Research on Dual-Motor Cross-Coupled Synchronous Control of Flexographic Printing Pressure Integrating Hertz Theory and Fuzzy PI

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

This paper addresses the control challenges of printing pressure in flexographic printing caused by dynamic loads, nonlinear coupling, and insufficient synchronization accuracy. It proposes a dual-motor cross-coupled synchronous control strategy that integrates Hertz contact theory and fuzzy PI control. The paper innovatively combines mechanical contact theory with intelligent control methods, constructing a complete closed-loop system from pressure modeling to control execution. Its three-layer control architecture reflects a hierarchical and systematic design, demonstrating strong theoretical depth and engineering practicality. The quantitative printing pressure model based on Hertz contact theory clearly expresses the mathematical relationship between pressure, structural parameters, and deformation, overcoming the limitations of traditional reliance on empirical thresholds. The control strategy is well-designed, highly adaptable, and supported by robust simulation data. The research problem has a clear engineering application background, the solution is comprehensively designed, the theoretical analysis is solid, and the simulation validation is thorough. Overall, the paper exhibits significant innovation and holds considerable academic value and engineering relevance for the field of high-precision industrial motion control.

Suggestions for Revision

1. The manuscript exhibits inconsistent usage of the full term "Permanent Magnet Synchronous Motor" and its abbreviation "PMSM." For instance, while the abstract and keywords employ the abbreviation, certain sections in the main text revert to the full term.
2. The conclusion section frequently uses qualitative phrases such as "significantly improved" or "effectively enhanced" without anchoring them to specific simulation data.
3. Lack of Justification for Fuzzy Rule Design Although complete fuzzy rule tables (Table 1 and Table 2) are provided, the manuscript omits an explanation of the rationale behind their design.
4. Punctuation throughout the text should be standardized. For example, "Abstract：" should be written as "Abstract:"; the usage of dashes, such as in "typical scenarios-such as load start-up," should be made consistent. Furthermore, the Abstract section requires further refinement and conciseness.
5. The INTRODUCTION section, typically structured into three to four parts, is suggested to be consolidated. It should clearly present the background, significance, identified problems, and the research content of this study. Please revise accordingly.
6. The references formatting needs to be adjusted for consistency, for example, [7], [17], [21] and [22], those references have some problems that need to be modified. And the references in the text are adjusted in order from smallest to largest. Please make the necessary corrections.
7. There are several formatting issues requiring modification. For instance, "R1 = 50mm" should be written as "R1 = 50 mm," with a space between the number and the unit. Additionally, the "caption 6" should be standardized as "Figure 6. Fuzzy control input-output mapping relationships: (a) ΔK_p; (b) ΔK_i." Please check and correct similar instances throughout the manuscript.
8. The text in Figures 9 and 11 is relatively vague, and there are marks/stains in the lower right corner of Figure 11. Please provide corresponding high-resolution images.
9. The derivation of Equation (1) relies on the assumption in Hertzian contact theory that the geometric radius of the contact region is much smaller than the radius of the roller.

Comments on the Quality of English Language

The English could be improved to more clearly express the research.

Author Response

Reviewer1

1. The manuscript exhibits inconsistent usage of the full term "Permanent Magnet Synchronous Motor" and its abbreviation "PMSM." For instance, while the abstract and keywords employ the abbreviation, certain sections in the main text revert to the full term.

Response: Thank you for your valuable suggestion. We have revised this issue accordingly. The full term is now provided when "PMSM." first appears in the keywords, and the abbreviation is used consistently thereafter. Except for the keywords section, all instances of the full term in the text have been replaced with the abbreviation to ensure consistent usage. The revised sections have been marked in red in the manuscript.

 

2. The conclusion section frequently uses qualitative phrases such as "significantly improved" or "effectively enhanced" without anchoring them to specific simulation data.

Response: Thank you for your suggestion. Following the reviewer's comment, we have thoroughly revised the conclusion section with quantitative data. The revised conclusion now explicitly references simulation results, for example: The simulation results indicate that under loaded conditions, the control performance of the single-motor three-loop vector control system degrades significantly. Specifically, the load regulation time extends to 0.40 s, exhibiting a pronounced response lag. Meanwhile, due to the inherent flexibility of the mechanical transmission system, the steady-state pressure fluctuation increases to 0.8%. These quantitative metrics fully demonstrate that the conventional control system is no longer capable of meeting the stringent requirements of high-precision printing processes in terms of pressure uniformity and stability.

 

3. Lack of Justification for Fuzzy Rule Design Although complete fuzzy rule tables (Table 1 and Table 2) are provided, the manuscript omits an explanation of the rationale behind their design.

Response: Thank you for pointing out the shortcomings in our initial manuscript. Following your suggestion, we have added a description of the principles for fuzzy rule design in Section 4.2, elaborating on the basic guidelines they follow. The relevant modifications have been highlighted in red in the Section 4.2

 

4. Punctuation throughout the text should be standardized. For example, "Abstract：" should be written as "Abstract:"; the usage of dashes, such as in "typical scenarios-such as load start-up," should be made consistent. Furthermore, the Abstract section requires further refinement and conciseness.

Response: Thank the reviewer for the careful and detailed comments. We have standardized the punctuation throughout the manuscript as suggested, and all changes have been marked in the revised version. The abstract has been further condensed, retaining the core elements of the study and emphasizing the integration of physical modeling and control algorithms. The revised sections have been marked in red in the abstract.

 

5. The INTRODUCTION section, typically structured into three to four parts, is suggested to be consolidated. It should clearly present the background, significance, identified problems, and the research content of this study. Please revise accordingly.

Response: Thank you very much for your valuable suggestion. Based on your comments, we have reorganized the introduction section, clearly dividing it into four paragraphs covering the research background and significance, the limitations of existing studies, and the content of this paper. The revisions have been marked in red for your review.

6. The references formatting needs to be adjusted for consistency, for example, [7], [17], [21] and [22], those references have some problems that need to be modified. And the references in the text are adjusted in order from smallest to largest. Please make the necessary corrections.

Response: Thank the reviewer for the careful verification of the reference format. We have thoroughly checked and standardized all references throughout the text as requested: Corrections to problematic references: For references [7], [17], [21], [22] and others with format issues, we have rechecked and supplemented the complete information to ensure that elements such as authors, titles, journal names, years, volumes, issues, page numbers, and DOIs are complete and the format is consistent; Unified formatting throughout the text: The punctuation, italics, abbreviations and other formats of all references have been standardized to ensure compliance with journal requirements; Citation order adjustment: We have also checked the citation order in the main text to ensure that all references are arranged in ascending order.

7. There are several formatting issues requiring modification. For instance, "R1 = 50mm" should be written as "R1 = 50 mm," with a space between the number and the unit. Additionally, the "caption 6" should be standardized as "Figure 6. Fuzzy control input-output mapping relationships: (a) ΔKp; (b) Δ" Please check and correct similar instances throughout the manuscript.

Response: Thank you very much for pointing out the formatting issues in the manuscript. We have carefully reviewed and revised the full text based on your suggestions, as follows: 

Regarding operator and unit spacing: We have checked and corrected all instances similar to "R1= 50mm" throughout the manuscript. Now, appropriate spaces have been added between numbers and units to ensure compliance with the standards. 

Regarding figure captions: We have reviewed all figure captions and revised the non-standard caption "Figure 6. Fuzzy control input-output mapping relationships." to the standard format "Figure 6. Fuzzy control input-output mapping relationships: (a)ΔKp; (b)ΔKi."

9. The derivation of Equation (1) relies on the assumption in Hertzian contact theory that the geometric radius of the contact region is much smaller than the radius of the roller.

Response: Thank you for your valuable comment. According to the Hertzian elastic contact theory, the contact between two cylinders forms a rectangular area whose geometric dimensions are much smaller than the radii of the rollers. We have added a relevant description in Section 2.1 and highlighted it in red for your review.

Reviewer 2 Report

Comments and Suggestions for Authors

Please refer to the attached document for details.

Comments for author File: Comments.pdf

Author Response

Reviewer2

1. Insufficient Isolation of the Core Control Contribution

The manuscript derives the motor speed dynamics (Equations (9) and (10)), in which the system behavior is decomposed into reference-tracking, disturbance-rejection,and cross-coupling terms. This formulation implies that the cross-coupling term plays a central role in synchronization performance.

Nevertheless, the simulation study in Section 5 compares only a single-motor control system with the proposed dual-motor system. As a result, the individual contribution of the cross-coupling strategy cannot be quantitatively distinguished from the performance improvement provided by the fuzzy PI controller alone.

To convincingly validate the claimed contribution, additional comparative simulations isolating the cross-coupling effect (e.g., coupling coefficient K=0 versus K>0) would be necessary.

Response: We thank the reviewer for the insightful suggestion and fully agree that this is crucial for validating the core innovation of our study. As the reviewer correctly pointed out, while our original simulation setup demonstrated the overall superiority of the proposed integrated system (combining fuzzy PI control with cross-coupling control), it did not effectively isolate the individual contributions of each component. Accordingly, we have added a new comparative simulation in Section 5.2, "Simulation of Dual-Motor Cross-Coupling Synchronization Control System." Specifically, while keeping the proposed dual-motor architecture and fuzzy PI controller unchanged, we conducted an additional simulation under the same disturbance conditions using an independent control strategy (i.e., coupling coefficient K=0). This result is compared with the existing simulation employing the cross-coupling control strategy (coupling coefficient K>0). This direct comparison clearly illustrates how the cross-coupling coefficients in Equations (9) and (10) actively promote bidirectional compensation compared to the non-coupled case, leading to faster disturbance rejection and lower steady-state synchronization error. The comparative results are presented in two curves, accompanied by an analysis in the main text to explicitly highlight the distinct contribution of the cross-coupling strategy.

 

2. Limited Representation of Practical Disturbance Conditions

While the introduction emphasizes dynamic load disturbances and nonlinear coupling effects inherent in roll-to-roll flexographic printing systems, the verification considers only constant loads and simple step-type disturbances.

Such disturbance models are significantly simplified compared to practical printing environments, where periodic disturbances related to roller rotation, eccentricity, backlash,and stochastic effects are commonly observed. Given the manuscript’s claims of fast disturbance compensation and high robustness, additional validation under more realistic,time-varying disturbance conditions appears necessary.

Response: We sincerely thank the reviewer for this professional and insightful comment. We agree that the step disturbance used in the original manuscript was an idealized simplification that did not fully capture the complex dynamics of the flexographic printing process. To rigorously validate the system's robustness under more realistic operating conditions, we have revised the manuscript by adding a new subsection, "5.3 System Robustness Verification under Time-Varying Periodic Disturbances."

The specific improvements are as follows:

• Construction of a Realistic Disturbance Model: We replaced the simple step signal with a composite time-varying disturbance model . This model systematically incorporates the factors mentioned by the reviewer:

Periodic Component: A sinusoidal termis introduced to simulate periodic fluctuations caused by roller eccentricity and gear meshing (with frequency coupled to the        motor speed).

Random Component: Bounded random noise is added to represent uncertainties such as mechanical vibrations and environmental noise.

Nominal Load: The Hertzian contact pressure step is retained as the baseline load.

• Comparative Simulation Validation: We conducted comparative simulations using this new model, with the results presented in the newly added Figure 17.

Under the independent control strategy: The system exhibited sustained sinusoidal oscillations (peak-to-peak amplitude of approximately 0.04 mm) with superimposed noise. This indicates that, without a coupling mechanism, mechanical eccentricity directly translates into synchronization error.

Under the proposed cross-coupling control strategy: The error curve remained flat and closely aligned with the zero-error line. Periodic fluctuations were effectively attenuated, and the synchronization error was suppressed to within 0.002 mm, fully validating the strong robustness of the proposed strategy under complex time-varying disturbances.

 

3. Insufficient Justification for the Feedback-Only Control Structure

The proposed control architecture relies exclusively on feedback-based control without incorporating feedforward compensation. In high-precision motion control systems,feedforward control is commonly employed to reduce phase lag and improve transient response.

Although the manuscript reports a fast response time without feedforward compensation,the underlying mechanism enabling this performance is not sufficiently explained. Further clarification—either through comparative analysis or theoretical justification—would strengthen the technical credibility of the proposed control structure.

Response: We sincerely thank the reviewer for this professional and constructive comment. We fully agree that feedforward control is indeed an effective standard method for reducing phase lag and improving transient response in high-precision motion systems.

However, the performance of feedforward control is highly dependent on the accuracy of the inverse model of the plant. In the revised manuscript, we have added a new Section 4.3, titled "Analysis of the Fast Response Mechanism without Feedforward," which provides a theoretical explanation for our choice of a pure feedback architecture and elucidates the mechanism by which it achieves fast response. The main arguments are as follows:

• Modeling Difficulty: The flexographic printing process involves significant Hertzian contact nonlinear stiffness and time-varying characteristics of consumables (e.g., gradual wear of the printing plate and viscoelastic changes in the mounting tape). Under such complexities, constructing an accurate and globally applicable feedforward inverse model is extremely challenging, and model mismatch could easily lead to reduced system robustness.
• Intrinsic Response Mechanism: The proposed fuzzy PI controller essentially functions as a nonlinear gain scheduler. As illustrated in Figure 6(a), when the system detects a large error (e.g., during startup or under sudden disturbance), the fuzzy logic rules rapidly adjust the proportional gain (Kp) to its peak value. This high-gain injection mechanism physically generates a strong instantaneous torque output, functionally emulating the "acceleration" effect of a feedforward signal. Consequently, it effectively compensates for the inherent phase lag of conventional fixed-gain linear PI control without relying on a potentially fragile system model, thereby ensuring fast response.

We believe this theoretical elaboration enhances the technical credibility of the proposed control structure and justifies our design choice. The new Section 4.3 provides a detailed theoretical Argumentation on how the nonlinear gain scheduling mechanism overcomes phase lag. The relevant modifications have been highlighted in red for your review.

Reviewer 3 Report

Comments and Suggestions for Authors

Fig.1 L is not well defined (what is it?)
 
Lines 118-120 state that only the deformation of the flexible plate is considered. However, Eq.3 and line 150 use the parameters of the impression cylinder. Which statement is true?

Please double-check Eq.7, something seems off with the power factors.
Also line 164 refers to lambda_max but eq.7 does not contain it. Alternatively, a simpler form of eq.7 can be obtained by keeping lambda_max (if it helps implementing the controller)

Fig.3 There is no reference to the "optical grating scale". What is it used for?

Lines 283-284: there is no omega present in Fig.7. Also k and K appear. Are they the same? Also value for k is just droped-in with no explanation for that very particular value.

Line 331: Paragraph 5.1: It is not clear how a single motor could maintain parallelism between the cylinders given the setup presented. There is no reference before of single motor actuation. According to lines 265-267 the idea of mechanical synchro was discarded by default. Then how it is supposed to control the pressure without affecting the hertzian contact patch geometry?
  
Lines 339-344: Response times are not consistent with result presented in Figure 10. How can there be "pressure fluctuation" if there is "no load"? 
It is not clear what no load / under load means. Paragraph conclusion is not supported by the presented data.

Lines 350-351 state that "single-motor disturbance" and "multiple disturbances" conditions were tested. What does "multiple disturbances" refer to, is it on a single motor or on both motors? 

Figures 12-15 are too small / low resolution. 
Results presented in text are not consistent with graphical representation in Figures 12-15.

Lines 364-368: Under load positioning response time for 0.4mm is 0.15s. 
Disturbance response time for 0.05mm is 0.3s. Please explain how can this be qualified as "excellent anti-disturbance performance"?

Figure 15 has no text reference and no explanation for the data presented. Is it somehow related to line 351?

According to lines 341-342 and 358-359, both single motor and dual motor synchro methods yield the same pressure fluctuation of +/-0.8%. This render Conclusion (2) and (3) false. Please explain.

Please fill-in Author Contributions and Funding.

Author Response

Reviewer3

1. The definition of L in Figure 1 is unclear (what does it represent?)

Response: Thank you for your valuable comments. Based on your suggestions, we have added an explanatory note for the symbol L in Section 2.1, line 110 of the article, clearly defining it as "the distance between the centers P1 and P2 of the two rollers" to make the description clearer and more accurate.

 

2. Lines 118-120 state that only the deformation of the flexible plate is considered. However, Eq.3 and line 150 use the parameters of the impression cylinder. Which statement is true?

Response: Thank you for your valuable question. In practical printing engineering applications, the rollers are typically made of stainless steel or aluminum alloy, whose elastic modulus is significantly higher than that of the flexible plate composed of photosensitive resin. Therefore, the deformation is considered to occur on the flexible plate wrapped around the plate cylinder. According to Hertzian contact theory, when the contact area between two cylinders involves two different materials, an equivalent treatment of the elastic modulus in the contact region is required. Consequently, the parameters of the impression roller are incorporated into the calculation.

 

3. Please check Equation 7 again; there seems to be an issue with the power/exponent factor.

Response: Thank the reviewer for the rigorous examination of Equation (7). We have re-derived and verified Equation (7):

Re-derivation and verification: According to the Hertz contact theory, after substituting Equations (1), (2), and (5) into Equation (6), we carefully re-derived the expression for the printing pressure P and confirmed the power relationships of each variable. The revised Equation (7) is:

4.λ_max is mentioned in line 164 but is not included in Equation 7. Alternatively, if. λ_maxcontributes to the implementation of the controller, a simpler version of Equation 7.  could also be obtained.e.

Response: Thank you for your correction. Equation (7) is an expanded form derived through further mathematical deduction based on the substitution of the aforementioned relevant formulas. The purpose of presenting this equation is to more intuitively illustrate the mathematical relationship between key parameters and printing pressure.

 

3. 3 There is no reference to the "optical grating scale". What is it used for?

Response: Thank you for your valuable question. Regarding the "optical grating scale," it is installed on the side of the guide rail to measure the actual displacement of the cylinder mount in real time. Serving as the feedback element of the position loop, the grating scale eliminates errors caused by the backlash of the screw drive, ensuring that the displacement detection accuracy reaches the submicron level. We have added the relevant modification in Section 3.1 and highlighted it in red.

 

6. Lines 283-284: there is no omega present in Fig.7. Also k and K appear. Are they the same? Also value for k is just droped-in with no explanation for that very particular value.

Response: Thank you for your valuable comment. Although ω does not appear in Figure 7, it represents the rotational speed and is an important parameter in Equations (9) and (10). We have re-described the notations for ω, k, and K, and provided an explanation for the value of k: k=0.1047 is an approximation of 2π/60 (conversion from revolutions per minute to radians per second). The relevant modifications have been highlighted in red in Section 4.4.

 

7.Line 331: Paragraph 5.1: It is not clear how a single motor could maintain parallelism between the cylinders given the setup presented. There is no reference before of single motor actuation. According to lines 265-267 the idea of mechanical synchro was discarded by default. Then how it is supposed to control the pressure without affecting the hertzian contact patch geometry?

Response: Thank you for your valuable comment. This part was not included in the manuscript due to an oversight. The "single-motor system" refers to the traditional "single motor+mechanical transmission shaft" configuration, and a description has been added in Section 5.1. By utilizing a rigid transmission shaft to simultaneously drive the screw mechanisms on both sides, the parallel movement of the cylinder mount is theoretically achieved, thereby maintaining the rectangular shape of the hertzian contact area.

8.Lines 339-344: Response times are not consistent with result presented in Figure 10. How can there be "pressure fluctuation" if there is "no load"? 

Response: Thank you for your correction. We have made two revisions to the main text: 1) ensuring that the textual description is strictly consistent with the data in Figure 10; 2) revising the imprecise expression "pressure fluctuation" under no-load conditions to the more accurate "steady-state error displacement." The relevant modifications have been marked in red in Section 5.1.

 

9. It is not clear what no load / under load means. Paragraph conclusion is not supported by the presented data.

Response: Thank you very much for your comments. We have adopted your suggestions to clearly define the concepts of "no-load" and "loaded" conditions. Meanwhile, we have re-examined and revised the conclusions in this paragraph based on the data in Figure 10 to ensure accuracy and data support. The revisions have been marked in red.

10. Lines 350-351 state that "single-motor disturbance" and "multiple disturbances" conditions were tested. What does "multiple disturbances" refer to, is it on a single motor or on both motors? 

Response: Thank you very much for raising this question. We have adopted your suggestion and added an explanation of the specific conditions of the "multiple disturbances" in Section 5.2: specifically, disturbances were applied simultaneously to the drive-side motor and the operation-side motor at t=1.5s and t=2.0s. The corresponding modifications in the text have been marked in red.

11. Figures 12-15 are too small / low resolution. 
    Results presented in text are not consistent with graphical representation in Figures 12-15.

Response: Thank you for your valuable comment. We have replaced Figures 12-15 with images in EMF format to ensure clarity and have confirmed that the presented results remain consistent with the descriptions in Figures 12-15.

12.Lines 364-368: Under load positioning response time for 0.4mm is 0.15s. 
Disturbance response time for 0.05mm is 0.3s. Please explain how can this be qualified as "excellent anti-disturbance performance"?

Response: We accurately understand and appreciate the reviewer’s insightful observation regarding the difference in response times.Unlike the rapid positioning phase (0.15 s), the disturbance rejection phase (0.30 s) involves the cross-coupling mechanism actively constraining the undisturbed motor to match the trajectory of the disturbed one. While this "cooperative deceleration" extends the absolute settling time, it strictly limits the synchronization error to ±0.002 mm throughout the process. We classify this as "excellent" performance because maintaining cylinder parallelism is the decisive factor for printing uniformity, effectively preventing quality defects even under asymmetric loads. We have revised the 'Single-Motor Disturbance' subsection in Section 5.2 to explicitly clarify that the extended recovery time (0.30 s) results from the system prioritizing synchronization over single-axis speed. This explanation justifies the performance metric based on the minimized relative error. The relevant modifications have been highlighted in red for your convenience.

 

13. Figure 15 has no text reference and no explanation for the data presented. Is it somehow related to line 351?

Response: Thank you very much for pointing out this omission. We indeed overlooked the corresponding description of Figure 15. To address this, we have added an analysis and explanation of the "displacement response curves under multiple disturbances" shown in this figure in Section 5.2, with the new content marked in red for your review.

14. According to lines 341-342 and 358-359, both single motor and dual motor synchro methods yield the same pressure fluctuation of +/-0.8%. This render Conclusion (2) and (3) false. Please explain.

Response: Thank you very much for pointing out this issue. Upon verification, there was a clerical error in the description of the pressure fluctuation amplitude in the original text, and the correct value should be ±0.3%. We have adopted your suggestion and corrected the relevant data to "±0.3%" in Section 5.2 of the main text and the conclusion section, with the revisions marked in red for your verification.

 

Reviewer 4 Report

Comments and Suggestions for Authors

1. All the figures starting from number 5 needs to be improved in terms of quality, font size and clarity.
2. For Figure 7, I think there is a mistake. The two summing points at the left has four connection, and one of them is going upward then forward to the summing point at the far right, however the lines doesn’t connect to any signal, in fact there is a small bridge at the signals of n1 and 2. I think that instead of bridge there at n1 and n2 there should be a connection point.
3. For Figure 11, it is better to have multiple sub-figures. First subfigure with the main one but with submasks to just show the main outline, and then other subfigure to show the details of each submask, this would make the presentation cleaner.
4. Equations should all have the same formatting and orientation.
5. The results section should be improved, providing more case studies, tables to compare performance of the proposed solution.

Author Response

Reviewer#4

1.All the figures starting from number 5 needs to be improved in terms of quality, font size and clarity.

Response:We fully agree with the reviewer's comment. Clear and well-presented figures are crucial for effectively communicating research findings. In the revised manuscript, we have carefully optimized all illustrations from Figure 5 to Figure 15. These improvements ensure high resolution, appropriately sized fonts, and clear lines and details, thereby enhancing overall readability and professionalism.

2.For Figure 7, I think there is a mistake. The two summing points at the left has four connection, and one of them is going upward then forward to the summing point at the far right, however the lines doesn’t connect to any signal, in fact there is a small bridge at the signals of n1 and 2. I think that instead of bridge there at n1 and n2 there should be a connection point.

Response:Thank you for your careful review of the figures. You are correct that the visual representation of the cross-coupled signal connections in Figure 7 was ambiguous and did not clearly indicate the actual convergence point of the signals. We have revised Figure 7 according to your feedback. The updated figure is provided below:

Figure 7. Schematic diagram of cross-coupled control.

 

3.For Figure 11, it is better to have multiple sub-figures. First subfigure with the main one but with submasks to just show the main outline, and then other subfigure to show the details of each submask, this would make the presentation cleaner.

Response:This is a very valuable suggestion. Following your recommendation, we will reconstruct Figure 11 into five subfigures: (a) Subfigure: presenting the overall system architecture and signal flow. (b) Subfigure: the single-motor core control subsystem. (c) Subfigure: the internal structure of the fuzzy PI controller. (d) Subfigure: the coordinate transformation and SVPWM generation subsystem. (e) Subfigure: the data observation and logging configuration. This layered presentation will significantly improve readability and allow readers to better understand the composite structure of the system. The optimized Figure 11 is as follows:

(a) Top-level architecture and synchronization control diagram.

(b) Single motor core control subsystem.

(c) Internal structure of the Fuzzy PI controller.

(d) Coordinate transformation and SVPWM generation subsystem.

(e) Data observation and recording configuration.

Figure 11. Flexographic printing pressure synchronization control simulation model.

4.Equations should all have the same formatting and orientation.

Response:Thank you for your comment. In the revised manuscript, we have thoroughly checked all mathematical formulas and processed them using professional equation editing tools. This ensures that they maintain consistent font, size, numbering style, and alignment throughout the entire document.

 

5.The results section should be improved, providing more case studies, tables to compare performance of the proposed solution.

Response:We sincerely appreciate your valuable comments. We have added Table 3: Comprehensive performance comparison of different control strategies under various working conditions at the end of Section 5.3. This table provides a detailed comparison of response time, pressure fluctuation, sync error under periodic disturbances, and operational stability. The relevant modifications have been highlighted in red in Section 5.3.

Table 3. Comprehensive performance comparison of different control strategies under various working conditions.

Key Performance Indicators	Traditional Single-Motor	Dual-Motor Independent Control	Proposed Intelligent Cross-Coupled Control
Response Time (Load)	0.40 s	Fails to Coordinate	≤ 0.30 s
Pressure Fluctuation	± 0.8%	N/A	± 0.3%
Sync Error (Periodic Disturbance)	N/A	≈ 0.04 mm	± 0.002 mm
Operational Stability	Low	Medium	99.50%

 

Round 2

Reviewer 2 Report

Comments and Suggestions for Authors

The authors have adequately addressed the reviewer comments and have revised the manuscript accordingly.

In the revised version, additional comparative simulations have been included to demonstrate the contribution of the cross-coupled control strategy clearly. The authors have also introduced more realistic disturbance conditions, including periodic disturbances and stochastic noise, which improves the robustness validation of the proposed method.

Furthermore, the explanation regarding the feedback-based control architecture has been clarified, enhancing the technical transparency of the control design.

Overall, the revisions have improved the clarity and technical completeness of the manuscript.
Therefore, the manuscript is recommended for acceptance in its current form.

Author Response

We greatly appreciate the reviewer’s careful comments and acceptance of our manuscript. Thank you for your time and professional review.

Reviewer 3 Report

Comments and Suggestions for Authors

The paper has been properly revised. Recommend accept.

Author Response

We greatly appreciate the reviewer’s careful comments and acceptance of our manuscript. Thank you for your time and professional review.

Reviewer 4 Report

Comments and Suggestions for Authors

The authors have address all of my comments; only the remaining point is as follows:

Figure 11 is now better, but the modification detailed in your reply letter of five sub-figures is not actually shown in the updated manuscript. I’m assuming you forgot to include it.

Author Response

Thank you for your correction. You are absolutely right. The plan we mentioned in the reply letter to split Figure 11 into five subgraphs was indeed not reflected in the previous update. We are deeply sorry for this. During the process of preparing the revision, we re-examined the subgraph scheme and found that after splitting the original figure, the information presented in each subgraph was too scattered, which instead weakened the important comparisons and connections between different parts and was not conducive to readers' understanding of the overall trend and core conclusion. Therefore, after careful evaluation, we decided to optimize the original figure instead of simply splitting it into subgraphs.
The revised Figure 11 and the corresponding text explanations have been updated in the manuscript.