Post on 18-Jan-2016
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