Discussion of Three Examples to Recent Results of Finite- and Fixed-Time Convergent Algorithms

TL;DR

This paper critically analyzes three examples from recent literature on finite- and fixed-time convergent algorithms, clarifying their relevance, limitations, and extensions to improve understanding and applicability of these algorithms.

Contribution

It clarifies the relevance of examples from recent works, extends differentiator algorithms to general cases, and confirms the validity of key theorems under practical gain selection.

Findings

The first example is shown to be irrelevant to the original results.

A method is provided to extend differentiator algorithms to the general case.

Theorem 1 remains valid in practical gain selection, and Theorem 2's fixed convergence time estimate holds.

Abstract

This note discusses three examples given in the recent technical correspondence paper [1], which addresses the results presented in [2,3,4]. It is shown that the first example ([1], Section 3) is irrelevant to the results of [2]. The second example ([1], Section 4) establishes a well-known fact that a continuous differentiator can exactly differentiate a signal, only if its second derivative is equal zero. This note provides a method to extend the algorithms presented in [3] to the general case. Finally, the third example ([1], Section 5) presents a particular case related to Theorem 1 of [4]. Theorem 1 of [4] remains, however, valid in the most practical case of selecting control gains. The result of Theorem 2 in [4] estimating the fixed convergence time holds as well.