CALIFORNIA INSTITUTE OF TECHNOLOGY. Control and Dynamical Systems CDS 270. Eugene Lavretsky Spring 2007

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1 CALIFORNIA INSTITUTE OF TECHNOLOGY Cotrol ad Dyamical Systems CDS 7 Eugee Lavretsky Sprig 7 Lecture 1 1. Itroductio Readig material: [1]: Chapter 1, Sectios 1.1, 1. [1]: Chapter 3, Sectio 3.1 []: Chapter 1, Sectios 1.1, 1..1 We cosider dyamical systems that are modeled by a fiite umber of coupled 1 st order ordiary differetial equatios (ODE-s): (,, ) = f t x u (1.1) I (1.1), t deotes time ad f is a vector field. We call (1.1) the state equatio, refer to x R as the system state, ad u of the state vector compoets is called the order of the system. Sometimes, aother equatio m R as the cotrol iput, (exteral sigal). The umber (,, ) y = h t x u (1.) p is also give, where y R deotes the system output. Equatios (1.1) ad (1.) together form the system state space model. A solutio x() t of (1.1) (if oe exists) correspods to a curve i state space, as t varies from ad iitial time to ifiity. This curve is ofte referred to as a state trajectory or a system trajectory. A special case of (1.1) (1.) is liear (affie i the cotrol iput) system

2 (, ) (, ) (, ) = f t x + g t x u y = h t x (1.3) Lettig x x1 x x T =, a special class of oliear cotiuous-time dyamics is give by systems i Bruovsky caoical form. 1 3 = x = x = f x + g x u y = h x (1.4) For liear time-variat (LTV) systems the state space model (1.1) (1.) is: () () = At x+ Btu y = C t x+ D t u (1.5) Fially, the class of liear time-ivariat (LTI) systems is writte i the familiar form: = Ax+ Bu y = Cx+ Du (1.6) If (1.1) does ot cotai a iput sigal u (, ) = f t x (1.7) the the resultig dyamics is called uforced. If i additio the fuctio f does ot deped explicitly o t, that is = f x (1.8) the the system dyamics is called autoomous or time-ivariat. Systems that do deped o time (explicitly) are called o-autoomous or time-variat. Defiitio 1.1 A poit x = x i the state space is a equilibrium poit of (1.8) if f ( x ) = (1.9)

3 I other words, wheever the state of the system starts at x, it will remai at x future times. for all The liear system = Ax has a isolated equilibrium poit at x = if det A, that is if A has o zero eigevalues. Otherwise, the system has a cotiuum of equilibrium poits. These are the oly possible equilibrium patters that a liear system may have. O the other had, a oliear system (1.8) ca have multiple isolated equilibrium poits. Lemma 1.1 Trajectories of a 1 st order autoomous ODE (assumig that they exist) are mootoic fuctios of time. Homework: Prove Lemma Existece ad Uiqueess Readig material: [1]: Chapter 1, Sectio 4.1 []: Chapter 3, Sectio 3.1 []: Appedix A []: Appedix C1 For the uforced system (1.7) to be a useful mathematical model of a physical system, it must be able to predict future states of the system give its curret state x at t. I other words, the Iitial Value Problem (IVP) = f ( t, x) = x x t (.1) must have a uique solutio. Homework Example.1 Prove that the IVP = + < x = 1 x, x 1 x, x has o solutios at all o the iterval t 1. Homework Example. Show that the IVP 3

4 3 = x x = has ifiitely may solutios, each of which is defied o R : 1 3 ( t a), t < a 7 x() t =, a t b 1 3 ( t b), t > b 7 where a < ad b > are arbitrary costats. If a = b= the there are solutios: 3 t x() t = ad x() t. 7 The existece ad uiqueess of IVP ca be esured by imposig appropriate costraits f txi, (.1). o the right had side fuctio We start by statig a sufficiet coditio for the IVP problem to admit a solutio which may ot be ecessarily uique. Theorem.1 (Cauchy / Peao Existece Theorem) f tx, is cotiuous i a closed regio If {(, ):, } B tx t t T x x R R R = (.) where T ad R are strictly positive costats, the there exists t < t 1 T such that the IVP has at least oe cotiuous i time solutio x( t ). I other words, cotiuity of f ( tx, ) i its argumets esures that there is at least oe solutio of the IVP i (.1). The above theorem does ot guaratee the uiqueess of the solutio. The key costrait f tx, satisfies the iequality that yields uiqueess is the Lipschitz coditio, whereby (, ) (, ) for all ( tx, ) ad ( t, y ) i some eighborhood of (, ) orm. For a vector x R f tx f ty Lx y (.3) t x. I (.3), i deotes a vector p, we cosider the class of p orms, defied by 4

5 1 p p x = xi, 1 p p < (.4) i= 1 ad x = max x (.5) 1 i i The ext theorem gives sufficiet coditios for the uique existece of a solutio. Theorem. (Local Existece ad Uiqueess) f tx, be piecewise cotiuous i t ad satisfy the Lipschitz coditio (.3) Let { } [ 1] x, y B= x R : x x r, t t, t (.6) The, there exists some δ > such that the state equatio = f ( t, x) with x( t ) has a uique solutio over [ t, t + δ ]. = x The key assumptio i the above theorem is the Lipschitz coditio (.3) which is t, x o the compact domai assumed to be valid locally, that is i a eighborhood of B i (.6). Oe may try to exted the iterval of existece ad uiqueess over a give time iterval [ t, t + δ ] by takig t t + δ as a ew iitial time ad x x( t + δ ) as a ew iitial t + δ, x t + δ the there exist state. If the coditios of the theorem are satisfied at ( ( )) δ > such that the IVP has a uique solutio over [ t δ, t δ δ] through the poit ( t δ, x( t δ )) existece of a uique solutio over the iterval [ t, t δ δ ] that passes + +. We piece together the solutios to establish the + +. This idea ca be repeated to keep extedig the solutio. However, i geeral the solutio caot be exteded idefiitely. I that case, there will be a maximum iterval [ t, T ), where the uique solutio exists. Homework: Read ad study the proof of Theorem 3.1, pp , []. Example.3 The IVP 5

6 = x x = 1 has a solutio () x t 1 = t 1 which is defied oly for t < 1 ad ca ot be exteded to R. Note that the fuctio f ( x) = x is locally Lipschitz for all x R, ad as t 1 the solutio has a fiite escape time, that is it leaves ay compact set withi a fiite time. The phrase fiite escape time is used to describe the pheomeo that a trajectory escapes to ifiity at a fiite time. Assumig that f ( x ) is globally Lipschitz, the ext theorem establishes the existece of a uique solutio over ay arbitrarily large iterval. Theorem.3 (Global existece ad Uiqueess) Suppose that f is piecewise cotiuous i t ad globally Lipschitz i x, that is the fuctio satisfies Lipschitz coditio (.3) [ ] x, y R, t t, t (.7) 1 The the IVP (.1) has a uique solutio over [, ] arbitrarily large. Homework: Read ad study the proof of Theorem 3., p. 93, []. t t 1, where the fial time 1 Sufficiet coditios i the above theorem are overly coservative. t may be Example.4 The IVP = x x 3 = x has a uique solutio x x t + 1 () = xt for ay iitial coditio x ad for all t. 6

7 Basically, if it is kow that IVP has a solutio that evolves o a compact domai the the solutio ca be exteded idefiitely. Theorem.4 (Global existece ad Uiqueess o a compact domai) Let f ( tx, ) be piecewise cotiuous i t, locally Lipschitz i x for all t ad all x i a domai D R. Let W D be a compact subset of D, x W, ad suppose it is kow that every solutio of the correspodig IVP lies etirely i W. The there is a uique solutio that is defied for all t. Homework: Read ad study the proof of Theorem 3.3, pp , []. Remark: There are extesios that deal with existece ad uiqueess of IVP-s whose system dyamics is discotiuous i x, (i.e., ot Lipschitz). Homework Example.5 The IVP 1 = x = x sg x + x x 1 = x does ot satisfy the sufficiet coditios for existece ad uiqueess of its solutio. Nevertheless, the IVP solutio does exist. Simulate the system startig from differet iitial coditios. Costruct phase portrait of the system ad argue that a) the maifold c( x) = x1+ x = is the system global attractor, b) all system trajectories reach maifold i fiite time, ad c) the solutio slides dow the maifold towards the origi. 3. Lyapuov Stability Readig material: [1]: Chapter 3, Sectios [1]: Chapter 3, Sectio 4.5 []: Chapter 4 Stability of equilibrium poits is usually characterized i the sese of Lyapuov. Alexader Michailovich Lyapuov, Russia mathematicia ad egieer who laid out the foudatio of the Stability Theory Results published i 189, Russia Traslated ito Frech, 197 Reprited by Priceto Uiversity, 1947 America Cotrol Egieerig Commuity Iterest, 196 s 7

8 Lyapuov stability theorems give sufficiet coditios for stability, asymptotic stability, ad so o. Statemets that establish ecessity of these coditios are called the coverse theorems. For example, it is kow that a equilibrium poit of a oliear system is expoetially stable if ad oly if the liearizatio of the system about that poit has a expoetially stable equilibrium at the origi. We will be mostly cocer with the d theorem of Lyapuov. We will use it to: a) derive stable adaptive laws for ucertai system, ad b) show boudedess of the system closedloop solutios eve whe the system has o equilibrium poits. Without a loss of geerality, we ll study stability of the origi for the autoomous system: where f ( x ) is locally Lipschitz i x ad = f x (3.1) f =. Defiitio 3.1 (local stability for autoomous systems) The equilibrium poit x = of (3.1) is stable if R >, r R >, x < r t, x t < R (3.) { } { () } ustable if it is ot stable (write formal defiitio similar to (3.)) r = r R ca be chose such that asymptotically stable if it is stable ad x r x( t) < lim = (3.3) t margially stable if it is stable but ot asymptotically stable, (write formal defiitio) expoetially stable if it is stable ad r, αλ, >, x < r t > : x t α x e λ t, (3.4) { } () 8

9 Figure 3.1: Stable System Figure 3.: Ustable System Basically, a equilibrium poit is stable if all solutios startig at earby poits stay earby; otherwise it is ustable. It is asymptotically stable if all solutios startig at earby poits ot oly stay earby, but also ted to the equilibrium poit as time approaches ifiity. Remark: Stabilizable systems are ot ecessarily stable. Note that by defiitio, expoetial stability implies asymptotic stability, which i tur implies stability. By defiitio, stability i the sese of Lyapuov defies local behavior of the system trajectories ear the equilibrium. I order to aalyze how the system behaves some distace away from the equilibrium, global cocepts of stability are required. Homework: Read Sectio 4.5 from []. Study Lyapuov stability for oautoomous systems. Defiitio 3. (global stability) If asymptotic (expoetial) stability holds for ay iitial states, the equilibrium poit is said to be globally asymptotically (expoetially) stable. Next, two mai theorems of Lyapuov are preseted. Theorem 3.1 (Lyapuov idirect method) Let x = be a equilibrium poit for the oliear system (3.1), where f : D R is cotiuously differetiable ad D is a eighborhood of the origi. Let The: f A= (3.5) ( x) x x = 9

10 the origi is asymptotically stable if Reλ i < for all eigevalues of A the origi is ustable if Reλ i > for at least oe of the eigevalues of A if at least oe of the eigevalues is o the jω axis, (i.e., the liearized system is margially stable), the othig ca be said about the origial oliear system behavior Before statig the d theorem of Lyapuov we eed to itroduce the cocept of positive defiite fuctios. Defiitio 3.3 Let D R be a eighborhood of the origi. A fuctio V ( x): D R is said to be (positive semidefiite) if locally positive defiite, if: V = ad V ( x) >, x D { } locally positive semidefiite, if: V = ad V ( x), x D { } locally egative defiite (semidefiite), if it is ot locally positive defiite (semidefiite) If i the above defiitio (semidefiite). D = R the the fuctio is globally positive (egative) defiite Theorem 3. (Lyapuov direct method) Let x = be a equilibrium poit for (3.1) ad let D R be a domai cotaiig the origi. If there is a cotiuously differetiable positive defiite fuctio V ( x): D R, whose time derivative alog the system trajectories is egative semidefiite i D V V V V ( x) = x = fi( x) = f ( x) x x x (3.6) i i= 1 i i= 1 the the equilibrium is stable. Moreover, if V ( x ) < i D { } asymptotically stable. i, the the equilibrium is Homework: Read ad study the proof of Lyapuov s Theorem, [, Theorem 4.1, pp ]. Defiitio 3.4 A cotiuously differetiable positive defiite fuctio V ( x ) satisfyig (3.6) is called a Lyapuov fuctio. Homework Example 3.1 Cosider the 1 st order ODE = c x 1

11 where c( x ) is locally Lipschitz o ( a, a) ad satisfies { c = } { c( x) x>, x : x ( a a) } Show that both V ( x) = c( y) dy ad V ( x) x x = are the Lyapuov fuctios ad cosequetly the origi is a asymptotically stable equilibrium (locally) of the system. Whe the origi x = is a asymptotically stable equilibrium of the system, we are ofte iterested i determiig its regio of attractio, (also called regio of asymptotic stability, domai of attractio, or basi). We wat to be able to aswer the questio: Uder what coditio will the regio of attractio be the whole space R? Defiitio 3.5 If the regio of attractio of a asymptotically stable equilibrium poit at the origi is the whole space R, the equilibrium is said to be globally asymptotically stable. Defiitio 3.6 A fuctio V : R R such that lim V ( x) = is called radially ubouded. x Theorem 3.3 (Barbashi-Krasovskii theorem) Let x = be a equilibrium poit for (3.1). Let V : R R be a radially ubouded Lyapuov fuctio of the system. The the equilibrium is globally asymptotically stable. Lecture 4. LaSalle s Ivariace Priciple Readig material: [1]: Chapter 3, Sectios []: Chapter 4, Sectio 4. We begi with a motivatig example. Example 4.1 (oliear pedulum dyamics with frictio) 11

12 Figure 4.1: Pedulum Dyamics of a pedulum with frictio ca be writte as: or, equivaletly i state space form: (4.1) MR θ + kθ + MgRsi θ = = x 1 = asi x bx 1 (4.) where x1 = θ, x = g k θ, a =, ad b =. We study stability of the origi x e =. R M R Note that the latter is equivalet to studyig stability of all the equilibrium poits i the T form: xe = ( π l ), l =, ± 1, ±, Cosider the total eergy of the pedulum as a Lyapuov fuctio cadidate. x1 x x V x = asi ydy+ = a 1 cos x + Potetial 1 Kietic (4.3) It is clear that V ( x ) is a positive defiite fuctio, (locally, aroud the origi). Its time derivative alog the system trajectories is: V x = asi x + x = bx (4.4) 1 1 The time derivative is egative semidefiite. It is ot egative defiite because V ( x ) = for x = irrespective of the value of 1 x. Therefore, we ca coclude that the origi is a stable equilibrium. 1

13 However, usig the phase portrait of the pedulum equatio (or just commo sese), we expect the origi to be a asymptotically stable equilibrium. Cosequetly, the Lyapuov eergy fuctio argumet fails to show this fact. O the other had, we otice that for the system to maitai V ( x ) = coditio, the trajectory must be cofied to the lie x =. Usig the system dyamics (4.) yields: x six x 1 1 Hece o the segmet π < x1 < π of the lie x = the system ca maitai the coditio oly at the origi V x t must decrease to V ( x ) = x =. Therefore, ( ) toward ad, cosequetly, x() t as t, which is cosistet with the fact that, due to frictio, eergy caot remai costat while the system is i motio. The forgoig argumet shows that if i a domai about the origi we ca fid a Lyapuov fuctio whose derivative alog the system trajectories is egative semidefiite, ad we ca establish that o trajectory ca stay idetically at poits where V ( x ) =, except at the origi, the the origi is asymptotically stable. This argumet follows from the LaSalle s Ivariace Priciple. Defiitio 4.1 A set M R is said to be a ivariat set with respect to (3.1) if: a positively ivariat set with respect to if: x M x t M, t R x M x t M, t Theorem 4.1 (LaSalle s theorem) Let Ω D R be a compact positively ivariat set with respect to the system dyamics (3.1). Let V : D R be a cotiuously differetiable fuctio such that V ( x() t ) i Ω. Let E Ω be the set of all poits i Ω where V ( x ) =. Let M E be the largest ivariat set i E. The every solutio startig i Ω approaches M as t, that is lim if x() t z = t z M dist ( xt (), M) Notice that the iclusio of the sets i the LaSalle s theorem is: M E Ω D R 13

14 I fact, the formal proof of the theorem (see [], Theorem 4.4, p. 18) reveals that all trajectories x() t are bouded ad approach a positive limit set L + M as t. The latter may cotai asymptotically stable equilibriums ad stable limit cycles. Remark 4.1 Ulike Lyapuov theorems, LaSalle s theorem does ot require the fuctio V ( x ) to be positive defiite. Most ofte, our iterest will be to show that x( t) as t. For that we will eed to establish that the largest ivariat set i E is the origi, that is: M = { }. This is doe by showig that o solutio ca stay idetically i E other tha the trivial solutio x() t. Theorem 4.1 (Barbashi-Krasovskii theorem) Let x = be a equilibrium poit for (3.1). Let V : D R be a cotiuously differetiable positive defiite fuctio o a domai D R cotaiig the origi, such that V x() t S = x D: V x = ad suppose that o other solutio i D. Let { } ca stay i S, other tha the trivial solutio x( t). The the origi is locally asymptotically stable. If, i additio, V ( x ) is radially ubouded the the origi is globally asymptotically stable. Note that if V ( x) is egative defiite the { } S = ad the above theorem coicides with the Lyapuov d theorem. Also ote that the LaSalle s ivariat set theorems are applicable to autoomous system oly. Example 4. Cosider the 1 st order system together with its adaptive cotrol law The dyamics of the adaptive gai ˆk ( t ) is = ax+ u ˆ u = k t x ˆk = γ x where γ > is called the adaptatio rate. The the closed-loop system becomes: 14

15 ( ˆ() ) = k t a x ˆ k = γ x The lie x = represets the system equilibrium set. We wat to show that the trajectories approach this equilibrium set, as t, which meas that the adaptive cotroller regulates x() t to zero i the presece of costat ucertaity i a. Cosider the Lyapuov fuctio cadidate where b (, ˆ) = ( ˆ ) V x k x k b γ > a. The time derivative of V alog the trajectories of the system is give by ( ˆ 1 ) ( ˆ ) ˆ ˆ ˆ V x, k = x + k b k = x k a + k b x = x b a γ Sice (, ˆ) V (, ˆ x k) is semi-egative, the set (, ˆ) : (, ˆ) V x k is positive defiite ad radially ubouded fuctio, whose derivative { } Ω c = x k R V x k c is compact, positively ivariat set. Thus takig Ω =Ω, all the coditios of LaSalle s Theorem are satisfied. The set E is give by ( ˆ) c {, : c } E = x k Ω x=. Because ay poit o the lie x = is a equilibrium poit, E is a ivariat set. Therefore, i this example M = E. From LaSalle s Theorem we coclude that every trajectory startig i Ω approaches E, as t, that is () xt as t. Moreover, sice (, ˆ) (, ˆ ) Ω c ca be chose large eough that ( ˆ) c V x k is radially ubouded, the coclusio is global, that is it holds for all iitial coditios costat c i the defiitio of Homework: simulate the closed-loop system from Example 4.. x, k ˆ test differet iitial coditios x k because the x, k Ω c. ru simulatios of the system while icreasig the rate of adaptatio γ > util high frequecy oscillatios ad / or system departure occurs. Try to quatify maximum allowable γ max as a fuctio of the iitial coditios. ru simulatios of the system while icreasig the cotrol time delay τ, that is usig cotrol i the form u( t) = kˆ ( t τ ) x( t τ ). Try to quatify maximum allowable time delay τ max, (as a fuctio of the iitial coditios ad rate of adaptatio), before the system starts to oscillate or departs. 15

16 5. Boudedess ad Ultimate Boudedess Readig material: []: Sectio 4.8 Cosider the oautoomous system where :, [ ) [, ) D, ad f D R (, ) = f t x (5.1) is piecewise cotiuous i t, locally Lipschitz i x o D R is a domai that cotais the origi x =. Note that if the origi is a equilibrium poit for (5.1) the by defiitio: f ( t,) =, t. O the other had, eve if there is o equilibrium at the origi, Lyapuov aalysis ca still be used to show boudedess of the system trajectories. We begi with a motivatig example. Example 5.1 Cosider the IVP with oautoomous scalar dyamics = x+ δ si t x t = a> δ > (5.) The system has o equilibrium poits. The IVP explicit solutio ca be easily foud ad show to be bouded for all t t, uiformly i t, that is with a boud b idepedet of t. I this case, the solutio is said to be uiformly ultimately bouded (UUB), ad b is called the ultimate boud, (prove it). Turs out, the UUB property of (5.) ca be established via Lyapuov aalysis ad x without usig the explicit solutio of the state equatio. I fact, startig with V ( x ) =, we calculate the time derivative of V alog the system trajectories. = = ( + δ si ) = + δ si + δ = ( δ) V x x x x t x x t x x x x It immediately follows that, V x < x > δ 16

17 =, or I other words, the time derivative of V is egative outside the set Bδ { x δ} equivaletly, all solutios that start outside of B δ will eter the iterval withi a fiite time, ad will remai withi the iterval bouds afterward. Formally, it ca be stated as follows. Choose δ c >. The all solutios startig i the set { } = Bc V x c B δ x c will remai therei for all future time sice V is egative o the boudary. Hece the solutios are uiformly bouded. Moreover, a ultimate boud of the solutios ca also be foud. Choose ε such that δ < ε < The V is egative i the aulus set { V ( x) c} V ( x() t ) will decrease mootoically i time util the solutio eters the set { V( x) ε} c ε, which implies that i this set. From that time o, caot leave the set because agai V is egative o its boudary x V ( x) = ε. Sice V ( x ) =, we ca coclude that the solutio is UUB with the ultimate boud x ε. Defiitio 5.1 The solutios of (5.1) are uiformly bouded if there exists a positive costat c, idepedet of t, ad for every a (, c), there is β = β( a) >, idepedet of t, such that x t a x t β, t t (5.3) globally uiformly bouded if (5.3) holds for arbitrarily large a uiformly ultimately bouded with ultimate boud b if there exist positive costats b ad c, idepedet of a, c, there is (, ) T = T a b, idepedet of t, such that t, ad for every x t a x t b, t t + T (5.4) 17

18 globally uiformly ultimately bouded if (5.4) holds for arbitrarily large a. Figure 5.1: UUB Cocept I the defiitio above, the term uiform idicates that the boud b does ot deped o t. The term ultimate idicates that boudedess holds after the lapse of a certai time T. The costat c defies a eighborhood of the origi, idepedet of t, such that all trajectories startig i the eighborhood will remai bouded i time. If c ca be chose arbitrarily large the the UUB otio becomes global. Basically, UUB ca be cosidered as a milder form of stability i the sese of Lyapuov (SISL). A compariso betwee SISL ad UUB cocepts is give below. SISL is defied with respect to a equilibrium, while UUB is ot. Asymptotic SISL is a strog property that is very difficult to achieve i practical dyamical systems. SISL requires the ability to keep the state arbitrarily close to the system equilibrium by startig sufficietly close to it. This is still too strog a requiremet for practical systems operatig i the presece of ukow disturbaces. The mai differece betwee UUB ad SISL is that the UUB boud b caot be made arbitrarily small by startig closer to the equilibrium or the origi. I practical systems, the boud b depeds o disturbaces ad system ucertaities. To demostrate how Lyapuov aalysis ca be used to study UUB, cosider a cotiuously differetiable positive defiite fuctio V ( x ). Choose < ε < c. Suppose that the sets Ω ε = V ( x) ε ad Ω c = V ( x) c are compact. Let { } { } { ε } c Λ= V x c =Ω Ω ε 18

19 ad suppose that it is kow that the time derivative of ( ) V x t alog the trajectories of the oautoomous dyamical system (5.1) is egative defiite iside Λ, that is ( ()) ( ) <, Λ, V x t W x t x t t where W x() t is a cotiuous positive defiite fuctio. Sice V is egative i Λ, a trajectory startig i Λ must move i the directio of decreasig V ( x() t ). I fact, it ca be show that i the set Λ the trajectory behaves as if the origi was uiformly asymptotically stable, (which it does ot have to be i this case). Cosequetly, the fuctio V ( x() t ) will cotiue decreasig util the trajectory eters the set Ω ε i fiite time ad stays there for all future time. Hece, the solutios of (5.1) are UUB with the ultimate boud b= max x. A sketch of the sets Λ, Ω c, Ω ε is show i Figure 5.. x Ω ε Figure 5.: UUB by Lyapuov Aalysis is derived ad show to be valid o a domai, which is specified i terms of x. I such cases, UUB aalysis ivolves fidig I may problems, the relatio V ( t, x) W( x) the correspodig domais of attractio ad a ultimate boud. This aalysis will be performed ext. Refereces 1. J.J. Slotie, W. Li, Applied Noliear Cotrol, Pretice Hall, H.K. Khalil, Noliear Systems, 3 rd Editio, Pretice Hall, New Jersey,. 19