Differences between PID, PI-D and I-PD 2

[ScottyUK]

This is a follow-up to an OT discussion which was developing in the PLC forum.

Peter Nachtwey offered to illustrate the differences between PID, PI-D and I-PD and I thought the discussion might be of interest to a wider audience. Over to you Peter...


----------------------------------

Sometimes I only open my mouth to swap feet...

 

[JLSeagull]

Look for web tutorials on Ziegler Nichols controller tuning. One example follows:

http://www.dei.unipd.it/~picci/Files/TEACHING/CONTROLLO_DEI_PROCESSI/AstromPID.pdf

PID is an acronym for Proportional, Integral and Derivative. Controllers respond to the error between a selected set point and the offset or error signal that is the difference between the measurement value and the set point. Optimum values can be computed based upon the natural frequency of a system. Too much feedback (positive feedback cause stability problems) causes increasing oscillation.

With proportional (gain) only control the output increases or decreases to a new value that is proportional to the error. This cannot bring you to the correct setpoint - only limits the amount of error. Higher gain makes the output change larger corresponding to the error.

Integral can be added to the proportional action to ramp the output at a particular rate thus bring the error back toward zero.

Derivative can be added as a momentary spike of corrective action that tails off. Derivative can be a bad thing with a noisy signal.

 

[ScottyUK]

Hi JL,

I'm familiar enough with PID controllers, but Peter appears to have some alternate implementations of P-I-D which can give different results. I'm curious what the differences are and how they behave.


----------------------------------

Sometimes I only open my mouth to swap feet...

 

[sreid]

The usual meanings are as follows;

PID All use the error signal as the input

PI-D Error signal operates P and I. The D sees the feedback signal only.

I-PD Error signal operates I only. The P and D see only the feedback signal.

 

[ScottyUK]

Ahh, now I see where the difference originates from. Thanks sreid. It will be interesting to see how the different types of controller respond.


----------------------------------

Sometimes I only open my mouth to swap feet...

 

[PNachtwey]

ZN is old school. I use a technique called pole placement. You can see it works well if you know the actuator or plant model. One can see the differences in the response in the sine wave .pdf. If I really wanted to follow the sine waves I would use feed forwards too. I just wanted to show how the different PID act on their own in response to a sine wave at the same frequency as the actuator ( we are talking motion control ) corner frequency.

I wanted a critically damped response so I place the 3 poles of the closed loop characteristic equation at -lambda.
I have formulas all worked out so I can calculate the gains given I know the actuator parameters. I chose a simple actuator for these examples. A simple hydraulic actuator model should have a gain, natural frequency and damping factor.

Note the Bode plots. I-PD has the lowest bandwidth, PI-D is next and PID has the highest band width an therefore does the best job of following the sine wave.

ftp://ftp.deltacompsys.com/public/NG/Mathcad%20-%20t1p1%20pid%20comparison%20sine.pdf

I used an integrator anti-wind up technique that works very well. I couldn't get the PID, or PI-D to over shoot in the step example although they will in some cases. The I-PD will not.

ftp://ftp.deltacompsys.com/public/NG/Mathcad%20-%20t1p1%20pid%20comparison%20step.pdf

In addition to the three forms of PID I have shown there is also the velocity or incremental form of PID which I haven't shown. The trick is to know what PID to use for what application. I use the full PID when following a motion profile but I like to use the I-PD otherwise. I like the I-PD when following a joy stick or as an inner loop PID because the target or reference is not smooth. This will cause the output to jump around because the reference noise is feed into the proportional and especially the derivative terms. This high frequency output noise will excite higher order and unmodeled 'features' that will ruin the response. Ideally one wants to keep the control output smooth. This is why higher resolution feedback is important.

There are other forms called two degree of freedom where the Kp and Kd gain in the forward path are not the same as in the feed back path. This can result in an extend bandwidth without having the actual position over shoot the target as in the sine wave example.

Peter Nachtwey

 

[ScottyUK]

Very good Peter. This thread would form the basis of a rather good FAQ if you have time. Thanks - I have learned something today.


----------------------------------

Sometimes I only open my mouth to swap feet...