Unknown Input Based Observer Synthesis for a Polynomial T-S Fuzzy Model System with Uncertainties

Transcription

1 Unknown Input Based Obseve Synthess fo a Polynomal -S Fuzzy Model System wth Uncetantes Van-Phong Vu Wen-June Wang Fellow IEEE Hsang-heh hen Jacek M Zuada Lfe Fellow IEEE Abstact hs pape poposes a new appoach based on the unknown nput method to synthesze the obseve fo polynomal akag-sugeno (-S) fuzzy system wth uncetantes In ths pape the uppe bounds of uncetantes ae not gven the effect of uncetantes s elmnated wthout desgnng an exta contolle Wth the ads of the non-common Lyapunov theoy Matlab s tools of the Sum-of-Squae (SOS) a new obseve s syntheszed n whch the obseve fom s completely dffeent fom the tadtonal obseve foms epoted n pevous papes he condtons fo the obseve synthess ae much elaxed the complexty of the desgn pocess s educed Fnally two llustatve examples ae pesented to demonstate the effectveness of the poposed method Index ems Uncetan polynomal -S fuzzy systems obseve synthess unknown nputs Sum of Squae (SOS) I INRODUION agak-sugeno (-S) fuzzy model []-[4] has eceved a geat deal of attenton n contol system eseach aea hs model has povded anothe way to solve the poblems of nonlnea contol systems heefoe many theoes contol desgn methods fo lnea contol systems can be appled n the -S fuzzy system In addton a new appoach fo desgnng an optmal coodnaton contolle based on the adaptve fuzzy dynamc pogammng game theoy fo solvng the consensus poblem of mult-agent dffeental games was studed n [5] hee was a book [6] to study the ype- fuzzy logc n detal a elated study to desgn an adaptve slde mode contolle fo the nteval ype- fuzzy systems was epoted n [7] In 9 a moe geneal fom of the -S fuzzy model called the polynomal -S fuzzy model has been ntoduced n [8] an nteval ype polynomal fuzzy model was nvestgated n [9] hs model allows the system Manuscpt eceved Octobe 9 6; evsed Febuay 9 7; accepted June 7 7 (oespondng autho: Wen-June Wang) V P Vu W J Wang ae wth the Depatment of Electcal Engneeng Natonal ental Unvesty Jhong-L 3 awan RO (e-mal: phongvv@hcmuteeduvn ;wjwang@eencuedutw) H hen s wth the Depatment of Electcal Engneeng Natonal Unted Unvesty Maol 3663 awan RO(e-mal: chc@nuuedutw) J M Zuada s wth the Depatment of Electcal ompute Engneeng Unvesty of Lousvlle Lousvlle KY49 USA (e-mal: jacekzuada@lousvlleedu) matces contanng polynomal foms n ts entes nstead of only constant foms Wth the suppotng Sum-Of-Squae (SOS) ools n Matlab [] the polynomal -S fuzzy system can be consdeed as an effectve tool fo modelng nonlnea contol systems Recently a lage numbe of studes focused on the polynomal -S fuzzy systems such as contolle desgn obseve desgn stablty analyss [8]-[4] Fo example the stablty analyses fo the polynomal -S fuzzy systems by employng the multple Lyapunov functon swtchng Lyapunov functon wee nvestgated n [3] [4] espectvely Besdes the contolle desgn obseve-based contolle desgn fo the polynomal -S fuzzy system wee studed n many papes such as [8] [] -[] A non-pd contol desgn fo a polynomal -S fuzzy system by usng contol Lyapunov functon Songtag s fomula was poposed n [8] he obseve-based contolles fo the polynomal -S fuzzy system wth mmeasuable pemse vaables wee syntheszed n [] [3] In [] the authos poposed a new appoach fo stablty analyss contolle desgn fo a geneal polynomal -S fuzzy system n whch the polynomal Lyapunov functon cdate does not need to satsfy any constant Addtonally the contolle synthess fo dscete tme polynomal -S fuzzy systems wthout wth delay tme was developed n [9] [] espectvely Fom the above evew t becomes obvous that the polynomal -S fuzzy system has been pad attenton nceasngly t extends the study scope lage than the conventonal -S fuzzy system does fo nonlnea contol systems In a wde ange of eal-lfe pactcal systems all o some of state vaables ae mmeasuable o dffcult to obtan by usng the measuement devces due to both techncal economc ssues hese states howeve ae eally necessay fo system supevson contolle desgn heefoe the obseve desgn was taken notce nceasngly Regadng the obseve synthess fo the polynomal -S fuzzy systems vaous methods have been pesented n the past few yeas [5] -[8] In [5] a synthess of both contolle obseve fo polynomal -S fuzzy system was poposed to guaantee the system stablty the state estmaton smultaneously In addton the method fo desgnng obseve contolle smultaneously fo both contnuous dscete tme polynomal -S fuzzy systems wee pesented n [7] [8] espectvely

2 In pactce thee ae a lage numbe of systems affected by many knds of uncetantes he pesence of uncetantes n the system makes the desgn of the obseve contolle fo the system much moe dffcult In ou lteatue suvey a vaety of studes dealng wth uncetan poblems of -S fuzzy system wee found n [9]-[33] howeve thee ae only few papes concenng the polynomal -S fuzzy system wth uncetantes Recently the obust contolle synthess fo the polynomal -S fuzzy system wth uncetantes was nvestgated n [] [34] n whch the uncetantes n these papes must satsfy the nom-bounded constants Regadng the obseve desgn fo uncetan polynomal -S fuzzy system moe ecently the papes [8] [35] poposed the method to synthesze the contolle obseve smultaneously hese papes not only elmnated the nfluence of uncetantes but also guaanteed the state estmaton eos appoachng to zeo asymptotcally It should be noted that the uncetantes n the above papes must be unde some bounds If the uppe bound of uncetantes s unknown o only obseve wthout contolle s desgned the methods n [8] [35] wll not wok o addess ths shotcomng we popose n ths pape a new appoach based on the unknown nput method to synthesze the obseve fo the uncetan polynomal -S fuzzy system whee the bounded constants of uncetantes ae not gven the contolle s not needed to be desgned smultaneously to elmnate the nfluence of uncetantes Recently thee have been seveal papes poposed the method to desgn obseve based on the unknown nput method [36]-[38] fo -S fuzzy system unknown nput polynomal -S fuzzy system n [39] [4] Unfotunately the esults n [39] [4] have two lmtatons he fst s t s dffcult to fnd the feasble soluton fo the paamete matces to desgn the obseve he detals wll be explaned n Secton II he othe s the poposed method usng the common Lyapunov functon cdate that often leads to a much consevatve esult In ode to ovecome the above dsadvantages ths pape poposes a new fom of obseve uses a non-common Lyapunov functon [4]-[4] to deve the condtons fo the obseve synthess he pape s oganzed as follows In Secton II we descbe the consdeed polynomal fuzzy system model wth uncetantes pont out the man poblems to be esolved n ths study In Secton III the man theoems obseve synthess pocedues ae poposed In Secton IV two examples ae pesented to llustate the effectveness of the syntheszed obseve Fnally a concluson s pesented n Secton V Notatons: A denote the postve defnte matx A ; A denotes the tanspose of matx A ; A denotes the nvese of A ; A denotes the Mooe-Penose pseudo-nvese of A ( nm A A A) A he symbol denotes the set of n m matces; I denotes the dentty matx; the astesk (*) denotes the tansposed elements of the symmetc matx II SYSEM MODEL AND PROBLEM DESRIPION A System model Let us consde the nonlnea system pesented as follows x f ( x u ) (a) y x (b) whee f s the nonlnea functon ncludng possbly uncetantes x s the state vecto s the vecto of possbly tme-vayng paamete he equaton (b) s a lnea output equaton wth constant matx s the output vecto On the bass of the secto nonlneaty method [43] suppose the nonlnea system () can be epesented as the class I polynomal -S fuzzy system as follows [7]: Rule IF () t : s Q s () t s x ( A ( ) A ( )) x Q s ( B ( ) B ( )) u y HEN (a) y x (b) whee s s the vecto of measuable pemse vaables j= s) s the fuzzy set s the numbe of pemse vaables Q j ( = ; s the numbe of ules s () t n Suppose xt () s m the unavalable state vecto ut () s the nput vecto p yt () s the output It s noted that s the measuable vaable that could be a functon of extenal vaable output nn /o tme he polynomal matces A( ) yt () B( ) nm () t ae known polynomal matces of the states nputs espectvely Moeove A( ) B( ) ae the uncetantes of A ( ) B ( ) espectvely he oveall uncetan polynomal -S fuzzy system nfeed fom the plant ules of () s descbed below: x ( ) ( A ( ) A ( )) x ( B( ) B( )) u y x whee s w ( ) Q ( ) j j n (3) w( ) w ( ) w ( ) ( ) ( ) w( )

3 3 Remak : As pesented n [7] the polynomal -S fuzzy system s classfed nto thee types (class I class II class III) In ths pape we only consde the class I polynomal -S fuzzy system whch has the polynomal system matces depend on the measuable tme-vayng vaable () t Remak : If () t s a constant the system matces A( ) B ( ) become constant matces A B the polynomal -S fuzzy system () becomes the conventonal -S fuzzy system Remak 3: he uncetantes on () t A( ) B( ) depend t means that the uncetantes ae moe pactcal compaed to the uncetantes n [] [8] [34] whch ae only dependent on tme In addton the bound condtons of these uncetantes ae not gven Remak 4: A nonlnea system can be tansfomed to a polynomal -S fuzzy system of lass I by usng the secto nonlneaty method f the system matx nput matx of the polynomal -S fuzzy system contan only the measuable vaables that could be a functon of extenal vaable output /o tme If some un-avalable state B vaables ae nsde /o then these systems ae consdeed as a polynomal -S fuzzy system of lass II III lass II III ae moe complcated to be studed whch ae not consdeed n ths pape he polynomal -S fuzzy of class I whch s an extenson of the geneal -S fuzzy system can sgnfcantly educe the numbe of fuzzy ules fo pesentng the ognal nonlnea because the pemse vaable has been put nsde the system matces [7] A Poposton []: If p( x( t )) s a SOS then p( x( t )) can be ewtten as polynomal n yt () B n ql x t whee ql ( x) l p( x) ( ( )) xt () the SOS t mples that p( x) B Poblem descpton A s a heefoe when p( x( t )) s detemned as Suppose that when all o some state vaables xt () of the polynomal -S fuzzy system (3) ae unavalable t s necessay to synthesze an obseve to estmate these unavalable state vaables Hence ths pape ams to desgn an obseve fo the system (3) to guaantee the estmated state vaables appoachng to eal states As dscussed befoe [8] [35] dealt wth the poblem of obseve desgn fo uncetan polynomal -S fuzzy systems Howeve n ode to elmnate the nfluences of uncetantes estmate state vaables smultaneously the methods n [8] [35] desgned an obseve-based contolle to acheve ths objectve In addton the uppe bounds of uncetantes have to be gven n advance; othewse t s nfeasble to desgn the obseve In ode to ovecome these dawbacks n ths pape we popose the new appoach based on the unknown nput method to synthesze an obseve fo the uncetan polynomal -S fuzzy system (3) If the unknown nput obseve fom (obseve (7) n [4]) s consdeed z ( ) N ( ) z G ( ) u L ( ) y xˆ z Ey (4) It s noted that the matx E n (4) s a constant matx t s had to satsfy the condton ( P S) R (() n heoem of [4]) whee S=PE snce P S ae constant matces whle the matx s the polynomal matces R Due to the above analyses ths study tes to popose a new appoach to synthesze a specfc fom of the obseve fo the uncetan polynomal -S fuzzy system Befoe the man devaton the followng two assumptons ae needed Assumpton : Assume the matchng condtons A ( ) D( ) A ( ) B ( ) D( ) B ( ) ae satsfed whee D( ) nq s a full nomal column qn ank matx (see [45]-[47]) A( ) qm B( ) ae tme-vayng uncetan matces whch depend on () t Assumpton : he matces D( ) ae full ow nomal column anks espectvely the nomal ank of ( D( )) s equal to the nomal ank of D( ) Remak 5: he Assumpton s necessay to tansfom the uncetantes to unknown nputs the Assumpton s needed to guaantee the exstence of geneal solutons of matx equaton whch wll appea n the poof of heoem III OBSERVER SYNHESIS Fstly on the bass of the Assumpton the uncetan polynomal -S fuzzy system s tansfomed to the unknown nput polynomal -S fuzzy system as follows Based on the Assumpton let us defne A ( ) x B ( ) u whee q q () t () t ae consdeed the unknown nputs heefoe the system (3) can be ewtten as follows: x ( ) A ( ) x B ( ) u D( ) D( ) y x Let be substtuted nto (5) we obtan (5)

4 4 x ( ) A ( ) x B ( ) u D( ) y x (6) Now the uncetan polynomal -S fuzzy system (3) has been tansfomed to unknown nput polynomal -S fuzzy system (6) heefoe the obseve wll be syntheszed to estmate unavalable states of the system (6) athe than of the system (3) Re-consde the fom of obseve (4) but wth a polynomal matx E( ) as follows z ( )[ N ( ) x G ( ) u L ( ) y](7a) xˆ z E( ) y (7b) he s the estmaton of the state vaable s xt ˆ( ) the state vecto of the obseve Let us defne the estmaton eo xt () zt () e x xˆ (8) substtutng (7b) nto (8) yelds e x z E( ) y ( I E( ) ) x z M ( ) x z whee M( ) I E( ) omputng the devatve of estmaton eo M( ) e x M ( ) x z () t et () (9) n (9) yelds () M( ) Fom () t s clealy seen that the tem x s () t vey complcated because of the tems of dffeentaton heefoe ths pape poposes a new obseve fom as follows xˆ ( ( )) ( ( )) ˆ t N t x G ( ) u L ( ) y F( ) y () nn nm n p whee N ( ) G( ) L( ) n p F( ) ae polynomal matces to be detemned late xt ˆ( ) s the estmaton of the eal state vaable xt () Fom (6) we have y x ( )[ A ( ) x B ( ) u D( ) ] Fom (8) we have () e x xˆ (3) Substtutng (6) () nto (3) yelds e ( ) A ( ) x B ( ) u D( ) ( ) N ( ) xˆ G ( ) u (4) L ( ) y F( ) y ombnng () (4) the obtaned esult s e ( ) A ( ) x B ( ) u D( ) N ( ) xˆ G ( ) u L ( ) y F( )[ A ( ) x B ( ) u D( ) ] ( ) A ( ) x F( ) A ( ) x L ( ) x N ( ) xˆ B ( ) u F( ) B ( ) u G ( ) u D( ) F( ) D( ) ( ) ( A ( ) F( ) A ( ) L ( ) N ( )) x N ( ) e ( B ( ) F( ) B ( ) G ( )) u ( D( ) F( ) D( )) (5) In the followng poof of the heoem an assumpton s needed Assumpton 3 [4]: Assume that ( ) k whee s a constant k heoem : Unde Assumptons 3 the estmaton eo (8) wth the obseve () conveges to zeo asymptotcally f thee exst polynomal matces F( ) N ( ) L( ) G ( ) symmetc matx condtons hold wth = P k such that the followng A ( ) F( ) A ( ) L ( ) N ( ) (6) B ( ) F( ) B ( ) G ( ) (7) D( ) F( ) D( ) (8) v ( P I) v s SOS (9) k k j j k v ( P N ( ) P P N ( ) ( ) I) v s SOS whee v v () ae vectos wth appopate dmensons that do not depend on () t ( ) at Poof: If the condtons (6)-(8) of heoem hold then (5) becomes k

5 5 e ( ){ N ( ) e} () Select the non- common Lyapunov functon as follows V ( ) e Pe () It s noted that f the condton (9) holds t means that the symmetc akng the devatve of Lyapunov functon esults n P k k k V ( ) e P e ( )[ e Pe e Pe] (3) Substtutng () nto (3) on the bass of the Assumpton 3 poduces Lemma : Let A S P R be matces wth pope szes he followng two nequaltes ae equvalent: ) R A P PA A S SA R P S A S P S SA S S ) S : Poof of Lemma : *) ) mplcaton ): We pe-multply post-multply () wth I A I A I A espectvely whch yelds A S SA R P S A S I R A P PA P S SA S S A *) ) mplcaton ): Fom ) we have k k j k j V e P e ( ){ e N ( ) P e j k k k j e P N ( ) e} e P e ( ) e { N ( ) P P N ( )} e j j j j k k j k ( ) e { N ( ) P P N ( ) P } e (4) It can be seen fom (4) that f the condton () holds then Vt ( ) t means that the estmaton eo (8) appoaches zeo asymptotcally he poof s completed Remak 5: he tem postve ( ) ae used n heoem to guaantee k P s postve defne matx k Pk N ( ) Pj Pj N ( ) s negatve defne matx nstead of sem-postve sem-negatve defne In ode to synthesze the obseve () all condtons (6)-() must be solved to detemne the paametes N ( ) G ( ) L ( ) F( ) of the obseve () Howeve the condton () s a polynomal BMI (Blnea Matx Inequalty) whch cannot be solved by usng SOS OOL n Matlab theefoe we need to tansfom t nto the polynomal Lnea Matx Inequaltes (LMIs) he followng heoem wll be wth polynomal LMI mn Lemma [44]: hee ae two matces A wth m n kn B suppose that BA A B hen any matx of the fom X BA Y( I AA ) s a soluton of XA B whee km Y s an abtay matx A s defned as ( A A A) A whch s the Mooe-Penose pseudo-nvese of A ( R A P PA) ( A S A S ) ( SA SA) ( A S A A S A) ( A SA A SA) A S SA R PA SA A S A A P A S A SA A S A A SA We can ewte (*) n the fom of matx as follows I A A S SA R P S A S I P S SA S S A hat leads to A S SA R P S A S P S SA S S heefoe ) ) ae equvalent he poof s completed heoem : Unde Assumptons 3 the estmaton eo (8) wth the obseve () conveges to zeo asymptotcally f thee exst polynomal matces F( ) N ( ) L( ) G P ( ) K( ) X( ) Q( ) symmetc matx such that the followng condtons hold fo = v ( P I) v s SOS (5) () (*) k v () ( ) I v s SOS j X ( ) X ( ) (6) (*)

6 6 () k k k k P ( A ( ) U ( ) A ( )) X ( ) ( V ( ) A ( )) K ( ) Q ( ) X ( )( A ( ) U ( ) A ( )) K( )( V ( ) A ( )) Q ( ) (7) () P X ( ) X ( )( A ( ) U( ) A ( )) j j K( )( V ( ) A( )) Q( ) (8) whee ae vectos wth appopate dmensons that v v do not depend on () t ( ( t)) at ( ( ( ))) (( ( ( ))) D t D t ( D( ))) ( D( )) U( ) D( )( D( )) (9) V( ) ( I ( D( ))( D( )) ) (3) K( ) X ( ) Y( ) (3) Q ( ) X( ) L ( ) (3) Moeove the paametes of obseve () ae computed as follows F( ) U( ) Y( ) V( ) (33) G ( ) B ( ) F( ) B ( ) (34) ( ( )) ( ( ))Q ( ( )) L t X t t (35) N ( ) A ( ) F( ) A ( ) L ( ) (36) Poof: Fom the condton () of heoem t can be nfeed that k Pk N ( ) Pj Pj N ( ) (37) k Employng the Lemma fo (37) wth slack vaable X( ) one obtans () (*) k () j X ( ) X ( ) whee () k k k k (38) P N ( ) X ( ) X ( ) N ( ) (39) P X ( ) X ( ) N ( ) (4) () j j Fom (8) we have F( ) D( ) D( ) (4) On the bass of the Lemma the geneal soluton of (4) s F( ) D( )( D( )) whee Y ( )( I ( D( ))( D( )) ) ( ( ( ))) (( ( ( ))) D t D t ( D( ))) ( D( )) (4) It s noted that the exstence of the geneal soluton (4) s guaanteed f only f the matces satsfy the Assumpton Let s denote D( (t)) U( ) D( )( D( )) (43) V( ) ( I ( D( ))( D( )) ) (44) Substtutng (43) (44) nto (4) yelds F( ) U( ) Y( ) V( ) (45) Fom (6) (45) one obtans N ( ) A ( ) ( U( ) Y( ) V ( )) A ( ) L ( ) A ( ) ( U( ) A ( )) ( Y( ) V ( ) A ( )) L ( ) Substtutng (46) nto (39) the obtaned esult s () k k k k P ( A ( ) ( U ( ) A ( )) ( Y ( ) V ( ) A ( )) L ( ) ) X ( ) X ( )( A ( ) ( U ( ) A ( )) ( Y ( ) V ( ) A ( )) L ( ) ) k P ( A ( ) U ( ) A ( )) X ( ) k k (46) ( V ( ) A ( )) Y ( ) X ( ) L ( ) X ( ) X ( )( A ( ) U ( ) A ( )) X ( ) Y ( )( V ( ) A ( )) X ( ) L ( ) Substtutng (46) nto (4) we have P X ( ) X ( )( A ( ) ( U( ) A ( )) () j j ( Y( ) V ( ) A ( )) L ( ) ) P X ( ) X ( )( A ( ) U( ) A ( )) j X ( ) Y( )( V ( ) A ( )) X ( ) L ( ) Let us defne (47) (48) K( ) X ( ) Y( ) (49) Q ( ) X( ) L ( ) (5) Substtutng (49) (5) nto (47) (48) we obtan

7 7 () k k k k P ( A ( ) U ( ) A ( )) X ( ) ( V ( ) A ( )) K ( Q ( ) X ( )( A ( ) U ( ) A ( )) K( )( V ( ) A ( )) Q ( ) (5) () P X ( ) X ( )( A ( ) U( ) A ( )) j j K( )( V ( ) A ( )) Q ( ) (5) Fom (5) (5) t s seen that (38) becomes () (*) k () j X ( ) X ( ) whee () k () j (53) ae expessed as (5) (5) espectvely It s seen that (53) s equvalent to (6) of heoem t s a polynomal LMI It means that the polynomal BMI () has been successfully tansfomed nto polynomal LMI n heoem he poof s completed It s noted that the polynomal LMI n heoem can be solved easly by usng SOS OOL of Matlab [46] he bef pocedue fo the obseve () synthess s pesented below: Step : heck the matces D( ) satsfy the Assumpton o not If yes we go to the next step If not ths method does not wok fo ths case Step : Fom (9) (3) U( ) V( ) ae obtaned Step 3: Resolve (5) (6) to obtan Q( ) then fom (3) we obtan Y( ) P j K( ) X( ) Step 4: he matces F( ) G( ) L( ) N ( ) ae obtaned fom (33)-(36) espectvely Step 5: he obseve () s syntheszed IV ILLUSRAIVE EXAMPLES In ths secton two examples ae llustated to pove the effectveness of the poposed method Example s a numecal example Example s an applcaton fo Inveted Pendulum A Example onsde a nonlnea system as follows Let the system (54) be tansfomed nto a polynomal -S fuzzy system whee of polynomal -S fuzzy system s the output y(t) hen () t x ( )[ A x B u] (55) y x 3 A y A y y B B cos( x) cos( x) ( ) ; ( ) We can see that sn( x) sn( x) ( ) x ( ) x then we assume 4 Suppose the system (55) s nfluenced by the uncetantes the bounded constants of these uncetantes ae unknown he system (55) s expessed n the followng fom x ( )[( A A ) x ( B B ) u] y x (56) whee cos / y sn( y ) / y A 5cos 5sn sn / y 4cos / y A 5sn 6cos cos( y ) / y sn( y ) / y B B 55cos( y ) 5sn( y ) ansfomng system (56) nto the unknown nput system yelds x ( )[ A x B u D ] (57) y x whee 3 x cos( x ) x x u x x x x y x (54) / y Dy ( ) 5 Step : We can see that D( y) ae full ow column anks nomal ank( D( (t))) nomal ank( D( ) thus the Assumpton s satsfed

8 8 Step : Fom (9) (3) U y /4 V ; Step 3: Resolve the constant (5) (6) of heoem usng the SOS OOL of Matlab We can obtan the values below P P K Y X y 6543 y35 Q y 59 y y y377 Q y 3987 y 34 Step 4 Step 5: he matx F( ) s obtaned fom (33) We obtan N ( ) fom (36) L( ) fom (35) G ( ) fom (34) y 463 y N( y) N( y) L y y y y / y 436 y y/ 4 57 y y y y y y y L G 4 y /4 ( G y) 4 y / 4 F y / y 435 y y y Hee we used Smulnk tools to cay out the smulaton he ntal states of the system x() -5 3 the estmated states ae xˆ() -5 5 he nput used fo smulaton s u sn Fg Real state x () t estmated state xˆ () t Fg Real state x () t estmated state xˆ () t Estmaton eo e () t Estmaton eo e () t Fg 3 Estmaton eos e () t e () t

9 9 Fgues -3 show the smulaton esults of the numecal example n whch the system (56) s petubed by the uncetantes A ( y) A ( y) B ( y) B ( y) Fgues depct the states the estmated sgnals he estmaton eo s llustated n Fg 3 It can be seen fom these fgues that the estmated states can appoach eal states asymptotcally Hence the poposed method s successful n syntheszng obseve fo polynomal -S fuzzy system wth uncetantes x x Remak 6: It s noted that f we use geneal -S fuzzy system to epesent the nonlnea system (54) both must cos( x ) be lneazed the local bound ange of must be gven hen thee wll be fou fuzzy ules n the -S fuzzy system Howeve f let a polynomal -S fuzzy system be pesent the ognal nonlnea system x wll be putted n the system matces theefoe the bound ange of does not need to know the numbe of fuzzy ules wll be educed to two It means that the polynomal -S fuzzy system wll epesent the system (54) be moe exactly ove global egon x x x Mc 8kg L m ae the mass of Pendulum at the length of the pendulum espectvely; a / ( m M ) p c ut () the contol nput foce mposed on the cat 7 7 x () t 8 8 In ode to educe computatonal buden the tem sn( x ) s x tan( x ) tx Applyng nonlnea secto to lneaze ths system the nonlnea system (58) s epesented as the followng polynomal -S fuzzy system It s noted that ( ) x y x t x x theefoe the esult of the nonlnea system can be expessed n the fom of (3) wth beng the output y(t) s 8578 t 5534 x ( x ) A x B u y x() t s () t (59) B Example In ths example we consde a pactcal dynamc model of an Inveted Pendulum on a cat [] he system s depcted n the followng fgue whee A a A a B f mn a B fmaxa fmn 5 fmax 7647 a f ( gt am Ly s ) a f ( gt am Ly s ) mn p he pemse vaables max p Fg 4 Inveted Pendulum on a cat he nonlnea equaton of the Inveted Pendulum on a cat s expessed as follows x x g sn( x ) am Lx sn( x )cos( x ) p () 4 L / 3 ampl cos ( x ) x t a cos( x ) u 4 L / 3 ampl cos ( x ) whee (58) x x x s a vecto of states s the angle (n adans) s the angula velocty m kg p ( x ) f ( x ) f max f mn f max cos( x ) f ( x ) 4 L / 3 am Lcos ( x ) p ( x ) ( x ) ; Suppose the (59) ae affect by the followng uncetantes then x ( x ) ( A A ) x ( B B ) u y x() t (6) y sn(y ) y cos A 55sn 6cos y cos y sn A 55cos sn

10 y cos( y ) B 5cos( y ) y sn( y ) B 5sn( y ) G (7767 y ) / 5 ansfom (6) nto unknown nput system wth y 7 7 Dy ( ) Wth x () t n ths pape we assume 5 Step : It s seen that the matces ae full ow column anks espectvely nomal ank( D( (t))) nomal ank( D( ) theefoe the Assumpton s satsfed Step : he matces ae calculated: U V D( y) y 8 y5537 L 3 6 y y y y357 L 3 63 y 4865 y Afte obtanng all paametes the obseve () s constucted he smulaton esults of the fuzzy obseve fo the system n Example wth nput u sn ntal values of the states x() 7*p/8 estmated states xˆ() -35*p/8 ae llustated as follows y /5 U V Step 3: he matces X ( y) P Q ( y) ae obtaned as follows K( y) Y( y) P P 8 Y K X( y) y 3 y 53 Q y 469 y y 985 y 53 Q 4 7 y 88 y 3 4 Fg 5 Real state x () t estmated state xˆ () t Step 4 Step 5: he paametes N ( y ) L ( y) ae computed G F he matces N N ( ) y ae expessed n (6) (6) espectvely y /5 (6 y ) / 5 F G Fg 6 Real state x () t estmated state xˆ () t he smulaton esults of the obseve synthess fo the pactcal dynamc system of Inveted Pendulum on a at whch s affected by uncetantes ae llustated n Fgs 5-7 Obvously Fg 5 Fg 6 show that estmaton states ˆx N 4 ( y ((447 y ) 79496)) / 5 893y 8 y y 9397 y 999 (6) N 8 ( y ((53 y ) 68645)) / y 4559 y y 48 y 9999 (6)

11 Estmaton eo e () t Estmaton eo e () t Estmaton eo e () t ˆx appoach asymptotcally to eal states x x espectvely In Fg 7 t s clealy seen that the estmaton eos convege to zeo asymptotcally e e Remak 7: ( ) k n Assumpton 3 s k k used n the above two examples Howeve n ths study the bound ange of some o all state vaables ae not known the bound of the uncetantes s not known ethe; hence the uppe bounds ae dffcult to estmate heefoe n ths study the values of n Example ae selected by tal eo If the chosen make the obseve synthess be feasble the smulaton s successful then the syntheszed obseve s effectve If the chosen ethe make the obseve synthess be not feasble o the smulaton s not successful we need to choose anothe couple Based on ou expeence f the feasble obseve s /o not obtaned we may decease to desgn t agan; howeve f the obseve synthess s feasble but the smulaton s not successful we may ncease We have to admt that tal eo method s not elable fo the obseve synthess t should be esolved n the futue wok /o Remak 8: It s woth notng that we assume that the uppe bounds of the uncetantes n Example Example ae unavalable theefoe the method n [] [8] wll fal to desgn obseve fo the system (56) (6) By usng the poposed method wth a new fom of the obseve () the obseve has been syntheszed successfully fo the uncetan polynomal -S fuzzy system n whch the bounded constants ae unknown the contolle dd not need to be desgned togethe wth an obseve fo the pupose of elmnatng the effects of uncetantes V ONLUSION hs pape has poposed a new appoach based on the unknown nput method to synthesze the obseve fo the uncetan polynomal -S fuzzy system On the bass of non-common Lyanpunov functon SOS technque two man theoems whch contan the condtons fo obseve Fg 7 Estmaton eos e () t e () t synthess have been deved hese condtons ae solved easly by usng SOS OOL of Matlab Fnally two examples have demonstated to pesent the effectveness of the poposed method he man chaactestcs of ths pape ae as follows () hee ae no uppe bound lmts fo uncetantes () he obseve synthess can be completed ndependently wthout desgnng contolle togethe () he obseve fom s new s notably completely dffeent fom othe exstng tadtonal fom of obseves AKNOWLEDGMEN hs wok was suppoted by the Gant MOS4--E-8-54-MY3 fom the Mnsty of Scence echnology of awan the coespondng autho D W J Wang completed ths wok when he was the vstng schola n the Unvesty of Lousvlle KY wth the Gant MOS 5-98-I-8-7 REFERENES [] akag M Sugeno Fuzzy dentfcaton of systems ts applcaton to modelng contol IEEE ans Syst Man yben vol 5 no pp [] K anaka M Sugeno Stablty analyss desgn of fuzzy contol systems Fuzzy Sets Syst vol 45 no pp Jan 99 [3] H G Zhang Y B Quan Modelng dentfcaton contol of a class of nonlnea systems IEEE ans Fuzzy Syst vol 9 no pp [4] H G Zhang D R Lu Fuzzy modellng fuzzy contol Bkhause Boston 6 [5] H G Zhang J L Zhang G H Yang Y H Luo Leade-based optmal coodnaton contol fo the consensus poblem of multagent dffeental games va fuzzy adaptve dynamc pogammng IEEE ans Fuzzy Syst vol 3 no pp [6] O astllo P Meln ype- fuzzy logc: theoy applcatons vol 3 Spnge-Velag Beln Hedelbeg 8 [7] H Y L J H Wang H K Lam Q Zhou H P Du Adaptve sldng mode contol fo nteval type- fuzzy systems IEEE ans Syst Man ybe:syst vol 46 no pp [8] K anaka H Yoshda H Ohtake Hua O Wang A sum-of-squaes appoach to modelng contol of nonlnea dynamcal systems wth polynomal fuzzy systems IEEE ans Fuzzy Syst vol 7 no 4 pp [9] B Xao HK Lam H Y L Stablzaton of nteval type- polynomal-fuzzy-model-based contol systems IEEE ans Fuzzy Syst vol 5 no pp [] A Papachstodoulou J Andeson G Valmobda S Pajna P Sele P A Palo SOSOOLS: Sum of Squaes Optmzaton oolbox fo MALAB Veson 3 6

12 [] K anaka H Ohtake H O Wang Guaanteed cost contol of polynomal fuzzy systems va a sum of squaes appoach IEEE ans Syst Man yben B yben vol 39 no pp [] M Naman H K Lam SOS-based stablty analyss of polynomal fuzzy-model-based contol systems va polynomal membeshp functons IEEE ans Fuzzy Syst vol 8 no 5 pp Oct [3] H K Lam M Naman H Y L H H Lu Stablty analyss of polynomal-fuzzy-model-based contol systems usng swtchng polynomal Lyapunov functon IEEE ans Fuzzy Syst vol no 5 pp [4] K Guelton N Manamann Duong D L Koumba-Emanwe Sum-of-squaes stablty analyss of akag-sugeno systems based on multple polynomal Lyapunov functons Intenatonal Jounal of Fuzzy Systems vol 5 no pp -8 Mach 3 [5] G R Yu L W Huang Y heng he SOS-based extended desgn of polynomal fuzzy contol the IEEE Intenatonal onfeence on Systems Man ybenetcs Octobe Seoul Koea pp [6] H K Lam S H sa Stablty analyss of polynomal-fuzzy-model-based contol systems wth msmatched pemse membeshp functons IEEE ans Fuzzy Syst vol no pp [7] J L Ptach A Sala V Ano losed-fom estmates of the doman of attacton fo nonlnea systems va fuzzy-polynomal models IEEE ans Fuzzy Syst vol 44 no 4 pp [8] R Fuqon Y J hen M anaka K anaka H O Wang An SOS-based contol Lyapunov functon desgn fo polynomal fuzzy contol of nonlnea systems IEEE ans Fuzzy Syst vol 3 no 9 pp -3 4 [9] Y J hen M anaka K anaka H Ohtake H O Wang Dscete polynomal fuzzy systems contol IE ontol heoy Appl vol 8 no 4 pp [] Y Y Wang H G Zhang Y Wang J Y Zhang Stablty analyss contolle desgn of dscete-tme polynomal fuzzy tme-vayng delay systems Jounal of the Fankln Insttute vol 35 pp [] Lu H K Lam Desgn of polynomal fuzzy obseve-contolle wth sampled-output measuements fo nonlnea systems consdeng unmeasuable pemse vaables IEEE ans Fuzzy Systvol 3 no 6 pp [] H K Lam L G Wu J Lam wo-step stablty analyss fo geneal polynomal-fuzzy-model-based contol systems IEEE ans Fuzzy Syst vol 3 no 3 pp [3] Lu H K Lam X J Ban X D Zhao Desgn of polynomal fuzzy obseve-contolle wth membeshp functon usng unmeasuable pemse vaables fo nonlnea systems Infomaton Scences vol pp [4] H G Han J Y hen H R Kam State dstubance obseves-based polynomal fuzzy contolle Infomaton Scences vol pp [5] K anaka H Ohtake Seo H O Wang An SOS-based obseve desgn fo polynomal fuzzy systems the Amecan ontol onfeence on O'Faell Steet San Fancsco A USA June 9 - July pp [6] K anaka H Ohtake M Wada H O Wang Y J hen Polynomal fuzzy obseve desgn: a sum of squaes appoach the Jont 48th IEEE onfeence on Decson ontol 8th hnese ontol onfeence Shangha PR hna Dec 9 pp [7] K anaka H Ohtake Seo M anaka HO Wang Polynomal fuzzy obseve desgns: a sum-of-squaes appoach IEEE ans Syst Man yben Pat B: yben vol 4 no 5 pp Oct [8] Y Y Wang H G Zhang J Y Zhang Y Wang An SOS-based obseve desgn fo dscete-tme polynomal fuzzy systems Int J Fuzzy Syst vol7 no pp [9] V Dang W J Wang L Luoh et al Adaptve obseve desgn fo the uncetan akag Sugeno fuzzy system wth output dstubance IE ontol heoy Appl vol 6 no pp [3] W J Wang V P Vu W hang H Sun S J Yeh A synthess of obseve-based contolle fo stablzng uncetan -S fuzzy systems J Intell Fuzzy Syst vol3 pp [3] H H ho LMI-based nonlnea fuzzy obseve-contolle desgn fo uncetan MIMO nonlnea system IEEE ans Fuzzy Syst vol 5 no7 pp [3] A Golab M Behesht M H Aseman H obust fuzzy dynamc obseve-based contolle fo uncetan akag Sugeno fuzzy systems IE ontol heoy Appl vol 6 no pp [33] V Dang W J Wang H Huang et al Obseve synthess fo the S fuzzy system wth uncetanty output dstubance J Intell Fuzzy Syst vol no 6 pp [34] H S Km J B Pak Y H Joo Robust stablzaton condton fo a polynomal fuzzy system wth paametc uncetantes the th Intenatonal onfeence on ontol Automaton Systems Jeju Isl Koea Oct pp 7- [35] K anaka M anaka Y J hen H O Wang A new sum-of-squaes desgn famewok fo obust contol of polynomal fuzzy systems wth uncetantes IEEE ans Fuzzy Syst vol 4 no pp 94-6 [36] V P Vu W J Wang Obseve synthess fo uncetan akag Sugeno fuzzy systems wth multple output matces IE ontol heoy Appl vol no pp [37] S J Yeh W hang WJ Wang Unknown nput based obseve synthess fo uncetan S fuzzy systems IE ontol heoy Appl vol 9 no7 pp [38] V P Vu W J Wang "Obseve desgn fo a dscete-tme -S fuzzy system wth uncetantes" the 5 IEEE Intenatonal onfeence on Automaton Scence Engneeng (ASE) Gothenbug Sweden Aug 5 pp 6-67 [39] A hban M hadl M M Belhaouane N B Baek Polynomal obseve desgn fo unknown nputs polynomal fuzzy systems: A Sum of Squaes appoach the 53d IEEE onfeence on Decson ontol Los Angeles alfona USA Dec 4 pp [4] A hban M hadl N B Baek A sum of squaes appoach fo polynomal fuzzy obseve desgn fo polynomal fuzzy systems wth unknown nputs Intenatonal Jounal of ontol Automaton Systems vol 4 no pp [4] K anaka Ho H O Wang A multple Lyapunov functon appoach to stablzaton of fuzzy contol systems IEEE ans Fuzzy Syst vol no 4 pp [4] S H Km P G Pak Obseve-based elaxed H contol fo fuzzy systems usng a multple Lyapunov functon IEEE ans Fuzzy Syst vol 7 no pp [43] K anaka H O Wang Fuzzy ontol Systems Desgn Analyss: A Lnea Matx Inequalty Appoach Hoboken NJ:Wley [44] A J Laub Matx Analyss fo Scentsts Engnees Sam 5 [45] P J Antsakls A N Mchel Lnea Systems chapte 7 Bkhause Boston 6 [46] Polynomal oolbox 3 Manual fo Matlab PolyX Ltd Septembe 9 [47] WS Levne he contol h book vol Jaco Publshng House 999 Van-Phong Vu eceved the BS degee n electcal engneeng fom Ha No Unvesty of Scences echnology Vetnam n 7 the MS degee n electcal engneeng fom Southen awan Unvesty of Scences echnology awan n Snce he has been a lectue at Ho h Mnh cty Unvesty of Educaton echnology Vetnam He s cuently pusung the PhD degee n electcal engneeng at Natonal ental Unvesty Hs eseach nteests ae fuzzy systems ntellgent contol obseve contolle desgn fo uncetan system

13 3 Wen-June Wang eceved the BS degee n the Depatment of ontol Engneeng fom Natonal hao-ung Unvesty Hsn-hu awan n 98; MS degee n the Depatment of Electcal Engneeng fom atung Unvesty ape awan n 984 Moeove he eceved the PhD degee n the Insttute of Electoncs fom Natonal hao-ung Unvesty of awan n 987 Pof Wang s pesently a ha Pofesso of Depatment of Electcal Engneeng He was the Dean of ollege of Electcal Engneeng ompute Scence Natonal ental Unvesty hung-l ty awan He was also a ha Pofesso the Dean of the Reseach Development Offce of Natonal ape Unvesty of echnology ape awan n 7~9 In 5~7 he was the Dean of ollege of Scence echnology Natonal h-nan Unvesty Pul Nanou awan Moeove Pof Wang obtaned the hono of IEEE Fellow n 8 IFSA Fellow n 7 Pof Wang has authoed o coauthoed ove 6 efeeed jounal papes 6 confeence papes n the aeas of fuzzy systems theoems obust nonlnea contol n lage scale systems neual netwoks etc Hs most sgnfcant contbutons ae the desgn of fuzzy systems the development of obotcs Hs othe eseach nteests nclude the aeas of obot contol neual netwoks patten ecognton etc echnology Intellgent omputng the edto of Spnge s Natual omputng Book Sees He was elected as the Pesdent of the IEEE omputatonal Intellgence Socety n 4-5 He now chas the IEEE AB Peodcals ommttee (- ) the IEEE AB Peodcals Revew ommttee (-3) a Lfe Fellow of the IEEE Hsang-heh hen eceved hs BS degee n engneeng scence fom the Natonal heng Kung Unvesty awan n hs MS PhD degees n electcal engneeng fom the Natonal ental Unvesty aoyuan awan n 5 9 Snce Aug 6 he joned the faculty of the Depatment of Electcal Engneeng Natonal Unted Unvesty Maol awan whee he s cuently an assstant pofesso Hs eseach nteests nclude mage pocessng compute vson atfcal ntellgence obotcs Jacek M Zuada eceved the MS PhD degees n electcal engneeng fom the echncal Unvesty of Gdansk Pol n espectvely He has publshed ove 4 joual confeence papes n vaous aeas Fom 998 to 3 he was the edto-n-chef of the IEEE ansactons on Neual Netwoks He was an assocate edto of the IEEE ansactons on cuts Systems Pat I Pat II seved on the edtoal boad of the Poceedngs of IEEE He s an assocate edto of Neual Netwoks Neuocomputng Schedae Infomatcae the Intenatonal Jounal of Appled Mathematcs ompute Scence the advsoy edto of the Intenatonal Jounal of Infomaton