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dc.contributor.authorRobertz, Daniel

This paper addresses systems of linear functional equations from an algebraic point of view. We give an introduction to and an overview of recent work by a small group of people including the author of this article on effective methods which determine structural properties of such systems. We focus on parametrizability of the behavior, i.e., the set of solutions in an appropriate signal space, which is equivalent to controllability in many control-theoretic situations. Flatness of the linear system corresponds to the existence of an injective parametrization. Using an algebraic analysis approach, we associate with a linear system a module over a ring of operators. For systems of linear partial differential equations we choose a ring of differential operators, for multidimensional discrete linear systems a ring of shift operators, for linear differential time-delay systems a combination of those, etc. Rings of these kinds are Ore algebras, which admit Janet basis or Gröbner basis computations. Module theory and homological algebra can then be applied effectively to study a linear system via its system module, the interpretation depending on the duality between equations and solutions. In particular, the problem of computing bases of finitely generated free modules (i.e., of computing flat outputs for linear systems) is addressed for different kinds of algebras of operators, e.g., the Weyl algebras. Some work on computer algebra packages, which have been developed in this context, is summarized.

dc.publisherSpringer Science and Business Media LLC
dc.subjectSystems of linear functional equations
dc.subjectSystems theory
dc.subjectControl theory
dc.subjectAlgebraic analysis
dc.subjectHomological algebra
dc.subjectLinear partial differential equations
dc.subjectJanet bases
dc.subjectGrobner bases
dc.titleRecent progress in an algebraic analysis approach to linear systems
dc.typeJournal Article
plymouth.journalMultidimensional Systems and Signal Processing
plymouth.organisational-group/Plymouth/Faculty of Science and Engineering
plymouth.organisational-group/Plymouth/Faculty of Science and Engineering/School of Engineering, Computing and Mathematics
plymouth.organisational-group/Plymouth/Users by role
plymouth.organisational-group/Plymouth/Users by role/Academics
dc.rights.embargoperiodNot known
rioxxterms.typeJournal Article/Review

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