Kalman–Yakubovič–Popov lemma for descriptor systems

Abstract

For a given descriptor realization of para-hermitian rational matrix Π(s), we present a generalization of Kalman–Yakubovič–Popov lemma, i.e. necessary and sufficient conditions for Π≥0 on the imaginary axis, in terms of an inequality with constant matrices. The result is quite general, since Π can have poles and zeros on the extended imaginary axis, hence the nonstrict inequality Π(jω)≥0, ω∈R can hold, instead of the strict inequality. Π can be singular. The descriptor realization is required to be only impulse controllable and controllable (or stabilizable and detΠ≠0). A spectral factorization of Π is given, by the above mentioned constant matrices. Three consequences of the generalized KYP lemma, and an illustrative numerical example are given.