Domain of Attraction

Remarks on the domain of attraction
x  f ( x ),
f (0 )  0,
x R
n
R A  { x 0  R : x ( x 0 , t 0 , t )  0 as t   }
n
Complete (total) domain of attraction
Estimate of Domain of attraction :
Rˆ A
Lemma : The complete domain of attraction is an open,
invariant set. Its boundary is formed by trajectories
5-1
Consider x  f ( x )
Let D  R n be such that
Is D in Rˆ A ?
V ( x)  0
and
V  0,  x  D , x  0
V  c1
V  c2
c 2  c1
D
V
might be positive

could escape from D
What is a good
Rˆ A ?
Consider
2
2
2
V  0 for all x1  x 2  a
V (x)  C  D
*
Rˆ A
D
c
5-2
Example
Ex:
x1   2 x1  x1 x 2
0
 
0
x 2   x 2  x1 x 2
1
 
2
0
Here   is asymptotic ally stable since
0
Thus let Q  I ,
PA  A P   I
f
x
 14
P  
0

T
A
x0
 2
 
 0
0 

 1
0
1
2
Since V ( x )  x Px
T
2
2
2
2
V ( x )   ( x1  x 2 )  ( 12 x1 x 2  x1 x 2 )
 x
Here
2

1
2
x1 x 2 
1
2
x 2,
2

 x
2
2
x1 x 2 x1  2 x 2
2
5
4
x
x1  2 x 2 
3
2
2
  x 2 (1 
5 x
5
4
2
x 2)
5-3
Example (Continued)
Thus V is negative
Note that
V c with
Therefore
definite
in a ball of radius
2
x Px   min ( P ) x 2 . Thus
T
r 
4
5
.
we can choose
level set
c   min ( P ) r .
2
c  0 . 79 
The set  c with
1
4

 
4
5
2
 0 .8 .
c  0 . 79 is the estimate
of the region
of attraction .
5-4
Zubov’s Theorem
x  f ( x ) with
Consider
there
exists
V :G  R
h:R
n
n
f ( 0 )  0 and suppose
 R
 R with the
following
properties
:
(i) V is continuous ly differenti able and positive definite in G and
0  V ( x )  1,  x  G  { 0}.
(ii) As x approaches
of G , or in case of unbounded
x   , lim V ( x )  1 .
G as
(iii) h is continuous
(iv) V ( x ) 
Then
the boundary
V
x
and positive
definite
n
on R .
f ( x )   h ( x )[1  V ( x )],  x  G .
x  0 is asymptotic ally stable and G is
the region
of attraction .
5-5
Example
In the region of the origin,
Hence
the origin
the region
V ( x ) is p.d. & V ( x ) is n.d.
is asymptotic
ally stable.
of attraction , we need to show
  { x  G | V ( x )  1} is invariant
Ex: x1   k ( x1 )  g 2 ( x 2 )
where
x 2   g 1 ( x1 )
To show
and x ( 0 )  G  x ( t )  0 as t   .
 z g ( ) d    as z   a or z  b
i
i
 0 i

 k ( 0 )  0 , zk ( z )  0 ,   a 1  z  b1
 g ( 0 )  0 , zg ( z )  0 ,   a  z  b
i
i
i
 i

D  { x  R :  a i  x i  bi } is the region
2
G is
that
for some positive
Show
that
constants
a i , bi
of attraction .
5-6
Example (Continued)
Solution: Let h ( x )  g 1 ( x1 ) k ( x1 ). Choose V  1  W 1 ( x1 )W 2 ( x 2 ).
Using

W1
 x1
Zubov' s theorem,
V
x
(i.e.,
W 2 (  k ( x1 )  g 2 ( x 2 )) 
W 2
x 2
f ( x )   h ( x )[ 1  V ( x )]),
W 1 (  g 1 ( x1 ))  g 1 ( x1 ) k ( x1 )[ W 1 ( x1 )W 2 ( x 2 )]  0
Then

W1
 x1

 g 1 ( x1 )W 1 ( x1 ) W 2 ( x 2 ) k ( x1 ) 
Then
W1
 x1
it is easy to
Thus our choice
W 2
x 2
W 1 ( x1 ) g 1 ( x 1 ) 
 is satisfied
see
  g 1 ( x1 )W 1 ( x1 ),

W 2
x 2
by the
W1
 x1

W 2 ( x2 ) g 2 ( x2 )  0
following

W 1 ( x1 ), W 2 ( x 2 )
  g 2 ( x 2 )W 2 ( x 2 )
of W will
V 1 e
 0x1 g 1 (  ) d   0x 2 g 2 (  ) d  
We see that V has the properties
: V (0)  0,
0  V ( x )  1,  x  D
and V ( x )  1 as x   D
5-7
Example (Continued)
And
  1 g 1 (  ) d   2 g 2 (  ) d 
0
V   g 1 ( x1 )  (  k ( x1 )  g 2 ( x 2 ))  g 2 ( x 2 )(  g 1 ( x1 ))   e 0

x
x



  g 1 ( x1 ) k ( x1 )[1  V ( x )]  0
All the conditions
V ( x ) is negative
V  0

x1 ( t )  0
of Zubov' s theorem
semi - definite.
g 1 ( x1 ) k ( x1 )  0

g 2 ( x2 )  0
By LaSalle' s theorem,


are satisfied
except th at
However
x1  0
x2  0
D is the region
of attraction .
5-8
The most
straight
to use quadratic
forward
but conservati ve method
to find
Rˆ A is
form
(1) Linearize
(2) Find
Q.F. Lyapunov
(3) Find the derievativ
traj ectory
(4) Find
e of the Lyapunov
of the nonlinear
function
along
the
system
V  0
D where
(5) Inscribe
function.
a level set of V in D . This
Analogous
procedure
could
be carried
(1) Find V

p.d. in D1
(2) Find V

n.d. in D 2
is an Rˆ A .
out using
the direct
Lyapunov
method.
(3) Find a level set of V in D1  D 2
This is an Rˆ A .
5-9
Advanced Stability Theory
Theory
for x  f ( x ) is generalize d here to
( ) x  f ( t , x )
(  ) x  f ( t , x )  g ( t , x )
( ) Stability
if  V
If stable, then V   (Converse
Theorem)
( ) Another t ype of stability
u
bounded
y
bounded ?
5-10
Stability of time varying systems

Stability of time varying systems
x  f ( t , x )
f : R  D  R
(1)
n
where
D  R .
n
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.
x  f ( x ) where
z  x  x (t )
*
*
x ( t ) is a solution
x  z  x (t )
*
z  x ( t )  f ( z  x ( t ))
Linearizat ioin
z  f ( z  x ( t ))  x ( t )  F ( t , z )
z  A ( t ) z 
*
*
*
*
F
z
z
z0
5-11
Stability
• Definition of stability
Definition
: 0 is an equilibriu
Definition
: The equilibriu
m of (1) if
m point
f (t ,0 )  0 ,
0 of (1) is stable
t  0
if    0 and  t 0  0 ,
  ( , t 0 )  0 such that
x ( x0 , t0 , t )   ,
if
The equilibriu
 t  t0
x 0   ( , t 0 )
m point
0 of (1) is uniformly
stable
if    0
  (  )  0 such that
x ( x0 , t0 , t )   ,
if
 t  t0
x 0   ( )
5-12
Example
Ex: x  ( 6 t sin t  2 t ) x
dx
 ( 6 t sin t  2 t ) dt
x
ln x
x
t
 t ( 6 t sin t  2 t ) dt
0
x0
ln x  ln x 0  6 sin t  6 t cos t  t  6 sin t 0  6 t 0 cos t 0  t 0
2
Then
x (t )  x (t0 ) e
2
2
[ 6 sin t  6 t cos t  t  6 sin t 0  6 t 0 cos t 0  t 0 ]
let c ( t 0 )  sup e
Hence
2
2
2
[ 6 sin t  6 t cos t  t  6 sin t 0  6 t 0 cos t 0  t 0 ]
t  t0
Then
x ( t )  x ( t 0 ) c ( t 0 ),
 t  t0
For any   0 , the choice   
the origin
Suppose
c (t0 )
shows that
is stable.
t 0 takes on successive
and suppose
x ( t ) is evaluated
values
t 0  2 n  , n  0 ,1, 2 , 
 seconds
later in each case.
5-13
Example (Continued)
Then
x ( t 0   )  x ( t 0 ) exp{ 6 sin( 2 n  1)  6 ( 2 n  1)  cos( 2 n  1)   ( 2 n  1)
2 }
 6 sin( 2 n  )  6 ( 2 n  ) cos( 2 n  )  ( 2 n  ) }
2
 x ( t 0 ) exp{ 6 ( 2 n  1)   ( 2 n  1)
2
 6(2n )  (2n ) }
 x ( t 0 ) exp{  (12 n  6 )  ( 4 n  4 n  1)
2
 x ( t 0 ) exp{ 24 n   4 n 
2

2
2
2
 12 n   4 n  }
2
2
 6 }
 x ( t 0 ) exp{  ( 24 n  4 n     6 )}
 x ( t 0 ) exp{  ( 4 n  1)( 6   )}
for x ( t 0 )  0 ,
This implies,
x (t0   )
  as n  
x (t0 )
Thus given
would
  0 , there is no  independen t of t 0 that
satisfy t he requiremen t uniformly
in t 0 .
5-14
Example (Continued)
The equilibriu
m point
0 of (1) is asymptotic
ally stable
if it is stable
and
  1 ( t 0 ) such that
x ( x 0 , t 0 , t )  0 as t   if
The equilibriu
uniformly
m point
stable
0 of (1) is uniformly
uniformly
m point
asymptotic
The equilibriu
asymptotic
ally stable
if it is
and   1 such that
x ( x 0 , t 0 , t )  0 as t   if
The equilibriu
x (t 0 )   1 (t 0 )
x (t 0 )   1
0 of (1) is globally
ally stable
m point
uniformly
asymptotic
ally stable
if it is
and  1   .
0 of (1) is exponentia
lly stable
   0 , M  0 and   0
such that
x ( x0 , t0 , t )  M x0 e
 ( t  t0 )
if
x0   ,
t  t0
5-15
Example (Continued)
The equilibriu
exponentia
m point
exponentia
lly stable
if it is
and    .
lly stable
For time - invariant
( ) stability
0 of (1) is globally
system
 uniformly
( ) stability
There is another class of systems where the same is true – periodic system.
x  f ( t , x ),
 T  0 such that
f (t  T , x )  f (t , x )
Like
x  f (sin t , x )
Reason : it is always possible to find
min  (  , t 0 )  0
t 0 [ 0 , T )
5-16
Positive definite function
• Positive definite function
Class
V (t , x )
K : all continuous
function
 ( z ) such that
 ( 0 )  0 and  ( ) is strictly
( x )
Class
K  : all continuous
function
 ( 0 )  0 ,  ( ) is strictly
x
Definition: A continuous
increasing
.
 ( z ) such that
increasing
, and
 ( r )   as r   .
function
V : R   R  R is l.p.d
n
if  r  0 and   K
V (t , x )   ( x )
 x  r and V ( t , 0 )  0 ,  t  0
if the above property
radially
unbounded
holds and r   , V is p.d and
if the above holds for   K  .
5-17
Decrescent
A continuous
function
V : R   R  R is decresent
n
if  r  0 and   K such that
V (t , x )   ( x )
 x  r , t  0
( x )
positive definite decrescent
V (t , x )
( x )
Thoerem: A continuous
function
V : R   R  R with V ( t , 0 )  0 ,  t is
n
l.p.d (p.d) if and only if  a l.p.d (p.d) W : R  R such that
n
V ( t , x )  W ( x ),  t ,  x  r (  x  R )
n
V ( t , x ) is radially
radially
unbounded
if this is satisfied
with W ( x )
unbounded
5-18
Decrescent (Continued)
V ( t , x ) is decrescent if and only if
sup sup V ( t , x )    p  [0, r ]
x p
t0
Proof : see Nonlinear systems analysis
Ex: V ( t , x1 , x 2 )  ( t  1)( x12  x 22 )  x12  x 22  p.d, radially unbounded,
not decrescent
V ( t , x1 , x 2 )  t ( x1  x 2 ) 
2
2
( x1  x 2 )( t  1)
2
V ( t , x1 , x 2 ) 
2
2
(t  2 )
2
( x1  x 2 )( t  1)
2
V ( t , x1 , x 2 ) 
not l.p.d, not decrescent
2
2
V
t
p.d, decrescent,
radially unbounded

p.d, not decrescent,
not radially unbounded
2
( x1  2 )
Finally V ( t , x ) 


V
x
f (t , x )
5-19
Stability theorem
• Stability theorem
Thoerem: The equilibriu
- stable
m point
0 of x  f ( t , x ) is
if  a continuous
such that
- uniformly
ly differenti
able l.p.d.f.
V (t , x )
V (t , x )  0 ,  t  t 0  x  r  0
stable
if the above
condition
holds
and V ( t , x )
if  continuous
ly differenti
is decrescent
- uniformly
asymptotic
l.p.d, decrescent
- globally
differenti
uniformly
ally stable
V ( t , x ) such that
asymptotic
able p.d, decrescent
V ( t , x ) such that
able
V ( t , x ) is l.n.d.f.
ally stable
and radially
if  a continuous
ly
unbounded
V ( t , x ) is n.d.f.
5-20
Stability theorem (Continued)
- exponentia
continuous
a x
p
lly stable if  a , b , c , r  0 and
p  1 and a
ly differenti able V ( t , x ) such that
 V (t , x )  b x
p
,  t  0,  x  r
and
V ( t , x )   c x
- globally
p
exponentia
,  t  0,  x  r
lly stable if the above
Proof : same as before,
holds
for r  
 ( x ) plays the role of old V ( x ).
5-21
Example
Ex:
y  y  ( 2  sin t ) y  0
x1  x 2
Mathieu eq.
x 2   x 2  ( 2  sin t ) x1
2
x  x  V (t , x )  x 
2
1
2
2
2
1
decrescent
x2
2  sin t
x2
( 2  sin t )
 
 x 
x2
3
positive definite
2
V 
2
2
1
2
(  cos t )  2 x1 x1 
4  2 sin t  cos t
( 2  sin t )
2
2 x2
2  sin t
x 2
x2  0,  t ,  x  R
2
2
0
Thus   is uniformly stable.
0
5-22
Theorem
Remark : LaSalle’s theorem does not work in general for time-varying
system. But for periodic systems they work. So (uniformly)
asymptotically stable.
C onsider x ( t )  f ( t , x )
w here f ( t , x )  f ( t  T , x ),  x  R ,  t  0
n
Theorem
Suppose V : R   R n  R is a continuously differentiable p.d.f and
radially unbounded with
V ( t , x )  V ( t  T , x ),  x  R ,  t  0
n
Define
S  { x  R : V ( t , x )  0,  t  0}
n
Suppose V ( t , x )  0,  t  0,  x  R n , and that S contains no nontrivial
trajectories. Under these conditions, the equilibrium point 0 is globally
asymptotically stable.
5-23
Example
Ex: y  ( a  b ( t ) cos y ) y  c ( t ) sin y  0
where a  0 , b ( t  T )  b ( t )
c ( t  T )  c ( t ), T  0
b ( t ), c ( t ) continuous ly differenti able.
b (t )  bM  a ,  t
max c ( t )  c M  
t
min c ( t )  c m  0
t
c  2 ( a  b M ) c m ,  t
Now
x1  x 2
x 2   [ a  b ( t ) cos x1 ] x 2  c ( t ) sin x1
5-24
Example (Continued)
Choose
1  cos x1 
x2
2
 V ( t , x1 , x 2 )  1  cos x1 
x2
2
 1  cos x1 
x2
2
2cM


   

2 c (t )
2cm
           

l.p.d
decrescent
x2
V ( t )  
2
2
2 c (t )
Obviously
2 c ( t )[ a  b ( t ) cos
x1 ]  c ( t ) 
2 c ( t )[ a  b ( t ) cos x1 ]  c ( t )  2 c m ( a  b M )  c
and 2 c m ( a  b M )  c  0 ,  t
So, V  0
Now we again
use the invariance
principle
V  0  x 2  0  x1  const.  c ( t ) sin x1  0  x1  0
 (Uniformly
) asymptotic
ally
stable
5-25
Instability Theorem (Chetaev)
• Instability Theorem (Chetaev)
The equilibriu
differenti
m point
0 of x  f ( t , x ) is unstable
ly
able V ( t , x ), set B r  { x  R : x  r }, an open set   B r
and a function
n
r  K such that
0  V ( t , x ),  t  0 ,  x  
sup sup V ( t , x )  
t
if  a continuous
x 
0  
V (t , x )  0 ,  t ,  x     B r
V  0

V  0
V  0
V  0
V ( t , x )  r ( x ),  t ,  x  
5-26
Linear time-varying systems and linearization

Linear time-varying systems and linearization
x  A ( t ) x
x R
n
x (t )   (t , t0 )x (t0 )
state transitio n matrix
d
dt
 (t , t0 )  A (t )  (t , t0 )
 (t0 , t0 )  I ,
Stability
analyze
 t0  0
of linear ti me varying
as that of nonlinear
system
is almost
as difficult
to
system.
5-27
Example
Ex:
x  A ( t ) x
  1  1 . 5 cos 2 t
A (t )  
  1  1 . 5 sin t cos t
1  1 . 5 sin t cos t 

2
 1  1 . 5 sin t 
det(  I  A ( t ))  0
1 , 2   0 . 25  j 0 . 25 7
 e 0 . 5 t cos t
But  ( t , 0 )  
0 .5 t
sin t
 e
looks like stable??
e
e
t
t
sin t 

cos t 
Thus  x 0 such that
x ( x 0 , t 0 , t )   as t  
5-28
Theorem
Theorem:
The equibrium
0 of x  A ( t ) x is stable
point
iff
sup  ( t , t 0 )   ,  t 0
t  t0
- uniformly
stable
iff
sup sup  ( t , t 0 )
t0  0
- (globally)
t  t0
uniformly
 (t , t 0 )
- (globally)
i
i

asymptotic
ally stable
iff the above
condition
holds
 0 as t   ,  t 0
uniformly
asymptotic
ally stable
iff
 r (t t )
0
 ( t , t 0 ) i  Ke
,  t  t0  0, K  0, r  0
              
exponentia lly stable
Proof : See Nonlinear systems analysis
5-29
Lyapunov function approach
• Lyapunov function approach
x  A ( t ) x
V ( t , x )  x P ( t ) x where
T
Then
2
c1 I  P ( t )  c 2 I ,  t , c i  0
2
c1 x  x P ( t ) x  c 2 x
     
T
p.d
decrescent
T
T
T
V ( t , x )  x P ( t ) x  x P ( t ) x  x P ( t ) x
T
T
T
T
 x A ( t ) P ( t ) x  x P ( t ) x  x P ( t ) A ( t ) x
 x
T
P ( t )  A
  x  (t ) x
T
i.e. V ( t , x )   c 3 x
T

(t ) P (t )  P (t ) A (t ) x
where
 (t )  c3 I ,  t , c3  0
2
Result : exponentia lly stability
Note that P ( t ) is defined
by a positive
definite
symmetric
solution
of
T
 P ( t )  P ( t ) A ( t )  A ( t ) P ( t )   ( t )
5-30
Theorem
Theorem: x  f ( t , x ), f : R   R n  R n
continuous ly diff.
f (t ,0 )  0 ,  t  0
lim sup
f1 ( t , x )
x  0 t0
A (t ) 
 0 where
x
 f (t , x )
x
is
bounded
f1 ( t , x )  f ( t , x )  A ( t ) x
t
x0
Then 0 is an exponentia lly stable eq. point of x  f ( t , x )
if it is exponentia lly stable for x  A ( t ) x .
Proof : See Nonlinear systems analysis
5-31
Converse (Inverse) Theorem & Invariance Theorem

Converse (Inverse) Theorem
• i) if  V  stable
• ii) (uniformly asymptotically exponentially) stable

 V
Invariance Theorem
 : positive


V :   R , V  0 in 


E  { x   : V  0}


M : largest invariant set in E

x ( x 0 , t 0 , t )  M as t   ,  x 0   
invariant
We can eliminate
set
indirect
how to define E
is not clear since
V is a function
of
t , x as well.
x  D
set E  { x  D : W ( x )  0} can be defined
analogous
case,
the uncertaint y by assuming
V ( t , x )   W ( x )  0
Then the
In time varying
of the LaSalle' s theorem
as before. Thus an
can be formed
as follows :
5-32
Theorem
Theorem : Let
D  { x  R : x  r } where
2
and locally
function
Lipschitz
x  f ( t , x ) is piecewise
in X , uniformly
continuous
in t
in t . Let V be a cont. diff.
such that
W1 ( x )  V (t , x )  W 2 ( x )
V
V
V ( t , x ) 

f ( t , x )   W ( x ),  t  0 ,  x  D
t
x
where W 1 ( ), W 2 ( ) are continuous
a continuous
positive
Then all solutions
bounded
positive
semidefini te function
of x  f ( t , x ) with
definite
functions
and W ( ) is
on D . Let   min W 1 ( x ).
x r
x ( t 0 )  { x  B r : W 2 ( x )   } are
and W ( x ( t ))  0 as t  
Proof : See Ch 4.3 of Nonlinear Systems
x ( t ) approaches
Therefore
E as t   since W ( x ( t ))  0 as t   .
the positive
limit set of x ( t ) is a subset
of E .
5-33