4. SHAFT SENSORLESS FORCED DYNAMICS CONTROL OF RELUCTANCE SYNCHRONOUS MOTOR DRIVES 4.. VECTOR CONTROLLED RELUCTANCE SYNCHRONOUS MOTOR DRIVES WITH PRESCRIBED CLOSED-LOOP SPEED DYNAMICS Abstact: A new spee contol system fo electc ves employng eluctance synchonous motos (RSM) s pesente. The basc contol system s smla to that pesente n secton fo PMSM, but fo completeness of the pesentaton, some epetton s entale. The usual vecto contol metho s complemente wth foce ynamc contol nstea of the PI contol foun n conventonal ves. Ths ntal stuy s estcte to spee contol wth the lnea fst oe ynamc moe n whch the close-loop system esponse s fst oe wth a pole locaton chosen by the use. To mpove obustness of the close-loop pefomance, an oute contol loop base on moel efeence aaptve contol (MRAC) s ae. Smulaton esults pesente show goo coesponence wth the theoy an pect substantal obustness mpovements wth the a of MRAC. 4.. Intoucton In contast to conventonal appoaches to electc ve contol, a new contol stategy fo eluctance synchonous motos (RSM) s pesente base on foce ynamc contol (FDC). The combne RSM an loa ae vewe as a multvaable plant, the contol, measuement an contolle vaables beng, espectvely, the stato voltages, the stato cuents, an the oto spee. Lke the systems escbe n pevous chaptes, a cuent contol loop s close va the powe electonc swtches so that the stato cuent emans become the contol vaables fo the FDC base spee contol loop. Ths emboes RSM vecto contol [] an geneates automatcally stato cuent emans such that the oto spee espons to the spee eman wth the pescbe ynamcs of the selecte ynamc moe (ef., Chapte ) whch, n ths case s the lnea fst oe moe.
The new RSM ve contol system emboes the block contol pncple [], the moton sepaaton pncple [] an slng moe contol [4]. The new RSM contol system compses two pats: a) the contol law compsng maste an slave contol laws aange n a heachcal stuctue [] an b) the state estmaton an flteng system, compsng a complementay set of two obseves, one use fo econstucton of the oto spee, an the othe fo extenal loa toue estmaton [5]. Fg.4.. shows the contol system stuctue an ts opeaton, the nvual blocks beng fully explane n the followng sectons. As n the pevously escbe IM an PMSM ves, ths ve may be nclue as an actuato n a lage scale contol scheme to whch lnea contol system esgn methos can be apple. The same appoach has also been nvestgate fo pemanent magnet synchonous moto ves [5], [6] an pelmnay expemental esults wee pesente n [7]. The maste contol law s opeate n the lnea fst oe ynamc moe n whch the oto spee s contolle wth lnea, fst oe closeloop ynamcs, the close-loop tme constant beng chosen by the contol system esgne. In ths ntal nvestgaton, the moto s assume to ve a g boy netal loa wth moment of neta, J, zeo fcton an subject to a constant extenal loa toue, Γ Le. Τ emane oto spee Maste Contol Law Flteng obseve Γ^L emane stato cuents I _, α_β I _ a,b,c tansf. I Ψ I s Magn. flux calculato Ψ Ψ emane theephase voltages Slave Contol Law U U U Swtchng table U U I I Pseuo-slng moe obseve U c Powe electonc ve ccut α_β _ I tansf. I I v e v e Angula velocty extacto extenal loa toue Γ Le Reluctance SM oto poston measue stato cuents θ
Fg. 4.. Oveall contol system block agam 4.. Contol System Development a) Moel of Moto an Loa The followng set of non-lnea ffeental euatons fomulate n the otatng, co-onate system, couple to the oto, escbe the RSM an fom the bass of the contol system evelopment: u = R + Ψ s + jψ s (4..) t s s s J = t p ( Ψ Ψ ) ΓL = c5 ( L L ) ΓL, (4..) whee, s = + j, u s =u + ju an Ψ s =Ψ + jψ ae, espectvely, the stato cuent, stato voltage an magnetc flux, s the oto velocty, p s the numbe of stato wnng pole pas, Γ L s the extenal loa toue, R s s the phase esstance, L an L ae the ect an uaatue phase nuctances an c 5 =p/. The ALA (axally lamnate ansotopc mateal) RSM paametes assume n ths stuy ae lste n the Appenx. The last tem n (4..) s obtane by notng that the magnetc flux components ae gven by Ψ =L ( ). an Ψ =L.. b) Maste contol law The basc phlosophy of the foce ynamc contol law evelopment s the fomulaton of lneasng functons, n ths case emboyng vecto contol, whch foce the nonlnea plant (.e., the RSM an ts loa) to obey specfe lnea close-loop ffeental euatons, whch n ths case s a fst oe euaton fo the oto spee yelng a ynamc esponse to a emane spee, (t), wth a pescbe tme constant, T. The oto spee theefoe s mae to satsfy: t = T ( ). (4..)
The oto spee lneasng functon s chosen to foce the non-lnea ffeental euaton (4..) to have the same esponse as the lnea euaton (4..). The lneasng functon s obtane smply by euatng the ght han ses of (4..) an (4..), as follows: [ 5( Ψ Ψ ) ΓL] = ( ) J c T. (4..4) In the contol law to be eve, estmates of the magnetc flux components, Ψ an the Ψ ae evaluate fom the known stato cuents, an, by the magnetc flux calculato whch takes nto account the vaatons of the ect nuctance, L as a functon of the cuent,, n the ect axs (see Appenx) whle the uaatue nuctance, L, s taken as a constant: ( ) Ψ = L an Ψ = L. (4..5) Mathematcally, thee ae nfntely many combnatons of an that may be chosen to satsfy (4..4). Ths enables the fst pat of the contol law to be fomulate on the bass of vecto contol []. Two vecto contol optons wll be pesente. The fst ensues that the maxmum toue pe unt of stato cuent s obtane. Ths eues the maxmum pactcable constant value, K, of the stato cuent component,, up to the base spee an not to allow ts eucton une the pescbe value when the RSM s le unnng. Above the base spee, s euce to ensue coect opeaton of the contol system s mantane by keepng the magntue of the back e.m.f. below the.c. lnk voltage. Thus: K fo < base = = (4..6a) K base K fo base Altenatvely, fo maxmum powe facto the cuent, can be etemne fom (4..6b) as: J T ( ) + Γ L =, c5 ( L L ) tan δ (4..6b)
whee δ s the angle of cuent vecto n, system. Euaton (4..6b) s base on a stana conton fo maxmum powe facto fo RSM [] n whch the moto s assume to otate at constant spee n one ecton, an so the tem, J ( ) +ΓL, woul not go negatve. The fact that t can o so n the ve T contol system pesente s taken nto account n the fomulaton of the maste contol law below. The maste contol law geneates the emane values of an, whch wll be enote espectvely by _ an _, on the assumpton that the nne cuent contol loop (slave contol law) ensues that an. Euaton (4..6a) o euaton (4..6b) ae then use to evaluate _ an then euaton (4..4) s solve fo _, yelng the secon pat of the maste contol law. Thus, usng the flux estmates fom euaton (4..5), an the loa toue estmate, $Γ L, fom the obseve of secton., the followng maste contol laws ae eve, one fo each of the two vecto contol optons: a) fo maxmum toue pe unt of stato cuent: K = ~ J T = c 5 ( ˆ ) ( L ~ ) L ~ K + Γˆ L (4..7a) b) fo maxmum powe facto: _ = ~ J T c 5 ( ˆ ) + Γˆ ( L ~ L ~ ) tan( δ) L (4..7b) _ = _ tan ( δ) The estmates of all constant paametes, p, use n any moel-base contol law cannot be known wth nfnte pecson, an ae theefoe enote by ~ p as fo the contol laws eve n the pevous chaptes.
c) The slave contol law The slave contol law closes the stato cuent contol loop an s the same as n all the ve contol systems pesente n the pevous chaptes. The sub-plant to be contolle hee s efne by euaton (4..), the contol vaables now beng u an u an the output vaables an to espon to the emane cuents _ an _. The slave contol law s the followng bang-bang contol law: ( ), j a, b, c u j = U s sgn j _ j =, (4..8) whee geneally the tansfomatons between the, components of the stato cuents an voltages an the coesponng thee-phase stato voltages an cuents ae gven by: z z = C S S C z z z. (4..9) A specal statng algothm wth constant cuent an emans comes nto play whle the magnetc flux nom, Ψ, buls up an contol s hane ove to the maste contol law of the pevous secton only once Ψ has sen above a pe-set mnmum theshol. As n the pevous ve contol systems pesente, the magnetc flux nom s efne as: Ψ = Ψ + Ψ. (4..) 4.. State Estmaton an Flteng The loa toue estmate, whch s necessay fo the maste contol algothm s gane n a smla way to that of [5] an [6] fo synchonous moto ves. Fst, a stato cuent vecto pseuo slng-moe obseve s fomulate fo geneaton of an unfltee estmate of the oto spee. Secon, the loa toue estmate eue by the maste contol law s pove by a stana obseve havng a smla stuctue to a Kalman flte, a ect measuement of loa toue beng assume to be unavalable. It shoul be note hee that ths loa toue an fltee spee obseve s entcal n fom fo all the ve contol systems, but the electcal toue nput to the eal tme moel s calculate usng a ffeent euaton fo each type of moto.
a) The pseuo slng moe obseve an angula velocty extacto The eal tme moel of the system s base on the stato cuent euaton (4..) fe by the measue stato voltages an stato cuents, but puposely usng only the tems wthout the oto spee,. Thus: + = e e s s v v L R L R u u L ~ L ~ t (4..) whee v e an v e ae the moel coectons an an ae estmates of an as n a conventonal obseve. The useful obseve outputs of the classcal slng moe obseve woul be the contnuous euvalent values of the aply swtchng vaables:. = max e e sgn V v v (4..) Rathe than compute ths usng a low pass flte, a pseuo-slng-moe obseve may be fome to obtan close appoxmatons to v e an v e by eplacng euaton (4..) wth (4..):, = sm e e K v v (4..) whee the gan, K sm, s mae as hgh as possble wthn the stablty lmt set by the samplng tme of the gtal pocesso. Fo lage K sm, the eos between eal moto cuents an fcttous obseve cuents ae ven almost to zeo, esultng n (4..4): = e e L ~ L ~ p L ~ L ~ p v v. (4..4) The ght han se contans the tems nvolvng that wee omtte fom the ealtme moel. An unfltee oto spee estmate,, can then be extacte fom
euaton (4..4). The component, v e of (4..4), has been foun to have lowe nose levels than v e an s theefoe use alone to geneate : L ~ ve = pl ~. (4..5) b) The Loa Toue Obseve Smple ect means of measung the extenal loa toue, Γ L ae not avalable. Ths s estmate by an obseve that s entcal n basc fom to those of the ve contol systems pesente n the pevous chaptes. The etals ae gven agan n the nteests of self-contanment of ths chapte an the fact that the electcal toue euaton ffes fom one moto type to the next. The poblem of loa toue estmaton s easly solve by teatng Γ L as a state vaable an nclung ts estmate n the eal tme moel of an obseve. If the stato cuent measuement nose s sgnfcant, then the system pefomance wll be mpove by usng the angula velocty estmate, ˆ, fom the obseve, whch s a fltee veson of. The obseve pesente pouces ths fltee angula velocty estmate, wthout ntoucng a ynamc lag, whch woul mpa the contol system pefomance, n a smla fashon to a Kalman flte. The obseve pouces also a fltee oto spee estmate an ths s use n the contol algothm as well as the loa toue estmate. The eal tme moel of the obseve s base on toue euaton (4..). The obseve coecton loop s actuate by the eo between the oto spee estmate,, fom the angula velocty extacto of the pevous secton an the estmate, ˆ, fom the eal tme moel: e = [ c ( Ψ Ψ ) Γˆ ] ˆ & = ~ 5 L + k e (4..6) J Γˆ & = k e L Γ ˆ
Snce ˆ s a fltee veson of t s use ectly n the maste contol law. Ths s a conventonal secon oe lnea obseve wth a coecton loop chaactestc polynomal, whch may be chosen va the gans, k an k Γ, to yel the ese balance of flteng between the nose fom the measuements of the cuents an an the nose fom the velocty measuement,. 4..4 Moel Refeence Contol Base Oute Loop To mpove the obustness of the ve contol system escbe n the pevous sectons an oute contol loop base on moel efeence aaptve contol (MRAC) may be fome. Ths s entcal to that pesente fo the PMSM ve an so the eae s efee to secton.. fo ths. 4..5 Smulaton Results All the smulaton esults ae pesente n Fg. 4.., Fg. 4.., Fg.4..4 an Fg. 4..5. They wee cae out wth a computatonal step of Δt=5e-5 s, whch coespons to a samplng feuency of khz fo gtal mplementaton. All the smulatons ae cae out wth zeo ntal state vaables an a step oto spee eman of = a/s. A step extenal loa toue of Γ L =,5 Nm eual to the nomnal moto toue s apple at t =,, beng zeo fo tme nteval t <, s. The esponse of the new basc foce ynamc contol system opeatng n the lnea fst oe ynamc moe s smulate fst to llustate the opeaton wth the two altenatve vecto contol optons of the maste contol law efne by euatons (4..7a) an (4..7b). The esults ae shown n Fg. 4.. fo maxmum toue pe unt stato cuent ( = const) an n Fg. 4..4 fo contolle to yel the maxmum powe facto. These ae then compae wth the coesponng esponses of the same contol system augmente by a MRAC base oute loop an the smulaton ae shown n Fg. 4.. an Fg. 4..5.
.5.5.7.6.5.4.5...5.5..5..5..5.4.4...5..5..5..5.4 a) = f(t), = f(t) b) Ψ = f(t), Ψ = f(t) [H] 4.8.6.4 -.8.6.4...5..5..5..5.4 [Vs] - -.5..5..5..5.4 c) L = f(t) ) α = f(t), β = f(t) 4.5 4.5.5 [Nm] -. -.4 -.6 -.8.5..5..5..5.4.5.5 -.5.5..5..5..5.4 e) Ψ α = f(t), Ψ β = (t) f) Γ el = f(t), Γ L = f(t), $Γ L = f(t) 8 6 4 [a/s] 9 8 7 6 5 4 [a/s] -.5..5..5..5.4.5..5..5..5.4 g) = f(t), $ = f(t) h) = f(t), = f(t) Fg. 4.. Vecto contol wth =const In all the fgues the subplots (a) an (b) show the emane an eal values of the cuent components an eal values of the magnetc flux components n the, oto fxe fame. The changes of the ect nuctance, L ue to changes n the ect cuent,, ae shown n subplot (c). Subplots () an (e) show the stato cuents an magnetc flux components vewe n the stato fxe α, β fame. Subplot (f) shows the ntally exponentally ecayng moto toue an the apple loa toue, Γ L togethe wth ts estmate Γ^L. The estmate values of the loa
Toue fom the obseve may just be seen to follow the step ncease n loa toue at t =, s wth a small ynamc lag accong to T s = 5 ms. Ths gves se to the small tansent eucton n the oto spee just afte t =, s. The spee estmate fom the flteng obseve togethe wth the eal spee esponse ae shown n subplot (g). Apat fom the tansent ue to a small lag n loa toue estmaton, the eue fst oe spee ynamcs wth a pescbe tme constant of T =,5 s s event fom the moto spee esponse, whch ae shown n subplot (h) togethe wth the eal one. 4.5.5.5.5 -.5.5..5..5..5.4.4. [H].7.6.5.4... [Vs] -..5..5..5..5.4 a) = f(t), = f(t) b) Ψ = f(t), Ψ = f(t) 4.8.6.4 -.8.6.4...5..5..5..5.4 [Vs] - -.5..5..5..5.4 c) L = f(t) ) α = f(t), β = f(t) 5 4 [Nm] -. -.4 -.6 -.8.5..5..5..5.4 -.5..5..5..5.4 e) Ψ α = f(t), Ψ β = (t) f) Γ el = f(t), Γ L = f(t), $Γ L = f(t) [a/s] 8 6 4 9 8 7 6 5 4 [a/s] -.5..5..5..5.4.5..5..5..5.4 g) = f(t), $ = f(t) h) = f(t), = f(t) Fg. 4.. Vecto contol wth =const an MRAC
.5.8 [Vs].5.6.4.5..4. -.5.5..5..5..5.4 [H] -..5..5..5..5.4 a) = f(t), = f(t) b) Ψ = f(t), Ψ = f(t).8.6.4 -.8.6..5..5..5..5.4 [Vs] - -.5..5..5..5.4 c) L = f(t) ) α = f(t), β = f(t) 4.5 4.5 [Nm].4..5 -. -.4 -.6 -.8 -.5..5..5..5.4 8.5.5 -.5.5..5..5..5.4 e) Ψ α = f(t), Ψ β = (t) f) Γ el = f(t), Γ L = f(t), $Γ L = f(t) [a/s] 9 8 7 [a/s] 6 4 -.5..5..5..5.4 6 5 4.5..5..5..5.4 g) = f(t), $ = f(t) h) = f(t), = f(t) Fg. 4..4 Cuent, angle contol fo maxmum powe facto The contol law paametes fo all the smulatons wee as follows: maste contol law close-loop tme constant: T =,5 s; obseve flteng tme constant: T s =,5 s; pseuo slng moe obseve moel coecton loop gan: K sm =6. In all these smulatons pefect matchng between the moto paametes an those assume n the contol laws an obseves ae assume.
.5. [Vs].5.8.6.5.5..5..5..5.4.4..4..5..5..5..5.4 a) = f(t), = f(t) b) Ψ = f(t), Ψ = f(t) [H].8.6.4 -.5..5..5..5..5.4 [Vs] - -.5..5..5..5.4 c) L = f(t) ) α = f(t), β = f(t) 6 5 4 [Nm].5 -.5 -.5..5..5..5.4 -.5..5..5..5.4 e) Ψ α = f(t), Ψ β = (t) f) Γ el = f(t), Γ L = f(t), $Γ L = f(t) 8 6 4 [a/s] 9 8 7 6 5 4 [a/s] -.5..5..5..5.4.5..5..5..5.4 g) = f(t), $ = f(t) h) = f(t), = f(t) Fg. 4..5 Cuent, angle contol fo maxmum powe facto wth MRAC A substantal mpovement of the ve pefomance fo both contol algothms can be seen when the MRAC oute loop s ae (Fg. 4.. an Fg. 4..5). Whle compensaton of the angula spee op ue to applcaton of nomnal loa toue takes appoxmately, s fo the basc algothms, ths s
compensate n,5 s by the MRAC oute contol loop. Also the absolute value of spee op s nealy fou tmes lowe wth the MRAC augmentaton, when compae wth the basc system combnng foce ynamc contol an the stana vecto contol. 4..6 Conclusons an Recommenatons The smulaton esults of the popose new contol metho fo electc ves employng RSM wth foce ynamcs show a goo ageement wth the theoetcal pectons. The only substantal epatue of the system pefomance fom the eal s the tansent nfluence of the extenal loa toue on the emane oto spee. Although ths effect s not too seous, ts conseable eucton wth the a of a MRAC base oute contol loop wee vefe. Some pelmnay nvestgatons to test the obustness to moto paamete msmatches show pomsng esults whch ae not publshe hee ue to space lmtatons, especally when the amount of esults s ouble when the MRAC oute contol loop s ae. Futhe nvestgatons of obustness, howeve, shoul be cae out, patculaly wth ega to ynamc loa paamete msmatches an tme vayng extenal loa toues. It s hghly esable to futhe nvestgate the popose contol stategy expementally wth a new ALA RSM escbe n [8]. 4..7 Refeences [] I., BOLDEA, A. S., NASAR: Vecto Contol of AC Dves. CRC Pess. Boca Raton, 99. [] S. V., DRAKUNOV, D. B., IZOSIMOV, A. G., LUK YANOV, V. A., UTKIN, V. I., UTKIN: The block contol pncple, I, II. Automaton an Remote Contol, Vol. 45, pp. 6-69, No. 5, 99. [] V. I., UTKIN: Metho of sepaaton of motons n obsevaton poblems. Automaton an Remote Contol, Vol. 44, pp. -8, No., 99. [4] V. I. UTKIN: Slng Moes n Contol an Optmsaton. Spnge-
Velag, Beln, 99. [5] S. J., DODDS, V. A., UTKIN, J., VITTEK: Self Oscllatng, Synchonous Moto Dve Contol System wth Pescbe Close-Loop Spee Dynamcs. Poceengs of n EPE Chapte Symposum, Nancy, Fance, June 996, pp. -8. [6] S. J., DODDS, S. J., J., VITTEK: Synchonous Moto Dve wth Pescbe Close-loop Spee Dynamcs Employng a Two-phase Oscllato. Poceengs of EDPE 96 confeence Vol., Hgh Tatas, Slovaka, Oct. 996, pp. 8-88 [7] S. J., DODDS, J., VITTEK, S., SEMAN: Implementaton of a Sensoless Synchonous Moto Dve Contol System wth Pescbe Close-Loop Spee Dynamcs. Poceengs of SPEEDAM 98 Symposum, Soento, Italy, June 998, pp. P4-5 P4- [8] M., LICKO, V., HRABOVCOVA: Desgn Aspects of Axally Lamnate Ansotopc Reluctance Synchonous Moto. Poceengs of TRANSCOM 99 confeence, secton, Unvesty of Zlna, Slovaka, May 999, pp. 9 45. Appenx Reluctance synchonous moto paametes: Nomnal voltage 8 V (fo Y) Stato esstance Rs = 8.6 Ω Nomnal cuent. A Qua. nuctance L = 6.8 mh Rate powe 4 W Numbe of polpas p = Invete c voltage 55 V Moment of neta J =. kgm Polynomal appoxmaton of the ect nuctance L ( ) n the wokng ange of stato cuents: L =,9.,755. +,4 [H; A] wth conton: f L <,45 then L =,45. Acknowlegements The authos wsh to thank the Euopean Commsson, Bussels, fo funng the INCO COPERNICUS pogamme No.9669 Ucove an Slovak Gant
Agency VEGA fo funng eseach pogamme No./6/99 whch enable us to pesent these esults.