Delay-dependent robust stabilization of uncertain neutral systems with saturating actuators

Transcription

1 Delay-depede robus sablzao of ucera eural syses wh saura acuaors Aar Haura a * Haah H Mchalska b Beo Boule c * Correpod auhor Phoe Fax Eal aar@ccllca abc McGll Cere for Ielle Maches McGll Uversy 48 Uversy Sree Moréal Québec Caada HA A7 b Eal chalsk@ccllca c Eal boule@ccllca Absrac he robus sablzao proble for eural syses wh saura acuaors s addressed here he syses cosdered are couous-e wh paraerc uceraes eer all he arces he syse represeao A saura corol law s desed ad a reo s specfed whch he sably of he closed-loop syse s esured he Lyapuov-Krasovsk fucoal s eployed he des s depede o he e-delay ad s rae of chae he coroller s cosruced ers of he soluo o a se of arx equales by eploy a descrpor odel rasforao of he syse Iroduco I rece years uch work has bee devoed o he aalyss ad syhess of corollers for sae-delayed syses wh or whou paraerc uceraes; see [] for earler work hs area hs eres s sroly ovaed by he fac ha delays ad uceraes are he wo os pora causes of sably Furherore boh delays ad uceraes occur frequely he checal ad process dusres whch provdes aoher reaso for he sudy of ew ad less coservave sably codos he corol des proble for he eeral case of eural (descrpor syses whch he e-delays ca appear boh he sae ad s dervave has so far obaed relavely lle aeo [] Ulke sple rearded syses eural syses are parcularly sesve o delays ad ca be easly desablzed [4] o he bes kowlede of he auhors sablzao of eural syses wh acuaor saurao has oly bee suded [5] However he coroller desed [5] s delay-depede (whch s relavely coservave he syse represeao cludes o uceraes ad he delays are cosdered kow ad e-vara I hs coex he corbuo of hs paper s as follows: A delay-depede robus sablzao proble s addressed here for eural syses full eeraly wh acuaors cosras ad or-bouded paraerc //$7 IEEE 59 uceraes eer all he arces he syse represeao I aep o ake he approach leas coservave a descrpor odel rasforao ad a Lyapuov-Krasovsk fucoal are eployed here as [6] he leas coservave represeao of acuaor saurao as show [7] s used here he value of he e-delay as well as s rae of chae are ake o accou he des ehod preseed whch furher reduces he coservaveess of he approach Adoped oao he follow oao s adoped For ay arx A he expressos A ad da{ A} deoe he raspose of A ad he daoal of A respecvely R s he desoal Eucldea space wh vecor or R s he se of all real arces ad for ay arx P R he equaly P > sfes ha P s syerc ad posve defe he sadard oao of q L ab s adoped for he space of all fucos [ ] f : R R q whch are Lebesque erable he square over he erval [ a b ] wh he sadard or sybol [ ] couous fucos x : [ a b] R supreu or λ ( P ad λ ( P he L X a b s used o deoe he space of all wh he sadard ax deoe he axal ad al eevalue of arx P respecvely he sybol Cd = C( [ d ] R deoes he Baach space of couous vecor fucos app he erval [ d] o R wh he opoloy of ufor coverece; φ = sup φ s he or of a fuco c d w φ C d he se d w Cd = { φ Cd ; φ < w c } where w s a posve real Proceeds of he Aerca Corol Coferece Dever Colorado Jue 4-6 C s defed by

2 uber For ay vecor ( ( v R sa v = s v u v u > = For ay x d deoes he resrco of x o he erval [ ] raslaed o [ d] ( ψ = x( + ψ forall ψ [ d] x Proble saee Cosder he follow syse: ha s ( = + ( + x Gx Ax Ax h Bu where he delays h ad are assued o be soe ukow fucos of e ad are bouded as: h h ad < for all ax ad for soe ve posve cosa u R syse sae vecor ad h x R ax s he corol pu he al codos for syse are ve as: + w x + ψ = φ ψ ψ dax φ R C d [ ] ax wh dax = ax ( h s he he ucera arces G A A ad B are ve by: G = G + G A= A+ A A = A+ A B = B+ B ( where he arces G A A ad B are real ad are assued o be kow exacly he arces G A A ad B are real-valued represe he or-bouded paraeer uceraes ad are assued o be of he follow for: A E G = LFE = LF B E A = LFE (4 b j j j where F R F R ad F R are real ucera possbly e-vary arces wh Lebesue easurable eres whch addoally ee he follow requrees: FF I FF I ad FF I he arces L L L E E E ad Eb are kow ad real ad characerze he way whch he ucera paraeers of F F ad F eer he oal arces G A A ad B Addoally delays h ad are assued o be couously dffereable wh her respecve raes of chae bouded as follows: h β h < β < for (5 where βh ad β are ve posve cosas Also A s bouded as follows: A A A A (6 ax ax where Aax s cosa ad kow 5 he follow s assued: Assupo : ( A A B + s sablzable Assupo : he pu vecor s subjec o aplude cosras e u U R wh U = { u R ; u u u u > = } (7 he follow addoal assupo [9] s eeded o eable he applcao of Lyapuov s secod ehod for he sably of eural syses: Assupo : he dfferece operaor D :X R R D x = x ( ] ve by ( ( G x s delay-depedely sable (e he hooeeous dfferece equao Dx = asypocally sable rrespecve of he delay he robus sablzao proble wh saura acuaors: Fd arx K R ad a se of al codos S R such ha he asypoc sably of he closedloop syse : x Gx ( = Ax + Ax ( h + BsaKx (8 s esured ha s φψ S ψ [ dax] syse (8 s locally asypocally sable Reark Sce o assupo s ade abou he sably of he ope-loop syse he proble a had s ha of local sably he lobal asypoc sably of syse (8 ca be addressed oly f he ope-loop syse s assued sable 4 Ma resuls he follow heore delvers he a resul of hs paper heore Cosder he syse (8 Suppose ha here exs -arces: P > P P S = S > U = U > W W W W 4 ad R = R R R = R > a arx K R a vecor αl R a posve scalar γ ad posve scalars ε ; = 4 whch sasfy he follow arx equales: j < P αl K αl K γu j = (9 = ( Proceeds of he Aerca Corol Coferece Dever Colorado Jue 4-6 s

3 < α l = wh Ψ Ψ h Φ W A ε E E j j ax 4 Ψj Ψ haxφ W4 A j hax hax haxr AW ε4e E AW 4 ( βh S + ε4e E GP GP P G PG = Φ Φ P L P L P L W L P L P L P L W4 L 4 = { ε ε ε ε } 4 ( β U εee + ( = da I I I I (4 where ( ( α ( α + W A + AW + S + ( ε + ε4 E E + ε ( E + EbΓ j ( αl K ( E+ EbΓ j ( αl K P P ( A A B ( α K P A W Ψ = A + A + BΓ K P + P A+ A + BΓ K j j l j l Ψ = Γ + j j l 4 Ψ =P P + U + h A R A ax ax ax Φ = W + P W + P Φ = W W + P 4 R R R = R = > R R Uder hese codos syse (8 s asypocally sable for ay al codo φ belo o he ball Φ ( σ where: Φ σ = φ w φ σ (5 { C d ax ; c } wh σ = γπ where hax π = ax λax( P + λax ( S ( βh haxλax ( S + λax ( U + haxλax ( Aax R Aax ( β (6 Reark I [7] a coparso bewee he dffere represeaos of sauraos used he leraure was ade coclud ha he dfferece clusos ehod used hs paper (see proof of heore Appedx for he hs for of represeao leads o he leas coservave des 5 Reark I [5] he proposed des s delay-depede boh he eural ad he rearded delays whle heore of hs paper he preseed coroller s delaydepede he eural delay ad delay-depede he rearded delay hs ves a less coservave des ha {5] he reaso for choos a delay-depede des for he eural delay s ha ulke sple rearded syses eural syses are parcularly sesve o delays ad ca be easly desablzed [4] Also [5] oly he uceray-free syse s reaed ad he delays are assued kow ad e-vara whch akes he reae he prese paper ore eeral Reark 4 I [7] a delay-depede des for rearded syses was preseed However he descrpor syse rasforao used here roduces addoal dyacs o he oral syse [4] ad hus akes he des ore coservave ha he oe proposed he prese paper Also he boud ehods used here (see proof of heore Appedx reduce furher he coservaveess as copared o [7] Reark 5 he a dffculy he applcao of heore s he presece of soe oleares he varables P P αl ad K hs ca be overcoe by us soe relaxao echques [56] e by era wh respec o αl ad K Iequales (9 ad ( he reduce o lear arx equales (LMIs all he rea varables he laer are easly solved us he echques of [8] 5 Cocluso he proble forulao eployed here s beleved o be he frs ad he os eeral cosdered so far wh referece o he delay-depede robus sablzao of eural syses wh saura acuaors Such syses are hhly releva o applcaos process dusry ad are lkely o be eployed here A ajor ovao of he approach adoped here s ha he sablz corol des s ade depede o boh he value of he e-delay as well as o s rae of chae Oo work s cocered wh he exeso of he prese work o he full robus H corol proble Appedx he follow leas wll prove helpful he sequel: Lea : [] Le he fuco z = f ( s dsdθ he z dz ( d b a θ s a soluo of he dffereal equao a = b a f b f s ds b Proceeds of he Aerca Corol Coferece Dever Colorado Jue 4-6

4 ( + b a f s ds a Lea : [] Le A L E ad F be real arces (possbly e-vary of approprae desos wh F sasfy FF I ; he he follow holds: - For ay scalar ε > P LFE + E F L P ε P LL P + εe E - For ay arx P > ad ay scalar ε > such ha ε I EPE > ( A+ LFE P( A+ LFE ( ε APA + APE I EPE EPA + εll - For ay arx P > ad ay scalar ε > such ha P ε LL > ( ε A + LFE P A + LFE A P LL A + ε E E Reark 6 Saee Lea as appled o A ca be used o choose boud A ax of (6 Lea : [] Assue ha a( s R ad b( s erable over arx R R holds: R are s he for ay posve defe ad ay arx M R he follow a s R RM a s b ( s a( s ds ds b s M R Γ b s where ( M R I R ( MR I Γ= + + (7 Proof of heore A locally equvale polyopc represeao for he olear syse (8 based o dfferece clusos [] s used lead o ( α x Gx = A+ BΓ x K x+ Axh where Γ( α ( x (8 s a daoal arx whose daoal elees are defed for = as: α u f ν > u ν ( x = f u ν u u f ν < u ν (9 wh ν = K x By defo oe es ( x < α x = x R Noe ha he scalar α deree of he ca be cosdered as a dcaor of he saurao u h ery of Hece he saller s 5 α he farher s x fro he reo of leary ( x S( u defed by { } ν S u = x R ; u = ( Sce hs paper addresses he proble of local sablzao we have o l x ad herefore o cosder a lower boud for α ( x herefore f we cosder ay copac se vecor α ( x α Sc R he for x Sc he copoes of ad a lower boud deoed: { c} αl = x ; x S = ha eas ha for ay x Sc oe es < αl α( x = whch allow o defe he vecors α l Frohsvecorwe ca defe he follow verex arces: A = A+ BΓ α K ( j j ( l where Γ ( α j l s a daoal arx whose daoal elees arbrarly ake he values or α l = Hece f x Sc he ẋ ca be deered fro he follow polyopc odel: ( = λ j j + ( x Gx A x Ax h (4 j= wh λj = λj j= Furherore s pora o oe ha he vecor αl allows o defe he polyhedral se as: ( ; u S u αl = x R ν = (5 α l Acually he se S( u αl coas Sc ad correspods o he axal se whch odel (4 represes syse (8 We cosder as a covee choce for he se Sc ellpsods eercally defed fro a syerc posve defe arx P as Ψ P γ = x R ; x Px γ (6 { } ( he sasfaco of assupo ( of he heore eas ha he ellpsod (6 s cluded he se S( u αl defed (5 where he vecor αl verfes he x ca be copued fro he polyopc syse (4 Equao (4 s wre s equvale descrpor for: x = y y = Gy( + λ jajx j = ( h + A x (7 Proceeds of he Aerca Corol Coferece Dever Colorado Jue 4-6

5 Us x h = x x s ds (Lebz-Newo h pers o re-wre (7 ye a ore racable for whou roduc addoal dyacs [4]: x = y = y + Gy( so ha for λ j j j = I E = h + A + A x A y s ds (8 he aueed syse s: x x E = + y y λ jaj + A I y G j= y ( s ds A h I (9 he follow Lyapuov-Krasovsk fucoal wll be used here: V = V + V+ V + V ( where x V = x y EP = x Px ( y V x ( τ Sx( τ dτ h V y ( τ Uy( τ dτ h ax ax ax + θ = ( = ( V y s A R A y s dsdθ wh = (4 P P = P P P > U = U > S = S > R = R > (5 Dfferea ( ad us (9 yelds I dv x = x y P d λ jaj + A I y j = + x y P y G h A x y P y( s ds (6 A boud for he las er of (6 wll be derved as follows: Defe x y P η = y ( s ds h A (7 Us Lea wh a( s y( s b( s = A ad wh = h ves: η x P y = 5 h x x y P ( M R+ I R ( MR+ I P ds y x + y ( s ds A h RMP y y s A R y s ds + h A x hax x y P ( M R I R ( MR I P y + + x + x x h A RMP y y ( s A R y( s ds (8 + h ax A as R > Dfferea V ad V of ( ad ( dv = x Sx ( h x ( h Sx( h d dv = y Uy ( y ( Uy ( d ad us he assupos S > U > ad he bouds (5 dv x Sx ( β h x ( h Sx( h (9 d dv y Uy ( β y ( Uy ( (4 d Apply Lea o V dv d = h y A R A y ax ax ax hax (4 y s Aax R Aax y s ds Fro (7 ad he fac ha x s square erable o D y L Uder [ follows ha ( [ Assupo he laer ples ha y L [ Eploy ( (6 (8 (9 (4 ad (4 whle oc ha sce R > ad fro boud (6 hax y ( s A R y( s ds A hax hax y s Aax R Aax y s ds = y s A R A A R A y s ds ax ax ad fally apply Lea o ers G A A ad B resuls a boud for V whch has he follow for: V ξ ξ where ξ s he areaed sae vecor ad ad are as ( ad (4 respecvely Cocer has ers volv Proceeds of he Aerca Corol Coferece Dever Colorado Jue 4-6

6 λ j ( ad so o uaraee V j= < < we eed o have By covexy he laer s uaraeed by hav j < for j = where j s as Suffce codos for uaraee ha V < ca hus be saed as follows: j < j = < (assupos of Lea By he Schur coplees lea codos ad are equvale o assupo (9 of he heore hus here exss π such ha V ( x π x ad herefore oe es V ( x V( x provded ha he odel (4 s vald ha s provded ha x Su ( α l Furherore follow [8] he Lyapuov fucoal defed ( ca be show o sasfy D V wh π λ ( P c π φ φ π φ (4 = ad hax π = ax λax( P + λax ( S ( βh h λ S + λ U + h λ A R A ( β ax ax ax ax ax ax ax Hece for φψ ψ [ d ] ( Φ oe es ax γ (4 x Px V x V x (44 herefore for ay al codo φ he ball Φ ( σ defed (5 he syse (8 verfes he codos of he V x s a local srcly Krasovsk heore [5] ad decreas Lyapuov fuco hus he asypoc sably of syse (8 s esured QED Refereces [] - De Souza C E & L X (995 LMI Approach o Delay-Depede Robus Sably ad Sablzao of Ucera lear Delay Syses he 4 h IEEE Coferece o Decso ad Corol -8 [] - Mahoud M (b Robus H corol of lear eural syses Auoaca [] - Hale J & Luel S (999 Effecs of sall delays o sably ad corol Vrje Uversy Aserda he Neherlads Rep WS-58 [4] - Loea H & owley S (996 he effec of sall delays he feedback loop o he sably of he eural syses Syses&Corol Leers [5] - arbourech S & Garca G (999 Sablzao of Neural Lear e-delay Syses wh Saura Acuaors Proceeds of he 8 h Coferece o Decso & Corol -6 [6] - Frda E & Shaked U ( A Descrpor Syse Approach o H Corol of Lear e- Delay Syses IEEE rasacos o Auoac Corol [7] - Goes da Slva J M arbourech S & Reao R ( Coservavy of ellpsodal sably reos esaes for pu sauraed lear syses IFAC Barceloa Spa [8] - Boyd S El Ghaou L Fero E & Balakrsha V (994 Lear Marx Iequales Syse ad Corol heory SIAM Sudes Appled Maheacs Phladelpha [9] - Hale J (977 Fucoal Dffereal Equaos New York: Sprer-Verla [] - K J H ( Delay ad s e-dervave Depede Robus Sably of e-delay Lear Syses wh Uceray IEEE rasacos o Auoac Corol 46 ( [] - De Souza C E & L X (999 Delay-depede robus H corol of ucera lear sae-delayed syses Auoaca 5 - [] - Park P (999 A delay-depede sably crero for syses wh ucera e-vara delays IEEE rasacos o Auoac Corol [] - Molchaov A P & Pyask E S (989 Crera of asypoc sably of dffereal ad dfferece clusos ecouered corol heory Syses & Corol Leers [4] - Charoov V & Melchor-Aular D ( O delay-depede sably codos Syses&Corol Leers [5] - Hale J K & Verduy Luel S M (99 Iroduco o fucoal dffereal equaos Appled Mah Sceces Sprer-Verla New York 99 [6] arbourech S & Goes da Slva Jr J M ( Syshess of corollers for couos-e delay syses wh saura corols va LMIs IEEE rasacos o Auoac Corol [7] arbourech S Peres P Garca G & Queec I ( Delay-depede sablzao of e-delay syses wh saura acuaors Proceeds of he 9 h Coferece o Decso & Corol 48-5 [8] Xu S La J Ya C & Verres E I ( A LMI approach o uaraeed cos corol for ucera lear eural delay syses Ieraoal Joural of Robus ad Nolear corol Proceeds of he Aerca Corol Coferece Dever Colorado Jue 4-6