On unimodular transformations of conservative L-systems

We study unimodular transformations of conservative $L$-systems. Classes ${\text{sM}}^Q$, ${\text{sM}}_{\kappa }^Q$, ${\text{sM}}_{\kappa }^{-1,Q}$ 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 ${\text{sM}}^{(-Q)}$, ${\text{sM}}_{\kappa }^{(-Q)}$, ${\text{sM}}_{\kappa }^{-1,(-Q)}$, 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.