Induction Motor Flux Estimation using Nonlinear Sliding Observers

Transcription

1 Jounal of Mathmatic and Statitic (): ISSN Scinc Publication Induction Moto Flux Etimation uing Nonlina Sliding Obv Hakiki Khalid Mazai Bnyounè and Djab Sid Ahmd Laboatoi d Automatiqu t d Analy d Sytèm (LAAS)ENSE-Oan BP El Mnaou Oan Algi Laboatoi d Dévloppmnt d Entaînmnt Elctiqu (LDEE)USOMB-Oan Abtact: A nonlina liding flux wa popod fo an induction moto It dynamic obvation o convg aymptotically to zo indpndntly fom th input h aim of thi wok wa to tudy th obutn of thi obv with pct to th vaiation of th oto itanc known to b a cucial paamt fo th contol h dynamic pfomanc of thi liding obv wa compad to that of Vgh obv via a imulation of an IM divn by U/F contol in opn loop Ky wod: Nonlina liding obv induction moto oto flux timation INRODUCION DC moto hav bn ud xtnivly in th induty bcau of th impl contol tatgi quid to achiv good pfomanc in vaiabl pd application Howv in compaion with thi countpat IM div DC div ult mo xpniv and l obut dvic not to mntion th maintnanc thy qui du to th commutato Bcau thy a highly nonlina thu quiing much mo contol complx algoithm IM div w aly ud in contol application in th pat Nowaday a a conqunc of th impotant pog alizd in nonlina contol thoy and pow lctonic th AC div by uing nw contol tchniqu hav povd to outpfom th DC on Among th tchniqu both fild ointd contol (FOC) and nonlina input-output dcoupling hav mgd a powful tool fo high pfomanc contol of induction machin [] h main dawback of both algoithm i th nd of flux no which a to b intd in th ai gap and involv a dign of th machin which duc liability and impli both additional cot and tchnological difficulti Fo thi aon flux obv hav bn widly invtigatd [] : thy a ath nitiv with pct to oto itanc vaiation Stating with [] oto itanc timato hav bn tudid [6] but mot contibution ly on implifying aumption and dfinit ult a till not compltly availabl inc no mthod appli whn th moto i in low-pd gim In th litatu val altnativ mthod xit fo th dign of diffnt obv tuctu: linaization by a chang of coodinat and output injction [7-9] vaiabl tuctu ytm [0] and Lyapunov-bad dign [] Howv clod-loop tability cannot b guaantd a pioi if th contol dign i bad on th paation pincipl which i vifid only fo lina ytm hu bad on contol thoy and noting that obvability i a dual poblm of contollability liding mod obv w dvlopd [-] hy div fom a tanpoition of th witching contoll [] to th poblm of tat obvation in nonlina ytm Sliding contol dign conit in dfining a witching ufac in th pha plan that i ndd attactiv by th action of th witching tm h dynamic i dtmind by th Fillipov olution concpt [] which indicat that th ytm dynamic bhaviou within th witching ufac can b dcibd a a pondd avag of th dynamic of ach id of th dicontinuou ufac Cla of liding mod obv: Conid th n th - od nonlina ytm: ( ) n q = f x u x R ; u = R () and fo convninc conid a vcto of maumnt: y = Cx y R () h ytm i aumd to b obvabl and th obv i dfind with th following tuctu : xˆ = fˆ ( xˆ y u) + KI () Coponding Autho: Hakiki Khalid Laboatoi d Automatiqu t d Analy d Sytèm (LAAS) ENSE-Oan l: Fax:

2 n wh xˆ R ˆf i ou modl of f K i n gain matix to b pcifid and I [gn( )gn( )gn( )] = () wh [ ˆ ] = S =Γ[ y Cx] () and Γ i q matix to b pcifid Dfining th o vcto y = C and = ( xxˆ ) on ha = f KI (6) wh f = f( x u) fˆ ( xˆ y u) h dimnional ufac S=0 will b attactiv if: i i < 0 i { } Duing th liding th witching tm in () i kping S 0 ; hnc fomally S 0 So I th quivalnt witching vcto [6] can b obtaind fom: ΓC( f KI ) = 0 o that: I = ( ΓCK) ΓC f (7) h matix Γ CK i inviibl with an appopiat choic fo Γ and K hu fom (6) and (7) th quivalnt dynamic on th ducd od manifold i givn by: x = ( I K( ΓCK) ΓC) f (8) with Γ C = 0 h tuctu of f mut b known bfo any futh analyi can b don J Math & Stat (): contant th mchanical pd and th numb of pol pai hfo tting x = ( ωφ ) α φβ iα iβ (9) i wittn in th fom: = α( xx xx ) αl αx = ax bx pxx = ax bx + pxx (0) = γx + γ x + γxx + γ v = γx + γ x γxx + γ v h Vgh obv modl which i a copy of th fit fou quation of (9) wh addd a coctiv tm du to a pdiction o i wittn in compact fom a [] : γ I ( M / c ) I ˆ ' + i ( / ) ( / ) ˆ M I I i = ˆ' 0 ( M / c) J ˆ φ φ ω 0 J ( L / c) ki + k wj + v ( iˆ i ) 0 + k I + k w J wh φ = φα φ β i = i i α β v = Vα V β 0 0 I = J c σ L L 0 = = 0 ˆ ' ˆ ' ( i φ ) a th divativ of ( iˆ ˆ φ ) timat of Vgh obv: h dynamic bhaviou of an ( i φ ) k i a cala and ω = pω i th induction moto woking und no atuation of it lctical pd of th oto h dynamic of th magntic cicuit can b dcibd in a fixd tato fnc ( αβ) fam [] by: obvation o = xˆ x i givn by : ( k γ ) I ( M / c ) I dφα / dt = bφα + aiα ωpφβ [ ( / )] ( / ) ' k + M I I dφβ / dt = bφβ + aiβ + ωpφ = α diα / dt = γ Vα γ iα + γ φα + γ ωφ (9) k J ( M / c) J + ω β k J J diβ / dt = γ Vβ γ iβ + γ φβ γ ωφβ If k and k a lctd uch that: k γ = k / and J dω / dt = p( M / L)[ iβ φα iα φβ ] L + fvω k + ( M / ) = k / withα = pm / J L ' th o dynamic bcom : = AQ () b = R / L = / σ = ( M / LL ) a= RM / L wh: ki (M/c)I A = γ = ( R / σl) + ( RM / σll) γ = MR / σll ki I γ / = Mp σ L L γ = / σ L and: ( / )I + ωj 0 Q = wh: ( φα φβ )( iα iβ ) L J σ 0 ( / fv )I ωj + h fdom that on ha in chooing k and k i R L R L M ω and p a pctivly th oto ud to plac th ignvalu of A in pai at abitay flux th tato cunt th toqu load th momnt location a i vifid by noting that th chaactitic of intia th lakag and ticky fiction cofficint polynomial of A i: th oto and tato winding itanc and inductanc th mutual inductanc th oto tim p ( + k) p+ k + k( M / c) 66

3 J Math & Stat (): If th ignvalu of A a p (twic) and p (twic) thn th ignvalu of th matix poduct in () can b hown to b: [( / ) ± jω ] p and [( / ) ± jω] p Hnc if th pd i (naly) contant th o dynamic i ( appoximatly) govnd by th ignvalu If it i tim-vaying w will attmpt a Lyapunov analyi Flux liding mod obv: h popod typ of liding mod bad obv of (9) can b wittn a: ˆ = α( xxˆ xx ˆ ) α ˆ L αx+ KI + q( xx) ˆ = ax bxˆ ˆ pxx + KI ˆ ˆ ˆ = ax bx + pxx + KI ˆ ˆ ˆ = γx + γ x + γxx + γ v+ KI ˆ ˆ ˆ = γx + γ x γxx + γ v + KI wh KiI = λiign( ) + λiign( ) fo i {} K i and q a th obv gain h liding ufac S i givn by: x xˆ 0 S = M = = x ˆ x 0 Stting ˆ i = xi xi fo i {} th obvation o dynamic i: = α( x x) KI q =b px KI = b + px KI = γ + γx KI = γ γx KI h tability analyi conit of dtmining K and K uch that th ufac S 0 i th attactiv hn K K K and q a dtmind uch that th ducd od ytm obtaind whn S 0 i locally tabl to 0 in th attactiv domain dfind a follow: S S Lt u conid th Lyapunov function V = uch that (S = 0 = = 0) with M a a gula matix h attactiv of liding ufac S= 0 i givn by: S 0 S V= S = S MW < 0 () t wh: W M λ λ = I with: K M ; λ λ = K δ 0 = 0 δ δ δ > 0 Fom th ingula ptubation thoy [] th dynamic of ω i uppod to b a low vaiabl with pct to th cunt and th flux dynamic h condition of attaction btwn and a dcoupld So () i obtaind within th t dfind by th following inqualiti: if > 0 thn <δ if < 0 thn >δ if > 0 thn <δ if < 0 thn >δ On th liding ufac S= 0 which i invaiant th vcto I I = i givn by: 0 δ 0 M I 0 = 0 and / δ I = = δ / δ Fom th dfinition of quivalnt vcto [6] on obtain th dynamic o aft a finit tim t 0 which i ducd to: (x x ) ( / / ) =α λ δ λ δ q λ λ H b = px H = λ λ px b hu: [ / / ] [ x x )] λ δ λ δ = α α and : λ λ q 0 = H + λ λ 0 q h ducd ytm of th obv o can b wittn a: ė i = q ii with q i > 0 and i= h obvation dynamic o i thn tabl with an appopiat choic of q i Digital imulation: A combination of th two obv pntd in Sction and i imulatd in Matlab/Simulink fo th imultanou timation of oto flux h combind timation i dpictd in th block diagam of Fig 67

4 J Math & Stat (): CONCLUSION Fig : Flux obv fo induction moto In thi tudy a novl combind chm fo concunt timation of th oto flux of an IM wa pntd h popod mthod i bad on two nonlina obv h imulation ult how a good pfomanc of th liding mod timation chm It ha bn hown that liding mod obv dign mthod bad on th pcibd fom of th Lyapunov function candidat can b uccfully applid h imulation ult a uggting that dign can b implmntd bad on th mchanical motion maumnt only thu avoiding flux vaiabl maumnt In addition th implicity of th algoithm mak it uitabl fo an on-lin implmntation In futh wok th autho intnd to tudy highod liding mod in both contol and obvation fo an induction machin and a pnumatic obot to mov th chatting ffct which i known to b th main dawback of th tandad liding mod ACKNOWLEDGEMENS Fig : Simulation ult A digital imulation of th popod combind chm illutat it bhaviou whn th moto i opating und U/F contol in opn loop with full load capability (h IM data in imulation a givn in th Appndix) Figu a how th fnc pd of th moto with th maud on and Fig b how th fit componnt of th oto flux Fom Fig c w vify that obvation o convg to zo in tady-tat opation but a tatd pviouly thy tnd to thi maximum valu in low-pd gim paticulaly at zo-coing Figu d illutat th wll-known innitivity popty of liding mod with pct to ditubanc Figu and f how th obutn of th two obv in ca of an intantanou oto itanc inca of 0% in th IM div h pnt ach wok ha bn uppotd by Scinc and chnology fo Safty in anpotation and fundd by th Euopan Union th Délégation Régional à la Rchch t à la chnologi th Minitè Délégué à l Enignmnt Supéiu t à la Rchch th Région Nod Pa d Calai (Fanc) and th Cnt National d la Rchch Scintifiqu Appndix Machin paamt otal Stato Inductanc 0 H otal Roto Inductanc 0076 H Mutual Inductanc 0099 H Stato Ritanc 6Ω Roto Ritanc 09 Ω Roto Intia (IM + load) 09 Kgm Numb of Pol pai Ratd magnitud Dict voltag 0 V Load oqu +7 & -7Nm Spd 0 pm Stato Flux 09 Wb Pow Kw Cofficint of ticky fiction 0008 Nm/d 68

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