Transfer Function for PID controller optimization

Question

Hi all,

a customer wants to know if they can use JMP for PID controller design and optimization.  Transfer Functions appear to be used for this (time to frequency transformation).

Does anybody have knowledge about this or even examples?

Thanks

Florian

Florian_Vogt · Answer

Thank you all @Chandramouli_R @Daniel_Valente @Jason_Wiggins !

Interesting twist for me :)  I will speak with the customer to learn more about their situation and if DoE might actually be what they need!

Jason_Wiggins · Answer

adding my 2 cents

DOE can be used to aid in PID controller design and optimization. So many fun things to consider here! I think how DOE can help depends on the design and optimization activities your customer is doing. Here are a couple of possibilities…

Assuming customer has knowledge of classical control theory, simulation software and equipment to test controller parameters:

In the simplest case, the customer has modeled the physical system and determined estimates for proportional (P), integral (I), and derivative (D) gains for the compensator transfer function. Estimates are chosen based on design goals like response time, overshoot and steady state error. Often a stability analysis will be performed to understand values for gains that may lead to instability in the system. The design work is often done in software like MATLAB. When the gains are applied to a controller on a system like a machine, the actual behavior is often different than expected and the parameters need tuning. This is where DOE comes in. A typical work flow might be

Design an experiment to find optimal PID gains. 
Factors: P, I, D, + any additional process vars
Limits: choose limits that contain the simulated PID gains and that avoid gains or combinations of gains that will lead to system instability. Use constraints if necessary.

Responses: Use design goals like resp time, overshoot, ss error given as responses in the experiment.

Run the experiment in a random order on the equipment and analyze results. Find optimal PID parameters. I attached a paper that gives a simple example of this idea. In this paper and in my past life, responses were based on summarizations of time series curves that are common in control theory for measures like response time, overshoot etc.. I did not have access to FDE in my past life when doing this type of work. I wish I did. I believe it would be much better to determine values of the PID gains that achieve an ideal time series curve. On a side note, I believe this would make a fantastic discovery paper if you customer wanted to try the approach!!!

The type of experiment design could be debated. Space filling design as Dan mentioned can work. Perhaps another possibility might be to use a classical design (or custom design if constraints are at play) using a log LaPlace transform of transfer function as the model? I have not tried this but seems possible. The closest thing to this idea I could find in the literature is a paper Taguchi wrote where he used orthogonal arrays to minimize signal to noise ratio of a PID controlled system. I attached the paper. If the customer is purely in simulation mode and exploring transfer function parameters that achieve optimal system behavior then the space filling design approach Dan mentioned definitely makes sense as the model is deterministic. I would also expect that more parameters would be included as PID gains are just compensator parameters.

Knowledge of control theory, transfer functions and access to simulation software is very helpful in understanding how to determine limits for gains in an experiment. It is not strictly necessary though. I believe this is rare for customers that are not designing controllers but are instead tuning a controller on an an existing machine. In this case, using the existing PID gains to establish limits works. Machines behave different over time due to wear and different factors. It may also be useful to include other factors that may alter the controllability of the machine in the DOE. My best estimation is that this is the most common application of DOE in PID tuning/optimization. It is certainly what I have done the most.

Hope this helps. I wish I still had machines to play with and knowledge of control theory a little fresher in my mind. I think this is a veritable playground for DOE!

applying-doe-in-performance-optimization-of-an-automated-position-control-system-a-collaborated-case-study-between-two-engineering-technology-courses.pdf

Taguchi PID.pdf

Daniel_Valente · Answer

As @Chandramouli_R -- JMP can be very helpful here. The space-filling designs and Gaussian process fitting can be very effective at building surrogate models of a system in a time-efficient manner, and then the profiler can be used to perform what-if analyses, additional Monte Carlo simulation, and communicate the very complex model to stakeholders.

I had an example in the MATLAB WP I wrote a while back, but I think that was taken down. I can dig it up.