1213_2_-KM40501 Control Lab#4

KM40501/ 1213(2)

4.1 Objective

• To understand proportional plus integral plus derivative

• To become familiar with proportional plus integral plus derivative (P.I.D) Controllers

4.2 Proportional and Derivative Control

Derivative feedback control involves using the rate of change of the error signal to reduce

overshoot. When a step input is applied, the error initially rises to a high value and then decreases

as the system nears alignment. The initial high rate of change of error results in the derivative

output producing a very short positive peak.

As the motor accelerates to maximum speed, the error signal begins to decrease. This results in the

derivative decreasing to a negative value, rising to zero when the motor has reached its maximum

speed and the error becomes constant. The process then reverses producing a negative

motor reverses at maximum overshoot.

In a proportional feedback system, the error signal is used to control the motor drive. The error

signal does not go negative (producing reverse torque) until the step input goes negative. The

important point is that due to the overshoot of the error signal, no reverse torque can be applied

via the motor to slow it down until after it has passed through alignment.

However a combination of error and its derivative becomes negative before alignment causing

motor to provide reverse torque and stop the overshoot. As the motor slows towards alignment the

derivative component drops towards zero. If too much derivative component is added the response

becomes slow. The best response depends on the application

tolerable in exchange for a reasonable response speed.

Expressed mathematically the motor control voltage, V is given by:

V = error + derivative of error

= V + dV/dt

When the two gain factors are added:

V = Kp( V + Kd(dV/dt) )

Where Kp is the proportional gain and Kd the derivative gain.

4.3 Proportional and Integral Control

The reason that in a purely proportional control system there must be a residual following error is

simple. As the motor is driven only by error,

system is static the error can be zero because there is no requirement to drive the motor. However

as soon as the motor is required to move there must be following error.

Lab 4: Positional Servo using P.I.D Control

Control Lab#4: Positional Servo using P.I.D Control 4

Control Lab#4: Positional Servo using P.I.D Control 4

To understand proportional plus integral plus derivative control action

To become familiar with proportional plus integral plus derivative (P.I.D) Controllers

Proportional and Derivative Control


a step input is applied, the error initially rises to a high value and then decreases


output producing a very short positive peak.

to maximum speed, the error signal begins to decrease. This results in the


speed and the error becomes constant. The process then reverses producing a negative

motor reverses at maximum overshoot.



oint is that due to the overshoot of the error signal, no reverse torque can be applied

via the motor to slow it down until after it has passed through alignment.

However a combination of error and its derivative becomes negative before alignment causing



becomes slow. The best response depends on the application but small amount of overshoot is

tolerable in exchange for a reasonable response speed.


= error + derivative of error

When the two gain factors are added:

is the proportional gain and Kd the derivative gain.

Proportional and Integral Control


simple. As the motor is driven only by error, if there were none the motor would stop! Hence if the


as soon as the motor is required to move there must be following error.

Positional Servo using P.I.D Control

Positional Servo using P.I.D Control 4-1

Positional Servo using P.I.D Control 4-1

control action.

To become familiar with proportional plus integral plus derivative (P.I.D) Controllers.


a step input is applied, the error initially rises to a high value and then decreases


to maximum speed, the error signal begins to decrease. This results in the


speed and the error becomes constant. The process then reverses producing a negative spike as the



oint is that due to the overshoot of the error signal, no reverse torque can be applied

However a combination of error and its derivative becomes negative before alignment causing the



but small amount of overshoot is


if there were none the motor would stop! Hence if the


Control Lab#4: Positional Servo using P.I.D Control 4-2


KM40501/ 1213(2)

Increasing the gain reduces the following error that is needed to keep the motor turning. For a fixed

speed, the signal required to drive the motor is fixed. Therefore the higher the gain, the smaller the

following error can be to provide that drive.

The faster the input changes, the faster the motor needs to go in order to follow the input. So, for a

fixed gain, the larger the following error must become to supply the drive. Suppose that the motor

is being driven in order to follow an increasing input. The following error is a constant value. A sum

of all the previous errors would be rising continuously and, if this component were added to the

motor drive signal, the motor would speed up and the following error reduce. This would in turn

make the integral component level off at a value just enough to keep the motor running at the

correct speed to make the error zero. The system always tries to maintains a state of zero following

error.

The important point is that now even though the following error may be zero, the motor can still be

driven by the integral component.


V = error + integral of error

= V + V dt

When gain factors are added:

V = Kp ( V + Ki (V dt) )

Where Kp is the proportional gain and Ki the integral gain.

4.4 Proportional, Derivative and Integral Control

The combination of the three terms (proportional, integral and derivative) can be thought of as

separate characteristics. Proportional, to provide the general error driven control signal. Integral, so

that there does not have to be a residual error to provide the control signal. Derivative, to give the

system stability and hence reduce overshoot.

However, in some ways the derivative and integral terms act against each other and are all

controlled by one overall gain, making the analysis much more involved.

The error control channel is like this:


V = error + integral of error + derivative of error

= V + V dt + dV/dt



KM40501/ 1213(2)

When gain factors are added then:

V = Kp ( V + Ki ( V dt) + Kd (dV/dt) )

Where

Kp = proportional gain

Ki = integral gain

Kd = derivative gain.

4.5 Sample Time

All practicals within this assignment allow the modification of the sample time. This parameter

controls how often the computer checks the state of the mechanical unit and makes adjustments.

Taking too few samples can inhibit the performance of the system, since the mechanical unit may

go beyond a set point without the computer noticing. Taking too many samples has the potential to

put a load on your computer, causing other processes to be potentially 'starved' of processor time.

When designing a controller, a comprimise between the required response and the available

processing resources of a digital controller must be met.

4.6 Practical 1 : Proportional Control with Derivative Action

#1 In this practical you will investigate the effect of adding a derivative component to the error

signal used to control the motor.

Note: In the previous assignment it became clear that increasing

the gain in order to reduce error caused the system to

become unstable, with a large overshoot in response to a

step input.

This can be corrected to some extent by adding a derivative

component to the error signal. This component is simply the

rate of change of error and, as the motor is driven by the

error, could be obtained from a tachometer.

However, in a computer controlled system such as this, it is

easiest to derive the signal directly from the error by

calculation.

#2 Here the derivative is generated by taking the difference between successive error values.

The sampling rate can be varied using the set sample time control box. Initially it is set to

100 milliseconds.

#3 Adjust the sample time and observe the behavior of the controller. Different time settings

will cause different effects. This diagram shows how the system blocks are configured for

this practical.



KM40501/ 1213(2)

#4 Make the appropriate patching on the DIGITAL UNIT 33-120 as shown Fig.4.2

Fig. 4.2: Patching diagram Positional servo using P.I.D control

#5 Use the square wave input. Set derivative gain to zero, increase the proportional gain and

observe the overshoot when the gain is high.

#6 Increase the derivative gain. Notice the overshoot reduces and the stability improves. Use

the Display box to select the display parameters and input excitation.

#7 Now use the triangle input. Note that following error reduces with high proportional gain,

but is slightly increased by the derivative component.

Proportional Gain

Derivative Gain



KM40501/ 1213(2)

#8 Set the proportional gain high and the derivative gain to zero. Observe the overshoot on the

measured signal. Increase the derivative gain. Explain how the derivative gain reduces the

overshoot on the measured signal.

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#9 Set the proportional gain to a mid-value and set the derivative gain to zero. Observe the

measured output. Now increase the derivative gain to a similar value as the proportional

gain. Explain why the response becomes slow with high derivative gain.

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#10 Given that proportional and derivative control will always gives a steady state error and has

a slow response, suggest some suitable applications.

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4.7 Practical 2: Proportional Control with Integral Action

Note: In this practical the effect of integral action is investigated.

In the previous assignment, it could be seen that unless the

system is static there is always a residual following error.

Increasing the gain reduces it but, in purely proportional

control, it can never be zero.

By adding a component consisting of the sum of all the

previous error samples, following error can be reduced

considerably. This component corresponds to the

mathematical integration process and hence is called

integral action.

Integral action has the disadvantage of slowing down the

general response and, while reducing the average

following error, will often increase its peak value. Only a

small amount of integral gain can be added before the

system becomes very unstable. The sample rate may be

adjusted as in the previous practical. This diagram shows

how the system blocks are connected for this practical.

#11 Patching diagram is same with the previous practical. Use the triangle wave input. With the

integral gain set to zero, observe the following error.

#12 Add integral action by increasing the integral gain slowly. Notice that the average error

decreases but as the motor reverses the response is worse.



KM40501/ 1213(2)

Note: Use the square wave input to observe the step response.

Use the Display box to select the display parameters and

input excitation.

Note only a little integral action can be added before the

system becomes unstable.

Proportional Gain

Derivative Gain

#13 Set the integral gain to zero. Increase the proportional gain and observe the following error.

Why does the following error occur? Why can`t the following error be reduced to zero when

there is no integral gain?

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#14 Set the proportional gain to maximum and increase the integral gain slowly. Why does the

system become unstable when the integral gain is increased?

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#15 Increase the rate of change of input using the control on the mechanical unit. Observe the

error and measured value. Set the rate of change of input back to a low value and apply the

brake on the side of the mechanical unit. Observe the error and measured value.

Do these results make proportional and integral control suitable for fast continual load

variations and high inertia applications?

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KM40501/ 1213(2)

4.8 Practical 3: Proportional Control with Integral and Derivative Action

Note: In this practical the ideas of proportional integral and

derivative control are combined into a PID or Three Term

Controller. In concept, by adding the characteristics of all

three components the best possible system performance

should be obtainable. Of course with three gains to set the

adjustment of such a system is more difficult and the

mathematical analysis quite involved.

However a reasonable approximation can be made by

simple experimentation, using your experience of the

characteristics of each term alone. Try adjusting the

sampling rate and see what effect it has on the system.

Remember that like many engineering problems there is no

one perfect solution. This diagram shows how the system

blocks are configured for this practical.

#16 Patching diagram is same with the two previous practical.

4.9 Full P.I.D Control

#17 Use both triangle and square wave inputs to investigate the effect of the three gain terms.

#18 Start with only proportional gain and then add derivative action. When the system is stable,

add a small amount of integral gain and observe carefully the effects. Adjust all three gains

and note their interactive nature.

Use the Display box to select the display parameters and input excitation.

Notice that the best step response is not accompanied by minimum following error.

Proportional Gain

Integral Gain

Derivative Gain



KM40501/ 1213(2)

#19 Describe the process which occurs when an input is applied to the motor, in terms of the

effects which the proportional, derivative and integral feedback have on the measured

output.

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#20 Compare the values which are set for the three gains in the practical with similar values set

in the math’s model. Is the measured value output the same for both cases?

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#21 What other factors will affect the practical which do not occur in the math’s model?

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Group Members: 1.______________________________ 2. _____________________________

3.______________________________ 4. _____________________________

5.______________________________ 6. _____________________________

4.1 Positional Servo using P.I.D Control

#7

Proportional Gain

Derivative Gain

#12

Proportional Gain

Derivative Gain

#18

Proportional Gain

Integral Gain

Derivative Gain

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1213_2_-KM40501 Control Lab#4