Section: New Results

Algebraic technique for estimation, differentiation and its applications

Participants : Cédric Join, Mamadou Mboup, Wilfrid Perruquetti, Rosane Ushirobira, Olivier Gibaru.

Elementary techniques from operational calculus, differential algebra, and noncommutative 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:

• The paper [30] proposes an algebraic method to fault diagnosis for uncertain linear systems. The main advantage of this new approach is to realize fault diagnosis only from knowledge of input and output measurements without identifying explicitly model parameters. Using tools and results of algebraic identification and pseudospectra analysis, the issues of robustness of the proposed approach compared to the model order and noise measurement are examined.

• The aim of [79] , [84] is to develop an algebraic approach to estimate human posture in the sagittal plane using inertial measurement unit providing accelerations and angular velocities. For this purpose the issue of the estimation of the amplitude, frequency and phase is addressed for a biased and noisy sum of three sinusoidal waveform signals on a moving time horizon. Since the length of the time window is small, the estimation must be done within a fraction of the signal's period. The problem is solved via algebraic techniques.

• An application of algebraic estimation approach for estimation of option pricing and dynamic hedging is given in [66] .

• A model-based online fault-diagnosis scheme for an electromagnetically supported plate is presented in [73] as an example of a nonlinear and open-loop unstable system. First, residuals for sensor as well as for actuator faults are generated using algebraic derivative estimators. Then, the robust detection and isolation of step-like sensor and actuator faults is presented.

• The paper [57] uses the extreme value theory for threshold selection in a previously proposed algebraic spike detection method. The algebraic method characterizes the occurrence of a spike by an irregularity in the neural signal and devises a nonlinear (Volterra) filter which enhances the presence of such irregularities.

• The papers [39] , [40] generalize the algebraic method from the integer order to the fractional order for estimating the fractional order derivatives of noisy signals. The proposed fractional order differentiator is deduced from the Jacobi orthogonal polynomial filter and the Riemann-Liouville fractional order derivative definition. Exact and simple formula for this differentiator is given where an integral formula involving Jacobi polynomials and the noisy signal is used without complex mathematical deduction. Hence, it can be used both for continuous-time and discrete-time models. The comparison between our differentiator and the recently introduced digital fractional order Savitzky-Golay differentiator is given in numerical simulations so as to show its accuracy and robustness with respect to corrupting noises.