Consider the time-varying system, (14.1)

Transcription

1 Leue 4 // Oulie Moivaio Equivale Defiiios fo Lyapuov Sabiliy Uifomly Sabiliy ad Uifomly Asympoial Sabiliy 4 Covese Lyapuov Theoem 5 Ivaiae- lie Theoem 6 Summay Moivaio Taig poblem i ool, Suppose ha x ( ) is a soluio of x = g(x ) as a efeee aeoy Le e= x x () The, e = x x () = g( g( x ()) = g( e+ x ()) g( x ()): = f(, e) If he equilibium e = of he eo equaio e = f(, e) is AS, he we say ha x as a efeee aeoy x ( ) The oigi e = of he ime-vayig sysem e = f(, e) oespods o he efeee soluio of he ime-ivaia sysem x = g(! Alhough he efeee sysem is auoomous, we have o solve a ime-vayig sysem fo a aig poblem Thee ae may poblems lie oholoomi sysems ha ae auoomous They eed a ime-vayig feedba o ool suh sysems, so hey ae ime-vayig sysems Noholoomi poblems have may appliaios i ool, lie igid obo models, udewae ship models, e This is a ho lie of eseah i ool ad appliaios I Mah iself, ime-vayig sysems ae a aual exesio of auoomous sysems Howeve, he hee impoa popeies of auoomous sysems ae o loge ue fo ime-vayig sysems So he sudy of ime-vayig sysems will be muh ompliaed ad ough as we expe Equivale Defiiios fo Lyapuov Sabiliy Coside he ime-vayig sysem x = f (,, (4) whee f is oiuous ad loally Lip i x o [, ) D, ad D R is a

2 domai oaiig he oigi, ad f(, ), ) Compaiso Fuio: Class K ad Class KL Defiiio 4 A oiuous fuio α : [, a ) [, ) is of lass K if i is sily ieasig ad α ( ) =, deoed by α K I is lass K if a = ad α () as, α K Defiiio 4 A oiuous fuio β : [, a ) [, ) [, ) is of lass KL if, fo eah fixed s, β (, s) is lass K w ad, fo eah fixed, β (, s) is deeasig w s ad β (, s) as s Example 4 a( ) = a sie α () = > I is of lass K, bu o of lass K + π sie limα( ) = < α () = fo ay > sie () = > Moeove, lim ( ) = α α ; hus, i is of lass K β (,) s = s + s sie β i fo ay > sie = > ( s + ) β = < s ( s + ) ad i Moeove, β (, s) as s Hee, i is of lass KL s β (, s) = e, fo >, is of lass KL Lemma 4 Le α () ad α () be of lass K fuios o [, a ), α () ad α () be of lass K 4 fuios, ad β (, s ) be of lass KL fuio The,

3 α is defied o [, a ( a)) ad is of lass K ; () α is defied o [, ) ad () is of lass K ; α α α is of lass K ; α α α 4 is of lass K ; s(, s) = α( βα ( ( ), s)) is of lass KL Rema 4 Class K ad KL fuios ae impoa ools i aalysis of oliea sysems ) Equivale Defiiios of Uifom Sabiliy Lemma 4 The oigi x = of (4) is uifomly sable (US i sho) α () K ad >, idepede of, suh ha x ( ;, α( x ),, x < ; (4) uifomly asympoially sable (UAS i sho) β (, s) KL ad >, idepede of, suh ha x ( ;, β ( x, ),, x < ; (4) uifomly globally asympoially sable (UGAS i sho) (4) is saisfied fo ay x R s expoeially sable (ES) if (4) is saisfied wih β (, s) = e γ s globally expoeially sable (GES) if (4) is saisfied wih β (, s) = e γ fo ay x R Rema 4 Fo ime-vayig sysems, GAS ad UGAS ae diffee Fo example, x x = + has a soluio x () = x + + I is GAS Howeve, i is o UGAS by oadiio If hee exiss β KL s fo all, + x ( ) = x β ( x, ) +

4 ould be saisfied, we would have + = β (, + ) + by aig x = ad = + > Sie β (, + ) as, his oadiio shows ha GAS ad UGAS ae diffee ) Some Impoa Auxiliay Resuls Lemma 4 Coside a sala equaio y α( y), y ( ) = y whee α() K is a loally Lip, defied o [, a ) Fo all y a, his equaio has a uique soluio defied fo all Moeove, y () σ ( y, ), whee s (, s) KL defied o [, a ) [, ) Rema 4 This ompaiso lemma is vey useful fo aalysis of Lyapuov sabiliy The poof iself is simply appliaio of he ompaiso piiple Lemma 44 Le V ( > be posiive defiie, whee x D R, ad B D whee > The, α (), ( ) α K, defied o [, ), suh ha α ( x ) V( α ( x ) (44) fo all x B Moeove, if D = R ad ( V is adially ubouded, he α () ad α () a be hose o be of lass K ad (44) holds fo all x R Rema 44 All he poofs of hese Lemmas a be foud i Noliea Sysems d ed by H Khalil, Peie Hall, Uppe Saddle Rive, NJ, Hee all ae omied 4) Lyapuov Theoem fo Time-Vayig Sysems Theoem 4 Le V :[, ) D R be of C suh ha 4

5 W ( V(, W ( ; (45) V V + fx (, ), (46) x fo all, ad all x D, whee W ( x ) ( =, ) ae posiive defiie The, x = of (4) is US If D = R, ad W ( ) x is adially ubouded, he he oigi is uifomly globally sable Poof Sie he deivaive of Vx (, ) alog aeoies of (4) is give by V V V (, = + f(,, x we hoose > ad > suh ha B D ad < mi W ( x ) The, x = { x B W ( } B Defie a ime-depede se Ω, by Ω = { x B Vx (, ) }, Sie W ( V (,, we have { ( ) } W x B W x, O he ohe had, V (, W ( yields W, { x B W( } Thus, { x B W( } Ω, { x B W( } B fo all These five esed ses ae sehed i Fig 4 Fig 4 5

6 Fo ay, ad ay x Ω, he soluio, x (;, x ) says i Ω, fo all beause V ( x, ( ;, ) Vx (, ( ;, ) V (, W( fo all Theefoe, x (;, B ad x (;, x ) is defied fo all By Lemma 44, hee exis α ad α K, defied o [, ], suh ha The, we have W ( α ( x ), W( α ( x ) α ( x( ;, x ) ) V(, x( ;, x )) V(, x ) W ( x ) α ( x ) Fom whih we olude ha whee If x ( ;, α ( α ( x )) = α( x ), α K by Lemma 4 Theefoe, he oigi is US by Lemma 4 D = R, α, α K Hee, α ad α ae idepede of > Sie W ( ) x is adially ubouded, we a hoose > suh ha x α ( ) The, x { x R W( } This shows ha he oigi is globally uifomly sable Theoem 4 Le V :[, ) D R be of C suh ha W ( V(, W ( ; (45) V V + f (, W (,, x D, (47) x whee W ( x ) ( =,, ) ae posiive defiie The, x = of (4) is UAS If D = R, ad W ( ) x is adially ubouded, he he oigi is UGAS Poof We go o wih he poof of Theoem 4, we ow ha x (;, B ad x (;, x ) is defied fo all Suppose ha x { x B W( } By Lemma 44, hee exis α K, defied o [, ], suh ha W( α ( x ) Hee, we have α ( x ) Vx (, ) α ( x ) ; 6

7 Vx (, ) α ( x ) Cosequely, V α ( x ) α ( α ( V)) = α( V) Assume, wihou loss of geealiy, ha α () is loally Lip Le y () be he soluio of y = α ( y), y ( ) = V (, By he ompaiso piiple, we have V( x, ( ;, ) y ( ), By Lemma 4, hee exiss s (, s) KL defied o [, a ) [, ) suh ha Vx (, ( ;, ) y ( ) σ ( V (, ), ), fo ay V (, [, ] Theefoe, ay soluio saig i Ω, saisfies he iequaliy x( ;, x ) α ( V(, x( ;, x ))) α ( σ( V(, x ), )) α ( σα ( ( x ), )) = β( x, ) By Lemma 4 i shows ha β (, ) is of lass KL fuio Thus, he iequaliy (4) is saisfied fo all x { x B W( }, whih implies ha x = is UAS If D = R, ( ) W x is adially ubouded, so is W ( ) x by (45) Theefoe, we a fid α K s W( α ( x ) Fo ay x R, we hoose > suh ha x α ( ) The, x { x R W( } The es of he poof is he same as he above pa fo showig UAS Rema 45 V (, saisfyig he lef iequaliy of (45) is said o be posiive defiie; ad saisfyig he igh iequaliy of (45) is said o be deese Vx (, ) saisfyig (45) ad (47) is alled a si Lyapuov fuio If W ( x ) is posiive semi-defiie i (47), V (, saisfyig (45) ad (47) is alled a Lyapuov fuio 7

8 Rema 46 { x B W( } a be ae as he esimae of he egio of aaio sie W ( x ) ( =, ) ae posiive defiie Rema 47 UGAS is oeed by ool fo ime-vayig sysems If α () =, we have he followig esul fo expoeial sabiliy Coollay 4 Suppose all he assumpios of Theoem 4 ae saisfied wih W( x, W( x, W( x fo >, ad > The, x = is ES Moeove, if he assumpios hold globally, he x = is GES Poof V ad V saisfy he iequaliies By he ompaiso lemma, Hee, x Vx (, ) x ; Vx x Vx (, ) (, ) V( x, ( )) V(, x ( ))exp{ ( )} V(, x ( ))exp{ ( )} V( x, ( )) x ( ) ( ) exp{ ( )} x x = ( ) ( ) exp{ ( )} Hee, he oigi is ES If all he assumpios hold globally, he above iequaliy holds fo all x ( ) R 8

9 Example 4 Coside x = ( + g ( )) x whee g () is oiuous ad g ( ) fo all Usig he Lyapuov fuio adidae V x = >, we obai ( ) x 4 4 = ( + ( )) <, x R V g x x, Hee, he oigi is UGAS Example 55 Coside x = x gx () x = x x whee g () is oiuously diffeeiable ad saisfies g( ) ad g ( ) g( ), Tae Vx (, ) = x + ( + g ( )) x, saisfyig x + x Vx (, ) x + ( + x ), x R The, Vx (, ) = x + xx (+ g () g ()) x, Usig he iequaliy + g( ) g ( ) + g( ) g( ), x x T V (, x + xx x = : = x Qx< x, x T whee Q is posiive defiie; hee, V (, is egaive defiie The oigi is GES 4 Covese Lyapuov Theoem Theoem 4 Le x = be equilibium fo he ime-vayig sysem x = f (,, (4) whee f :[, ) D R is oiuous ad loally Lip i x o D = { x R x < } Assume ha fo ay (, [, ) D, he soluio x (;, x ) of (4) saisfies 9

10 x ( ;, β ( x, ), x D,, (48) whee D = { x R x <, β (,) < } D The, hee exiss a oiuous fuio Vx (, ) of C ha saisfies α ( x ) Vx (, ) α ( x ) ; (49) Vx (, ) α ( x ), (4) whee α K ( =,, ) Moeove, if (48) holds globally, he, (49) ad (4) hold globally, whee α K Poof Fis of all, fo ay x D, x (;, D fo all D is a suiable egio of aaio Fo ay β (,) s KL, hee exiss α, γ K, ( K fo global ase) s α( β( s, )) γ( s)exp{ } (4) Rema 48 (4) is a vey useful esimae fo aalysis of Lyapuov sabiliy I was show by E Soag Sie my poof of his ovese heoem is oivial, diffee fom he adiioal oe My poof is oloal osaied Defie Vx (, ) as follows Vx (, ) : = sup α( x ( + ; x, ) )exp{ }, whee x ( + ; x, ) is a soluio of (4) Taig = yields Meawhile, by (4), we have V (, α( xx ( ;, ) ) = α( x ) : = α ( x ) V(, sup αβ ( ( x, ))exp{ } γ( x )supexp{ } γ( x ) : = α( x ) The deivaive of Vx (, ) alog aeoies of (4) is give V ( hx, ( hx ;, )) Vxx (, ( ;, )) V ( hx, ) Vx (, ) Vx (, ) = lim = lim, h h h h whee x= x ( + h ;, Sie

11 V(, : = sup α( x( + h+ ; + h, )exp{ } = sup α( x ( + h+ ; x, ) )exp{ } sup α( x ( + ; x, ) )exp{ }exp{ h} = Vx (, )exp{ h}, we have exp{ h} V ( hx, ) Vx (, ) Vx + (, ) = lim lim Vx (, ) = Vx (, ) h h h h α( x ) : = α( x ) If (48) holds globally, obviously α K, heefoe, (49) ad (4) hold globally If (4) is a auoomous sysem, we have x ( ; x, ) = x ( ; The, V(, : = sup α( x ( + ; x, ) )exp{ } = sup α( x ( ; )exp{ } = V(, is idepede of Rema 49 Compae o he adiioal poof, his poof is oloal ad o so ompliaed Alhough he smooh equieme of f is deeased, he smooh of Vx (, ) is also deeased Howeve, his disadvaage a mae up by usig he mehod povided by Yuada Li, ED Soag ad Yua Wag fo he followig pape Yuada Li, ED Soag ad Yua Wag, A Smooh Covese Lyapuov Theoem fo Robus Sabiliy SIAM J Cool ad Opimizaio, vol 4, o pp 4-6, Ivaiae- lie Theoem I Theoem 4, if (47) is posiive semi-defiie, ie V V + fx (, ) W ( x ), x whee W ( Le S = { x R W( = } The bes we a do is ha he

12 aeoies of (4) appoahes S Wha addiioal odiios we a expe fo UAS is o lea so fa Howeve, he osuio of a si Lyapuov fuio based o his Lyapuov fuio is eely a eseah ho lie Lemma 45 (Babala s Lemma) Le ϕ : R R be a uifomly oiuous fuio o [, ) Suppose ha lim ϕ( s) ds exiss ad fiie The, ϕ() as Rema 4 Babala s Lemma iself is ohig wih (4) Howeve, i plays impoa ole bu havig shoomigs fo UAS of (4) whe i oes aeoies of (4) I has seveal vaiaios I hope you do a suvey o Babala s Lemma if you ae ieesed i his opi Theoem 44 (LaSalle-Yoshizawa) Le V :[, ) D R be of C suh ha W ( V(, W ( ; (4) V V + f (, x ) W ( x ),, x D, (4) x whee W ( x ) > ( =, ) ae posiive defiie ad W ( ) x is posiive semi-defiie o D The, all aeoies x (;, x ) of he sysem (4) wih x { x B DW ( }, whee < mi W ( x ), ae bouded ad saisfy x = lim W( x ( ;, ) = (44) Moeove, if all he assumpios hold globally ad W ( ) x is adially ubouded The saeme is ue fo all x R Poof Simila o he poof of Theoem 4, i a be show ha x { x B W( } Ω, x (;, x ) W, { x B W( } B, fo all Hee, x ( ;, fo all Sie Vx (, ), Vx (, ) is o-ieasig o alog he aeoies x (;, x ) ad bouded below by zeo Theefoe, lim V(, x ( ;, ) exiss ad fiie fo eah (, [, ) { x B W( }

13 Now, I implies, W ( x( s;, x )) ds V ( s, x( s;, x )) ds = V(, x ) V(, x( ;, x )) lim ( ( ;, )) (, ) W x s x ds x x = <, fo eah (, x ) Nex, we eed o pove ha W( x ( ;, x )) is uifomly oiuous i o [, ) Whe x { x B W( } x (;, B, f(, x ) saisfies loal Lipshiz i x o he ompa se B, we deoe is Lipshiz osa as L The, fo ay,, we have ( ;, ) ( ;, ) (, ( ;, )) = (, ( ;, )) (, ) x x x x f s x s x ds f s x s x f s ds L x( s;, x )) ds L < ε ε wheeve < δ = Theefoe, x (;, x ) is uifomly oiuous o L [, ) So is W( x ( ;, x )) beause W ( x ) is oiuous o he ompa se B, so i is uifomly oiuous o B The, appliaio of Babala s Lemma yields lim W( x ( ;, ) = fo eah, x { x B W( } (44) If all he assumpios hold globally ad W ( ) x is adially ubouded, fo ay give x R, hee exiss > s x { x B W( } Ω, The es of he poof is he same wih he loal This omplees he poof Rema 4 Thee is a puzzle fo (44) Is i uifomly ovege wih espe o he iiial ime? This is o sue fom he above poof hough Babala s Lemma How o give a igoous poof is ieesig!! O give a oue-example o show ha saisfyig Babala s Lemma is o uifomly ovege!! Please pay aeio o his issue beause i is fudameal i aalysis of ime-vayig sysems Rema 4 lim W( x ( ;, ) = x ( ;, S= { x B W( = } as

14 Ω(, S, whih is o eessaily ivaia! I is big diffee fom he ase of auoomous sysems 6 Summay Class K ad KL fuios ae useful ools i sabiliy aalysis of oliea sysems Uifom asympoi sabiliy a be poved usig V (, ha may depede o Howeve, hee should exis α, α, adα K, suh ha α( x ) V (, α( x ) ; V (, α ( x ) So he boudig fuios α ae idepede of Fo he Ivaiae-lie ase, hee is a boad ope spae o exploe fo eseah Hope you daw some aeio Homewo Show Lemma 4 Show Lemma 44 Show Lemma 45 (Babala s Lemma) 4