Design of observer for one-sided Lipschitz nonlinear systems with interval time-varying delay

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1 WSEAS RANSACIONS o SYSES a CONROL Waju Lu Yal Dog Ska Zuo Desg of observer for oe-se Lpscz olear syses w erval e-varyg elay WANJUN LIU YALI DONG SHIKAI ZUO Scool of Scece aj Polyecc Uversy aj 8 CHINA ogyl@vp.sa.co Absrac: - s paper eals w sae observer esg proble for a class of olear yacal syses w erval e-varyg elay a sasfes e oe-se Lpscz coo. e syses uer coserao clue e well-sue Lpscz syse as a specal case a possess ere avaages w respec o coservaveess. By cosrucg a ovel Lyapuov-Krasovsk fucoal a ulzg free-wegg arces approac ovel suffce coos a guaraee e observer error coverge asypocally o zero are obae for a class of olear yacal syses w erval e-varyg elay ers of lear arx equales. e copug eo for observer ga arx s gve. Fally llusrave exaples are gve o eosrae e effecveess of e propose eo. Key-Wors: - Nolear observers; Oe-se Lpscz coo; Quarac er-boueess; Asypoc sably; Ierval e-varyg elay Irouco Over e las ecaes a large aou of researc acves ave focuse o observer esg for olear yacal syses as ca be sow roug e vas leraure s fel. s opc was spre by e fac a sae esao ca be use for corol agoss a supervso purposes. Recely oer applcaos suc as sycrozao a pu recovery coucao syses ave becoe oes of e eergg a eresg researc areas [-]. I s well kow a e exsece of e elay a syse ay cause sably or ba syse perforace close feeback syses. e eelay peoeo ay be ecouere ay praccal syses suc as e AIDS epec arcraf sablzao cecal egeerg syses ferre grg oel aual corol eural ework uclear reacor populao yac oel rollg ll sp sablzao a syses w lossless rassso les. Hece rece years ave wesse vesgao of sably aalyss a observer esg for e-elay syses [-]. e coveoal Lpscz coo s cooly use e exsg olear observer esg. However a ajor lao e exsg resuls for Lpscz olear syses s a os of e work oly for aequaely sall values of e Lpscz cosa [8]. I orer o overcoe s lao e so-calle oe-se Lpscz coo was frs rouce [9] for olear sae esao. Several eresg works o sae observers a sablzao for s ype of syses were evelope [-]. I [] e proble of observer esg for oe-se Lpscz o-lear syses was vesgae by usg e lear arx equaly (LI approac. Suffce coos a esure e exsece of observers for oe-se Lpscz o-lear syses were esablse a expresse ers of lear arx equales (LIs. Sae observer esg for a geeral class of olear scree-e syses a sasfes e oe-se Lpscz coo was cosere []. e proble of observer-base sablzao for lear syses w paraeer equaly was cosere []. A esg eoology was propose ascrbe o a jucous use of e faous Youg relao. Soe ew elay epee suffce coos esurg e sably for ucera eural syse w scree a srbue elays were obae ers of lear arx equales (LIs []. Usually e rage of elays cosere os of e exsg refereces s fro zero o a upper bou []. I pracce owever e elay ay vary a rage for wc e lower bou s o resrce o beg zero []. However o e bes of our kowlege few resuls ave bee gve o e E-ISSN: -8 8 Volue

2 WSEAS RANSACIONS o SYSES a CONROL Waju Lu Yal Dog Ska Zuo suy of observers for oe-se Lpscz olear syses w erval e-varyg elay a soe specal oleares. s ovaes our prese researc. I s paper we coser e olear observer esg for oe-se Lpscz syses w erval e-varyg elay a oleares. By usg Lyapuov-Krasovsk fucoal a free-wegg arces approac we esabls ovel suffce coos for a class of olear yacal syses w erval e-varyg elay ers of lear arx equales wc guaraee e observer error coverges asypocally o zero. e copug eo for observer ga arx s propose. s paper s orgaze as follows. I Seco e syse escrpo soe assupos a leas are gve a e correspog observer s rouce. I Seco soe suffce coos a guaraee e observer error coverges asypocally o zero for a class of olear yacal syses w erval e-varyg elay are presee. I Seco sulao exaples are gve o sow e perforaces of our eo. Fally cocluso s raw a rearks are ae Seco. Noaos. R eoes e -esoal real Euclea space. R represes e se of all real arces. s e er prouc R a s gve xy R e xy = xy x s e raspose of x. eoes e Euclea or R. For a square arx S S > ( S < eas a e arx s posve efe (egave efe. I syerc block arces we use a asersk o represe a er uce by syery. I represes a ey arx of approprae eso. Proble saee a prelares Coser e followg ucera olear syses w erval e-varyg elay x ( = Ax ( Bx ( ( Φ( x ( u f( x ( x ( ( u ( y ( = Cx ( Dx ( ( x R s e sae vecor u R s e p corol pu y R s e easure oupu AB R a p CD R are kow arces e couous fucos f( x ( x ( ( u a Φ ( x ( u are ukow a eoe e olear ers w respec o x a u. e elay ( s e varyg a sasfes ( ( < ( a are cosas represeg respecvely e lower a upper bous of e elay s a cosa. Our objecve s o erve suffce coos a esure e sably of observer error yacs. e followg assupos are ae rougou s paper. Assupo Φ ( x ( u a f( x ( x ( ( u are oe-se Lpscz w respec o x a u.e. Φ( x ( u Φ( x ˆˆˆ ( u x ( x ( ρ x ( x ( ( f( x ( x ( ( u f( x ˆˆ( x ( ( u x ( x ˆˆ( ρ x ( x ( ρ ˆ x ( ( x ( ( p for ay xx ˆ R y R u R ρ ρ ρ are e oe-se Lpscz cosas wc ca be posve or egave. Assupo Φ ( x ( u a f( x ( x ( ( u are quarac er-boueess w respec o x a u.e. ( Φ( x ( u Φ( x ˆˆ( u ( Φ( x ( u Φ( x ( u γ x ( x ˆˆ( Φ( x ( u Φ ( x ( u ( β ˆ x ( x ( ( f( x ( x ( ( u f( x ˆˆ( x ( ( u ( f( x ( x ( ( u f( x ˆˆ( x ( ( u ( x ( x ˆˆ( β x ( ( x ( ( β γ ˆ x ( x ( f( x ( x ( ( u f( x ˆˆ( x ( ( u γ ˆ x ( ( x( ( f( x ( x ( ( u f( x ˆˆ( x ( ( u ( β β β γ γ a γ are real scalars. Lea [] For ay cosa arx R = > scalar η > vecor fuco ω: [ η] R suc a e egraos cocere are well efe e η η ω( s s ω( s s ( η η ω ( s ω( s s. E-ISSN: -8 8 Volue

3 WSEAS RANSACIONS o SYSES a CONROL Waju Lu Yal Dog Ska Zuo Lea (Scur Coplee [8]. Gve cosa arces Σ Σ a Σ w approprae esos Σ =Σ a Σ =Σ e Σ ΣΣ Σ < f a oly f Σ Σ Σ Σ < or. < (8 Σ Σ Σ Σ Reark Ulke e well-kow Lpscz coo e cosas ρ ρ ρ β β β γ γ γ ca be posve egave or zero. I ao f e fucos Φ ( x ( u a f( x ( x ( ( u are Lpscz e ey are also bo oe-se Lpscz a quaracally er-boue bu e coverse s o rue. e oe-se Lpscz coo proves a less coservave coo a e classcal Lpscz oe. e cocep of quarac er-boueess s very useful o prove racable LI sably coos. e observer of syse ( s efe by e followg for x ˆˆˆˆ ( = Ax ( Bx ( ( Φ( x ( u f( x ˆˆ( x ( ( u Ly ( Cx ˆˆ( Dx ( ( (9 eoe e ( = x ( x ˆ( e e observer error yacs s govere by e ( = ( A LCe ( ( BLDe ( ( ( Φ f Φ Φ( x ( u Φ( x ˆ( u a f f( x ( x ( ( u f( x ˆˆ( x ( ( u. a Resuls I s seco we prese LI coos for e observer syess probles of oe-se Lpscz olear syses w erval e-varyg elay. eore Assue a assupo a are sasfe. For gve scalars a observer ga arx L e observer error yacs ( s asypocally sable f ere exs scalars ε ( = a arces P > X Q > ( = N = N N N N = Y of approprae esos suc a e followg LIs ol: Π Π < ( Π X Y N Q N Q ( φ X Y N S Q Q Π = Q -Q S Q S Q S Q Π = Q Π = Q Q S = [ A LC B LD I I ] φ φ φ φ φ I φ = ( ε I ε I φ = A P PA C L P PLC Q Q Q N N ( ερ ερ εβ εβ I φ = PB PLD N ε φ = P N I ε φ = P N I φ =( Q ( ε ρ ε β I. Proof. Le us coose e Lyapuov-Krasovsk fucoal caae as follows: V( = V( V( V( E-ISSN: -8 8 Volue

4 WSEAS RANSACIONS o SYSES a CONROL Waju Lu Yal Dog Ska Zuo V ( = e ( Pe( = ( V ( e ( s Q e( s s e ( s Q e( s s e = e e λ e λ ( s Q e( s s V ( e ( s Q e( s s ( s Q e( s s ( s Q e( s sλ ( s Q e( s sλ e V ( = e ( ( A LC P P( A LC e( e ( P( B LD( e ( e ( PΦ e ( Pf V ( = e ( Qe ( e ( Qe ( e ( Qe ( e ( Qe ( ( ( e ( ( Qe ( ( ( e ( Qe( e ( [ Q Q Q ] e ( e ( Qe( ( e ( ( Qe ( ( e ( Qe( V ( = e ( Qe ( e ( Qe ( e ( s Q e( s s e ( s Q e( s s e ( Q e ( ( ( e s Q e s s e Q e e s Q e s s ( ( ( (. ( Defe ξ ( = e ( e ( ( Φ f. e e followg equaos ol for ay arces a N w approprae esos: ξ ( e ( e ( ess ( = ( ξ ( N e ( e ( ess ( =. oreover for arces X a Y w approprae esos we ave ξ ( Xξ( ξ ( Xξ( s= ( ξ ( Yξ ( ξ ( Yξ ( s=. O e oer a for ay scalars ε ( = f follows fro (-( a ε ρe ( e( e ( Φ ε ρe ( e ( ρe ( ( e ( ( (8 e ( f ε βe ( e( γe ( ΦΦ Φ ε βe ( e ( βe ( ( e ( ( (9 γ e ( f γ e ( ( f f f. Fro (-( we ave V( = V ( V ( V ( e ( ( A LC P P( A LC e( e( PB ( LDe ( ( e( PΦ e ( Pf e ( Q Q Q e( [ ] e ( Qe( e ( Q e( ( e ( ( Qe ( ( e ( Qe ( e ( Qe ( e ( Qe ( e ( Qe ( e ( s Q e( s s e ( s Q e( s s e e ( s Q e( s s e ( s Q e( s s. ( Fro (-( follows a V ( e ( ( A LC P P( A LC e( e( PB ( LDe ( ( e( PΦ e ( Pf e ( Q Q Q e( e ( Qe( [ ] ( e ( ( Qe ( ( e ( Q e( e ( Qe ( e ( Qe ( ( Qe ( e ( Qe ( e ( s Q e( s s e ( s Q e( s s e ( s Q e( s s e ( s Q e( s s E-ISSN: Volue

5 WSEAS RANSACIONS o SYSES a CONROL Waju Lu Yal Dog Ska Zuo ξ ( e ( e ( ess ( ξ ( N e ( e ( ess ( ξ ( Xξ ( ξ ( Xξ ( s ξ ( Yξ ( ξ ( Yξ ( s ρe ( e( e ( Φ ρe ( e( e ( f ρ e ( ( e ( ( βe ( e( γe ( ΦΦ Φ ε [ βe ( e ( βe ( ( e ( ( γ e ( f γ e ( ( f f f]. ( Fro ( we ge V ( e ( ( A LC P P( A LC Q Q Q e ( e ( PΦ ] e ( P( B LD e( ( e ( Pf e ( Qe( e ( Q e( e ( e ( ( Qe ( ( e ( Qe ( e ( Qe ( ( Qe ( e ( Qe ( e ( s Q e( s s e ( s Q e( s s ξ ( e ( e ( ess ( ξ ( N e ( e ( ess ( ξ ( Xξ ( ξ ( Xξ ( s ξ ( Yξ ( ξ ( Yξ ( s ρe ( e( e ( Φ ρe ( e( e ( f ρ e ( ( e ( ( βe ( e( γe ( ΦΦ Φ βe ( e ( βe ( ( e ( ( γ e ( f γ e ( ( f f f = ξ ( Θ S Q S S Q S S QS S QS ξ ( X ξ ( s ξ( ss Q Y N ξ ( s ξ( ss N Q ( ξ ( = ξ ( e ( e ( ξ ( s = ξ ( e ( s S = [ S ] φ X Y N Θ= Q. Q Fro ( ( a Lea we ge a V< ( for all e (. s coplees e proof. Reark eore proves e erval evaryg elay epee sably crera for gve ga arx L wc ca be llusrae by Exaple e ex seco. However our esg goal s o f a observer ga arx L suc a e error yacs ( s asypocally sable. We ca erve e followg eore. eore Assue a assupo a are sasfe. For gve scalars > > a e observer error yacs ( s asypocally sable f ere exs scalars ε ( = a arces P > Q > ( = of approprae esos suc a e followg LI ols: = < ˆ ( I = ε I ε I = A P PA R C C R Q Q Q Q Q ( ερ ερ εβ εβ I ε = PB R D = P I E-ISSN: Volue

6 WSEAS RANSACIONS o SYSES a CONROL Waju Lu Yal Dog Ska Zuo ε = P I = ( Q ( ε ρ ε β I Q Q ˆ = Q. Q If ( ols a feasble soluo e e ga arx L s gve by L= P R. Proof. Coose e Lyapuov-Krasovsk fucoal caae as follows: V( = V( V( V( V ( = e ( Pe( = ( V ( e ( s Q e( s s e ( s Q e( s s e ( s Q e( s s = V ( e ( s Q e( s s e ( s Q e( s s. We ave V ( = e ( ( A LC P P( A LC e( e ( P( B LD e( ( e( PΦ e( Pf V ( = e ( Qe ( e ( Qe ( e ( Qe ( e ( Qe ( ( ( e ( ( Qe ( ( e ( Q e( [ Q] e ( Q Q e ( e ( Qe( ( e ( ( Qe ( ( e ( Qe( V ( = e ( Qe ( e ( Qe ( a By Lea we ave e ( s Q e( s s e ( s Q e( s s. e ( s Q e( s s e s s Q e s s ( ( e ( s Q e( s s e( s s Q e( s s e V ( e ( Qe ( e ( Qe ( e( s s Q e( s s e( s s Q e( s s. Ag e ers o e lef ses of (8-(9 o e e ervave of V ( yels V ( e ( ( A LC P P( A LC e( e ( P( B LD e( ( e( PΦ e( Pf e ( [ Q Q Q] e ( e ( Qe ( ( e ( ( Qe ( ( e ( Qe( e ( Qe ( e ( Qe ( e( s s Q e( s s e( s s Q e( s s ρe ( e( e ( Φ ρe ( e ( ρe ( ( e ( ( e ( f βe ( e( γe ( ΦΦ Φ βe ( e ( βe ( ( e ( ( γ e ( f γ e ( ( f f f = ζ ( ζ( ( ζ ( = e ( e ( ( Φ f e ( e ( e ( s s e ( s s. Le L= P R. e follows fro ( a V< ( for all e ( prove a e LI ( ols a feasble soluo. e proof s coplee. Reark I soe exsg leraure e rage of varyg elay cosere ay exsg E-ISSN: -8 9 Volue

7 WSEAS RANSACIONS o SYSES a CONROL Waju Lu Yal Dog Ska Zuo refereces s fro zero o a upper bou. We =.e. e e-varyg elay sasfes ( ( we ca oba e followg resul. Corollary Assue a assupo a are sasfe. Coser syse ( w =. For gve scalars a observer ga arx L e observer error yacs ( s asypocally sable f ere exs scalars ε ( = a arces P > N = N N N N Y Q > ( = of approprae esos suc a e followg LIs ol: φ Y N S Q S Q Q < Q Q Y N N Q ( φ φ φ φ φ I φ = ε I ε I S = A LC B LD I I φ [ ] = A P PA C L P PLC Q Q N N ( ερ ερ εβ εβ I φ = PB PLD N ε φ = P N I ε φ = P N I φ = ( Q ( ε ρ ε β I. Proof. Le us coose e Lyapuov-Krasovsk fucoal caae as follows: V( = V( V( V( V ( = e ( Pe( = ( V ( e ( s Q e( s s e ( s Q e( s s = e λ V ( e ( s Q e( s s ( s Q e( s sλ. Usg e slar eo sow e proof of eore e equaly ( ca be obae a e eale proof s oe. Corollary Assue a assupo a are sasfe. Coser syse ( w =. For gve scalars a e observer error yacs ( s asypocally sable f ere exs scalars ε ( = a arces P > O > ( = of approprae esos suc a e followg LI ols: I ε I < ε I O O ( = A P PA R C C R O O O ( ερ ερ εβ εβ I ε = PB R D = P I ε = P I =( O ( ε ρ ε β I. If ( ols a feasble soluo e e ga arx L s gve by L= P R. Proof. Slar o e proof of eore e coo ( Corollary ca be obae a e eale proof s oe. Reark We f( x ( x ( ( u = a Φ ( xu = e syse ( ca be wre as x ( = Ax ( Bx ( ( ( y ( = Cx ( Dx ( (. e observer of syse ( s efe by e followg for xˆˆˆ ( = Ax( Bx( ( (8 L( y Cxˆˆ ( Dx( (. Deoe e ( = x ( x ˆ( e e observer error yacs s govere by e ( = ( A LCe ( ( BLDe ( (. (9 E-ISSN: -8 9 Volue

8 WSEAS RANSACIONS o SYSES a CONROL Waju Lu Yal Dog Ska Zuo For syse (9 we ave e followg corollary. Corollary Assue a assupo a are sasfe. For gve scalars a e observer error yacs (9 s asypocally sable f ere exs arces P > a Q > ( = of approprae esos suc a e followg LI ols: Σ Σ Σ Q Q < Q Q ( Σ = A P PA R C C R Q Q Q Q Q Σ = PB R D Σ = ( Q. If ( ols a feasble soluo e e ga arx L s gve by L= P R. Proof. Slar o e proof of eore e coo (9 Corollary ca be obae a e eale proof s oe. Illusrave exaples I s seco we prese wo uercal exaples orer o sow e superory of our esg eo. Exaple. Coser e ucera olear syses ( w e followg paraeers: A= B= C = [ ] D= [ ] Φ ( x ( u = u (.s( x. x( x x f =. x( x x l =. =. =. =.9 γ =. γ =.9 γ =. ρ =. ρ =. ρ =. β =.8 β =. β =... eore ca be apple o esg a observer a solvg e LIs ( a ( yels P= Q..8 = Q =.. Q = Q = Q. 98. = Q = Q 899. =.8.98 R = 8.. ake L = ( [ ] = N = X X Y Y X = Y X = Y ε = ε = ε = ε = X = X = E-ISSN: -8 9 Volue

9 WSEAS RANSACIONS o SYSES a CONROL Waju Lu Yal Dog Ska Zuo X = Y = Y = Y = Accorg o eore e esao error ( coverges asypocally owars zero. Le e pu u ( = s(. For sulao we se al coos as follows: x ( = (. xˆ ( = (.. Fgs. a sow e sulao resuls for saes x ( k a x ( k respecvely. Fro e sulao we kow a e effec of sae rajecory rackg s sasfacory. Fg.. Resposes of sae x ( k a e esae of x ( k Exaple Fg.. Resposes of sae x ( k a e esae of x ( k Exaple Exaple Coser e e-varyg elay syse ( w e followg paraeers.9... A= B..9 =.. C = [.. ] D=. For gve =. = =.9 ρ =. ρ =. ρ =. β =.8 β =. β =. γ =. γ =.9 γ =. solvg e LIs ( yels P = Q = Q = Q = Q = Q =.9.89 R = [.. ] e e ga arx s gve by 9.9 L= P R =.. For sulao we se al coos as follows: x ( = (. xˆ ( = (.. E-ISSN: -8 9 Volue

10 WSEAS RANSACIONS o SYSES a CONROL Waju Lu Yal Dog Ska Zuo Fgs. a sow e sulao resuls for saes x ( k a x ( k respecvely. Fro e sulao we kow a e effec of sae rajecory rackg s sasfacory. Fg.. Resposes of sae x ( k a e esae of x ( k Exaple Fg.. Resposes of sae x ( k a e esae of x ( k Exaple Cocluso I s paper for a geeral class of couous-e olear yacal syse a sasfes e oese Lpscz coo w erval e-varyg elay we propose ew observer esg approac. Soe ovel suffce coos esurg e exsece of sae observer are esablse for e class of olear yacal syses w erval e-varyg elay ers of lear arx equales. e copug eo for observer ga arx s presee. Soe sulao exaples are gve o sow e effcecy of e propose approac. Refereces: []. Darouac Lear fucoal observers for syses w elays sae varables: e scree-e case IEEE ras. Auo. Corol. Vol. No. pp. 8-. [] Y. Wag L. Xe C. E. e Souza Robus corol of a class of ucera olear syses. Sys. Corol Le. Vol. 9 No. 99 pp [] Z. Wag Y. Se X. Zag Q. Wag Observer esg for scree-e escrpor syses: A LI approac Syses & Corol Leers Vol. No. pp [] A. Zeouce. Bouayeb Observer syess eo for Lpscz o-lear scree e syses w e elay: a LI approac. Appl. a. Copu. Vol. 8 No. pp [] Y. Dog J. Lu S. e: Observer esg for a class of olear scree-e syses w e-elay. Kybereka. Vol. 9 No. pp. -9 [] S. Ya A.GalpUlsoya P. W. Nelsob Desg of observer-base feeback corol for eelay syses w applcao o auoove powerra corol Joural of e Frakl Isue Vol. No. pp. 8-. [] Y. Dog H. Wag Y. Wag Desg of observers for olear syses w H perforace aalyss aeacal eos e Apple Sceces Vol. No. pp. 8-. [8]. Abbaszae H. arquez Nolear observer esg for oe-se Lpscz syses : Proceegsof e Aerca Corol Coferece Balore USA pp [9] G. Hu Observers for oe-se Lpscz olear syses IAJ. a. Corol If. Vol. No. pp. 9. [] G. Hu A oe o observer for oe-se Lpscz o-lear syses IA J. a. Corol If. Vol. No. 8 pp. 9. []. Xu G. Hu Y. Zao Reuce-orer observer esg for oe-se Lpscz olear syses IAJ. a. Corol If. Vol. No. 9 pp. 99. [] W. Zag H.-S. Su Y. Lag Z.-Z. Ha Nolear observer esg for oe-se Lpscz syses: a lear arx equaly approac IE Corol eory a Applcaos Vol. No. 9 pp. 9. []. Beallouc.Bouayeb. Zasazsk Observer esg for oe-se Lpscz E-ISSN: -8 9 Volue

11 WSEAS RANSACIONS o SYSES a CONROL Waju Lu Yal Dog Ska Zuo scree-e syses Syses & Corol Leers Vol. No. 9 pp [] H. Kelouf A. Zeouce F. Beouee. Bouayeb O LI coos o esg observer-base corollers for lear syses w paraeer uceraes Auoaca Vol. 9 No. pp. -. [] H. Ce Y. Zag Y. Zao Sably aalyss for ucera eural syses w scree a srbue elay Apple aeacs a Copuao Vol. 8 No. pp. -. [] W. Zag X.-S. Ca Z.-Z. Ha Robus sably crera for syses w erval evaryg elay a olear perurbaos Joural of Copuaoal a Apple aeacs Vol. No. pp []. Sye Al O expoeal sably of eural elay ffereal syse w olear uceraes Cou Nolear Sc Nuer Sula Vol. No. pp. 9-. [8] H. r. Alee A eoryless sae observer for scree e-elay syses IEEE ras. Auo. Corol Vol. No. 99 pp. -. E-ISSN: -8 9 Volue