Matrix Fraction Description in Large Scale MIMO Descriptor Systems: Matrix Polynomials Approaches

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

• In the abstract and introduction, more clearly highlight how the proposed MFD approach for descriptor systems fundamentally differs from existing methods, and why it is more effective.

•  The introduction is rich in references but could be more logically structured by first stating the problem, then summarizing limitations of current solutions, and finally presenting your contributions.

•  Ensure all mathematical symbols are defined upon first use (e.g., P(λ), G~(λ), and remain consistent throughout.

•  A small motivating numerical example in the preliminaries could help illustrate the challenges the proposed method overcomes.

• When discussing controllability of descriptor systems, briefly relate it to the design goals and stability guarantees of the proposed MFD method.

• The new normalization formula is an important contribution; provide an intuitive explanation alongside the derivation so readers can grasp its purpose without deep prior knowledge.

•  For the MFD-based gain computation, present the algorithm in a structured pseudocode format with input–output definitions to make it reproducible.

•  State clearly the assumptions (e.g., system regularity, controllability conditions) under which the closed-form MFD derivations are valid.

•  Include at least one detailed numerical experiment comparing the proposed method against traditional approaches, showing computation time and accuracy benefits.

• The conclusion could be strengthened by summarizing key numerical results and suggesting concrete extensions (e.g., robustness analysis, stochastic systems).

Comments on the Quality of English Language

The manuscript is generally well-written, but some sentences are lengthy and could be simplified for clarity. Minor grammatical adjustments and improved flow between sections would enhance readability.

Author Response

Responses to the comments of Reviewer 1

Paper: Matrix Fraction Description in Large Scale MIMO Descriptor Systems: Matrix Polynomials Approaches

The authors would like to express their gratitude and thankful to the reviewer for the helpful comments and careful review, which helped to improve the quality of the revised version of the paper. The reviewer's comments have been adopted into the updated paper, and the responses to each comment are provided below.

Note: The modifications in the revised paper have been highlighted by yellow colour.

We thank the reviewer for the constructive feedback. The introduction has been slightly restructured for improved flow, while the research design remains unchanged as it is already appropriate. Clarifications were added to the methods for completeness, and the results section was streamlined for clearer presentation. The discussions are directly supported by the results, and figures and tables have been carefully revised to ensure clarity. 

Comments and Suggestions for Authors

In the abstract and introduction, more clearly highlight how the proposed MFD approach for descriptor systems fundamentally differs from existing methods, and why it is more effective.

Answer: We thank the reviewer for this remark. In the revised manuscript (Abstract, p.1, lines 7–14), we have explicitly highlighted how the proposed MFD differs from conventional companion linearizations and Weierstrass–Kronecker reductions. The advantages in terms of closed-form derivation and preservation of descriptor structure are now clearly stated.

The introduction is rich in references but could be more logically structured by first stating the problem, then summarizing limitations of current solutions, and finally presenting your contributions.

Answer: We agree. The Introduction has been reorganized (pp.2–4): the problem statement is presented first, followed by a discussion of the limitations of existing methods, and finally a concise presentation of the contributions. 

Ensure all mathematical symbols are defined upon first use (e.g., P(λ), G~(λ), and remain consistent throughout.

Answer: All key symbols are now defined at their first occurrence (see Sec. 2, p.6–7), and notations have been harmonized throughout the paper.

A small motivating numerical example in the preliminaries could help illustrate the challenges the proposed method overcomes.

Answer: We sincerely thank the reviewer for this constructive suggestion. We agree that a motivating example could be illustrative. However, to maintain the flow and mathematical rigor of the preliminaries section, we preferred to keep it focused on the theoretical framework. A numerical validation already appears in the results section (Lines 647–680, 771-784, and 830-837), where the advantages of the proposed method are demonstrated in practice. We believe this placement avoids redundancy while still ensuring that the reader gains clear insight into the challenges addressed and the improvements achieved.

When discussing controllability of descriptor systems, briefly relate it to the design goals and stability guarantees of the proposed MFD method.

Answer: We clarified the relation between controllability and stability guarantees of the MFD-based design in Sec. 3.1 (p.15–16), linking structural properties with the design objectives.

The new normalization formula is an important contribution; provide an intuitive explanation alongside the derivation so readers can grasp its purpose without deep prior knowledge.

Answer: Alongside the derivation, an intuitive explanation is provided (Sec. 4.3, p.20), highlighting that normalization eliminates impulsive modes and ensures admissibility.

For the MFD-based gain computation, present the algorithm in a structured pseudocode format with input–output definitions to make it reproducible.

Answer: As suggested, the gain computation is now presented in a structured pseudocode format with input–output specification (Sec. 4.3, p.21-23).

State clearly the assumptions (e.g., system regularity, controllability conditions) under which the closed-form MFD derivations are valid.

Answer: The assumptions under which the derivations hold (regularity, controllability) are explicitly stated in Sec. 4 (lines 571-572  and lines 736-740).

Include at least one detailed numerical experiment comparing the proposed method against traditional approaches, showing computation time and accuracy benefits.

Answer: We added a detailed comparison study (Sec. 4.5, p.25–26) showing accuracy and runtime versus traditional formulations.   

The conclusion could be strengthened by summarizing key numerical results and suggesting concrete extensions (e.g., robustness analysis, stochastic systems).

Answer: The conclusion (p.27) has been strengthened by summarizing the key numerical outcomes and suggesting extensions such as robustness and stochastic settings.

Comments on the Quality of English Language The manuscript is generally well-written, but some sentences are lengthy and could be simplified for clarity. Minor grammatical adjustments and improved flow between sections would enhance readability.

Answer: We carefully revised sentence structures and transitions to enhance clarity and readability across sections.

Also, kindly please see the attached file. 

Sincerely

Associate Prof. Dr. Abdel-Nasser Sharkawy, the corresponding author

on behalf of all the authors

Author Response File: Author Response.docx

Reviewer 2 Report

Comments and Suggestions for Authors

The paper is written in a good way. I have curious to know some points and want to elobrate more in the paper:

• For readers it is important to mention explicitly the assumptions on the descriptor system, such as regularity of the matrix pencil and absence of impulsive modes, to clarify the scope of applicability.
• Symbols are confusing and I do recommend for more detailed elaboration. Include a concise table summarizing the symbols, matrices, and variables used in the paper for quick reference.
• Comparative analysis is necessary, add a short paragraph/table comparing the proposed closed-form expression with well-known formulations for matrix transfer functions in standard state-space systems, highlighting similarities and differences.
• A detailed comment on the computational cost of the proposed methods relative to traditional eigen structure assignment or transfer function computation methods is required.
• Include a short discussion on how sensitive the proposed closed-form solution is to perturbations in EEE, AAA, or the feedback matrix.
• Add one or two sentences on potential application areas, such as large-scale power systems or mechanical multibody systems, where the proposed techniques would be particularly useful.
• Include a schematic block diagram showing the relation between the descriptor system, feedback, and resulting transfer function for better intuition.
• Mention how the method behaves or requires adaptation when EEE is singular or rank-deficient.
• Add a remark on how the proposed closed-form expression and normalization may influence closed-loop stability analysis in the frequency domain.

Author Response

Responses to the comments of Reviewer 2

Paper: Matrix Fraction Description in Large Scale MIMO Descriptor Systems: Matrix Polynomials Approaches

The authors would like to express their gratitude and thankful to the reviewer for the helpful comments and careful review, which helped to improve the quality of the revised version of the paper. The reviewer's comments have been adopted into the updated paper, and the responses to each comment are provided below.

Note: The modifications in the revised paper have been highlighted by yellow colour.

 

Comments and Suggestions for Authors

The paper is written in a good way. I have curious to know some points and want to elaborate more in the paper: For readers it is important to mention explicitly the assumptions on the descriptor system, such as regularity of the matrix pencil and absence of impulsive modes, to clarify the scope of applicability.

Answer: We thank the reviewer. We have now explicitly listed the key assumptions (regularity of the pencil, admissibility, controllability of the triple ) in Sec. 4 (p.15 lines 571-572 and p.21 lines 736-740).

 

Symbols are confusing and I do recommend for more detailed elaboration. Include a concise table summarizing the symbols, matrices, and variables used in the paper for quick reference.

Answer: To avoid confusion, we added a concise Table (after Sec. 5, p.27) summarizing all symbols, matrices, and variables used.  

 

Comparative analysis is necessary, add a short paragraph/table comparing the proposed closed-form expression with well-known formulations for matrix transfer functions in standard state-space systems, highlighting similarities and differences.

Answer: A short comparison study with a table have been added in Sec. 4.5 (p.25-26), contrasting the proposed closed-form expression with classical state-space MTF derivations.

 

A detailed comment on the computational cost of the proposed methods relative to traditional eigen structure assignment or transfer function computation methods is required.

Answer: We included a discussion in Sec. 1 (p.2) and Sec. 4.5 (p.25-26), noting that the proposed method avoids linearizations and reduces computational cost, especially in large-scale systems.

 

Include a short discussion on how sensitive the proposed closed-form solution is to perturbations in E, A, or the feedback matrix.

Answer: A brief remark was added (p.25-26), discussing sensitivity to perturbations in , , and feedback gain .

 

Add one or two sentences on potential application areas, such as large-scale power systems or mechanical multibody systems, where the proposed techniques would be particularly useful.

Answer: Two potential application domains (large-scale power networks, mechanical multibody dynamics) are mentioned in Sec. 5 (p.26).

 

Include a schematic block diagram showing the relation between the descriptor system, feedback, and resulting transfer function for better intuition.

Answer: We thank the reviewer for this valuable remark. Indeed, a schematic block diagram would offer additional intuition. Nevertheless, since the derivation already directly links the descriptor system, the feedback structure, and the resulting transfer function in closed form (see Sec. 4, pp. 15–24), we chose to present this connection through the equations themselves. This choice preserves the algebraic focus of the paper and avoids over-simplifying a framework intended for large-scale MIMO systems. We hope the step-by-step construction in Sec. 4 provides sufficient clarity for readers.

 

Mention how the method behaves or requires adaptation when  is singular or rank-deficient.

Answer: We have added a short remark (Sec. 4.3, p.20) on how the method adapts when  is singular or rank-deficient.  

 

Add a remark on how the proposed closed-form expression and normalization may influence closed-loop stability analysis in the frequency domain.

Answer: A new remark in (lines 724-731) explains how the normalization ensures frequency-domain admissibility and facilitates stability checks.  

Also, Kindly please see the attached file.

Sincerely

Associate Prof. Dr. Abdel-Nasser Sharkawy, the corresponding author

on behalf of all the authors

Author Response File: Author Response.docx