A Modified ADRC Scheme Based on Model Information for Maglev Train

Comments 1: In line 15, it is suggested that the “ controller” is revised as “control”.

Response 1: Thank you for pointing this out. I agree with this comment. Therefore, I have updated text in the manuscript, this change can be found in line 15.

Comments 2: Lines 54 to 67 provide a summary of the approach for addressing non-minimum phase systems, but how to take the model information into account? What is "model information"? 

Response 2: Thank you for pointing this out. I agree with this comment. Therefore, I have revised the text in the manuscript to explain the meaning of "model information", this change can be found from lines 56 to 58.

Comments 3: The abbreviation of ADRC has already been explained in the previous text, so there is no need to explain the abbreviation of ADRC in line 68. 

Response 3: Thank you for pointing this out. I agree with this comment. Therefore, I have revised the text in the manuscript, this change can be found in line 69.

Comments 4: It is recommended to replace "can be obtained" with "can be obtained as" in line 144.

Response 4: Thank you for pointing this out. I agree with this comment. Therefore, I have revised the text in the manuscript, this change can be found in line 147.

Comments 5: It is recommended to remove "Formula (4)." in line 148. 

Response 5: Thank you for pointing this out. I agree with this comment. However, I have removed Formula (3), so Formula (4) has changed into Formula (3).

Comments 6: The word "Where" at the bottom of the formula should be changed to "where," without indentation. 

Response 6: Thank you for pointing this out. I agree with this comment. Therefore, I have revised the text in the manuscript, this change can be found in line 130,149,173,185,191,224.

Comments 7: In Figure 7, the maximum value of the vertical coordinate of some pictures is not marked. In addition, the expression of the disturbance in Figure 7 is not clear. What kind of disturbance is it? What is the physical quantity of the disturbance? Should disturbance have units? 

Response 7: Thank you for pointing this out. I agree with this comment. Therefore, I have revised the figure in the manuscript, this change can be found in Figure 7. However, the disturbance consists of model bias and external disturbances, so there are no units for the disturbance.

Comments 8: It is recommended to reverse the direction of the arrows in Figure 10. 

Response 8: Thank you for pointing this out. I agree with this comment. Therefore, I have revised the figure in the manuscript, this change can be found in Figure 10.

1. Point-by-point response to Comments and Suggestions for Authors

Response 1: Thank you for pointing this out. I agree with this comment. Therefore, I have emphasized that non-minimum phase is referred to unstable systems instead of unstable zero dynamics in the manuscript, this change can be found in lines 47-51.

Comments 2: The linearized version must be expressed in terms of \Delta z = z-z_0 and \Delta i = i-i_0. Also, the third and second rows from equation (2) are not equivalent [where is term F_d?]. In (2) the F_d term is absent, while in (4) this term appears again. You could present the system as a multiple-input and single-output and use a transfer matrix in (3).

Response 2: Thank you for pointing this out. I agree with this comment. Therefore, I have revised the formula (2) in the manuscript, according to formula (2), the state space equation with disturbance F_d can be obtained, this change can be found in formula (2).

Comments 3: Please explain how LESO and MESO from (6) and (9) are obtained → the paper is not self-contained.

Response 3: Thank you for pointing this out. I agree with this comment. In general, there is a standard form for LESO, and a reference is added to explain it, this change can be found in lines 167-169..And I have revised the text to explain what improvements has MESO made based on LESO, this change can be found in lines 181-182.

Comments 4: Definition from (12) is not rigorous: k is not properly defined when e_1=0 (you should extend the definition of k for e_1=0 using the continuity property.

Response 4: Thank you for pointing this out. I agree with this comment. Therefore, I have revised the formula in the manuscript to extend the definition when e_1=0, this change can be found in formula(12).

Comments 5: Theorem 1 is in the same spirit as Theorem 1 from [20] and the proof is not original at all (moreover, it is wrong: how can you compare complex numbers and how can you divide by s+alpha without any concerns regarding the sign of the inequality?) Also, if you take a look at the theorem, the term P^2 does not appear and for me is not clear how P^2 appears on the fourth row of (19). Take a better look at Khalil’s book [H. Khalil, “Nonlinear Systems”] where he defines exponential stability and you will find how you can prove your inequality more rigorously and correctly. Moreover, (15) is a BMI that is not convex and I think that LMI conditions could be imposed. 

Response 5: Thank you for pointing this out. I agree with this comment. Theorem 1 is cited to explain why MESO is stable, and the proof is not original. Therefore, I have removed the proof, this change can be found in lines 216-225.

Comments 6: Regarding the numerical results: please explain the tools used to find the parameters. Also, I consider that the parameters could be optimized according to certain rules. Maybe Theorem 1 could be followed by a controller design procedure. 

Response 6: Thank you for pointing this out. I agree with this comment. Therefore, I have explained the rules used to find the parameters which referred to the paper by Gao, it can be found in line 276.

Comments 7: ESO is not defined at first usage [in abstract];

Response 7: Thank you for pointing this out. I agree with this comment. Therefore, I have given the definition of ESO at first usage in the manuscript, this change can be found in line 16.

Comments 8: line 137: what coefficient? What K means?

Response 8: Thank you for pointing this out. I agree with this comment. Therefore, I have revised the text in the manuscript, this change can be found in lines 138-139.

Comments 9: in Figure 2 is used \Phi while in text is used \phi → please be consistent;

Response 9: Thank you for pointing this out. I agree with this comment. Therefore, I have revised the figure in the manuscript, this change can be found in Figure 2.

Comments 10: Figures from Fourth Section: use Time[s], z[mm], and so on because “/” is confusing and used with double meaning in your figures.

Response 10: Thank you for pointing this out. I agree with this comment. “/” is used with double meaning because of “dz/dt”. Therefore, I have revised the figure in the manuscript, this change can be found in Figure 7 and Figure 9.

2. Response to Comments on the Quality of English Language

Point 1: line 83: an modified → a modified;

Response 1: Thank you for pointing this out. I agree with this comment. I have revised the text in the manuscript, this change can be found in line 84.

Point 2: lines 128-129: comma after each equation and “where” instead of “Where” (the equations must be included in the sentences as well); this issue should be addressed in a similar manner after eq. (3), (7), (9), (10) etc. Punctuation in equations seems to be a problem.

Response 2: Thank you for pointing this out. I agree with this comment. I have revised the text in the manuscript.

Point 3: line 133: comma instead of point after “disturbing force”;

Response 3: Thank you for pointing this out. I agree with this comment. I have revised the text in the manuscript, this change can be found in line 134.

Point 4: lines 138-139: pay attention to parenthesis;

Response 4: Thank you for pointing this out. I agree with this comment. I have revised the text in the manuscript, this change can be found in lines 140-141.

Point 5: line 188: avoid starting a phrase with a symbol;

Response 5: Thank you for pointing this out. I agree with this comment. I have revised the text in the manuscript, this change can be found in line 191.

Point 6: line 211: after comma use “the” instead of “The” or use “.”;

Response 6: Thank you for pointing this out. I agree with this comment. I have revised the text in the manuscript, this change can be found in line 214.

Comments 1: First, the comment “In Figure 7, the maximum value of the vertical coordinate of some pictures is not marked.” is still not addressed.

Response 1: Thank you for pointing this out. I agree with this comment. Therefore, I have marked the maximum value of the vertical coordinate in Figure 7 and Figure 9.

Comments 2: Secondly, in fact, the disturbance observed have physical meaning, so there is corresponding physical quantity and unit. According to (2), fd is a force and its unit should be N.

Response 2: Thank you for pointing this out. I agree with this comment. Therefore, I have marked the unit of fd, this change can be found from in Figure 7 and Figure 9.

Comments 3: Finally, s1, s2, and s3 in figure 7 do not give corresponding explanations.

Response 3: Thank you for pointing this out. I agree with this comment. Therefore, I have revised the text in the manuscript, this change can be found in lines 297-300.

Comments 1 The vast majority of my comments have been successfully addressed. However, I think that the nonconvex optimization problem from (14)-(15) could be written in a convex manner. But, the results are fair.

Response 1: Thank you for pointing this out. I agree with this comment. Therefore, I have revised the convergence conditions for the MESO, which refers paper[23], this change can be found in Formula (14)-(15).