Discrete-Time Modeling and Input-Output Linearization of Current-Fed Induction Motors for Torque and Stator Flux Decoupled Control

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1 BULGARIAN ACADEMY OF SCIENCES CYBERNEICS AND INFORMAION ECHNOLOGIES Volue 4 No Sofia 24 Pin ISSN: 3-972; Online ISSN: DOI:.2478/cai-24- Dicee-ie Modeling and Inpu-Oupu Lineaizaion of Cuen-Fed Inducion Moo fo oque and Sao Flux Decoupled Conol Sanilav Enev Faculy of Auoaic echnical Univeiy of Sofia 8 Kl. Ohidi S. Sofia Eail: enev@u-ofia.bg Abac: In hi pape an exac dicee-ie odel of he inducion oo in a cuen-fed ode including ao flux coponen i deived and validaed. he equaion of he oo ae wien in a fae aligned wih he oo elecical poiion which eul in a linea ie-invaian ye. Baed on he deived exac dicee-ie epeenaion of he oo dynaic an inpu-oupu lineaizing conol law i deigned fo decoupled oque and ao flux conol. he applied deign echnique led o a non-ivial ill ueful definiion of he elecoagneic oupu of he oo. Siulaion eul ae peened howing ha he aied pefoance i obained ha i no coupling exi beween he oupu and he iniial deign poble of conolling a nonlinea ineacing IO ye i educed o a poble of conolling wo linea and decoupled SISO ye wih iple dynaic. Keywod: Inpu-oupu lineaizaion inducion oo dicee-ie nonlinea conol.. Inoducion he inducion oo i pobably he o widely ued elecic achine in induy. he elaed conol poble nown fo i difficuly ha eceived lage aenion in he cienific lieaue. Diffeen oluion wee found he o enowned being 28

2 he o-called field-oiened conol [ 2 3]. Nowaday in one of i any vaian and odificaion i i in induial pacice whee high dynaic pefoance i equied. Anohe appoach ubjec of cienific eeach allowing poenially fo upeio pefoance and abolue decoupling beween he flux and oque ahe han only aypoic (ha i unde conan flux condiion) a in he field-oiened conol cae i he inpu-oupu lineaizaion baed conol [3-9]. Only few inpu-oupu lineaizaion deign ae baed on a odel of he oo conaining ao flux coponen and defining he quaed ao flux a an oupu [8 9] while o of he define he oo flux (quaed) a he elecoagneic oupu of he oo and ue he epecive decipion of he achine. I i ue howeve ha he ain opeaional apec of he oo ae oe clealy expeed in e of he oo flux agniude hu oe eaily elaed o conol goal. Fo field-oiened deign hough depie hee popeie and even he addiional advanage of he oo flux oiened conol ove he ao flux oienaion chee fo a conol pefoance pepecive he lae chee i pobably oe ofen ued in induial dive becaue i poee oe ipleenaion-elaed advanage e.g. lowe eniiviy of he flux eiaion odel. he inpu-oupu lineaizaion appoach can poenially eliinae oe of he ao-flux oienaion chee diadvanage uch a he eidual inpu coupling while eaining i advanage. In all cae he deign i ypically pefoed uing coninuou-ie decipion of he oo while he conol law i ipleened uing digial device being inheenly a dicee-ie poce. hi ae he a of poving and guaaneeing he abiliy of he oveall ye (ineconnecion of wo nonlinea ye) vey difficul pacically ipoible. In hi apec a conol law deigned on a dicee-ie odel will poenially eliinae hi poble. he deign of he conol law diecly in he dicee-ie doain fo volage-coand ode applicaion i.e. uing he coplee odel of he achine elie alo on appoxiaion ince he decipion doe no allow exac diceizaion. Fo he cuen-fed ode of he opeaion howeve exac dicee-ie epeenaion can be obained when he equaion of he oo ae wien in a fae aligned wih he oo elecical poiion. Fo he oo flux conol cae a dicee-ie field-oiened conol law i popoed in [ ] and abiliy condiion ae deived baed on an exac diceeie odel of he oo dynaic. In [2 3] an inpu-oupu lineaizaion deign baed on he ae dicee-ie decipion i popoed and validaed. In hi pape he ao flux conol cae i conideed. Fi an exac diceeie odel of he inducion oo in a cuen-fed ode including ao flux coponen i deived. hen an inpu-oupu lineaizing and decoupling conol law i deigned uing hi exac decipion. In Secion 4 he popoed conol law along wih he oo equaion i odeled in Malab Siulin envionen and oe iulaion eul ae peened. 29

3 2. he ao-flux odel Unde he coon aupion fo yeical conucion inuoidal diibuion of he field in he ai-gap and lineaiy of he agneic cicui he equivalen full-ode wo-phae odel of he achine expeed in he wo-phae ao-fixed α-β fae wih ao flux linage and ao cuen a ae i given a follow: () 3 i i i i n i n ( l ) ( l ) u p p i i n i n ( l ) ( l ) u p p (2) np ( i i ) whee u ( ) u ( ) ae he ao volage i ( ) i ( ) he ao cuen ( ) ( ) he ao flux linage () he oo peed () he oo elecoagneic oque. he paaee in he odel ae defined a follow: 2 ( l l ) ( l l ) ( l l )( l l ) whee l i he ao phae winding inducance winding inducance inducance n p u u he ao phae winding eiance l he oo phae he oo phae winding eiance he uual he nube of pole-pai. In a cuen-fed ode he ao cuen ae efficienly ued a conol inpu o he achine. In ode o achieve uch ode of opeaion he inoducion of a cuen conol chee i equied. Seveal ype exi he ain one uing highgain ypically PI cuen conol loop [3 4] feedfowad chee [3] and o ofen hyeei elay loop [4]. In [4] a hoough oveview of he cuen conolle fo hee-phae invee can be found. In ode o inoduce he cuen-fed ode of opeaion baed on hi decipion of he oo he ao volage ae eliinaed in he flux equaion by expeing he fo he cuen dynaic. hu we have: (3) which i pu in a aix fo a n l i l n i l i p p n l i l n i l i p p (4) i l wih i [ i i ] φ [ ] l. (5) wih φ M φ M i i he aice in (4) can be decopoed a: M I np J M l I l n J J I. i p If he oo peed i conideed a a paaee in he decipion i i een ha (4) epeen a linea ie-vaying ye wih ao cuen a inpu and fluxe

4 a ae and an addiional oupu he oo oque being a nonlinea funcion of he inpu and ae. A i can be een he cuen deivaive appea alo in he flux deivaive expeion i.e. a diec feed-hough beween he inpu and oupu exi. 3. Dicee-ie epeenaion of he ao-flux odel he exac dicee-ie epeenaion of he oo dynaic i obained in he fae aligned wih he elecical oo poiion (and oaing wih he elecical oo peed). Le u define he cuen and flux veco in he conideed fae (denoed by ubcip indice *A(B) ) i [ ia ib ] [ A B ] φ and he anfoaion aix beween he ao-fixed and he oaing fae a co( p ) in( p ) (6) n n in( n ) co( n ) wih being he oo angula poiion. Fo he anfoaion aix we have: (7). p I i alo noed ha he veco agniude ae peeved by he anfoaion. By ubiuing he cuen and he flux veco in (4) a i i φ φ we obain: d d ( φ ) M φ M ii l ( i ) d d which wih he developed deivaive i ewien a (8) φ φ M φ M i li li. i Given (7) and he following popeie: p e J ej np Je np J ( ) equaion (8) i educed o (9) φ φ li li. I i een ha in hi fae he ie-vaying feaue of he dynaic i eliinaed and he ye i linea ie-invaian. Fo he oque we have: ( ) np ( i i ) npi ( ) Jφ ( ) () n i ( ) Jφ ( ) n i ( ) Jφ ( ). p p Exac dicee-ie decipion of he oo dynaic i obained fo (9) auing ha he ao cuen ae held conan duing he apling peiod i.e. auing zeo-ode hold a he inpu: i ( ) i ( ) con fo i he apling peiod. In ode o obain he dicee-ie decipion he oluion of (9) i fi conideed. Fo hi pupoe he flux equaion i decopoed a: 3

5 () Fo we have ψ ψ l ( ) i φ ψ l i ( ) ( ) ψ ( ) ψ ( ) e l ( ) e i ( ) d and uliaely ( ) ( ) (2) ψ ( ) ψ ( ) e l ( )( e ) i. hu fo () i i obained fo he flux: ( ) ( ) (3) φ ( ) ( ) ( φ e l e ) i ( ) l i ( ). Fo he oque we have wih ( ) ( ) npi ( ) ( ) ( ) Jφ e (4) np A ib B ia ( ) ( ( ) ( ) ( ) ( )). Finally fo he flux i given by (5) φ ( ) φ ( ) e l ( e ) i ( ) l i ( ). hough he oo oque i a nonlinea funcion of he ae he paicula funcion ucue along wih he conideed inpu exciaion (zeo-ode hold on he ao cuen in he oaing fae) enable i exac econucion fo i aple by a fo of he exponenial hold. hi in un give he poibiliy o obain an exac dicee-ie epeenaion of he peed dynaic auing i linea a follow: (6) J c L wih J oen of ineia efeed o he oo; c vicou ficion coefficien. ( ) e he anfe funcion of he hold i. he oveall diceizaion chee i illuaed in Fig... Fig.. Diceizaion chee he peed a he apling inan i obained a (7) cj cj e e cj ( ) l ( ) e ( ) ( ) e ( ) d J c J In cae when he load oque aifie l ( ) l ( ) fo ( ) ha i i i conan duing he apling peiod we have: 32.

6 (8) cj cj ( ) e l e ( ) d l( ) J c (fo c we have J ( ) l ). In hee cae a diffeence equaion can be wien fo he oo peed. Equaion (5) (4) (7) and (8) epeen he exac dicee-ie odel of he ye. 4. Inpu-oupu lineaizing conol law deign he heoeical foundaion of he feedbac lineaizaion and he baic conol deign echnique exended fo dicee-ie ye can be found in [5 6]. Hee i will be only noed ha he baic ucue of a conol ye uing uch conol law coni of wo loop an inne one in which he lineaizaion i achieved and an oue linea loop whee he linea conolle aibue he deied dynaic of he oveall ye. he odel ued hee a bai fo he conol law deign i given by: x ( ) e x ( ) l ( e ) x ( ) l u ( ) (9) x ( ) e x ( ) l ( e ) x ( ) l u ( ) x ( ) u ( ) 3 x ( ) u ( ) 4 2 wih [ x ( ) x ( ) x ( ) x ( )] [ ( ) ( ) i ( ) i ( )] A B A B [ u ( ) u2 ( )] [ ia ( ) ib ( )]. A een odel (5) i augened by wo addiional ae vaiable x ( ) 3 and x 4 ( ) and hei epecive equaion by adding a delay of one apling peiod a each inpu. hu a decipion in a ae-pace fo i obained. Alo in hi way one apling peiod i allowed fo calculaion which ende he conol law ealizable ince he conol deign echnique applied will eul in a aic ae feedbac. Fo he conol deign he inducion oo i noally conideed a a IO-ye wih he oque oo peed o poiion being he ain oupu of echanical naue and he flux agniude (quaed) a he econd oupu of elecoagneic naue. Hee he conolled quaniie ae defined a follow: y ( ) ( ) np ( x ( ) x4 ( ) x2 ( ) x3 ( )) (2) 2 2 y ( ) x ( ) x ( ) x ( ) x ( ) e ( x ( ) x ( )) he fi oupu y ( ) i defined a he oo oque. hu he peed dynaic (being linea) i no accouned fo in he decoupling conol which ae i iple and oe obu becaue i doe no include he echanical paaee. he econd oupu y ( ) 2 i defined afe he following odificaion aing 2 2 fo he expeion fo he quaed ao flux x ( ) x2( ). Fi he peviou value of each coponen ae inoduced o ha he euling conol law becoe an affine funcion of he new exenal inpu vaiable. hu he lef e in he expeion i obained. hen a coecion e popoional o he peviou value 33

7 of he flux quae i added o ha he abilizaion of y ( ) 2 guaanee phyically accepable egie of he oo opeaion and uliae abilizaion of he ao flux. I hould be noed ha he inu ign i ipoan he value of he coefficien 34 e i choen o ha he euling expeion fo he conol law ae iplified. In he eady-ae (a a conan peed) ( ) 2 he following way [2]: y i elaed o he quaed ao flux in 2 2 ( ) φ ( ) (co( l ) ) y e whee l i he elecical lip peed. Since he apling peiod i ypically ao in he illiecond ange and he lip peed i geneally low (ypically a ingle-digi pecenage of he oo peed) i can be aued wih aifacoy peciion ha co( ) and l 2 (2) y e ( ) φ ( ) ( ). 2 he expeion fo boh oupu ae wien fo a (22) and he inpuoupu decipion i pu in he veco-aix fo (23): (22) (23) wih (24) y ( ) n ( e x ( ) l ( e ) x ( )) u ( ) p 2 4 n ( e x ( ) l ( e ) x ( )) u ( ) p 3 2 y ( ) x ( ) x ( ) x ( ) x ( ) e ( x ( ) x ( )) l ( e )( x ( ) x3 ( ) x2 ( ) x4 ( )) l x u l x 2 u 2 b ( ) b ( ) y ( ) b ( ) b ( ) u ( ) 2 y ( ) a( ) b ( ) b ( ) u ( ) B ( ) he decoupling aix and b2 ( ) b22 ( ) p p ( ) ( ) ( ) ( ); a( ) l ( e )( x ( ) x ( ) x ( ) x ( )) b ( ) n ( e x ( ) l ( e ) x ( )) b ( ) n ( e x ( ) l ( e ) x ( )) By inoducing he conol inpu a b ( ) l x ( ) b ( ) l x ( ). u ( ) v ( ) B u ( ) v ( ) a( ) ( ) 2 2 wih v ( ) v 2 ( ) being he new inpu ignal he inpu-oupu elaion fo he ye obained ae given by (25) y ( ) v ( ) y ( ) v ( ) 2 2 A een fo he equaion no coupling exi beween he wo oupu which delay hei epecive inpu ignal by one apling peiod. I i alo noiced ha.

8 he obained inpu-oupu dynaic i of econd ode while he iniial decipion i of fouh o ha econd ode inenal dynaic unobevable fo he oupu exi. I u be uch ha he ye ae eain bounded while conolling he oupu ohewie he conol law would be unueful. Hee he non-ivial definiion of he econd oupu peven he diec defining of a coplee novel e of coodinae and he epecive anfoaion o ha he inenal abiliy analyi can be pefoed. In ode o do o he odel u be exended by wo addiional vaiable o ha he econd oupu becoe a aic funcion of ae. No inenal inabiliie wee obeved duing iulaion of he ye behaviou he foal analyi howeve will be a ubjec of a fuue eeach. I u be noed ha in he coninuou-ie cae wih he ao flux quae a a ye oupu he inenal dynaic can be expeed in e of he oo flux coponen which guaanee inenal abiliy when inpu-oupu abiliy i achieved. In ode o inoduce he conol law he aix inveion in (24) u be poible i.e. he decoupling aix u be non-ingula. A dicuion on he way o enue i i given in he following line. he following ubiuion ae inoduced fo noaional ipliciy: e x l e x3 ( ) ( ) ( ) ( ) 2 e x2 l e x4 ( ) ( ) ( ) ( ). he deeinan of he aix i given by de B ( ) n l ( x ( ) ( ) x ( ) ( )). (26) p 2 2 he inveibiliy condiion (equieen fo a non-zeo deeinan of he decoupling aix) epeen he equieen fo non-oogonaliy of he ao flux veco φ ( ) and he veco φ ( ) [ ( ) ( )]. 2 Noe ha a φ ( ) l φ ( ) l ( φ ( ) l i ( )). hu in he aypoic coninuou-ie cae he condiion i educed o pevening he ao and oo flux veco fo becoing ohogonal which a poined ou in [9] can be elaed o he axial available oque of he achine fo i cuen elecoagneic ae. Moe icly if he axial available oo oque i no equied ha i if he oque aifie (27) n ( l l ) φ φ ax p ax wih being he axial oque he ealizabiliy of he conol law i guaaneed. he oo oque expeed in e of he ao and oo fluxe i given by ax np( ll ) φ Jφ and i obained when he wo flux veco ae ohogonal. I hould be noed ha only he ic equaliy i no allowed. Alo noally ax i uch highe han he aed achine oque fo he noinal opeaion egie o ha he inveibiliy condiion doe no ipoe ubanial liiaion on he achievable pefoance. he inigh gained fo he coninuou-ie cae can be applied fo he dicee-ie cae conideed hee. An appoach o enue he ealizabiliy of he 35

9 popoed conol law ay coni of a dynaically auaing ignal v ( ) (i can be viewed a a oque efeence) o ha de B( ) eain negaive. 5. Siulaion eul o validae he popoed dicee-ie odel of he inducion oo and he lineaizing conol law a e of iulaion wee conduced. he following odel wa ipleened in Malab Siulin envionen. Fig. 2. Siulaion odel he inducion achine paaee ued in he iulaion ae aen fo [7] and ae a follow:.52 Ω l.375 H.7 Ω l.323 H.3 H n p 2 J 2.4 g.. he noinal powe and he aed oque ae P no 37 W and no 24 N.. he value of he vicou ficion coefficien i e 4 o c N... he apling peiod wa e o. Soe of he eul ae hown in Fig. 3. All anien confi he validiy of he dicee-ie decipion of he oo ince all ignal poduced by he odel (hown held duing apling peiod) ach exacly he oo fluxe and he peed (he deail hown in he ebedded gaph). Alo he expeced conol pefoance i confied ince no coupling i obeved beween he oupu and each oupu ac i epecive inpu wih a delay of one apling peiod a given by (25). 36

10 Fig. 3. Siulaed anien No oue loop wee inoduced and he wo inpu vaiable ae ued a efeence fo he epecive oupu. he eady ae value of v ( ) 2 wee geneaed fo he deied agniude of he quaed ao flux by uing (2). A een he acual quaed flux agniude eache he deied efeence value hu poving ha he cale faco in (2) i coec. I hould be noed ha ligh deviaion above he obained in hi way efeence value will be obeved ince he lip peed i no zeo. hee a well a he ipple obeved in he flux agniude ae inignifican fo a pacical poin of view. Inignifican ipple ae obeved alo in he oo oque. On he ohe hand lage ipple ae obeved in he econd oupu of he oo hough hi i no elevan o he conol o he opeaion egie of he oo. A een a ceain dynaic lag i obeved beween 37

11 he acual quaed flux value and he econd oupu efeence he eling-ie of he flux eeing o be pacically unaffeced by he efeence ie-ie. Finally a a he oque ep efeence i applied a all deviaion in he quaed flux agniude i een which how coupling beween hee quaniie hough again a obviou he apliude i inignifican. A load oque of N. i applied a.6. Fig. 4 how anien wih an oue conol loop inoduced in he flux ubye. he conolle i deigned baed on he following aupion: φ φ y ( ) x ( ) x ( ) e ( x ( ) x ( )) ( ) e ( ). hu a dicee-ie anfe funcion can be defined and he oue loop i configued a given in Fig. 5. he epecive vaiaion of v ( ) 2 and y ( ) 2 ae hown in he gaph o he igh. he oque and peed epone ae he ae a in Fig. 3. Fig. 4. Siulaed anien Fig. 5. Oue conol loop configuaion 38

12 6. Concluion In hi pape an exac dicee-ie odel of he inducion oo in a cuen-fed ode including ao flux coponen i deived and validaed. A above enioned in he pacical eup he cuen-fed ode i foced by addiional cuen conol loop. In ode he dicee-ie epeenaion o hold exacly he ao cuen applied o he oo (he cuen in he α-β fae o a lea hei efeence value) u vay beween he apling inan unle he oo peed i zeo a een by he coodinae anfoaion beween he wo fae. hi will equie a highe pefoance cuen conol which in un will call fo fae and poibly oe coplex cuen conol loop a well a highe apling ae in he poiion ignal acquiiion channel. Baed on he deived exac dicee-ie epeenaion of he oo dynaic an inpu-oupu lineaizing and decoupling conol i deigned fo oque and ao flux agniude conol. he applied deign echnique equie a non-ivial definiion of he elecoagneic oupu of he oo. Howeve a well pecified odificaion eul in a ueful oupu definiion which i oivaed by hoough analyi and dicuion. he ajo benefi of he popoed chee i ha he abiliy analyi of he cloed-loop ye i ivial ince no appoxiaion ae done in any age of he deign. Of coue pecie cuen conol i aued fo a dicee-ie odel of he oo ha hold exacly. Soe iulaed anien ae peened howing ha he aied pefoance i obained ha i no coupling exi beween he oupu and he iniial deign poble of conolling he nonlinea ineacing IO ye i educed o a poble of conolling wo linea and decoupled SISO ye wih iple dynaic. he popoed conol law calculaion equie ao fluxe which in a pacical eup will equie he ipleenaion of a ceain flux eiaion chee in he oveall conol ye ucue. hi epeen a whole epaae eeach field. he volage odel i he ivial choice hough i i nown fo he pue inegaion-elaed dif poble. Hee a diffeen odel aie no including pue inegaion alhough he oo eiance eun in he expeion hu binging bac he elaed vaiabiliy iue. he fuue eeach ay focu fi on including he cuen deviaion fo hei deied value in he foal eup and udying he induced effec. Of coue fuhe iulaion udie wih oe deailed odel accouning he diffeen pocee peen in a pacical ipleenaion uch a cuen conol noie flux eiaion paaee vaiabiliy ae planned in ode o validae he applicabiliy of he popoed conol law in a pacical eup and uliaely lead o expeienal validaion on a phyical ebed. 39

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