Exponential Stability of Gradient Systems with Applications to Nonlinear-in-Control Design Methods

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1 Poeedigs of he 7 Ameia Cool Cofeee Maio Maquis Hoel a imes Squae New Yo Ciy, USA, uly -3, 7 FC4. Eoeial Sabiliy of Gadie Sysems wih Aliaios o Noliea-i-Cool Desig Mehods Eugee Lavesy, Chegyu Cao, ad Naia Hovaimya Absa Eoeial sabiliy aalysis fo gadie sysems is he imay fous of his ae. Suffiie odiios ae deived ha guaaee eoeial sabiliy fo boh auoomous ad aamee-deede gadie sysems. hese odiios equie boudedess of sigula values of a aobia mai, uifomly i he sysem sae sae. he eoed heoeial esuls ae subsequely alied o desig aig oolles fo a lass of oliea-i-ool dyamial sysems. he desig is aied ou usig ime-sale seaaio ehiques. I I. INRODUCION his ae, we ese suffiie odiios fo eoeial sabiliy of dyamial sysems whose igh had side eeses a gadie maig. I aiula, we fis ivesigae eoeial sabiliy of he auoomous gadie sysems i he fom: f ( (. = f ( = V ( V( whee X R is he sysem sae veo, V ( gadie of a sala diffeeiable fuio V.5 f ( is he =, defied i a oe domai X R, ad f : X R is a oiuously diffeeiable ma. I is fuhe assumed ha hee eiss a veo eq X suh ha V ( eq =. hus by he defiiio, eq eeses he sysem equilibium. Wihou loss of geealiy, we sudy eoeial sabiliy of he oigi, ha is eq =. Lae i he ae, we iodue ad aalyze uifom eoeial sabiliy of he aameedeede gadie sysems: f (, θ = f (, θ = V (, θ (. whee V(, θ θ R is a veo of osa aamees. his maeial is based uo wo suoed by he Uied Saes Ai Foe ude Coas No. FA955-4-C-47 ad FA E. Lavesy is wih he Boeig Comay, Huigo Beah, CA 9647, eugee.lavesy@boeig.om C. Cao is wih AOE Deame, Vigiia eh, Blasbug, VA 46-3, hegyu@v.edu N. Hovaimya is wih AOE Deame, Vigiia eh, Blasbug, VA 46-3, hovaim@v.edu Noe ha usig V ( as he Lyauov fuio, oe a easily esablish asymoially sabiliy of (.. Howeve, we emhasize ha i is o he asymoi bu ahe he eoeial sabiliy whih is of seial iees o us. he esuls eoed i his ae ovide a heoeial basis fo he oosed i his ae desig of oliea-i-ool dyamial sysems. Sabiliy aalysis of hese mehods is based o ihoov heoem [], [, heoem., ], whee eoeial sabiliy of he so-alled edued ad bouday-laye sysems is show o be suffiie o guaaee oveall losed-loo sysem sabiliy. o eieae, ou moivaio fo aalyzig he lass of gadie sysems is imaily dive by ool desig oblems fo oliea-i-ool dyamial sysems. hese sysems wee osideed i [] ad hei ool desigs wee efomed usig ime-sale seaaio mehods, []. Suffiie odiios wee saed ha guaaeed eisee of dyami aig oolles asymoially ovegig o hei oesodig ideal / uow dyami ivesio based ool soluios. I his ae, we will eed / ela hese suffiie odiios. Afe eesive lieaue seah, eees [3, 4, 5] wee ideified o oai mos ee ad eleva esuls wih egads o eoeial sabiliy aalysis of gadie sysems. I [4, 5], he auhos aalyze asymoi ad eoeial sabiliy of he so-alled globally ojeed dyami sysems ha wee oigially iodued i [3]. Iside a edefied oma domai, hese sysems beome of gadie ye. I [4], suffiie odiios fo eoeial sabiliy of hese gadie sysems ae fomulaed, assumig ha he gadie ma f ( = V ( is sogly moooe wih modulus µ, [6]. he lae imlies osiive-defiieess of he ma f ( aobia ( =, uifomly i X. I his ae, we ove eoeial sabiliy ude a muh weae assumio, whih equies ha he sigula values of he aobia ae fiie ad bouded away fom zeo, uifomly i X. I [5, heoem 5. 39], he auhos laim fiie ime ovegee fo wha hey alled a eual ewo model, ude some mild assumios. Ufouaely, he oof of hei heoem 5 is ivalid. Fis, Mea Value heoem is ioely alied o a veo fuio, [5, eq. 6,. 36]. Seod, i is saed ha if he sysem sas a a o-equilibium he he veo om of he igh had side of (9 o. 37, alled G( z, is uifomly bouded away fom zeo. his is lealy ioe, sie as he sysem sae aoahes he /7/$5. 7 IEEE. 63 Auhoized liesed use limied o: UNIVERSIY OF CONNECICU. Dowloaded o Febuay, 9 a 9:5 fom IEEE Xloe. Resiios aly.

2 FC4. equilibium, he igh had side eds o zeo, (usig oiuiy agume. Beause of hese, fiie-ime ovegee agumes eseed i [5,. 36], ae ivalid ad so is he mai laim of heoem 5. Cosequely, he quesio of a fiie ime ovegee fo gadie sysems emais oe. he es of he ae is ogaized as follows. Seio II eses basi defiiios ad fas, whih ae subsequely used houghou he ae. Ou s mai esul (heoem 3. o he eoeial sabiliy fo auoomous gadie sysems is eseed i Seio III. Suffiie odiios fo uifom eoeial sabiliy of aamee-deede gadie sysems ae saed i Seio IV. Based o hese esuls, Seio V eses ool desig mehod fo oliea-i-ool sysems. his desig osiues ou d heoeial oibuio, (heoem 5.. he ae eds wih olusios give i Seio VI. II. PRELIMINARIES Defiiio. Le λi ( i deoe he i h eigevalue of a mai ad le m A R. he, he sigula values of A ae he mi (, m oegaive umbes ( A,, mi (, m( A, whee ( A fo all i,, mi (, m =. i ( ( λi A A, m = λi A A, m (. Lemma. m Le f ( : R R be a diffeeiable ma, defied i a oe eighbohood X of R. Le s R. he fo ay X, he followig eeseaio aes lae: f ( + s( f ( = f ( + ds ( (. Moeove, if he aobia of f is oiuous i X he fo ay, X hee mus eis sala bouded fuios h ξi = ξi (, suh ha i m, he i omoe fi ( of f ( a be wie as: fi ( + ξi (, ( fi ( = fi ( + ( (.3 Poof: Diely follows fom Mea Value heoem []. Defiiio. A auoomous gadie sysem is a dyamial sysem of he fom V ( V ( (.4 = = V D R R whee : is wie oiuously diffeeiable sala fuio of he veo agume R. A aamee-deede gadie sysem is defied as = V, θ (.5 ( whee θ R is a fied veo of aamees. Rema. Le V( = f ( f (, whee f : X R R is a oiuously diffeeiable fuio i he oe domai X. he, aodig o (.4, he dyamis f ( = V ( = f ( (.6 osiue a auoomous gadie sysem. Lemma. he followig eeseaio is valid fo ay osigula mai A R ad fo ay veo R : (.7 A ma ( A mi ( A whee mi ( A, ma ( A ae he miimum ad he maimum sigula values of A ad deoes ay Eulidia veo om of. Ne, we eall few basi defiiios ad ivese fuio heoems fom [8] ad [9]. Le X ad Y be wo oemy subses of R. Wih ay osiive iege, le C deoe he se of all imes oiuously diffeeiable maigs bewee oe saes of R. Defiiio.3 he ma f ( : X Y is alled a homeomohism of X oo Y if f is oiuous, oe-o-oe, f ( X = Y, ad he ivese ma f eiss ad oiuous. Defiiio.4 he ma f ( : X Y is oe if he ivese image f ( U { X : f ( U } = (.8 of ay oma se U Y is oma. Rema. Le A ad B be ay olleio of oma subses of Y ad X, eseively. he i [9,. 999 ], i is show ha f is oe if ad oly if fo eah S A, hee is a B suh ha f ( s S imlies s. Moeove, i is show ha if f is a oiuous ma of R oo iself, he f is oe if ad oly if i is adially ubouded, ha is if lim f (. Defiiio.5 he ma f ( : X Y is alled a C diffeomohism of X oo Y if f C, he ma is a homeomohism of X oo Y, ad f C. 64 Auhoized liesed use limied o: UNIVERSIY OF CONNECICU. Dowloaded o Febuay, 9 a 9:5 fom IEEE Xloe. Resiios aly.

3 FC4. he followig heoem is due o Hadamad, [9]. I eses suffiie ad eessay odiios fo a ma f o be a C diffeomohism. I he heoem, he ma aobia h i, j is defied as he ( mai (, whose ( f ( eleme i j eeses a aial deivaive of he i h fuio omoe wih ese o he j h omoe of he veo agume. Geealizaios of he Hadamadye global ivese fuio heoem a be foud i []. heoem. (Hadamad Le f C. he f is a C diffeomohism of X oo Y if ad oly if de ( fo all X ad f is oe. III. EXPONENIAL SABILIY FOR AUONOMOUS GRADIEN SYSEMS Le f : X Y defied i a oe domai oe domai Y R. C ma X R wih is values i he be a oiuously diffeeiable ( Assumio 3. he ma f is oe ad hee eis osiive osas mi, ma suh ha < mi i ( ( ma < (3. i, X R, whee ( i is he i h sigula value of he mai aobia (. Lemma 3. he double-iequaliy (3. is equivale o he followig oe: i, X R < i ( ( < (3. ma i mi Rema 3. he assumed bouds i (3. guaaee ha he aobia is osigula, uifomly i. Howeve, his mai may have eigevalues wih osiive ad / o egaive eal as, as well as eigevalues o he imagiay ais. Hee, his odiio is less osevaive ha he suffiie odiio fo he eoeial sabiliy eoed i [4]. Defie symmei osiive-defiie mai P( = ( ( (3.3 he he eigevalues of P( ae he squaed sigula values of he aobia (, ha is λi ( ( i ( ( P =, i =,, (3.4 Cosequely, (3. is equivale o: < λ P < (3.5 ( ( mi ma ad, heoe ( mi P ma (3.6 he mai esul of his seio is saed e. heoem 3. Le f ( : X Y be a oiuously diffeeiable ma, defied i a oe domai X R, X, ad wih is values i he oe domai Y R, Y. Suose ha f =. Also suose ha Assumio 3. holds. Le ( ( deoe he ma aobia. he he oigi is he uique i X eoeially sable equilibium of he auoomous gadie sysem =( f ( (3.7 Poof: he fa ha he oigi is he uique sysem equilibium follows diely fom (3. ad he assumed elaio f ( =. hus, oly eoeial sabiliy emais o be ove. owads ha ed, oside he Lyauov fuio adidae: V( = f ( = f ( f ( (3.8 Fis, we deive a ue boud o he Lyauov fuio whih will be used lae o i he oof. Usig (. wih =, ha is alyig he Mea Value heoem o he ma f (, ad subsiuig he esul io (3.8, gives: (3.9 V ( = f ( f ( = ( s ds ( s ds ma he ime deivaive of V( ( evaluaed alog he sysem (3.7 ajeoies is V ( ( ( ( ( (3. = f f P ( hus, he dyamis ae sable. Moeove, wih he hel of he LaSalle Ivaiae heoem [] oe a show asymoi sabiliy of he oigi. Howeve, ou iees is i showig eoeial sabiliy. owads ha ed, we use (3.5, (3.6, (3., ad wie: V ( =f ( P( f ( mi ( P( f ( mi V ( (3. Alyig he Comaiso Lemma [, Lemma 3.4,. 3], yields he ue boud: mi mi V ( ( e V ( ( = e V (3. whee ( V X deoes ay iiial odiio of he sysem sae. Usig he ue boud fom (3.9, esuls i he eoeial ae of deay fo he values of he Lyauov fuio, ha is: mi V( ( e ( ma mi = e V (3.3 Equivalely, (3. a be wie as: mi f ( ( e V (3.4 A his mome, oe eeds o ove ha as, o oly f eoeially ovege o zeo he fuio values ( ( 65 Auhoized liesed use limied o: UNIVERSIY OF CONNECICU. Dowloaded o Febuay, 9 a 9:5 fom IEEE Xloe. Resiios aly.

4 FC4. bu also he values of he sae veo ( ed o he oigi eoeially fas. Fom (3.4 i follows ha mi fi ( ( e V (3.5 fo all i. hus, hee mus eis eoeially deayig fuios { ( ( } mi gi = O e i= suh ha f ( ( = g( = ( g ( g ( (3.6 Due o Assumio 3. ad heoem., he ivese ma f ( i eiss ad i is oiuously diffeeiable a evey oi X. I ohe wods, he ma f ( i eeses a C diffeomohism of X oo Y. Cosequely, he sysem of equaios (3.6 a be wie as: ( = f ( g( (3.7 Alyig (. o he ivese ma above, while eaig as a aamee, yields: f ( s g( ( = f ( g( = ds g( (3.8 g whee s deoes he iegaio vaiable. Ne, we deive lowe ad ue bouds fo he om of he ivese ma aobia. Diffeeiaig he ideiy f ( f ( g = g (3.9 wih ese o g, esuls i: ( f f ( g f ( g = I (3. g heoe, f ( g (3. = ( ( = f ( g g ad usig Lemma 3., gives uifom bouds fo he aobia of he ivese ma f ( i : f ( g (3. ma g mi Fuhemoe, usig (3.8 ad (3., yields: f ( s g( V mi ( ds g( (3.3 e g mi Fially, subsiuig he ue boud fom (3.9 io (3.3 fo V, esuls i: ma ( mi e ( (3.4 mi ad oves ha he sysem sae ( oveges o he oigi eoeially fas, as. Rema 3. Lieaizaio of he sysem dyamis (3.7 aoud he oigi a be easily efomed esulig i he oesodig liea dyamis: = ( ( = A (3.5 A hus, if he aobia evaluaed a he oigi is osigula, de, he liea sysem dyamis ae loally ( eoeially sable. Comaig his odiio wih he suffiie odiios i heoem 3., i beomes lea ha he fome is a subse of he lae. IV. UNIFORM EXPONENIAL SABILIY FOR PARAMEER- DEPENDEN GRADIEN SYSEMS Le X Θ R R ad Y R be oe domais ad le f : X Θ Y be a C ma, wih ese o. Coside a aamee-deede gadie sysem of he fom: f (, θ (4. f (, θ = whee θ Θ R eeses a osa veo of aamees. Assume ha = eq is he sysem uique equilibium f (, eq θ = (4. defied uifomly i θ. he followig defiiio fom [] iodues he oio of uifom eoeial sabiliy. Defiiio 4. he equilibium oi = eq X of he aameedeede gadie sysem (4. is eoeially sable, uifomly i θ Θ R, if hee eis osiive osas, γ, ad ρ suh ha he soluios of (4. saisfy ( eq ( eq e γ (4.3 fo all iiial odiios ( eq < ρ, fo all fied aamees θ Θ R, ad fo all. heoem 4. Le f : X Θ Y be a oe oiuously diffeeiable ma, ad le (, θ deoe he ma aobia omued wih ese o he sysem sae veo. If i ad (, θ X Θ R R < mi i ( (, θ ma < (4.4 he he equilibium oi = eq X of he aameedeede gadie sysem (4. is globally (i X eoeially sable, uifomly i θ Θ R. Poof: If eq he iodue ew vaiable y = eq. I his ase, oe eeds o ove eoeial sabiliy of he oigi, whih is easily show followig he ea same agumes used duig he oof of heoem 3.. V. APPLICAION: NONLINEAR-IN-CONROL DESIGN FOR KNOWN DYNAMICS USING IME-SCALE SEPARAION Coside he followig ow sysem dyamis: ( = A( + Bg( u + B ( (5. y( = C( m whee R is he dimesioal sae veo, u R m m is he ommaded ool iu, g ( u : R R is a ow oliea ool ma, ad u( = g( u (5. 66 Auhoized liesed use limied o: UNIVERSIY OF CONNECICU. Dowloaded o Febuay, 9 a 9:5 fom IEEE Xloe. Resiios aly.

5 FC4. is he m dimesioal ool iu. Also i (5., y is he dimesioal egulaed ouu, ( is he dimesioal eeal ime-vayig ommad, ad maies A, B, B, C ae ow ad of he oesodig dimesios. Moeove, i is assumed ha ( A, B is oollable, ( C, A is obsevable, ad = m. he ool goal of iees is bouded aig, ha is oe eeds o fid ommaded sigal u suh ha he sysem ouu y as eeal ommad, wih bouded eos ad i he esee of he ool olieaiies i g ( u. houghou he es of he ae, i is assumed ha a u = g( u is a globally iveible ma, ad b he ivese ma u = g ( u is o ow aalyially. I essee, he uavailabiliy of he ivese ma osiues he mai ool hallege. Rema 5. Le ey = y deoe he ouu aig eo sigal. Moivaio fo osideig a sysem dyamis i he fom of (5. omes fom he fa ha ofe, i aial aliaios, he oolle is eeed o a osa ommads ad, a he same ime, eje osa disubaes. his is ow as he sevomehaism oblem. Is ool soluio a be foud by a iludig ey he iegaed ouu aig eo eyi = io he sysem s dyamis ad b desigig a Pooioal + Iegal (PI oolle. he lae ovides ye losed-loo sysem esose ad solves he sevomehaism oblem. Negleig he ool olieaiy g ( i ad assumig osa eeal ommads, ideal oolle gais K a be foud suh ha: u = K = g( u (5.3 If he ivese ma g ( i was ow he he ideal ommaded ool u ould have bee se o: u = g ( u (5.4 Sie i geeal g ( i I m m ad i is assumed ha his ivese ma is o ow aalyially, he sysem dyamis a be wie as: = ( A+ BK + B g( u K + B A g ( u (5.5 o, equivalely, = A + B( g( u K + B (5.6 whee A eeses he ideal / desied losed-loo sysem mai. hese dyamis a be eoveed if u is hose o ael he ool defiiey sigal, g( u K =, whih imlies ha u g ( K =. Ufouaely, his oeaio ao be efomed sie he ivese ma ( g i is assumed o be uavailable. I ode o iumve his siuaio, he ool iu u will be foud o miimize he udesiable effes of he ool defiiey sigal g ( u K o he sysem dyamis (5.5. owads ha ed, oside he oblem of miimizig he os fuio V( u = g( u K mi (5.7 u wih ese o he ommaded ool iu u. he os fuio gadie is: g( u V ( u = ( g( u K (5.8 u Usig (5.8, he os fuio miimizaio a be efomed usig o-lie vesio of he gadie dese mehod: ε u = V ( u (5.9 whee < ε is a small osiive osa. Subsiuig (5.8 io (5.9, yields he dyami oolle: g( u ε u = ( g( u K (5. u Combiig (5. wih (5.6, gives he losed-loo sysem dyamis: = A + B( g( u K + B (5. g( u ε u = g( u K ( u hese dyamis ae i he fom of he sadad sigula eubaio model fom []. I suh a oe, he oigial sysem sae ad he ool ommad u eese he slow ad he fas omoes of he lose-loo sysem, eseively. Assumio 5. he ool ma g : X Y is C ad oe. Also, he g ( u sigula values of he ma aobia g ( u = u saisfy he double-iequaliy: < mi ( ( g i g u ma < (5. g i m, u m X R, wih some osiive osas ad mi ma g g. Assume ha is osa. Seig ε = i (5., gives eoeially sable edued sysem: = A + B (5.3 Fom (5.3, i follows ha he sysem equilibium is: eq = A B (5.4 Le e= eq. he e = A e (5.5 Iodue he hage of vaiables w= u u, whee u = g ( K is he ideal uow ool ommad. Is owledge is o equied oly is eisee will be elied 67 Auhoized liesed use limied o: UNIVERSIY OF CONNECICU. Dowloaded o Febuay, 9 a 9:5 fom IEEE Xloe. Resiios aly.

6 FC4. o duig sabiliy aalysis. Followig [], he boudaylaye sysem is defied as: g( u + w w = g( u + w K (5.6 u gu ( o, equivalely: dw g( u + w = ( g ( u + w g( u (5.7 dτ u he bouday-laye dyamis i (5.7 ae eessed i he fas τ ime sale wih = ε τ. hese dyamis ae alulaed by seig ε =, while eaig u as a fied aamee. he e saeme is a saighfowad aliaio of heoem 4. o he bouday-laye dyamis (5.7 ad as suh, i is saed wihou a oof. heoem 5. Ude Assumio 5., he equilibium w = of he bouday-laye sysem (5.7 is globally eoeially sable, uifomly i u. he bouday-laye dyamis (5.7 eese a aameedeede gadie sysem, wih ese o u. Hee, he os (5.7 is he sysem Lyauov fuio. Is aiula fom is emloyed o esablish global eoeial sabiliy of he sysem equilibium w =, i τ ime sale, uifomly i u. Moeove, sie boh he edued ad he boudaylaye dyamis ae globally eoeially sable, oe a aly ihoov heoem [], [] ad sae he mai esul of his seio. heoem 5. Coside he sysem dyamis i (5., oeaig ude he dyami oolle (5., wih he feedba gai mai K hose suh ha A = A+ BK is Huwiz. Suose ha Assumio 5. is ue. he hee eiss ε > suh ha fo all ε < ε, ajeoies of he losed-loo sysem (5. saisfy he followig asymoi elaios, uifomly i : A ( = eq + e ( ( eq + O( ε (5.8 u ( = u + w + O( ε ε Moeove, fo a give ε > hee eis wo osiive osas ad γ suh ha, fo all. w ε e γ ε Rema 5. Wihou loss of geealiy, oe a assume ha he eee sysem DC gai is uiy: CA B = Im m. I his ase, usig he s asymoi elaio fom (5.8, gives: A y( = ( + C( e ( ( eq + O( ε ( + O( ε (5.9 Cosequely, he dyami oolle (5. ovides bouded aig of he eeal osa ommad by he sysem ouu y (, wihi he O( ε aig eo, as. A he same ime, he ool ommad sigal u oveges o a O( ε eighbohood of he ideal / uow ommad u = g ( K : u ( g ( K O( ε + (5. hus, heoem 5. solves he sevomehaism oblem i he esee of ow ool olieaiies whose ivese ma is aalyially uow. VI. CONCLUSIONS I his ae, we eseed suffiie odiios fo eoeial sabiliy of gadie sysems. Seifially, i was show ha if he oesodig ma f ( was oe ad sigula values of is aobia wee uifomly bouded away fom ad, he he sysem equilibium was guaaeed o be globally eoeially sable. hese esuls wee subsequely alied o efom bouded aig desig fo a lass of oliea-i-ool dyamial sysems. ACKNOWLEDGMEN he auhos would lie o ha Pof. Chisohe Byes fo his isighful ommes egadig global ivese maig oblems. REFERENCES [] N. Hovaimya, E. Lavesy, C. Cao, Dyami Ivesio of Muliiu Noaffie Sysems via ime-sale Seaaio, I Poeedigs of Ameia Cool Cofeee, Mieaolis, MN, 6. [] H.K. Khalil. Noliea Sysems, 3 d Ediio, Peie Hall I.,. [3]. L. Fiesz, D. H. Besei, N.. Meha, R. L. obi, ad S. Gajlizadeh, Day-o-day dyami ewo disequilibia ad idealized avele ifomaio sysems, Oeaios Res., vol. 4,. 36, 994. [4] X. B. Gao, Eoeial sabiliy of globally ojeed dyami sysems, IEEE as. Neual New., vol. 4, o., , 3. [5] X. B. Gao, L.-Z. Liao, ad L. Qi, A ovel eual ewo fo vaiaioal iequaliies wih liea ad oliea osais, IEEE as. Neual Newos, vol. 6, o. 6, , 5. [6]. M. Oega ad W. C. Rheibold, Ieaive Soluio of Noliea Equaios i Seveal Vaiables. New Yo: Aademi, 97. [7] M. W. Hish ad S. Smale, Diffeeial Equaios, Dyamial Sysems, ad Liea Algeba. Aademi Pess, New Yo, 974. [8] F. F. Wu ad C. A. Desoe, Global ivese fuio heoem, IEEE as. Ciui heoy, vol. C-9,. 99-, 97. [9] I. W. Sadbeg, Global ivese fuio heoems, IEEE as. Ciui heoy, vol. CAS-7, , 98. [] A. N. ihoov, O he deedee of he soluios of diffeeial equaios o a small aamee, Ma. Sboi N. S. (64,. 93 4, (Russia MR 9:588, 948. [] C. I. Byes ad A. Lidquis, Ieio Poi Soluios of Vaiaioal Poblems ad Global Ivese Fuio heoems, Ieaioal oual of Robus ad Noliea Cool, vol. 6,. 8, 6. [] S. Hayi, Neual Newos: A Comehesive Foudaio, d ediio, Peie Hall I., 999. [3] R. Sae ad.. Sloie, Gaussia ewos fo die adaive ool, IEEE asaios o Neual Newos, 3(6, , 99. [4]. B. Pome ad L. Paly. Adaive oliea egulaio: esimaio fom he Lyauov equaio. IEEE as. Auom. Co., 37(6, , Auhoized liesed use limied o: UNIVERSIY OF CONNECICU. Dowloaded o Febuay, 9 a 9:5 fom IEEE Xloe. Resiios aly.