The humblest controller: a thermostat just switches the heat ON when it’s too cold and OFF when it’s too warm. To stop it chattering on-off forever it uses a HYSTERESIS band — a dead-zone around the setpoint — so the temperature gently sawtooths within a range. It’s bang-bang control, and it’s everywhere from fridges to furnaces. Slide the band width and watch the swing.
A thermostat is a bang-bang (on–off) controller with HYSTERESIS: the heater turns on when temperature falls below setpoint − h and off when it rises above setpoint + h. Without the band (h=0) it would switch infinitely fast (chatter) at the setpoint; the hysteresis band trades tight regulation for fewer switch cycles, so the temperature oscillates in a sawtooth of width ~2h around the setpoint. It is the simplest possible feedback loop — no proportioning, just a threshold with memory — and it demonstrates the core of self-regulation: sense, compare to a setpoint, act. A fail-loud self-check throws unless the temperature settles into oscillation within the ±h band. ◆ real control theory, node-verified.
An idealised symmetric heat/cool with fixed rates (the exact hysteresis-band oscillation); real systems add thermal lag that widens the swing and duty-cycle effects — the sense-compare-act loop and the band tradeoff are exact.