classical pid controllers1

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Classical PID Controllers

Eng R. L. NkumbwaSchool of Technology

Copperbelt University

2010


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PID Controllers

This note examines a particular control structure that has becomealmost universally used in industrial control.

It is based on a particular fixed structure controller family, the so-called PID controller family.

These controllers have proven to be robust and extremelybeneficial in the control of many important applications.

PID stands for:

P (Proportional)

I (Integral)

D (Derivative)


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General Control System Arrangement
http://upload.wikimedia.org/wikipedia/commons/4/43/PID_en.svg


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What is Classical Control?

Classical control involves the choice of a

suitable controller in the transfer functionGc(s) so that closed performance meets the

specifications as in figure on the previous

slide.


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Historical Note

Early feedback control devices implicitly or explicitly

used the ideas of proportional, integral, and

derivative action in their structures. However, it was

probably not until Minorskys work on ship steering

published in 1922, that rigorous theoretical

consideration was given to PID control.

This was the first mathematical treatment of the typeof controller that is now used to control almost all

industrial processes.


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The Current Situation

Despite the abundance of sophisticated

tools, including advanced controllers, theProportional, Integral, Derivative (PID

controller) is still the most widely used in

modern industry, controlling more that 95%

of closed-loop industrial processes.


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So, what is a Controller?

A controller is a device that generates an

output signal based on the input signal itreceives.

The input signal is actually an error signal,

which is the difference between the

measured variable and the desired value, orset point.

See figure below for the Controller.


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Heat Exchanger Control System


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Structure of a Controller


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Structure of a Controller

This input error signal represents the amountof deviation between where the processsystem is actually operating and where theprocess system is desired to be operating.

The controller provides an output signal tothe final control element, which adjusts theprocess system to reduce this deviation.

The characteristic of this output signal isdependent on the type, or mode, of thecontroller.


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Modes of Controllers

The mode of control is the manner in which a controlsystem makes corrections relative to an error that

exists between the desired value (set point) of acontrolled variable and its actual value.

The mode of control used for a specific applicationdepends on the characteristics of the process beingcontrolled. For example, some processes can beoperated over a wide band, while others must bemaintained very close to the set point.

Deviationis the difference between the set point of aprocess variable and its actual value. This is a key

term used when discussing various modes of control.


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Four Modes of Controllers

Four modes of control commonly used for

most applications are: Proportional (P)

Proportional plus Reset (PI)

Proportional plus Rate (PD)

Proportional plus Reset plus Rate (PID)

Other Authors state it differently as follows


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Four Modes of Controllers


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Proportional Controllers (P)

Each mode of control has characteristicadvantages and limitations.

The modes of control are discussed in thisand the next several sections of this module.

In theproportional (throttling) mode, there is

a continuous linear relation between value ofthe controlled variable and position of thefinal control element.

In other words, amount of valve movement is

proportional to amount of deviation.


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Proportional Controllers

Three terms commonly used to describe the

proportional mode of control areproportional

band, gain and offset.

Proportional band, (also called throttling range), is

the change in value of the controlled variable that

causes full travel of the final control element.

Gain, also called sensitivity, compares the ratio ofamount of change in the final control element to

amount of change in the controlled variable.

Mathematically, gain and sensitivity are reciprocal to

proportional band.


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Proportional Controllers

Offset, also called droop, is deviation that

remains after a process has stabilized. Offset isan inherent characteristic of the proportional

mode of control. In other words, the proportional

mode of control will not necessarily return a

controlled variable to its set point. Proportional control is also referred to as

throttling control.


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Reset or Integral Controller (I)

Integral control describes a controller in whichthe output rate of change is dependent on themagnitude of the input.

Specifically, a smaller amplitude input causes aslower rate of change of the output.

This controller is called an integral controllerbecause it approximates the mathematicalfunction of integration.

The integral control method is also known as

reset control.


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Definition of Integral Control

A device that performs the mathematical

function of integration is called an integrator. The mathematical result of integration is called

the integral.

The integrator provides a linear output with a

rate of change that is directly related to theamplitude of the step change input and a

constant that specifies the function of

integration.


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Example of an Integral Output for a

Fixed Input


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Integral Flow Rate Controller


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Integral Flow Rate Controller


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Properties of Integral Control

The major advantage of integral controllers is that they

have the unique ability to return the controlled variable

back to the exact set point following a disturbance.

Disadvantages of the integral control mode are that it

responds relatively slowly to an error signal and that it

can initially allow a large deviation at the instant the

error is produced. This can lead to system instability and cyclic operation.

For this reason, the integral control mode is not normally

used alone, but is combined with another control mode.


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Proportional Plus Reset Control (PI)

Proportional plus reset control is a

combination of the proportional and integralcontrol modes.

This type control is actually a combination of

two previously discussed control modes,

proportional and integral. Combining the two modes results in gaining

the advantages and c o m p e n s a t i n g f o r

t h e disadvantages of the two individual

modes.


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Characteristics of the PI

The main advantage of the proportional control mode

is that an immediate proportional output is produced

as soon as an error signal exists at the controller asshown in Figure below.

The proportional controller is considered a fast-acting

device.

This immediate output change enables the proportionalcontroller to reposition the final control element within a

relatively short period of time in response to the error.

See below figure for the response of the PI Control


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Eng R. L. Nkumbwa, Copperbelt

University, School of Technology25


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Disadvantages of the

Proportional Control

The main disadvantage of the proportional control mode

is that a residual offset error exists between the

measured variable and the set point for all but one set ofsystem conditions.

The main advantage of the integral control mode is that

the controller output continues to reposition the final

control element until the error is reduced to zero. This results in the elimination of the residual offset error

allowed by the proportional mode.


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Disadvantages of the

Proportional Control

The main disadvantage of the integral mode is

that the controller output does not immediately

direct the final control element to a new position

in response to an error signal.

The controller output changes at a defined rate

of change, and time is needed for the finalcontrol element to be repositioned.


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PI Equations


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PI Characteristics


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Characteristics of the PI

The combination of the two control modes is

called the proportional plus reset (PI) control

mode.

It combines the immediate output characteristics

of a proportional control mode with the zero

residual offset characteristics of the integralmode.


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Example of the PI for the Plant

Heat Exchanger


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Effects of Disturbance on Reverse

Acting Controller


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Effects of Disturbance on Reverse

Acting Controller

By adding the reset action to the proportional

action the controller produces a larger output

for the given error signal and causes a greater

adjustment of the control valve.

This causes the process to come back to the

set point more quickly. Additionally, the resetaction acts to eliminate the offset error after a

period of time.


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Reset Wind-Up

Proportional plus reset controllers act to eliminate the offset errorfound in proportional control by continuing to change the outputafter the proportional action is completed and by returning thecontrolled variable to the set point.

An inherent disadvantage to proportional plus reset controllers isthe possible adverse effects caused by large error signals.

The large error can be caused by a large demand deviation orwhen initially starting up the system.

This is a problem because a large sustained error signal willeventually cause the controller to drive to its limit, and the result iscalled "reset windup."

Because of reset windup, this control mode is not well-suited forprocesses that are frequently shut down and started up.


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Proportional plus Rate control (PD)

Proportional plus rate control is a control mode in which a

derivative section is added to the proportional controller.

Proportional plus rate describes a control mode in which

a derivative section is added to a proportional controller.

This derivative section responds to the rate of change of

the error signal, not the amplitude; this derivative action

responds to the rate of change the instant it starts.

This causes the controller output to be initially larger in

direct relation with the error signal rate of change.


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PD Controller


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PD Controller


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Proportional plus Rate control (PD)

The higher the error signal rate of change, thesooner the final control element is positioned to

the desired value. The added derivative action reduces initial

overshoot of the measured variable, andtherefore aids in stabilizing the process sooner.

This control mode is called proportional plus rate(PD) control because the derivative sectionresponds to the rate of change of the errorsignal.


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Definition of the Derivative Control

A device that produces a derivative signal iscalled a differentiator. Figure below shows the

input versus output relationship of adifferentiator.

The differentiator provides an output that isdirectly related to the rate of change of the input

and a constant that specifies the function ofdifferentiation.

The derivative constant is expressed in units ofseconds and defines the differential controlleroutput.


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Derivative Output for a Constant Rate

of change


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Rate Control Output


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Rate Control Output

The differentiator acts to transform a changing

signal to a constant magnitude signal as shown

in Figure above.

As long as the input rate of change is constant,

the magnitude of the output is constant.

A new input rate of change would give a newoutput magnitude.


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Rate Control Output

Derivative cannot be used alone as a controlmode.

This is because a steady-state input produces azero output in a differentiator.

If the differentiator were used as a controller, theinput signal it would receive is the error signal.

As just described, a steady-state error signalcorresponds to any number of necessary outputsignals for the positioning of the final controlelement.


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Rate Control Output

Therefore, derivative action is combined with

proportional action in a manner such that the

proportional section output serves as the

derivative section input.

Proportional plus rate controllers take advantage

of both proportional and rate control modes.


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Proportional Action

As seen in Figure below, proportional action

provides an output proportional to the error.

If the error is not a step change, but is slowly

changing, the proportional action is slow.

Rate action, when added, provides quick

response to the error.


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Example of the Proportional Plus rate

Control

To illustrate proportional plus rate control, we will

use the same heat exchanger process that has

been analyzed in previous chapters (see Figure

below).

For this example, however, the temperature

controller used is a proportional plus ratecontroller.


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Proportional Plus rate Control for the

Heat Exchanger


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Effects of Disturbance


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Industrial Applications

Proportional plus rate control is normally used with largecapacity or slow-responding processes such astemperature control.

The leading action of the controller output compensatesfor the lagging characteristics of large capacity, slowprocesses.

Rate action is not usually employed with fast respondingprocesses such as flow control or noisy processesbecause derivative action responds to any rate of changein the error signal, including the noise.

Proportional plus rate controllers are useful withprocesses which are frequently started up and shut downbecause it is not susceptible to reset windup.


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Proportional Plus Integral Plus

Derivative Control (PID)

Proportional plus reset plus rate controllers combineproportional control actions with integral and derivativeactions.

For processes that can operate with continuous cycling,the relatively inexpensive two position controller isadequate.

For processes that cannot tolerate continuous cycling, aproportional controller is often employed.

For processes that can tolerate neither continuouscycling nor offset error, a proportional plus resetcontroller can be used.

For processes that need improved stability and cantolerate an offset error, a proportional plus rate controlleris employed.


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Proportional Plus Integral Plus

Derivative Control (PID)

However, there are some processes that

cannot tolerate offset error, yet need good

stability.

The logical solution is to use a control mode

that combines the advantages of proportional,

reset, and rate action. This chapter describes the mode identified as

proportional plus reset plus rate, commonly

called Proportional-Integral-Derivative (PID).


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Controller Action

When an error is introduced to a PID controller,

the controllers response is a combination of the

proportional, integral, and derivative actions, as

shown in Figure below.


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Controller Action

Assume the error is due to a slowly increasing measuredvariable. As the error increases, the proportional action ofthe PID controller produces an output that is proportionalto the error signal.

The reset action of the controller produces an outputwhose rate of change is determined by the magnitude ofthe error.

In this case, as the error continues to increase at asteady rate, the reset output continues to increase its rateof change.

The rate action of the controller produces an outputwhose magnitude is determined by the rate of change.When combined, these actions produce an output as

shown in Figure above.


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Controller Action

As you can see from the combined action

curve, the output produced responds

immediately to the error with a signal that is

proportional to the magnitude of the error

and that will continue to increase as long as

the error remains increasing.


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PID Controller Response Curves


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PID Response Curves

Figure above demonstrates the combinedcontroller response to a demand disturbance.

The proportional action of the controller stabilizesthe process.

The reset action combined with the proportionalaction causes the measured variable to return tothe set point.

The rate action combined with the proportionalaction reduces the initial overshoot and cyclicperiod.


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PID Controller


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PID Controllers Equations


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PID Controller Characteristics


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Design of PID Controllers

Based on the knowledge of P, I and D

trial and error manual tuning

simulation


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Tuning of PID Controllers

Because of their widespread use in practice, we

present below several methods for tuning PID

controllers. Actually these methods are quite old and date back

to the 1950s.

Nonetheless, they remain in widespread use today.

In particular, we will study Ziegler-Nichols Oscillation Method

Ziegler-Nichols Reaction Curve Method

Cohen-Coon Reaction Curve Method


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Ziegler-Nichols Method

Ziegler-Nichols method

based on a open-loop process based on a critical gain


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Ziegler-Nichols (Z-N)

Oscillation Method

This procedure is only valid for open loop stable

plants and it is carried out through the following

steps:

Set the true plant under proportional control, with a

very small gain.

Increase the gain until the loop starts oscillating.

Note that linear oscillation is required and that it

should be detected at the controller output.


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Ziegler-Nichols (Z-N)

Oscillation Method

Record the controller critical gain Kp = Kcand the

oscillation period of the controller output, Pc.

Adjust the controller parameters according to nextslide; there is some controversy regarding the PID

parameterization for which the Z-N method was

developed, but the version described here is

applicable to the parameterization of standardform PID.


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Ziegler-Nichols tuning using the

oscillation method


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Reaction Curve


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Reaction Curve Based Methods

A linearized quantitative version of a simple

plant can be obtained with an open loop

experiment, using the following procedure:

1. With the plant in open loop, take the plant

manually to a normal operating point. Say that

the plant output settles at y(t) = y0 for a constant

plant input u(t) = u0.

2. At an initial time, t0, apply a step change to

the plant input, from u0 to u

(this should be in

the range of10 to 20% of full scale).


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Reaction Curve Based Methods

3. Record the plant output until it settles to the

new operating point. Assume you obtain the curve

shown on the next slide. This curve is known astheprocess reaction curve.

In the figure on next page, m.s.t. stands for

maximum slope tangent.

4. Compute the parameter model as follows


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Plant Response


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Plant Response


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Z-N Tuning Using Reaction Curve


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PI Z-N Tuned Reaction Curve


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Observation

We see from the previous slide that the

Ziegler-Nichols reaction curve tuning method

is very sensitive to the ratio of delay to timeconstant.


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Cohen-Coon Reaction

Curve Method

Cohen and Coon carried out further studies

to find controller settings which, based on the

same model, lead to a weaker dependenceon the ratio of delay to time constant.


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Cohen-Coon Tuning Reaction Curve


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Lead-Lag Compensators

Closely related to PID control is the idea of

lead-lag compensation. The transfer function

of these compensators is of the form:


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