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Prooving Identities with Computer‐Algebra ‐‐ Example: Algebraic Time‐Derivative Estimation
Author(s) -
Reger Johann,
Jouffroy Jérôme
Publication year - 2008
Publication title -
pamm
Language(s) - English
Resource type - Journals
ISSN - 1617-7061
DOI - 10.1002/pamm.200810905
Subject(s) - algebraic number , linear algebra , derivative (finance) , algebra over a field , lti system theory , invariant (physics) , mathematics , linear system , computer science , pure mathematics , mathematical analysis , geometry , financial economics , economics , mathematical physics
For the case of continuous–time systems, this note contributes a detailed proof relating the so–called algebraic approach to time–derivative estimation, as proposed by Fliess and co–workers, to classical results from linear estimation theory. The proof is based on a modern computer–algebra proof technique that, in the main, resorts to the celebrated algorithm by Wilf and Zeilberger in the multisum case. As a result of the proof, the algebraic approach to time–derivative estimation is traced back, equivalently, to state estimation using the reconstructibility Gramian of the dynamic system, here, with respect to a particular nilpotent time–invariant input–free linear system. Additionally, the close relationship of the algebraic approach with least–squares time–derivative estimation is pointed out. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)