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LQRegulatorGains
AffineStateSpaceModel
OutputResponse
StateResponse
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Control Systems
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System Modeling
Levitate a Ball about a Nominal Position
Design a control system for an electromagnet
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The magnetic field generated by the electromagnet is controlled by the voltage input v.
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:
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c
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s
t
a
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m
b
a
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k
m
a
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c
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t
a
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r
a
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c
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a
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x
p
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b
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I
n
[
1
]
:
=
e
q
n
s
=
m
′
′
x
[
t
]
g
m
-
k
2
i
[
t
]
2
x
[
t
]
,
v
[
t
]
r
i
[
t
]
+
l
′
i
[
t
]
;
A set of numerical values for the model parameters:
I
n
[
2
]
:
=
p
a
r
s
=
{
r
1
0
,
m
0
.
0
5
,
k
1
,
l
0
.
0
5
,
g
9
.
8
,
x
0
0
.
5
}
;
Solve for the value of
x
0
=
0
.
5
:
I
n
[
3
]
:
=
i
0
=
x
0
m
g
k
,
v
0
=
r
x
0
m
g
k
/
.
p
a
r
s
O
u
t
[
3
]
=
{
0
.
3
5
,
3
.
5
}
An affine state space model of the system:
I
n
[
4
]
:
=
a
s
s
m
=
A
f
f
i
n
e
S
t
a
t
e
S
p
a
c
e
M
o
d
e
l
[
e
q
n
s
,
{
{
x
[
t
]
,
x
0
}
,
x
'
[
t
]
,
{
i
[
t
]
,
i
0
}
}
,
{
{
v
[
t
]
,
v
0
}
}
,
x
[
t
]
,
t
]
/
.
p
a
r
s
O
u
t
[
4
]
=
{
x
[
t
]
,
0
.
5
}
x
.
1
[
t
]
0
x
.
1
[
t
]
2
0
.
0
.
4
9
-
2
i
[
t
]
2
x
[
t
]
0
{
i
[
t
]
,
0
.
3
5
}
-
2
0
.
(
-
3
.
5
+
1
0
i
[
t
]
)
2
0
.
x
[
t
]
0
The balls falls down without feedback control:
I
n
[
5
]
:
=
O
u
t
p
u
t
R
e
s
p
o
n
s
e
[
{
a
s
s
m
,
{
R
a
n
d
o
m
I
n
t
e
g
e
r
[
{
1
,
1
0
}
]
,
0
,
0
}
}
,
0
,
{
t
,
0
,
5
}
]
;
P
l
o
t
[
%
,
{
t
,
0
,
5
}
]
O
u
t
[
5
]
=
Set the voltage as the sole feedback input:
I
n
[
6
]
:
=
s
s
p
e
c
=
"
I
n
p
u
t
M
o
d
e
l
"
A
f
f
i
n
e
S
t
a
t
e
S
p
a
c
e
M
o
d
e
l
[
a
s
s
m
,
A
u
t
o
m
a
t
i
c
,
A
u
t
o
m
a
t
i
c
,
A
u
t
o
m
a
t
i
c
,
N
o
n
e
]
,
"
F
e
e
d
b
a
c
k
I
n
p
u
t
s
"
1
O
u
t
[
6
]
=
I
n
p
u
t
M
o
d
e
l
{
x
,
0
.
5
}
x
.
1
0
x
.
1
2
0
.
0
.
4
9
-
2
i
2
x
0
{
i
,
0
.
3
5
}
-
2
0
.
(
-
3
.
5
+
1
0
i
)
2
0
.
x
0
,
F
e
e
d
b
a
c
k
I
n
p
u
t
s
1
Compute the regulator controller:
I
n
[
7
]
:
=
=
L
Q
R
e
g
u
l
a
t
o
r
G
a
i
n
s
s
s
p
e
c
,
0
.
1
0
0
0
0
.
1
0
0
0
0
.
1
,
{
{
5
}
}
,
"
D
a
t
a
"
O
u
t
[
7
]
=
S
y
s
t
e
m
s
M
o
d
e
l
C
o
n
t
r
o
l
l
e
r
D
a
t
a
D
e
s
i
g
n
:
l
i
n
e
a
r
q
u
a
d
r
a
t
i
c
r
e
g
u
l
a
t
o
r
»
F
e
e
d
b
a
c
k
i
n
p
u
t
s
c
o
u
n
t
:
1
Obtain the closed-loop system:
I
n
[
8
]
:
=
c
s
y
s
=
[
"
C
l
o
s
e
d
L
o
o
p
S
y
s
t
e
m
"
]
/
/
S
i
m
p
l
i
f
y
O
u
t
[
8
]
=
{
x
,
0
.
5
}
x
.
1
x
.
1
9
.
8
-
2
0
.
2
i
2
x
{
i
,
0
.
3
5
}
-
7
0
.
0
2
1
3
-
2
1
2
.
5
6
6
i
+
2
8
8
.
8
3
9
x
+
2
0
.
u
.
1
+
4
6
.
2
1
7
5
x
.
1
1
.
x
T
h
e
b
a
l
l
i
s
l
e
v
i
t
a
t
e
d
t
o
t
h
e
e
q
u
i
l
i
b
r
i
u
m
p
o
i
n
t
x
0
:
I
n
[
9
]
:
=
s
r
=
S
t
a
t
e
R
e
s
p
o
n
s
e
[
{
c
s
y
s
,
{
0
.
8
,
0
,
0
}
}
,
0
,
{
t
,
0
,
2
}
]
;
P
l
o
t
{
%
〚
1
〛
,
0
.
5
}
,
{
t
,
0
,
2
}
,
p
l
o
t
O
p
t
s
O
u
t
[
9
]
=
The velocity of the ball and current of the electromagnet:
I
n
[
1
0
]
:
=
T
a
b
l
e
P
l
o
t
j
1
,
{
t
,
0
,
2
}
,
p
l
o
t
O
p
t
s
,
{
j
,
T
h
r
e
a
d
[
{
s
r
〚
2
;
;
3
〛
,
{
x
'
,
i
}
}
]
}
O
u
t
[
1
0
]
=
Obtain the controller model:
I
n
[
1
1
]
:
=
c
m
=
[
"
C
o
n
t
r
o
l
l
e
r
M
o
d
e
l
"
]
O
u
t
[
1
1
]
=
(
1
.
u
.
1
+
1
.
(
-
7
.
0
0
1
0
6
-
0
.
6
2
8
3
0
7
i
+
1
4
.
4
4
1
9
x
+
2
.
3
1
0
8
8
x
.
1
)
)
The control effort:
I
n
[
1
2
]
:
=
O
u
t
p
u
t
R
e
s
p
o
n
s
e
[
c
m
,
J
o
i
n
[
{
0
}
,
s
r
]
,
{
t
,
0
,
2
}
]
;
P
l
o
t
[
%
,
{
t
,
0
,
2
}
,
P
l
o
t
R
a
n
g
e
A
l
l
]
O
u
t
[
1
2
]
=
Related Symbols
LQRegulatorGains
AffineStateSpaceModel
OutputResponse
StateResponse
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