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

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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

    E-mail: [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 proportional-integral-derivative 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 set-point. 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 three-term 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 set-point, 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 panel-mounted 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 set-point (SP). The

    input to the process (the water valve position) is called the

    manipulated variable (MV). The difference between the temperature

    measurement and the set-point 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 speed-up the

    process by opening up the hot water valve more-and-more 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 set-point 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 set-point 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 (off-set) : Because a non-zero error is required to drive it, a

    proportional controller generally operates with a steady-state error, referred to

    as droop(off-set). 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 set-point 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 closed-loop

    system performs or compromises as desired.

    Some processes have a degree of nonlinearityand so parameters

    that work well at full-load conditions don't work when the process

    is starting up from no-load; 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 set-points, 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 set-point

    change varies depending on the application. Two basic requirements are regulation(disturbance

    rejectionstaying at a given set-point) and commandtracking(implementing set-point 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 set-point if, for example, this would be unsafe. Other

    processes must minimize the energy expended in

    reaching a new set-point.

    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 trial-and-error, 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 non-steady-state (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 set-point more quickly; however, some systems

    cannot accept overshoot, in which case an over-dampedclosed-loop 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 set-point 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 set-point 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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    2-Overshooting 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 set-point. 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

    Time-response of the system is often approximately known. Then it is an

    advantage to simulate this time-response 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 set-point . The result can then be used

    instead of the integral function. This also achieves a control error of zero in thesteady-state, 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 closed-loop 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 (proportional-integral 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

    Dead-band

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    Dead-band

    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 dead-bandin

    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 dead-bandto 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 dead-band range). The calculated output must leave the

    dead-band before the actual output will change.

    Dead-band 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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    Set-point 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 set-point

    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 set-point). 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.

    Set-point ramping:In this modification, the set-point 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.

    Set-point 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 steady-state control errors. This affects the controller's set-point 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

    1-While 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 feed-forward

    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 feed-forward

    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 low-pass 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 set-point value.

    They also have difficulties in the presence of non-linearities , may

    trade-off 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 non-linear systems (such as HVAC( Heating

    Ventilation-Air Conditioning or High-Voltage Alternating

    Current)systems. Also non-isothermal chemical reactors and

    distillation columns and other non-isothermal 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 low-pass filterin order to remove

    higher-frequency noise components. However, low-passfiltering 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/Low-pass_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/Low-pass_filterhttp://en.wikipedia.org/wiki/Low-pass_filterhttp://en.wikipedia.org/wiki/Low-pass_filterhttp://en.wikipedia.org/wiki/Noise
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    Improvements

    Feed-forward : The control system performance can be improved by combining

    the feedback(or closed-loop) control of a PID controller with feed-forward(or

    open-loop) 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 feed-forward 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 set-point (SP) and

    the actual value of the process variable (PV). Since the feed-forward 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 set-point and the feedback value. Working together,

    the combined open-loop feed-forward controller and closed-loop 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 feed-forward, 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 set-point

    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 set-point, 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 temperature-controlled 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 set-point and the measured temperature of the heater.

    Its output controls the actual heater to stay near this set-point.

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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 closed-loop 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

    Pseudo-code

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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/PID-process-control-a-Cruise-Control-examplehttp://www.codeproject.com/Articles/36459/PID-process-control-a-Cruise-Control-examplehttp://www.codeproject.com/Articles/36459/PID-process-control-a-Cruise-Control-examplehttp://www.codeproject.com/Articles/36459/PID-process-control-a-Cruise-Control-examplehttp://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 Pseudo-code 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 set-point(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 re-use 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 set-point 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/Digital-to-analog_converterhttp://en.wikipedia.org/wiki/Analog-to-digital_converterhttp://en.wikipedia.org/wiki/Analog-to-digital_converterhttp://en.wikipedia.org/wiki/Analog-to-digital_converterhttp://en.wikipedia.org/wiki/Analog-to-digital_converterhttp://en.wikipedia.org/wiki/Digital-to-analog_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 = pre-act (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.

    Log-Log 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 feedback-level 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 (liquid-level

    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[h-hss ]/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 self-regulatory 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.(h-hss)] hh-hss=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[h-hss ]/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) non-linear 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 non-linear 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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