  In this paper, we consider the computation of controlled invariant sets (CIS)
of discrete-time nonlinear control affine systems. We propose an iterative
refinement procedure based on polytopic inclusion functions, which is able to
approximate the maximal controlled invariant set to within a guaranteed
precision. In particular, this procedure allows us to guarantee the invariance
of the resulting near-maximal CIS while also computing sets of control inputs
that enforce the invariance. Further, we propose an accelerated version of this
procedure which refines the CIS by computing backward reachable sets of
individual components of set unions, rather than all at once. This reduces the
total number of iterations required for convergence, especially when compared
with existing methods. Finally, we compare our methods to a sampling-based
approach and demonstrate the improved accuracy and faster convergence.
