A Finite-Time Control Design for the Discrete-Time Chaotic Logistic Equations

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

In this paper, a new finite-time controller for the chaotic logistic equation is designed by using analog and discrete-time algorithms. The following comments are given to further improve the quality of the paper (Some minor issues can be found in the attached file):

1. The contribution of this paper should be introduced with more details in the Abstract.

2. In Section 3, more details about the chaotic logistic equation should be given, for example, what is the definition of chaotic logistic equation?

3. In line 93, the authors briefly introduce the PWM technique, it is suggested to give a reference.

4. What is the advantage of using finite-time control for the chaotic logistic equation?

 

5. More details should be given in Conclusion.

Comments for author File: Comments.pdf

Comments on the Quality of English Language

Author Response

Thank you very much for your assistance in the review of our article. Below are our replies to each of your feedback inputs.

1) Input: “The contribution of this paper should be introduced with more details in the Abstract.”

Reply: We have expanded the Summary section by adding green text.

 

2) Input: “In Section 3, more details about the chaotic logistic equation should be given, for example, what is the definition of chaotic logistic equation?”

Reply: To complete the main attributes of a chaotic system and why we are using it, we have added the following text to our new paper version. Evidently, new references were also added. See the text in green:

“Furthermore, the control of chaotic systems has been a topic of research due to its possible applications, such as the synchronization of chaotic systems, the control of nonlinear systems in chaotic behavior, etc.  [10-12]. This is because chaotic systems are nonlinear deterministic systems that exhibit complex and unpredictable behavior .”   

 

3) Input: “In line 93, the authors briefly introduce the PWM technique, it is suggested to give a reference.”

Reply: We have typed an overview of the PWM modulation and added a new figure, the number 3. See the text in the green of our new paper version in the paragraph previous to equation (10).

 

4) Input: “What is the advantage of using finite-time control for the chaotic logistic equation?”

Reply: Finite-time control belongs to the field of robust controllers. This is the main intention of using it. However, we believe that this input is also related to the answer given in point 2).

 

5) Input: “More details should be given in Conclusion.”

Reply: We have added the following paragraph that we consider important for the field of chaotic systems theory:

“Finally, another important point to conclude is that when observing the behavior of the unstable system, it seems to present better random behavior than the uncontrolled case.”

In the Discussion section, we also improved it. Please see the text in green color.

Thank you very much for your inputs.

Reviewer 2 Report

Comments and Suggestions for Authors

This paper is devoted to the development of a finite-time experimental platform for a chaotic logistic equation using analog and discrete algorithms. The paper contains interesting research, but the paper needs to be rewritten.

1) Why was the parameter r = 3.6 chosen in the logistic map formula? If you look at the bifurcation diagram of the logistic map, you can see that the values ​​for r = 3.6 will be approximately in the range of (0.3:0.6) & (0.75:0.9), that do not cover the entire range from 0 to 1, as would be the case, for example, with r = 4.

2) Please use "forward Euler method" instead of "first Euler method".

3) formula (4). First, why open the parentheses when writing a finite-difference scheme? It's just confusing. Where did the value 2.6 come from? Looks wrong. It is better to use the general view of the logistics map and write in the formula not the selected value 3.6, but the variable (r).

4) formula (10)-(11). z_k shouldn't have been in the formula.

5)  "In programming, we use 258" I think this is precisely because you chose the wrong value for the r parameter, which does not give you a distribution from 0 to 1. You choose a larger parameter to level out this difference. 

6) The results are not described at all. It is necessary to add an explanation for each picture and draw conclusions. Expand the conclusions, add numerical results, scientific novelty, scope, and directions for further research.

 

 

minor issues:

1) Line 90:

"linearized system ( 9)-( 7) is " replace on "linearized system (7)-(9) is "

"From the above Corollary" replace on "From the above Corollary 1"

2) In Figure 3 there are 2 black curved arrows drawn by hand, redraw them.

3) Line 112. (u = u + 1) formulas look like exponentiation, describe the explanation in the text, without references.

Author Response

Thank you very much for your assistance in the review of our article. Below are our replies to each of your feedback inputs.

1) Input. “Why was the parameter r = 3.6 chosen in the logistic map formula? If you look at the bifurcation diagram of the logistic map, you can see that the values ​​for r = 3.6 will be approximately in the range of (0.3:0.6) & (0.75:0.9), that do not cover the entire range from 0 to 1, as would be the case, for example, with r = 4.”

Reply: The chaotic logistic map has been extensively studied. For instance, in the link https://en.wikipedia.org/wiki/Logistic_map#cite_note-May,_Robert_M_1976-2, it is said that the logistic map presents chaos if the parameter r is between 3.57 and 4.  However, because we have a programmable experimental platform, other values of this parameter ​​can be used. This is left for future work on the synchronization of two chaotic logistic systems.

 

2) Input. “Please use "forward Euler method" instead of "first Euler method"”

Reply: Done. Thank you.

 

3) Input. “formula (4). First, why open the parentheses when writing a finite-difference scheme? It's just confusing. Where did the value 2.6 come from? Looks wrong. It is better to use the general view of the logistics map and write in the formula not the selected value 3.6, but the variable (r).”

Reply:  Equation (4) has been verified. On the other hand, we have slightly modified the logistic map presentation using the parameter r, and then using its corresponding value. We believe that in this way the mathematical manipulations are more evident.

 

4) Input. “formula (10)-(11). z_k shouldn't have been in the formula.”

Reply: We are sorry for this simple-to-spot typographical error. It has already been corrected and verified. Thank you so much. Please, forgive us.

 

5) Input. “"In programming, we use 258" I think this is precisely because you chose the wrong value for the r parameter, which does not give you a distribution from 0 to 1. You choose a larger parameter to level out this difference. ”

Reply: We are sorry for the misunderstanding. This value sets the base period of the PWM signal. We have clarified this in the paper too. See the label of Figure 6.

 

6) Input. “The results are not described at all. It is necessary to add an explanation for each picture and draw conclusions. Expand the conclusions, add numerical results, scientific novelty, scope, and directions for further research.”

Reply: We have expanded the discussion section by adding numerical data on the settling-time convergence of the stable closed-loop systems and some comments for the unstable case too. Especially, we also add, among others,  the following text:

“On the other hand, realizing a chaotic system or enhancing it is a control design named “anti-control” or “chaotification” of dynamical systems [x5]. It is an important topic in secure communication based on chaotic signals [x5]. If we observe the unstable closed-loop system, see Figures 8 and 10, it seems that the unstable control design, our controller improves the chaoticity of the system.”

[x5] Chen, Hsien-Keng, and Ching-I. Lee. "Anti-control of chaos in rigid body motion." Chaos, Solitons and Fractals 21.4 (2004): 957-965.

 

7) Input: “minor issues:

1) Line 90:

Input: "linearized system ( 9)-( 7) is " replace on "linearized system (7)-(9) is "

Reply: Done, thank you.

 

Input: "From the above Corollary" replace on "From the above Corollary 1"

Reply: Done, thanks.

 

2) Input: “In Figure 3 there are 2 black curved arrows drawn by hand, redraw them.”

Reply: Done, thank you.

 

3) Line 112. (u = u + 1) formulas look like exponentiation, describe the explanation in the text, without references.”

Reply: This is due to the line programming of the Pic-microcontroller. To clarify, we have added the next text to our new paper version:

“Related to the programming setting u=u+1 or u=u-1, in mathematical format, these are equivalent to u(k+1)=u(k)+1 and u(k+1)=u(k)-1, respectively.”

Thank you for your inputs.

Reviewer 3 Report

Comments and Suggestions for Authors

This paper deals with a finite-time control design for the discrete-time chaotic logistic equation. I would like to point out following.

1.       The description of object is ambiguous even though authors mentioned about the goal of this paper in introduction. What is novelty? It is not clear.

2.       The description of previous work is weak. please add their advantage and drawback using table as possible as.

3.       What is control object in logistics map? It is not clear.

4.       What is input in the system and also what is control result applied input?

5.       Can you show the result by figure for before and after control?

6.       Please rewrite this paper to easily understand by reader.

Comments on the Quality of English Language

Please check again English grammar.

Author Response

Thank you very much for your assistance in the review of our article. Below are our replies to each of your feedback inputs.

1) Input. “The description of object is ambiguous even though authors mentioned about the goal of this paper in introduction. What is novelty? It is not clear.”

Reply: We have improved the paper presentation. Please see the new paper version along with the green text.

 

2) Input. “ The description of previous work is weak. please add their advantage and drawback using table as possible as”

Reply: To the best of the authors' knowledge, controlling the logistics map by using analog electronics and nonlinear control design theory appears to be recent. We have commented on this issue in the Conclusions section.

 

3) Input. “What is control object in logistics map? It is not clear.”

Reply: The control of chaotic systems has been a topic of research due to its possible applications, such as the synchronization of chaotic systems, the control of nonlinear systems in chaotic behavior, etc.  [x1-x3]. This is because chaotic systems are nonlinear deterministic systems that exhibit complex and unpredictable behavior.   

Additional references

[x1] Hua, Changchun, and Xinping Guan. "Adaptive control for chaotic systems." Chaos, Solitons & Fractals 22.1 (2004): 55-60.

[x2] Yassen, M. T. "Controlling chaos and synchronization for new chaotic system using linear feedback control." Chaos, Solitons & Fractals 26.3 (2005): 913-920.

[x3] Park, Ju H. "Controlling chaotic systems via nonlinear feedback control." Chaos, Solitons & Fractals 23.3 (2005): 1049-1054

We have commented on these comments on the new paper version. Please see the text in green.

 

4) Input. “What is input in the system and also what is control result applied input?”

Reply: We have improved the presentation of Figure 4 to highlight the closed-loop trajectory. The experimental results validate the controllability of the closed-loop system having a settling time of about 2.5 seconds. Also, the unstable case is an important topic related to the anti-control of dynamical systems. This due to realizing a chaotic system or enhancing is a control design named “anti-control” or “chaotification” of dynamical systems [x5] is an important topic in secure communication based on chaotic signals [x5]. If we observe the unstable closed-loop system, see Figure x-y, it seems that the unstable control design improves the chaoticity of the system.

[x5] Chen, Hsien-Keng, and Ching-I. Lee. "Anti-control of chaos in rigid body motion." Chaos, Solitons & Fractals 21.4 (2004): 957-965.

We have commented on these topics in the new paper version. Please see the text in green.

 

5) Input. “Can you show the result by figure for before and after control?”

Reply: Traditionally, in the controlled system, the performance of the control action is also evident when you show the system response for the uncontrolled stage and then the controlled phase. We follow this format, especially when you have a system that is so sensitive to the initial condition, as are the chaotic systems.

 

6) Input. “ Please rewrite this paper to easily understand by reader.”

Reply:  We realized it and expanded it according to the other reviewers too.

 

Thank you for your input. 

Round 2

Reviewer 1 Report

Comments and Suggestions for Authors

In the response letter, only some of the questions have been answered, however, some minor issues I put in the attached files are not mentioned, please address them before publication.

Comments for author File: Comments.pdf

Comments on the Quality of English Language

Good

Author Response

Reply letter:

“In the response letter, only some of the questions have been answered, however, some minor issues I put in the attached files are not mentioned, please address them before publication.”

Answer:  We regret having skipped these important comments. Next are the corresponding replies.

1. “There are some grammatical or typing errors, the authors should check the paper carefully. For example, in line 86, it should be “sgn” but not “sng” .“

Reply: We are sorry about the typo mistake. These were corrected. See the blue text in the new document version.

2. “ What is the “sw” in the top of Figure 4?”

Reply: “SW” is the reset switch of the microcontroller. We have added this comment to the figure’s label.

3. “The literature review is not enough in the Introduction Section and there are only few references. It seems that the controller designed in this paper is a sliding mode control with finite-time stability, or finite-time sliding mode controller. Therefore, the authors should introduce more latest published articles that focus on the finitetime control or sliding mode control. (In this case, I would like to suggest some recent publications that focus on this area, such as Sliding Mode Control of NeutralPoint-Clamped Power Converters with Gain Adaptation, Adaptive-Gain SecondOrder Sliding Mode Control of NPC Converters Via Super-Twisting Technique; Cascade Control of Grid-Connected NPC Converters via Sliding Mode Technique).”

Reply: Thank you for the input. We have added the following paragraph and updated the reference list (see section five in blue text in the new paper version):

Finite-time convergence of controlled dynamical systems has been studied for several years. Nowadays, some contributions have been made. For instance, by using the super-twisting algorithm [f1] in combination with a proportional-integral controller along with an exponent gain observer, a new sliding mode control design for neutral-point-clamped power converters was proposed in [f2]. In [f3] the authors designed a bilateral continuous finite-time adaptive terminal sliding mode controller for a teleoperation system. In [f4] a new finite-time sliding mode control algorithm was reported. One way or the other, in all the cited references, the signum function is presented in their mathematical development to ensure finite-time convergence and robustness of the closed-loop systems. We have also used it.

[f1] Zargham, F., and A. H. Mazinan. "Super-twisting sliding mode control approach with its application to wind turbine systems." Energy Systems 10.1 (2019): 211-229.

[f2] Shen, Xiaoning, et al. "Sliding Mode Control of Neutral-Point-Clamped Power Converters with Gain Adaptation." IEEE Transactions on Power Electronics (2024).

[f3] Wang, Jingwen, et al. "Control of time delay force feedback teleoperation system with finite time convergence." Frontiers in Neurorobotics 16 (2022): 877069.

[f4] Mishra, Jyoti P., Xinghuo Yu, and Mahdi Jalili. "Arbitrary-order continuous finite-time sliding mode controller for fixed-time convergence." IEEE Transactions on Circuits and Systems II: Express Briefs 65.12 (2018): 1988-1992.

4. “The title of Figure 4 should be modified, the authors should give a name of this experimental platform but not using “our experimental platform”. “

Reply: Thank you for this comment. We have used “experimental platform” instead of “our experimental platform”.

Thank you for your input.

 

Reviewer 2 Report

Comments and Suggestions for Authors

The paper has been reworked, but I have some minor comments: 
- The chaotic regime in the logistic map does start at 3.6, but the difference in this parameter is in the range of values ​​that the map produces. This is clearly visible in the bifurcation diagram. It would be nice to include it in the paper. From Figure 1, you can see that the discrete solution of the logistic system with parameter 3.6 is inside the interval between 0.3 and 0.9, not between 0 and 1. Thus, some accuracy is lost.

- Update your bibliography, almost all references are quite old. The study of chaos theory is developing very actively, you can find many new studies.

I believe that after some minor corrections, the article could become an excellent contribution to the MDPI

 

Author Response

Reply letter:

The paper has been reworked, but I have some minor comments:
1- Input: “The chaotic regime in the logistic map does start at 3.6, but the difference in this parameter is in the range of values ​​that the map produces. This is clearly visible in the bifurcation diagram. It would be nice to include it in the paper. From Figure 1, you can see that the discrete solution of the logistic system with parameter 3.6 is inside the interval between 0.3 and 0.9, not between 0 and 1. Thus, some accuracy is lost.”

Reply: Yes, but r=3.6 is inside of an interval where the logistic map presents chaos at r=3.57 (https://www.stsci.edu/~lbradley/seminar/logdiffeqn.html).  In the experimental platform, there are some losses of “accuracy” too. One is due to the microcontroller using the 8-bit architecture, and the other is the electronic stages. For instance, the electronic block carrying the digital-to-analog conversion absorbs some energy. See Figure 4. The resistor of 3.3 kilo-ohms has some voltage drop between its terminals. These are some items of loss of accuracy, all these are a justification of the system’s response inside of the expected interval. However observing our experimental results, parameter robustness is evidenced. Moreover, in the unstable case, chaotic enhancement is highlighted as notable.

2. Input: “Update your bibliography, almost all references are quite old. The study of chaos theory is developing very actively, you can find many new studies.”

Reply: Thank you. We have added the following paragraphs and updated the reference list (see section five in blue text in the new paper version):

Finite-time convergence of controlled dynamical systems has been studied for several years. Nowadays, some contributions have been made. For instance, by using the super-twisting algorithm [f1] in combination with a proportional-integral controller along with an exponent gain observer, a new sliding mode control design for neutral-point-clamped power converters was proposed in [f2]. In [f3] the authors designed a bilateral continuous finite-time adaptive terminal sliding mode controller for a teleoperation system. In [f4] a new finite-time sliding mode control algorithm was reported. One way or the other, in all the cited references, the signum function is presented in their mathematical development to ensure finite-time convergence and robustness of the closed-loop systems. We have also used it.

[f1] Zargham, F., and A. H. Mazinan. "Super-twisting sliding mode control approach with its application to wind turbine systems." Energy Systems 10.1 (2019): 211-229.

[f2] Shen, Xiaoning, et al. "Sliding Mode Control of Neutral-Point-Clamped Power Converters with Gain Adaptation." IEEE Transactions on Power Electronics (2024).

[f3] Wang, Jingwen, et al. "Control of time delay force feedback teleoperation system with finite time convergence." Frontiers in Neurorobotics 16 (2022): 877069.

[f4] Mishra, Jyoti P., Xinghuo Yu, and Mahdi Jalili. "Arbitrary-order continuous finite-time sliding mode controller for fixed-time convergence." IEEE Transactions on Circuits and Systems II: Express Briefs 65.12 (2018): 1988-1992.

3. Input: “I believe that after some minor corrections, the article could become an excellent contribution to the MDPI”

Reply: Thank you so much.

Reviewer 3 Report

Comments and Suggestions for Authors

I think this paper is well revised according to reviewer's point out. Thus, I would like to decide as an "accept"

Comments on the Quality of English Language

Please check again English grammar.

Author Response

Reply letter

1. Input: “I think this paper is well revised according to reviewer's point out. Thus, I would like to decide as an "accept".”

Reply: Thank you so much.