State dependent disturbances in computation of forward reachable sets and minimal invariant sets

We consider linear systems with additive exogenous signals, whose range is state-dependent and is constrained in a polytope in the state-disturbance space. We observe that these systems are equivalent (i) to a linear inclusion defined as projection of a polytopic set on the state space, and (ii) to piecewise polytopic affine dynamics, which is convex on the support of the disturbance set. We use these descriptions to characterise the reachable sets and consequently characterise the minimal Robust Positively Invariant (mRPI) set. Our results can be used in a variety of applications.