Image Processing Reference
In-Depth Information
Let us begin with a linear system
G
represented in the following state-space equation or
state-space representation (Rugh, 1996):
: x
[
k
+
1
]=
Ax
[
k
]+
Bu
[
k
]
,
G
(1)
[
]=
[
]+
[
]
=
y
k
Cx
k
Du
k
, k
0,1,2,....
n is called the state
[
] R
The nonnegative number k denotes the time index. The vector x
k
[
] R
[
] R
G
vector, u
k
is the input and y
k
is the output of the system
. The matrices A
n
×
n , B
n
×
1 , C
1
×
n ,and D
R
R
R
R
are assumed to be static, that is, independent of the
(
)
G
time index k . Then the transfer function G
z
of the system
is defined by
) 1 B
(
)
=
(
+
C
G
z
:
C
zI
A
D ,
z
.
(
)
The transfer function G
z
is a rational function of z of the form
b n 1 z n 1
b n z n
+
+ ··· +
b 1 z
+
b 0
G
(
z
)=
.
z n
+
a n 1 z n 1
+ ··· +
+
a 1 z
a 0
] } k = 0 of the system
Note that G
(
z
)
is the Z -transform of the impulse response
{
g
[
k
G
with the
initial state x
[
0
]=
0, that is,
k = 0 g [ k ] z k
k = 1 CA k 1 Bz k .
G
(
z
)=
=
D
+
To convert a state-space equation to its transfer function, one can use the above equations or
the MATLAB command tf . On the other hand, to convert a transfer function to a state-space
equation, one can use realization theory which provides a method to derive the state space
matrices from a given transfer function (Rugh, 1996). An easy way to obtain the matrices is to
use MATLAB or Scilab with the command ss .
Example 1. We here show an example of MATLAB commands. First, we define state-space matrices:
>A=[0,1;-1,-2]; B=[0;1]; C=[1,1]; D=0;
>G=ss(A,B,C,D,1);
This defines a state-space (ss) representation of
G
with the state-space matrices
01
, B
0
1
, C
11 , D
A
=
=
=
=
0.
1
2
The last argument 1 of ss sets the sampling time to be 1.
To obtain the transfer function G
) 1 B
(
z
)=
C
(
zI
A
+
D ,wecanusethecommand tf
>> tf(G)
Transfer function:
z+1
-------------
z^2 + 2 z + 1
Sampling time (seconds): 1
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