  In this paper, we first propose a method that can efficiently compute the
maximal robust controlled invariant set for discrete-time linear systems with
pure delay in input. The key to this method is to construct an auxiliary linear
system (without delay) with the same state-space dimension of the original
system in consideration and to relate the maximal invariant set of the
auxiliary system to that of the original system. When the system is subject to
disturbances, guaranteeing safety is harder for systems with input delays.
Ability to incorporate any additional information about the disturbance becomes
more critical in these cases. Motivated by this observation, in the second part
of the paper, we generalize the proposed method to take into account additional
preview information on the disturbances, while maintaining computational
efficiency. Compared with the naive approach of constructing a higher
dimensional system by appending the state-space with the delayed inputs and
previewed disturbances, the proposed approach is demonstrated to scale much
better with the increasing delay time.
