Posicast controllers are used to dampen the oscillations of a stable system(to improve transient response and reduce overshoot).By stable system, I refer 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.

A posicast controller usually bifurcates the response into transient and steady state parts, and adjusts input step accordingly.For example, in the above image, the controller, which is called a posicast half cycle controller, applies a step reference of 1/(1+delta) till the response reaches peak value and then immediately starts applying the original step input of 1

Footnotes