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Related Symbols
StateSpaceModel
SystemsModelDelay
BodePlot
SystemsModelDelayApproximate
OutputResponse
PIDTune
Related Categories
Control Systems
Engineering
Mathematics
System Modeling
Model and Control the Cutting Process of a Lathe
Example Notebook
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The cutting process of a lathe contains time delays:
I
n
[
1
]
:
=
I
n
[
2
]
:
=
e
q
n
=
m
x
'
'
[
t
]
+
c
x
'
[
t
]
+
k
x
[
t
]
-
α
(
f
[
t
]
+
x
[
t
]
-
x
[
t
-
τ
]
)
/
.
τ
2
π
ω
;
Construct a symbolic state-space model of the process:
I
n
[
3
]
:
=
s
s
m
=
S
t
a
t
e
S
p
a
c
e
M
o
d
e
l
e
q
n
,
{
x
[
t
]
,
x
'
[
t
]
}
,
f
[
t
]
,
x
[
t
]
,
t
,
l
a
b
e
l
s
O
u
t
[
3
]
=
f
x
0
1
0
x
'
-
k
-
α
+
α
2
π
ω
m
-
c
m
-
α
m
x
1
0
0
Set numerical values for the model's parameters:
I
n
[
4
]
:
=
p
a
r
s
=
{
m
0
.
7
5
,
ω
3
,
α
1
0
,
k
0
.
1
,
c
5
,
τ
3
}
;
s
s
m
n
=
s
s
m
/
.
p
a
r
s
O
u
t
[
4
]
=
f
x
0
1
0
x
'
1
.
3
3
3
3
3
-
1
0
.
1
+
1
0
2
π
3
-
6
.
6
6
6
6
7
-
1
3
.
3
3
3
3
x
1
0
0
Its frequency response exhibits chattering typical of a time-delay system:
I
n
[
5
]
:
=
B
o
d
e
P
l
o
t
[
s
s
m
n
,
P
l
o
t
L
a
y
o
u
t
"
H
o
r
i
z
o
n
t
a
l
G
r
i
d
"
,
I
m
a
g
e
S
i
z
e
S
m
a
l
l
]
O
u
t
[
5
]
=
Obtain the approximate, delay-free system:
I
n
[
6
]
:
=
s
s
m
n
0
=
S
y
s
t
e
m
s
M
o
d
e
l
D
e
l
a
y
A
p
p
r
o
x
i
m
a
t
e
[
s
s
m
n
,
0
]
O
u
t
[
6
]
=
f
0
1
.
0
-
0
.
1
3
3
3
3
3
-
6
.
6
6
6
6
7
1
.
x
-
1
3
.
3
3
3
3
0
0
The approximate, delay-free system exhibits no chattering in its frequency response:
I
n
[
7
]
:
=
B
o
d
e
P
l
o
t
[
s
s
m
n
0
,
P
l
o
t
L
a
y
o
u
t
"
H
o
r
i
z
o
n
t
a
l
G
r
i
d
"
,
I
m
a
g
e
S
i
z
e
S
m
a
l
l
]
O
u
t
[
7
]
=
Compare the step-response of the time-delay and delay-free systems:
I
n
[
8
]
:
=
o
r
=
T
a
b
l
e
[
O
u
t
p
u
t
R
e
s
p
o
n
s
e
[
s
y
s
,
U
n
i
t
S
t
e
p
[
t
]
,
{
t
,
0
,
3
0
}
]
,
{
s
y
s
,
{
s
s
m
n
,
s
s
m
n
0
}
}
]
;
P
l
o
t
%
,
{
t
,
0
,
3
0
}
,
p
l
o
t
O
p
t
s
O
u
t
[
8
]
=
Design a PI controller for the delay-free system to stabilize the movement of the lathe:
I
n
[
9
]
:
=
p
i
d
=
P
I
D
T
u
n
e
[
s
s
m
n
0
,
"
P
I
"
,
"
D
a
t
a
"
]
O
u
t
[
9
]
=
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
:
Z
i
e
g
l
e
r
N
i
c
h
o
l
s
P
I
D
»
P
a
r
a
l
l
e
l
p
a
r
a
m
e
t
e
r
s
:
{
-
2
.
9
9
,
-
5
.
9
6
,
0
.
}
Evaluate its response to a step input:
I
n
[
1
0
]
:
=
O
u
t
p
u
t
R
e
s
p
o
n
s
e
[
p
i
d
[
"
R
e
f
e
r
e
n
c
e
O
u
t
p
u
t
"
]
,
U
n
i
t
S
t
e
p
[
t
]
,
{
t
,
0
,
1
0
}
]
;
P
l
o
t
[
%
,
{
t
,
0
,
1
0
}
,
P
l
o
t
R
a
n
g
e
A
l
l
]
O
u
t
[
1
0
]
=
Source Metadata
Citation:
Gu, Kharitonov and Chen 2003, Stability of Time-Delay Systems, page 2
Related Symbols
StateSpaceModel
SystemsModelDelay
BodePlot
SystemsModelDelayApproximate
OutputResponse
PIDTune
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