INFOTEH-JAHORINA, Vol. 3, Ref. D-1, p , March 2003.

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1 INFOTEH-JAHORINA, Vl., Ref. D-, p. 6-66, ach. APPLICATION OF INTERNAL ODELS IN THE DESIGN OF DIGITALLY CONTROLLED ELECTRICAL DRIVES * PRIENA UNUTRAŠNJIH ODELA U PROJEKTOVANJU DIGITALNO UPRAVLJANIH ELEKTROOTORNIH POGONA lć R. Stjć, Faclt f Electcal Engneeng, Belga lan S. atjevć, Faclt f echancal Engneeng, Kagjevac Abstact - The sgn f a cntllng stcte th ntenal mls f dgtall cntlled electcal dves s gven. In the cntl ptn f the stcte, the nmnal ml f the plant and ml f mmeasable extenal dstbances ae ncld t mpve the sstem bstness and t extact the expected class f dstbances. Patcla attentn s pad t the sgn f IPACT (Intenal l Pncple and Cntl Tgethe stcte f dgtall cntlled dves and a stable mdfcatn f the stcte s ppsed n t mpve the sstem pefmance and t facltate sstem snthess. Sadžaj Daje se pjektvanje pavljačke stkte sa ntašnjm mlma a dgtaln pavljane elektčne pgne. U pavljačkm l stkte nse se nmnaln ml bjekta pavljanja ml spljnjeg nemeljvg pemećaja sa cljem pvećanja bstnst sstema elmnacje čekvane klase spljnjh pemećaja. Psebna pažnja je psvećena pjektvanj IPACT (Intenal l Pncple and Cntl Tgethe stkte dgtaln pavljanh pgna p čem pedlaže pgdna mdfkacja stkte kja mgćava pvećanje kvalteta pnašanja sstema pjednstavljje snte sstema.. INTRODUCTION In man applcatns f cntlled electcal dves the hgh dnamcal pefmance and capablt f the sstem t eject the nflence f extenal dstbances n the steadstate vale f the cntlled vaable ae eqed. In the tackng egme, the tackng e s t be edced t the level f sens esltn, n the pesence f the genealed dstbance that cmpses the extenal dstbance and ncetantes f plant paametes. These eqements can be acheved b the sgn f IPACT cntllng stcte [- 4] sted f the sgn f speed- and pstn-cntlled electcal dves. The IPACT stcte has the mets f bth the stctes based pn the IP (Intenal l Pncple and IC (Intenal l Cntl [,5]. As t s knn, IP means the nclsn f dstbance ml nt the cntllng stcte n t cmpensate effects f expected class f extenal dstbances n the sstem tpt the sstem cntlled vaable (angla speed pstn f the mt shaft. The IC stcte s nt stable f dstbance ejectn bt t enables the achevement f the bst sstem stablt and hgh dnamc pefmance. In ths pape, the cnventnal IPACT stcte s scbed and then ts mdfcatn s ppsed f applcatn t the stctal sgn f dgtall cntlled electcal dves. It ll be shn that the applcatn f pedctve fltes nstead f dstbance bseve, ncldng the ml f dstbance, gves the same even bette sstem pefmance. Unlke the stcte th dstbance bseve, the altenatve cntl stcte, ppsed n ths pape, s smple and th a smalle nmbe f tnng paametes thn the ntenal mls b hch the bstness, flteng ppetes, and hgh dnamc pefmance f the sstem can be easl adjsted.. PRINCIPLE OF ABSORPTION Sppse that kth sample f an extenal dstbance ( t ma be temned b a fnte nmbe m f pevs samples. Then, the dstbance s egla and ma be scbed b extaplatn eqatn [4] ( kt D ( (( k T ( hee D ( s the pedctn plnmal f m -. Relatn ( s called the eqatn f extaplatn pedctn [4] and t ma be etten as ( ( D ( ( hee ntes the -tansfm f dstbance. Relatn ( s called cmpensatn eqatn and FIR flte s the havng the plse tansfe fnctn ( D ( absptn flte the cmpensatn plnmal [4]. Absptn flte Φ ( D ( s sgned f a knn class f dstbances and ts mplse espnse becmes ntcall eqal t e afte n samplng nstants, hee n m. Hence, the cmpensatn eqatn ( ma be cnsed as the absptn cndtn f a gven class f dstbances. The cndtn can be expessed as ( Φ ( (, a t kt (g Φ T ( The extaplatn plnmal D( s temned b an ap nfmatn abt dstbance ( t [4, 6], nevetheless, t s smpl eslved as nm Φ ( n (, ( n ( (4 ( * Ths pape s an abdged vesn f the pape " APPLICATION OF INTERNAL ODELS IN THE DESIGN OF DIGITALLY CONTROLLED ELECTRICAL DRIVES " pblshed n the jnal Electncs. 6

2 In the case f a stchastc dstbance s(t, absptn flte (4 shld sppess as mch as pssble effects f dstbance n the sstem tpt. Ths, f a l feqenc dstbance s(t, hch can be geneated b dble ntegatn f the hte nse, an apppate chce f absptn flte s Φ s ( ( that cespnds t absptn f lnea (amp dstbance [, 7]. In majt f pactcal applcatns an apppate chce mght be D(. Accdng t (4, pedctn plnmal D( ejects amp dstbances; bt, t enables als the extactn f sl vang dstbances and even sppessn f the effects f l feqenc stchastc dstbances.. IPACT STRUCTURE In the IPACT stcte shn n Fg., the cntllng pcess s gven b ts plse tansfe fnctn b plnmals P ( and Q (, and the pcess ad-tme gven b ntege k. Wthn the cntl ptn f the stcte n Fg. (shad pat t ntenal mls ae ncld: the t-npt nmnal plant ml Fg.. IPACT cntllng stcte W k P ( Q ( ( explctl and the dstbance ml embedd nt the dscete flte A ( /. Bth the ntenal nmnal plant ml and dstbance ml s teated as the dstbance estmat. The cntl ptn has t cntl lps that can be sgned npenntl. The mn lcal cntl lp s sgned b the ppe chce f plnmals A ( and C (, hle plnmals P ( and P ( n the man cntl lp ae temned t acheve the sed sstem set pnt espnse. F a mnmal phase plant, the ppe chce f plnmal P ( s R( P ( []. Un the nmnal cndtns ( P ( P (, Q( Q ( and f R( P (, the clsed-lp tansfe fnctns ( / ( and ( / ( ae easl ved fm Fg. as (5 and ( ( Q ( ( ( Q k [ A( ] k [ Q ( + P ( k P ( k ( + P ( (6. (7 In vte f (7, the sstem set-pnt espnse can be adjsted b temnng apppate plnmals P ( and P ( accdng t the sed sstem clsed lp tansfe fnctn ( / ( G (. Then, the absptn f an extenal dstbance and the speed f dstbance tansent espnse ae adjsted b chsng the stcte and paametes f the dstbance estmat... Elmnatn f dstbance Fm (6, the stead-state e n the pesence f a knn class f extenal dstbance (t ll becme e f lm( Q ( k [ A( ] k [ Q ( + P ( (. (8 In the case f stable plnmal C ( and the plant f nnmnmal phase, Q ( lm k (9 [ Q ( + P ( ] and then the elatn (8 s edced t lm( k [ A( ] (. ( As shn late, the stable plnmal C ( s t be chsen accdng t the sed speed f dstbance ejectn and the eqed gee f sstem bstness and then plnmal A ( s temned t satsf elatn (. Accdng t the pncple f absptn, t s pssble t sgn the bseve estmat that ejects an knd f expected dstbances. T ths end, cns the class f dstbances havng the -tansfm ( N ( / D (. Then, elatn ( s satsfed f the fllng Dphantne eqatn hlds A( + B ( Φ( k ( hee Φ( epesents the absptn plnmal temned b Φ( D (. F example, t the plnmal and snsdal dstbances ( m ( t d t and ( t sn ωt cespnd espectvel Φ( m+ ( and Φ( hee T s s the samplng ped. cs ω Ts +, 6

3 A nqe sltn f the Dphantne eqatn, hch plas a ccal le n the sgn pcede f the bseve estmat, ppsed n ths pape, des nt exst [8]. Relatn ( s a lnea eqatn n plnmals A ( and B (. Geneall, the exstence f the sltn f Dphantne s eqatn s gven n [9]. Accdng t [9], thee alas exsts the sltn f ( f A ( and B ( f the geatest cmmn fact f plnmals k and Φ( dvs plnmal C ( ; then, the eqatn has man sltns. The patcla sltn f ( s cnstaned b the fact that the cntl la mst be casal,.e., g A ( g. Hence, afte chsng a stable plnmal C ( and gees f plnmals A ( and B (, and nsetng the absptn plnmal Φ( that cespnds t an expected extenal dstbance, plnmals A ( and B ( ae calclated b eqatng ceffcents f eqal fm the left- and ght-hand ss f eqatn (. Plnmal A ( btaned b slvng ( gaantees the absptn f the expected class f dstbances, hle the chce f C ( affects the speed f dstbance ejectn, sstem bstness, and senstvt th espect t measng nse. Gd flteng ppetes and the sstem effcenc n dstbance ejectn ae the mtall cnflctng eqements. Theefe, t edce the nse cntamnatn, the l-pass dgtal flte ma be ntdced t mdf the ntenal ml f the dstbance nt A( Af ( A ( ( hee A f ( / epesents the plse tansfe fnctn f the l-pass flte and A ( s a plnmal that satsfes ( and ths ncls mplctl the ntenal ml f dstbance. The le banddth f the l-pass flte cespnds t a hghe gee f sstem bstness and vce vesa []. Accdng t [], cmplex dstbances eqe hghe f plnmal A ( and t ll fthe edce sstem bstness th espect t msmatches f plant paametes... Paamete settng The man cntl lp f the sstem f Fg. s sgned t acheve the sed set-pnt espnse temned b the sstem clsed-lp tansfe fnctn k H ( G (. ( K ( Accdng t (7, the sed clsed-lp tansfe fnctn s acheved f the fllng ntt hlds k k P ( H (. (4 Q ( + k P ( K ( T satsf (4, t s fst necessa t slve the Dphantne eqatn Q ( k + P ( T( K ( (5 f plnmals P ( and T ( and then t temne the secnd plnmal f the man cntl lp f the sstem f Fg. as P ( T( H (. (6 hee T ( n (5 s chsen as a stable plnmal. Recall that, f a mnmal phase plant, R( P (. The chaactestc plnmal K ( s ead fm ( t ma be temned b the sed clsed-lp sstem ple spectm. T mpve the sstem bstness th espect t ncetantes f plant paametes, plnmal K ( ma be extend b facts n ( b, b.9. (7 At the begnnng, the vales f b and ntege n ae t be chsen as small as pssble and then the can be nceased gadall ntl the eqed cten f bst stablt s satsfed. At the same tme, plnmal P ( shld be mdfed nt P n ( b ( n ( b (8 t save the acheved set-pnt espnse and t keep nchanged the stead-state vale f the sstem tpt. 4. ODIFIED IPACT STRUCTURE Fg. shs the mdfed IPACT stcte f the cntl plants tht tanspt lags, hch ma be appled f stctal sgn f dgtall cntlled electcal dves [,]. Sgnal mled the nflence f lad tqe n sstem tpt (angla speed pstn f the mt shaft. Fg.. dfed IPACT cntllng stcte The cntl plant f the stcte n Fg. s gven b ts nmnal plse tansfe fnctn W k P ( ( (9 Q ( hch s sed as a t-npt ntenal plant ml thn the cntl ptn f the stcte. Sgnal ε estmates the 6

4 nflence f genealed dstbance n the sstem tpt. Uncetantes f plant mlng ma be aqatel scbed b the mltplcatve bnd f ncetantes α (ω [] W( W ( ( + δ W( (a jωt δw( e α( ω, ω [, π T]. (b Then, the sstem f Fg. satsfes the cndtn f bst stablt f the nmnal plant s stable and f the fllng neqalt hlds α( ω < Q ( P ( R ( ( P ( + P ( + Q ( P ( D( e jωt, ω [, π T] ( The bst sstem pefmance s acheved b the peatn f the lcal lp f the stcte n Fg.. Namel, the le f lcal lp s t sppess as mch s pssble the effects f genealed dstbance n the sstem tpt. Accdng t the pncple f absptn, t s necessa t ncl, nt the cntl pat f the stcte, the ntenal ml f dstbance havng the npt ε. In the case f cntl plant tht the tanspt lag, the ntenal ml f dstbance s edced t the pedctn plnmal D (. In Tspkn s ks, mst feqentl the pedctn plnmal D ( ( s ppsed [, 4]. Ths plnmal cespnds t lnea dstbances bt t effectvel ejects dffeent classes f sll vang dstbances, t, especall n the case f small samplng ped [, 5]. Accdng t the standad pcede f IPACT stcte sgn [5], n the case f mnmm phase plants, R( P ( ( s t be adpted. The man cntl lp f the stcte n Fg. s sgned t acheve the sed plse tansfe fnctn G ( f the clsed-lp sstem. Namel, b eqatng ntcall the sed G ( th P ( G ( (4 Q ( + P ( ne can easl temne the plnmals P ( and P ( and ths the stcte sgn s cmpleted. In the stcte f Fg. the enc tectng the angla speed pstn s nt ndcated. When the elve t dgtal cnvete (R/D f lmted esltn s appled, the measng sgnal s cntamnated b qantatn nse [], hch pdces the flctatn f cntl vaable and lsses n the mt. The pedctve flte n the lcal lp nceases the nse and that makes the sstem me senstve t qantatn f the speed and pstn. Theefe, the stcte f Fg. s mdfed b ncldng the extend bseve, as s shn n Fg.. The bseve s extend b the ml f dstbance t enable the estmatn f angla speed n the case f the pesence f a cnstant sll vang dstbance. Fg.. dfed IPACT stcte f dgtall cntlled speed sevmechansm In ths pape, f the extactn f dstbance, the smple plnmal pedct s appled n the lcal mn lp f the stcte f Fg., nstead f ntenal ml f dstbance sed n the dna IPACT stcte f Fg.. Geneall, the pedctve flte s fned as an algthm that estmates fte vales f the npt sgnal and sppesses the nse cntamnatn []. The elatvel smple fms f dgtal pedctve fltes cespndng t plnmal dstbances ae teated. Sppse that sgnal ε(... plnmal k a + a k+ + a k a k ε( k a + a k a k a k ma be mled b (5 hee ceffcents a ae nknn eal cnstants. F example, the plse tansfe fnctn f Netn s pedct, hch estmates sgnal (5 th pedctn hn f p samples (.e.!( ε k + p has the fm p p H ( (. (6 Ths flte estmates sample!( ε k + p b + pecedng samples ε( k. In the patcla case f and p, flte (6 becmes ntcal t pedctn plnmal (. Geneall, hen an electcal dve s n cnsatn (cntl plant has mnmal tanspt lag t s alas p. Flteng ppetes f the dffeent Netn s fltes ae llstated n Fg. 4. Feqenc chaactestcs f Fg. 4 sh that nse cmpnents n the sgnal ae nceased hen the f the flte becmes geate. Theefe a lnea appxmatn f sgnal ( ma be adpted as an aqate, fm the standpnt f nse senstvt. LSN (Lnea Smthed Netn pedct [], btaned b mpvng classcal Netn s, passes th dffeence f npt sgnal thgh the l-pass dgtal flte S(. If the sgnal s aqatel mled b nth plnmal, then the th dffeence f sgnal s cnstant and the necesst f ts flteng s evnt. In a geneal case, LSN pedct s gven b plse tansfe fnctn p p p H, LSN ( ( + S( ( (7 64

5 D(e -jωt D( , ( D( , ( D( - - -, ( ωt Fg. 4. Feqenc chaactestcs f Netn s pedctve flte f p and, and F fthe smplfcatn, the l-pass dgtal flte S( ma be adpted as a dgtal eqvalent f the smplest l-pass analge flte T( + S( (8 T s + T + T + ( T T f s T + havng nl ne tnng paamete T f f a clea phscal meanng. B nceasng the vale f T f the bette flteng ppetes and hghe sstem bstness ae acheved, especall thn the l feqenc band. On the the hand, th geate T f the speed f dstbance absptn s edced and vce vesa D(e -jωt - e -jωt ωt f f T f.5 T f. T f. Fg. 5. Feqenc chaactestcs f LSN pedct and pedctn plnmal F cmpasn, the feqenc chaactestcs f the fst pedctn plnmal and LSN pedct ae llstated n Fg ILLUSTRATIVE EXAPLE The effcenc f LSN pedct th T f.s, hen cmpaed th the applcatn f pedctn plnmal D(, n the case f IPACT stcte f pstnng sevmechansm s llstated b Fgs. 6 and 7. In the sevmechansm, the 6-bts D/A cnvete and -bts R/D cnvete ae appled. The eslts f smlatn ns gven n Fg. 6 (a and (b and Fg. 7 (a and (b ae btaned b LSN pedct hle the eslts n Fg. 6 (c and (d and Fg. 7 (c and (d ae accmplshed b the pedctn plnmal. Ntce that LSN flte sppesses the effects f qantatn nse n the cntl vaable and slghtl sls dn the speed f dstbance ejectn. Samplng ped T.s s assmed. The cntl plant s DC mt U4T havng the electmagnetc gan fact K 4.8 and mechancal tme cnstant T m.s. The sed clse-lp sstem tansfe fnctn s specfed b t cnjgate cmplex ples th ndamped natal feqenc ω n.5 ad/s and elatve dampng ceffcent ζ a.5 pavljanje [V] pavljanje [V] c.5.5 θ [ad] 5 b θ [ad] 5 d Fg. 6. Opeatn f IPACT stcte n the absence f tqe dstbance 4 p [Nm] a b 4 pavljanje [V] pavljanje [V] p [Nm] c d.5.5 θ [ad] θ [ad] Fg. 7. Opeatn f IPACT stcte n the pesence f tqe dstbance The effcenc f LSN pedct applcatn as a pedctn thn the IPACT stcte f the speedcntlled electcal dve s llstated n Fgs. 8 and 9. The same DC mt U4T and T.s, as n the case f the pstnng sevmechansm, s appled. Accdng t the 65

6 ppsed pcede, the sed clsed-lp sstem tansfe fnctn G ( ( s specfed and then the fllng plnmals f the cntl stcte ae calclated P ( 765., Q (. 746, R( 765., P ( ( and P ( The sstem smlatn s pefmed hen 6-bts D/A and -bts R/D cnvetes ae appled (Fg. 8. The standad vatn f the dffeence beteen tpt sgnals geneated th and tht qantatn nse, n the case f IPACT stcte th LSN pedct (Fg. 8, s % less than n the case f the bseve based stcte. B nceasng tme cnstant T f, qantatn nse s me sppessed bt, at the same tme, the speed f dstbance ejectn s sled dn. Hence, the mplementatn f LSN pedct nstead f pedctn plnmal gave appxmatel the same eslts as n the case f bseve mplementatn n IPACT stcte, bt the stcte th LSN pedct s sgnfcantl smple. Ftheme, b tnng paamete T f t s pssble, n a smple a, t adjst sstem dnamc ppetes, sppessn f qantatn nse, and t mpve bst stablt f the sstem ω [ad/s] p [Nm] Fg. 8. Respnses f IPACT stcte f speed-cntlled sevmechansm th LSN pedct (T f.s 6. CONCLUSION The sgn pcede f IPACT stcte f dgtall-cntlled speed and pstn sevmechansms has been gven. It as shn that the set-pnt espnse f the stcte and speed f dstbance ejectn cld be adjsted npenntl. Instead f the sgn f dstbance estmat thn the lcal lp f the stcte, as n the case f basc IPACT stcte, dffeent pedcts ae empled, f the ppse f dstbance extactn. Ths altenatve appach has seveal advantages: the elatvel eas settng f cntlle paametes, adjstable speed f dstbance ejectn, and cntl f sstem bst stablt. 6. REFERENCES [].S. atjevć, Ravj nvh stkta dgtaln pavljanh elektmtnh pgna ndstjskh pcesa, Dktska dsetacja, ašnsk fakltet Kagjevc,. [].R. Stjć,.S. atjevć Pmena pedktvnh fltaa mest ntašnjeg mla pemećaja IPACT stkte, Zbnk XLVI Knf. ETRAN-a, Teslć,. [] _.З. Цζпкин, Синтез робастно оптимал{нζх систем управлени обψектами в услови х ограничено неопре-делености, Автом. и Телемех., Н о 9, с. 9-59, 99. [4] Ya.Z. Tspkn and U. Hlmbeg, Rbst stchastc cntl sng the ntenal ml pncple and ntenal ml cntl, Int. J. Cntl, vl. 6, N 4, P 89-8, 995. [5].R. Stjć, Lj.S. Daganvć, and.s. atjevć, Sve and man feates f cntl stctes th ntenal mls (nvted pape n Seban, Pc. f XLIII ETRAN Cnf., vl. I, pp. -, Zlatb, 999, Ygslava. [6] Ya.Z. Tspkn and P.V. Nahdn, Rbst cntns cntl sstems th ntenal mls, Cntl The and Advanced Technlg, vl. 9, N., pp. 59-7, 99. [7].R. Stjć,.S. atjevć, Sppessn f nse cntamnatn n cntl sstems th ntenal mls, Electncs, vl., n., pp. -4, 998. [8] K. Ogata, Dscete-Tme Cntl Sstems, Pentce-Hall, Inc., Engled Clffs, Ne Jese, 995. [9] K.J. Åstöm and B. Wttenmak, Cmpte Cntlled Sstems: The and Desgn, Pentce-Hall, Inc., Engled Clffs, Ne Jese, 984. [] Stjć, atjevć., Daganvć Lj., A bst Smth pedct mdfed b ntenal mls f ntegatng pcess th ad tme, IEEE Tansactns n Atmatc Cntl, Vl. 46, N 8, pp. 9-98, Agst. [] a., Zaf E., Rbst Pcess Cntl, Engled Clffs, Ne Jese, Pentce Hall, 989. [] S. Välvta, S.J. Ovaska and Oll Van, Plnmal pedctve flteng n cntl nstmentatn: A eve, IEEE Tans. n Indstal Electncs, vl. 46, N 5, pp ,