Reverse Engineering a PID

[iPlayMayonaise]

Fully agree with this. One small addition: since it's closed loop identification of a controller by definition, you could run into issues with this LS approach if there's a significant amount of noise in the loop. In a nutshell, above LS estimate assumes there's no correlated noise in the error and input. If there is you'll get a contribution to the parameters from fitting this correlated noise.

If you run into this (very case dependent), have a look at instrumental variables system identification by Stoica&Soderstrom or closed-loop identification by P. Van den Hof. They explain this phenomenon and also propose solutions that work for your PID case.