Towards an effective study of the algebraic parameter estimation problem

International audienceThe paper aims at developing the first steps toward a symbolic computation approach to the algebraic parameter estimation problem defined by Fliess and Sira-Ramirez and their coauthors. In this paper, within the algebraic analysis approach, we first give a general formulation of the algebraic parameter estimation for signals which are defined by ordinary differential equations with polynomial coefficients such as the standard orthogonal polynomials (Chebyshev, Jacobi, Legendre, Laguerre, Hermite, ... polynomials). Based on a result on holonomic functions, we show that the algebraic parameter estimation problem for a truncated expansion of a function into an orthogonal basis of L_2 defined by orthogonal polynomials can be studied similarly. Then, using symbolic computation methods such as Gröbner basis techniques for (noncommutative) polynomial rings, we first show how to compute ordinary differential operators which annihilate a given polynomial and which contain only certain parameters in their coefficients. Then, we explain how to compute the intersection of the annihilator ideals of two polynomials and characterize the ordinary differential operators which annihilate a first polynomial but not a second one. These results, which are at the core of the algebraic parameter estimation, are implemented in the Non-A package built upon the OreModules software