experiment 13 - digital pid controller_a

8/10/2019 Experiment 13 - Digital Pid Controller_a

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EXPERIMENT 13

DIGITAL PID CONTROLLER

13.1 OBJECTIVE OF THE EXPERIMENT

a. To test a proportional controller using square wave input.

b. To test a Proportional Integral controller using a step input.

c. To test proportional derivative controller using a sinusoidal input.

d. To study the gain and time constant of processes.

e. To study PID controller response

f. To study closed loop tuning of process

g. To study the networking of controllers

h. Cascade control

i. Feed forward control

. !upervisory control

k. Direct Digital Control

13.2 INTRODUCTION

Frequency response is a measure of the ability of a circuit or a system to respond or transmit

input signals of various frequencies that are applied to it. Frequency response measures the

ability of the device to respond to changes in the input that are changing with respect to time. It

therefore measures the dynamic characteristics of the system as against the first four

e"periments in this series that measures the static characteristics like accuracy and resolution of

the device. Frequency response can also be used as a technique for parameter estimation of

unknown system. #y determining the frequency response of unknown system we can determine

the order of the system as well as its dynamic parameters like time constant$ gain and delay time.

%&.'.% !inusoidal response of a f irst order system

Consider a first order system represented by the transfer function$

1)(

+==

s

K

X

YsG

where

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($) * input and output variables respectively$ in the +aplace domain

, * system gain

* time constant

s * +aplace operator

If the input in the time domain " is a sinusoidal signal such that$

tAX sin=

then$ the corresponding +aplace domain input is given by

22

+

=

s

AX

!ubstituting and simplifying we can obtain the frequency response of the system. - simpler

method for determining the frequency is to substitute s *and simpify the resulting comple"

function comple" function. sing the technique we get$

+=

jjG

1

1)(

/n rationali0ing by multiplying the numerator and denominator by 1%23 and separating the real

and imaginary factors we get

222211

1)(

+

+

=

jjG

The comple" numbers in the rectangular form a4b can be converted to polar form by the

relationships$

a

bzangleandbaz 122 tan=+=

Converting into polar form using$ we get

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)(tan

1

1

1

22

=

+

=

anglePhase

ARratioAmplitue

The above analysis shows that when the transfer function of a system is known$ we candetermine its frequency response by substituting for s and then after rationali0ing the comple"

function convert it to the polar form and determine the magnitude and argument 1angle3 of the

comple" number in the polar form. The substitution of for the +aplace transform operator s is

the result of the De 5oivre6s theorem that relates the e"ponential functions with trigonometric

functions.

%&.'.' #ode diagrams

The frequency response of s system is presented in the form of the #ode diagrams. For obtaining

the #ode diagram of a system its response at a number of frequencies from * 7 to very large

values of is determined and the -8 and determines at each value of . Then log -8 is plotted

against log and is plotted against log . These two graphs are called the #ode diagrams of a

system. -n e"ample of a theretical #ode diagram of a first order system given by %91%4's3 is

given by Figure %&.%.

Figure %&.% #ode diagram of a first order system 1T*'s3

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From figure %&.% we can summarise the following characteristics of frequency response of a first

order system.

The bode diagram consists of two graphs:

a. +ogarithm of the -8 logarithm of the frequency versus

b. Phase angle versus logarithm of the frequency

The unit of the frequency is radians9s or ;0. The -8 is the output amplitude divided by the input

amplitude.


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Consider a system whose transfer function is given by the equation$

)12.0)(1(

1)(

++=

sssG

The equation represent the transfer function of a second order system obtained by combining

two first order systems of time constants % and 7.' respectively. The overall gain of the system is

represented by the numerator and is %. The frequency response of this second order system can

be obtained as before by substituting for s and then writing the comple" number in the polar

form.

-lternatelt we can combine the frequency response of the two component first order systems to

determine the overall second order system response. The graphical method for combining two

first order system responses to determine the second order response is discussed in

Coughnanowr 1%>?%3.- simple method for generating frequency response is to use software like

5atlab. Figure %&.' is generated using 5atlab software. Frequency response of higher order

cascaded like &rd$ @thand Athorder system can be generated in a similar manner.

Figure %&.' Frequency response of second order system 1T%* %s$ T'* 7.'s3

The response of the 'ndorder system as shown in Figure %&.' satisfies the following conditions.

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a. The low frequency asymptote is a hori0ontal line at -8 *%

b. The high frequency asymptote has a slope of 2'

c. If the cis the frequency at which the two asymptotes intersect and B is the amplitude

ratio of the frequency response curve at this frequency$ then the following equations are

satisfied:

221

21

1

1

c

c

TT

GTT

=

=+

d. If %and 'are the frequencies at which the phase is 29@ and 29' respectively$ then the

following equations are satisfied:

2

2

21

2

2

121

1

1

=

=+

TT

TT

The equation gives us methods for determining the time constants of a second order system from

its frequency response diagrams.

%&.'.@ Controllers

The controller used in the present e"periment is the )!2%7 5icroprocessor based controller.

This controller can be used as proportional controller$ Proportional Integral Controller$Proportional Derivative Controller or Proportional Integral Derivative controller. The controller

equations are as follows:

+

+=++

+=+

+=+

==

dt

dEdtEEKYcontrollerDerivativeIntegralopotional

dt

dEEKYcontrollerDerivativeopotional

dtEEKYcontrollerIntegraloportional

EKYcontrolleropotional

XXEError

D

I

c

Dc

I

c

c

R

1Pr

Pr

1Pr

:Pr

,

( * process variable 1controlled variable3

(8 * setpoint variable

) * controller output

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* error

,c * proportional gain

TI * integral time

TD * derivative time

%&.'.A 8esponse of Controlled to different inputs

The proportional controller can be studied by making a square wave input. For a proportional

controller the response to a square wave is shown in Figure %&.&

Figure %&.& 8esponse of a proportional controller to a square wave input 1Proportional gain ,c*'3

The response of a Proportional Integral controller to a step input is shown in Figure %&.@

Figure %&.@ 8esponse of Proportional Integral controller to a step input

The proportional derivative controller cannot be tested using a step input.


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The equation of a proportional derivative controller is given by

)9,c* TDs4%

!ubstituting s*we can determine its frequency response as

)9,c* TD4%

Therefore$

)(tan

1(

1

22

D

D

T

TAR

=

+=

- typical response curve of the proportional derivative controller is shown in Figure %A.A. The

value of TDcan be determined from the relationship$ TD*%9c.

Figure %&.A #ode diagram of Proportional Derivative controller$ TD* %

13.3 EXPERIMENTAL APPARATUS

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The equipment required for performing the e"periments are shown in Table %'.E.

Table %A.E quipment required

Unit nam M!"# N!

PID controller )okogawa 5odel )! %7

Function generator )okogawa 5odel FB%'7

Paperless recorder )okogawa 5odel D(%7@

%&.&.% )!%7 PID controller

The )!%7 controller an able to carry out fle"ible control and arithmetic operations which are

required for process control and have following features:

i. Display$ setting and operations of I9/ value$ various constants$ and built in control

functions can be controlled easily from the full dot +CD and key switches on the front

panel

ii. Trend display of process variable 1P3 is possible.

iii The built in adustable set point filter can provide a better response to set point changes.iv Communication functions can be installed to enable easy connection with a distributed

process control system or computer

v The self diagnosis function can be used to check the operation of the instrument and the

status of the input and output signal lines.

%&.&.' FB%'7 Function generator

The FB%'7 !ynthesi0ed Function Benerator is used to provide the sinusoidal function required

for testing the system. The model 7E7%' synthesi0ed function generator can provide voltages in

the frequency range % ;0 to ' 5;0. It can provide sine$ square$ tringuler$ ramp and pulse

functions. There are two channels which are independent. The output can be adusted

continuously in the range of %7 .

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%&.&.& D(%72@ Paperless recorder

The D(%7@ Paperless recorder is used to record the result of the testing. The measured data can

also be save to e"ternal storage media such as floppy disks$ 0ip disks and -T- flash memory

cards.

The data that have been saved to an e"ternal storage medium can be displayed on a PC by

using the standard software that comes with the package. The data can also be loaded onto the

D(%7@ to be displayed.

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13.$ PROCEDURE

%&.@.% Beneral outlook of the control panel

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%&.@.' quipment connections

Figure %&. quipment Connections

#efore the set up of the e"periment$ please read the following Instruction 5anual for the

operation of each button9key function and how to do the data setting.

a3 I5%#C%27% model )!%7 single loop programmable controller

b3 I57E7%%27% Fg%'7 !ynthesi0ed Function Benerator

c3 I57@+7%-7%27% 5odel D(%7@ D-G!T-TI/H D(%77.

d3 -ppendi" in this manual.

%&.@.& Proportional Controller 1P#%3

%. Connect the equipment as shown in Figure %&.. For the recording device we can use the

+8 recorder$ (2)2t plotter or the ' channel /scillograph or the paperless recorder. In the

present e"periment we will be using the paperless recorder D(%7@ together with the

recorder software.

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'. -t the )!%7 controller please make sure P7%$ P7'$ P7&$P7@$ P7A$ P7E and P%7 1in

Tuning 5enu$ PT 8B3 are set to 0ero. -lso set -CT% 1in HB.5H%$ C/HFIB'3 to

DI8 1Direct -ction

&. For calibrating the Proportional band 1P#%3 at )!%7 1in Tuning 5enu$ PID %3$ set the

integral time 1TI%3 to >>>> and derivative time 1TD%3 to 0ero.

@. Choose square wave function at the FB %'7 function generator. The frequency is not

important in this e"periment. Choose any convenient frequency 1small value should be

chosen3. 8efer to table %&.? for the DC voltage setting for the data of P#% set. -lso set

time period to E7 sec.

A. !et the )!%7 to 5anual mode 153$ 5*7 and at )!%7 1Tuning 5enu$ PID %3 set the

proportional band 1P#%3 to @77J.Turn on the output key of FB%'7 and make sure the

input )!%7 at base value before change it to -T/ mode 1-3. Determine the input and

output value and enter in table %&.?. Calculate the system gain and compare it with the

set value as shown in Figure %&.>.E. 8epeat step 1v3 for others data of P#% as shown in table %&.?.

Table %&.? Proportional band of the controller

!et FB%'7 to square wave with high level * @.77 and low level to %.77

Controller

P#% set

1J3

input Difference

in*1%2

b%3

out Difference

out*1'2

b'3

Bain 1B3

out9in

-ctual Bain

%779P#b% % b' '

@77

&A7

&77

'A7

'77

%A7

%77

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!et FB%'7 to square wave with high level * '.77 and low level to %.77

Controller

P#% set

1J3

input Difference

in*1%2

b%3

out Difference

out*1'2

b'3

Bain 1B3

out9in

-ctual Bain

%779P#b% % b' '

%77

?7

E7

A7

@7

&7

'A

Figure %&.> Proportional gain of the controller

. ary the input voltage to the proportional controller from% volts to ' volt keeping the P#%

at %77J. Determine the output

?. Change the input to & volts and repeat..8epeat the e"periment in steps till the input

voltage is @ volts. nter the data in Table %&.%7.

%& 8epeat the e"periment with the P#% set at A7J and %A7J. Plot the graph between the input

voltage and the output voltage of the controller Figure %&.%%

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Table %&.%7 Input2/utput relationship in a proportional controller

P#%77J

input Difference

in*1%2b%3

out Difference

out*1'2b'3

Bain 1B3

out9in

-ctual Bain

%779P#b% % b' '

P#A7J

input Difference

in*1%2

b%3

out Difference

out*1'2

b'3

Bain 1B3

out9in

-ctual Bain

%779P#b% % b' '

P#%A7

input Difference

in*1%2

b%3

out Difference

out*1'2

b'3

Bain 1B3

out9in

-ctual Bain

%779P#b% % b' '

Figure %&.%% Proportional controller output change with input

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%&.@.@ Proportional Integral Controller

%@ !et the P# at @77J and TI at %77 s. 8ecord the output. The output will show a step change

due to the proportional contribution and ramp change due to the integral contribution.

%A The value of the step change in the output has a value of ,C- and the slope of the ramp is

given by 1,C-9TI3. 5easure the value of - from the base and the slope dy9d" and enter in

table %&.%'. 8epeat the e"periment keeping the Pb constant at @77J and decreasing the TI

values to >7$ ?7$ 7$ E7$ A7$ @7$ &7$ '7 and %7 s.

%E Draw a graph between the measured and set value of TI. Figure %&.%&.

TI setting

1!3

/ffset 1-3

1base3

/ffset -

1%3

Bradient # D(9D) /ffset -

%2base

TI meas 1!3

D( D)

Figure %&.%& 5easured and e"perimental values of TI.

%&[email protected] Proportional Derivative controller

% - step change cannot be applied to the Proportional derivative controller because of the

presence of the derivative term will saturate the output. The proportional derivative controller

can be tested using the sinusoidal output of the FB%'7 function generator

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%? !et the controller for Proportional Derivative operation. !et the P#*%77J $ TI * >>>>s and

TD*%A s.

%> !et the FB frequency to 7.7% ;0.!et DC voltage high level at &.'A and low level at '.A .

Plot the input and output sinusoid.

'7 How set the TD%*@ s$ frequency at 7.7% ;0. nter the data in the table %'.%@. Increase the

frequency in steps till a frequency of % ;0 is reached.

'% Plot the -82bode diagrams and determine the corner frequency.TD is given as %9c.

'' 8epeat the e"periment for several values of TD and enter in table %'.%A. Plot a graph

between the e"perimental value and the set value of the derivative time Figure %'.%E.

Table %&.%@ Derivative controller values

Input frequency ;0 Input 1C; %3 /utput 1C; '3 -8*'9%

5a"

5in

-mplitude

%

5a"

5in

-mplitude

'

7.7%7.7A

7.%7

7.'7

7.&7

7.@7

7.A7

7.E7

7.7

7.?7

7.>7

%.77

Table %&.%A Derivative !et alue and 5easure alue

TD % !et

sec

From #ode Diagram

;0

Time 5easure 1%9F3

sec

7

@

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%&[email protected] Process Characteristics

5ost processes

Can be modelled by equations of the following form in the +aplace domain. 5any processes may

be higher order than second order plus delay time model. The present e"periment illustrates the

e"perimental determination of the model characteristics$ gain , and Time constant T and delay

time .

)1)(1(

)1)(1(

1

1

21

21

++=

++=

+=

+=

sTsT

Ke

X

Y

timedelaorder!econd

sTsT

K

X

Y

order!econd

Ts

Ke

X

Y

timedelaorder"irst

Ts

K

X

Y

order"irst

s

s

( * input

) * outputT$ T%$ T' * time constants

* delay time

s * +aplace operator

13.% PROCEDURE

Table %&.%E quipment required

Unit nam M!"# N!

!ynthesised Function generator )okogawa 5odel FB%'7

Programmable controller )okogawa 5odel )!%7

Paperless recorder )okogawa 5odel D(%7@

Process simulator !oftware program in )!%7

%&.A.% FB%'7 Function generator

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The FB%'7 !ynthesi0ed Function Benerator is used to provide the sinusoidal function required

for testing the system. The model 7E7%' synthesi0ed function generator can provide voltages in

the frequency range % ;0 to ' 5;0. It can provide sine$ square$ tringuler$ ramp and pulse

functions. There are two channels which are independent. The output can be adusted

continuously in the range of %7 .

%&.A.' D(%72@ Paperless recorder

The D(%7@ Paperless recorder is used to record the result of the testing. The measured data can

also be save to e"ternal storage media such as floppy disks$ 0ip disks and -T- flash memory

cards.

The data that have been saved to an e"ternal storage medium can be displayed on a PC by

using the standard software that comes with the package. The data can also be loaded onto the

D(%7@ to be displayed.

%&.A.& Process simulator

The process simulator is a software where it was programmed into the )!%7 controller. The

software process simulator schematic diagram Figure %&.% provides the first$ second and third

order system used in this present study. The process gain 1,3 is set to '.

Figure %&.% !oftware Process simulator schematic diagram

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%&.A.@ quipment connections

Figure %&.%? quipment connections

%. !et the time constant P7% * %A and P/' * P/& * 7 1THIHB 5H$ PT 8B3. !et at

-CT% 1in HB.5H %$ C/HFIB '3 to 8! 18everse -ction3

'. Put the )!%7 to 5anual mode and set the output 5 * A7J at )!%&E set to Cascade

mode C

&. Change the controller output 5 to 7J.

@. 8ecord the process system input$ ( 1C;7%3 and the process system output )$ 1C;7'3

A. nter the data as shown in table %&.%>

Table %&.%> 8esponse of first order process

Process /utput

Time 1sec3 /utput 1J3 Incomplete response 1J3

7

A

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%7

%A

'7

'A

&7

&A

@7@A

A7

AA

E7

EA

7

A

?7

%&.A.A Data reduction$ results and graphs

%. Plot the output of the system against time.

'. Determine the final value of the curve$ )ma"

&. The value of the input change was taken as from A7J to 7J and equal to '7J. ;ence

gain , *)ma"9'7

@. Plot the percent incomplete response. The percent incomplete respoose is

PI8 * 1)2)7391)ma"2)73


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13.%.&.1 PROCEDURE

%. Connect the equipment as shown in figure %&.'7

Figure %&.'7 quipment connections

'. For third order system in the software process simulator we need to set P7%$ P7' andP7&.

&. -t )!%7 in 5anual mode 153 perform the following:

Process simulator : Disturbance point is not connected. !et the disturbance value

1P%73 *'7$ Time constant P7%$ P7' and P7& are set to %7

)!%7 controller set P*!*A7J and 5*A7J$ P#%*7J$ TI*'7s and TD%*As.

@. !et the )!%7 to -uto mode.


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Figure %&.'' To calculate the value of -$ Tset$ D$ P and

Table %&.'& ffect of P#$ TI and TD on system response

P# TI TD 5a"

amplitude

- 1J3

!ettling

time

1min3

Decay

ratio

D

/scillation

Frequency

T 1min3

rror

J

7 '7 A

%A7 '7 A

&7 '7 A

7 &7 A

7 %' A

7 '7 &7

7 '7 7

7 E7 7

7 E 7

7 >>>> A

7 >>>> E7

7 >>>> &

7 >>>> 7

%A7 >>>> 7

&7 >>>> 7

13.& OPTIMUM PID SETTINGS

For this e"periment also the same equipment as used in %&.A.E.

%&.E.% P8/CD8

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%. !et the following conditions in the processK /peration mode of )!%7 controller to

manual mode 153. Disconnect the disturbance point and set P*!*A7J and 5*A7J.

P#%*%77J$ TI%*>>>>$ TD%*7. Time constant of the process simulator$

P/%*P7'*P7&*%7.

'. !et the operation of )!%7 to -uto mode.

&. Produce a disturbance in the process by increasing the ! to 7J and bringing it quickly

back to A7J

@. if the response curve quickly dampens out 1Figure %'.''3 then the P# is too high. 8educe

the P# and repeat step &. Continue the process till the P and 5 continuously oscillate.

A. Hote the values of P# which is called the ultimate value$ P#u. Hote the corresponding

period of the oscillations which is called the P.

E. !et the value of P#$ TI and TD using the recommended values given in table %A.'&.

Table %&.'& Liegler Hichols !ettings

5ode P# TI TD

P 'P#u

PI '.'P#u 7.?&P

PID %.P#u 7.AP

. !et the above values of the controller settings and determine the response of the system

to a disturbance. Check whether a quarter decay ratio response is obtained. If not slightly

adust the parameter obtained from the tabe %&.'& til a M decay ratio is obtained.

%&.E.' Liegler Hichols second method.

%. !et the following conditions in the processK /peration mode of )!%7 controller to

manual mode 153. Disconnect the disturbance point and set P*!*A7J and 5*A7J.

P#%*%77J$ TI%*>>>>$ TD%*7. Time constant of the process simulator$

P/%*P7'*P7&*%7.

'. Increase 5 by %7J. The response curve will be obtained as shown in Figure %'.'@.

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Figure %&.'@ Typical reaction curve for step change in manual output

&. Draw a tangent to the curve at the point of infle"ion. Determine the values of + * time in

seconds between the step change and the point where the tangent crosses the intial

value of the controlled variable * sec

T * time constant * sec

@. The system gain ,

A. The controller settings are given in table %&.'A.

Table %&.'A Liegler Hichols !ettings5ode P# TI TD

P %77,+9T

PI %%7,+9T &.&+

PID ?&,+9T '+ 7.A+

. !et the above values of the controller settings and determine the response of the system

to a disturbance. Check whether a quarter decay ratio response is obtained. If not slightly

adust the parameter obtained from the tabe %&.'E til a M decay ratio is obtained

13.' CASCADE CONTROL OF PROCESS

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In cascade control two control loops are nested one inside the other. The output of the T&A7

controller becomes the set point of the )!%7 controller. The T&A7 gets an e"ternal setpoint.

The structural diagrams of the connection is shown in figure %&.'.

Figure %&.' Cascade control structure connection

%&..% P8/CD8

%. Connect the equipment as shown in figure %'.'. T&A7 is primary controller variable

while )!%7 is the secondary controlled variable. The )!&E auto manual selector

provides the signal input to the )!%7 and )!%&A auto set selector provides the signal

input to the T&A7

'. !et the P7%$ P7'$ P7& and P7E of the process simulator to %7 each. !et P%7 to '7

&. -t the T&A7 controller$ set P#$ TI and TD at the tuned values obtained from %'.E.

@. !et the )!%&E and )!%&A to cascade mode

A. !et )!%7 to manual mode and set 5*A7J. !et the )!%7 P#%*%A7$ TI%*%7 and

TD%*'7

E. !et )!%7 to -uto

. Introduce a disturbance by changing ! to 7J at )!%7. 8ecord the output 1C;7'3 of

the system

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?. #ring the ! value to A7J. !et the operation of T&A7 controller to -uto

>. How bring the )!%7 controller to cascade mode

%7. Introduce a disturbance by connecting the disturbance point a few second and

disconnect it . 8ecord the output of the system

%%. Tuning the cascade control system

a. if the output of )!%7 controller takes longer time to stable. Try to change the

P#$ TI and TD of )!%7 to get M ratio

b. If changing the PID setting still not satisfy. Try to change the PID setting of T&A7

controller

Figure %&.'? 8esponse of the system to cascade control

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13.( FEED FOR)ARD CONTROL

Feedforward is a method in which the correction to a load disturbance enters the system. In

normal feedback system the effect of the disturbance has to be seen at the P before the

correction can be introduced. IH FF we measure the value of the disturbance and introduce anapproprtiate correction to the 5. The structure in Figure %'.'?.

Figure %&.'? !ingle loop mode

%&..% quipment connections

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Figure %&.'> quipment connections

%&.?.% P8/CD8

%. Connect the equipment as shown in figure %A.'>.

'. !et the time constant P7'*P7&*%7. !et the P7%*%A and P7A*'7

&. !et the P#$TI and TD values at the optimum values obtained in section %'.E.

@. !et the feed forward gain$ P/@*@.

A. !et the output of )!%&E$ 5*7

E. Changing the )!%&E to %7J and by introduce the disturbance with perform the

connection of feed forward point. 8ecord the output of the system.

. !et the feed forward gain$ P/@*%

?. 8epeat step E. 8ecord the output. Compare the response with that obtained in step

There will be improvement. Change the FFB to %.' and repeat the e"periment. Continue

increasing the FFB in steps of 7.% till effect of the disturbance is removed from the

process output

>. ven after the FFB is tuned as in step ?. There will be temporary small effect on the

output. This can be removed by introducing the dynamic element in the feed forward

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path. Increase the feedforward lag F+B time constant to %s. 8epeat step E. Continue

increase the F+B step of % s till the effect of the disturbance is completely removed from

the output.

13.* REVIE) +UESTIONS

%. >%.

experiment 13 - digital pid controller_a

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