^{1}

^{2}

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Edited by: Muhammad Javaid, University of Management and Technology, Lahore, Pakistan

Reviewed by: Lin Zhao, Qingdao University, China; Peijun Wang, Anhui Normal University, China; Shumin Ha, Shaanxi Normal University, China

This article was submitted to Mathematical Physics, a section of the journal Frontiers in Physics

This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

In this paper, neural network (NN) control of the fractional-order Duffing system (FODS) by using a backstepping method within finite time in the presence of input saturation has been investigated. A fractional-order filter with an order lying on the interval (1,2) was used to estimate the virtual input together with its fractional derivative, and this showed that the estimation error tends to a small region in some finite time. Fractional-order law is designed for the parameter of the NN, and an adaptive NN controller was given. The proposed method drives the tracking error, tending to an arbitrary small region within a finite time. The simulation results verify the validity of the proposed method.

It is a well-known fact that classical differential operators are local operators and cannot describe some complex properties. For example, Brownian motion, viscoelastic materials, anomalous diffusion, and irregular fluctuations in the turbulent velocity field have memory problems. Fractional-order differential operators are non-local and can well-characterize memory, genetic, and global correlation in the real world. The physical process is an important tool for describing physical processes and complex mechanics [

On the other hand, it is well-known that chaos control is a research hot topic and has some successful applications. With the in-depth study of chaotic systems, people began to try to migrate the synchronization method of integer-order chaotic systems to the synchronization of fractional-order chaotic systems (FOCSs). This natural idea is not easy to implement. For this reason, some people try to use the Laplace transform method and time-frequency domain transformation method. By solving the ^{α} by using the Laplace transform method, finite time control was investigated in Tavazoei and Haeri [

Inspired with above discussion, we will address the finite time NN control of the fractional-order Deffuing system (FODS) with input saturation. Take some related works, such as Liu et al. [

The NN with three layers is expressed as

where _{+} denote the amount of neurons three layers (input, middle, and output), _{ji} denotes a weight whose value is on the interval [−1, 1]. Usually, φ(·) can be defined by

Then, the NN is given as

with ^{n} is unknown, then it can be approximated by the NN as

with ε(ℏ) denoting the optimal approximation error, where

with

The

with Γ(·) representing Euler's function, and the

where

Lemma 1.

Lemma 2. _{2} < 1. _{2} ^{*}.

Lemma 3. _{1}, _{2} > 0, 0 < _{3} < 1,

^{n}.

Lemma 4.

_{1} _{2}

The integer-order Duffing system is written as

where _{1}(_{2}(

in which sat :

with _{r} > 0, _{l} < 0. Denoting the term that exceeds the saturation limiter as γ(

For the target, let _{1}(_{d}(_{1}(_{1}(_{d}(

with

where β_{2}(_{1}, _{1} > 0, and β(0) = 0. Using Lemma 4, we can estimate α(

Thus, (15) and Lemma 4 imply that

where

with

It follows from (10), (11), (12), and (16) that

with Θ(

Let the optimal parameter of NN be

To meet the control objective, we can design the compensated signal as

with _{2}, _{2} > 0. Then, let us construct the final input as

where σ, _{2} > 0, and

Then, (24) implies

Define

According to (18), (25), and (26), we have

The fractional-order adaptation law is

with κ_{1}, κ_{2} > 0.

The following theorem provides a conclusion for the discussion.

_{1}(

Then, based on (27), (28), and (29), we obtain

Then, (30) implies

If

On the contrary, if

Thus, it follows from (32) and (33) that

Substituting (34) into (31) yields

with _{min} = min{κ_{1}, κ_{2}}. As a result, (35) can be arranged as

According to (36) and Lemma 3, when _{1}(

in some finite time. Since _{1}(_{1}(_{1}(_{2}(_{2}(_{2}(_{1}(_{2}(

where

where _{1}, _{2}}, _{1}, _{2}}, _{1}(_{2}(

Remark 1.

Remark 2.

In system (10), let parameters _{1}(0) = −1.2, _{2}(0) = 1.2. When

Chaotic phenomenon of uncontrolled FODS (10).

In the simulation, let _{1}(0) = 2, _{2}(0) = 0, and let the reference signal be

The design parameters are _{1} = _{2} = 0.9; _{1} = _{2} = 1, _{1} = _{2} = 1, κ_{1} = κ_{2} = 1, ν = 0.70, _{1} = _{2} = 1. The NN uses _{1}(_{2}(^{81}. The saturation parameters are _{l} = −5, _{r} = 5.

Then, the simulation results can be seen in _{1}(_{2}(

Simulation1 results in _{1}(_{2}(_{1}(_{2}(_{d}(_{1}(

Virtual input and NN parameters in _{1}(_{2}(

Comparison results.

To show the rapid convergence speed of the proposed high-order filter, some comparative simulation results will be given here. Noting that in Liu et al. [

This paper addressed the finite time control of an unknown disturbed FODS in the presence of input saturation. By using the backstepping technique, a high order fractional filter with the order lying on (1,2) is proposed, and thus, the virtual input and its fractional derivative can be approximated. It is proven that the filter's approximation error can be enough small and can converge to the small region in some finite time. Then, an adaptive NN controller is given. The stability is proven strictly. In addition, the robustness of the proposed method is shown in simulation results. Our future research directions including how to design sliding mode surface for FODS and how to construct a high-order filter.

All datasets generated for this study are included in the article/supplementary material.

HL and XZ contributed the conception of the study. HL wrote the study and organized the literature. XZ wrote the simulation programs. All authors contributed to the manuscript revision, read, and approved the submitted version.

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.