control theory application areas?

Can some of the pros please let me know of the fields where one can put to use stuff like: pole placement, compensation, root locus.. and also model identification, LQG controller disgn, Kalman filtering, neural networks...?
I hear there are some heavy control problems involved in car engine control nowadays?

The "mathematical" type things can be used in PLC's if the application can use them... else, why try?

It's certainly true that a PLC is basically a CPU. It can manipulate a lot of mathematical functions and can deal with a lot of "event-manipulation". But, in terms of some of the things that you listed, the PLC is simply way too restricted; at least, in repect to what a "free-running" CPU can do.

You would do far better to use most of those things in a "real" CPU, ie, a PC or Embedded Processor designed for the purpose. The PLC is just plain too restrictive in what it will do.

It might seem to many that the PLC can do anything... (it can do a hell of a lot) but that really ain't so; at least, not when compared to a "free running" CPU.

"Free Running" means, at "machine-level", that is, at the processor level without that layer of interface that arbitrarily (by vendor design) controls what can and can not be done.