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Corrige HEI Control systems 2008 TC

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iFEEDBACK CONTROL SYSTEMS: TD n°2 CORRECTION 1. Step Responsek From: U(1) u(p) 212 k k θ (p) 10 1 + τ p2 101 + τ p p 18 6 θ (p) λH (p) = = 4u(p) p(1 + τ ...

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Abdel AITOUCHE, 2007-2008, FEEDBACK CONTROL SYSTEMS
FEEDBACK
CONTROL SYSTEMS: TD n°2 CORRECTION
1.
)
1
)(
1
(
)
(
)
(
)
(
2
1
p
p
p
p
u
p
p
H
τ
τ
λ
θ
+
+
=
=
2.1
2.2.1
We neglect
the pole
1
1
τ
.
2.2.2
)
1
(
2
p
p
k
H
BOa
τ
λ
+
=
2.2.3
1
1
1
2
2
+
+
=
p
k
p
k
H
BFa
λ
λ
τ
2.2.4
2
τ
λ
ω
k
n
=
2
2
1
λτ
ξ
k
=
2.2.5
0208
,
0
:
2
1
2
=
=
k
AN
k
λτ
2.2.6
heures
t
t
r
n
r
10
3
=
=
ξω
2.2.7
C
D
e
D
°
=
=
=
10
)
(
%
33
,
4
%
%
2
1
θ
ξ
ξπ
2.4.1
dB
MG
MG
k
MG
2
,
27
9
,
22
2
1
2
1
+
=
τ
λτ
τ
τ
3.1
3.2
)
1
(
1
p
p
k
H
BO
τ
λ
+
=
1
1
1
2
1
+
+
=
p
k
p
k
H
BF
λ
λ
τ
3.3
208
,
0
:
2
1
1
=
=
k
AN
k
λτ
3.4
heure
t
t
r
n
r
1
3
=
=
ξω
3.5
C
D
e
D
°
=
=
=
10
)
(
%
33
,
4
%
%
2
1
θ
ξ
ξπ
3.6 Controller PD allows accelerating the dynamic of system 10
times faster than the controller P for a same accuracy and a same
overshoot.
4.1
4.2
1
1
1
)
(
)
(
1
+
=
=
=
p
p
H
p
p
H
T
BF
BO
d
λ
λ
τ
4.3
25
,
1
:
3
3
1
1
0
1
=
=
=
=
k
AN
k
k
k
t
r
τ
τ
λ
4.4.1
4.4.2
4.4.3
1
1
1
1
)
(
2
2
2
+
+
=
p
k
k
p
k
k
p
H
i
i
BF
λ
4.4.4
1
2
2
3
τ
k
k
i
=
0025
,
0
=
i
k
4.4.5
mn
t
t
r
n
r
20
3
=
=
ξω
C
D
e
D
°
=
=
=
10
)
(
%
33
,
4
%
%
2
1
θ
ξ
ξπ
4.5
Cascade Control allows obtaining a step time response three
times less than a controller PD, thirty time less than a
controller P with the same accuracy and the same overshoot.
.
2
1
τ
1
1
τ
0
Tim e (s ec .)
Amplitude
S tep Response
0
1400
2800
4200
5600
7000
0
2
4
6
8
10
12
From: U(1)
To: Y(1)
Amplitude
S tep Res pons e
0
500
1000
1500
2000
2500
3000
0
0.005
0.01
0.015
0.02
0.025
0.03
From: U(1)
To: Y(1)
u(p)
)
(
p
θ
p
k
0
p
k
1
1
1
τ
+
p
k
2
2
1
τ
+
ref
θ
)
(
p
θ
)
(
p
H
k
ref
θ
)
(
p
θ
)
1
1
(
2
+
p
p
k
k
i
λ
ref
θ
)
(
p
θ
1
ϑ
BF
H
p
p
T
k
i
i
)
1
(
+
p
k
2
2
1
τ
+
ref
1
θ
)
(
1
p
θ
)
1
(
1
1
0
p
p
k
k
τ
+
k
)
1
(
p
T
d
+
ref
θ
)
(
p
θ
)
(
p
H
)
1
(
p
T
k
d
+