Minimal positive realizations of transfer functions with real poles

A standard result of linear-system theory states that a single-input-single-output (SISO) rational nth-order transfer function always has a state-space realization of the same order. In some applications, one is interested in having a realization with nonnegative entries (i.e., a positive system) and it is known that such constraints may lead to a minimal order positive realization of order much greater than the transfer function order. In this technical note, necessary and sufficient conditions for a third-order transfer function with distinct real poles to have a third-order positive realization are given: these conditions are expressed in terms of lower bounds for the first three samples of the impulse response and therefore are very easy to check. This result is an extension of a previous result for transfer functions with distinct real positive poles.