Prooving Identities with Computer‐Algebra ‐‐ Example: Algebraic Time‐Derivative Estimation

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)