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ENSC 483 HW#5 Solutions by M. Saif
1
Simon Fraser University
School of Engineering Science
ENSC 483  Modern Control Systems
1.
Electrical circuit–
Obtain the state space description of the network shown in Figure 1. Use the
voltages across the capacitors and the current through the inductor as the state variables. Note that
v
1
is the input and
v
2
is the output.
Figure 1: An electrical network
Solution:
dv
C
1
dt
=
1
C
1
i
C
1
dv
C
2
dt
=
1
C
2
i
C
2
di
L
dt
=
1
L
v
L
R
1
i
R
1
+
v
C
1
=
v
1
R
2
i
R
2
+
R
3
i
L
+
v
L
=
v
C
1
R
3
i
L
+
v
L
=
v
C
2
v
2
=
v
C
2
i
R
1
=
i
C
1
+
i
R
2
i
R
2
=
i
L
+
i
i
=
i
C
2
Now if we take
v
C
1
;
v
C
2
, and
i
L
as our state variables, we get:
˙
v
C
1
˙
v
C
2
˙
i
L
=

1
C
1
‡
1
R
1
+
1
R
2
·
1
C
1
R
2
0
1
C
2
R
2

1
C
2
R
2

1
C
2
0
1
L

R
3
L
v
C
1
v
C
2
i
L
+
1
R
1
C
1
0
0
v
1
v
2
= [0 1 0]
v
C
1
v
C
2
i
L
2.
MassSpring Systems
– Obtain the diﬀerential equations for the system shown in Figure 2. Draw a
block diagram for this system and ﬁnd the transfer function
H
(
s
) =
X
2
(
s
)
F
(
s
)
. Write the state variable
formulation of this system using the state variables:
z
1
=
x
1
,z
2
= ˙
x
1
,z
3
=
x
2
,z
4
= ˙
x
2
.
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2
Figure 2: A massspring system.
Figure 3: Electrical analogue of the massspring system
ENSC 483 HW#5 Solutions by M. Saif
3
Solution:
Draw the electrical analog of the massspring system as in Figure 3. From it we get,
‰
M
1
s
2
X
1
(
s
) + (
Ds
+
K
1
)(
X
1

X
2
) =
U
(
s
)
M
2
s
2
X
2
(
S
) + (
D
1
s
+
K
1
)(
X
2

X
1
) +
K
2
X
2
= 0
The block diagram representation of the system in terms of the input
U
(
s
)
, and output
X
2
(
s
)
is shown
in Figure 4. The transfer function of it is
H
(
s
) =
Ds
+
K
1
M
1
M
2
s
4
+ (
M
1
+
M
2
)
Ds
3
+ (
M
1
K
1
+
M
2
K
1
+
M
1
K
2
)
s
2
+
K
2
Ds
+
K
1
K
2
Finally, the state space representation of the system is
˙
z
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This note was uploaded on 10/09/2009 for the course ENSC 1166 taught by Professor Mehrdadsaif during the Spring '07 term at Simon Fraser.
 Spring '07
 MehrdadSaif

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