The Adjugate Method: Reassignment with System Classification

Round 1

Reviewer 1 Report

I am responding to MDPI request for me to provide you with a report in connection with the submitted manuscript “The Adjugate Method: Reassignment with System Classification” by Omar M. E. El-Ghezawi. I am pleased to do so.

In this article, the author has theoretically studied the adjugate method for eigenvalue-eigenvector assignment for some important cases. More specifically, insights have been gained through the admissible pair approach to eigenstructure reassignment concerning the cases of single, multi-input, controllable, and uncontrollable systems. Among others, he concluded that the case Zi =0 indicates that reassignment is entangled with an assigned eigenvector being an open loop one. The case Wi =0 indicates the associated eigenvalue λi is an uncontrollable one. The research topic is very interesting, the presentation is very good and their results appear to be correct. The only comments are the following:

1)     Z_i and W_i have to be introduced in abstract. It is not specified in the abstract section.

2)     What is the software used in his calculations?

3)     For the studied system, could he make any mention to any numerical methods to solve more demanding eigenstructure systems? (e.g. genetic algorithms SoftwareX 10, 100355, 2019). Most probably, the studied system for small matrices and using matlab programming it works. If the system has large sized matrices like quantum structures, the method will not be realizable.

4)     What is the importance of the current work in engineering structures?

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Author Response

First Reviewer

1)Z_i and W_i have to be introduced in abstract. It is not specified in the abstract section.

The incident has been resolved by inserting “the closed loop eigenvector companion” within the abstract as a highlighted text in yellow in the revised manuscript.

W_i is already defined in the abstract as “eligible closed loop eigenvectors.”

 

2)     What is the software used in his calculations?

         Unless misunderstanding you, it is MATLAB.      

 

3)     For the studied system, could he make any mention to any numerical methods to solve more demanding eigenstructure systems? (e.g. genetic algorithms SoftwareX 10, 100355, 2019). Most probably, the studied system for small matrices and using matlab programming it works. If the system has large sized matrices like quantum structures, the method will not be realizable.

You are right. However, pursuing such an approach in our treatment will take us far beyond our main intention   of presenting a conceptual formulation and derivation of the acclaimed results. In relation to our specific treatment of the subject matter, problems of numerical efficiency, programming environments and  genetic algorithms are left as  new approaches in future works.

Additionally, readers may find previous works by Kautsky, Nichols, and Van Dooren interesting. They have utilized freedom in the solution to optimize the condition of the closed-loop eigenvalue-eigenvectors. Besides, well known transformations based on unitrary matrices or block Hessenberg transformation are also a viable option.

 

4)     What is the importance of the current work in engineering structures?

Engineering structures were probably one of the earliest engineering disciplines to utilize eigenstructure theory and design methodologies; covering the following areas in mechanical and aeronautical structures:

Three-axis dynamic flight motion, multibody aircraft dynamics, damping and stiffness requirements, large scale systems, trade-offs between sensitivity and stability, vibration cancellation and suppression, and large space structures ‎to mention but a few.

Further details concerning the issues above are detailed in references [6], [9-13]

Reviewer 2 Report

This paper presents a novel method for eigenvalue asignment for linear time-invariant control systems. The paper is quite well written but can be improved. Some issues are the following:

1. Nomenclature section is suggested. For example in line 126 z_i=I_m is confusing, an identity matrix assigned to a vector?

2. In lema 3 line 190, why is +-B?

3. In section 6, line 310 and 311 author claims "This is possible guided by the fact that controllability is invariant under state feedback, so tests developed based on the closed loop information are equally valid". This must be further elaborated.

4. At the conclusions section, in line 591, author claim "It is sufficient for multi-input systems". Please justify.

Formating issues: 

1. Bullets in lines 104-119 must be aligned. The same for lines 435-447.

2. In lines 319-325, bullets are required.

3. Line 390, Please use programming style for the commands and adjust limits.

 

1. Please, re-check english and grammar.

2. In line 380, please avoid words like "Horrendous".

Author Response

Please see the attachment.

Author Response File: Author Response.pdf