Fuzzy neural network-based chaos synchronization for a class of fractional-order chaotic systems: an adaptive sliding mode control approach

Abstract

This paper deals with chaos synchronization problem between two different uncertain fractional-order (FO) chaotic systems with disturbance based on FO Lyapunov stability analysis method. A T–S fuzzy neural network model as a universal approximator is constructed to approximate those uncertain terms and unknown parameters. An adaptive sliding mode control scheme is established, and the adaptive sliding mode control design procedure is proposed, which not only guarantees the stability and robustness of the proposed control method, but also guarantees that the external disturbance on the synchronization error can be attenuated. Finally, simulation results show applicability and feasibility of the proposed control strategy.

1 Introduction

In recent years, fractional-order systems (FOSs) have been receiving intensive studies because the underlying facts about the fractional differentiation are significantly different from those of the integer-order counterparts. Also, numerous physical systems are described more accurately by FO differential equations, for instance dielectric polarization, electromagnetic waves and viscoelastic systems [1], heat and fluid flow processes [2], flight control in uncrewed aerial vehicles [3], and so on. Thus, several results related to stability analysis and control design of FO systems have been reported [4,5,6,7,8]. Many FO control methods including FO sliding mode control [5], FO neural network control [9], FO adaptive control [10, 11], and so on, have been derived by integrating some classic control approaches and FO operators. Despite so, researchers are still interested in the FO systems and FO control design due to many interesting dynamical behaviors such as existence of limit cycle and chaotic behaviors have been observed in FO nonlinear systems [12, 13], which leads the researchers obviously to the investigation of chaotic characteristic, chaos suppression and synchronization of such systems [14, 15]. It should be pointed out that the control techniques for integer-order systems may not to be extended directly to the FO cases. As a result, the chaotic characteristic analysis and chaos synchronization for FO systems is still an attractive and challenging problem, which motivates the work.

A lots of efforts have been devoted to the study of chaotic synchronization control for FO chaotic systems over past few decades due to its potential applications in secure communication and control processing [16,17,18,19,20,21,22,23]. To mention a few, in [16], for instance, two FO hyperchaotic systems can be synchronized based on Lyapunov stability theorem by using direct adaptive interval type 2 fuzzy active sliding control approach. In [20], the impulsive synchronization of fractional-order discrete-time chaotic systems is discussed. In addition, the adaptive synchronization problem of fractional-order memristor-based neural networks with time delay is investigated in [24] by combining the adaptive control, linear delay feedback control, with a fractional-order inequality. The problem of synchronization of FO complex-valued neural networks with time delays is discussed in [25]. The investigation of FO chaos synchronization meets technical obstacles due to the fact that the actual systems have uncertain parameters or uncertain system structure, or it is difficult to be described accurately. One of the most efficient solutions to such problem is probably adaptive control scheme based on fuzzy neural network, which is extensively used for approximating uncertain terms or unknown parameters, which is just as done in [16,17,18, 23, 26]. It should be noted that the considered system structure in [16] is known and fuzzy network is used only to approximate the upper bound of disturbance. However, it is difficult to design a controller with the above-mentioned technique when neither the system structure nor the upper bound of disturbance is known. A typical issue we have to face is the issue of approximating the unknown parameters of the considered system. Therefore, how to overcome this obstacle to design a controller for chaos synchronization of FO systems is a open problem, which is another motivation of this work.

To the best of our knowledge, chaos synchronization problem between two different uncertain FO chaotic systems with disturbance has still not been fully investigated in the existing studies. Inspired by the results in [17], chaos synchronization problem between two different uncertain FO chaotic systems based on adaptive fuzzy sliding mode control technique is investigated in this paper. The main contributions of this work are listed as follows.

• (1) A adaptive sliding mode control technique is established for chaos synchronization of uncertain FO chaotic systems.

• (2) A T–S fuzzy neural network model as a universal approximator is constructed to approximate the unknown functions and disturbances, and FO adaptive laws are designed to update the neuro-fuzzy parameters.

This paper is organized as follows: In Sect. 2, an introduction to fractional derivative and its relation to the approximation solution and some lemmas are addressed. Problem formulation and a brief introduction of the fuzzy neural networks are presented in Sect. 3, and in this section, a controller is designed by sliding mode method with fuzzy neural network model used as an approximator. In Sect. 4, the synchronization error and system stability are analyzed by a FO Lyapunov approach. In Sect. 5, application of the proposed method on FO expression of chaotic system is investigated and the results of numerical simulation are shown. The conclusion is given in Sect. 6.

2 Basic definition and preliminaries for fractional-order systems

Fractional calculus can describe and model real objects more accurately than classical “integer-order” methods. Fractional calculus is denoted by $_aD^q_t$, which is a notation for taking both the fractional integral and derivative in an expression defined as

$$\begin{aligned} {}_{a} D_{t}^{q}=\left\{ {{ \begin{array}{ll} \frac{\mathrm{d}^{q}}{\mathrm{d}t^{q}}, &{}\quad {}q>0 \\ 1, &{}\quad {}q=0 \\ \int \nolimits _{a}^{t}\left( {\mathrm{d}\tau }\right) ^{q}, &{}\quad {}q<0. \\ \end{array}}} \right. \end{aligned}$$

(1)

There are three most common definitions of fractional differential integral in the literature. One of them is Caputo’s derivative, which is used in this paper, defined as:

$$\begin{aligned} {}_{a} D_{t}^{q}f\left( t\right) =\frac{1}{\varGamma \left( {m-q} \right) }\int \nolimits _{a}^{t}\frac{f^{m}\left( \tau \right) }{\left( {t-\tau }\right) ^{1-\left( {m-q}\right) }}\mathrm{d}\tau , \end{aligned}$$

(2)

where $m-1<q<m$ and $\varGamma$ is the Gamma function.

The numerical simulation of a fractional differential equation is not simple as that of an ordinary differential equation. In our case, the fractional Adams–Bashforth–Moulton method is chosen as a representative numerical scheme [27]. To explain this method, the following differential equation is considered

$$\begin{aligned} \left\{ {{\begin{array}{ll} D_{t}^{q}y\left( t\right) =r\left( {t,y\left( t\right) }\right) , &{}\quad 0\le t\le T, \\ y^{\left( k\right) }\left( 0\right) =y_{0}^{\left( k\right) }, &{}\quad k=0,1,\ldots ,m-1. \\ \end{array}}} \right. \end{aligned}$$

(3)

The solution of Eq. (3) is equivalent to Eq. (4)

$$\begin{aligned} y\left( t\right)= & {} \sum \limits _{k=0}^{\lceil q\rceil -1} y_{0}^{\left( k\right) }\frac{t^{k}}{k!}\nonumber \\&+\,\frac{1}{\varGamma \left( q\right) }\int \nolimits _{0}^{t}\left( {t-s} \right) ^{q-1}r\left( {s,y\left( s\right) }\right) \mathrm{d}s. \end{aligned}$$

(4)

Setting $h=T/N$, $t_n=nh$, $n=0,1,\dots ,N$, Eq. (4) can be discretized as follows:

$$\begin{aligned} \begin{aligned} y_{h}\left( {t_{n+1}}\right)&= \sum \limits _{k=0}^{\lceil q\rceil -1} y_{0}^{\left( k \right) }\frac{t^{k}}{k!}+\frac{h^{q}}{\varGamma \left( {q+2} \right) }r\left( {t_{n+1} ,y_{h}^{p}\left( {t_{n+1}}\right) } \right) \\&\quad +\,\frac{h^{q}}{\varGamma ({q+2})}\sum \limits _{j=0}^{n}a_{j,n+1} r({t_{j},y_{h}({t_{j}})}), \end{aligned} \end{aligned}$$

(5)

where

$$\begin{aligned} y_{h}^{p}\left( {t_{n+1}}\right)= & {} {} \sum \limits _{k=0}^{\lceil q\rceil -1} y_{0}^{\left( k\right) }\frac{t^{k}}{k!} +\frac{1}{\varGamma \left( q \right) }\sum \limits _{j=0}^{n}b_{j,n+1} r({t_{j},y_{h}({t_{j}})}),\nonumber \\ a_{j,n+1}\quad= & {} \left\{ {{ \begin{array}{lll} n^{q+1}-\left( {n-q}\right) \left( {n+1}\right) ^{q}, &{}\quad {}j=0 \\ \left( {n-j+2}\right) ^{q+1}+\left( {n-j}\right) ^{q+1}&{}{}\\ \ \quad -2 \left( {n-j+1}\right) ^{q+1}, &{}\quad {}1\le j\le n\\ 1, &{}\quad {}j=n+1 \\ \end{array}}}\right. \nonumber \\ \end{aligned}$$

(6)

and

$$\begin{aligned} b_{j,n+1} =\frac{h^{q}}{q}( ( {n+1-j} ) ^{q}-( {n-j}) ^{q}) . \end{aligned}$$

The approximation error is described as

$$\begin{aligned} \mathop {\max }\limits _{j=0,1,\ldots ,N} \left| {y\left( {t_{j}} \right) -y_{h}\left( {t_{j}}\right) } \right| =0({h^{p}}), \end{aligned}$$

where $p=\min (2,1+q)$.

In the rest of this paper, the operator $D^q$ denotes the Caputo’s fractional differential operator of order q. The following lemmas will be used later.

Lemma 1

[28] For the FO system (7)

$$\begin{aligned} D^qx(t)=f(x(t)), \end{aligned}$$

(7)

with $q\in (0,1)$, $x\in {\mathbb {R}}^n$ is the state vector and $x=\mathbf{0 }$ is the equilibrium point of (7), the equilibrium point of system (7) is stable if

$$\begin{aligned} x(t)f(x(t))\le 0, \quad \forall {x\ne 0}. \end{aligned}$$

And the equilibrium point is asymptotically stable if

$$\begin{aligned} x(t)f(x(t))<0, \quad \forall {x\ne 0}. \end{aligned}$$

Lemma 2

[28] Let $x(t)\in {\mathbb {R}}^n$ be continuous and differentiable. Then, the following inequality holds for any time instant $t\ge t_0$

$$\begin{aligned} \frac{1}{2}D^qx^2(t)\le x(t)D^qx(t), \quad \forall {q}\in (0,1). \end{aligned}$$

Lemma 3

[29] For a given linear autonomous FO system

$$\begin{aligned} D^qx(t)=Ax(t), \end{aligned}$$

(8)

the zero solution to system (8) is asymptotically stable if all eigenvalues $\lambda _i,(i=1,2,\ldots ,n)$ of matrix A satisfy

$$\begin{aligned} |\mathrm{arg}(\lambda _i)|>q\frac{\pi }{2}, \end{aligned}$$

where $0<q\le 1$.

Lemma 4

[30] The trivial solution of system (7) is asymptotically stable if there exists a positive definite Lyapunov function V(x(t)) such that $D^qV(x(t))<0$ for all $t>0$.

3 Problem formulation and controller design

Consider the synchronization of the following two N-dimension FO chaotic systems, the master system:

$$\begin{aligned} \left\{ \begin{array}{ll}D^qx_i=x_{i+1}, &{} \quad 1\le i\le n-1 \\ D^qx_n=g(x,t), &{} \end{array} \right. \end{aligned}$$

(9)

and the slave system:

$$\begin{aligned} \left\{ \begin{array}{ll}D^qy_i=y_{i+1}, &{} \quad 1\le i\le n-1 \\ D^qy_n=f(y,t)+d(t)+u(t), &{} \end{array} \right. \nonumber \\ \end{aligned}$$

(10)

where $x_i, y_i\in {R}, (i=1,2,\ldots ,n)$ are system states, $x=[x_1, x_2, \ldots , x_n]^\mathrm{T}$, $y=[y_1, y_2, \ldots , y_n]^\mathrm{T}$, f(y, t) and g(x, t) are unknown but bounded functions, d(t) is unknown bounded disturbance, u(t) is the controller to be designed.

Define the synchronization error as

$$\begin{aligned} e_i=y_i-x_i, \quad (i=1,2,\ldots ,n); \end{aligned}$$

(11)

then, the following error dynamic is obtained

$$\begin{aligned} \left\{ \begin{array}{l}D^qe_1=e_2, \\ D^qe_2=e_3, \\ \vdots \\ D^qe_n=f(y,t)-g(x,t)+d(t)+u(t), \end{array} \right. \end{aligned}$$

(12)

where $\mathbf{e }=[e_1,e_2,\dots ,e_n]^\mathrm{T}$ is the error vector.

Consider the following sliding mode function

$$\begin{aligned} s=-\mathbf{c }\mathbf{e }=-(c_1e_1+c_2e_2+\cdots +c_{n-1}e_{n-1}+e_n),\nonumber \\ \end{aligned}$$

(13)

where $\mathbf{c }=[c_1,c_2,\ldots ,c_{n-1},1]$ is a constant vector to be designed satisfying condition in Lemma 3 so that the sliding mode surface vanishes quickly. The process can be classified into two phases: One is the approaching phase with $s\ne 0$ and the other is the sliding phase with $s=0$. Our objective is to design the control effort such that the sliding mode condition $sD^qs\le -\eta _{\varDelta }|s|$ ($\eta _{\varDelta }$ is a given positive constant) is satisfied which, by Lemma 1, can guarantee that the trajectory of the synchronization error vector $\mathbf{e }$ moves from approaching phase to sliding phase.

In view of (12) and (13), the following equation holds

$$\begin{aligned} \begin{aligned} D^qs&=-\mathbf{c }D^q\mathbf{e }\\&=-[c_1e_2+c_2e_3+\ldots +c_{n-1}e_n+f(y,t)\\&\quad -\,g(x,t)+d(t)+u(t)]. \end{aligned}\nonumber \\ \end{aligned}$$

(14)

Consequently, the control effort can be designed, by the idea of equivalent control, as

$$\begin{aligned} \left\{ \begin{array}{l}u=u_{eq}+u_{d}, \\ u_{eq}=-\sum ^{n-1}_{i=1}{c_ie_{i+1}}-f(y,t)+g(x,t), \\ u_d=\eta \mathrm{sign}(s), \end{array} \right. \end{aligned}$$

(15)

where $\eta \ge \eta _\varDelta >0$, f(y, t), g(x, t) and $\eta$ are all unknown. These unknown functions and parameter make the control effort (15) not implementable. In order to identify these variables and parameters, a T–S fuzzy neural network model which includes a fuzzy logic system and a neural network is introduced as shown in Fig. 1.

Fig. 1

Configuration of the T–S fuzzy neural networks

Fuzzy logic systems address the imprecision of input and output variables directly by defining them with fuzzy numbers (and fuzzy sets) that can be expressed in linguistic terms. The basic configuration of the T–S fuzzy neural network (FNN) system includes a fuzzy rule base, which consists of a collection of fuzzy if–then rules in the following form:

$$\begin{aligned} R^{(i)}:if \ x_i \ is \ A_1^i \ \mathrm{and} \ \ldots \ \mathrm{and} \ x_n \ \mathrm{is} \ A_n^i \ \mathrm{then} \ y \ \mathrm{is} \ B^i, \end{aligned}$$

where $A_1^i,A_2^i,\ldots A_n^i \$ and $B^i$ are fuzzy sets. The output of the fuzzy logic system with central average defuzzifier, product inference and singleton fuzzifier can be expressed as

$$\begin{aligned} y(X)=\frac{\sum \nolimits _{i=1}^h{{\bar{y}}^i} \left( \prod \nolimits _{j=1}^n{\mu _{A_j^i(x_j)}}\right) }{\sum \nolimits _{i=1}^h \left( \prod \nolimits _{j=1}^n{\mu _{A_j^i(x_j)}}\right) } =\theta ^T\xi (X), \end{aligned}$$

(16)

where h is the number of fuzzy rules, $\mu _{A_j^i(x_j)}$ is the membership function value of the fuzzy variable which is commonly selected as Gaussian function and $\theta ^T=[{\bar{y}}_1^T,{\bar{y}}_2^T,\ldots ,{\bar{y}}_h^T]$ is an adjustable parameter vector. Fuzzy basis function vector is defined as

$$\begin{aligned} \xi ^i(X)=\frac{\prod \nolimits ^n_{j=1} \mu _{A^i_j(x_j)}}{\sum \nolimits ^h_{i=1}\left( \prod \nolimits ^n_{j=1} \mu _{A^i_j(x_j)}\right) }. \end{aligned}$$

(17)

The layer 1 and layer 2 of this network are used for input and membership function, respectively, and layer 3 is used for product fuzzy basis function vector. Network output with adjustable parameter vector can be obtained in layer 4, namely

$$\begin{aligned} y(X)=\theta ^T \xi (X). \end{aligned}$$

(18)

The above fuzzy logic system can uniformly approximate any well-defined nonlinear function over a compact set to any degree of accuracy by the universal approximation theorem [31]. Also it is straightforward to show that a multi-output system can always be approximated by a group of single-output approximation systems.

Hence, the approximation values of f(y, t), g(x, t) and $\eta$ via the T–S fuzzy neural network can be expressed as:

$$\begin{aligned} {\hat{f}}\left( y,{{\theta }_{y}} \right)= & {} \hat{\theta }_{y}^{T}{{\xi }_{y}}\left( y \right) , \nonumber \\ {\hat{g}}\left( x,{{\theta }_{x}} \right)= & {} \hat{\theta }_{x}^{T}{{\xi }_{x}}\left( x \right) , \nonumber \\ \hat{\eta }\left( e,{{\theta }_{e}} \right)= & {} \hat{\theta }_{e}^{T}{{\xi }_{e}}\left( e \right) . \end{aligned}$$

(19)

Substituting (19) into (15), the control effort can be rewritten as

$$\begin{aligned} u=-\underset{i=1}{\overset{n-1}{\sum }}\,{{c}_{i}}{{e}_{i+1}}-{\hat{f}}\left( y,t \right) +{\hat{g}}\left( x,t \right) +\hat{\eta }\mathrm{sign}\left( s \right) .\nonumber \\ \end{aligned}$$

(20)

Define optimal parameter estimations of f(y, t), g(x, t) and $\eta$ as

$$\begin{aligned} \begin{aligned}&\theta _{f}^{*}=\underset{{{{\underset{\scriptscriptstyle -}{\theta _{f} }}}}\in {{\varOmega }_{f}}}{\mathop {\arg \min }}\,\left[ \underset{y\in {{\varOmega }_{y}}}{\mathop {\sup }}\,|{\hat{f}}(y\mid {{\theta }_{f}})-f(y,t)| \right] , \\&\theta _{g}^{*}=\underset{{{{\underset{\scriptscriptstyle -}{\theta _{g} }}}}\in {{\varOmega }_{g}}}{\mathop {\arg \min }}\,\left[ \underset{x\in {{\varOmega }_{x}}}{\mathop {\sup }}\,|{\hat{g}}(x\mid {{\theta }_{g}})-g(x,t)| \right] , \\&\theta _{\eta }^{*}=\underset{{{{\underset{\scriptscriptstyle -}{\theta _{e } }}}}\in {{\varOmega }_{\eta }}}{\mathop {\arg \min }}\,\left[ \underset{e\in {{\varOmega }_{e}}}{\mathop {\sup }}\,|\hat{\eta }(e\mid {{\theta }_{\eta }})-\eta | \right] , \\ \end{aligned} \end{aligned}$$

(21)

and some constraint sets of suitable bounds as

$$\begin{aligned}&{{\varOmega }_{f}}=\{f\mid |f|\le {{M}_{f}}\},\quad {{\varOmega }_{g}}=\{g\mid |g|\le {{M}_{g}}\},\nonumber \\&{{\varOmega }_{\eta }}=\{\eta \mid \eta \le {{M}_{\eta }}\}, \quad {{\varOmega }_{y}}=\{y\mid |y|\le {{M}_{y}}\},\nonumber \\&{{\varOmega }_{x}}=\{x\mid |x|\le {{M}_{x}}\},\quad \ {{\varOmega }_{e}}=\{e\mid |e|\le {{M}_{e}}\}, \end{aligned}$$

(22)

separately, where $M_f,M_g,M_{\eta },M_y,M_x,M_e$ are uncertain positive constants. The block diagram of the controlled system is illustrated in Fig. 2.

Fig. 2

Block diagram of the controlled system

4 Synchronization error and system stability analysis

Consider the Lyapunov function candidate

$$\begin{aligned} V=\frac{1}{2}s^2+\frac{1}{2r_f}{\tilde{\theta }}_f^T {\tilde{\theta }}_f+\frac{1}{2r_g}{\tilde{\theta }}_g^T {\tilde{\theta }}_g+\frac{1}{2r_{\eta }} {\tilde{\theta }}_{\eta }^T{\tilde{\theta }}_{\eta }, \end{aligned}$$

(23)

where $r_f, r_g$ and $r_{\eta }$ are positive constants and ${\tilde{\theta }}_f,{\tilde{\theta }}_g$, and ${\tilde{\theta }}_{\eta }$ are defined as

$$\begin{aligned} \begin{aligned}&{\tilde{\theta }}_f=\hat{\theta }_f-\theta ^*_f, \\&{\tilde{\theta }}_g=\hat{\theta }_g-\theta ^*_g, \\&{\tilde{\theta }}_{\eta }=\hat{\theta }_{\eta }-\theta ^*_{\eta }. \\ \end{aligned} \end{aligned}$$

(24)

Taking the q-order derivative of (23) with respect to time and utilizing Lemma 2 and (24), we get

$$\begin{aligned} \begin{aligned} D^qV&\le sD^qs+\frac{1}{r_f}{\tilde{\theta }}_f^T D^q{\tilde{\theta }}_f \\&\quad +\frac{1}{r_g}{\tilde{\theta }}_g^T D^q{\tilde{\theta }}_g + \frac{1}{r_{\eta }}{\tilde{\theta }}_{\eta }^T D^q\tilde{\theta }_{\eta } \\&\le sD^qs+\frac{1}{r_f}\tilde{\theta }_f^T D^q(\hat{\theta }_f-\theta _f^*) \\&\quad +\,\frac{1}{r_g}\tilde{\theta }_g^T D^q(\hat{\theta }_g-\theta _g^*) \\&\quad +\, \frac{1}{r_{\eta }}\tilde{\theta }_{\eta }^T D^q(\hat{\theta }_{\eta }-\theta _{\eta }^*) \\&\le sD^qs+\frac{1}{r_f}\tilde{\theta }_f^T D^q\hat{\theta }_f +\frac{1}{r_g}\tilde{\theta }_g^T D^q\hat{\theta }_g \\&\quad + \frac{1}{r_{\eta }}\tilde{\theta }_{\eta }^T D^q\hat{\theta }_{\eta }. \\ \end{aligned} \end{aligned}$$

(25)

In view of (13), (15) and (20), (14) can be further expressed as

$$\begin{aligned} \begin{aligned} D^qs&= -\left( f(y,t)-f^*(y,t)+f^*(y,t)-{\hat{f}}(y,t) \right) \\&\quad +\,g(x,t)-g^*(x,t)+g^*(x,t)\\&\quad -\,{\hat{g}}(x,t) -d(t)-\hat{\eta }\mathrm{sign}(s), \\ \end{aligned} \end{aligned}$$

(26)

where

$$\begin{aligned} \begin{aligned}&{{f}^{*}}\left( y,{{\theta }_{f}} \right) ={{\theta }^{*}}_{y}^{T}{{\xi }_{y}}\left( y \right) , \\&{{g}^{*}}\left( x,{{\theta }_{g}} \right) ={{\theta }^{*}}_{x}^{T}{{\xi }_{x}}\left( x \right) , \\&{{\eta }^{*}}\left( e,{{\theta }_{\eta }} \right) ={{\theta }^{*}}_{\eta }^{T}{{\xi }_{e}}\left( e \right) , \\ \end{aligned} \end{aligned}$$

(27)

are optimal estimations of functions f, g and $\eta$.

Define the optimal approximation error as

$$\begin{aligned}&{{d}_{d}}\left( t \right) = -\left( f\left( y,t \right) -{{f}^{*}}\left( y,t \right) \right) \\&\quad +\left( g\left( x,t \right) -{{g}^{*}}\left( x,t \right) \right) . \end{aligned}$$

With (19) and (27), (26) is rewritten as

$$\begin{aligned} \begin{aligned} {{D}^{q}}s&= -\left( {{\theta }^{*}}_{y}^{T}-\hat{\theta }_{y}^{T} \right) {{\xi }_{y}}\left( y \right) +\left( {{\theta }^{*}}_{x}^{T}-\hat{\theta }_{x}^{T} \right) {{\xi }_{x}}\left( x \right) \\&\qquad +\,{{d}_{d}}\left( t \right) -d\left( t \right) -\left( \hat{\theta }_{e}^{T}{{\xi }_{e}}\left( e \right) \right) \mathrm{sign}(s). \end{aligned} \end{aligned}$$

(28)

Substituting (28) into (25), the following inequality holds

$$\begin{aligned} \begin{aligned} {{D}^{q}}V&\le s\left( {{{\tilde{\theta }}}_{f}}^{T}{{\xi }_{y}}\left( y \right) -{{{\tilde{\theta }}}_{g}}^{T}{{\xi }_{x}}\left( x\right) +{{d}_{d}}\left( t \right) -d\left( t \right) \right) \\&\quad -\,s\left( \tilde{\theta }_{\eta }^{T}{{\xi }_{e}}\left( e \right) signs+{{\theta }^{*}}_{\eta }^{T}{{\xi }_{e}}\left( e \right) signs\right) \\&\quad +\,\frac{1}{{{r}_{f}}}{{{\tilde{\theta }}}_{f}}^{T}{{D}^{q}}{{{\hat{\theta }}}_{f}}+\frac{1}{{{r}_{g}}}{{{\tilde{\theta }}}_{g}}^{T}{{D}^{q}}{{{\hat{\theta }}}_{g}}+\frac{1}{{{r}_{\eta }}}{{{\tilde{\theta }}}_{\eta }}^{T}{{D}^{q}}{{{\hat{\theta }}}_{\eta }} \\&\le {{{\tilde{\theta }}}_{f}}^{T}\left( \frac{1}{{{r}_{f}}}{{D}^{q}}{{{\hat{\theta }}}_{f}}+s{{\xi }_{y}}\left( y \right) \right) \\&\quad +\,{{{\tilde{\theta }}}_{g}}^{T}\left( \frac{1}{{{r}_{g}}}{{D}^{q}}{{{\hat{\theta }}}_{g}}-s{{\xi }_{x}}\left( x \right) \right) \\&\quad +\,{{{\tilde{\theta }}}_{\eta }}^{T}\left( \frac{1}{{{r}_{\eta }}}{{D}^{q}}{{{\hat{\theta }}}_{\eta }}-\left| s \right| {{\xi }_{e}}\left( e \right) \right) \\&\quad +\,s\left( {{d}_{d}}\left( t \right) -d\left( t \right) \right) -\left( {{\theta }^{*}}_{\eta }^{T}{{\xi }_{e}}\left( e \right) \right) \left| s \right| . \end{aligned} \end{aligned}$$

(29)

Now, defining the adaptive laws of adjustable parameter vectors as

$$\begin{aligned} \begin{aligned}&{{D}^{q}}{{{\hat{\theta }}}_{f}}=-{{r}_{f}}s{{\xi }_{y}}\left( y \right) , \\&{{D}^{q}}{{{\hat{\theta }}}_{g}}={{r}_{g}}s{{\xi }_{x}}\left( x \right) , \\&{{D}^{q}}{{{\hat{\theta }}}_{\eta }}={{r}_{\eta }}\left| s \right| {{\xi }_{e}}\left( e \right) , \\ \\ \end{aligned} \end{aligned}$$

(30)

and substituting (30) into (29), we have

$$\begin{aligned} \begin{aligned} {{D}^{q}}V&= s\left( {{d}_{d}}\left( t \right) -d\left( t \right) \right) -\left| s \right| {{\theta }^{*}}_{\eta }^{T}{{\xi }_{e}}\left( e \right) \\&= \left| s \right| \left| {{d}_{d}}\left( t \right) -d\left( t \right) \right| -\left| s \right| {{\eta }^{*}}. \\ \end{aligned} \end{aligned}$$

(31)

It is the fact, by the assumptions in (22) that $d_d$ is bounded which together with the assumption that disturbance d(t) is bounded, one can obtain the following inequalities with some positive constant $\eta ^*$

$$\begin{aligned} \begin{aligned}&\left| {{d}_{d}}\left( t \right) -d\left( t \right) \right| -{{\eta }^{*}}<0, \\&{{D}^{q}}V\le \left| s \right| \left( \left| {{d}_{d}}\left( t \right) -d\left( t \right) \right| -{{\eta }^{*}} \right) <0. \end{aligned} \end{aligned}$$

(32)

Therefore, ${{D}^{q}}V<0$ holds, which means that, by Lemma 4, asymptotical stability of system (12) is guaranteed and synchronization of systems (9) and (10) can be achieved with controller (20) and the adaptive law (30). Hence, the following result is obtained.

Theorem 1

Consider two N-dimension FO chaotic systems, master system (9) and slave system (10), and the control effort of the slave system is given in (15) with the fuzzy adaptive laws in (30). Then, under the effect of controller (20), the resulting closed-loop system is global asymptotically stable and the synchronization error will converge to zero asymptotically.

Remark 1

It is worth noting that the undesired chattering phenomenon is unavoidable when the sign function is introduced to controller design in (15) to estimate the effect caused by external disturbances and approximation errors. To solve the problem, the sigmoid function with good smoothness is used to replace the sign function, so that it can not only realize the function of sign function but also restrain the chattering phenomenon.

5 Simulation examples

In this section, we will consider two examples to illustrate the effectiveness of our proposed design method.

Example 1

Consider the chaos synchronization of two nonidentical FO uncertain chaotic systems with the order $q=0.98$, which is chosen from [32].

Master system: FO Duffing–Holmes system

$$\begin{aligned} \left\{ \begin{array}{ll}D^qx_1=x_{2}, &{} \\ D^qx_2=x_1-ax_2-x_1^3+b\cos (t)+d_{x}(t). &{} \end{array} \right. \end{aligned}$$

(33)

Slave system: FO Gyros system

$$\begin{aligned} \left\{ \begin{array}{ll}D^qy_1=y_{2}, &{} \\ D^qy_2=-\epsilon ^2\frac{(1-\cos y_{1})^2}{(\sin y_{1})^3}\\ -\beta \sin y_1 -\psi _1y_1-\psi _2y_2^3\\ \,\,\,\,\,\,\,+(\beta +\lambda \sin (\omega t))\sin y_1+u+d_{y}(t). &{} \end{array} \right. \end{aligned}$$

(34)

The parameters are chosen as

$$\begin{aligned} \begin{array}{lllll}a=0.25, &{} b=0.3, &{} \epsilon =10, &{} \beta =1, \psi _{1}=0.5, \\ \psi _{2}=0.05, &{} \omega =2, &{} \lambda =35, &{} r_{f}=r_g=r_{\eta }=50. \end{array} \end{aligned}$$

(35)

Under the selected parameters, the equilibrium of system (33) is unstable, which can be verified by using the first Lyapunov method. In fact, there are two real eigenvalues of the Jacobian matrix, one of which is positive, in the equilibrium point of system (33). Also, the maximum Lyapunov exponent of system (33) is computed as 0.0773, which shows the FO Duffing–Holmes system is chaotic. The Lyapunov exponent spectrums are shown in Fig. 3.

Fig. 3

Lyapunov exponent spectrums of system (33)

For the purpose of numerical simulations, it is assumed that the external disturbances are $d_x(t)=0.1\cos (3t)$ in the master system and $d_y(t)=0.2\cos (t)$ in the slave system, respectively. The initial values of the master and slave systems are set as $x(0)=[0.3,-0.2]^\mathrm{T}$ and $y(0)=[-0.1,0.2]^\mathrm{T}$, respectively.

The phase portrait of the FO Duffing–Holmes master system and FO Gyros slave system without control effort is depicted in Fig. 4, which shows the dynamic behaviors of the two systems are different. The membership functions for $x_i$ and $y_i,(i=1,2)$ are selected as the following Gaussian functions:

$$\begin{aligned} {\mu }A^i_1{(x_i)}= & {} \exp [-(x_i+4)^2], \nonumber \\ {\mu }A^i_2{(x_i)}= & {} \exp [-(x_i+2)^2], \nonumber \\ {\mu }A^i_3{(x_i)}= & {} \exp [-(x_i)^2], \nonumber \\ {\mu }A^i_4{(x_i)}= & {} \exp [-(x_i-2)^2], \nonumber \\ {\mu }A^i_5{(x_i)}= & {} \exp [-(x_i-4)^2]. \end{aligned}$$

(36)

Refer to remark 1. By computing the adaptive laws in (30) and the estimated values of variables and parameters in (19), with sliding mode surface $s=-150e_1-e_2$, the control effort of the slave system can be obtained as

$$\begin{aligned} u(t)=-c_1e_2-{\hat{f}}(y,t)+{\hat{g}}(x,t)+\hat{\eta }\mathrm{simoid}(-3s).\nonumber \\ \end{aligned}$$

(37)

Under the control effort of (37), the phase portrait of the synchronization performance of the master and slave systems is shown in Fig. 5. Figures 6 and 7 show the synchronization error dynamics $e_1$ and $e_2$, respectively. Figures 8 and 9 show the trajectories of the states $x_1,y_1$ and $x_2,y_2$, respectively. It is noticeable that the tracking error (or synchronization error) converges to zero asymptotically. The dynamic behavior of sliding mode surface s(t) is shown in Fig. 10. Obviously, the value of switching surface converges to zero speedily. It is also noticeable that the chattering phenomenon, which is usually considered as a drawback of conventional sliding mode control, does not appear in our design.

For the sake of comparison, systems (33) and (34) are also simulated by using the method proposed in [32] with the same uncertainties and external disturbances. The simulation results are shown in Fig. 11. To compare with our method, the better synchronization result can be obtained from the perspective of the synchronization time of states by utilizing our method, which further highlights the validity and superiority of our control method.

Fig. 4

Phase portrait of master and slave systems without control. a FO Duffing–Holmes system and b FO Gyros system

Fig. 5

Phase portrait of Duffing–Holmes master system and Gyros slave system with control

Fig. 6

Error dynamics of the states $x_1$ and $y_1$

Fig. 7

Error dynamics of the states $x_2$ and $y_2$

Fig. 8

Synchronization behaviors of the states $x_1$ and $y_1$

Fig. 9

Synchronization behaviors of the states $x_2$ and $y_2$

Fig. 10

Dynamic behavior of sliding surface

Fig. 11

Errors dynamics of Duffing–Holmes master system and Gyros slave system with the method in [32]

Example 2

Let us consider two different uncertain fractional-order Jerk chaotic systems with the order $q=0.95$ as follows.

Master system:

$$\begin{aligned} \left\{ \begin{array}{ll}D^qx_1=x_{2}, &{} \\ D^qx_2=x_3, &{} \\ D^qx_3=a_{21}x_3-x_2+x_1+b_{21}\sinh (x_1) &{} \end{array} \right. \end{aligned}$$

(38)

Slave system:

$$\begin{aligned} \left\{ \begin{array}{ll}D^qy_1=y_{2}, &{} \\ D^qy_2=y_3, &{} \\ D^qy_3=a_{22}y_3-y_2+y_1\\ \qquad +b_{22}\sinh (y_1)+u(t)+d(t) &{} \end{array} \right. \end{aligned}$$

(39)

Choose the parameters as

$$\begin{aligned}&a_{21}=0.6, \quad b_{21}=-0.5, \quad c_1=35, \quad a_{22}=0.68,\\&b_{22}=-0.54, \quad c_2=20, \quad r_f =r_g=r_\eta =50. \end{aligned}$$

Under these parameters, the maximum Lyapunov exponent of system (38) is computed as 0.0601, which shows this system is chaotic and surely unstable. In addition, the external disturbance is set as $d(t)=0.05\sin (2t)$ and the initial conditions of the master and slave systems are chosen as $x(0)=[-0.81, -1.02, -0.5]^\mathrm{T}$ and $y(0)=[0.8, 1.2, 0.5]^\mathrm{T}$, respectively. The membership functions for $x_i$ and $y_i,(i=1,2,3)$ are selected as the following Gaussian functions:

$$\begin{aligned} {\mu }A^i_1{(x_i)}= & {} \exp [-(x_i+3)^2],\nonumber \\ {\mu }A^i_2{(x_i)}= & {} \exp [-(x_i+1)^2], \nonumber \\ {\mu }A^i_3{(x_i)}= & {} \exp [-(x_i)^2], \nonumber \\ {\mu }A^i_4{(x_i)}= & {} \exp [-(x_i-1)^2], \nonumber \\ {\mu }A^i_5{(x_i)}= & {} \exp [-(x_i-3)^2]. \end{aligned}$$

(40)

The control effort of the slave system can be obtained as

$$\begin{aligned} u(t)=-c_1e_2-c_2e_3-{\hat{f}}(y,t)+{\hat{g}}(x,t)+\hat{\eta }\mathrm{sign}(s).\nonumber \\ \end{aligned}$$

(41)

For free of control input, the phase portrait of the master and slave systems is given in Fig. 12 and the 2-D phase trajectories of different variables are shown in Figs. 13, 14 and 15, respectively. Under the proposed control effort, the 3-D phase portrait of synchronization of the master and slave systems is shown in Fig. 16, and the 2-D phase portrait of synchronization of the master and slave systems is shown in Figs. 17, 18 and 19, respectively. Figures 20, 21 and 22 show the error dynamics. Figures 23, 24 and 25 show the synchronization behaviors of state variables of the master and slave systems. It is noticeable that tracking errors converge to zero asymptotically, which verifies the effectiveness of our proposed control design method.

Fig. 12

3-D phase portrait of the Jerk chaotic master and slave systems without control

Fig. 13

2-D projection of the master and slave systems on $(x_1, x_2)$-plane and $(y_1 ,y_2)$-plane without control

Fig. 14

2-D projection of the master and slave systems on $(x_1, x_3)$-plane and $(y_1 ,y_3)$-plane without control

Fig. 15

2-D projection of the master and slave systems on $(x_2, x_3)$-plane and $(y_2 ,y_3)$-plane without control

Fig. 16

3-D phase portrait of the jerk chaotic master and slave systems with control

Fig. 17

2-D projection of the master and slave systems on $(x_1, x_2)$-plane and $(y_1 ,y_2)$-plane with control effort

Fig. 18

2-D projection of the master and slave systems on $(x_1, x_3)$-plane and $(y_1 ,y_3)$-plane with control effort

Fig. 19

2-D projection of the master and slave systems on $(x_2, x_3)$-plane and $(y_2 ,y_3)$-plane with control effort

Fig. 20

Error dynamic of the states $x_1$ and $y_1$

Fig. 21

Error dynamic of the states $x_2$ and $y_2$

Fig. 22

Error dynamic of the states $x_3$ and $y_3$

Fig. 23

Synchronization behavior of the state variables $x_1$ and $y_1$

Fig. 24

Synchronization behavior of the state variables $x_2$ and $y_2$

Fig. 25

Synchronization behavior of the state variables $x_3$ and $y_3$

6 Conclusion

In this paper, an FO adaptive sliding mode control is proposed for chaos synchronization between two different uncertain FO chaotic systems with disturbance based on FO Lyapunov approach. A adaptive sliding mode control technique is established for chaos synchronization of FO chaotic systems with external disturbances. A T–S fuzzy neural network model is introduced to approximate the uncertain terms and unknown upper bounds of some parameters. Furthermore, the corresponding adaptive laws are designed to tune online a set of free parameters. Two simulation examples, one of which consists of two nonidentical FO uncertain chaotic system, are given to demonstrate the effectiveness of the proposed method. Simulation results show that an asymptotical synchronization of the master and the slave systems can be achieved. On the other hand, how to eliminate the chattering caused by sign function in another way as well as to design a different sliding mode surface so that the systems have better synchronization effect is one of our primary works in the future.