Positive realization of difference equations

Abstract

The problem treated here is that of realization of an nthorder linear difference equation <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">d(D)y = 0</tex> describing free responses of a physical system in the form <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">x(k + 1)=Ax(k), y(k)=c'x(k)</tex> , 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.