Review Reports - The Synchronization of a Class of Time-Delayed Chaotic Systems Using Sliding Mode Control Based on a Fractional-Order Nonlinear PID Sliding Surface and Its Application in Secure Communication

Round 1

Reviewer 1 Report

There are a few issues and concerns that need to be addressed in a major revision.

1. The paper title is too long.

2. Literature review needs to be revised in a way that has a clear logic thread. Rather than piling up references from difference research groups, the underlying ideas and the evolvement should be explained in-depth.

3. As per Assumption 1 and Assumption 2, can you explain in details in your case how to determine the boundaries of the signals in practice? They will influence the conservativeness of control but can you explain in what extent?

4. The resolutions of the results figures especially Figs. 2-7 and 9-10 are poor. The legends therein are unreadable.

5. Recent advancement in secure communication should be analyzed and compared, such as in article Secure data transmission and trustworthiness judgement approaches against cyber-physical attacks in an integrated data-driven framework.

Author Response

Comments of Reviewer #1

-----------------------------------------------------------------------------------------------------------------------------------------

The authors would like to cordially appreciate the respectable reviewer for his/her deep concentrations and appropriate comments. In the following, we separately answer to each comment of the reviewer.

-----------------------------------------------------------------------------------------------------------------------------------------

Comment 1: The paper title is too long.

Answer: Thanks to the respected reviewer, the title of the paper was changed as follows:

Synchronization a class of Time-Delayed Chaotic Systems using Sliding Mode Control based on Fractional-Order Nonlinear PID Sliding surface and Its Application in Secure Communication

-----------------------------------------------------------------------------------------------------------------------------------------

Comment 2: Literature review needs to be revised in a way that has a clear logic thread. Rather than piling up references from difference research groups, the underlying ideas and the evolvement should be explained in-depth.

Answer: The extra parts of the article were removed and the following explanations were added to the end of the introduction section:

The following innovations have been presented in the study that was carried out in order to synchronize two uncertain chaotic systems:

1- Using the NLFOPID sliding surface instead of conventional sliding surfaces.

2- Existence of unknown time delays

3- Being unknown of the Limits of uncertainty and disturbance.

Accordingly, based on the above concepts, using the appropriate Lyapunov function and update rules, a control mechanism has been proposed, which can overcome the problem of unknown time delay, uncertain uncertainty, and uncertain disturbances by properly adjusting the controller parameters.

-----------------------------------------------------------------------------------------------------------------------------------------

Comment 3: As per Assumption 1 and Assumption 2, can you explain in details in your case how to determine the boundaries of the signals in practice? They will influence the conservativeness of control but can you explain in what extent?

Answer: In this regard, the following matters are brought to the notice of the honorable reviewer:

Explanations of assumption 1:

Generally, bounded disturbance and uncertainty are assumed because it is a common method in control problems. Uncertainty is usually defined as , which we have defined in this article as a positive function , which is more general.
In such problems, the limits of uncertainty and disturbance are generally assumed to be known, which are unknown in the mechanism proposed in this article, and their limits are estimated according to the performance of the propound control system.

Explanations of assumption 2:

a) In order to extract the delay from the functions and , it is necessary to assume that the functions and have Lipchitz conditions.
b) We also note that in chaotic systems, the state variables have a limited range and are smooth. Therefore, they have a bounded derivative. So the state vector is also a Lipsheets function.

-----------------------------------------------------------------------------------------------------------------------------------------

Comment 4: The resolutions of the results figures especially Figs. 2-7 and 9-10 are poor. The legends therein are unreadable.

Answer: All the figures of the article were revised and their quality increased.

-----------------------------------------------------------------------------------------------------------------------------------------

Comment 5: Recent advancement in secure communication should be analyzed and compared, such as in article secure data transmission and trustworthiness judgment approaches against cyber-physical attacks in an integrated data-driven framework.

Answer: Also, thanks for the careful opinion of the honorable reviewer. It is recalled that the main problem in this article is to propose a control mechanism for synchronizing the uncertain chaotic systems with disturbance and unknown time delays.

In this paper, the subject of secure communication is a sub-problem, the purpose of which is to present the application of the proposed mechanism in this subject. Secondly, the evaluation indicators of secure communication, such as entropy, etc., are generally for images. In this article, the message is in the form of a signal, which is generally evaluated as RMS, which is done in the article.

-----------------------------------------------------------------------------------------------------------------------------------------

We thank the reviewer very much for his/her valuable comments and suggestions that help us greatly improve this paper.

-----------------------------------------------------------------------------------------------------------------------------------------

Author Response File: Author Response.docx

Reviewer 2 Report

please see the attachment.

Comments for author File: Comments.pdf

Author Response

Comments of Reviewer #2

--------------------------------------------------------------------------------------------------------------------------

-----------------------------------------------------------------------------------------------------------------------------------------

This paper dealt with the issues of synchronization and secure communication for a class of uncertain chaotic systems. A fractional-order controller was developed by adaptive sliding mode control. But, this paper is not well written and has many problems. Some comments are given as follows.

Comment 1: The title of this paper is too tedious and needs to be condensed.

Answer: Thanks to the respected reviewer, the title of the paper was changed as follows:

Synchronization a class of Time-Delayed Chaotic Systems using Sliding Mode Control based on Fractional-Order Nonlinear PID Sliding surface and Its Application in Secure Communication

-----------------------------------------------------------------------------------------------------------------------------------------

Comment 2: Are the time delays constants? How do you understand the first equation of (21)?

Answer: Due to the constancy of the time delay limit, the time delay of the system changes step by step with time, which is described in section 6 of the article. also:

which is applied in the article.

-----------------------------------------------------------------------------------------------------------------------------------------

Comment 3: For the controller (20), the estimates of the disturbances, time delays and uncertainty are obtained from (21). Please explain how you can get such signals from (21).

Answer: Thanks for your helpful comment for improving the quality of our article. The relation (21) was reviewed and corrected as follows:

where:

------------------------------------------------------------------------------------------------------------------------------------------

Comment 4: In (21), the parameters b1 and b2 should be defined firstly.

Answer: With respect to the opinion of the respected reviewer, the item mentioned in the article was corrected. b1 and b2 are positive and unknown numbers that determine the uncertainty limits.

------------------------------------------------------------------------------------------------------------------------------------------

Comment 5: In simulations, the transition time is t = s 4.5 . How do you determine it?

Answer: By applying the control signal at the moment t=0 and observing and checking the simulations, it is clear that at the moment t=4.5s, a transient time has passed and the system has reached its steady state (this time may be used for simulation with the conditions the other primary is changed).

Therefore, the time t=4.5s is chosen to send the message signal. In future research, which will be focused on "finite-time synchronization", it can be guaranteed that the synchronization will be done completely at a certain time.

------------------------------------------------------------------------------------------------------------------------------------------

Comment 6: In Page 14, the title and narration of Table 1 are confused.

Answer: While thanking the esteemed reviewer for her accuracy, the problem raised was resolved and highlighted for convenience.

------------------------------------------------------------------------------------------------------------------------------------------

Comment 7: The format of the references is not uniform.

Answer: Thanks to the honorable reviewer, the references were completely reviewed and corrected.

------------------------------------------------------------------------------------------------------------------------------------------

Comment 8: Some sentences are incorrect. For example,

- the first sentence of Abstract “…, a novel control approach is presented for robust synchronization a class of chaotic systems with …”

- below (1), the sentence “In which, is the fractional order operator with a complex value.”

Answer: While thanking the reviewer, all items were corrected.

------------------------------------------------------------------------------------------------------------------------------------------

Comment 9: Some expressions do not conform to the usual statement, even easy to cause confusion. For example

- The function f ( x.t ) should be f ( x, t ) , and f ( x ( t -τ1 ).t) should be f ( x ( t -τ1 ),t). There are many such problems.

- Below (2), the initial time t0 should be t0.

- Many formulas don't have commas or periods after them.

- Following many formulas, the word “Where” should be “where”.

Answer: While thanking the respected reviewer, the mentioned items were reviewed and corrected.

--------------------------------------------------------------------------------------------------------------------------

Author Response File: Author Response.docx

Reviewer 3 Report

1. Chaotic systems have delays, so they may not behave chaotically. How is their chaotic behaviour guaranteed?

2. What is the difference and superiority of the FONPID sliding surface with the PID sliding surface?

3. How is equation 12 obtained from Lipschitz functions f and g. Do state vectors play a role in that?

4. In Figure 5, the amplitude of the control signal is high at the initial time. What is your solution to reduce it?

5. Parameters ϵ_1 and ϵ_2 are fixed in the proposed method. Can these values change with time? In this case, does the proposed method have the ability to deal with it? How?

6. There is a chattering control function. What measures do you take to reduce it?

Author Response

Comments of Reviewer #3

--------------------------------------------------------------------------------------------------------------------------

-----------------------------------------------------------------------------------------------------------------------------------------

Comment 1: Chaotic systems have delays, so they may not behave chaotically. How is their chaotic behavior guaranteed?

Answer: There are different methods to determine the chaotic nature of the system, including the Lyapunov diagram and the phase diagram of the system, etc. In this article, the analysis and interpretation of the phase diagram is chosen to determine the chaotic nature of the system.

------------------------------------------------------------------------------------------------------------------------------------------

Comment 2: What is the difference and superiority of the FONPID sliding surface with the PID sliding surface?

Answer: First, if , the sliding surface will change from NLFOPID to FOPID. Therefore, NLFOPID is more general and has a higher degree of freedom than FOPID.

Secondly, it has been proven in [55] that using NLFOPID will have better control performance in chaotic systems.

------------------------------------------------------------------------------------------------------------------------------------------

Comment 3: How is equation 12 obtained from Lipschitz functions f and g. Do state vectors play a role in that?

Answer: Thanks to the honorable reviewer, state variables of chaotic systems have limited range and soft changes (bounded derivative). It means that the state vector is also a Lipschitz function. It was also corrected in the article.

Similar to the above relations, it is also true for the function .

------------------------------------------------------------------------------------------------------------------------------------------

Comment 4: In Figure 5, the amplitude of the control signal is high at the initial time. What is your solution to reduce it?

Answer: In order to reduce the amplitude of the signal, the saturation operator can be used on the control signal. Of course, the error convergence speed to zero will decrease.

------------------------------------------------------------------------------------------------------------------------------------------

Comment 5: Parameters ϵ_1 and ϵ_2 are fixed in the proposed method. Can these values change with time? In this case, does the proposed method have the ability to deal with it? How?

Answer: If , then can be considered as part of the uncertainty and summed with the total uncertainty.Therefore, relation (32) is rewritten as follows:

Where : which has the same structure as before. It is done in the same way for the slave system.

------------------------------------------------------------------------------------------------------------------------------------------

Comment 6: There is a chattering control function. What measures do you take to reduce it?

Answer: The accuracy of the honorable reviewer's opinion is appreciated and he informs that the chattering is caused by the sign function, which can be replaced by another function such as . In addition, by adjusting the control parameters, the chattering problem can be reduced to an acceptable level.

------------------------------------------------------------------------------------------------------------------------------------------

We thank the reviewer very much for his/her valuable comments and suggestions that help us greatly improve this paper.

------------------------------------------------------------------------------------------------------------------------------------------

All changes and corrections are highlighted in the revised article.

Best regard

Mohammad Rasouli, Assef Zare *, Majid Halaji, Roohallah Alizadehsani
8 December 2022