Quantitatively hyper-positive real rational functions II

Colloquially, quantitatively Hyper-Positive functions form a sub-family of the “Strictly Positive real” functions, where in addition state-space realization always exists and the limit at infinity is non-singular. In scalar terminology, such function maps the right-half of the complex plane into a sub-region of the right-half plane, which can be contained in a finite disk. Moreover, under inversion this disk is mapped onto itself. In [9] we showed that these functions appear in the “absolute stability” Lurie-type problems and characterized these families through their state-space realization, in the spirit of the Kalman-Yakubovich-Popov Lemma. Here, we present additional characterizations, i.e. as rational functions and through the corresponding kernels. Subsequently, a description of all solutions of the tangential Nevanlinna-Pick interpolation problem, in this framework, is presented.