On Nonnegative Realizations

Abstract

Problems of nonnegative finite dimensional realizations of a nonnegative impulse response (or, equivalently, Markov sequence) or of a given rational transfer function are studied. We first apply the nonnegative factorization and nonnegative rank approach introduced by Cohen and Rothblum [4], and employed to nonnegative realizations by [6]. Theorem 3 proves that the finiteness of the so called c+Rank of a matrix formed from the Markov coefficients is a necessary and sufficient condition for the nonnegative realizability of the nonnegative Markov sequence (when the coefficients are c × b nonnegative matrices).

We show that if a matrix-valued transfer function has only nonnegative and simple poles with nonnegative coefficient matrices (residues), then (nonnegative realizations exist, and) the order p of any nonnegative-minimal (MP) realization lies between the sum of the ranks and the sum of the nonnegative ranks of the residues. We present an example showing that both inequalities simultaneously may be strict.

The work of the second author was supported by the Hungarian National Scientific Grant OTKA No. T-030042.