  In this paper we consider the problem of computing control invariant sets for
linear controlled systems with constraints on the input and on the states. We
focus in particular on the complexity of the computation of the N-step
operator, given by the Minkowski addition of sets, that is the basis of many of
the iterative procedures for obtaining control invariant sets. Set inclusions
conditions for control invariance are presented that involve the N-step sets
and are posed in form of linear programming problems. Such conditions are
employed in algorithms based on LP problems that allow to overcome the
complexity limitation inherent to the set addition and can be applied also to
high dimensional systems. The efficiency and scalability of the method are
illustrated by computing in less than two seconds an approximation of the
maximal control invariant set, based on the 15-step operator, for a system
whose state and input dimensions are 20 and 10 respectively.
