Stability Analysis of a Sliding-Mode Speed Observer during Transient State

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1 Poceedng of he 5h WA In. Conf. on Inuenaon Meaueen Ccu and ye Hangzhou Chna Apl (pp35-40 ably Analy of a ldng-mode peed Obeve dung anen ae WIO ANGUNGONG AAWU UJIJON chool of leccal ngneeng Inue of ngneeng uanaee Unvey of echnology Unvey Avenue Muang Dc Nahon achaa HAILAND hp:// Abac: - he ecen adapve ldng-ode peed obeve [] able n he Lyapunov ene unde a conan peed condon. Dung anen ae h obeve ay becoe oenaly unable becaue echancal dynac could ponenly appea n he e devave of Lyapunov funcon (V. In effec V ay be ehe pove o negave. A feable analy of he anen ably of h obeve o deene wo poan oluon accodng o he quadac nequaly concenng angula acceleaon of he oo. A a conequence he heoe of Laalle nvaan e wa eployed o explan ably cenao nce po endng of anen ae up o eady ae. oe ulaon eul ae hown o ndcae whehe h obeve able dung he anen ae. Key-Wod: - Inducon oo; peed obeve; anen ae; Quadac nequaly; Invaan e; Lyapunov funcon Noenclaue B f eq Vcou fcon coeffcen (N /ad ao o upply fequency (Hz quvalen cuen n popoon o elecoagnec oque ao cuen veco α β α and β coponen of ao cuen (A J Moen of nea (g K uface gan ax Inegal gan p p p p Popoonal gan K oque conan L oo elf-nducance (H L ao elf-nducance (H M Muual nducance (H p Nube of pole Coe lo eance (Ω oo eance (Ω ao eance (Ω lp uface veco e lecoagnec oque (N L Load oque (N U o Coecon veco V Lyapunov o cala funcon v ao volage veco α β Coponen n fxed ao coodnae σ oal leaage faco oo flux lnage veco α β α and β coponen of oo flux (Wb ao angula velocy (ad/ec Mechancal haf peed (ad/ec o p leccal oo (angula peed (ad/ec oo (angula acceleaon o deceleaon (ad/ec Angula velocy of lp (ad/ec l Inoducon he ecen developen of an adapve ldng-ode peed obeve [] povde a paccal obeve ha able n he Lyapunov ene unde alo conan peed of oo evoluon. Dung anen ae he oo acceleaon ( no zeo and ably ha no ye been guaaneed. he obeve hu uable fo eady-ae opeaon uch a pape ll ubbe exuon ec. h laon could be ovecoe f he obeve ably n anen ae wee guaaneed. Whou h l he applcaon of he obeve fo evo conol would be poble. h acle nvegae anen ably of he adapve ldng-ode obeve va Laalle heoe. Fly decpon of elecng a Lyapunov funcon fo he PI adapve law paed. econdly dealed nvegaon and dcuon on anen ably of he obeve va an nvaan e ae peened. hdly ulaon eul ae peened o vefy he cla.

2 Poceedng of he 5h WA In. Conf. on Inuenaon Meaueen Ccu and ye Hangzhou Chna Apl (pp35-40 Lyapunov Funcon fo PI Adapve Law In he eale wo [] an poveen of paaee eaon nde he adapve ldng-ode peed obeve wa accoplhed by fou popoonal plu negal (PI adapve law nead of olay negal eleen. In ode o enue ably a Lyapunov funcon canddae wll be choen and exaned. Ou peed obeve fo a hee phae nducon oo (IM ang coe-lo no accoun wen a follow: ( d î d  ˆ ˆ î  Bv Dˆ Uo ( ( dˆ d Â î ˆ ˆ  Dˆ ( whee he eanng of each e gven n Appendx. Fou PI adapve law unnng paallel o he obeve concuenly eae ao oo and coe-lo eance and oo angula peed of he nducon oo. hee PI law ae denoed copacly a ˆ ˆ ˆ p (3 p p p (4 (5 ˆ (6 whee ( î î ( ˆ î ( ˆ Mî ( ŝ ˆ ( L ŝm ˆ ( ˆ ˆ J I L and whole PI gan u be only pove value ( p p p p > 0 and > 0. When an negal ype of fou adapve echan eplaced by he PI poon an enhanced Lyapunov funcon eleced and wen n V L σl ( ( ( ( 0 p p p ML p.(7 Fo plfyng he devave of h Lyapunov funcon a poduc [] beween he anpoe of he uface veco and he devave of he ae wh epec o e expeed a σl {( A K e ( A D e U } L ML o.(8 ubequenly dffeenang he Lyapunov funcon along e bng foh V σl L ML ( ( ( ( p p p p p p p p p p p p.(9 By ubung he poduc n q. (8 and fou adapve law n q. (3 o (6 no q. (9 and aung ha and ae alo conan n copaon wh ye dynac of ae vaable hen V uncaed no a hoe fo a {( A K e ( A D e U } V o p p p (0 p 0 σl L ML whee ( A K e ( A D { e U } 0 o. q. (7 and q. (0 gnfy ha f an nducon oo conneced n algnen wh load oae a a conan haf peed he peed obeve alway and uually ean able becaue V negave edefne. Dung anen ae he oo angula acceleaon o deceleaon ( no zeo he

3 Poceedng of he 5h WA In. Conf. on Inuenaon Meaueen Ccu and ye Hangzhou Chna Apl (pp35-40 obeve ay o ay no be able n he Lyapunov ene becaue V ay be pove o negave. he nex econ nvegae he obeve ably dung anen ae. 3 Invegaon egadng ably dung anen ae Wheneve he ae of change of he oo peed vae ye echancal dynac wll nfluence V. h yeld V p σl {( A K e ( A D e U } p L ( p p ML and he oo peed expeed by J pk p o ( eq L ( B B B whee eq β α α β and K ( 3p 4( M L. A an oucoe he Lyapunov funcon n q. (7 and devave n q. ( becoe cala funcon. By upeedng q. ( no q. ( one could oban J Ω V V (3 B B whee p p ( ˆ ( B ( ˆ Ω Ω e L e L p and V {( A K e ( A D e Uo} p p p. p σl L ML q. (3 pecbe ha n addon o q. (0 wheea he oo peed changng he eaed oo peed ( ˆ he echancal paaee of he ye and he dffeence beween elecoagnec oque ( e K eq and load oque ( L affec ably of he peed obeve hough he funcon V. hen n ode o elucdae he condon fo anen ably q. (3 eaanged a an nequaly J Ω B V 0.(4 whee a vaable whch u be fuhe eolved o oluon. ubequenly Ineq. (4 ewen a Ω B V 0.(5 J J Whn a peod of e f Ω Ω 4 B JV J J o Ω Ω 4 B JV Ineq. (5 J J ue and V n q. ( negave. hu he peed obeve able. Howeve f Ω Ω 4 B JV Ω Ω 4 B JV J J J J Ineq. (5 becoe fale and V n q. ( becoe pove. In addon he oluon u be eal. hen he obeve becoe unable. Hence Ω 4B JV epeen a dcnan. Ω Ω 4 B JV and J J Ω Ω 4 B JV and fo he uppe J J bound and he lowe bound of epecvely. he e Ω ( J ean a d-pon quany beween he uppe and he lowe bound. Whle an nducon oo wh load a oang fo andll o changng peed due o dubance he dcnan he uppe bound he lowe bound and he d-pon quany of ae alo vayng along he nananeouly ˆ he echancal dynac and q. (0. heefoe he peed obeve ay be ehe able o unable dependng upon he locaon of whehe conaned whn he bound. When e elape adequaely and he oo-load ye oae wh deceang uccevely ll beng le han a val level he dcnan becoe alle and coence o be negave. egadng h he dance beween he uppe and he lowe bound wll becoe naowe and hen he wo bound ee ogehe a a pon n e befoe hey vanh. o f fuhe lowe owad zeo (.e. he oo peed end o be conan. V becoe

4 Poceedng of he 5h WA In. Conf. on Inuenaon Meaueen Ccu and ye Hangzhou Chna Apl (pp35-40 connuouly eely negave. Hence he peed obeve becoe able becaue Ω Ω 4 0 JBV (6 J 4J Ineq.(6 Ineq.(5 n aangeen of copleng he quae ue. Wheea he obeve ene eady ae V decay o zeo. 4 xplanaon Concenng ably va an Invaan e In ode o ae ably of he peed obeve nce po endng of anen ae up o coplee eady ae a ho evew of he nvaance pncple abued o Laalle [] gven. Conde an auonoou ye x ρ ρ f (7 ( x whee x ρ a ae vaable veco. heoe: Le V be a pove cala funcon wh connuou f devave fo he ye n q. (7. Le Ξ C be he egon o e conanng all ebe of x ρ uch ha V 0. Le Ξ O be he egon o e whoe ebe ae all x ρ afyng he condon ha V 0 only. Moeove le Ξ I be he lage nvaan egon o e whn Ξ O. Owng o Ξ I Ξ O Ξ C hen evey ajecoy of x ρ ognang n Ξ C appoache Ξ I a e pae uffcenly long whee V u be a Lyapunov funcon canddae. Accodng o he heoe wo eo equaon [] beween he oo-load ye and he peed obeve dealng wh obanng hee e of Ξ C Ξ O and Ξ I can be expeed a e ( A D e ( A D ˆ Uo e A A e î (8 ( A D e A î ( A D ˆ A e.(9 Unde he uaon of 0 he dcnan and V becoe negave ulaneouly. Bede he dcnan u be fuhe negave uccevely. heeafe when he oo peed eache eady ae V noally negave edefne. By h eaon a e of Ξ C gven a Ξ C { e e f 0 hen Ω 4 B J V 0 o f 0 hen V 0 } (0 whee a e of colun veco wh any wo eal nube. Fo q.(7 and q. (0 whle becoe equal o zeo V a deceang funcon of (.e. V( V(0. When e goe by adequaely long ( 0 e and 0 a well a and convege o he coepondng conan value n eady ae. hu a e of Ξ I wen a Ξ I { e e 0 e V 0 } 0 e 0 e 0.( hough Ineq. (6 a he nance of 0 when he dcnan declne o zeo equal a d-pon quany a well a V equal zeo oenaly. In ohe even V end o zeo n eady ae. heeby a e of Ξ O gven a Ξ O { e e f 0 hen Ω 4 B JV 0 Ω and Υ Ξ I J.( Caued by Ξ I Ξ O Ξ C e e Ξ C ove ono e e Ξ I a. 5 ulaon eul ulaon ae caed ou o vefy ably of he peed obeve dung anen ae. Accodng o dec-on-lne ang a he nal nan of e ( 0 he oo pevouly de-enegzed a andll conneced decly o a 0 V 50 Hz hee-phae ac nuodal upply []. An acual load oque uppoed o be conan. he peed obeve eceve eauable ao volage and cuen n ode o on-lne updae ao oo and coe-lo eance a well a eae oo angula peed and flux lnage of he nducon oo. he eulan dcnan he uppe bound and he lowe bound

5 Poceedng of he 5h WA In. Conf. on Inuenaon Meaueen Ccu and ye Hangzhou Chna Apl (pp35-40 of and he d-pon quany beween wo hee bound ae evealed n Fg. whle he devave of V (V n q. (3 hown n Fg.. Becaue canno be a coplex nube he quae oo of he abolue value of he dcnan copued n leu of he quae oo of he dcnan. hu wheneve h dcnan becoe negave connuouly whou he wo above bound beng eanngle fo exaple nce he pon K of e n Fg Concluon h acle ha hown ha when he oo peed change due o e-pon change o load dubance he adapve ldng-ode peed obeve becoe unable fo a ho oen befoe eganng ably. In pacce h unable peod can be hoen by nceang he gan of he PI adapve law uch ha he dffeence beween oo dynac and obeve dynac ae deceaed. Howeve cae u be aen ha hgh gan do no aplfy chaeng noe and haonc o ha peed eaon eve nceaed. o oban opu gan ll an open queon. Pevou ude [][3] have hown ha he obeve alway able wh pove PI-gan unde eady ae oo opeaon Fg. he dcnan he uppe bound and he lowe bound of and he d-pon quany x Fg. V dung anen ae When he oo peed nceae o deceae V can be pove o negave. Dung anen ae V and V ay ocllae. A e goe by he ocllaon of de down and deceae connually. V how a la paen and fnally V 0 fo exaple nce he pon K of e n Fg.. Appendx : Meanng of ach e n he peed Obeve he ybol ^ ndcae he eaed value o veco. he eanng of he ybol ued clafed n he Noenclaue and he ace of he peed obeve ae a follow: [ î î ] [ ˆ ˆ ] α ˆ β î α β Â âi M ˆ ˆ I L L B σ σl I Â ˆ Mˆ I ˆ J Â I L L Â Â Dˆ ŝˆ ˆ I ˆ L p ˆ ( L ŝm ˆ Dˆ I l ˆ ŝ ML 0 0 I J 0 π f 0 M σl σ L > 0 L and L > 0. M U o he coecon veco lad o copel he eaon eo o zeo []. Le he ache beween he eaed and he acual veco a well a beween he eaed and he acual paaee be eα α îα e eβ β îβ

6 Poceedng of he 5h WA In. Conf. on Inuenaon Meaueen Ccu and ye Hangzhou Chna Apl (pp35-40 e e e α β α β ˆ ˆ α β ˆ ˆ ˆ and ˆ. Acnowledgeen he auho ae hanful fo he gan fo he negy Polcy and Plannng Offce he Mny of negy haland and he hell Cenennal ducaon Fund on he 00h annveay of hell copany n haland a well a he fnancal uppo fo uanaee Unvey of echnology (U. efeence: [] angungong W. and ujjon. Adapve ldng-mode peed-oque Obeve. WA anacon on ye Vol.5 No Mach pp [] Khall K.H. (000. Nonlnea ye. ngapoe: Pence-Hall hd don. [3] Kojabad H.M. Chang L. and Doawa. ffec of Adapve PI Conolle Gan on peed aon Convegence and Noe a enole Inducon Moo Dve. In Poceedng of I Canadan Confeence on leccal and Copue ngneeng (CCC Vol. 003 pp Moneal Canada: 4-7 May.