Lurie Control Systems Applied to the Sudden Cardiac Death Problem Based on Chua Circuit Dynamics

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

This paper introduces a Chua circuit with delay, and proposes a novel control design technique based on Lurie-type control systems theory combined with mixed-sensitivity H∞ methodology.  The proposed controller enables precise regulation of HR in Chua’s chaotic circuit. The presented theoretical frame-work, supported by numerical simulations, demonstrates the effectiveness of the conceptualization. Before it can be accepted for publication, the following comments should be addressed.

1） The definition of absolute stability should be given.

2) Similar issue have been investigated. Please see, a looped functional method to design state feedback controllers for Lurie networked control systems. The authors are suggested to make a comparison and discussion.

3) What is the size of delay to affect the  systems's performance？Delay is regarded as a source of poor performance. However, it is reported in [Applied Mathematics and Computation, 374 (2020) 125041] that proper delays in a system may improve the performance of the system.  It would be better to make some discussion.

4) There are some minor typos and grammar mistakes in the text.  The authors are suggested to double check and correct them. 

Comments on the Quality of English Language

The quality of English language is acceptable. However, there are some minor grammar mistakes in the text.  The authors are suggested to check the paper carefully the polish it.. 

Author Response

General Comments: This paper introduces a Chua circuit with delay, and proposes a novel control design technique based on Lurie-type control systems theory combined with mixed-sensitivity H∞ methodology.  The proposed controller enables precise regulation of HR in Chua’s chaotic circuit. The presented theoretical frame-work, supported by numerical simulations, demonstrates the effectiveness of the conceptualization. Before it can be accepted for publication, the following comments should be addressed.

Response: Thank you for your thorough and detailed review!

 

Comment 1: The definition of absolute stability should be given.

Response: The definition of absolute stability has been given. See Definition 1.

 

Comment 2: Similar issue have been investigated. Please see, a looped functional method to design state feedback controllers for Lurie networked control systems. The authors are suggested to make a comparison and discussion.

Response: The introduction includes a brief discussion of the recent work:

[37] Wang, W., & Zeng, H. B. (2023). A looped functional method to design state feedback controllers for lurie networked control systems. IEEE/CAA Journal of Automatica Sinica, 10(4), 1093-1095.

In addition, other references dealing with Lurie's problem have been presented. The reworded text in the introduction is as follows:

“Lurie's problem [27] emerged in the 1940s in the context of automatic aircraft control, introducing the concept of  “absolute stability” for nonlinear systems. Over time, it has been explored in various domains, including chaos theory [28,29], \(L_2\)-stability [30], \(\mu\)-analysis [31,32], uncertain [33] and switched systems [34], as well as aeronautical applications [35]. Recent studies have leveraged Lyapunov functions and linear matrix inequalities (LMI) to refine stability conditions [28,36]. [37] introduced a loop-functional approach to stabilize networked Lurie control systems with network-induced delays, ensuring absolute stability through state feedback control. In contrast, this work applies Lurie's control theory to the prevention of sudden cardiac death (SCD), integrating Chua’s circuit dynamics with mixed sensitivity \(H_{\infty}\) control to regulate heart rate. A recent review of applications of Lurie's Problem in the medical and biological fields can be found at [21].”

 

Comment 3: What is the size of delay to affect the  systems's performance？Delay is regarded as a source of poor performance. However, it is reported in [Applied Mathematics and Computation, 374 (2020) 125041] that proper delays in a system may improve the performance of the system.  It would be better to make some discussion.

Response: In the context of this work, the use of delay is aimed at modeling the SCD in order to represent a biological response, at least in the initial instants. With regard to the work cited, in fact, the delay can improve or worsen the modeling. However, in terms of performance, the controller maintains its performance for any value of delay (verified by simulations). This validates the theorem, where the condition establishes a robust controller. Therefore, in terms of the controlled system, performance is not affected (for this specific system), since the controller developed respects the condition of the theorem guaranteeing performance. The following text has been inserted in the discussion section of the paper.

“Regarding the impact of delay on performance, it is generally assumed to be a source of performance degradation. However, as reported in \cite{Zeng2020}, appropriate delays can actually enhance system performance. In this work, delay is used to model the SCD, aiming to produce an artificial temporal response that more closely resembles a biological one, so that its adjustment (either increasing or decreasing it) can improve the model’s ability to accurately represent a biological system. Furthermore, after implementing control, simulations confirmed that the developed controller maintains its performance regardless of the delay value, validating Theorem \ref{theo:controller3}, whose condition ensures controller robustness. Thus, in the controlled system, performance remains unaffected, as the designed controller satisfies the theorem’s condition, guaranteeing robustness in performance.”

 

Comment 4: There are some minor typos and grammar mistakes in the text.  The authors are suggested to double check and correct them. 

Response: Thank you for your comment. We have carefully reviewed the manuscript and corrected all minor typos and grammar mistakes.

 

Comment 5: Comments on the Quality of English Language

The quality of English language is acceptable. However, there are some minor grammar mistakes in the text.  The authors are suggested to check the paper carefully the polish it.

Response: Thank you for your feedback. We have carefully revised the manuscript and corrected any grammar mistakes to improve the quality of the English language.

Reviewer 2 Report

Comments and Suggestions for Authors

This manuscript proposes a control design method based on Lurie control system, which is applied to Chua circuit model to solve the problem of sudden cardiac death. The theoretical framework of the article is relatively complete. It combines nonlinear control theory and robust control technology, proposes a new controller design method, and verifies its effectiveness through numerical simulation. The innovation of the article lies in combining Lurie control theory with Chua circuit, introducing time delay system, and expanding the existing theoretical framework. Overall, the article has high academic value, especially in the cross-application of biomedical engineering and control system.

 

I have some suggestions for the author's reference.

 

1. Some of the formulas in the article are missing numbers.

 

2. The vertical axis titles of Figure 7-10 are missing.

 

3. The article mainly verifies the effectiveness of the controller based on numerical simulation, but lacks verification of actual physiological data. It is recommended that in future work, it should be verified in combination with actual physiological data (such as HR, BP, etc.) to further prove the feasibility of the model in practical applications.

 

4. The control method proposed in the article is expected to be applied to medical devices, but its feasibility in actual hardware implementation is not discussed. It is recommended that in future work, how to integrate the controller into actual medical devices should be explored.

I think this manuscript can be accepted after minor revision.

Author Response

General Comments: This manuscript proposes a control design method based on Lurie control system, which is applied to Chua circuit model to solve the problem of sudden cardiac death. The theoretical framework of the article is relatively complete. It combines nonlinear control theory and robust control technology, proposes a new controller design method, and verifies its effectiveness through numerical simulation. The innovation of the article lies in combining Lurie control theory with Chua circuit, introducing time delay system, and expanding the existing theoretical framework. Overall, the article has high academic value, especially in the cross-application of biomedical engineering and control system.

I have some suggestions for the author's reference.

Response: Thank you for your thorough and detailed review!

 

Comment 1: Some of the formulas in the article are missing numbers.

 Response: The numbers have been inserted into the equations (2), (5-8), (14), (18-22), (23), (25-29), (31-34), (37), (40-47) and (48).

 

Comment 2: The vertical axis titles of Figure 7-10 are missing.

Response: The titles have been inserted. 

 

Comment 3: The article mainly verifies the effectiveness of the controller based on numerical simulation but lacks verification of actual physiological data. It is recommended that in future work, it should be verified in combination with actual physiological data (such as HR, BP, etc.) to further prove the feasibility of the model in practical applications.

Response: In the conclusion, the following text was inserted below.

“Additionally, refinement and validation using real physiological data for the delayed Chua circuit are necessary, as this study primarily focused on the development of the controller. Future work should incorporate actual physiological data, such as heart rate (HR) and blood pressure (BP), to further verify the feasibility of the model in practical applications.”

 

Comment 4: The control method proposed in the article is expected to be applied to medical devices, but its feasibility in actual hardware implementation is not discussed. It is recommended that in future work, how to integrate the controller into actual medical devices should be explored.

Response: We've put the answer to your question in the last two paragraphs of the conclusion, as shown below.

“Future research should advance the integration of Chua's circuit into a medical device for blood pressure control. Initially, the goal is to develop an experimental platform with electronic components to implement both Chua's circuit and the controller. The implementation can be carried out using the Hardware-In-The-Loop (HIL) methodology \cite{Gaspar2024}, allowing interaction between analog and computational electronic systems. This approach will contribute to expanding knowledge, reducing costs, and enhancing the scalability of bench experiments, representing an innovative strategy for treating arrhythmias and preventing fatal events.

After conducting hardware experiments and their proper validation, a prototype can be developed for future \textit{in vivo} experiments. This prototype would integrate advanced detection, dynamic control, and precise stimulation to stabilize heart rhythm, marking a significant step at the intersection of biomedicine and dynamical systems theory.”

Reviewer 3 Report

Comments and Suggestions for Authors

Manuscript ID: Eng-3429168

Title:   Lurie Control Systems Applied to the Sudden Cardiac Death Problem Based on Chua Circuit Dynamics

This paper introduces a Chua circuit with delay and proposes a novel control design technique based on Lurie-type control systems theory combined with the mixed-sensitivity (S/KS/T) methodology.

Reviewer’s Comments and Suggestions:

1. Plagiarism Concern: The iThenticate report shows a 14% similarity, which is not acceptable. Please revise the text to ensure originality.
2. Based on the Introduction, it appears that the model proposed by Osaka ([6]) includes a modified Chua’s circuit that simulates its relationship, while this paper focuses on controlling this chaos using robust control techniques. This makes the study a case of "chaos control" rather than a fundamentally new concept. The claim that this work "presents a novel approach for designing controllers that could potentially offer new advancements in the future that avoid SCD" is unclear. Additionally, Osaka’s paper ([6]) was published in 2011 and has been cited only once. How does this justify the claim of "potential future advancements"? Please clarify the novelty.
3. The literature review (lines 41-54) is inadequate. The Introduction should be expanded to include a more comprehensive discussion of relevant research.
4. It is uncommon for a paper to reference theorems and equations in the Introduction. Please provide an example of a similar study where this is done to justify its inclusion.
5. Equation (2) represents the delayed system, but no proof is provided to confirm that the system still exhibits chaotic behavior for specific parameter values and initial conditions. Please include such proof.
6. In line 140, the sensitivity function and the complementary sensitivity function are defined. However, KS is not clearly explained. Please define it.
7. Several parameters are missing from the section "Theoretical Basis". For example, please define wb ​, ​, and Wm​.
8. In Sections 4.2.1 and 4.2.2, the controller is designed with a value of 0.9432. However, sensitivity and complementary sensitivity Bode diagrams and system response plots are not provided, even though they are typically included in similar studies. Please include them.
9. What is the purpose of using a single reference (sinusoidal signal)? In control systems, the equilibrium point is usually used as the reference value. Please clarify.
10. Robust control typically implies resistance to disturbances, noise, and uncertainties. However, no robustness tests are presented in the study. What is the superiority of the proposed controller in this regard? Please provide robustness validation.

Author Response

General Comments:  This paper introduces a Chua circuit with delay and proposes a novel control design technique based on Lurie-type control systems theory combined with the mixed-sensitivity (S/KS/T) methodology.

Response: Thank you for your thorough and detailed review!

 

Comment 1: Plagiarism Concern: The iThenticate report shows a 14% similarity, which is not acceptable. Please revise the text to ensure originality.

Response: We thank you for your review and reaffirm that our work is original and the result of our own research. However, in view of your observation regarding the similarity index, we have revised and restructured some parts of the text in order to try to reduce this percentage, without compromising the clarity, precision or scientific integrity of the study. In addition, the revision of the work itself and the inclusion of the reviewers' suggestions can lead to a natural reduction in this percentage. We believe that the changes made ensure greater originality of the manuscript, and we remain available for any further clarification.

 

Comment 2: Based on the Introduction, it appears that the model proposed by Osaka ([6]) includes a modified Chua’s circuit that simulates its relationship, while this paper focuses on controlling this chaos using robust control techniques. This makes the study a case of "chaos control" rather than a fundamentally new concept. The claim that this work "presents a novel approach for designing controllers that could potentially offer new advancements in the future that avoid SCD" is unclear. Additionally, Osaka’s paper ([6]) was published in 2011 and has been cited only once. How does this justify the claim of "potential future advancements"? Please clarify the novelty.

Response: To better respond to the reviewer's comment, let's divide it into 2 sub-comments:

Comment 2.1: Based on the Introduction, it appears that the model proposed by Osaka ([6]) includes a modified Chua’s circuit that simulates its relationship, while this paper focuses on controlling this chaos using robust control techniques. This makes the study a case of "chaos control" rather than a fundamentally new concept.  

Response: Regarding what makes this study novel, it is particularly Theorem 3 and the presentation of the Chua's circuit with delay to simulate HR, SNA and BP preceding SCD. This theorem is new in the literature, although we acknowledge that it is conceived using well-known techniques from robust control theory. Furthermore, Theorem 3 can be applied to other types of systems beyond Chua's circuit. It applies to a broader range of systems known in the literature as Lurie-type systems, which are described in Section 3.1. In fact, in this paper, we present a control system capable of controlling Chua's chaotic circuit (Eq. (1) from Osaka); however, it can also be applied to any Lurie-type SISO system. To clarify the reviewer's concern, we have reformulated the third bullet point in the introduction with the following text:

“The main contribution of this work is the presentation, demonstration, and validation of Theorem \ref{theo:controller3}. This theorem introduces a novel approach for controller synthesis based on the $\mathcal{H}_\infty$ mixed sensitivity technique for time-delay systems. Furthermore, the theorem is not limited to the control of Chua’s circuits with or without delay but extends to a broader class of dynamical systems, specifically Lurie-type SISO systems with or without delay and nonlinearities mapped by sector conditions. This generalization significantly expands the applicability of the proposed method, enabling its use in various systems beyond the Osaka model, thereby reinforcing its originality and contribution to the field of robust control of chaotic or nonlinear systems.”

In the presentation of the delayed Chua circuit, it is highlighted that this approach is an original contribution of the paper, as mentioned in the first bullet point:

 “Presentation of the model (\ref{eq:chua_ckt_delay}) that includes the time delay in relation to that presented by \cite{Osaka2011} represented by the system (\ref{eq:chua_ckt});”

Comment 2.2: The claim that this work "presents a novel approach for designing controllers that could potentially offer new advancements in the future that avoid SCD" is unclear. Additionally, Osaka’s paper ([6]) was published in 2011 and has been cited only once. How does this justify the claim of "potential future advancements"? Please clarify the novelty.

Response: Although the article by Osaka [6] has only one citation, according to the Scimago Journal & Country Rank (SJR) (reference year 2011), it was published in a journal classified as Q1 in the field of Engineering (Miscellaneous). This indicates that Osaka's work provides original contributions of high impact to the scientific community, in addition to presenting a high technical level. The fact that Osaka’s paper [6] has only been cited once may be attributed to the complexity of the engineering concepts for healthcare professionals and, conversely, the medical approach for engineers. Addressing this topic effectively requires a highly qualified multidisciplinary team with expertise in both fields, as well as a foundational understanding of engineering for physicians and cardiology for engineers. In this regard, our team, composed of systems and control engineers, a biomedical engineer, a general medical doctor, and a cardiologist, recognized significant value in Osaka’s research [6] and its potential contributions. This motivated us to expand on Osaka's work [6], introducing a new model with delay and integrating a control strategy that could, in the future, evolve into a novel approach for monitoring and intervention devices aimed at preventing SCD.

With regard to “potential future advances”, this term was used in a context where Osaka's work only presented a mathematical model of a heart condition that can lead to sudden death. On the other hand, our work introduces a controller to prevent SCD, a fact that we consider may in the future generate advances to prevent SCD. In order to improve the clarity of the paragraph referred to in the introduction, we have changed it to the following: 

“In this context, while Osaka’s study focused solely on the mathematical modeling of a cardiac condition that may lead to sudden death, our approach goes further by proposing a control mechanism aimed at mitigating SCD. This contribution has the potential to drive future advancements in the prevention of SCD.”

Furthermore, we have included in the conclusion some directions for future research, exploring ways to implement this approach in hardware and eventually progress to in vivo experiments.

“Future research should advance the integration of Chua's circuit into a medical device for blood pressure control. Initially, the goal is to develop an experimental platform with electronic components to implement both Chua's circuit and the controller. The implementation can be carried out using the Hardware-In-The-Loop (HIL) methodology \cite{Gaspar2024}, allowing interaction between analog and computational electronic systems. This approach will contribute to expanding knowledge, reducing costs, and enhancing the scalability of bench experiments, representing an innovative strategy for treating arrhythmias and preventing fatal events.

After conducting hardware experiments and their proper validation, a prototype can be developed for future \textit{in vivo} experiments. This prototype would integrate advanced detection, dynamic control, and precise stimulation to stabilize heart rhythm, marking a significant step at the intersection of biomedicine and dynamical systems theory.”

 

Comment 3: The literature review (lines 41-54) is inadequate. The Introduction should be expanded to include a more comprehensive discussion of relevant research.

Response: It was included a more comprehensive discussion of relevant research:

“Lurie's problem [27] emerged in the 1940s in the context of automatic aircraft control, introducing the concept of  “absolute stability” for nonlinear systems. Over time, it has been explored in various domains, including chaos theory [28,29], \(L_2\)-stability [30], \(\mu\)-analysis [31,32], uncertain [33] and switched systems [34], as well as aeronautical applications [35]. Recent studies have leveraged Lyapunov functions and linear matrix inequalities (LMI) to refine stability conditions [28,36]. [37] introduced a loop-functional approach to stabilize networked Lurie control systems with network-induced delays, ensuring absolute stability through state feedback control. In contrast, this work applies Lurie's control theory to the prevention of sudden cardiac death (SCD), integrating Chua’s circuit dynamics with mixed sensitivity \(H_{\infty}\) control to regulate heart rate. A recent review of applications of Lurie's Problem in the medical and biological fields can be found at [21].”

 

Comment 4: It is uncommon for a paper to reference theorems and equations in the Introduction. Please provide an example of a similar study where this is done to justify its inclusion.

Response: The purpose of referencing the equations and the theorem in the introduction was to make clear to the reader the main contributions of the paper. We have other papers that use this formalism and have been well accepted in high-impact scientific journals, please see:

https://doi.org/10.1016/j.neucom.2025.129967

https://doi.org/10.7717/peerj-cs.2516

https://doi.org/10.1016/j.jfranklin.2024.01.044

 

Comment 5:  Equation (2) represents the delayed system, but no proof is provided to confirm that the system still exhibits chaotic behavior for specific parameter values and initial conditions. Please include such proof.

Response: We appreciate the reviewer's comment. However, in this work, we have chosen not to include a detailed analysis of the dynamics of the delayed Chua circuit or its validation with experimental data. Our main focus was to present, demonstrate, and validate the control strategy applied to both the chaotic Chua system proposed by Osaka [6] and the delayed Chua system.

It is important to highlight that we do not claim in the paper whether the delayed Chua system is chaotic or not, although its initial response suggests such behavior. Given the complexity of this analysis, we believe it is more appropriate to explore it in future works, where we can further investigate the system’s dynamics and its experimental validation. We include this information in the first paragraph of the introduction:

“Future work should incorporate actual physiological data, such as heart rate (HR) and blood pressure (BP), to further verify the feasibility of the model in practical applications, as well as present a detailed analysis of the dynamics of the delayed Chua circuit presented in this work.”

 

Comment 6:  In line 140, the sensitivity function and the complementary sensitivity function are defined. However, KS is not clearly explained. Please define it.

Response: The term KS refers to the product of the controller transfer function K and the sensitivity function S. Specifically, KS represents the control effort, meaning the controller's response to disturbances and noise in the system. The sensitivity function S is defined as S = (I + GK)^{-1} \), where G is the plant of the system and K is the controller. The term KS is then used to evaluate and limit the control effort within a certain frequency range. In the paper this information was inserted as follows:

“The term $KS$ represents the control effort, defined as the product of the controller transfer function $K$ and the sensitivity function $S$, reflecting the controller's response to disturbances and noise.”

 

Comment 7: Several parameters are missing from the section "Theoretical Basis". For example, please define wb, ​ and Wm​.

Response: In fact, the elements of equation (16) have not been defined. Thank you for your comment. The following has been inserted in the text:

where:

$M$: Robustness margin (or robustness parameter), which defines an upper limit for the sensitivity function, balancing performance and robustness. Typically, $M > 1$. 

$w_b$: Bandwidth crossover frequency, which determines the frequency at which the sensitivity function is approximately 1. It is related to system performance in terms of response speed. 

$E$: Defines error attenuation at low frequencies and the transition to high frequencies, adjusting the decay rate of the weight $W_p$. 

We can't find the term Wm in the text. Perhaps you referred to the term W_{IM} which was already defined in the paper as:

“The weight $ W_{IM}$ is used for modeling the uncertainty and is defined by the uncertainty model of the plant family.”

 

Comment 8: In Sections 4.2.1 and 4.2.2, the controller is designed with a value of 0.9432. However, sensitivity and complementary sensitivity Bode diagrams and system response plots are not provided, even though they are typically included in similar studies. Please include them.

Response: Bode diagrams of the sensitivity and complementary sensitivity functions were inserted, as well as Bode diagrams for analyzing stability robustness and performance robustness. In addition, conditions (17) and (18) have been given which make this analysis possible via Bode diagrams.

 

Comment 9: What is the purpose of using a single reference (sinusoidal signal)? In control systems, the equilibrium point is usually used as the reference value. Please clarify.

Response:  Although, in conventional control systems, the equilibrium point is often used as a reference, in the specific context of this work, we are dealing with cardiac responses, which are inherently dynamic and cyclical phenomena.

The choice of a sinusoidal input signal as a reference is motivated by the fact that heart rate variability, as well as various physiological dynamics associated with the cardiovascular system, exhibit natural oscillatory characteristics. Therefore, using a sinusoidal reference allows us to evaluate the controller's performance under conditions that better represent the physiology of the problem, ensuring a more realistic and relevant analysis for medical applications.

To make this clear in the text, we have inserted the following remark:

“Remark 2. In conventional control systems, the equilibrium point is often used as a reference. However, in the context of this study, where we deal with cardiac responses, a sinusoidal reference signal is more appropriate. This choice is justified by the fact that heart rate variability and various physiological dynamics of the cardiovascular system exhibit natural oscillatory behavior. By using a sinusoidal reference, we ensure that the controller's performance is evaluated under conditions that better reflect the physiological characteristics of the problem, making the analysis more realistic and relevant for medical applications.”

 

Comment 10: Robust control typically implies resistance to disturbances, noise, and uncertainties. However, no robustness tests are presented in the study. What is the superiority of the proposed controller in this regard? Please provide robustness validation.

Response: The approach used aims to solve a non-linear control problem using robust control techniques. Note that in Theorems 1 and 3 the nonlinearity “f” is replaced by a parametric uncertainty “a”, which maps the non-linear function f of Lurie's problem (also of Chua's circuit) onto the sector [k_1, k_2]. Therefore, the controller is robust to the variation of parameter “a”. As the non-linear function of Chua's circuit is defined, the result of the time response itself is a validator for the technique employed. However, as requested by the reviewer, stability robustness and performance robustness can be verified in the Bode diagrams inserted in the paper.

In order to clarify this issue, we have included Remark 1 in the paper:

Remark 1. It is important to note that the controller ensures RS and RP when varying the parameter “a”, which varies from k_1 to k_2 and replaces the nonlinearity “f”. Thus, this theorem proposes a robust control approach to solve a non-linear control problem. The same observation also applies to Theorem 3.

Round 2

Reviewer 1 Report

Comments and Suggestions for Authors

The revised version is greatly improved. This paper can be accepted after a minor revision.

Minor comment:

1) The stability and stabilization of Lurie system were also investigated in [DOI: 10.1016/j.amc.2025.129455]. It is suggested to address it and give a comparison with this approach. 

2) Some references have inconsistent formatting. 

Author Response

Comment 1: The stability and stabilization of Lurie system were also investigated in [DOI: 10.1016/j.amc.2025.129455]. It is suggested to address it and give a comparison with this approach. 

Response 1. Thank you very much for the suggestion. We have included a paragraph in the discussion section with comparisons between the two studies, as well as suggestions for future investigations.

The text included is as follows:

“This work focuses on the continuous-time case with delay; however, in the discrete domain, future investigations could follow the approach proposed by \citep{wang2025}, who address the stabilization of Lurie systems using sampled-data control within a hybrid continuous-discrete structure. While the present study applies $\mathcal{H}_\infty$ control to regulate the dynamics of the Chua circuit for biomedical applications, \citep{wang2025} propose an iterative method (CCLI) to compute discrete controller gains, taking into account delays and data loss. This comparison suggests that discrete-time methods could be adapted to control delayed physiological circuits, especially in implantable devices operating with sampled signals. As future directions, we propose: (i) exploring the application of sampled-data control to biomedical circuits with delays, and (ii) developing hybrid control strategies that combine robust continuous-time techniques with discrete-time structures to achieve higher efficiency in real physiological systems.”

Comment 2: Some references have inconsistent formatting. 

Response 2: The references have been revised and placed according to the MDPI template.

 

Reviewer 3 Report

Comments and Suggestions for Authors

My initial decision was based on the perspective of chaotic systems. There is a significant improvement in the revised manuscript. The authors have provided detailed responses to my concerns and comments. The study is acceptable in its current form.

I have only one final suggestion: highlighting the revised sections in yellow would make it easier to track the changes.

Author Response

Comment: My initial decision was based on the perspective of chaotic systems. There is a significant improvement in the revised manuscript. The authors have provided detailed responses to my concerns and comments. The study is acceptable in its current form.

I have only one final suggestion: highlighting the revised sections in yellow would make it easier to track the changes.

Response: Dear Reviewer, we sincerely appreciate your thoughtful comments and careful review of the manuscript. We are very pleased to know that the revisions made have been satisfactory and that the study is now acceptable in its current form.

We are grateful for your suggestion to highlight the revised sections. In this last round, we made sure to highlight all the changes, and we will certainly continue to follow this practice in future submissions to make it easier to track modifications.

Once again, thank you for the time and valuable feedback, which has greatly contributed to improving the quality of our work.