keh process dynamics and control 2 - Åbo akademikeh process dynamics and control 2–9 m å f 1 h f...
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KEH Process Dynamics and Control 2–1
KEH Process Dynamics and Control 2–2
t u(t)y(t)
u
y
KEH Process Dynamics and Control 2–3
q x,
p1 p2 .
q
q x p1 p2
x p1 p2
KEH Process Dynamics and Control 2–4
Figure 2.2. Schematic of a control valve.
Figure 2.3. Block diagram.
Valve
KEH Process Dynamics and Control 2–5
( ) dy t u t
KEH Process Dynamics and Control 2–6
KEH Process Dynamics and Control 2–7
Figur 2.5. Process diagram for flow control.
KEH Process Dynamics and Control 2–8
h
F1 F2 h
h
F2 F1
KEH Process Dynamics and Control 2–9
nivå/inström utström/nivåF1 h F2
styrvariabel
F1
F2h
nivå/inström
nivå/utström
F1
h
F2
Kp > 0
Kp < 0
+
+
styrvariabel
störning
F1
F2h
The block diagram also illustrates what
is meant by a positive and negative gain
KEH Process Dynamics and Control 2–10
v = 1 m/s
TC
60 m
1
2i
rånga
vätska
KEH Process Dynamics and Control 2–11
Examples of open-loop control applications:
bread toaster
idle-speed control of (an old) car engine
KEH Process Dynamics and Control 2–12
KEH Process Dynamics and Control 2–13
KEH Process Dynamics and Control 2–14
Temp.
sensor
Controller Heater
Temp.
sensor
Controller Heater
KEH Process Dynamics and Control 2–15
F1 F2
F1 10 l/min.
V = 1000 liters.
F1 F2
KEH Process Dynamics and Control 2–16
F2 = 10 l/min
h
F2 = F1.
KEH Process Dynamics and Control 2–17
FC
10 l/min
F1
F2
V hFC
10 l/min
FC
10 l/min
F1
F2
V h
1000 l
FC
10 l/min
F1
F2
V hFC
KEH Process Dynamics and Control 2–18
y
r ym y
Controller Controlled system
Measuring device
v
y
Output signalControl signal
u
ym
Measured value
Control error
e
Comparator
+–
r
Setpoint
Disturbance
KEH Process Dynamics and Control 2–19
KEH Process Dynamics and Control 2–20
T
Kp P
T
i a
P
i P
ii p a
d
dT K P
t
pK
KEH Process Dynamics and Control 2–21
Kc P0
Kc > 0
.
p 0K id / d 0t
r
pK T
c r i 0( )P K P
0Pi a P i
ia
i
a
P
Kc = 0
ϑr
Kc = 0,
P0 = 0
Kc > 0)
Kc = 1/ Kp
P0
KEH Process Dynamics and Control 2–22
p c pi r a 0
p c p c p c
1
1 1 1
K K KP
K K K K K K
i a p 0K P
i r a p 00,5 0,5 0,5K P
r a !
i r
Kc ϑr
P0 , i.e. if Kc→ ,
Kp
KEH Process Dynamics and Control 2–23
i
a i r
a
i r
KEH Process Dynamics and Control 2–24
ϑ2 60 m
v = 1 m/s ϑ1
ϑi
ṁ
ϑ2 t +1 ϑ1 t ϑi t Kpṁ(t)
t Kp
ϑ2 ṁ
ṁ(t) = Kc(ϑr ϑ2 t ṁ0
Kc ṁ0
KEH Process Dynamics and Control 2–25
2 r
ϑ2 t +1 ϑi t KpKc(ϑr ϑ2 t Kpṁ0
Δϑ2 t +1 Δϑi t KpKc(ϑr Δϑ2 t Kpṁ0
t 0
Δϑi,step ϑi
Δϑ2 1 Δϑi,step Δϑ2 2 Δϑi,step KpKcΔϑ2 1 1 - KpKc Δϑi,step
t k
KEH Process Dynamics and Control 2–26
i 2( , )
2 i p c r 2 p 0( )K K K m
i i i( ) ( )t t
2 2 2( ) ( ) ,t t
1
2 p c i,step
0
( ) ( )k
j
j
k K K
KpKc 1
KpKc = 1, Δϑ2 Δϑi,step Δϑi,step
KpKc 1 ,
k → ∞ KpKc 1
Δϑ2 k 0,5Δϑi,step k → ∞ Δϑ2 0.
KEH Process Dynamics and Control 2–27
i,step2
p c
( )1
kK K
KEH Process Dynamics and Control 2–28
KEH Process Dynamics and Control 2–29
u(t) e(t)
.
u0 Kc Ti
Td
KEH Process Dynamics and Control 2–30
c d 0i 0
1 d ( )( ) ( ) ( )d
d
te t
u t K e t e t T uT t
Td = 0.
Ti = ∞ Td = 0 ( Ti ≠ 0 !).
Kc = 0)
KEH Process Dynamics and Control 2–31
c i d 0
0
d ( )( ) ( ) ( )d
d
te t
u t K e t K e t K ut
KEH Process Dynamics and Control 2–32
t ts u(t) e(t) t ts
e(ts) = 0
e(t)
x(t)
KEH Process Dynamics and Control 2–33
c 0i 0
1( ) ( ) ( )d
t
u t K e t x t uT
0
( ) ( )d
t
x t e t
KEH Process Dynamics and Control 2–34
KEH Process Dynamics and Control 2–35