Section: New Results

Algebraic technique for estimation, differentiation and its applications

Elementary techniques from operational calculus, differential algebra, and non-commutative algebra lead to a new algebraic approach for estimation and detection. It is investigated in various areas of applied sciences and engineering. The following lists only some applications:

• Design of a stabilizing feedback based on acceleration measurements and an algebraic state estimation method has been proposed in [54] .

• An extension of the algebraic differentiation method to fractional derivatives calculation in continuous and discrete time has been studied in [88] and [89] respectively. Applications to identification and parameter estimation of fractional linear systems have been considered in [67] , [68] .

• Smoothing noisy data with spline functions is well known in approximation theory. Smoothing splines have been already used to deal with the problem of numerical differentiation. In [43] , we extend this method to estimate the fractional derivatives of a smooth signal from its discrete noisy data. We begin with finding a smoothing spline by solving the Tikhonov regularization problem. Then, we propose a fractional order differentiator by calculating the fractional derivative of the obtained smoothing spline.

• In [81] , we apply an algebraic method to estimate the amplitudes, phases and frequencies of a biased and noisy sum of complex exponential sinusoidal signals. The obtained estimates are integrals of the noisy measured signal: these integrals act as time-varying filters.