Bilinear realizability is equivalent to existence of a singular affine differential i/o equation
type: Item
instance of: scholarly article
title: Bilinear realizability is equivalent to existence of a singular affine differential i/o equation
zbMATH Open document ID: 0657.93010
DOI: 10.1016/0167-6911(88)90057-6
published in: Systems \& Control Letters
publication date: 1988-01-01T00:00:00Z
review text: The paper discusses the following realization problem. Given an affine input-output equation of the form \[ \sum^{k}_{i=1}a_ i(u(t),u^ 1(t),...,u^{(k-1)}(t))y^{(i)}(t)=b(u(t),...,u^{(k-1)}(t)), \] under which conditions does there exist a state space system \[ \Sigma:\quad \dot x=f(x)+g(x)u,\quad y=h(x) \] realizing the same input-output pairs. It is shown that under the assumption that there does exist some state- space realization for the affine input-output equation, then this realization can immersed into a bilinear control system \[ \dot z=Az+(Bz)u,\quad y=Cz. \] Conversely, whenever the state space system \(\Sigma\) can be immersed into a bilinear system then the input-output pairs satisfy an affine input-output equation. An analogous result is stated and proved in terms of so-called generating series of a control system, an analog of the well-known Volterra kernels of a nonlinear system.
Mathematics Subject Classification ID: 93B15
Mathematics Subject Classification ID: 93C10
Mathematics Subject Classification ID: 93C15
Mathematics Subject Classification ID: 93B27
zbMATH DE Number: 4074887
zbMATH Keywords: realization problem
zbMATH Keywords: affine input-output equation
zbMATH Keywords: bilinear control system
zbMATH Keywords: generating series
Wikidata QID: Q115363789
reviewed by: H. Nijmeijer
MaRDI profile type: Publication
full work available at URL: 0167-6911(88)90057-6
OpenAlex ID: W2039189339
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author: Eduardo D. Sontag
description: scientific article; zbMATH DE number 4074887