Jounl of Comput Scinc 8 (1): 17-171, 1 ISSN 1549-3636 1 Scinc Publictions Dsign of Spd Contoll fo Pmnnnt Mgnt Synchonous Moto Div Using ntic Algoithm Bsd Low Od Systm Modlling 1 K. Ayy, 1 K. Rmsh nd. uusmy 1 Dptmnt of Elcticl nd Elctonics Engining, Vll Collg of Engining nd Tchnology, Eod, Tmilndu, Indi Dptmnt of Elcticl nd Elctonics Engining, Bnnimmn Institut of Tchnology, Sthymnglm, Eod, Tmilndu, Indi Abstct: Poblm sttmnt: In this study, Modl Od Rduction (MOR) mthod is poposd fo ducing high od systm into low od systm. Spd contoll dsign is cid out to th low od systm by ntic Algoithm (A) ppoch nd this contoll is usd to high od systm. Appoch: This study is usd to find solution to givn objctiv function mploying diffnt pocdus nd computtionl tchniqus. Th poblm chosn is tht of low od systm modlling usd in dsign of spd contoll fo Pmnnnt Mgnt Synchonous Moto (PMSM) div. Rsults: ntic Algoithm obtins btt contoll vlus tht flcts th chctistics of th oiginl high od systm nd th pfomnc vlutd using this mthod compd with th xisting ppoximtion mthod. Conclusion: Pfomnc of this Spd contoll hs bn vifid though Simultion using MATLAB pckg. Kywods: Pmnnnt Mgnt Synchonous Moto (PMSM), div, gntic lgoithm, low od systm modlling, spd contoll INTRODUCTION Duing th ly 195s schs studid volutiony systms s n optimistion tool, with n intoduction to th bsics of volutiony computing (Sivnndm nd Dp, 9). Until 196s volutiony systms ws woking in plll with ntic Algoithm (A) sch. At this stg, volutiony pogmming ws dvlopd with th concpts of volution, slction nd muttion. Hollnd (199) intoducd th concpt of ntic Algoithm s pincipl of Chls Dwinin thoy of volution to ntul biology. Th woking of gntic lgoithm stts with popultion of ndom chomosoms. Th lgoithm thn vluts ths stuctus nd llocts poductiv oppotunitis such tht chomosoms, which hv btt solution to th poblm, giv mo chnc to poduc. Whil slcting th bst cndidts, nw fitt offsping poducd nd instd nd th lss fit is movd. Th xchng of chctistics of chomosoms tks plc-using optos lik cossov nd muttion. Th solution is dfind with spct to th cunt popultion. A option bsiclly dpnds on th Schm thom. As cognizd s bst function optimiss nd is usd bodly in pttn discovy, img pocssing, signl pocssing nd in tining Nul Ntwoks. Mny contol systm pplictions, such s stllit ltitud contol, fight icft contol, modl-bsd pdictiv contol, contol of ful injctos, utomobil spk tim, possss mthmticl modl of th pocss with high od, du to which th systm dfind bcoms complx. Ths high od modls cumbsom to hndl (Sivnndm nd Dp, 9). As sult, low od systm modlling cn b pfomd, which hlps in llviting computtionl complxity nd implmnttion difficultis involvd in th dsign of contolls nd compnstos fo high od systms. Futh, th dvlopmnt nd usg of mico contolls nd micopocssos in th dsign nd implmnttion of contol systm componnts hs incsd th impotnc of low od systm modling (Psd, ; 3; 3b). Thus, in this study, ntic Algoithm is usd indpndntly to high od systms nd suitbl low od systm is modlld (Sivnndm nd Dp, 9). Th vilbility of modn Pmnnt Mgnts (PM) with considbl ngy dnsity ld to th dvlopmnt of dc mchins with PM fild xcittion in th 195 s. Intoduction of PM to plc lctomgnts, which hv windings nd qui n xtnl lctic ngy souc, sultd in compct dc mchins (Islm t l., 11). Th synchonous mchin, with its convntionl fild xcittion in th Cosponding Autho: Ayy, K., Dptmnt of Elcticl nd Elctonics Engining, Vll Collg of Engining nd Tchnology, Eod, Tmilndu, Indi 17
oto, is plcd by th PM xcittion; th slip ings nd bush ssmbly dispnsd with. With th dvnt of switching pow tnsisto nd silicon-contolld ctifi dvics in lt pt of 195s, th plcmnt of th mchnicl commutto with n lctonic commutto in th fom of n invt ws chivd. Ths two dvlopmnts contibutd to th dvlopmnt of PM synchonous nd bushlss dc mchins (Islm t l., 11). Th mtu of th dc mchin nd not b on th oto if th mchnicl commutto is plcd by its lctonic vsion. Thfo, th mtu of th mchin cn b on th stto, nbling btt cooling nd llowing high voltgs to b chivd: significnt clnc spc is vilbl fo insultion in th stto. Th xcittion fild tht usd to b on th stto is tnsfd to th ot with th PM pols. Ths mchins nothing but n insid out dc mchin with th fild nd mtu intchngd fom th stto to oto nd oto to stto spctivly (Pilly nd Kishnn, 1989). In this study contins dsign of spd contoll fo pmnnt mgnt synchonous mchins using ntic lgoithm bsd low od modlling. Spd contoll dsign: Th dsign of th spdcontoll is impotnt fom th point of viw of impting dsid tnsint nd stdy stt chctistics to th spd-contolld PMSM div systms (Islm t l., 11). A popotionl-plus-intgl contoll is sufficint fo mny industil pplictions; hnc, it is considd in this wok. Slction of th gin nd tim constnts of such contoll (Tlbi t l., 7) by using th symmticoptimum pincipl is stightfowd if th d xis stto cunt is ssumd to b zo (Wllc, 1994). In th psnc of d xis stto cunt, th d nd q cunt chnnls coss-coupld nd th modl is non-lin, s sult of th toqu tm. Und th ssumption tht th d xis cunt bing zo (i.., i ds =), thn th systm bcoms lin nd smbls tht of sptly-xcitd dc moto with constnt xcittion (Shm t l., 8). Fom thn on, th block-digm divtion, cunt loop ppoximtion, spd-loop ppoximtion nd divtion of th spd contoll by using symmtic optimum idnticl to thos fo dc moto div spd contoll dsign. Block digm divtion: Th moto q xis voltg qution with th d xis cunt bing zo bcoms (Shm t l., 8): J. Comput Sci., 8 (1): 17-171, 1 wh, th lctomgntic toqu is givn by Eq. 3: 3 P T =. λ fiqs (3) And if th lod is ssumd to b fictionl, thn Eq. 4: 1 1 m V qs = (R s Lqp)iqs ωλ f (1) Cunt loop: This inducd-mf loop cosss th q xis And th lctomchnicl Eq. is: cunt loop nd it could b simplifid by moving th pick-off point fo th inducd-mf loop fom spd to P (T T 1 ) = Jp ω B 1ω () cunt output point. This givs th cunt-loop tnsf function (Shm t l., 8) fom Fig.. 171 T = B ω (4) Which, upon substitution, givs th lctomchnicl Eq. 5 s: (Jp B ) ω =. λ i = K.i 3 P t f qs t qs Th fictionl toqu cofficint is Eq. 6: P Bt = Bl B1 (6) And toqu constnt is Eq. 7: 3 P K t =. λ f Th Eq. 1 nd 5, whn combind into block digm with th cunt-nd spd-fdbck loops ddd (Shm t l., 8) shown in Fig. 1. Th invt is modld s gin with tim lg (Tlbi t l., 7) by Eq. 8-1: (s) Wh: K T K 1 st (5) (7) in = (8) V in dc in =.65 (9) V cm in 1 = (1) f c wh, V dc is th dc-link voltg input to th invt (Islm t l., 11), V cm is th mximum contol voltg nd f c is th switching (ci) fquncy of th invt. Th inducd mf du to oto flux linkgs,, is Eq. 11: =λ ω (V) (11) f
J. Comput Sci., 8 (1): 17-171, 1 Fig. 1: Block digm of th sppd-contolld PMSM div Fig. : Cunt contoll Fig. 3: Spd-contol loop 17
J. Comput Sci., 8 (1): 17-171, 1 This inducd mf loop cosss th q xis cunt loop nd it could b simplifid by moving th pickoff point fo th inducd-mf loop fom spd to cunt output point. This givs th cunt-loop tnsf function fom Fig. s Eq. 1 nd 13: i (s) K K (1 st ) i * (s) H K K (1 st ) (1 st ) qs in m = qs c in m in Wh: { KK b (1 st )(1 st m) } q m m b t m f R s R s Bt Bt (1) 1 L 1 J K = ;T = ;K = ;T = ;K = K K λ (13) This cunt-loop tnsf function (Kishnn nd Rmswmi, 1974) is substitutd in th dsign of th spd contoll s follows. Spd contoll: Th spd-contol loop is shown in Fig. 3. Div pmts: Th PMSM div systm pmts s follows: R s = 1.4Ω, L d =.56H, L q =.9H, λ f =.1546 Wb- Tun, B t =.1 N-m/d/sc, J =.6 kg-m, P = 6, f c = khz, V cm = 1V, H ω =.5 V/V, T ω =. sc, H c =.8 V/A, V dc =85V. Fom th bov div pmts th following vlus obtind: Invt: in,k in = 18.55 V/V; Tim constnt, T in =.5 sc. Moto (lcticl): in, K =.7143; Tim constnt, T =.64 sc. Inducd mf loop: Toqu constnt, K t =.87 N.m/A Mchnicl gin, K m = 1 d/s/nm; Mchnicl Tim constnt, T m =.6 sc. K b = K t K m λ f = 3.6. s s... s ( s) = b b s b s... b s b s n 1 1 n 1 n 1 n 1 n 1 n (14) (S) cn b xpndd into pow sis bout S = of th fom Eq. 15-17 (Shmsh, 1975): ( ) = (15) s c c s c s Wh: c 1 = (16) b And: k 1 ck = k b jc k j, k > (17) b j= 1 With: dk = k > n 1 Th d i dictly popotionl to th tim momnts of th systm, ssuming th systm is stbl Eq. 18 (Shmsh, 1975): d d s d s... d s 1 1 1 ( s) = 1 1s s... 1s s (18) Thn fo R (s) to b Pd ppoximnt of (S), th following Eq. 19 nd obtind: d = c (19) d = c c 1 1 1 = c c c 1 1 1 = c c 1 () Poposd mthod of modl duction: Th poposd mthod of modl duction is Coss Multipliction of Polynomils Modl od duction mthod. It consists of th following stps in th systm ppoximtion pocss. Stp-1: Th dnominto nd numto polynomil constnt tms in th ducd od modl obtind though Pd ppoximtion: Th tnsf function of high od (n th ) is considd s Eq. 14: b 173 c Fom th Eq. 16 nd : d = b = (1) Fom th Eq. 1, lt Eq. : d = ()
J. Comput Sci., 8 (1): 17-171, 1 Stp-: Th unknown cofficints of diffnt pows of s mining in ducd od modl dtmind: Th givn high od systm tnsf function is qutd nd coss multiplid with k th od gnl tnsf function. This pocss yilds (n) qutions with (-1) unknown ducd od tnsf function cofficints. This stp is simil to th modl od duction mthod poposd in Mnigndn t l. (5), wh th vlus of o d kpt s qul to 1 ispctiv of th systm condition to obtin th vlus of unknown cofficints in th ducd od modl tnsf function. But in this poposd mthod, th vlus of nd d obtind though Pd ppoximtion mthod s dtild in stp-1. This lds to btt systm ppoximtion s compd to th modl od duction mthod poposd by Eq. 3 nd 4 Mnigndn t l. (5): s s... s b b s b s... b s b s n 1 1 n 1 n 1 n 1 n 1 n d d s d s... d s = s s... s s 1 1 1 1 1 1 n 1 ( 1s s... n 1s ) 1 ( 1s s... 1s s ) n 1 n = ( b b1s b s... b n 1s b ns ) 1 ( d d 1s d s... d 1s ) (3) (4) Th cofficints of sm pow of s on both sid of th Eq. 4 qutd with ch oth (Rmsh t l., 8) nd is givn by Eq. 5: = b d n 1 n 1 = b d b d n 1 1 n n 1 1 n (5) = b d b d b d 1 1 1 1 = b d b d 1 1 1 1 = b d Th (n) st of Eq. 5 is solvd with th vlus of d, obtind in (). This lds to hv diffnt st qutions fo solving th mining unknown pmts. Bsd on th optiml ISE vlu, th unknown vlus slctd nd th sultnt ducd od modl is obtind s Eq. 6 nd 7: d d s d s... d s 1 1 1 ( s ) = 1 1s s... 1s s If = (6) Dp, 9): Consid, th tnsf function of high od (n th ) s: s s... s ( s) = b b s b s... b s b s n 1 1 n 1 n 1 n 1 n 1 n Th gnl fom of th tnsf function of scond od systm in th s-domin cn b psntd s: T T S 1 ( s) = i s ζω ns ωn (8) wh, ζ is th dmping tio nd ω n is th undmpd ntul fquncy of oscilltion in d/sc. Th vlus of T 1 nd T cosponding to Eq. 8 cn b computd s T 1 = T g nd T = S ω. Wh, th tnsint gin g n (T g ) nd stdy stt gin (S g ) computd s: Tg = nd Sg = b b n 1 n By using poposd scnio-1, th ducd od modl obtind in stp-/stp-3 is modifid in to n initil fom s Eq. 9: d d1 s A A1s i ( s) = = 1 B B1s s Wh: s s d d A = = T, A = = T, 1 1 1 B = nd B = 1 1 (9) Th unit stp input tim spons of th initil scond od ppoximnt i (S) is nlyzd with comput pogm nd its chctistics notd. Th cumultiv o indx J using th intgl squ o of th unit stp tim sponss of th givn high od systm (s) psntd by Eq. 15 nd th initil scond od ppoximnt i (S) psntd by Eq. 9 is clcultd. Th cumultiv o indx J is clcultd using th fomul Eq. 3: N (3) t = J = [y(t) y (t)] d d1s ( s) = (7) wh, y(t) is th output spons of th high od s 1s systm t th N th instnt of tim, y (t) is th output spons of th scond od modl t th N th instnt of Stp-3: Th cumultiv o indx (J) fo initil tim nd N is th tim intvl in sconds ov which ducd od modl is clcultd (Sivnndm nd th o indx is computd. 174
J. Comput Sci., 8 (1): 17-171, 1 Fig. 4: Block digm of th A bsd PID contoll of th systm Tbl 1: Pmts of A A popty Vlu/mthod Popultion siz 6 Mximum Numb of gntions Pfomnc indx/fitnss function Mn squ o Slction Mthod Nomlizd omtic slction Pobbility of slction.5 Cossov mthod Aithmtic cossov Numb of cossov points 3 Muttion mthod Unifom muttion Muttion pobbility.1 Tbl : A bsd PID contoll gin vlus in pmts K p K i K d in vlus 17.7713 31.7933.47 Aft giving th bov pmts to A th PID contolls cn b sily tund nd thus systm pfomnc cn b impovd (Thoms nd Poongodi, 9). Stp-4: Find th PID Contoll Constnts using A: A cn b pplid to th tuning of PID contoll gins to nsu optiml contol pfomnc t nominl opting conditions. Th block digm fo th nti systm is givn blow in Fig. 4 nd lso th gntic lgoithm pmts chosn (Thoms nd Poongodi, 9) fo th tuning pupos shown blow in Tbl 1.Th constnts K p, K i nd K d dtmind using ntic Algoithm (A) ppoch (Mhony t l., ). Th Contoll dsign fo sultnt ducd od modl will closly mtch with th cosponding high od modl. Aft giving th bov pmts to A th PID contolls cn b sily tund nd thus systm pfomnc cn b impovd. Spd contoll dsign by poposd mthod: Tnsf function Appoch: Oiginl High od systm without spd contoll nd filt: Lt (s) b th tnsf function of th oiginl high od systm. Th tnsf function of PMSM div systm without spd contoll nd filt is s follows Eq. 31: 1 6 5 7. 7 8 s 7 6 3. (s ) =. 5 7 6 s. 4 s 4 3 4. s 7.7 7 8 s 3 4.6 3 (31) Rducd od systm: Lt (s) b th tnsf function of th ducd od systm (Poton, 1997). Th tnsf function of th ducd od systm of PMSM div by th ppliction of poposd mthod is s follows Eq. 3: 337.95s 763. (s) = 8.1s 47.59s 34.63 Fom Eq. 9, th ducd od modl is obtind s Eq. 33 (Rvichndn, 7): 394.4s 336.56 (s) = s 5.796s 4. (33) Spd contoll: To obtin n optimum tnsint spons of th systm, PID contoll is chosn with tnsf function Eq. 34 (Kuo nd olnghi, 3; Ogt, 1): Ki c (s) = K p Kds (34) s Wh: K p = Popotionl gin K i = Intgl gin K d = Divtiv gin Th Vlus of K p, K i nd K d obtind by ntic Algoithm (A) ppoch. Th sultnt K p, K i nd K d vlus tbultd s shown in Tbl. Spd contoll dsign fo ducd systm: Th block digm of ducd systm with spd contoll is shown in Fig. 5. wh, K p, K i nd K d vlus 17.7713, 31.7933 nd.47 spctivly. Spd contoll dsign fo oiginl systm: Th K p, K i nd K d vlus of spd contoll of Oiginl systm is sm s tht of ducd systm. Using this vlu th spd contoll of oiginl systm is don. Fig. 6 shows th block digm of oiginl systm with spd contoll. Dsign spcifictions: Th systm is tstd with unit stp input nd th dsign pocdu is followd bsd on th following dsign spcifictions: Mximum pk ovshoot = lss thn 3% Sttling tim = lss thn 3 sc Stdy stt o = % (ssumd fo optimum spons) Bfo pocding on to th simultion, th stting vlus of th pmts of contoll (3) dducd using nwly poposd pocdus 175
J. Comput Sci., 8 (1): 17-171, 1 Fig. 5: Block digm of ducd systm with spd contoll Fig. 6: Block digm of oiginl systm with spd contoll Fig. 7: Stp spons of oiginl systm without spd contoll 176
J. Comput Sci., 8 (1): 17-171, 1 Fig. 8: Stp spons of ducd systm without spd contoll Fig. 9: Stp spons of oiginl systm with spd contoll Simultion sults: Th ffctivnss of th nwly poposd schm fo th dsign of PID Spd contoll fo PMSM divs dmonsttd using comput simultions. Th systm is simultd fo stp input using MATLAB-SIMULINK softw with nd without contolls (Chpmn, ). Th output sponss of th bov simultion studis givn in th following Figus. Stp spons of oiginl systm: Th stp spons of th PMSM div systm is shown in Fig. 7. Stp spons of ducd systm: Th stp spons of th ducd od PMSM div systm is shown in Fig. 8. Stp spons of oiginl systm with contoll: Th stp spons of th PMSM div systm with Spd contoll by poposd mthod is shown in Fig. 9. Compd with PMSM div with convntionl spd contoll it givs btt pfomnc s listd in Tbl 3. Stp spons of ducd systm with contoll: Th stp spons of th ducd od PMSM div systm with Spd contoll by poposd mthod is shown in Fig. 1. Stp spons of oiginl systm with convntionl contoll: Stp spons of oiginl systm with convntionl spd contoll of PMSM div is shown in Fig. 11. 177
J. Comput Sci., 8 (1): 17-171, 1 Fig. 1: Stp spons of ducd systm with spd contoll Fig. 11: Stp spons of oiginl systm with convntionl spd contoll MATERIALS AND METHODS poblms.th us of A mthods in th dtmintion of th diffnt contoll pmts is ffctiv du to Fo n idl contol pfomnc by th PID thi fst convgnc nd sonbl ccucy.this contoll, n ppopit PID pmt tunning is wok th pmts of th PID spd contoll is ncssy (Oi t l., 8). Mostly usd PID contoll tund using ntic lgoithm. tunning mthods fo div contols Zigl-Nichols mthod nd symmtic optimum tunning mthod.ths RESULTS tunning mthods vy simpl, but cnnot gunt In this study, th pfomnc of PMSM div to b lwys ffctiv. To sumount this inconvninc, with MOR bsd spd contoll is vlutd on th optimiztion pocdu my b usd fo th btt bsis of is tim, sttling tim nd mximum dsign of contolls. ovshoot. Th pfomnc of th div systm with ntic lgoithm (A) mthods hv bn MOR bsd contoll hs bn impovd s compd widly usd in contol pplictions.th A mthod with th convntionl PI spd contoll (Singh, 6). hv bn mployd succssfully to solv complx Tbl 3 givs th spons of th div systm. 178
Tbl 3: Compision of stp sponss of th systm with nd without contoll fo both poposd nd convntionl mthod of spd contoll Ris tim Sttling tim Mximum Css (t ) sc (t s) sc ovshoot (%) Oiginl systm.443.79. 1-14 without contoll Rducd systm.444.788.967 without contoll Oiginl systm with.7.193.677 poposd contoll Rducd systm with.98.115.663 poposd contoll Oiginl systm with.648.376 3.8 convntionl contoll Th simultion sult (Fig. 9) shows tht th spons of th PMSM div systm with MOR bsd spd contoll is btt s compd to convntionl mthod (Tbl 3). Th spd contol loop of th div is simultd with Convntionl contoll; in od to comp th pfomncs to thos obtind fom th spctiv MOR bsd div systm (Rhmn, 3). DISCUSSION Th dynmic nd stdy stt pfomnc of th MOR bsd spd contoll fo pmnnt mgnt synchonous moto div is much btt thn th Convntionl PI spd contoll. All th compisons fo th diffnt css tbultd in Tbl 3. CONCLUSION Th modl od duction mthod poposd in this study givs btt ppoximtd ducd od modl fo th givn PMSM div systm. Bcus of this w gt th ducd od systm pfomnc s clos s possibl to th high od systm spons. This will sult in duction in dsign cost nd systm complxity. Th mthod poposd in this study pplid fo th Spd contoll dsign of PMSM div. This study focuss on th duction of modls it minimizs th complxity involvd in dict dsign of PID Spd Contoll. Th ppoximt vlus fo PID Contoll pmts clcultd fom th ntic lgoithm ppoch nd suitbly tund to mt th quid pfomnc spcifictions. Th tund vlus of ths contoll pmts ttchd with th oiginl systm nd its closd loop spons fo unit stp input is found to b in good ccod with th spons of ducd od modl. REFERENCES Chpmn, S.J.,. MATLAB Pogmming fo Engins. 4th Edn., Cngg Lning, Stmfod, ISBN-1: 97849544493, pp: 567. J. Comput Sci., 8 (1): 17-171, 1 179 Hollnd, J.H., 199. Adpttion in Ntul nd Atificil Systms. 1st Edn., MIT Pss, Cmbidg, ISBN-1: 6581116, pp: 11. Islm, S., F.I. Bkhsh, M. Khushd, S. Ahmd nd A. Iqbl, 11. A novl tchniqu fo th dsign of contoll of vcto-contolld pmnnt mgnt synchonous moto div. Pocdings of th Annul IEEE Indi Confnc, Dc. 16-18, IEEE Xplo Pss, Hydbd, pp: 1-6. DOI: 1.119/INDCON.11.6139558 Kishnn, T. nd B. Rmswmi, 1974. A fstspons DC moto spd contol systm. IEEE Tns. Ind. Appli., pp: 643-651. DOI: 1.119/TIA.1974.34914 Kuo, B.C. nd F. olnghi, 3. Automtic Contol Systms. 8th Edn., John Wily nd Sons, Nw Yok, ISBN: 471134767, pp: 64. Mhony, T., C.J. Downing nd K. Ftl,. ntic Algoithm fo PID Pmt Optimiztion: Minimizing Eo Citi. Univsity of Stcthclyd. Mnigndn, T., N. Dvjn nd S.N. Sivnndm, 5. DESIN of PID contoll using ducd od modl. Acdmic Opn Int. J. Ogt, K., 1. Modn Contol Engining. 5th Edn., Pntic-Hll, Boston, ISBN-1: 978136156734, pp: 894. Oi., A., C. Nkzw, T. Mtsui, H. Fujiw nd K. Mtsumoto t l., 8. PID optiml tuning mthod by Pticl Swm Optimiztion. Pocdings of th SICE Annul Confnc, Aug. -, IEEE Xplo Pss, Tokyo, pp: 347-3473. DOI: 1.119/SICE.8.46556 Pilly, P. nd R. Kishnn, 1989. Modling, simultion nd nlysis of pmnnt-mgnt moto divs. I. Th pmnnt-mgnt synchonous moto div. IEEE Tns. Industy Appli., 5: 65-73. DOI: 1.119/8.5541 Poton, A., 1997. Modl duction tchniqus in tokmk modlling. Pocdings of th 36th IEEE Confnc on Dcision nd Contol, Dc. 1-1, IEEE Xplo Pss, Sn Digo, CA, pp: 3691-3696. DOI: 1.119/CDC.1997.6543 Psd, R.,. Pd typ modl od duction fo multivibl systms using outh ppoximtion. Comput. Elct. Eng., 6: 445-459. DOI: 1.116/S45-796()-1 Psd, R., 3. Impovd Pd ppoximnts fo multivibl systms using stbility qution mthod. J. Institution Eng. 84: 161-165. DOI: 1.413/377-63.48531 Psd, R., 3b. Lin modl duction using th dvntgs of Mikhilov cition nd fcto division. J. Instit. Eng., 84: 9-1.
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