PID is implemented the integral term will grow until the temperature stays steady.
The "I" term tends to make the setpoint and measured value agree. In an analog controller, the power supply rails become limits. When you implement PID in software, you have to prevent "reset wind up". There is no point for the I term to get bigger that 100%, so you just clamp it there, otherwise the recovery time can get very long.
"D" or the derivative term tends to prevent overshoot.
The P or is the gain of the system. As P gets bigger the control either gets tighter or the system will tend to oscillate.
A very simple explanation is here:
https://www.csimn.com/CSI_pages/PIDforDummies.html
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One of the other issues is just how are you controlling the power. This is where things get really tricky. We can talk about bang-bang control which typically gives you lots of overshoot. We can control the number of power line cycles on and off. This tends to reduce RFI. 5 on, 5 off is 50%. if you really want finer control, the phase angle firing is used, It's somewhat better to have 0 to 100% be proportional to V^2 and not V. For a heater, (V^2)/R is proportional to power and usually R is relatively constant except for a light bulb. In that case, the 0-100% gizmo is matched to a Tungsten load. These are some of the finer points. R vs T is incorporated into the power controlling device.
I've done a fair bit with PID. Where I worked and before my time, "they" used a controller that operated into variacs to control 30 V custom tantalum heaters and had analog meters to measure some repeatability of V and I. I said enough of this nonsense and we then used a DC power supply. One thing it lacked was the ability to measure power directly, but unlike the phase angle fired system, power was power and not just used to check the process reproducibility.
We had to change to some sort of non-proprietary control like 0-5 V, 4-20 mA, 0-20 mA anyway because of obsolescence.
We did some cool stuff with PID. In one case, I had an environmental chamber that had it's own temperature controller but we wanted to control the temperature of a sample within the chamber, so my -100 to +100 % was the +-10 degree proportional band around a constantly changing setpoint. e.g. 100% meant you set the setpoint of the chamber controller to the measured value of the chamber plus 10 degrees.