Image Processing Reference
In-Depth Information
On the other hand, suppose that we have a transfer function at first:
>> z=tf('z',1);
>> Gz=(z^2+2 * z+1)/(z^2+0.5 * z+1);
Thefirstcommanddefinesthevariable z of Z -transformwith sampling time 1, and the second
command defines the following transfer function:
z 2
+
2 z
+
1
1 .
To convert this to state-space matrices A , B , C ,and D , use the command ss as follows:
G
(
z
)=
z 2
+
+
0.5 z
>> ss(Gz)
a=
x1
x2
x1
-0.5
-1
x2
1
0
b=
u1
x1
1
x2
0
c=
x1
x2
y1
1.5
0
d=
u1
y1
1
Sampling time (seconds): 1
Discrete-time model.
These outputs shows that the state-space matrices are given by
, B
1
0
, C
1.5 0 , D
0.5
1
10
A
=
=
=
=
1,
with sampling time 1.
Note that the state-space representation in Example 1 is minimal in that the state-space model
describes the same input/output behavior with the minimum number of states. Such a system
is called minimal realization (Rugh, 1996).
We then introduce a useful notation, called packed notation (Vidyasagar, 1988), describing the
transfer function from state-space matrices as
: A B
C D
) 1 B
G
(
z
)=
C
(
zI
A
+
D
=
(
z
)
.
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