This may or may not be relevant, but ...

I was involved for some time with TEM hot stages (1300 deg C). During
development, a lot of time was spent trying to come up with a temperature
controlles that would control the temperature :) without the stage moving.
With a lateral mechanical stability requirement of a about 1 nanometre per
minute, no closed loop solution was ever found.

Finally, the best approach turned out to be the simplest - a very stable
power supply for the heater and a thermo couple. To start with, a quick
calibration run was done to establish what current was required to get a
specific temperature. Subsequently, it was just necessary to dial up the
required current and wait for the stage to stabilise - about 5 minutes.

Mechanical stability was excellent. The main disadvantage was that setting
a very precise temperature was tricky but with a bit more care in
establishing a calibration curve, +- 5 deg could be set, which was often
acceptable when aiming at temperatures around 1300 deg C. For a stage
operating at lower temperatures, I would expect to be able to set a
temperature within a degreee or so.

Regards,
Larry Stoter