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