Proportional + Derivative Controller output waveform

ilovescience85

1. Homework Statement

FIGURE 5 shows a proportional plus derivative controller that has a

proportional band of 20% and a derivative action time of 0.1 minutes.

Construct the shape of the output waveform for the triangular input

waveform shown, if the input rises and falls at the rate of 4 units

per minute.

2. Homework Equations
Unsure

3. The Attempt at a Solution
I have calculated the gain by using the PB to calculate a figure of 5 for the gain. Would I be right in saying that for and input change of 1 units there would be an output change of 5 units? The integral action time is 6 seconds so after 12 seconds the output will have increased to 10 units, is this right? Am no sure how to sketch this as a waveform? Any help would be greatly appreciated.

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donpacino

Gold Member
Your calculation of the proportional gain is correct. it has a gain of 5, however this is independant of time or what happened before

you say it is a PD controller (proportional + derivative), but then in your attempt you are talking about integrals?
Your derivative portion will simply multiply the action by the rate of change.

ilovescience85

Sorry that is a typo it should say derivative action time.

donpacino

Gold Member
http://cs4hs.cs.pub.ro/wiki/_media/roboticsisfun/chapter5/pdcontroller.png?w=800

above shows another implementation of a PD controller.
The important thing to note is that the proportional and derivative paths are in parallel. that means that they simply sum together, and the output of one does not effect the output of the other. So you compute the output of the proportional path (which as you said is the input with a gain of 5) plus the output of the derivative path (which the action multiplied by the rate of change) does that make sense?

if you are having trouble, try sketching the proportional and derivative paths independent of each other, then add them

ilovescience85

I have these diagrams in the learning material (see attached) but i wouldn't class this as a waveform.

So a input change of 1 unit would produce a output change of 5 units, then every 6 seconds this output would increase by 5 due to the derivative action?? Then after 30 seconds the input would begin to decrease and the reverse would happen?

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donpacino

Gold Member
no

the change has no effect on the proportional gain, it is independent of time. in your case the proportional output will allways be the current input values multiplied by 5.

example.
pro gain=2

input=1, pro output=2
input=2, pro output=4
input=100, pro output=200

the derivative portion will always be the action multiplied by the derivative.
example
derivative action=0.5
rate of change=1, der output=0.5
rate of change=-1, der output=-0.5
rate of change=2, der output=1

so in our example if you line that increases at a rate of 5 units/sec starting at 0,

the derivative output will always be 2.5 (because the rate of change will stay the same)
the proportion output will be 0 at t=0, 10 at t=1, 20 at t=2, etc. does that make sense?\

now try your problem again

ilovescience85

Right so I get that proportional output will be,
input output
0 0
1 5
2 10
so over the minute step change the input will rise 2 units and then fall 2 units? Equating to a proportional output of 0 initially followed by 5, 10, 5, 0?

Sorry I'm not following the derivative part. I understand that the derivative action of 0.1 multiplied by the rate of change will give you the derivative output, is the rate of change for my example 4 units / 60 seconds?

Accept my apologise it's a new subject for me and I'm doing it distance learning which doesn't help.

donpacino

Gold Member
It is fine, no apology necessary....

the derivative poprtion of the controller simply take the derivative of the input (rate of change) and multiplies it by the action (whihc is just a multiplier value). lets say the derivative action was 1, just to make it simple. before the input starts to rise, the rate of change is 0 units/second. So the output due to the derivative is zero. Then when the input starts to rise, the derivative is 4 units/sec (this is given). so simply multiply the 4 units/sec by the derivative action to find the output due to the derivative controller.
does that make sense?

side note:
When you are designing PD, PID, or PD controllers, you would change your action and proportional band values based on the system to get the output responses that you want.

ilovescience85

Ok so when you say multiply the rate of input change i.e.4 units/minute in my example by the derivative action do you mean the derivative action time? So 4 units/minute x 01. minutes to give an initial output increase of 0.4 units? Then a proportional increase of 5 units per 1 unit of input?

Do I need to show the controller output reducing? After the initial 30sec input increase of 2 units the input will start to decrease back to 0.

donpacino

Gold Member
Ok so when you say multiply the rate of input change i.e.4 units/minute in my example by the derivative action do you mean the derivative action time? So 4 units/minute x 01. minutes to give an initial output increase of 0.4 units? Then a proportional increase of 5 units per 1 unit of input?

Do I need to show the controller output reducing? After the initial 30sec input increase of 2 units the input will start to decrease back to 0.
Yes. after the input starts to ramp down, the derivative will be negative and the gain will start to decrease.

Morton11

Hi guys, I have been struggling on this question for my assignment but after looking through your posts it has increased my understanding so thank you for that. However, I am finding it difficult to picture how the waveform should look. Any advice would be greatly appreciated, thank you.

rude man

Homework Helper
Gold Member
Hi guys, I have been struggling on this question for my assignment but after looking through your posts it has increased my understanding so thank you for that. However, I am finding it difficult to picture how the waveform should look. Any advice would be greatly appreciated, thank you.
Ideally, run the input and transfer function through a simulator like Simulink.

If you want to do it in closed form you'd have to determine the Laplace or Fourier transform of the input, then multiply by the transfer function F(jw) or F(s), then invert the product output. Or, of course, via convolution in the time domain.

I assume this is a single pulse. If this is an ifinite series of such pulses the problem can be handled similarly but the math is gooier.

"Proportional + Derivative Controller output waveform"

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