Adaptive Neural Network Sliding Mode Control for Nonlinear Singular Fractional Order Systems with Mismatched Uncertainties

Round 1

Reviewer 1 Report

The authors have investigated the issue of adaptive sliding mode control (SMC) for mismatched uncertain system of fractional order differential equations. The also have applied RBF to obtain solution. The paper is interesting but need major revision.

I suggest to revise it according the following comments and suggestion:

1-The English most be improved carefully.

2- Page 3, Line 83,  "The $\alpha$th Caputo fractional derivative" must be changed to " The $\alpha$th order Caputo fractional derivative ".

3- I suggest to use " System of fractional order differential equations (SFDEs) " instead of "Uncertain singular fractional order systems (SFOSs)." Because the fractional order differential equations are singular.

4- They must add few lines regarding to novelty and advantages of the presented methods respect to other exiting method.

Author Response

1. The English most be improved carefully.
Reply: Thank you very much for your careful reading. We have corrected the grammatical
mistakes and typos throughout the manuscript. The revised text is marked in blue.
2. Page 3, Line 83, ”The αth Caputo fractional derivative” must be changed to ” The αth
order Caputo fractional derivative ”.
Reply: Thank you very much for your comments. According to your suggestion, we have
changed our description.
”The αth Caputo fractional derivative” −−− > ”The αth order Caputo fractional derivative”
3. I suggest to use ” System of fractional order differential equations (SFDEs) ” instead of
”Uncertain singular fractional order systems (SFOSs).” Because the fractional order differential
equations are singular.
Reply: Thank you very much for your comments. In this manuscript, we study the admissibility of singular fractional order systems, The derivative matrix contains the uncertainty ∆E.
“System of fractional order differential equations (SFDEs)” does not highlight the uncertain
characteristics of the SFOSs. In addition, many papers use the expression like ”Uncertain
1
singular fractional order systems (SFOSs)”, such as [16], [18], [19], [22], [24]. So, after careful
consideration, we still use “Uncertain singular fractional order systems (SFOSs)”.
4. They must add few lines regarding to novelty and advantages of the presented methods
respect to other exiting method.
Reply: The novelty and advantages of the presented methods respect to other exiting method
are shown as follows:
• The novel necessary and sufficient condition for the admissibility of SOFSs is provided in this manuscript. The conditions proposed in Lemma 4 is strict LMIs which is more efficient
and general than other theorems [19, 24, 41] because fewer variables are introduced and the
complex calculation is avoided successfully.
• In [14, 21], the admissibility problems are investigated for singular systems. However, the proportional-plus derivative state feedback controller is designed so that the system is
normalizable. This control method is essentially a normal system solution approach instead of
a singular system solution approach. In this manuscript, the novel SFOS solution approach is
proposed. • Based on RBF neural network method, b f(t,x(t)) is constructed to estimate the nonlinear term f(t,x(t)). The restricted assumption in [41] is removed.
We have discussed the novelty and advantages in Introduction, Remark 1 and Remark 3.
The added text in the manuscript is listed as follows:
Remark 3. In [14, 21], the admissibility problems are investigated for singular systems.
However, the proportional-plus derivative state feedback controller is designed so that the
system is normalizable. This control method is essentially a normal system solution approach
instead of a singular system solution approach. In this paper, the novel SFOS solution approach
is proposed.
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Author Response File: Author Response.pdf

Reviewer 2 Report

The reviewer's comments are attached. 

Comments for author File: Comments.pdf

Author Response

1. The writing is generally good but still needs to be improved. Many grammatical issues exist
throughout the paper. For example, in line 9: the validity of designed procedures should be the
validity of the designed procedures; in line 16, to FOSs. a basic theorem should be to FOSs.
A basic theorem; and many others in lines 44, 52, 56, 57, 59, 63, 106, etc. So I recommend
proofreading the paper when revising it.
Reply: Thank you very much for your carefully reading. We have corrected the grammatical
mistakes and typos in the manuscript. The modified text is marked in blue.
2. According to the topic of this paper, the authors need to supplement some related references
for completeness.
Reply: Thank you very much for your comments. We have added the following related refer
ences in Introduction to make our manuscript more perfect.
Nguyen, A.T.; Xuan-Mung, N.; Hong, S.K. Quadcopter adaptive trajectory tracking control:
a new approach via backstepping technique. Appl. Sci. 2019, 9, 3873.
Xuan-Mung, N.; Hong, S.K. Robust backstepping trajectory tracking control of a quadrotor
with input saturation via extended state observer. Appl. Sci. 2019, 9, 5184.
Xuan-Mung, N.; Hong, S.K. Improved altitude control algorithm for quadcopter unmanned
aerial vehicles. Appl. Sci. 2019, 9, 2122.
Xuan-Mung, N.; Hong, S.K. Barometric altitude measurement fault diagnosis for the im
provement of quadcopter altitude control. In Proceedings of the 19th International Conference
on Control, Automation and Systems, Jeju, Korea, 15–18 October 2019.
3. From line 109, an RBF network structure with three hidden layers is introduced. The
description of this neural network needs to be amended to make it comprehensible. The au
thors may need to provide more information about the number of nodes in each layers, the
optimization algorithm used for training, etc.
Reply: Thank you very much for your comments. The proposed adaptive neural network
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approach does not involve neural network training. A simple adaptive neural network control
approach is utilized to estimate the nonlinear terms of SFOSs in this manuscript. We set b f(t,x(t)) =c WT(t)h(x(t)) to approximate f(t,x(t)). The adaptive law is designed as ˙ c W(t) = ωh(x(t))sT(t)G B to update the adaptive matrix c W(t). In order to ensure that the input value x(t) of the network is within the effective range of the Gaussian function, we choose the appropriate value of cji and µj (j = 1,··· ,m i = 1,···n) according to the actual range of the input value of the network input x(t). The RBF neural network approximate accuracy is
closely related to the number of neuron nodes. In order to improve the approximate accuracy,
we can select suitable number of neural nodes. In this manuscript, we choose m = 5, µj = 0.2. cji is uniformly distributed in [−2, 2]. This adaptive neural network control approach is also utilized in [42, 43, 46]. We added a description of the parameters in the manuscript, The added
text is listed as follows:
The neural network parameters are selected as m = 5, µj = 0.2, and cji is uniformly distributed in [−2, 2].
4. In the equation (42), the author presented some maths with matric traces. sT(t)G Bf WT(t)h(x(t)) = Tr(sT(t)G Bf WT(t)h(x(t))) = Tr(f WT(t)h(x(t))sT(t)G B). However, the basis of these transformations is not provided, which makes it not clear. I rec
ommend the author to provide some related references to make it convey to a broader range of
readers, such as: Robust adaptive formation control of quadcopters based on a leader-follower
approach.
Reply:Thank you very much for your comments. Tr(X) denotes the trace of matrix X. So, we have Tr(sT(t)G Bf WT(t)h(x(t))) = Tr(f WT(t)h(x(t))sT(t)G B). Thus, the equation sT(t)G Bf WT(t)h(x(t)) = Tr(sT(t)G Bf WT(t)h(x(t))) = Tr(f WT(t)h(x(t))sT(t)G B) is obtained. According to your suggestion, We have added the following related reference:
Xuan-Mung, N.; Hong, S.K. Robust adaptive formation control of quadcopters based on a
leader- follower approach. International Journal of Advanced Robotic Systems doi.org/10.1177/17
29881419862733.
5. In line 116, the system uncertainty is assumed as: x1 sin(x1(t)). What is the reason for
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this assumption? Is this type of uncertainty popular in real-world applications? These points
should be addressed.
Reply: Thank you very much for your comments. f(t,x(t)) is the system nonlinearity, not
the uncertainty. So we assume that f(t,x(t)) = x1 sin(x1(t)) is reasonable. This assumption is
very common in the SMC papers [35, 37, 41, 44].

Reviewer 3 Report

The article investigates the issue of adaptive SMC for mismatched uncertain SFOSs. Furthermore, the authors developed necessary and sufficient condition for the admissibility of SFOSs. To my opinion, the article presents interesting results. However, the authors should revise the editing to improve paper readability. Here are some suggestions to improve paper readability: 

• Although the paper is well written, the simulation results section could be improved. The simulation result section should include another example to show a variety.

• The conclusion should be written in detail by stressing the importance of the proposed theory. Please show when the presented approach is applicable, what are the pros and cons of using it, when it should be preferred over the other existing theory etc. A detailed illustration of why adaptive SMC for mismatched uncertain SFOSs is a powerful tool in applications would increase the interest of the reader.

Author Response

1.  
2. 2. The conclusion should be written in detail by stressing the importance of the proposed
   theory. Please show when the presented approach is applicable, what are the pros and cons of
   using it, when it should be preferred over the other existing theory etc. A detailed illustration
   of why adaptive SMC for mismatched uncertain SFOSs is a powerful tool in applications would
   increase the interest of the reader.
   Reply: The strict LMI approach is developed to both judge admissibility and design sliding
   mode controller. It is worth noting that the Lemma 4 obtained in this manuscript can be
   regarded as a natural extension of Lyapunov stability from normal integer orders systems to
   singular fractional order systems with the consistent format. The conditions of Theorem 1 is
   giving without including complex calculations [24] and without involving non-strict inequality
   [19, 41]. We have discussed the advantages in Remarks 1 and 3. .By designing sliding surface,
   the mismatched uncertainty does not exist in the derivative matrix of the sliding mode dynamic.
   So adaptive SMC for mismatched uncertain SFOSs is a powerful tool. We have modified our
   Conclusion as follows:
   • This paper investigates the issue of adaptive SMC for mismatched uncertain SFOSs. The new necessary and sufficient condition for the admissibility of SFOSs is developed, which is
   strict LMIs. The integral sliding mode surface with expanded dimension is constructed so that
   mismatched uncertainty does not exist in the derivative matrix of the sliding mode dynamic.
   By RBF neural network method, the adaptive control law is devised to make SFOSs satisfy the
   reaching condition. The restrictive assumption that the nonlinearity f(t,x(t)) is norm bounded
   is removed. In the further, the issues of SMC for SFOSs with time delay will be studied.
   Although the paper is well written, the simulation results section could be improved. The
   simulation result section should include another example to show a variety.
   Reply: Thank you very much for your comments. Your suggestion is helpful to improve the
   quality of the manuscript. In this manuscript, we have studied the adaptive sliding mode
   control problem for a class of uncertain singular fractional order systems with fractional order
   0 < α < 1. We consider the other case in Example 1. When system (1) is not subject to control
   law (35), system (1) is unstable. the following figure shows the state response of system (1)
   without SMC law. The added text in the manuscript is listed as follows

Reviewer 4 Report

The paper deal with the problem of sliding mode control of some singular fractional order system with uncertainties and propose to use neural network approach within this scope.
The paper is well written and clearly presents the context, the goals, and the results of the study.
However, talking about the usage of neural network formalism, it looks like it is used only as a form of approximate solution representation as only the simplest type of neural network is used and nothing is said about its training. So, some remarks should be added on training the neural network at least in the simulation examples section.

Some minor issues should also be addressed to make the text clearer:
L54, "nonlinear term f_i" - as it is f_i, did you mean nonlinear terms?
L78 - changing full stops to commas around the definition of a and b can enhance readability;
L80 - The type of fractional derivative is not defined after (1), only later on L83 without clear connection with (1). Consider rewriting this fragment;
L82 - please add the dimension of the unknown function f;
L85, "(3) is denoted as the triple (E, A)." It is clearly a tuple, not triple. Please rewrite this sentence. Check also capital letter in its beginning;
L100-101, "solving the unknown antisymmetric matrix" - did you mean solving a system of linear algebraic equations with such a matrix? Please clarify this sentence;
L111, "structure with three hidden layers" - This neuron network seem to have no hidden layers, only one input and one output layer;
L121, "it is note that" - did you mean "it should be noted that" or something similar? please, rewrite this sentence;

Consider also changing sentences "The proof is ended" with the corresponding sign.
As figures 2-5 show fast convergence, consider using logarithmic scale for the time axis. It can possibly improve the representation of the results.

Author Response

1. The paper is well written and clearly presents the context, the goals, and the results of
the study. However, talking about the usage of neural network formalism, it looks like it is
used only as a form of approximate solution representation as only the simplest type of neural
network is used and nothing is said about its training. So, some remarks should be added on
training the neural network at least in the simulation examples section.
6
Reply: Thank you very much for your comments. The proposed adaptive neural network
approach does not involve neural network training. A simple adaptive neural network control
approach is utilized to estimate the nonlinear terms of SFOSs in this manuscript. We set b f(t,x(t)) =c WT(t)h(x(t)) to approximate f(t,x(t)). The adaptive law is designed as ˙ c W(t) = ωh(x(t))sT(t)G B to update the adaptive matrix c W(t). In order to ensure that the input value x(t) of the network is within the effective range of the Gaussian function, we choose the appropriate value of cji and µj (j = 1,··· ,m i = 1,···n) according to the actual range of the input value of the network input x(t). The RBF neural network approximate accuracy is
closely related to the number of neuron nodes. In order to improve the approximate accuracy,
we can select suitable number of neural nodes. In this manuscript, we choose m = 5, µj = 0.2. cji is uniformly distributed in [−2, 2]. This adaptive neural network control approach is also utilized in [42, 43, 46]. We added a description of the parameters in the manuscript, The added
text is listed as follows:
The neural network parameters are selected as m = 5, µj = 0.2, and cji is uniformly distributed in [−2, 2].
2. Some minor issues should also be addressed to make the text clearer:
L54, “nonlinear term fi”- as it is fi, did you mean nonlinear terms?
L78 - changing full stops to commas around the definition of a and b can enhance readability;
L80 - The type of fractional derivative is not defined after (1), only later on L83 without
clear connection with (1). Consider rewriting this fragment;
L82 - please add the dimension of the unknown function f;
L85, ”(3) is denoted as the triple (E, A).” It is clearly a tuple, not triple. Please rewrite
this sentence. Check also capital letter in its beginning;
L100-101, ”solving the unknown antisymmetric matrix” - did you mean solving a system of
linear algebraic equations with such a matrix? Please clarify this sentence;
L111, ”structure with three hidden layers” - This neuron network seem to have no hidden
layers, only one input and one output layer;
7
L121, ”it is note that” - did you mean ”it should be noted that” or something similar?
please, rewrite this sentence;
Reply: Thank you very much for your comments. Your suggestion is helpful to improve the
quality of the manuscript. We have carefully studied your opinion and made the following
changes in the manuscript.
• “nonlinear term fi”−−− > “nonlinear terms fi(t,x(t))”. • “. a = sin(aπ 2),b = cos(aπ 2). ”−−− > “, a = sin(aπ 2),b = cos(aπ 2),”. • We have moved the “ The αth order Caputo fractional derivative of f(t) is defined as
Dαf(t) =
1 Γ(n−α)∫ t 0
(t−τ)α+1−nf(n)(τ)dτ, where n−1 < α < n, n ∈N+ and Γ(·) is the Gamma function.” into Notations. • “f(t,x(t)) represents the nonlinear term”−−− > “f(t,x(t))∈Rl represents the nonlinear term”.
• “considering”−−− > “Considering”, “the triple (E,A)” −−− > “the triple (E,A,α)”. • “require solving the unknown antisymmetric matrix X2” −−− > “involve the unknown antisymmetric matrix X2”. • Fig. 1 in the manuscript shows the RBF network structure. The x =[xi]on the left of Fig. 1 is the input of the network. y(t) on the right of Fig. 1 is the output of the network. The hidden layer of the network is h(x(t)) = [h1(x(t)) h2(x(t)) ··· hm(x(t))]T in the middle of Fig. 1.
• “it is note that” −−− >“It is easy to see that”.
3. Consider also changing sentences ”The proof is ended” with the corresponding sign.
Reply: Thank you very much for your comments. According to your suggestion, we have
changed sentences ”The proof is ended” with the corresponding sign.
4. As figures 2-5 show fast convergence, consider using logarithmic scale for the time axis. It
can possibly improve the representation of the results.
8
Reply: Thank you very much for your comments. We have studied your suggestion carefully.
In the manuscript, as can be seen from Fig. 2, The tendency of x(t) to converge to 0 is obvious.
The following figure uses a logarithmic scale as the time axis. However, the state response of
x(t) per second cannot be clearly displayed. In addition, it takes a long time for the simulation
to show good results. So we do not consider using logarithmic scale for the time axis for Figs.
2–5.

Round 2

Reviewer 2 Report

The authors have given satisfactory answers to all of my concerns. I have no more comments.