Positive realization of difference equations

The problem treated here is that of realization of an nthorder linear difference equationd(D)y = 0describing free responses of a physical system in the formx(k + 1)=Ax(k), y(k)=c'x(k), where the elements of matrix A and vector c are restricted to be nonnegative to reflect physical constraints. The specific problem treated here are realizability conditions, and characterizations of minimal realizations. These problems are discussed in detail through a geometric approach, specifically through the convex analysis. It is shown that the necessary and sufficient condition for realizability and the minimal dimension are completely characterized by a convex cone derived from the difference equation. A matrix equation generating all possible realizations is obtained, and then the canonical structure of minimal realizations is derived.