Novel Frequency-Based Approach to Analyze the Stability of Polynomial Fractional Differential Equations

Round 1

Reviewer 1 Report

This paper attempts to explore an innovative stability analysis tool for the study of polynomial fractional systems, which is of great interest to the reviewers. However, a careful reading of the entire article reveals structural confusion, unclear logic, imperfect proofs of the main conclusions, and a lack of foundation in part, while part of the lemma does not give a source. Unfortunately, the authors point out that LMI, a widely used analytical tool, has limitations, so the SOS of this paper is considered more promising, but no numerical treatment is given, whereas LMI has powerful numerical treatment tools. Some of the problems in this paper are:

1. The paper is crudely written and contains errors such as P2 line 61-62, P3 line 114, 119, P4 line 169, etc.

2. Whether the generalized inverse of A^{+} exists for all matrices A, it is well known that there are many generalized inverses.

3, P5, the proof of Theorem 3 cannot be seen as obtaining necessary and sufficient conditions.

4, How to get formula (9) from formula (8), which is important for this paper. The reference [9] given by the authors does not give this conversion process.

5. This paper lacks numerical values or examples to verify its effectiveness, which is very important for the validity of the control method.

Author Response

This paper attempts to explore an innovative stability analysis tool for the study of polynomial fractional systems, which is of great interest to the reviewers. However, a careful reading of the entire article reveals structural confusion, unclear logic, imperfect proofs of the main conclusions, and a lack of foundation in part, while part of the lemma does not give a source. Unfortunately, the authors point out that LMI, a widely used analytical tool, has limitations, so the SOS of this paper is considered more promising, but no numerical treatment is given, whereas LMI has powerful numerical treatment tools. Some of the problems in this paper are:

1. The paper is crudely written and contains errors such as P2 line 61-62, P3 line 114, 119, P4 line 169, etc.

    typos were corrected and the article carefully checks and was revised in terms of grammar.

2. Whether the generalized inverse of A^{+} exists for all matrices A, it is well known that there are many generalized inverses.

As stated in the article:

Matrix  denotes the generalized inverse of matrix A which satisfies and

 

3, P5, the proof of Theorem 3 cannot be seen as obtaining necessary and sufficient conditions.

A typographical error occurred and was corrected.

4, How to get formula (9) from formula (8), which is important for this paper. The reference [9] given by the authors does not give this conversion process.

It is explained in the reference [9] in formulas 13,14,15. (section 3.1)

5. This paper lacks numerical values or examples to verify its effectiveness, which is very important for the validity of the control method.

In this article, an analytical method for stability analysis is presented and the necessary conditions for stability are obtained by mathematical relations.

Author Response File: Author Response.docx

Reviewer 2 Report

This study proposed a necessary and sufficient condition for polynomial FOS stability using a frequency based model. Despite the passable quality of the proposed research, a few problems need to be addressed before potential publication.

  1. Improvement on the text and proofreading are required on this manuscript. A few examples that must be addressed include:  The paragraphing from Line 60 to Line 63;  Put citation elsewhere instead of paragraph title on Line102; Formatting to include Section 3 (i.e. where is section 3?); Also rename the Stability analyze section; it is the only subsection for Main theorem; and the wording should be 'analyses' as a noun; Keep the consistency of using 'notation' 1 through 6;   2. Revising Figure 1. More information is required on the caption of  this figure. Also, show the concept of Figure 1 in the beginning of the paper (probably in the Introduction section) to let readers have a general idea of this proposed method.    3. Strengths and future research/Conclusion section Need to elaborate the limitations. Admittedly, the optimization requires lots of computational tasks. However, more information should be provided in detail on this shortcoming regarding computational volume, such as estimation on the computing or results from others' works. More information regarding the detail about the uncertainty in this method is required to. Text that goes through Figure 1 is required in this section. Address this figure in the text to help readers better understand the flow of the proposed method.   Proofreading focusing on the grammar and wording is also required. Then it is recommended for another revision on this manuscript.    

Author Response

Reviewer2 answer:

 

1. Improvement on the text and proofreading are required on this manuscript. A few examples that must be addressed include:  The paragraphing from Line 60 to Line 63;  Put citation elsewhere instead of paragraph title on Line102; Formatting to include Section 3 (i.e. where is section 3?); Also rename the Stability analyze section; it is the only subsection for Main theorem; and the wording should be 'analyses' as a noun; Keep the consistency of using 'notation' 1 through 6;  

typos were corrected and the article carefully checks and was revised in terms of grammar.

2. Revising Figure 1. More information is required on the caption of  this figure. Also, show the concept of Figure 1 in the beginning of the paper (probably in the Introduction section) to let readers have a general idea of this proposed method.   

in the Introduction section, fig 1 is explained and highlighted.

3. Strengths and future research/Conclusion section Need to elaborate the limitations. Admittedly, the optimization requires lots of computational tasks. However, more information should be provided in detail on this shortcoming regarding computational volume, such as estimation on the computing or results from others' works. More information regarding the detail about the uncertainty in this method is required to. Text that goes through Figure 1 is required in this section. Address this figure in the text to help readers better understand the flow of the proposed method.   Proofreading focusing on the grammar and wording is also required. Then it is recommended for another revision on this manuscript.  

The main goal of this article is to present an analytical method based on the optimization method for fractional order models in the form of frequency response. This method can convert it into an optimization problem, and by changing the solution of the optimization problem, the stability of the fractional order system can be improved. Here, the uncertainty in the system is not taken but in future research, the presented method can be developed with uncertainty.

The strengths and future research/Conclusion section are more explain.

Author Response File: Author Response.docx

Round 2

Reviewer 1 Report

This version of the draft has not changed substantially from the previous version, and several of the reviewers' questions have not been completely answered. In particular, the lack of numerical or practical examples to verify the effectiveness of the proposed method in this paper is a extremely vital matter for control systems, and it is regrettable that reviewers cannot accept control methods that have not been verified.

Author Response

Reviewer1:

This version of the draft has not changed substantially from the previous version, and several of the reviewers' questions have not been completely answered. In particular, the lack of numerical or practical examples to verify the effectiveness of the proposed method in this paper is a extremely vital matter for control systems, and it is regrettable that reviewers cannot accept control methods that have not been verified.

Thanks for reviewer.

According to the presented method, an example was added to the article. In this example, the fraction order system is polynomial and the Lyapunov function is determined to prove its stability. In addition, matrices and  coefficients obtained from solving optimization problem 14 are also presented.

Author Response File: Author Response.docx

Reviewer 2 Report

Questions are addressed; no more further comments.

Author Response

Reveiwer2:

Accept

Thanks for your acceptance.

Author Response File: Author Response.docx

Round 3

Reviewer 1 Report

I have no questions about this manuscript.