Lesson 8
Stability
linear system by meiling CHEN
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Stability
• Asymptotically stable
– Consider a system represent in state space:
x  Ax x(0)  x0
if
x(t )  0 as t  
• Bounded-Input Bounde-Output stable
– (Input-output stability)
u(t )  N    y(t )  M  
for all bounded input
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Stability condition
• Asymptotically stable
• All the eigenvalues of the system have
negative real parts (i.e. in the LHP)
• BIBO stable
• All the transfer function poles be in the
LHP
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N ( s)
Cadj[sI  A]B
1
T ( s) 
 C ( sI  A) B 
D( s )
sI  A
D( s)  0
Find all the transfer function poles
Characteristic function of the system
sI  A  0
Find all the system eigenvalue
In the absence of pole-zero cancellations, transfer function
poles are identical to the system eigenvalues, hence BIBO
stability and asymptotically stability are equivalent.
If the system is controllable and observable, then BIBO
stable is equal to asymptotically stable.
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Stability testing method
• Asymptotically stable
• BIBO stable
• Routh-Hurwitz criterion
• Root locus method
• Nyquist criterion
• ....etc.
linear system by meiling CHEN
5
Lyapunov stability
A state xe of an autonomous system is called an equilibrium
state, if starting at that state the system will not move from it in
the absence of the forcing input.
x  f ( x(t ), u (t ), t )
f ( xe ,0, t )  0,
example
1  1
0
x  
x   u (t )

 2  3 1
let
t  t0
x2
Equilibrium point
u (t )  0
x1
1   x1e 
 x1e  0
0
 2  3  x   0   x   0

  2e 
 2e   
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example
0 0   2 
x  
x   u (t )

0  2  1 
let
u (t )  0
 x1e  k 
0 0   x1e 
0  2   x   0   x    0 

  2e 
 2e   
x2
x1
Equilibrium line
linear system by meiling CHEN
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Definition: An equilibrium state xe of an autonomous system is
stable in the sense of Lyapunov if for every   0 , exist a ( )  0
such that x0  xe    x(t , x0 )  xe   for t  t0
x2


xe
x0
linear system by meiling CHEN
x1
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Definition: An equilibrium state xe of an autonomous system is
asymptotically stable if
(i) It is stable and
(ii) exist  a  if x0  xe   a  x(t )  xe  0, as t  
x2

a
xe

x0
linear system by meiling CHEN
x1
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Asymptotically stable in the large
( globally asymptotically stable)
(1) The system is asymptotically stable for all the initial states x(t0 ) .
(2) The system has only one equilibrium state.
(3) For the LTI system, asymptotically stable and globally
asymptotically stable are equivalent.
linear system by meiling CHEN
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Lyapunov’s function
A function V(x) is called a Lyapunov function if
v ( x )
(i) V(x) and x are continuous in a region R containing the
i
origin (E.S.)
(ii) V(x) is positive definite in R.
(iii) relative to a system x  f (x) V (x) along the trajectory of
the system is negative semi-definite in R.
A scalar function V(x) is positive (negative) definite if
(i) V(0)=0
(ii) V(x)>0 (<0) for x  0
A scalar function V(x) is positive (negative) semidefinite if
(i) V(0)=0 and V(x)=0 possibly at some x  0
(ii) V ( x)  0( 0)
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A function is not definite or semidefinite in either sense
is defined to be indefinite.
example
V1 ( x1 , x2 , x3 )  x12  2 x22  2 x1 x2  x32
V1 ( x1 , x2 , x3 )  ( x1  x2 ) 2  x22  x32
positive definite
V2 ( x1 , x2 , x3 )  x14  x24  x32
positive definite
V3 ( x1 , x2 , x3 )  x12  x22  3x32  x34
if
x3  3
V3 ( x1 , x2 , x3 )  x12  x22  x32 (3  x32 )
if
x3  3 p.s.d.
linear system by meiling CHEN
p.d.
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V4 ( x1 , x2 , x3 )  ( x1  x2 ) 2  x32
V5 ( x1 , x2 , x3 )  x1  x2  x32
p.s.d.
indefinite
Quadratic form
n
n
V ( x)   aij xi x j
i 1 j 1
 x1
x2
 a11 a12  a1n   x1 
a
x 
21
2



 xn 
 
  

 
an1 an 2  ann   xn 

 X T AX
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 a11 a12  a1n 
a

 21

 



an1 an 2  ann 
1
1


a
(
a

a
)

(
a

a
)
11
12
21
1n
n1 

2
2
1

 (a12  a21 )
 Symmetry
 2
 matrix Q



1
1

ann
 2 (a1n  an1 ) 2 (a2 n  an 2 ) 

1
Q  ( A  AT )
2
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Q: symmetric matrix V ( x)  xT Qx then
(1) Q is p.d.
(2)Q is n.d.
V(x) is p.d.
V(x) is n.d.
(3)Q is p.s.d.
V(x) is p.s.d.
(4)Q is n.s..d.
V(x) is n.s..d.
(5)Q is indefinite.
V(x) is indefinite.
(6) Q is p.d.
eigenvalues of Q are positives
(7) Q is n.d.
eigenvalues of Q are negatives
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Sylvester’s criterion
A symmetric n n matrix Q is p.d. if and only if all its n
leading principle minors are positive.
Definition
The i-th leading principle minor Qi i  1,2,3,, n of an n n
matrix Q is the determinant of the i  i matrix extracted from
the upper left-hand corner of Q.
 q11
Q  q21
 q31
q11
Q2 
q21
q12
q22
q32
q21
q22
q13 
q23 
q33 
Q1  q11
Q3  Q
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Remark
(1) Q1 , Q2 , Qn are all negative
Q is n.d.
(2) All leading principle minors of –Q are positive
Q is n.d.
example
V ( x)  2 x12  4 x1 x3  3x12  6 x2 x3  x32
 x1
 x1
x2
x2
2
x3 0
0
2
x3 0
2
0 4  x1 
3 6  x2 
0 1  x3 
0 2  x1 
3 3  x2 
3 1  x3 
Q1  2  0
Q2  6  0
Q3  24  0
Q is not p.d.
linear system by meiling CHEN
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Lyapunov’s method
Consider the system x  f ( x), f (0)  0
If in a neighborhood R about the origin a Lyapunov
function V(x) can be found such that V (x) is n.d. along the
trajectory then the origin is asymptotically stable.
Consider linear autonomous system
x  Ax if
x  0  E.S.
A 0
Let Lyapunov function V ( x)  xT px
 V ( x)  x T Px  xT Px
 xT AT Px  xT PAx
 Q  ( AT P  PA)
If Q is p.d. then V (x) is n.d.
 xT ( AT P  PA) x
  xT Qx
x  0 is asymptotically stable
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example
let
0 1
x  
x

 1  1
E.S.  x  0
QI
AT P  PA   I
0  1  p11 p12   p11 p12   0 1   1 0 




1  1  p



p
p
p

1

1
0

1

  12
 

22 
22  
 12
 p11 p12  1 3 1 
P
 


p
p
1
2
2


22 
 12
p11  3  0
P 50
P is p.d.
System is asymptotically stable
The Lyapunov function is:
V ( x)  xT Px  1 (3x12  2 x1 x2  2 x22 )
2
V ( x)  ( x12  x22 )
linear system by meiling CHEN
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