I use Levenberg-Marquardt.
Auto tuning is more difficult because you must have a model for each
non-linearity.
The following link shows how a hydraulic actuator with a non-linear valve
can be modeled. The valve has two gain segments which are clearly seen.
It is also possible to model the valve as a odd order polynomial with an
offset to model valve with curved responses. All the models must be tested
and the one with the lowest error is USUALLY the right one.

formatting link

Linear temperature systems are easy. I haven't done it yet but properly
modeling a heat exchanger and taking into account the log mean temperature
difference shouldn't be a problem.

Peter Nachtwey

Peter, I had some trouble getting the LMTD calc to converge, and sought some
assistance in a newsgroup, may have been this one or else chem eng related.
Charlie W, a regular poster, suggested some small fiddle factors in the
calc, that solved the problem and I've been grateful for his assistance ever
since. Here's the equation without commentary, if you want any more info
revert to here and I can send you more of my code.

Qlmtd = U

*** A *** (at1 - at2 + 1.0E-7 *(at1+at2))

* / (log (at1 / *at2) +
2.0E-7) ;