Synthesis Methodology for Discrete MIMO PID Controller with Loop Shaping on LTV Plant Model via Iterated LMI Restrictions

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

Dear Author

The paper aims to present a methodology for synthesizing discrete MIMO PID controllers. The methodology is designed to synthesize a Linear Time-Invariant (LTI) controller for a Linear Time-Varying (LTV) plant model, ensuring robust stability of the synthesized system. In general, the article was able to convey the desired concepts to the reader, but the following points are suggested to improve the article:

1- Overall, the abstract effectively communicates the paper's focus, methodology, and potential contribution to the field of digital control systems. But you can add more information about the results obtained from this research for the better understanding of the reader in this section.

2- The references used in this article are somewhat old, it is recommended to use more recent references, preferably 2022 onwards.

3- The English language used in the article needs minor editing

In general, the article is well written and the explanations are complete, but making the mentioned corrections will help to improve the quality of the article.

Author Response

Dear Reviewer,

I sincerely thank you for taking the time to read through this manuscript and for your diligent efforts in reviewing it. Your suggestions have helped me improve the paper, and I genuinely appreciate your contribution to my work. Below are my responses.

 

Comment:

1- Overall, the abstract effectively communicates the paper's focus, methodology, and potential contribution to the field of digital control systems. But you can add more information about the results obtained from this research for the better understanding of the reader in this section.

Response to comment:

I have revised the abstract accordingly. The results obtained on discrete MIMO PID controllers have been elaborated upon in Section 4.7 and in the Conclusions section. Additionally, a paragraph has been included in the Conclusions section where I discuss my experience using MIMO PID for plasma magnetic control systems in tokamaks.

 

Comment:

2- The references used in this article are somewhat old, it is recommended to use more recent references, preferably 2022 onwards.

Response to comment:

In the Conclusions section, I have added references to recent works where I applied the methodology for synthesizing controllers in the plasma magnetic control system in tokamaks. Furthermore, several new references have been included in the Introduction.

Some old references serve as primary sources. My belief is that it is better to refer to primary sources, thus showing respect to the pioneers.

 

Comment:

3- The English language used in the article needs minor editing

Response to comment 3:

I have made the necessary adjustments to improve the clarity and coherence of the language used in the article.

 

Comment:

In general, the article is well written and the explanations are complete, but making the mentioned corrections will help to improve the quality of the article.

Response to comment:

Once again, I extend my gratitude for your constructive feedback.

 

Major revisions to the text of the paper have been highlighted in color.  

 

Reviewer 2 Report

Comments and Suggestions for Authors

The introdution part is not persuasive.The necessity of employing MIMO PID in MIMO control systems is not explained in detail. Additionally, MPC or other model-based control technologies are with decoupling function but they aren't mentioned in this paper. Could you explain the advantage?

Based on the MIMO PID tuning methodology for continuous-time systems introduced by Boyd et al. in reference [1], this manuscipt presents a novel version for discrete-time control systems, including extra derivations and calculations. 1. Some improvement could be used in the introduction. The necessity of discrete controller synthesis is explained in 1.4 while while an introduction to related work is lacking. 2. Elaborate theory is shown in the main part and it benefits the acceptance of the reader. However, the article dwells too much on existing theories, hindering readers from understanding the innovativeness of this paper. I suggest that the author should adjust the proportion of exposition appropriately.

 

Comments on the Quality of English Language

It is challenging to follow the narrative of some of the central part of the manuscript.  

Author Response

Dear Reviewer,

I sincerely thank you for taking the time to read through this manuscript and for your diligent efforts in reviewing it. Your suggestions have helped me improve the paper, and I genuinely appreciate your contribution to my work. Below are my responses.

 

Comment:

The introdution part is not persuasive. The necessity of employing MIMO PID in MIMO control systems is not explained in detail. 

 

Response to comment:

I have thoroughly revised the Introduction. The necessity of using MIMO PID controllers primarily stems from the need to conduct control loop decoupling. This aspect has been further emphasized with references to two additional works.

Comment:

Additionally, MPC or other model-based control technologies are with decoupling function but they aren't mentioned in this paper. Could you explain the advantage?

Response to comment:

Unfortunately, I am not an expert in model predictive control and cannot fairly assess its strengths and weaknesses, especially regarding control loop decoupling.

My considerations regarding the comparison of my methodology with other approaches were written after the conclusion in the "Limitations" section.

My main area of expertise is in developing plasma control systems in tokamaks. The need for synthesizing MIMO controllers in such a control system was the primary motivation for developing a new method for synthesizing discrete MIMO PID controllers.

At the request of another reviewer, I added several references to works where this method was applied for plasma control in tokamaks.

Several years ago, I attempted to apply MPC for plasma shape control systems. I used the Model Predictive Control Toolbox in MATLAB for synthesis but failed to achieve acceptable control performance.

 I'm unsure if this should be mentioned in the article. My experience with MPC applies only to one class of plants - plasma in tokamaks. My personal experience by no means suggests that model predictive control has global shortcomings compared to MIMO PID.

I am confident that there is a class of plants for which the application of MPC would be significantly better.

If you are an expert in MPC and are interested in comparing MIMO PID and MPC for several controlled plants, I am willing to participate in this joint research. If you are interested, please email me after this paper is published.

 

Comment:

1. Some improvement could be used in the introduction. The necessity of discrete controller synthesis is explained in 1.4 while while an introduction to related work is lacking.

Response to comment:

I have extensively revised the introduction to address this concern.

Comment:

2. Elaborate theory is shown in the main part and it benefits the acceptance of the reader. However, the article dwells too much on existing theories, hindering readers from understanding the innovativeness of this paper. I suggest that the author should adjust the proportion of exposition appropriately.

Response to comment:

Existing theories and concepts are only addressed in Section 2. It occupies approximately 3.5 pages, less than 10% of the total volume.

In Sections 3 and 4, they are mentioned very briefly at the beginning of each section.

Additionally, a paragraph describing some basic concepts was removed from the "Discussion" section.

 

I have made the necessary adjustments to improve the clarity and coherence of the language used in the article.

Once again, thank you for your valuable insights and suggestions.

 

Major revisions to the text of the paper have been highlighted in color.

 

Reviewer 3 Report

Comments and Suggestions for Authors

Mathematics-2832318

 

1. The abstract section is incomplete. For example, the authors should provide some parameters such as the results of their method.

2. The novelty of the article is not clear at all.

3. The materials presented in many sections are basic and can be removed

4. Comparison of the method with new methods has not been done.

5. The performance of the controller should be checked in the presence of disturbance and uncertainty.

6. Fuzzy controller can be used for better performance:

https://doi.org/10.1016/j.aej.2015.12.003

7. Please add a section about "literature review". In this section, it is better to have a tabular summary of the paper's review to give readers a better understanding of the research done in this field from 2021-2023. In this section, some articles should be presented in the form of text, and the rest of the papers should be summarized in a table.

8. Please add a section about "limitation of study".

9. In the Conclusion section, please explain more about future works. This section requires further discussion. For example, you can discuss new deep learning methods such as transformers, attention learning, and etc.

10. English language is acceptable in general, but there are some errors that should be corrected.

Comments on the Quality of English Language

Moderate editing of English language required

Author Response

Dear Reviewer,

I sincerely thank you for taking the time to read through this manuscript and for your diligent efforts in reviewing it. Your insights have been instrumental in refining the paper, and I sincerely appreciate your contribution to my work. Below are my responses to your comments.

 

Comment:

1. The abstract section is incomplete. For example, the authors should provide some parameters such as the results of their method.

Response to comment:

The abstract has been revised.

 

Comment:

2. The novelty of the article is not clear at all.

Response to comment:

I have emphasized the novelty of the study at the beginning of the Introduction section to provide clarity on its original contributions.

 

Comment:

3. The materials presented in many sections are basic and can be removed

Response to comment:

In the "Introduction" section, at the request of another reviewer, a basic PID controller formula has been added. I apologize, but this was one of the main requirements from the other reviewer.

I agree that basic concepts are presented in Section 2. I have tried to be as concise as possible in discussing them. It occupies approximately 3.5 pages, less than 10% of the total volume. I believe it's better when basic concepts are consolidated in one section rather than scattered throughout the text.

In the remaining sections, basic concepts are provided very briefly. A paragraph describing basic concepts was removed from the "Discussion" section.

 

Comment:

4. Comparison of the method with new methods has not been done.

Response to comment:

I have presented my thoughts on comparing different methods in a new section titled "Limitations of the Study."

 

Comment:

5. The performance of the controller should be checked in the presence of disturbance and uncertainty.

Response to comment:

It has been done. The impact of parametric disturbances is demonstrated. Synthesis occurs on an array of plant models for each example. Figures 11 and 13 depict the singular values of the control system and both models. Additionally, step responses for the control system with each model in the array are shown in Figures 10 and 12.

The influence of external output disturbances on the closed-loop system is best assessed through the singular values of the transfer function W(z) across the entire frequency range. This is fully illustrated in Figures 11 and 13, where the singular values of the function W(z) are provided.

Conducting modeling with a predefined input disturbance signal allows the assessment of the disturbance's influence only at the frequencies contained in that signal. This does not enable a full evaluation of the external disturbance's impact.

 

Comment:

6. Fuzzy controller can be used for better performance:

https://doi.org/10.1016/j.aej.2015.12.003

Response to comment:

This depends on many factors. There are no universal controllers, just as there are no universal methods for their tuning.

The consideration of fuzzy controllers goes beyond the scope of my paper. I am not a expert in fuzzy controllers, and I do not plan to work with them in the future.

I have read the survey https://doi.org/10.1016/j.compind.2010.10.001

I have no doubt that there are classes of plants for which fuzzy controllers will perform better than PID controllers.

General considerations regarding the comparison of controllers of different types are provided in the "Limitations of the study" section.

 

Comment:

7. Please add a section about "literature review". In this section, it is better to have a tabular summary of the paper's review to give readers a better understanding of the research done in this field from 2021-2023. In this section, some articles should be presented in the form of text, and the rest of the papers should be summarized in a table.

Response to comment:

The literature review on MIMO PID synthesis methods is comprehensively presented in section 1.2. References to all key works in this direction are provided. It is worth noting that there are not many works on the synthesis of discrete MIMO PID controllers at the moment. Perhaps there will be more in the future. I have also added new references for 2021-2023.

I do not currently see the possibility of presenting the literature review in a tabular format.

 

Comment:

8. Please add a section about "limitation of study".

Response to comment:

Thank you very much for this suggestion. I have added a section titled "Limitations of the study" at the end of the paper. This suggestion has helped improve the article, and I hope that adding this section will address many questions about my paper.

 

Comment:

9. In the Conclusion section, please explain more about future works. This section requires further discussion. For example, you can discuss new deep learning methods such as transformers, attention learning, and etc.

Response to comment:

Future work has been described in the "Discussion" section.

I am not an expert in deep learning methods, and I do not plan to work with them in the future.

I do not have ideas on how to use deep learning methods to tune MIMO PID controllers in the frequency domain with loop shaping.

In any case, deep learning methods are not related to the topic of this study.

 

Comment:

10. English language is acceptable in general, but there are some errors that should be corrected.

Response to comment:

I have made adjustments to improve the language quality throughout the manuscript.

 

Once again, I appreciate your insightful comments and suggestions, which have significantly contributed to enhancing the manuscript's clarity and depth.

 

Major revisions to the text of the paper have been highlighted in color.  

Reviewer 4 Report

Comments and Suggestions for Authors

The paper is well written and presented. Can be further improved if:

1) Try to give more insights into the equations. Some physical interpretation can be provided.

2) Line 464, "Introduction" please change to "introduction"

3) I understand that it is a theoretical paper, if there is any possibility of real implementation or atleast give some insights into  how real implementation can be done would be worth mentioning.

4) A section on comparison with related methodologies can also be provided towards the end of the paper.

5) There are many acronyms used, for example LMI in the abstract section. Please provide a section elaborating these acronyms.

 

Comments on the Quality of English Language

The quality of English is at par with what is expected from a mathematical paper.

Author Response

Dear Reviewer,

I sincerely thank you for taking the time to read through this manuscript and for your diligent efforts in reviewing it. Your suggestions have helped me improve the paper, and I genuinely appreciate your contribution to my work. Below are my responses.

 

Comment:

The paper is well written and presented. Can be further improved if:

1) Try to give more insights into the equations. Some physical interpretation can be provided.

Response to comment:

This article is theoretical and falls within the field of control science. The signals in Figure 5 may have physical interpretations. Therefore, the specific physical meaning of some equations only emerges after applying the method to a real plant.

Nevertheless, in the problem formulation in Section 3, a possible physical interpretation of some equations was provided.

 

Comment:

2) Line 464, "Introduction" please change to "introduction"

Response to comment:

Thank you for catching that oversight. The change has been made accordingly.

Comment:

3) I understand that it is a theoretical paper, if there is any possibility of real implementation or atleast give some insights into  how real implementation can be done would be worth mentioning.

Response to comment:

Certainly. Real-world implementation possibilities are a crucial aspect to consider. As someone actively involved in developing digital plasma magnetic control systems in tokamaks, I have added references to my works applying this method at the end of the "Conclusions" section to provide insights into potential real-world applications.

Comment:

4) A section on comparison with related methodologies can also be provided towards the end of the paper.

Response to comment:

Unfortunately, I am not aware of other methodologies where the tasks of synthesizing discrete MIMO PID controllers on LTV object models, loop shaping, and having similar performance criteria were simultaneously addressed.

Most previous works primarily focused on the synthesis of continuous MIMO PID controllers. Regarding works dedicated to the synthesis of discrete MIMO PID controllers, I have added references. It is worth noting that the problem formulation in those works significantly differs from what I have done.

I have added a "Limitations" section at the end of the article where I have outlined my considerations regarding conducting comparisons.

Comment:

5) There are many acronyms used, for example LMI in the abstract section. Please provide a section elaborating these acronyms.

Response to comment:

At the end of the paper, between the "Funding" and "Appendix" sections, there is a section called "Abbreviations" where all abbreviations are listed. Additionally, the expansion of all abbreviations is provided in the main text of the paper upon their first mention.

 

 

Once again, I am grateful for your insightful comments, which have undoubtedly contributed to improving the manuscript's clarity and comprehensibility.

 

Major revisions to the text of the paper have been highlighted in color.  

Reviewer 5 Report

Comments and Suggestions for Authors

1-It is better reducing abbreviations in the title in order to be clear.

2- In abstract all author must explain all abbreviations appeared.

3- In abstract the author must write the model used in this study (examples) and finally add the criteria's and bound used.

4-In introduction section the paper need to be enhanced with more studies related to the theory of paper subject (LMI Model).

5- In section 1.1 add the basic PID equation then explain the meaning of its variables.

6- Add the reference of each figure and equation appeared in the study.

7-In section 7 the author must analyze the effect of the items appeared on section 6 for the two examples chose.

8- Some figures appeared without legend try to add it on figure needed.

9-Several references are very old try to replace them with more recent studies.      

Comments on the Quality of English Language

Need to be improved

Author Response

Dear Reviewer,

I sincerely thank you for taking the time to read through this manuscript and for your diligent efforts in reviewing it. Your suggestions have helped me improve the paper, and I genuinely appreciate your contribution to my work. Below are my responses.

 

Comment:

1-It is better reducing abbreviations in the title in order to be clear.

Response to comment:

I understand your concern regarding abbreviations in the title. However, the abbreviations used in the title accurately reflect the content of the paper and are standard in the field of control theory. I've explained the rationale behind each abbreviation and why expanding them would result in an unwieldy title.

 

MIMO (Multiple Input Multiple Output) - a widely known abbreviation used in various fields (communication, networks, control, etc.). The term "multivariable" in this context is considered inappropriate.

PID (Proportional Integral Derivative) - a widely known abbreviation.

LMI (Linear Matrix Inequality) - also a widely known abbreviation, used both in control theory and convex optimization theory.

LTV (Linear Time-Varying) - although this is also a well-known abbreviation in control theory (along with LPV - Linear Parameter Varying), it could have been replaced with "non-stationary plant" (it would be incorrect to write "non-stationary plant model" here), but I have come to the reasoned conclusion that this would be incorrect. The LTV class is only a special case of possible models for non-stationary plants.

Unfortunately, I cannot offer anything better. The current title accurately reflects the content of the work, and it is important to preserve it.

Comment:

2- In abstract all author must explain all abbreviations appeared.

Response to comment:

In the paper, there is a section called "Abbreviations," where all abbreviations are explained. Additionally, all abbreviations are clarified at the beginning of the "Introduction" section and further as they are used in the text. I hope this is sufficient.

Journal guidelines specify that the abstract must adhere to a volume limitation - "A single paragraph of about 200 words maximum." Unfortunately, I cannot expand on the abbreviations there for the third time.

 

Comment:

3- In abstract the author must write the model used in this study (examples) and finally add the criteria's and bound used.

Response to comment:

The model is described. It's an LTV (Linear Time-Varying) plant model. Details about the model are provided in Section 3.1.

Comment:

4-In introduction section the paper need to be enhanced with more studies related to the theory of paper subject (LMI Model).

Response to comment:

I have found another study related to the discrete MIMO PID synthesis method. It has been added to Section 1.2. It is worth noting that there are not many studies on this topic at present.

 

Comment:

5- In section 1.1 add the basic PID equation then explain the meaning of its variables.

Response to comment:

Thank you for the suggestion. I have included the basic PID equation in Section 1.1 and provided an explanation of the variables to improve clarity and comprehension.

 

Comment:

6- Add the reference of each figure and equation appeared in the study.

Response to comment:

It has been done. References to the equations used have been provided.

All figures in the article were created by me specifically for this paper and do not require references to sources. References to each figure are included in the text of the paper.

If there is anything else remaining, please provide specific examples.

 

Comment:

7-In section 7 the author must analyze the effect of the items appeared on section 6 for the two examples chose.

Response to comment:

This has been added.

 

Comment:

8- Some figures appeared without legend try to add it on figure needed.

Response to comment:

The legend has been added to Figure 6.

 

Comment:

9-Several references are very old try to replace them with more recent studies.      

Response to comment:

While I appreciate the suggestion, I have retained some older references as they serve as primary sources in the field. My belief is that it is better to reference primary sources, thus showing respect to the pioneers. I spent a lot of time searching for primary sources.

At the request of another reviewer, references to several new works related to the approach used in the article have been added.

Additionally, several new references have been added to the Introduction section.

 

 

Thank you once again for your thorough review and valuable feedback.

 

Major revisions to the text of the paper have been highlighted in color.

 

Round 2

Reviewer 2 Report

Comments and Suggestions for Authors

The revised version manuscript makes a valuable contribution to the field of control systems engineering. With minor revisions and additional elaboration in certain sections.

Reviewer 3 Report

Comments and Suggestions for Authors

Minor editing of English language required

Comments on the Quality of English Language

Minor editing of English language required