Characterizing families of positive real matrices by matrix substitutions on scalar rational functions

This paper concerns the characterization of positive real matrices generated by substitutions (of the Laplace variable s) in scalar rational transfer function by matrix positive real functions. Our main results are restricted to both strongly strictly positive real matrices arid strictly bounded real matrices. As a way to illustrate our main results, we also include here a partial extension of both the Kalman-Yakubovich-Popov lemma (for strongly strictly positive real systems of zero relative degree) and the circle criterion (for strictly positive real systems of zero relative degree).