Two Position Controllers:

A two position controller is a device which has two operating conditions: completely on or completely off.

Below figure displays the input to output, features waveform for a two position controller which switches from its "OFF" state to its "ON" state while the measured variable raise above the setpoint. On the other hand, it switches from its "ON" state to its "OFF" state while the measured variable decreases below the setpoint. This device gives an output determined through whether the error signal is above or below the setpoint.  A magnitude of the error signal is above or below the setpoint.  A magnitude of the error signal past in that point is of no concern to the controller.

Figure: Two Position Controller Input/Output Relationship