Stabilization of Networked Control Systems with Variable Delays and Saturating Inputs

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1 Sablzao of Newored Corol Syses wh Varable Delays ad Saurag Ipus M. Mahod Kaleybar* ad R. Mahboob Esfaa* (C.A.) Absrac:I hs paper, less coservave codos for he syhess of sac saefeedbac coroller are roduced o sablze ewored corol syses subec o acuaor saurao. Boh of he daa loss ad laecy whch deerorae he perforace of he closed-loop syse are odeled as he varable delays. wo dffere echques are eployed o por acuaor saurao he coroller desg procedure. he ovely of he proposed schees s o ulze a proved Lyapuov-Krasovs fucoal, freewegh arx ad paraeer ug ehod o oba ore effce codos o deere sae-feedbac ga for a cosraed syse whch s corolled over he coucao ewor. Moreover, opzao proble s forulaed order o fd he larges possble esae for he rego of araco correspodg o axu allowable delays. Nuercal exaples are preseed o deosrae he ouperforace of he suggesed approaches copared o he exsg resuls he leraure. Keywords: Ipu Saurao, Lear Marx Iequaly (LMI), Newored Corol Syses, Varable Delay. Iroduco Newored Corol Syse (NCS) s a feedbac srucure where he corol loop s closed hrough a coucao ewor. he advaages of NCSs such as low cos ad sple sallao ad aeace ae he ore ad ore popular ay real-world applcaos cludg dusral auoao ad ulage syses []. However, he presece of coucao ewor he corol loop coplcaes he aalyss ad desg of he corol syse. Ma ssues are he delay ad dropou of daa paces whch occur whe sesors, acuaors ad coroller exchage forao across he ewor. he desg of NCSs wh cosderg he effecs of daa delay ad dropou has bee suded by ay researchers [-4]. O he oher had, physcal cosras, especally acuaor saurao are ecouered praccal syses [5]. So, he pu saurao he aalyss ad syhess of e-delay syses has araced recely ay aeos [6-6]. I [6], sablzg coroller was desged for NCSs wh acuaor saurao ad saplg perod varao. A couous fucoal whose values a he saplg sas cocdes wh a dscree-e Lyapuov Iraa Joural of Elecrcal & Elecroc Egeerg, 4. Paper frs receved 7 Aug. ad revsed for 8 Feb. 4. * he Auhors are wh he Depare of Elecrcal Egeerg, Sahad Uversy of echology, abrz, Ira. E-als: ahboob@su.ac.r ad _ahod@su.ac.r. fuco s ulzed o derve suffce codo ers of lear arx equaly (LMI) for copug he sablzg coroller. I [7], sablzao proble of ewored sochasc syses subec o acuaor saurao was suded. he oleary of acuaor saurao was odeled as a covex polyope of lear syses. I [8], he proble of desgg sae-feedbac sablzg coroller ad elargg he coroller doa of araco s forulaed as a opzao proble wh LMI cosras. I [9], sple suffce LMI codos are derved for sablzao of syses wh polyopc uceray for regoal sablzao of syses wh sapled-daa sauraed sae-feedbac va descrpor approach. he regoal sablzao ad H corol proble have bee suded [] by cobg he descrpor odel rasforao ad Moo's equaly whch s used o ge a less coservave boud for he cross ers. Usg a ew Lyapuov- Krasovs fucoal ad geeralzed secor relao, codos were exraced [] for he coroller desg, ag a he elargee of he rego of araco, as well axzg he upper boud of he saplg perod. I [], sablzao proble of eural delay syses he presece of corol saurao s solved based o he descrpor approach ad he use of a odfed secor codo. hs paper preses less coservave procedures o syhess asypocally sablzg sae-feedbac coroller for ewored corol syses subec o 96 Iraa Joural of Elecrcal & Elecroc Egeerg, Vol., No., Jue 4

2 acuaor saurao. he NCS odel developed [] s adoped ad he oleary of acuaor saurao s acled wo ways: I he frs approach, he saurao s represeed by a covex polyope of lear syses. I he secod schee, geeralzed secor codo (deceralzed dead zoe oleary) s used o hadle he saurao effecs. he ey deas he proposed ehods are frs, o use a proved Lyapuov- Krasovs fucoal ad secod, o ulze he freewegh arx ad paraeer ug ehods for exracg proved codos o oba he coroller ga for ewored syses wh saurag pu, ag a elargg he esae of he rego of araco ad axzg axu allowable delay boud. A challegg ssue he coroller syhess for he olear processes s sablzg he closed-loop syses whle achevg he larges possble doa of araco,.e. elargee he se of al saes for whch he asypoc covergece of he syse raecores o he org s esured. hus, hs oe, a copuaoally racable opzao proble wh LMI cosras s forulaed o fd a less coservave esae for he doa of araco. Sulao resuls deosrae ha he desged coroller leads o larger doa of araco whle creases he axu allowable delay boud. he paper s orgazed as follows: I seco, he NCS odel s descrbed ad he he proble of eres s explaed. Seco preses soe prelary facs whch wll be used he dervao of he a resuls of he paper. he proposed procedures o deere coroller ga, axu allowable delay ad doa of araco are derved seco 4. I seco 5, uercal exaples are gve o llusrae he superory of he proposed ehods copared o he exsg resuls he leraure. Seco 6 cocludes he paper. Noaos: R deoes he desoal Eucldea space wh vecor or ad R s he se of all real arces. he oao P> (P ) eas ha P s syerc ad posve defe (posve se defe). he subscrp sads for arx rasposo. Co{} sybolzes he covex hull. dag{} s used as a ellpse for bloc-dagoal arx. σ () deoes he larges sgular value of he arx. he sybol * shows he syerc ery a syerc arx. Fally, he space of couously dffereable vecor fuco over [-η, ] s represeed by C [-η, ]. Proble Saee A ypcal ewored corol syse s show Fg., where he coroller, sesor ad he acuaor are assued o be separaed ad coeced hrough a coucao ewor. he corolled syse s lear ad e vara, sesor s e-drve ad coroller ad acuaor are eve-drve. I he cosdered ZOH x ( ) = Ax( ) + Bu( ) τ ca sa ewor, all he daa are luped ogeher o oe pace ad rased a he sae e (sgle pace rassso) ad he se paces are e saped. he coroller ad acuaor always use he ew daa paces ad dscard he old oes. Whe a old daa pace arrves, s deal wh as a pace loss. A zeroorder-hold s placed he pu of he pla ad he pu s zero before he frs coroller pace arrves. Regardg he above assupo o he NCS, he followg equaos ca descrbe he closed-loop syse behavor: x () = Ax() + Bu() () u( ) = sa( Kx ( τ )), [, ) h+ τ + h+ τ + () where x () R ad u () R are he sae ad he corol vecors, respecvely. A ad B are wo cosa arces wh approprae desos. K s he sae feedbac ga arx. h sads for he saplg perod. =,,, s he uber of he corols whch ac o he syse, s a eger deog he saplg sa of he sae feedbac correspodg o he -h effecve corol. rassso delay ad loss duced by he ewor s coposed of wo pars: sesor-o-coroller τ sc ad coroller-o-acuaor τ ca. Sce he coroller s sac, hese wo values ca be luped ogeher as τ = τ, sc + τca where e-varyg τ represes he ewor-duced delay ad dropou a he sa h. he fuco sa( ): R R deoes sadard saurao: sa ( u ) = sg ( u ) ( u, u ) wh u = ax( u). he closed-loop syse odel Eqs. ()-() ca be represeed as x () = Ax() + B sa( Kx( h)) for [ h+ τ, ) + h+ τ +. Now, by defo of τ () = h, [ h+ τ, ) + h+ τ +, hs relao ca be rewre as Eq. (): x () = Ax() + B sa( Kx( τ ())) () whch s a couous-e syse wh delayed pu. Noe ha he varyg delay s bouded as follows: τ τ( ) ( ) h+ τ η (4) + + Coroller Fg. Scheac Dagra of he Sauraed NCS. x( ) τ sc Mahod Kaleybar & Mahboob Esfaa: Sablzao of Newored Corol Syses 97

3 Furherore, he al codo for syse of Eq. () s a couous dffereable fuco whch s show as he followg: x = φ( θ ), θ [ η,] (5) Brefly, he NCS s odeled as he olear edelay syse Eq. (), where he varable delay characerzed Eq. (4) represes boh of he daa loss ad laecy he ewor. he proble of eres s o deere he sae-feedbac gak such ha he coroller Eq. () reders he closed-loop syse of Eqs. ()-() asypocally sable; as well a esae of he doa of araco s obaed. Prelares I hs seco, soe useful facs whch are eeded o solve he explaed proble are recalled. Frs, he sably heore of e-delay syse s preseed ad aferward, esseal defos ad relaos o forulae he acuaor saurao are revewed. heore (Lyapuov-Krasovs): Suppose ha f aps a bouded se froc [-η, ] o a bouded se o R, ad α, α, α : R R are couous, o-decreasg fucos wh α ()=α ()=α ()= adα (s)>, α (s)> for s>. If here exss a couous fucoal V : C [ η,] R such ha α ( x()) V α ( sup x( ) ), V <α ( x( )) (6) [ η,] he he equlbru of Eq. () s sable. If, addo, α (s)> for s>, he s asypocally sable. Defo [7]: Le be he h row of he arx K, a polyhedro rego L( K ) he sae space s defed as follows: { } L( K) = x R : x u, =,,,. (7) Furherore, a ellpsod E he sae space s characerzed as he followg: E ( P,) = { x R : x Px } (8) where, P R s a posve defe arx. Defo [7]: he se ν cosss of all dagoal arces whose dagoal elees are eher or ; so, he uber of ebers ν are. Le he arx D,,,, = be a eber of he se ν, ad defe: D = I D. I s clear ha he arx D s also a eber of ν,.e. D, D ν. I he Lea, based o he defos ad, he saurao fuco of vecors belog o a polyhedro rego s descrbed as a covex cobao of well-defed verces. Lea [7]: Le KH, R are gve; for all desoal vecor x L( H ), he followg holds: sa ( Kx) Co{ D Kx+ D Hx, =,, } (9) Hece, sa ( Kx) ca be expressed as follows: sa( Kx) = λ ( D K+ D H) x () = whch, λ = ad λ. = Defo [8]: Deceralzed dead-zoe oleary s he vecor fuco ψ whch s defed as follows: ψ ( Kx) = Kx sa( Kx ) () Lea [8]: Cosder he fuco ψ ( Kx) defed Eq.(). For x R f x L( KH ), he followg s hold: ψ ( Kx) U( ψ( Kx) Hx ) () for ay dagoal posve defe arx U R. he resul of Lea whch s ow as geeralzed secor codo wll be ulzed laer o rasfor he desg codos o LMI for. Defo 4: Le ϕ(, x ) be he sae raecory ofhe syse of Eq. (), sarg fro he al fuco x C [ η,] ; he doa of araco of he org s defed as he followg: S = { x C [ η,] : l ϕ(, x ) = } () Furherore, a esae of he doa of araco Χ DOA S ca be obaed as follows: Χ DOA = { x S :ax x δ, ax x δ} (4) by axzg posve scalars δ ( =,). 4 Ma Resuls I hs subseco, he suffce codos are derved o deere sae-feedbac ga o asypocally sablze he syse of Eq. (). Based o he covex represeao of saurao fuco Lea, desg codo s roduced heore o oba he coroller ga. he resul of heore s used Corollary o deere he larges possble esae of he doa of araco. I heore, usg he propery of deceralzed dead zoe oleary Lea, aoher crero s derved o oba he coroller ga ad correspodg doa of araco. heore : Gve scalars η > ad p, =,,4 he syse of Eq. () wh he ewored eoryless sae-feedbac coroller Eq. () s asypocally 98 Iraa Joural of Elecrcal & Elecroc Egeerg, Vol., No., Jue 4

4 sable f here exs arces P = P >, Q = Q >, R = R >, G, Y ad osgular arx Φ = Ω of approprae desos such ha he followg arx equales hold: Φ <, =,,..., (5) us gs, s =,,..., sp (6) where Φ = Ω wh: Q R R P * R R Ω = (7) * * η R * * * Q R AX B( DY+ DG) X pax pb( DY+ DG) px Ω = (8) pax pb( DY+ DG) px p4ax p4b( DY+ DG) p4x ad g s s he s-h row of G; Furherore, K = YX ad H = GX. A esae of he doa of araco s he for of Eq. (4) wh δ ad δ sasfyg: ( δ σ( X PX ) + η σ( X QX ) ) η (9) δ σ + ( X RX ) Proof: Regardg he Lea, he closed-loop syse Eq. () s represeed as a ore racable for of Eq. (): x () = Ax() + λ B( D K+ D H) x( h) () = provded ha x L( H ), where ad = λ =. herefore, he syse equao verex s as follows: x () = Ax() + A x( h) () whch A = B( DK+ D H ) for =,,. I wha follows, he dervave of a approprae eergy fucoal o every verex of he syse, represeed Eq. () s se o be egave. Iproved Lyapuov-Krasovs fucoal caddae s cosdered as follows: V() x () Px() x () s Qx ( s) ds = + η + η x () s Rx () s ds dθ () η + θ λ whch, P= P >, Q= Q > ad R= R > are o be deered. Calculag he e dervave of V () alog he raecores of he syse of Eq. () yelds o: V () = x () Px () + x Qx() x ( η) Qx( η) + η x () Rx () η x () s Rx () () s ds η o oba desg codo ers of arx equales, frs, a quadrac upper boud s derved for he egral er V (). o hs ed, he followg relao s used: η x () s Rx () s ds = η η h η x () s Rx () s dsη x () s Rx () s ds h (4) O he oher had, regardg he Jese Lea [], he followg equales hold: η x ( s) Rx ( s) ds (5) h [ x( ) x( h)] R[ x( ) x( h)] h η x () s Rx () s ds η [ x( h) x( η)] R[ x( h) x( η)] (6) So, subsug Eqs. (5) ad (6) Eq. (4) resuls he followg upper boud for V (): V x ( ) Px ( ) + x ( ) Qx( ) x ( η) Qx( η) [ x( h) x( η)] R[ x( h) x( η)] (7) x () Rx() [ x() x( h)] R[ x() x( h)] + η Now, le us defe ξ() = [ x(), x( ), (), ( )] h x x η ; s obvous ha for ay arx M, he followg relao s rue: ξ () Mx [ () Ax() B( D K+ D Hx ) ( h)] = (8) I should be oed ha M s a free-wegh arx whch s eced upper boud of V () o crease he degree of freedo he fal desg codo o reduce he coservaveess of he obaed suffce crero. Addg Eq. (8) o Eq. (7) ad arragg he obaed relao yelds o: V () ξ () Φ ξ() (9) where Φ= Ω+ Ω + Ω ad Q R R P * R R Ω = * * η R * * * QR () Ω = MA MB ( D K + D H ) M. () Mahod Kaleybar & Mahboob Esfaa: Sablzao of Newored Corol Syses 99

5 If Φ <, he Lyapuov-Krasovs heore esures ha he syse of Eq. () ad cosequely, he syse of Eq. () s asypocally sable. he equaly codo Φ < s a olear arx equaly whch s rasfored o LMI by chagg varable echque. For hs purpose, frs, he arx M s paroed as follows: = 4 M M M M M () Aferward, le M =M, M =p M, M =p M, M 4 =p 4 M,X=M ad Z=dag(X,X,X,X). Defe: Φ = Z Φ Z = Ω + Ω + Ω, where, Ω = Z ΩZ P = X P X, Q = X Q X, R = X R X, Y= K X ad G= H X. he equaly Φ < ples ha Φ <. Brefly, hs proves he suffcecy of he codo of Eq. (5) o asypoc sably of he closed-loop syse. Eployg he Lea o derve Eq. () requres ha x L( H ) s assured. I he followg, codo s derved o guaraee he belogg of he sae o he eoed rego. Le he ellpsod E( P,) s a subse of he rego L( H ), so he followg equaly s sasfed: hx u ( + xpx) u, =,, () Sce: u h hx u( + xpx) = [ ± x] P ± x (4) he followg holds: u h P (5) If boh sdes of he above equaly pre ad pos ulpled sulaeously wh dag(, I X) ad s raspose respecvely, he equaly of Eq. (6) s obaed wh g = h X. Fally, a esae of he doa of araco he for of Eq. (4) s copued. Fro V () <, follows ha V( x ) < V( x ) ad herefore for > : x() P x() < V( x ) < V( x ) (6) Regardg Eq. (4), he followg equales hold: V ( x ) ax φθ ( ) ( σ( P) + ησ( Q)) θ [ η,] η + ax φθ ( ) σ( R) (7) θ [ η,] η δ ( σ( P) + ησ( Q)) + δ σ( R) So, f: η δ ( σ( P) + ησ( Q)) + δ σ( R ) (8) he, for all he al fucos belog o Χ DOA Eq. (4), he raecores of he closed-loop syse rea he ellpsod E( P,) L( H ) ad he polyhedro represeao of saurao fuco s vald. heore gves a syseac approach o deere coroller ga K va feasble soluo of equales Eq. (5) ad Eq. (6) afer ug of he paraeers p, =,,4; f he coroller ga s ow a-pror, he codos of Eq. (5) ad Eq. (6) ca be used for he sably aalyss of he cosraed ewored syse of Eq. (). he deals are expressed Corollary whch preses LMI codos o chec he sably of he closed-loop. Rear: I coras o [9] ad [], he Lyapuov- Krasovs fucoal cosdered Eq. (), coas egral er of sae ad double egral er of sae rae. Moreover, Eq. (5) ad Eq. (6), Jese equaly s eployed o aa gher boud for he egral phrases. I addo, dfferely fro [9], freewegh arx s corporaed dervave of eergy fucoal va Eq. (8). hese gredes lead o proved desg codos copared o [9] ad [] whch wll be llusraed laer seco 5. Corollary : Le K,H R be gve. he closedloop syse of Eq. () s asypocally sable f here exs arces P >, Q >, R > ad M such ha he followg LMIs hold: Φ <, =,,..., (9) us hs, s =,,..., (4) sp where, Φ= Ω+ Ω + Ω wh: Q R R P * R R Ω = (4) * * η R * * * QR ad Ω = [ MA MB( D K+ D H) M ]. (4) Based o he resul of Corollary, a opzao proble wh LMI cosras s forulaed o oba a large esae of he doa of araco. o splfy he procedure, s assued ha δ = δ = δ ax ad paraeers w >, =,, are roduced o boud he arces P, Q ad R o ge a less coservave esae of he doa of araco. Followg he copuao of he arces K ad H usg heore, he subseque opzao proble s solved va YALMIP oolbox, o aa a axal esae of he doa of araco. Iraa Joural of Elecrcal & Elecroc Egeerg, Vol., No., Jue 4

6 γ s.. codos of Corollary (4) wip wiq wir where γ = w+ ηw +.5η w. hus, he radus of axal esae of he doa of araco s copued as: δ ax (44) σ( P) + ησ( Q) +.5 η σ( R) I heore, geeralzed secor codo preseed Lea s eployed o oba a ew syhess codo for sablzg sae-feedbac coroller ga. heore : Gve scalars η > ad p, =,,4, he syse of Eq. () wh he ewored eoryless sae-feedbac coroller Eq. () s asypocally sable f here exs P = P >, Q = Q >, R = R >, G, Y, dagoal posve defe U ad osgular arx Φ = Ω of approprae desos such ha he followg arx equales hold: Φ <, (45) us ys gs, s =,,..., (46) sp where Φ = Ω wh. Q R R P * R R G Ω = * * η R * * * Q R * * * * U (47) AX BY X BU pax pby px pbu Ω = pax pby px p BU p4ax p4by p4x p4bu whch y s ad g s are he s -h row of Y ad G, respecvely. Moreover, K = YX ad H= GX. A esae of he doa of araco s he for of Eq. (4) wh δ ad δ sasfyg: ( ( X PX ) + ( X QX )) δ σ ησ η δσ X + ( RX ) (48) Proof: he sech of proof rus alog he les of heore. Regardg Defo, he closed-loop syse of Eq. () s represeed as follows: x () = Ax() + BKx( h) Bψ ( Kx( h)) (49) Iproved Lyapuov-Krasovs fucoal s desgaed as Eq. () ad s dervave o he raecores of he syse of Eq. (49) s forced o be egave. I wha follows, free-wegh arx M s defed o be corporaed he upper boud of V () o reduce he coservaveess of he fal desg codo. Le us defe ξ() = [ x(), x( h), (), ( ), ( ( ))] x x η ψ Kx h ad le M be of he for: M = M M M M4 (5) I s obvous ha he followg equao holds: ξ Mx [ ( ) Ax( ) BK( x( h)) + Bψ( Kx( h))] = (5) O he oher sde, by Lea, he followg relao s rue: ψ ( Kx) U( ψ( Kx) Hx) (5) provded ha x L ( K H ). Icludg Eqs. (5) ad (5) he upper boud of V Eq. (7) yelds o: V x ( ) Px ( ) + x ( ) Qx( ) x ( η) Qx( η) + η x () Rx () [ x() x( h)] R[ x() x( h)] [ x( h) x( η)] R[ x( h) x( η)] (5) ψ ( Kx( h) U[ ψ( Kx( h) Hx( h)] + ξ Mx [ ( ) Ax( ) BK( x( h)) + Bψ ( Kx( h))] whch ca be rearraged as V () ξ () Φ ξ() ; where Φ= Ω+ Ω + Ω wh: Q R R P * R R H U Ω = * * η R (54) * * * Q R * * * * U Ω = [ MA MBK M MB] If Φ <, he Lyapuov-Krasovs heore guaraees ha he syse of Eq. (49) s asypocally sable. he codo Φ < s olear; hus by he chagg varable ehod, s rasfored o LMI codo. Le M = M, M = p M, M = p M, M = p M, 4 4 X= M ad Z= dag ( X,X,X,X ). Defe: Φ = Z Φ Z = Ω wh Ω = Z Ω Z, =,, P = XPX, Q = XQX, R = XRX, Y= K X, G= H X, U = U. he codo Φ < ples ha Φ <. hs proves he suffcecy of he codo Eq. (45) o asypoc sably of he closed-loop syse. he res of proof s he sae as heore ad oed for he sae of brevy. Mahod Kaleybar & Mahboob Esfaa: Sablzao of Newored Corol Syses

7 Corollary : Le K, H R be gve. he closed-loop syse of Eq. () s asypocally sable f here exs arces P >, Q>, R >, dagoal U > ad M such ha he followg LMIs hold: Φ < us s hs, sp s =,,..., (55) (56) where Φ = Ω + Ω + Ω, wh: Q R * Ω = * * * Ω = MA [ R P R * η R * * * * MBK M he opzao proble for doa of araco s slar o Eq. (4). 5 Illusrave Exaple o llusrae he advaages of he proposed ehods, a coparave uercal exaple s preseed. Exaple: Cosder he syse of Eq. () wh he followg arces [9], []:. A =.6,.5 B =. (58) ad u = u = 5. he resuls are suarzed able, where δ ax sads for he radus of doa of araco ad η ax s he axu aaable η whch was defed Eq. (4). he ehod of [9] leads o he feedbac ga K = [ ] o sablze he closed-loop syse for he η =.75, ad he se of adssble al codos s gve by a ellpsod Ε( P,) wh.9.86 P = R H U Q R * U MB ] (57) ( 59) he larges crcle ca be cluded hs ellpsod s of radus.56 whch s approxaely sx es saller ha he oe obaed fro heore (.9999 /.56 6). he approach of [] yelds o he feedbac ga K = [ ] wh η =.75, ad he correspodg se of adssble al codos s gve by a ellpsod Ε( P,) wh P =.7.9. (6) he larges crcle ca be cluded hs ellpsod s of radus. whch s approxaely e es saller ha he oe obaed fro heore (. 999/. 9). Moreover, by he proposed ehods axu allowable η ax ca be creased up o. 5 whch s cosderably coparable wh he η ax obaed fro approaches [9] ad []. Fgs. ad llusrae he covergece of sae raecores o he org, by usg he corollerr obaed fro heore for wo dffere values of η. Fg. Sae raecores ad sably ball (η =..5 sec). able Sably ball radus ad correspodg coroller ga. Mehod ( secod) δ ax K η ax =.5.48 [.8.46] heore η = [ ] ] η ax =.5.85 [.8.46] ] heore η = [ ] [9] [] η = ax. η = ax [.696.5] 75. [ ] ] Fg. Sae raecores ad sably ball (η =..75 sec). Iraa Joural of Elecrcal & Elecrocc Egeerg, Vol., No., Jue 4

8 he er ellpse Fg. shows he esae of he doa of aracos. he ouer ellpse Fg. shows he ellpsod xpx β, as see all raecores beg o he perphery of he er ellpse ever leave he ouer ellpsod ad ed up a he org. Fgs. ad ogeher wh he forao able clarfy ha here s verse relao bewee η ad δ ax. 6 Cocluso I hs paper wo less coservave crera have bee preseed o syhess sablzg coroller for ewored corol syse subec o pu saurao. I he frs ehod, he sauraed lear syse has bee represeed wh a se of lear syses ebedded wh a covex polyope. I he secod ehod, acuaor saurao has bee acled va a geeralzed secor codo. Furherore, a esae of doa of araco has bee obaed hrough he LMI opzao. Illusrave exaple deosraes he ouperforace of he suggesed ehods copared o he exsg approaches he leraure. Refereces [] A. Feredua, H. Lesa, C. Lucas, M. Lehoe ad M. M. Norda, A Syse Approach o Iforao echology Ifrasrucure Desg for Uly Maagee Auoao Syses, Iraa Joural of Elecrcal ad Elecroc Egeerg, Vol., No., pp. 9-4, 6. [] C. Peg, Y.-C. a ad M. O. ade, Sae Feedbac Coroller Desg of Newored Corol Syses wh Ierval e-varyg Delay ad Noleary, I. Joural of Robus ad Nolear Corol, Vol. 8, No., pp. 85-, 8. [] B. ag, G.-P. Lu ad W.-H. Gu, Iprovee of Sae Feedbac Coroller Desg for Newored Corol Syses, IEEE ras. o Crcus Syses II, Express Brefs, Vol. 55, No. 5, pp , 8. [4] R. A. Gupa ad M-Y. Chow, Newored Corol Syse, Overvew ad Research reds, IEEE rasacos o Idusral Elecrocs, Vol. 57, No. 7, pp ,. [5] M. Mabood, M. H. Ashar Lar ad M. Alyar Shoorehdel, A Uder Load Servo Acuaor Idefcao ad Coparso bewee he Resuls of Dffere Mehods, Iraa Joural of Elecrcal ad Elecrocs Egeerg, Vol. 8, No., pp. 7-,. [6] A. Seure ad J. M. Goes da Slva Jr., Newored Corol: ag o Accou Saple Perod Varaos ad Acuaor Saurao, Proceedg of he 8 h IFAC World Cogress, pp. -6,. [7] Z. Xaoer,. Hag ad L. Guopg, Sablzao of Newored Sochasc Syses Subec o Acuaor Saurao, Proceedgs of 6 h Chese Corol Coferece, pp. -7, 7. [8] Z. Zuo, Y. Wag ad G. Zhag, Sably Aalyss ad Coroller Desg for Lear edelay Syses wh Acuaor Saurao, Proceedgs of he 7 Aerca Corol Coferece, pp , 7. [9] AE. Frda, A. Seure ad J-P Rchard, Robus Sapled-daa Sablzao of Lear Syses: a Ipu Delay Approach, Auoaca, Vol. 4, No. 8, pp , 4. [] E. Frda, A. Pla ad U. Shaed, Regoal Sablzao ad H Corol of e Delay syses wh saurag acuaors, Ieraoal Joural of Robus ad Nolear Corol, Vol., No. 9, pp ,. [] E. Frda, A Refed Ipu Delay Approach o Sapled-daa Corol, Auoaca, Vol. 46, No., pp. 4-47,. [] A. Seure ad J. M. Goes da Slva Jr., ag o Accou Perod Varaos ad Acuaor Saurao Sapled-daa Syses, Syses ad Corol Leers, Vol. 6, No., pp. 86-9,. [] J. M. Goes da Slva Jr., A. Seure, E. Frda ad J. P. Rchard, Sablzao of Neural Syses wh Saurag Corol Ipus I. Joural of Syses Scece, Vol. 4, No. 7, pp. 9-,. [4] Z. Zuo, D. W. C. Ho, Y. Wag ad C. Yag, A New Approach for Esag he Doa of Araco for Lear Syses wh e-varyg Delay ad Saurag Acuaors, Proceedgs of Seveh Asa Corol Coferece, pp , 9. [5] J. M. Goes da Slva Jr. ad S. arbourech, A-wdup Desg wh Guaraeed Regos of Sably for Dscree-e Lear Syses, Syses ad Corol Leers, Vol. 55, No., pp. 84-9, 6. [6] J. Su, G. P. Lu, J. Che ad D. Rees, Iproved Sably Crera for Lear Syses wh e Varyg Delay, IE Corol heory ad Applcao, Vol. 4, No. 4, pp ,. [7]. Hu, Z. L ad B. M. Che, A Aalyss ad Desg Mehod for Lear Syses Subec o Acuaor Saurao ad Dsurbace, Auoaca, Vol. 8, No., pp. 5-59,. [8] S. arbourech, J. M. Goes da Slva Jr. ad G. Garca, Delay-depede A-wdup Sraegy for Syses wh Saurag ad Delayed Oupus, Ieraoal Joural of Robus ad Nolear Corol, Vol. 4, No. 7, pp , 4. Mahod Kaleybar & Mahboob Esfaa: Sablzao of Newored Corol Syses

9 Masod Mahod Kaleybar receved hs B.Sc. degree fro Islac Azad Uversy of abrz, abrz, Ira, 8, ad M.Sc. degree fro Sahad Uversy of echology, abrz, Ira,, boh Elecrcal Egeerg. Hs curre research eress are aalyss ad desg of ewored corol syse. Reza Mahboob Esfaa receved hs B.Sc. degree fro Depare of Elecrcal Egeerg, Sahad Uversy of echology, abrz, Ira ; He receved he M.Sc. ad Ph.D. degrees fro Arabr Uversy of echology (ehra Polyechc) 4 ad 9, respecvely. He has held faculy poso a he Elecrcal Egeerg Depare, Sahad Uversy of echology, sce. Hs research eress clude aalyss ad corol of e-delay ad ewored syses. 4 Iraa Joural of Elecrcal & Elecroc Egeerg, Vol., No., Jue 4