Open Access. Powered by Scholars. Published by Universities.®
Physical Sciences and Mathematics Commons™
Open Access. Powered by Scholars. Published by Universities.®
Articles 1 - 2 of 2
Full-Text Articles in Physical Sciences and Mathematics
Convex Cones Of Generalized Positive Rational Functions And Nevanlinna-Pick Interpolation, Daniel Alpay, Izchak Lewkowicz
Convex Cones Of Generalized Positive Rational Functions And Nevanlinna-Pick Interpolation, Daniel Alpay, Izchak Lewkowicz
Mathematics, Physics, and Computer Science Faculty Articles and Research
Scalar rational functions with a non-negative real part on the right half plane, called positive, are classical in the study of electrical networks, dissipative systems, Nevanlinna-Pick interpolation and other areas. We here study generalized positive functions, i.e with a non-negative real part on the imaginary axis. These functions form a Convex Invertible Cone, cic in short, and we explore two partitionings of this set: (i) into (infinitely many non-invertible) convex cones of functions with prescribed poles and zeroes in the right half plane and (ii) each generalized positive function can be written as a sum of even and odd parts. …
The Positive Real Lemma And Construction Of All Realizations Of Generalized Positive Rational Functions, Daniel Alpay, Izchak Lewkowicz
The Positive Real Lemma And Construction Of All Realizations Of Generalized Positive Rational Functions, Daniel Alpay, Izchak Lewkowicz
Mathematics, Physics, and Computer Science Faculty Articles and Research
We here extend the well known Positive Real Lemma (also known as the Kalman-Yakubovich-Popov Lemma) to complex matrix-valued generalized positive rational function, when non-minimal realizations are considered. All state space realizations are partitioned into subsets, each is identified with a set of matrices satisfying the same Lyapunov inclusion. Thus, each subset forms a convex invertible cone, cic in short, and is in fact is replica of all realizations of positive functions of the same dimensions. We then exploit this result to provide an easy construction procedure of all (not necessarily minimal) state space realizations of generalized positive functions. As a …