Robust Dynamic Output Feedback Second-Order Sliding Mode Controller for Uncertain Systems

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1 Ieraioal Joural of Corol, Auoaio, ad Syses (3 (5: DOI.7/s ISSN: eissn:5-49 hp:// Robus Dyaic Oupu Feedbac Secod-Order Slidig Mode Coroller for cerai Syses Jeag-Li Chag Absrac: his paper addresses he proble of desigig a dyaic oupu feedbac slidig ode corol algorih o sabilize a liear MIMO ucerai syse havig relaive degree wo. Iroducig a suiable dyaic copesaor io he slidig variable, he addiioal degree of freedo ca be used o robusly guaraee he closed-loop syse sabiliy oce he syse is i he slidig ode. A odified asypoically sable secod-order slidig ode corol is aalyzed ad he proposed coroller ca obai he real secod-order slidig ode. Fially, he feasibiliy of he proposed ehod is illusraed by a uerical exaple. Keywords: Dyaic oupu feedbac, relaive degree wo, secod-order, slidig ode.. INRODCION revious researches -7] have coceraed o desigs for oupu feedbac corollers via slidig ode echique o sabilize ulivariable plas wih ached uceraiies. Early o, Za ad Hui ] developed a algorih ha uses he eigesrucure ehod o desig a oupu-depede slidig variable for ucerai syses. Kwa 3] preseed a adaped dyaic oupu feedbac coroller o reove wo ajor liis fro he schee of Za ad Hui s ehod ]. Furher, Edwards ad Spurgeo 4] have syhesized oupu feedbac corollers for ucerai syses wih referece o he ideas of slidig ode. Of he basis of aalyzig saic oupu feedbac slidig ode corol desig, wo codiios preseed here are used for checig for he exisece of a sable coroller. he firs is ha he syse us be iiu phase. he secod is a ra codiio i which he relaive degree of he rasfer fucio arix is oe. For a echaical syse usig he posiio iforaio oly, he saic oupu feedbac slidig ode corol algorih cao be direcly ipleeed, because of he lac of he ra codiio. Hece, hese wo ipora codiios lii he pracical applicaios of he aboveeioed approaches. he cocep of high order slidig ode as he geeraio of coveioal slidig ode has bee recely developed. For exaple, he case of secodorder slidig ode correspods o he corol acig o he secod derivaive of he slidig variable. Several such secod-order slidig ode algorihs have bee preseed i hese papers 8-3]. Leva 8,9] preseed he wisig algorih o sabilize secod-order oliear Mauscrip received February 3, ; revised February 7, 3; acceped May 8, 3. Recoeded by Edior Yoshio Oha. Jeag-Li Chag is wih he Depare of Elecrical Egieerig, Orieal Isiue of echology, Bachiao, New aipei Ciy, aiwa (e-ail: jlchag@ee.oi.edu.w. syses bu used owledge of he oupu-derivaive. Barolii ] developed a opiized versio of he wisig algorih. he super wisig algorih 8,9] does o require he oupu derivaive o be easured bu i has bee origially developed ad aalyzed for syse wih relaive degree oe. A robus exac fiie ie covergece differeiaor is proposed i ], which is based o his coroller. Frida e al. 3] applied he siilar echique o cosruc he velociy observer for echaical syses. A aleraive oupu feedbac secod-order slidig ode coroller for relaive degree wo MIMO syses is proposed i his paper. he developed corol algorih does o iclude ay explici differeiaor. We firs propose a odified secod-order slidig ode corol i which i ca guaraee he global asypoically sabiliy ad does o require he derivaive of he slidig variable. A suiable dyaic copesaor is iroduced io he slidig variable i which he effec of he derivaive acio ca be obaied by he addiioal copesaor. Oce he syse is i he slidig ode, he addiioal degree of freedo ca be used o robusly sabilize he closed-loop syse ad obai he desired syse perforace. he proposed corol law heoreically provides he real secod-order slidig ode.. ROBLEM FORMLAION Cosider a ucerai syse ha saisfies he ached codiio of he for ( x ( = Ax( + B u( + d x,, y( = Cx(, where x R, u R, d R, ad ( p y R are he sae posiio vecor, he corol forces, he uow ached disurbace vecor ad he oupu vecor, respecively. Wihou loss geeraliy, we assue ha ra( C = p ad ra( B = where p. Suppose ICROS, KIEE ad Spriger 3

2 Robus Dyaic Oupu Feedbac Secod-Order Slidig Mode Coroller for cerai Syses 879 ha he pairs (A, B ad (A, C are sabilizable ad deecable, respecively. If he wo codiios ( he riple (C, A, B is iiu phase ad ( ra(cb = hold, Spurgeo ad Edwards 4] have show ha a saic oupu-depede slidig variable ca be desiged o sabilize he reduced-order syse. Whe he echaical syse uses posiio easuree oly, he rasfer arix fucio has relaive degree wo, so coveioal saic oupu feedbac slidig ode corol ehods - 4] cao be direcly ipleeed i echaical syses wihou usig velociy easurees. I his paper, we cosider a dyaic oupu feedbac slidig ode corol algorih i which he proposed procedure is capable of beig used i he syse wih relaive degree wo. A odified robus globally asypoically sable secodorder slidig ode is preseed ad discussed. Iroducig a addiioal dyaic copesaor io he slidig variable ad usig he cocep of secod-order slidig ode, boh robus sabiliy of he closed-loop syse ad exeral disurbace aeuaio ca be guaraeed oce he syse is i he slidig ode. Before iroducig he proposed ehod, he followig assupios are ade hroughou his paper. Assupio : he ached disurbace d(x, has he upper boud. Assupio : he arix CAB is of full ra. Assupio 3: Syse ( is iiu phase. 3. GLOBAL SABILIY OF ERRBED SECOND-ORDER SYSEMS I his secio, we prese a preliiary resul ha will be useful i desigig he coroller. Cosider he followig syse: σ ( + lσ ( + l σ( = sig σ( + f(, ( where σ R, l, l ad are posiive cosas desiged by he user. Moreover, f ( is a exeral perurbaio wih he boud f( η, (3 where η > is a ow cosa. Lea : Cosider he uperurbed syse as σ ( + lσ ( + l σ( = sig σ(. (4 If he roos of he characerisic equaio s + ls + l = are sable, he he wo variables σ( ad σ ( asypoically coverge o zero for >. roof: Firs, we choose he paraeers l ad l such ha he roos of he characerisic equaio, s + ls + l =, are locaed i he lef-half plae. Assue ow for sipliciy ha he iiial codiios are σ = ad σ >. hus he rajecory eers he half-plae σ ( > (quadra I. Whe σ ( >, we have σ ( + lσ ( + lσ( = ad obai is equivale poi as ( σσ, = ( / l,. Sice he roos of he characerisic equaio are all sable, he curve will hi he axis σ ( = i fiie ie. Le he rajecory of syse (4 iersec ex ie wih he axis σ ( = a he poi σ (. he he rajecory eers he halfplae σ ( < (quadra III. Whe σ ( <, we have σ ( + lσ ( + lσ( = ad obai is equivale poi as ( σσ, = ( / l,. Sice he syse is sable, i follows ha he syse will hi he axis σ ( = i fiie ie. herefore, is soluios cross he axis σ ( = fro quadra II o quadra I, ad fro quadra IV o quadra III. Afer gluig hese pahs alog he lie σ ( =, we obai he phase porrai of he syse, as show i Fig.. he we choose a Lyapuov fucio as σ ( l σ ( V ( = + + σ (, ad he obai is ie derivaive as ( ( ( ( ( ( V ( = σ ( lσ ( lσ( sigσ( + lσ σ = lσ (. + sig σ σ Fro he above equaio, we ca obai ha he variables σ ( ad σ ( asypoically coverge o zero. he proof of he lea is fiished. heore : Cosider syse ( wih saisfyig (3. If he paraeers l ad l, ad he gai are chose o saisfy he followig codiios: l σ ( < l < ad > η, (5 4 he he wo variables σ ( ad σ ( asypoically coverge o zero. roof: Sice he paraeers l ad l are chose o saisfy he codiio (5, he roos of he characerisic l σ ( σ σ ( equaio, s + ls+ l =, are sable. Whe σ ( >, equaio ( becoes σ ( + lσ ( + lσ( = + f(. Le v = σ + ad v = v = σ. I follows ha l v v ( = + =Φ ( + f(, l l v( f( v b l σ ( > Fig.. hase pahs of he secod-order syse. σ

3 88 Jeag-Li Chag where v = v v, Φ=, l l Wrie he above dyaic equaio as Φ Φτ v( = e v( + e b f( τ dτ. ad b =. Sice wo paraeers l ad l are chose o saisfy (5, we ow ha i has wo disic real roos λ, = α, β where β > α >, l = α + β ad l = αβ. der his codiio, we have e Φ α β α β ( βe αe ( e e β α β α =. αβ α β α β ( e e ( αe βe α β α β he upper boud of v ( ca be cosruced as α η ατ βτ v( Ce + e e dτ β α α η β α α η = Ce + = Ce +, β α αβ l where C > is a cosa. I follows ha α σ η v ( = ( + / l C e + / l. (6 Equaio (6 shows ha he ball of radius r = η / l wih ceer locaed a ( / l, is a aracor B s. Siilar o he wor, we have, whe σ ( <, he ball of radius r, wih ceer locaed a ( / l, is aoher aracor B s. Choose he gai o saisfy he iequaliy > η ad he we have η η r = > ad + r = + <. l l l l l l I follows fro he above wo iequaliies ha he wo aracors B s ad B s do o iersec each oher, ad he behavior of he perurbed syse ( will be qualiaively siilar o he behavior of he oial syse (4. herefore, he perurbed syse coverges o he origi i he sae way of he oial syse ad he codiio ha σ ( ad σ ( asypoically go o zero ca be guaraeed. We coplee he proof of his heore. Give a slidig variable σ, however ideal slidig ode is achieved by eas of a corol sigal swichig a ifiie frequecy, which cao be aaied i real plas. I was prove 8] ha he bes possible slidig accuracy aaiable wih discree easurees is σ ( = O( ad σ ( = O(, (7 where he saplig ierval is > ad he agiude of a variable v is said o be of order O if v li ad li v =, (8 where is a ieger. We defie ha O = O(. Moreover, he codiio (7 is also called he real secod-order slidig ode. Lea 4]: If g is coiuously differeiable wih respec o is all argues ad saisfies Gdiff dg( sup = O(, d he for ay uber ζ > ad ζ >, ζ ζ = O, we have + g( ζ g( ζ = O(. heore : Cosider he perurbed syse as (. If he codiios of heore hold ad he variable σ is sapled wih a cosa saplig ierval, he, afer a fiie ie, he syse esures he esablishe of a real secod-order slidig ode, i.e., σ ( = O( ad σ ( = O(, where he saplig ierval is sufficiely sall such + ha he wo operaios O + O O ad O O( O are hold. roof: Based o he resul of heore, we ow ha he behavior of he perurbed syse ( will be qualiaively siilar o he behavior of he oial syse (3. Le σ ( = O( where, decided i he laer, is a posiive cosa. I follows fro Lea + + ha σ ( = O( ad σ ( = O(. sig wo approxiaio operaios, we ca obai σ + lσ + l σ σ + l σ + l σ + + = O + O( O + O( O O. Sice sig( σ = O(, we fro he above equaio have = ad he obai σ ( = O( ad σ ( = O(. he proof of his heore is fiished. 4. O FEEDBACK CONROLLER DESIGN Chag 5] has proposed a secod-order slidig ode corol for MIMO syses, where a addiioal dyaics is iposed o he slidig variable, bu he resulig coroller requires he derivaive of he slidig variable. I his secio, we develop a odified secodorder slidig ode coroller usig oupu feedbac oly i which he proposed algorih does o require he derivaive of he slidig variable. A oupu-depede ID slidig variable wih wo idepede gai paraeers is iroduced o obai he effec of he derivaive acio hroughou he addiioal copesaor. he proposed corol law ca obai he desired syse perforace oce he syse is i he slidig ode. p Sice Assupio holds, he arix F R which is of full ra should be foud such ha he arix

4 Robus Dyaic Oupu Feedbac Secod-Order Slidig Mode Coroller for cerai Syses 88 FCAB is iverible. Le L R be he posiive defiie diagoal arix give by L = diag( l,, l ad desig he slidig variable as w ( = Lw( LFy( + KDy( + K y( τ dτ, (9 s( = Fy( + w(, where w = w w ], s = s s ], ad he gai q q arices K D R ad K R are desiged i he laer. Fro (9, we have + = + D + s ( L s( Fy ( K y( K y( τ dτ. ( aig he ie derivaive of ( ad subsiuig he syse dyaics io i ca obai ( D s( + L s ( = FCA + K CA x( + K y( he coroller is desiged as ( d( x + FCAB u( +,. (( D u( = FCAB FCA + K CA xˆ ( + K y( + L s( + Ksig s(, ( ( where L = diag( l,, l, K = diag(,,, ad sig(( s = sig( s ( sig( s (]. he esiaio saes are geeraed by he followig Lueberger observer ẋ ˆ = ˆ Ax + ( + ( ( ˆ( Bu L y Cx. (3 Sice he pair (A, C is deecable, a gai arix p L R should be foud o sabilize he arix A LC. Subsiuig he corol ipu ( io (, we obai s( + L s ( + L s( = Ksig s( + f(, (4 where f ( = ( FCA + K DCA x ( + FCABd( x, ad x = x ˆ x is he esiaio error. he dyaic equaio of x is give by x ( = ( A LC x ( + Bd( x,. (5 Sice he arix A LC is Hurwiz ad he ached disurbace is bouded, he esiaio error x ( has he upper boud ad hece, he vecor f( is also bouded. Le he vecor f ( = f f ] have he upper boud ad he obai fi( ηi for i =,, where η i > is a ow cosa. We firs express syse (4 as a se of secod-order syses wih he for s ( + l s ( + l s( = sig s ( + f (. (6 i i i i i i Sice he sigal f i ( has he upper boud, we choose he paraeers l i, l i ad i are chose o saisfy he followig codiios: li l i < ad i > ηi, for i =,,. (7 4 Based o he resul of heore ad (7, we ow ha he syse (6 saisfies he codiio of real secod-order slidig ode ad hece, ca obai fro he cocep of equivale corol = ( + D Ksig s( FCA K CA x ( eq + FCABd( x,. he he corol ipu i he slidig ode becoes ( u ( = FCAB K y( eq ( FCA + K DCA x d( x +,. (8 (9 Subsiue his er (9 io syse ( o obai he closed-loop syse i he slidig ode as ( ( D x = A B FCAB FCA + K CA + K C x ( = BGC x, ( A C where C =, A = A B( FCAB FCA, ad G = CA ( FCAB K K ]. Fro (, we ow ha he D esiaio error dyaics ad he ached disurbace do o affec he syse behavior. Alhough syse ( has relaive degree wo, he proposed algorih ca obai he effec of he desired differeiaor ad he addiioal degree of freedo is capable of sabilizig syse (. Lea 3: Cosider he followig syse x ( = ( A B( FCAB FCA x( + Bu(, y( = Cx(. ( If Assupios o 3 hold, he syse ( is iiu phase. roof: I his proof, we shall show ha he ivaria zeros of syses ( ad ( are he sae. Firs, λi A + B( FCAB FCA B C λi A B I =. C ( FCAB FCA I If λ R is a ivaria zero of syse (, he i follows ha λi A + B( FCAB FCA B ra < +. C Hece, here exis vecors x ad g such ha he followig equaio is saisfied: λi A + B( FCAB FCA B x =. C g

5 88 Jeag-Li Chag Fro he above equaio, we have ( λi A + B( FCAB FCA x + Bg = ad Cx =. Muliplyig he equaio ( λi A + B( FCAB FCA x + Bg = fro he lef by C ad usig he codiios Cx = ad CB = ca be show ha CAx =. Hece, we ca obai ha he riple λ R, x, g saisfy he followig equaio: λi A + B FCAB FCA B x =, C g ad coclude ha λ R is also a ivaria zero of syse (. If λ R is a ivaria zero of syse (, he here exis vecors x ad g saisfyig λ I A + B FCAB FCA B x =. C g Fro he above equaio, we have ( λi A+ B( FCAB FCA x + Bg = ad Cx =. Le g 3 = ( FCAB FCA x + g ad he obai λ 3 I A B x =. C g I follows ha λ R is also a ivaria zero of syse (. Fro he above aalysis, we ca coclude ha he ivaria zeros of syse ( ad syse ( are equivale. As a resul, syse ( is iiu phase. We coplee he proof of his lea. Lea 4: If Assupios o 3 hold, he he pairs ( AB, ad ( AC, are sabilizable ad deecable, respecively. roof: Sice sae feedbac cao chage he corollabiliy, fro A = A B( FCAB FCA, we ca coclude ha he pair ( AB, is sabilizable. I follows fro Lea 3 ha si A B si A B ra = ra C C = + s C +. Sice ra( CB = ra( FCAB =, he realizaio ( CAB,, ca be wrie as 4] A A B A =, B =, C= C C], A A where he arix B R is iverible ad he p arix C R has full ra. Hece, si A B ra C A si A = ra +. C C I follows ha A si A ra = C C Re( s >. Fro liear algebraic heory ad he above ra codiio, we ca obai si A A ra ra s = Re( s >. si A = A I A C C C As a resul, he pair ( AC, is deecable. We coplee he proof of he lea. CB Fro CB = ad Lea 3, we ca ow ha CAB syse ( is iiu phase ad has relaive degree oe. Whe he syse is iiu phase ad has relaive degree oe, Schuacher 6] have show ha i ca be sabilized by direc oupu feedbac aloe. As a resul, he saic oupu feedbac desig echiques 7,8] ca be used o search he gai arix G such ha he arix A BGC is Hurwiz. Havig desiged he arix G, we ca fro G = ( FCAB K D K ] obai ha he paraeers K D ad K. Sice he arix A BGC is Hurwiz, we ca fro ( obai ha he syse perforace saisfies he followig propery: y( as. 5. NMERICAL EXAMLE o deosrae he desig echiques, a ivered pedulu echaical syse is cosidered i his secio, where he values of he physical paraeers are he sae as hose i Edwards s boo 4]. Le r, ṙ, θ, ad θ be he syse saes ad assue ha oly r ad θ are available for easuree. Moreover, he uow ached disurbace is se as d( =.5si(π +.5cos(5. For his case, he coveioal saic oupu feedbac slidig ode corollers -4] cao be successfully C ipleeed. Sice = I4, we firs assig he CA desired eigevalues of he closed-loop syse A BK K ] as {, 3 ± i,.5} o obai K D = D ] ad K = ]. Choosig L = 5 ad L =, we desig he slidig variable as s = ] y + w, w = 5 w ] y ] y( τ dτ. he coroller usig he sig fucio is give by u = ] xˆ ] y( + s + 6 sig s,

6 Robus Dyaic Oupu Feedbac Secod-Order Slidig Mode Coroller for cerai Syses 883 where he observer is desiged as ẋ ˆ = x ˆ Bu( + y ( he siulaio is carried ou a a fixed sep size of. illisecods ad he iiial saes are se as x ( =. ]. Figs. -3 show he resposes of he syse oupu ad he slidig variable, respecively. he phase plo of he slidig variable is show i Fig. 4. he seady sae error i he slidig variable, as show i Fig. 5, is of he order of 8 ad ha of ṡ ( is of he order of 4. As ca be see fro hese figures, he proposed ehod produces he real secod-order slidig ode Figs. 6-8 gives he siulaio resuls usig he sauraio fucio sa(, s ε isead of he sig fucio where ε =.5. he resposes of he syse oupu ad he corol ipu are illusraed i Figs. 6 ad 7, respecively. Fig. 8 shows ha he syse is fially cosraied i he slidig layer. Alhough he dyaic oupu feedbac corol law raises corol coplexiy ad requires addiioal sofware, he proposed corol schee globally guaraees he robus sabiliy of he closed-loop syse ad he propery of disurbace aeuaio i he proposed algorih is evide. Wihou usig he velociy feedbac, he proposed corol law ca sabilize he ivered pedulu echaical syse very well. F Fig. 4. hase plo usig he sig fucio. Z Z F Fig. 5. Resposes of s( ad ṡ ( usig he sig fucio for 4 9. Fig. 6. Syse oupus usig he sauraio fucio. W Fig.. Syse oupus usig he sig fucio. Fig. 7. Corol ipu usig he sauraio fucio. F Fig. 3. Resposes of s( ad ṡ ( usig he sig fucio. Fig. 8. Slidig variable usig he sauraio fucio.

7 884 Jeag-Li Chag 6. CONCLSION I his paper we have proposed a odified secodorder slidig ode corol algorih for a MIMO ucerai syse wih relaive degree wo. Iroducig a suiable dyaic copesaor io he slidig variable, he addiio degree of freedo ca be used o sabilize he closed-loop syse oce he syse is i he slidig ode. sig he developed slidig ode coroller, i is show ha he real secod-order slidig ode ca be guaraeed. Fially, a ivered pedulu usig posiio easuree oly is used o deosrae he algorih. he siulaio resuls deosrae ha he proposed corol schee exhibis reasoably good syse perforace. REFERENCES ] S. H. Za ad S. Hui, Oupu feedbac variable srucure corollers ad sae esiaors for ucerai/oliear dyaic syses, IEE roc. D, Corol heory ad Applicaios, vol. 4, o., pp. 4-49, 993. ] S. V. Yallapragada, B. S. Hec, ad J. D. Fiey, Reachig codiio for variable srucure corol wih oupu feedbac, Joural of Guidace, Corol ad Dyaics, vol. 9, o. 4, pp , ] C. M. Kwa, O variable srucure oupu feedbac Coroller, IEEE ras. Auoaic Corol, vol. 4, o., pp , ] C. Edwards ad S. K. Spurgeo, Slidig Mode Corol heory ad Applicaio, aylor & Fracis, Lodo, ] M. C. ai, Observer-based adapive slidig ode corol for robus racig ad odel followig, Ieraioal Joural of Corol, Auoaio, ad Syses, vol., o., pp. 5-3, 3. 6] Q. Kha, A. I. Bhai, S. Iqbal, ad M. Iqbal, Dyaic iegral slidig ode for MIMO ucerai oliear syses, Ieraioal Joural of Corol, Auoaio, ad Syses, vol. 9, o., pp. 5-6,. 7] H. C. ig, J. L. Chag, ad Y.. Che, Applyig oupu feedbac iegral slidig ode coroller o ucerai ie-delay syses wih isached disurbaces, Ieraioal Joural of Corol, Auoaio, ad Syses, vol. 9, o. 6, pp.56-66,. 8] A. Leva, Slidig order ad slidig accuracy i he slidig ode corol, Ieraioal Joural of Corol, vol. 58, o. 6, pp , ] A. Leva, High-order slidig ode, differeiaio ad oupu-feedbac corol, Ieraioal Joural of Corol, vol. 76, o. 9-, pp , 3. ] Z. Sog ad H. Li, Secod-order slidig ode corol wih bacseppig for aeroelasic syses based o fiie-ie echique, Ieraioal Joural of Corol, Auoaio, ad Syses, vol., o., pp. 46-4, 3. ] G. Barolii, A. Ferrara, ad E. sai, Chaerig avoidace by secod-order slidig ode corol, IEEE ras. o Auoaic Corol, vol. 43, o., pp. 4-46, 998. ] A. Leva, Robus exac differeiaio via slidig ode echique, Auoaica, vol. 34, o. 3, pp , ] J. Davila, L. Frida, ad A. Leva, Secodorder slidig-ode observer for echaical syses, IEEE ras. o Auoaic Corol, vol. 5, o., pp , 5. 4] J. X. Xu, F. Zheg, ad. H. Lee, O sapled daa variable srucure corol syses, Variable Srucure Syses, Slidig Mode ad Noliear Corol, K. D. Youg ad. Ozguer, Eds, vol. 47, pp.69-9, Sprig-Verlag, Lodo, ] L. W. Chag, A MIMO slidig corol wih a firs-order plus iegral slidig codiio, Auoaica, vol. 7, o. 5, pp , 99. 6] J. M. Schuacher, Alos sabiliy subspaces ad high gai feedbac, IEEE ras. o Auoaic Corol, vol. 9, o. 7, pp. 6-68, ] J. Gadewadiar, F. L. Lewis, L. Xie, V. Kucera, ad M. Abu-Khalaf, araeerizaio sabilizig H saic sae-feedbac gais: applicaio o oupu-feedbac desig, Auoaica, vol. 43, o. 9, pp , 7. 8] V. Syros, C. Abdallah,. Dorao, ad K. Grigoriadis, Saic oupu feedbac - a survey, Auoaica, vol. 33, o., pp. 5-37, 997. Jeag-Li Chag received his B.S. ad M.S. degrees i Corol Egieerig, his h.d. degree i Elecrical ad Corol Egieerig fro Naioal Chiao ug iversiy, aiwa, R.O.C., i 99, 994, ad 999, respecively. He was wih he Mechaical Research Laboraory, Idusrial echology Research Isiue, aiwa, durig I 999, he joied he Depare of Elecrical Egieerig, Orieal Isiue of echology, as a Assisa rofessor. He is currely a rofessor ad a Dea of Research ad Develope. His research ieress iclude slidig ode corol, oio corol, ad sigal processig.