Multirate Digital Control for Fuzzy Systems: LMI-Based Design and Stability Analysis

Transcription

1 506 Internatonal Do Wan Journal m, Jn of Control, Bae Park, Automaton, Young Hoon an Joo, Systems, an Sung vol. Ho 4, no. m 4, pp , August 006 Multrate Dgtal Control for Fuzzy Systems: LMI-Base Desgn an Stablty Analyss Do Wan m, Jn Bae Park*, Young Hoon Joo, an Sung Ho m Abstract: hs paper stues an ntellgent gtal control for nonlnear systems wth multrate samplng. It s worth notng that the multrate control esgn s aresse for a gven nonlnear system represente by akag-sugeno (-S fuzzy moels. he man features of the propose metho are that t s prove that the suffcent contons for stablzaton of the screte-tme -S fuzzy system n the sense of Lyapunov stablty crteron, whch s can be formulate n the lnear matrx neualtes (LMIs; an the stablty propertes of the trval soluton of the gtal control system can be euce from that of the soluton of ts scretze versons. An example s prove for showng the feasblty of the propose metho. eywors: Lnear matrx neualty, multrate gtal control, stablty analyss, akag-sugeno fuzzy system.. IRODUCIO Many nustral control systems consst of an analog plant an a gtal controller nterconnecte va A/D an D/A converters. Owng to the recent evelopment of the mcroprocessor an ts nterfacng harware, the gtal controller has playe an mportant role n controllng robot manpulator, chemcal process, an arcraft atttue. However, such applcatons have often severe nonlneartes, whch thus post atonal ffcultes to the gtal control esgn. So far, varous gtal control technues have been consstently pursue wth tremenous effort by many researchers n the fel. One of them s to synergetcally merge the lnear gtal control scheme [-3] an the akag-sugeno (-S fuzzy-moel-base control technology [4-] whch proves a way to acheve gtal control [4-7,] for nonlnear control systems. Drawng upon recent progress n the -S fuzzymoel-base gtal control, t s observe that a number of mportant works have use a snglerate Manuscrpt receve July, 005; accepte Aprl, 006. Recommene by Etoral Boar member Guang-Hong Yang uner the recton of Etor Shuzh Sam Ge. hs work was supporte n part by the orea Scence an Engneerng Founaton (Project number: R Do Wan m an Jn Bae Park are wth the Department of Electrcal an Electronc Engneerng, Yonse Unversty, Snchon-ong, Seoaemun-gu, Seoul 0-749, orea (emals: {wkm, jbpark}@control.yonse.ac.kr. Young Hoon Joo an Sung Ho m are wth the School of Electronc an Informaton Engneerng, unsan atonal Unversty, San 68, Mryong-ong, unsan, Chonbuk , orea (e-mals: {yhjoo, shkm}@kunsan.ac.kr. * Corresponng author. controller [4-7,] to meet the stablty reurements. he gtal control problem was conucte as a stablzng the scretze moel of contnuous-tme -S fuzzy plant n [4-6] an a stablzng the jumpe fuzzy system n []. However, strctly speakng, the problem reformulaton s ncorrect because ther scretze moel has the approxmaton error, whch s rectly proportonal to the samplng tme. One gets better exactly scretze moel f one can make A/D an D/A conversons faster. But faster A/D an D/A conversons mean hgher cost n mplementaton. In aton, the gtal control system s hybr system nvolvng contnuous-tme an screte-tme, but ther scusson n [4-6] only contane the stablty of the gtal control system n the screte-tme oman. A multrate control approach [3-6] can be an alternatve. Interestngly, avantages of applyng faster A/D an D/A conversons are obtane by usng A/D an D/A at fferent rates. Furthermore, n [5], stablty analyss between the multrate gtal control system an the screte-tme control system was well tackle. At ths pont, we attempt the multrate control for -S fuzzy system that has not yet been fully tackle uner ths framework. Motvate by the above observatons, we evelop an ntellgent multrate control for a class of nonlnear systems uner the hgh spee D/A converter. he man contrbuton of ths paper s two-fol. Frst, we erve some suffcent contons n terms of the lnear matrx neualtes (LMIs, such that the eulbrum pont s a globally asymptotcally stable eulbrum pont of the screte-tme fuzzy moel erve by the fast scretzaton n the sense of Lyapunov stablty crteron. Secon, we show that f the screte-tme control system s globally asymptotcally stable, so s

2 Multrate Dgtal Control for Fuzzy Systems: LMI-Base Desgn an Stablty Analyss 507 the resultng gtal control system. hs paper s organze as follows: In Secton, we formulates the gtal control problem of the fuzzy system wth multrate-samplng. In Secton 3, synthess an analyss of multrate gtal control system are prove. In Secton 4, the chaotc Lorenz system s use to emonstrate the effectveness of the propose metho. hs paper s conclue n Secton 5.. PROBLEM SAEME In the followng, let an ' be the samplng pero an the control upate pero, respectvely. For convenence, we take ' = / for a postve nteger, where s an nput multplcty. hen, t = k + l' for k 0 an l [0, -], where the nexes k an l ncate samplng an control upate nstants, respectvely. Conser a nonlnear gtal control system escrbe by x (= t f( x(, t u ( t ( for t [ k + l, k + l +, ( k, l 0 [0, ] n where xt ( s the state vector, an u ( t = m (,, u k l s the multrate gtal control nput. he control actons are swtche wth an. Moreover, the gtal control sgnals are fe nto the plant wth the eal zero-orer hol. o facltate the control esgn, we wll evelop a smplfe moel, whch can represent the local lnear nput-output relatons of the nonlnear system. hs type of moels s referre as -S fuzzy moels. he fuzzy ynamcal moel corresponng to ( s escrbe by the followng IF-HE rules [4-]: R : IF z( t s about Γ an an zp( t s about Γp, HE xt ( = Axt ( + Bu (, t ( where R, I = {,,, }, s the th fuzzy rule, zh(, t h Ip ={,,, p}, s the h th premse varable, an Γh,( h, I Ip, s the fuzzy set. hen, gven a par ( x( t, u ( t, usng the center-average efuzzfcaton, prouct nference, an sngleton fuzzfer, the overall ynamcs of ( has the form x (= t A( θ( t x( t + B( θ( t u ( t, (3 where A( θ( t= θ ( (, ( ( = ( (, = z t A B θ t θ z t B w (( z t p θ((= z t, w((= z t ( ( Γ h= h zh t an w (( z t Γ h ( zh ( t s the grae of membershp of zh ( t n Γ h. he possbly tme-varyng parameter vector θ belongs to a convex polytope Θ, where Θ := θ =, 0 θ. = It s clear that as θ vares nse Θ, A( θ ( t an B( θ ( t range over a matrx polytope [ A( θ( t, B( θ( t] Co {( A, B, I }, where Co enotes the convex hull. In ths note, the stablzaton of the polytopc moel (3 s euvalent to the smultaneous stablzaton of ts vertces ( A, B, I. he man problem n ths paper s to esgn the multrate feeback controller such that the eulbrum pont of (3 s a globally asymptotcally stable n the sense of Lyapunov stablty crteron. he system (3 s a hybr system nvolvng both contnuous-tme an screte-tme. hs makes the tratonal synthess an analyss methoologes usng purely screte-tme an contnuous-tme formulatons ffcult to apply. Because of these ffcultes, we frst erve the suffcent contons to globally asymptotcally stablze the eulbrum pont of the scrtze verson of (3, an then we show that the stablty of the trval soluton of (3 wth the multrate feeback controller. 3. MAI RESULS 3.. Fast scretzaton of fuzzy system o evelop the scretze verson of (3, we apply the fast scretzaton technue [4] to (3. In specfc, we frst erve a multrate scretze verson of (3, an then we apply a screte-tme lftng technue to the multrate screte-tme moel. Connectng the fast-samplng operator an the fasthol operator wth [ k + l, k + l +, ( kl, 0 [0, ], to (3 leas the multrate scretetme plant moel. Assumpton : Suppose that the frng strength θ ( z( t for t [ k + l, k + l + s θ ( z( k + l. hen, the nonlnear matrces θ = ( z ( t A an θ = ( z ( t B of (3 can be approxmate as the pecewse constant matrces A( θ ( k + l an B( θ ( k + l, respectvely. Obvously, we assume that θ( z( t = θ ( z( k + l f s suffcently small.

3 508 Do Wan m, Jn Bae Park, Young Hoon Joo, an Sung Ho m he next result presents the multrate scretze verson of (3. Proposton : he multrate screte-tme moel of (3 can be approxmate by xk ( + l + G( θ ( k+ l xk ( + l + H( θ ( k + l u ( k + l for t [ k l, k l ( , ( kl, 0 [0, ], where G( θ( k + l = θ ( z( k + l G, H( θ( k θ, ( + l = ( z( k + l H =( G =exp A, an H G I A B. Proof: he exact soluton to (3 at t = k + l + s xk ( + l + = Φ ( k + l +, k + l x( k + l k + l + k + l + Φ ( k + l +, τ B( θτ ( u ( τ τ. Uner the Assumpton, the state transton matrx Φ (, satsfes Φ ( t0, t0 = θ ( ( z k A an t Φ(, tt= Φ(, tt Φ ( t, t. hen, we get 0 0 xk ( + l + =exp A( ( k + l x( k + l ( θ k + l + ( Aθ k + l k + l + τ + exp ( ( ( k + l B( θ( k + l u ( k + l τ =exp A( ( k l x( k l ( θ + + ( exp ( A( θ ( k l I + + A ( θ( k + l B( θ( k + l u ( k + l. (5 However, (5 s not represente n the polytopc structure unlke the general fuzzy system. For ths reason, (5 s approxmate as follows: xk ( + l + =exp θ( zk ( + l A xk ( + l + exp θ( zk ( + l A I θ( z( k + l A θ( z( k + l Bu ( k + l ( θ (( zk+ l exp A xk ( + l θ ( ( + (( z k + l exp A I A B u ( k + l. (6 herefore, we obtan (4, the approxmately scretze verson of (3. In the propose scretzaton metho, two approxmatons are performe as follows: ( θ ( exp A( ( k + l G( θ( k + l, (7 G( θ( k + l I A ( θ( k + l B( θ( k + l (8 H( θ ( k + l. o analyze these, ntrouce approxmaton error efne by ( θ e = exp A( ( k + l G( θ( k + l. Applyng aylor seres expanson from the rghthan se gves e = θ ( ( z k + l A n! + +! ( = Ο. θ ( zk ( l A In the same manner, approxmaton error n (8 s ( θ ( e = G( ( k + l I A ( θ( k + l B( θ( k + l H( θ( k + l = Ο. (0 ( From (0 an (, approxmaton error clearly goes to zero as approaches zero. o transform the system (4 nto the snglerate one, we nvoke the screte-tme lftng. Assumpton : Suppose that the frng strength θ ( z( t, for t [ k, k + s θ ( z( k. It s reasonable that θ((= z t θ (( z k f s suffcently small. Proposton : Gven the system (3 for l [0, ], a lfte sample nput u ( k u ( k + u ( k = u ( k + leas a lfte system m (

4 Multrate Dgtal Control for Fuzzy Systems: LMI-Base Desgn an Stablty Analyss 509 x( k + G ( θ( k x( k + H ( θ( k u ( k (3 for t [ k, k, k 0 +, where n n G( θ( k= G ( θ( k, H( θ( k = [ G ( θ( k H( θ( k, G ( θ( k H( θ( k, n m, H( θ ( k]. Proof: Uner Assumpton, the exact soluton to (3 evaluate at t = k + s x( k + = Φ ( k +, k x( k k + (4 + Φ ( k +, τ B( θ( k u ( τ τ. k Applyng the screte-tme lftng, Φ ( k +, k k + an Φ ( k +, τ B( θτ ( u ( τ τ of (4 can k be represente by Φ ( k +, k= Φ ( k +, k + Φ ( k +, k + Φ ( k +, k an k + k ( k + Φ + k k + k + k + k + Φ 0 ( 0 = Φ ( (5 Φ k +, τ B( θ( τ u ( τ τ = ( k, τ τb( θ( k u ( k + Φ ( k +, τ τb( θ( k u ( k, + + Φ ( k +, τ τb( θ( k u ( k, = Φ ( ( α αb( θ( k u ( k +Φ Φ ( α αb( θ( k u ( k Φ ( α αb( θ( k u ( 0 k + (6 respectvely. From Proposton, we know that Φ( G( θ ( k, (7 Φ ( α α B( θ ( k H( θ ( k. (8 0 herefore, substtutng (7 an (8 to (5 an (6, respectvely, we can obtan the lfte system ( An LMI approach to multrate gtal control esgn In ths subsecton, we convert the multrate gtal control problem to the solvablty of LMIs. For the system (3, we conser the followng multrate feeback controller uk ( = ( θ ( k xk ( (9 l an have the lfte control nput represente as uk ( = ( θ ( k xk (, (0 where ( θ( k=[ 0 ( θ( k, ( θ( k,, m n ( θ ( k ], 0( θ( k= θ ( ( = z k 0, an l ( θ( k= 0( θ( k ( G( θ( k + H( θ( k 0 l ( θ ( k. he close-loop system wth (3 an (0 s x( k +,0 ( G ( θ( k + H ( θ( k ( θ( k x( k = G ( θ ( k x( k, c where Gc = G( θ( k + H( θ( k 0( θ( k. he next theorem proves the suffcent contons for the stablzaton n the sense of the Lyapunov asymptotc stablty for (3. heorem : he gven system (3 uner (0 s globally asymptotcally stable n the sense of Lyapunov stablty crteron f there exst Q = Q > 0 an constant matrces F such that Q * < 0, [, ], GQ + HF Q Q * GQ HFj GjQ HjF < 0, < j [, ], Q where * enotes the transpose element n symmetrc poston. Proof: For the system (3, choose the Lyapunov functon as V( x( k = x ( k Px( k. (4 hen, the rate of ncreases of V( x( k s ΔV( x( k = V( x( k +,0 V( x( k = x( k+,0 Pxk ( +,0 xk ( Pxk ( ( c θ c θ = x ( k G ( ( k PG ( ( k P x( k. (5 Suppose that Gc ( θ( k PGc( θ( k P< 0, then (5 s obvously negatve efnte. ote that c G ( θ( k PG ( θ( k P c

5 50 Do Wan m, Jn Bae Park, Young Hoon Joo, an Sung Ho m j ( 0j j j 0 j= θ (( zk θ (( zk G+ H + G + H 4 ( 0j j j 0 4 PG+ H + G + H P (( + 0 ( + 0 θ zk G H PG H P G + H0j + Gj + Hj0 + θ( ( θj( ( < j = ( ( z k z k P G + H0j + Gj + Hj0 P. (6 hus, f the followng two neualtes are negatve efnte, then the controlle system ( s globally asymptotcally stable. ( ( G + H P G + H P<, ( G + H0j + Gj + H j0 P G + H0j + Gj + H j0 P < 0. (8 Applyng Schur complement an the congruence transformaton wth ag P, I to (7 an (8, an lettng Q= P an F = 0P yels the LMIs ( an (3. herefore, ( s globally asymptotcally stable. Remark : In heorem, the snglerate control problem s a specal case of the multrate one wth = Stablty analyss he stablty of (3 controlle by (9 can be etermne solely by the nformaton at the samplng an control nstants. From that reason, we wll analyze the global asymptotc stablty of the close-loop system of (3 an (9 at the control nstants from stablty at the samplng nstants. hen, we wll conclue that (3 controlle by (9 s globally asymptotcally stable base on the stable propertes at the samplng an control nstants. In a preparatory stage to analyze the stablty, we nee to show that (3 an (4 are locally controllable of whch efnton s as follows: Defnton : For the controller synthess, t s assume that the fuzzy system s locally controllable, that s, ( A, B are controllable. he followng mofe results are extensons of Lemma 6 an Lemma 8 n [5]. Lemma : Assume that G satsfes that for every egenvalue of G, none of ponts λ exp π k j, =,,,, k s an egenvalue of G. If the system (4 s locally controllable, so s (3. Base on Lemma an Lemma, the next theorem conclues global asymptotcal stablty for (4 controlle by (0 from global asymptotcal stablty of the close-loop system wth (3 an (0. heorem : If the lfte system (3 controlle by (0 s globally asymptotcally stable, so s the multrate screte-tme system (4 wth (0. Proof: Euaton (4 at l = 0 s x( k + = G( θ( k x( k + H ( θ( k u( k. hen, we compute (4 at l = as xk ( + = G( θ ( k+ G( θ ( k xk ( + G( θ( k + H( θ( k u( k + H( θ ( k + u( k +. Proceeng forwar, we can realy obtan (9 at the bottom of ths page, for l = >. herefore, for l, t follows that all [0, ] x ( k = G ( θ x( k + H ( θ u ( k, (30 where xk ( + xk ( xk ( = +, x( k + G ( θ= θ ( ( = = = z k + xk ( + = G( θ( k + G( θ( k + G( θ( k x( k + G( θ( k + G( θ( k + G( θ( k + H( θ( k u( k + G( θ ( k + G( θ( k + G( θ( k + H( θ( k + u( k H( θ ( k + u( k + (9

6 Multrate Dgtal Control for Fuzzy Systems: LMI-Base Desgn an Stablty Analyss 5 θ ( zk ( + θ ( ( zk G, n H ( θ= θ( z( k + = = θ ( zk ( + θ ( zk ( H, n G G G G n =, GG G H n = H G H 0 0 H 0 GG G H GG G H H (3 n whch (,,, I I. hen, t follows from (30 an (0 that x ( k G ( θ + H ( θ ( θ( k x( k ( ( G = = = j= + H n j x( k. n (3 ote that G n + H n j s nepenent of k. herefore, x( k, l s globally asymptotcally convergent at a control nstant l n [ k +, k + ]. hs result shows that (3 controlle by (9 s globally asymptotcally stable at every ntersample ponts from stablty of the close-loop system of (3 an (0. he next theorem eals wth the stablty of (3 controlle by (9 base on Lemma an heorem. heorem 3: If (3 controlle by (9 s globally asymptotcally stable, so s the close-loop system of (3 an (9. Proof: It follows from (4 an (0, for t [ k + l, k + l + that x( t exp( A( θ ( k + l ( t k l x( k + l t k + l + exp( A( θ( k + l ( t τ B( θ( k + l l ( θ( k x( k τ exp ( A( θ ( k + l x( k + l + exp( A( θ ( k + l B( θ( k + l l ( θ( k x( k exp A x( k + l + exp A B lj j= x( k. (33 exp A exp A ote that ( an ( are nepenent of k an l. herefore, we conclue that the close-loop system of (3 an (9 s globally asymptotcally stable. 4. COMPUER SIMULAIOS Conser the Lorenz euaton x ( t σx ( t + σx ( t, (34 x ( t = rx( t x( t x( t x3( t x 3( t x( t x( t bx3( t where ( σ, rb, = (0,8,8/3. he fuzzy system corresponng to (34 s gven by x (= t A( θ ( t x( t, (35 where the membershp functons for all x [ x mn, x max ] = [ 0,30] are x ( t + x x ( t x Γ ( ( =, ( ( =, max mn x t Γ x t xmax xmn xmax xmn an the local system matrces are σ σ 0 σ σ 0 A = r x, A = r x. mn max 0 xmn b 0 xmax b Fg. shows the trajectory of the fuzzy system (35. Fg.. rajectory of the chaotc lorenz system.

7 5 Do Wan m, Jn Bae Park, Young Hoon Joo, an Sung Ho m Suppose u( k, l = l( x( k s the multrate gtal control law that globally asymptotcally stablzes the eulbrum x =[0] n of the closeloop system x (= t A( θ( t x( t + B( θ( t ( θ( k x( k, (36 where the nput matrces preservng the local controllablty of the system are arbtrarly chosen as B = B = 0. 0 l o get a feel for the tme response of the snglerate gtal control system n the long samplng tme, let us smulate the system (36 wth =. Applyng heorem yels the gtal gan matrces, as 0 [ ] [ ] = = Fg.. me responses of the close-loop system uner the snglerate gtal controller for = 0.0s (sol, = 0.04s (sol wth crcle, an = 0.08s (sol wth bullet. for = 0.0s, 0 0 [ ] [ ] = = for =0.04s, an 0 0 [ ] [ ] = = for = 0.08s. ote that we cannot obtan the feasble gtal gans n the case of = when >0.09s. Fgs. an 3 show the trajectory of the snglerate gtal control system uner state feeback. he ntal contons are x 0 = [ 0 0 0]. In fact, the snglerate gtal control system cannot cover the ntersample behavor an the scretzaton error. hese two factors may estroy the stablty of the system as ncreases. Fgs. an 3 report the behavor of the snglerate gtal control system, where the response uner state feeback evates from the eulbrum pont as ncreases. hs s the mpact of the both ntersample behavor an scretzaton error. We can overcome the both ntersample response an scretzaton error by ncreasng. When =, the multrate gtal control gans are taken as 0 0 [ ] [ ] = = Fg. 3. rajectores of the close-loop system uner the snglerate gtal controller for = 0.0s (sol, = 0.04s (sol wth crcle, an = 0.08s (sol wth bullet. for for =0.0s, 0 0 [ ] [ ] = = =0.04s, an 0 0 [ ] [ ] = = for = 0.08s. Fgs. 4 an 5 show the trajectores of the multrate gtal control system uner state feeback. he control u ( t s shown on a shorter samplng pero /. he same fgures report that all trajectores for three fferent values of are gue to the eulbrum ponts at orgn. ote that we ncrease to 0. where nfeasblty s

8 Multrate Dgtal Control for Fuzzy Systems: LMI-Base Desgn an Stablty Analyss 53 Fg. 4. me responses of the close-loop system uner the multrate gtal controller ( = for = 0.0s (sol, = 0.04s (sol wth crcle, an = 0.08s (sol wth bullet. Fg. 6. me responses of the close-loop system uner the multrate gtal controller ( = 0 for = 0.0s (sol. Fg. 5. rajectores of the close-loop system uner the multrate gtal controller ( = for = 0.0s (sol, = 0.04s (sol wth crcle, an = 0.08s (sol wth bullet. etecte n the snglerate gtal control. When =0, the multrate gtal control gans are taken as 0 = [ ] = [ ] for = 0.s. Fgs. 6 an 7 show the tme responses an the trajectores of the multrate controlle systems. he stablzablty of the gven system can be well guarantee. It s note that the propose metho guarantees the stablty of the controlle system n much wer range of samplng pero than the snglerate gtal metho n whch may fal to stablze the system especally for relatvely longer samplng pero, whch s major avantage of the propose metho. hs s because the propose multrate control can reuce the mpact of the both ntersample behavor an scretzaton error as ncreases. Fg. 7. rajectores of the close-loop system uner the multrate gtal controller ( = 0 for = 0.0s (sol. 5. CLOSIG REMARS hs paper propose the multrate control esgn usng the LMI approach for the fuzzy system. Some suffcent contons were erve for stablzaton of the scretze moel va the fast scretzaton. he stablty of the gtal control system was also guarantee. he numercal example concretely emonstrate the avantage of the propose multrate control metho over the snglerate control metho, whch mples the potental of the propose metho for relable gtal nustral applcatons. REFERECES [] W. Chang, J. B. Park, H. J. Lee, an Y. H. Joo, LMI approach to gtal reesgn of lnear tmenvarnt systems, IEE Proc., Control heory Appl., vol. 49, no. 4, pp , 00. [] H. J. Lee, J. B. Park, an Y. H. Joo, An effcent observer-base sample-ata control: Dgtal reesgn approach, IEEE rans. Crc. Syst. I, vol 50, no., pp , 003.

9 54 Do Wan m, Jn Bae Park, Young Hoon Joo, an Sung Ho m [3] B. C. uo, Dgtal Control Systems, Rnehart, Wnston, an Holt, ewyork, 980. [4] Y. H. Joo, L. S. Sheh, an G. Chen, Hybr state-space fuzzy moel-base controller wth ual-rate samplng for gtal control of chaotc systems, IEEE rans. Fuzzy Syst., vol. 7, no. 4, pp , 999. [5] W. Chang, J. B. Park, Y. H. Joo, an G. Chen, Desgn of sample-ata fuzzy-moel-base control systems by usng ntellgent gtal reesgn, IEEE rans. Crc. Syst. I, vol 49, no. 4, pp , 00. [6] H. J. Lee, H. m, Y. H. Joo, W. Chang, an J. B. Park, A new ntellgent gtal reesgn for -S fuzzy systems: Global approach, IEEE rans. Fuzzy Syst., vol., no., pp , 004. [7] W. Chang, J. B. Park, an Y. H. Joo, GA-base ntellgent gtal reesgn of fuzzy-moel-base controllers, IEEE rans. Fuzzy Syst., vol., no., pp , 003. [8] H. O. Wang,. anaka, an M. F. Grffn, An approach to fuzzy control of nonlnear systems: Stablty an esgn ssues, IEEE rans. Fuzzy Syst., vol. 4, no., pp. 4-3, 996. [9]. anaka,. osak, an H. O. Wang, Backng control problem of a moble robot wth multple tralers: Fuzzy moelng an LMI-base esgn, IEEE rans. Syst. Man, Cybern. C, vol. 8, no. 3, pp , 998. [0] Y. Y. Cao an P. M. Frank, Robust H sturbance attenuaton for a class of uncertan screte-tme fuzzy systems, IEEE rans. Fuzzy Syst., vol. 8, no. 4, pp , 000. []. anaka an H. O. Wang, Fuzzy Control Systems Desgn an Analyss: A Lnear Matrx Ineualty Approach, John Wley & Sons, Inc., 00. [] H. atayama an A. Ichkawa, H control for sample-ata fuzzy systems, Proc. of Conf. Amercan Control Conference, vol. 5, pp , 003. [3] L. S. Hu, J. Lam, Y. Y. Cao, an H. H. Shao, A lnear matrx neualty (LMI approach to robust H sample-ata control for Lnear Uncertan Systems, IEEE rans. Syst. Man, Cybern. B, vol. 33, no., pp , 003. [4]. Chen an B. Francs, Optmal Sample-Data Control Systems, Sprnger-Verlag, 995. [5] B. A. Francs an.. Georgou, Stablty theory for lnear tme-nvarant plants wth peroc gtal controllers, IEEE rans. Automat. Contr., vol. 33, no. 9, pp , 988. [6] L. S. Sheh, W. M. Wang, J. Ban, an J. W. Sunkel, Desgn of lfte ual-rate gtal controllers for X-38 vehcle, Journal of Guance Contr. Dynamcs, vol. 3, pp , 000. Do Wan m receve the B.S. an M.S. egrees n Electrcal an Electronc Engneerng from Yonse Unversty, Seoul, orea, n 00 an 004, respectvely. He s currently workng towar a Ph.D. egree n the Department of Electrcal an Electronc Engneerng, Yonse Un-versty, Seoul. orea. Hs current research nterests nclue stablty analyss n fuzzy systems, hybr ynamcal systems, fuzzy-moel-base control, an gtal reesgn. Jn Bae Park receve the B.S. egree n Electrcal Engneerng from Yonse Unversty, Seoul, orea, n 977 an the M.S. an Ph.D. egrees n Electrcal Engneerng from ansas State Unversty, Manhattan, n 985 an 990, respectvely. Snce 99 he has been wth the Department of Electrcal an Electronc Engneerng, Yonse Unversty, Seoul, orea, where he s currently a Professor. Hs research nterests nclue robust control an flterng, nonlnear control, moble robot, fuzzy logc control, neural networks, genetc algorthms, an Haamar-transform spectroscopy. He ha serve as Vce- Present for the Insttute of Control, Automaton, an Systems Engneers. He s servng as an Etor-n-Chef for the Internatonal Journal of Control, Automaton, an Systems.

10 Multrate Dgtal Control for Fuzzy Systems: LMI-Base Desgn an Stablty Analyss 55 Young Hoon Joo receve the B.S., M.S., an Ph.D. egrees n Electrcal Engneerng from Yonse Unversty, Seoul, orea, n 98, 984, an 995, respectvely. He worke wth Samsung Electroncs Company, Seoul, orea, from 986 to 995, as a Project Manager. He was wth the Unversty of Houston, Houston, X, from 998 to 999, as a Vstng Professor n the Department of Electrcal an Computer Engneerng. He s currently an Assocate Professor n the School of Electronc an Informaton Engneerng, unsan atonal Unversty, orea. Hs major nterest s manly n the fel of ntellgent control, ntellgent robot, fuzzy moelng, genetc algorthms, an nonlnear systems control. He s servng as Vce-Present for the Journal of Fuzzy Logc an Intellgent Systems (FIS ( an Drector for the ransactons of the orean Insttute of Electrcal Engneers (IEE ( an for the Insttute of Control, Automaton an Systems Engneers (ICASE (006. Sung Ho m receve the B.S., M.S., an Ph.D. egrees n Electrcal Engneerng from orea Unversty, Seoul, orea, n 984, 986, an 99 respectvely. He was wth the Hroshma Unversty, Japan, from 995 to 996, as a Vstng Professor n the Department of Inustral an Systems Engneerng. Currently, he s a Professor of the School of Electronc an Informaton Engneerng, unsan atonal Unversty, unsan, orea. Hs current research an teachng actvtes are n the area of ntellgent control an web-base remote montorng an fault agnostc system usng fuzzy logc an neural network.