Posicast controllers are used to dampen the oscillations of a stable system(to improve transient response and reduce overshoot). By stable system, this refers to systems with manageable overshoot which satisfy BIBO stability.

They were initially used as feed forward dampeners, especially by Otto J Smith in the 1950s who actually invented this class of controllers. Modern posicast controllers are available in both feedforward and feedback schemes and are widely used in power electronics.

The above explanation is from Control Systems Engineer on Quora