Kalman–Yakubovič–Popov lemma for descriptor systems

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