The Kalman-Yakubovich-Popov Lemma for discrete-time positive linear systems

A theorem of alternatives on the feasibility of linear matrix inequalities (LMIs) is used in order to provide a simple proof of the Kalman-Yakubovich-Popov (KYP) Lemma for discrete-time positive linear systems. It is further shown that for some classes of positive linear systems the KYP Lemma can also be equivalently stated in terms of an associated system matrix (which is only composed by the four system matrices) by requiring its spectral radius being smaller than one. A recursive method, to determine whether a positive matrix is or is not Schur, is obtained as an application of the aforementioned equivalence.