Periodic orbits from /spl Delta/-modulation of stable linear systems

The /spl Delta/-modulated control of a single input, discrete time, linear stable system is investigated. The modulation direction is given by c/sup T/x where c /spl isin/R/sup n//spl bsol/{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 /spl Delta/-modulated orbit.