Eigenstructure through Matrix Adjugates and Admissible Pairs
Round 1
Reviewer 1 Report (Previous Reviewer 3)
I can not change my previous opinion because the paper can not change its main idea - structure
Author Response
Obviously, my two- page reply concerning your first comments and suggestions were not convincing to you.
Regardless, reading your second revision; your sole (comment, suggestion for authors) and your checking of all of the six issues as (must be improved) entail a request to a dramatic and a reformed writing of an almost new paper. This can be done, time permitting, knowing specifically what is in the mind of the reviewer. To do that, I wish you were more specific and provided more guidance.
Hence, due to the lack of being specific, I guess, you’re leaving me with nothing I can do, change, or improve even to my revised manuscript. How could I?
Lastly, thank you for your time.
Reviewer 2 Report (Previous Reviewer 1)
The revision includes all my comments, it's better now.
1. The contribution of the paper is significantly improved.
2. The key advantage of the proposed approach over the traditional null space general method is explained clearly.
3. Authors have improved the recent work by citing the latest work.
4. The role of the open loop characteristic polynomial is now well justified in the proposed approach.
5. The organization of the paper is improved.
6. The authors have elaborated on dimensionality reduction theoretically.
7. However, there are some formatting issues, for example, 10.Conclusion -> 10. Conclusion
Author Response
I have done my best and improved my manuscript in the light of your previous comments and suggestions. Thank you.
I do my best to improve my manuscript every time I read it. However, what a manuscript is currently at, is the result of the author’s vision (at its best) the time it is read, unless Enlighted and /or guided by the others.
Finally, your concern of some formatting issues has been recognized and the necessary action taken.
Reviewer 3 Report (Previous Reviewer 2)
The Authors have adequately addressed all of my comments. I have no additional or further comments, and can recommend acceptance of the manuscript in its current form.
Author Response
You’re leaving me with nothing to do and nothing to say except, “ it was my pleasure”. Thank you for your time and effort.
Reviewer 4 Report (New Reviewer)
This paper studies the eigenstructure assignment method and proposed a method namely the adjugate method for the determination of the permissible closed-loop eigenvector subspaces and the companion input-subspace in control theory. The method is supported by four control system examples. The work is exciting and well and in fact over-explained. My observations for this work are as follows:
1. The abstract should be concise and point towards the actual contribution.
2. Eq. (29) is a piece of code and not an equation. Therefore, avoid writing equation numbers.
3. The author should include an example in which the repeated roots are on an imaginary axis. Such a typical example will attract the readers.
Author Response
I did my best in my first revised manuscript, but that was in the absence of your comments and suggestions.
I wish you further elaborated on all issues that can be improved. Or, you are probably a supporter of the following quote “Perfection is the enemy of progress”?
Nevertheless, in the light of your comments and suggestions dispatched on 11/1/2022:
- The abstract has been trimmed. Please refer to the revised manuscript.
- Point taken and that piece of code is now unnumbered.
- Point taken into consideration. I envisaged treating your request as an addendum to example 3 for the following two reasons:
- The system in example 3 is fourth order (even order), thus, permitting assignment of two repeated complex eigenvalues. Two cases have been considered: assignment of i, i, -i, and -i and a case of dead-beat control where all four eigenvalues are located at the origin.
- To avoid oversizing the paper; by not including a new fifth example worked out from the scratch as hinted by your comment.
Round 2
Reviewer 1 Report (Previous Reviewer 3)
I do not have any further comments.
Author Response
Reply to the First Reviewer
I did all that can be done to improve the manuscript in its two rounds of revision. I cannot envisage any further improvement. I would if I could. This is a natural consequence whenever a reviewer abstains from pinpointing to what and where are all those issues of a paper that must be improved. My paper is not intended to be an MSc thesis either in length or in eigenstructure inclusiveness.
Reviewer 4 Report (New Reviewer)
Responses are ok from my side.
Author Response
Reply to the Fourth Reviewer
I understand that my first improvement recommended by you has been met, evident by their implementation by Electronics in the manuscript latest version.
I did all that can be done to improve the manuscript in its two rounds of revision. I cannot envisage any further improvement. I would if I could.
I wish you further elaborated and pinpointed to those issues that can be improved.
Thank you for your time and effort.
This manuscript is a resubmission of an earlier submission. The following is a list of the peer review reports and author responses from that submission.
Round 1
Reviewer 1 Report
In this paper, the authors reconsider the traditional eigenstructure assignment method and develop an approach through matrix adjugates that enable a more eloquent decomposition compared to the lumped solution usually obtained through matrix null spaces. The problem discussed in the paper is interesting. However, it requires major revision. Following are a few comments regarding this work.
The contribution of the paper is not clearly highlighted. The key advantage of the proposed approach over the traditional null space general method needs to be explained clearly.
Recent work is missing in the problem formulation. section, like the following.
Gu, D. K., Wang, R. Y., & Liu, Y. D. (2021). A parametric approach of partial eigenstructure assignment for high-order linear systems via proportional plus derivative state feedback. AIMS Mathematics, 6(10), 11139-11166.
Duan, G. R. (2019). Circulation algorithm for partial eigenstructure assignment via state feedback. European Journal of Control, 50, 107-116.
Denton, P., Parke, S., Tao, T., & Zhang, X. (2022). Eigenvectors from eigenvalues: a survey of a basic identity in linear algebra. Bulletin of the American Mathematical Society, 59(1), 31-58.
Prewett, P. (2022). Matrices. In Foundation Mathematics for Science and Engineering Students (pp. 29-46). Springer, Cham.
It is not clear how the authors have determined the permissible closed loop eigenvector subspaces?
The role of the open loop characteristic polynomial needs to be justified in the proposed approach.
How the closed loop eigenvectors associated with repeated eigen values are explicitly computed by means of differentiation as opposed to null space?
The claim that the adjugate of a matrix is unique leading to a naturally scaled admissible pair needs a provable explanation.
How the proposed approach reduces dimensionality theoretically? It requires description.
The presentation of the paper needs to be improved.
Reviewer 2 Report
The refereed manuscript is devoted to theoretical study of the eigenstructure through Matrix adjugates and admissible 2 pairs. The authors found that the adjugate method proves a worthy method for exposing internal structures within eigenstructure assignment problem. With regard to the lumped solution as given by traditional methods, the adjugate method offers decoupled solutions for each component of the admissible pair, which is considered advantageous (w, z). The adjugate method attests itself a formula-based method. It is explicit and independently determines both wi and zi. This is considered an advantage since closed forms methods are pedagogically favorable. Besides, such independent determination of the admissible pair can be looked upon as a form of parallelism which is always desirable. The manuscript is interesting and with potential for publication. However, some issues have to be attended before the acceptance of the manuscript.
1- I find this paper reasonably well written, but too sharp: the model is given, the numerical results are presented and the summary is finally exposed. The paper mostly consists of a lengthy and cumbersome listing of formulas to represent the model and of the syntetic report of the numerical results.
2- It will be better understanding and interesting for reading the manuscript if some practical application (example) of the study is presented in the manuscript.
3- The references are not enough, especially in the method section.
4- It will be better if the authors should use scientific references instead of Wikipedia.
Reviewer 3 Report
The article uses some basic linear algebra to establish its goals.
Overall the novelty is unclear since the mathematics used and the example are simple.
There are also no realistic applications to help the reader understand any contribution.
