2 - Transfer function.pdf

Transfer Functions

• Consider the tank heating system shown in Figure 1. A liquid enters

the tank with a flowrate Fi (ft3/min), and a temperature Ti, where it is

heated with steam (having a flowrate Q, lb/min). Let F and T be the

flowrate and temperature of the stream leaving the tank. The tank is

considered to be well stirred, which implies that the temperature of the

effluent is equal to the temperature of the liquid in the tank.

• The control objective of this heater is to keep the effluent temperature

T at a desired value. Draw a P& ID and block diagram for this control

system.

Unit 1: Process Control Loop

Process control loop

I/P Process

Sensor

Transmitter

Controller

Transducer

Control valve

4-20 mA

1-5 Vdc

PID

Fuzzy logic

4-20 mA

3-15 psig

dP cell

Capacitance

Radar, Sonic

Magnetic

Resistance

IR/Laser

Pressure

Flow

Level

Temperature

pH

Linear

Equal percentage

© Abdul Aziz Ishak, Universiti Teknologi MARA Malaysia (2009)


Process

Sensor

Transmitter

Controller

Transducer

Control valve

Pressure

Flow

Level

Temperature

pH

dP cell

Capacitance

Radar, Sonic

Magnetic

Resistance

IR/Laser

4-20 mA

1-5 Vdc

Field/profibus

PID

Fuzzy logic

4-20 mA

3-15 psig

Linear

Equal percentage

CONTROLLER

CONTROL

VALVE

PROCESS

TRANSMITTER

PV

SP

Process control loop


*SP = set point

*PV = process value


CO

NT

RO

LLE

R

CO

NT

RO

L

VA

LV

E

PR

OC

ES

S

PV SP

Simulation mode

Process control loop: The Block Diagram


TRANSMITTER

• Convenient representation of a linear, dynamic model.

• A transfer function (TF) relates one input and one output:

systemx t y t

X s Y s

The following terminology is used:

x

input

forcing function

“cause”

y

output

response

“effect”

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Transfer Functions

The TF model enables us to determine the output response to

any change in an input.

Definition of the transfer function:

Let G(s) denote the transfer function between an input, x, and an

output, y. Then, by definition

where:

Y sG s

X s

Y s y t

X s x t

L

L

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Figure 2.3 Stirred-tank heating process with constant holdup, V.

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Example: Stirred Tank Heating

System

Transfer Functions for a Process

Equation (1) is the energy balance of the stirred-tank heating

system, assuming constant liquid holdup and flow rates:

(2-36)i

dTV C wC T T Q

dt

Suppose the process is at steady state:

0 (2)iwC T T Q

Subtract (2) from (1):

(3)i i

dTV C wC T T T T Q Q

dt

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(1)

But,

(4)i

dTV C wC T T Q

dt

where the “deviation variables” are

, ,i i iT T T T T T Q Q Q

0 (5)iV C sT s T wC T s T s Q s

Take L of (4):

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4

At the initial steady state, T′(0) = 0.

Rearrange (5) to solve for

1

(6)1 1

i

KT s Q s T s

s s

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where

1and

VK

wC w

0 (5)iV C sT s T wC T s T s Q s

(s)T(s)G(s)Q(s)(s)=GT i

21

1

(6)1 1

i

KT s Q s T s

s s

(s)T(s)G(s)Q(s)(s)=GT i

21

1

(6)1 1

i

KT s Q s T s

s s

K (gain) – it describes how far the output will travel

with the change of the input.

(time constant) – describes how fast the output moves

in response to a change in the input.

*The time constant must be positive and it must have units of

time

*If a process has a large K, then a small change in the

input will cause the output to move a large amount. If a

process has a small K, the same input change will move

the output a small amount

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Order of transfer function

)(1

)( sXs

KsY

• General first order transfer function

• General second order transfer function

)(12

)(22

sXss

KsY

)(1

)(0

sXs

KesY

st

• First-order-plus-dead-time (FOPDT)

Response with time delay

to=Time delay/dead time

X(t)

Y(t)

t=0 t=t0

•All first order systems forced by a step function will have

a response of this same shape.

Step response for first order system

To calculate the gain and time constant from

the graph

x

yK

Gain,

Time constant, – value of t which the response is

63.2% complete

Transfer Functions for a Transmitter

KT = transmitter gain

H(s) Process variable

PV(s)

Transmitter output

C(s)

1

K

)(

)()( T

ssPV

sCsH

T

T = transmitter time constant

For proportional control, the controller output is proportional to

the error signal,

(8-2)cp t p K e t

where: controller output

bias (steady-state) value

controller gain (usually dimensionless)c

p t

p

K

Proportional Control

is controller output when the error is zero

-The proportionality is given by the controller gain, Kc

-The controller gain determines how much the output

from the controller changes for a given change in error

p

Transfer Functions for a Controller

Transfer function

20

Integral Control

For integral control action, the controller output depends on the

integral of the error signal over time,

0

1* * (8-7)

τ

t

I

p t p e t dt

where , an adjustable parameter referred to as the integral time

or reset time, has units of time.

τI

•Integral control action is normally used in conjunction with

proportional control as the proportional-integral (PI) controller :

0

1* * (8-8)

τ

t

cI

p t p K e t e t dt

τI

•The corresponding transfer function for the PI controller in is

given by

•The PI controller has two parameters, Kc and

Transfer function

Derivative Control

•The function of derivative control gives the controller the

capability to anticipate where the process is heading by

calculating the derivative error

• Thus, for ideal derivative action,

τ (8-10)D

de tp t p

dt

where , the derivative time, has units of time. τD

Proportional-Integral-Derivative (PID) Control

Now we consider the combination of the proportional, integral,

and derivative control modes as a PID controller.

Form of PID Control

The form of the PID control algorithm is given by

0

1* * τ (8-13)

τ

t

c DI

de tp t p K e t e t dt

dt

The corresponding transfer function is:

Transfer function

Transfer function for Valve

2 - Transfer function.pdf

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