Null controllable region of LTI discrete-time systems with input saturation

We present a formula for the extremes of the null controllable region of a general LTI discrete-time system with bounded inputs. For an nth order system with only real poles (not necessarily distinct), the formula is simplified to an elementary matrix function, which in turn shows that the set of the extremes coincides with a set of trajectories of the time-reversed system under bang–bang controls with n−2 or less switches.