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thank you for your interesting post.
I am not sure how to create a fuzzy PID controller, but I am definitely very interested in it.
I am not sure if there is some guide towards selecting the linguistic terms based on the properties of the functions (e.g. Gaussian, Triangle), S-Norms and T-Norms, or defuzzifiers, unless you are already interested in specific properties of the components.
In general, a basic Mamdani controller is made up of Triangles as linguistic terms, Maximum or AlgebraicSum as accumulation, Centroid as a defuzzifier, Minimum or AlgebraicProduct as T-Norms, and Maximum or AlgebraicSum as S-Norms. There are many decisions involved in the design of a fuzzy controller, for which obtaining the best performance can be considered as a combinatorial optimization problem. Not only for these design variables, but also for the rules. For combinatorial optimization problems there are many methods available nowadays. You may want to revise the literature on Genetic Fuzzy Systems, for example.
For fuzzy PID controllers in fuzzylite, I have not seen one yet. I think you would be better off searching for fuzzy PID controllers without fuzzylite, and then we could discuss how to adapt them to fuzzylite. It should be straight-forward.
After seeing the pseudocode of a PID here, I think you might be looking for a Takagi-Sugeno controller, but I have not thought thoroughly about it.
I hope this helps you a bit on your quest. Let me know if I could be of help once you have your ideas clear on fuzzy PID controllers.