On Strictly Positive Real Improper Transfer Matrices

In this article, we consider linear systems having real rational transfer matrices that may be improper. We investigate the properties of strictly positive real and weakly strictly positive real systems and their connections. Our first contribution is to establish a necessary and sufficient condition for a rational and possibly improper transfer matrix to be strictly positive real. Besides stability, this condition only involves the properties of the transfer matrix in the imaginary axis and a geometric relation between the feedthrough term and the residue at the infinity of the transfer matrix. Our condition connects some of the properties of the system with those of its proper part. This leads to our second contribution: integrate, in the improper setting, all the several conditions derived in the literature for the proper case that discriminate weakly strictly positive real and strictly positive real systems.