Hierarchical Intelligent Sliding Mode Control: Application to Stepper Motors

Transcription

1 Heachcal Intellgent Sldng Mode Contol: Applcaton to Steppe Moto Benado Rncon Maque Aleande G Louanov and Edga N Sanche Membe IEEE Abtact: In th pape one method of obut contol baed on ldng mode and fuy logc technque peented It combne heachcal hgh gan contol wth fuy logc to degn a dcontnuou contol fo nonlnea multvaable ytem n ode to elmnate chatteng n peence of dtubance The popoed appoach then ued to degn a obut contolle fo a teppe moto Smulaton eult ae peented to llutate the applcablty of the appoach I INTRODUCTION A mple and good technque fo obut contol the ldng mode contol [9 ] nce t compoed of two clea tep: ft electon of a ldng uface uch that a ldng mode uaton on th uface obut n peence of dtubance and econd degn of a dcontnuou contol whch table the pojecton moton of the cloed loop ytem on the ldng uface ubpace It nown that the ldng mode moton nvaant wth epect to dtubance whch atfe the matchng condton [] Thee ae two cae fo contollng ytem wth unmatchng condton: meaued dtubance and unmeaued one Fo the econd cae the poblem can be olved ung hgh gan contol but t can poduce chatteng [] due to defect on the contol devce In th pape we popoe a new contol cheme ung combnaton of the ldng mode contol bloc contol [] and fuy logc contol [3] technque to elmnate chatteng on the cloed-loop ytem fo both matched and unmatched unnown dtubance Note that the ldng mode fuy logc contolle wa nvetgated fo ytem wth matched dtubance n [ ] Th pape oganed a follow: In ecton the popoed method whch combne the Sldng Mode and Fuy Logc technque hown The tablty condton fo the Sldng Mode ae etablhed and the gan ae defned baed on the dtubance bound In ecton 3 an applcaton eample peented ung the popoed method to pemanent magnet teppe moto contol and n Wo uppoted by Conacyt Meco unde Gant 3696A B R M Autho wth the FIME of Colma Unvety Campu Coqumatlán Col 84 Méco e-mal: bm@ucolm A G L Autho wth Guadalajaa Unvety Campu Lago de Moeno al 4746 Méco e-mal: lou@gdlcnvetavm E N S Autho wth the CINVESTAV IPN Guadalajaa Unt P O Bo Guadalajaa al 4455 Méco e - mal: ecton 4 the eult of mulaton of th eample ae hown Fnally ome elevant concluon ae tated II CONTROL METHOD Let conde a multple nput multple output (MIMO nonlnea ytem ubject to dtubance f( + B(u +G(w ( yh( n m whee t ( t R the tate vecto u R the contol nput bounded a u ma ma > q w R epeent an etenal dtubance whch p n unnown but bounded y R the output f ( R and n m B ( R ae uffcently mooth and bounded functon n q G ( R an unnown but bounded functon and f ( We aume that thee et a nonlnea tanfomaton that educe the ytem ( to the o-called Bloc Contollable Fom wth dtubance [7]: f( + B( + G( w f( + B( + + G( w ( f( + B( u+ G( w y wth ( T ( T B and f ( and B ( ae uffcently mooth and bounded functon Fo electomechancal ytem the followng tuctue fuently condeed n n n m whee n the vecto e coepondng to tate and m the nput dmenon If the dtubance w atfe the o called matchng condton that thee a cala functon λ( uch that G( B( λ( then the ldng mode moton nvaant wth epect to etenal dtubance [] The am of th pape to degn a dcontnuou contol whch povde obutne fo the cloed-loop ytem and chatteng fee moton n peence anche@gdlcnvetavm

2 of both matched and unmatched dtubance The contol pocedue cont of the followng tep: Suppoe that the output y ued to tac the efeence gnal ef Ung the bloc contol technque [7] we ntoduce the followng ecuve tanfomaton: ef Φ ( ef (3a f( + b( ef + Φ ( ef (3b Φ( ef + B( + + f( + Φ ( ef (3c Φ+ ( + ef whee ( T a new vaable vecto > ( ( T ( ( - T ef ef ef ef ( ef ef ef B B B The tanfomaton (3a-(3c educe the ytem ( to the followng deed fom: + + G( w (4a + ef ef + + G ( w (4b f ( ef + B ( ef u+ G ( ef w (4c whee ( T f( ef a contnuou and bounded functon B B B and B In ode to geneate ldng mode n (4a-(4c a natual choce fo the wtchng functon (3c Then the deed dynamc of the cloed-loop ytem fo the cae of unnown w can be elected a B( ef f ( ef B ( ef + G ( ef w B( ef > (5 Fom (4c and (5 a dcontnuou contol tategy can be obtaned a: B( ef u (6 B ( ef In ode to deve the tablty condton we ue a potve defnte functon V ( Then fom V ( ef ( ef + ( ef B f G w we can obtan u ( w (7 ef whee the uvalent contol u defned a a oluton of (4c uch a u B ( ef f ( ef + G ( ef w (8 Unde condton (7 the tate convege to the uface and the ldng mode moton occu on th uface n a fnte tme Th moton decbed by the followng ( th ode ytem: + + G( w (9a + ef ef + + G ( w (9b + G ( ef w (9c If G ( on ( bounded then tanfomaton (3a- (3c bounded too theefoe G( ef w n (9a-(9c bounded The followng aumpton on the bound of the unnown tem n (4a-(4c tated: A Thee et potve contant q j and d uch that G( ef w q + d (a G ( ef w q + q + d (b G3 ( 3 ef 3 w q q3 + q + d3 (c ( G ( w q + q + d ef j j j j 4 (d To acheve the obutne popety wth epect to unnown but bounded uncetanty the contolle gan n have to be choen n a heachcally way fom the mallet to the bgget Thu nce G ( ef w n (a doe not depend on the value of th coeffcent can be choen o hgh that the tem n (9a wll be domnant By the bloc lneaaton pocedue the tem G( ef w n (b depend on but not on Then fo fed the appopate choce of value povde the domnaton of tem fo the econd bloc of (9b and o on In ode to etablh the ued heachy fo the contol gan whch enue tablty of the ldng mode moton (9a-(9c we chooe a Lyapunov functon canddate V fo the ytem (9a-(9c a a um of Lyapunov functon canddate fo the each bloc of (9a-(9c namely V V V n and calculate the devatve V tep by tep fom the ft bloc to the lat bloc of (9a-(9c At the ft tep dffeentatng the Lyapunov functon canddate V along the tajectoe of (9a and ung aumpton A namely (a we get V + + G( ef w ( q d whch negatve n the egon

3 d > + q q Theefoe the tate ultmately ente the doman of ubpace defned by ( α + β whee the paamete α and β defned a ( q α and β αd ae potve f the followng condton hold: > q ( At the econd tep followng mla lne to thoe of the ft bloc the devatve V of the Lyapunov functon canddate V calculated along the tajectoe of the econd bloc of (9b unde condton (a (b and ( gven by V + [ 3 + G( ef w] ( q 3 q d ( q q α 3 q β d whch negatve f ( q q α 3 q β d > Hence the tate ultmately ente the doman of the ubpace ( 3 defned by α β 3 and conuently α β3 whee the cala paamete α 3 β3 α 3 and β 3 defned a α ( q q α 3 3 α 3 qβ + d 3 αβ3 + β β ( α 3 αα 3 and β ae potve f the value of and atfy the followng nualte > q and > q + qα ( Poceedng along the ame pocedue fo the ( th bloc of the ytem (9a-(9c then the convegence doman of the ubpace ( α + β α + β (3 α + β α whee α α α j j ( ( q q α j j j j β j j α j β + β j At the net tep tang agan the devatve of the Lyapunov functon V along the tajectoe of the th bloc of (9a-(9c and ung (a-(d we obtan V + [ + + ( ef ] j + + q + q + d ( + j j j j Ung now (3 we can majoe V a ( V ( q j q j jα j ( + j j q jβ j d Fom th uaton t follow that α β + (4 whee the paamete and ( α + α j q j j q j j β α β ( + + j j q j j d 4 n ae potve f the condton ( > q + j j q jα j (5 hold Subttuton of (4 n (3 gve the followng et of nualte fo the ubpace ( + : + α + + β + α β + (6 α β + α + β whee α j + α j α + and β j α j β + + β j + j 4 n At the lat tep we have the doman of convegence of the ubpace ( n defned by the followng nualte: α n n + β n n Thee epeon ae ued to evaluate the devatve fo the Lyapunov functon canddate V n n along the tajectoe of (9c that V n n n+ n Gn( n ef n w n j n n+ n ( qn n+ j j q j j + dn ( n q q α ( j ( n ( j q j jβ j n d n n n j j j n n + + If n choen uch that the condton ( n n > qn n + j j qn j j n α (7

4 hold then we obtan V n α n wth potve α n β V + n V n n q n n n ( n n ( n j j qn jα j n and β n j n + j j qn j β d n By the Compaon Lemma [6] we have t γ ep α ( t t h (8 [ ] n ( n n n + n whee γ n n n ( t hn and Thu lm t up n ( t n β n h n α n h (9 Theefoe ung the obtaned uppe (8 and ultmate (9 bound on the oluton n ( t and the nualte (6 and gong bac fom the ( th bloc to the ft bloc of (9a-(9c we can fnd tep-by-tep uppe etmaton and ultmate bound on the oluton t ( t ( n ( n3 t In ode to educe the effect of the unnown dtubance acton n (5 that enue nualte ( (5 and (7 fo gven bound (a-(d and epectvely nceae the egon of the ldng mode tablty (7 t needed to nceae the value of the contolle gan n n (6 Th hgh gan howeve can poduce chatteng due to ome defect of the contol devce To olve th poblem we popoe to adjut the value of the gan n dependng on the value of and the uvalent contol u (8 u B ( ef f ( ef + G ( ef w Fo the cae of unnown dtubance the value of u can be obtaned by flteng the contol gnal a [9] τ u + u u The n gan modfcaton can be done by ung a fuy logc cheme Fo the cae of a bounded contol the value of n begn wth n n ma and then a tend to eo the value of n deceae moothly up to n n mn avodng chatteng The bloc dagam of the cloed-loop ytem wth bloc tanfomaton and ldng mode fuy logc contolle (SMFLC BC peented n the Fgue The dagam cont of the followng pat: Bloc Contol Th bloc tanfom tate to the new coodnate T BC : uch that T BC ( whee the map T BC defned by (3a-(3c and compute the value of the followng vaable: n and u B ( ef f ( ef + G ( ef w Fuy Contolle: Th bloc ue only two nput: In and In u and detemne the gan n uch that to atfy the tablty condton (7 Th contolle cont of the followng component: Input Nomalaton whch cale nput Fufcaton whch tanfom the cp nput value n fufed value F : In Ln uch that F( In Ln( whee In In a cp nput value defned on the unvee of dcoue In and Ln( the coepondng fufed nput value Infeence Mechanm whch ue the followng type ule: Rule m: If (Ln( and Ln( Then CR(j whee CR(j the coepondng d gan value fo the ule conuent Defufcaton whch baed on the weghted mean defufcaton method [3] to poduce a cala value d calculated a ef y Bloc Contol ef -y Cp Nomaled Contolle Input u Input Nomalaton In In Fufed Input Fufcaton Infeence Mechanm Defufed Fuy Contolle Concluon Output Rule-Bae Defufcaton d Output Denomalaton Contolle Output n Sldng Mode Contolle u w G B Σ f τ + Fuy Contolle Sytem Model h( Fgue Bloc Dagam

5 ne ne j j ( j CR( j ( d ne LAnt ne LAnt ( j whee LAnt ( j mn( LIn( LIn( the peme quantfcaton of the actve ule and ne fo the fuy et e Denomalaton whch multple nomaled fuy contolle output ( wth denomalaton facto (cale n d cale uch that the ytem (5 tay table Sldng Mode Contolle: whch mplement (6 to obtan u wth [ m] and m n III STEPPER MOTOR CONTROL In th ecton we apply the popoed cheme to contol a pemanent magnet teppe moto It mathematcal model gven by dθ ω dt dω [ Kma n( Nθ + Kmb co( Nθ Bvωτl ] dt ( db [ Rb Kmω co( Nθ + ub ] dt L da [ Ra + Kmω n( Nθ + ua ] dt L whee θ the angula poton; ω the haft peed; a and b ae the cuent n phae A and B epectvely; u a and u b ae the voltage n phae A and B epectvely; the moment of neta; R and L ae the etance and nductance n each of the phae wndng N the numbe of oto teeth K m the moto toque contant B v the vcou fcton and τ l peent the load toque petubaton Selectng the followng tate vaable θ ω 3 b and 4 a the nput a u u b and u ua the ytem ( epeented a a bloc contollable ytem contng of tee bloc and ubject to the unnown dtubance τ l w [ ] [ ] [ ] [ a + b( 3 b( 4] gw 3 cb ( a33 u + b 4 3 cb ( a 4 4 u Km Km whee b( co( N b( n( N g B a R a 3 a 4 c and b L L L Suppoe that the output y ued to tac the efeence gnal ef Followng the bloc tanfomaton pocedue ft we defne the tacng eo a and ef ef ( Then a deed dynamc fo ntoduced a + (3 Solvng ( and (3 fo we obtan + ef ef Then ( a + b ( b ( 3 4 ef ef gw Fom th uaton and the deed dynamc + 3 gw we have 3 f 3 ( + b ( 3 b ( 4 + ϕ( t whee 3 a new vaable f + + a ( ( 3 ϕ ( t ef + ef ef In ode to have a nonngula tanfomaton we ntoduce a new vaable 4 4 b ( 3 b ( 4 uch that the mat b( b( B ha full b( b( an In ode to obtan the contol acton ft we defne the wtchng functon a f3( b( b( 3 + b( b( 4 ϕ ( t + Then the pojecton moton on the ubpace govened by f 3 ( u ϕ ( t + bb + f ( u 4 T whee ( 3 4 ; f 3 ( and f 4 ( ae contnuou functon The contol tategy elected a u 3gn( B u 4gn( and the ldng mode tablty condton ae b3 > f 3 ( + ϕ ( t and b 4 > f 4 ( Unde thee condton the tate convege to the ldng manfold and when th manfold

6 eached the ldng mode moton decbed by the econd ode ytem wth unnown nonvanhng petubaton + w + In ode to educe the dtubance nfluence we apply the popoed ldng mode fuy logc contol cheme fo adjutng the gan 4 uch that 4 u wth u > IV SIMULATION RESULTS In th ecton mulaton eult ae peented fo the Pemanent Magnet Steppe Moto wth paamete: 6 L mh R 8 4Ω 36 Kg m 4 m 5V / ad N 5 B Nm / ad The mamal uppled voltage u V Fgue and 3 how the behavo of the tate ( 3 4 a well a of the new tate ( 3 4 Fgue dplay Bloc Contol Tacng (BCT eult n peence of dtubance (5 eg % of nomnal toque wthout Fuy Logc Contol (FLC thee we obeve chatteng and dtubance effect On the othe hand Fgue 3 dplay BCT wth Fuy Logc Contol (FLC eult ung the popoed appoach whee the chatteng educed V CONCLUSIONS A can be een the popoed cheme pefomance encouagng Fo the mulaton we aume that the dtubance can not be meaued whch an eteme tuaton Howeve the popoed heachcal ldng mode contol appoach wth the fuy logc contol mpove the ytem behavo educng chatteng and guaanteeng tablty VI REFERENCES Aleí M Vtte (994 Adaptve ldng mode contol of poton evo ytem IFAC Wohop on Sldng Mode Smolence Slovaa Septembe 7- pp Dajenovc B (969 The nvaance condton n vaable tuctue ytem Automatca; 5: Danov D Hellendoon H Renfan M (996 An Intoducton to Fuy Contol Spnge-Velag USA 4 Ha QP Nguyen QH Rye DC Duant-Whyte HF ( Fuy ldng-mode contolle wth applcaton IEEE Tan Ind Electon; 48: Kayna O Ebatu K Etugul M ( The fuon of computatonally ntellgent methodologe and ldng-mode contol-a uvey IEEE Tan Ind Electon; 48: 4-7 t at e X t at e 5-5 tme ( - 5 Xef X X X3 X4 Xef X X X3 X4 tme ( Khall HK (996 Nonlnea Sytem nd ed Pentce-Hall New-eey 7 Louanov AG (998 Nonlnea bloc contol wth ldng mode Automaton and Remote Contol; 597: Palm R Danov D Hellendoon H (997 Model Baed Fuy Contol - Fuy Gan Schedule and Sldng Mode Fuy Contolle Spnge-Velag Gemany 9 Utn VI (99 Sldng Mode n Contol Optmaton Spnge-Velag USA Utn VI Guldne Sh (999 Sldng Mode Contol n Electomechancal Sytem Taylo and Fanc Wong LK Leung FHF Tam PKS ( A fuy ldng contolle fo nonlnea ytem IEEE Tan Ind Electon; 48: tme Fgue 3: BCT wth FLC and dtubance tme ( 3 4 Fgue : BCT wth dtubance wthout FLC