974. [8]. Valáš, P. tnbar, onlnar ontrol o ltboy ysts, Eroch, sabon, 999, pp [9] II /, ynchronos otors FK7, Gnraton, ongraton anal /, 697-D6-P4. [] K. la, D. Voš, Explct Gnralz Prctv lgorths or p ontrol o P Drvs. IEO, stra,, 6 pp. []. Wang, ol Prctv ontrol yst Dsgn an Iplntaton Usng, prngr, 9. []. Kvasnca, J. öbrg,. Hrcg,. ra, an. Far, "ow coplxty polynoal approxaton o xplct P va lnar prograng," ontrol onrnc, pp h athor apprcats n spport o th Grant gncy o th zch Rpblc by th grant o. GP//47 ontrol an Paratr Intcaton o Elctrc Drvs nr rtcal Opratng ontons.

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Rotaton Knematcs, Moment of Inerta, and Torque Mathematcally, rotaton of a rgd body about a fxed axs s analogous to a lnear moton n one dmenson. Although the physcal quanttes nvolved n rotaton are qute