Auotomodeling rational mnemofunctions and their link to an analytical representation of distributions

Mnemofunctions of the form f(x/ε), where f is the proper rational function without singularities on the real line, are considered in this article. Such mnemofunctions are called automodeling rational mnemofunctions. They possess the following fine properties: asymptotic expansions in the space of distributions can be written in explicit form and the asymptotic expansion of the product of such mnemofunctions is uniquely determined by the expansions of multiplicands.Partial fraction decomposition of automodeling rational mnemofunctions generates the so-called sloped analytical representation of a distribution, i.e. the representation of a distribution by a jump of the boundary values of the functions analytical in upper and lower half-planes. Sloped analytical representation is similar to the classical Cauchy analytical representation, but its structure is more complicated. The multiplication rule of such representations is described in this article.

Realization problem for positive linear systems with time delay

The realization problem for positive single-input single-output discrete-time systems with one time delay is formulated and solved. Necessary and sufficient conditions for the solvability of the realization problem are established. A procedure for computation of a minimal positive realization of a proper rational function is presented and illustrated by an example.

A Computational Method for Obtaining the z-Transform from the s-Transform

This paper presents a method for the numerical determination of the z-transform and the modified z-transform of a proper rational function expressed in the Laplace transform complex variables. The method is easy to implement on a computer using any high-level language and does not necessarily require the poles of the s-domain function to be found. The method is equally applicable to single-loop and multi-loop systems.