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28 March 2024
 
  » arxiv » 1608.8583

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On unimodular transformations of conservative L-systems
Sergey Belyi ; Konstantin Makarov ; Eduard Tsekanovskii ;
Date 30 Aug 2016
AbstractWe 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.
Source arXiv, 1608.8583
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