T
H
R
E
E
Modeling
in the Time Domain
SOLUTIONS TO CASE STUDIES CHALLENGES
Antenna Control: State-Space Representation
For the power amplifier,
E
a
(s)
V
p
(s)
=
150
s+150
. Taking the inverse Laplace transform, e
a
.
+150e
a
=
150v
p
.
Thus, the state equation is
e
a
•
= −
150e
a
+
150v
p
For the motor and load, define the state variables as x
1
=
θ
m
and x
2
=
θ
.
m
. Therefore,
x
.
1
= x
2
(1)
Using the transfer function of the motor, cross multiplying, taking the inverse Laplace transform,
and using the definitions for the state variables,
x
.
2
= -
1
J
m
(D
m
+
K
t
K
a
R
a
) x
2
+
K
t
R
a
J
m
e
a
(2)
Using the gear ratio, the output equation is
y = 0.2x
1
(3)
Also, J
m
= J
a
+5(
1
5
)
2
= 0.05+0.2 = 0.25, D
m
= D
a
+3(
1
5
)
2
= 0.01+0.12 = 0.13,
K
t
R
a
J
m
=
1
(5)(0.25)
= 0.8, and
1
J
m
(D
m
+
K
t
K
a
R
a
)
= 1.32. Using Eqs. (1), (2), and (3) along with the previous values, the
state and output equations are,
x
.
=
0
1
0
-1.32
x
+
0
0.8
e
a
; y =
0.2
0
x

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