caa a22 4500 606199632 CHVBK 20210128100939.0 cr unu---uuuuu 210128e20150401xx s 000 0 eng 10.1007/s11045-014-0280-9 doi (NATIONALLICENCE)springer-10.1007/s11045-014-0280-9 Robertz D. School of Computing and Mathematics, Plymouth University, 2-5 Kirkby Place, Drake Circus, PL4 8AA, Plymouth,, United Kingdom aut Recent progress in an algebraic analysis approach to linear systems [Elektronische Daten] [D. Robertz] 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. Springer Science+Business Media New York, 2014 Systems of linear functional equations nationallicence Systems theory nationallicence Control theory nationallicence Algebraic analysis nationallicence Homological algebra nationallicence Linear partial differential equations nationallicence Janet bases nationallicence Gröbner bases nationallicence Multidimensional Systems and Signal Processing Springer US; http://www.springer-ny.com 26/2(2015-04-01), 349-388 0923-6082 26:2<349 2015 26 11045 https://doi.org/10.1007/s11045-014-0280-9 text/html Onlinezugriff via DOI BK010053 XK010053 XK010000 Metadata rights reserved Springer special CC-BY-NC licence nationallicence 1 research-article jats NATIONALLICENCE NATIONALLICENCE NL-springer NATIONALLICENCE 856 40 https://doi.org/10.1007/s11045-014-0280-9 text/html Onlinezugriff via DOI NATIONALLICENCE 100 1- Robertz D. School of Computing and Mathematics, Plymouth University, 2-5 Kirkby Place, Drake Circus, PL4 8AA, Plymouth,, United Kingdom aut NATIONALLICENCE 773 0- Multidimensional Systems and Signal Processing Springer US; http://www.springer-ny.com 26/2(2015-04-01), 349-388 0923-6082 26:2<349 2015 26 11045