Neural Network Command Filtered Control of Fractional-Order Chaotic Systems

An adaptive neural network (NN) backstepping control method based on command filtering is proposed for a class of fractional-order chaotic systems (FOCSs) in this paper. In order to solve the problem of the item explosion in the classical backstepping method, a command filter method is adopted and the error compensation mechanism is introduced to overcome the shortcomings of the dynamic surface method. Moreover, an adaptive neural network method for unknown FOCSs is proposed. Compared with the existing control methods, the advantage of the proposed control method is that the design of the compensation signals eliminates the filtering errors, which makes the control effect of the actual system improve well. Finally, two examples are given to prove the effectiveness and potential of the proposed method.


Introduction
e fractional calculus equations are increasingly used to describe problems in optical and thermal systems, material and mechanical systems, signal processing, system identification, control, robotics, and other applications. e theory of fractional calculus has also been paid more and more attention by scholars at home and abroad, and especially, the fractional differential equations abstracted from practical problems have become the research focus of many mathematicians [1][2][3]. Bao and Cao [4] proved that fractionalorder differential equations can well simulate the dynamics of various chemical materials and special materials in practical applications. e traditional integer-order differential equation can be regarded as a special model of fractional-order differential equation. More information on fractional-order differential integration can be found in [5,6]. In recent years, many achievements have been made in the study of fractional-order calculus equations, which are used to study the problems of stability analysis and control of fractional-order strictly feedback nonlinear systems. For example, Li et al. [7] proposed a Lyapunov direct method for fractional-order nonlinear systems. Boroujeni and Momeni [8] proposed a fractional-order state observer for fractional-order nonlinear systems. Yang and Chen [9] presented a finite-time stabilization method for fractional-order switching systems. Adaptive control methods had been used in the control of fractional-order nonlinear systems [5,6,10]. Huang et al. [11] proposed a fuzzy feedback control method for uncertain fractional-order chaotic systems (FOCSs). erefore, the research on the stability and control of FOCSs is becoming more and more popular. e backstepping control was proposed for the first time to obtain asymptotic tracking and global stability of strictly feedback nonlinear systems [12]. e backstepping technique has always been a powerful tool for the design of controllers for nonlinear dynamic systems [13,14]. However, the classical backstepping control has two obvious disadvantages: one is the explosion of terms caused by the repeated derivation of virtual control signals [15][16][17], and the other is certain functions in the systems with uncertain parameters must be linear [18][19][20]. Li et al. [20] presented a fuzzy adaptive output feedback control method to improve the tracking effect of the systems. In addition, Shao-Cheng Tong et al. [21] proposed a backstepping method based on command filters to solve the problem of item explosion caused by repeated derivation of the virtual controller. en, the command filtered backstepping control was extended to the adaptive case in [22]. In each step of the control design, the output of the command filter was used to approximate the differential coefficient of the virtual control signal, which eliminates the problem of item explosion. Note that the error compensation mechanism was designed to eliminate errors generated by the command filters [23]. However, the above works considered only the cases of systems with unknown parameters which are constant and systems without unknown parameters. In addition, the above research results only consider the case of integer-order systems, but the research results of the backstepping control method based on command filtering are relatively few for fractional-order strict feedback systems. In this paper, a backstepping control method based on command filtering is presented for a class of fractional-order strictly feedback nonlinear systems.
Based on the radial-basis-function neural network (NN), an adaptive control method based on approximation theory was proposed to deal with nonlinear systems with uncertain parameters [24][25][26][27][28] or NN [29][30][31] approximation. e neural network control based on the backstepping control method of command filtering is used to solve nonlinear problems in uncertain systems. It can solve the nonlinear system which does not meet the matching condition and the certain functions are nonlinear [32][33][34][35]. For a class of FOCSs, an adaptive controller via the backstepping technology similar to [36,37] was designed in [38]. However, because it is difficult to solve the fractional-order derivative of quadratic function, the adaptive backstepping control technology of integerorder nonlinear system cannot be directly applied to the fractional nonlinear system. For FOCSs, an adaptive backstepping control method was proposed in [39], and the stability of the controller was analyzed by the integerorder Lyapunov method. erefore, for fractional-order nonlinear systems, how to design an adaptive backstepping controller via Lyapunov stability theory is an urgent problem to be solved.
According to the above discussion, an NN adaptive backstepping control method based on command filtering is proposed for FOCSs in this paper. e NN is used to deal with unknown functions in the system. e command filters are proposed to solve the problem of item explosion caused by repeated derivation of virtual controllers, and compensation signals are designed to eliminate the errors caused by the command filter. Based on NN approximation theory, an adaptive backstepping method based on command filtering is presented. e results show that this control method can adjust the tracking error to the small neighborhood of origin and ensure that all the closed-loop system signals have the limit boundedness of half blade uniformity. e main advantages of this approach over current results can be summarized as follows: (i) e command filtered adaptive NN backstepping control can solve two problems of classical backstepping of a class of fractional-order nonlinear systems and reduce the calculation burden.
(ii) An error compensation mechanism is introduced to overcome the shortcomings of the dynamic surface method, and the tracking error is controlled in the small neighborhood of zero.
e rest of this article is organized as follows: Section 2 is about the fuzzy logic system, fractional calculus description, and the fractional-order nonlinear systems. In Section 3, detailed controller design and stability analysis are given. In Section 4, simulation results are given to verify the effectiveness of the backstepping adaptive control method based on command filtering. Finally, Section 5 is the conclusion of this paper.

Preliminaries of Fractional-Order Calculus.
In recent several decades, the relatively common two kinds of fractional-order calculus are Riemann-Liouville (RL) fractionalorder calculus and Caputo fractional-order calculus. e nice property of the Caputo fractional-order differential equation is that its integral value at zero makes sense. erefore, this paper will adopt the definition of Caputo fractional-order calculus. e fractional-order calculus operators can be regarded as a broader concept of integral calculus operators.
us, the definition of the fractional-order integral can be expressed as [ where Γ(·) is the gamma function and Γ(s) � +∞ 0 t s− 1 e − t dt. Consequently, the fractional-order derivative operator is represented by [ (2) e Laplace transform of (2) can be expressed as where X(s) is the Laplace transform of X(t).
In the following sections, we examine only the case of β ∈ (0, 1).
e Mittag-Leffler function can be expressed as with β, c > 0, and z ∈ C. e Laplace transform of (5) can be described as Lemma 1 (see [1]). If there is a complex number c and two real numbers β (0 < β < 1) and ϖ, where ϖ satisfies the following condition: 2 Computational Intelligence and Neuroscience then, the following formula is true for all integers n ≥ 1: when |z| ⟶ ∞ and ϖ ≤ |arg(z)| ≤ n.
Definition 2 (see [7]). Suppose w: [0, c) ⟶ I is a continuous function. If the continuous function is strictly increasing and w(0) � 0, then it is said to be class K.

Preliminaries of Radial-Basis-Function NN. Let h(χ(t))
be a continuous unknown nonlinear function defined over a compact set Ω. en, there exists a radial-basis-function NN to appropriate the unknown nonlinear function h(χ(t)) as follows: where h: R n ⟶ R (n ∈ N) is a continuous mapping, , ψ 2 (χ(t)), . . . , ψ N (χ(t))] T ∈ R N is a regressor, and N is the number of NN nodes. e regression variable ψ ı (χ(t)) selected in this paper is the Gaussian radial-basis-function . . , n) represent the centers of the Gaussian function, and ι ı ∈ R + represent the widths of the Gaussian function. erefore, according to the above notation, the optimal estimate of h(χ(t)) can be expressed as in which ϵ( and θ * is the optimal constant weight vector and satisfies the following condition: Let in which θ is the parameter estimation error. According to the properties of the radial-basis-function NN, it is assumed that the optimal approximation error remains bounded. For any ϵ, there exists a sufficient number of NN nodes such that where ϵ ı is a known positive constant and ı � 1, 2, . . . , N. en, one can obtain in which ı � 1, 2, . . . , N.

FOCSs.
Consider the following FOCSs: where β(0 < β < 1) is the system commensurate order, . . , n) are unknown smooth nonlinear functions, d(t) ∈ R is an unknown external disturbance, and u(t) ∈ R and y(t) ∈ R are control input and control output of the fractional-order strictly feedback nonlinear systems.

Adaptive NN Backstepping Control Design
Let in which x d is a known reference signal and x i,c (i � 2, 3, . . . , n) are the command filters. e command filtering mechanism is defined as follows: where κ i is a positive constant and α i (t) and x i+1,c (t) (i � 1, 2, . . . , n − 1) are the input signal and the output signal in the command filtering mechanism, respectively. It is worth noting that command filtering produces errors that add to the burden of getting better tracking results. To solve this problem, an error compensation mechanism is presented to overcome the errors (x i+1,c − α i ) generated during the filtering process. en, the error compensation mechanism is defined as where k 1i > 0 are design parameters and λ i (i � 1, 2, . . . , n − 1) are the error compensating signals. e compensated tracking error signals e i (t) are designed as in which i � 1, 2, . . . , n.
Our goal is to design the control input u of the nonlinear systems such that the control output y(t) of the system to track x d and the compensated tracking error of the nonlinear systems e 1 (t) eventually converge to a sufficiently small region of origin. en, the virtual control functions α i (i � 1, 2, . . . , n) are constructed as where k 2i > ϵ i (i � 1, 2, . . . , n − 1) are design parameters and ϵ i will be determined later.
e controller u(t) is designed as follows: in which d(t) is the estimation of d, k 2n > ϵ n is a design parameter, and ϵ n will be determined later. en, the backstepping control algorithms of the fractional-order nonlinear system are shown as follows.
Step 2. Consider the second Lyapunov function v 2 (x) as follows: According to Lemma 4, differentiating v 2 (x) can gain Computational Intelligence and Neuroscience Substituting (27) into (41), we can obtain where k 12 > 0 and k 22 > ε 2 are design parameters. Choose the Lyapunov function V 2 (x) as follows: e design of adaptive law is as follows: where ρ 2 and c 2 are positive design parameters. Because θ * 2 is a constant, then we have en, according to Lemma 4 and formulas (42) and (44), we can have According to formulas (45) and (46), the above equation can be reduced to in which k 2 � min k 1 , 2k 12 , c 2 and Θ 2 � Θ 1 + (c 2 / 2ρ 2 )θ * T 2 θ * 2 are two positive constants.
Step 3. i (i � 3, 4, . . . , n − 1). Consider the ith Lyapunov function v i (x) as follows: Substituting (27) into (49), we can obtain where k 1i > 0 and k 2i > ϵ i are design parameters. Choose the Lyapunov function V i (x) as follows: e design of adaptive law is as follows: where ρ i and c i are positive design parameters. Because θ * i is a constant, then we have en, according to Lemma 4, we can have in which V n (s) and Q(s) are Laplace transforms of V n (t) and q(t), respectively, and V n (0)is the initial condition. According to (6) and (71), it can be represented as where * is the convolution operator. If sine q(t) and t − 1 E β,0 (− k n t β ) are both nonnegative functions, then q(t) * t − 1 E β,0 (− k n t β ) ≥ 0 can be obtained. Hence, the following inequality holds on: Because arg(− k n t β ) � − π, | − k n t β | ≥ 0, for all t ≥ 0 and β ∈ (0, 2) and according to Lemma 2, then there exists a positive constant A such that According to (75), one can obtain Consequently, for every η > 0, there exists a constant t r > 0, such that t > t r holds on, then one can obtain In addition, according to Lemma 1, one can obtain in which the integer n is chosen as 1. According to (78), for every η > 0, there exists a constant t b > 0, such that t > t b holds on, one obtains en, we can select appropriate design parameters such that (Θ n /n) ≤ (η/3). erefore, from (74), (77), and (79), the following inequality can be obtained: According to (80) and the definition of V n (t), all signals in the closed-loop system are bounded and the tracking error e 1 (t) tends to an arbitrary small region of the origin, i.e., for all t > max t r , t b , such that (1/2)e 2 1 (t) ≤ η.

Simulation Results
In this section, two examples will be given to demonstrate the effectiveness of the adaptive neural network backstepping control based on the command filtering.
Example 1. e fractional-order Duffing's Oscillator chaos system is as follows: 3 cos t are unknown functions. Let the fractional-order β � 0.95 and the initial conditions x 1 � 0.21 and x 2 � 0.13. In addition, it can be seen from Figures 1 and 2 that the nonlinear system is unstable when both the controller u(t) and the disturbance signal d(t) of the system are zero. e known smooth reference signal x d (t) and the unknown external disturbance signal d(t) are chosen as sin(t) and sin(t) + cos(t), respectively. e parameters to be designed are selected as follows: k 11 � k 12 � 1, k 21 � k 22 � 1, ρ 1 � ρ 2 � 5, c 1 � c 2 � 0.5, ξ 1 � ξ 2 � 1, and κ 1 � 50. In the process of designing the controller, sign(·) is used to represent arctan (10) to avoid the chattering phenomenon.

Computational Intelligence and Neuroscience
When the above initial conditions are met, the simulation of the nonlinear system (81) is shown in Figures 3-7.
As can be seen from Figures 3 and 4, the state variables x 1 (t) and x 2 (t) of the system are tracked by signals x d (t) and x 2c (t) and the tracking effect is relatively well. As shown in Figure 5, the tracking error e 1 (t) rapidly converges to a relatively small neighborhood of the origin. From Figure 6, the control input is large at first, then decreases rapidly, and when the system reaches stability, it is controlled in a relatively small neighborhood. As shown in Figure 7, the adaptive laws of the system are bounded and converge rapidly to the neighborhood of the origin. Consequently, the effectiveness of the adaptive neural network backstepping control method based on command filtering for a class of classical fractional-order nonlinear strictly feedback systems.

Example 2.
e fractional-order Arneodo's chaos system is as follows: where β � 0.97, If the input and disturbance of the system satisfy u(t) � 0 and d(t) � 0, the state variables of the system are shown in Figures 8 and 9. e known smooth reference signal x d (t) and the unknown external disturbance signal d(t) are sin(t) and 2 sin(t)cos(t), respectively. e parameters to be designed are selected as follows:k 11 � 40, k 12 � 25, k 13 � 30, In Example 2, there are three radial-basis-function neural networks. In the first radial-basis-function NN, the input variable is x 1 (t) and four Gaussian functions evenly distributed within the interval [− 3, 3] are designed. e second radial-basis-function uses x 1 (t) and x 2 (t) as input variables. Same as the first radial-basis-function, each input variable corresponds to four Gaussian functions evenly distributed on [− 3, 3]. e third radial-basis-function uses x 1 (t), x 2 (t), and x 3 (t) as input variables. Same as the before radial-basis-functions, four Gaussian functions in every  x 2 ( t ) x 1 ( t ) Figure 8: e uncontrolled fractional-order Arneodo's chaos system (81).

Conclusion
In this paper, an adaptive neural network backstepping control method based on command filtering is proposed for a class of fractional-order strictly feedback nonlinear systems. e command filtering technology is used to deal with the explosive of terms in the traditional backstepping technology, and the error compensation mechanism is introduced to overcome the shortcomings of the dynamic surface method. e simulation results of the fractionalorder nonlinear strictly feedback system show the effectiveness of the proposed method. e future research will be adaptive neural network control of fractional-order nonlinear strictly feedback system with disturbance observer based on command filtering.  Computational Intelligence and Neuroscience 13

Data Availability
All the datasets generated for this study are included within the article.

Conflicts of Interest
e author declares that there are no conflicts of interest regarding the publication of this paper.