Modal differentiator design (Q1578489)

In the book by \textit{S. V. Emel'yanov} and \textit{S. K. Korovin} [New types of feedback, Moskow, Nauk (1997)], the binary system theory is used to design a high-frequency noise tolerant differentiator for a wide class of signals. However, the realization of such a differentiator requires optimization of the smoothing filter parameter. The aim of the authors is to propose a method of modal differentiator design that ensures high-quality multiple differentiation for a wide signal class \(MW^m[t_0,+\infty)\) without any optimization. Here \(M= \text{const}> 0\), \(m\in N= \{1,2,\dots\}\), \(t_0\geq 0\), and \(MW^m[t_0, +\infty)\) is the set of all Carathéodory solutions \(f\in C^{m-1}[t_0, +\infty)\) of the differential equation \(f^{(m)}(t)= \varphi(t)\), where \(\varphi(t)\) is a measurable function satisfying the inequality \(|\varphi(t)|\leq M\) for every \(t\geq t_0\). Applying the transfer function method due to the second author of the present paper given in [Avtom. Telemekh. No. 5, 49-55 (1995; Zbl 0925.93297)] and an existence and uniqueness theorem of \textit{A. F. Filipov} [Differential equations with discontinuous right-hand side, Moscow, Nauk (1985)], it is proved that a device with a transfer function of the form \[ W_{dk}(p)= {p^kL(p)\over D(p)},\quad k= 0,1,\dots, m, \] where \(D(p)\), \(L(p)\) are certain polynomials given herein, is an asymptotic differentiator for \(MW^m[t_0, +\infty)\) signals. Further, the relative order of the transfer function of this differentiator is \(s- k\) \((k= 0,1,\dots, m-1)\), which ensures that the differentiator obtained is noise-tolerant. To convey the effectiveness of the result, an example with the signal class \(MW^3[0, +\infty)\) is also discussed in detail for the case when \(k= 2\).