The rate at which your house warms while the heat is on is an indirect measure of the effect of both the temperature of the outside of your walls and the amount

The rate at which your house warms while the heat is on is an indirect measure of the effect of both the temperature of the outside of your walls and the amount of solar radiation coming through your windows. As others have indirectly pointed out, for the range of circumstances a given house is going to encounter, the linearizing assumptions in a standard PID controller are probably sufficient. If tuned properly, the "differential" portion of the PID controller will shut down the heater earlier when it's warmer out and/or there's more solar energy coming through your windows.