ISSN 0005-1179, Automation and Remote Control, 2015, Vol. 76, No. 11, pp. 2110–2123. © Pleiades Publishing, Ltd., 2015. Original Russian Text © S.V. Emel’yanov, A.P. Afanas’ev, 2015, published in Avtomatika i Telemekhanika, 2015, No. 11, pp. 27–42. LINEAR SYSTEMS Signal Differentiation in Automatic Control Systems S. V. Emel’yanov and A. P. Afanas’ev Institute for Information Transmission Problems (Kharkevich Institute), Russian Academy of Sciences, Moscow, Russia National University of Science and Technology MISiS, Moscow, Russia Lomonosov State University, Moscow, Russia e-mail: isa@isa.ru, apa@isa.ru Received October 31, 2014 Abstract—Realization of the traditional approaches to differentiation of signals in the automatic control systems is hindered by the negative effect of gain on the accuracy of differentiation and stability to high-frequency noise. An approach was proposed based on representing the consid- ered signal as a power function with coefficients that are calculated by repeated integration and approximate the derivative of the original signal. At that, the accuracy of differentiation and the level of noise immunity can be improved by using more integrators, rather than increasing the gains. DOI: 10.1134/S0005117915110024 1. INTRODUCTION Differentiation of signals in the automatic control systems represents one of the most impor- tant problems. Traditionally, used are analog differentiators [1–3] or digital differentiation circuits [4, 5]. Circuits based on the wavelet transformations [7] are actively used along with the classical approach [6] to the noisy signals. The present paper proposes to realize signal differentiation by restoring the signal under consideration in the functional form. We recall that by the signal is meant the function with values defined over a time segment limited from right by the current time interval. The signal under consideration is restored in [8] as a segment of the power series and as a sum of the trigonometric functions. Signal is restored in the neighborhood of the measured point. An approximation in the form of a power series can be conveniently used to calculate the derivatives. The main result established in the present paper is a formula representing the signal under consideration as a power function with coefficients computed by repeated integration and having the sense of approximation relative to the operation of differentiation. The resulting formula is compared with the Taylor formula, and the accuracy with which the coefficients of the obtained formula approximate the derivatives is ascertained. The paper proposes a block diagram enabling one to restore not only the considered signal, but its derivatives as well. The computational characteristics of this lock diagram are examined. 2. FORMULATION OF THE PROBLEM OF SIGNAL DIFFERENTIATION The function f (t) will be said to be a signal if given is the function f (s):[t 0 ,t] → R 1 , where t 0 is the point of the starting observation and t is the current time instant. The following operations are defined over the signal: 1. Signal multiplication by the numbers kf (t) (or amplification by the factor of k). 2. Addition and subtraction of signals and functions f (t) ± x(t). 2110