An Overview of the Euler-Type Universal Numerical Integrator (E-TUNI): Applications in Non-Linear Dynamics and Predictive Control

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

The manuscript presents an overview of the Euler-Type Universal Numerical Integrator (E-TUNI) methodology, emphasizing its theoretical foundations, mathematical formulation, and applications in non-linear dynamics and predictive control. The topic is relevant to numerical analysis and computational modeling, and the paper compiles a considerable amount of theoretical background.

The strengths of the manuscript include the broad survey of existing literature on Universal Numerical Integrators, a detailed derivation of the mathematical framework, and illustrative computational examples. The authors attempt to address both theoretical justification and practical applications, which is commendable.

However, the manuscript suffers from several weaknesses that limit its clarity and scientific impact. The introduction is overly focused on historical and bibliographic exposition and does not clearly highlight the research gap or specific contributions in concise form. The research design and methodology description are difficult to follow and some sections appear more like lecture notes than a research article, which affects readability. The results are not presented in a concise, comparative manner, and the practical validation of E-TUNI remains limited. The conclusions largely restate the content rather than critically discussing implications, limitations, and possible extensions.

Author Response

Dear Reviewer #1,

Attached is a PDF file with detailed descriptions of my revisions to this article.

Sincerely,

Prof. Dr. Paulo M. Tasinaffo.

Author Response File: Author Response.pdf

Reviewer 2 Report

Comments and Suggestions for Authors

REVIEW OF

An Overview of the Euler-Type Universal Numerical Integrator (E-TUNI): Applications in Non-linear Dynamics and Predictive Control

BY

Paulo M. Tasinaffo, Gildárcio S. Gonçalves, Johnny C. Marques, Luiz A. V. Dias, Adilson M. da Cunha

 

The article is a continuation of the authors’ attempts to systematize the knowledge accumulated in connection with the numerical integration of ordinary differential equations. Or, if you will, of functions approximated by autonomous differential systems, which is essentially the same thing. This article, like the previous one, in fact recounts how this works, using simple ideas and methods that have been established for more than 250 years. Moreover, the reviewed article is primarily a continuation of work [19]. The reviewer is not inclined to give a strict definition such as “development,” “supplement,” etc. Work [19] is different: in [19] the emphasis is placed on the applied development and validation of the E-TUNI methodology for backward integration. The reviewed article is, to a greater extent, a theoretical and methodological review and a formalization of the general context of E-TUNI. The article offers nothing new; there are no scientific results in the article. This article, together with [19], is merely a systematized presentation of what has long been obtained and studied.

 

At the same time, the reviewer has no significant comments, since the text is prepared quite neatly, well systematized, and written in good English.

 

Before publication, the text requires careful revision on the following points:

 

1. Both articles (the reviewed one and [19]) are based on the same mathematical justification. This justification is given in theorems and properties in the first part of the article. Most (almost all) of this material is repeated in the reviewed article, practically verbatim, including the proofs. In the reviewed article, this part should be revised in order to:

• eliminate the repeated technical details (replace them with a reference “proved in …”);
• supplement it with an indication of the original source (replace with a reference “formulated in …”).

 

2. The article lacks a formal problem statement. There are many words, notations, and an in-depth review (starting from Euler himself), but the essence of the problem is not explicitly stated. The only thing we find is: “The original contributions of this paper are as follows: (i) it presents an overview of how E-TUNI works to solve dynamics and control problems.” However, there is no actual formulation of “dynamics and control problems.” The authors should clearly state the problem, and there is a place for it — at the end of Section 2. It should contain the traditional mathematical statement: given — required!

 

3. Statements such as “Thus, in [16] it is shown that it is necessary to know the partial derivatives … to solve the problem of optimization given by the equation iteratively (10) through the Kalman filter.” are unacceptable. [16] is a set of theses in Portuguese of more than 230 pages. Where exactly and why should the reader look for the essence of such statements? The authors should either accompany all such statements (there are several, not just the one cited) with a precise indication of the page and equation number, or supplement the article with a description of the method.

 

4. Example 2. Extremely unfortunate notation: after formula (37) it says “m (rocket mass),” while in line 522 it says “m = 1 and m = 15.”

 

5. “each unit of time equals 58.2 days.” — what unit, what time?

 

6. The quality of Figure 5, including the “blurry” labels, is unacceptable.

 

7. How was the Kalman filter applied to the nonlinear model (37)? This comment has already been mentioned above. In this example, given the limited information, it is appropriate to present the formulas used directly.

 

The reviewer is confident that the authors will address the comments and produce a good article.

Author Response

Dear Reviewer #2,

Attached is a PDF file with detailed descriptions of my revisions to this article.

Sincerely,

Prof. Dr. Paulo M. Tasinaffo.

Author Response File: Author Response.pdf

Round 2

Reviewer 1 Report

Comments and Suggestions for Authors

The authors have made substantial revisions to the manuscript, addressing the concerns raised in the previous review. The introduction was restructured to better emphasize the research gap and contributions, and the methodology section was streamlined by removing redundant proofs while referencing original sources. A new section clarifying the problem formulation has been added, and an additional simulation example provides stronger practical validation. The conclusions were rewritten to be more critical and forward-looking. Overall, the revised version shows clearer organization, improved readability, and more robust validation.

Reviewer 2 Report

Comments and Suggestions for Authors

All done