NONNEGATIVE REALIZATIONS

Lorenzo Farina
Dipartimento di Informatica e Sistemistica
Universita' di Roma 'La Sapienza'
Via Eudossiana 18, 00184 Roma, Italy
e-mail: farina@dis.uniroma1.it

Given a strictly proper rational transfer function H(z), the triple (A,b,c) is said to be a nonnegative realization if $H(z)=\sum_{k\geq 1}cA^{k-1}bz^{-k}$ with (A,b,c) nonnegative, i.e. with nonnegative entries. The nonnegative realization problem consists of providing answers to the questions:

1.

The existence problem Is there a nonnegative realization (A,b,c) of some finite dimension N and how may it be found?

2.

The minimality problem What is a minimal value for N?

3.

The generation problem How can we generate all possible positive realizations?

In this talk, after presenting different approaches to tackle this problem and reviewing the literature in this area of research, the solution to problem 1 will be presented together with some preliminary results on problem 2.