Speed Control of BLDC Motor by Using Tuned Linear Quadratic Regulator

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1 Inernaional Journal of Scienific Engineering and Research (IJSER) ISSN (Online): , Impac Facor (4): 3.5 Speed Conrol of BLDC Moor by Using uned Linear Quadraic Regulaor Vishnu C.S., Riya Mary Francis Deparmen of Elecrical and Elecronics Engineering, KM College of Engineering, Kollam rofessor, Deparmen of Elecrical and Elecronics Engineering, KM College of Engineering, Kollam Absrac: Brushless Direc Curren moors are one of he moor ypes rapidly gaining populariy. he major problem in BLDC drive sysem is ha some disurbances are originaed in he drive which will resuls in errors and reduces he sabiliy of he sysem. Convenional conroller is used o conrol he speed of he moor, bu he response of he sysem is affeced by seady sae error and represens a poor ransien reponse.o regulae he speed of he moor a desired speed is an imporan applicaion in auomoive indusries. So we use a Linear Quadraic Regulaor conroller o regulae he speed and posiion of he moor. he sae variables and conrol variables of he BLDC drive sysem are synhesized in his paper. he main objecive of his paper is o formulae he conrol law which resuls in minimum performance inde. his paper spolighs boh he design and simulaion of opimal conrol sysems for BLDC moor drive sysem. his opimal design will reduce he burden of edious compuaions in conrol engineers. his opimal design helps o realize he BLDC sysem wih pracical componens which will provide he desired operaing performance. Keywords: Brushless DC (BLDC) moor drives, Linear Quadraic Regulaor (LQR), Sae Variables, erformance Inde, Conrol Variables.. Inroducion Brushless Direc Curren (BLDC) are becoming prominen as he demand for efficiency, precise speed and orque conrol, reliabiliy and ruggedness increases. BLDC provide high efficiency and eemplary precision of conrol when compared o convenional moors. he mos imporan among hem are he lower mainenance due o he eliminaion of he mechanical commuaor and brushes [], []. hey are more efficien and have lower roor losses due o he absence of field windings. his drive can be used for variable speed applicaions like Elecrical Vehicles, Roboics ec. Modeling and simulaion of BLDC moor drive are described in [], []. A mahemaical model of MSM is given in [3]. A fuzzy ID conroller of BLDC moor drive is implemened Using digial signal processor in []. A phase locked observer is proposed o erac he speed and posiion of he moor [5]. ae-won chun [7] proposed a hyseresis comparaor o compensae he phase delay of he back emf consan. A novel digial conrol echnique for Brushless DC moor drives is given in [8], [9]. Anand Sahyan e.al [4] presened an FGA-based novel digial conrol scheme for BLDC moor drives. A ransfer funcion for he BLDC drive is derived in his paper. hey have no invesigaed an opimal conroller for he BLDC sysem. In his paper, a LQR regulaor sysem is designed for he digially WM conrolled BLDC moor drive sysem. he srucure of his paper is as follows. Secion describes abou he digial model of BLDC drives. he sae variable feedback of he BLDC machine is presened in secion 3. Secion 4 spolighs he design of LQR. Secion 5 discuses abou he uning of LQR. Simulaion resuls is given in secion 6.conclusion and fuure scope discuss in secion 7 and 8 respecively.some discussions on i. Secion VI is he Conclusion par.. Modelling of BLDC Moor he modelling of disillaion column can be fundamenal, empirical and hybrid modeling. he fundamenal modeling gives he complee idea abou he process dynamics, empirical one uses only inpu oupu daa and no informaion abou he inner dynamics. Bu he hybrid modeling combines he advanages of boh fundamenal and empirical modeling. he fundamenal modelling is commonly used which can be simulaed and undersand he column dynamics. he speed of a BLDC moor can be conrolled by changing he applied volage across he moor phases. his can be achieved by pulse ampliude modulaion, WM or hyseresis conrol. Anoher mehod of speed conrol involves sensor less echniques. An FGA-based novel digial WM conrol scheme for BLDC moor drives have been presened in [4]. Fig. shows he block diagram for digial WM conrol for a BLDC moor drive sysem. A conroller has been designed in his paper. he orque equaion is given by, dω Τem J B d L () Where em, ω(), B, J and L denoe elecromagneic orque, roor angular velociy, viscous fricion consan, roor momen of ineria and load orque respecively. em I () em I (3) dω I J B d L (4) where K = orque consan and I=average curren Volume 3 Issue 8, Augus 5 Licensed Under Creaive Commons Aribuion CC BY aper ID: IJSER of 4

2 Inernaional Journal of Scienific Engineering and Research (IJSER) ISSN (Online): , Impac Facor (4): 3.5 For he purpose of analysis, he digial conroller was considered equivalen o a proporional conroller wih high gain and sauraion. Figure : Block Diagram for Digial WM Conrol for a BLDC Moor Drive Sysem he ransfer funcion for a BLDC moor is given by [4], K ( s) V ( s) JRa B BRa KKe s s he sae variable equaion for his BLDC drive is given by, (6) BRa k e JRa B u (7) Arranging in mari form we ge, BRa k e JRa B k u (8) he oupu equaion is given by, y (9) his is in he form, A Bu () y C () C k e where A BRa JRa B B ; 3. Sae variable Feedback he design of a sae feedback BLDC moor conrol sysem is based on a suiable selecion of a feedback sysem srucure. he sabiliy of BLDC moor drive sysem is a major concern. If he sae variables are known, hen hey can be uilized o design a feedback conroller so ha he inpu becomes U=KX. I is necessary o measure and uilize he sae variables of he sysem in order o conrol he speed of he BLDC moor. his design approach of sae variable feedback conrol gives sufficien informaion abou he sabiliy of he BLDC drive sysem. he design of a feedback conrol sysem for BLDC drive using sae variables are discussed in his secion [], []. ; (5) Volume 3 Issue 8, Augus 5 Licensed Under Creaive Commons Aribuion CC BY he vecor differenial equaion of BLDC drive sysem is given in equaion (8). We will choose a feedback conrol sysem so ha, u( ) k k () hen he equaion (6) and (7) becomes, BRa k e k JRa B k (3) Arranging in mari form, we ge, which is in he form k e BRa k JRa B k A k ( A k) H (4) H k e (5) where, BRa k JRa B k Le k and deermine a suiable value for k so ha he performance inde is minimized. o minimize he performance inde J, consider he following wo equaions, (6) & (7) J X Xd X ( ) X () (6) H BRa k e JRa B k k e H I (7) BRa JRa B k (8) obain, JRa BRa k e JRa B k JRa B k BRa k e (9) JRa () JRa k e BRa k e JRa B k k e BRa k e () J () o minimize as a funcion of k, aper ID: IJSER of 4

3 herefore k JRaB Inernaional Journal of Scienific Engineering and Research (IJSER) ISSN (Online): , Impac Facor (4): 3.5 J Se k JRaB BRa k k k BRa k k k J Ra JRaB B e e (3) k (4) k.499 (5) J min.47 (6) he sysem mari H obained for he compensaed sysem is, H (7) he feedback conrol signal is obained as, u.5 (8) his compensaed sysem is considered o an opimal sysem which resuls in a minimum value for he performance inde. he simulaion of his compensaed sysem is lised below and shown in figure 5 & figure 6. he BLDC drive sysem parameers are shown in able able : BLDC Drive arameers SL. NO. arameer Symbol Uni Value. Saor Winding Ra Ω.4. Resisance Saor Winding H Inducance Roor ineria J Kg-m Moor Viscous Fricion Coefficien orque Consan Velociy Consan B Nm/rad/sec Nm/Amp Vols/rad B Ra k ke J J Ra B J k Ra k ke J B J Ra B J (3) Solving we obain he following hree equaions, B Ra k e (3) J J J J Ra B J B Ra k e J (33) J Ra B J J (34) Solving hese hree equaions we ge, (35) he opimal feedback gain mari is obained as, k R B (36) k (37) u k.98 (38) 4. Linear Quadraic Regulaor (LQR) his secion deals wih he design of a sable conrol sysem for BLDC drive based on quadraic performance indees. he main advanage of using he quadraic opimal conrol scheme is ha he sysem designed will be sable, ecep in he case where he sysem is no conrollable. he mari is deermined from he soluion of he mari Riccai equaion. his opimal conrol is called he Linear Quadraic Regulaor (LQR) [], []. Le assume =, he conrol law u =.98 his conrol signal yields an opimal resul for any iniial sae under he given performance inde. Figure () shows he block diagram for opimal conrol of he BLDC drive sysem. he opimal feedback gain mari k can be obained by solving he following Riccai equaion for a posiive-definie mari. A A BR B Q (9) Q (3) Le Volume 3 Issue 8, Augus 5 Licensed Under Creaive Commons Aribuion CC BY Figure : Opimal Conrol of he BLDC Drive Sysem aper ID: IJSER of 4

4 X X X Inernaional Journal of Scienific Engineering and Research (IJSER) ISSN (Online): , Impac Facor (4): uning of Q & R Mari in LQR.4 In LQR he cos funcion which is o minimized is (39) he wo marices Q and R are seleced by he design engineer by rial and error mehod. Generally speaking, selecing a large value for Q requires he value of J o be small. On he oher hand, selecing a large value for R, he conrol inpu u mus be smaller o keep value of J small One should selec value of Q o be posiive semidefinie and R o be posiive definie. his means ha he scalar quaniy is always posiive or zero a each ime. he Q & R mari is uned by rial & error mehod. he rial & mehod is done by MALAB coding s. he bes value of he Q & R mari is calculaed by checking he sep response of he sysem. he bes value of Q= & R=[] By uning Q & R mari he value of =.5 =.39,he conrol law U = (4) 6. Simulaion Resuls and Discussions MALAB sofware package is used o deermine he response of he sysem. uning of he Q & R mari is done by separae coding. he regulaion of speed and rae of change of speed is deermined wih and wihou uning of Q & R mari and he racking of he moor is also deermined. he Simulink model of he sysem shown in figure Figure 4: Regulaion of speed wihou uning Figure 5: Regulaion of rae of change of speed wih uning Figure 3: Regulaion of rae of change of speed wihou uning Figure 6: Regulaion of speed wih uning From figure 4 and figure 5 he sysem is regulaed a. sec, bu he sysem consiss of large no of overshoo and undershoo. he sysem is no precisely regulaed a his condiion. From figure 5 and 6 he sysem is regulaed a. sec wihou any oscillaions. he sysem is compleely conrolled and he racking of he moor is shown in figure 7.he moor is racked a raed speed a 5.5 sec. Volume 3 Issue 8, Augus 5 Licensed Under Creaive Commons Aribuion CC BY aper ID: IJSER of 4

5 speed (rpm) Inernaional Journal of Scienific Engineering and Research (IJSER) ISSN (Online): , Impac Facor (4): Figure 7: racking of BLDC moor a raed speed 7. Conclusion In his paper a sae variable feedback sysem was designed for BLDC drive sysem o achieve he desired sysem response. Also, an LQR sysem was designed for BLDC drive which resuls in a minimum value for he performance inde. he LQR design provides an opimal sae feedback conrol minimizes he quadraic sae error and conrol effor his opimal conrolled BLDC drive sysem resuls in a minimum value for he performance inde. Also, he conrol law given by equaion (4) yields opimal resul for any iniial sae under he given performance inde. Boh he ransien and seady sae response of he sysem is improved wih LQR conroller. his design based on he quadraic performance inde yields a sable conrol sysem for he BLDC drive sysem References [].illay and R.Krishnan, Modelling, Simulaion, and Analysis of ermanen Magne Drives, ar I. he ermanen Magne Synchronous Moor Drive IEEE rans. Indus. Applicaions, vol.5, No.,pp 65-73,Mar/Apr [].illay and R.Krishnan, Modelling, Simulaion, and Analysis of ermanen Magne Drives, II. he BLDC Moor Drive IEEE rans. Indus. Applicaions, vol.5, No.,pp 74-79,Mar/Apr [3] R.Krishnan, Elecric Moor Drives, Modelling, Analysis and Conrol, HI Learning rivae Limied, New Delhi, 9. [4] Anand Sahyan, N.Milivojevic,Y-J Lee and M.Krishnamoorhy, An FGA-Based Novel Digial WM Conrol Scheme for BLDC Moor Drives IEEE rans. on Indusrial Elecronics,Vol.35, pp Augus 9. [5] Liviu Ioan Iepure, Ion Boldea Hybrid I-f Saring and Observer-Based Sensorless Conrol of Single-hase BLDC-M Moor Drives, In. J Adv Manuf echnol. 8: , May 5. IEEE ransacions on indusrial elecronics, vol. 59, no. 9, sepember [6] R. Venkiaraman,B. Ramaswami, hyrisor Converer- Fed Synchronous Moor Drive, Elecric Machines and Elecromechanics, 6: , 98. Volume 3 Issue 8, Augus 5 Licensed Under Creaive Commons Aribuion CC BY [7] ae-won chun, Sensorless conrol of BLDC moor Drive for auomoive fuel pump using a hyseresis comparaor, IEEE rans. on Indusrial Elecronics, Vol.55,No.9, pp , Sep 8. [8] F.Rodigeuz,.Desai and A.Emadi, A Novel Digial Conrol echnique for rapezoidal Brushless Dc Moor Drives, in roc. Elecron, echnol. Conf., Chicago, IL, Nov. 4. [9] F.Rodigeuz, and A.Emadi, A Novel Digial Conrol echnique for Brushless Dc Moor Drives: Conducionangle conrol, in roc. IEEE In. Elec. Mach. Drives Conf., pp 38-34, May 5. [] Kasuhiko Ogaa, Modern Conrol Engineering (Inernaional Ediion) renice- Hall of India, New Delhi 988 [] Richard C.Dorf, Modern Conrol Sysems, Addison Wesley Longman Inc., USA, 998. [] R. Shanmugasundram, K. Muhammad Zakariahand,, N. Yadaiah Implemenaion and erformance Analysis of Digial Conrollers for Brushless DC Moor Drives, IEEE ransacions On Mecharonics, Vol. 9, No., February 4 Auhor rofile Vishnu C S was born in Kerala, India in 6//99. He received B.ech degree in Elecrical and Elecronics Engineering from Baselios Mahews II College of Engineering,Kollam,Kerala in 3.Currenly, he is pursuing his M ech degree in Indusrial Insrumenaion & Conrol from KM College of Enginnering Kollam. His research ineress include Conrol heory and Elecrical Drives. aper ID: IJSER of 4