A Kalman-Yakubovich-Popov Lemma with affine dependence on the frequency

This paper revisits the problem of checking feasibility of a given matrix inequality with rational dependence on a real variable, omega, often interpreted as frequency. In the case the frequency variable is allowed to assume arbitrary values, the Kalman-Yakubovich-Popov (KYP) Lemma provides an equivalent formulation of this problem as a linear matrix inequality (LMI). When the frequency lies within a finite or semi-infinite range, generalizations of the KYP Lemma provide equivalent formulations as a pair of LMIs. All such tests have a particular form in which a constant, i.e. frequency independent, coefficient matrix, Theta, is used to parametrize the frequency domain inequality (FDI). Previous results showed how one of these LMI tests can be modified to render a sufficient test for a given FDI in which Theta(omega) is an affine function of omega. The main contribution of the present paper is to present a construction that proves such test is also necessary. Many interesting results are presented along the way related to the case when Theta(omega) is quadratic.