Fault-tolerant Output Feedback Control for a Class of Multiple Input Fuzzy Bilinear Systems

Transcription

1 Sesos & asduces Vol 7 Issue 6 Jue 04 pp 47-5 Sesos & asduces 04 by IFSA Publshg S L hp://wwwsesospoalco Faul-olea Oupu Feedbac Cool fo a Class of Mulple Ipu Fuzzy Blea Syses * YU Yag WAG We School of Eleccal Egeeg Laog Uvesy of echology 00 Cha * el: * E-al: a_yuyag@6co Receved: 9 Apl 04 /Acceped: 0 May 04 /Publshed: 0 Jue 04 Absac: A fuzzy faul-olea cool schee s poposed fo a class of ulple pu fuzzy blea syses wh ueasuable saes Fs a fuzzy deecve obseve s desged o deec he faul Based o he developed deecve obseve a fuzzy coolle s poposed o guaaee he closed-loop syse whou fauls s asypocally sable he a fuzzy adapve dagosc obseve s desged o esae he faul A las he faul esao s used o faul accoodao ad a faul-olea coolle s desged Based o Lyapuov sably s poved ha he faul-olea coolle esues he closed-loop syse wh fauls s asypocally sable A exaple s povded o show he effecveess of he poposed ehod Copygh 04 IFSA Publshg S L eywods: Fuzzy blea syse Fuzzy faul-olea coolle Fuzzy deecve obseve Fuzzy adapve dagosc obseve Faul accoodao Ioduco Blea syses ae specal olea syses whch ae placed a eedae level bewee olea ad lea syses he blea syses ca descbe olea syses wh a bee pefoace ha lea oes whle blea syses ae sple ha he geeal olea syses Moeove hee ae ay physcal syses odeled by blea syses such as boegeeg bochesy uclea egeeg ad socoecoocs heefoe ay cool ehods have bee poposed fo of blea syses Dug he pas decade fuzzy cool has becoe a poweful cool ehod fo coplex olea syses aog whch fuzzy cool based o -S odel s oe of he os popula ad covee ool [-4] -S fuzzy blea odel s a specal exeso whch was fs poposed by L ad sa o odel olea syses [5] whch fuzzy coolle based o -S fuzzy blea odel was poposed fo olea syses Howeve he cool ga was equed o be ow whch s escve o he applcaos pacce I [6] oupu feedbac coolle was developed fo fuzzy blea syses wh e delay ad he cool ga was obaed by oducg a auxlay ax Howeve he fuzzy syses [6] wee equed o be sgle pu Fuzzy cool ehod was poposed fo fuzzy blea syses wh ulple pus [7] ude he codo ha he saes wee assued o be easuable Howeve pacce soe sae vaables of ages ay be ueasuable I s well ow ha soe fauls acuaos o sesos ca degade he cool pefoaces ad eve esul caasophc accdes heefoe he eseach o faul-olea cool has aaced oe ad oe aeo ece yeas [8 9] Howeve o hp://wwwsesospoalco/hml/diges/p_h 47

2 Sesos & asduces Vol 7 Issue 6 Jue 04 pp 47-5 he bes of ou owledge hee ae few esuls o he faul-olea cool fo he fuzzy blea syses wh ulple pu ad ueasuable saes Movaed by he above obsevaos hs pape we cosde he faul-olea cool poble fo he ulple pu fuzzy blea syses wh ueasuable saes Fuzzy deecve obseve s desged o esae he ueasuable saes ad deec he faul Based o he developed deecve obseve fuzzy coolle s developed o guaaee he syses whou fauls asypocally sable I he case of faul occuece fuzzy adapve dagosc obseve s desged o esae he faul fo faul accoodao Based o he faul esao he faul-olea coolle s desged o esue ha he closed-loop syse wh fauls s asypocally sable A exaple s povded o vefy he valdy of he poposed faul-olea cool ehod he es of he pape s ogazed as follows Seco povdes poble foulao Seco povdes he desg of fuzzy obseve ad fuzzy coolle Seco 4 povdes a uecal exaple Seco 5 gves he cocluso Poble Foulao Cosde a class of olea syses descbed by he followg -S fuzzy blea odel: R : If ξ s F ad ξ s F he x () Ax () Bu () ux () () () y () Cx () whee s he ube of fuzzy ules ξ [ ξ ξ ] s he fuzzy pese vaable veco F ( p ) s he fuzzy se p x() R () [ ] q u u u R y R ae he sae veco cool pu ad oupu u() u A B C ae soe ow aces wh appopae desos ( A B C ) s locally coollable ad obsevable hough sgleo fuzzfcao cee aveage defuzzfcao ad poduc feece he global odels of () he pesece of acuao fauls ae gve by x () μ [ Ax() BM() u() y () μ Cx () au() x ()] () whee l F p F p p μ μ / μ ebeshp fuco of l s he p ξ F p wh μ 0 μ M () s he acuao ga wh espec o u () If hee s o faul he acuao M () I ; ohewse M () dag{ a a } a wh 0< ( ) he cool obecve of hs pape s o desg a fuzzy faul-olea coolle based o he fuzzy deecve obseve ad he fuzzy adapve dagosc obseve o guaaee he closed-loop syse wh fauls asypocally sable he fuzzy deecve obseve s used o deec he fauls whle he fuzzy adapve dagosc obseve s used o esae he faul fo faul accoodao he Desg of Fuzzy Obseves ad Fuzzy Coolles he Desg of Fuzzy Deecve Obseve ad Fuzzy Coolle I he case of o faul he syse ( M () I ) desg fuzzy deecve obseves by usg paallel dsbued copesao (PDC) echque as R : If ξ s F ad ξ s F he x () Ax () Bu () u () x() L ( y( ) y( )) y C x () () q whee x() R y() R ae he sae veco ad oupu of he obseves L s he obseve ga o be desged he he global odel of he syses ae gve by x () μ [ Ax () Bu() u () x() L ( y( ) y ( ))] y () μ C x () (4) Usg PDC echque he global odel of fuzzy coolle based o he developed obseve s desged as 48

3 Sesos & asduces Vol 7 Issue 6 Jue 04 pp 47-5 u () μγ x () x ( ) x( ) μγ sθ μγ cos θ x ( ) (5) whee γ > 0 s he desg paaee R ( ) s he coolle ga o be deeed sθ x() x( ) x( ) cos θ π π θ [ ] x ( ) x ( ) Defe he obseve eo as e() x() x() he he closed-loop syse equao ad obseve equao ae descbed by whee x () μμ [ A B γ cosθ B γ cos θ e ( ) γ s θ ] x( ) μμ B ae he h colus of B e () μμ[( A L C ) γ s θ ] e ( ) (6) (7) he followg heoe saes he suffce codo ha syses (6) ad (7) ae globally asypocally covege heoe : Suppose ha hee exs paaees γ > 0 ε > 0 aces P > 0 P > 0 ad aces L ( ; ) sasfyg he followg LMIs he oupu feedbac coolles (5) sablze he fuzzy blea syse () whou fauls ad he obseve syse (4) Ω Ω () () 0 () < Ω () P () 0 < 0 εi whee Ω XA A X Iε () Ω [ M B X ] () Ω dag{ ε I ε I} ; () A P P A C G GC () () ε B B εi ; εγ M B M B M B X [ X X] ε [ ] ε I blocdag{ ε I ε I} X P M X G PL poof: Choose Lyapuov fuco as (8) (9) V x () Px() e () Pe () (0) he devave of (0) alog (6) ad (7) s V μμ x () {[ A B γ cosθ γ s θ ] P P[ A B γ γ s θ ]} x( ) μμ x ( ) cosθ P B γ cos θ e ( ) μμe (){[( A L C ) P[( A L C ) γ s θ ] P γ s θ ]} e ( ) () 49

4 Sesos & asduces Vol 7 Issue 6 Jue 04 pp 47-5 By usg Youg's equales oe has γ cosθ B P PB γ cosθ cos P B B s P P s s P xpb γ θ e x P x εe B B e s P P s P ε γ θ ε γ γ θ γ θ ε γ θ ε cos ε γ γ θ γ θ ε γ ε whee ε > 0 s he desg paaee he we have whee V x () Φ x() e () Ψ e () () y a* x bφ A P PA ε P ε B B ) () ( Ψ ( A L C ) P P( A LC) P ε ε B B ( ) (4) Pe- ad pos-ulplyg X P by () he we ge (8) ad (9) by applyg Schu coplee foula ae he oupu eo of he obseve () y( y) y () μ Ce () as he ceo of faul deeco 0 < f o faul; l ( ) 0 > f faul whee f s he faul occuece e (5) xˆ () μ [ Axˆ() BMˆ () u() aˆu () xˆ() L( y( ) ˆy( ))] ˆy () μ Cx ˆ() (6) ˆ() ˆ() ae he sae veco q whee x R y R ad oupu especvely â s he esao of faul a L s obseve ga o be desged Defe he dagosc obseve eo as e () x () x ˆ() Le () y () ˆy () μ Ce () deoe he oupu eo of he dagosc obseve he oe has whee e () μμ[( A LC) e () BM () u() W ()] (7) M () M () M ˆ() W () W () Wˆ () a u () x() aˆu () xˆ() he obecve of faul dagoss s o fd a algoh o guaaee ha l ( ) 0 l Mˆ ( ) M( ) he followg heoe saes he sably of he fuzzy adapve dagosc obseve heoe : Cosdeg he syse (7) f hee exs ax P > 0 sasfyg he followg LMI PA A P GC C G < (8) 0 whee G PL he adapve laws of ˆ() ˆ () ae updaed by M W ˆ( ) μ () () (9) M E u he Desg of Fuzzy Adapve Dagosc Obseve By usg PDC echque he fuzzy adapve dagosc obseve s desged as follows whee ˆ () μ () (0) W E 50

5 Sesos & asduces Vol 7 Issue 6 Jue 04 pp 47-5 > 0 > 0 E C B P E C P he l 0 l Mˆ ( ) M( ) Poof: Choose he followg Lyapuov fuco he followg heoe saes he suffce codos fo he sably of he syse wh faul heoe : Cosdeg he syses (4) pesse excao u () f he followg codo holds V e () Pe () [ M ( ) M ( )] [ W ( ) W ( )] he devave of () alog (7) s () x () μ λ ax 4 4 B u MM ( P ) λ ( P ) () () ˆ () μ[ λ ( Q ) Ω] 4 (5) V ()[ ( ) A LC P e () P[ BM () u() [ M ( ) Mˆ ( )] e P A LC e W ()] μμ ( ) ] ( ) [ W ( ) Wˆ ( )] μ () whee Ω λ ( ) () () ˆ () P u M M ax 4 he l M ( ) 0 he closed-loop syse wh faul s asypocally sable Poof: he syse (4) s asypocally sable f M () 0 heefoe fo ay gve Q 0 > hee exss P 4 > 0 sasfyg PA A P Q (6) 4 4 Subsug (8) (9) (0) o he above equao oe has V < 0 I s poved ha l ( ) 0 l Mˆ ( ) M( ) he Desg of Fuzzy Faul-olea Coolle Based o he developed fuzzy adapve dagosc obseve ad he esao of faul Mˆ( ) he fuzzy faul-olea coolle s desged as () () () () R u Mˆ u he closed-loop syse wh fauls s descbed by x () μ [ Ax () Bu () u () x() BM Mˆ u () () () u MM ˆ () () () x ()] (4) whee A s he syse ax of closed-loop syse whou faul Cosde Lyapuov fuco V x () Px() (7) 4 he devave of (7) alog (4) s μ () () x () P[ BM () Mˆ () u() 4 u() MM () ˆ () x ()] V x Qx μ[ ( λ ( Q ) P u ()) x() 4 P Bu M Mˆ x 4 μ () () () ()] (8) Choosg Q o sasfy ( ) >Ω we have λ Q 5

6 Sesos & asduces Vol 7 Issue 6 Jue 04 pp 47-5 f x () > V < 0 (9) () () ˆ () λ ( Q ) Ω P Bu M M 4 Usg he fac ha λ we oba P4 x V λax P4 x ( ) ( ) ( ) ( ) V 4 4λ ( P ) ax 4 λ ( P ) 4 μ () () () B u MMˆ [ μλ ( Q ) Ω] 4 (0) (5) guaaees he above equaly holds Ude he codo of u () pesse excao he closedloop syse (4) s asypocally sable ad l M ( ) 0 4 Sulao I hs seco a sulao exaple s gve o vefy he valdy of he developed faul-olea coolle We cosde a fuzzy blea e-delay syse wh he followg f-he ules [0] I sulao γ γ 005 ε ( ; ; ) Based o heoe ad heoe we ge he soluos of LMIs (8) (9) ad (8) as follows P P P [ ] [ ] [ ] [ ] L [8008;94] L [668; 005] L [0; 8] L [ 050; 0] Gve he faul M () 05 > s whch eas ha he acuao gas los 50\% afe s he oupu eos of he syse ae show Fg fo whch ca be see ha he eos ae ozeo whe he faul occus he saes ad esao saes of he syse whou faul accoodao ae show Fg fo whch ca be see ha he syse s o covege R : If x s F he x Ax Bu ux y Cx; R : If x s F he x Ax Bu ux y Cx whee A 0 A 0 B B C C [ 0] 0 Choose fuzzy ebeshp fucos as cosx ( x ) ( x ) ( x ) μ μ μ L L L Fg he aecoes of oupu eos Fg he saes (sold le) ad esaed saes (doed le) whou faul accoodao 5

7 Sesos & asduces Vol 7 Issue 6 Jue 04 pp 47-5 he esao of faul s show Fg fo whch we ca see ha he adapve fuzzy obseve ca esae he faul coecly he saes ad esao saes of he syse afe faul accoodao ae show Fg 4 fo whch we ca see ha afe faul accoodao he closedloop syse s asypocally sable exaple s gve o llusae he valdy of he poposed faul-olea cool appoach Acowledgees hs wo s paally suppoed by he Scefc Reseach Fud of Laog Povcal Educao Depae (L044) ad pa by Reseach Foudao of Laog Uvesy of echology fo Youg eaches (X0) Refeeces Fg he faul M () (sold le) ad s esao (doed le) Fg 4 he saes (sold le) ad esaed saes (doed le) afe faul accoodao 5 Coclusos hs pape poposes a oupu feedbac faulolea cool law fo fuzzy blea syses wh ueasuable saes Fuzzy adapve deecve obseve s desged o deec he faul Based o he developed fuzzy adapve deecve obseve a oupu feedbac fuzzy coolle s poposed o sablze he syse whou fauls he fuzzy adapve dagosc obseve s desged o esae he fauls he case of faul occuece he esaed fauls ae used o faul accoodao ad he desg of faul-olea coolle he fuzzy faul-olea coolle esues ha he closed-loop syse fauls s asypocally sable Fally a [] aag ad M Sugeo Fuzzy defcao of syses ad s applcaos o odelg ad cool IEEE asaco o Syses Ma Cybeecs Vol 5 Issue 985 pp 6- [] aaa M Sugeo Sably aalyss ad desg of fuzzy cool syse Fuzzy Ses ad Syses Vol 45 Issue 99 pp 5-56 [] B Che X P Lu Fuzzy guaaeed cos cool fo olea syses wh e-vayg delay IEEE asaco o Fuzzy Syses Vol Issue 005 pp 8-49 [4] S C og W Wag L J Qu Decealzed obus cool fo ucea -S fuzzy lage-scale syses wh e-delay Ieaoal Joual of Iovave Copug Ifoao ad Cool Vol Issue 007 pp [5] H S L S H sa -S fuzzy blea odel ad fuzzy coolle desg fo a class of olea syses IEEE asaco o Fuzzy Syses Vol 5 Issue 007 pp [6] Y Ju Oupu feedbac cool desg fo aag- Sugeo fuzzy blea e-delay syses Poceedgs of he Cofeece o 'Syses Ma ad Cybeecs Isabul uey 0- Ocobe 00 pp [7] J M L G Zhag C X Du Robus H cool fo a class of ulple pu fuzzy blea syse wh uceaes Cool heoy ad Applcaos Vol 6 Issue 009 pp 98-0 [8] H Yag M Saoswec B Jag J Y Lu Faul olea coopeave cool fo a class of olea ul-age syses Syses & Cool Lees Vol 60 Issue 4 0 pp 7-77 [9] H Wu M Z Ba Sochasc sably aalyss ad syhess fo olea faul olea cool syses based o he -S fuzzy odel Ieaoal Joual of Iovave Copug Ifoao ad Cool Vol 6 Issue 9 00 pp [0] Z Sh J M L L L Yag Adapve fuzzy cool of olea syses based o ulple pus -S fuzzy blea odel Poceedgs of he Cofeece o Chese Cool Hefe Cha 5-7 July 0 pp Copygh Ieaoal Fequecy Seso Assocao (IFSA) Publshg S L All ghs eseved (hp://wwwsesospoalco) 5