Review Reports

Round 1

Reviewer 1 Report

The authors described an approach related to constrained nonlinear MIMO systems subjected to bounded disturbances. They investigated a robust model predictive control with a LPV-IO model, and a recurrent neural network algorithm to deal with the online optimization problem. Results of a comparison of that approach to the linearized robust model predictive control and linear parameter varying – robust model predictive control in a nonlinear alkylation of benzene process showed an interesting computational effectiveness. That approach was deeply analyzed, and it is the strong point of the paper.

The authors presented a comprehensive literature review. However, the research domain is really extensive due to its importance, and there are other interesting works not mentioned in the paper. Methods were selected and discussed adequately, experimental part of the work supported with results and discussion is presented correctly.

Several issues should be addressed by the authors before it is printed:

1. Please, consider a slight revision of section 1. Introduction. There are several statement in which achievements of the research are reported (e.g., lines 93-102). They are expected at the end, in the concluding part of the paper.
2. Lines 134-135 “the terminal penalty matrix being designed to ensure stability” – could you expand on that?
3. Line 280 “Figure 1. RNN framework” – if possible, improve quality of the figure.
4. Lines 324-325 “a third-degree polynomial interpolation produced a good fit for constructing the global LPV-IO model” – is it possible to specify “how good”?

Author Response

To whom it may concern,

We would like to thank the editor and reviewers for their thoughtful comments and constructive suggestion, which help us improve this manuscript's quality. We have modified the paper based on their suggestions and comments, and we believe the revised version addresses their concerns. In this response, we first provide a summary of changes. We then present our itemized responses with reviewers' comments in italics and our responses in a standard font with blue color.

Summary of changes:

• We modified the Abstract and Conclusion sections.
• We modified Figure1, Figure 5, and Figure 6 to address reviewers' comments regarding the quality of the figures.
• We reviewed the entire manuscript and addressed all grammatical issues.
• We analyzed and discussed the results in detail.

 

Reviewers' comments:

Reviewer #1:

1. Please, consider a slight revision of section 1. Introduction. There are several statement in which achievements of the research are reported (e.g., lines 93-102). They are expected at the end, in the concluding part of the paper.

I removed those statements and added them to the conclusion as follows:

"An RMPC with an LPV-IO model is investigated in this paper, in which an RNN algorithm solves the real-time optimization problem. The study had three contributions. Firstly, An RNN-based optimization algorithm was developed to offer global convergence and decline the online computational load. Secondly, free control moves are added to the constant control gain to maintain the closed-loop stability when facing bounded disturbances. Lastly, concerning previous studies for MPC with LPV, the proposed method inherently enjoys a shrunken conservatism degree due to finding the larger possible terminal region, using free control moves, and the global solution of the optimization problem."

2. Lines 134-135 "the terminal penalty matrix being designed to ensure stability" – could you expand on that?

The cost function of a linear MPC typically include a term for control increments ( ) and a term for error (e) as follows:

 

In contrast, in a nonlinear MPC, usually a new term, called terminal cost, will be added to the cost function to guarantee stability:

 

A terminal region is considered for this terminal cost to limit the final value of the state variables over the prediction horizon. In our paper, an ellipsoidal invariant set was chosen for the terminal region, as depicted in equation 13:

(13)

The terminal region was calculated offline. As stated in equations 14 and 15, the maximum value of  is chosen in order to reduce the conservatism.

3. Line 280 "Figure 1. RNN framework" – if possible, improve quality of the figure.

 

Figure 1 is replaced with a new paper with better quality.

 

4. Lines 324-325 "a third-degree polynomial interpolation produced a good fit for constructing the global LPV-IO model" – is it possible to specify "how good"?

 

Firstly, transfer functions were identified by the MATLAB system identification toolbox. These transfer functions are called local models. Secondly, a global model is then obtained from the polynomial interpolation of those local models. Two metrics are used to assess the estimated model, namely Mean Square Error (MSE) and Akaike's Final Prediction Error (FPE).  The MSE and FPE for the global model (three-degree polynomial) were 0.032 and 0.058 in succession, and for the local LPV models, the range of MSE was between 0.004 and 0.062, and FPE changed from 0.040 to 0.081.

 

 

Author Response File: Author Response.pdf

Reviewer 2 Report

The text of the article is clear and easy to read. The mathematical background is well explained and the results prove the claims of the paper. The overall impression of the article is good. In order to enhance the article quality, I suggest the following remarks be taken into account:

+ Authors should extend the Abstract and its contribution should be explained simply and clearly.

+ Format mistakes need to be fixed: font size in Introduction (Paragraph 2, 3, 5, and 6)

+ In Figures 5 and 6, Legend should be added to realize the differences between all cases.

+ The presentation of the paper can be improved and the quality of some figures should be enhanced, i.e, Figure 1.

+ The explanations and analysis of simulation results should be enriched to show the validity of the data.

+ The reviewers recommend that more future work should be added to the Conclusion Section.

+ There are a few grammatical errors, some sentences are long and hard to understand. But those are minor.

Author Response

To whom it may concern,

We would like to thank the editor and reviewers for their thoughtful comments and constructive suggestion, which help us improve this manuscript's quality. We have modified the paper based on their suggestions and comments, and we believe the revised version addresses their concerns. In this response, we first provide a summary of changes. We then present our itemized responses with reviewers' comments in italics and our responses in a standard font with blue color.

Summary of changes:

• We modified the Abstract and Conclusion sections.
• We modified Figure1, Figure 5, and Figure 6 to address reviewers' comments regarding the quality of the figures.
• We reviewed the entire manuscript and addressed all grammatical issues.
• We analyzed and discussed the results in detail.

 

Reviewers' comments:

Reviewer #2:

1. Authors should extend the Abstract and its contribution should be explained simply and clearly.

According to the Journal template, the abstract should be less than 200 words. That is why we couldn't state our contributions in detail. However, we revised the Abstract such that it conveys the main contributions of the paper clearly.

2. Format mistakes need to be fixed: font size in Introduction (Paragraph 2, 3, 5, and 6)

This issue is addressed. Thank you for your comment.

3. In Figures 5 and 6, Legend should be added to realize the differences between all cases.

Legends are added to Figures 5 and 6.

4. The presentation of the paper can be improved and the quality of some figures should be enhanced, i.e, Figure 1.

Figure 1 is replaced with a new paper with better quality.

5. The explanations and analysis of simulation results should be enriched to show the validity of the data.

We provided more analysis and discussions in Section 6, focussing on trajectory tracking, disturbance rejection, and optimization algorithms. Meanwhile, we did a close analysis of Figures 7 to 9.

6. The reviewers recommend that more future work should be added to the Conclusion Section.

Future works are added to the conclusion as follows:

"The future works will focus on showing the stability of the LPV-IO system with multiplicative disturbances. Moreover, the conservatism can be further declined by ellipsoidal invariant sets with polyhedral invariant sets."

7. There are a few grammatical errors, some sentences are long and hard to understand. But those are minor.

The entire paper has been scrutinized for its language aspects. The following issues were thus revised to provide a linguistically legible version:

• Grammatical issues were corrected.
• Stylistic aspects and punctuation were ameliorated to make the text more effective and accurate.
• Discourse markers, all ambiguous cohesive references, and time confusions were revised to enhance readability.

 

Author Response File: Author Response.pdf