Characterization of strictly positive real multivariable systems

In the literature are given several equivalent conditions for the strict positive realness property of transfer function matrices. Recently, it is shown that some of these equivalent conditions are not fully matched with the definition of strict positive realness. On the other hand, there is a necessary condition in the recent statements which is difficult to check and this difficulty is strongly increased when the number of system poles or the dimension of the transfer function matrix of the system is increased. In addition, there is not a correct statement for characterizing strictly positive real (SPR) improper transfer function matrices yet. In this paper, we remove these deficiencies and derive a more complete version of necessary and sufficient conditions for characterizing SPR multivariable systems. For this purpose, we use Taylor series expansion and Laurent series expansion simultaneously. To illustrate the ease and comprehensiveness of our results, we present several illustrative examples.