An algorithmic approach to the algebraic parameter estimation problem

The parameter estimation problem — extendedly studied in control theory and signal processing — aims at estimating unknown constant parameters of a dynamical system or a control system through system observations that can be affected by perturbations and noises. For instance, from the observation of the sum of sinusoids perturbed by an unknown bias and noise, the parameter estimation problem aims at recovering the amplitude, frequency and the phase of each sinusoid of the sum.

In 2003, Fliess and Sira-Ramirez proposed a new approach to the parameter estimation problem, called thereafter the algebraic parameter estimation problem. Within this approach, the perturbations are supposed to follow known dynamical models but with unknown parameters, and the parameters to be estimated (e.g., system parameters, initial conditions) are obtained as closed-form solutions expressed by iterative indefinite integrals of the observations and the noises only (i.e., without using the unknown parameters of the dynamical models defining the external perturbations). This approach yields real-time estimators (contrary to the asymptotical estimators in the standard literature) and the iterative indefinite integrals naturally filter the effect of the noise.

The first goal of the talk is to give a short overview of the algebraic parameter estimation problem and to illustrate it with explicit examples (e.g. numerical differentiation, finite sums of exponentials or sinusoids). We then explain how to extend the approach to the case of signals defined by linear ordinary differential equations (ODEs) with polynomial coefficients (e.g., holonomic functions, orthogonal polynomials, truncated expansions of functions onto orthogonal bases of L^2). Moreover, using computer algebra methods (e.g., Gröbner bases for noncommutative polynomial rings of ordinary differential operators, elimination theory), we describe a general methodology for the algebraic parameter estimation problem and show how some of the steps can be made algorithmic and implemented in Maple based on the packages inttrans, OreModules and NonA. We finally illustrate our approach by showing how the coefficients and the initial conditions of an ODE, with polynomial coefficients and without singularity at the origin, can be explicitly expressed by means of iterative indefinite integrals of a generic solution of this ODE.

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