he st Interntionl Symposium on Automtion nd Rootics in Construction nd Mining (ISARC 04) Nonliner Fltness-Bsed Controller for Permnent Mgnet-Excited Synchronous Motor Phm m hnh nd Nguyen D. ht Fculty of Electricl nd Electronics Engineering Vietnm Mritime University Vietnm Fculty of Engineering nd Informtion echnology Sydney NSW 007 Austrli E-mil: phmtmthnh@vimru.vn ht.nguyendinh@uts.edu.u Astrct: his pper ddresses the prolem of nonliner discrete-time fltness-sed controller design for Permnent Mgnet-Excited Synchronous Motor (PMSM). o eliminte the sttic errors of the system stte vriles nd consider the nonliner chrcteristics of PMSM cscded fltnesssed control scheme is proposed. Simultion results re provided to illustrte the effectiveness of the proposed control structures in terms of etter performnce. eyords: Fltness-sed control; Permnent Mgnet-Excited Synchronous Motor (PMSM); nonliner control; rel-time control. NOMENCAURE Symol Unit Description A B N S Mtrices of model f Nonliner function s r H Sttor rotor inductnce m H Mutul inductnce R s R r Ω Sttor rotor resistnce u u V dq components sttor voltge i i A dq components of sttor current s r s Sttor rotor time constnts H d xis q xis inductnce s rd/s Sttor circuit velocity Mechnicl rotor velocity rd Angle of flux orientted s coordinte system otl lekge fctor y p W Pole flux J kg m orque of inerti m W Nm od torque z p Numer of pole pirs Introduction With the dvntges of superior poer density high performnce motion control ith fst speed nd enhnced ccurcy Permnent Mgnet-Excited Synchronous Motors (PMSM) hve een incresingly used in rootics precision mchining nd mny utomtion processes. herefore the prolem of control design for PMSM hs received considerle ttention. Hoever there remin interesting questions s to ho to design controller so tht the sttic errors of the system stte vriles re minimized in hich the nonliner chrcteristics of the PMSM re tken into ccount. he concept of differentilly flt nonliner systems s first introduced y Fliess et l 99. he system is considered to e flt if set of outputs cn e found such tht ll sttes nd inputs cn e determined from these outputs ithout integrtion. he min purpose of the fltness-sed control method is first to design n open-loop nominl control corresponding to the predicted trjectory of the flt output. hen feedck control l is pplied to stilize the rel trjectory round the predicted trjectory of the flt output. he fltness sed control hs een recognized s promising method to del ith nonliner systems (see e.g. evine 009 nd references therein). Bsed on the comintion of the nturl energy dissiption properties of the permnent mgnet stepper motor system ith its differentil fltness property nonliner feedck controller s proposed in Rmirez 000. o minimize the copper loss t ll operting points of the PMSM hierrchicl fltness-sed control scheme s developed in Delleu et l. 004. In Dnnehl nd Fuchs 006 nonliner differentil fltness-sed control s proposed to the induction mchine fed y voltge source converter in hich the fltness-sed control s used for the inner current nd outer flux nd speed loops. Another pproch for drive systems ith elsticlly coupled lods s reported y homsen nd Fuchs 00. By using fuzzy logic technique to eliminte the effects of the time-vrying nonlinerities of n induction
AUOMAION AND CONRO motor fuzzy differentil fltness-sed controller s developed y Fn nd Zhng 0. In Houri et l. 0 ne differentil fltness-sed control method s presented for three-phse inverter ith n C filter. Recently qusi-continuous implementtion of structurl nonliner controller sed on directdecoupling for PMSM s reported in hnh nd Qung 0. It should e noted tht in the forementioned ppers the prolem of nonliner fltness-sed control hs not een fully investigted nd the minimiztion prolem of the system stte vrile sttic errors hs not received considerle ttention. hese fcts hve een motivted us to the present study. In this pper the prolem of nonliner discrete-time fltness-sed controller design for PMSM is investigted. By considering the nonliner chrcteristics of PMSM control strtegy sed on fltness theory is proposed to suppress the system stte sttic errors. Simultion results re given to illustrte the effectiveness of the proposed pproch. he pper is orgnized s follos. Section presents nonliner fltness-sed controller design method for PMSM. Simultion results for stndrd PMSM re given in Section. Section 4 concludes the pper. Nonliner fltness sed controller design for PMSM. Fltness sed control As mentioned in Fliess et l. 994 nd Fliess et l. 995 the min property of differentil fltness is tht the stte nd input vriles cn e directly expressed ithout integrting ny differentil eqution in terms of the flt output nd finite numer of its derivtives. herefore the trjectory of input cn e determined from desired trjectory of flt output. he generl fltness-sed control structure consists of nominl feedforrd controller comined ith feedck stilizing controller s shon in Figure. In this structure the feedck controller is crucil of importnce to compenste the effects of externl disturnces nd model uncertinties.. Fltness sed controller for PMSM o eliminte the system stte sttic errors nd consider the nonliner chrcteristics of PMSM cscde nonliner fltness-sed control structure is proposed in Figure. As shon in Figure the cscde control structure includes to loopss hich re coupled to ech other. he outer loop (speed loop) consists of proportinlintegrl controller (PI-controller) nd current Feedforrd lock hile the inner loop (current loop) contining nother PI lock comined ith voltge Feedforrd lock. Here the current controller for the current loop is first designed to gurntee tht i i i i sufficiently fst ith respect to the vritions of the desired trjectories hich ill e chieved y the mechnicl susystem (speed loop). hen speed controller is synthesized. It should e noted tht these PI locks re used to compenste the current nd speed sttic errors.... Sttor current controller Motivted y Qung nd Dittrich 008 e consider the continuous-time model current of PMSM s follos ìï ïx =() f x H()=() xu f x h() xu h() xu h() xu í () ï ïî y= g( x) here - Stte vector: x= [ x x x ] = éi i J ê s ú - Input vector: u= [ u u u ] = éu u ê s ú - Output vector: y= [ y y y] = [ x x x ] [ ] [ ] f( x) =-c x - d x 0 () H ( x) = h ( x) h ( x) h ( x) () [ ] [ ] h ( x) = 0 0 ; h ( x) = 0 0 é = y ê - - p ú h ( x) x x [ ] [ ] gx ( ) = g( x) g( x) g( x) = x x x (4) nd temporry prmeters: = ; = ; c= ; d =. Note tht the functions f(.) nd H(.) in eqution () re nonliner in nture the ordinry differentil eqution () cnnot e solved exctly nd hence the exct form of the discrete-time differentil eqution is difficult to otin. herefore to otin the discrete-time current model of PMSM ylor s series expnsion is used x( k ) = x( k) x( t) X( ) t= k (5)
he st Interntionl Symposium on Automtion nd Rootics in Construction nd Mining (ISARC 04) l dy d y u Q y... l dt dt dx f xu dt l du d u y F xu... l dt dt Figure. he generl fltness-sed control structure i _ ref i ref i _ f i j s e i _ ff s m W j s e s Figure. Cscded control structure of fltness-sed control of PMSM here is smpling period nd X ( ) is the higherorder terms of the ylor s series expnsion hich cn e expressed s follos ï ìï i ( k ) = (- c) i ( k) u ( k) i ( k) ( k) s íi ( k ) = (- d) i ( k) u ( k) - i ( k) ( k) -y ( k) s p s () ( n) n X ( ) = x ( k )... x ( k ) J ( k ) = J ( k) ( k) s s s ï! n! ïî (6) (8) ( n ) n x ( z) z Î ( k k ). ( n )! here = ; = ; c= ; d =. As the smpling period in the dvnced electric drive systems is very smll the higher-order terms in Note tht the nonliner chrcteristics in the current eqution (6) cn therefore e neglected. By sustituting model of PMSM (8) is considered in terms of the () into (5) the discrete-time current model of PMSM is products eteen the stte vriles (current otined s components i ( k ) i ( k )) nd input vrile ( ( k) ). q s Fom (8) y using the property of differentil fltness x( k ) = x( k) f( x( k)) H( x( k)) u( k) X( ) (7) heres the trjectory inputs u ( k ) nd u ( k ) _ ff _ ff yk ( ) = gxk ( ( )). cn e directly determined from desired trjectory of Eqution (7) cn e reritten in the form of flt outputs i ( k ) i ( k ) nd ( k). Controllers for s voltge feedforrd lock is proposed s
AUOMAION AND CONRO é u ( k) = i ( k ) -(-c) i ( k) - i ( k) ( k) _ ff s ê ú é u k i k d i k i k k k ê ú ( ) = ( ) -(- ) ( ) ( ) ( ) y ( ). _ ff s p s (9) From (9) it cn e seen tht the coupling effects cused y decoupling current component i nd i cn e eliminted y - i ( k ) ( k ) nd s i ( ) ( ) k s k terms. Denote the inding time constnts = nd Rs = the control prmeters of feedck controller Rs (PI controller) in the current loop is therefore determined s = - R = p_ d s i_ d e e d d = - R = p_ q s i_ q e e q q here e e re positive sclrs such tht d q e > 0 e > 0. d q... Speed controller he motion eqution of PMSM is considered s d J = z éy i ( - ) i -m dt ê ú p p W. () Similrly y dopting the discretiztion pproximtion (5) the discrete-time speed model of the PMSM is otined s follos J [ ( k ) - 4 ( k- ) ( k- ) ] = = z éy i ( k) ( - ) i ( k) -m. p p W ê ú () Bsed on the property of differentil fltness the controller of the current feedforrd lock is otined s J é ( k ) 4 ( k ) ( k ) mw ( ) - - - i k = _ ff z éy i ( k) p p ( - ) ê ú (4) nd feedck controller (PI controller) i ( k) = i ( k- ) r é ( k) ( k) _ f _ f 0 - r é ( k-) -( k-). 0 Finlly the speed controller cn e otined s (5) i ( k) = i ( k) i ( k) (6) _ ff _ f J é ( k ) 4 ( k ) ( k ) mw ( ) - - - i k = z éy i ( k) p p ( ) ê - ú i ( k- ) r é ( k) ( k) r é ( k ) ( k ) _ f 0 - - - - 0 (7) here hus the current controller is otined s i _ r =. 0 p_ s u ( k) u ( k ) r éi ( k) i ( k) r éi ( k ) i ( k ) = - _ f _ f 0 i _ d - i _ d - - - i _ u ( k) = u ( k- ) r éi ( k) - i ( k) r éi ( k-) -i ( k-) _ f _ f 0 i _ q i _ q r =. - s p_ (0) nd re respectively the proportionl gin p_ i_ here nd integrl gin of PI controller in the speed loop. i_ d i_ q r =. ; r =. 0 i_ d p_ d si 0 i_ q p_ q si () i_ d i_ q r =. - ; r =. - Simultion results _ i d si p_ d _ i q si p_ q nd si is the smpling time for current loop. In this ork to verify the effectiveness of the proposed control strtegy e consider stndrd PMSM ith the folloing prmeters s shon in le. le. PMSM prmeters Prmeter Vlue Rted output poer P rted 04kW Rted voltge U rted 0V Rted current.7a Instntneous pek current 8.A Rted torque.7nm Numer of poles p p 8
he st Interntionl Symposium on Automtion nd Rootics in Construction nd Mining (ISARC 04) Rted rotted speed 000rpm Sttor resistnce R s.5 Equivlent inerti J 000kgm Sttor inductnce s 065H Severl scenrios ere considered for simultion to ssess the cpility of the proposed controller. Here the speed of PMSM ill e controlled ith constnt lod nd this lod ill e involved fter 0.5sec. Cse : he reference speed is set t 57. (rd/s)-forrd speed nd -57. (rd/s)-reverse speed nd the motor ill reverse rottion t 0.s. he constnt lod involves t 0.5sec. Simultion results re shon in Fig -8. Cse : he reference speed is set from 57. (rd/s) to 4. (rd/s) nd the motor ill reverse rottion t 0.7s. he constnt lod lso involves t 0.5sec. Simultion results re given in Fig 9-4. Figure 9. Speed Response - Sttor Current Response error_omeg[rd/s].0 error_omeg.8.6.4..0 0.8 0.6 0.4 0. -0. -0.6-0.8 -.0 0. 0.4 0.6 0.8.0 Figure 0. Speed Error 0.4 error_i 0. Figure. Speed Response.0 error_omeg.8.6.4..0 0.8 0.6 0.4 0. -0. -0.6-0.8 -.0 0. 0.4 0.6 0.8.0 error_omeg[rd/s] Figure 4. Speed Error Figure. Response of component i nd i error_i[a] -0. -0.6-0.8 -.0 -. -.4 0. 0.4 0.6 0.8.0 Figure. Error of i error_i[a] 0.6 0.5 0.4 0. 0. 0. -0. -0. error_i error_i[a] 6 5 4 0 - - - -4-5 error_i Figure 5. Current Response -0. 0. 0.4 0.6 0.8.0 Figure 6. Error of i Figure. Response of hree-phse current -6 0. 0.4 0.6 0.8.0 Figure 4. Error of i Figure 7. Response hree-phse Current error_i[a] Figure 8. Error of i It cn e seen tht the motor speed trcked the desired speed fter 0. sec. he trcking error curves for oth cses re depicted in Fig 4 6 8 0 4. It is pprent tht the trcking performnce of the proposed method in this study is good hen the sttic errors including the speed nd current errors converge into zero fter sec. It is lso oserved tht the trcking errors re still ithin n cceptle level even hen reversing the rottion nd strting-up ith constnt lod.
AUOMAION AND CONRO 4 Conclusion In this pper the prolem of nonliner fltnesssed controller design of PMSM hs een ddressed. Bsed on ylor series expnsion nd y using the property of differentil fltness to controllers of the current nd speed loops re proposed to eliminte the sttic errors. Simultion results re provided to illustrte the fesiility of the proposed pproch. References [] E.C. Anene U.O. Aliyu J. evine nd G.. Venygmoorthy. Fltness-sed feedck lineriztion of synchronous mchine model ith sttic excittion nd fst turine vlving. IEEE Poer Engineering Society Generl Meeting mp F pp. -6 007. [] E. Delleu nd A.M. Stnkovic Fltness-sed hierrchicl control of the PM synchronous motor. Proceeding of the 004 Americn Control Conference Boston Msschusetts pp. 65-70 004. [] H.S. Rmirez. A pssivity plus fltness controller for the permnent mgnet stepper motor. Asin Journl of Control vol. no. pp. -9 000. [4] J. Dnnehl nd F.W. Fuchs. Fltness-sed control of n Induction Mchine Fed vi Voltge Source Inverter-concept Control Design nd Performnce Anlysis. IEEE Industril Electronics IECON 006- nd Annul Conference Pris pp. 55-50 006. [5] J. evine. Anlysis nd Control of Nonliner Systems: A Fltness-sed Approch. Springer 009. [6] M. Fliess J. evine P. Mrtin nd P. Rouchon. Fltness nd defect of nonliner systems: introductory theory nd exmples. CAS internl report A-84 994. [7] M. Fliess J. evine P. Mrtin nd P. Rouchon Fltness nd defect of non-liner systems: Introductory theory nd exmples. Interntionl Journl of Control vol. 6 no. 6 pp. 7 6 995. [8] P. Mrtin R.M. Murry nd P. Rouchon. Fltness systems equivlence nd trjectory genertion CDS echnicl report CDS 00-008 00. [9] N.P. Qung nd J.A. Dittrich. Vector control of three-phse AC mchines: system development in the prctice. Springer 008. [0] S. homson nd F.W. Fuchs. Fltness sed control of drive systems ith resonnt lods. he 6 th Annul Conference on IEEE Industril Electronics Society Glendle AZ pp. 0-5 00. []. Fn nd. Zhng. Fuzzy sed fltness control of n induction motor. Procedi Engineering vol. pp. 7-76 0. [] M. Fliess J. evine P. Mrtin nd P. Rouchon. On differentilly flt nonliner systems. In Proceedings of the IFAC-Symposium on Nonliner Control Systems Bordeux pp. 408-4 99. [] A. Houri H. Renudineu nd J. Mrtin. Fltness sed control of three phse inverter ith output C filter. IEEE rnsctions on Industril Electronics pp. 890-897 0. [4] Z.H. Qu Roust Control of Nonliner Uncertin Systems Wiley: Ne York 998. [5] H. het nd M. Aydi. Fltness-sed trjectory genertion for induction mchine control. Interntionl Conference on Electricl Engineering nd Softre Applictions Hmmet pp. -6 0. [6] P.. hnh nd N.P. Qung. Qusi-continuous implementtion of structurl nonliner controller sed on direct-decoupling for Permnent Mgnet Synchronous Motor. IEEE Interntionl Conference on Control Automtion nd Informtion Sciences Nh rng Viet Nm pp. 54-59 0.