Synchronization of Fractional Chaotic Systems with Time-Varying Perturbation

Abstract

Aiming at the synchronization problem of fractional time-varying perturbation systems, an improved WRBF neural network was proposed based on the wavelet function and radial basis function (RBF). Then, the adaptive controller and updated law are derived based on the WRBF network. It is used to approximate functions and adjust the corresponding parameters in the controller. Based on Lyapunov and Barbalat stability theory, the synchronization of a fractional system with time-varying perturbation is proved effectively.

1. Introduction

With the continuous development of technology, the application of fractional calculus in engineering practice and physics has become a hot topic [1,2,3,4,5]. With the upsurge of studying fractional calculus which has been extended to chaotic systems, fractional chaotic systems have seen broad applications in image encryption [6], electrical circuit [7], secure communication [8], biomedical [9], neural network [10], and other fields. Synchronization is one branch of nonlinear science [11], its application scope includes quantum communication [12], biological neural network [13], physical internet [14], etc. Various synchronization control methods have been proposed, which mainly include parameter identification method in [15], hybrid projective combination synchronization method in [16], adaptive sliding mode control approach in [17], Lyapunov-based approach in [18], H^∞ synchronization approach in [19], fuzzy feedback control method in [20], modified projective function synchronization method in [21], active parameter identification method [22], etc. The above methods are all about synchronization of fractional systems. Whether they are suitable for fractional uncertain chaotic systems with external disturbances remains to be studied.

From the practical perspective, the fractional chaotic system is of practical significance for investigating dynamical behavior and its synchronization. In Ref. [23], a parameter identification controller is designed to obtain the synchronization of isomer system in the case of multiple disturbance terms, which realizes combined synchronization in a finite time. Ref. [24] uses the idea of multi-switch sliding mode control to directly eliminate nonlinear terms and realizes the combined synchronization of fractional system with external interference, but its control cost is relatively high. In Ref. [25], the inverse matrix method was used to realize passive synchronization of a fractional memristive neural network. However, the controller needs to know the coefficient matrix of the linear part between the driving and response systems, which is highly dependent on the system model. When there are uncertainties or external disturbances, the method finds it difficult to achieve synchronization. Ref. [26] uses a fuzzy control method to realize fixed-time synchronization. However, the design of this controller is complicated, which is not conducive to its implementation in engineering. In Ref. [27], a discrete time-varying observer was designed to obtain the impulse synchronous control of fractional systems in the case of multiple disturbance terms. The observer realized impulse synchronization. In addition, all the above control methods are based on uncertain or unknown systems, or there is aging and decay of components in the operation process of the system. The above factors make the ideal model. On the one hand, the RBF neural network has been proven to approximate any continuous function [28,29]; based on the above considerations, this manuscript mainly studies the synchronous control of uncertain fractional chaotic systems based on an improved RBF neural network and Wavelet function, that is, wavelet radial-based function neural network. The asymptotic synchronization of a fractional uncertain chaotic system is realized. The novelty in the manuscript under review is on the focus of two parts:

(1)

An improved RBF neural network with a wavelet function is proposed.

(2)

Based on the Barbalat lemma, an adaptive WRBF neural network controller and adaptive law are designed. The designed controller can obtain the synchronization of fractional systems under the condition of external disturbance.

2. Preliminaries

Definition 1 

([30]). The Caputo transformation is described as

$${}_0{}^{C}D_t^{-q}f(t)={\displaystyle {\int }_0^t{(t-\tau )}^{q-1}}f(\tau )d\tau /\mathsf{\Gamma }(q),$$

(1)

The fractional derivative of Caputo [30] is

$${}_0{}^{C}D_t^{q}f(t)={\displaystyle \frac{1}{\mathsf{\Gamma }(n-q)}}{\displaystyle {\int }_0^t\frac{f^{(n)}(\tau )}{{(t-\tau )}^{q+1-n}}}d\tau ,$$

(2)

where $\mathsf{\Gamma }(z)={\displaystyle {\int }_0^{\infty }{t}^{z-1}}{e}^{-t}dt$, and n denotes n-order.

The structure of WRBF [29,30,31,32] is depicted in Figure 1.

$H_i(x)$ is given as

$$H_i(x)=\left(1-{‖{\displaystyle \frac{x-{\mu }_i}{{\sigma }_i^2}}‖}^2\right)\exp \left(-{\displaystyle \frac{{‖x-{\mu }_i‖}^2}{2{\sigma }_i^2}}\right),$$

(3)

The output $y_j$ is given as

$$y_j={\displaystyle \sum _{i=1}^L{w}_{ji}}{H}_i(x)=W_j^{T}H(x),$$

(4)

$$\widehat{Q}=Q+{\epsilon }_{\Delta }={\displaystyle \sum _{j=1}^m{\omega }_{i,j}}{H}_{i,j}(x)+{\epsilon }_{\Delta }=WH(x)+{\epsilon }_{\Delta },i=1,2,\cdots ,n,$$

(5)

where $x\in R^n$, $y\in R^m$, $H(x)$ is the output vector, $L$ denotes the number of hidden layers, $W_j={[w_{j1},w_{j2},\cdots ,w_{jq}]}^T$, $m$ denotes the number of output layers, and ${\mu }_i$ and ${\sigma }_i$ express the center and width, respectively. Figure 2 shows the responses of WRBF and RBF. The unknown function $Q(t,X)$ will be estimated by the WRBF neural network, in which according to Equation (14), the error of estimation can be expressed as $W_i{H}_i+{\epsilon }_{\Delta }-\widehat{W}H=\tilde{W}H+{\epsilon }_{\Delta }$.

3. Synchronization Controller Design

3.1. Description of Synchronization Control Problems

Two fractional systems are considered.

Driven system

$$D_t^{q}x=f(x)+d_m.$$

(6)

Response system

$$D_t^{q}y=g(y)+d_s+u.$$

(7)

where $d_m$ and $d_s$ are external disturbances, $u(t)$ is the controller. The error $e=y-x$. $e={[e_1,e_2,\cdots ,e_n]}^T$. When $\underset{t\to \infty }{\lim }‖e‖=0$, that is, when the synchronization error asymptotically approaches zero, Systems (6) and (7) can realize synchronization.

3.2. Controller Design with WRBF

The error

$$D_t^{q}e=g(y)-f(x)+d_s(t)-d_m(t)+u(t),$$

(8)

Let

$$Q=g(y)-f(x)+d_s(t)-d_m(t),$$

(9)

where $Q$ is an unknown nonlinear function; then, Equation (8) can be rewritten as

$$D_t^{q}e=Q+u(t),$$

(10)

The uncertain function $Q$ is approximated by Equation (5) of the WRBF neural network.

$${\widehat{Q}}_i=W_i^{T}H,$$

(11)

where $W_i$ is the weight vector of the WRBF network. $W_i^{*}$ expresses its optimal estimation parameter, and the optimal estimation is

$${\widehat{Q}}_i^{*}=W_i^{*T}H,$$

(12)

And defining ${\overline{W}}_i=W_i-W_i^{*}$, one obtains

$${\dot{\overline{W}}}_i={\dot{W}}_i,$$

(13)

The synchronization controller is designed as

$$u_i(t)=-W_i^{T}H-{\eta }_i\mathrm{sgn}[e_i(t)]-{\delta }_i{D}_t^{q-1}{e}_i(t),$$

(14)

where ${\eta }_i$ is the optimal constant and ${\delta }_i$ is the estimated value of feedback gain ${\delta }_i^{*}$. The integer order adaptive laws, the estimated value of the upper bound of optimal approximation error, and the estimated value of feedback gain are, respectively, as follows:

$${\dot{W}}_i=k_i[D_t^{q-1}{e}_i(t)]H,$$

(15)

$${\dot{\eta }}_i=m_i{D}_t^{q-1}\left|e_i(t)\right|,$$

(16)

$${\dot{\delta }}_i=n_i{[D_t^{q-1}{e}_i(t)]}^2,$$

(17)

where $k_i,m_i,n_i>0$ are the adjusting parameters of the adaptive update laws, respectively.

Theorem 1.

Given the initial conditions, under the action of the designed adaptive WRBF neural network controller (14) and adaptive laws (15)–(17), the synchronization control of (6) and (7) can be realized.

Proof. 

The synchronization controller (14) can be substituted into Equation (10), and one has

$$D_t^q{e}_i(t)=Q_i-W_i^{T}H-{\eta }_i\mathrm{sgn}[e_i(t)]-{\delta }_i{D}_t^{q-1}{e}_i(t),$$

(18)

According to Equations (5) and (18), one obtains

$$D_t^q{e}_i(t)={\epsilon }_i-{\widehat{W}}_i^{T}H-{\eta }_i\mathrm{sgn}[e_i(t)]-{\delta }_i{D}_t^{q-1}{e}_i(t),$$

(19)

where ${\epsilon }_i={\varphi }_i-{\widehat{\varphi }}_i$.

Define Lyapunov function as follows:

$$V={\displaystyle \frac{1}{2}}{\displaystyle \sum _{i=1}^n{[D_t^{q-1}{e}_i(t)]}^2}+{\displaystyle \frac{1}{2}}{\displaystyle \sum _{i=1}^n\frac{1}{k_i}}{\overline{W}}_i^T{\overline{W}}_i+{\displaystyle \frac{1}{2}}{\displaystyle \sum _{i=1}^n\frac{1}{m_i}}{({\eta }_i-{\epsilon }_i^{*})}^2+{\displaystyle \frac{1}{2}}{\displaystyle \sum _{i=1}^n\frac{1}{n_i}{({\delta }_i-{\delta }_i^{*})}^2},$$

(20)

where ${\epsilon }_i^{*}$ and ${\delta }_i^{*}$ are constants. The derivative of both sides of Equation (20), one has

$$\dot{V}={\displaystyle \sum _{i=1}^n[D_t^{q-1}{e}_i(t)]}[D_t^q{e}_i(t)]+{\displaystyle \sum _{i=1}^n\frac{1}{k_i}}{\overline{W}}_i^T{\dot{\overline{W}}}_i+{\displaystyle \sum _{i=1}^n\frac{1}{m_i}({\eta }_i-{\epsilon }_i^{*}){\dot{\eta }}_i}+{\displaystyle \sum _{i=1}^n\frac{1}{n_i}({\delta }_i-{\delta }_i^{*})}{\dot{\delta }}_i.$$

(21)

When Equation (19) and update laws (15)–(17) are substituted into Equation (21), one has

$$\begin{array}{l}\dot{V}={\displaystyle \sum _{i=1}^n[D_t^{q-1}{e}_i(t)]}\left({\epsilon }_i-{\overline{W}}_i^{T}H-{\eta }_i\mathrm{sgn}(e_i(t))-{\delta }_i{D}_t^{q-1}{e}_i(t)\right)+{\displaystyle \sum _{i=1}^n{\overline{W}}^T\left[\left(D_t^{q-1}{e}_i(t)\right)H\right]}+\\ {\displaystyle \sum _{i=1}^n({\eta }_i-{\epsilon }_i^{*})\left[D_t^{q-1}\left|e_i(t)\right|\right]}+{\displaystyle \sum _{i=1}^n({\delta }_i-{\delta }_i^{*}){\left[D_t^{q-1}{e}_i(t)\right]}^2},\\ ={\displaystyle \sum _{i=1}^n{\epsilon }_i[D_t^{q-1}{e}_i(t)]}-{\displaystyle \sum _{i=1}^n[D_t^{q-1}{e}_i(t)]{\overline{W}}_i^T}H-{\displaystyle \sum _{i=1}^n{\eta }_i\left[D_t^{q-1}\left|e_i(t)\right|\right]}-{\displaystyle \sum _{i=1}^n{\delta }_i{\left[D_t^{q-1}{e}_i(t)\right]}^2}\\ +{\displaystyle \sum _{i=1}^n\left[\left(D_t^{q-1}{e}_i(t)\right){\overline{W}}_i^{T}H\right]}+{\displaystyle \sum _{i=1}^n({\eta }_i-{\epsilon }_i^{*})\left[D_t^{q-1}\left|e_i(t)\right|\right]}+{\displaystyle \sum _{i=1}^n({\delta }_i-{\delta }_i^{*}){\left[D_t^{q-1}{e}_i(t)\right]}^2}\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em} ={\displaystyle \sum _{i=1}^n{\epsilon }_i[D_t^{q-1}{e}_i(t)]}-{\displaystyle \sum _{i=1}^n{\epsilon }_i^{*}\left[D_t^{q-1}\left|e_i(t)\right|\right]}-{\displaystyle \sum _{i=1}^n{\delta }_i^{*}{\left[D_t^{q-1}{e}_i(t)\right]}^2},\end{array}$$

(22)

As $0\le \left|{\epsilon }_i\right|\le {\epsilon }_i^{*}$, $D_t^{q-1}{e}_i(t)\le D_t^{q-1}\left|e_i(t)\right|$,

$$\begin{array}{l}\dot{V}\le {\displaystyle \sum _{i=1}^n{\epsilon }_i^{*}[D_t^{q-1}{e}_i(t)]}-{\displaystyle \sum _{i=1}^n{\epsilon }_i^{*}\left[D_t^{q-1}\left|e_i(t)\right|\right]}-\delta {\displaystyle \sum _{i=1}^n{\delta }_i^{*}{\left[D_t^{q-1}{e}_i(t)\right]}^2}\\ =-{\displaystyle \sum _{i=1}^n{\delta }_i^{*}{\left[D_t^{q-1}{e}_i(t)\right]}^2}\\ =-{\displaystyle \sum _{i=1}^n\delta {\left[D_t^{q-1}{e}_i(t)\right]}^2}\le 0,\end{array}$$

(23)

where $\delta =\min \left\{{\delta }_1^{*},{\delta }_2^{*},\dots ,{\delta }_n^{*}\right\}$. One obtains

$${\displaystyle \frac{1}{2}}{\displaystyle \sum _{i=1}^n{\left[D_t^{q-1}{e}_i(t)\right]}^2}\le V(t),$$

(24)

Thus, ${\displaystyle \frac{1}{2}}{[D_t^{q-1}{e}_i(t)]}^2\le V(t)\le V(0)$, $D_t^{q-1}{e}_i(t)$ is bounded. Similarly, ${\overline{W}}_i$, ${\eta }_i$, and ${\delta }_i$ are bounded. The integral of the inequality $\dot{V}(t)\le -{\displaystyle \sum _{i=1}^n\delta {[D_t^{q-1}{e}_i(t)]}^2}$, one obtains

$$V(t)-V(0)\le -\delta {\displaystyle {\int }_0^t{\displaystyle \sum _{i=1}^n{[D_t^{q-1}{e}_i(\tau )]}^2}}d\tau \le -\delta {\displaystyle {\int }_0^t{[D_t^{q-1}{e}_i(\tau )]}^2}d\tau ,$$

(25)

According to Equation (25), $\underset{t\to \infty }{\lim }{[D_t^{q-1}{e}_i(\tau )]}^{2}d\tau \le V(0)/\delta -\underset{t\to \infty }{\lim }V(t)/\delta <\infty$, that is, $D_t^{q-1}{e}_i(t)\in L_2$. As such, one has

$$‖D_t^q{e}_i(t)‖\le ‖{\overline{W}}_i^T‖‖H‖+{\epsilon }_i^{*}+\left|{\eta }_i\right|+\left|{\delta }_i\right|\left|D_t^{q-1}{e}_i(t)\right|,$$

(26)

According to Equation (3), the function $‖H‖$ is bounded and $D_t^q{e}_i(t)$ is bounded. $D_t^{q-1}{e}_i(t)\in L_2\cap L_{\infty }$ and $D_t^q{e}_i(t)\in L_{\infty }$. According to the Barbalat theorem, one has

$$\underset{t\to \infty }{\lim }{e}_i(t)=0,$$

(27)

Thus, the synchronization error asymptotically approaches zero. The proof is finished.

4. Simulation

For simulation, let the fractional chaotic system be

$$\left\{\begin{array}{l}{D}_t^q{x}_1=-a{x}_1+x_2{x}_3+d_{m1}(t)\hfill \\ D_t^q{x}_2=c{x}_1-a{x}_1{x}_3+d_{m2}(t)\hfill \\ D_t^q{x}_3=x_1{x}_2-b{x}_3+d_{m3}(t)\hfill \end{array}\right.,$$

(28)

When the parameters $q=0.91$, $a=5$, $b=2$, $c=34$, initial value $x(0)={[1,2,3]}^T$. Equation (28) appears chaotic in Figure 3. The bifurcation diagram (BD) of $q$ is shown in Figure 4.

As the time-varying perturbation is added, the driven system can be described as

$$\left[\begin{array}{c}{D}_t^q{x}_1\\ D_t^q{x}_2\\ D_t^q{x}_3\end{array}\right]=\left[\begin{array}{c}-5x_1+x_2{x}_3\\ 34x_1-5x_1{x}_3\\ x_1{x}_2-2x_3\end{array}\right]+\left[\begin{array}{c}0.1\sin (t)+\cos (t)\\ 0.1\sin (t)+\cos (t)\\ 0.1\sin (t)+\cos (t)\end{array}\right],$$

(29)

and the response system with time-varying perturbation is given as

$$\left[\begin{array}{c}{D}_t^q{y}_1\\ D_t^q{y}_2\\ D_t^q{y}_3\end{array}\right]=\left[\begin{array}{c}-5y_1+y_2{y}_3\\ 34y_1-5y_1{y}_3\\ y_1{y}_2-2y_3\end{array}\right]+\left[\begin{array}{c}0.2\sin (t)+\cos (2t)\\ 0.2\sin (t)+\cos (2t)\\ 0.2\sin (t)+\cos (2t)\end{array}\right]+\left[\begin{array}{c}{u}_1(t)\\ u_2(t)\\ u_2(t)\end{array}\right].$$

(30)

In numerical simulation, the hidden layer neuron is set to 9, $\mu =$[−2, −1.5, −1, −0.5, 0, 0.5, 1, 1.5, 2], and the width ${\sigma }_j=0.7$, ($j=$1,2,……9). Using controller (14) and the adaptive update laws (15)–(17), the adjusting parameters of the adaptive law $k_i=1$, $m_i=1$, $n_i=0.1$, and the weights of the RBF neural network $W_i(0)=1$. The initial values of the adaptive law ${\eta }_i(0)=5$, ${\delta }_i(0)=0.1$. From Figure 5, it is observed that the synchronization error converges quickly, which indicates that the designed WRBF neural network has a good approximation effect.

5. Conclusions

In summary, an improved RBF neural network using a wavelet function for time-varying perturbation systems was proposed. A controller was designed. In stability analysis, the derivative of the corresponding Lyapunov function was directly used to avoid the fractional derivative, and the error system asymptotically reached zero by using the correlation lemma. The controller designed in this paper is independent of the system model and can realize synchronous control under completely unknown external disturbance.