Sliding Mode Speed Observer for the Induction Motor Drive with Different Sign Function Approximation Forms and Gain Adaptation

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1 Ukazuje ię od 99 oku a'3 Ogan Stowazyzenia Elektyków Polkich Wydawnictwo SIGA-O Sp. z o.o. Gzegoz ARCHAŁA, eea ORŁOWSKA-KOWALSKA Politechnika Wocławka, Intytut azyn, apędów i Poiaów Elektycznych Sliding ode Speed Obeve fo the Induction oto Dive with Diffeent Sign Function Appoiation Fo and Gain Adaptation Abtact. hi pape deal with ign function continuou appoiation fo influence on the pefoance of the claical Sliding ode Obeve (SO). Satuation and igoid function ae ued to educe the chatteing, intoduced due to the ign function uage. he influence of the filte tie contant i alo being widely conideed. Finally, two iple ethod of the obeve gain adaptation ae peented and decibed. Siulation and epeiental tet, obtained uing a DSP-baed yte, ae hown to illutate peented iue. Stezczenie. W atykule opiano wpływ zatoowania ciągłej apokyacji funkcji znaku na pacę klaycznego ślizgowego obewatoa pędkości. Funkcje naycenia oaz funkcja igoidalna użyte zotały w celu edukcji zjawika chatteingu, któy powtaje na kutek wykozytania funkcji znaku. Szeoko pzebadany zotał ównież wpływ tałej czaowej filtu. Zapezentowane zotały także dwie pote etody adaptacji wzocnienia obewatoa. Opiywane zagadnienia zotały zilutowane pzy poocy badań yulacyjnych i ekpeyentalnych. (Ślizgowy obewato pędkości dla napędu indukcyjnego z óżnyi funkcjai apokyującyi oaz adaptacją wzocnieni. Keywod: liding ode obeve, induction oto dive, chatteing, continuou appoiation. Słowa kluczowe: obewato ślizgowy, napęd indukcyjny, chatteing, ciągła apokyacja. Intoduction he induction oto dive due to thei low cot and opeational eliability ae becoing oe and oe coon in the induty. Pefect dynaic pefoance of the dive can be aued by one of the field-oiented contol ethod. Additional cable eduction can be aued by a peed etiation. Speed etiation can be alo applied in application whee the peed eno can be daaged []. he contol tuctue wa the fit liding ode application aea fo induction oto. he liding ode (S) a a ain pat of vaiable tuctue yte theoy wa fit popoed by Izoiov in []. hi theoy can be alo ued fo the oto/tato flu and peed obeve deign like [3]. Sliding ode Obeve (SO) allow obtaining eential feedback value in the vecto contol tuctue (e.g. DFOC o DC). Siple hadwae ipleentation, obutne ove pecified ange of oto paaete uncetaintie, ditubance and eaueent noie, eventually fat dynaic epone entail till gowing publication nube in the field of the S application. odel Refeence Adaptive Syte type SO i popoed in [4]. Sliding ode contol and obeve both woking iultaneouly wee intoduced in [5]. he adaptive [6], [7] and the econd-ode [8] appoache wee alo peented. A new obeve deigned in a ynchonou fae i intoduced in [9]. An ecellent oveview of eiting (till ) SO type fo diffeent oto can be found in []. Few publication wee publihed in ode to copae SO with diffeent etiation ethod odel Refeence Adaptive Syte (RAS) baed obeve in [], [], etended Kalan filte [3] and, togethe with Luenbege tyle obeve, in [4]. In ecent yea pecial conideation ha been alo focued on the I paaete etiation ethod utilizing the liding-ode appoach. In [5] tato and oto winding eitance etiation ethod i peented. Ieauable oto paaete can be alo etiated, a in [6], [7], alo with the econd-ode liding ode [8]. he Luenbege-liding ode type obeve wa popoed fo tato eitance and oto tie contant etiation [9]. All of the above-entioned olution take the advantage of the ign function in the S algoith. heefoe lowvaiable etiated ignal, i.e. oto peed o winding eitance paaete becoe the high fequency, dicontinuou value. Due to thee apid change, called the chatteing phenoenon, the equivalent value ut be etacted. hee ae any diffeent way to educe the chatteing in the field of the vaiable tuctue contol yte, what wa copaed in []. In cae of the etiation, a low-pa filte ee to be the iplet olution. Howeve, thee appeaed an idea of applying a continuou appoiation intead of the ign function in ode to eliinate o at leat to educe the chatteing. he atuation function wa fit popoed fo the nonlinea plant contol in [] and futhe invetigated in []. he igoid function ha been popoed in [3]. hee two idea can be applied in liding ode obeve theoy a well the atuation function o a igoid function can eplace the ign function uccefully. In liteatue any autho only efe to thi poibility, while thi pape tie to look oe inightfully into thee poble. he tandad SO [] will be ued to illutate decibed iue. he pape i oganized in eight chapte. he I atheatical odel i peented fit. hen, the peed obeve deign i intoduced. et, the filtation poble of the etiated peed i decibed, the filte tie contant PRZEGLĄD ELEKROECHICZY, ISS 33-97, R. 89 R a/3

2 influence i invetigated. Sign function continuou appoiation fo and iple, linea adaptation ethod ae peented aftewad. Load toque value influence on pefoance of the obeve and hot concluion ae hown at the end of the pape. All decibed iue ae illutated by iulation and epeiental tet. atheatical odel of the Induction oto atheatical odel of the induction oto i eential fo the Sliding ode Obeve deign, a the obeve i one of the algoithic peed etiation ethod. he odel of the quiel-cage induction oto can be deived uing the coonly-known auption [], [4] (3 phae yety, agnetizing lineaity etc.) in a tationay efeence fae and pe unit yte: voltage equation: d () u i () d i j cuent-flu equation: (3) i i (4) i i and the equation of otion: d e o (5) * (6) I i i i e whee: u =u + ju, i =i + ji, = +j pace vecto of the tato voltage, cuent and flu, i =i + ji, = +j pace vecto of the oto cuent and flu,,,,,, induction oto paaete: tato and oto winding eitance, eactance, agnetizing eactance and echanical tie contant,, e, o oto peed, electoagnetic and load toque, eult of the pe unit yte intoduction, =/(f ), f noinal fequency. oto noinal value, it paaete and bae value neceay to ake the conveion between phyical and pe unit yte ae hown in the Appendi. Speed and oto flu liding-ode etiato deign Peented Sliding ode Obeve (SO) i baed diectly on the I equation hown in peviou ection ()- (4). Roto flu etiation equation can be obtained cobining () and (4): d (7) i j he tato cuent vecto dynaic can be epeed fo (),(3) and (4), a follow: (8) d i u i i j In the obeve equation echanical peed i eplaced by the etiated one, and an additional vaiable i intoduced. hi fo a lightly odified veion of []. Etiated oto flu vecto: dˆ ˆ ˆ i j ˆ ˆ (9) whee: ^ indicate etiated quantitie. Speed etiation, ince it elie on the tato cuent etiation eo, deand the tato cuent vecto to be etiated, and thi can be ealized uing: () ˆ d i u ˆ i ˆ i ˆ ˆ ˆ ˆ j whee: total leakage facto= /( ). In the above equation (9), (), the etiated peed and the additional facto ˆ take the advantage of the ign function. he dicontinuou ignal of the peed can be calculated by: () ˆ K ign whee: K poitive contant. Speed witching function i a follow: iˆ i ˆ iˆ i ˆ () he auiliay vaiable can be intoduced to educe the eo caued by itaken deteination of the oto tie contant: (3) ˆ K ign whee: K poitive contant, and the witching function of the auiliay vaiable: iˆ i ˆ iˆ i ˆ (4) A the auiliay paaete i ued only in the SO tuning, thee i no need to eplace the ign function in (3) by it continuou appoiation. If the tato flu vecto coponent ae needed (e.g. in a DC tuctue) they can be etiated by: (5) i u Flu calculation Cuent-baed iulato (eq. 9) Cuent etiation (eq. ) ˆi ˆ ˆ i ˆ ˆ i f ( ˆ, i ) (eq. 5) Switching function, (eq.,3) ign(), at(), ig() ˆ, ˆ Speed calculation ˆ ( p), fil ˆ ( p) f p Low-pa filte Fig.. Block diaga of the peented liding-ode peed obeve he block diaga of the decibed SO i peented in Fig.. Vecto of tato cuent and voltage ae the input vaiable of the obeve. Voltage value ae etiated uing the DC-link voltage and the invete tanito poition (not hown in the figue). Stato o oto flu vecto and etiated peed ae the output of the obeve. Etiated, high-fequency ignal of peed and ae ued fo tuning the flu and cuent etiato. Howeve, thi high-fequency peed ignal i unuitable fo ue and analyi, theefoe it ut be filteed. ˆ ˆ ˆ,fil PRZEGLĄD ELEKROECHICZY, ISS 33-97, R. 89 R a/3

3 Filtation poble in S Obeve he dicontinuou peed etiation ignal () obtained uing the ign function:, whee (6) ign( ), whee, et, et, et, et f =. -.5 f = et et.5.5 f =. - et.5.5 f =.5 et.5.5 t [] i inconvenient a a cloe-loop feedback in the peed contol algoith (e.g. field-oiented ethod) a well a in diagnotic o fo field-weakening opeation (a the peed infoation i equied in the field-weakening algoith).. f = f = f = f = t [] Fig.. Speed etiation poce fo diffeent filte tie contant: eal and etiated peed, peed etiation eo, ign function, K =, epeiental eult he equivalent value of the eal peed ignal of the SO ut be etacted fo the dicontinuou ignal, uing a low-pa filte. he iplet filte to ue i the fit-ode tanfe function a follow: ˆ (7), fil ( p) ˆ ( p) p whee: f filte tie contant, the filte ettling tie =3 f and p Laplace opeato. he filte tie contant influence on the pefoance of the dive, duing epeiental tet, i hown in Fig.. It can be een, that the bigge the tie contant the alle the ocillation level. Howeve, the dynaical eo duing fat peed eveion becoe bigge. In uch ituation a copoie i neceay, taking into account the dynaic of the dive and it application. he ocillation level can be educed, fo a pecific tie contant, uing the ign function continuou appoiation. Sign function appoiation fo he ign function (6) can be eplaced with it appoiation a continuou fo. hee ae two diffeent appoache poible, i.e. atuation and igoid function. he atuation function can be decibed a follow: (8), whee at( ) ign( ), whee hee eit a geat vaiety of the igoid type function. Soe of the ae lited below (noalized in uch way that thei value ae between ±): (9 ig ( ) e f (9 ig ( ) tanh( / ) (9c) ig 3 ( ) actan (9d) ig 4 ( ) (9e) ig ( ) 5 Anothe paaete appea,, defining the lope of the at() and ig() function. ign(), at(), ig 4 () ig(), ign() - ign() ig 4 (), =.5 at(), = ig 4 () ig () ig () ig 5 () ign() ig 3 () Fig.3. Sign function continuou appoiation fo: a copaion of ign, at and one of the igoid function, diffeent igoid function fo the ae value of PRZEGLĄD ELEKROECHICZY, ISS 33-97, R. 89 R a/3 3

4 In Fig. 3a a copaion of the ign, atuation and igoid function i peented. Set of igoid function i peented in Fig. 3b. Depite the ignificant divegence between entioned chaacteitic fo the Fig. 3b, fo pecific value (diffeent fo each function) the function becoe vey iila. In uch ituation the peed obeve pefoance i pactically the ae fo each of the above function it can be hown in epeiental tet. If one fo of (9) i to be choen, the ig 4 () ee to be the optial olution coputation ae the iplet then. Siulation eult Speed etiation eult obtained duing iulation tet ae peented in Fig. 4. It can be een (Fig. 4 that the high fequency ignal ut be filteed, when the ign function i ued., et, et,fil et, ign() et - et,fil.5 t []..4. at() et.5 t []. Fig.4. Etiation poce duing iulation tet: uing ign function, uing atuation function Filteed etiated peed follow the eal peed alot eactly. Uage of the continuou appoiation, the at() function in thi cae, allow eliinating the filte copletely (Fig. 4. Howeve, thi ituation can be obeved only duing iple iulation tet. Epeiental tet Epeiental tet indicate clealy that in a eal dive coplete filte eliination i ipoible. It can be noticed in, et, et, et, et et =K ign( ), K = - et.5.5 et =K ig(,), =., K = - et.5.5 et =K ig(,), =.5, K = - et.5.5 et =K ig(,), =, K = - et.5.5 t [] et et et et Fig. 5. Ocillation of the etiated, non-filteed peed can be educed ignificantly (Fig. 5 uing the ign function appoiation. Howeve, the bigge lope of the igoid function,, the bigge dynaical eo appea. hi can be een clealy in Fig. 5c. Again, the choice of the paaete ut be a copoie between the ocillation level and dynaical eo. When copaing ign and igoid function with all paaete, it can be een that the aiu dynaical eo eain the ae while the ocillation level i educed. Laboatoy tial indicate alo that uing any type of the appoiation function the atuation o the igoid one with iila lope (a in Fig. 3, give alot the ae eult. Gain adaptation ethod fo liding-ode obeve It ee clea that fo a contant value of the obeve gain K in () the ocillation level of the etiated peed i the ae fo whole peed ange. It ean that fo all peed value the vibation becoe elatively lage. When K i et too high the dive can becoe even untable. Ocillation level of etiated value can be educed if the entioned gain i diffeent fo diffeent opeation point. One of poible iple ethod of K adapting i to change it with the efeence peed: ef () K K K whee: K, K poitive contant. he obeve opeation uing thi adaptation ethod i hown in Fig.6 (econd line). he undeied ituation can appea when thee eit a big diffeence between actual (etiated) and efeence peed, e.g. duing apid eveion, a afte about 5.5. A lage, hot etiation eo can be een in Fig. 6c (econd line). hi poble can be olved uing econd adaptation ethod. Alo, when thee i no infoation about the efeence peed e.g. in taction dive with diect toque contol without any oute contol loop, the paaete K can be adapted diectly with the etiated peed: et () K K K Speed etiation fo thi adaptation ethod i hown in Fig. 6 (thid line). hee i no longe lage etiation eo duing apid peed change. et et,fil et et,fil et et,fil et et,fil t [] t [] t [] t [] t [] Fig.5. Speed etiation fo ign function and igoid function (diffeent value of paaete ): eal and etiated peed, non-filteed and filteed etiated peed, c) peed etiation eo, epeiental eult, f =.5 c) 4 PRZEGLĄD ELEKROECHICZY, ISS 33-97, R. 89 R a/3

5 et =K ign( ), K =. c) et, - - et et =K ign( ), K =.+.* ef et - - et et,fil et, - - et et =K ign( ), K =.+.* et et - - et et,fil et, - - et t [] et - - et et,fil t [] t [] Fig.6. Speed etiation fo diffeent obeve gain adaptation ethod, fo contant gain (fit line), changed with the efeence (econd line) and etiated peed (thid line): eal and etiated peed, filteed and non-filteed peed, c) peed etiation eo, epeiental eult he adaptation contant K, K can be found quite eaily - K hould be et lightly above one and K hould be geate than zeo. he lat condition ut be fulfilled when the adaptation ethod () i applied when K = the peed etiation i unealizable. In Fig. 6 the diffeence between contant and adaptive gain appoache i hown. hi diffeence can be een epecially fo the low peed ange the geate the peed the oe iila the tanient becoe. It i clealy connected with the equation ()-(). Load toque influence Load toque influence on the claical Sliding ode Obeve opeation i peented in Fig. 7. wo diffeent ituation ae peented with noinal and without load toque (etiated electoagnetic toque i hown in Fig. 7c). In both cae etiated peed follow the eal peed alot ideally. he obeve wa teted uing the igoid function with the obeve gain adaptation accoding to the equation (). It can be een that the etiation eo (Fig. 7 i lightly bigge with noinal toque opeation than unde the idle unning. Analogou concluion can be ade fo the eaining apect of the obeve deigning filte tie contant and ign function appoiation paaete influence etiated peed ocillation becoe lightly bigge in cae of the full load toque opeation. et et,, et =K ig(,), =., K =.+.* et - et 3 et =K ig(,), =., K =.+.* et - 3 t [] et t [] c) et et e e t [] Fig.7. Load toque influence on the peed obeve opeation, full (fit line) and no load opeation (econd line), epectively: eal and etiated peed, peed etiation eo, c) etiated oto toque Concluion Seveal apect of the liding ode peed obeve opeation wee dicued in thi pape and illutated uing iulation and epeiental eult. Fit, the influence of diffeent filte tie contant on the peed etiation eo value wa hown. It wa hown that the bigge the tie contant the ocillation level of the etiation eo becoe alle. Howeve, thee appea iultaneouly a ignificant dynaic eo duing fat tanient. heefoe the choice of the tie contant ut be a copoie, taking into account the dynaic of the dive and the etiated peed ocillation. Second invetigated poble wa the influence of two iple ign function appoiation fo: atuation and igoid function on the pefoance of the etiato. hee i alot no diffeence in opeation of thee two function, howeve the choice of the lope paaete becoe an inteeting iue. Uage of the ign function appoiation with all allow to deceae the value of ocillation with the ae dynaical eo, copaing to the ign function. PRZEGLĄD ELEKROECHICZY, ISS 33-97, R. 89 R a/3 5

6 Howeve, togethe with inceaing lope the dynaical eo i augenting. It wa hown that thee i a poibility to eliinate the low pa filte of the etiated peed only duing iulation tet. Finally, two obeve gain adaptation ethod wee intoduced. Fit of the utilized linea adaptation with the efeence peed, howeve lage eo can appea when the eal peed i ignificantly diffeent than the efeence one. Second of the eliinated entioned dawback of the peviou ethod, and the obeve gain wee dependent on the etiated peed value. Appendi Bae value fo conveion fo the phyical [ph.u.] to pe unit yte ae hown in able. Induction oto paaete and it noinal value ae peented in able and able 3, epectively. echanical tie contant of the dive: () b J p p whee: J oent of inetia, p p pole pai, bae value ae peented in able 3. able. Bae Value Powe [kw] oque [] Speed [p] Voltage [V] Cuent [A] S b =3/U b I b b =p p S b b n b =6f /p p U b = U I b = I Fequency [Hz] Velocity [ad/] Flu [Wb] Ipedance [ f b =f b =f b U b b b U b b able. oto noinal value Powe oque Speed Voltage Cuent 3. [kw].46 [] 4 [p] 4 [V] 4. [A] Fequency Stato flu Roto flu 5 [Hz] 34.6 [Wb].8 [Wb].8.73 b able 3. Paaete of the I equivalent cicuit R R X X X 7. [] 5.4 [ 67.8 [ 9.8 [ 9.8 [ REFERECES [] Ołowka-Kowalka., Senole Induction oto Dive, Wocław Univeity of echnology Pe, (3). [] Izoiov D.B., atic B., Utkin V. et al., Application of liding ode in poble of electical achine contol Doklady Akadeii auk SSSR, 4 (978), n. 4, , (in Ruian). [3] Izoiov D.B., Sliding-ode onlinea State Obeve of an Induction oto, ocow: auka, (983), [4] Coanecu., Xu L.Y., Sliding-ode RAS peed etiato fo enole vecto contol of induction achine, IEEE an. on Ind. Elect., 53 (6), n., [5] Benchaib A., Rachid A., Audezet E. et al., Real-tie lidingode obeve and contol of an induction oto, IEEE an. on Ind. Elect., 46 (999), n., [6] uini., Petella R., Paailiti F., Adaptive liding-ode obeve fo peed-enole contol of induction oto, IEEE an. on Induty Application, 36 (), n.5, [7] Zheng Y.H., Fattah H.A.A., Lopao K. A., on-linea adaptive liding ode obeve-contolle chee fo induction oto, Int. Jounal of Adaptive Contol and Signal Poceing, 4 (), n.-3, [8] Solva S., Le V., Ghane. et al., Senole econd ode liding ode obeve fo induction oto, IEEE Int. Conf. on Contol Application, (), [9] Lacu C., Boldea I., Blaabjeg F., A cla of peed-enole liding-ode obeve fo high-pefoance induction oto dive, IEEE an. on Ind. Elect., 56 (9), n.9, []Yan Z., Utkin V., Sliding ode obeve fo electic achine - an oveview, 8th IECO Annual Conf. of the IEEE Ind. Elect. Society, (), [] Lin F.J., Wai R.J., Kuo R.H. et al., A copaative tudy of liding ode and odel efeence adaptive peed obeve fo induction oto dive, Electic Powe Syte Reeach, 44 (998), n.3, [] Khate.., Zaky.S., Yain H. et al., A copaative tudy of liding ode and odel Refeence Adaptive Speed obeve fo induction oto dive, th EPCO Int. iddle Eat Powe Syte Conf., (6), [3] Chen F., Dunnigan.W., Copaative tudy of a Sliding-ode Obeve and Kalan filte fo full tate etiation in an induction achine, IEE Poceeding - Electic Powe Application, 49 (), n., [4] Zhang Y., Zhao Z., Lu. et al., A copaative tudy of Luenbege Obeve, Sliding ode Obeve and Etended Kalan Filte fo enole vecto contol of induction oto dive, IEEE Enegy Conveion Conge and Epoition, (9), [5] Picadi C., Scibilia F., Sliding-ode obeve with eitance o peed adaptation fo field-oiented induction oto dive, 3nd IECO Annual Conf. on IEEE Ind. Elect., (6), [6] Poca A.B., Keyhani A., Sliding-ode flu obeve with online oto paaete etiation fo induction oto, IEEE an. on Ind. Elect., 54 (7), n., [7] Dediyok A., Yan Z., Guven. et al., A liding ode peed and oto tie contant obeve fo induction achine, 7th IECO Annual Conf. of the IEEE Ind. Elect. Society, (), [8] Rao S., Bu., and Utkin V., Siultaneou State and Paaete Etiation in Induction oto Uing Fit- and Second-Ode Sliding ode, IEEE an. on Ind. Elect., 56 (9), n.9, [9] ayee Haan S.., Huain I., A Luenbege-Sliding ode Obeve fo Online Paaete Etiation and Adaptation in High-Pefoance Induction oto Dive, IEEE an. on Induty Application, 45 (9), n., [] Sabanovic A., Vaiable Stuctue Syte With Sliding ode in otion Contol - A Suvey, IEEE an. on Ind. Infoatic, 7 (), n., -3. [] Slotine J.J., Saty S.S., acking contol of non-linea yte uing liding uface, with application to obot anipulato, Int. Jounal of Contol, 38 (983), n., [] Buton J.A., and Zinobe A. S., Continuou appoiation of vaiable tuctue contol, Int. Jounal of Syte Science, 7 (986), n.6, [3] Aboino G., Celentano G., Gaofalo F., Vaiable tuctue odel efeence adaptive contol yte, Int. Jounal of Contol, 39 (984), n.6, [4] Kaziekowki., Kihnan R., Blaabjeg F., Contol in Powe Elect. - Selected Poble, USA: Acadeic Pe An ipint of Elevie Science, (). 6 PRZEGLĄD ELEKROECHICZY, ISS 33-97, R. 89 R a/3