Fractional-Order Sliding Mode Control Method for a Class of Integer-Order Nonlinear Systems

Round 1

Reviewer 1 Report

 

In this paper the authors address the stabilisation of a class of nonautonomous nonlinear systems.

They derive a fractional stability theorem based on a fractional-order Lyapunov inequality. Then, a fractional-order sliding surface, is proposed. Finally, a fractional-order sliding-mode-based control is designed.

Simulation results demonstrate the applicability and efficiency of the proposed method.

 

In my opinion the paper is interesting. However, some improvements could be implemented:

• References to pioneer works about fractional SMC are missing (ex. Those by Tenreiro Machado)
• Check line 65;
• How does the results compare with integer order controllers?
• The results are presented in nice figures, but their analysis relies on visual perception. The authors should use some quantitative criteria to assess the effectiveness of the proposed method.

 

 

Author Response

Dear reviewer,

I have revised the content of the manuscript you pointed out one by one.

1. I added the work of Tenreiro Machado in Introduction;
2. The simulations of three integer SMC were added for comparison of the proposed guidance law;
3. Several tables were added to quantitatively show the simulation results.

Thank you very much for your kind suggestion to my work!

Reviewer 2 Report

The article proposes a fractional order stability theorem was established, which can be utilized as a fractional-order Lyapunov direct method. Also, a novel fractional order sliding surface was proposed, together with a fractional-order sliding mode-based control law for a class of integer-order nonlinear systems.   The proposed method is applied to the missile-target interception problem with impact angle constraints. To this reviewer, the paper results are quite interesting and relevant to the field of fractional order control since they may be applied to the control of nonlinear systems described by integer order differential equations. Some points that may improve the paper presentation follow.

1. The authors should explain why the $sgm(.)$ is not used in all the terms that involve the $sgn(.)$ function in control law (52), and not just in one term.  It is also advisable, to make such a change in the example to evaluate how the use of the $sgm(.)$ function affects the closed loop system behavior.

2. The English and the edition should be revised and corrected in detail.

 

Author Response

Dear reviewer,

I have revised the content of the manuscript you pointed out one by one.

1. I revised the term $|z|^a sgn(z)$ to $sig(z)^a$ all of the paper in order not to confused with the singular term $b sgn(z)$,  which can be replaced to $b sgm(z)$ to omit the singularity of the guidance law.
2. Some spelling and gramma errors were corrected.

Thank you very much for your kind suggestion to my work!

Reviewer 3 Report

Manuscript presents interesting results regarding process control using fractional calculus. The manuscript needs to be improved in the follwoing sense:

1) A comparison with integer order sliding control in order to show that the proposed fractional calculus based strategy is actually providing some gain over the existing/classical techniques.

2) The authors are encouraged to provide some control loop performance index in order to adequately compare the different fractional order alpha.

Author Response

Dear reviewer,

I have revised the content of the manuscript you pointed out one by one.

1. The simulations of three integer SMC were added for comparison of the proposed guidance law;
2. Several tables were added to quantitatively show the simulation results.

Thank you very much for your kind suggestion to my work!

Reviewer 4 Report

The authors studied  the problem of stabilization of a class of non autonomous nonlinear systems. The paper is scientifically sound. However,  the authors are recommended to show clearly the  contributions in the last paragraph of the paper ( line 45 to 51) and show the improvement over the current state-of-the art. Some section numbers are missing.

Other comments  can be found in the attached manuscript.

Comments for author File: Comments.pdf

Author Response

Dear reviewer,

I have revised the content of the manuscript you pointed out one by one.

1. Section number in the last paragraph of Section 1 was revised;
2. The  contributions of the paper in Section 1 was rewritten.
3. Figure and table environment were changed as the reviewer advised.

Thank you very much for your kind suggestion to my work.

Round 2

Reviewer 3 Report

In my particular opinion, the manuscript can now be accepted.