Aspects on analysis and synthesis of linear discrete systems over the finite field Fq

This note presents a method for synthesizing a static state feedback controller for multiple-input linear systems over discrete sets. The method allows of a complete setting of the cyclic subspaces of the closed loop system by specifying invariant polynomials of the closed loop system. To this end, fundamentals of finite field theory and basic elements of the polynomial approach are developed towards a means of analysis of the cyclic behavior of the system, which, with regard to synthesis issues, results in an algorithm for setting the invariant polynomials of the system dynamics in the closed loop system. The basic ideas are illustrated in an example.