Stability Analysis of Simultaneous Estimation of Speed and Stator Resistance for Sensorless Induction Motor Drives

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1 Poceedng of the 4 th Intenatonal Mddle Eat Powe yte Confeence (MEPCON 0), Cao Unvety, Egypt, Decebe 9-, 00, Pape ID 80. tablty Analy of ultaneou Etaton of peed and tato Retance fo enole Inducton Moto Dve Mohaed. Zaky Electcal Engneeng Dept., Faculty of Engneeng, Mnoufya Unvety, hebn El-o (5), Mnoufya, Egypt. Mob ; Fax: , e-al: zaky78@yahoo.co) Abtact Th pape peent tablty analy of a Multple- Input-Multple-Output (MIMO) yte of ultaneou peed and tato etance etato fo enole nducton oto dve. The ntablty poble of the Adaptve Flux Obeve (AFO) n the egeneatve ode at low peed wthout feedback gan deontated. Degn of the obeve feedback gan to guaantee tablty, epecally n the egeneatve ode at low peed popoed. Cloed-loop contol yte of the ndependent ue of the two etato ae developed. All gan of the adaptve PI contolle, baed on Root-Locu ethod, ae elected to povde good tackng pefoance and fat dynac epone. Pefoance of AFO ung the popoed feedback gan and adaptve PI contolle gan, wth a enole Indect Feld Oented (IFO) contolled nducton oto dve, vefed by ulaton and expeental eult. The eult how a good poveent n both the convegence and tablty, epecally n the egeneatve ode at low peed, whch uppot the valdty of the popoed analy. Index Te enole contol, tablty analy, adaptve law degn, feedback gan degn M I. INTRODUCTION ANY eeach effot have been devoted fo yeldng bette peed etaton of enole nducton oto dve ung AFO []-[]. Howeve, thee a well known untable egon encounteed at low peed wth egeneatng load. Th untable egon of AFO can be educed by pope degn of both the obeve feedback gan and adaptve law ung eveal technque [4]-[9]. Recently, nueou attept have been ade to pove the pefoance of peed-enole nducton oto dve, epecally at low peed, by dentfyng tato etance togethe wth peed. tato etance etaton of utot potance fo accuate opeaton of enole nducton oto dve n low peed egon. eveal onlne tato etance adaptaton chee have been popoed [7], [9], [0]. peed and tato etance etato ay be table when degned ndependently. Howeve, ultaneou ue of the two etato ay lead to ntablty unde oe opeatng condton, epecally n the egeneatve ode at low peed [7]-[9]. Theefoe, tablty analy of ultaneou etaton of peed and tato etance fo peed-enole nducton oto dve a elevant and vey challengng topc. Howeve, t can be ad that tablty analy of ultaneou peed and tato etance etato ha eceved vey lttle attenton n the pat eeach wok. tablzng the Model Refeence Adaptve yte (MRA)- baed etato fo cobned peed and tato etance acheved by adjutng the adaptve law [7]. tablty analy of the peed etaton of AFO and plfcaton of the tuctue of a enole contol yte by ean of the decouplng contol ha been popoed [8]. The ntablty poble fo ultaneou etaton of oto peed and tato etance ung aveage technque ha been olved [9]. The adaptaton PI gan fo ultaneou etato, whch ae alo condeed an potant paaete fo pecfyng the etaton poce of AFO, need to be degned to gve quck tanent epone and good tackng pefoance. In th pape, tablty analy of ultaneou etaton of peed and tato etance fo enole nducton oto dve peented. Cloed-loop contol yte of peed and tato etance etato ae developed fo both ndependent and ultaneou ue. tablty analy of the coplete MIMO yte fo ultaneou ue of the two etato tuded. The ntablty poble n the egeneatve ode at low peed deontated. Degn of the feedback gan popoed to guaantee the tablty, epecally n the egeneatve ode at low peed. tablty of each etato teated a ngle-input-ngle-output (IO) yte analyzed ung Root-Locu ethod. Degn of the adaptve PI contolle gan fo each etato peented to gve quck tanent epone and good tackng pefoance. The coplete enole dve yte pleented ung DP-D0 contol boad. Fnally, ulaton and expeental eult ae peented to conf the valdty of the popoed analy. II. MATHEMATICAL MODEL FOR INDUCTION MOTOR AND ADAPTIVE FLUX OBERVER The block daga fo the popoed enole IFO contolled nducton oto dve hown n Fg.. 64

2 ˆ PI * λ d IFOC θ DP Contol Platfo * abc abc Fg. Block daga of enole ndect feld oented contol yte. Cuent Regulato ˆ Pule tato etance & oto peed Etato IGBT Invete A - Vdc - B C qd v qd Hadwae Voltage and Cuent eno ab abc to dq Tanfo IM v ab DC Gen. A. Dynac Model of the Inducton Moto The nducton oto can be epeented by t dynac odel expeed n the tatonay efeence fae n te of the tato cuent and oto flux a follow []: p = A A λ b v pλ = A A λ whee, T = d q, λ = λd λ q and A = ai, A = ci dj, A = ei, A = ε A, b = bi 0 0 I =, J = 0 0 R L L a =, c =, d =, e = σl σlt L εt ε T σll L L ε=, b =, σ=, T = L σl LL R B. Adaptve Flux Obeve Wth efeence to () and (), the adaptve flux obeve can be contucted a []: T ˆ ˆ ˆ ˆ ˆ p = A A λ b v ˆ ˆ ˆ ˆ ˆ pλ = A Aλ (4) The etaton chee of oto peed and tato etance ae la to the one ued n []. ˆ ˆ T = ( P I dt) e J λ (5) ˆ e Tˆ R = PR dt (6) whee, ˆ T ˆ T ˆ ˆ, ˆ ˆ, ˆ d = q λ = λd λ q e = P, I, PR and ae adaptve gan of the peed and tato etance etato, epectvely. The ybol ^ () () () gnfe the etated value and vaable. the feedback gan atx defned a = I J. ˆ A, ˆ A ˆ, A ae the coeffcent n whch and R ae eplaced by etated peed ˆ and tato etance ˆ R et n the obeve. III. MIMO TABILITY INVETIGATION A. Eo Equaton of tato Cuent and Roto Flux The tate eo equaton of tato cuent and oto flux ae obtaned by ubtactng () and () fo () and (4), epectvely a follow: ˆ ˆ pe = ( A ) e Ae λ ΔAλ ΔA (7) ˆ peλ = Ae Ae λ ΔAλ (8) ˆ ΔA ΔA Whee, e λ = λ λ, Δ A = A ˆ A =. ΔA ΔA If the oto peed and tato etance ae condeed a vaable paaete, aung no othe paaete vaaton, the eo atx ΔA can be expeed a follow: ΔRI ΔJ Δ A =, Δ A =, Δ A = 0, Δ A =Δ J σl ε whee, Δ R ˆ, ˆ = R R Δ = nce the tato cuent eo ued to etate the oto peed and tato etance, the elatonhp between the peed eo and tato etance eo deved a follow: Takng Laplace tanfo fo (7) and (8) gve ˆ ˆ I ( A ) e = Ae λ ΔAλ ΔA (9) ˆ [ I A ] eλ = Ae ΔAλ (0) Fo whch, the tato cuent eo equaton n te of both the peed eo and tato etance eo can be expeed a follow [9]: e = e e () whee, 65

3 ˆ e = G () ( ˆ Jλ ) () ˆ e = G () ( ˆ R R R) () and A G() = I ( A A ) A A ε ε (4) () [ I A] A GR = I ( A A ) A A σl ε (5) The oto peed and tato etance etato, gven by (5), (6), (4) and (5), ae epeented by the MIMO cloed-loop contol yte a hown n Fg.. peed and tato etance etato ay be table when degned ndependently. Actually, thee an effect fo each etato whch appea a a dtubance on the othe one. Theefoe, tablty can not be guaanteed fo ultaneou ue of the two etato and need to be nvetgated. In the next ecton, tablty fo ultaneou etato wll be nvetgated takng the effect of each etato on the othe nto condeaton. Aftewad, the tanfe functon of the peed and tato etance etato teated a IO contol yte ae deved to elect the gan of adaptve PI contolle. B. Tanfe Functon of peed Etato plfyng (4) gve G () = I ( ai aj ) ( ai a4j ) ε (6) whee, R L a = σl σlt L T a = R L L a = T σl σlt L εt R L L a4 = T σl σlt L εt Then, the eo equaton expeed a: R Δ R Δ G R () G () e e e e e e PR G cr () P G c () Fg. Block daga of MIMO cloed-loop contol yte fo ultaneou etaton of peed and tato etance. I ˆR 0 ed G() G () e = ˆ q G() G () λ ( ˆ ) (7) The te J λ ˆ ha t coponent only on the q-ax, theefoe, the tanfe functon that ae elated to th nput and ae elevant to the cloed-loop tablty ae G () and G (). Takng nto account that the nput to the nonlnea feedback block eque only the output eo on the q-ax, e q, to calculate λ ˆ, t concluded that only the gnal n the q- eq ax fo the cloed loop yte. Th ean that only G () nvolved n the tablty analy of the peed etaton yte. Theefoe, the eo equaton can be wtten a follow: e () ˆ ( ˆ q = G λ ) (8) The tanfe functon G () can be calculated fo (6) a: G ( a a) () = ε 4 g g g g4 whee, g = a g = a a a g = aa aa 4 g = a a 4 4 C. Tanfe Functon of tato Retance Etato plfyng (5) to obtan the G R () a follow: GR() = G() G() σl T (9) (0) whee G () and G () can be obtaned fo (6) a follow: ( a a) G() = 4 g g g g4 ( a a4) G() = 4 g g g g4 ubttutng n (0) gve f f f GR() = σ L 4 g g g g 4 () whee, a a f = a, f = a a, f = a4 T T T The ultaneou peed and tato etance etato, whch ae hown n Fg., have been condeed fo tablty analy. ultaneou ue of the two etato ay lead to the poblty of ntablty unde oe opeatng condton, epecally at low peed egeneatve ode. Theefoe, the tablty of the coplete MIMO yte ut be 66

4 nvetgated. The tablty analy pefoed by fndng the locaton of the open-loop pole and zeo of the open-loop tanfe functon fo the MIMO yte. Fo Fg., the elatonhp between the yte output ˆ and R ˆ and the eo Δ, Δ R, whch epeent the fowad tanfe functon of the MIMO yte, deved a follow: ˆ ˆ ˆ ˆ λ GG c λ GG c R Δ = ˆ R ˆ ˆ ˆ R GcRG GcRG Δ λ R The chaactetc equaton can be obtaned fo () a: () ˆ ˆ ˆ λ GcG λ Gc G R I = 0 () ˆ ˆ ˆ λ GcRG GcRG R whee, I Gc = P, GcR = PR ae the tanfe functon of the adaptve PI contolle fo oto peed and tato etance, epectvely. Equaton () can be educed to ˆ ˆ λ G G G G = 0 (4) c cr R Fo whch, 4 n n n n 4 n5 = 0 εσl 5 g 4 g g g 4 whee, σl ˆ ˆ n = λ P ε PR σl ˆ ˆ n = λ ap I ε ( fpr ) σl ˆ ˆ n = λ a a ε f f σl ˆ ˆ n 4 = λ ai ε ( fpr f) n =ε ˆ f 5 P I PR (5) Root-Locu plot of ultaneou ue of the two etato at ad/ec wth zeo feedback gan = 0 hown n Fg.. It obeved that the yte ha one zeo (ZR) n the ght hand de of the -plane and one pole (PR) at the ogn. A a conequence, the pole (PR) at the ogn wll ove towad the untable zeo (ZR) a hown n Fg. (b) and the peed etaton poce wll be untable egadle of the adaptaton PI gan. A a eult, the two etato becoe untable n the egeneatve ode at low peed. The ulaton eult of Fg. 4 deontate the dvegence of the etated peed wth zeo feedback gan = 0 n the egeneatng ode at low peed (5 ad/ec, T L = - N) due to the untable zeo. Th vefe the analytcal eult of Fg.. IV. DEIGN OF FEEDBAC GAIN The peented analytcal eult ung Root-Locu ethod and alo the ulaton eult eveal that thee an untable egon n the egeneatve ode at low peed fo ultaneou ue of the peed and tato etance etato wth zeo feedback gan. Th clealy obeved a a eult of the extence of an untable zeo on the ght-hand de of the -plane. Thu, t vey potant to elnate th ntablty poble and guaantee the tablty of the peed etaton n all opeatng condton, epecally n the egeneatve ode at low peed by agnng the feedback gan value. The feedback gan hould be uch that all zeo of the nueato of the tanfe functon (5) le n the left-hand de of the - plane. Thu, the condton of table zeo can be deduced ung Routh-Huwtz cteon of (6). n n n n n = (6) It found that the condton of tablty ung Routh tablty ctea ae deduced a gven n (7). n > 0 n > 0 nn nn 4 > 0 (7) n4 ( nn nn4) nn5 > 0 n 5 > 0 The uffcent condton of feedback gan to ake the yte zeo le n the left-hand de of the -plane and to guaantee the tablty condton of (7) a follow: = 0 R L L T < σl σltl εt T (8) The condton n (8) ae valdated by nvetgated nuecally the locaton of the cloed loop pole fo vaou opeatng ode of the dve fo a ange of peed tatng at ± 0. up to ± 50 ad/ec fo the oto paaete gven n Appendx B. Root-Locu plot of ultaneou ue of the two etato at ad/ec wth the popoed feedback gan (8) hown n Fg. 5. It obeved that the yte becoe table n the egeneatng ode at low peed. It evdent that the untable zeo oved to the left hand de and theefoe, the pole at the ogn wll ove towad the table zeo and the obeve tablty guaanteed. The effectvene of the popoed feedback gan poved by the analytcal eult a hown n Fg. 5. The feedback gan wa adjuted nde the obeve accodng to (8) and the ae ulaton eult wee taken at the ae condton a Fg. 4 to deontate the effect of the popoed feedback gan on tablty. The ulaton eult of Fg. 6 deontate a good convegence of the etated peed n the egeneatng ode at low peed (5 ad/ec, T L = - N) wth the popoed feedback gan (8). 67

5 V. DEIGN OF ADAPTIVE PI GAIN FOR PEED AND TATOR REITANCE ETIMATOR Adaptve PI gan of peed and tato etance etato ae eponble fo enung good convegence of the etated paaete, and conequently, tablty of the dentfcaton yte fo the two etato nvetgated. Theefoe, the a of th ecton to degn the adaptve PI gan fo thee etato, baed on tablty analy, ung Root-Locu ethod. Fo Fg., the block daga of cloed-loop oto peed and tato etance etato teated a IO yte by gnong the effect of each etato on the othe ae hown n Fg. 7 and 9, epectvely. PR ZR PR PR ZR ZR PR Fg. 5 Root Locu of MIMO cloed-loop contol yte at ad/ec dung egeneatng ode. Zoo n the donant pole and zeo wth the popoed feedback gan. PR4 ZR4 ZR ZR ad/ec PR5 ZR PR ad/ec (a) PR q PR q ZR q ZR (b) PR ZR PR Fg. Root Locu of MIMO cloed-loop contol yte at ad/ec dung egeneatng ode. (a) Coplete Root-Locu, (b) Zoo n the donant pole and zeo ( = 0). ad/ec q q q ad/ec Fg. 4 ulaton eult howng the ntablty poble at low peed egeneatng opeaton at 5 ad/ec and T L = - N wthout the feedback gan ( = 0). Fg. 6 ulaton eult howng the behavo wth the popoed feedback gan at egeneatng ode low peed opeaton at 5 ad/ec and T L = - N. Root locu plot of G () at ad/ec hown n Fg. 8; zoo n the donant pole and zeo. The tanfe functon G () ha fou pole and thee zeo. All ae located on the left-hand de of the -plane except one zeo located at the ogn. The othe two zeo ae -5.8 and -50, epectvely. The fou pole of G () ae coplex conjugate located at -5.8 ± j0.65 and -50 ± j4.5, epectvely. The locaton of the cloed-loop tanfe functon pole chaacteze the contol yte dynac. The locaton of the PI contolle zeo hould be on the eal ax at -5.8 to ake ue that the fat dynac cloed-loop pole ae alway located to the left of the donant conjugate pole, guaanteeng fat tanent epone and good tackng pefoance. Theefoe, P taken of lage value and I P hould equal the negatve eal pat of the donant pole (The value P = 500 and = 5.8 ae ued). Th allow fo fat exponentally I P peed eo decayng wth te wthout ovehoot o undehoot. Root-Locu plot of G R () at ad/ec ung the popoed feedback gan hown n Fg. 0; zoo n the donant pole and zeo. la to the peed etato, the adaptve PI contolle gan ae degned. The tanfe functon G R () ha fou pole and thee zeo. All ae located on the left-hand 68

6 de of the -plane. To nvetgate both the fat dynac epone and good convegence between the actual and etated value of the tato etance, the PI contolle zeo elected. Th can be acheved by choong PR ha lage value and PR equal the negatve eal pat of the donant pole. Theefoe, PR ae choen a 00 and = 5.8. PR Δ G () Fg. 7 Block daga of cloed-loop oto peed etato. P G c () I well a the peed contolle gan of IFO contol yte ae gven n the appendx. Alo, the adaptve PI contolle gan of peed and tato etance etato ae gven n the appendx. aple of the expeental eult ae peented to deontate the ntablty poble wth zeo feedback gan and t eedy ung the popoed feedback gan n the egeneatve ode at low peed. Expeental eult have been caed out to deontate the untable behavo n the low peed egeneatng ode a hown n Fg. wth peed efeence et at 5 ad/ec and a load toque about - N.. It obvou that the etated peed dvegence fo the eaued peed and the etated cuent dffe fo the eaued one. The ae expeental eult wee taken wth the popoed feedback gan (8) to llutate the tablty of the etated vaable n the low peed egeneatng ode at 5 ad/ec peed efeence and a load toque about - N. a hown n Fg.. The eult of Fg. and Fg. ae taken at fequency of.65 Hz and upply voltage of.54 Volt. The obtaned eult howed the effectvene of the popoed feedback gan and conf the valdty of the analytcal eult of Fg. 5. The peented ulaton and expeental eult how the ablty of the AFO to povde an accuate peed etate n the low peed egeneatng ode ung the popoed feedback gan (8) and the elected PI gan. Alo, thee eult vefy the valdty of the theoetcal analy. Fg. 8 Root locu plot of G () fo peed etato at ad/ec. Zoo n the donant pole and zeo wth the popoed. R Δ R G R () PR G cr () Fg. 9 Block daga of cloed-loop tato etance etato. ˆR [.5 ad/ec/dv] q [ /dv] [ ec/dv] [ad/ec] [] q Fg. 0 Root Locu of G R () fo tato etance etato at ad/ec dung egeneatve ode. Zoo n the donant pole and zeo wth the popoed feedback gan. VI. EXPERIMENTAL REULT The expeental yte cont of an nducton oto ntefaced wth a dgtal contol boad D0 baed on a Texa Intuent TM0C Dgtal gnal Poceo (DP). The nducton oto coupled wth a dc geneato fo loadng. The atng and paaete of the nducton oto a Fg. Expeental eult howng the ntablty poble at low peed egeneatng opeaton at 5 ad/ec and T L = - N wth = 0. 69

7 [.5 ad/ec/dv] q [ /dv] ad/ec [ ec/dv] vecto T v = [ vd vq] tato voltage vecto B. Inducton Moto Paaete:- TABLE I Inducton Moto Paaete Rated powe (w) 50 R (oh) Rated voltage (volt) 80 R (oh).6840 Rated cuent ().5 L (H) 0.45 Rated peed (p) 400 L (H) 0.45 Rated fequency (Hz) 50 L (H) 0.44 Nube of pole 4 Toque (N.) TABLE II Contolle Paaete q peed Contolle Popotonal gan 0.7 Integal gan 0.0 Fg. Expeental eult dung low peed egeneatng opeaton at 5 ad/ec and T L = - N wth the popoed feedback gan. VII. CONCLUION In th pape, tablty analy of ultaneou ue of peed and tato etance etato fo enole nducton oto dve ha been peented. The ntablty poble n the egeneatve ode at low peed fo ultaneou ue of the two etato ha been deontated. It ha been concluded that tablzaton of each ndependent peed and tato etance etato nuffcent to enue the tablty when ued ultaneouly. The obeve feedback gan ha been degned to guaantee the tablty n the egeneatve ode at low peed. Value of the adaptve PI contolle gan of the two etato baed on Root-Locu ethod have been detened. The valdty of all popoed gan electon ha been vefed and aeed by ulaton and expeental eult. The eult have been hown an poveent n both the convegence and tablty n the egeneatve ode at low peed. Moeove, the popoed gan ae ot utable fo pactcal pleentaton and ake good pefoance. APPENDIX A. Lt of ybol L Magnetzng nductance L, L tato and oto elf nductance R tato etance T Roto te contant, ˆ Actual and Etated Roto peed Leakage coeffcent σ T = [ ], ˆ = [ ˆ ˆ ] T Actual and etated tato cuent d q d q vecto T λ = [ λd λq], ˆ [ ˆ ˆ T λ = λd λq] Actual and etated oto flux TABLE III Adaptve PI Gan of peed and tato Retance Etato peed Etato P 500 tato Retance Etato I 900 PR REFERENCE [] H. ubota,. Matue, and T. Nakano, DP-baed peed adaptve flux obeve of nducton oto, IEEE Tan. Ind. Appl., vol. 9, no., pp , Ma./Ap. 99. [] G. Yang and T. H. Chn, Adaptve-peed dentfcaton chee fo a vecto-contolled peed enole nvete-nducton oto dve, IEEE Tan. Ind. Appl., vol. 9, no. 4, pp , Jul./Aug. 99. [] Y. Ln and C. Chen, Adaptve peudoeduced-ode flux obeve fo peed enole feld-oented contol of IM, IEEE Tan. Ind. Electon., vol. 46, no. 5, pp , Octobe 999. [4] H. ugoto, One povng eaue of tablty n egeneaton opeaton of peed enole vecto contol nducton oto yte ung adaptve obeve of econday agnetc flux, n Poc. Int. Powe Electonc Conf. (IPEC), Tokyo, Japan, vol., pp , 000. [5] H. ubota, I. ato, Y. Taua,. Matue, H. Ohta, and Y. Ho, "Regeneatng-ode low-peed opeaton of enole nducton oto dve wth adaptve obeve", IEEE Tan. on Ind. Appl., vol. 8, no. 4, pp , Jul./Aug. 00. [6]. uwankawn and. angwongwanch, "Degn tategy of an adaptve full-ode obeve fo peed-enole nducton-oto dve-tackng pefoance and tablzaton", IEEE Tan. on Ind. Electon., vol. 5, no., pp. 96-9, Feb [7] M. Rahed and A. F. tonach, "A table back-ef MRA-baed enole low peed nducton oto dve nentve to tato etance vaaton", IEE Poc. Elect. Powe Appl., vol. 5, no. 6, pp , Nov [8]. uwankawn and. angwongwanch, "A peed-enole IM dve wth decouplng contol and tablty analy of peed etaton", IEEE Tan. on Ind. Electon., vol. 49, no., pp , Ap. 00. [9] M. aeja and. angwongwanch, "Aveagng analy appoach fo tablty analy of peed-enole nducton oto dve wth tato etance etaton", IEEE Tan. on Ind. Electon., vol. 5, no., pp. 6-77, Feb [0] M.. Zaky, M. M. hate,.. hokalla, and H. A. Yan, "Wde peed ange etaton wth onlne paaete dentfcaton chee of enole nducton oto dve", IEEE Tan. on Ind. Electon., vol. 56, Iue 5, pp , May