Hi,
you need to find out a mathematical relationship between phase angle and heating power...
You could do this with tests / measurement or by a known mathematical / physical relationship.
We don´t know what your input to the PID control loop is:
* is it voltage
* is it current
* is it apparent power
* is it true power
* is it anything else..
And it seems your PID output is the phase angle for phase control.
But i bet it is far away from being linear.
The unlinearities make a control loop difficult, because it´s hard to make an unlinear control loop stable.
One solution is to make it slow. Not with additional delays, but with low feedback. Especially at the I and P part. (here I don´t think you need the D part at all).
The other solution is to compensate the unlinearities. This needs to analyze your whole regulation system loop. Form each step into mathematical formulas and build the inverse function of the formulas.
Really a lot of mathematics. With sine, square, integrals, logarithm, differential equations, and square root.... (I think in this order).
****
I´ve done this before for a very precise and fast phase control system. A lot of work. Weeks of physics and mathematics.
A lot of software for the linearisation.
But in the end it works incredibly good. Fast and precise. (Btw: controlled by an ATMeaga128).
But the result was a constant loop gain of the whole system. Independent of setpoint, phase control angle and load resistance.
It is mathematically predictable stable for every situation (i can think of).
****
In your case it seems you have an about predictable laod resistance. This makes - at least one part - easy.
Instead of mathematics you could make some tests and find out:
* heating power
* rise in temperature
for phase angle from 0 to 180° in steps of 5° (the smaller the steps the more precise)
Analyze the curve. There are steep parts and flat (almost horizontal) parts.
Your system needs to be stable from the flat parts to the steep parts. (Either by beeing slow and unprecise or by continously adjusting the P I D parameters)
Decide how you want to manage this.
Klaus