Optimal Speed Control of Induction Motor Based on Linear Quadratic Regulator Theory

Transcription

1 IEEE INDICON Opimal Speed Conrol of Inducion Moor Baed on Linear Quadraic Regulaor Theory Manoj Swargiary Deparmen of Elecrical Engineering, NIT Durgapur Durgapur 73209, India manojargu@gmail.com Abrac In hi paper, he opimal peed conrol, baed on Linear Quadraic Regulaor (LQR) heory, hrough aor field orienaion (SFO) of an inverer fed inducion moor i addreed. The volage ource inverer, behaving a an acuaor, i operaed uing pace vecor pule widh modulaion (SVPWM) echnique. The ae feedback yem i augmened wih inegral oupu error o enhance reference racking and diurbance rejecion crieria. In addiion, a feed forward gain i inroduced o enure uniy dc gain in cae of reference racking. The LQR ynhei i carried ou for he whole yem conidering he inegral oupu error a an addiional ae. I i inereing o noe ha he rucure of he conrol law ued here reemble he claical proporional inegral conrol law. Keyword Inducion moor (IM); Linear regulaor (LQR); Speed conrol; Vecor conrol /5/$ IEEE quadraic I. INTRODUCTION In recen year, he ue of inducion moor (IM) in indurie a variable peed ervo drive increae due o i co effecivene in inallaion a well a in mainenance. However, he mooh conrol of peed in IM i no ha eay a DC moor, which impede he ue IM drive. Wih he advancemen of vecor conrol echnique for IM drive, he peed conrol become relaively mooher. Among he differen vecor conrolled peed regulaion echnique of IM, he field oriened conrol (FOC) i recognized a he be. There are differen FOC echnique namely a aor field orienaion (SFO), roor field orienaion (RFO), air gap field orienaion (AFO) ec. In hi work, SFO i conidered a i i ineniive o he roor circui and alo aor reiance and curren can be eaily and mo accuraely meaured and o doe he eimaion of aor flux[]-[3]. In hi paper, aor field oriened peed conrol i accomplihed baed on he opimal conrol echnique, linear quadraic regulaor (LQR). To obain he field orienaion, LQR heory i implemened o an IM muli inpu muli oupu (MIMO) model in ynchronouly roaing reference frame. For racking of reference peed in condiion of diurbance and model uncerainy, inegral error i inroduced a an addiional ae. Then he LQR ynhei i carried ou for he whole yem along wih hi addiional Jayai Dey and Tapa Kumar Saha Deparmen of Elecrical Engineering, NIT Durgapur Durgapur 73209, India deybiwajayai@gmail.com and apaahani@gmail.com ae. Thi work i ynheized and imulaed in MAT LAB environmen. Thi paper i organized a follow. The mahemaical model of he IM i decribed in ecion II, wherea a brief decripion of propoed conrol cheme i preened in ecion III. Secion IV and V deal wih he reul and obervaional concluion, repecively. II. IM MIMO MODEL The mahemaical equaion of IM in he ynchronouly roaing ( d q) frame can be wrien a equaion () [2],[0]. F 3P ωr+ ( φdq i φqd i ) T l ωr J 4J J φ d ωeφq Ri d v + d d φq = ωφ e d Ri q+ v q d id τr ωr τ + τr φd φq id ωiq vd i q σl σl σ σl ωr τr τ + τr φd + φq ωid iq + vq σl σl σ σl () Where, v, v, φ, φ, i, i are he d q componen d q d q d q of aor volage, aor flux linkage and aor curren repecively and ωr, ωe, ω are he roor, ynchronou and lip peed repecively. The ymbol R and L are he aor reiance and inducance, wherea Rr and L r are ha of he roor circui. L m i magneizing inducance. The aor and roor conan γ, γ r and σ are given in equaion (2). R Rr L γ =, γr =, σ = (2) L L L L 2 m r r Taking x () a he ae vecor, u () a he inpu, y () a he oupu vecor and d () a he exernal diurbance o he

2 yem, he ae pace nonlinear MIMO model of he IM i conruced a follow. x () = f( x (), u (), d (),) y () = Cx () which i linearized abou he nominal operaing poin ( xn, un, d n ) o obain he linear model a given below. Where, x () = Ax () + Bu () + Ed () y () = Cx () x () = [ ωφ φ i i ] u () = [ ω v v ] y () = [ ωφ φ ] d () = T l r d q d q e d q r d q Where, A, B, C, E are he linear yem marix, conrol marix, oupu marix and diurbance marix, repecively and are obained following mall ignal approximaion echnique a follow. f (.) A = x ( x, u, d ) n n n F 3P 3P 3P 3P iqn idn φqn φdn J 4J 4J 4J 4J 0 0 ωen R 0 0 ωen 0 0 R =, τ r ωrn τ + τr φqn ω n σl σl σl σ ωrn τ r τ + τr φdn ωn σl σl σl σ φqn 0 φdn 0 f (.) B = = u, 0 0 ( xn, un, dn) σ L 0 0 σ L J f (.) 0 E = =, C = d 0 ( xn, un, dn) The nonlinear IM model in () i linearized around nominal operaing poin which are choen o be (3) (4) ω rn = 50 r/, φ dn = 2.4 Wb, φ qn = 0, idn = 6 A, i qn = 0, ω en = 57 r/ Wih he above aumpion he yem marix A, inpu marix B and he diurbance marix E ake he form A = , B = , E = III. CONTROLLER SYNTHESIS The ae feedback conrol can be achieved eiher by uing variou pole placemen echnique or by uing opimal quadraic error minimizaion echnique. In hi paper, he laer menioned echnique i implemened in he conex of linear quadraic regulaor heory. The advanage of uing quadraic error minimizaion echnique i ha, he conrol inpu i proporional o he quared error, hu if he error i high, minimizaion i faer or vice vera. To implemen LQR he performance index/crieria (PI) [9] i aken a ( ( ) J = x Qx ( ) + u ( ) Ru ( )) d LQR (5) Where, Q 0 and R 0 are he weigh marice for ae x () and conrol inpu u () repecively. Thee wo are uer defined variable and he dynamic performance of a yem can be adjued depending on hee wo weigh marice. The conrol inpu for he yem (4) i conidered o be. u () = Gx () + Vr () (6) Where G i he eady ae opimal feedback gain marix i given by G= R B P (7) 2

3 Where P i he eady ae oluion of he algebraic Riccai equaion The conroller rucure i a hown below: () P = Q+ AP () + P () A PBR () BP () (8) A feed forward gain, V, i inroduced o make he eady ae error zero [4] by mean of making command reference o oupu dc gain uniy. V can be derived from (4) and (6) a follow: From (4) and (6), x ( ) can be wrien a x () = ( A BG) x() + BVr() (9) And hu he ranfer marix from r () o y () can be wrien a Y() = C( I A + BG) BV (0) R () So, o make he DC gain uniy i.e, Y() lim = 0 R () V = ( C( A BG) B) () For diurbance rejecion and o ackle wih model uncerainy an inegral pah for oupu error i alo provided, ε () = (() r y()) d (2) I i inroduced a addiional ae and i augmened wih he ae pace IM model. The new augmened ae pace model become () () x A 0 x( ) B 0 = u () r () ε + + C 0 ε ( ) 0 I (3) Now, he conrol law for (3) wih repec o he performance index (5) can be wrien a x () u () = [ g g2] + Vr () ε() (4) From he obervaion of he conrol law, i can be aed ha hi conrol law reemble he claical proporional plu inegral conrol law wih g a proporional gain and g 2 a he inegral gain. Fig : Block diagram of he propoed conroller. IV. SIMULATION AND RESULTS The chemaic diagram of propoed work i hown in he Fig 2. The ae feedback gain ynhei and imulaion work i carried ou in MAT LAB environmen. The co-ordinae ranformaion angleθ e, for converion of elecrical quaniie from α β o d qcomponen i obained from he one of he conroller oupu, ω e, following he mahemaical expreion θ e = ω e d (5) A volage model baed eimaor i ued o eimae he wo aor flux componen in he aionery( α β ) reference frame from he equaion φ = ( v R i ) d (6) Where, φ = φ + jφ, v = v + jv, i = i + ji d q d q d q A MOSFET wo level volage ource inverer (VSI) i aken for acuaion. To examine he effecivene of he propoed conroller wo e are carried ou, one for diurbance rejecion and he oher i for he peed reference racking. For boh he e reference φd, φ q are e a 2.4 Wb and 0 Wb repecively. The weigh marice Q and R in he performance crieria for he augmened yem are choen a Q q r =, R = q r 8x8 3x

4 Where, q, q4, q5 are e a 00, wherea o make he flux repone faer correponding weigh q 2 and q 3 are e a 0000, he weigh aociaed wih he inegral error q6, q7, q 8 are e a. The weigh aociaed wih he conrol inpu r, r 2, r 3 are e a 50. The VSI inver he dc link volage which i e a 500V, baed on he α β componen of he conrol inpu volage, hrough a wo level SVPWM pule generaor wih wiching frequency 0 khz. The conrol cheme i hown below: A. Diurbance Rejecion Here mechanical load change i conidered a he only ource of diurbance. The load diurbance rejecion i an imporan conrol objecive ince here will be no pracical yem where he load o he inducion moor can be kep conan. For he e of diurbance rejecion of he propoed work, he e IM i operaed a 00 r/ peed reference keeping he aor fluxe φd & φq a heir reference value 2.4 Wb and 0 Wb, repecively. The moor i ared wih zero load orque and hen a 3 econd he load orque i uddenly increaed o 8 N-m and again a 6 load orque i reduced o 3 N-m. The repone i hown in Fig 4. I i oberved from Fig 3 ha he propoed conrol cheme achieve perfec diurbance rejecion wih a eling ime of 2. Fig 2: The propoed conrol cheme TABLE : MOTOR SPECIFICATION No of Pole 4 Power.5 kw Volage 380 V Operaing Frequency 50 Hz Saor/Roor Reiance 6.3/3.6 Ω Saor/Roor Inducance H Muual Inducance H Moor Ineria Nm 2 Fricion Coefficien Nm The repone wih no load condiion and peed reference 00r/ alongwih aor flux refrence φd & φq a 2.4 Wb and 0 Wb, repecively i hown in Fig 3. The conrol cheme i hown o work aifacorily wih a eling ime of 2. Fig 3: Nominal repone wih no load a 00r/ peed reference. 4

5 The peed curve ha undergone an underhoo and overhoo of approximaely 2% in boh he cae wih a udden increae and decreae of load orque, repecively. The flux φd & φ q are alo mainained a heir reference value 2.4 Wb and 0 Wb, repecively. B. Reference Tracking The main objecive of deigning a conroller i o obain a deired repone of he yem wih aking he phyical conrain ino conideraion. In he preen cae, operaing he moor a differen peed level and a conan d q fluxe i primary objecive along wih he diurbance rejecion. The reul are preened in he Fig 5. The moor i ared wih no load and he whole racking examinaion i performed under no load condiion.the reference peed o be racked i e iniially a 00 r/ and hen a 5 a ep change in reference peed from 00 r/ o 50 r/ i inroduced and again a 0 he reference peed i brough back o 00 r/. From he obervaion of he repone curve, i can be verified ha, he conroller i able o rack he reference command efficienly. Fig 4: Enlarged elecromagneic orque, peed, and flux V Time wih ep change in mechanical load. Fig 5: Enlarged Speed, flux and elecromagneic orque V Time wih ep change in peed reference. 5

6 The propoed conroller i hown o achieve perfec peed reference racking uccefully. I i inereing o noe ha he flux φ & φ have alo been recovered back o d q heir reference value 2.4 Wb and 0 Wb, repecively. V. CONCLUSION In hi paper, a full ae feedback conrol of IM MIMO yem i addreed baed on opimal LQR heory. The propoed conrol law mimic he claical proporionalinegral conrol law. Therefore, i ha he goodne of he claical proporional inegral conrol a well a he comfor of modern ime domain robu conrol. The effecivene of he propoed conroller i verified in MAT LAB imulaion environmen. The reul obained are found aifacory. Reference [] O.S. Ebrahim M.F. Salem P.K. Jain M.A. Badr "Applicaion of Linear Quadraic Regulaor Theory o he Saor Field-Oriened Conrol of Inducion Moor." IET Elecr. Power Appl., 200, Vol. 4, I. 8, Pp [2] M. Popecu, Inducion Moor Modelling for Vecor Conrol Purpoe, Helinki Univeriy of Technology, Laboraory of Elecromechanic, Repor, Epoo 2000, 44 P. [3] M. Negm, J. Bakhahwain, Shwehdi M.: "Speed Conrol Of A Three- Phae Inducion Moor Baed on Robu Opimal Preview Conrol Theory",IEEE Tran. Energy Conver., 2006,2, (), Pp [4] M. Ruderman, J. Kreek, F. Hoffmann, T. Berram "Opimal Sae Space Conrol of Dc Moo", Proceeding of The 7h World Congre The Inernaional Federaion Of Auomaic Conrol Seoul, Korea, July 6-, [5] Ch. Naga Koi Kumar, S.Sundeep, Ch.Durga Praad, "Opimal Conrol of Inducion Moor wih Kalman Filer" ICAESM -202 Pp [6] H. Broeck "Analyi And Realizaion of Pule Widh Modulaor Baed on Space Vecor", IEEE Tran. Ind Appl, 988,24,() Pp [7] Mohamed Bouak And Kamel Jarray, "A High Performance Senor Le Indirec Saor Flux Orienaion Conrol of Inducion Moor Drive", IEEE Tranacion on Indurial Elecronic, Vol. 53, No., February [8] Omanecu M., Xu L., Bazel T.: "Decoupled Curren Conrol Of Senorle Inducion-Moor Drive by Inegral Sliding Mode", IEEE Tran. Ind. Elecron., 2008,55, (), Pp [9] M. Gopal, "Modern Conrol Syem Theory", New Age Inernaional Publiher, Third Ediion 204. [0] B. K Boe, "Modern Power Elecronic And Ac Drive", Prenice Hall Pr,