# Proportional, Integral and Derivative

1. Dec 28, 2011

### ssana

Dear all,
I am working on PID loops now a days and need to know the exact description of PID control.
From TOHO TTM-X04 temperature controller, i learnt about P and I, but still i can't understand about D (Differential). As you know that this temperature controller shows manipulated output in %.
Below are the observations i learnt about P and I,

Proportional band only check the difference between actual value and set value and decide how much output in percentage is required to reduce this difference.
Example:
If P=1%
Present Value = 80 deg C
Set Value = 100 deg C
Difference = 20 deg C
Manipulated Output = 80% (assume)

If P=2%
Present Value = 80 deg C
Set Value = 100 deg C
Difference = 20 deg C
Manipulated Output = 60% (assume)

Conclusion is that manipulated output reduces as we increase the value of P in %.

Integral band increases the output if the difference remains unchanged.
Example:
If P=2% and I=300 Seconds or 5 minutes.

At 0th second of control time:
Present Value = 80 deg C
Set Value = 100 deg C
Difference = 20 deg C
Manipulated Output = 60% (assume)

At 300th second of control time:
Present Value = 80 deg C
Set Value = 100 deg C
Difference = 20 deg C
Manipulated Output = 100% (assume)

Conclusion is that output increases gradually and becomes full 100% in 300 seconds if present value remains 80 deg C.

Kindly share information about differential, as i am still struggling in getting clear information about differential.

Thanks to all

2. Dec 28, 2011

### jim hardy

the controller operates on the error, ie input minus setpoint.

proportional is k X error
integral is a term proportional to integral(error)

derivative just adds a term proportional to d/dt(error)

a picture is worth a thousand words ,,, see if these illustrated tutorials help.

http://www.jashaw.com/pid/tutorial/pid3.html [Broken]

http://en.wikipedia.org/wiki/PID_controller

do you use Laplace transform to analyze circuits?

what nailed this for me was to derive the controller'stransfer function

and then its response to step input

Last edited by a moderator: May 5, 2017
3. Dec 28, 2011

### ssana

Dear Jim Hardy,
Thanks for information, Your provided literature is very easy to understand. Before this i only observed the behavior of derivative that output will not down to 0% even after reaching setpoint if derivative has some value, but not clearly understand about its working inside.
This literature might clear this concept. I will observe the behavior of controller again after reading this literature.
Thanks

4. Dec 29, 2011

### jim hardy

indeed output is sum of three terms:

proportional term: k X error , this term is the proportional part

AND
integral term, see contents of brackets:

[another k X integral of (error), this k accounts for integral time setting
integral is often but not always expressed in "repeats per time interval"
where the integral action adds to the output an amount equal to the error X k every time interval
sometimes it's expressed as the time interval to add an amount equal to the error.

sometimes it's multiplied also by the proportional k, sometimes not - designer's choice.

if your error is 5%, and time interval is 1 minute, the output will 'creep' (integrate) 5% per minute. In some controllers it's also multiplied by the proportional K and in others it's not , designer's choice. - as you see there's permutations in the terminolgy.. so get the concepts working in your head you'll figurre out the variants as you encounter them.

observe that integral means output can hold at any value so long as error is zero and that's a change from straight proportional. i attribute it to "initial condition" or "integration constant" we encountered when doing integrals in math homework.]

AND
derivative term, again see brackets:

[another k X rate-of-change of (error) , called the derivative or rate term

sometimes it too is multiplied by proportional gain sometimes not.

Derivative actually measures the rate of change of error and adds to output in proportion.
but since a true differentiator amplifies noise by a lot, the rate term gets slowed down by a low pass filter so it has Laplace transfer function ts/(ts+1), or s/(s+1/t).
if you multiply that by step 1/s , or by ramp 1/s^2 , and look up inverse Laplace you'll see the dynamics of the rate term.
it's a scary looking equation in time, so i suggest you graph it for both step and ramp. it's really simple to envision in a mind-picture but i'm having difficulty finding concise words for you.]

the trick to getting your mind around controllers is to rigorously study how the controller responds to step and ramp inputs, but open loop- Get to where your brain does it automatically BUT don't consider the effect of controller reaqching around through process and changing your input just yet.. master open loop controller response first.
then go ahead and add to that mental picture what your controller output will do to the process, which reaches around and changes your input, driving error hopefully toward zero.

closed loop is a really different animal - have you studied G/(1+GH) yet?
anything is possible. beware of quadratics in denominator.

when you define the H of your process and use G for your controller you can write the closed loop equations and study them.
i used to do that in Basic on my TI-99.
i used a for-next loop with time as index and turned output into a string variable of spaces with an asterisk at value of output.
set printer for 132 character line length and scale output so max value just fills a line, run it and turn paper sideways and one has a graph of output vs time.
That is how i plotted graphs in 1980's when we had only 5X7 dot matrix printers and text screens. probably excel or something does it now, just i could never bring myself to trust excel's math.. arcsin i think it was blew up on me so i went back to Basic and wrote a Taylor series for it.

i hope you enjoy feedback controls. the math was thought up by Descartes then shelved for a few hundred years. The Germans revived it in WW2 for their rockets.

see Willis Eschenbach's Thermostat Hypothesis for an application to climate, timely today.
http://wattsupwiththat.com/2010/07/24/willis-publishes-his-thermostat-hypothesis-paper/
will get you there. (i immodestly refer you to last comment...)

glad to see somebody interested and working hard.

old jim

Last edited: Dec 29, 2011