Minimal positive realizations of transfer functions with positive real poles (Q2724333)

Given a strictly proper rational transfer function \(H(z)\), the triple \(\{A, b,c^T\}\) is said to be a positive realization if NEWLINE\[NEWLINEH(z)= \sum_{k\geq 1} c^T A^{k-1} bz^{-k},NEWLINE\]NEWLINE where \(A\), \(b\) and \(c^T\) all have nonnegative entries.NEWLINENEWLINENEWLINEThe existence problem (is there a nonnegative realization of some finite dimension \(N\) and how may it be found) has been solved in the nineties.NEWLINENEWLINENEWLINEThe authors concentrate on the difficult minimality problem (what is the minimal value for \(N\)?) and solve this problem in a constructive way for a third-order transfer function with distinct real positive poles. In this manner, they give necessary and sufficient conditions for such a transfer function to have a third-order positive realization.