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Passivity theory and the positive real lemma have been recognised as two of the cornerstones of modern systems and control theory. As digital control is pervasive in virtually all control applications, developing a general theory on the discretetime positive real lemma appears to be an important issue. While for minimal realisations the relations between passivity, positiverealness and existence of solutions of the positive real lemma equations is very well understood, it seems fair to say that this is not the case in the discretetime case, especially when the realisation is nonminimal and no conditions are assumed on left and/or rightinvertibility of the transfer function. The purpose of this study is to present a necessary and sufficient condition for existence of solutions of the positive real equations under the only assumption that the state matrix A is asymptotically stable.
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