  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.
