Digital Signal Processing Reference
In-Depth Information
Example 8.13 gives two state models for the same system. If a different trans-
formation matrix
P
(that has an inverse) had been chosen, a third model would re-
sult. In fact, for each different transformation
P
that has an inverse, a different state
model results. Thus, an unlimited number of state models exists for a given system
transfer function. The choice of the state model for a given system can be based on
the natural state variables (position, velocity, current, voltage, etc.), on ease of
analysis and design, and so on.
To check the state model developed in Example 8.13, we now derive the trans-
fer function.
EXAMPLE
8.14
Transfer function for the system of Example 8.13
The transformed state equations of Example 8.13 are given by
v
#
(t) =
A
v
(t) +
B
v
u(t) =
-20
-3
1
2
B
R
B
R
v
(t) +
u(t)
-1
and
y(t) =
C
v
v
(t) = [3
1]
v
(t).
From (8.56), the transfer function of this system is given by
H
v
(s) = C
v
(s
I
-
A
v
)
-1
B
v
.
(s
I
-
A
v
)
-1
.
First, we calculate
Now,
10
01
-20
-3
s + 2
0
s
I
-
A
v
= s
B
R
-
B
R
=
B
R
.
-1
3
s + 1
Therefore,
det (s
I
-
A
) = (s + 2)
(s + 1) = s
2
+ 3s + 2.
Then, letting
det (s
I
-
A
) =¢(s)
for convenience, we have
≥
s
+
1
0
adj (s
I
-
A
)
det (s
I
-
A
)
=
¢(s)
(s
I
-
A
)
-1
¥
=
,
-3
¢(s)
s
+
2
¢(s)
and the transfer function is given by
H
v
(s) =
C
v
(s
I
-
A
v
)
-1
B
v
≥
s
+
1
0
¢(s)
1
2
= [3
1]
¥
B
R
-3
¢(s)
s
+
2
¢(s)
