Periodic orbits from Δ-modulation of stable linear systems

The Δ-modulated control of a single input, discrete time, linear stable system is investigated. The modulation direction is given by c^Tx where c ∈Rⁿ/{0} is a given, otherwise arbitrary, vector. We obtain necessary and sufficient conditions for the existence of periodic points of a finite order. Some concrete results about the existence of a certain order of periodic points are also derived. We also study the relationship between certain polyhedra and the periodicity of the Δ-modulated orbit.