Controllable regions of LTI discrete-time systems with input saturation

In this paper, we present a formula to compute the vertices of the (null) controllable regions for general LTI discrete-time systems with bounded inputs. For n^th order systems with only real poles (not necessarily distinct), the formula is simplified to an elementary matrix function, which can be used to show that the set of vertices coincides with a class of time responses of the time-reversed system to bang-bang controls with n-2 or less switches. For second-order systems with a pair of complex conjugate poles, a closed form formula to compute the vertices is provided; the set of vertices can also be obtained from the steady state response of the time-reversed system to a periodic or near periodic bang-bang control. The influence of the sampling period on the controllable regions is clearly demonstrated with some examples. A preliminary investigation is made on the existence of nonlinear controllers and the nonexistence of linear controllers to achieve certain stabilization tasks