Application of Higher Order Derivatives of Lyapunov Functions in Stability Analysis of Nonlinear Homogeneous Systems

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1 Poceedgs of the Itetol MultCofeece of Egees d Comute Scetsts 009 Vol II IMECS 009, Mch 8-0, 009, Hog Kog Alcto of Hghe Ode Detes of Lyuo Fuctos Stblty Alyss of Nole Homogeeous Systems Vhd Megol, d S. K. Y. Nesh Abstct--he Lyuo stblty lyss method fo ole dymc systems eeds oste defte fucto whose tme dete s t lest egte sem-defte the decto of the system s solutos. Howee megg the both oetes sgle fucto s chllegg ts. I ths e some le combto of hghe ode detes of the Lyuo fucto wth o-egte coeffcets s esulted. If the esultt summto s egte defte d ll the detes e decescet the the zeo equlbum stte of the ole system s symtotclly stble. If the hghe ode tme detes of the Lyuo fucto e ot welldefed, the some well-defed smooth fuctos my be used sted. I ths cse le combto of tme detes of ll fuctos, wth o-egte coeffcets, must be egte defte. he ew codtos e the efomed to be led fo stblty lyss of ole homogeeous systems. Some exmles e eseted to descbe the och. Idex ems--ole systems, stblty lyss, Lyuo fuctos, hghe ode detes, homogeeous systems. NOMENCLAUE A ge om o x(t,t 0,x 0) A tjectoy sttg t x(t 0) = x0 u he udele ble mes ecto qutty m V: ecto fucto of dmeso m (VF) φ K ( φ K ) φ s fucto of clss K(K fty) [] () (t,x) he -th totl tme dete of (t,x) fuc. b comoet-wse equlty I. INODUCION Cosde the followg -dmesol outoomous dymc system wth zeo equlbum stte (ZES): x = f(t,x) t 0, x () he dtge of the Lyuo method s the use of Lyuo fuctos (LF) o eegy le fuctos. he Lyuo stblty lyss method fo ufom symtotc stblty (u..s.) of ZES of ole dymc systems eeds loclly decescet (LD)loclly oste defte fucto (LPDF) (t, x) whose tme dete (t,x) s egte Musct eceed August 0, 008. Both uthos e wth the Electcl Egeeg Detmet, Aboueyh Buldg, Amb U. of ech. (eh Polytechc), eh, I. Vhd Megol (coesodg utho, hoe: (+98) ; fx: (+98) ; e-ml: hd_mey@yhoo.com, megol@ut.c.). S. K. Y. Nesh (hoe: ; e-ml: sh@ut.c.). defte the decto of the system s solutos. Whe the dete s egte semdefte, stblty the th symtotc stblty (.s.) follows. Whe the comlexty of ole system s cesed, selectg sutble LF hg t lest egte sem-defte dete s olg ts, See [] d []. Gudeso [] cosdeed the stblty lyss of (), usg LF (t,x) wth the equlty (m) (m ) (t,x) g m (t,,,, ), fo some oste tege m, whee ll the hghe ode detes () (t,x) wee comuted wth esect to tme t log the tjectoes of (). S/he comed ths equlty by ole co-system (m) (m ) u (t) = g m (t,u,u,,u ). If the m g m () s of clss W (o-decesg) d the co-system hs.s. ZES the the ZES of () s lso.s. he method uses secl Vecto Lyuo fuctos (VLF) V(t,x) = [ (t,x), (t,x),, m(t,x)] () ( ) defg (t,x) (t,x), fo =,,m, but oly the fst comoet of V(t,x) s oste defte fucto (PDF) d the othe comoets mght be defte. hs s dffeet fom ody VLFs wth ll oste sem-defte comoets d geetg le combto m = (t,x), > 0 whch s PDF, [4] d [5]. We cll the VLF used by Gudeso [], detes ecto Lyuo fucto (DVLF). he we geelze the defto d efe to y ecto fucto V(t,x) DVLF s f s hg fst comoet whch s PDF (t,x) d the emde comoets e ossbly defte fuctos. Butz [6] cosdeed the utoomous system x = f(x) togethe wth LPDF (x) stsfyg (x) + (x) + (x) < 0, x 0 ( 0 fo =,) d cocluded.s. of the ZES. he eous eseches [7] d [8] used dffeetl equlty V AV fo DVLF to lyse the stblty of ZES of (). he A mtx ws cotollble cocl fom wth Huwtz chctestc equto,.e. det(si A ) = 0. I [9] we exteded the esult of Butz [6] to lyse the u..s. of ZES of () usg the hghe ode tme detes of tme yg LPDF (t,x),.e. f m m (d dt ) (t,x) s egte defte whe ll = m 0, the the ZES of () s u..s.. he ew method could te ce of the cses whee the LPDF (t,x) d/o the systems e ot smooth eough d the hghe ode ISBN: IMECS 009

2 Poceedgs of the Itetol MultCofeece of Egees d Comute Scetsts 009 Vol II IMECS 009, Mch 8-0, 009, Hog Kog tme detes of the LF e ot well-defed. We cosdeed moe geel fom of dffeetl equlty fo DVLF V(t,x), d the ZES of () would be u..s. s f m s m = (t,x) s egte defte d (t,x) < α( x ) fo α K d =,,,m. Now let us cosde the homogeeous ole systems whch cot wde gou of ole systems, d he bee oul dug the lst thee decde. A ce oety bout homogeeous systems s tht, they ct some how betwee le d ole systems. Also lot of subjects coceg the ole systems fst he bee led to homogeeous systems o e most elted to them, such s: cotollblty d locl oxmto [0], exoetl stblzto [], cotol by ddg owe tegto techque [], d fte tme stblzto []. he fst theoem of ths e summzes the m esults of [9] bout the hghe ode tme detes of LF wthout oof. he we focus o the lctos of ths theoem to stblty lyss of homogeeous ole systems. hs theoem s show to be useful oly fo stblty lyss of ole zeo degee homogeeous systems, hece ew theoem fo geel ole homogeeous systems s deeloed. We ssume the ede s fml wth the Lyuo stblty methods [-]. hs e s ogzed s follows. he elmy deftos d esults bout homogeeous systems e ge secto II. he m theoem o stblty lyss of homogeeous systems s eseted Secto III. Some exmles e ge IV. Cocludg ems e ge Secto V. II. HE PELIMINAY DEFINIIONS AND ESULS A. he Hghe Ode me Detes of LF If fucto (t,x) d the ole system () e smooth eough, the the hghe ode totl tme detes () (t,x), fo =,, log the solutos of () e (0) comuted tetely, usg ( (t,x) = (t,x) ) () ( ) ( ) (t,x) [ x] f(t,x) + t () Defto [9]: A bty scl fucto (t,x) (my be defte). s clled loclly decescet (LD) f thee exst > 0 d α K such tht fo eey x < (t, x) < α( x ) (4). s clled globlly decescet (GD) f (4) stsfes globlly. I the followg geel theoem fo lyzg the stblty of () s toduced. heoem [9]: Cosde the m-ecto C fucto V(t,x) of the fom () wth the followg oetes:. he fst comoet (t,x) of V(t,x) s dlly ubouded (U) d PDF,.e. (t,0) = 0, t 0 d thee exsts some φ K, such tht: (t, x) φ ( x ) x, t 0 (5). All the (t,x) comoets e GD,.e. thee exst α K fo =,,m such tht (t, x) α( x ) x, t 0 (6) ) If the followg dffeetl equlty stsfes fo ll (t,x) log the solutos of (): (7) 0 0 j 0 0 m, m,m 0 m m m m,m mm m φ( x ) whee φ K wth the dom of D φ = [0, + ) d A = [ j] m m s lowe tgul mtx wth the followg oetes: = 0, f < j (8) j > 0, f = j 0, f > j the the ZES of () s globlly ufomly symtotclly stble (g.u..s.). b) If the boe codtos hold oly loclly,.e. fo x < fo ge > 0 the the ZES of () s u..s. Coolly [9]: Cosde the smooth eough tme yg system () d smooth eough U d PDF (t,x). If the hghe ode detes () (t,x) fo ll = 0,,,m e GD d thee exst 0 fo =,,m d φ K such tht m () (t,x) φ ( x ), = x (9) he the ZES of () s g.u..s. ( ) Poof: use the heoem wth (t,x) (t,x) fo =,,m. em : Fo m = the boe coolly s educed to the Lyuo dect method fo the stblty lyss of the ZES of (). B. he Homogeeous Systems Cosde fucto : d the ecto feld f(t,x) of the ole system (), we befly ecll the oto of homogeety fo d f fom [4]: Fo sequece of oste weghts = (,, ), d o-egte ble λ 0, dlto s defed s le m λ (x) ( λ x,, λ x ). he the (t,x) fucto d the f(t,x) ecto feld e defed to be homogeeous of ode wth esect to (w..t.) the dlto λ, f (t, λ x) = λ (t,x) d f (t, λx) = λ λf (t,x) esectely. I ths cse we befly defe d f e - homogeeous of ode d symbolze wth H d f. he secl weghts = (,,,) e efeed s stdd weghts, hece (t,x) d f(t,x) e sd to be stdd homogeeous of ode f (t, λx) = λ (t,x) d f (t, λx) = λ + f (t,x) esectely. ISBN: IMECS 009

3 Poceedgs of the Itetol MultCofeece of Egees d Comute Scetsts 009 Vol II IMECS 009, Mch 8-0, 009, Hog Kog Fo mx the -homogeeous -om s defed by, = ( x ). It s cle tht, H, whle ths s ot tue om, becuse t does t stsfy the tgul equlty. Cosdeg -homogeeous LF the Lyuo dect method fo the stblty lyss of ge -homogeeous ecto feld s usul ts the ltetue [5]. I the followg we cocette o the lctos of hghe ode tme detes of -homogeeous LFs to - homogeeous systems. Exmle [9]: Cosde the followg ole dymc system: x = x (0) x = x x( b+ x x + x ) wth the followg metes = 0., b =. () whch s cotuous t x = x = 0 d hs ZES. hs system s obously of the stdd zeo ode homogeeous fom. Let us ewte the dymc equto (0) the ol coodte fom (t) d (t) usg x = cos C d x = s S : = S [S + bs + C ( )] () = S C (b + S )SC Usg the LF cddte (x) = x + x =, oe hs (x) = = S [S + bs + C ( )] () whch s defte fo the mete lues (), d thus the Lyuo dect method fls og g.u..s. of ZES usg ths LF. he hghe ode tme detes of (x) fucto would be s follows: (x) = (x) + (x) = [ C.74C (4) C +.8C + C + S (9.6.68C 7.C.4C )] 4 ( x ) = [ C C 0.48C 7.C C + 4.4C.96C 6.8C + S ( C C.8C C + 9.8C +.78C.6C 8C )] (5) hese detes wll be used the stblty lyss. Note () tht (x) fo ech = 0,,, s eodc fucto oly of. It ws show [9] fo =, =.4, = tht () (x) < 0 (6) = () Hece = (x) s egte defte d the codtos of Coolly e stsfed fo the ole utoomous system (0) d thus the ZES s g.u..s. I the boe exmle the ole system ws homogeeous of zeo ode, d ll the hghe ode detes of LF (x) (see eq. ()-(5)) wee homogeeous of ode two. he heomeo of sme ode of homogeety fo () (x) s ot ccdetl, d t s cosequece of the followg mott fct bout the - homogeety: Lemm [0]: If the fucto (t,x) H d the ecto feld f(t,x) w..t. some dlto λ, the the scl multlcto f +, d the totl tme dete of log the solutos of f,.e. (t,x) H +. heefoe by () ducto [(t,x)] f (t,x) + d (t,x) H + fo =,,. I the eous exmle (x) H d f (x) 0, d () thus (x) H + 0 fo =,,. heefoe y le combtos of () (x) fo seel e homogeeous fuctos of ode two, d we could esly obt ths sg usg the ol coodte. Howee the followg em shows some dffcultes fo homogeeous ole systems of ode > 0. em : If the ole system s homogeeous of ode > 0, the the hghe ode detes of homogeeous LF (x) e homogeeous of dffeet ode d we c ot esly deteme the sg of the le combtos. Moeoe ths cse t could be show tht f m () (x) < 0 x 0 d = 0 the (x) 0 ey e the og, becuse the fst dete domtes the othe detes ey smll eghbohood of zeo (see Lemm ). hus the Lyuo dect method s useful fo ths cse, d the heoem s megless fo homogeeous ole systems of ode > 0. III. HE MAIN ESULS It ws show the eous secto tht the heoem s ot useful fo stblty lyss of ole homogeeous systems of ode > 0. He we do some smll chges heoem d me t useful fo stblty lyss of ole homogeeous systems of bty ode. Fo smlcty we cosde oly utoomous cse,.e. the followg ole system: x = f(x), x (7) Let f fo some > 0 (7) d x s ge homogeeous om w..t. ge dlto λ, defe the followg ole system: f(x) x, x 0 x = f(x) = (8) 0, x = 0 It s cle tht f 0 d f(x) s cotuous t zeo. he descbed mg fom ole system (7) to the ole system (8) ws fst used [4] fo mlemetg the t homogeeous coes stblty lyss, but we use ths mg fo dffeet uose. Lemm : he ZES of (7) s g.u..s. ff the ZES of (8) s g.u..s. Poof: It s cle fom the defto tht y ole homogeeous system of o-egte ode such s (7) d (8) hs ZES. Moeoe sce x 0 fo x 0, the the ole system (7) hs ot y o-zeo equlbum ot ff the othe system (8) hs ot ethe. Moeoe the soluto cues of both systems cocde wth ech othe, but wth dffeet eloctes t ech ot. Hece ISBN: IMECS 009

4 Poceedgs of the Itetol MultCofeece of Egees d Comute Scetsts 009 Vol II IMECS 009, Mch 8-0, 009, Hog Kog we cosde the soluto cues of both systems s emeteztos of ech othe. he hse otts of the two systems e equlet d some qultte efomces such s g.u..s. of ZES e equlet. Now cosde ge C fucto g(x), we wt to come the tme dete g(x) log the solutos of (7) d (8) t ech ot x. hs s smly doe, by usg (), (7) d (8). Let t d t be the tme bles (7) d (8) esectely, d thus dg(x) dt = [ g(x) x] f(x) = (9) [ g(x) x] f (x) x = dg(x) x dt Let us ew tht both systems (7) d (8) e equlet usg the sme stte ecto x d the ble tme sclg (deedg o stte), becuse dx dt = f(x) = f (x) x = (dx dt) x dt = x dt (0) he eltosh (0) shows the eltty of tme sclg two systems. It deeds o the homogeeous om of the stte ecto x. Also (0) ges ew teetto of (9). Sce f 0, the heoem my be helful og the g.u..s. of ZES of (8). A. he M heoem he followg theoem coces the stblty lyss of (7), d uses (0) d Lemm ts oof. heoem : Cosde the ole homogeeous system (7) ( f ) d m-ecto C fucto V(x) = [ (x), (x),, m(x)]. If the followg codtos e stsfed:. (x) s U, PDF.. All (x) e GD d H fo =,,m.. the followg dffeetl equlty stsfes fo detes log the solutos of (7): j 0 0 x () m, m,m 0 m m m m,m mm m m+ whee m+ (x) H s PDF d A = [ j] m m s mtx wth the oety (8), the the ZES of (7) s g.u..s. Poof: Usg (9) yelds (d dt) x d dt fo the = tme detes of ech (x) log the solutos of (7) d (8). Moeoe mlemetg H d f yelds (d dt) H + fo =,,m. Ech tem () s homogeeous fucto of ode +. Ddg () by x d usg (d dt) x = d dt esults the followg eltosh, comoet se: A dv(x) dt [ (x),, (x), (x)] () m m+ Hece the codtos of heoem e stsfed fo g.u..s. of ZES of (8). Usg Lemm esults the g.u..s. of ZES of (7). B. he homogeeous ol coodte Although the heoem s lcble fo bty ode homogeeous systems, but we eed some desgg tools to fd the useful (x) fuctos fo ge ole system. I the eous exmle we used the ol coodte. he usul ol coodte s useful oly fo stdd ole homogeeous systems, but ot fo geel homogeety. Hee ew ol coodte w..t. ge weghts = (, ) fo = s toduced. We desgte to ech ot x = [x,x ] (, ) s - ol coodte. Cosdeg ge -homogeeous om, let () x = C x = S Defg u [C,S ] we he u, (C ) + (S ) = d u = x =, d thus x, = d H. Moeoe ech (x) H d f(x) could be decomosed s: (x) = (u ) (4) f(x) = f(u ) he decomosto of d s ey mott d wll be used ths e. Dffeettg () w..t. tme d solg fo d we obt: c s ( ) 0 c 0 x = 0 ( ) s c x 0 s (5) Usg (7) d (4) we he x = f(x) = f(u ) = f(u ). Substtutg ths to (5) yelds: + c s ( ) 0 c 0 = f(u ) ( (6) 0 s c ) 0 s he lst equto s the -ol dffeetl equto of the ole system wth f(x). he exteso of -ol coodtes to > s stghtfowd; Just set x = C fo =,, whee = C =. C. heoem Imlemetto A mott questo s : How to use the -ol coodtes to mlemet the heoem? Aswe: Usg the ssumto f(x) d the -ol coodtes we he x,=. I ou ocedue of mlemetg the heoem, we Costuct () oe ow fte othe. Let us be t the th teto,.e. (x) j fo j=,,, e eously defed d we m to fd (x) fo j=,, d costuct the 'th ow of + d j j j j = + (),.e. (x) (x) o equletly ISBN: IMECS 009

5 Poceedgs of the Itetol MultCofeece of Egees d Comute Scetsts 009 Vol II IMECS 009, Mch 8-0, 009, Hog Kog (x) (x) (7) j + j j( ) = + Accodg to the ssumto (x) j H, (x) j H +, theefoe fo j =,, the fuctos + j 0 ( (x) ) H (7) e deedet of,.e. they e ow eodc fuctos oly of. Hece usg umecl methods such s lottg (x) + esus, oe c fd le j 0 combto of them d ew fucto ( + (x) ) H such tht (7) stsfes. IV. SOME EXAMPLES Exmle : he ole system x x (8) x = x s stdd homogeeous of ode two,.e. f. Smlly to Exmle we chge (8) to the ol dffeetl equtos. Whe = = d = e used, the -ol coodtes fo stdd homogeety, cocde wth the usul ol coodtes. Substtutg u = [C,S ], =, = =, d = fo (6) we he: 0 C S C (9) = S C 0 S Let x = to ly heoem fo ths exmle. Sttg wth (x) = x + x =, we use the followg secl fom of () fo stblty lyss: (0) 0 0 = 0 0 m m m m,m mm m m+ Hece fo =,,,m (x) (x) (x) () + (x) = Substtutg (9) to () we he fo =,,,m C S C () + (x) = S C S All the hghe ode detes e well-defed, C eeywhee d belog to H, e.g. (x) C () (x) = = C S S We cosdeed the mete lues 0. A =, d comuted (x) usg () s well. Although (x) s PDF, but fo ths metes (x) s ot egte defte, d thus the Lyuo dect method fls to oe g.u..s. of ZES of (8). We he foud umeclly tht (x) (x) 4(x) s egte fucto oly of (see (7) fo ou method). Lettg m =, the eltosh (0) (d thus ()) s stsfed. Moeoe ll (x) e GD, d thus the ZES of (8) s g.u..s. Exmle : he ole system x 0 x x + x x = 5 0 x = (4) x x + x x s -homogeeous of ode two w..t. weghts = (, ) = (,),.e. f, becuse ( λx ) + ( λ x ) f( λ x) = 5 ( λx ) + ( λx ) ( λ x ) 0 λ x+ x = λ λ f(x) = 5 λ 0 λ x+ x x We use the -ol coodto (x,x ) = ( cos, s ) d the -homogeeous om 6 6,6 x = x + x = fo ths system. Substtutg (, ) = (,), = 6, u [ = C,S ] d = to (6) the we obt the -ol dffeetl equto s follows: / 0 C S C (5) = C 0 S C S he PDF 6 6 6,6 = + = (x) x x x d the equto (0) wll be used to stblty lyss of ZES of (4) usg heoem. substtutg (5) to () we wll he fo =,,,m / C S C (6) + (x) = C S C S All the hghe ode detes e well-defed, C eeywhee d belog to H 6. We he cosdeed the mete lues 0 A =, d fo ths metes (x) s ot egte defte, theefoe the Lyuo dect method fls to oe g.u..s. of ZES of (4). Lettg m =, (x) = (x) fo =, e comuted. he we he foud umeclly tht + (x) (x) 4(x) s egte fucto oly of (see (7) fo ou method), d thus (x) + 60 (x) 4(x) s egte defte. Moeoe ll (x) e GD, d thus the ZES of (4) s g.u..s. V. CONCLUSION he ew method toduced ths e s befly summzed s follows: Suose the.s. of ZES of ge homogeeous dymc system usg the Lyuo dect method s ude cosdeto. Fst oe tes to guess the coect homogeeous LF cddte wth egte defte fst ode dete. If the fst ode LF dete ws ot egte defte, the the Lyuo dect method s fled usg the ge LF, ee f the LF cddte s chose ey exetly. ISBN: IMECS 009

6 Poceedgs of the Itetol MultCofeece of Egees d Comute Scetsts 009 Vol II IMECS 009, Mch 8-0, 009, Hog Kog By the use of heoem, some oxmtos of the hghe ode tme detes of the LF e used to comeste the ole of o-egte defteess of the LF fst ode dete the stblty lyss. Some exmles e ge to show the ldty of the och. EFEENCES [] M. Vdysg, Nole Systems Alyss, Petce Hll, d Ed, 99. [] H. K. Khll, Nole systems, hd ed., 00. []. W. Gudeso, A Comso Lemm fo Hghe Ode jectoy Detes, Poceedgs of the Amec Mthemtcl Socety, Vol. 7, No., , 97. [4] V. M. Mtoso, Method of ecto Luo fuctos of tecoected systems wth dstbuted metes (suey) ( uss), Atomt elemeh, ol.,. 6 75, 97. [5] S. G. Neseso, W.M. Hddd, O the stblty d cotol of ole dymcl systems ecto Lyuo fuctos, IEEE scto o Automtc Cotol, ol. 5, o., 006. [6] A. Butz, Hghe ode detes of Luo fuctos, IEEE sctos o Automtc Cotol (Coesodece), Vol. 4,. -, 969. [7] V. Megol, d S. K. Y. Nesh, Exteso of Hghe Ode Detes of Lyuo Fuctos Stblty Alyss of Nole Systems, Acceted to the Amb Joul of Scece d echology, 008. [8] V. Megol, d S. K. Y. Nesh, A New heoem o Hghe Ode Detes of Lyuo Fuctos, Acceted to the ISA sctos, Elsee, 008. [9] V. Megol, d S. K. Y. Nesh, Hghe Ode detes of Lyuo Fucto Aoch fo Stblty Alyss of Nole Systems, submtted to Systems d Cotol Lettes, 008. [0] H. Hemes, Nlotet oxmtos of cotol systems d dstbutos, SIAM Joul Cotol Otmzto, ol. 4, o. 4,. 7-76, July 986. [].. M Closey,.M. Muy, Noholoomc systems d exoetl coegece: Some Alyss tools, Poceedg of the d Cof. o Decso d Cotol, , S Atoo, exs, Dec 99. [] C. Q, W. L, A cotuous feedbc och to globl stog stblzto of ole systems, IEEE s. O Automtc Cotol, Vol. 46, No. 7, , July 00 [] S.P. Bht, D.S. Beste, Fte-tme stblty of homogeeous systems, Poc. Amec Cotol Cof.,. 5-54, Jue 997. [4] M. Kws, Homogeeous feedbc stblzto, : New eds Systems heoy, G. Cote, A. M. Pedo d B. Wym eds., Pogess Systems d Cotol heoy, ol. 7 (99) [5] L. ose, Homogeeous Lyuo fucto fo homogeeous cotuous ecto feld, Systems d Cotol Lettes 9, , Noth-Holld, 99. ISBN: IMECS 009