Fractional Sliding Mode Nonlinear Procedure for Robust Control of an Eutrophying Microalgae Photobioreactor

Round 1

Reviewer 1 Report

There are few innovations in this article. However, there is no contribution in theory. The research background is novel and meaningful, but only simulations are given. It is suggested to carry out practical experiments. The convergence of the method is not proved. ${d_{\max }}$ in remark 1 should be ${\delta _{\max }}$. Formula expression should be rigorous. There are various errors improper expressions in formula (2), (10), (18). The definition of Caputo fractional order derivative and integral is wrong and has no formula number. If Grunwald-Letnikov derivative is used, it is better to give its definition. As for the influence of the value of $\beta $ on the convergence rate, if $\beta = 0.9$ is a certain critical value, then it should be proved. If it is the critical value shown in the simulations, it requires more results near 0.9. Simulation in 5.1 should give the values of $\beta $ and other parameters.

Author Response

Please see the pdf, in this is the letter of response to each reviewer and the PDF of the new version of the paper, as well as the statement to each question, point, etc. of each reviewer.

Author Response File: Author Response.pdf

Reviewer 2 Report

The paper is very well-written and it presents an interesting application of control design to a biological problem. The presentation is well-organized and easy to follow for readers. A fractional-type controller is designed which guarantees convergence to the sliding surface under a class of modelling disturbances. The results are compared via simulations to the design of an internal model control. I highly recommend the acceptance of this paper.

Author Response

Please see the pdf, in this is the letter of response to each reviewer and the PDF of the new version of the paper, as well as the statement to each question, point, etc. of each reviewer.

Author Response File: Author Response.pdf

Reviewer 3 Report

The fractional order strategy  proposed on line 118 is not correct. First, what is  the value of \beta? Second,  (13) does not follow from (12).

Additionally, there are many unclear parts in the paper. For example:

-The inequalities on line 3 page 4 to be explained, more exactly the inequality after the arrow ->

- Why is the inequality of line 83 satisfied?

- Figure 1 is graphed for particular values of the three functions. Which are they?Why for example \mu(.) at 0 is 20? Which software did you use to graph Figure 1?

- nothing is written about the values of beta, but at the same time. in the simulations only values on (0,1) are taken. Why?

- what are the bounds in the integral in (9)?

Author Response

Please see the pdf, in this is the letter of response to each reviewer and the PDF of the new version of the paper, as well as the statement to each question, point, etc. of each reviewer.

Author Response File: Author Response.pdf

Round 2

Reviewer 1 Report

There are no other comments.

Author Response

Thanks you

Reviewer 3 Report

The revised version of the paper is still not acceptable for publication The authors need to be very careful and to read the written many times before submitting.

-        The integral on line 106 is not correct. Something is missing.

In (12)  beta is using but nothing is written about its value. Also, the power of s means  the derivative, I guess fractional derivative. First, on line 106 the fractional derivative is defined and another notation is used. Second, but not the last,  the notation D^{-\beta) is not defined. On line 120 it is written “ the fractional derivative of order (-beta) in (12) is equivalent to  its fractional integral”, but the ere is no (-beta) in (12). There is beta. Again, I’ll ask the authors to prove that (13) follow from (12). Also, if the power means a fractional derivative, the what does power (-1) means in (15). I don’t accept the answer “The fact that the fractional derivative of order (?b) of s1+b is given by equation (15) has been proven in some previous papers published in the literature,” This answer is not correct answer of my question. Oppositely.

-About the answer of my second comment- it is written  “The inequality mentioned is a reasonable assumption, reflecting experimental” If it is an assumprtion, then it has to be written as an assumption. Now it seems the inequality on line 83 follow from the previously written.

- About the answer of my third comments- it is written” constant values a1, a2, a3 are arbitrary.” Again it is not a correct answer. When a software is used, the constants have to have particular values. But the authors did not gave them.

- About the answer of my last comment- I don’t any bounds of the integral in (9) written in the paper.

-        Many part in the paper are written mathematically incorrectly. See for example (16). The integral has no bounds. The same is about (20)

As an overall,  this paper is written very badly on mathematical point of view. It is full with unclear parts. Also, it has no novelty on application point of view neither on computer simulation point of view.

 

Definitely, paper is not acceptable for publication. 

Author Response

Dear Reviewer

In the attached PDF you will find answers to all your points, we thank you very much in advance for your help.

Greetings

Author Response File: Author Response.pdf