Download - 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)+
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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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Block Diagram Reduction
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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Block Diagram Reduction
Gd
Gc Gp
Gf
R
D
C
Derive the TransferFunction Expression for C/R and C/D
++
+_