Is it not possible to forecast temperature changes rather than installing a bunch of extra sensors? If the device knows its location and has an internet connection, it can examine the day's weather forecast, know where the sun will shine, and know how cloudy it will be.
Good point. I wonder if the temperature sensor in the thermostat would be able to help correct that error.
E.g., if the thermostat expected a temperature of 75 F at 12 PM but it was actually 78 F, then you know the house received a lot of sunlight and can correct accordingly.
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.
