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```Eigenvalues, Zeros and Poles
x  Ax  Bu
y  Cx  Du ,
x R ,
x ( 0 )  x0
yR ,
n
(1)
r
uR
m
Definition: Roots of the characteristic polynomial of a system are
called eigenvalues of the system.
characteristic
p (  )  det(  I  A )  (   1 )(    2 )...(    n )
polynomial
eigenvalues  i , i  1,2,..., n .
X ( s )  [ sI  A ]
n
x (t ) 
  ie
i 1
it
,
1
x0 
adj ( sI  A )
det[ sI  A ]
 i ˆ R i x 0
nx1
n
x0 

i 1
nxn
Ri
( s  i )
x0
Transfer function of the system (1):
H (s) 
Y (s)
 C ( sI  A )
1
B  D
U (s)

C [ adj ( sI  A )] B  D [det( sI  A )]
[det( sI  A )]
In general H ( s ) is a matrix. H ij ( s ) is then the transfer function from the
input U j ( s ) to the output Y i ( s ) .
After cancelling the common terms in nominator and denominator,
one gets
W (s)
H ij ( s ) 
 (s)
Definition:
Poles of H ij ( s ) (or the system (1)) are the roots of  ( s ).
Zeros of H ij ( s ) (or the system (1)) are the roots of W ( s ).
Result: Poles of a system are a subset of system eigenvalues.
Stability
x ( t )  Ax ( t )  Bu ( t )
y ( t )  Cx ( t )  Du ( t )
Zero input stability:
x ( t )  Ax ( t ),
x (t 0 )  x 0
(2)
Definition: (Equilibrium) A constant solution x ( t )  x d , t  t 0 of the
system (2) is called an equilibrium of the system x  f ( x ) .
How to find x d ?
What is x d for
linear systems?
Solve 0  f ( x d ) !
If A is invertible then x d is zero. Otherwise, there
are infinitely many equilibria. Solve 0  Ax d .
Definition (Norm): Let V be a vector space (a space with appropriate
addition and multiplication by scalars operations). Norm of a vector is
defined as a positive valued function that satisfies.
x 0
. :V  R


x0
x   x
x  y  x  y ,  x, y  V
Generalization of
the notion ...........
Definition: (Lyapunov stability)
Let x d be an equilibrium of the system x ( t )  f ( x ( t ))
The equilibrium is called Lyapunov stable if for every arbitrary small
  0 , there exists a  (  ) such that
x ( t 0 )  x d   ( , t 0 )

x (t )  x d   ,  t  t 0
Definition: (Asymptotic stability)
Let x d be a Lyapunov stable equilibrium of the system x ( t )  f ( x ( t ))
Then x d is asymptotically stable if there exists a   0 such that
x(t0 ) - xd < d

lim x ( t )  x d  0 ,  t  t 0
t 
Theorem:
The zero equilibrium of the system x ( t )  Ax ( t ) is
Lyapunov stable if and only if
 (t , t 0 )  c ,
Proof:  (if)
Solution:
 t  t0
x (t )   (t , t 0 ) x 0 ,  t  t 0
x (t )   (t , t 0 ) x 0   (t , t 0 )
 (t , t 0 )  c ,
Norm property
 t  t0 
x (t )   (t , t 0 ) x 0   (t , t 0 )
 (  , t 0 ) ˆ
 (only if)
x0


c
x0  c x0
x (t )  
Let x d  0 be Lyapunov stable but  ( t , t 0 ) not bounded.
Then there exist a  ji ( t , t 0 ) which is not bounded. Choose
x 0  0
...
0
i.
1
0
...
0
T
Then,
 x ( t )   ( t , t 0 ) x 0   ji ( t , t 0 ) 
 ji ( t , t 0 )  c  x ( t )  
 (t , t 0 )  c
Theorem:
The zero equilibrium of the system x ( t )  Ax ( t ) is
Lyapunov stable if and only if lim  ( t , t 0 )  0 ,  t  t 0
t 
Proof: Similar to the previous proof.
Theorem:
1)
For the system
let the
x ( t )  Ax ( t ) ,
eigenvalues of the system be  i , i  1, 2 ,... n . Then,
x ( t )  Ax ( t ) is Lyapunov stable
 Re  i  0 , i  1,2,...., n
and eigenvalues with
multiplicity one.
2)
Re  i  0
have
x ( t )  Ax ( t ) is asymptotically stable  Re  i  0 , i  1,2,...., n
x ( t )  Ax ( t )  Bu ( t ),
x (t 0 )  0
y ( t )  Cx ( t )  Du ( t )
Definition: (BIBO sability) The system
is bounded input
bounded output (BIBO) stable if and only if output of the system is
bounded for all bounded inputs.
Theorem:
is BIBO stable
 All poles of
have negative real part.
Theorem:
is asymptotically stable

is BIBO stable.
Example:
Find the equilibria and analyze the stability for the following system!
é
2
-x
x
+
x
2 1
1 -1
ê
x=
ê
-x2 x1
ë
ù
ú
ú
û
Example:
Find the transfer function
for the following system!
Is the system
a) Lyapunov stable?
b) asymptotically stable?
c) BIBO stable?
 2

x t   0

 0
2
3
0
1

1  x t  

1
y t   1 1 1  x t 
1 
 
1  u t 
 
 0 
```
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