The Study on Direct Adaptive Fuzzy Controllers

Transcription

1 Iteratoal Joural of Fuzzy Systes, Vol., No.3, Septeber The Study o Drect Adaptve Fuzzy Cotrollers Shu-Feg Su, Jua-Chh Chag, ad Sog-Shyog Che Abstract Drect adaptve fuzzy cotrollers have bee proposed ad dscussed the lterature. Eve though such cotrollers have bee prove to be effectve, our study, soe pheoea are observed. I ths paper, those probles of usg adaptve fuzzy cotrollers for ukow olear systes are reported. The role of a paraeter atrx requred the Lyapuov equato s dscussed. It ca be foud that eve though the Lyapuov sythess approach has already prove that the syste s asyptotcally stable, the paraeter atrx stll eeds to be selected approprately besde of the syetrc postve defte property. Aother ssue s that t ca also be foud our sulatos that ths kd of adaptve fuzzy cotrollers does ot coverge to a fxed cotroller as assued the lterature, but adaptvely adjusts ts paraeters accordg to the errors. As a cosequece, whe the cosdered syste has sesory ose, the syste ay gradually becoe ustable. Ways of restrag such a ubouded pheoeo are proposed. Fro our sulatos, the proposed approaches ca have ce perforace. Keywords: Adaptve Fuzzy Cotrol, Lyapuov Theore, Stablty.. Itroducto Lgustc fuzzy odels are excellet hadlg ucerta forato ad ca also be prove to be uversal approxators []-[3]. I fact, fuzzy systes have deostrated good perforace varous real applcatos as reported the lterature [], []. Fuzzy cotrol has bee a very actve research feld the fuzzy couty ad also has show prosg results varous applcatos [9]-[], [7]. Adaptve fuzzy cotrol [7], [7]-[] s a portat ethodology fuzzy cotrol. Eve Correspodg Author: Shu-Feg Su. s wth the Departet of Electrcal Egeerg, Natoal Tape Uversty of Techology, Tawa (also wth Natoal Tawa Uversty of Scece ad Techology). E-al: su@oro.ee.tust.edu.tw Mauscrpt receved May, ; accepted Septeber,. though adaptve fuzzy cotrol has bee prove to be effectve, our study, soe pheoea are observed. Two kds of adaptve fuzzy cotrol ca be foud the lterature. I adaptve fuzzy cotrol, a adaptve cotrol law s derved to esure the covergece of the adaptve cotrol. If such a adaptve law s for the cotroller tself [7], ths cotroller s referred to as a drect adaptve fuzzy cotroller the lterature. O the other had, f the adaptve law s to odel certa ters the cosdered systes [], the obtaed cotroller s referred to as a drect adaptve fuzzy cotroller. I ths study, soe pheoea drect adaptve fuzzy cotrol are observed ad reported. I the fuzzy cotrol aalyss process, usually, the Lyapuov stable codto s eployed to guaratee the syste stablty []-[7]. The Lyapuov theore [] s to defe a eergy-lke Lyapuov fucto ad the theore states that the syste s asyptotcally stable whe the Lyapuov fucto s coverget. I the desg process of usg fuzzy adaptve cotrollers for olear systes, the Lyapuov stable codto s also eployed to guaratee the stablty of a syste [7]. I our study of fuzzy adaptve cotrollers for olear systes, we foud that eve though the Lyapuov theore has guarateed the stablty of the cotrol syste, f certa paraeters are ot properly selected, the syste ay stll becoe ustable. We shall dscuss ths ssue ths paper. Aother ssue reported the paper s that t ca also be foud our sulato that the cotroller does ot coverge to a fxed value but vares wth trajectores. It s because whe there exst errors, the cotroller wll chage those paraeters accordg to the derved adaptve law. Ths sees cotracto to what has bee claed the lterature [7]. Ths chagg behavor stulates our terest the followg proble. Ca the adaptve cotroller survve whe there exst sesory errors? I our sulato, t ca be foud that whe the adaptve cotroller s eployed for a ustable syste wth sesory ose, the output error wll drve the cotrol TFSA

2 S. -F. Su, et al.: The Study o Drect Adaptve Fuzzy Cotrollers ubouded. Thus, f o costrats are used, the syste wll gradually becoe ustable. I ths paper, two ways of restrag cotrollers are proposed to resolve ths proble. Sulatos show that both approaches ca resolve the ustable proble.. Drect Adaptve Fuzzy Cotrollers Fuzzy cotrol s a cotrol of process through fuzzy lgustc descrptos. Recetly, t has bee successfully appled to ay practcal probles [9]-[]. Usually, fuzzy cotrollers ca acheve better cotrol perforaces tha covetoal cotrollers do. Cotrol desg ay face ucertates occurrg systes or eve ukow systes. Thus, t s strogly desred to have adaptve fuzzy cotrollers. I the followg, the dervato of the drect adaptve fuzzy cotroller s troduced. Bascally, the troducto of the drect adaptve cotrol ths secto s proposed [7]. I [7] ad other slar adaptve fuzzy cotrol approaches, the ebershp fuctos are always assued to be fxed ad the cosequeces of fuzzy rules are vewed as adjustable paraeters. I ths study, we also assue the sae stuato. Thus, the adaptve process cosdered s to detfy those paraeters used to defe the cosequeces of fuzzy rules. Now cosder a th-order olear syste of the for ( ) ( ) x = f( x, x&,..., x ) + bu () y = x where f s a ukow cotuous fucto, b s a postve ukow costat, ad u R s the put of the syste, ad y R s the output of the syste. Assue that the state vector T ( ) T x = ( x, x,..., x ) = ( x, x&,..., x ) R s easurable. The cotrol objectve s to force y to follow a gve bouded referece sgal y. Hece, t s to detere a feedback cotrol u, whch s a costructed fro a tuable paraeter vector θ. If θ ca be adjusted approprately, the feedback cotrol u ca approach the optal feedback cotrol u *. Defe the trackg error as e y y. Whe e approaches zero, the th-order olear syste ca track the referece sgal well. If the th-order olear syste s kow (the cotue fucto f ad the postve costat b are kow), the the so-called perfect feedback cotrol s ( ) T u* = [ f( x) + y + c e] () b ( ) T where e = (,,..., e e& e ) R ad c = ( c, c,..., c ) T R. Whe the coeffcets c, c,..., c are properly chose, the t ples that l t et ( ) =. I practce, t s dffcult to kow the syste equato, especally for coplcated systes. Whe f ad b are ukow, the perfect cotrol u * caot be obtaed. The adaptve fuzzy cotroller proposed [7] s to develop a way of approachg ths optal feedback cotrol. The dea s to adaptvely approach the perfect cotrol by a fuzzy syste. The used fuzzy syste s to perfor a appg fro the curret states to the desred put u. The appg cossts of a set of fuzzy IF-THEN rules. I our study, the l-th rule s of the for (l) ( l) (l) R : IF x s F ad... ad x s F THEN u= θ (l), (3) ( l) where x,, x are the state varables, F,, (l) F are the correspodg fuzzy labels, ad (l) θ s the correspodg output value for the l-th rule. By usg the product operatos for the cojucto relatos the prese parts of fuzzy rules, the output of a fuzzy syste cosstg of N rules s obtaed as: u = N () l θ µ ( l ) F l = = N l = = ( ( x )) ( µ ( x )) ( l ) F = θ T ξ (x) () where µ l ) ( x ) s the ebershp degree of x F ( belogg to the fuzzy label () () ( N ) ] T (l) ( N ) ] T F, θ = [ θ, θ, L, θ, ad ξ = [ ξ, ξ, L, ξ ad s referred to as the regressve vector. Here the supscrpt T for a vector s the traspose of the vector, ad µ ( l ) ( x F ) ξ s called the fuzzy bass ( l) = = N ( l = = µ ( l ) F ( x )) fucto []. Now, defe the optal paraeter vector θ * as θ* = arg[sup u ( x θ ) u* ] () θ R x R c ad the u approxato error s T ω = uc( x θ*) u* = θ* ξ( x) u*. () Also, defe Θ = θ * θ. The desred adaptve law ca be derved a Lyapuov stablty sese. Defe a Lyapuov fucto as [7] T b T V = e Pe+ Θ Θ (7) γ where γ s a postve costat, ad P s a postve syetrc defte atrx [][3]. Let p be the last colu of P. Fro [7], the adaptve law s () ()

3 Iteratoal Joural of Fuzzy Systes, Vol., No.3, Septeber obtaed as & T θ = γe pξ( x) () T T Wth such a adaptve law, V& = e Qe e PBω ad the Lyapuov theore states that the syste s stable. I the above dervato, t s requred that T T e PBω < e Qe. As dcated [], whe there are eough rules to descrbe a fuzzy syste, the u approxato error ω wll be very sall. It s qute T T reasoable to assue that e PBω < e Qe. Hece, V & s egatve. Therefore, t ca be claed that the syste s stable [7]. 3. Study of Drect Adaptve Fuzzy Cotrollers I ths sesso, we shall report our observatos about eployg the above adaptve fuzzy cotroller for two dfferet systes. The frst oe s the syste used [7] ad the secod oe s a drastc ustable syste. The frst exaple cosdered s a frst-order ustable syste [7] descrbed as xt () e x& () t = + u() t (9) xt () + e y = x It s easy to verfy that the syste s ustable f there s o cotrol put. The cotrol objectve s to regulate the output of the syste to the org. The sae as that [7], sx fuzzy sets are defed over the terval [ 3, 3] wth ebershp fuctos as µ ( e ) = NL /( + exp(( e+ ))), µ ( e ) = NM exp( ( e+.) ), µ NS () e = exp(( e+.)), µ BS ( e) = exp( ( e.) ), µ () e = BM exp( ( e.)), ad µ BL( e) = /( + exp( ( e ))). Those ebershp fuctos are also show Fg. for llustrato. The followg values are used; x () =, γ =, b=, the saplg rate s., ad θ ()' s are zeros. Frst, we wat to study the effects of usg varous values for the paraeter P. It should be oted that P s a postve defte atrx used the Lyapuov fucto (Eq. (7)). Sce the cosdered syste s a frst-order syste, P becoes a scalar. Dfferet values are used here (P=,,, ad, respectvely). Fgures are the sulato results. Fg. shows the actual output y ad desred output y, ad Fg. shows the cotrol u for these cases. It ca be foud that whe P s large, the speed of the covergece s fast ad the value of the cotrol u s also large. It sees o proble here. Nevertheless, t ca be foud that the ext exaple, whe P s large, the syste ay becoe ustable p= p= p= p= p= p= p= p= Fgure. Fuzzy ebershp fuctos defed over the state space. Fgure. The actual output y ad the desred output ad the cotrol u, wth the dfferet P=,,,. y, Now, the cotrol objectve s to cotrol the actual output y of the ustable syste to track a desred trajectory y =s(t). The used paraeters are x () =, γ =, b=, ad the tal θ ()' s are zeros. I order to

4 S. -F. Su, et al.: The Study o Drect Adaptve Fuzzy Cotrollers 3 acheve better trackg perforace, we set P=. The sulato results are gve Fgures 3. Fg. 3 shows the the actual output y ad the desred output y. Fg. 3 s the cotrol u, ad Fg. 3 s the adaptve behavor of the paraeter θ. I the dervato of the adaptve fuzzy cotrol [7], t s assued that there exsts a optal cotrol as (). However, fro the above sulato results, t ca be foud that the cotrol sees ot coverget to a fxed values or to a lt cycle. Istead, the trajectores see dverget. Fro the adaptve law, the paraeters are tued f errors exst. I other words, the syste wll attept to chage ts cotrol law wheever there exst errors. The cotroller s to adaptvely tue the paraeters to reduce errors foud stead of usg the foud cotroller to elate errors. Ths ca be see our sulato that whe the syste s traed for a perod of te ad we stop the adaptve echas, the syste becoe ustable. I other words, the leared cotroller caot cotrol the syste. The secod exaple cosdered s also a frst-order ustable syste, but t s ore drastcally ustable tha the syste s Exaple. The cosdered syste s x x& = e + u () y = x The a objectve of the cotroller s to regulate the output of the drastc ustable syste to the org. The tal values for x, θ, γ, ad sx ebershp fuctos are the sae as that used prevous exaple. Now, the perforaces of usg varous P s are aga observed. Fg. shows the results for usg P=. It ca be see that the drect adaptve fuzzy cotroller fal to cotrol the syste. Fg. shows the result of usg P=. Now, the cotroller ca cotrol the drastc ustable syste to the org. I our sulatos, t ca be foud that the syste ca coverge for P, but dverge for P tal x () =. I other words, there exsts a lower boud for P. It should be oted that the dervato secto states that as log as P s postve the syste wll be stable a Lyapuov sese. However, fro ths result, t s easy to fd that the proposed adaptve algorth caot stablze the syste whe P s sall. Fro the adaptve law, t ca be foud that P serves as a role lke a learg costat a tradtoal learg algorth. If P s sall, t eas the tug effect s sall. But, ths exaple, the syste possesses drastc behavor. Thus, f the tug aout s too sall to atch up the drastc behavor of the syste, the syste wll certaly go ustable. I cocluso, whe we desg a drect adaptve fuzzy cotroller, the paraeter P of the Lyapuov equato (7) ust be selected approprately; otherwse, the cotroller caot have stable cotrol for ustable systes. Fally, we use the dfferet tal x () =.,.,.7,.,.9, to test the sae syste. The correspodg P s that wll ake the syste stable or ustable are tabulated Table y y Fgure 3. The actual output y ad the desred output y, the cotrol u, ad the value of the θ of the cosequece wth sx ebershp fuctos.

5 Iteratoal Joural of Fuzzy Systes, Vol., No.3, Septeber 3 3 y y y y Fgure. The actual output y s dverget for P= ad coverget for P=. Fro the above dscusso, t ca be see that f there are errors, the syste wll tue the paraeters. Now we cosder a real scearo whch there exst sesory errors; say r, whch s called a exteral ose ths study. Now, the easured error s e= y y r. The regulato proble (.e., to drve the output of ths frst-order syste to the org y =) ad the trackg proble (.e., to track a desred trajectory y = s( t) ) are both cosdered. Frst, cosder the frst exaple ad assue that P=, the exteral ose r =. rad(), ad the other paraeters are the sae as those used the above sulato for Exaple. It ca be see that the syste caot coverge. Fgures (d) show the output y, the cotrol u, the error e, the tued paraeters, respectvely, for the regulato case. Fgures (d) show the results for the trackg case. It s evdet that due to the exstece of a ozero error, there s a varato value the adaptve law. The, the paraeter θ s tued by the varato value. Such a pheoeo ca be see Fg. (d) ad Fg. (d). Sce the value θ s related to the cotrol u, the cotrol u s the dverget. Fg. ad Fg. show the behavor (d) Fgure The actual output y, the cotrol u, the error e, ad (d) the sx paraeter θ of the sple fuzzy cotroller, wth r =. rad(), P=, ad o boud.

6 S. -F. Su, et al.: The Study o Drect Adaptve Fuzzy Cotrollers Sce the dvergece behavor s drve by ubouded cotrol, whch s owg to the ubouded tug for those paraeters, a sple way of resolvg ths proble s to costra those paraeters. Two ethods are proposed to costra those paraeter tug. Ths frst oe s to set a boud M θ for all paraeters. The dea s that f all paraeters θ s do ot exceed the boud M θ, the the paraeters ca be tued accordgly. If soe of the paraeters exceed M θ, these paraeters are set to be equal to the boud M θ ad the others stll follow the adaptve law to adjust. I the sulato, we set M θ = ±. Fgures 7 (d) show the output y, the cotrol u, the error e, ad the sx tued paraeters, respectvely for the regulato case ad Fgures (d) show the results for the trackg case. Fro those sulatos, t s evdet that the cotroller ca deed acheve the desred goal soe exted. However, fro Fg. 7 (d) ad Fg. (d), t ca be foud that the cotroller evetually becoes a bag-bag cotroller. Thus, we attept to resolve ths stuato by aother approach (d) Fgure The actual output y ad the desred output y, the cotrol u, the error e, ad (d) the sx θ of the pure fuzzy cotroller wth r =. rad(), P=, ad o boud

7 Iteratoal Joural of Fuzzy Systes, Vol., No.3, Septeber (d) Fgure 7 The actual output y, the cotrol u, the error e, ad (d) the sx paraeters θ of the pure cotroller wth r =. rad(), P=, ad M θ =±. Now, we defe a boud M θ for the or of θ stead of for each paraeter. The dea s slar to the above except that f the or of θ exceeds the boud M θ, the or of θ s shruk to be equal to M θ. The update paraeters θ ca be calculated as follows: M θ ew = θ * θ () old θold I the sulato, we set M θ = ±. Fgures 9 (e) show the output y, the cotrol u, the error e, the tued paraeters, ad the or of the paraeters, respectvely for the regulato case. Fgures (e) show the results for the trackg case (d) Fgure The actual output y, the cotrol u, the error e, ad (d) the sx θ of the pure cotroller wth r =. rad(), P=, ad M θ =±

8 S. -F. Su, et al.: The Study o Drect Adaptve Fuzzy Cotrollers (d) (e) Fgure 9 The actual output y, the cotrol u, the error e, (d) the sx paraeters θ of the pure cotroller, ad (e) the or of the all paraeters θ wth r =. rad(), P=, ad M θ =± Fgure The actual output y, the cotrol u, the error e, (d) the sx θ of the pure cotroller, ad (e) The or of the all paraeters θ wth r =. rad(), P=, ad M θ = ±.

9 Iteratoal Joural of Fuzzy Systes, Vol., No.3, Septeber Fro the above, t s evdet that whe the tued paraeters are bouded as show Fg. 7 (d) ad (d) or Fg. 9 (d) ad (d), the cotrol u ca be bouded as show Fg. 7,, 9, ad. As a result, the actual output y of the syste coverges as show Fg. 7,, 9 ad. The sus of the absolute errors of the above regulato cases (as show Fg., 7, ad 9 ) are 9.9,.3, ad.9, respectvely. Table lsts the coparsos of the sus of the absolute errors of Fg.,, ad, by cycles. Fro those results, t ca be cocluded that the approach of usg bouds for the ors of the tued paraeters ca have better perforaces. Now wth costrats the tued paraeters, the perforaces of usg dfferet values for P are studed. Here, we use the boud for the or of the tued paraeters. Table 3 shows the results for the regulato case. The sallest error s observed whe P=. Table shows the results of the trackg case. The sallest error s observed whe P=. Fro Tables 3 ad, t ca be foud that whe P s too large or too sall, the resultat errors both are large. Whe there s o sesory ose, the larger s P, the better s the perforace. Whe there exst ose, a large P ay result a large aout of tug but ca have fast coverget speed the begg stage. Whe P s sall, although there s a sall error the adaptve process, the coverget speed s slow the begg stage. Thus, the paraeter P ust be selected approprately real applcatos.. Coclusos I ths paper, the drect adaptve fuzzy cotroller s studed. The adaptve cotroller s desged based o the Lyapuov sythess approach. I ths paper, we dscussed the varato of a paraeter requred the Lyapuov ethod. We foud that the paraeter atrx P ust be selected approprately besde of satsfyg the syetrc postve defte property. I our sulatos, t ca be foud that the cotroller does ot coverge to a fxed value but vares wth trajectores. If we use the orgal cotroller for a ustable syste wth sesory ose, the output error wll always drve the cotrol ubouded. I our study, two ways of restrag cotrollers are eployed to resolve ths proble. Sulatos show that both approaches ca resolve the ustable proble. Hopefully, wth our dscusso ad research, soe sghts for the desg of drect adaptve fuzzy cotrollers ca be otced. We beleve that there are soe ew deas that ca provde hts for developg ew approaches for adaptve fuzzy cotrollers.. Refereces [] L. X. Wag ad J. M. Medel, Fuzzy bass fuctos, uversal approxato, ad orthogoal least squares learg, IEEE Tras. Neural Networks, vol. 3, pp. 7, 99. [] J. J. Buckley, Sugeo type cotrollers are uversal cotrollers, Fuzzy Sets ad Systes, vol. 3, pp , 993. [3] J. L. Castro, Fuzzy logc cotrollers are uversal approxators, IEEE Tras. o Systes, Ma, ad Cyberetcs, vol., o., pp ,. [] K. Taaka ad M. Sao, A robust stablzato proble of fuzzy cotrol systes ad ts applcato to backg up cotrol of a truck-taler, IEEE Tras. o Fuzzy Systes, vol., o., pp. 9-3, 99. [] C. L. Che, P. C. Che, ad C. K., Che, Aalyss ad desg of fuzzy cotrol syste, Fuzzy Sets ad Systes, vol. 7, pp. -, 993. [] K. Taaka ad M. Sugeo, Stablty aalyss ad desg of fuzzy cotrol systes, Fuzzy Sets ad Systes, vol., pp. 3-, 99. [7] L. X. Wag, Stable adaptve fuzzy cotrol of olear systes, IEEE Tras. Fuzzy Syst., vol., pp. -, 993. [] C. T. Che, Lear Syste Theory ad Desg, 9. [9] T. Takag ad M. Sugeo, Fuzzy detfcato of systes ad ts applcatos to odelg ad cotrol, IEEE Tras o Sst. Ma ad Cb., vol., o, pp. -3, 9. [] S. Takasha, Exaples of Fuzzy Theory Applcato ostly Japa, Trgger, 99. [] Y. J. Che, Fuzzy sldg ode cotroller desg: drect adaptve approach, Cyber. Syst., vol. 3, o., pp. 9-7, 999. [] H. Yg, The Takag-Sugeo fuzzy cotrollers usg the splfed lear rules are olear varable ga cotrollers, Autoatca, vol. 3, o., pp. 7-7, 99. [3] R. C. Dorf ad R. H. Bshop, Moder Cotrol Systes, Addso-Wesley, 99. [] G. C. Goodw ad D. Q. Maye, A paraeter estato perspectve of cotuous te odel referece adaptve cotrol, Autoatca, vol. 3, pp. 7-7, 97. [] D. G. Lueberger, Lear ad Nolear Prograg, Readg, MA: Addso- Wesley, 9. [] J. Y. Che, ad C. C. Wog, Ipleetato of the Takag-Sugeo odel-based fuzzy cotrol usg a adaptve ga cotroller, IEE Proc. Cotrol Theory Appl., vol. 7, o., pp. 9-,

10 S. -F. Su, et al.: The Study o Drect Adaptve Fuzzy Cotrollers 9. [7] D. L. Tsay, H. Y. Chug, ad C. J. Lee, The adaptve cotrol of olear systes usg the Sugeo-type of fuzzy logc, IEEE Tras. Fuzzy Syst., vol. 7, o., pp. -9, 999. [] L. X. Wag, Stable adaptve fuzzy cotrollers wth applcato to verted pedulu trackg, IEEE Tras. Syst., Ma, Cyber., vol., o., pp. 77-9, 99. [9] Y. G. Leu, T. T. Lee, ad W. Y. Wag, Observer-based adaptve fuzzy-eural cotrol for ukow olear dyacal systes, IEEE Tras. Syst., Ma, Cyber., vol. 9,o., pp. 77-9, 999. [] L. X. Wag, Adaptve Fuzzy Systes ad Cotrol: Desg ad Stablty Aalyss. Eglewood Clffs, NJ: Pretce-Hall, 99. [] S.-S. Che, S.-F. Su, ad T.-T. Lee, Drect adaptve odel referece fuzzy cotrollers, Chese Autoatc Cotrol Coferece,. [] T. K. Y ad C. S. G. Lee, Fuzzy odel-referece adaptve cotrol, IEEE Tras. Syst., Ma, Cyber., vol., o., pp. -, 99. [3] S.-S. Che, S.-F. Su, ad Y.-C. Chag,, The stable trackg adaptve fuzzy cotrol of olear dyac systes usg the takag-sugeo fuzzy logc, Proceedgs of the 7 th Cof. O Artfcal Itellgece ad Applcatos, pp. -7. [] R. B. Mcla, M. A. Heso, ad M. Potta, Drect adaptve cotrol of partally kow olear systes, IEEE Tras. o Neural Networks, vol., o. 3, pp. 7-7, 999. [] S.-F. Su ad K.-Y. Che, Fuzzy herarchcal data fuso etworks for terra locato detfcato probles, IEEE Trasactos o Systes, Ma, ad Cyberetcs, Part B: Cyberetcs, vol. 3, o., pp ,. [] S.-F. Su ad S.-R. Huag, Applcatos of odel-free estators to the stock arket wth the use of techcal dcators ad o-deterstc features, Joural of the Chese Isttute of Egeers, vol., o., pp. -3, 3. [7] S.-F. Su ad W.-J. Wag, Fuzzy cotrol applcatos- Scag the ssue, Specal Issue o Fuzzy Cotrol Applcatos Iteratoal Joural of Coputer Applcatos Techology, Guest Edtors: Shu-Feg Su ad We-Jue Wag,. Shu-Feg Su receved the B.S. degree electrcal egeerg, 93, fro Natoal Tawa Uversty, Tawa, R.O.C., ad the M.S. ad Ph.D. degrees electrcal egeerg, 99 ad 99, respectvely, fro Purdue Uversty, West Lafayette, IN. He s a Professor of the Departet of Electrcal Egeerg, Natoal Tawa Uversty of Scece ad Techology, Tawa, R.O.C. He s also a Professor wth the Departet of Electrcal Egeerg, Natoal Tape Uversty of Techology ad s curretly the Secretary-Geeral at that Uversty. He has publshed ore tha 9 refereed joural ad coferece papers the areas of robotcs, tellget cotrol, fuzzy systes, eural etworks, ad o-dervatve optzato,. Hs curret research terests clude eural etworks, fuzzy odelg, ache learg, vrtual realty sulato, tellget trasportato systes, data g, ad tellget cotrol. Jua-Chh Chag receved the M.S. degrees electrcal egeerg, fro Natoal Tawa Uversty of Scece ad Techology, Tawa, R.O.C. Sog-Shyog Che receved the B.S. degree ad the M.S. degree fro the Natoal Tawa Isttute of Techology, Tape, Tawa, ad the Ph.D. degree electrcal egeerg fro the Natoal Tawa Uversty of Scece ad Techology, Tape, Tawa 99, 99, ad 3, respectvely. He s curretly a assocate Professor wth the Departet of Electrcal Egeerg, Ch-M Isttute of Techology, Tawa, R.O.C. Hs curret research terests clude olear cotrol, fuzzy odelg ad cotrol, ad robust adaptve cotrol.