Algorithm for positive realization of transfer functions

The aim of this brief is to present a finite-step algorithm for the positive realization of a rational transfer function H(z). In comparision with previously described algorithms we emphasize that we do not make an a priori assumption on (but, instead, include a finite step procedure for checking) the non- negativity of the impulse response sequence of H(z). For primitive transfer functions a new method for reducing the pole order of the dominant pole is also proposed

Polynomial and rational inequalities on Jordan arcs and domains

In this paper we prove an asymptotically sharp Bernstein-type inequality for polynomials on analytic Jordan arcs. Also a general statement on mapping of a domain bounded by finitely many Jordan curves onto a complement to a system of the same number of arcs with rational function is presented here. This fact, as well as, Borwein-Erd\'elyi inequality for derivative of rational functions on the unit circle, Gonchar-Grigorjan estimate of the norm of holomorphic part of meromorphic functions and Totik's construction of fast decreasing polynomials play key roles in the proof of the main result.Comment: Minor typos corrected, DOI adde

A lowerbound on the dimension of positive realizations

A basic phenomenon in positive system theory is that the dimension N of an arbitrary positive realization of a given transfer function H(z) may be strictly larger than the dimension n of its minimal realizations. The aim of this brief is to provide a non-trivial lower bound on the value of N under the assumption that there exists a time instant k0 at which the (always nonnegative) impulse response of H(z) is 0 but the impulse response becomes strictly positive for all k > k0. Transfer functions with this property may be regarded as extremal cases in positive system theory

Embedding graphs having Ore-degree at most five

Let $H$ and $G$ be graphs on $n$ vertices, where $n$ is sufficiently large. We prove that if $H$ has Ore-degree at most 5 and $G$ has minimum degree at least $2n/3$ then $H\subset G.$Comment: accepted for publication at SIAM J. Disc. Mat

Order bound for the realization of a combination of positive filters

In a problem on the realization of digital ¯lters, initiated by Gersho and Gopinath [8], we extend and complete a remarkable result of Benvenuti, Farina and Anderson [4] on decomposing the transfer function t(z) of an arbitrary linear, asymptotically stable, discrete, time-invariant SISO system as a di®erence t(z) = t1(z) ¡ t2(z) of two positive, asymptotically stable linear systems. We give an easy-to-compute algorithm to handle the general problem, in particular, also the case of transfer functions t(z) with multiple poles, which was left open in [4]. One of the appearing positive, asymptotically stable systems is always 1-dimensional, while the other has dimension depending on the order and, in the case of nonreal poles, also on the location of the poles of t(z). The appearing dimension is seen to be minimal in some cases and it can always be calculated before carrying out the realization