Synchronisation of 6D hyper-chaotic system with unknown parameters in the presence of disturbance and parametric uncertainty with unknown bounds

Abstract

In this paper, adaptive-sliding mode control method is proposed for synchronisation of two 6D hyper-chaotic systems in the presence of external disturbance and parametric uncertainty and unknown parameters in the slave system. In the first section of this paper, two 6D integer order hyper-chaotic systems in the presence of external disturbance signal, parametric uncertainty and unknown parameters in the slave system are studied. In the second part of this paper, after identifying chaos in fractional order dynamic of the mentioned system, synchronisation of two 6D fractional derivative hyper-chaotic systems in the presence of external disturbance signal, parametric uncertainty and unknown parameters in the slave system is investigated, in which fractional order Riemann-Liouville derivative is used; a new fractional order sliding surface is defined for the hyper-chaotic system to determine the proper active control. Proper adaptive control laws are used to estimate the uncertainty bound, unknown disturbance signal and system parameters. Stability of the closed-loop control system is proved using Lyapunov theory in both modes. Simulation results in MATLAB show the desired application of the proposed controllers in the presence of disturbance and parametric uncertainty.

KEYWORDS:

• 6D hyper-chaotic system
• fractional order system
• adaptive-sliding mode control

1. Introduction

Chaos theory has important features like nonlinear dynamic and sensitivity to initial condition. The main characteristic of the chaotic systems is sensitivity to initial conditions. Edward Lorenz published results of differential system calculations of atmospheric thermal displacement and observed that in a certain range of factors, without the involvement of random elements or external disturbance, irregular fluctuations emerge in the system response. He concluded that a small change in the initial condition of the equations predicting atmospheric condition, results in fluctuations in system response and considerable changes in the results. Chaos has attracted attention in many sciences including secure communication (Belkhouja, Du, Mohamed, Al-Ali, & Guizani, 2018), nonlinear circuits (Chithra & Raja Mohamed, 2017), power electronics (Kumar, Banerjee, Kumar, & Kumar, 2018; Zribi, Alrifai, & Smaoui, 2018), robotics (Zang, Iqbal, Zhu, Liu, & Zhao, 2016), etc.

Lyapunov exponent was used before the emergence of chaos theory for specifying stability of linear and nonlinear systems. Lyapunov exponent is calculated through measuring the extension or contraction of a system. In fact, in this method, the mean speed with which two-point transition paths which were close at the beginning and deviate from each other exponentially, is calculated. Various methods have been proposed for calculating Lyapunov exponent among which Jacobin matrix method can be mentioned in which a negative Lyapunov exponent indicates convergence rate and a positive Lyapunov exponent indicates divergence rate. Chaotic systems of higher dimensions are called hyper-chaotic systems. Hyper-chaotic systems which have at least one positive Lyapunov exponent have more irregular and complicated behaviour compared to the chaotic systems which are used in many applications including secure communication (Liu et al. 2018) and cryptography (Lagmiri, Amghar, & Sbiti, 2017).

Fractional calculus improves the efficiency of the control systems in two ways: (a) improving modelling quality. (b) Improving the efficiency of the controllers. Fractional calculus helps accurate modelling of the processes and control systems. It is obvious that using a more accurate model of the system, a more appropriate controller can be designed for controlling the intent system and process.

There are many fractional order systems with fractional order dynamic in which chaos and hyper-chaos phenomena are observed. Like Lu integer order hyper-chaotic system (Wang, 2015) and Lu fractional order hyper-chaotic system (Huang, Wang, Zhang, & Guo, 2018), Lorenz integer order hyper-chaotic system (Chen, 2018; Yang & Bai, 2017) and Lorenz fractional order hyper-chaotic system (Khamsuwan, Sangpet, & Kuntanapreeda, 2018), Liu integer order hyper-chaotic system (Wang, 2016) and Liu fractional order hyper-chaotic system (Qiang, Chong-Xin, Lei, & Da-Rui, 2013) which have attracted attention of researchers.

Various studies in recent decades have been conducted on synchronisation of integer order hyper-chaos like Daltzis, Volos, Nistazakis, Tsigopoulos, and Tombras (2018), and Lu et al. (2018) in which synchronisation of hyper-chaotic systems using adaptive control has been studied. In Ahmad, Saaban, Ibrahim, and Shahzad (2018), synchronisation of hyper-chaotic system using active control has been investigated. In Rajagopal, Vaidyanathan, Karthikeyan, and Srinivasan (2018), synchronisation of hyper-chaotic systems using adaptive sliding mode control has been studied. In the study by Vaidyanathan et al. (2018), Vaidyanathan et al. (2018), and Vaidyanathan (2017), synchronisation of hyper-chaotic systems with adaptive backstepping control has been proposed. In Wu, Xiong, Hu, Feng, and Li (2018), synchronisation of hyper-chaotic systems with sandwich control has been investigated. In the presented papers, dimension of chaotic systems is less than 6, but in recent years, researchers have focused considerably on fractional order chaotic systems. But, fewer studies have been conducted on fractional order hyper-chaotic systems among which Noghredani and Balochian (2015) study can be mentioned which has investigated synchronisation of fractional order hyper-chaotic systems using sliding mode control. In Khan and Tyagi (2017), and Khan and Tyagi (2018), two fractional order chaotic systems with adaptive-sliding mode control have been synchronised. In Zhou, Liu, Cao, and Li (2019), two fractional order neural network systems with fuzzy control have been synchronised in Caputo definition of fractional derivative. In Wang, Ni, and Yang (2018), fractional order hyper-chaotic systems with fuzzy control have been synchronised. In Khan, Khattar, and Agrawal (2018), and Liu, Chen, Li, Xiang, and Xu (2017), feedback control has been used for synchronisation of fractional order systems. In Fu and Liu (2018), fractional order hyper-chaotic systems with adaptive-fuzzy control have been synchronised and in Liu et al. (2018), synchronisation of two identical fractional order neural networks with full/under-actuation and Caputo definition of fractional derivative is presented via fractional order sliding mode controller, and a sufficient condition is given to seek the control gain matrix by solving LMI.

In many of these studies, synchronisation of fractional order hyper-chaotic systems which their dimension is less than 6 has been investigated.

One important application of chaos in information exchange is security and protection. In this communication, chaotic systems can be used to increase the security of the exchanged information. Since the behaviour of chaotic systems is close to the behaviour of complicated and noisy systems, information can be hidden in such systems with high reliability. The main idea of information exchange using chaotic systems is based on embedding information in the transmitter which creates a chaotic signal. In secure communication, high order chaotic systems and fractional order chaotic systems can be used due to higher complexity. There are many studied on the application of chaos and hyper-chaos in secure communication (Djennoune, Bettayeb, & Al-Saggaf, 2019; Ouannas, Bendoukha, Volos, Boumaza, & Karouma, 2019).

In most studies of chaos and synchronisation, authors have used sliding mode controller design for synchronising integer order or fractional order chaotic and hyper chaotic systems in the presence of disturbance, uncertainty and unknown parameters which are not simultaneous. Also, fractional order chaotic systems which their order is 6 or higher than 6 have not been synchronised.

But in this paper, not only 6D like-Lorenz fractional order chaotic system is identified but also two similar 6D like-Lorenz chaotic systems in the presence of disturbance and bounded uncertainty with unknown bounds and unknown system parameters are synchronised using adaptive-sliding mode controller in which, stability and estimation of unknown parameters of the systems are proved using Lyapunov theorem. Efficiency of the proposed method is shown in a practical application which is considered to be secure communication.

Briefly, main contributions of this paper are as follows:

• Identification of new fractional order dynamic of the 6D hyper-chaotic systems

• Synchronisation of identified 6D fractional order hyper-chaotic systems

• Proper adaptive control laws are used to estimate uncertainty and disturbance signal bounded with unknown bounds and unknown system parameters for 6D fractional order hyper chaotic system.

In this paper, synchronisation of 6D hyper-chaotic systems in the presence of disturbance and parametric uncertainty and unknown parameters in the slave system is investigated in two integer order and fractional order dynamic modes. To this end, adaptive-sliding model control along with the Lyapunov theorem is used to design the proper controller. Adaptive-sliding mode control is designed as the control method for synchronisation due to its advantages including simple and easy implementation, fast response, good transient performance and robustness against system uncertainty and disturbance. The stability of synchronisation control is proved using Lyapunov theorem in which states of the slave system converge to states of the master system asymptotically. The general structure of this paper is as follows: in Section 2, the dynamic model of the 6D hyper-chaotic model is introduced and synchronisation of two 6D integer order hyper-chaotic systems in the presence of uncertainty with disturbances of unknown bound and unknown parameters is described. In Section 3, dynamic model of the 6D fractional order hyper-chaotic system is introduced and synchronisation of two 6D fractional order hyper-chaotic systems in the presence of uncertainty with disturbances of unknown bound and unknown parameters is described. Results of numerical simulation using MATLAB are presented in Section 4. Finally, the paper is concluded in Section 5.

2. Introduction of the dynamic model of the 6D hyper chaotic system

In 2009, a linear feedback controller and a nonlinear feedback control were proposed for 5D Lorenz hyper-chaotic system (Hu, 2009). (1) $\{\begin{array}{l}{\dot{x}}_1=a(x_2-x_1)+x_4\\ {\dot{x}}_2=c{x}_1-x_1{x}_3-x_5-x_2\\ {\dot{x}}_3=x_1{x}_2-b{x}_3\\ {\dot{x}}_4=d{x}_4-x_1{x}_3\\ {\dot{x}}_5=h{x}_2\end{array}$(1) Considering system (1), a novel 6D chaotic system similar to Lorenz system is presented which is formed by adding linear feedback and nonlinear feedback to the above system which has four positive Lyapunov exponents. Therefore, one version of the 6D hyper-chaotic system is considered as follows: (2) $\{\begin{array}{l}{\dot{x}}_1=a(x_2-x_1)+x_4\\ {\dot{x}}_2=c{x}_1-x_1{x}_3+x_5-x_2\\ {\dot{x}}_3=x_1{x}_2-b{x}_3\\ {\dot{x}}_4=d{x}_4-x_1{x}_3\\ {\dot{x}}_5=-h{x}_2+x_6\\ {\dot{x}}_6=-k_1{x}_1+k_2{x}_2\end{array}$(2) where $X=[x_1,x_2,x_3,x_4,x_5,x_6{]}^T$ is the state vector of the system and $a,b,c,d,h,k_1,k_2$ are constants of the system.

System (1) is a hyper-chaotic system. Because it has more than two Lyapunov exponents. To calculate Lyapunov exponent with according to (Vallejo & Sanjuan, 2017) we have:

Jacobian matrix of system (1) is as follows: $J=\left[\begin{array}{cccccc}-a& a& 0& 1& 0& 0\\ c-x_3& -1& -x_1& 0& 1& 0\\ x_2& x_1& -b& 0& 0& 0\\ -x_3& 0& -x_1& d& 0& 0\\ 0& -h& 0& 0& 0& 1\\ -k_1& k_2& 0& 0& 0& 0\end{array}\right]$Lyapunov exponents and Lyapunov exponent graph of the 6D hyper chaotic system are as follows. $\begin{array}{rl}{\lambda }_{L{E}_1}& =-22.6839,{\lambda }_{L{E}_2}=11.3461,{\lambda }_{L{E}_3}=0.3259,{\lambda }_{L{E}_4}=0.0119,{\lambda }_{L{E}_5}=-2.6667,{\lambda }_{L{E}_6}\\ & =2.0000\end{array}$

In addition, considering disturbance and uncertainty in parameters, the slave system is described as follows:

(3) $\{\begin{array}{l}{\dot{y}}_1=\overline{a}(y_2-y_1)+y_4+w_1+\mathrm{\Delta }{f}_1+u_1\\ {\dot{y}}_2=\overline{c}{y}_1-y_1{y}_3+y_5-y_2+w_2+\mathrm{\Delta }{f}_2+u_2\\ {\dot{y}}_3=y_1{y}_2-\overline{b}{y}_3+w_3+\mathrm{\Delta }{f}_3+u_3\\ {\dot{y}}_4=\overline{d}{y}_4-y_1{y}_3+w_4+\mathrm{\Delta }{f}_4+u_4\\ {\dot{y}}_5=-\overline{h}{y}_2+y_6+w_5+\mathrm{\Delta }{f}_5+u_5\\ {\dot{y}}_6=-{\overline{k}}_1{y}_1+{\overline{k}}_2{y}_2+w_6+\mathrm{\Delta }{f}_6+u_6\end{array}$(3)

in which, $U^T=[u_1,u_2,u_3,u_4,u_5,u_6{]}^T$ is the control signal, $Y^T=[y_1,y_2,y_3,y_4,y_5,y_6{]}^T$ is state variable vector of the slave system, $||\mathrm{\Delta }{f}_i||\le {\gamma }_i,i=1,\dots ,6$ is an uncertain bounded function of the states with unknown bound and $||w_i||\le {\beta }_i,i=1,\dots ,6$ is the bounded disturbance input with unknown bound and $\overline{a},\overline{b},\overline{c},\overline{d},\overline{h},{\overline{k}}_1,{\overline{k}}_2$ are unknown parameters of the slave system.

Figures 1 and 2 show the hyper chaotic system and Lyapunov exponent graph for the parameter set of $a=10,b=8/3,c=28,d=2,h=8.8,k_1=1,k_2=0.9$, respectively.

Figure 1. Images of Lyapunov exponent graph of the hyper chaotic system.

Figure 2. Images of attractor of the hyper chaotic system.

In this part, synchronisation of two 6D hyper chaotic systems with unknown parameters, Input uncertainty and disturbance with unknown bound is presented. Input disturbance bound and parametric uncertainty are unknown. Therefore, synchronisation might be difficult. Consider the master and slave systems (2) and (3), respectively. Defining error as $e_i=y_i-x_i,i=1,\dots ,6$:

(4) $\{\begin{array}{l}{\dot{e}}_1=\overline{a}(y_2-y_1)-a(x_2-x_1)+e_4+\mathrm{\Delta }{f}_1+w_1+u_1\\ {\dot{e}}_2=\overline{c}{y}_1-c{x}_1-y_1{y}_3+x_1{x}_3-e_2+e_5+\mathrm{\Delta }{f}_2+w_2+u_2\\ {\dot{e}}_3=-\overline{b}{y}_3+b{x}_3+y_1{y}_2-x_1{x}_2+\mathrm{\Delta }{f}_3+w_3+u_3\\ {\dot{e}}_4=\overline{d}{y}_4-d{x}_4-y_1{y}_3+x_1{x}_3+\mathrm{\Delta }{f}_4+w_4+u_4\\ {\dot{e}}_5=-\overline{h}{y}_2+h{x}_2+e_6+\mathrm{\Delta }{f}_5+w_5+u_5\\ {\dot{e}}_6=-{\overline{k}}_1{y}_1+k_1{x}_1+{\overline{k}}_2{y}_2-k_2{x}_2+\mathrm{\Delta }{f}_6+w_6+u_6\end{array}$(4)

If error dynamic of Equation (4) is asymptotically stable through determining control signal $u_i$, all states of the slave system converge to states of the master system. But the problem is the presence of uncertainty, disturbance and more important unknown parameters of the slave system and disturbance bound and uncertainty bound. In order to solve this problem, synchronisation of two hyper-chaotic systems in the presence of unknown parameters and disturbance and uncertainty with unknown bounds is solved using the proposed adaptive-sliding mode control.

Sliding mode control is a nonlinear control method that guarantees control strategy over uncertainties. In this way, stability is achieved by keeping the system modes on the sliding surface.

Disturbance and parametric uncertainty bounds of the slave system are estimated using adaptive control.

Now, considering the following sliding surface, (5) $s_i=e_i+\int k{e}_{i}i=1,\dots ,6$(5) Gives

(6) $\{\begin{array}{l}{\dot{s}}_1=\overline{a}(y_2-y_1)-a(x_2-x_1)+e_4+k{e}_1+\mathrm{\Delta }{f}_1+w_1+u_1\\ {\dot{s}}_2=\overline{c}{y}_1-c{x}_1-y_1{y}_3+x_1{x}_3-e_2+e_5+k{e}_2+\mathrm{\Delta }{f}_2+w_2+u_2\\ {\dot{s}}_3=-\overline{b}{y}_3+b{x}_3+y_1{y}_2-x_1{x}_2+k{e}_3+\mathrm{\Delta }{f}_3+w_3+u_3\\ {\dot{s}}_4=\overline{d}{y}_4-d{x}_4-y_1{y}_3+x_1{x}_3+k{e}_4+\mathrm{\Delta }{f}_4+w_4+u_4\\ {\dot{s}}_5=-\overline{h}{y}_2+h{x}_2+e_6+k{e}_5+\mathrm{\Delta }{f}_5+w_5+u_5\\ {\dot{s}}_6=-{\overline{k}}_1{y}_1+k_1{x}_1+{\overline{k}}_2{y}_2-k_2{x}_2+k{e}_6+\mathrm{\Delta }{f}_6+w_6+u_6\end{array}$(6)

and assuming $\dot{s}=0$, we have: (7) $u_{eq}=\{\begin{array}{l}{u}_1=-(\tilde{a}+a)(e_2-e_1)-e_4-k{e}_1\\ u_2=-(\tilde{c}+c)e_1-e_5+e_2+y_1{y}_3-x_1{x}_3-k{e}_2\\ u_3=(\tilde{b}+b)e_3-y_1{y}_2+x_1{x}_2-k{e}_3\\ u_4=-(\tilde{d}+d)e_4+y_1{y}_3-x_1{x}_3-k{e}_4\\ u_5=(\tilde{h}+h)e_2-e_6-k{e}_5\\ u_6=({\tilde{k}}_1+k_1)e_1-({\tilde{k}}_2+k_2)e_2-k{e}_6\end{array}$(7)

In which, $\tilde{a},\tilde{b},\tilde{c},\tilde{d},\tilde{h},{\tilde{k}}_1,{\tilde{k}}_2$ are estimation error of the parameters of the slave system.

On the other hand, the control signal of the proposed controller is considered as follows. (8) $u_i=u_{e{q}_i}-(r{s}_i+\rho \mathrm{sgn}(s_i))-({\overline{\gamma }}_i+{\overline{\beta }}_i)$(8)

In which, $\rho$ and $r$ are positive real constants and: (9) $\begin{array}{rl}\tilde{a}& =\overline{a}-a,\tilde{b}=\overline{b}-b,\tilde{c}=\overline{c}-c\\ \tilde{d}& =\overline{d}-d,\tilde{h}=\overline{h}-h,{\tilde{k}}_1={\overline{k}}_1-k_1,{\tilde{k}}_2={\overline{k}}_2-k_2\\ \dot{\tilde{a}}& =\dot{\overline{a}},\dot{\tilde{b}}=\dot{\overline{b}},\dot{\tilde{c}}=\dot{\overline{c}},\dot{\tilde{d}}=\dot{\overline{d}},\dot{\tilde{h}}=\dot{\overline{h}},{\dot{\tilde{k}}}_1={\dot{\overline{k}}}_1,{\dot{\tilde{k}}}_2={\dot{\overline{k}}}_2\\ {\tilde{\gamma }}_i& ={\overline{\gamma }}_i-{\gamma }_i,{{\tilde{\beta }}_i={\overline{\beta }}_i-\beta }_i,{\dot{\tilde{\gamma }}}_i={\dot{\overline{\gamma }}}_i,{\dot{\tilde{\beta }}}_i={\dot{\overline{\beta }}}_i,i=1,\dots ,6\end{array}$(9)

And estimation rules are as follows: (10) $\begin{array}{rl}\dot{\tilde{a}}& =-s_1(x_2-x_1),\dot{\tilde{b}}=s_3{x}_3,\dot{\tilde{c}}=-s_2{x}_1,\dot{\tilde{d}}=-s_4{x}_4\\ \dot{\tilde{h}}& =s_5{x}_2,{\dot{\tilde{k}}}_1=s_6{x}_1,{\dot{\tilde{k}}}_2=-s_6{x}_2\\ {\dot{\tilde{\gamma }}}_i& =s_i,{\dot{\tilde{\beta }}}_i=s_i,i=1,\dots ,6\end{array}$(10)

Theorem 1

If the control rule (8), where parameters $\overline{a},\overline{b},\overline{c},\overline{d},\overline{h},{\overline{k}}_1,{\overline{k}}_2$ are unknown constants of the chaotic slave system and estimation rule Equations (10), then all states of system (3) will converge to those of system (2).

Proof: Suppose that the Lyapunov function is considered as (11) which is a positive definite function. Given the Lyapunov stability theorem, to prove the stability of the error dynamic (4), it should be shown that the derivative of the Lyapunov function is negative, so according to selective s, u proves the asymptotic stability using the Lyapunov stability.

The proposed Lyapunov function is as follows: (11) $V=\frac{1}{2}\left(\sum _{i=1}^6(s_i^2+{\tilde{\gamma }}_i^2+{\tilde{\beta }}_i^2)+{\tilde{a}}^2+{\tilde{b}}^2+{\tilde{c}}^2+{\tilde{d}}^2+{\tilde{h}}^2+{{\tilde{k}}^2}_1+{{\tilde{k}}^2}_2\right)$(11)

And its derivative is as follows. (12) $\begin{array}{rl}\dot{V}& =\sum _{i=1}^6{s}_i{\dot{s}}_i+\tilde{a}\dot{\tilde{a}}+\tilde{b}\dot{\tilde{b}}+\tilde{c}\dot{\tilde{c}}+\tilde{d}\dot{\tilde{d}}+\tilde{h}\dot{\tilde{h}}+{{\tilde{k}}^{}}_1{{\dot{\tilde{k}}}^{}}_1+{{\tilde{k}}^{}}_2{{\dot{\tilde{k}}}^{}}_2\\ & +{\tilde{\gamma }}_1^{}{\dot{\tilde{\gamma }}}_1^{}+{\tilde{\gamma }}_2^{}{\dot{\tilde{\gamma }}}_2^{}+{\tilde{\gamma }}_3^{}{\dot{\tilde{\gamma }}}_3^{}+{\tilde{\gamma }}_4^{}{\dot{\tilde{\gamma }}}_4^{}+{\tilde{\gamma }}_5^{}{\dot{\tilde{\gamma }}}_5^{}+{\tilde{\gamma }}_6^{}{\dot{\tilde{\gamma }}}_6^{}\\ & +{\tilde{\beta }}_1^{}{\dot{\tilde{\beta }}}_1^{}+{\tilde{\beta }}_2^{}{\dot{\tilde{\beta }}}_2^{}+{\tilde{\beta }}_3^{}{\dot{\tilde{\beta }}}_3^{}+{\tilde{\beta }}_4^{}{\dot{\tilde{\beta }}}_4^{}+{\tilde{\beta }}_5^{}{\dot{\tilde{\beta }}}_5^{}+{\tilde{\beta }}_6^{}{\dot{\tilde{\beta }}}_6^{}\end{array}$(12)

Using Equation (5), we have (13) $\begin{array}{rl}\dot{V}& =s_1(\overline{a}(y_2-y_1)-a(x_2-x_1)+e_4+k{e}_1+\mathrm{\Delta }{f}_1+w_1+u_1)\\ & +s_2(\overline{c}{y}_1-c{x}_1-y_1{y}_3+x_1{x}_3-e_2+e_5+k{e}_2+\mathrm{\Delta }{f}_2+w_2+u_2)\\ & +s_3(-\overline{b}{y}_3+b{x}_3+y_1{y}_2-x_1{x}_2+k{e}_3+\mathrm{\Delta }{f}_3+w_3+u_3)\\ & +s_4(\overline{d}{y}_4-d{x}_4-y_1{y}_3+x_1{x}_3+k{e}_4+\mathrm{\Delta }{f}_4+w_4+u_4)\\ & +s_5(-\overline{h}{y}_2+h{x}_2+e_6+k{e}_5+\mathrm{\Delta }{f}_5+w_5+u_5)\\ & +s_6(-{\overline{k}}_1{y}_1+k_1{x}_1+{\overline{k}}_2{y}_2-k_2{x}_2+k{e}_6+\mathrm{\Delta }{f}_6+w_6+u_6)\\ & +\tilde{a}\dot{\tilde{a}}+\tilde{b}\dot{\tilde{b}}+\tilde{c}\dot{\tilde{c}}+\tilde{d}\dot{\tilde{d}}+\tilde{h}\dot{\tilde{h}}+{{\tilde{k}}^{}}_1{{\dot{\tilde{k}}}^{}}_1+{{\tilde{k}}^{}}_2{{\dot{\tilde{k}}}^{}}_2\\ & +{\tilde{\gamma }}_1^{}{\dot{\tilde{\gamma }}}_1^{}+{\tilde{\gamma }}_2^{}{\dot{\tilde{\gamma }}}_2^{}+{\tilde{\gamma }}_3^{}{\dot{\tilde{\gamma }}}_3^{}+{\tilde{\gamma }}_4^{}{\dot{\tilde{\gamma }}}_4^{}+{\tilde{\gamma }}_5^{}{\dot{\tilde{\gamma }}}_5^{}+{\tilde{\gamma }}_6^{}{\dot{\tilde{\gamma }}}_6^{}\\ & +{\tilde{\beta }}_1^{}{\dot{\tilde{\beta }}}_1^{}+{\tilde{\beta }}_2^{}{\dot{\tilde{\beta }}}_2^{}+{\tilde{\beta }}_3^{}{\dot{\tilde{\beta }}}_3^{}+{\tilde{\beta }}_4^{}{\dot{\tilde{\beta }}}_4^{}+{\tilde{\beta }}_5^{}{\dot{\tilde{\beta }}}_5^{}+{\tilde{\beta }}_6^{}{\dot{\tilde{\beta }}}_6^{}\end{array}$(13)

According to $||\mathrm{\Delta }{f}_i||\le {\gamma }_i$ and $||w_i||\le {\beta }_i,i=1,\dots ,6$, the relations would be as follows: (14) $\begin{array}{rl}\dot{V}& \le s_1(\overline{a}(y_2-y_1)-a(x_2-x_1)+e_4+k{e}_1+{\gamma }_1+{\beta }_1+u_1+\overline{a}(x_2-x_1)-\overline{a}(x_2-x_1))\\ & +s_2(\overline{c}{y}_1-c{x}_1-y_1{y}_3+x_1{x}_3-e_2+e_5+k{e}_2+{\gamma }_2+{\beta }_2+u_2+\overline{c}{x}_1-\overline{c}{x}_1)\\ & +s_3(-\overline{b}{y}_3+b{x}_3+y_1{y}_2-x_1{x}_2+k{e}_3+{\gamma }_3+{\beta }_3+u_3+\overline{b}{x}_3-\overline{b}{x}_3)\\ & +s_4(\overline{d}{y}_4-d{x}_4-y_1{y}_3+x_1{x}_3+k{e}_4+{\gamma }_4+{\beta }_4+u_4+\overline{d}{x}_4-\overline{d}{x}_4)\\ & +s_5(-\overline{h}{y}_2+h{x}_2+e_6+k{e}_5+{\gamma }_5+{\beta }_5+u_5+\overline{h}{x}_2-\overline{h}{x}_2)\\ & +s_6(-{\overline{k}}_1{y}_1+k_1{x}_1+{\overline{k}}_2{y}_2-k_2{x}_2+k{e}_6+{\gamma }_6+{\beta }_6+u_6+{\overline{k}}_1{x}_1-{\overline{k}}_1{x}_1+{\overline{k}}_2{x}_2-{\overline{k}}_2{x}_2)\\ & +\tilde{a}\dot{\tilde{a}}+\tilde{b}\dot{\tilde{b}}+\tilde{c}\dot{\tilde{c}}+\tilde{d}\dot{\tilde{d}}+\tilde{h}\dot{\tilde{h}}+{{\tilde{k}}^{}}_1{{\dot{\tilde{k}}}^{}}_1+{{\tilde{k}}^{}}_2{{\dot{\tilde{k}}}^{}}_2\\ & +{\tilde{\gamma }}_1^{}{\dot{\tilde{\gamma }}}_1^{}+{\tilde{\gamma }}_2^{}{\dot{\tilde{\gamma }}}_2^{}+{\tilde{\gamma }}_3^{}{\dot{\tilde{\gamma }}}_3^{}+{\tilde{\gamma }}_4^{}{\dot{\tilde{\gamma }}}_4^{}+{\tilde{\gamma }}_5^{}{\dot{\tilde{\gamma }}}_5^{}+{\tilde{\gamma }}_6^{}{\dot{\tilde{\gamma }}}_6^{}\\ & +{\tilde{\beta }}_1^{}{\dot{\tilde{\beta }}}_1^{}+{\tilde{\beta }}_2^{}{\dot{\tilde{\beta }}}_2^{}+{\tilde{\beta }}_3^{}{\dot{\tilde{\beta }}}_3^{}+{\tilde{\beta }}_4^{}{\dot{\tilde{\beta }}}_4^{}+{\tilde{\beta }}_5^{}{\dot{\tilde{\beta }}}_5^{}+{\tilde{\beta }}_6^{}{\dot{\tilde{\beta }}}_6^{}\end{array}$(14)

Now, according to Equations (8), (9) and (10), the relations can be rewritten as follows: (15) $\begin{array}{rl}\dot{V}& \le s_1(-r{s}_1-\rho \mathrm{sgn}(s_1))+s_2(-r{s}_2-\rho \mathrm{sgn}(s_2))\\ & +s_3(-r{s}_3-\rho \mathrm{sgn}(s_3))+s_4(-r{s}_4-\rho \mathrm{sgn}(s_4))\\ & +s_5(-r{s}_5-\rho \mathrm{sgn}(s_5))+s_6(-r{s}_6-\rho \mathrm{sgn}(s_6))\\ \dot{V}& \le (-r{s}_1^2-\rho |s_1|)+(-r{s}_2^2-\rho |s_2|)+(-r{s}_3^2-\rho |s_3|)+(-r{s}_4^2-\rho |s_4|)\\ & +(-r{s}_5^2-\rho |s_5|)+(-r{s}_6^2-\rho |s_6|)\\ & forr\ge 0,\rho \ge 0\Rightarrow \dot{V}\le 0\end{array}$(15)

By choosing, values of ρ and r greater than zero, $\dot{V}$ becomes negative and Lypanov’s stability condition is met.

Remark 1

the control parameter $k$ gives us the privilege to synchronise the states in less time by regulating this parameter. And the larger the selection, the faster the synchronisation will be. The parameters $r$ and $\rho$ are also other control parameters, whose main condition in the Theorem 1, is that they must be selected greater than zero, otherwise the derivative of the Lyapunov function will not be negative and no stability will be achieved.

3. 6d fractional order hyper chaotic system

In this paper, Reimann–Liouville fractional order derivative is used. Definition of Riemann–Liouville from the fractional derivative is as follows: ${}_a^{RL}{D}_t^{\alpha }f(t)=\frac{d^m}{d{t}^m}\{\frac{1}{\mathrm{\Gamma }(m-\alpha )}{\int }_a^t{(t-\tau )}^{m-\alpha -1}f(\tau )d\tau \},\alpha \in {\mathbb{R}}^{+}$where a and t are lower and upper bounds of the fractional integral in the definition of this operator and $\mathrm{\Gamma }$ is the gamma function. $\mathrm{\Gamma }$ is described as follows: $\mathrm{\Gamma }(z)={\int }_0^{\mathrm{\infty }}{t}^{z-1}{e}^{-t}dt$

In the following, a 6D fractional order nonlinear system is introduced: $D_t^{\alpha }x=f(x),x(0)=C$

In which, $f(x)$ is a vector function of the state variables $X=[x_1,x_2,x_3,x_4,x_5,x_6{]}^T$ and $\alpha$ $(\alpha \in (0,1])$ is the fractional order derivative and $C=[c_1,c_2,c_3,c_4,c_5,c_6{]}^T$ are initial values and $D_t^{\alpha }$ is the Riemann–Liouville fractional order differentiation. Consider the 6D dynamic as follows: (16) $\{\begin{array}{l}{D}_t^{\alpha }{x}_1=a(x_2-x_1)+x_4\\ D_t^{\alpha }{x}_2=c{x}_1-x_1{x}_3+x_5-x_2\\ D_t^{\alpha }{x}_3=x_1{x}_2-b{x}_3\\ D_t^{\alpha }{x}_4=d{x}_4-x_1{x}_3\\ D_t^{\alpha }{x}_5=-h{x}_2+x_6\\ D_t^{\alpha }{x}_6=-k_1{x}_1+k_2{x}_2\end{array}$(16) where $\alpha$ $(\alpha \in (0,1])$ is the fractional order of differentiation and $X=[x_1,x_2,x_3,x_4,x_5,x_6{]}^T$ is the state vector of the system and $a,b,c,d,h,k_1,k_2$ are constants of the system.

Therefore, considering the disturbance and uncertainty of the parameters, the slave system is described as follows: (17) $\{\begin{array}{l}{D}_t^{\alpha }{y}_1=\overline{a}(y_2-y_1)+y_4+w_1+\mathrm{\Delta }{f}_1+u_1\\ D_t^{\alpha }{y}_2=\overline{c}{y}_1-y_1{y}_3+y_5-y_2+w_2+\mathrm{\Delta }{f}_2+u_2\\ D_t^{\alpha }{y}_3=y_1{y}_2-\overline{b}{y}_3+w_3+\mathrm{\Delta }{f}_3+u_3\\ D_t^{\alpha }{y}_4=\overline{d}{y}_4-y_1{y}_3+w_4+\mathrm{\Delta }{f}_4+u_4\\ D_t^{\alpha }{y}_5=-\overline{h}{y}_2+y_6+w_5+\mathrm{\Delta }{f}_5+u_5\\ D_t^{\alpha }{y}_6=-{\overline{k}}_1{y}_1+{\overline{k}}_2{y}_2+w_6+\mathrm{\Delta }{f}_6+u_6\end{array}$(17)

In which, $U^T=[u_1,u_2,u_3,u_4,u_5,u_6{]}^T$ is the control signal, $Y^T=[y_1,y_2,y_3,y_4,y_5,y_6{]}^T$ is the vector of state variables of the slave system, $||\mathrm{\Delta }{f}_i||\le {\gamma }_i,i=1,\dots ,6$ are uncertain bounded functions of the states with unknown bound and $||w_i||\le {\beta }_i,i=1,\dots ,6$ are bounded disturbance input with unknown bound and $\overline{a},\overline{b},\overline{c},\overline{d},\overline{h},{\overline{k}}_1,{\overline{k}}_2$ are unknown parameters of the slave system. Also, the following relationships are given.

Figure 3 shows the fractional derivative of the hyper chaotic system for the parameter set of $a=7,b=8/3,c=33,d=2,h=9,k_1=0.75,k_2=0.8,\alpha =0.\text{89}$

Figure 3. Images of attractor of the fractional hyper chaotic system.

Consider master and slave systems (16) and (17), respectively. Defining error as $e_i=y_i-x_i,i=1,\dots ,6$, we have: (18) $\{\begin{array}{l}{D}_t^{\alpha }{e}_1=\overline{a}(y_2-y_1)-a(x_2-x_1)+e_4+\mathrm{\Delta }{f}_1+w_1+u_1\\ D_t^{\alpha }{e}_2=\overline{c}{y}_1-c{x}_1-y_1{y}_3+x_1{x}_3-e_2+e_5+\mathrm{\Delta }{f}_2+w_2+u_2\\ D_t^{\alpha }{e}_3=-\overline{b}{y}_3+b{x}_3+y_1{y}_2-x_1{x}_2+\mathrm{\Delta }{f}_3+w_3+u_3\\ D_t^{\alpha }{e}_4=\overline{d}{y}_4-d{x}_4-y_1{y}_3+x_1{x}_3+\mathrm{\Delta }{f}_4+w_4+u_4\\ D_t^{\alpha }{e}_5=-\overline{h}{y}_2+h{x}_2+e_6+\mathrm{\Delta }{f}_5+w_5+u_5\\ D_t^{\alpha }{e}_6=-{\overline{k}}_1{y}_1+k_1{x}_1+{\overline{k}}_2{y}_2-k_2{x}_2+\mathrm{\Delta }{f}_6+w_6+u_6\end{array}$(18)

Now, synchronisation of two fractional order hyper-chaotic systems in the presence of disturbance and uncertainty with unknown bounds and unknown parameters is presented using adaptive-sliding mode control and bounds of disturbance and parametric uncertainty of the slave systems are estimated using the adaptive rules.

Now, considering the following sliding surface, (19) $s_i=\int D_t^{\alpha }{e}_{i}i=1,\dots ,6$(19) Gives (20) $\{\begin{array}{l}{\dot{s}}_1=\overline{a}(y_2-y_1)-a(x_2-x_1)+e_4+\mathrm{\Delta }{f}_1+w_1+u_1\\ {\dot{s}}_2=\overline{c}{y}_1-c{x}_1-y_1{y}_3+x_1{x}_3-e_2+e_5+\mathrm{\Delta }{f}_2+w_2+u_2\\ {\dot{s}}_3=-\overline{b}{y}_3+b{x}_3+y_1{y}_2-x_1{x}_2+\mathrm{\Delta }{f}_3+w_3+u_3\\ {\dot{s}}_4=\overline{d}{y}_4-d{x}_4-y_1{y}_3+x_1{x}_3+\mathrm{\Delta }{f}_4+w_4+u_4\\ {\dot{s}}_5=-\overline{h}{y}_2+h{x}_2+e_6+\mathrm{\Delta }{f}_5+w_5+u_5\\ {\dot{s}}_6=-{\overline{k}}_1{y}_1+k_1{x}_1+{\overline{k}}_2{y}_2-k_2{x}_2+\mathrm{\Delta }{f}_6+w_6+u_6\end{array}$(20)

We have the following through mathematical manipulation: (21) $u_{eq}=\{\begin{array}{l}{u}_1=-(\tilde{a}+a)(e_2-e_1)-e_4\\ u_2=-(\tilde{c}+c)e_1-e_5+e_2+y_1{y}_3-x_1{x}_3\\ u_3=(\tilde{b}+b)e_3-y_1{y}_2+x_1{x}_2\\ u_4=-(\tilde{d}+d)e_4+y_1{y}_3-x_1{x}_3\\ u_5=(\tilde{h}+h)e_2-e_6\\ u_6=({\tilde{k}}_1+k_1)e_1-({\tilde{k}}_2+k_2)e_2\end{array}$(21) In which $\tilde{a},\tilde{b},\tilde{c},\tilde{d},\tilde{h},{\tilde{k}}_1,{\tilde{k}}_2$ are estimation errors of parameters of the slave system. On the other hand, the control signal of the proposed controller is considered as follows. (22) $u_i=u_{e{q}_i}-(r{s}_i+\rho \mathrm{sgn}(s_i))-({\gamma }_i+{\beta }_i),i=1,\dots ,6$(22)

In which, ρ and r are positive real constants and: (23) $\begin{array}{rl}\tilde{a}& =\overline{a}-a,\tilde{b}=\overline{b}-b,\tilde{c}=\overline{c}-c\\ \tilde{d}& =\overline{d}-d,\tilde{h}=\overline{h}-h,{\tilde{k}}_1={\overline{k}}_1-k_1,{\tilde{k}}_2={\overline{k}}_2-k_2\\ \dot{\tilde{a}}& =\dot{\overline{a}},\dot{\tilde{b}}=\dot{\overline{b}},\dot{\tilde{c}}=\dot{\overline{c}},\dot{\tilde{d}}=\dot{\overline{d}},\dot{\tilde{h}}=\dot{\overline{h}},{\dot{\tilde{k}}}_1={\dot{\overline{k}}}_1,{\dot{\tilde{k}}}_2={\dot{\overline{k}}}_2\\ {\tilde{\gamma }}_i& ={\overline{\gamma }}_i-{\gamma }_i,{\tilde{\beta }}_i={\overline{\beta }}_i-\beta ,{\dot{\tilde{\gamma }}}_i={\dot{\overline{\gamma }}}_i,{\dot{\tilde{\beta }}}_i={\dot{\overline{\beta }}}_i,& i=1,\dots ,6\end{array}$(23) And estimation rules are as follows: (24) $\begin{array}{rl}\dot{\tilde{a}}& =-s_1(x_2-x_1),\dot{\tilde{b}}=s_3{x}_3,\dot{\tilde{c}}=-s_2{x}_1,\dot{\tilde{d}}=-s_4{x}_4\\ \dot{\tilde{h}}& =s_5{x}_2,{\dot{\tilde{k}}}_1=s_6{x}_1,{\dot{\tilde{k}}}_2=-s_6{x}_2,{\dot{\tilde{\gamma }}}_i=s_i,{\dot{\tilde{\beta }}}_i=s_i,i=1,\dots ,6\end{array}$(24)

Theorem 1

By applying the control laws in Equation (22) to the system of Equation (17) and adaptive laws of Equation (24), all system states (17) converge to system (16), asymptotically.

Proof: Suppose that the Lyapunov function is considered as (25) which is a positive definite function. Given the Lyapunov stability theorem, to prove the stability of the error dynamic (18), it should be shown that the derivative of the Lyapunov function is negative, so according to selective s, u proves the asymptotic stability using the Lyapunov stability.

The proposed Lyapunov function is as follows: (25) $\begin{array}{rl}V& =\frac{1}{2}\sum _{i=1}^6{s}_i^2+{\tilde{a}}^2+{\tilde{b}}^2+{\tilde{c}}^2+{\tilde{d}}^2+{\tilde{h}}^2+{{\tilde{k}}^2}_1+{{\tilde{k}}^2}_2\\ & +{\tilde{\gamma }}_1^2+{\tilde{\gamma }}_2^2+{\tilde{\gamma }}_3^2+{\tilde{\gamma }}_4^2+{\tilde{\gamma }}_5^2+{\tilde{\gamma }}_6^2+{\tilde{\beta }}_1^2+{\tilde{\beta }}_2^2+{\tilde{\beta }}_3^2+{\tilde{\beta }}_4^2+{\tilde{\beta }}_5^2+{\tilde{\beta }}_6^2\end{array}$(25)

And its derivative is as follows. $\begin{array}{rl}\dot{V}& =\sum _{i=1}^6{s}_i{\dot{s}}_i+\tilde{a}\dot{\tilde{a}}+\tilde{b}\dot{\tilde{b}}+\tilde{c}\dot{\tilde{c}}+\tilde{d}\dot{\tilde{d}}+\tilde{h}\dot{\tilde{h}}+{{\tilde{k}}^{}}_1{{\dot{\tilde{k}}}^{}}_1+{{\tilde{k}}^{}}_2{{\dot{\tilde{k}}}^{}}_2\\ & +{\tilde{\gamma }}_1^{}{\dot{\tilde{\gamma }}}_1^{}+{\tilde{\gamma }}_2^{}{\dot{\tilde{\gamma }}}_2^{}+{\tilde{\gamma }}_3^{}{\dot{\tilde{\gamma }}}_3^{}+{\tilde{\gamma }}_4^{}{\dot{\tilde{\gamma }}}_4^{}+{\tilde{\gamma }}_5^{}{\dot{\tilde{\gamma }}}_5^{}+{\tilde{\gamma }}_6^{}{\dot{\tilde{\gamma }}}_6^{}\\ & +{\tilde{\beta }}_1^{}{\dot{\tilde{\beta }}}_1^{}+{\tilde{\beta }}_2^{}{\dot{\tilde{\beta }}}_2^{}+{\tilde{\beta }}_3^{}{\dot{\tilde{\beta }}}_3^{}+{\tilde{\beta }}_4^{}{\dot{\tilde{\beta }}}_4^{}+{\tilde{\beta }}_5^{}{\dot{\tilde{\beta }}}_5^{}+{\tilde{\beta }}_6^{}{\dot{\tilde{\beta }}}_6^{}\end{array}$

Using Equations (20), we have (26) $\begin{array}{rl}\dot{V}& =s_1(\overline{a}(y_2-y_1)-a(x_2-x_1)+e_4+\mathrm{\Delta }{f}_1+w_1+u_1)\\ & +s_2(\overline{c}{y}_1-c{x}_1-y_1{y}_3+x_1{x}_3-e_2+e_5+\mathrm{\Delta }{f}_2+w_2+u_2)\\ & +s_3(-\overline{b}{y}_3+b{x}_3-y_1{y}_2+x_1{x}_2+\mathrm{\Delta }{f}_3+w_3+u_3)\\ & +s_4(\overline{d}{y}_4-d{x}_4-y_1{y}_3+x_1{x}_3+\mathrm{\Delta }{f}_4+w_4+u_4)\\ & +s_5(-\overline{h}{y}_2+h{x}_2+e_6+\mathrm{\Delta }{f}_5+w_5+u_5)\\ & +s_6(-{\overline{k}}_1{y}_1+k_1{x}_1+{\overline{k}}_2{y}_2-k_2{x}_2+\mathrm{\Delta }{f}_6+w_6+u_6)\\ & +\tilde{a}\dot{\tilde{a}}+\tilde{b}\dot{\tilde{b}}+\tilde{c}\dot{\tilde{c}}+\tilde{d}\dot{\tilde{d}}+\tilde{h}\dot{\tilde{h}}+{{\tilde{k}}^{}}_1{{\dot{\tilde{k}}}^{}}_1+{{\tilde{k}}^{}}_2{{\dot{\tilde{k}}}^{}}_2\\ & +{\tilde{\gamma }}_1^{}{\dot{\tilde{\gamma }}}_1^{}+{\tilde{\gamma }}_2^{}{\dot{\tilde{\gamma }}}_2^{}+{\tilde{\gamma }}_3^{}{\dot{\tilde{\gamma }}}_3^{}+{\tilde{\gamma }}_4^{}{\dot{\tilde{\gamma }}}_4^{}+{\tilde{\gamma }}_5^{}{\dot{\tilde{\gamma }}}_5^{}+{\tilde{\gamma }}_6^{}{\dot{\tilde{\gamma }}}_6^{}\\ & +{\tilde{\beta }}_1^{}{\dot{\tilde{\beta }}}_1^{}+{\tilde{\beta }}_2^{}{\dot{\tilde{\beta }}}_2^{}+{\tilde{\beta }}_3^{}{\dot{\tilde{\beta }}}_3^{}+{\tilde{\beta }}_4^{}{\dot{\tilde{\beta }}}_4^{}+{\tilde{\beta }}_5^{}{\dot{\tilde{\beta }}}_5^{}+{\tilde{\beta }}_6^{}{\dot{\tilde{\beta }}}_6^{}\end{array}$(26)

According to $||\mathrm{\Delta }{f}_i||\le {\gamma }_i$ and $||w_i||\le {\beta }_i,i=1,6$, the relations would become as follows: (27) $\begin{array}{rl}\dot{V}& \le s_1(\overline{a}(y_2-y_1)-a(x_2-x_1)+e_4+{\gamma }_1+{\beta }_1+u_1+\overline{a}(x_2-x_1)-\overline{a}(x_2-x_1))\\ & +s_2(\overline{c}{y}_1-c{x}_1-y_1{y}_3+x_1{x}_3-e_2+e_5+{\gamma }_2+{\beta }_2+u_2+\overline{c}{x}_1-\overline{c}{x}_1)\\ & +s_3(-\overline{b}{y}_3+b{x}_3-y_1{y}_2+x_1{x}_2+{\gamma }_3+{\beta }_3+u_3+\overline{b}{x}_3-\overline{b}{x}_3)\\ & +s_4(\overline{d}{y}_4-d{x}_4-y_1{y}_3+x_1{x}_3+{\gamma }_4+{\beta }_4+u_4+\overline{d}{x}_4-\overline{d}{x}_4)\\ & +s_5(-\overline{h}{y}_2+h{x}_2+e_6+{\gamma }_5+{\beta }_5+u_5+\overline{h}{x}_2-\overline{h}{x}_2)\\ & +s_6(-{\overline{k}}_1{y}_1+k_1{x}_1+{\overline{k}}_2{y}_2-k_2{x}_2+{\gamma }_6+{\beta }_6+u_6+{\overline{k}}_1{x}_1-{\overline{k}}_1{x}_1+{\overline{k}}_2{x}_2-{\overline{k}}_2{x}_2)\\ & +\tilde{a}\dot{\tilde{a}}+\tilde{b}\dot{\tilde{b}}+\tilde{c}\dot{\tilde{c}}+\tilde{d}\dot{\tilde{d}}+\tilde{h}\dot{\tilde{h}}+{{\tilde{k}}^{}}_1{{\dot{\tilde{k}}}^{}}_1+{{\tilde{k}}^{}}_2{{\dot{\tilde{k}}}^{}}_2\\ & +{\tilde{\gamma }}_1^{}{\dot{\tilde{\gamma }}}_1^{}+{\tilde{\gamma }}_2^{}{\dot{\tilde{\gamma }}}_2^{}+{\tilde{\gamma }}_3^{}{\dot{\tilde{\gamma }}}_3^{}+{\tilde{\gamma }}_4^{}{\dot{\tilde{\gamma }}}_4^{}+{\tilde{\gamma }}_5^{}{\dot{\tilde{\gamma }}}_5^{}+{\tilde{\gamma }}_6^{}{\dot{\tilde{\gamma }}}_6^{}\\ & +{\tilde{\beta }}_1^{}{\dot{\tilde{\beta }}}_1^{}+{\tilde{\beta }}_2^{}{\dot{\tilde{\beta }}}_2^{}+{\tilde{\beta }}_3^{}{\dot{\tilde{\beta }}}_3^{}+{\tilde{\beta }}_4^{}{\dot{\tilde{\beta }}}_4^{}+{\tilde{\beta }}_5^{}{\dot{\tilde{\beta }}}_5^{}+{\tilde{\beta }}_6^{}{\dot{\tilde{\beta }}}_6^{}\end{array}$(27)

Now, according to Equations (22), (23) and (24), the relations can be rewritten as follows: (28) $\begin{array}{rl}\dot{V}& \le s_1(-r{s}_1-\rho \mathrm{sgn}(s_1))+s_2(-r{s}_2-\rho \mathrm{sgn}(s_2))\\ & +s_3(-r{s}_3-\rho \mathrm{sgn}(s_3))+s_4(-r{s}_4-\rho \mathrm{sgn}(s_4))\\ & +s_5(-r{s}_5-\rho \mathrm{sgn}(s_5))+s_6(-r{s}_6-\rho \mathrm{sgn}(s_6))\end{array}$(28) (29) $\begin{array}{rl}\dot{V}& \le (-r{s}_1^2-\rho |s_1|)+(-r{s}_2^2-\rho |s_2|)+(-r{s}_3^2-\rho |s_3|)+(-r{s}_4^2-\rho |s_4|)+(-r{s}_5^2-\rho |s_5|)\\ & +(-r{s}_6^2-\rho |s_6|)forr\ge 0,\rho \ge 0\Rightarrow \dot{V}\le 0\end{array}$(29)

By choosing values of ρ and r greater than zero, $\dot{V}$ becomes negative and the Lypanov’s stability condition will be met.

4. Simulation

In this section, numerical simulation for three examples of synchronisation of two 6D integer order and fractional order hyper-chaotic systems with unknown parameters in the presence of disturbance and parametric uncertainty with unknown bounds is presented. Examples 1 and 2 show the performance of the proposed method and Example 3 shows the application of the proposed method in secure communication. Simulation results are obtained using MATLAB software with sampling time 0.00001 s.

Example 1

Consider systems (1) and (2). Simulation is performed with the following initial conditions: $[x_1,x_2,x_3,x_4,x_5,x_6{]}^T=[1,0,1,1,0,1{]}^T$, $[y_1,y_2,y_3,y_4,y_5,y_6{]}^T=[4,-50,3,4,5,-2{]}^T$, and $a=10,b=8/3,c=28,d=2,h=8.8,k_1=1,k_2=0.9$. The parameters used in the designed controller are $r=50,\rho =20,k=10$. Uncertainty and chaos bounded disturbance applied to the system are $\mathrm{\Delta }{f}_i=A\sin (y_i)\cos (y_i),i=1,\dots ,6$ and $w_i=A\sin (t),0.1<A<1,i=1,\dots ,6$.

Figures 4 and 5 show the results of synchronisation of the two hyper chaotic 6D systems after applying the controller Equation (8), representing the performance of the proposed controller. Figure 6 shows control signals of Equation (8) and Figure 7 shows the estimates of $\overline{a},\overline{b},\overline{c},\overline{d},\overline{h},{\overline{k}}_1,{\overline{k}}_2$ using Equation (10).

Figure 4. Master and slave system states after applying the proposed controller in Equation (8) for the 6D hyper chaotic system.

Figure 5. Synchronisation error after using the controller Equation (8).

Figure 6. Control signals using the controller Equation (8).

Figure 7. Estimates of $\overline{a},\overline{b},\overline{c},\overline{d},\overline{h},{\overline{k}}_1,{\overline{k}}_2$ using Equation (10).

Example 2

Consider the fractional derivative systems (16) and (17) with $\alpha =0.84,0.89$. Simulation is performed with the following initial conditions:

$[x_1,x_2,x_3,x_4,x_5,x_6{]}^T=[1,2,1,1,0,1,1{]}^T$, $[y_1,y_2,y_3,y_4,y_5,y_6{]}^T=[4,4,3,4,5,-2{]}^T$, and $a=7,b=8/3,c=33,d=2,h=9,k_1=0.75,k_2=0.8$. The parameters used in the designed controller are $r=80,\rho =80,k=10$. Uncertainty and chaos bounded disturbance applied to the system are $\mathrm{\Delta }{f}_i=A\sin (y_i)\cos (y_i),i=1,\dots ,6$ $w_i=A\sin (t),0.1<A<1,i=1,\dots ,6$.

Figures 8 and 9 show the results of synchronisation of the two fractional order hyper chaotic 6D systems after applying the controller as per Equation (22), representing the performance of the proposed controller. Figure 10 shows control signals of Equation (22) and Figure 11 estimates of $\overline{a},\overline{b},\overline{c},\overline{d},\overline{h},{\overline{k}}_1,{\overline{k}}_2$ with using Equation (24).

Figure 8. Master and slave system states after applying the proposed controller in Equation (22) for the 6D fractional order hyper chaotic system.

Figure 9. Synchronisation error after using the controller in Equation (22).

Figure 10. Control signals using the controller in Equation (22).

Figure 11. Estimates of $\overline{a},\overline{b},\overline{c},\overline{d},\overline{h},{\overline{k}}_1,{\overline{k}}_2$ using Equation (24).

Example 2

In this example, the mentioned chaotic system is used for secure communication. In the proposed scheme, the message signal $M(t)$ is covered by variables of the chaotic system and then it is transmitted. Its simplified block diagram is shown in Figure 12. Encoding and decoding equations in the transmitter and receiver are as follows. $D_t^{\alpha }x=f(x)x_{out}=x_2+M(t)$and $D_t^{\alpha }y=f(y)+u{y}_{out}=y_2$

Figure 12. Block diagram of a communication system.

In which, $x_{out}$ and $y_{out}$ are outputs the master and slave systems, respectively. $f(x)$ and $f(y)$ are secondary sides of equations of the 6D like Lorenz fractional order hyper chaotic systems of (16) as the master system and (17) as the slave system with α = 0.89. u is the control signal and $M(t)$ is the message signal combined with state variable $x_2$ and considered as the output of the master system. After synchronisation and calculating error as follows, we have: $\underset{t\to \mathrm{\infty }}{lim}{x}_{out}-y_{out}=\underset{t\to \mathrm{\infty }}{lim}(x_2+M(t)-y_2)=\underset{t\to \mathrm{\infty }}{lim}(-e_2+M(t))=\hat{M}(t)$

Simulation is performed with the following initial conditions: $\begin{array}{rl}[x_1,x_2,x_3,x_4,x_5,x_6{]}^T& =[3,1,2,8,-2,3{]}^T,\\ [y_1,y_2,y_3,y_4,y_5,y_6{]}^T& =[3,6,3,1,7,-1{]}^T,\end{array}$and

$a=7,b=8/3,c=33,d=2,h=9,k_1=0.75,k_2=\mathrm{0.8.}$The parameters used in the designed controller are $r=80,\rho =80,k=10$. Uncertainty and bounded disturbance applied to the system are $\mathrm{\Delta }{f}_i=A\sin (y_i),i=1,\dots ,6\text{and}{w}_i=A\sin (t),0.1<A<1,i=1,\dots ,6.$Figure 13 shows the transmitted message signal $M(t)$. Figure 14 shows chaotic state variable $x_2$. Figure 15 shows output of the master system. As can be seen, by adding the message signal, behaviour of $x_2$ has remained chaotic. Output signal of the system is received at the receiver and by applying the control law in Equation (22) which is shown in Figure 16, the slave system is synchronised with the master system and output of the slave system is received and compared with the received signal to extract the message signal. Figure 15 shows control signals for initial values of Example 3. Figure 17 shows the difference of the transmitted message signal $M(t)$ and the extracted message signal $\hat{M}(t)$. As can be seen, after synchronisation, the difference of $M(t)$ and $\hat{M}(t)$ becomes zero. In order to eliminate mismatch between the two signals at the beginning of the signal, the message signal is usually transmitted with delay.

Figure 13. The transmitted message signal $M(t)$.

Figure 14. Behaviour of state variable $x_2$.

Figure 15. $x_{out}=x_2+M(t)$ signal.

Figure 16. Control signals using the controller in Equation (22) with $\alpha \text{ = 0}\text{.89}$ and $[x_1,x_2,x_3,x_4,x_5,x_6{]}^T=[3,1,2,8,-2,3{]}^T$, $[y_1,y_2,y_3,y_4,y_5,y_6{]}^T=[3,6,3,1,7,-1{]}^T$.

Figure 17. $M(t)-\hat{M}(t)$ signal.

In most studies on secure communication, high order chaotic systems are less used for synchronisation of two chaotic systems or chaotic systems in the presence of disturbance, uncertainty and unknown parameters are used. For example:

In Li and Wu (2019), 3D fractional order chaotic system without disturbance, uncertainty and known parameters have been studied. In Djennoune et al. (2019), 3D chaotic system without disturbance and uncertainty has been studied in the context of secure communication. In Ouannas et al. (2019), 4D hyper-chaotic fractional order system without disturbance and uncertainty in the presence of known parameters has been studied.

5. Conclusion

The main purpose of this paper is to design an adaptive-sliding mode controller for 6D integer order and fractional order hyper-chaotic systems in the presence of parametric uncertainty and disturbance with unknown bounds and unknown system parameters. First, the proposed controller is designed for 6D integer order system, then the chaos is observed in the system with fractional order dynamic and an adaptive-sliding mode controller is designed. In order to achieve this goal and synchronise two fractional order and integer order hyper-chaotic systems, sliding control methods which are robust against uncertainty is used. In order to identify the disturbance bounds and unknown parametric uncertainty and system parameters, adaptive controller is used and stability of the chaotic system is proved using Lyapunov theorem. Simulation results in MATLAB show the efficiency of the designed controller. Some of the open problems in this context are: the investigation of chaotic fractional order systems with orders greater than 6 and solution of the problem in presence of measurement noise, synchronisation of two 6D hyper chaotic systems with less control signals with respect to system’s dimensions, and the presence of the uncertainty and disturbance in slave system.

Disclosure statement

No potential conflict of interest was reported by the author(s).