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
)
.