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

8/10/2019 2014 Part 7 with HW8 PID+Tunning++of+Process+Dynamics+and+Control+(7)(1)

1/76

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]


2/76

PID Controller


3/76

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.


4/76

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


5/76

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)


6/76

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


7/76

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


8/76

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


9/76

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


10/76

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


11/76

PID controller theory


12/76

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)


13/76

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


14/76

Integral term


15/76

PV vs. time diagram for different values of integral gain

Plot of PV vs time, for three values of Ki(Kpand Kdheld constant)

d


16/76

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:


17/76


18/76

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


19/76

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


20/76

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)


21/76

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


22/76

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


23/76

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


24/76

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


25/76

Effects of increasinga parameter independently


26/76

Comment about previous slide


27/76

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


28/76

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.


29/76

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)


30/76

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)


31/76

PI Controller(2)


32/76

PI Controller(3). Without derivative action

Dead-band


33/76

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


34/76

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


35/76

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


36/76

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


37/76

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


38/76

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


39/76

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


40/76

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


41/76

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


42/76

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.


43/76

Alt ti N l t d PID F


44/76

Alternative Nomenclature and PID Forms

Reciprocal and % gain


45/76

Reciprocal and % gain

B i d i i i PV


46/76

Basing derivative action on PV

Basing proportional action on PV


47/76

Basing proportional action on PV

Laplace form of the PID controller


48/76

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


49/76

PID Pole Zero Cancellation

Di t i l t ti


50/76

Discrete implementation

Pseudo-code


51/76

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


52/76

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)


53/76

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


54/76

in radians/sec on the horizontal axis


55/76

in radians/sec on the horizontal axis

A Typical Bode Diagram


56/76

A Typical Bode Diagram

Effect of PID on Bode Diagram as shown in previous slide


57/76

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


58/76

PID

Gc(s)

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


59/76

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


60/76

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:


61/76

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


62/76

Linearization of a non-linear equation


63/76

Keep the problem linear


64/76

Keep the problem linear

Simple Introduction to Block Diagram Algebra


65/76

Simple Introduction to Block Diagram Algebra

Simple block diagram


66/76

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)


67/76

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


68/76

1/(A.s)

1/(A.s)

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

More General Form of Previous Slide


69/76

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


70/76

Application for the tank problem


71/76

For PID controller


72/76

For PID controller

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


73/76

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)


74/76

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.


75/76

HW8


76/76

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

Documents