Lecture Introduction PID Controllers2010[1]

8/8/2019 Lecture Introduction PID Controllers2010[1]

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Control of Continuous Process

Lecturer:

Dr. Shallon Stubbs


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Types of Process

Discrete Processes

A discrete process consists of distinct operations

with a definite condition for initiating each

operation. Discrete process operations can be

grouped into two categories those that can be

initiated by time and those which are initiated by

an event.


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Types of Process

Continuous Process

A continuous process has uninterrupted inputs

and outputs measures and controls continuous

process variables. The output is maintained at

some desired set-point by continuously adjusting

one or more input to the system.


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Modes of Control

Two-Position Control Mode

Multi-position controllers

Proportional Control Mode (P) Integral Control mode (I)

Derivative Control Mode (D)

PI,PD, PID Control Modes


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PID Controller


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On/off (Two-Position) Mode

This is the cheapest and simplest controlmode. The controller output has only twopossible states, depending on the sign of

the error. Most two position controllers have a

neutral zone to prevent chattering. The

neutral zone is an hysteresis region setupabout the zero error where there is nochange in the control action.


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ON-OFF Controller with Nuetral zone

A liquid level control system linearly converts a displacement of 2 to 3 minto a 4 20 mA signal. A relay serves as the two-position controller. Theliquid level must stay within 2.3 to 2.5 m, what would be the upper and lowerlimit of the nuetral zone in mA.


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Factors that contribute to cycling

(chattering)1) Small Capacitance of the system: Example a

tank of smaller cross-sectional area would

exhibit more drastic changes in height with

volumetric flowrate disturbances.

2) Large Dead-time lag of the process: Refers to

delay between changes in the process variable

being recognized and control action.

3) Large Load changes: regularly and or suddenchanges in the disturbance variables of the

system.


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Multiposition Control Mode

This is a logical extension of the two-

position control-mode in which several

intermediate settings of the controller

output is possible between its upper most

and lower most state.

This is to reduce the cycling behavior

associated with the two-position mode.


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Proportional Control Mode

The proportional control mode changes

the output of the controller proportionally

to the size of the error signal

Controller Equation:

p = kpe + po

Kp proportional gain (%per unit %error)

e percentage error

p0 controller nominal output (zero-error output)


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Proportional Mode


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Direct and Reverse Acting Mode

Reverse Action: In reverse acting mode a

positive error results in an increase of the

controllers output, however this actually

corresponds to the measured variable

falling below the setpoint.

P = KpxE + Po -E = S.P M.V

P = KpxE + Po +E = S.P M.V


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Direct and Reverse Acting Mode

Direct Action: In Direct acting mode a

negative error results in a increase in the

controllers output, hence the output

adjusts in the same direction as the

feedback signal is changing.

P = -KpxE + Po -E = S.P M.V

P = -KpxE + Po +E = S.P M.V


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Proportional Band and Proportional

Gain Proportional Band (PB) is the amount of

error that is required to result in a 100%

change in the controllers output

Proportional Gain (Kp) is the % change in

the controllers output per unit change in

error.

Kp = 100/PB PB = 100/Kp


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Proportional-Mode Offset Error

Problem

One problem with the proportional controller is that it

cannot completely eliminate the error caused by a load

change. A residual error is always required to maintain

the final control element (the valve) at some position

other than corresponding to the controller output po.

This limits the use of the P-controller to only a few cases,

particularly those where a manual reset of the operating

point is possible to eliminate offset.

Proportional control generally is used in processes withminimal load changes or with moderate to small process

lag times which allows for large Kp (i.e very small

proportional band setting).


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Proportional-Mode Offset Error

Problem

Consider the proportional

mode level-control system of

Figure 2b. valve A is linear,

with a flow scale factor 10m3/h

percent controller output. Thecontroller output is nominally

50% with a constant of Kp =

10% per%error. A load change

occurs when flow through

valve B changes from 500

m3/h to 600m3/h . Calculate thenew controller output and

offset error.


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Integral Mode

The integral changes the output of the controllerby an amount proportional to the integral of theerror.

Thus even if the size of the error is constant, thecontroller response would increase with timeduration of the error.

The integral action is essential achieved by

summing the error over time, multiplying thatsum by a the integral gain, and adding thepresent controller output.


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Integral Mode

Control mode Equation:

Where KI is the integral gain or integral rate (per unit time)

! 0pedtKp I


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PI Control Mode

The PI mode controller combines the integraland proportional mode action

The integral mode eliminates the offset error

associated with the proportional-mode action notbe able to adjust the zero-error state of thecontroller to accommodate for load changes

The integral mode provides reset action

because it will continue to adjust the controlleroutput until the error is reduced to zero andwhatever changes it makes to the outputremains even after the error is eliminated.


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PI Control Mode Equation

p = kpe + KpKI + p0

Or

p = kpe + + p0

where KI = 1/Ti and Ti is referred to as the

integral time. This is the time taken for the

integral mode to repeat the action of the

proportional mode. The integral time isgiven in units of time therefore the integral

gain is given in per unit time.

edt

edtT

K

i

p

1


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Response of Proportional Plus Integral

Mode Controller


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Derivative Control Mode

The derivative mode changes the output of thecontroller proportionally to the rate of change ofthe error signal

The derivative mode is an attempt to anticipatethe error by observing the rate of change of theerror and advancing the control action in aneffort to combat the expected error.

The derivative mode is never used alone. When

used in a PD or PID controller, the gain isusually chosen to be small to avoid large suddenchanges in the output response due to a rapidrate of change of the error.


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Derivative Control Mode


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Summary of characterisitics:

1. If the error is zero or constant the modeprovides no output adjustment.

2. If the error is changing with time, themode adjust the output by KD% for a unitrate of change of the error.

3. The direction in which the output is

adjusted is dependent upon the whetherthe error is increasing or decreasing withtime.


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Proportional Derivative Mode

The derivative control mode is sometimes usedwith the proportional mode to reduce thetendency for oscillations and allow a higherproportional gain setting.

The Proportional mode action will response tothe immediate value of the error while thederivative mode response to the future value ofthe error.

The anticipatory action of the derivative modemakes PD and PID controller suitable forprocess with sudden load changes that produceexcessive errors.


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Proportional Plus Derivative Mode

The equation for the controller is given by:

Where KD is the derivative gain (time)

Kp proportional gain

The derivative gain KD may be interpreted

as the time advance into the future forwhich the error size is anticipated.

oDpp pdt

deKKeKp !


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PID Control Mode

The PID control mode is a combination ofthe proportional, integral, and derivativecontrol modes.

The integral mode is used to provide resetaction (eliminate the offset error due toload changes).

The derivative mode reduces the tendencytowards oscillation and providesanticipatory control action.


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PID Controller

The equation of a three-mode controller is givenby:

opIppp

dt

deedtep !

_ a

! dt

tdekkdttekktekLtpL

dpIpp)(

The Transfer function of the Controller is obtain by carrying

out the laplace transformation of the above equation:

)()(

)()( ssEkks

sE

kksEksP dpIpp !

)()( sskks

kkks

dp

Ip

p

!

Gc(s)R(s)

C(s)

P(s)E(s)+

-

Where R(s) Setpoint, C(s) Control Variable, and Gc(s) = P(s)/E(s)


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Block Diagram Reduction

The block diagram ofFigure 1 can be simplified to derive the closed-looptransfer function of the system as follows:

Error = reference feedback variable

E=SP Cm (i)

Controller output = error x controller transfer functionV = Gc

.E (ii)

Manipulated variable = controller output x manipulating element TF

M= Gm. V (iii)

Controlled variable C = manipulating variable x Process TF

C = GPM (iv)

Combining eqn (ii), (iii) and (iv):

C = GmGcGPE let G= GmGcGp (Forward transfer function)

C = G E (v)


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Feedback variable = Controlled variable x Transmitter transfer function

Cm = C.H (vi)

Since C= GEand E= SP Cm, then

C = (SP - Cm).G (vii)

Substituting eqn (vi) into eqn (vii) gives

C = (SP C.H).G

C + CGH= SPG

C(1 + GH) = SPG GH

G

SP !

1

CThe close-loop TF is therefore:

GcGmGp

H

SP C

Cm

+

-Gcl

SP C


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Gd

Gc Gp

Gf

R

D

C

Derive the TransferFunction Expression for C/R and C/D

++

+_

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Lecture Introduction PID Controllers2010[1]