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SPEED CONTROLLER DESI GN FOR A VECTOR- CONTROLLED PERMANENT MAGNET SYNCHRONOUSMOTOR DRI VE WTH PARAMETER VARI ATI ONS

J uan C. Bal daUni versi t y of ArkansasFayet t evi l l e, AR 72701

ABSTRACT - I f cl assi cal t echni ques such ast he root l ocus, Bode pl ots and Nyqui stdi agrams are used for desi gni ng f i xed-st ructure speed cont rol l ers f or ac dr i ves,t he desi gn woul d normal l y be done around anomnal val ue of t he cont rol l ed pl ant .General l y, a sensi t i vi t y anal ysi s woul dsubsequentl y be done t o ensure t hat t hedesi gn speci f i cati ons are met when t he pl antparameters change. Thi s procedure can and hasworked wel l . I n t hi s paper an al t ernat i ve i sproposed where the parameter var i at i ons ar ei ncl uded at t he out set of t he desi gn task.The Ni chol s chart l ends i t sel f r at her wel l t ot hi s appl i cat i on si nce i t r epresent s bot hmagni t ude and phase i nformati on on a si ngl edi agram By usi ng thi s al t ernat i ve, i t may bepossi bl e t o r educe t he overal l t i me needed t ocompl ete t he desi gn. The par t i cul ar t echni quei s cal l ed Quant i t ati ve Feedback Theory ( QFT)whi ch i s used i n conj unct i on w t h the Ni chol schart . Thi s paper present s t he basi cs of QFTand shows how i t can be used f or t he desi gnof f i xed- str ucture cont rol l ers f orparamet er - sensi t i ve pl ant s i n conj unct i onw t h t he Ni chol s char t . A desi gn i s present edand ver i f i ed experi ment al l y.

I . I NTRODUCTI ONThe permanent magnet synchronous mot or (PMSMwhi ch by def i ni t i on here has a si nusoi dalback em [l], i s establ i shi ng i t sel f as aser i ous compet i t or t o the vector - cont rol l edi nduct i on motor ( I M and t he dc motor f orhi gh per f ormance speed and posi t i onappl i cat i ons. Thi s i s part l y due to t hei ncreased t orque to i ner t i a rat i o and powerdensi t y [2,3] when compared t o the I M or t hedc mot or , i n t he f r acti onal t o 30 HP range.Thi s has been made possi bl e by t he use ofhi gh resi dual f l ux densi t y/hi gh coerci vi t ypermanent magnets. Curr ent r esearch i ntoreduci ng t he temperature dependence andi ncreasi ng t he thermal capabi l i t y of magnet sw l l probabl y i ncrease t he penet rat i on of t hePMSM dr i ve i n t he servo i ndust ry.The hi gh per f ormance at t ai nabl e f r omthe PMSMhas prompted ori gi nal research i nt o t hedesi gn and perf ormance of t he enti re motordr i ve i ncl udi ng the motor [4], posi t i on andspeed f eedback [ 5], i nver t er , curr ent , speedand posi t i on cont rol l ers. The appl i cat i on ofa PMSM t o an el ect r i c vehi cl e has beenexamned [6] whi l e hi gh speed operat i on hasal so been i nvest i gat ed [7 ,8] .

Pragasen Pi l l ayUni versi t y of Newcast l eEngl and, NE1 7RU, UK

I n order t o extract t he best perf ormance f r oma gi ven machi ne, t he proper desi gn of t hespeed and cur rent cont rol l ers ( f or a speedservo) i s i mpor t ant . However al l dr i ves areparameter sensi t i ve t o some degree.Tradi t i onal met hods of cont rol l er desi gn i nt he f r equency domai n use a nomnal val ue oft he pl ant parameters. The ef f ects of changesi n t he parameters can be subsequent l y checkedby a sensi t i vi t y anal ysi s [9,101. Thi s methodcan work wel l , however an al t ernat i ve i sproposed i n thi s paper f or t he desi gn off i xed- st ructure speed cont rol l er s [ll, 31 f ora hi gh per f ormance dr i ve. Thi s consi st sessent i al l y of i ncl udi ng i nf ormat i on on thedr i ve parameter vhr i ati ons at t he out set sot hat t he per f ormance cr i t eri a can be met f romt he begi nni ng, w t hout havi ng to subsequent l ydo a sensi t i vi t y anal ysi s. Thi s shoul d r educet he ef f or t and t i me of t he desi gner w t houtcomprom si ng t he desi gn.Whereas desi gn t ool s l i ke t he r oot l ocus,Bode pl ot s and Nyqui st di agrams have beenused extensi vel y i n cont rol l er desi gn, t heNi chol s chart l ends i t sel f rather wel l t o t hepart i cul ar probl emof r epresent i ng paramet ervar i at i ons i n a dri ve. The Ni chol s char tr epresents both phase and magni t udei nformati on on t he same di agramunl i ke Bodepl ot s whi ch represent t hemseparat el y.Thi s paper demonst r at es how af r equency-domai n techni que known a sQuanti t at i ve Feedback Theory' may be appl i edt o the desi gn of f i xed- st ructure speedcont rol l er s f or a vector- cont rol l ed PMSMdri ve where t he motor parameters vary bet weenknown l i mt s. The paramet er vari at i ons can becaused by changes i n temperature, curr entl evel or operat i ng f r equency. I n r obot i cappl i cat i ons, changes i n i ner t i a occur aswel l . Fi nal l y, anot her sour ce of uncer t ai nt yi s the fact t hat t he parameters of t heNomnal Pl ant are normal l y measured( cal cul at ed) w t h a cer t ai n er r or whi ch i sgeneral l y expressed as a percent age of t henomnal ' parameter val ue.The ai m of t hi s paper i s two- f ol d. Fi rst l y,t he el ement s of QFT are revi sed and i t i st hen shown how QFT can be appl i ed t o thedesi gn of f i xed- st ruct ure cont rol l er s f orparamet er - sensi t i ve pl ants. I n t hi s case, t hepaper i s t ut or i al i n nat ure. Fur t hermore, aspeed cont rol l er desi gn i s present ed andver i f i ed exper i ment al l y. Thi s i s the secondcont r i but i on of t hi s paper.

90KH 2935-5/90/ooo(M163$01 .WO1990EE E

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Figure 1 Schematic of a permanent magnet synchronous motor.11. MATHEMATICAL MODEL - TRANSFER FUNCTION OFA PMSM DRIVE WITH PARAMETER VARIATIONSMathemat ica l Mode lS i m p l i f i c a t i o n s o c c u r i n t h e m od el in g o f aPMSM unde r ve c t o r c on t r o l wh ic h r e s u l t i n al i n e a r t r a n s f e r f u n c t i o n be tw ee n t h e o u t p u tand commanded spe eds . Thi s can be der iv edfrom th e PMSM d ,q eq ua ti on s a s f o l l o w s :

whe r e vd r vq a re t h e d , q a x i s v o l t a g e s ; hd,h,a re t h e d , q a x i s f l u x l in k a g e s ; id, , a r e t h ed , q a x i s c u r r e n t s , P i s t h e num be r o f p o lep a i r s ; p i s t h e d e r i v a t i v e o p e r a t or ; L,, L,a re t h e d , q a x i s i n d u ct a n ce s , T, i s t h ee l e c t r i c t o r q u e , T, i s t h e l o a d t o rq u e , B i st h e d am pin g c o e f f i c i e n t a n d J i s t h e momento f i n e r t i a . T he e l e c t r i c spe e d a, i s r e l a t e dt o t h e m e ch a n i ca l s p e e d th r o u g h t h e n um be r ofp o l e p a i r s . F i n a l l y ha , i s t h e m u tu a l f l u xb et w ee n t h e m ag ne t a nd t h e s t a t o r d u e t o t h emagnet and

h = L i9 9 9 (7 )E q u a t i o n s (1) t o ( 3 ) a re n o n l i n e a r b u t i fv e c t o r c o n t r o l [ 3] i s u se d t o f o r c e id o b ez e r o l t h e n u s i n g ( 6 ) a n d (7), (1) t o ( 3 )r e d u c e t o

where Kt is t h e to r qu e c on s t a n t o f t h e PMSM;t h e a b o ve e q u a t i o n s a re v e r y s i m i l a r t o t h a tof a dc motor . It i s o n l y n e c e s sa r y t oi n c l u d e t h e d y na m ic s o f i, i n t h e m o del s i n c et h e e l e c t r i c t o r q u e d e p en d s o n l y o n i,. Frome q u a t i o n s (8), (10) and ( 4 ) t h e l i n e a r b lo ckd i ag r am i n F i g . 1 can be drawn.A PMSM s p e e d s e r v o d r i v e is o b t a i n e d f r o mF i g . 1b y i n c l u d i n g t h e s p e ed a nd c u r r e n tc o n t r o l l e r s as shown i n F i g . 2 ( No te t h a t a l lcommanded values a r e i n d i c a t e d w i t h a n I*).The c u r r e n t c o n t r o l l e r i s u se d t o e n s u r e t h a tt h e a c t u a l c u r r e n t t r a c k s t h e c ommandedc u r r e n t w h i l e t h e s pe ed c o n t r o l l e r d o e s t h esame f o r t h e s p ee d . C u r r e nt a n d s p e e df e e dba c k i s nor m a l ly u se d a s shown i n F i g . 2 .Fo r a g i v e n c o n f i g u r a t i o n o f t h e c u r r e n tc o n t r o l l e r , t h e b lo c k diagrams t o t h e r i g h to f t h e d o t t e d l i n e c o n s t i t u t e t h e c o n t r o l l e dp l a n t P ( s ) a n d t h e a i m o f t h i s p a p er i s t h ep r o p e r d e s i g n o f G , ( s ) u n de r u n c e r t a i n t i e s an dp a ra m e te r v a r i a t i o n s i n P (5) .n

F i g u r e 2 Schematic of a permanent magnet synchronous motor dr ive .164

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Parameter Vari ati onsAssumng t hat t he cur rent cont rol l er gai nsare set cor rect l y, paramet er uncer t ai nt i esare t hen produced i n G(s) , G , ( s ) , G(s) andG6(s) whi ch r epresent t he machi ne as a resul tof changes i n t he temperat ure, cur rent l evel ,i nert i a and operat i ng f requency. For exampl e,changes i n t he t emperature af f ect t he st atorr esi st ance Re and magnet f l ux l i nkage hafwhi ch i n t urn af f ects q. Changes i n thecur rent can sat urat e L, whi l e changes i nf requency can af f ect Rs but general l y to al esser ext ent t han changes i n t he t emperature[ l o] . I n robot i cs, i ner t i a changes al so occurwhi ch w l l af f ect G, ( s ) . The cri t eri a f orchoosi ng t he range of var i at i on f or t heparameters i s presented i n Ref . [lo] andi ncl ude t he fl ux l oss coef f i ci ent of t hemagnet and the degree of saturati on f or t hemachi ne. Thi s paper has al so i ncl uded t het ol erance i n t he Nomnal Pl ant paramet er sdue t o measurement ( cal cul at i on) err ors. Theranges of paramet er var i ati ons are gi ven i nTabl e I as a f ract i on of t he nomnal val uewhi ch i s at ambi ent t emperature.

blboa3a2

TABLE I . PARAMETER VARI ATI ONS FOR AC DRI VE

Nomnal M ni mum Maxi mum1941783. 10 470513. 74 3814859. 208. 7865et09 2. 129et09 1. 7262e+10

1. 0 1. 0 1. 09771. 01 7504. 11 17479. 11

Transf er Funct i onFromFi g. 2, t he pl ant t r ansf er f uncti on P ( s )i s gi ven by

-~al 44703871. 0 33647342. 0 80291919. 0a 7026177. 70 2702403. 20 12548858. 0

nr 9 )I;(S)

bl s + bos3 + a2s2 + al s + a-=

whi ch can be at t ai ned f r om bl ock di agramreduct i on or Mason s r ul e/ si gnal f l owgraphs.Froman anal ysi s of t he parameter vari ati onsshown i n Tabl e I , t he vari at i ons of t hecoef f i ci ent s of P ( 3 ) are as f ol l ows:

111. BASI CS OF QFTThi s secti on onl y r evi ews some of t he mai nt heoret i cal concept s of QFT si nce i t has beenextensi vel y publ i shed el sewhere [ l l - 131.Consi der t he regul ati ng f eedback syst emshowni n Fi g. 2 whose out put equat i on i s gi ven by:

0. 90 5 R, 5 1. 1 (0. 12 ohms)0. 63 5 haf L 1. 1 (1. 513 V/ rad/ sec)

Numeri cal val ues are present ed as f ract i onsof nom nal val ues whi ch are bet ween brackets.

where n , (s ) , @(s) are t he Lapl ace t ransf ormsof or and a: respecti vel y, L ( s ) = G (s)P(s)i s t he l oop t r ansf er f uncti on, and P(s) i st he pl ant whose parameter vari ati ons areshown i n Tabl e 11.

An anal ysi s of (12) shows that t he desi gnobj ect i ve i s that t he pl ant out put n,(s)f ol l ows t he i nput si gnal n:(s) as cl osel y aspossi bl e. Si nce one of t he ai ms of t hi s paperi s to i l l ustrate the f easi bi l i ty of t he QFTapproach to desi gn a f i xed- st ructure speedcont rol l er f or ac dri ves, t he desi gn t ask hasbeen si mpl i f i ed by not consi der i ng t hei nf l uence of any sensor noi se or di st urbance.However, i t must be cl ear t hat t hi s does notrepresent a l i mt at i on of t hi s desi gnt echni que si nce any spur i ous si gnal can al sobe consi dered [131.The desi gn i s done on t he Ni chol s chart whi chi s shown i n Fi g. 3; t he x- axi s represent sphase ( i n degrees) , he y- axi s represent smagni t ude ( i n dB) and a background ( dot t edl i nes) i s i ncl uded whi ch cor responds t oconst ant phase ( or magni t ude) curves of[L (s) ( 1 +L ( s ) ] . The Ni chol s chart present st he advant age t hat t he l oop t r ansf er f unct i onL ( j w ) = G ( j w) P(j w) may be easi l y drawn f roma know edge of P(j o) by j ust addi ng t hemagni t ude ( i n dB ) and the phase ( i n degrees)of G (j o) t o t he magni t ude and phase of P ( j o )f or each rel evant f requency ( t he poi nt P( j o)i s t hen t r ansl ated but not rotated) .The uncer t ai nt y of P(s) i s consi dered i n thesame way as i n the t racki ng desi gns ofHorow t z [11, 12] . The regul at or probl emi s t omai nt ai n [ L ( s )/ (1+L (s) [ bel ow a speci f i edl i m t at speci f i ed f requenci es for al lpossi bl e P(s) ( see Tabl e 11) . Of al l P ( s ) ,t he desi gner usual l y chooses an arbi t r ar yP,(s) whi ch cor responds t o the so- cal l ed Nom nal Pl ant .

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60dB

30

0

-30-

0 dB ;0 . 5 dB

-3 dB-6 dB

-2 0 dB. , ........................... ... . ,Iio -270 -180 - 90DEGREES

F i g u r e 3 N i c ho l s c h a r t w i t h t h e p l a n t t e m p l a te f o r1.0 r a d / s .T he ' p l a n t t e m p l at e s ' a r e t h e n f i t t e d b et we enc e r t a i n IL ( j o ) (1+L ( j o ) ( B c u r v e s ( b o u n d a r i e s )a r o un d t h e N y q u i st p o i n t (-180, 0 d B ) a sshown i n t h e n e x t s e c t i o n . A ' p l a n t t e m p l a t e'is t h e l o c i o f p o i n t s c o rr es p o nd i ng t o a l lP ( j o ) c o n s id e r ed i n t h e u n c e r t a i n t y r a n g e(shown Table 11) f o r a g i v e n f r e q u e n c y o. Duet o s p ac e l i m i t a t i o n s , o n l y o ne t y p i c a lt e m p l a t e o f P ( jw ) f o r w = 1 . 0 r a d / s ( 0 . 1 5 9 Hz)i s shown i n F i g . 3 a s a s h ad e d a r e a ; t h ep o i n t c o rr e sp o nd i ng t o t h e 'Nomina l P l a n t ' i sr e p r e s e n t e d b y t h e l e t t e r ' N ' .I n a QFT d e s i g n , t h e s t r u c t u r e o f t h ec o n t r o l l e r i s n o t f i x e d a p r i o r i b u t e v o l ve sa s p a r t o f t h e d e s i gn p r o ce s s . I n t h i s way iti s g u a r a n t e e d t h a t n o o v er - o r u n d e r -d e s ig no f t h e c o n t r o l l e r o c c u r s . D i sa d va n ta g es o ft h e QFT a re t h a t it i s n o t a l wa ys p o s s i b l e t oe s t a b l i s h t h e u n c e r t a i n t y r an g e o f P ( s )q u a n t i t a t i v e l y a nd c o r r e l a t i o n b et we en t i m e -a n d f r eq u en c y -d o m ai n s p e c i f i c a t i o n s i s a s o r to f a n a r t ; b u t s o i s much of des ign.

I V . SYSTEMATIC DESIGN OF THE SPEED CONTROLLER

T he d e s i g n t a s k c h o s en was t o s e l e c t a G , ( s )w hic h e n s u r e s t h a t t h e c l o s e d l o o p r e s p on s eI L ( j w ) / ( l + L ( j w ) ) ( L . 0 d B ( 1 3 )f o r a l l f r e q u e n c i e s b e l o w 1 5 0 . 0 r a d / s a n d f o ra l l p o s s ib l e P ( j o ) (see T a b l e 11). The 1 . 0 d Bc r i t e r i o n w a s c ho se n a r b i t r a r i l y a nd t h es t e a d y - s t a t e s p ee d e r r o r m us t b e z e r o s i n c et h i s p l a n t i s a s p e e d s e r v o . A l t h o ug h t h e s es p e c i f i c a t i o n s w e r e s e t a r b i t r a r i l y , t h ed e s i g n e r ca n s p e c i f y o t h e r c o n s t r a i n t sa c co r di n g t o t h e a p p l i c a t i o n of t h e d r i v e .

The f o l l o w i n g s t e p s a r e c a r r i e d o u t i n o r d e rt o a c h ie v e t h e d e s i g n s p e c i f i c a t i o n s :

a ) O b ta in t h e t e m p la t e s o f P ( s ) f o r t h er a n g e o f f r e q u e n c i e s w hi ch a re ofi n t e r e s t (see F i g . 3 f o r t h e t e m pl a tec o r re s p o nd i n g t o 1 .0 r a d / s ) .

b ) De te r m ine a bounda r y o f t he 'Nom ina lP l a n t ' P o ( j o ) w i t h G , ( j o ) = 1 . 0 i n t h eN i c h o ls c h a r t a t e a c h r e l e v a n t f r eq u e n cyW . T h i s i s d on e b y s h i f t i n g t h e t e m p l a tec o r re s p o n di n g t o a s p e c i f i c f r e qu en c y w(see F i g . 3 ) a r o u n d t h e 1 . 0 dB c u r v es u ch t h a t NO POINT wi th in th e t e m pla t ef a l l s i n s id e t h e 1 .0 dB c u r v e (sees e c t i o n 111: n o r o t a t i o n o f t h e t em p l a tem u st o cc u r , o n l y v e r t i c a l a n d /o rh o r i z o n t a l d i s p la c e m en t s ) . The po in tw i t h i n t h e t e m p l a t e w hi ch c o r r e sp o n d s t oPo ( jo) ( N i n F ig . 3 ) t r a c e s o u t aboundary of L o ( j o ) a t t h e c o n s id e r edf r e que nc y W . F i g u r e 4 d e p i c t s som e o f t h eb o u n d ar i e s t h u s o b t a in e d w i t h t h es p e c i f i c fr eq u en c y b e i ng e n c i r c l e d .F i n a l l y , t h e 1 . 0 dB c u r v e i s c o n s i d e r e dh e r e b ec a us e t h i s i s one of t h e d e s i gns p e c i f i c a t i o n s (see ( 1 3 ) ) .

c ) Design a c o n t r o l l e r f o r t h e 'N om in alP l a n t ' P o ( s ) u s i n g t h e N i ch o l s c h a r t . Thed e s i g n mu st g u a r a n t ee t h a t t h e v a l u e oft h e l o op t r a n s f e r f u n ct i o n L , ( j o ) ( i . e .P o ( j o ) a nd d e s i g n e d G , ( j o ) ) f o r a nyp a r t i c u l a r f re q ue n cy o i s o u t s i d e t h eb o u nd a ry f o r t h a t f r e q u e n cy . T hen a n yo t h e r L ( j o ) ( i . e . f o r a ny o t h e r P ( j o 1w i t h i n T a b l e 11) a l s o m e e t t h e d e s i gns p e c i f i c a t i o n s a t t h a t p a r t i c u l a rf r e que nc y o. T h i s i s t r u e b e ca u se t h eb o u n d a r i e s o f P o ( o) w e r e o b t a i n e d s u c ht h a t NO POINT of t h e temp la t e ( a l lc o n s i d e r e d P ( j w ) ) f a l l s w i t h i n t h e 1 . 0 d Bc u r v e f o r t h e c o n s i d e r e d f r e q u en c y(Remember t h a t t h e m ag n it ud e o f L ( jo ) i nt h e N ic h ol s c h a r t i s e q u a l t o t h e sum o ft h e m a g ni t ud e s o f P ( j o ) a n d G l ( j o ) i n dB,a n d t h e p h a s e o f L ( jo ) i s t h e sum o f t h ep h a s e s o f P ( o) and G I ( j w ) i n d e g re e s ) .I n o t h e r w o rd s, i f t h e 'N om in al P l a n t 'P o ( s ) w i t h t h e d e s i gn e d c o n t r o l l e r G,(s)s a t i s f i e s a l l s p e c if i ca t io n s , it i s t h e nc o n c l u d e d t h a t a l l P ( s ) i n T ab le I1 musta l s o s a t i s f y t h e s p e c i f i c a t i o n s [ll, 21 .

F i g u r e 4 shows t h e l o c i o f t h e l o o p t r a n s f e rf u n c t i o n L o ( s ) f o r t h e 'N om in al P l a n t ' w i t h :

fi r (9 ) 5 (1+ s / 5 )( 1 4 )l ( S ) =7fi ,(s) s (1 + s/600) (1+ s/2000)

where G , ( s ) i s t h e r e s u l t o f f o l lo w i n g t h ea bo v e s t e p s i n a n i n t e r a c t i v e m an ne r. Thel ow -p as s f i l t e r s a t 60 0 a n d 2 00 0 r a d / s a r er e q u i r e d t o r e d u c e h i g h f r e q ue n c y n o i s e . Froma n a n a l y s i s o f F i g . 4 , it c a n b e n o t e d t h a tt h e p o i n t L , ( j l O ) j u s t s a t i f i e s t h e b ou nd aryf o r o = 1 0 . 0 r a d / s ; all o t h e r r e l e v a n t p o i n t sa mp ly s a t i s f y t h e d e s i g n r e q ui r em e n ts .

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. . '. . .." . . . , .. ' ' ,

HPra t ed speed (rpm)r a t e d t o r q u e (N-m)Kt (N-m/amps RMS)

R- @ 25C ( a )bac k emf (Vll/rpm)

.dB..dB...dB...., . . . .dB

8.9 9.41350 145047.1 46.22.39 2.180.144 0.111.08 1.165

-30 ! I / I I-360 -270 -180 -90 DEG 0F i g u r e 4 1. 0 dB des ign l i m i t s ( s e e (13)) a t 10 , 100 an d150 r a d / s t o g e t h e r w i t h t h e l o op t r a n s f e r

f u n c t i o n f o r t h e ' No mi na l P l a n t ' Po ( s ) and thec o n t r o l l e r G1(s) ( s e e ( 1 4 ) ) .

F i n a l l y , t h i s d e s i g n p r o ce d u re i s s u i t a b l ef o r co m pu te r im p l em e n ta t io n i n a n i n t e r a c t i v em an ne r; t h e u s e r c a n g r a p h i c a l l y n o t e i f ac o n t r o l l e r s a t i s f i e s t h e s p e c i f i c a t i o n s an dm o di f y t h e d e s i g n i f n e c e s s a r y .

F i g u r e 5 S t a r t - u p s p e e d r e s p o n s e w i t h t h e d e s i g n e ds p e e d c o n t r o l l e r .

BODE PLOTSI I f

V. RESULTSI n o rd e r t o t e s t t h e d e s i g n t ec h n i q u e , ad e s i g n was done on a m otor w i th pa r a m e te r sp r e se n te d unde r m o to r A i n T a b le I11 b u ta c tua l ly im p le m e n te d on m oto r B which hadd i f f e r e n t p a ra m et e rs w i t h i n t h e u n c e r t a i n t yr a n ge c o n s i d e r e d i n T a b l e 11.

O I Fc7

pi j150 , LOG (a) r a d / s

TABLE 111. PMSM DATAF i g u r e 6 Bode p l o t s of t h e m o t or d r i v e w i t h t h e d e s i gn e d

s p e e d c o n t r o l l e r (see ( 1 4 ) ) .

1 Parameter 1 Motor A 1 Motor B [

The s p ee d c o n t r o l l e r G, 5) d e s c r i b e dp r e v i o u s l y w a s im p le m en ted us ing a na lo gc i r c u i t s . W it ho ut a ny f u r t h e r t u n i n g , t h ec o n t r o l l e r w a s t e s te d by commanding a sp eedof 520 rpm . The s t a r tu p spe e d t r a n s i e n t i sshown i n F i g . 5. There i s a s l i g h t o v e rs h oo tw hi ch w as a l l o w ed f o r i n t h e d e s i g n .

F i g u r e 6 s ho ws t h e Bode p l o t s o f t h e e n t i r ed r i v e i n c l u d in g t h e c o n t r o l l e r ; t h e s pe e dloop ba ndwid th i s a p p r o x i m a t e l y 250 r a d / s . I fa h igher bandwidth i s n e ed e d, t h i s c a n b ei n c l u d e d a s p a r t o f t h e s p e c i f i c a t i o n s .

A key t e s t of t he pe r f o r m a nce o f a s p e e ds e r v o i s i t s r e s po n s e t o a l o a d t o r q u ed i s t u r b a n c e . I n d e e d i f r e j e c t i o n c a p a b i l i t i e st o a l o a d t o r q u e d i s t u r b a n c e a re j ud ge d t o b eof p r i m e i m po rt a nc e, t h e s p e e d c o n t r o l l e rd e s ig n c an b e do ne w it h t h i s s p e c i f i c a t i o n i nmind. To examine th e per formance of t h e speedc o n t r o l l e r t o l o a d t o r q u e d i s t u rb a n c es , t h et r a n s f e r f u n c t i o n of (a,/T,) w a s deve loped andi t s B ode p lo t s a re shown i n F i g . 7 . The higha t t e n u a t i o n , ev e n a t 1 . 0 r a d / s and whichi n c r e a s e s as t h e f r e q u e n c y o f T, i s i n c r e a s e d ,a t t e s t s t o t h e e x t re m el y go od l o a d r e j e c t i o nc a p a b i l i t i e s o f t h e s pe e d c o n t r o l l e r . Theh i g h a t t e n u a t i o n i m p l i e s t h a t t h e a c t u a lspe e d doe s no t r e spond w e l l t o l o a d t o r qu ei n p u t s , w h i c h i s h i g h l y d e s i r a b l e .I n a d d i t i o n t o t h e r e s po n se o f t h e s p ee d t oa c ha n gi n g l o a d t o r q u e i n p u t , t h e s t e a d y -s t a t e e r r o r i n t h e s pe ed du e t o a s t e p i n pu ti n t h e l o ad t o rq u e w a s examined. For a 1 . 0p . u . i n p u t t o r q u e , t h e r e i s a r e d u ct i o n i n2.9% i n t h e f u l l l o ad s pe ed .

167

7/22/2019 pmsm speed control

6/6

-3 0F4

W

HzE3-90

BODE PLOTS

060 1 1 LOG (U) r a d / sF i g u r e 7 Frequency re sponse of t h e m ot or d r i v e t o l o a d

d i s t u r b a n c e s (Q TL ) .

Zero steady- state er ror i s di f f i cul t i f noti mpossi bl e t o obt ai n due to t he pract i caldi f f i cul t i es of i mpl ement i ng a purei ntegrator. I n practi ce, i t i s general l ynecessary to i mpl ement t he i nt egrator pol ecl ose t o the jw axi s and so t he purei nt egrat or i s actual l y i mpl ement ed as a l owpass f i l t er and hence w t h some steady-st at eerror.

VI . CONCLUSI ONSThi s paper has i l l ust r at ed how Quant i t at i veFeedback Theory can be used i n conj unt i onw t h t he Ni chol s char t t o desi gn a f i xed-st ructur e speed cont rol l er f or a PMSM dr i ve.Parameter var i ati ons were t aken i nto accounti n t he i ni t i al stages of t he desi gn whi ch i sconduct ed such t hat even i n the worst case,t he desi gn speci f i cati ons are met. Parametervar i at i ons due t o t emperature, saturat i on,operat i ng f requency and change i n i nert i awere consi dered. A systemati c procedure f ort he cont rol l er desi gn was al so gi ven.I n order t o check t he abi l i t y of t hecont rol l er t o per f orm i n t he presence ofparameter var i ati ons, t he desi gn wasconduct ed w t h one set of motor par ametersbut i mpl emented on another motor whoseparameters were w t hi n the uncert ai nt y rangeof t he fi rst . The pract i cal resul t s weresat i sf actory t hus val i dat i ng the desi gnprocedure and i mpl ement ati on. The speed l oopBode pl ot s and l oad rej ect i on capabi l i t i eswere al so examned and found t o be accept abl eas was t he st eady- st ate speed err or due tot he practi cal di f f i cul t i es of i mpl ement i ng apure i nt egrat or .

VI I . ACKNOWLEDGEMENTSThe aut hors are grat ef ul f or t he hardwaresupport r ecei ved f r om I ndustr i al Dri ves,Radf ord (VA).

168

Dr. Pragasen Pi l l ay acknow edges t he Sci enceand Engi neer i ng Research Counci l , U K. , f orsupport i ng hi s r esearch at t he Uni versi t y ofNewcast l e. Dr . J uan C. Bal da acknow edges thesof t ware support f r omUni versi t y of Nat al andCl emson Uni versi t y and t he f i nanci al supportf r omArkansas Sci ence & Technol ogy Aut hori t y.VI I I . REFERENCES

G Pf af f , A, Wescht a and A. Wck,' Desi gn and Experi ment al Resul t s of aBrushl ess ac Servo Dri ve' , 1981 AnnualMeet i nq I EEE- I AS. , 1981, pp. 692- 697.R. Kr i shnan, ' Sel ecti on Cr i t er i a f orSer vo Motor Dri ves' , I EEE Trans. on I A,P. Pi l l ay and R. Kr i shnan, ' Model i ng,Anal ysi s and Si mul at i on of a Hi ghPer f ormance, Vect or- Cont rol l ed, PMSMDri ve' , 1987 Annual Meet i nq I EEE- I AS,1987, pp. 253-261.N Boul es, ' Predi cti on of no l oad Fl uxDensi t y D st r i but i on i n Permanet MagnetMachi nes' , 1984 Annual Meet i ns I EEE-m, 1984, pp. 464- 473.E.K. Persson, ' Brushl ess DC Motor - ARevi ew of t he St at e of t he Art ' , Conf .Rec. 1981 Motorcon Conf . , pp. 1- 16.B. K. Bose, A Hi gh Perf ormanceI nver t er - Fed Dr i ve Systemof an I nt er i orPermanent Magnet Synchronous Machi ne' ,1987 Annual Meet i nq I EEE- I AS. , 1987, pp.T. M J ahns, ' Fl ux Weakeni ng Regi meOperat i on of an I nt eri or PermanentMagnet Synchronous Motor Dri ve' , 1986Annual Meet i nq I EEE- I AS, 1986, pp.814- 823.T. Sebast i an and G. R. Sl emon, ' Operati ngl i mt s of I nver t er - Dr i ven PermanentMagnet Mot or Dr i ves' , 1986 AnnualMeet i nq I EEE- I AS, pp. 800- 805.G . F . Frankl i n, J . D Powel l and A.Emam- Naci ni , ' Feedback Cont rol ofDvnamc Svstem' , ( book).R. Kri shman and P. Pi l l ay, ' Paramet erSensi t i vi t y i n Vector Cont rol l ed ACMotor Dri ves' , 1987 I EEE I ECON, pp.252-218.I . Horow t z and M Si di , ' Synthesi s ofFeedback Syst ems w t h Large Pl antI gnorance f or Prescr i bed Ti me-Domai nTol erances' , I nt. J . Cont rol , Vol . 16,NO. 2, 1972, pp. 287-309.I . Horow tz, S. O dak and 0. Yani v, ' AnI mport ant Propert y of Non- M ni mumPhaseMul t i pl e- I nput - Mul t i pl e- Out put FeedbackSystem' , I nt . J . Cont rol , Vol . 44, No.E. Ei t el berg, J . C. Bal da, E. S. Boj e andR. G Har l ey, St abi l i zi ng SSROsci l l at i ons w t h a Shunt ReactorCont rol l er f or Uncer t ai n Level s ofSeri es Compensat i on' , I EEE Transact i onson Power Svstems, Vol . 3 , No. 3, August

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