New results on improper (weakly) strictly positive real systems

In this article we consider the class of rational positive real transfer matrices without any properness assumption. In this setting we establish a high frequency condition that is necessary and sufficient for weakly strictly positive real transfer matrices to be strictly positive real. This is an important problem that has been addressed in several previous works in the literature. So far, however, the large majority of results are limited to the case of proper transfer matrices and very little attention has been devoted to the general case of possibly improper transfer matrices. The latter is of great interest both because the impedance matrix of a passive multi-port electrical network may be improper and because PID controllers are frequently improper and positive real. This paper gives a contribution in this specific area by providing a high frequency condition holding for improper transfer matrices. We complete our analysis of possibly improper positive real transfer matrices by establishing that weakly strictly positive realness is invariant under inversion as much as (strictly) positive realness. Finally, we present a key relationship that enable us to generate a sequence of weakly strictly positive real transfer matrices such that the smaller eigenvalue of the corresponding spectral densities exhibits high-frequency behavior that converges to zero faster and faster. Some illustrative examples are provided to support the results.