A thermostat with a deadband does nothing while the temperature is ‘close enough’ — it only acts at the edges of a tolerance band. That stops it chattering on and off, but it means the state NEVER settles exactly on the setpoint; it wanders inside the band forever. It’s the load-bearing gap made literal: you trade exactness for stability. Slide the band width.
A deadband (or hysteresis band) controller applies no correction while the error |e| is within a tolerance ±b, acting only past the edges. This eliminates limit-cycling chatter around a switching threshold, but guarantees a nonzero steady-state error: the state drifts within the band and is never driven exactly to the setpoint. Widening b buys more stability and a larger residual gap; narrowing it approaches chatter. A fail-loud self-check throws unless the settled state stays within the band yet never converges to a single setpoint value. ◆ real control theory, node-verified.
The plant and disturbance are a simple model; the trade — a deadband removes chatter at the cost of a residual error it can never close — is exact. This IS hysteresis in control: the switch-on and switch-off thresholds differ.