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The paper is dealing with the concept of positive real and strictly positive real properties of linear time-invariant systems. Although strict positive realness is the most important subset of positive real systems and extensively used in stabilizing control systems, there are inconsistencies in the literature for its necessary and sufficient conditions. Also, many frequency properties of such systems are less attended by the scholars. In this paper, a comprehensive discussion on definitions, properties, lemmas, and theorems related to multi-input multi-output positive real and strictly positive real systems will be presented. Further, several frequency domain properties are presented to characterize these functions. Also, three equivalent high frequency condition presented in the literature for strictly positive real systems are compared. The main results have been supported via several illustrative examples.

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.

The necessary and sufficient conditions for (strict) positive realness of strictly proper transfer functions are derived in the above-mentioned paper refid="ref1"/. Presenting a counter-example, we show that there are inconsistencies between definitions stated in refid="sec2"Section II and main results proved in section III of refid="ref1"/. Also, the required corrections in the main results of refid="ref1"/ are presented. Note however that the results in refid="ref1"/ are correct for the definition of SPR in the Circuit Theory sense. [ABSTRACT FROM AUTHOR]