EXACT MODEL MATCHING AND DISTURBANCE REJECTION FOR GENERAL LINEAR TIME DELAY SYSTEMS VIA MEASUREMENT OUTPUT FEEDBACK

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1 EXACT ODEL ATCHIN AND DISTURBANCE REJECTION FOR ENERAL LINEAR TIE DELAY SYSTES VIA EASUREENT OUTUT FEEDBACK Fotis N. Kouboulis Halkis Istitut of Tchology Dpatt of Autoatio saha Evoia Halki 344, c kouboulis@tihal.g og E. aagiotakis Halkis Istitut of Tchology Dpatt of Autoatio saha Evoia Halki 344, c paagiotakis@tihal.g Abstact Th pobl of xact odl atchig with siultaous distubac jctio (EDR fo gal utal ulti-dlay syst via popotioal alizabl cotoll fdig back asuabl distubacs ad asut output is studid h. Th dsid odl is assud to b lft-ivtibl. Fo this pobl th followig two ajo issus a solvd: Th cssay ad sufficit coditios fo th pobl to hav a alizabl solutio ad th gal aalytical xpssio of th cotoll atics solvig th pobl. Th sults hav b applid to cotol a ital cobustio (IC gi. Noclatu EDR Exact odl atchig with siultaous Distubac Rjctio LR {} Lft ultipl of th agut atix with gad to th Right bializabl uitaizig tasfoatio RR {} Right ultipl of th agut atix with gad to th Right bializabl uitaizig tasfoatio τ i oit dlays ( i =,..., q lialy idpdt ov th atioal ubs Fild of al ubs ( s Fild of atioal fuctios of s with al cofficits ( Fild of atioal fuctios of th lts of with al cofficits ( Rig of alizabl atioal fuctios of th lts of with al cofficits Fild of atioal fuctios of s with cofficits i ( Th tasf atix of th dsid odl ( ak [ ] Rak of th agut atix ov ( Rak [ ] Rak of th agut atix ov. Itoductio Th class of ti dlay systs is of paticula itst i both syst thoy ad cotoll dsig (s [] ad th fcs thi. This typ of dsciptios is usually t i otio systs as wll as i chical pocsss paticulaly thos of distibutd fo. H, th gal class of lia utal ulti-dlay difftial systs [] is studid. Th Distubac Rjctio (DR pobl of ti dlay systs has b studid i th past fo th spcial cas of tadd sigl dlay systs i [3]. Also th pobl has b studid (ud coditios ad usig a gotic appoach i [4] fo oal systs with lts of th syst atics dfid i a ig, big icipal Idal Doai, ad with a cotoll atix havig its lts i th sa ig. All sults i th fild ai towads alizabl cotoll aly cotolls ot ivolvig pdictios. Th solutio of DR fo gal utal ulti-dlay ad tadd ulti-dlay syst via a popotioal alizabl ulti-dlay cotoll, has b divd i [5] fo th cas of lftivtibl systs Th pobl of Exact odl atchig (E has attactd cosidabl atttio fo th spcial cas of tadd sigl dlay systs i [3], [6] ad [7]. All sults i th fild also focus towads a alizabl cotoll. Not that i [8] th pobl has b solvd fo utal sigl dlay syst via alizabl but ot popotioal (dyaic cotolls. Th solutio of E fo gal lft-ivtibl utal syst via a popotioal alizabl cotoll, ot stictd to b polyoial, has b divd i []. I pactic asuabl ad/o o-asuabl distubacs a usually t. So th pobl of E has to b cobid to that of DR. Fo systs ot X/5/$. 5 IEEE

2 ivolvig dlays th cobid pobl has b studid i [9] ad []. Th cotibutio of th pst pap cosists i stablishig th followig aspcts of th EDR pobl fo gal utal ulti-dlay systs with asuabl ad/o o-asuabl distubacs ad fo lft-ivtibl idal odl usig a popotioal alizabl asut output fdback law: Th cssay ad sufficit coditios fo th solvability of th pobl ad th gal solutio of th spctiv alizabl cotoll. Th ai cotibutio of th pst appoach is that it is quit gal thus uifyig th solutio of sval tasf fuctio dsig pobls. I paticula, th pst sults cov th solutio of th EDR pobl fo th gal class of utal ulti-dlay systs via popotioal alizabl stat o pfoac output cotoll, i th psc o ot of asuabl ad/o o asuabl distubacs. Th pst sults also cov th solutio of th EDR pobl fo th gal class of tadd ultidlay systs via a popotioal alizabl cotoll, as a spcial cas. Oth spcial typs of utal o tadd dlay systs (f.. cosuat dlays a also covd by th pst sults. Th pst sults a divd o th basis of a pu algbaic appoach yildig aalytic foula of th gal xpssio of th cotoll atics ad th solvability coditios. Fially, ot that th pst sults ca dictly b usd to dsig odl followig adaptiv cotolls fo distibutd idustial pocsss dscibd by ti dlay odl usig static asut output fdback cotolls.. athatical Backgoud dot th st of ultivaiabl atioal fuctios of s,, s q τ (o o copactly of = xp( sτ xp( sτq, wh [] xp[] = dots th xpotial of th agut quatity. Lt dot th st of atioal fuctios of s with cofficits atioal fuctios of. Ths two sts a claly filds. Lt, ( s dot th st of al ubs ad th st atioal fuctios of s with al cofficit spctivly. A ultivaiabl atioal fuctio of Lt ( xp ( sτ,, xp( sτq blogs to ( (i.. a atioal fuctio that is said to b alizabl ([], [5] if o pdictos a quid fo its alizatio. This is foally statd as: th liit of th atioal fuctio of, fo s tdig to ifiity, is fiit. Th st of ultivaiabl alizabl atioal fuctios of is dotd by (. Claly ( is a ig. Th followig La has b stablishd i [] bi- La.: Lt N ( ( atic lt ( ad Ξ(, with Ξ( alizabl, i.. Ξ( li { ( }: fiit ad Ξ( b of full ow ak ov, th it holds that: th xist two ivtibl ( Ξ li { ( Ξ } ( : fiit is alizabl is alizabl, havig th popty ( N( Ξ ( = I a (. Explicit foula fo ( ad Ξ( a giv i []. Th abov tasfoatio, big of gat ipotac fo th study of alizability issus fo ti dlay syst has b calld ight bi-alizabl ad uitaizig tasfoatio []. Haft, ( Ξ( of th atix N ( ( will b calld th Lft ad th Right ultipls big of full ow ak ov,with gad to th Right bi-alizabl uitaizig tasfoatio ad thy will b dotd by LR N RR N, spctivly. { ( } ad { ( } Th coditios fo th xistc of alizabl atics i th ag of a o-hoogous ap with lts, will b pstd. Cosid th ap i ( ( = Θ( ( ( F N V (. wh, Θ( is a abitay γ a atix, N ( is a a β fixd full ow ak atix ad V ( is a fixd γ β atix. Th lts of N (, V ( ad Θ( blog to (. Fo th ap (., it will b ivstigatd ud which coditios th xist a Θ F is alizabl, i.. such that ( such that ( th lts of F ( blog to (. To pst th, coditios fo th lia ap (. to b alizabl, dfi * V ( = V ( RR { N( } I (.3a β a I a ** V ( = V ( RR { N( } (.3b Fo th lia ap (. ad upo xtdig th sults of [] to th pst cas th followig tho as wll as th followig coollay ca b stablishd

3 Tho.: Th xist a Θ( ( =Θ( ( ( ( big of full ow ak ov ( * ( is alizabl, i.. li V ( : fiit such that th ap F N V is alizabl, with N * V, iff Coollay.: If Tho. hold th th gal Θ F is fo of ( yildig alizabl ( ( ** ( V ( Θ = Λ + LR { N( },wh γ a ( Λ ( is a abitay alizabl F is atix. Th gal fo of all alizabl ( * F ( ( ( RR { ( } = Λ V N. 3. Solutio of th EDR pobl fo oasuabl distubacs 3..obl foulatio ad pliiay sults Th focd spos of th gal class of lia utal ulti-dlay difftial systs [], [5] is govd by th followig st of quatios sx s A X s B U s D U D ( = ( ( + ( ( + ( ( Y ( s = C( X( s (3. wh Xs (, Us (, UD( s ad Ys ( dot th Laplac tasfo of th stat vcto x( t, th iput vcto ut (, th o asuabl distubac vcto ud ( t ad th pfoac output vcto p yt (, spctivly. Th lts of A(, B (, D ( ad C ( ( τ, blog to th fild. Th dlays,, τq a cosidd to b atioally idpdt []. Lt Ψ ( s b th Laplac tasfo of th vcto ψ( t dotig th asut outputs of th syst Ψ ( s = L( X( s (3. H, th dsig goal is that of Exact odl atchig, with siultaous Distubac Rjctio (EDR aly to div a closd loop syst with iput-output tasf fuctio big qual to that of a dsid odl ad siultaously liiat th ifluc of th distubacs. Th fdback is poposd to b of popotioal typ, i.. U( s = K( Ψ ( s + ( Ω( s (3.3 wh Ω ( s is th vcto of xtal iputs ad K ad wh th lts of th atics ( ( blog to th fild ( ( is ivtibl ov (. Th atix (gulaity of th fdback law i od to su th idpdc of th closd loop syst iputs. Substitutig th cotol law (3.3 to th syst (3. th pobl of xact odl atchig with siultaous distubac jctio is foally statd as i th followig quatio ( ( ( ( ( ( ( ( (, C si A B K L (3.4 B D s = p wh H is th odl s tasf atix ad I is th -disioal uitay atix. Equatio (3.4 foulats th pobl i a oal syst dsciptio. It is assud that th odl s tasf atix is lft ivtibl, i.. ( Rak = (3.5 ( p, wh Rak [] dots th ak of th agut atix ov th fild. To avoid coflict Rak [] should ot b cofusd with th oal ak [] dotig th ak of th agut atix ov th fild (. Although th lts of th atics A(, B (, D ( ad C ( a ot stictd to b alizabl atioal fuctios i, th ipltability of th cotoll quis that th lts of th atics K ( ( b alizabl (i.. li K ( li ( : fiit. Dfi ( B( D( ad : fiit ad =. Equatio (3.4 ca b witt as follows ( ( ( = (, ( C si A H s { I K( L( A( ( } (3.6 Du to th lft ivtibility of th odl s tasf atix, th ivtibility of th atix ( ad th latio (3.6 it is adily cocludd that th iput-output tasf atix of th op-loop syst is lft-ivtibl i. This is claly a cssay coditio fo th pobl at had to b solvabl. Futho, fo

4 (3.4, (3.6 ad th ivtibility of ( th followig cssay coditios a divd fo th solvability of th EDR pobl Rak C( A( ( =Rak C( A ( Β( Rak = (3.7 Coditios (3.7 togth with th odl s lft ivtibility iply th syst s lft ivtibility i.. Rak C( A( B ( = ( Ncssay ad sufficit coditios Th lft-ivs of th tasf fuctio atix of th dsid odl, lt ˆ (, s, ay adily b coputd i a a siila to that i [], []. This lft ivs satisfis th followig latio I ˆ = (3.9 ( p Usig (3.9 quatio (3.6 ay b witt i th followig way Π = ( { I K( L( si A( ( } (3. wh Π ˆ = ( p ( + C si A (3. ( ( ( ( Th qualitis (3.9 ad (3. a bfitd by th coditios i (3.5 ad (3.7. Claly, ud (3.5 ad (3.7 th pobl is ducd to that of solvig (3. fo K. ( ad ( Equatio (3. is a quatio i (. This quatio will b solvd to div th alizabl popotioal asut output fdback that achivs EDR. I paticula xpadig Π i gativ pows of s, i.. σ (, σ ( ( + Q ( s + Π s = Q s + + Q s (3. ad upo quatig lik pows of s i both sids of th dsig quatio (3., quatio (3. ay b bok dow to th followig st of quatios σ Q ( s Q ( σ + + s ( ( ( K( R( = ( Q( (3.3a Q Q = (33b (3.3c wh (, ( Q Q Q a giv by th latios ( Q (, Q ( Q ( ad wh I = = I R( = L ( ( A ( ( (3.4a Q( = Q( Q ( (3.4b Fo th divatio of th abov sults th ad ( s as wll as isoophis btw ( btw ( ad has b usd I what follow so dfiitios will b pstd i od to stablish th cssay ad sufficit coditios fo th xistc of a alizabl asut output fdback cotoll of th fo (3.3 such that th closd loop syst satisfis (3.4. To this d, lt η=ak R( ( + η. Also lt J b a colu slctio atix slctig th lialy idpdt colus of R (, i. ak R( J = η. Lt b a ow aagt atix, odig th lialy idpdt ows of R( J, i.. S ( } η R ( J = (3.5 W ( } η i such a way that ak S ( = η. Th ows of W S, i.. ( a dpdt upo th ows of ( W ( =S ( S( wh S ( is th dpdc atix ( S ( W ( = S(. Fially spcify th atics N ( ad V ( appaig i Sctio fo th pst cas to b: N ( W ( = S( I η (3.6a

5 V ( Q ( Q( J = S( ( η (3.6b wh ow it holds that a = η, β = ad γ = Such spcificatios a pittd sic th atix at had W ( S( I η is of full ow ak. Th atics Ξ { W S I η } { η } ( =RR ( ( (3.7 ( =LR ( ( W S I (3.8 ca also b spcifid o th basis of th foulas divd i []. Th basd o latios (.3a ad b ad upo usig th spcificatios (3.6a ad b th followig atics ay also b spcifid * V ( Q ( Q( J = S( ( η { { ( ( }} RR W S I η I η (3.9a ** V ( Q ( Q( J = S( ( η I { RR { ( ( }} η W S I η (3.9b Basd upo th abov dfiitios ad spcificatios th followig tho will b stablishd. Tho 3.. Th xact odl atchig pobl with siultaous distubac jctio of gal utal ulti-dlay systs (3. ad fo lft-ivtibl odl (3.5, is solvabl via a alizabl popotioal asut output fdback cotoll (3.3, if ad oly if th followig coditios a satisfid: (i Rak C( A( ( = =Rak C( A( B ( = (ii Qj ( ( j R( (iii ak =ak R( Q ( (iv dt Q ( / (v Q ( (vi ( is alizabl ( li Q ( Q fiit { } is * (vii th atix V ( * (i.. li V ( dfid i (3.9a, is alizabl is fiit oof. Coditios (i-(v a claly cssay ad sufficit fo th pobl s solvability via cotolls ov ( as ca asily b divd fo latio (3.7 ad fo th dsig quatios (3.3a,b,c To stablish th cssay ad sufficit coditios fo th pobl s solvability, via alizabl cotoll assu that coditios (i-(v a satisfid ad ivstigat if th xist a alizabl solutio i th st of th cotolls ov ( dscibd by (3.3a,b,c (which possibly ivolv pdictos. Fo (3.3b ad coditios (iv - (v, obsv that th solutio of is uiqu ad it is giv by th latio ( ( = Q (, (3. Thus coditio (vi is th cssay ad sufficit. With coditio fo th xistc of a alizabl ( gad to K (, fist obsv that sic coditio (iii is assud to b satisfid th gal solutio of (3.3c fo K ( ov ( (which possibly ivolv pdictos is giv by th followig latio K ( = Q ( Q( J S( ( η ( W ( +Θ S( I η (3. wh us was ad of latio (3.5, th dfiitios Θ is a bfo that, latio (3. ad wh ( ( η abitay atix. This gal solutio fo ( K ad upo usig th spcificatios (3.6a ad (3.6b ay b witt i th followig copact fo ( = Θ( ( ( K N V, (3. Applyig Tho. to (3. ad upo usig (3.9a, coditio (vii is divd to b cssay ad sufficit K. fo th xistc of a alizabl ( 3.3. al Fo of th Ralizabl Cotolls If th coditios of Tho 3., a satisfid th st K ad of all alizabl cotoll atic lt ( (., th gal fo of Θ( K (, is dtid to b, will b divd. To this d, usig Coollay, yildig alizabl

6 ( ** ( ( ( Θ = Λ + V (3.3 wh Λ ( ** atix ad wh V ( ad ( is a abitay ( η alizabl a giv by th latios (3.9b ad (3.8, spctivly. Substitutig latio (3.3 i (3. th followig tho ay adily b stablishd. Tho 3.. If coditios (i-(vii of Tho 3. a satisfid, th th gal aalytical xpssios of K th alizabl cotoll atics ( ad ( i (3., lt ( ad K (, solvig th EDR pobl fo gal utal ulti-dlay systs ad fo lft-ivtibl odl a: K ( = Q ( (3.4a ( = Q ( Q( J S( ( η ** V ( ( W ( + S( I η ( ( ( ( W +Λ S I η (3.4b wh Ξ( is giv by latio (3.7, ( ** giv by latio (3.8, V ( is is giv by th latio (3.9b ad wh th oly f paats a th lts of th ( η abitay alizabl atix ( Λ 4. Solutio of th EDR pobl fo asuabl ad o-asuabl distubacs 4..obl foulatio ad pliiay sults Th focd spos of th gal class of lia utal ulti-dlay difftial systs is govd by th followig st of quatios sx s = A X s + B U s ( ( ( ( ( ( ˆD ( ( D ( ( ( ( + D U + D U (4.a Y s = C X s (4.b wh Xs ( is th Laplac tasfo of th stat vcto x( t, Us ( is th Laplac tasfo of th iput vcto ut (, Uˆ ( s is th Laplac tasfo of D th asuabl distubac vcto uˆd ( t, U D( s is th Laplac tasfo of th o-asuabl distubac vcto u D ( t ad Ys ( is th Laplac p tasfo of th pfoac output vcto y( t. A B D D Th lts of (, (, (, ( ad C (, blog to th fild (. Th dlays τ,, τq a cosidd to b atioally idpdt. Lt ψ( t dot th asut output of th syst. Th Ψ ( s = L( X( s (4. wh Ψ ( s is th Laplac tasfo of ψ( t L ( (. ad H, th dsig goal i as i Sctio 3, that of xact odl atchig ad distubac jctio. Th fdback is poposd to b of popotioal typ, i.. U s = K Ψ s + K U s + Ω s ˆD ( ( ( ( ( ( ( (4.3 wh Ω ( s is th vcto of xtal iputs ad K K wh th lts of th atics (, ( ad ( blog to th fild (. Th atix ( is ivtibl ov ( fo th sa asos as i Sctio 3. Substitutig th cotol law (4.3 to th syst (4. th pobl of xact odl atchig with siultaous distubac jctio is foally statd as i th followig quatio ( ( ( ( ( C A B K L B( ( B( K( + D( D( = p p (4.4 wh H is th odl s tasf atix ad I is th -disioal uitay atix. Equatio (4.4 foulats th pobl i a oal syst dsciptio. It is also assud that (s Sctio 3 th odl s H is lft ivtibl, i.. tasf atix ( ( Rak = (4.5 Although th lts of th atics A( (, D (, D ( ad C (, B a ot stictd to b alizabl atioal fuctios i

7 quis that th lts of th atics K ( K ( ad ( b alizabl (i.. li K ( fiit, li K ( : fiit ad li ( : fiit. Dfi ( = B( D ( D ( th ipltability of th cotoll, :. Equatio (4.4 ca b witt as follows ( ( ( = (, ( { I K ( (4.6 C si A H s ( ( ( ( } K L si A Du to th lft ivtibility of th odl s tasf atix, th ivtibility of th atix ( ad th latios (4.4 ad (4.6 it is adily cocludd that th tasf atix of th op-loop syst is lft-ivtibl i. This is claly a cssay coditio fo th pobl at had to b solvabl. Futho, fo (4.6 ad th ivtibility of ( th followig cssay coditios a divd fo th solvability of th EDR pobl Rak C( A( ( =Rak C( A( B ( Rak = (4.7 Coditios (4.7 togth with th lft ivtibility of th odl claly cov th lft ivtibility of th op loop syst i.. Rak C( A( B ( = ( Ncssay ad sufficit coditios Th lft-ivs of th dsid odls tasf fuctio, lt ˆ (, s, ay adily b coputd as i Sctio 3 I ˆ = (4..9 ( p Followig Sctio 3 quatio (4.6 ay b witt as follows ( Π = { I K ( K ( L( A( ( } (4. wh Π ( p ( + + = Hˆ C si A ( ( ( ( (4. Th qualitis (4.9 ad (4. a bfitd by th coditios i (4.5 ad (4.7. Claly, ud (4.5 ad (4.7 th pobl is ducd to that of solvig (4. fo K (, K ( ad (. Equatio (4. is a quatio i (. This quatio will b solvd to div th alizabl popotioal asut output fdback ad asuabl distubac fdback that achivs EDR i a a siila to that i Sctio 3. Siilaly to Π, i.. Sctio 3 xpadig ( Π = ( σ ( ( = Qσ s + + Q s + Q s + (4. ad quatig lik pows of s i (4. yilds σ ( ( Q σ s + + Q s (4.3a Q ( Q ( Q3 ( = (4.3b ( = ( K ( ( ( ( K R Q (4.3c = Q Q Q 3 a giv by th latios I Q ( = Q (, Q ( = Q ( I Q3 ( = Q ( I ad wh R( = L ( ( A ( ( (4.4a Q( = Q( Q ( (4.4b wh (, ( ad ( Lt η=ak R( ( + + η. Also lt J b a colu slctio atix slctig th lialy R, i. idpdt colus of ( ak R( J = η. Lt b a ow

8 aagt atix, odig th lialy idpdt ows of R( J, i.. S ( } η R ( J = (4.5 W ( } η i such a way that ak S ( = η. Th ows of W S, i.. ( a dpdt upo th ows of ( W ( =S ( S( wh S ( dpdc atix S( W ( = S( Fially th atics N ( ad V ( spcifid to b is th (. a N ( W ( = S( I η (4.6a V ( Q ( Q( J = S( ( η (4.6b wh ow it holds that a = η, β = ad γ = N is of full ow ak. Th atics Th atix ( Ξ { W S I η } { η } ( =RR ( ( (4.7 ( =LR ( ( W S I (4.8 ca also b spcifid o th basis of th foulas divd i []. Th basd o latios (.3a ad b ad upo usig (4.6a ad b th followig latios a divd * V ( Q ( Q( J = S( ( η { { ( ( }} RR W S I η I η (4.9a ** V ( Q ( Q( J = S( ( η I { RR { ( ( }} η W S I η (4.9b Basd upo th abov dfiitios ad spcificatios th followig tho will b stablishd. Tho 4.. Th xact odl atchig pobl with siultaous distubac jctio of gal utal ulti-dlay systs (4. is solvabl fo lftivtibl odl (4.5, via a alizabl popotioal asut output ad asuabl distubac fdback cotoll (4.3, if ad oly if th followig coditios a satisfid: (i Rak C A =Rak C( A( B ( = (ii Qj ( ( j ( ( ( ( R( (iii ak =ak R( Q ( (iv dt Q ( / (v Q3 ( (vi ( is alizabl ( li Q ( Q fiit * (vii th atix V ( * (i.. li V ( is fiit (viii th atix Q ( Q ( li Q ( Q ( { } is dfid i (4.9a, is alizabl is alizabl (i.. is fiit oof. Coditios (i-(v a claly cssay ad sufficit fo th pobl s solvability via cotolls ov ( as ca asily b divd fo latio (4.7 ad fo (4.3a,b,c To stablish th cssay ad sufficit coditios fo th pobl s solvability, via alizabl cotoll assu that coditios (i-(v a satisfid ad ivstigat if th xist a alizabl solutio i th st of th cotolls ov (. To ivstigat this call th poof of Tho 3. as wll (4.3a,b,c. Sic coditios (iv ad (v a satisfid th followig uiqu solutio fo ( ad K ( is divd ( = Q (, (4.a K( = Q( Q(, (4.b Thus coditio (vi is th cssay ad sufficit coditio fo th xistc of a alizabl ( ad coditio (viii is th cssay ad sufficit coditio K. With gad fo th xistc of a alizabl ( to K (, ad siilaly to th poof of Tho 3. ad upo usig (4.5, (4.a, (4.6a ad (4.6b th followig latio is divd ( = Θ( ( ( K N V, (4. Applyig Tho. to th ap (4. ad upo usig (4.9a, coditio (vii is divd to b cssay ad K. sufficit fo th xistc of a alizabl (

9 4.3 al Fo of th Ralizabl Cotoll atics If th coditios of Tho 4., a satisfid th st K K of all alizabl cotoll atics (, ( ad (, lt (, ( ad (, K K will b divd. To this d, usig Coollay., th gal fo of Θ(, yildig alizabl K (, is dtid to b ( ** ( ( ( Θ = Λ + V (4. is a abitay ( η alizabl wh Λ ( ** atix ad wh V ( ad ( a giv by th latios (4.9b ad (4.8, spctivly. Siilaly to Sctio 3 th followig tho ay adily b stablishd. Tho 4.. If coditios (i-(viii of Tho 4. a satisfid, th th gal alizabl aalytical xpssios of th alizabl cotoll atics K K, K (, ( ad ( i (4.3, lt ( (, ad ( K solvig th EDR pobl fo gal utal ulti-dlay systs ad lftivtibl odl, a: ( = ( Q ( ( ( ( K = (4.3a Q Q J S ( η ** V ( ( W ( + S( I η ( ( ( ( W +Λ S I η (4.3b ( = ( ( K Q Q (4.3c wh Ξ( is giv by latio (4.7, ( ** giv by latio (4.8, V ( is is giv by th latio (4.9b ad wh th oly f paats a th lts of th ( η abitay alizabl atix ( Λ 5.Applicatio to a IC Egi A highly siplifid two-iput (idl by-pass valv opig ad spak advac two-output (gi spd ad itak aifold pssu idl spd cotol odl fo IC gis will b usd. I paticula cosid th followig stat-spac dsciptio of th liaizd odl [] x ( θ = Ax ( θ + Ax ( θ θd + Bu( θ + Dτf ( θ ω( θ a ( θ x ( θ = p ( θ, u ( θ = δ( θ τ J ω Kτ Jω A = Ka ηυv, A d ε ω = 4πV Kδ Jω J ω B = Kε ω, D = wh, ω is th gi spd ( ad/s, p is th itak aifold pssu ( ka, δ is th spak advac ( dg, a is th thottl opig ( dg, θ is th cak agl ( ad, θ d is th iductio to pow dlay ( ad, τ f is th distubac toqu ( N, τ is th t gi toqu ( N, K ε is th cofficit latig th thottl opig to dlayd aifold pssu ( ka/s/dg, K τ is th cofficit latig gi toqu to aifold pssu ( ka/s/dg, K δ is th cofficit latig gi toqu to spak tiig ( N/dg, J is th gi itia ( N-s / ad, η υ is th volutic 3 fficicy, V is th gi displact ( ad d V 3 is th aifold volu (. Th subscipt dots oial opatig poit ad dots icts fo th oial opatig poit. Th pfoac output is x ( θ whil th asut output is ω( θ. I ay cass th distubac τ f is cosistd of two pats i.. τf = τˆ f + τ f wh τ ˆf is th kow o asuabl pat ad τf is th o-asuabl pat. It ca b obsvd that th ifluc of τ ˆf to th gi spd ca b jctd via appopiat cotoll. Howv th ifluc of τ f to th gi spd caot b jctd. It ca oly b attuatd if th ods latig τ f to th gi spd a sufficitly stabl. To achiv such a quit as wll as appopiat cotol of th gi spd without affctig th itak aifold pssu th idal odl is chos to b H sθd ; > sθd Kτη Vd υ s + Jω( s + ( 4πVs + ηυvd = ηυv d 4πVs ηυv + d

10 Th dsig quit is that of EDR i. th tasf fuctio atix latig th xtal iputs to th outputs to b qual to whil th tasf fuctio atix latig τ ˆf to th outputs of th closd loop syst to b qual to zo. To satisfy th dsig quits th sults i Sctio 4 a applid to th pst cas with J ω D ( = ad D ( =. It ca b obsvd that th cssay ad sufficit coditios a satisfid ad th gal xpssio of th cotoll atics solvig th pobl is ηυv dω 4πVK = Jω =, K K δ δ a ω K ( = J ω τε Kδω ε (, K ( Substitutio of th divd cotoll to th op-loop syst yilds d ω( θ Kτ = ω( θ + p ( θ θd dt J ω + ω ( θ + τf J ω d p ( θ ηυvd ηυvd = p ( θ + ω ( θ dt 4πV 4πV wh ω ( θ ad ω ( θ a th two xtal iputs. Fo ω ( θ = ad fo zo iitial coditios w gt li ω( θ = li ω ( θ + li τ f. If θ θ jω θ is gat tha th ifluc of th ukow pat of J ω th toqu to th gi spd is lss tha. li τ. 6. Coclusios θ I this pap, th EDR pobl fo gal utal ulti-dlay systs with asuabl ad/o oasuabl distubacs ad fo lft-ivtibl odl, via alizabl popotioal asut output fdback, has xtsivly b solvd. Th cssay ad sufficit coditios fo th pobl to hav a solutio hav b stablishd (Tho 3. ad Tho 4.. Th gal aalytical xpssio of th f alizabl popotioal cotoll atics has b divd (Tho 3., Tho 4.. All abov sults cotibut to th divatio of adaptiv cotolls fo utal ti dlay systs ad paticulaly thos dscibig distibutd idustial pocsss. Th EDR fo gal utal ulti-dlay syst via alizabl popotioal cotoll, without th stictio of lft ivtibl odl is cutly ud ivstigatio. Rfcs [] V.B. Kolaovskii, J.-. Richad, "Editoial: So w tds i th study of ti-dlay systs", athatics ad Coputs i Siulatio, vol. 45, pp. 9-, 998. [] F. N. Koubouli. E. aagiotakis ad. N. aaskvopoulo "Exact odl atchig of Lft Ivtibl Nutal Ti Dlay Systs", ocdigs of th 3 th ditaa Cofc o Cotol ad Autoatio (5 ISIC-ED, pp , 5. [3]. alab, R. Rabah, "Stuctu at ifiity, odl atchig ad distubac jctio, of lia systs with dlays" ' Kybtica, vol. 9, pp , 995. [4]. Cot ad A.. do, "Th Distubac Dcouplig pobl fo systs ov a ig", SIA J. of Cot. ad Opti., vol. 33, pp , 995. [5]. N. aaskvopoulo F. N. Kouboulis ad. E. aagiotaki "Distubac Rjctio of Lft Ivtibl Nutal Ti Dlay Systs", subittd. [6].Cot ad A..do, "odl atchig pobls fo systs ov a ig ad applicatios to dlay difftial systs", I oc. 3 d IFAC Cof. Systs Stuctu ad Cotol, Nat Fac, pp , 995. [7]. icad, J. F. Lafay ad V. Kuca, "odl atchig fo lia systs with dlay" oc. 3 th IFAC Wold Cogs Sa Fasisco, USA, Vol D, pp , 996. [8]. icad, J. F. Lafay ad V. Kuca, "odl atchig fo lia systs with dlays ad -D Systs", Autoatica, vol. 34, pp 83-9,998. [9]. N. aaskvopoulo F. N. Kouboulis ad K.. Tziaki "Distubac Rjctio with Siultaous Exact odl atchig of alizd Stat Spac Systs", ocdigs of th Scod Euopa Cotol Cofc (ECC 93, pp55-59, 993 [] F.N. Kouboulis ad.. Skapti "Robust Distubac Dcouplig with Siultaous Exact odl atchig via Static asut Output Fdback", i Coputs ad Coputatioal Egiig i Cotol, Wold Scitific ad Egiig Socity s Elctical ad Coput Egiig Si pp 78-9, 999 []. N. aaskvopoulo F. N. Kouboulis ad D.F.Aastasaki "Exact odl atchig of alizd Stat Spac Systs", J. Optiizatio Thoy ad Appl., vol. 76, pp 57-85, 993. [] Xiaoqiu Li ad Stph Yukovich, "Slidig od Cotol of Dlayd Systs with Applicatio to Egi Idl Spd Cotol", IEEE Tas. Cot. Syst. Tch., vol. 9, pp 8-8,.