Computing the state difference equations for discrete overdetermined linear mD systems

We derive an algorithm that computes the state difference equations for a given set of poles of linear discrete overdetermined autonomous mmD systems. These difference equations allow the realization of the dynamical system by means of delay, multiplication and addition elements in simulation diagrams. In doing so we generalize the classical Cayley–Hamilton theorem to multivariate polynomial ideals and provide a system theoretic interpretation to the notion of polynomial ideals, leading monomials and Gröbner bases. Furthermore, we extend the problem to include poles at infinity and so arrive at a description of overdetermined descriptor systems. This results in a new state space description of autonomous mmD descriptor systems. In addition, we discuss the separation of the state variables of singular mmD systems into a regular and singular part. A sufficient condition under which these two state vector parts can be interpreted as a forward evolving regular part and a backward evolving singular part is given. The robustness and efficiency of the developed algorithms are demonstrated via numerical experiments.