keh process dynamics and control 2 - Åbo akademikeh process dynamics and control 2–9 m å f 1 h f...

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