It's a little bit disappointing when I later realized that most of heater/air conditioner basically just use on/off control with a deadband. I guess the limiting factor here isn't the controller but the actuator part, but still a let-down to see them being so primitive.
- With a home thermostat you probably want it to reach the desired temperature as quickly as possible. PID will never reach the setpoint as fast as bang-bang control since bang-bang control will always use the maximum possible power.
- PID might help you avoid overshooting the desired temperature but that's not really an important issue for a home furnace. In fact it might be better if it overshoots the desired temperature a bit on very cold days.
- The air in the home acts like a big thermal battery that helps negate the need for a sophisticated controller.
PID works very good for a wide variety of situations but it's important to remember it is not always the best tool for every job. In that way it is kind of like "quicksort".
If the only thing you can do is turn something on and off, a bang-bang controller is the only thing that works.
In a real system this is exactly how MOSFETS are used to control a voltage; you would almost never use an analog amplifier for high power because of the losses, so voltage control is binary on/off, but that doesn't restrict you to just bang-bang control despite the GP's statement.
In fact, an electric heater can and often will be controlled via a PWM method with a continuous controller rather than via hysteresis.
It's a natural choice when your furnace can't throttle, or can but operates most efficiently at one output level anyway.
Since the system always pushes as hard as it can, it automatically reaches the setpoint in the minimum possible time. It's almost like a PID controller with infinite gain. But the tradeoff is jitter -- all the error gets pushed up to the highest frequency bands. As long as you don't care about that in your system, it's a great choice and effortless to tune.
It can also be a bad choice if there's any cost to changing or toggling the output, like if your furnace wears out from switching on and off a lot.
Properly set, this dampens the swings of temperature in the rooms as the heat gain is more closely matched to the heat loss, increasing comfort and economy of the system.
Edit: It's called a bang-bang controller [0]
[0]: https://en.wikipedia.org/wiki/Bang%E2%80%93bang_control