Improved Stability Criteria on Linear Systems with Distributed Interval Time-Varying Delays and Nonlinear Perturbations
Round 1
Reviewer 1 Report
The authors present a stability criterion for linear systems with interval time-varying state delay, non-linear instantaneous and delayed perturbations and distributed time-varying delays. The new stability criterion is based on an original Lyapunov-Krasovskii functional that can be seen as an extension of those proposed in refs. [20, 23]. Using this new functional, sufficient conditions for stability of the linear system are derived. These new conditions prove to be less conservative than many already proposed ones.
The paper is globally well-written, and the results are validated using numerous numerical examples and extensively compared with results from other works. The theorem and its proof are well structured. The quality of the paper is quite good. Thus, I would recommand it for publication.
I would just have minor suggestions for the authors:
1- Given the complexity of the terms involved, it would help the reader if few words about the philosophy behind the functional and the proof were added. For example, it is not clear why Z1 and Z2 terms involved in V5(t) (line 152) are treated differently, while they could be grouped since the beginning of the proof.
2- Some acronyms are used without being properly defined in the text: LKF, LMI.
3- It seems to me that the first condition of equation (13) is strictly equivalent to the second one. If it is the case, removing the redundancy could ease understanding.
To summarize, the work is of good quality both concerning the content and the shape. The expression involved are a bit heavy, but it seems unavoidable. Thus, I would recommend the paper for publication.
Author Response
Author's Reply to the Review Report (Reviewer 1)
Comments and Suggestions for Authors
The authors present a stability criterion for linear systems with interval time-varying state delay, non-linear instantaneous and delayed perturbations and distributed time-varying delays. The new stability criterion is based on an original Lyapunov-Krasovskii functional that can be seen as an extension of those proposed in refs. [20, 23]. Using this new functional, sufficient conditions for stability of the linear system are derived. These new conditions prove to be less conservative than many already proposed ones.
The paper is globally well-written, and the results are validated using numerous numerical examples and extensively compared with results from other works. The theorem and its proof are well structured. The quality of the paper is quite good. Thus, I would recommend it for publication.
I would just have minor suggestions for the authors:
1- Given the complexity of the terms involved, it would help the reader if few words about the philosophy behind the functional and the proof were added. For example, it is not clear why Z1 and Z2 terms involved in V5(t) (line 152) are treated differently, while they could be grouped since the beginning of the proof.
Answer: We improved V5(t) (line 148) and we used differently estimation V5(t) because of increasing the number of parameters. Moreover, we improved main results part (line 107-147).
2- Some acronyms are used without being properly defined in the text: LKF, LMI.
Answer: We added “linear matrix inequalities (LMIs)” in Abstract part (line 5) and delete LKF.
3- It seems to me that the first condition of equation (13) is strictly equivalent to the second one. If it is the case, removing the redundancy could ease understanding.
Answer: We delete some inequality in equation (13) (line 78).
To summarize, the work is of good quality both concerning the content and the shape. The expression involved are a bit heavy, but it seems unavoidable. Thus, I would recommend the paper for publication.
Author Response File: 
Author Response.docx
Reviewer 2 Report
The paper is so interesting and technically with a good scientific sound. I have checked all calculations working so well. Anyway two aspects should be improved:
- The scientific aim of the paper is too technical and a focused audience. The authors should explain better the scope. The introduction and the conclusions must be improved.
- The related literature must be studied deeply with a better presentation in the introduction
The paper can be reconsidered after revision
Author Response
Author's Reply to the Review Report (Reviewer 2)
Comments and Suggestions for Authors
The paper is so interesting and technically with a good scientific sound. I have checked all calculations working so well. Anyway two aspects should be improved:
- The scientific aim of the paper is too technical and a focused audience. The authors should explain better the scope. The introduction and the conclusions must be improved.
Answer: We improved Introduction and Conclusions part (line 11-36, 210-216).
- The related literature must be studied deeply with a better presentation in the introduction.
Answer: We improved Introduction (line 23-36).
The paper can be reconsidered after revision
Author Response File: Author Response.docx
Reviewer 3 Report
See attached file.
Comments for author File: 
Comments.pdf
Author Response
Author's Reply to the Review Report (Reviewer 3)
Comments and Suggestions for Authors
Title of the manuscript: ”Improved stability criteria on linear systems with distributed interval time-varying delays and nonlinear perturbations”
Author: Jitsin Piyawatthanachot, Narongsak Yotha and Kanit Mukdasai
Review Report: Reconsider after major revision.
In this paper, the authors are studied stability of linear systems with distributed time-varying delays and nonlinear perturbations The researched problem in this work is topical. The ideas are not clear.
Here are some comments on the paper.
- Please highlight your contribution in the introduction.
Answer: We improved Introduction (line 20-36).
- Please indicate in the introduction the difficulties that have been overcome in solving the problem.
Answer: We improved Introduction (line 30-36).
- Please expand the introduction by explaining in more detail the development of the problem you are researching.
Answer: We improved Introduction (line 34-50).
- Please show that the conditions of Theorem 1 are satisfied for each of the examples.
Answer: See in Examples part.
- The formulas in the article are long and vague. Please make appropriate assumptions to make the article easier to read.
Answer: See in line 59-65, 107-147.
Author Response File: Author Response.docx
Round 2
Reviewer 2 Report
The paper in the revised version is fully acceptable
Reviewer 3 Report
Accept in present form.
