ON SOME USEFUL LINKS BETWEEN EXTERNAL POSITIVITY, KALMAN-YAKUBOVICH-POPOV LEMMA AND POSITIVE REAL LEMMA FOR LINEAR TIME-INVARIANT SYSTEMS WITH CONSTANT POINT DELAYS

This paper deals with the conditions on external positivity of linear time-invariant systems subject to a finite number of constant point delays. The relations and comparisons between the state and output of the whole delayed system with those of its associated delay-free and delay-free dynamics auxiliary systems are investigated. "Ad hoc" versions of the Kalman-Yakubovich-Popov Lemma and of the Positive Real Lemma for the general delayed system are established and proved. The first one is seen to be related to an "a priori" prescribed boundedness of the transfer matrix norm in the H-infinity sense while the second one is related to the passivity property of a modified time-delay system with attenuated output, equivalently, to the positive realness of its associate input-output operator. Most of the stated and proved results are independent of the sizes of the delays. However, the obtained "ad hoc" Positive Real Lemma is, in general, dependent on the delays since the attenuation gain is piecewise constant time-varying and dependent on the delays.