Fixed-Time Fractional-Order Sliding Mode Control for UAVs under External Disturbances
Round 1
Reviewer 1 Report
The paper considered the fixed-time tracking control with fractional-order dynamics of a quadrotor subjected to external disturbances. However, the manuscript has the main issues. Are you sure that Lemmas 1and 2 are suitable for fractional-order system? In my view, the definition of fixed time of integer system is not applicable for fractional order system. Therefore, the results are questionable.
Minor editing of English language required
Author Response
Manuscript ID: fractalfract-2595800
Subject: Revision and resubmission of manuscript fractalfract-2595800
Dear Editor, Reviewers,
We appreciate you and the reviewers for your precious time in reviewing our paper entitled ‘Fixed-time Fractional-order sliding mode controller for quadrotor UAVs under external disturbances’ and providing valuable comments. It was your valuable and insightful comments that led to possible improvements in the current version. The authors have carefully considered the comments and tried our best to address every one of them. We hope the manuscript after careful revisions meets your high standards. The authors welcome further constructive comments if any. Below we provide the point-by-point responses. All modifications in the manuscript have been highlighted in red.
Sincerely,
Benaddy Abdellah, PhD [email protected]
MIET Laboratory, Faculty of Science and Technology.
Hassan First University of Settat, Morocco
Response to Reviewer 1:
[General Comment] The paper considered the fixed-time tracking control with fractional-order dynamics of a quadrotor subjected to external disturbances. However, the manuscript has the main issues. Are you sure that Lemmas 1 and 2 are suitable for fractional-order system? In my view, the definition of fixed time of integer system is not applicable for fractional order system. Therefore, the results are questionable.
Response : Thank you again for your positive comments and valuable suggestions to improve the quality of our manuscript. The fixed-time stability is guarantee based on Lemma 1 and 2; any solution of the system unique as long as x(t) ≠ 0, but the proven fixed-time stability guarantee the uniqueness of the solution after the reaching of the origin and the convergence time is upper bounded by T < Tmax proving in equations 29 and 38 . In addition the fixed time control is appropriate for fractional-order systems as mentioned in the previous articles like [1, 2, 3, 4].
References:
[1] Labbadi, M.; Boubaker, S.; Djemai, M.; Mekni, S.K.; Bekrar, A. Fixed-Time Fractional-Order Global Sliding Mode Control for 328 Nonholonomic Mobile Robot Systems under External Disturbances. Fractal and Fractional 2022, 6, 177.
[2] Idrissi, M.; Salami, M.; Annaz, F. A Review of Quadrotor Unmanned Aerial Vehicles: Applications, Architectural Design and 332 Control Algorithms. Journal of Intelligent and Robotic Systems 2022, 104.
[3] Huang, S.; Wang, J. Fixed-time fractional-order sliding mode control for nonlinear power systems. Journal of Vibration and Control 334 2020, 26, 1425–1434.
[4] Ni, J.; Liu, L.; Liu, C.; Hu, X. Fractional order fixed-time nonsingular terminal sliding mode synchronization and control of 348 fractional order chaotic systems. Nonlinear Dynamics 2017, 89, 2065–2083.
Author Response File: 
Author Response.pdf
Reviewer 2 Report
Overall this paper presented something new to UAV control, it will be more convincing to add experimental results.
It is also important to tell in advance the finite time bound in advance. How to make the estimate using what knowledge?
I wish to see the results reproducible by readers, and it is good to use Matlab UAV toolbox.
Gust wind is important, please add discussion on this specific disturbance.
When in low altitude, we know UAV can be affected by the ground reflection effect, please add discussion on this.
please add . to end the equations as a part of a sentence.
Author Response
Manuscript ID: fractalfract-2595800
Subject: Revision and resubmission of manuscript fractalfract-2595800
Dear Editor, Reviewers,
We appreciate you and the reviewers for your precious time in reviewing our paper entitled ‘Fixed-time Fractional-order sliding mode controller for quadrotor UAVs under external disturbances’ and providing valuable comments. It was your valuable and insightful comments that led to possible improvements in the current version. The authors have carefully considered the comments and tried our best to address every one of them. We hope the manuscript after careful revisions meets your high standards. The authors welcome further constructive comments if any. Below we provide the point-by-point responses. All modifications in the manuscript have been highlighted in red.
Sincerely,
Benaddy Abdellah, PhD [email protected]
MIET Laboratory, Faculty of Science and Technology.
Hassan First University of Settat, Morocco
Response to Reviewer 2:
[Comment 1] Overall, this paper presented something new to UAV control; it will be more convincing to add experimental results.
Response: Thank you again for your positive comments and valuable suggestions to improve the quality of our manuscript. Due to an incomplete drone test bench and a lack of elements that would allow us to support the research with experimental results, the demonstration has been limited to theoretical and simulation results in this manuscript. However, we can conduct the experiment results, and you have access to the materials that we'll be using in Remark 4 and Figure 10.
[Comment 2] It is also important to tell in advance the finite time bound in advance. How to make the estimate using what knowledge?
Response: Thanks for your comment; the suggested correction has been made. We have added some knowledge to the finite/fixed time in definitions 1 and 2 page 4.
[Comment 3] I wish to see the results reproducible by readers, and it is good to use Matlab UAV toolbox.
Response: Thanks for your suggested comment; the same mathematical model is used for modeling and control development. The model of UAV used in this paper can be implemented easily in Matlab/Simulink to reproduce the simulation results.
[Comment 4] Gust wind is important; please add discussion on this specific disturbance.
Response: Thanks for your comment;
In many applications the wind gust information can be obtained using unmanned aerial vehicles (UAVs) using wind sensors and algorithm of estimation. This technique can be useful as feedback for robust control as well as weightless substitute [5]. The estimated gust wind disturbance affecting the quadrotor is modeled as the input distributing the vehicle trajectory.
[Comment 5] When in low altitude, we know UAV can be affected by the ground reflection effect, please add discussion on this.
Response: Thanks for your comment; the suggested discussion as mentioned in Remark 3.
The wind distributions and mass variation affect the quadrotor trajectory, the ground effect is an additional problem that needs to be considered. In many works in the literature [6] the authors successfully demonstrated the robustness of their algorithm through experiments without considering the influence of the ground effect when the quadrotor flights near the earth and did not offer any proves of such scenarios. In the case of altitude control, the impact of the ground effect has often been neglected.
Remark: Many works demonstrate in the literature [6] successfully demonstrated the robustness of their algorithm through experiments without considering the influence of the ground effect and did not offer any proves of such scenarios. Therefore, the impact of the ground effect has often been neglected.
[Comment 6] Please add . to end the equations as a part of a sentence.
Response: Thank you. At the finish of the equations, we added the dot.
References:
[5] A. Asignacion, S. Suzuki, R. Noda, T. Nakata and H. Liu, "Frequency-Based Wind Gust Estimation for Quadrotors Using a Nonlinear Disturbance Observer," in IEEE Robotics and Automation Letters, vol. 7, no. 4, pp. 9224-9231, Oct. 2022.
[6] Sanchez-Cuevas P, Heredia G and Ollero A (2017) Characterization of the aerodynamic ground effect and its influence in multirotor control. International Journal of Aerospace Engineering 2017: 1–17.
Author Response File: Author Response.pdf
Reviewer 3 Report
Please look for the suggestion in the attached file.
Comments for author File: Comments.pdf
no
Author Response
Manuscript ID: fractalfract-2595800
Subject: Revision and resubmission of manuscript fractalfract-2595800
Dear Editor, Reviewers,
We appreciate you and the reviewers for your precious time in reviewing our paper entitled ‘Fixed-time Fractional-order sliding mode controller for quadrotor UAVs under external disturbances’ and providing valuable comments. It was your valuable and insightful comments that led to possible improvements in the current version. The authors have carefully considered the comments and tried our best to address every one of them. We hope the manuscript after careful revisions meets your high standards. The authors welcome further constructive comments if any. Below we provide the point-by-point responses. All modifications in the manuscript have been highlighted in red.
Sincerely,
Benaddy Abdellah, PhD [email protected]
MIET Laboratory, Faculty of Science and Technology.
Hassan First University of Settat, Morocco
Response to Reviewer 3:
[Comment 1] In the abstract “backstopping sliding mode control” Please revise this statement.
Response: Revised accordingly.
[Comment 2] Please carefully check the English writing.
Response: During proofreading the whole manuscript, grammatical errors, typos, and artificial phrasing were fixed. Despite having carefully inspected the revised manuscript, the authors would like to apologize if there still are a few errors being overlooked by chance or due to our limited English language ability.
[Comment 3] The introduction of the fixed-time stability is short. Please add more recent research works for the citation of the paper.
Response: Thanks for your comment; the fixed time stability is discussed in the revised paper.
‘The finite-time stability ensures that the system’s trajectories converge to the desired target after some finite time, and the finite time is called the settling time or the convergence time. In addition to the advantages of finite time stability, fixed time stability involves finite time stable systems for which the minimum bound of the settling-time function is guaranteed to be independent of the system's initial conditions and can a priori be adjusted [7]. In order to regulate controllable nonlinear systems with matching disturbances, fixed-time sliding mode controllers were devised [8, 9]. The quadrotor system can converge to a stable range within an upper-bound convergence time thanks to the fixed-time stability, regardless of the initial operational states [10, 11]. Therefore, the distributed fixed-time control methods have been proposed in [12] to produce improved performances for the power system, whereas the fixed-time control approach can better overcome these drawbacks. However, the nonlinear system's convergence time under the fixed-time control approach cannot be estimated directly and must be calculated by a sophisticated estimation function that is based on tuning parameters [13]. The control of fractional-order systems using a fixed time controller was proposed in [14]. Following the research presented above, an appropriate SMC can be built to more effectively establish fractional-order fixed-time control.’
[Comment 4] The introduction fractional-order controller should be more illustrated by using recent papers as the references of the paper.
Response: Thanks for your constructive comments; the fractional-order controller is discussed in the revised paper.
‘In [15], fractional-order chaotic systems can be controlled and synchronized using a fractional-order fixed-time nonsingular terminal sliding mode. Additionally, fractional order controllers can increase the stability of dynamical systems [16].’
[Comment 5] Please check the Ref. 3 and Ref. 19.
Response: Thanks for your comment; the suggested correction has been made.
[Comment 6] The recent two years published papers should be considered for the references.
Response: Thanks for your comment; the suggested correction has been made.
[Comment 7] The wind should be more complicated than the sine wave form.
Response: Thank you for your comment.
In general, the wind is modeled as a sum of sine wave functions, therefore the model of wind disturbance is close to the form of smooth sine wave functions which vary around a value and increase and decrease smoothly versus time. The step or linear complicated functions are not a reel behavior of the wind effect.
[Comment 8] Please compare the proposed of this paper with the double phase of fixed-time stability of the “double phases fixed-time sliding mode control”
Response: Revised accordingly.
[Comment 9] The assumptions should be added into the paper for the reality of using the control system. For example, the disturbances should be bounded.
Response: Thanks for your comment; the suggested correction has been made. Additionally, the quadrotor dynamic model in this research is created under the following assumptions. (i) The effect of the ground is neglected. (ii) The vehicle structure is symmetrical. (iii) Both the blades and the vehicle frame are rigid. (iv) The disturbances should be bounded. (v) The torques and thrust generated by the rotor speeds are proportional to the square of the rotor rotating speeds.
[Comment 10] The SMC design should be more detail analyzed by using the switching and equivalent controls.
Response: Revised accordingly to your comment.
[Comment 11] How to overcome the chattering. Please compare with existing paper “Chattering-free sliding mode control-based disturbance observer”
Response: Revised accordingly.
[Comment 12] Carefully solve the Eq. (32) for reader easy to understand the robustness characteristics of the utilized SMC.
Response: Thanks for your comment; the suggested correction has been made.
[Comment 13] Summary: The stability and sliding mode control design should be carefully checked.
Response: Thanks for your comment; the suggested correction has been made.
If there are any other modifications we could make, we would like very much to modify them and we really appreciate your help. Thank you very much for your help.
References:
[7] Lee, J.; Haddad, W.M. Fixed time stability and optimal stabilisation of discrete autonomous systems. International Journal of 352 Control 2022, 96, 2341–2355.
[8] Olguin-Roque, J.; Salazar, S.; González-Hernandez, I.; Lozano, R. A Robust Fixed-Time Sliding Mode Control for Quadrotor UAV. 354 Algorithms 2023, 16, 229. 355 21.
[9] Giap, V.N.; Nguyen, Q.D.; Trung, N.K.; Huang, S.C. Time-varying disturbance observer based on sliding-mode observer and 356 double phases fixed-time sliding mode control for a T-S fuzzy micro-electro-mechanical system gyroscope. Journal of Vibration and 357 Control 2022, 29, 1927–1942.
[10] Ni, J.; Liu, L.; Liu, C.; Hu, X.; Li, S. Fast Fixed-Time Nonsingular Terminal Sliding Mode Control and Its Application to Chaos 359 Suppression in Power System. IEEE Transactions on Circuits and Systems II: Express Briefs 2017, 64, 151–155. 360 23.
[11] Su, Y. Comments on “Fixed-time sliding mode control with mismatched disturbances” [Automatica 136 (2022) 110009]. Automatica 361 2023, 151, 110916. 362 24.
[12] Wang, Z.; Wang, J.; Scala, M.L. A Novel Distributed-Decentralized Fixed-Time Optimal Frequency and Excitation Control 363 Framework in a Nonlinear Network-Preserving Power System. IEEE Transactions on Power Systems 2021, 36, 1285–1297. 364 25.
[13] Zeng, T.; Ren, X.; Zhang, Y. Fixed-Time Sliding Mode Control and High-Gain Nonlinearity Compensation for Dual-Motor Driving 365 System. IEEE Transactions on Industrial Informatics 2020, 16, 4090–4098. 366 26.
[14] Shirkavand, M.; Pourgholi, M. Robust fixed-time synchronization of fractional order chaotic using free chattering nonsingular 367 adaptive fractional sliding mode controller design. Chaos, Solitons and Fractals 2018, 113, 135–147. 368 27. Bhat, S.P.; Bernstein, D.S. Finite-Time Stability of Continuous Autonomous Systems. SIAM Journal on Control and Optimization 369 2000, 38, 751–766.
[15] Ni, J.; Liu, L.; Liu, C.; Hu, X. Fractional order fixed-time nonsingular terminal sliding mode synchronization and control of 348 fractional-order chaotic systems. Nonlinear Dynamics 2017, 89, 2065–2083. 349 18.
[16] Aghababa, M.P. A fractional sliding mode for finite-time control scheme with application to stabilization of electrostatic and 350 electromechanical transducers. Applied Mathematical Modelling 2015, 39, 6103–6113.
Author Response File: Author Response.pdf
Round 2
Reviewer 2 Report
the revision is good.
Author Response
Thank you again for your positive comments and valuable suggestions to improve the quality of our manuscript.
Reviewer 3 Report
I have no further comments. The paper can be accepted as it is.
Author Response
Thank you again for your positive comments and valuable suggestions to improve the quality of our manuscript.
