5-1

Domain of Attraction

Remarks on the domain of attraction

Complete (total) domain of attraction

Estimate of Domain of attraction :

Lemma : The complete domain of attraction is an open,

invariant set. Its boundary is formed by trajectories

0 0 0

( ), (0) 0,

{ : ( , , ) 0 as }

n

nA

x f x f x R

R x R x x t t t

ˆAR

5-2

Consider

Let be such that and

Is in ?

What is a good ?

( )x f xnD R ( ) 0V x 0, , 0V x D x

D ˆAR

ˆAR

ˆAR

c

*( )V x C D

D

D

1V c2V c

2 1c c

might be positiveV

could escape from D

Consider22

221 allfor 0 axxV

5-3

Example

2122

2111 2

xxxx

xxxx

Ex:

2

1

0

0

10

02 since stableally asymptotic is

0

0 Here

0xx

fA

21

41

0

0,let Thus PIPAPAIQ T

PxxxV T)( Since

2121212

2

2212

212

122

21

2

)()()(

xxxxx

xxxxxxxV

2212

221

21 52, Here xxxxxx

)1(24

52

2

3

2452

2xxxx

5-4

Example (Continued)

. radius of ball ain definite negative is Thus5

4rV

.)( with

set level choosecan weThus .)( that Note2

min

2

2min

rPcV

xPPxx

c

T

.8.079.0 Therefore2

54

41 c

.attraction of

region theof estimate theis 79.0 with set The cc

5-5

Zubov’s Theorem

: properties following with the:

: exists there

suppose and 0)0( with )(Consider

RRh

RRGV

fxfx

n

n

. )],(1)[()()(

.on definite positive and continuous is

.1)(lim , as

unbounded of casein or , ofboundary theapproaches As

}.0{ ,1)(0

and in definite positive and abledifferentily continuous is

GxxVxhxfxV

Rh

xVxG

Gx

GxxV

GV

xV

n

(i)

(ii)

(iii)

(iv)

.attraction ofregion the

is and stableally asymptotic is 0Then Gx

5-6

Example

. as 0)()0( and invariant is }1)(|{

thatshow toneed we,attraction ofregion the

is that show To stable.ally asymptotic isorigin theHence

n.d. is )( & p.d. is )( origin, theofregion In the

ttxGxxVGx

G

xVxV

Ex:

)(

)()(

112

2211

xgx

xgxkx

iiii

z

iii

bzazzgg

bzazzkk

bzazdg

,0)( ,0)0(

,0)( ,0)0(

or as )(

where 11

0

ii ba , constants positive somefor

.attraction ofregion theis }:{ Show 2iii bxaRxD

5-7

Example (Continued)Solution: ).()(1 Choose ).()()(Let 2211111 xWxWVxkxgxh

0)]()()[()())(())()((

)]),(1)[()( (i.e., theorem,s Zubov'Using

22111111112212

x

2

2

1

1

xWxWxkxgxgWxgxkW

xVxhxf

x

W

x

W

V

0)()()()()()()()(

Then

222211111221111 1

1

2

2

1

1

xgxWxgxWxkxWxWxg x

WxW

xW

)(),( following by the satisfied is see easy to isit Then 2211 xWxW

)()(),()( 22221111 2

2

1

1 xWxgxWxg xW

xW

1

02

0 21 )()(1

will of choiceour Thusx x dgdgeV

W

DxxV

DxxVVV

as 1)( and

,1)(0,0)0( : properties thehas that see We

5-8

Example (Continued)

And

0)](1)[()(

))()(())()(()(

111

)()(

112222111

2

02

1

01

xVxkxg

exgxgxgxkxgV

xxdgdg

However definite. - semi negative is )(

atexcept th satisfied are theorems Zubov'of conditions theAll

xV

.attraction ofregion theis theorem,sLaSalle'By

00)(0)(

00)()(0

2221

1111

D

xxgtx

xxkxgV

5-9

form quadratic use to

is ˆ find tomethod veconservatibut forwardstraight most The AR

.ˆan is This .in ofset level a Inscribe (5)

0 where Find (4)

system nonlinear theofectory traj

thealongfunction Lyapunov theof ederievativ theFind (3)

function. Lyapunov Q.F. Find (2)

Linearize (1)

ARDV

VD

method. Lyapunovdirect theusingout carried be could procedure Analogous

21

2

1

in ofset level a Find (3)

in n.d. Find (2)

in p.d. Find (1)

DDV

DV

DV

.ˆan is This AR

5-10

Advanced Stability Theory

tohere dgeneralize is )(for Theory xfx

stability of ypeAnother t )(

Theorem) (Converse then stable, If

if Stability )(

),(),( )(

),( )(

V

V

xtgxtfx

xtfx

u y

bounded bounded ?

5-11

Stability of time varying systems

Stability of time varying systems

. where:

),(nn RDRDRf

xtfx

(1)

f is piecewise continuous in t and Lipschitz in x.

Origin of time varying : (i) parameters change in time.

(ii) investigation of stability of trajectories of time invariant system.

solution a is )( where)( * txxfx

),()())((

))(()(

)()(

**

**

**

ztFtxtxzfz

txzftxz

txzxtxxz

z

z

FztAz

z 0

)(

ioinLinearizat

5-12

Stability

• Definition of stability

: Definition

0,0)0,( if (1) of mequilibriuan is 0 : Definition ttf

such that 0),(

,0 and 0 if stable is (1) of 0point mequilibriu The

0

0

t

t

),( if

,),,(

00

000

tx

ttttxx

such that 0)(

0 if stable uniformly is (1) of 0point mequilibriu The

000 ,),,( ttttxx

)( if 0 x

5-13

Example Ex:

20000

20 cos6sin6cos6sin6lnln

)2sin6(ln

)2sin6(

)2sin6(

00

ttttttttxx

dttttx

dttttx

dx

xtttx

tt

x

x

Then ]cos6sin6cos6sin6[0

20000

2

)()( ttttttttetxtx Hence ]cos6sin6cos6sin6[

0

20000

2

0

sup)(let tttttttt

ttetc

Then 000 ),()()( tttctxtx

stable. isorigin the

thatshows )( choice the,0any For 0tc

case.each in later seconds evaluated is )( suppose and

,2,1,0,2 valuessuccessiveon takes Suppose 0 0

tx

nntt

5-14

Example (Continued)

Then

)}6)(14(exp{)(

)}6424(exp{)(

}6424exp{)(

}412)144()612(exp{)(

})2()2(6)12()12(6exp{)(

})2()2cos()2(6)2sin(6

})12()12cos()12(6)12sin(6exp{)()(

0

0

220

22220

220

2

200

ntx

nntx

nntx

nnnnntx

nnnntx

nnnn

nnnntxtx

ntx

tx

tx

as )(

)(

,0)(for implies, This

0

0

0

.in uniformly t requiremen hesatisfy t would

that oft independen no is there,0given Thus

0

0

t

t

5-15

Example (Continued)

)()( if as 0),,(

such that )(

and stable isit if stableally asymptotic is (1) of 0point mequilibriu The

01000

01

ttxtttxx

t

1000

1

)( if as 0),,(

such that and stable uniformly

isit if stableally asymptoticuniformly is (1) of 0point mequilibriu The

txtttxx

. and stableally asymptotic uniformly

isit if stableally asymptoticuniformly globally is (1) of 0point mequilibriu The

1

00)(

000 , if ),,(

such that

0 and 0,0 stablelly exponentia is (1) of 0point mequilibriu The

0 ttxexMttxx

M

tt

5-16

Example (Continued)

. and stablelly exponentia

isit if stablelly exponentiaglobally is (1) of 0point mequilibriu The

stability )(uniformly stability )(

system invariant -For time

There is another class of systems where the same is true – periodic system.

),(sin

),(),(

such that 0),,(

xtfx

xtfxTtf

Txtfx

Like

Reason : it is always possible to find

0),(min 0),0[0

tTt

5-17

• Positive definite function

Positive definite function

. as )(

and ,increasingstrictly is )( ,0)0(

such that )(function continuous all : K Class

.increasingstrictly is )( and 0)0(

such that )(function continuous all :K Class

rr

z

z

.for holds above theif unboundedradially

and p.d is , and holdsproperty above theif

0 ,0)0,( and )(),(

and 0 if

l.p.d is :function continuousA

K

Vr

ttVrxxxtV

Kr

RRRV n

Definition:

( )x

( , )V t x

x

5-18

Decrescent

0 , )(),(

such that and 0 if

decresent is :function continuousA

trxxxtV

Kr

RRRV n

( )x

( )x( , )V t x positive definite decrescent

unboundedradially

)( with satisfied is thisif unboundedradially is ),(

such that : (p.d) l.p.d ifonly and if (p.d) l.p.d

is ,0)0,( with :function continuousA

xWxtV

RRWa

ttVRRRVn

n

( , ) ( ), , ( )nV t x W x t x r x R

Thoerem:

5-19

Decrescent (Continued)

ifonly and if decrescent is ),( xtV

0

sup sup ( , ) [0, ]x p t

V t x p r

Proof : see Nonlinear systems analysis

2 2 2 21 2 1 2 1 2

2 21 2 1 2

2 2 21 2

1 2 2

2 2 21 2

1 2 21

( , , ) ( 1)( )

( , , ) ( )

( )( 1)( , , )

( 2)

( )( 1)( , , )

( 2)

V t x x t x x x x

V t x x t x x

x x tV t x x

t

x x tV t x x

x

p.d, radially unbounded,not decrescent

not l.p.d, not decrescent

p.d, decrescent,radially unbounded

p.d, not decrescent,not radially unbounded

( , ) ( , )V V

V t x f t xt x

Ex:

Finally

5-20

Stability theorem

• Stability theorem

n.d.f. is ),( such that ),(

unboundedradially and decrescent p.d, abledifferenti

lycontinuous a if stableally asymptotic uniformly globally -

l.n.d.f. is ),( such that ),( decrescent l.p.d,

abledifferentily continuous if stableally asymptotic uniformly -

decrescent is

),( and holds condition above theif stable uniformly -

0 ,0),( such that

),( l.p.d.f. abledifferentily continuous a if stable -

is ),( of 0point mequilibriu The

0

xtVxtV

xtVxtV

xtV

rxttxtV

xtV

xtfx

Thoerem:

5-21

Stability theorem (Continued)

).( old of role theplays )( before, as same : Proof

for holds above theif stablelly exponentiaglobally -

,0 ,),(

and

,0 ,),(

such that ),( abledifferentily continuous

a and 1 and 0,,, if stablelly exponentia -

xVx

r

rxtxcxtV

rxtxbxtVxa

xtV

prcba

p

pp

5-22

Example

positive definitedecrescent

Mathieu eq.

0

0

0)sin2( ytyy Ex:

Thus is uniformly stable.

122

21

)sin2( xtxx

xx

3sin2),(

222

1

222

122

21

xx

t

xxxtVxx

2222

22

112

22

, ,0)sin2(

cossin24

sin2

22)cos(

)sin2(

Rxtxt

tt

xt

xxxt

t

xV

5-23

Theorem

Remark : LaSalle’s theorem does not work in general for time-varying system. But for periodic systems they work. So (uniformly) asymptotically stable.

Consider ( ) ( , )

where ( , ) ( , ), , 0n

x t f t x

f t x f t T x x R t

TheoremSuppose is a continuously differentiable p.d.f and radially unbounded with

Define

Suppose , and that contains no nontrivialtrajectories. Under these conditions, the equilibrium point 0 is globally asymptotically stable.

( , ) ( , ), , 0nV t x V t T x x R t

{ : ( , ) 0, 0}nS x R V t x t ( , ) 0, 0, nV t x t x R S

RRRV n :

5-24

Example

Ex:

1212

21

sin)( ]cos)([

Now

, )(2

0)(min

)(max

,)(

able.differentily continuous )(),(

0 ),()(

)()( ,0 where

0sin)()cos)((

xtcxxtbax

xx

tcbac

ctc

ctc

tabtb

tctb

TtcTtc

tbTtba

ytcyytbay

mM

mt

Mt

M

5-25

Example (Continued)

stableally asymptotic )(Uniformly

00sin)(const.00

principle invariance theuseagain weNow

0 So,

,0)( 2 and

)( 2)(]cos)()[(2Obviously

)(]cos)()[(2)(2

)(

1112

1

12

22

xxtcxxV

V

tcbac

cbactcxtbatc

tcxtbatctc

xtV

Mm

Mm

decrescentl.p.d

2cos1

)(2cos1),,(

2cos1

Choose2

21

22

121

22

1mM c

xx

tc

xxxxtV

c

xx

5-26

Instability Theorem (Chetaev)

• Instability Theorem (Chetaev)

xtxrxtV

BxtxtV

xtV

xtxtV

Kr

BrxRxBxtV

xtfx

r

xt

rn

r

, ),(),(

, ,0),(

0

),( sup sup

,0 ),,(0

such that function a and

set open an },:{set ),,( abledifferenti

lycontinuous a if unstable is ),( of 0point mequilibriu The

0V

0V

0V0V

5-27

Linear time-varying systems and linearization

Linear time-varying systems and linearization

system. nonlinear of that as analyze

todifficult asalmost is system varyingmelinear ti of Stability

0 ,),(

),()(),(

)(),()(

)(

000

00

00

matrixn transitiostate

tItt

tttAttdt

d

txtttx

RxxtAx n

5-28

Example

tttxx

x

tete

tetet

j

tAI

ttt

ttttA

xtAx

tt

tt

as ),,(

such that Thus

cossin

sincos)0,(But

stable?? like looks 725.025.0

0))(det(

sin5.11cossin5.11

cossin5.11cos5.11)(

)(

00

0

5.0

5.0

2,1

2

2

Ex:

5-29

Theorem

stablelly exponentia

0 ,0 ,0 ,),(

iff stableally asymptotic uniformly (globally) -

, as 0),(

holds condition above theiff stableally asymptotic uniformly (globally) -

),( sup sup

iff stable uniformly -

,),(sup

iff stable is )( of 0point equibrium The

0)(

0

00

00

00

0

00

0

rKttKett

tttt

tt

ttt

xtAx

ttr

i

i

ittt

tt

Theorem:

Proof : See Nonlinear systems analysis

5-30

Lyapunov function approach

• Lyapunov function approach

)()()()()()(

ofsolution symmetric definite positive aby defined is )( that Note

stabilitylly exponentia :Result

),( i.e.

0 , ,)( where)(

)()()()()(

)()()()()(

)()()(),(

)(Then

0 , ,)( where)(),(

)(

23

33

22

21

21

decrescentp.d

ttPtAtAtPtP

tP

xcxtV

ctIctxtx

xtAtPtPtAtPx

xtAtPxxtPxxtPtAx

xtPxxtPxxtPxxtV

xcxtPxxc

ctIctPIcxtPxxtV

xtAx

T

T

TT

TTTT

TTT

T

iT

5-31

Theorem

Theorem:

.)(for stablelly exponentia isit if

),( ofpoint eq. stablelly exponentiaan is 0Then

bounded is ),(

)(

)(),(),( where0),(

suplim

0 ,0)0,(

diff.ly continuous

: ),,(

0

11

00

xtAx

xtfx

tx

xtftA

xtAxtfxtfx

xtf

ttf

RRRfxtfx

x

tx

nn

Proof : See Nonlinear systems analysis

5-32

Converse (Inverse) Theorem & Invariance Theorem

Converse (Inverse) Theorem• i) if stable

• ii) (uniformly asymptotically exponentially) stable

Invariance Theorem

V

V

, as ),,(

in set invariant largest :

}0:{

in 0 , :

set invariant positive :

000

xtMttxx

EM

VxE

VRV

well.as ,

offunction a is

sinceclear not is

define tohow

case, varyingIn time

xt

V

E

: follows as formed becan theoremsLaSalle' theof analogousindirect

an Thus before. as defined becan }0)(:{set Then the

0)(),(

assumingby y uncertaint theeliminatecan We

xWDxE

DxxWxtV

5-33

Theorem

ttxW

xWBxtxxtfx

xWD

WWW

DxtxWxtfx

V

t

VxtV

xWxtVxW

VtX

txtfxrxRxD

r

rx

as 0))(( and bounded

are })(:{)( with ),( of solutions allThen

).(minLet .on function tesemidefini positive continuous a

is )( and functions definite positive continuous are )( ),( where

,0 ),(),(),(

)(),()(

such thatfunction

diff. cont. a be Let .in uniformly ,in Lipschitzlocally and

in continuous piecewise is ),( where}:{Let

20

1

21

21

2

Theorem :

Proof : See Ch 4.3 of Nonlinear Systems

. ofsubset a is )( ofset limit positive theTherefore

. as 0))(( since as approaches )(

Etx

ttxWtEtx