Sliding Mode Control for Robust Stabilization of Uncertain Input-Delay Systems

Transcription

1 98 CSE: he siue of Corol, uoaio ad Syses Egieers, KORE Vol, No, Jue, Slidig Mode Corol for Robus Sabilizaio of Ucerai pu-delay Syses Youg-Hoo Roh ad Ju-Ho Oh bsrac: his paper is cocered wih a delay-depede slidig ode schee for he robus sabilizaio of ipu-delay syses wih bouded ukow uceraiies slidig surface based o a predicor is proposed o iiize he effec of he ipu delay he, a robus corol law is derived o esure he exisece of a slidig ode o he surface ipu-delay syses, uceraiies give durig he delayed ie are o direcly corolled by he swichig corol because of causaliy proble of he hey ca ifluece he sabiliy of he syse i he slidig ode Hece, a delay-depede sabiliy aalysis for reduced order dyaics is eployed o esiae axiu delay boud such ha he syse is globally asypoically sable i he slidig ode uerical exaple is give o illusrae he desig procedure Keywords: ucerai ipu-delayed syses, slidig ode corol, delay-depede sabiliy, robus sabilizaio, predicor-based corol Mauscrip received: Sep 3, 999, cceped: Jue, Youg-Hoo Roh: Digial ppliace Research Laboraory, LG Elecroics c, Korea Ju-Ho Oh: Dep of Mechaical Egieerig, KS roducio ie delays ca be foud i various egieerig syses such as cheical processes, peuaic/hydraulic syses, biological syses, ad ecooic syses hese delays cab be frequely a source of isabiliy ie delay also liis he achievable badwidh ad he use of high gai feedback oher ajor proble i real-world syses is he robus corol whe here is uceraiy i he syses Several auhors deal wih he corol proble of he ie-delay syses via predicor-based corollers [4][8][] Predicorbased corollers iclude a predicor o copesae for ie delay, ad so overcoe he effec of he ie delay Uder a predicor-based coroller, herefore, a ie-delay syse ca be rasfored io a delay-free syse i which he delay is eliiaed fro he closed loop syse his approach eables us o characerize he desig procedure by he delay-free syse However, if here are uceraiies, he ucerai iedelay syse is hardly rasfored io a ucerai delayfree syse because of causaliy proble of he uceraiy ers Recely, robus sabiliy ad robus sabilizaio for iedelay syses have received cosiderable aeio he sabiliy crieria ca be classified io wo caegories accordig o he depedece o he size of delays; delayidepede crieria[][][5] ad delay-depede crieria [8][6] Oe of robus sabilizaio echiques for ucerai ie-delay syses is o use he eoryless sae feedback corol May resuls ca be foud i he lieraure; he Reccai equaio approaches [], he liear arix iequaliy (LM) approaches [8], he H corol heory [7] hese approaches do o cosider copesaio for ipu delay slidig ode corol (SMC) has aracive feaures such as fas respose ad good rasie respose [3][6][4] is also isesiive o uceraiy i syse Oher SMC schees are proposed for ucerai liear syses wih sae delay oly [7][3] However, heir ehods cao esure he robus sabilizaio of ucerai ipu-delay syses because heir corollers do o use ay predicor o copesae for he ipu delay slidig ode corol wih a sae predicorbased slidig surface has bee proposed for he robus sabilizaio of ucerai liear ipu-delay syses [] he sae predicor is applied o copesae for he delay of he syse i he slidig ode pu-delay syses have a delayed corol loop ad so are sabilized by he corol ipu afer he delayed ie Hece, if here are ukow uceraiies eerig he syse, i is o easy o solve he robusess proble because of lack of causaliy of he hese ypes of syses, here usually exiss a delay boud τ such ha he syses are sabilizable for ay τ saisfyig τ τ his paper deals wih delay-depede codiio for robus sabilizaio of ucerai ipu-delay syses wih a predicorbased SMC proposed i he lieraure [] robus corol law is derived o esure he exisece of a slidig ode ad o overcoe he effecs of he delay ad uceraiy i he slidig ode dyaics he here is a axiu delay boud for robus sabilizaio of he ucerai ipu-delay syse uder he corol Delay-depede sabiliy aalysis for reduced order dyaics i he slidig ode is eployed o esiae he delay boud Syse descripio Le us cosider a liear ucerai syse wih ipu delay described by x &( ) = x () Bu ( τ ) f( x (),) f ( x ( τ ),) () where x R, u( ) R ad τ ([, ), R) are he sae vecor, he ipu vecor ad he delay ie, respecively, ad, B are cosa arices wih appropriae diesios he ukow uceraiies f ( x ( ), ) ad f( x ( τ ), ) represe he oliear perurbaios wih respec o he curre sae ad he delayed sae, respecively addiio o (), he iiial codiios are give by ) = x, x ( θ) = φ( θ), u ( θ) = v( θ), τ θ () where x( θ) = θ) ad u( θ) = u( θ) is assued ha he syse is corollable, ie, rak [,exp( sτ ) B] = for ay s, ad he saes are available for feedback We also assue ha he uceraiies f, f:r R R saisfy he achig codiio, ie,

2 rasacio o Corol, uoaio, ad Syse Egieerig Vol, No, Jue, 99 where f ( ), ) = Be ( ), ) f ( x ( τ), ) = Be( x ( τ), ) e ( ), ) ρ ) e ( τ ), ) ρ τ) (3) (4) he, he equivale corol is obaied by u SMC τ ( θ τ) [ Se B] S[ ) Ze Bu( θ) dθ] eq = τe [ Se B ] S x = τ (9) for posiive cosas ρ, ρ > Here deoes he or pu-delay syses have a delayed corol loop ad so are sabilized by he corol ipu afer he delayed ie Hece, if here are ukow uceraiies ha eer he syse, i is o easy o esure he robus sabilizaio because of lack of iforaio ad causaliy of he d ipu-delay syses are o also corollable for he iiial ie, [ τ,] Durig he iiial ie, ozero iiial saes ad uceraiies ca affec he sabiliy of he syses hese ypes of syses i is usually eeded ha he codiio, v( θ) L(( τ,), R ) exiss S ( Se ) σ ) τ B SB φ φsg(σ) ( Se ) ( s) g ˆδ τ B S - u u u() eq e τ B Be sτ f (, ), τ)) ( s ) Be sτ ) Desig of slidig ode coroller wih delay copesaio We cosider a predicor (predicive sae), x R as follows () = () ( τ θ) ( θ ) τ x x e Bu dθ (5) he slidig surface is defied as σ = Sx = (6) whereσ = σ,, σ R ad S S S = R,, is oed ha he proposed slidig surface icludes he sae predicor, ad so yields slidig ode dyaics ha copesaes for he ipu delay is assued ha arix S ad B are of full rak ad Se τ B is o-sigular he, he arix S is chose such ha he dyaics o he slidig surface has he desired closed-loop behaviors fer selecig he slidig surface, he ex sep is o choose he corol law such ha i saisfies he codiio for he exisece of he slidig ode; σ &< σ his codiio esures ha he corol law will force syse rajecories oward he slidig surface i fiie ie ad aiai he o he surface aferwards We cosider he followig corol srucure of he for show i fig u ()= u u (7) where u eq is a equivale corol for he oial syse of () wihou he uceraiy ad u N is a swichig corol o overcoe he uceraiies of he syse he equivale corol law u eq is derived by &σ = for he oial syse of () he derivaive of σ alog he oial syse of () is eq [&( ) ( τ θ) ( ) τ ( ) τ & σ = Sx e Bu θ dθ e Bu Bu( τ )] ( τ θ) τ [ ( ) ( θ) θ ( )] τ = Sx e Bu d e Bu N (8) Fig Block diagra of he proposed SMC he equaio (9) represes a sae predicor-based feedback corol ha copesaes for he delay of he oial syse of () Now, we eed o eliiae he effec of he uceraiies i spie of he delays, ad also o force he syse rajecories oward he desiged slidig surface he, he swichig corol u N is chose by ( se u = N τ B) SBB S σ ˆ( δ x, ) B S σ if B S σ oherwise () where δ( x, ) = ρ x β, for ρ = ρ ρ q, q>, β >, is he upper boud o he or of he luped uceraiy of he syse he swichig corol () is deeried o reove he effec of ipu delay ad uceraiy by he slidig surface he uceraiy give a ie ca o be cacelled by he swichig corol uil he ie, τ, because of he delayed ipu iplies ha he uceraiies give i he syse are sequeially cacelled by he correspodig swichig corols afer each delay ierval Reark :Sice he corol ipu eers he syse wih he ierval of delay, τ, he reachig oio of he slidig ode is geeraed afer he delay, ad so he slidig ode exiss a > s τ for iiial ie order o esure he exisece of he slidig ode, we cosider he ie derivaive of σ alog he ucerai ipudelay syse () as ( θ τ) τ [ ( ) ( ) ( ) τ & σ = Sx e Bu θ dθ e Bu () f( ), ) f( τ ), )] he dyaics () is cosraied o he slidig surface he dyaics (), he uceraiy give a ie is cosidered bu he uceraiies (doed arrows) give durig he delayed period are o cosidered because of lack of causaliy of he as show i fig is said ha he

3 CSE: he siue of Corol, uoaio ad Syses Egieers, KORE Vol, No, Jue, dyaics () is i ideal slidig ode Now we are ready for he followig Fig Corol propery of he proposed SMC schee heore : f he corol law (7) is used for he syse (), he a slidig ode always exiss, ie, he dyaics (), is asypoically sable Proof: We choose he Lyapuov fucio V( σ, ) = σ σ () he ie derivaive of V alog he rajecories of he syse () is ( ) V& θ τ τ = σ S[ ) e Bu( θ) dθ e Bu( ) τ (3) f( ), ) f( τ ), )] Subsiuig (3) ad (7) io he above equaio yields & τ V= σ [ Se BuN SBe { ( x ( ), ) e ( x ( τ), )}] (4) Subsiuig () io he above equaio yields { ( ( ), ) ( ( ), ) } & (5) V B S σ ρ x β e x e x τ Fro he Razuikhi heore, x ( θ ) < q ), q >, τ θ, he e ( τ), ) ρ τ) ρ q ) (6) hus τ g τ θ = ( ) e τ Uceraiies [ f (, ), τ ))] g Bu( θ) dθ ) ) SMC e ( ), ) e( τ), ) e e ρ ) (7) u() τ where ρ = ρ ρq > We ca fially obai he followig iequaliy: V& β B S σ < (8) for σ Sice β is posiive, σ as Fro he heore he slidig ode of he dyaics () alog he slidig surface σ = always exiss i he fiie ie V Global sabiliy of syse We see ha he ideal slidig ode dyaics icludes uceraiy give a ie, However, he uceraiies give durig he delayed period, which are o cosidered i he ideal slidig ode dyaics, ca ifluece he sabiliy of he acual syse i he slidig ode hough hey will be cacelled ou afer each ierval of he delay is eeded o ivesigae he effec of he upo he ideal slidig ode dyaics We firs rasfor he ucerai ipu-delay syse () o a ucerai delay-free syse by differeiaig he sae predicor (5) alog he rajecories of he syse () as follows x& τ () = ) e Bu() f( ),) f( τ ),) (9) Sice he syse (9) cosiders oly he uceraiy a ie, he relaio f( ), τ ) f( τ), τ) () τ = e { f( x ( ), ) f( x ( τ ), )} is derived he, he delay-free syse (9) is rewrie as x& τ τ = x e Bu() e { f( ), τ ) () f ( τ), τ)} Subsiuig (3) io he above equaio yields x& = x Bu %() Be %{ ( x (), τ ) e( x ( τ), τ)} () ~ where B = e τ B Le us cosider rasforaio arix as B% = (3) B % where B% z R is osigular d defie z = x = z where z R, z R he, he syse is represeed by z& () = z () z () (4) z& () = z () z () B% { u() e ( z( ), τ ) e ( z( τ), τ)} where = Le us defie a arix K 3 4 ad cosider he followig cosrai z (5) R ( ) = Kz (6) Sice (, B ) is a corollable pair, he arix pair (, ) is also corollable Cobiig (4) ad (6) give he reduced order dyaics of he syse (9) i he slidig ode as z&( ) = ( K) z() (7) is said ha he dyaics (7) is i ideal slidig ode geeral, he arix K is chose so ha assigs eigevalues of he reduced order syse (7) o he lef-half plae We ow will ivesigae he sabiliy of he reduced order dyaics of he acual syse wih uceraiies ha are o cosidered i he ideal slidig ode dyaics Le us defie Z Z = x = where = e τ he, he reduced order dy- Z aics of he acual syse i he slidig ode, which

4 rasacio o Corol, uoaio, ad Syse Egieerig Vol, No, Jue, icludes uceraiies give for a period of he delay, is rewrie as ) z& () = ( K) z() f( z( θ ),) ) (8) f ( z ( θ τ), ) where = = ad he uceraiies give for τ θ are obaied by ) (9) f( z( θ ), ) = f z( s), s ds τ K ) f( z( θ τ ), ) f z( s ), s ds τ K τ (3) = is show ha he uceraiy ers of he above reduced order dyaics ca o saisfy achig codiio due o he causaliy proble he equaio (8) is also closed-loop dyaics of he acual syse () uder he corol law (7) heore : f he Lyapuov equaio P( K) ( K) P = Q (3) ad he followig codiio λ ( Q) τρ PB > i (3) K are saisfied for he posiive cosa ρ = ρ ρq, q >, ad posiive defiie arix P ad posiive defiie syeric arix Q, he syse () uder corol law (7) is asypoically sable Proof: Cosider he Lyapuov fucio V = z Pz (33) he derivaive of he above equaio alog he rajecories of he syse (8) is give by & V = z Pz& z& Pz ) ) λ ( Q) z P( f f ) z (34) i where λ i () deoes he iiu eigevalue of arix () is easily show ha ) ) P( f f) Pf( z( s), s) τ K Pf( z( s τ ), s) ds K (35) V & λi( Q) τρ PB z K (38) he syse (8) is asypoically sable if he codiio (3) is saisfied; he here is a sufficiely sall q > such ha codiio (3) is saisfied Cosequely, he syse () uder he corol law (7) is asypoically sable he proof is copleed Fro heore i is possible o esiae a axiu delay boud, τ ha guaraees he asypoic sabiliy of he syse () i he slidig ode as follows: λi ( Q) < τ τ = ρ PB K (39) is poied ou ha he sae predicor-based SMC proposed here is a powerful ool for robus sabilizaio of ucerai ipu-delay syses, while he sae-based ehod show i he previous work [3][6][7][4] is icapable of dealig wih he ipu-delay syses addiio, eployig τ =, he sae predicor-based SMC is reduced o he saebased oe V llusraive exaple order o illusrae he procedure of he proposed SMC schee, we cosider a usable pla as follows: x &() = x () u ( τ ) 3 ( e ( x ( ), ) e ( x ( τ ), )) (4) where a iiial codiio, ) = [ 6 ], ad he oliear paraeer perurbaios are give by e( ),) = 5 x()si( x()) e ( τ ), ) = 3 x ( τ )si( x ( τ )) ca be easily see ha ρ = 5, ρ = 3 he desig objecive is o deerie he corol law ad axiu delay boud τ ha robusly sabilizes syse (4) By he desig procedure, assigig 5 as he eigevalue of he reduced order dyaics (8) resuls i K = 5 Choosig q = ad = yields PB = ad = 5 99 for K PB { Z ( s) Z ( s ) } ds ρ ρ τ (36) K τ Fro he Razuikhi heore [5] we obai ) ) P( f f) τρ PB z( ) K where ρ = ρ ρ q, q > he we obai (37)

5 CSE: he siue of Corol, uoaio ad Syses Egieers, KORE Vol, No, Jue, Fig 3 Siulaio resuls of he proposed SMC for delay ie τ = 5 sec P =, Q = Fro (39) he axiu delay boud τ = 84 is obaied We selec τ = 8 ad S = [ ] for uerical siulaio he siulaio resuls are show i Fig 3-6 hey show ha he proposed SMC esures he robus sabilizaio of ipu-delay syse (4) wih he oliear paraeer perurbaios Fig 4 Siulaio resuls of he proposed SMC for delay ie τ = sec Fig 5 Siulaio resuls of he proposed SMC for delay ie τ = sec ad f (, ))= Fig 6 Siulaio resuls of he proposed SMC for delay ie τ = sec ad f (, τ )) = is also show ha he reachig oio of he slidig ode is geeraed afer a period of delay, τ he slidig surface diverges rapidly durig he iiial ie, τ, bu i is bouded i he give delay ie Oce i reaches zero ad he aiais i s he delay ie is geig larger, he syse respose is geig faser bu overshoo is geig larger is show i Fig 5-6 ha he uceraiy er of curre sae affecs he syse respose ore ha he uceraiy er of delayed sae V Coclusios his paper, we have proposed a delay-depede slidig ode corol for robus sabilizaio of ucerai ipu-delay syses Our ehod uses a predicor o copesae for he ipu delay robus corol law is derived o esure he exisece of he slidig ode he proposed schee has a reachig oio of slidig ode a s > τ ad is depede o he size of he delay axiu delay boud for robus sabilizaio is esiaed by he delay-depede sabiliy aalysis of he reduced order dyaics i he slidig ode he siulaio resuls have show ha he proposed ehod effecively corolled he ipu-delayed syse wih oliear paraeer perurbaios he proposed schee, where τ =, ca be applied o syses wihou ipu delay Refereces [] E Cheres, S Gua, ad Z J Palor, Sabilizaio of ucerai dyaic syses icludig sae delay, EEE ras uoa Corol, vol C-34, pp 99-3, 989 [] H H Choi ad M J Chug, Meoryless sabilizaio of ucerai dyaics syses wih ie varyig delayed saes ad corols, uoaica, vol 3, pp , 994 [3] R DeCarlo, S H Zak, ad G P Mahews, Variable srucure corol of oliear ulivariable syses: uorial, Proceedigs of EEE, pp -3, 988 [4] Y Fiagedz ad E Pearso, Feedback sabilizaio of liear uooous ie lag syses, EEE ras

6 rasacio o Corol, uoaio, ad Syse Egieerig Vol, No, Jue, 3 uoa Corol, vol C-3, pp , 986 [5] J K Hale ad S M Verduy Luel, roducio o fucioal differeial equaios, Spriger-Verlag, New York, 993 [6] J Y Hug, W Gao, ad J C Hug, Variable srucure corol: survey, EEE ras o dusrial Elecroics, vol 4, pp -, 993 [7] J Koshkouei ad S Ziober, Slidig ode iedelay syses, Proc of Workshop o VSS, okyo, pp 97-, 996 [8] W H Kwo ad E Pearso, Feedback sabilizaio of liear syses wih delayed corol, EEE ras uoa Corol, vol C-5, pp 66-69, 98 [9] X Li ad C E Souza, Delay-depede robus sabiliy ad sabilizaio of ucerai liear delay syses: a liear arix iequaliy approach, EEE ras uoa Corol, vol C-4, pp 44-48, 997 [] Z Maiius ad W Olbro, Fiie specru assige proble for syse wih delays, EEE ras, uoa Corol, vol C-5, pp 66-69, 979 [] Y H Roh ad J H Oh, Robus sabilizaio of ucerai ipu-delay syses by slidig ode corol wih delay copesaio, uoaica, vol 35, pp , 999 [] J C She, B S Che, ad F C Kug, Meoryless sabilizaio of ucerai dyaic delay syses: Riccai equaio approach, EEE ras uoa Corol, vol C-36, pp , 99 [3] K K Shyu ad J J Ya, Robus sabiliy of ucerai ie-delay syses ad is sabilizaio by variable srucure corol, era J Corol, vol 57, pp 37-46, 993 [4] S K Spurgeo ad R Davies, oliear corol sraegy for robus slidig ode perforace i he presece of uached uceraiy, era J Corol, vol 57, pp 7-3, 993 [5] J H Su, Furher resuls o he robus sabiliy of liear syses wih a sigle ie delay, Syses Corol Le, vol 3, pp , 994 [6] J Su ad C G Huag, Robus sabiliy of delay depedece for liear ucerai syses, EEE ras uoa Corol, vol C-37, pp , 99 [7] L Yu, J Chu ad H Su, Robus eoryless H coroller desig for liear ie delay syses wih or-bouded ie varyig uceraiy, uoaica, vol 3, pp , 996 Youg-Hoo Roh Youg-Hoo Roh was bor i Korea o pril, 96 He received he BS ad MS degrees i echaical egieerig fro ha Uiversiy i 984 ad 986, ad he PhD degree fro KS i 999, respecively He is ow a seior researcher i LG Elecroics c sice 988 His research eress iclude robus corol, oliear corol, robus sabilizaio of ie-delay syses, ework corol, ad syse ideificaio ad corol i idusrial appliaces Ju-Ho Oh Ju-Ho Oh was bor i korea o Oc 3, 954 He received he BS ad MS degrees i echaical egieerig fro Yosei Uiversiy i 977 ad 979, ad he PhD degree fro Uiv of Califoria, Berkeley i 985, respecively He was a researcher i Korea oic Eergy Research siue fro 979 o 98 Fro , he was a visiig professor i Uiversiy of exas, usi Sice 985, he has bee a Professor of Mechaical Egieerig i KS His Mai eress iclude opial corol, eural ework, fuzzy corol ad slidig ode corol, ad robo corol such as quadruped walkig robo, ierface of sesors ad syses