Lyapunov Stability. Aleksandr Mikhailovich Lyapunov [1] 1 Autonomous Systems. R into Nonlinear Systems in Mechanical Engineering Lesson 5

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1 Joh vo Neuma Perre de Fermat Joseph Fourer etc 858 Nolear Systems Mechacal Egeerg Lesso 5 Aleksadr Mkhalovch Lyapuov [] Bor: 6 Jue 857 Yaroslavl Russa Ded: 3 November 98 Bega hs educato at home Graduated from St Petersburg Uversty 88 (3 years old) Got marred wth Natala Rafalova Secheov whe he was 9 years old Attaed doctorate degree 89 (35 years old) aught at Kharkov uversty utl 9 aught at St Petersburg uversty utl 97 aught at Uversty Odessa utl 98 Hs wfe suffered from tuberculoss ad ded o 3 October 98 Later that day Lyapuov shot hmself ad ded three days later a hosptal (6 years old) He s famous flud mechacs ad for stablty theorems for olear aalyss Other terestg mathematcas: Davd Hlbert Lyapuov Stablty So far we have three methods to determe stablty property of t drectly the equlbrum pot Frst we ca solve for the soluto hs mght ot be possble for some olear systems Secod we ca learze the olear system aroud ts equlbrum pot ad the study the egevalues of the matr A hs method ca produce wrog result whe plat s subjected to perturbato hrd we ca use umercal methods such as those the Matlab software to produce the phase portrat or the soluto plot Eve f t seems coveet umercal methods are oly for case-by-case bass ad usually produce o meagful results By usg the Lyapuov stablty theorems we ca determe the stablty property of the equlbrum pot of olear systems wthout havg to solve for the soluto It s such a powerful method because we ca use t to desg cotroller that esures stablty of the closed-loop system Lesso 5 6 ad 7 cota Lyapuov stablty Lyapuov stablty theorems eable us to determe stablty property of the equlbrum pot = f whch s of the system I lesso 5 the system s the form a autoomous system I lesso 6 the system s the form = f t = f u t I lesso 7 the system s the form Autoomous Systems Cosder a autoomous system = f where f : D R s a locally Lpschtz map from a doma D R R to Copyrght 7 by Wtht Chatlataagulcha

2 Ay equlbrum pot ca be shfted to the org va a chage of varables Suppose ad cosder a chage of varables y= he dervatve of y s gve by where y = = f = f y+ g y g = I the ew varable y the system has equlbrum at the org herefore wthout loss of geeralty we wll always assume that f = ad study the stablty of the org = f ( ) satsfes Defto 4 he equlbrum pot ) stable f for each s = of = f δ = δ ε > such that ε > there s δ < t < ε t ) ustable f t s ot stable 3) asymptotcally stable f t s stable ad δ ca be chose such that δ ( t) < lm = t I 89 Lyapuov showed that certa fuctos ca be used to determe stablty of a equlbrum pot heorem 4 Let D dfferetable fucto such that ad = be a equlbrum pot for = f R be a doma cotag he = s stable Moreover f = Let V : D R be a cotuously V ( ) = ad V( ) > { } D () D () the = s asymptotcally stable 858 Nolear Systems Mechacal Egeerg Lesso 5 ( ) < { } D (3) Proof Frst we prove the stable part of the theorem Sce B statemet = s stable meas for each ε > there s = > such that δ δ δ ε < t < ε t we process ths ested-quatfers statemet from left to rght to have Object: ε Property: ε > δ = δ ε > such that Somethg happes: there s ( ) < δ ( t) < ε t Object: δ Property: δ > Somethg happes: δ < t < ε t Usg the forward-backward method we have B: For each δ < t < ε t ε > there s δ = δ ε > such that A: Choose a real umber ε (Choose method) B: here s < t < t δ δ ε = > such that δ A: Costruct a real umber δ (Costructo method) B = R r D { } A3: Let ( ε] r { ( r )} Ω = R V m V B A4: Let β β = r ε Copyrght 7 by Wtht Chatlataagulcha

3 Sce ( ) ( ) t V t V β t we have that ay trajectory startg Ω stays A5: Let Bδ R δ V A6: herefore δ β { β} Ω for all future tme = < Ω < t < r ε t Sce A6 = B proof of the frst part s ow completed β β B δ Ω β B r D 858 Nolear Systems Mechacal Egeerg Lesso 5 whch cotradcts the assumpto that c> herefore c= ad the proof s completed he fucto V( ) satsfyg () s called Lyapuov fucto caddate If V( ) also satsfes () or (3) t s called Lyapuov fucto A fucto V( ) satsfyg codto () that s V ( ) = ad V( ) > for s sad to be postve defte If stead t satsfes V for t s sad to be postve semdefte A fucto V( ) s sad to be egatve defte or egatve semdefte f V s postve defte or postve semdefte respectvely heorem 4 ca the be rephrased as follows: the org s stable f there s a cotuously dfferetable postve defte fucto V( ) so V that s egatve semdefte ad t s asymptotcally stable f s egatve defte A eample of V( ) whose sg defteess ca be checked s V = P= pj j = j= Secod we prove the asymptotc stablty part of the theorem We proceed as follows B: ( t) as t B: ( ) A: V ( t) c V t as t as t We wat to show that c= Usg cotradcto suppose c> there ests Bd Ω c γ = ma d r we have Let where P s a real symmetrc matr V( ) s postve defte (postve semdefte) f ad oly f all egevalues of P are postve (oegatve) whch s true f ad oly f all the leadg prcpal mors of P are postve (oegatve) V If = P s postve defte (postve semdefte) we say that the matr P s postve defte (postve semdefte) ad wrte P> P If the sg defteess caot be determed we say t s defte ( ) ( ( τ) ) τ ( ) t V t = V + V d V γt< 3 Copyrght 7 by Wtht Chatlataagulcha

4 Eample : [] Fd the sg defteess of V = a + + a a Nolear Systems Mechacal Egeerg Lesso 5 Ufortuately there s o systematc method to fd the Lyapuov s fuctos Mostly the Lyapuov fucto caddates are eergy fuctos I other cases t s a matter of tral ad error he followg four eamples preset some deas for fdg the Lyapuov fuctos Eample : [] Cosder the equato = g where g( ) s locally Lpschtz o ( a a) g( ) = ; g > ad ( a a) asymptotcally stable ad satsfes Show that the org s Soluto Sce 3 [ ] a V = a = P a 3 we have that the leadg prcpal mors of P are herefore a a ad a( a 5 ) V s postve defte f a> 5 he egatve defteess ca be foud smlarly by workg o P 4 Copyrght 7 by Wtht Chatlataagulcha

5 858 Nolear Systems Mechacal Egeerg Lesso 5 Eample 3: [] Cosder the pedulum equato wthout frcto = = as Soluto A eample of g( ) s gve below Determe the stablty property of the equlbrum pot at the org g a a Cosder a Lyapuov fucto caddate V = g y dy Over the doma D= ( a a) V ( ) = ad V( ) > for all herefore Lyapuov fucto caddate Its dervatve s V s cotuously dfferetable V = g = g < D { } V s a vald By heorem 4 we ca coclude that the org s asymptotcally stable Soluto A atural Lyapuov fucto caddate s the eergy fucto V ( ) = ad he dervatve of V( ) s V = a( cos ) + V s postve defte over the doma π < < π = a s + = hus codtos of heorem 4 are satsfed ad we coclude that the org s stable 5 Copyrght 7 by Wtht Chatlataagulcha

6 Eample 4: [] Cosder aga the pedulum equato but ths tme wth frcto = = as b Determe the stablty property of the equlbrum pot at the org 858 Nolear Systems Mechacal Egeerg Lesso 5 V = P+ a( cos ) p p = [ ] a( cos ) p p + Its dervatve ca be foud to be = ( ) s s + ( ) + ( ) a p ap We eed to choose p p ad s egatve defte ad V p p b p p b p so that For V( ) to be postve defte we eed V s postve defte p > ad p p p > Soluto Frst we try ( cos ) ( / ) fucto caddate We have V = a + as a Lyapuov = a s + = b s egatve semdefte herefore we ca oly coclude that the org s stable However we kow that the org s deed asymptotcally stable he eergy fucto faled to show ths fact We the try aother Lyapuov fucto caddate p to be egatve defte we eed to choose = ad For V p = bp to cacel the terms a( p ) s ad ( p ) whch are sg defte ake p = b / the = ab s b he term s > for all π < < π { π π} D R = < < we have that p b akg V s postve defte ad s egatve defte over D hus from heorem 4 the org s asymptotcally stable 6 Copyrght 7 by Wtht Chatlataagulcha

7 Varable gradet method s a method to fd a Lyapuov fucto a backward maer he method has the followg steps: ) Let g V V V = V = scalar fucto f ad oly f the Jacoba matr [ g / ] that s g j g s a gradet of a s symmetrc g j = j ad j= (4) 858 Nolear Systems Mechacal Egeerg Lesso 5 Eample 5: [] Cosder a secod-order system = = h a where a> h( ) s locally Lpschtz h ( ) = ad y y ( b c) for some postve costats b ad yh y > for all c Use the varable gradet method to coclude the asymptotc stablty of the org ) V V = f = g f (5) 3) Usg tegrato of gradet vector we have = V g y dy = = ( ) ( ) = g y dy ( ) g y dy g y dy g y dy 4) Choose parameters of g( ) to satsfy (4) to make from (5) egatve defte (egatve semdefte) ad to make V( ) from (6) postve defte (6) 7 Copyrght 7 by Wtht Chatlataagulcha

8 858 Nolear Systems Mechacal Egeerg Lesso 5 he et theorem states codtos for the org to be globally asymptotcally stable whch meas startg at ay R the trajectory wll approach the org as t Soluto α + β ry g γ + δ δ are to be determed = where α β γ ad g eeds to satsfy g g = = g g h( ) + a < = V g y dy ( ) ( ) = g y dy + g y dy > Followg the soluto page of [] we arrve at δ ka ka V = + δ h( y) dy ka whch satsfes all equatos above over a doma D= R b< < c herefore the org s asymptotcally stable { } heorem 4 (Barbash-Krasovsk) Let = be a equlbrum pot for = f Let V : R R be a cotuously dfferetable fucto such that V ( ) = ad V > V < the = s globally asymptotcally stable Proof See page 4 of [] Eample 6: [] Cosder aga the system the prevous eample but ths tme let yh( y) > y Prove that the org s globally asymptotcally stable Soluto δ ka ka he Lyapuov fucto V = + δ h( y) dy ka s postve defte for all R ad radally ubouded V = aδ k aδ k h s egatve he dervatve defte for all R sce < k< herefore the org s globally asymptotcally stable 8 Copyrght 7 by Wtht Chatlataagulcha

9 If the org s a globally asymptotcally stable equlbrum pot of a system t must be the uque equlbrum pot of the system For f there were aother equlbrum pot the trajectory startg at that pot would rema there for all t herefore global asymptotc stablty s ot studed for multple equlbra systems lke the pedulum equato 858 Nolear Systems Mechacal Egeerg Lesso 5 he followg theorem states codtos for org to be ustable heorem 43 Let = be a equlbrum pot for = f V : D R be a cotuously dfferetable fucto such that ad V( ) > for some wth arbtrarly small U = B V > ad suppose that ( ) > { r } s ustable Proof See page 5 of [] Eample 7: [] Cosder a secod-order system where = + g = + g ad g k g k Let V = Defe a set U he = are locally Lpschtz fuctos a eghborhood D of the org Show that the org s ustable Soluto = t ca be show that Usg V ( ) whe r ( k) 3 k = k > < / therefore we ca coclude that the org s ustable 9 Copyrght 7 by Wtht Chatlataagulcha

10 he Ivarace Prcple he varace prcple s useful whe we wat to coclude s oly egatve asymptotcal stablty of the org whe semdefte I prevous eample of a pedulum wth frcto we foud that usg eergy fucto as Lyapuov fucto caddate leads to = b beg egatve semdefte s egatve everywhere ecept o the le = where ( ) = However whe we look closely we see that o trajectory ca stay o the le = uless = We ca see ths fact from t = t = s t = t = herefore V ( t ) must decrease toward ad the t as t We ca therefore coclude that f a doma about the org we ca fd a Lyapuov fucto whose dervatve alog the trajectores of the system s egatve semdefte ad f we ca establsh that o trajectory ca stay detcally at pots where ( ) = ecept at the org the the org s asymptotcally stable We eed the followg deftos A set p s sad to be a postve lmt pot of such that ( t) t f there s a sequece { t } wth t as p as he set of all postve lmt pots of t s called the postve lmt set of ( t ) A set M s sad to be a varat set wth respect to = f f ( ) M ( t) M t R A set M s sad to be a postvely varat set wth respect to = f f ( ) M ( t) M t he followg lemma s useful provg subsequet theorem 858 Nolear Systems Mechacal Egeerg Lesso 5 = f s bouded ad belogs to D Lemma 4 If a soluto ( t ) of for t the ts postve lmt set L + s a oempty compact varat set Moreover ( t ) approaches L + as t heorem 44 (LaSalle s heorem) Let Ω D be a compact set that s postvely varat wth respect to = f Let V : D R be a cotuously dfferetable fucto such that Ω Let E be the set of all pots Ω where ( ) = Let M be the largest varat set E he every soluto startg Ω approaches M as t Proof See page 8 [] Whe our terest s showg that ( t) as t we eed to establsh that set M s the org he followg two corollares are drect cosequeces of heorem 44 Corollary 4 Let = be a equlbrum pot for = f Let V : D R be a cotuously dfferetable postve defte fucto o a doma D cotag the org = such that D Let S = D = ad suppose that o soluto ca stay detcally { } S other tha the trval soluto ( t) he the org s asymptotcally stable Corollary 4 Let = be a equlbrum pot for = f Let V : R R be a cotuously dfferetable radally ubouded postve defte fucto such that V for all R Let S = R = ad suppose that o soluto ca stay detcally { } S other tha the trval soluto ( t) he the org s globally asymptotcally stable Copyrght 7 by Wtht Chatlataagulcha

11 Eample 8: [] Cosder the system = = h h 858 Nolear Systems Mechacal Egeerg Lesso 5 Eample 9: [] Cosder aga the same system prevous eample y hs tme let a= ad h ( z ) dz as y where h( ) ad h are locally Lpschtz ad satsfy ( ) = > ad y ( a a) h yh y y Show that the org s asymptotcally stable Soluto Usg V = h ( y) dy+ = h s egatve semdefte S = D = ad sce we have { } Let = = = Hece S = { D = } Let ( t ) be a soluto that belogs detcally to = = = = V h S We see that t t h t t herefore the oly soluto that ca stay S s the trval soluto ( t ) = hus the org s asymptotcally stable Soluto We have that V = h ( y) dy+ Usg smlar dervato as prevous eample the org s globally asymptotcally stable s radally ubouded Copyrght 7 by Wtht Chatlataagulcha

12 I summary LaSalle s theorem has four features ) It relaes the egatve defteess requremet of Lyapuov s ca be egatve semdefte to coclude asymptotcal theorem stablty of the org ) It gves a estmate of the rego of attracto whch s the set Ω that ca be ay compact varat set 3) It ca be used cases where the system has a equlbrum set rather tha a solated equlbrum pot 4) he fucto V( ) does ot have to be postve defte 858 Nolear Systems Mechacal Egeerg Lesso 5 he followg two eamples llustrate the thrd ad fourth features Eample : [] Cosder a frst-order system y = ay+ u together wth the adaptve cotrol law u= ky k = γ y γ > akg = y ad = k the closed-loop system s = a = γ Show that as t trajectores approach the equlbrum set whch s the le = Soluto Cosder a Lyapuov fucto caddate where b> a he dervatve s V b γ = + ( ) = b a Sce V( ) s radally ubouded we take Ω=Ω = ad M = E= { Ω = } { } c R V c herefore from heorem 44 we coclude that every trajectory startg Ω approaches E as t c c Copyrght 7 by Wtht Chatlataagulcha

13 Eample : [] Cosder the eural etwork system Lesso he state equatos are gve by 858 Nolear Systems Mechacal Egeerg Lesso 5 = h j j g + I C j R for = where the state varables are the voltages at the amplfer outputs ca oly take values the set { M M} H = R V < < V he fuctos g : R ( V V ) are sgmod fuctos = M dg h = > ( VM VM) du u g I are costat curret puts R > ad C > Assume the symmetry codto j = j s satsfed Show that every trajectory approaches the set of equlbrum pots as t M Soluto From the symmetry property = the vector whose compoet s j j g + I j R j j th s a gradet vector of a scalar fucto By tegrato smlar to what we have doe the varable gradet method t ca be show that ths scalar fucto s gve by V = + g ( y) dy I j j j R hs fucto typcally s ot postve defte 3 Copyrght 7 by Wtht Chatlataagulcha

14 Rewrte the state equatos as he dervatve of V( ) s = h C V V = h = C V herefore set M = E Moreover V = = = cotas equlbrum pots Costruct the set { M M } ( ε) R ( V ε) ( V ε) Ω = where ε > s arbtrarly small he set Ω s closed ad bouded ad ths set It remas to show that Ω s a postvely varat set Lettg g be a specfc form of the sgmod fucto g ( u ) λπu ta V M = λ> π V M d It ca be show that ( ) = < for VM ε < VM dt herefore trajectores startg Ω stays Ω for all future tme From heorem 44 every trajectory approaches the set of equlbrum pots as t 858 Nolear Systems Mechacal Egeerg Lesso 5 3 Lear Systems ad Learzato he followg theorem states codtos for the org of = A to be stable or asymptotcally stable heorem 45 he equlbrum pot = of = A s stable f ad oly f all egevalues of A satsfy Reλ ad for every egevalue wth Reλ = ad algebrac multplcty q rak( A λ I) = q where s the dmeso of he equlbrum pot = s (globally) asymptotcally stable f ad oly f all egevalues of A satsfy Reλ < Proof See page 34 of [] Matr A whose all egevalues satsfy Reλ < s called Hurwtz matr or stablty matr Asymptotc stablty of the org ca also be vestgated by usg the Lyapuov s method Let V = P be a quadratc Lyapuov fucto caddate where P s a real symmetrc postve defte matr We have = + = ( + ) = P P PA A P Q where Q s a symmetrc matr defed by PA+ A P= Q (7) If Q s postve defte we ca coclude by heorem 4 that the org s asymptotcally stable he equato (7) s called Lyapuov equato he followg theorem states the result above heorem 46 A matr A s Hurwtz; that s Reλ < for all egevalues of A f ad oly f for ay gve postve defte matr Q there ests a postve defte symmetrc matr P that satsfes the Lyapuov equato 4 Copyrght 7 by Wtht Chatlataagulcha

15 PA+ A P= Q Moreover f A s Hurwtz the P s the uque soluto of PA+ A P= Q Proof See page 36 of [] Eample : [] A lear system has A= Use heorem 46 to show that the org s asymptotcally stable 858 Nolear Systems Mechacal Egeerg Lesso 5 he followg theorem uses learzato to vestgate the stablty propertes of = f Note that the theorem does ot say aythg about the case whe Reλ because learzato fals to determe stablty of the equlbrum pot whe Reλ = due to perturbato heorem 47 (Lyapuov s drect method) Let = be a equlbrum = f where f : D R s pot for the olear system cotuously dfferetable ad D s a eghborhood of the org Let Soluto Choose Q p = p ad let P= p p equato (7) ca be rewrtte as p p = p he Lyapuov f A= = he he org s asymptotcally stable f Reλ < for all egevalues of A he org s ustable f Reλ > for oe or more of the egevalues of A Proof See page 39 of [] Eample 3: [] Ivestgate the stablty of the two equlbrum pots at π of the pedulum equato ( ) = ad = = as b 5 5 We the have P= 5 Sce P s postve defte (ts prcpal mors are 5 ad 5) from heorem 46 the org s asymptotcally stable 5 Copyrght 7 by Wtht Chatlataagulcha

16 Lesso 5 Homework Problems Nolear Systems Mechacal Egeerg Lesso (problem oly) (problem oly) 48 Homework problems are from the requred tetbook (Nolear Systems by Hassa K Khall Pretce Hall ) Refereces [] [] Nolear Systems by Hassa K Khall Pretce Hall Soluto For ( ) ( ) = we have f A = = = = a b For all a b > the egevalues satsfy Reλ < herefore the pot ( ) s asymptotcally stable For ( ) ( π ) = we evaluate the Jacoba at = = hs s equvalet to performg a chage of varables z = π z = to shft the equlbrum pot to the org ad evaluatg the Jacoba at z= π A f = = a b = π = For all a> ad b there s oe egevalue the ope rght-half π s ustable plae herefore the pot 6 Copyrght 7 by Wtht Chatlataagulcha