On unimodular transformations of conservative L-systems

TL;DR

This paper investigates unimodular transformations of conservative L-systems, introducing classes of impedance functions, establishing invariance under these transformations, and developing a coupling framework with explicit controllers, supported by illustrative examples.

Contribution

It introduces new classes of impedance functions for L-systems, proves their invariance under unimodular transformations, and develops a coupling method with explicit controllers.

Findings

Classes of impedance functions are invariant under unimodular transformations.

A coupling theorem for L-systems and F-systems is established.

Explicit forms of controllers for transformations are derived.

Abstract

We study unimodular transformations of conservative $L$-systems. Classes $\sM^Q$, $\sM^Q_\kappa$, $\sM^{-1,Q}_\kappa$ that are impedance functions of the corresponding $L$-systems are introduced. A unique unimodular transformation of a given $L$-system with impedance function from the mentioned above classes is found such that the impedance function of a new $L$-system belongs to $\sM^{(-Q)}$, $\sM^{(-Q)}_\kappa$, $\sM^{-1,(-Q)}_\kappa$, respectively. As a result we get that considered classes (that are perturbations of the Donoghue classes of Herglotz-Nevanlinna functions with an arbitrary real constant $Q$) are invariant under the corresponding unimodular transformations of $L$-systems. We define a coupling of an $L$-system and a so called $F$-system and on its basis obtain a multiplication theorem for their transfer functions. In particular, it is shown that any unimodular transformation of a given $L$-system is equivalent to a coupling of this system and the corresponding controller, an $F$-system with a constant unimodular transfer function. In addition, we derive an explicit form of a controller responsible for a corresponding unimodular transformation of an $L$-system. Examples that illustrate the developed approach are presented.