2014 part 7 with hw8 pid+tunning++of+process+dynamics+and+control+(7)(1)
TRANSCRIPT

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ECH 3112 PROCESS CONTROL AND
INSTRUMENTATION
Presentation(7):PID Tuning
Prof. Dr. Said Salah Eldin Elnashaie,
Chemical and Environmental Engineering Department,UPM, Malaysia
And
Chemical and Biological Engineering Department,University of British Columbia(UBC), Vancouver , Canada
Email: [email protected] ; hp: 0122243234
mailto:[email protected]:[email protected] 
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PID Controller

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Hints About PID Tuning
1. Higher Proportional Gain, faster will be the system response, but if it is too
high, the system can become unstable and oscillations will occur.
2. The higher the Integral Gain (or lower the Integral Time Constant), fasterwill be the system response, but if it is too high, then again system can become
unstable and oscillations will occur. If you make the integral gain zero (or
integral time constant too large), then the set point will never be reached and
there will always be some error. Integral term in the PID makes the steady state
error zero.3. Derivative Time Constant introduces damping effect, and tends to provide
stability to the system by limiting the rate of change of the system variable.
What needs to be done is you will have to keep 2 of the terms constant and
slowly increase the 3rd term till the system starts to become oscillatory. But
again, this is just a theoretical guideline, you will have to practically tune thesystem based on your own observation and judgment.

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PID Controller Tuning
A proportionalintegralderivative controller (PID controller) is a generic
control loopfeedback mechanism(controller) widely used in industrial control
systems. A PID controller calculates an "error" value as the difference between
a measured process variableand a desired setpoint. The controller attempts to
minimize the error by adjusting the process control inputs.
The PID controller calculation algorithminvolves three separate constant
parameters, and is accordingly sometimes called threeterm control: theproportional, the integraland derivativevalues, denoted P,I ,and D.Simply
put, these values can be interpreted in terms of time: Pdepends on the present
error, Ion the accumulation of pasterrors, and Dis a prediction of future
errors, based on current rate of change. The weighted sum of these three
actions is used to adjust the process via a control element such as the position of
a control valve, a damper, or the power supplied to a heating element.
http://en.wikipedia.org/wiki/Control_loophttp://en.wikipedia.org/wiki/Feedback_mechanismhttp://en.wikipedia.org/wiki/Controller_(control_theory)http://en.wikipedia.org/wiki/Industrial_control_systemhttp://en.wikipedia.org/wiki/Industrial_control_systemhttp://en.wikipedia.org/wiki/Process_variablehttp://en.wikipedia.org/wiki/Setpoint_(control_system)http://en.wikipedia.org/wiki/Algorithmhttp://en.wikipedia.org/wiki/Proportionality_(mathematics)http://en.wikipedia.org/wiki/Integralhttp://en.wikipedia.org/wiki/Derivativehttp://en.wikipedia.org/wiki/Control_valvehttp://en.wikipedia.org/wiki/Damper_(flow)http://en.wikipedia.org/wiki/Damper_(flow)http://en.wikipedia.org/wiki/Control_valvehttp://en.wikipedia.org/wiki/Derivativehttp://en.wikipedia.org/wiki/Integralhttp://en.wikipedia.org/wiki/Proportionality_(mathematics)http://en.wikipedia.org/wiki/Algorithmhttp://en.wikipedia.org/wiki/Setpoint_(control_system)http://en.wikipedia.org/wiki/Setpoint_(control_system)http://en.wikipedia.org/wiki/Setpoint_(control_system)http://en.wikipedia.org/wiki/Process_variablehttp://en.wikipedia.org/wiki/Industrial_control_systemhttp://en.wikipedia.org/wiki/Industrial_control_systemhttp://en.wikipedia.org/wiki/Controller_(control_theory)http://en.wikipedia.org/wiki/Feedback_mechanismhttp://en.wikipedia.org/wiki/Control_loop 
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Advantages of PID
In the absence of knowledge of the underlying process, a PID controller has
historically been considered to be the best controller.
By tuning the threeparameters in the PID controller algorithm, the controller can provide control
action designed for specific process requirements. The response of the
controller can be described in terms of the responsiveness of the controller to
an error, the degree to which the controller overshootsthe setpoint, and the
degree of system oscillation. Note that the use of the PID algorithm for control
does not guarantee optimal controlof the system or system stability.
Some applications may require using only one or two actions to provide the
appropriate system control. This is achieved by setting the other parameters to
zero. A PID controller will be called a PI, PD, P or I controller in the absence ofthe respective control actions. PI controllers are fairly common, since derivative
action is sensitive to measurement noise, whereas the absence of an integral
term may prevent the system from reaching its target value due to the control
action.
http://en.wikipedia.org/wiki/Overshoot_(signal)http://en.wikipedia.org/wiki/Optimal_controlhttp://en.wikipedia.org/wiki/Optimal_controlhttp://en.wikipedia.org/wiki/Overshoot_(signal) 
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Programmable Logic Controllers (PLCs)
Most modern PID controllers in industry are
implemented in programmable logic controllers(PLCs) or as a panelmounted digital controller.
Software implementations have the advantages
that they are relatively cheap and are flexible withrespect to the implementation of the PID algorithm.
PID temperature controllers are applied in
industrial ovens, plastics injection machinery, hotstamping machines and packing industry.
http://en.wikipedia.org/wiki/Programmable_logic_controllerhttp://en.wikipedia.org/wiki/Programmable_logic_controllerhttp://en.wikipedia.org/wiki/Programmable_logic_controllerhttp://en.wikipedia.org/wiki/Programmable_logic_controller 
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Control loop basics(1)
A familiar example of a control loop is the action taken when
adjusting hot and cold faucets (valves) to maintain the water at a
desired temperature. This typically involves the mixing of two
process streams, the hot and cold water. The person touches the
water to sense or measure its temperature. Based on this feedback
they perform a control action to adjust the hot and cold water valves
until the process temperature stabilizes at the desired value.
The sensed water temperature is the process variableor process
value (PV). The desired temperature is called the setpoint (SP). The
input to the process (the water valve position) is called the
manipulated variable (MV). The difference between the temperature
measurement and the setpoint is the error (e) and quantifies
whether the water is too hot or too cold and by how much.
http://en.wikipedia.org/wiki/Process_variablehttp://en.wikipedia.org/wiki/Process_variable 
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Control loop basics(2)
After measuring the temperature (PV), and then calculating the
error, the controller decides when to change the tap position (MV)
and by how much. When the controller first turns the valve on, it
may turn the hot valve only slightly if warm water is desired, or it
may open the valve all the way if very hot water is desired. This is an
example of a simple proportional control. In the event that hot water
does not arrive quickly, the controller may try to speedup the
process by opening up the hot water valve moreandmore as time
goes by. This is an example of an integral control.
Making a change that is too large when the error is small isequivalent to a high gain controller and will lead to overshoot. If the
controller were to repeatedly make changes that were too large and
repeatedly overshoot the target, the output would oscillatearound
the setpoint in either a constant, growing, or decaying sinusoid.
http://en.wikipedia.org/wiki/Oscillatehttp://en.wikipedia.org/wiki/Sine_wavehttp://en.wikipedia.org/wiki/Sine_wavehttp://en.wikipedia.org/wiki/Oscillate 
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Control Basics(3)
If the oscillations increase with time then the system is unstable,
whereas if they decrease the system is stable. If the oscillations
remain at a constant magnitude the system is marginally stable.
In the interest of achieving a gradual convergence at the desired
temperature (SP), the controller may wish to dampthe anticipated
future oscillations. So in order to compensate for this effect, the
controller may elect to temper its adjustments. This can be thought
of as a derivative control method.
http://en.wikipedia.org/wiki/Marginal_stabilityhttp://en.wikipedia.org/wiki/Dampinghttp://en.wikipedia.org/wiki/Dampinghttp://en.wikipedia.org/wiki/Marginal_stability 
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Control Basics (4)
If a controller starts from a stable state at zero error (PV = SP), then
further changes by the controller will be in response to changes in othermeasured or unmeasured inputs to the process that impact on the process,
and hence on the PV. Variables that impact on the process other than the
MV are known as disturbances. Generally controllers are used to reject
disturbances and/or implement setpoint changes. Changes in feed watertemperature constitute a disturbance to the faucet temperature control
process.
In theory, a controller can be used to control any process which has a
measurable output (PV), a known ideal value for that output (SP) and aninput to the process (MV) that will affect the relevant PV. Controllers are
used in industry to regulate temperature, pressure, flow rate, chemical
composition, speedand practically every other variable for which a
measurement exists.
http://en.wikipedia.org/wiki/Temperaturehttp://en.wikipedia.org/wiki/Pressurehttp://en.wikipedia.org/wiki/Flow_ratehttp://en.wikipedia.org/wiki/Chemicalhttp://en.wikipedia.org/wiki/Speedhttp://en.wikipedia.org/wiki/Speedhttp://en.wikipedia.org/wiki/Chemicalhttp://en.wikipedia.org/wiki/Flow_ratehttp://en.wikipedia.org/wiki/Pressurehttp://en.wikipedia.org/wiki/Temperature 
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PID controller theory

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Proportional term: The proportional term produces an output value that is proportional
to the current error value. The proportional response can be adjusted by multiplying the
error by a constant Kp, called the proportional gain constant. The proportional term is given
by:
Plot of PV vs time, for three values of Kp(Kiand Kdheld constant)

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A high proportional gain results in a large change in the output for a given
change in the error. If the proportional gain is too high, the system can become
unstable (see the section on loop tuning). In contrast, a small gain results in a
small output response to a large input error, and a less responsive or less
sensitive controller. If the proportional gain is too low, the control action may
be too small when responding to system disturbances. Tuning theory and
industrial practice indicate that the proportional term should contribute the
bulk of the output change.
Droop (offset) : Because a nonzero error is required to drive it, a
proportional controller generally operates with a steadystate error, referred to
as droop(offset). Droop is proportional to the process gain and inversely
proportional to proportional gain of the controller. Droop may be mitigated by
adding a compensating bias termto the setpoint or output, or corrected
dynamically by adding an integral term.
http://en.wikipedia.org/wiki/PID_controllerhttp://en.wikipedia.org/wiki/Biasinghttp://en.wikipedia.org/wiki/Biasinghttp://en.wikipedia.org/wiki/PID_controller 
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Integral term

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PV vs. time diagram for different values of integral gain
Plot of PV vs time, for three values of Ki(Kpand Kdheld constant)
d

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Derivative term
Plot of PV vs time, for three values of Kd(Kpand Kiheld constant)
.The magnitude of the contribution of the derivative term to the overall control action is termed the derivative gain, K
d.
The derivative term is given by:

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Loop tuning (1)
Tuningcontrol loop is adjustment of its control parameters
(proportional band/gain, integral gain/reset, derivative gain/rate) tooptimum values for the desired control response. Stability (bounded
oscillation) is a basic requirement, but beyond that, different systems
have different behavior, different applications have different
requirements, and requirements may conflict with one another. PID tuning is a difficult problem, even though there are only 3
parameters and in principle is simple to describe, because it must
satisfy complex criteria within the limitations of PID control. There
are accordingly various methods for loop tuning, and more
sophisticated techniques are the subject of patents; this section of the
course describes some traditional manual methods for loop tuning.
http://en.wikipedia.org/wiki/PID_controllerhttp://en.wikipedia.org/wiki/PID_controller 
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Loop tuning(2)
Designing and tuning a PID controller appears to be conceptually
intuitive, but can be hard in practice, if multiple (and often
conflicting) objectives such as short transient and high stability
are to be achieved. Usually, initial designs need to be adjusted
repeatedly through computer simulations until the closedloop
system performs or compromises as desired.
Some processes have a degree of nonlinearityand so parameters
that work well at fullload conditions don't work when the process
is starting up from noload; this can be corrected by gain
scheduling(using different parameters in different operating
regions). PID controllers often provide acceptable control usingdefault tunings, but performance can generally be improved by
careful tuning, and performance may be unacceptable with poor
tuning.
http://en.wikipedia.org/wiki/Nonlinear_systemhttp://en.wikipedia.org/wiki/Gain_schedulinghttp://en.wikipedia.org/wiki/Gain_schedulinghttp://en.wikipedia.org/wiki/Gain_schedulinghttp://en.wikipedia.org/wiki/Gain_schedulinghttp://en.wikipedia.org/wiki/Nonlinear_system 
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Stability
If the PID controller parameters (the gains of the
proportional, integral and derivative terms) are chosenincorrectly, the controlled process input can be unstable,
i.e., its output diverges, with or without oscillation, and
is limited only by saturation or mechanical breakage.
Instability is caused by excessgain, particularly in thepresence of significant lag.
Generally, stabilization of response is required and the
process must not oscillate for any combination of process
conditions and setpoints, though sometimes marginal
stability(bounded oscillation) is acceptable or desired
http://en.wikipedia.org/wiki/Divergence_(computer_science)http://en.wikipedia.org/wiki/Oscillationhttp://en.wikipedia.org/wiki/Laghttp://en.wikipedia.org/wiki/Marginal_stabilityhttp://en.wikipedia.org/wiki/Marginal_stabilityhttp://en.wikipedia.org/wiki/Marginal_stabilityhttp://en.wikipedia.org/wiki/Marginal_stabilityhttp://en.wikipedia.org/wiki/Laghttp://en.wikipedia.org/wiki/Oscillationhttp://en.wikipedia.org/wiki/Divergence_(computer_science) 
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Optimum behavior
The optimum behavior on a process change or setpoint
change varies depending on the application. Two basic requirements are regulation(disturbance
rejectionstaying at a given setpoint) and commandtracking(implementing setpoint changes, SERVO).
SRVO is how well the controlled variable tracks the
desired value. Specific criteria for command tracking
include rise timeand settling time. Some processes must
not allow an overshoot of the process variable beyondthe setpoint if, for example, this would be unsafe. Other
processes must minimize the energy expended in
reaching a new setpoint.
http://en.wikipedia.org/wiki/Rise_timehttp://en.wikipedia.org/wiki/Settling_timehttp://en.wikipedia.org/wiki/Settling_timehttp://en.wikipedia.org/wiki/Rise_time 
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Overview of methods
There are several methods for tuning a PID loop. The most
effective methods generally involve the development of some form
of process model, then choosing P, I, and D based on the dynamic
model parameters. Manual tuning methods can be relatively
inefficient, particularly if the loops have response times on the
order of minutes or longer.
The choice of method will depend largely on whether or not the
loop can be taken "offline" for tuning, and on the response time
of the system. If the system can be taken offline, the best tuning
method often involves subjecting the system to a step change in
input, measuring the output as a function of time, and using thisresponse to determine the control parameters.
Ch i t i th d

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Choosing a tuning method
Method Advantages Disadvantages
Manual tuning No math required; online. Requires experienced personnel
ZieglerNichols Proven method; online. Process upset, some trialanderror, very aggressive tuning
Software tools Consistent tuning; online or Some cost or training involved
offline  can employ computer
automated control system design
(CAutoD) techniques; may include
valve and sensor analysis; allows
simulation before downloading; can
support nonsteadystate (NSS) tuning.
CohenCoon Good process models Some math; offline;only good for first
order processes
http://en.wikipedia.org/wiki/CAutoDhttp://en.wikipedia.org/wiki/CAutoD 
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Manual tuning (reds are added to the slide and see next slide)
If the system must remain online, one tuning method is to first set Ki and Kd
values to zero. Increase the Kp until the output of the loop oscillates, this Kp,
will be called Ku then the Kpshould be set to approximately half of that valueKufor a "quarter amplitude decay" type response. Then increase KIuntil any
offset is corrected in sufficient time for the process. However, too much KI will
cause instability. Finally, increase Kd, if required, until the loop is acceptably
quick to reach its reference after a load disturbance. However, too much will
cause excessive response and overshoot. A fast PID loop tuning usuallyovershoots slightly to reach the setpoint more quickly; however, some systems
cannot accept overshoot, in which case an overdampedclosedloop system is
required, which will require a setting significantly less than half that of the
setting that was causing oscillation.[citat ion needed]
Eff t f i i t i d d tl
http://en.wikipedia.org/wiki/Overdampinghttp://en.wikipedia.org/wiki/Wikipedia:Citation_neededhttp://en.wikipedia.org/wiki/Wikipedia:Citation_neededhttp://en.wikipedia.org/wiki/Overdampinghttp://en.wikipedia.org/wiki/Overdampinghttp://en.wikipedia.org/wiki/Overdamping 
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Effects of increasinga parameter independently

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Comment about previous slide

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Modifications to the PID algorithm
Basic PID algorithm presents some challenges in control applications
that have been addressed by minor modifications to the PID form.
1 Integral windup: One common problem resulting from the ideal PID
implementations is integral windup, where a large change in setpoint occurs (say
a positive change) and the integral term accumulates an error larger than the
maximal value for the regulation variable (windup), thus the system overshootsand continues to increase as this accumulated error is unwound. This problem can
be addressed by:
Initializing the controller integral to a desired value
Increasing the setpoint in a suitable ramp
Disabling the integral function until the PV has entered the controllable region
Preventing the integral term from accumulating above or below pre
determined bounds
http://en.wikipedia.org/wiki/Integral_winduphttp://en.wikipedia.org/wiki/Integral_windup 
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2Overshooting from known disturbances
For example, a PID loop is used to control the temperature
of an electric resistance furnace where the system has
stabilized. Now when the door is opened and something
cold is put into the furnace the temperature drops below
the setpoint. The integral function of the controller tendsto compensate this error by introducing another error in
the positive direction. This overshoot can be avoided by
freezing of the integral function after the opening of thedoor for the time the control loop typically needs to reheat
the furnace.

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Replacing the integral function by a model based part
Timeresponse of the system is often approximately known. Then it is an
advantage to simulate this timeresponse with a model and to calculate some
unknown parameters from the actual response of the system. The model with its
experimentally obtained parameters can estimate the dynamic behavior of the
system. With this estimation, it is possible to calculate the required manipulated
variable by the error of the PV and the setpoint . The result can then be used
instead of the integral function. This also achieves a control error of zero in thesteadystate, but avoids integral windup and can give a significantly improved
control action compared to an optimized PID controller. This type of controller
does work properly in an open loop situation which causes integral windup with
an integral function. This is an advantage if, for example, the heating of a furnace
has to be reduced for some time because of the failure of a heating element, or if
the controller is used as an advisory system to a human operator who may not
switch it to closedloop operation. It may also be useful if the controller is inside a
branch of a complex control system that may be temporarily inactive.
PI t ll (1)

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PI controller(1)
Basic block of a PI controller
A PI Controller (proportionalintegral controller) is a
special case of the PID controller in which the derivative
(D) of the error is not used.The controller output is given
by ( see next slide)

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PI Controller(2)

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PI Controller(3). Without derivative action
Deadband

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Deadband
Many PID loops control a mechanical device (for example, a valve).
Mechanical maintenance can be a major cost and wear leads to
control degradation in the form of either stictionor a deadbandin
the mechanical response to an input signal. The rate of mechanical
wear is mainly a function of how often a device is activated to make
a change. Where wear is a significant concern, the PID loop may
have an output deadbandto reduce the frequency of activation of
the output (valve). This is accomplished by modifying the controller
to hold its output steady if the change would be small (within the
defined deadband range). The calculated output must leave the
deadband before the actual output will change.
Deadband is provided as a range the control action woks only
outside it
http://en.wikipedia.org/wiki/Stictionhttp://en.wikipedia.org/wiki/Deadbandhttp://en.wikipedia.org/wiki/Deadbandhttp://en.wikipedia.org/wiki/Deadbandhttp://en.wikipedia.org/wiki/Deadbandhttp://en.wikipedia.org/wiki/Deadbandhttp://en.wikipedia.org/wiki/Deadbandhttp://en.wikipedia.org/wiki/Deadbandhttp://en.wikipedia.org/wiki/Deadbandhttp://en.wikipedia.org/wiki/Stiction 
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Setpoint step change
The proportional and derivative terms can produce excessive movement in the output when
a system is subjected to an instantaneous step increase in the error, such as a large setpoint
change. In the case of the derivative term, this is due to taking the derivative of the error,which is very large in the case of an instantaneous step change. As a result, some PID
algorithms incorporate the following modifications:
Derivative of the process variable: In this case the PID controller measures the derivative of
the measured process variable(PV), rather than the derivative of the error. This quantity is
always continuous (i.e., never has a step change as a result of changed setpoint). For this
technique to be effective, the derivative of the PV must have the opposite sign of the
derivative of the error, in the case of negative feedback control.
Setpoint ramping:In this modification, the setpoint is gradually moved from its old value
to a newly specified value using a linear or first order differential ramp function. This
avoids the discontinuitypresent in a simple step change.
Setpoint weighting:It uses different multipliers for the error depending on which element
of the controller it is used in. The error in the integral term must be the true control error to
avoid steadystate control errors. This affects the controller's setpoint response. These
parameters do not affect the response to load disturbances and measurement noise.
http://en.wikipedia.org/wiki/Process_variablehttp://en.wikipedia.org/wiki/Discontinuity_(mathematics)http://en.wikipedia.org/wiki/Discontinuity_(mathematics)http://en.wikipedia.org/wiki/Process_variable 
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Limitations of PID control
1While PID controllers are applicable to many control
problems, and often perform satisfactorily without anyimprovements or even tuning, they can perform poorly in
some applications, and do not in general provide optimal
control. The fundamental difficulty with PID control is
that it is a feedbacksystem, with constantparameters, and
no direct knowledge of the process, and thus overall
performance is reactive and a compromisewhile PID
control is the best controller with no model of the process,
better performance can be obtained by incorporating a
model of the process.
2 The most significant improvement is to incorporate feedforward
http://en.wikipedia.org/wiki/Optimal_controlhttp://en.wikipedia.org/wiki/Optimal_controlhttp://en.wikipedia.org/wiki/Optimal_controlhttp://en.wikipedia.org/wiki/Optimal_controlhttp://en.wikipedia.org/wiki/Optimal_controlhttp://en.wikipedia.org/wiki/Optimal_control 
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2 The most significant improvement is to incorporate feedforward
control with knowledge about the system, and using the PID only to
control error. Alternatively, PIDs can be modified in more minor
ways, such as by changing the parameters (either gain schedulingindifferent use cases or adaptively modifying them based on
performance), improving measurement (higher sampling rate,
precision, and accuracy, and lowpass filtering if necessary), or
cascading multiple PID controllers.
PID controllers, when used alone, can give poor performance when
the PID loop gains must be reduced so that the control system does
not overshoot, oscillate or huntabout the control setpoint value.
They also have difficulties in the presence of nonlinearities , may
tradeoff regulation versus response time, do not react to changing
process behavior (say, the process changes after it has warmed up),
and have lag in responding to large disturbances.
Li it
http://en.wikipedia.org/wiki/Gain_schedulinghttp://en.wikipedia.org/wiki/Hunting_oscillationhttp://en.wikipedia.org/wiki/Hunting_oscillationhttp://en.wikipedia.org/wiki/Gain_scheduling 
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Linearity
Another problem faced with PID controllers is that they are
linear, and in particular symmetric. Thus, performance of PIDcontrollers in nonlinear systems (such as HVAC( Heating
VentilationAir Conditioning or HighVoltage Alternating
Current)systems. Also nonisothermal chemical reactors and
distillation columns and other nonisothermal chemical
engineering processes) is variable. For example, in temperature
control, a common use case is active heating (via a heating
element) but passive cooling (heating off, but no cooling), soovershoot can only be corrected slowlyit cannot be forceddownward. In this case the PID should be tuned to be over
damped, to prevent or reduce overshoot, though this reduces
performance (it increases settling time).
N i i d i ti
http://en.wikipedia.org/wiki/HVAC_control_systemhttp://en.wikipedia.org/wiki/HVAC_control_systemhttp://en.wikipedia.org/wiki/HVAC_control_systemhttp://en.wikipedia.org/wiki/HVAC_control_systemhttp://en.wikipedia.org/wiki/HVAC_control_systemhttp://en.wikipedia.org/wiki/HVAC_control_systemhttp://en.wikipedia.org/wiki/HVAC_control_systemhttp://en.wikipedia.org/wiki/HVAC_control_systemhttp://en.wikipedia.org/wiki/HVAC_control_systemhttp://en.wikipedia.org/wiki/HVAC_control_system 
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Noise in derivative
A problem with the derivative term is that small amounts
of measurement or process noisecan cause large amountsof change in the output. It is often helpful to filter the
measurements with a lowpass filterin order to remove
higherfrequency noise components. However, lowpassfiltering and derivative control can cancel each other out,
so reducing noise by instrumentation is a much better
choice. Alternatively, a nonlinear median filtermay be
used, which improves the filtering efficiency and practical
performance. In some cases, the differential band can be
turned off in many systems with little loss of control. This
is equivalent to using the PID controller as a PI controller.
Improvements
http://en.wikipedia.org/wiki/Noisehttp://en.wikipedia.org/wiki/Lowpass_filterhttp://en.wikipedia.org/wiki/Median_filterhttp://en.wikipedia.org/wiki/PID_controllerhttp://en.wikipedia.org/wiki/PID_controllerhttp://en.wikipedia.org/wiki/Median_filterhttp://en.wikipedia.org/wiki/Lowpass_filterhttp://en.wikipedia.org/wiki/Lowpass_filterhttp://en.wikipedia.org/wiki/Lowpass_filterhttp://en.wikipedia.org/wiki/Noise 
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Improvements
Feedforward : The control system performance can be improved by combining
the feedback(or closedloop) control of a PID controller with feedforward(or
openloop) control. Knowledge about the system (such as the desired accelerationand inertia) can be fed forward and combined with the PID output to improve the
overall system performance. The feedforward value alone can often provide the
major portion of the controller output. The PID controller can be used primarily
to respond to whatever difference or errorremains between the setpoint (SP) and
the actual value of the process variable (PV). Since the feedforward output is not
affected by the process feedback, it can never cause the control system to oscillate,
thus improving the system response and stability. For example, in most motion
control systems, in order to accelerate a mechanical load under control, more force
or torque is required from the prime mover, motor, or actuator(e.g.: the actingdiaphragm part of the control valve. The PID loop in this situation uses the
feedback information to change the combined output to reduce the remaining
difference between the process setpoint and the feedback value. Working together,
the combined openloop feedforward controller and closedloop PID controller
can provide a more responsive, stable and reliable control system.
O i
http://en.wikipedia.org/wiki/Feedbackhttp://en.wikipedia.org/wiki/Feed_forward_(control)http://en.wikipedia.org/wiki/Feed_forward_(control)http://en.wikipedia.org/wiki/Feed_forward_(control)http://en.wikipedia.org/wiki/Feed_forward_(control)http://en.wikipedia.org/wiki/Feedback 
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Other improvements
In addition to feedforward, PID controllers are often
enhanced through methods such as PID gain scheduling(changing parameters in different operating conditions),
fuzzy logicor computational verb logic. Further practical
application issues can arise from instrumentation
connected to the controller. A high enough sampling rate,
measurement precision, and measurement accuracy are
required to achieve adequate control performance.
Another new method for improvement of PID controller is
to increase the degree of freedom by using fractional order.
The order of the integrator and differentiator add
increased flexibility to the controller.
Cascade control
http://en.wikipedia.org/wiki/Gain_schedulinghttp://en.wikipedia.org/wiki/Fuzzy_logichttp://en.wikipedia.org/w/index.php?title=Computational_verb_logic&action=edit&redlink=1http://en.wikipedia.org/w/index.php?title=Computational_verb_logic&action=edit&redlink=1http://en.wikipedia.org/wiki/Fuzzy_logichttp://en.wikipedia.org/wiki/Gain_scheduling 
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Cascade control
One distinctive advantage of PID controllers is that two PID
controllers can be used together to yield better dynamic
performance. This is called cascaded PID control. In cascade control
there are two PIDs arranged with one PID controlling the setpoint
of another. A PID controller acts as outer loop controller, which
controls the primary physical parameter, such as fluid level orvelocity. The other controller acts as inner loop controller, which
reads the output of outer loop controller as setpoint, usually
controlling a more rapid changing parameter, flow rate or
acceleration. It can be mathematically proven that the workingfrequency of the controller is increased and the time constant of the
object is reduced by using cascaded PID controllers .
See example next slide
Example of cascade PID described in previous slide

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Example of cascade PID described in previous slide
For example, a temperaturecontrolled circulating bath has two
cascade PID controllers, each with its own thermocouple sensor. The
outer controller controls temperature of water using a thermocouple
located far from the heater where it accurately reads temperature of
the bulk of the water. The error term of this PID controller is the
difference between the desired bath temperature and measured
temperature. Instead of controlling the heater directly, outer PID
controller sets a heater temperature goal for the inner PID
controller. The inner PID controller controls the temperature of the
heater using a thermocouple attached to the heater. The innercontroller's error term is the difference between this heater
temperature setpoint and the measured temperature of the heater.
Its output controls the actual heater to stay near this setpoint.

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Alt ti N l t d PID F

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Alternative Nomenclature and PID Forms
Reciprocal and % gain

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Reciprocal and % gain
B i d i i i PV

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Basing derivative action on PV
Basing proportional action on PV

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Basing proportional action on PV
Laplace form of the PID controller

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Laplace form of the PID controller
Sometimes it is useful to write the PID regulator in
Laplace transformform:
Having the PID controller written in Laplace form and
having the transfer functionof the controlled system
makes it easy to determine the closedloop transfer
function of the system.
PID Pole Zero Cancellation
http://en.wikipedia.org/wiki/Laplace_transformhttp://en.wikipedia.org/wiki/Transfer_functionhttp://en.wikipedia.org/wiki/Transfer_functionhttp://en.wikipedia.org/wiki/Laplace_transform 
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PID Pole Zero Cancellation
Di t i l t ti

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Discrete implementation
Pseudocode

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Pseudo code
Here is a simple software loop that implements a PID algorithm:[X]
previous_error = 0
integral = 0
start:error = setpoint  measured_value (or opposite)
integral = integral + error*dt
derivative = (error  previous_error)/dtoutput = Kp*error + Ki*integral + Kd*derivative
previous_error = error
wait(dt)
goto start
See next slide
Reference[x]:^ Jump up to: ab"PID process control, a "Cruise Control"
example". CodeProject. 2009. Retrieved 4 November 2012.
On the Pseudo code in the Previous slide
http://en.wikipedia.org/wiki/PID_controllerhttp://en.wikipedia.org/wiki/PID_controllerhttp://en.wikipedia.org/wiki/PID_controllerhttp://en.wikipedia.org/wiki/PID_controllerhttp://www.codeproject.com/Articles/36459/PIDprocesscontrolaCruiseControlexamplehttp://www.codeproject.com/Articles/36459/PIDprocesscontrolaCruiseControlexamplehttp://www.codeproject.com/Articles/36459/PIDprocesscontrolaCruiseControlexamplehttp://www.codeproject.com/Articles/36459/PIDprocesscontrolaCruiseControlexamplehttp://en.wikipedia.org/wiki/PID_controllerhttp://en.wikipedia.org/wiki/PID_controllerhttp://en.wikipedia.org/wiki/PID_controllerhttp://en.wikipedia.org/wiki/PID_controller 
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On the Pseudocode in the Previous slide
In this example, two variables that will be maintained within the
loop are initializedto zero, then the loop begins. The current erroris
calculated by subtracting the measured_value(the process variable
or PV) from the current setpoint(SP). Then, integraland derivative
values are calculated and these and the errorare combined with
three preset gain termsthe proportional gain, the integral gain andthe derivative gainto derive an outputvalue. In the real world, thisis D to A convertedand passed into the process under control as the
manipulated value (or MV). The current error is stored elsewhere
for reuse in the next differentiation, the program then waits until dtseconds have passed since start, and the loop begins again, reading
innew values for the PV and the setpoint and calculating a new
value for the error.
Manufacturers names for gains as discussed earlier
http://en.wikipedia.org/wiki/Initialization_(programming)http://en.wikipedia.org/wiki/Integralhttp://en.wikipedia.org/wiki/Derivativehttp://en.wikipedia.org/wiki/Digitaltoanalog_converterhttp://en.wikipedia.org/wiki/Analogtodigital_converterhttp://en.wikipedia.org/wiki/Analogtodigital_converterhttp://en.wikipedia.org/wiki/Analogtodigital_converterhttp://en.wikipedia.org/wiki/Analogtodigital_converterhttp://en.wikipedia.org/wiki/Digitaltoanalog_converterhttp://en.wikipedia.org/wiki/Derivativehttp://en.wikipedia.org/wiki/Integralhttp://en.wikipedia.org/wiki/Initialization_(programming) 
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Manufacturers names for gains as discussed earlier
Manufacturers of PID controllers use different names to identify the
three modes. These equations show the relationships:
P Proportional Band= 100/gain
I Integral= 1/reset (units of time)
D Derivative= rate = preact (units of time)
Depending on the manufacturer, integral or reset action is set ineither time/repeat or repeat/time. One is just the reciprocal of the
other. Note that manufacturers are not consistent and often use reset
in units of time/repeat or integral in units of repeats/time. Derivative
and rate are the same.
Choosing the proper values for P, I, and D is called "PID Tuninga discussed earlier.
LogLog and Semi log papers for Bode Plots
http://www.expertune.com/tutor.aspxhttp://www.expertune.com/tutor.aspxhttp://www.expertune.com/tutor.aspxhttp://www.expertune.com/tutor.aspxhttp://www.expertune.com/tutor.aspxhttp://www.expertune.com/tutor.aspx 
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in radians/sec on the horizontal axis

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in radians/sec on the horizontal axis
A Typical Bode Diagram

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A Typical Bode Diagram
Effect of PID on Bode Diagram as shown in previous slide

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Effect of PID on Bode Diagram as shown in previous slide
With a PID controller AR has a dip near the center of the frequency
response. Integral action gives the controller high gain at low
frequencies, and derivative action causes the gain to start rising after
the "dip". At higher frequencies the filter on derivative action limits
the derivative action. At very high frequencies (above 314
radians/time; the Nyquist frequency) the controller phase and AR
increase and decrease quite a bit because of discrete sampling. If the
controller had no filter the controller amplitude ratio would steadily
increase at high frequencies up to the Nyquist frequency (1/2 the
sampling frequency). The controller phase now has a hump due tothe derivative lead action and filtering. The time response is less
oscillatory than with the PI controller. Derivative action has helped
stabilize the loop.
PID

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PID
Gc(s)
Going Back to the Tank Level Control ProblemC t l C fi ti f th Si l T k P bl

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Control Configurations for the Simple Tank Problem
First Configuration: we look at this simple feedbacklevel control using the
tank level h as the measured variable and the output flow rate F as the
control (or manipulated) variable as shown in Figure 5.18 ( next slide),where Fi is the input variable (disturbance), F is the output variable
(control or manipulated variable), and h is the output variable (liquidlevel
measurement). The model equation is given by:
AC. (dh/dt ) = Fi  F (5:15) & St.St. :AC. (dhss/dt ) = Fiss  F ss (5:17) ;
(dhss/dt )=0.0. Define deviation variables by subtracting 5.17 from 5.15 toget: AC . (d[hhss ]/dt ) = [Fi  Fiss ][F  Fss] (5.18).
Define the deviation variables as : y =( h hSS) ; d = (Fi FiSS) ; u = (F  FSS).
Thus Eq. (5.18) can be rewritten in terms of deviation variables as:
AC
. (dy/dt )= du ; This equation can be rearranged to give :dy/dt =(1/AC). (d u) (5:19). If the simple control law is a proportionalcontrol on F; that is F = FSS+ K.(h  hSS) (5:20). Where K is the
proportional controller gain. Equation (5.20) can be rearranged as:
F  FSS= K.(h  hSS) or u= k.y ; i.e.: dy/dt =(1/AC). (d K.y) .
Input Disturbance next slide
Input Disturbance For Feedback with Proportional Controller

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Input Disturbance For Feedback with Proportional Controller
Cross sectional area Ac (m2)
Fi(m3/sec) Set point, h=hss
h (height keep constant)
Fig.5.18 F (m3/sec)
Proportional
Controller
Let us suppose that Fiincreases from FiSS to Fi = FiSS + , then we haved = FiSS + FiSS = . Thus, Eq. (5.19) becomes : dy/dt =(1/AC)[K.y] (5.21)with the initial condition at , t = 0, y = 0 (which means that at t = 0, the system is at its
steady state). If is a constant, then we can write (/AC)=a1 and (K/AC) = a2 .Thus, Eq.(5.21) becomes : (dy/dt )= a1  (a2 . y ) (5:22) which can be rewritten as
dy/(a1(a2.y)) =dt ; On integrating the above equation, we get:(1/a2). ln(a1  a2 .y)=t +Const . (integration constant from initial conditions)
Const.= (1/a2). ln(a1 ). After some simple manipulations, we get
y =(a1/a2 ). [ 1 exp( a2 . t)] (5:23) check this eq. 5:23 and confirm if it is correct
This closed loop with K=0, does not show the self regulatory behavvior of open loop shown
earlier because F= Fss
Simplifications in the earlier tank level control problem:

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Simplifications in the earlier tank level control problem:
We did not take the control action on the C of the valve but onoutlet flow rate directly:
F = FSS+ Kc.(h  hSS), therefore not taking into consideration the selfregulatory effect of F dependence over h even without controller.
Here we will have : F = FSS+ Kc.(h  hSS), (1)
C= Css + Kc.(h  hSS), (2)Between (1) and (2) we get: F= Csshss+ [Css+ Kc.(hhss)] hhhss=e and if we want to use PID controller we should have:
F= Csshss+ [Css+ {Kc.e +KI. e.dt +KD . de/dt}]. h (3) This is F for the unsteady equation where the PID controller is
controlling F through e deviating from zero ( notice that: Csshss=Fss)AC . (d[hhss ]/dt ) = [Fi  Fiss ][F  Fss] (5.18). Becomes using (3):
(de/dt ) =(1/Ac){[Fi  Fiss ][Css+{Kc.e +KI. e.dt+KD. de/dt}].h}(5.18a) nonlinear equation see next page. Notice that e is the same
deviation of the state variable h we called y before.
Linearization of a non linear equation

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Linearization of a nonlinear equation

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Keep the problem linear

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Keep the problem linear
Simple Introduction to Block Diagram Algebra

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Simple Introduction to Block Diagram Algebra
Simple block diagram

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Simple block diagram
Open loop, no control Qi(s)
Fig. 5.37
+
Q0(s)  H(s)
(1/A.s)
Closed Loop (Including the Effect of the Feedback Control)

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(Kc+(KI/s)+Kd.s)

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1/(A.s)
1/(A.s)
(Kc+(KI/s)+Kd.s)
More General Form of Previous Slide

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Call the process Transfer Function(TF): Gp(s) and the
disturbance TF: Gd(s) and the state variable in the s
domain (H(s)) as X(s), C(s) input to the open loopprocess and D(s) is input disturbance. Then the open
loop equation in the s domain is thus:
X(s)= Gd(s) . D(s)  Gp(s). C(s) (5.47) Controller Equation: C(s)= Gc(s). X(s) (5.48)
Where Gc(s) is TF of controller. From Equations 5.47 &
5.48 we get: X(s)= Gd(s) . D(s)  Gp(s). Gc(s). X(s)
Which can be rearranged as: X(s){1+ Gp(s) . Gc(s)}= Gd(s). D(s)
Thus: X(s) = [Gd(s). D(s)]/[{1+ Gp(s) . Gc(s)}] (see Fig.5.41,p. 406))
X(s) = G(s). D(s), G(s)= [Gd(s)]/[{1+ Gp(s) . Gc(s)}]=overall TF
Application for the tank problem

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Application for the tank problem

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For PID controller

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For PID controller
In the 4th line from top in the previous slide, correct in the LHS of

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In the 4 line from top in the previous slide, correct in the LHS of
the equation: G(s).D)s) to G(s).D(s)

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Notice
Notice also in the slide before the last the
RHS is multiplied by D(s). And G(s).D(s) is
not the overall T.F. , the overall T.F. is G(s)
only.

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HW8

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