7/29/2019 Chapter 2 Stability Domain
1/22
C h a p t e r 2
S t a b i l i t y d o m a i n c o n c e p t s
2 . 1 I n t r o d u c t o r y c o m m e n t s
I n o r d e r t o g e t c o m p l e t e i n f o r m a t i o n a b o u t t h e c a u s a l i t y b e t w e e n i n i t i a l s t a t e s a n d
s y s t e m s m o t i o n s , c o n c e p t s o f d o m a i n s o f v a r i o u s s t a b i l i t y p r o p e r t i e s w e r e i n t r o -
d u c e d . I n t h e f r a m e w o r k o f t h e L y a p u n o v s t a b i l i t y t h e n o t i o n o f a t t r a c t i o n d o m a i n
o f t h e o r i g i n w a s d e n e d b y Z u b o v 2 1 8 ] a n d H a h n 9 4 ] , a n d n o t i o n s o f t h e s t a b i l -
i t y d o m a i n a n d t h e a s y m p t o t i c s t a b i l i t y d o m a i n w e r e d e n e d b y G r u j i c 6 4 ] , 6 5 ] ,
6 8 ] { 7 0 ] , 7 2 ] , a n d u s e d b y G r u j i c e t a l . 8 8 ] { 9 0 ] . T h e c o n c e p t o f p r a c t i c a l s t a b i l i t y
d o m a i n s w a s i n t r o d u c e d b y G r u j i c 7 0 ] .
I n t h e l i t e r a t u r e ( e . g . L a S a l l e a n d L e f s c h e t z 1 2 1 ] a n d Z u b o v 2 1 8 ] ) t h e n o t i o n o f
\ r e g i o n o f a s y m p t o t i c s t a b i l i t y " h a s b e e n u s e d i n t h e s e n s e o f t h e a t t r a c t i o n d o m a i n .
I n w h a t f o l l o w s t h e d i e r e n c e b e t w e e n t h e m w i l l b e c l a r i e d .
2 . 2 D o m a i n s o f L y a p u n o v s t a b i l i t y p r o p e r t i e s
2 . 2 . 1 T h e n o t i o n o f d o m a i n
T h e t e r m \ d o m a i n " d e n o t e s a s e t t h a t c a n b e , b u t n e e d n o t b e , o p e n o r c l o s e d .
D o m a i n s o f L y a p u n o v s t a b i l i t y p r o p e r t i e s w i l l b e c a l l e d f o r s h o r t \ L y a p u n o v
s t a b i l i t y d o m a i n s " i n a g e n e r a l s e n s e i n c o r p o r a t i n g d o m a i n s o f s t a b i l i t y , o f a t t r a c t i o n
a n d o f a s y m p t o t i c s t a b i l i t y . I n t h e c l o s e r s e n s e t h e n o t i o n \ L y a p u n o v s t a b i l i t y
d o m a i n " w i l l b e u s e d f o r t h e d o m a i n o f s t a b i l i t y ( f o r s h o r t , t h e s t a b i l i t y d o m a i n ) .
L y a p u n o v s t a b i l i t y d o m a i n s w i l l b e s t u d i e d h e r e i n i n t h e f r a m e w o r k o f t i m e -
i n v a r i a n t c o n t i n u o u s - t i m e n o n l i n e a r s y s t e m s g o v e r n e d b y
d X
d t
= f ( X ) ( 2 . 1 )
w i t h p o s s i b l y c e r t a i n s p e c i c f e a t u r e s t h a t w i l l b e d e s c r i b e d w h e n t h e y a r e n e e d e d .
I n t h e l i t e r a t u r e ( e . g . L a S a l l e a n d L e f s c h e t z 1 2 1 ] ) t h e n o t i o n \ r e g i o n " h a s b e e n
u s e d f o r a n o p e n c o n n e c t e d s e t . W e d o n o t w i s h a p r i o r i t o i m p o s e s u c h a r e s t r i c t i o n
2004 by Chapman & Hall/CRC
7/29/2019 Chapter 2 Stability Domain
2/22
o n t h e l a r g e s t s e t o f i n i t i a l s t a t e s a p p r o p r i a t e f o r a c o r r e s p o n d i n g s t a b i l i t y p r o p e r t y .
T h e r e f o r e , w e s h a l l u s e t h e t e r m \ d o m a i n " i n g e n e r a l r a t h e r t h a n \ r e g i o n " . I n c a s e
a d o m a i n i s o p e n a n d c o n n e c t e d t h e n w e c a n a l s o c a l l i t a \ r e g i o n " .
2 . 2 . 2 D e n i t i o n s o f s t a b i l i t y d o m a i n s
T h e d e n i t i o n s o f s t a b i l i t y d o m a i n s w e r e i n t r o d u c e d t o c o m p l y w i t h t h e d e n i t i o n s
o f s t a b i l i t y o f a s t a t e a n d o f a s e t 6 4 ] , 6 5 ] , 6 8 ] , 6 9 ] , 7 0 ] , 7 2 ] .
D e n i t i o n 2 . 1 ( a ) T h e s t a t e X = 0 o f t h e s y s t e m ( 2 . 1 ) h a s t h e s t a b i l i t y d o m a i n
d e n o t e d b y D
s
i f a n d o n l y i f b o t h
( i ) f o r e v e r y " 2
7/29/2019 Chapter 2 Stability Domain
3/22
W e a r e n o w i n t e r e s t e d o n l y i n s t a b i l i t y o f X = 0 . L e t "
=
p
2 = 2 . F o r a n y
" 2 0 "
] , t h e m a x i m a l ( " ) d e n o t e d b y
M
( " ) ( D e n i t i o n 1 . 2 , S e c t i o n 1 . 1 . 1 ) o b e y s
M
( " ) = " . F o r a n y " 2 "
+ 1 t h e m a x i m a l
M
( " ) = "
H o w e v e r
k X ( t X
0
) k < " 8 t 2
>
>
>
>
:
B
"
" 2 0 "
B
"
\ S
" 2 "
S
" 2 + 1
w h e r e
S
=
X X 2
>
>
>
>
:
B
"
" 2 0 "
B
"
\ S
" 2 "
S
" 2 + 1
s o t h a t
D
s
=
D
s
( " ) " 2
7/29/2019 Chapter 2 Stability Domain
4/22
x 2
x 1
S
SeO
Ds() = B
x 2
x 1
Ds() = B
a ) b )
" 2 0 "
) D
s
( " ) = B
"
" = "
=
p
2
2
) D
s
( "
) = B
"
x 2
x 1
Ds( )
B
=
x 2
x1
Ds()
B=B
c ) d )
" 2 "
) D
s
( " ) = B
"
\ S
" = ) D
s
( ) = B
"
\ S
x2
x 1
Ds()=Ds
B
x 2
x 1
Ds=S
e ) f )
" 2 + 1 ) D
s
( " ) = S
D
s
= D
s
( " ) " 2
7/29/2019 Chapter 2 Stability Domain
5/22
x2
x1
A
2
11 4-1-4
-1
-2
F i g u r e 2 . 2 : T h e s t a t e p o r t r a i t o f t h e s y s t e m X = ( 1 ; x
1
; x
2
) ( 4 ; x
1
; 2 x
2
) X . T h e
s h a d e d a r e a t o g e t h e r w i t h i t s b o u n d a r y r e p r e s e n t s t h e s e t A
E x a m p l e 2 . 2 L e t t h e s e c o n d o r d e r s y s t e m ( 2 . 1 ) b e s p e c i e d b y
d X
d t
= ( 1 ; x
1
; x
2
) ( 4 ; x
1
; 2 x
2
) X
W e a r e i n t e r e s t e d i n s t a b i l i t y o f t h e s e t A
A =
X X 2
7/29/2019 Chapter 2 Stability Domain
6/22
x 2
x 1
A2
1
1 4-1-4
-1
-2
0
S4
N(,A)=Ds(,A)
]0,25]5
a)
x2
x1
2
1
1 4-1-4
-1
-2
0=
= 255
b)
N(25,A)=Ds(25,
A)5 5
=
25
5
x2
x1
2
1
1 4-1-4
-1
-2
0
N(,A)
c)
N(,A)S4=Ds(,A)
x 2
x 1
2
1
1 4-1-4
-1
-2
0
d)
S4=Ds(,A)=Ds(A)
N(,A)
F i g u r e 2 . 3 : T h e d e p e n d e n c e o f t h e s u b s e t D
s
( A ) o f D
s
( A ) o n 2 0 + 1 ( E x a m p l e 2 . 2 ) :
a ) D
s
( A ) = N ( A ) = N ( A ) \ S
4
2 0
2
p
5
5
b ) D
s
(
2
p
5
5
A ) = N (
2
p
5
5
A ) = N (
2
p
5
5
A ) \ S
4
=
2
p
5
5
c ) D
s
( A ) = N ( A ) \ S
4
2
2
p
5
5
3
d ) D
s
( A ) = S
4
2 3 + 1
H e n c e D
s
( A ) = D
s
( A ) 2 0 + 1 = S
4
w h e r e
S
4
=
X X 2
7/29/2019 Chapter 2 Stability Domain
7/22
3
2
1
01 2 3
-1
-2
-3
-1-2-3 x1
x2
F i g u r e 2 . 4 : T h e s t a t e p o r t r a i t o f t h e s y s t e m ( E x a m p l e 2 . 3 ) d e s c r i b e d v i a t h e p o l a r c o o r -
d i n a t e s = k X k = a r c t a n
x
2
x
1
b y = ; ( 1 ;
2
) ( 4 ;
2
) ( 9 ;
2
) = ; 1
s o t h a t t h e s t a b i l i t y d o m a i n D
s
( A ) o f t h e s e t A i s f o u n d a s D
s
( A ) = S
4
D
s
( A ) =
X X 2
7/29/2019 Chapter 2 Stability Domain
8/22
I t i s n o w o b v i o u s i n v i e w o f F i g . 2 . 4 t h a t
k X ( t X
0
) k < " i
8
>
>
>
>
>
>
>
>
>
:
k X
0
k < " " 2 0 1
k X
0
k 1 " 2 1 2
k X
0
k < " " 2 2 3
k X
0
k 3 " 2 3 + 1
T h i s m e a n s
D
s
( " ) =
8
>
>
>
>
>
>
>
>
>
:
B
"
" 2 0 1
B
1
" 2 1 2
B
"
" 2 2 3
B
3
" 2 3 + 1
w h i c h y i e l d s
D
s
= D
s
( " ) " 2 0 + 1 ] = ( B
"
" 2 0 1 ] ) ]
B
1
B
"
" 2 2 3
B
3
= B
1
B
1
B
3
B
3
=
B
3
H e n c e , t h e s t a b i l i t y d o m a i n D
s
o f X = 0 o f t h e s y s t e m e q u a l s c o m p a c t c i r c l e
B
3
D
s
=
B
3
= f X X 2
7/29/2019 Chapter 2 Stability Domain
9/22
x2
x10
F i g u r e 2 . 5 : T h e s t a t e p o r t r a i t o f t h e s y s t e m : X = ( ; + x
1
+ x
2
) X ( E x a m p l e 2 . 4 ) .
E x a m p l e 2 . 4 L e t t h e s y s t e m c o n s i d e r e d i n E x a m p l e 2 . 1 b e r e a n a l y z e d ,
d X
d t
= ( ; + x
1
+ x
2
) X 2
7/29/2019 Chapter 2 Stability Domain
10/22
x2
0 x1
Da
F i g u r e 2 . 6 : T h e a t t r a c t i o n d o m a i n D
a
o f X = 0 o f t h e s y s t e m : X = ( ; + x
1
+ x
2
) X
T h e y a r e i n t e r r e l a t e d s o t h a t D
a
i s t h e i n t e r i o r o f D
s
a n d D
s
i s t h e c l o s u r e o f D
a
h e n c e , D
a
i s s u b s e t o f D
s
D
a
=
D
s
D
s
=
D
a
D
s
D
a
E x a m p l e 2 . 5 L e t
( k X
0
k ) =
8
>
:
0 i k X
0
k 1
k X
0
k + 1
k X
0
k ; 1
i k X
0
k > 1
a n d t h e s y s t e m m o t i o n s b e d e n e d b y
X ( t X
0
) = e x p ( ; t ) 1 + ( X
0
) t X
0
T h e y s a t i s f y t h e i n i t i a l c o n d i t i o n , X ( 0 X
0
) X
0
, a n d a r e c o n t i n u o u s i n t 2
7/29/2019 Chapter 2 Stability Domain
11/22
||X(
t;X
0)||
1
1
1< ||X0
||^
2||X
0||+1
0t
||X0||+1
||X0||-1
||X0|| exp(- )2
||X0||+1
||X(t;X0)||
^
F i g u r e 2 . 7 : T h e n o r m o f m o t i o n s d e n e d b y X ( t X
0
) = 1 + ( X
0
) t X
0
e x p ( ; t ) ( X
0
) = 0
k X
0
k 1 a n d ( X
0
) =
k X
0
k + 1
k X
0
k ; 1
k X
0
k > 1 ( E x a m p l e 2 . 5 ) .
B ) k X
0
k > 1 y i e l d s k X ( t X
0
) k = e x p ( ; t )
1 +
k X
0
k + 1
k X
0
k ; 1
t
k X
0
k . I n t h i s c a s e ,
m a x k X ( t X
0
) k t 2
7/29/2019 Chapter 2 Stability Domain
12/22
D
s c
( " ) =
8
7/29/2019 Chapter 2 Stability Domain
13/22
x2
2
A
4x1
- 4
- 2
1
- 1
- 1 10
D(A)
F i g u r e 2 . 8 : T h e a t t r a c t i o n d o m a i n D
a
( A ) o f t h e s e t A = f X X 2
7/29/2019 Chapter 2 Stability Domain
14/22
x2
x1
A
Da(A)
2
-2
-2
1
-1-1 1
0
-3 2 3
-3
3
a)
x2
x1
A
Ds(A)
2
-2
-2
1
-1-1 1
0
-3 2 3
-3
3
b)
F i g u r e 2 . 9 : a ) T h e d o m a i n D
a
( A ) o f a t t r a c t i o n o f t h e s e t A = f x x 2
7/29/2019 Chapter 2 Stability Domain
15/22
I n v i e w o f D e n i t i o n 2 . 5 , t h e s t a t e X = 0 o f t h e s y s t e m h a s a l s o t h e a s y m p t o t i c
s t a b i l i t y d o m a i n D
D = D
s
\ D
a
=
X X 2
7/29/2019 Chapter 2 Stability Domain
16/22
D e n i t i o n 2 . 6 ( a ) A s e t A
7/29/2019 Chapter 2 Stability Domain
17/22
o b e y t h e e s t i m a t e
k X ( t X
0
) k k X
0
k e x p ( ; t ) 8 t 2
7/29/2019 Chapter 2 Stability Domain
18/22
C o m m e n t 2 . 4 T h e s t a t e X = 0 o f t h e s y s t e m
d X
d t
= ( ; + x
1
+ x
2
) X 2
7/29/2019 Chapter 2 Stability Domain
19/22
2 . 2 . 6 D e n i t i o n s o f a s y m p t o t i c s t a b i l i t y d o m a i n s o n N
( )
( )
L e t t h e s y s t e m ( 2 . 1 ) b e o f t h e L u r i e f o r m ( S e c t i o n 1 . 1 . 5 ) ,
d X
d t
= A X + B f ( w ) ( 2 . 2 a )
w = C X + D f ( w ) ( 2 . 2 b )
I n o r d e r t o e x p l a i n t h e n e e d f o r t h e s t u d y o f d o m a i n s o f a s y m p t o t i c s t a b i l i t y
o n N
i
( ) w e p r e s e n t t h e f o l l o w i n g s i m p l e e x a m p l e .
E x a m p l e 2 . 1 3 L e t n = 1 a n d
d X
d t
= ; s i n X :
T h i s s y s t e m h a s i n n i t e l y m a n y e q u i l i b r i u m p o i n t s l o c a t e d a t X = k , w h e r e k
i s a n y i n t e g e r . H e n c e , X = 0 o b v i o u s l y i s n o t a s y m p t o t i c a l l y s t a b l e i n t h e l a r g e .
H o w e v e r , i t i s a s y m p t o t i c a l l y s t a b l e w i t h t h e d o m a i n o f a s y m p t o t i c s t a b i l i t y D =
; . O v e r S = D t h e n o n l i n e a r i t y f f ( X ) = ; s i n X , b e l o n g s t o t h e f a m i l y
N
1
( L M S ) f o r L = 0 1 ] a n d M = ; 1 1 . I f t h e s y s t e m i s e m b e d d e d i n t o t h e c l a s s
o f L u r i e s y s t e m s ( 2 . 2 ) , t h e n w e c a n s p e a k o n l y a b o u t a s y m p t o t i c s t a b i l i t y o f X = 0
f o r a p a r t i c u l a r f ( ) o r f o r a n y f 2 N
1
( L M S ) , o r f o r a n y f 2 N
0
( L S ) . T h i s
m e a n s t h a t w e c a n l o o k o n l y f o r t h e a s y m p t o t i c s t a b i l i t y d o m a i n f o r a p a r t i c u l a r f
e . g . f ( X ) = ; s i n X , o r f o r e v e r y f 2 N
1
( L M S ) , o r f o r e v e r y f 2 N
0
( L S )
L e t D
f
d e n o t e t h e a s y m p t o t i c s t a b i l i t y d o m a i n o f X = 0 o f t h e ( L u r i e )
s y s t e m ( 2 . 2 ) f o r a p a r t i c u l a r n o n l i n e a r i t y f
D e n i t i o n 2 . 9 T h e s t a t e X = 0 o f t h e s y s t e m ( 2 . 2 ) h a s t h e s t r i c t ] a s y m p t o t i c
s t a b i l i t y d o m a i n o n N
i
( L M S ) , w h i c h i s d e n o t e d b y D
i
( L M S ) D
i c
( L M S ) ]
i f a n d o n l y i f
a ) i t h a s t h e s t r i c t ] a s y m p t o t i c s t a b i l i t y d o m a i n D
f
D
f
c
] f o r e v e r y f 2
N
i
( L M S )
a n d
b ) D
i
( L M S ) = \ D
f
f 2 N
i
( L M S ) i s a n e i g h b o u r h o o d o f X = 0
D
i c
( L M S ) = \ D
f
c
f 2 N
i
( L M S ) i s a c o n n e c t e d n e i g h b o u r h o o d
o f X = 0 ] , r e s p e c t i v e l y .
T h i s d e n i t i o n w a s i n t r o d u c e d i n 6 8 ] , 6 9 ] . I t c a n b e e x t e n d e d t o s e t s a s f o l l o w s :
D e n i t i o n 2 . 1 0 A s e t A
7/29/2019 Chapter 2 Stability Domain
20/22
a n d
b ) D
i
( L M A S ) = \ D
f
f 2 N
i
( L M A S ) i s a n e i g h b o u r h o o d o f t h e
s e t A D
i c
= \ D
f
c
f 2 N
i
( L M A S ) i s a c o n n e c t e d n e i g h b o u r h o o d o f
t h e s e t A ] , r e s p e c t i v e l y .
2 . 3 D o m a i n s o f p r a c t i c a l s t a b i l i t y p r o p e r t i e s
2 . 3 . 1 D e n i t i o n s o f d o m a i n s o f p r a c t i c a l s t a b i l i t y
B y f o l l o w i n g 7 0 ] a n d S e c t i o n 1 . 2 . 2 w e a c c e p t t h e f o l l o w i n g d e n i t i o n f o r t h e
s y s t e m ( 2 . 3 ) :
d X
d t
= f ( X i ) f
7/29/2019 Chapter 2 Stability Domain
21/22
2 . 3 . 2 D e n i t i o n s o f d o m a i n s o f p r a c t i c a l c o n t r a c t i o n
w i t h s e t t l i n g t i m e
A s f o r p r a c t i c a l s t a b i l i t y d o m a i n s , w e r s t i n t r o d u c e t h e n o t i o n o f t h e d o m a i n o f
p r a c t i c a l c o n t r a c t i o n w i t h s e t t l i n g t i m e f o r t h e s y s t e m ( 2 . 3 ) ( s e e S e c t i o n 1 . 2 . 3 ) .
D e n i t i o n 2 . 1 3 T h e s y s t e m ( 2 . 3 ) h a s t h e d o m a i n o f p r a c t i c a l c o n t r a c t i o n w i t h t h e
s e t t l i n g t i m e
s
w i t h r e s p e c t t o f X
F
I g , w h i c h i s d e n o t e d b y D
p c
(
s
X
F
I ) i f
a n d o n l y i f b o t h
a ) i t s m o t i o n s o b e y
X ( t X
0
i ) 2 X
F
f o r e v e r y ( t i ) 2 T
s
I
p r o v i d e d o n l y t h a t X
0
2 D
p c
(
s
X
F
I )
a n d
b ) t h e i n t e r i o r
D
p c
(
s
X
F
I ) o f D
p c
(
s
X
F
I ) i s n o n - e m p t y .
W h e n
s
X
F
a n d I a r e p r e s p e c i e d t h e n w e m a y w r i t e D
p c
i n s t e a d
o f D
p c
(
s
X
F
I )
D e n i t i o n 2 . 1 4 A s e t A ( o f s t a t e s o f t h e s y s t e m ( 2 . 3 ) ) h a s t h e d o m a i n o f p r a c t i c a l
c o n t r a c t i o n w i t h t h e s e t t l i n g t i m e
s
w i t h r e s p e c t t o f X
F
I g , w h i c h i s d e n o t e d
b y D
p c
(
s
X
F
I A ) , i f a n d o n l y i f b o t h
a ) t h e s y s t e m m o t i o n s o b e y
X ( t X
0
i ) 2 X
F
f o r e v e r y ( t i ) 2 T
s
I
p r o v i d e d o n l y t h a t X
0
2 D
p c
(
s
X
F
I A )
a n d
b ) D
p c
(
s
X
F
I A ) i s a n e i g h b o u r h o o d o f t h e s e t A
W h e n
s
X
F
a n d I a r e k n o w n t h e n w e m a y r e p l a c e D
p c
(
s
X
F
I A )
b y D
p c
( A )
C o m m e n t 2 . 6 I f w e a r e i n t e r e s t e d i n t h e d o m a i n o f p r a c t i c a l c o n t r a c t i o n o f a
s t a t e X
t h e n w e m a y u s e D e n i t i o n 2 . 1 4 w i t h A = f X
g
2 . 3 . 3 D e n i t i o n s o f d o m a i n s o f p r a c t i c a l s t a b i l i t y
w i t h s e t t l i n g t i m e
I n v i e w o f t h e p r e c e d i n g d e n i t i o n a n d t h e n o t i o n o f p r a c t i c a l s t a b i l i t y w i t h s e t t l i n g
t i m e ( S e c t i o n 1 . 2 . 4 ) w e a c c e p t t h e f o l l o w i n g d e n i t i o n :
2004 by Chapman & Hall/CRC
7/29/2019 Chapter 2 Stability Domain
22/22
D e n i t i o n 2 . 1 5 T h e s y s t e m ( 2 . 3 ) h a s t h e d o m a i n o f p r a c t i c a l ( c o n t r a c t i v e ) s t a -
b i l i t y w i t h t h e s e t t l i n g t i m e
s
w i t h r e s p e c t t o f X
A
X
F
I g , w h i c h i s d e n o t e d b y
D
p
(
s
X
A
X
F
I ) , i f a n d o n l y i f
a ) b o t h
1 ) X ( t X
0
i ) 2 X
A
f o r e v e r y ( t i ) 2
Top Related