Transfer Functions

What is transfer function?-is an algebraic expression for dynamic relation between a selected input and output of the process model-can only be derived for a linear model because Laplace Transform can be applied only to linear equations.

What is Laplace transforms?

-method to convert the ODE (representing the dynamicbehaviour of process output variables) into algebraic equations-integral transforms that transforms a function of time [ f(t) ] into a function of complex variables s, F(s)

 • 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/PProcess

Sensor

Transmitter

Controller

Transducer

Control valve

4-20 mA1-5 Vdc

PIDFuzzy logic

4-20 mA3-15 psig

dP cellCapacitanceRadar, SonicMagneticResistanceIR/Laser

PressureFlowLevelTemperaturepH

LinearEqual percentage

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

Unit 1: Process Control Loop

Process

Sensor

Transmitter

Controller

Transducer

Control valve

PressureFlowLevelTemperaturepH

dP cellCapacitanceRadar, SonicMagneticResistanceIR/Laser

4-20 mA1-5 VdcField/profibus

PIDFuzzy logic

4-20 mA3-15 psig

LinearEqual percentage

CONTROLLER

CONTROLVALVE

PROCESS

TRANSMITTER

PV

SP

Process control loop

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

*SP = set point*PV = process value

Unit 1: Process Control Loop

CO

NTR

OLL

ER

CO

NTR

OL

VALV

E

PR

OC

ES

S

PVSP

Simulation mode

Process control loop: The Block Diagram

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

TRANSMITTER

• Convenient representation of a linear, dynamic model.• A transfer function (TF) relates one input and one output:

systemx t y tX s Y s

The following terminology is used:x

input

forcing function

“cause”

y

output

response

“effect”

Cha

pter

4Transfer Functions

The TF model enables us to determine the output response to any change in an input.

Type of forcing function: Step, Impulse, Ramp, SinusoidalType of response :Step, Impulse, Ramp, Sinusoidal

Credit: Process Systems Analysis and Control, Coughanowr and Leblanc

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

LL

Cha

pter

4

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

Page 27

Cha

pter

4Example: 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)idTV C wC T T Qdt

Suppose the process is at steady state:

0 (2)iwC T T Q

Subtract (2) from (1):

(3)i idTV C wC T T T T Q Qdt

Cha

pter

4

(1)

But,

(4)idTV C wC T T Qdt

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):

Cha

pter

4

At the initial steady state, T′(0) = 0.Because T’(0) = T-Ts= Ts- Ts =0

Rearrange (5) to solve for

1 (6)1 1 i

KT s Q s T ss s

Cha

pter

4

where

1 and VKwC 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 ss s

(s)T(s)G(s)Q(s)(s)=GT i 21

1 (6)1 1 i

KT s Q s T ss s

K (gain) – it describes how far the output will travel with the change of the input.

Gain, K = Δ y Δ x*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

50= 100 20

0.05= 1 20

MV

Credit: controlguru.com

(time constant) – describes how fast the output moves in response to a change in the input

Large values of mean a slow process responseSmall values of mean a fast response

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

MV

Credit: controlguru.com

Cha

pter

4Order of transfer function

)(1

)( sXsKsY

• General first order transfer function

• General second order transfer function

)(12

)(22

sXss

KsY

)(1

)(0

sXs

KesYst

• First-order-plus-dead-time (FOPDT)

aka first-order lag and single exponential stage

LAPLACE TRANSFORM • integral transforms that transforms a function of time, f(t)

into a function of complex variables s, F(s)

We often abbreviate this to

In short,

The step / unit step function, S(t)

f(t) = S(t)

*Laplace transform only work in the range of 0<t<∞

So, S(t)=0 is not taken as t<0

Thus, S(t)= 1 when t>0.

L S(t) = L {1} =

First Order Derivative

Higher Order Derivative

• a new variable, is introduced.

Eg 1.

Eg 2.

Exponential Functions

Ramp FunctionsEg 1.

What is Response with Time Delay/Dead Time?

X(t) Y(t)

ΔX= Xf – X0ΔY= Yf – Y0

X(t) Y(t)

• In a process when MV changes, the PV only responds after a certain time has elapsed.

• This elapsed time is known as the dead time/time delay. • Similarly, the response of the PV is delayed when the MV changes

back

X0 Y0

X(t)=MVY(t)=PV

XfYf

X(t)=MVY(t)=PV

X(t)

Y(t)

• A typical example here is a belt conveyor, where there is a certain time delay before a change in the chute feed rate is recorded at the measurement location.

• Another example is when someone tell you a joke/explanation, only after a while you crack a laugh/ understand.

Do you get it what it meant by Response with Time Delay/Dead Time??

• The relationship between process value x and manipulating variable y is as follows:

Gain, K = Δ y Δ x

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 ( a type MV function) will have a response of this same shape.

Step response for first order system

ΔY= Yf – Y0 = ???

To calculate the gain and time constant from the graph

xyK

Gain,

Time constant, – value of t which the response is 63.2% complete

= ?? 10

1. A sensor measures and transmits the current value of the process variable, PV, back to the controller 2. Controller error at current time t is computed as set point minus measured process variable, or e(t) = SP – PV3. The controller uses this e(t) in a control algorithm to compute a new controller output signal, CO4. The CO signal is sent to the final control element (e.g. valve, pump, heater, fan) causing it to change5. The change in the final control element (FCE) causes a change in a manipulated variable6. The change in the manipulated variable (e.g. flow rate of liquid or gas) causes a change in the PV

The goal of the controller is to make e(t) = 0 in spite of unplanned and unmeasured disturbances. Since e(t) = SP – PV, this is the same as saying a controller seeks to make PV = SP.

Transfer Functions for a Transmitter

KT = transmitter gain

H(s)Process variable

PV(s)

Transmitter output

C(s)

1K

)()()( T

ssPVsCsH

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) valuecontroller gain (usually dimensionless)c

p t

pK

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

LaplaceTransform

2. Introduce deviation variable, into

1

1

1.

3.

How to obtain Transfer Function for a Proportional Controller?

For process with self-regulation,

Examples of proportional processes are:- mechanical gearing without slip- mechanical transmission by lever- transistor (collector current Ic

follows the base current IB with virtually no delay)

46

Integral ControlFor integral control action, the controller output depends on the integral of the error signal over time,

0

1 * * (8-7)τ

t

Ip 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

tc

Ip t p K e t e t dt

Function of integral term:While the proportional term considers the current size of e(t) only at the time of the controller calculation;The integral term considers the history of the error/how long or how far the PV has been (away) from the SP over time

E(t) continually changes size and sign as time passes

Integral of each shaded portion has the same sign as the error.

The total integral sum grows as long as e(t) is positive andshrinks when it is negative.

It calculates accumulation of error over time.

τI• The PI controller has two parameters, Kc and

Transfer function

11.

2. Introduce deviation variable: p’(t) = into 1

3. p’(t) =

How to obtain Transfer Function for a Proportional-Integral Controller?

Proportional-Integral processes usually found in aircraftand liquid level in tank. Rowing a kayak, canoe, sampan byyour hand roughly mimicks the PI controller

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

Transfer function

• controller output is equal to the nominal value p as long as the error is constant (that is, as long as de/dt = 0).

• Derivative action is based on how rapidly the error is changing, not the magnitude of the error or how long the error has persisted. It is based on the slope of the error

versus time curve at any instant in time.

• Therefore, a rapidly changing error signal will induce a large derivative response. “Noisy” error signals cause significant problems for derivative action because of the rapidly changing slope of the error caused by noise.

• Derivative control should be avoided in these situations unless the error signal can be filtered to remove the noise.

For example, suppose that a reactor temperature increases by 10 oc in a short period of time, say, 3 min. This clearly is a more rapid increase in temperature than a 10 oc rise in 30 min, and it could indicate a potential runaway situation for an exothermic reaction. If the reactor were under manual control, an experienced plant operator would anticipate the consequences and quickly take appropriate corrective action to reduce the temperature. Such a response would not be obtainable from the proportional and integral control modes discussed so far. Note that a proportional controller reacts to a deviation in temperature only, making no distinction as to the time period over which the deviation develops. Integral control action is also ineffective for a sudden deviation in temperature, because the corrective action depends on the duration of the deviation

Transfer function for Valve

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

tc D

I

de tp t p K e t e t dt

dt

The corresponding transfer function is:

Transfer function