On an algebraic method for derivatives estimation and parameter estimation for partial derivatives systems

International audience— In this communication, we discuss two estimation problems dealing with partial derivatives systems. Namely, estimating partial derivatives of a multivariate noisy signal and identifying parameters of partial differential equations. The multivariate noisy signal is expressed as a truncated Taylor expression in a small time interval. An algebraic method can be then used to estimate its partial derivatives in the opera-tional domain. The same approach applies for the parameter identification problem. The main objects in the aforementioned algebraic method are particular differential operators called annihilators. Stafford's theorem can be applied to formalize the left ideal in the Weyl algebra formed by the annihilators. In the spatial domain, iterated integrals present in the estimates yield noise filtering

Algebraic estimation in partial derivatives systems: parameters and differentiation problems

International audienceTwo goals are sought in this paper: namely, to provide a succinct overview on algebraic techniques for numerical differentiation and parameter estimation for linear systems and to present novel algebraic methods in the case of several variables. The state-of-art in the introduction is followed by a brief description of the methodology in the subsequent sections. Our new algebraic methods are illustrated by two examples in the multidimensional case. Some algebraic preliminaries are given in the appendix

A MAPLE package for integro-differential operators and boundary problems

We present a Maple package for computing in algebras of integro-differential operators. This provides the appropriate algebraic setting for treating boundary problems [7] for linear ordinary differential equations symbolically. They allow to formulate a boundary problem - a differential equation and boundary conditions - but they are also expressive enough for describing its solution via an integral operator, which is called Green's operator. Our package provides procedures for algebraic operations on integro-differential operators as well as for solving LODEs with general boundary conditions, given a fundamental system. The implementation was tested in Maple 11, 12 and 13. It is available with an example worksheet at http://www.risc.jku.at/people/akorpora/index.html

An Algorithm for Converting Nonlinear Differential Equations to Integral Equations with an Application to Parameter Estimation from Noisy Data

International audienceThis paper provides a contribution to the parameter esti-mation methods for nonlinear dynamical systems. In such problems, a major issue is the presence of noise in measurements. In particular, most methods based on numerical estimates of derivations are very noise sen-sitive. An improvement consists in using integral equations, acting as noise filtering, rather than differential equations. Our contribution is a pair of algorithms for converting fractions of differential polynomials to integral equations. These algorithms rely on an improved version of a re-cent differential algebra algorithm. Their usefulness is illustrated by an application to the problem of estimating the parameters of a nonlinear dynamical system, from noisy data. In Engineering, a wide variety of information is not directly obtained through measurement. Various parameters or internal variables are unknown or not mea-sured. In addition, sensor signals are very often distorted and tainted by mea-surement noises. To simulate, control or supervise such processes, and to extract information conveyed by the signals, a system has to be identified and parame-ters and variables must be estimated. Most of traditional estimation methods are related to asymptotic statistics. However, there exist some difficulties that have been long known as inherent to these existing methods. Among them, two im-portant limitations can be pointed out: these methods apply essentially to linear systems and they are noise sensitive due to the use of numerical derivation. The parameter estimation problem has been tackled by many different approaches in control theory. Algebraic techniques to this end were notably introduced in the works by M. Fliess et al. [8, 15, 7, 9, 6] and inspired for instance, algebraic methods for the parameter estimation of a multi-sinusoidal waveform signal from noisy data [22]