Robust Stabilization of Uncertain Nonlinear Systems via Fuzzy Modeling and Numerical Optimization Programming

Transcription

1 Iteatoal Robust Joual Stablzato of Cotol, of Uceta Autoato, Nolea ad Systes, va vol Fuzzy 3, o Modelg, pp 5-35, ad Nuecal Jue 5 Optzato 5 Robust Stablzato of Uceta Nolea Systes va Fuzzy Modelg ad Nuecal Optzato Pogag Jogbae Lee, Chag-Woo Pak*, Ha-Gyeog Sug, ad Joohog L Abstact: hs pape pesets the obust stablty aalyss ad desg ethodology of the fuzzy feedback leazato cotol systes Ucetaty ad dstubaces wth kow bouds ae assued to be cluded the akag-sugeo (S) fuzzy odels epesetg the olea plats L obust stablty of the closed syste s aalyzed by castg the systes to the dagoal o bouded lea dffeetal clusos (DNLDI) foulato Based o the lea atx equalty (LMI) optzato pogag, a uecal ethod fo fdg the axu stable ages of the fuzzy feedback leazato cotol gas s also poposed o vefy the effectveess of the poposed schee, the obust stablty aalyss ad cotol desg exaples ae gve Keywods: L obust stablty, feedback leazato, fuzzy cotol, lea atx equaltes, akag-sugeo fuzzy odel INRODUCION A fuzzy odel has excellet capablty a olea syste descpto ad s patculaly sutable fo the coplex o uceta syste [] By usg ths popety of the fuzzy odels, the eseach o the fuzzy feedback leazato schee has bee coducted because the oleaty ca be effcetly odeled ad caceled by fuzzy logc syste [-8] Sce the dea of the fuzzy feedback leazato cotol based o akag-sugeo (S) odels was peseted [], vaous kds of obust [7-8] ad adaptve techques [3-5] have bee appled to the fuzzy feedback leazato cotol Whle the adaptve fuzzy feedback leazato guaatees Lyapuov stablty the pesece of ucetaty, t has soe pactcal ltatos due to ts coplex stuctues Fo a pactcal pot of vew, obust appoach s oe sutable fo the fuzzy feedback leazato to ovecoe the ucetaty [6-8] he stablty aalyss was ade the fequecy doa Mauscpt eceved Mach 5, 4; evsed Jauay 6, 5; accepted Febuay, 5 Recoeded by Edtoal Boad ebe J Youg Cho ude the decto of Edto J Bae Pak Jogbae Lee, Chag-Woo Pak, ad Ha-Gyeog Sug ae wth the Koea Electocs echology Isttute, 4-4 B/D 93,Yakdae-Dog, Wo-Gu, Pucho-S, Kyugg-Do, 4-734, Koea (e-als: {leeb, dcwpak, sughg}@ketek) Joohog L s wth School of Electcal Egeeg ad Copute Scece, Hayag Uvesty, 7, Sa dog, Sag-Rok Gu, Asa, Kyugg-do, Koea (e-al: hl@ hayagack) * Coespodg autho [6] ad the obust stablty codto ad desg ethod usg ultvaable ccle cteo have bee peseted [7] Howeve, they based o gaphcal stablty aalyss, thee exst soe dffcultes beg appled to the cotol pobles dectly O the othe had, he lea atx equalty (LMI) theoy s a ew ad fast gowg feld ad a valuable alteatve to the aalytcal ethod [9,] A vaety of pobles asg syste ad cotol theoy ca be educed to a few stadad covex o quascovex optzato pobles volvg the LMI Specfcally, fo a class of fuzzy cotol pobles whch s dffcult to solve aalytcally, the LMI techques ca affod the pactcal solutos I the ecet papes [-4], ts applcablty to the fuzzy cotol was show clealy I ode to obta the uecal solutos fo the fuzzy feedback leazato cotol systes, LMI based obust stablty codto whch ca be solved uecally fo the fuzzy feedback leazato egulato has bee peseted [8] Howeve, the oly stablty aalyss was doe ad the desg pobles wee ot hadled I addto, the tasfoato of the closed syste to LMI fo eeded soe coplex pocedue such as a loop tasfoato I ths pape, we study a cotolle desg as well as uecal stablty aalyss fo the obust fuzzy feedback leazato cotol systes usg S fuzzy odel S fuzzy odel based cotol has bee extesvely studed up to ow [,] because t ca epeset a olea equato wth a sall ube of ules [] o aalyze the obust stablty of the fuzzy feedback leazato cotol, we assue that the

2 6 Jogbae Lee, Chag-Woo Pak, Ha-Gyeog Sug, ad Joohog L ucetaty s cluded the odel stuctue wth kow bouds Fo these stuctued ucetaty, the L obust stablty of the closed syste s aalyzed by applyg the LMI based covex optzato ethod he stablty pobles ae cast to dagoal o bouded lea dffeetal clusos (DNLDI) ad a geealzed egevalue poble (GEVP) I the cotolle desg pat, based o the aalyss ethods, we peset a systeatc uecal ethod fo fdg the axu stable ages of the fuzzy feedback leazato cotol gas hs pape s ogazed as follows Secto dscusses the fuzzy feedback leazato cotol schee ad Secto 3, the uecal stablty aalyss ad desg ethod ae peseted he effectveess of the poposed aalyss ad desg schee s llustated though the detaled sulato, aely, the balacg of a veted pedulu o a cat Secto 4 Fally cocludg eaks ae collected Secto 5 PROBLEM FORMULAION he fuzzy odel epesets a olea syste wth the followg fo of fuzzy ules -th plat ule: ( ) l a a x () IF x s M ad x s M ad ad x s M ( ) he x ( + ) + ( b + b ) u+ d,,,3,, ( ) x whee x [ xx,,, ] s the state vecto whch s assued to be avalable ad a, a () t R, b, b () t R ad d R deotes ukow exteal dstubace whch belogs to L space such that dt () dt < () Also, M s the fuzzy set ad s the ube of fuzzy ules Also, a (t) ad b (t) deote the o-bouded te-vayg odelg ucetates fo syste ad put atces, espectvely he S fuzzy odel ca be feed as ( ) x a a x h ( ){( + ()) t x+ ( b + b ()) t u} + d, whee w ( x) ( ) ( x) ( ), ( x) w M x h w ( x ) (3) M (x (-) ) s the gade of ebeshp of x (-) M It s assued ths pape that w ( x),,,,, w ( x) > heefoe, h ( x),,,,, h ( x) Fo (3) to be cotollable, Uc h ( x ) b fo x a ceta cotollablty ego R s equed If ths cotollablty equeet s satsfed ad thee s o ucetaty (3), that s, a, b, d,the followg fuzzy feedback leazato cotolle (4) ca cacel the oleaty of (3) ad acheve exact leazato (5) u ( ) x d h( x)( ad a ) x, (4) h( x) b a x, (5) whee we use the sae a, b ad h ( x ) wth the fuzzy odel (3) fo all ad a d R s chose such that the exact leazed syste (5) s asyptotcally stable I pactcal applcato, howeve, ucetaty ad dstubaces ae evtable heefoe, the exact leazato caot be acheved Hece, fo the obust stablty, cosde the followg cotol law (6), R h d h( x) b ( a + ( x)( a a )) x u, (6) whee ar R s the appeded put vecto ode to educe the dstubace, whch coes fo the ucetates By substtutg (6) to (3), the closed loop syste ca be wtte as (7) ( ) x ad x+ ar x+ h( x) a x h( x) b + { h( )( ad ar a) } d x + x + h( x) b ad x+ an() t x+ d (7) whee a () t a + h( x) a () t (8) N R

3 Robust Stablzato of Uceta Nolea Systes va Fuzzy Modelg ad Nuecal Optzato 7 h( x) b + { h( x)( a + a a )} h( x) b d R I the ext secto, the obust stablty aalyss ad the desg of a fo (7) s peseted R 3 ROBUS SABILIY ANALYSIS AND DESIGN OF FEEDBACK LINEARIZAION CONROL 3 Robust stablty aalyss I ode to gve the uecal L stablty codto, the closed syste (7) s cast to Dagoal Noboud Lea Dffeetal Iclusos (DNLDI) DNLDI s a lea syste wth scala, uceta ad te-vayg feedback gas, each of whch s bouded by oe he DNLDI foulato of the closed syste (7) s gve by whee xax+bp+w, p V(t)Cx, zdx, () A R, ad ad ad3 ad B R, c c C c3 R, () c δ() t δ (t) δ3, δ an () t f c δ () t c f c costat : a () t c (,,, ) (3) N o equvaletly, pp xccx, (4) D I R, p R, z R, w R (5) d Reak : I (), c (,,, ) ca be ay o-egatve eal scala satsfyg the costat (3) o C ca be ay dagoal postve sedefte atx satsfyg the costat (4) Note that c ca be set to, oly f thee s o ucetaty the coespodg a, e an () t I Appedx A, the selectg ethod of c (,,, ) fo the stablty aalyss s poposed I (5), w s the ukow exteal dstubace put whch belogs to L space such that w w dt < (6) ad z s the output whch s the sae as the state x heoe [9]: he syste () s L stable ad ts L ga () s less tha γ f thee exst P>ad τ such that A P + PA + D D + τc C PB P BP τ I (7) γ P I Poof: Now, suppose thee exst a quadatc fucto V ( x) x Px, P>, ad γ such that fo all t, d V dt ( x) z z w w + γ x (A P+PA+D D)x+ x PBp+ Pw γ w w (8) fo all x ad p satsfyg pp xccx Usg the S-pocedue of LMI techques[9], (8) s equvalet to the exstece of P ad τ satsfyg A P+PA+D D+ τc C PB P BP τ I γ P I o show the L ga () s less tha γ, we tegate (8) fo to, wth the tal codto x(), to get ( ( )) + ( γ ) V x z z w w dt (9)

4 8 Jogbae Lee, Chag-Woo Pak, Ha-Gyeog Sug, ad Joohog L Sce V( x ( )), ths ples z w < γ () heefoe, fo the heoe, we ca obta the uppe boud o the L ga by solvg the followg EgeValue Poble (EVP) ze γ P>, τ, AP+PA+DD+ τcc PB P () BP τ I P γ I Based o the heoe, the aalyss pocedue ca be suazed as follows Step : Cast the closed syste (7) to DNLDI () Step : Select c (,,,) as Appedx A Step 3: Check the stablty codto of heoe hs ca be easly doe by solvg the feasblty poble Step 4: If thee exsts a feasble EVP soluto γ, the the closed syste s obust stable L sese ad L ga s less tha γ Also, we ca easly exted the deved put-output stablty codto to Lyapuov stablty fo the ufoced syste by the followg lea Lea : x s a globally attactve equlbu of the ufoced syste of the closed loop syste (7) (e, d) f thee exst P> ad τ whch satsfy LMI () A P+PA+D D+ τc C PB P BP τ I γ P I () Poof: he poof of ths lea wll be gve Appedx B 3 Robust stable desg Ou poble s that of deteg the L obust stablty age of a R (,,, ) whch ca ata the L ga of the closed syste (7) wth the specfed uppe boud γ ax Fo the costat (3), c ca be egaded as the uppe boud o an () t (,,,) whch was deved Appedx A heefoe, ode to detee a obust stable age o a R, we eed to fd the lagest possble c fo whch heoe holds wth γ γax should be obtaed hs ca be obtaed by solvg the followg optzato poble (3) axze c, c,, c subect to P>, τ, A P+PA+D D+ τc C PB P B P τ I (3) P γ ax I Howeve, t s dffcult to solve the ultple paaete optzato poble (3) staghtfowad Istead, by splttg (3) to the sgle paaete optzato pobles (4) fo each, t s possble to deve the feasble soluto of (3) fo the solutos of (4) axze c subect to P>, τ, A P τ +PA+D D+ C C PB P (4) B P τ I, P γ ax I whee C dag(,,, c,,,) If we defe λ c, the optzato poble (4) ca be vewed as the Geealzed Ege-Value Poble (GEVP) (5) axze λ subect to τ P>, λ A P τ +PA+D D+ C C PB P B P τ I,(5) P γ ax I whee E C c hus, the above GEVP ca be easly solved by well-establshed LMI optzato techques [] Deote the solutos of GEVP (5) as λ (,,, ) he, the solutos of the optzato poble (4) ca be wtte as c λ (,,,) Now, t should be oted that c, c,, c ca ot be a feasble soluto of the optzato poble (3) Fo, c c C c3 c C, (6)

5 Robust Stablzato of Uceta Nolea Systes va Fuzzy Modelg ad Nuecal Optzato 9 C (,,,,,, ) whee dag c, t ca ot be - - guaateed (7) holds hus, soe odfcatos ae eeded to obta a feasble soluto he odfed C ca be wtte as c c C c3 c τc τc τ c τ τc whee ( τc ), τ τ deotes τ coespodg to (7) λ o c (,,,) I heoe, t s show that heoe holds fo C (7) heoe : Fo C (7), thee exsts P> ad τ whch satsfy the LMI L stablty codto as A P + PA + D D + τc C PB P BP τ I (8) P γ axi Poof: Sce c (,,,) s the soluto of the optzato poble (4), the followg holds fo all A P + +PA+D D τc C PB P B P τ I, (9) P γ axi whee P deotes P coespodg to λ o c (,,,) Hece, fo the popety of the egatve sedefte atx (3) also holds A P + +PA+D D τ C C PB P B P τ I (3) P γ axi By eaagg the suatos, (3) becoes P P τ P P A ( ) + ( ) A+ D D+ ( ) C C ( ) B ( ) P τ B ( ) ( ) I P ( ) γ ax I Usg the popety of that τ τ C C ( τ C )( C) (3) C, t ca be easly show holds Eployg (3), (3) ca be wtte as (3) τ τ P P P P A ( ) + ( ) A+ D D+ ( C )( C) ( ) B ( ) P τ B ( ) ( ) I P ( ) ( γ ax I) (33) Let us choose P τ P ad τ (34) Usg (7), (34) ad (33) ca be expessed as AP+PA+DD+ τ C C PB P BP τ I (35) P γ ax I heefoe, fo C (7), thee exsts P> ad τ whch satsfes LMI L stablty codto (8) Sce heoe holds fo C (7), c, c,, c ca be a feasble soluto of the optzato poble (3) heefoe c (,,, ) ca be used as the lagest possble c (,,, ) fo whch heoe holds hus, usg the adssble bouds of an () t wth espect to a R, the obust stable age of a R ca be expessed by the followg set epesetato (36) a ax ( ) R ar + a t ax b + (ax + ) (36) ad ar a c b,,, he cotol desg pocedue s suazed as follows Step : Cast the closed loop syste (7) to DNLDI ()

6 3 Jogbae Lee, Chag-Woo Pak, Ha-Gyeog Sug, ad Joohog L Step : Solve the GEVP (5) Step 3: Fd the stable age (36) of a R fo (7) Step 4: Select pope a R the set (36) 4 SIMULAIONS C Cosde the poble of balacg ad swg-up of a veted pedulu o a cat show Fg he equatos of oto [3] fo the pedulu ae x x, (37) x f( x) + g( x) u+ d 4/3 l alcos( x ) gs( x ) alx s( x ) / acos( x ) u + dt (), whee [ x x ] x ad x deotes the agle ( adas) of the pedulu fo the vetcal, ad x s the agula velocty g98/s s the gavty costat, s the ass of the pedulu, M s the ass of the cat, l s the legth of the pedulu, u s the cotol foce appled to the cat ( Newtos) d(t) s the exteal dstubace ad a We choose, + M kg, M 8kg ad l the sulato he dyac equatos (37) ca be appoxated by the followg two fuzzy ules [] ad the ebeshp fuctos used ths fuzzy odel ae show Fg Rule : IF x s about HEN x ( a+ a) x+ ( b+ b) u + d π π Rule : IF x s about ± ( x < ) (38) HEN x ( a + a ( x+ ( b + b ( u + d (38) ca be feed as { } x a a x h ( ) ( + ) x+ ( b + b ) u + d, whee ( ) w w( ) M( x ( x) x ), h ( x) w ( x) g a [ 79 ] 4/3 l al, g a [ 935 ], π(4 l/3 alβ ) a b 765 4/3 l al, aβ b 5 4/3 l alβ (39) ad We assue that a, a, b, b ae ukow but bouded as follows: - a, -5 a 5, - a, -5 a 5, - b, - b I the followg aalyss ad desg secto, we use the feedback leazato cotol law as R h d h ( x) b ( a + ( x)( a a )) x u (4) ad the, the closed loop syste by substtutg (4) to (39) yelds x a x + a () t x + d, (4) d N whee a N() t a R + h( x) a () t h ( x) b { ( )( )} + h x ad + a R a h( x) b Fg he veted pedulu syste Fg Mebeshp fuctos 4 Robust stablty aalyss he obust stablty of the feedback leazato cotol syste (4) wth a d [- -] ad a R [-3-3] s aalyzed Step : Repeset the closed syste (4) to the DNLDI (4) xax+bp+w, p (t)cx, z Dx, (4) whee

7 Robust Stablzato of Uceta Nolea Systes va Fuzzy Modelg ad Nuecal Optzato 3 A c,,, B C c (43) an () t δ () t f c (t), ( ) ( ) δ t c δ t f c costat: an c (,) o equvaletly, P P x C Cx Step : Select c (, ) by ax ax b c ar + a + b ax ( a + a a ) 888, d R ax ax R + ( ) + c a a t b b ax ( ad + ar a ) 468 Step 3: Solve the followg GEVP (44) usg the teo pot ethod of LMI techques [,5] Mze λ P>, τ, A P +PA+D D+ τ C C PB P B P τ I (44) P γ I As a feasble GEVP soluto, we obta γ 38 wth τ 3 ad P (45) Step 4: Sce thee exsts a feasble EVP soluto γ >, the closed loop syste (4) s obust stable L sese ad L ga s less tha γ 38 4 Robust stable desg Cosde the desg poble fo a R,,, fo the feedback leazato cotol syste (4) wth a d [- -] Step : Cast the closed loop syste to DNLDI hs step s the sae as step the aalyss pat Step : Solve the GEVP (5) fo, Usg the GEVP solve [5], we have Fo : λ 33333, c 547, τ 5, P (46) Fo : λ , c 357, τ 5, P (47) whee γ γax was specfed Step 3: Fd the stable age of a R fo C Copute C as C C τ c C τ c τ (48) heefoe, the obust stable ages of ca be expessed by (49) ax b ar + ax a + b ar (ax ad+ ar a) 865 (49) ax b ar + ax a + b ar (ax ad + ar a ) 6667 Step 4: Select pope a R the set (49) Fg 3 shows the ego of a R ad a R fo the obtaed (49), whee we choose the paaetes as a R -5 ad a R -8 I the copute sulato, as a dstubace dt () whch belogs to L space, the sgal show Fg 4 s used Fgs 5 ad 6 llustate the sulato esults whch the tal codto s zeo I Fgs 7 ad 8, the eegy of the dstubace ad the output ae plotted wth espect to te, espectvely L o of the put ad output ca be coputed as Fg 3 Rego of a R ad a R

8 3 Jogbae Lee, Chag-Woo Pak, Ha-Gyeog Sug, ad Joohog L w dt d() t dt w w, (5) z z zdt x x dt 858 (5) hus L ga s sup w z w 858 (5) he sulato esults llustate that the closed syste (4) s obust stable L sese ad L ga s less tha whch s specfed the desg pocedue, Step Also, ode to aalyze the Lyapuov stablty, the sulato esults fo the ufoced syste, e dt () ad the tal codto x [ ] ae peseted Fgs 9 ad Fg 7 e vaato of exteal dstubace eegy Fg 8 e vaato of output eegy Fg 4 Exteal dstubace Fg 5 Sulato esult of state Fg 9 Sulato esult of state x (ufoced syste) Fg 6 Sulato esult of state Fg Sulato esult of state x (ufoced syste)

9 Robust Stablzato of Uceta Nolea Systes va Fuzzy Modelg ad Nuecal Optzato 33 5 CONCLUSIONS I ths wok, we have peseted the LMI-based L obust stablty aalyss ad desg ethod fo the fuzzy feedback leazato cotol systes he plat was epeseted by well-kow S fuzzy odel ad the aalyss ad desg pobles was uecally solved by castg the closed loop syste to DNLDI ad GEVP fo I the exaples, the fuzzy feedback leazato cotolle was developed effcetly ad the valdty of the poposed aalyss ad desg schee was show APPENDIX A Although c (,,, ) ca be ay postve eal scala satsfyg the costat (3), c (,,, ) should be chose as the u uppe boud fo a t to avod the cosevatve aalyss I ode to () N obta the u uppe boud fo an () t, (8) s wtte the copoet fo as (A) a () t a + h( x ()) t a () t (A) N R h( x) b d R h( x) b + h( x)( a + a a ), (,,, ) he, the followg equalty (A) holds fo all whch we used basc assupto, h ( x ) ad ax h ( x ( t )) a () t a + h( x()) t a () t + N R h( x) b h x t ad + ar a h( x) b x ( ( ))( ) (A) he secod ad thd tes the ght sde of (A) satsfy (A3) ad (A4) h( x) a ax a ( t ) (A3) h( x) b h( x) b d R h( x)( a + a a ) (A4) ax b (ax ad + ar a ) ax b he, the followg equalty holds fo all a () t a + ax a () t N R heefoe, we choose ax b + (ax ad + ar a ) ax b C a + ax a R ax b + (ax ad + ar a ) ax b,,, fo less cosevatve stablty aalyss (A5) APPENDIX B o pove Lea, we eed the followg heoe heoe 3 [6]: Cosde the syste, x Ax() t + Be(), t y() t Cx() t, e() t u() t Φ[, t y ()] t, (B) l whee x () t R, u () t R, y() t R ad A, B, C ae atces of copatble desos ad l Φ : R+ R R satsfes Φ (, t ), t If the followg thee codtos ae satsfed, x s a globally equlbu of the ufoced syste ) s globally Lpschtz cotuous; e, thee exsts a fte costat µ such that l Φ(, t y) Φ(, t y) µ y y, t, y, y R ) the pa (A,B) s cotollable, ad the pa (C,A) s obsevable ) the foced syste s L stable Poof: Poof of ths theoe ca be foud [6] I ode to pove Lea, the closed loop syste (7) s expessed as (B), whee A, ad ad ad3 a d B, (B)

10 34 Jogbae Lee, Chag-Woo Pak, Ha-Gyeog Sug, ad Joohog L C, an() t an Φ() t an3, u an d he, sce a N s bouded fo all ad t, we ca assue that Φ () t µ fo all t, whee µ s a fte costat heefoe, the followg equalty holds fo all t ad fo all y,y Φ() t y Φ() t y Φ()( t y y) (B3) Φ() t y y µ y y heefoe, Φ() t s globally Lpschtz cotuous ad the pa (A,B) ad the pa (C,A) ca be easly show to be cotollable ad obsevable espectvely, depedet of a d Fally, f thee exst P> ad τ whch satsfy the LMI (B4) A P + PA + D D + τc C PB P B P τ I (B4) γ P - I the, the foced syste (e d ) s L stable by heoe heefoe, by heoe 3, s a globally attactve equlbu of the ufoced syste of (7) (e d ) APPENDIX C S -pocedue of LMI theoy [9] Let F,, Fp be quadatc fuctos of the vaable ξ R such that F( ζ) ζ ζ + uζ + v,, p We cosde the followg codto o F,, Fp F ( ξ ) fo all ξ such that F ( ξ ),,,, p (C) Obvously, f thee exsts τ,, τ p such that p fo all ξ, F ( ζ) τf( ζ), (C) the (3) holds o equvaletly (3) holds p u u τ v v u u (C3) REFERENCES [] akag ad M Sugeo, Fuzzy detfcato of systes ad ts applcatos to odelg ad cotol, IEEE as o Systes, Ma ad Cybeetcs, vol 5, o, pp 6-3, 985 [] M Sugeo, Fuzzy Cotol, Nkagoubyou- Shbu-sha, okyo, 988 [3] Y W Cho, C W Pak, J H K, ad M Pak, Idect odel efeece adaptve fuzzy cotol of dyac fuzzy state space odel, IEE Poceedgs-Cotol heoy ad Applcatos, vol 48, o 4, pp 73-8, July [4] K Fschle ad D Schode, A poved stable adaptve fuzzy cotol ethod, IEEE as o Fuzzy Systes, vol 7, o, pp 7-4, 999 [5] D L say, H Y Chug, ad C J Lee, he adaptve cotol of olea systes usg the Sugeo-type of fuzzy logc, IEEE as o Fuzzy Systes, vol 7, o, pp 5-9, 999 [6] C C Fuh ad P C ug, Robust stablty aalyss of fuzzy cotol systes, Fuzzy Sets ad Systes, vol 88, o 3, pp 89-98, 997 [7] H J Kag, C Kwo, C H Lee, ad M Pak, Robust stablty aalyss ad desg ethod fo the fuzzy feedback leazato egulato, IEEE as o Fuzzy Systes, vol 6, o 4, pp , 998 [8] C W Pak, H Y Kag, Y H Yee, ad M Pak, Nuecal obust stablty aalyss of fuzzy feedback leazato egulato based o lea atx euqalty appoach, IEE Poceedgs- Cotol heoy ad Applcatos, vol 49, o, pp 8-88, [9] S Boyd, Lea Matx Iequaltes Systes ad Cotol heoy, SIAM, Phladelpha, 994 [] Y Nesteov ad A Neovsky, Iteo-Pot Polyoal Methods Covex Pogag, SIAM, Phladelpha, 994 [] H O Wag, K aaka, ad F G Gf, A appoach to fuzzy cotol of olea syste: stablty ad desg ssues, IEEE as o Fuzzy Systes, vol 4, o, pp 4-3, 996 [] K aaka, Ikeda, ad H O Wag, Robust stablzato of a class of uceta olea systes va fuzzy cotol: quadatc stablzablty, cotol theoy, ad lea atx equaltes, IEEE as o Fuzzy Systes, vol 4, o, pp -3, 996 [3] E K, H J Kag, ad M Pak, Nuecal stablty aalyss of fuzzy cotol systes va quadatc pogag ad lea atx equaltes, IEEE as o Fuzzy Systes, vol 9, o 4, pp , 999 [4] E K ad D K, Stablty aalyss ad sythess fo a affe fuzzy syste va LMI ad ILMI: dscete case, IEEE as o Systes, Ma ad Cybeetcs, vol 3, o pp 3-4,

11 Robust Stablzato of Uceta Nolea Systes va Fuzzy Modelg ad Nuecal Optzato 35 [5] P Gahet, A Neovsk, A Laub, ad M Chlal, LMI Cotol oolbox, he MathWoks, Ic, Natck, 995 [6] M Vdyasaga, Nolea Syste Aalyss, Petce-Hall, Eglewood Clffs, 993 [7] A Isdo, Nolea Cotol Systes II, Spge Velag, Lodo, 999 cotol Ha-Gyeog Sug eceved BS, MS degees Mechacs fo Hayag Uvesty 986, 995 ad PhD degee Mechacs fo Aou Uvesty 3 Sce 99, he has bee a Pcpal Reseache Koea Electocs echology Isttute Hs eseach teests ae obotcs, pecso oto desg ad tellget Jogbae Lee eceved BS, MS ad PhD degees Electocs fo Hayag Uvesty 99, 994 ad 4 Sce 995, he has bee a Maageal Reseache Koea Electocs echology Isttute Hs eseach teests ae hua copute teface, tellget cotol ad teleopeatg syste Chag-Woo Pak eceved BS degee Electocs fo Koea Uvesty 997, ad MS ad PhD degees Electocs fo Yose Uvesty 999 ad 3, espectvely Sce 3, he has bee a Seo Reseache Koea Electocs echology Isttute Hs eseach teests ae obotcs, tellget cotol ad hua-copute teface Joohog L eceved the BS degee Electocs fo Seoul Uvesty 979, MS degee Electocs fo KAS 98 ad PhD degee Electocs fo Uvesty of Iowa 986 Sce 99, he has bee a Pofesso the Electcal ad Copute Depatet of Hayag Uvesty Hs eseach teests ae syste egeeg, obotcs, ad olea syste