By Frank Owen PhD PE polyXengineering Inc San Luis Obispo California This is just one strategy for tuning a pid There are many strategies to follow to tune a PID The most common aims for using a PID are ID: 704198
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Slide1
Pid controllerpole/zero cancellation
By Frank Owen, PhD, PE
polyXengineering, Inc.
San Luis Obispo, CaliforniaSlide2
This is just one strategy for tuning a pid
There are many strategies to follow to tune a PID
The most common aims for using a PID are:
Reduce or eliminate steady-state errorMake response faster without causing more overshoot
The strategy described here is completely differentWe have a 2nd-order system whose dynamics we do not likeWe cancel the system’s dynamics out and replace them with different dynamics
Let’s see how to do this…Slide3
2
nd
–order with complex poles
Open-loop poles:
System has two asymptotesNo matter the value of KP , the system oscillatesBesides this, system has ess when given a step inputTo be complete, let’s specify that the open-loop system has a gain of 1 (KOL = 1) Aim:Make system non-oscillatory with a time constant of 1 second
Eliminate or reduce
e
ssSlide4
2
nd
–order with complex poles
A PID controller has:
A pole at the originTwo zeros, which can be placed at any locationA gain, which can be used to move the closed-loop poles along the revised root locus(See the video “Beyond P-only control via root locus”)…so the open-loop system with the PID controller is now just an integrator and a gainPlace the two zeros on top of the OL poles to cancel themPlace the two zeros on top of the OL poles to cancel them
Put the pole at the origin
The revised RL is just a path running from the origin along an asymptote at
180
°
…Slide5
2
nd
–order with complex poles
Now adjust the gain to move the closed-loop pole out to the left so that T = 1 second.
Adjust gain to move closed-loop pole to T = 1(pole at -1/T)-1/1The final step is to calculate KP , KI , and KD that give us these results…Slide6
2
nd
–order with complex poles
For the graphical solution, we draw the vectors from all poles and zeros to the desired closed-loop pole.
-1/1Mp2 , Mz2 Mp3
M
p1
, M
z1
Recall that the formula for the evaluation of GOL at the desired location is:
…but M
p1
= M
z1
…and M
p2
= M
z2
…also
and
. So these all cancel in the calculation. (This just shows that the zeros cancel the poles.) So we are left with:
Slide7
2
nd
–order with complex poles
To meet the magnitude criterion,
. But the question is, what is
?
-1/1
Let’s put the PID controller together. We added
For this part of the controller to have a gain of 1, we need to divide it by 13. Note that this does not change the location of the zeros.Slide8
2
nd
–order with complex poles
Thus the PID without its gain (or, otherwise stated, with a gain of 1) is
-1/1
From the last slide,
, but the original problem statement was that K
OL
= 1. Thus the overall PID is 1.
The complete PID controller is thus:
Slide9
2nd–order with complex poles
We match this with the standard form:
Thus
Slide10
The Problem with Pole/zero cancellation
The system’s open-loop poles can drift during operation (for example, hydraulic fluid warming up during operation) or over time (kinematic or friction changes). Thus the cancellation of the poles with the controller zeros is not complete.
Part of your study, if pole/zero cancellation is used, may be to assess the effect of this pole drift.Slide11
That’s all folks!
Fin
©
polyXengineering
, Inc.San Luis Obispo, Californiawww.polyxengineering.com