Control systems 2008 Tronc commun Hautes Etudes d'Ingénieur (Lille)

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Examen du Supérieur Hautes Etudes d'Ingénieur (Lille). Sujet de Control systems 2008. Retrouvez le corrigé Control systems 2008 sur Bankexam.fr.
Publié le : samedi 9 août 2008
Lecture(s) : 38
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Nombre de pages : 2
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HEI 43 TC 2007-2008
Control of Temperature
.
The controlled temperature
θ
of indirect heating is represented by the following figure:
u(t) represents the voltage control of valve,
q(t )
the
inlet flow in exchanger and
θ
1
the output
temperature of exchanger.
Let the following relations:
(t)
k
=
dt
(t)
d
+
(t)
q(t)
k
=
dt
(t)
d
+
(t)
).d
u(
k
=
q(t)
1
2
2
1
1
1
1
t
0
o
θ
θ
τ
θ
θ
τ
θ
τ
τ
We assume that all initial conditions are null. Numerical applications
:
τ
1
=600s,
τ
2
=6000s,
k
2
=1, k
1
=20 S.I., k
o
=2.10
-4
S.I.
We take:
λ
= k
o.
k
1
.k
2
.
1
. Give the box diagram of the system and deduce the transfer function of the exchanger H(p)
(input u and
θ
).
2. Proportional Controller
u(t) =G
r
ε
(t)= G
r
.(
θ
ref
(t) -
θ
(t)) where
θ
ref
(t) is the setpoint or reference.
2.1
Give the box diagram of the closed loop system
2.2
To determine k, we neglect a pole of the system.
2.2.1
From the numerical values, which is the pole to be neglected?
2.2.2
Give the expression of the approximated open loop transfer noted W
Oa
.
2.2.3
Compute from W
Oa
the approximated closed loop transfer W
Fa
.
2.3.4
W
Fa
will be written as normalized second order:
1
2
1
1
0
2
2
0
+
+
p
p
ω
ξ
ω
. Give
ξ
and
ω
0
.
2.2.5
Compute the proportional gain G
r
in order to obtain a damping coefficient un coefficient
2
2
=
ξ
.Numerical Application..
2.2.6
Determine the settling time (5%) of the controlled system . Numerical Application.
θ
1
q
u
θ
Exchanger
pomp
Plate
valve
2.2.7
Compute the final value, the overshoot in step response of
θ
ref
of 10° and plot the
approximate curve.
In the following problem, we do not take account of the approximations of W
O
.
2.4
The proportional gain in is now 0,02 .
2.4.1
Determine the gain margin. Conclude?
3. Derivative-Proportional Controller
:
θ
(p))
(p)
p)(
θ
T
(1
G
u(p)
ref
d
+
=
r
.
T
d
is compensated by the biggest time constant.
3.1
Give the box diagram of the system?.
3.2
Compute the closed loop transfer function.
3.3
Compute the proportional gain value in order to obtain a damping coefficient
2
2
=
ξ
.
3.4
Determine the settling time ( 5%) of the controlled system . Numerical Application
3.5
Compute the final value, the overshoot in step response of
θ
ref
of 10°
and plot the
approximate curve.
3.6
Compare the proportional action with the proportional-derivative action?
4. Cascade Controller.
The temperature
θ
1
is controlled firstly with a PID controller: u(p) = G
r
.(1+T
d
.p).(
θ
1ref
(p) -
θ
1
(p)).
4.1
Give the block diagram of part of the system.
4.2
Choose T
d
in order that
θ
θ
1
1
(p)
(p)
ef
r
will be a first order system.
4.3
Choose G
r
in order to obtain a settling time of
θ
1
(t) equal to
τ
1
when
θ
1ref
(t) is a step
setpoint.
4.4
A second loop is introduced with a controller PI.
(
)
(p)
(p)
p
p)
T
+
(1
G
=
(p)
i
i
1
θ
θ
θ
ref
ref
4.4.1
Give the block diagram of the whole plant.
4.4.2
We choose T
i
=
τ
2
. Reduce this block diagram.
4.4.3
Show that the system is a second order.
4.4.4
Compute the proportional gain G
i
in order to obtain a damping coefficient un coefficient
2
2
=
ξ
.
4.4.5
Determine the settling time (5%) of the controlled system. Numerical Application
4.4.6
Compute the final value, the overshoot in step response of
θ
ref
of 10°
and plot the
approximate curve.
4.5
Conclude about the advantage of the cascade controller?
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