Loop Tuning

I have no idea what you are trying to do, tune for, what two angles(?) make a ratio of tangents and I have no idea what L might mean except maybe length. Usually Ti indicates the metal Titanium but I doubt that you are trying to attempt metal transmutation or alchemy.

Please provide us some more information.

Nice paper.

This might be what the OP is talking about. I did not see the equation in question in that paper but I did not look very hard either.

You are exactly right: the OP's equation is not part of the Ziegler and Nichols' original recipe for PID tuning. It was only because I saw the "R" and "L" terms that I concluded he was doing open-loop tuning.

In practice I've found that either Ziegler-Nichols tuning method -- open-loop or closed-loop -- provides mediocre control quality at best, and for this reason there's probably a wide range of formulae yielding tuning parameter values that will more or less work.

Another factor explaining variations in tuning formulae is the range of units you find in modern controllers for the integral (reset) term: repeats per minute, seconds per repeat, etc. Also, you have a wide range of algorithm options such as parallel versus series (interacting) versus "ISA", whether or not derivative acts on error or PV alone, whether or not proportional action works on error or PV alone, etc. Many of these options simply didn't exist in 1942.

On a more general note, most loop tuning problems in my experience are caused by field instrument troubles, such as valve stiction, and therefore cannot be "tuned away" in the PID controller. Also, technicians and engineers alike generally lack a strong conceptual grasp of how and why PID is supposed to control a loop, so they end up plugging numbers into formulae expecting to get usable P, I, and D parameter values. Most techs I've worked with don't know how to do anything but blind trial and error tuning, while most engineers I've worked with don't know anything but theoretical solutions based on first having a Laplace transform function of the process (good luck with that!).

One of my heroes in this field is Michael Brown, a South African control engineer who writes extensively on the subject of PID loop optimization, and whose "Case History" documents are priceless:

http://www.controlloop.co.za/

I know I didnt provide enough information but I was hoping someone would have recognised the formula, I did more research and found what I was looking for,the above formula was from Cohen Coon, I think he improved Ziegle-nichols method of open loop Tuning.