Application of the butterworth polynomials for the design of modal differentiators

Abstract

The article considers a method for designing automatic modal differentiators using normalized Butterworth polynomials. The synthesis of modal differentiators is reduced to the construction of a tracking control system for an object which is a series-connected integrating network. The poles of automatic modal differentiators are the roots of the Butterworth polynomials. The Butterworth polynomial is the denominator of the Butterworth filter. The roots of the Butterworth polynomial are located on a circle of a certain radius equidistant from each other in the left half plane of the complex plane. The radius of the circle is determined by the cutoff frequency of the Butterworth filter. The constructed modal differentiators provide asymptotically exact noise-immune differentiation of a fairly large class of signals. The class of differentiatable signals is defined by a differential inequality and contains a set of continuously differentiable functions with a bounded higher-order derivative. The class of differentiable signals includes logarithmic, exponential and trigonometric functions, and algebraic polynomials. It should be noted that the class of differentiable signals is not limited to the above functions. Modal differentiators are immune to high-frequency noise. The bandwidth of the signals can be set by choosing an appropriate cutoff frequency of the Butterworth filter. The article provides a comparative analysis of modal differentiators constructed using the Butterworth polynomials and differentiators whose poles form a geometric sequence. Amplitude-frequency and phase-frequency characteristics were used to analyse the differentiators. An example of constructing a first-order differentiator is given. The result of the differentiation of a low-frequency harmonic signal was considered in the time domain. The differentiators proposed in this article can be used for the synthesis of high-quality automatic control systems, as well as for solving a wide range of problems regarding automatic differentiation.