An example of how positivity may force realizations of 'large' dimension

A standard result of linear system theory states that an SISO proper ration al transfer function of degree n always has a realization of dimension n. I n some applications one is interested in having a realization with nonnegat ive entries and it is known that, when the dominant poles display a specifi c pattern, forcing nonnegativity leads to a system which is not jointly rea chable and observable. In this paper, we show that the minimal dimension of a positive realization may be 'large' even in the case of a single dominan t pole. More precisely, we provide a family of transfer functions, each of which is of degree n = 3, such that for any integer N greater than or equal to 3 the corresponding member of the family admits a minimal positive real ization of state space dimension not smaller than N. (C) 1999 Elsevier Scie nce B.V. All rights reserved.