I'm not sure whether I'm reading too much into this or even looking at it right, but maybe someone can help me out.
Say I've got a heater modeled as a collection of caps (heat capacity) and resistors (thermal resistance) pumped up with a current (representing power), Iw. Temperature is represented by the output voltage, Vo. To fake the input power with a current,
Iw = Vin^2/R
... that doesn't work so well in a V(s)/V(s) transfer function.
I guess this can be linearized by treating small excursions of Vin from a given operating point.
It got me thinking.
I noticed this:
Call the transfer function
Vo G(s) = --- Iw
d Vo ----- = sG(s) d Iw
d P d Iw d ( Vin^2 ) 2 Vin --- = ---- = ---- | ----- | = ----- d Vin d Vin d Vin ( R ) R
d Vo d Iw d Vo 2 s G(s) Vin ----- ---- = ----- = ------------ d Iw d Vin d Vin R
Vo 2 G(s) Vin --- = ---------- Vin R
I think the Vin on the right side would be the operating point and the Vin on the left, the error voltage derived input to the heater.
In closed loop form by itself, that'd be
Vo 2 G(s) Vin --- = -------------- Vin R + 2 G(s) Vin
To better see what's up, say the thermal model was a simple capacitor, 1/sC. We'd have
2 Vin------------ RCs + 2 Vin
So if the temp (Vo) is zero and the command is Vin degrees, Vin is the error and the pole is way out to the left and moves in as the error decreases.
Am I looking at this right, so far? Dead on, slightly sideways, or dead wrong?
Is it the pole moving in that keeps the system from reaching the command temp?
Since the pole is sometimes way out in left plane, I'd be tempted to put a pole (integrator) somewhere closer in. I've read that the integral part (and the P and D) of a PID is used in this system, but if the pole moving in is what keeps the system from reaching the set point, that would be wrong. The proportional part that's most responsible for getting it there, right?
So what is the significance of this moving pole and is it the reason for using the I in PID, i.e., a purposely placed pole?
TIA