1 Lyapunov Stability Theory

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1 Lyapuov Stablty heory I ths secto we cosder proofs of stablty of equlbra of autoomous systems. hs s stadard theory for olear systems, ad oe of the most mportat tools the aalyss of olear systems. It may be eteded relatvely easly to cover oautoomous systems ad to provde a strategy for costructg a stablzg feedback cotroller. I the sequel we preset the results for tme varat systems. hey may be derved for tme varyg systems as well, but the essetal dea s more accessble for the tme varat case.. Autoomous Systems & Stablty Cosder the autoomous system where f : D R s a locally Lpschtz map from a doma D R to R. Suppose D s a equlbrum pot of the system; that s f (. Our goal s to characterze ad study the stablty of. For coveece, we state all deftos ad theorems for the case whe the equlbrum pot s at the org of R ; that s,. here s o loss of geeralty dog so because ay equlbrum pot ca be shfted to the org va a chage of varables. Suppose ad cosder the chage of varables y. he dervatve of y s gve by y f ( y + g( y I the ew varable y, the system has equlbrum at the org. herefore, wthout loss of geeralty, we wll assume that f ( satsfes org. Defto.: he equlbrum pot. stable f, for each. Ustable f t s ot stable of f ( ε >, there s δ δ ( ε f ad study the stablty of the s > such that 3. Asymptotcally stable f t s stable ad δ ca be chose such that f δ < t < ε, t δ ( t < lm t

2 Defto.: he system f ( stable about f s called (locally epoetally asymptotcally s asymptotcally stable about. f (. δ >, M >, λ > such that. Lyapuov s st or Drect Method t < δ t > t λ( t t t Me t Let : D R be a cotuously dfferetable fucto defed a doma D R that f deoted cotas the org. he rate of chage of alog the trajectores of by (, s gve by δ δ δ δ δ δ f f hs s called the total or absolute dervatve of. he absolute dervatve of alog the trajectores of a system s depedet o the φ t, s the system s equato. Hece, soluto of f ( that starts at tal state at tme t, the Eample.: Cosder the system d t dt φ (, t ( + f f ( f Wth the caddate Lyapuov fucto: λ + λ Calculate the total dervatve of : λ, λ the ( [ ] λ ( λ + λ + λ + 4λ 3 (

3 herefore, f ( s egatve, wll decrease alog the soluto of f (. We are ow early ready to state Lyapuov s stablty theorem. he followg defto wll be ecessary Defto.3: A doma D ( t D t t. ( t f D R s called varat for the system f ( heorem.: (Lyapuov s Frst or Drect Method Let f ad be a equlbrum pot for : D R be a cotuously dfferetable fucto such that. ( ad ( > D \{ }. ( D he, s stable. Moreover, f D R be varat, ad let 3. ( < D \ { } the s asymptotcally stable Remarks. If ( above holds, the s called locally postve defte lpd. If oly ( D \, the s locally sem-postve defte., { }. If ( ad ( hold, the s called a Lyapuov Fucto for the system f (. Proof of heorem 5. Proof of stablty: We cosder the level sets of the Lyapuov fucto. ε >, choose r (, ε ] Let such that { r, } Ω β { Br, β}. Choose β (, α. Defe B r D R. Let α m (. r

4 It holds that f ( β ( t β Further δ > such that δ ( Ω Ω t because ( ( ( t < t β Bδ Ωβ B r < < β δ β β ( < δ ( t ε t > B Ω t Ω t Br Proof of asymptotc stablty: We costruct a proof by cotradcto. Let v( t ( t * ( t. Covergece of (t to a fed pot mples that ( t * codto (3 v( v( t, but assume that. But by <. hs s a cotradcto. hus *, showg that all trajectores coverge to the org, thus the system s asymptotcally stable. Eample. (ctd Recall that 3 ( λ λ + 4λ Cosder that λ λ. he the total dervatve s gve by ( ( g( g( ( he the total dervatve s guarateed to be egatve wheever g ( >.

he level sets of, where < wll be varat. hus the red crcle above s level, ad wth ths crcle, <, so we coclude that the org s locally asymptotcally stable. heorem.

5 he level sets of, where < wll be varat. hus the red crcle above s level, ad wth ths crcle, <, so we coclude that the org s locally asymptotcally stable. heorem.: (Istablty result Suppose (, C '( D < δ such that ( >. Let, { r } ad that δ > r > such that Br U B >. Suppose ( > U. he s ustable. Proof { r } Note that the set U B ( D ad > s coected due to cotuty of. Furthermore, the org must be o the border of U, as there ests elemets of U whch are arbtrarly close to. Note that as ( > for U, trajectores startg U wll ot leave U o the border defed by (, rather they wll leave va B r. hus there are trajectores startg arbtrarly close to the org that wll leave a ball of a gve radus, provg stablty of the pot. Note that the requremets o ( are ot as strct o the requremets o a Lyapuov fucto. Eample.: he set U for ( ( s show the followg dagram., the ( > Cosder the system system s ustable. for U, provg that the

6 Eample.3: Mass-Sprg System Cosder a mass m o a sprg eertg a force whe moto, as gve by: Fe δ ad F F ( d F d ad subject to a frctoal force F e F( m δ Where s defed as the equlbrum, the pot where there s o force eerted by the sprg. he the system becomes ( m F δ ad wth, ad m we obta F ( δ f ( Cosder the Lyapuov fucto F s ds + he ( F (, ad ( δ δ F + F he system s stable but we ca t prove asymptotcally stablty because. I order to prove stablty we eed a more geeral result LaSalle s Ivarace Prcple.

7 .3 he Ivarace Prcple heorem.3: (Lasalle s Ivarace Prcple Let Ω D be a compact set that s postvely varat wth respect to f (. Let : D R be a cotuously dfferetable fucto such that ( Ω. Let E be. Let M be the largest varat set E. he the set of all pots Ω where ( every soluto startg Ω approaches M as t. Eample.3 (ctd: Mass-Sprg System Applyg ths otato to the eample above, we obta E, : {( } M { } hus, trajectores must coverge to the org, ad we have prove that the system s asymptotcally stable. Remark: La Salle s result ca also be used to fd lmt cycles.

8 .4 Lear Systems ad Learzato I ths secto we prove stablty of the system by cosderg the propertes of the learzato of the system. Before provg the ma result, we requre a termedate result. Re λ > Defto.4: A matr A s called Hurwtz f Cosder the system A. We look for a quadratc fucto P where P P >. he P + P A P + PA Q If there ests Q Q > such that A P + PA Q, the s a Lyapuov fucto ad s globally stable. hs equato s called the Matr Lyapuov Equato. We formulate ths as a matr problem: Gve Q postve defte, symmetrc, how ca we fd out f there ests P P > satsfyg the Matr Lyapuov equato. he followg result gves estece of a soluto to the Lypuov matr equato for ay gve Q. heorem.4: For A R the followg statemets are equvalets: A has all egevalues left of the j-as For all Q Q > there ests P P > satsfyg A P + PA Q. Proof outle We make a costructve proof. For a gve Q Q >, cosder the followg caddate soluto for P : P e A t hat P P > follows from the propertes of Q. Note that the tegral wll coverge f ad oly f A s a Hurwtz matr. We ow show that P satsfes the matr Lyapuov equato: Qe At dt

9 A hs establshes the result. P + PA A t At A t At [ A e Qe + e Qe ] e d dt A t A t At [ e Qe ] Qe At dt Q A dt hs theorem has a terestg terpretato terms of the eergy avalable to a system avalable. If we say that the eergy dsspated at a partcular pot phase space s gve by q( Q - meag that a trajectory passg through s loosg q( uts of eergy per ut tme, the the equato ( P, where P satsfes the matr Lyapuov equato gves the total amout of eergy that the system wll dsspate before reachg the org. hus ( P measures the eergy stored the state. Wth ths result we are a posto to prove the followg result: heorem.5: (Lyapuov s Secod or Idrect Method Let f where f : D R s a cotuously be a equlbrum pot for dfferetable ad D s a eghborhood of the org. Let f A he. he org s asymptotcally stable f (. he org s ustable f Re Re λ < for all egevalues of A λ > for oe or more of the egevalues of A Proof If A s Hurwtz, the there ests P P fucto of the learzed system. > so that P s a Lyapuov he f ( A + g ( where g for he ( f ( P + Pf ( ( PA + A P + Pg ( Q + Pg ( <?

10 As we coverge to, we eter the rego where Q >> Pg(, yeldg ( <, ad so s locally asymptotcally stable. hs proves pot of the theorem. Re λ > for some. For smplcty cosder λ real, ad let v be the assocated egevector. he locally a trajectory startg at ( εv for o prove pot : Cosder that λ suffcetly small ε wll be gve by e t ( ε v, so that t grows epoetally. hus the org must be ustable. For λ comple, cosder the assocated comple cojugate egevalues, ad the assocated egevectors. Locally ths also forms a epoetally growg trajectory, showg stablty of the org. Note. he theorem does ot say aythg whe Re λ wth Re λ for some. I ths case learzato fals to determe the stablty of the equlbrum pot, ad further aalyss s ecessary. hs s llustrated by the followg eample: Eample.4:. 3 a, a > s ustable. 3 a, a > s asymptotcally stable I both cases the matr A of the learzed system s the same: he mult-dmesoal result whch s relevat here s the Ceter Mafold heorem. hs theorem s beyod the scope of ths course, however the result s appromately that aroud ay equlbrum pot of a olear system, the space may be splt to:. A epoetally ustable mafold: A varat set correspodg to the egevalues of the learzed system wth postve real part. A epoetally stable mafold: A varat set correspodg to the egevalues of the learzed system wth egatve real part 3. he Ceter Mafold: A varat set correspodg to the egevalues of the learzed system wth egatve real part, whch may cota asymptotcally stable or ustable sub-mafolds, but ehbtg o epoetal covergece or dvergece.

11 .5 Coverse heorems Cosder the followg problem: Let be asymptotcally stable. Does there est a Lyapuov fucto for the system? he closest aswer to ths questo may be provded by the followg system: heorem.6: Let be locally epoetally asymptotcally stable for the system f. he there ests a Lyapuov fucto : D for the system such that: 3 C C C C 3 4 where C, C, C3, C 4 > Proof Idea Aalogously to the costructo of a soluto to the matr Lyapuov equato, a Lyapuov Fucto s costructed by: ( φ ( t, φ( t, dt Where φ ( t, s the soluto to the system dfferetal equatos, defg the trajectory startg at at tme t. Due to local epoetal stablty, the tegral may be show to coverge locally. By boudg the rate of growth of the tegral away from, the propertes may be prove. See Khall, 3 rd edto, heorem 3., p 49-5 Determg a Lyapuov fucto may also be epressed as the result of solvg a partal dfferetal equato: Cosder a locally postve defte fucto q(. he f there ests a soluto ( to the partal dfferetal equato gve by: ( ( q( he ( s a Lyapuov fucto for the system, ad the equlbrum must be stable.

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

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