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

A standard result of linear system theory states that an SISO proper rational transfer function of degree n always has a realization of dimension n. In some applications one is interested in having a realization with nonnegative entries and it is known that, when the dominant poles display a specic pattern, forcing nonnegativity leads to a system which is not jointly reachable 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 dominant pole. More precisely, we provide a family of transfer functions, each of which is of degree n=3, such that for any integer N>3 the corresponding member of the family admits a minimal positive realization of state space dimension not smaller than N.