On the Analysis and Optimal Control Design of BLDC Drives

Transcription

1 Inernaional Journal of Elecrical Engineering. ISSN Volume 6, Number 5 (), pp.59-5 Inernaional Research ublicaionhouse hp:// On he Analysis and Opimal Conrol Design of BLDC Drives B. Bijukumar and V. Ramachandran Assisan rofessor, Elecrical and Elecronics Engineering, Baselios Mahews II College of Engineering, Kollam, Kerala-695. Head of Elecronics and Insrumenaion Engineering K S R College of echnology, iruchengode id:bkbijuzion@gmail.com Absrac Now a day s Brushless DC moor drives are becoming more popular in indusrial applicaions. 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. 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 opimal conrol signal which resuls in minimum performance inde. his paper spolighs he design of opimal conrol sysems for BLDC moor drive sysem based on quadraic performance indees. his sae-space approach o BLDC sysem analysis reduces he compleiy of he mahemaical epressions. 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. Key words: Brushless DC (BLDC) moor drives, Opimal Conrol, Sae Variables, erformance Inde, Conrol Variables INRODUCION Developmen of Brushless DC moor drives has yielded increases efficiency, power facor and reliabiliy. A BLDC machine has several inheren advanages over he machine ypes. 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.

2 5 B. Bijukumar and V. Ramachandran Modeling, simulaion and analysis of ermanen magne synchronous moor drive are described in [], []. A mahemaical model of MSM is given in []. A fourswich BLDC moor drive is implemened using digial signal processor in []. he precise speed conrol of a permanen magne synchronous moor is given in [5]. Ching-sai an and Emily Fang [7] proposed a LL-assised inernal model adjusable speed conroller for BLDC moor drives. R.Venkiaraman and B.Ramaswami [6] designed a curren conroller and speed conroller for BLDC drive sysem. An accurae predicion of he sysem behavior is obained wih he help of digial simulaion. 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, an opimal conrol sysem is designed for he digially WM conrolled BLDC moor drive sysem. For he opimizaion of conrol sysem, sae variables and conrol variables are formulaed in his paper. he BLDC performance characerisics close o he opimal conrol sysem is synhesized by he use of opimal conrol heory. he srucure of his paper is as follows. Secion describes abou he digial WM conrol of BLDC drives. he sae variable feedback of he BLDC machine is presened in secion. Secion 4 spolighs he opimal sysem wih conrol energy consideraions. Secion 5 discuses abou he quadraic opimal conrol in BLDC machine. Conclusion is given in secion 6. DIGIAL WM CONROL OF BLDC DRIVES 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 Τ em J B L d ω () 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 () dω I J Bω L (4) d where K orque consan and Iaverage curren

3 On he Analysis and Opimal Conrol Design of BLDC Drives 5 For he purpose of analysis, he digial conroller was considered equivalen o a proporional conroller wih high gain and sauraion. Fig. Block Diagram for Digial WM Conrol for a BLDC Moor Drive Sysem he ransfer funcion for a BLDC moor is given by [4], K ω( s) V ( s) JRa BLa BRa KKe s s (5) he sae variable equaion for his BLDC drive is given by, & (6) ( BRa k e ) ( JRa BLa) u & (7) Arranging in mari form we ge, & BRa k e JRa BLa & ( ) ( ) u he oupu equaion is given by, y [ ] (8) (9) his is in he form, & A Bu () y C ()

4 5 B. Bijukumar and V. Ramachandran Where A ( BRa k e ) ( JRa BLa) ; B ; C [ ] SAE VARIABLE FEEDBACK he sabiliy is a major problem in BLDC moor drive sysem. he design of a sae feedback BLDC moor conrol sysem is based on a suiable selecion of a feedback sysem srucure. If he sae variables are known, hen hey can be uilized o design a feedback conroller so ha he inpu becomes uk. 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 [], []. 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 ) ( JRa BLa k ) k e & () Arranging in mari form, we ge, & BRa k e k & which is in he form where, BRa ( ) ( JRa BLa k ) & H H k e ( k ) ( JRa BLa k ) (4) (5) Le k and deermine a suiable value for k so ha he performance inde is minimized.

5 On he Analysis and Opimal Conrol Design of BLDC Drives 5 o minimize he performance inde J, consider he following wo equaions, X Xd X ( ) X () H H I J (6) (7) ( BRa k e ) ( ) JRa BLa k (8) BRa k k k JRa BLa k e ( ) ( ) Compleing mari muliplicaion, addiion and solving, we obain, ( JRa BRa ) ( JRa BLa k ) k e (9) ( JRa BLa k ) ( BRa k e ) JRa ( JRa k e ) () ( BRa k e ) ( JRa BLa k )( BRa ) () J k e k e () o minimize as a funcion of k, J Se k herefore k JRaBLa ± ( JRaBLa) ( BRa k k k )( BRa k k k ) J Ra JRaBLa B La e e () (4) k k.499 (5)

6 54 B. Bijukumar and V. Ramachandran 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. OIMAL SYSEM WIH CONROL ENERGY CONSIDERAIONS his secion considers he ependiure of conrol signal energy for he BLDC drive sysem as shown in Fig.. o accoun for he ependiure of he opimal energy of he conrol signal he performance inde ( X IX u u) scalar weighing facor and I ideniy mari []. We will choose a feedback conrol sysem so ha, k u k ki k J λ d is chosen where λ is he (9) herefore he mari Q is, Q I k k I k I k I λ λ λ () Le X ( ) [ ] H H Q ( λk )I () J () o simplify he algebra wihou any loss, le k k k. & H ( BRa k ) ( ) k e k JRa BLa k ()

7 On he Analysis and Opimal Conrol Design of BLDC Drives 55 ( BRa ) ( JRa BLa ) ( BRa ) ( JRa BLa ) ( λk ) k e k e k k k k ( λk ) (4) Compleing mari muliplicaion, addiion and solving we obain, ( JRa BLa k )( λk ) ( BRa k e k )( λk ) ( BRa k e k ) ( JRa BLa k ) (5) ( λk ) (6) ( BRa k e k ) ( λk )( BRa k e k ) ( JRa BLa k )( BRa k e k ) (7) J (8) o obain minimum performance inde, se dj dk we ge, 5 4 ak bk ck dk ek f (9) where, 4 a 8k λ (4) 4 b 8JRaBLak λ 8BRa λ 8 k e λ 8 JRaλ BLaλ 8BRa k e λ λ (4)

8 56 B. Bijukumar and V. Ramachandran c 4 JRaBLa λ RaBk 4 k e JRa 4BJRa λ 4B LaRa λ ( Ra Rak ) ( ) k e ( BRak BRak ) 4BRak λ k e 4 λ J Ra B La BLak k e J RaLa B k e 4 d J Ra B La JRaBLa 8B ( ) k J Ra λ k B La λ k 8J Ra B B Ra k ke B Ra k ke B Ra λ k λ k ke 8B Ra k ke J La λ 8B 8B Ra k ke La k λ 8J Ra k BLa k λ 6J Ra k λ 6B La k λ 4 k B Ra λ 4 k λ ke J La λ k k e 6BRak λ 6k k λ 4Jk Ra λ 4 k B e La λ 8J Ra B ke J Ra k B La k λ La k B La λ k k e Ra k ke La λ J La k B Ra k k 4 ke k e Ra k ke Jλ 4B Ra k 4 ke Jλ 8B Ra k ke La λ 8B Ra k ke J La λ (4) (4)

9 On he Analysis and Opimal Conrol Design of BLDC Drives 57 e ( J Ra B La JRaBLa) ( 4BJRa λ 4B LaRaλ 4k ke Bk ke ) J λ 4 J λ 4J B Ra La k 4JB Ra La k 4B Ra k ke J La 4B Ra k ke B 4 Ra k ke ( B k ) Ra ke B Ra k ke 4J Ra La λ 4J Ra Bλ 4J BJ Ra λ k ke 4 La λ Ra 4B La Ra λ 4B La λ k ke k f ( J Ra B La JB Ra La) ( J ) ( B ) Ra k ( J ) La k k B La k ke B Ra k Ra k ke La B Ra ke k he BLDC drive sysem parameers are shown in able able : BLDC Drive arameers SL. NO. arameer Symb. Uni Value. Saor Winding Resisance Ra Ω.4 (44) (45). Saor Winding Inducance La H.66. Roor ineria J Kg-m Moor Viscous Fricion Coefficien B Nm/rad/sec orque Consan k Nm/Amp. 6. Velociy Consan ke Vols/rad.8 When λ Solving he above equaions we ge, k k.87k.9 k 5.6 (46)

10 58 B. Bijukumar and V. Ramachandran Applying Lin s Mehod he eqn.(46) becomes, k.998 k.868 k.6 4 k ( )( ) (47) QUADRAIC OIMAL CONROL his secion discusses abou 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) [], []. 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 (49) Le μ ( μ ) Q (5) ( B ) Ra k ke J La Ra B La J La ( J ) ( B ) ( J ) Ra k ke J La [] Ra B La J La μ (5) Solving we obain he following hree equaions, ( B R a k e) J J L L a a (5)

11 On he Analysis and Opimal Conrol Design of BLDC Drives 59 J La ( J ) ( ) R a J La B La ( J ) B R a J La k e R a B L a μ J L a J L a (5) (54) Solving hese hree equaions we ge, μ μ (55) he opimal feedback gain mari is obained as, k R B (56) k u [ ] μ k ( μ ).98 (57) (58) 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. Fig. Opimal Conrol of he BLDC Drive Sysem

12 5 B. Bijukumar and V. Ramachandran CONCLUSION In his paper a sae variable feedback sysem was designed for BLDC drive sysem o achieve he desired sysem response. Also, an opimal conrol sysem was designed for BLDC drive which resuls in a minimum value for he performance inde. o accoun for he ependiure of energy and resources, he conrol signal is ofen included in he performance inegral. From he foregoing analysis, he value of K is obained as.49 so ha he performance inde is minimized. he minimum value of performance inde is obained as.47. his opimal conrolled BLDC drive sysem resuls in a minimum value for he performance inde. Also, he conrol law given by equaion (58) yields opimal resul for any iniial sae under he given performance inde. 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 65-7, 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 [] 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.5, pp 4-49 Augus 9. [5] Chung-Feng Jeffrey Kuo, Chih- Hui Hsu recise Speed Conrol of a ermanen Magne Synchronous Moor, In. J Adv Manuf echnol. 8: , May 5. [6] R. Venkiaraman, B. Ramaswami, hyrisor Converer-Fed Synchronous Moor Drive, Elecric Machines and Elecromechanics, 6: 4-449, 98. [7] Ching-sai an, Emily Fang, A hase-locked-assised Inernal Model Adjusable-Speed Conroller for BLDC Moors, IEEE rans. on Indusrial Elecronics, Vol.55, No.9, pp45-45, 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: Conducion-angle conrol, in roc. IEEE In. Elec. Mach. Drives Conf., pp 8-4, May 5. [] Kasuhiko Ogaa, Modern Conrol Engineering (Inernaional Ediion) renice- Hall of India, New Delhi 988

13 On he Analysis and Opimal Conrol Design of BLDC Drives 5 [] Richard C.Dorf, Modern Conrol Sysems, Addison Wesley Longman Inc., USA, 998. [] A.H.Niasar, H.Moghbeli and A.vahedi, Implemenaion of Four Swich Brushless DC Moor Drive based on MSLF47 DS, IEEE In. Conf. on Signal rocessing and Communicaions, p.p -5 Nov. 7.

14 5 B. Bijukumar and V. Ramachandran Auhors Biography Bijukumar.B was born in Kerala, India in 985. He received he Diploma in Elecrical Engineering from Sae Board of echnical Educaion, Kerala, India in 5. hen he obained he B.ech degree in Elecrical and Elecronics Engineering from Mahama Gandhi Universiy, Idukki, Kerala in 8. He is pos graduaed in a Anna Universiy, Coimbaore. He had worked as lecurer in Cochin Universiy and Kerala Universiy in 9 and respecively. He is currenly working as an Assisan rofessor in he deparmen of EEE a Baselios Mahews II College of Engineering, Kerala from June onwards. His eaching and research ineress includes ower Elecronics, Conrol heory and Elecrical Drives. Venkaram Ramachandran received he B.E and M.E degrees in Elecrical Engineering from he Universiy of Madras, Madras, India in 97 and 974 respecively. He obained h.d in Indusrial Drives and Conrol in 8. He was wih he faculy of Elecrical Engineering a A.C College of Engineering and echnology, Karaikudi, India from 974 o 987. He joined as a Secion Grade Lecurer in 987. Since 994, he has been an Assisan rofessor in he Deparmen of Elecrical Engineering. His eaching and research ineress include ower Elecronics, Elecrical Drives and Conrol Sysems.