Nonlinear Control of an Induction Motor Using a Reduced-Order Extended Sliding Mode Observer for Rotor Flux and Speed Sensorless Estimation

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1 Engineeing, 00,, doi:0.436/eng Publihed Online Octobe 00 ( Nonlinea Contol of an Induction Moto Uing a Reduced-Ode Extended Sliding Mode Obeve fo Roto Flux and Speed Senole Etiation Abtact Olivie Aeu, Michel Abaka Kouacou *, Theophile Roch Oi, Zié Yéo, Malandon Koffi, Xuefang in-shi Dépateent Génie Electique et Electonique, Intitut National Polytechnique Houphouët Boigny (INP HB), BP 093 Yaououko Côte d Ivoie Univeité de yon, AMPERE, INSA yon, CNRS UMR 5005, Villeubanne 696, Fance E-ail: abakaci@yahoo.f Received July 3, 00; evied Augut 9, 00; accepted Augut, 00 Thi aticle popoe an innovative tategy to the poble of non-linea etiation of tate fo electical achine yte. Thi ethod allow the etiation of vaiable that ae difficult to acce o that ae iply ipoible to eaue. Thu, a copaed with a full-ode liding ode obeve, in ode to educe the execution tie of the etiation, a educed-ode dicete-tie Extended liding ode obeve i popoed fo on-line etiation of oto flux, peed and oto eitance in an induction oto uing a obut feedback lineaization contol. Siulation eult on Matlab-Siulink envionent fo a.8 kw induction oto ae peented to pove the effectivene and high obutne of the popoed nonlinea contol and obeve againt odeling uncetainty and eaueent noie. Keywo: Robut Nonlinea Contol, Induction Moto, Reduced-Ode Extended Sliding Mode Obeve, Paaete Etiation. Intoduction The induction oto (IM) becoe vey popula fo otion contol application due to it eaonable cot, iple and eliable contuction. Howeve, the contol of IM i poved vey difficult ince the dynaic yte ae non linea, the electic oto vaiable ae not uually eauable o the tanduce ae expenive (uch a toque, eed, flux tanduce) and the phyical paaete ae often ipeciely known o vaiable. Fo intance, the oto eitance dift with the tepeatue of the oto cuent fequency. Thi natually tuctue of non-linea and ultivaiable tate of IM odel induce the ue of the non-linea contol etho and in paticula the obut feedback lineaization tategy [-3] to peit a decoupling, aue a good dynaic pefoance and tability of the IM. Howeve, a vaiation of the oto eitance can induce a tate-pace coupling which can induce a degadation of the yte. In ode to achieve bette dynaic pefoance, an on-line etiation of oto fluxe, peed and oto eitance i neceay. An appoach popoed in [4,5] to etiate with ucce the tate vaiable in an IM i the ue of the full-ode Sliding Mode Obeve (SMO). Thi latte, built fo the dynaic odel of the IM by adding coecto gain with witching te, i ued to povide not only the uneauable tate vaiable etiation (oto fluxe and peed) but alo the etiation of the eauable paaete (tato cuent). Howeve the deteination of the eauable paaete etiation ipoe oe etiation algoith vey long and uually ophiticated with an inceae of the coputational volue. Theefoe, in ode to educe the accuacy and the coputation ate of the etiation algoith, the eaued paaete etiation i not neceay. Thu a Reduced-Ode Dicete-Tie Extended Sliding Mode Obeve (RDESMO) fo the IM i peented in thi pape to olve only and pecially the poble of the uneauable paaete etiation (oto fluxe, peed and oto tie contant). So afte a bief eview of the IM odel, a obut input-output lineaization and decoupling contol of induction oto i peented in Section. A educed-ode

2 84 dicete-tie Extended Sliding Mode Obeve (ESMO) i developed in Section 3. Section 4 decibe the iulation eult caied out on a.8 kw IM dive yte. Finally, concluion ae uaized in Section 5.. Induction Moto Model and Robut Feedback Contol By auing that the atuation of the agnetic pat and the hyteei phenoenon ae neglected, the claical dynaic odel of the IM in a (d, q) ynchonou efeence fae can be decibed by [6]: V V R. I R. I d dt d dt.. ; d q.. I.. I (.a) I I ; d I d q I q (.b) The load echanical equation i: J d f Ce C whee Ce p. ( d. q. ) p dt J (.c) The application of (.a) to (.c) etun a yte of fifth-ode non-linea diffeential equation, with a tate vaiable the tato cuent (I, I ), the oto fluxe ( d, q ) and the oto pulation ( ): x c fc( xc) gc. u whee x c = [I I d q ] t, u = [V V ] t.... d.. q.... d.. q fc( xc)... d l. q ;.. l. d. q p f p..( d. q. ). C.. J J J 0. 0 g. c () ; ( ) ( ). ; T T ; =.. Moeove, by chooing a otating efeence fae (d, q) o that the diection of axe d i alway coincident with the diection of the oto flux epeentative vecto (field oientation), it i well known that thi oto field oientation in a otating ynchonou efeence fae ealize: d = = Contant and q = 0 (3) Thu the dynaic odel of the IM, copleted with the output equation, can be ewitten a: x fx () gu. ; y =[h (x) h (x)] t = [ ] t with x = [I I ] t, u = [V V ] t f ( x)... p p.... C. J J 0. g f J ;. Fo the expeion () and (3), one can wite: (4) d.. dt l. with I p. C e. (5) Thi elation (5) how that the dynaic odel of the IM can be epeented a a non-linea function of the oto tie contant. A vaiation of thi paaete can induce, fo the IM, a lack of field oientation, pefoance and tability. Thu, to peeve the eliability, obutne pefoance and tability of the yte unde paaete vaiation (in paticula the oto tie contant vaiation) and ditubance, we can ue a obut feedback lineaization tategy to egulate the oto tate. A a atte of fact, we can ee that the yte (4) ha elative degee = = and can be tanfoed into a linea and contollable yte by choen [7]: a uitable change of coodinate given by:

3 85 z = h (x); z = f h (x); z 3 = h (x); z 4 = f h (x); the feedback lineaization contol having the following fo: g f g f f g f g f f h ( x) h ( x) v h ( x) u h ( x) h ( x) v h ( x) whee v and v ae the new input of the obtained decoupled yte and two obut contolle C() to povide a good egulation and convegence of the oto flux ( ) and peed ( ). On the othe hand, in ode to ipoe afte a cloed loop a econd ode dynaic behaviou defined by H(), the contolle C() can be choen by [8]: J(). H() C () ; J () ; J( ) ( t 0 ) H() whee the eal t 0 i an adjuting poitive paaete. The block diaga tuctue fo the contol of (, ) i a follow: ( Ref, Ref ) - C() Feedback algoith H() Futheoe, a the contol of an IM geneally equied the knowledge of the intantaneou flux of the oto that i not eauable, a full-ode SMO built fo the odel () by adding coecto gain with witching te i widely ued [7,9] with ucce fo on-line etiation at one and the ae tie of oto tie contant, fluxe, cuent o peed. The equivalent value of the witching function depen on the cuent eo given by the diffeence between the etiated cuent to thei eal o eaued value. Howeve, a the cuent ae aleady eauable, thei etiated value ae not theefoe neceay. Thu in the next ection, in ode to educe the execution tie of the obevation with epect to the oto tie contant vaiation, a RDESMO i popoed to povide only the uneauable paaete etiation (oto fluxe, oto tie contant and peed). And the witching te of thi educed obeve will be only function of the eauable paaete (voltage, cuent ). 3. Reduced-Ode Dicete-Tie Extended Sliding Mode Obeve 3. Reduced-Ode Sliding Mode Obeve (6) et u conide the dynaic odel of the IM given by u Moto odel the yte (). Aue that aong the tate vaiable, the tato cuent (I, I ) ae eauable, theefoe thei on-line etiation i not neceay. Thu fo the expeion (), in ode to etiate only the oto flux ( d, q ) and peed ( ), a educed dienional tate vecto defined by X = [ d q ]T =[x x x 3 ], T can be intoduced. The coeponding educed-ode tate pace equation becoe: x () t J x (), t () t Whee (t) = [I, I ]T i the new input.. xωl. x.. I J x(), t () t ωl. x. x..i p f p..( I. x I. x). C. x3. J J J (7) The fact that the tate vecto only conit of the oto flux and peed offe an advantage naely the eduction of the coputational volue and coplexity. Thu the oto flux and peed can be oe eaily and apidly etiated. Denote x, x and x the etiate of the fluxe 3 d, q and. The Reduced-ode SMO i a copy of the odel (7) by adding coecto gain with witching te: x σ. x ωl. x σ.. I Γ. Ι x ωl. xσ. x σ.. I Γ. Ι p f x p..( I. x I. x ). C. x Γ. Ι J J J whee, and 3 ae the obeve gain. The witching I i defined a ign( ) I ; S M. Z ; ign( ) (9).. x3 M. x 3. whee Z ~ i a function depending on the paaete eaueent (tato cuent, voltage ). Setting x x x, the etiation eo dynaic i given by: x σ. x ωl. x Γ. Ι x ωl. x σ. x Γ. Ι (0) f x3 p..( x. I x. I ). x3 Γ3. Ι. J J With the following obeve gain atice given by the expeion (), the etiation eo ~ x, ~ x, ~ x con- 3 (8)

4 86 vege to zeo. l n 0 l. ; 0 n 3. I. I p. p.. J. J () whee and n ae poitive adjuting paaete which play a citical ole in the tability and the velocity of the obeve convegence. 3.. Reduced-Ode Extended Sliding Mode Obeve In ode to etiate the oto tie contant, a educed dienional extended tate vecto defined by X e = [ d q ]T =[x x x 3 x 4 ] T ha been intoduced with = R /. The coeponding educed-ode extended tate pace equation becoe: x () t J x (), t () t whee e e e x4. xωl. x x4.. I ωl. x x4. x x4.. I Je x (), t () t p f p..( I. x I. x). C. x3. J J J () whee peent the low vaiation of. The popoed Reduced-ode ESMO i: x x4. x ωl. x x4.. I Γ. Ι x ωl. xx4. x x4.. I Γ. Ι p f x p..( I. x I. x ). C. x Γ. Ι x 4 Γ4. Ι J J J (3) whee i,, and 3 ae epectively defined by (9) and (). To deteine obeve gain 4, it can be uppoed that the obevation eo of the fluxe convege to zeo. The etiation eo of the fluxe x i xi x i 0 (i =,) ae then given by: 0 x4. x x4. x l. x. I. x4. I (4) 0. l x x4. x x4. x. I. x4. I By eplacing the expeion of and in (4), the etiation eo dynaic of the oto tie contant i given by: x. I x x I x x. I x We can ee that thi eo dynaic i locally and exponentially table by choen: T. I x I x with > 0 (5) The paaete i adjuted with epect to oto tie contant etiation. Finally, fo the expeion () and (5), it can be een that thee ae thee poitive adjuting gain:, n and which play a citical ole in the tability and the velocity of the obeve convegence. Thee thee adjuting paaete ut be choen o that the educed obeve atifie obutne popetie, global o local tability, good accuacy and conideable apidity. In ode to ipleent the educed-ode ESMO algoith in a DSP fo eal-tie application, the coeponding educed-dienion tate pace equation defined in () ut be dicetized uing Eule appoxiation (t ode). Thu the new dicete-tie vaying odel epeented by a function depending on the tato cuent i given by: xe ( k ) xe Te. J x ( ) ( ). ( ), ( ) e e k xe k Te q xe k v k ye xe with q xe, v. d l. q.. l. d. q.. ; p f p.. I. d I. q. C.. J J J 0 (6) whee k ean the kth apling tie, i.e. t = k.te with Te the adequate apling peiod choen without failing the tability and the accuacy of the dicete-tie odel. The popoed RDESMO can be defined by the following equation: x ( k ) x T. q x, v G. I (7) e e e e whee the pediction vecto i: xe ( k ) xe Te. qxe, v with x ( ) ( ) ( ) ( ) ( ) T e k d k q k k k The witching vecto I (k), deduced fo the continuou cae given by (9), can be witten a:

5 87 whee ign( ) I with ign( ) S Te. M. Z ( k ) l M, l (8) zd z d ( k ) zk ( ) zq ( k ) z q ( k ) et u intoduce the eaue vecto z (k) = [z d (k), z q (k)]t witten a follow: z T.. and d d d e q z T.. q q q e d Fo the electical Equation (.a) of the IM, an appoxiate (t ode) dicete-tie elation of the fluxe i given by: T ( ). ( ). ( ). e. zd k V k R I k I ( k ) I Te.. I and Te... zq ( k ) V R. I I ( k ) I Te.. I z d d ( k ) d T e.. q (9) zq ( k ) q ( k ) q Te.. d The popoed gain atix epeentation G(k), deduced fo the continuou cae given by () and (5), can be defined a follow (dicete-tie appoach): T. T. T.(.. I ) T.(.. I ) e e l Te. l T. e ( ) ( ) k. I. I.. 3 Te. p. Te p.. J J 4 e d e q G k Once the fluxe ae etiated, it i eay to deduce the etiated toque defined by: C ( ). ( ). ( ) e k p d k k q. () 4. Siulation Reult In ode to veify the feaibility of the popoed RDESMO, the iulation on SIMUINK fo Mathwok ha been caied out fo a.8 kw induction oto contolled with a obut lineaization via feedback algoith (Figue ). The noinal paaete of the induction oto ae given in the Table. The RDESMO i iplanted in a S_function uing C language. In ode to evaluate it pefoance and effectivene, the copaion between the obeved tate vaiable and the iulated one have been ealized fo eveal opeating condition with the peence of about 5% noie on the iulated cuent (I, I ) o peed. Thu, uing a apling peiod T e =, the iulation ae ealized at fit in the noinal cae with the noinal paaete of the induction oto (Table ) and then, in the econd cae, with 50% vaiation of the noinal oto (0) tie contant ( =.5 n) in ode to veify the oto tie contant tacking and flux etiation. Figue and Figue 3 how the iulation eult fo a tep input of the oto peed and flux. One can ee that in both noinal (Figue (a), (c)) o non-noinal (Figue 3(a ), 3(c )) cae, the etiated value of fluxe and toque convege vey well to thei iulated value. The obeved fluxe (Figue (a)) indicate the good oientation ( d i contant and q convege to zeo) which i due to a favoable oto tie contant etiation (Figue (b), 3(b )). The etiated toque (Figue (c)) i in good ageeent with the iulated value. Once the fluxe ae etiated, we can deduce the algoith of the feedback lineaization contol (Figue ). The wavefo how the good uncoupling between the oto flux and the peed becaue a tep vaiation in d (Figue (a) and Figue 3(a )) can not geneate a peed change (Figue (d) and Figue 3(d )). Thu the field oientation and the ynthei of obut lineaization and decoupling contol ae well veified. All thoe eult how the atifying tuning, the excellent pefoance of the obut decoupling contol and RDESMO againt oto eitance vaiation and petubation o noie.

6 88 dref ω Ref dob Roto Flux Contolle Roto peed Contolle ω Ob I I Reduced Ode V Dicete tie Extended Sliding V Mode Obeve (RDESMO) ω I I V ω Ob ineaization ω Ob ω Algoith ω V dob Ob σ Ob I I Toque etiation ω Ob I I ω l etiation Induction oto Model In the (d,q) ynchonou efeence fae noie C eob ω l ω Ob I I d ω ω Figue. Siulation chee of the yte. Figue. (a, b, c, d): Noinal cae (R = Rn). Figue 3. (a, b, c, d ): Non noinal cae (R =.5Rn).

7 89 Table. Noinal paaete of the induction oto. P n =.8 kw Un = 0/380 V In = 0.8/ A p = Fn = 50 Hz n = 40 p Jn = 0.5 kg.n/ fn = 0.05 N.. /ad R n = 5.7 R n =.475 n = H n = 0.6 H fn = H n = 0.6 H 5. Concluion We have hown in thi pape that a obut feedback lineaization tategy and RDESMO ae ued to peit a egulation and obevation fo the Induction oto tate in ode to aue a good dynaic pefoance and tability of the global yte. In ode to educe the obevation execution tie, thi RDESMO, baed on the full -ode SMO pinciple, peit only and pecially fo the econtuction of the paaete non eauable in an IM (the fluxe, peed and the oto tie contant etiation). The inteeting iulation eult obtained on the induction oto how the effectivene, the convegence and the tability of thi obut decoupling contol and RDESMO againt oto eitance vaiation, eaued noie and load. Thu, in ode to validate the obutne of thi non-linea contol and RDESMO, Expeiental eult on a teting bench fo a.8 kw induction oto will be peent in the next eeach poject. 6. Refeence [] K. B. Mohanty, N. K. De and A. Routay, Senole Contol of a ineaized and Decoupled Induction Moto Dive, National Powe Syte Confeence, NPSC, Khaagpu, India, Decebe 00, pp [3] R. Yazdanpanah, J. Soltani and G. R. A. Makadeh, Nonlinea Toque and Stato Flux Contolle fo Induction Moto Dive Baed on Adaptive Input-Output Feedback ineaization and Sliding Mode Contol, Enegy Conveion and anageent, Vol. 49, No. 4, 008, pp [4] P. A. Bogdan and A. Keyhani, Sliding-Mode Flux Obeve with Online Roto Paaete Etiation fo Induction Moto, IEEE Tanaction on Indutial Electonic, Vol. 54, No., Apil 007, pp [5] A. Dediyok, Speed-Senole Contol of Induction Moto Uing a Continuou Contol Appoach of Sliding-Mode and Flux Obeve, IEEE Tanaction on Indutial Electonic, Vol. 5, No. 4, Augut 005, pp [6] D. F. Benad and. Jean-Paul, Identification et obevation de actionneu électique, Vol. &, Hee, Pai, 007. [7] O. Aeu, Z. Yeo, M. Koffi, X. in-shi, C. T. Haba and G.. ou, Robut Decoupling Contol and Extended Sliding Mode Obeve fo an Induction Moto, New -Phy Che, Vol. 48, July 009, pp. -8. [8] J. C. Doyle, B. A. Fanci and A. R. Tannenbau, Feedback Contol Theoy, Maxwell MacMillan Intenat, New Yok, 99. [9] M. Tuini, R. Petella and F. Paailiti, Adaptive Sliding Mode Obeve fo Speed Senole Contol of Induction Moto, IEEE Tanaction on Induty Application, Vol. 36, No. 5, Septebe/Octobe 000, pp [] D. I. Kein, I. J. Ha and M. S. Ko, Contol of Induction Moto via Feedback ineaization with Input-Output Decoupling, Intenational Jounal of Contol, Vol. 5, No. 4, 989, p Noenclatue C e, C l : Electoagnetic and load toque, N.. f: fiction coefficient, N./ad. I, I, I : Stationay fae (d,q)-axi tato cuent and oto agnetizing cuent, A. J: inetia, kg..,,, f : oto, tato, utual and leakage inductance, H. p: pole pai nube. R, R : tato and oto efeed eitance,. T e, T, T : apling peiod, oto and tato tie contant (T = /R = /σ ; T = /R ),. V, V : Stationay fae d- and q-axi tato voltage, V. d, q : d-q coponent of oto fluxe, Wb., : d-q coponent of tato fluxe, Wb.,, l : tato, oto and lip pulation (o peed), ad/.