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For modeling and controlling dynamic phenomena it is important to establish with higher accuracy some significant quantities corresponding to the dynamic system. For fast phenomena, such significant quantities are represented by the derivatives of the received signals. In case of advanced computer modeling, the received signal should be filtered and converted into a time series corresponding to the estimated values for the dynamic system through a sampling procedure. This paper will show that present-day methods for computing in a robust manner the first derivative of a received signal (using an oscillating system working on a limited time interval and a supplementary differentiation method) can be extended to the robust computation of higher order derivatives of the received signal by using a specific set of second-order oscillating systems (working also on limited time intervals) so as estimative values for higher-order derivatives are to be directly generated (avoiding the necessity of additional differentiation or amplifying procedures, which represent a source of supplementary errors in present-day methods).

For modeling and controlling dynamic phenomena it is important to establish with higher accuracy some physical quantities corresponding to the dynamic system. Usually this procedure is based on signal processing method applied upon the signal received from the dynamic system, implying some filtering methods (for noise rejection). In case of advanced computer modeling, the filtered signal should be converted further into a time series corresponding to the estimated values for the dynamic system through a sampling procedure. Many times these filtering and sampling devices consist of lowpass filters represented by asymptotically stable systems, the sampling moment of time being set after the transient regime of the filtering device has passed.

However, for fast phenomena, significant quantities are represented by the derivatives of the received signals. Usually the derivatives of a received signal

The average value of the first derivative can be approximated by

As a consequence, the filtering device should reject the noise (supposed to present fast variation as compared to the variations of the useful part

However, such structures are very sensitive at the random variations of the integration period (for unity-step input, the signal, which is integrated, is equal to unity at the sampling moment of time). Even if we use oscillators with a very high accuracy, such random variations will appear due to the fact that the integration is performed by an electric current charging a capacitor. This capacitor must be charged at a certain electric charge

The disadvantage of using asymptotically stable systems (previously mentioned) can be avoided by using an oscillating second-order system having the transfer function

However, this method presents a major disadvantage: the filtering devices can generate an electronic voltage corresponding to

A general mathematical method for obtaining the derivatives of the useful part

The output of this signal processing system for an input corresponding to

This means that it can be written as

The whole procedure can continue by analyzing the output

The analysis of the action of transfer function

We must point the fact that certain limitations appear as the order

(i) For a great number of alternating functions, it is quite possible for the maximum value of the sum of alternating functions of angular frequencies

(ii) Taylor series for

This paper has presented a possibility of obtaining the derivatives of the received electrical signal using a filtering device consisting of a sequence of certain oscillating second-order systems and an integrator. The oscillating systems are working on a time period for filtering a received electrical signal, with initial null conditions. The output of this system is integrated over a time period corresponding to a multiple of all time periods of the second-order systems which are part of the signal processing device (at the end of this period the integrated signal being sampled). The influence of all alternating components is rejected due to the integration performed on a multiple of all time periods, and thus the final result corresponds to the integration of a constant function which is proportional to the derivative having to be estimated. The proposed method has shown that present-day methods for computing in a robust manner the first derivative of a received signal (using an oscillating system working on a limited time interval and a supplementary differentiation method) can be extended to the robust computation of higher-order derivatives of the received signal by using a specific set of second-order oscillating systems (working also on limited time intervals) so as estimative values for higher order derivatives are to be directly generated (avoiding the necessity of additional differentiation or amplifying procedures, which represent a source of supplementary errors in present-day methods). It can be used for decreasing the phase delay for signal processing methods, but without using a weighted sum of real and filtered derivatives of the received signal, as in [

In future studies, the analysis will continue by trying to use nonlinear dynamical equations able to generate practical test functions for estimating in robust manner and with greater accuracy the derivatives of the signal transmitted by dynamic systems (see [