1 Introduction

Hyperchaotic systems are high-dimensional chaotic systems with complex behavior, which have more than one positive Lyaponuv exponent. Since, these systems exhibit a behavior as noise-like, unpredictability and highly sensitive to the initial conditions, are more useful in some applications such as secure communications and encryption. Also, the fractional systems contain fractional orders, and this feature leads to more complicated chaotic behaviors which can be used in applications such as secure communications. By utilizing the fractional calculus techniques, some fractional-order chaotic/hyperchaotic systems have been identified such as fractional-order chaotic/hyperchaotic Chen system [1, 2], fractional-order hyperchaotic Novel system [3] and fractional-order hyperchaos Lorenz system [4]. In [5], many systems which behave in a chaotic manner are reviewed. Also in [6,7,8], the some fractional discrete systems which can exhibit chaos behavior are introduced. In the chaos-based secure communication schemes, to recover the message from the transmitter, master-slave synchronization must be completely achieved. In recent years, many types of chaos synchronization approaches have been proposed, such as projective synchronization, adaptive control, robust synchronization and active pinning control [9,10,11,12].

In many practical applications, the accurate dynamics of the systems are not available or are perturbed by the external disturbances, time delays, time-varying parameters, etc. To cope with the uncertainties and perturbations, many robust controllers have been presented. For instance in [13, 14], the \(H_\infty \) control problem is investigated for the fractional-order linear systems. The mean-square exponential stability problem is studied in [15, 16]. In [17], based on the matrix’s singular value decomposition, some sufficient conditions for quadratic stability of the uncertain fractional-order linear systems are presented. The feedback stabilization of the fractional-order systems using linear matrix inequality technique is studied in [18, 19]. The problem of the exponential synchronization of the discrete-time neural networks with mixed time delays, actuator saturation and failures, by using Lyapunov functional approach is investigated in [20]. The robust sliding mode control of the fractional-order systems is proposed in [21, 22]. In [23], a robust model predictive control scheme is presented to control the fractional-order discrete-time systems.

To deal with the uncertainties in the dynamics of the system, some approaches have been presented based on the approximation property of the fuzzy neural networks (FNNs). In [24], a fuzzy sliding mode controller is proposed. A states feedback method by using LMI technique, based on fuzzy model, is proposed in [25] to stabilize the fractional-order chaotic systems. The generalized projective synchronization by estimating unknown nonlinear functions using fuzzy systems is presented in [26]. An adaptive fuzzy controller with \(H_\infty \) synchronization performance is studied in [27].

The main problem in the design of fuzzy controller is the problem of “curse of dimensionality”. By increasing the input variables of a fuzzy neural network, the number of rules exponentially increases. Since the hyperchaotic systems are high-dimensional nonlinear system, FNNs cannot be adequately applied to estimate the uncertain functions in the dynamic of these systems. One approach to cope with this problem can be self-evolving FNNs.

For this purpose, some self-evolving type-2 FNNs have been proposed. For instance recently in [28], a SE-T2FNN is presented, the rule database of which is initially empty, and all rules are automatically grown. In [29], a self-evolving recurrent type-2 fuzzy radial basis function network is presented, where new neurons and rules are generated based on a clustering algorithm. In [30], a hierarchical SE-T2FNN is presented in which its new rules and antecedent part are generated by using \(\varepsilon \)-completeness criterion (this means that the firing strength of at least one fuzzy rule for any input within the operating range is not less than \(\varepsilon \)) and its consequent part is designed by using ant-colony optimization method. In [31], the rule database is modified in response to the controller’s performance.

Singleton type-2 FNNs are used in all of the mentioned self-evolving algorithms, in which the linguistic and input numerical uncertainties are handled only by MFs. The input uncertainties arise from noise and inaccuracy of sensors, observers and input devises. The non-singleton type-2 FNNs are considered in [32,33,34]. In these papers, some learning algorithms are developed for tuning free parameters of non-singleton type-2 FNNS.

Fig. 1
figure 1

Proposed control block diagram for the i-th subsystem

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Motivated by the discussions above, in this paper, a new self-evolving non-singleton type-2 FNN is presented. The proposed self-evolving algorithm in this paper is simple, and it can be applied to the high-dimensional problems. The proposed SE-NT2FNN has only one rule initially. New MFs are added or replaced, or the existing MFs are changed such that \(\varepsilon \)- completeness criterion is satisfied. The dynamic of FOHS is assumed to be unknown and the proposed SE-NT2FNNs are employed to estimate the unknown functions. To eliminate the effect of approximation error, a LMI-based robust controller is combined with the output feedback control scheme. Some advantages of the proposed method are as follows:

  • The dynamic of fractional-order chaotic system is assumed to be unknown; then the proposed controller can be applied to a wide class of FOHS.

  • A new self-evolving non-singleton type-2 fuzzy neural network is presented, in which the new MFs and rules are added based on the simple algorithm, and unimportant rules are deleted. Furthermore, input uncertainties are handled by using non-singleton fuzzification.

  • The effect of approximation error and external disturbance is eliminated by a proposed LMI-based robust controller.

  • Since triangular type-2 MFs are employed, in each sample time, a few number of MFs are activated for each input; furthermore, interpretability of the FNN is increased.

The remaining of this paper is organized as follows. In Sect. 2, system description and problem formulation are presented. The proposed non-singleton type-2 fuzzy neural network is presented in Sect. 3. The proposed self-evolving algorithm is introduced in Sect. 4. Stability analysis is presented in Sect. 5. The simulation results of the controlled fractional-order hyperchaotic systems are shown in Sect. 6. Finally, conclusions remarks are presented in Sect. 7.

2 Problem formulation and system description

The following class of FOHS is considered as slave system:

$$\begin{aligned} \begin{aligned} {D^{{\alpha _1}}}{y_1}&= {f_1}\left( {{y_1},\ldots ,{y_n}} \right) + {d_1}(t) + {u_1}(t)\\&\vdots \\ {D^{{\alpha _n}}}{y_n}&= {f_n}\left( {{y_1},\ldots ,{y_n}} \right) + {d_n}(t) + {u_n}(t) \end{aligned} \end{aligned}$$
(1)

where \({f_i}\,,i = 1,2,\ldots ,n\) are unknown but bounded functions, \({d_i}(t)\,,i = 1,2,\ldots ,n\) are bounded external disturbance, \({u_i}\,,i = 1,2,\ldots ,n\) are control signals, \(\underline{y} = {\left[ {{y_1},{y_2},\ldots ,{y_n}} \right] ^\mathrm{T}}\) are the outputs of slave system and \(0< {\alpha _i} < 1\,\,,\,i = 1,\ldots ,n\) are the fractional derivatives orders. \({D^\alpha }y_i\) is the fractional derivative of \(y_i\). The regular definitions for fractional derivatives are: Grünwald–Letnikov, Riemann–Liouville and Caputo definitions. For instance, the Caputo definition is given as:

$$\begin{aligned} {D^\alpha }y = \frac{1}{{\Gamma (m - \alpha )}}\int _0^t {\frac{{{y^m}(\tau )}}{{{{(t - \tau )}^{\alpha - m + 1}}}}\mathrm{d}\tau } \end{aligned}$$
(2)

where m is integer so that \(m - 1< \alpha < m\) and \(\Gamma ( \cdot )\) is Gamma function (\(\Gamma \left( t \right) = \int _0^\infty {{x^{t - 1}}} {e^{ - x}}\mathrm{d}x\)).

The master system is considered as follows:

$$\begin{aligned} \begin{aligned} {D^{{\beta _1}}}{x_1}&= {g_1}\left( {{x_1},\ldots ,{x_n}} \right) \\&\vdots \\ {D^{{\beta _n}}}{x_n}&= {g_n}\left( {{x_1},\ldots ,{x_n}} \right) \end{aligned} \end{aligned}$$
(3)

where \({g_i}\,,i = 1,2,\ldots ,n\) are unknown but bounded functions, \(\underline{x} = {\left[ {{x_1},{x_2},\ldots ,{x_n}} \right] ^\mathrm{T}}\) are the outputs of master system, and \(0< {\beta _i} < 1\,\,,\,i = 1,\ldots ,n\) are the fractional derivatives orders. The synchronization errors are defined as \({e_i} = {y_i} - {x_{i\,}}\,,\,i = 1,\ldots ,n\). The control objective is to design controllers \({u_i}\,,i = 1,2,\ldots ,n\) such that \(\left\| {{e_i}} \right\| \rightarrow 0\,\,as\,\,t \rightarrow \infty \).

The proposed control block diagram for i-th subsystem is shown in Fig. 1. As shown in Fig. 1, the unknown function in the dynamics of the slave systems is estimated by the proposed self-evolving fuzzy system, and by using this fuzzy system, an error feedback controller is designed [see Eq. (6)]. Then, the synchronization problem [see Eq. (8)] is rewritten as a standard \(H_\infty \) problem [see Eqs. (22, 23)]. To show the robustness of the proposed control scheme and the stability analysis, a compensator is designed based on the \(H_\infty \) control technique [see \({u_{{s_i}}}\) in Eq. (23)], by using the results of [14].

The i-th subsystem is considered as follows:

$$\begin{aligned} {D^{{\alpha _i}}}{y_i} = {f_i}\left( {{y_1},\ldots ,{y_n}} \right) + {d_i}(t) + {u_i}(t) \end{aligned}$$
(4)
Fig. 2
figure 2

The proposed secure communication scheme

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As shown in Fig. 1, the term \({f_i}\left( {{y_1},\ldots ,{y_n}} \right) + {d_i}(t)\) in (4) is estimated by SE-NT2FNN \({\hat{f}_i}\). The approximation error is defined as follows:

$$\begin{aligned} {\varepsilon _i} = {f_i}\left( {{y_1},\ldots ,{y_n}} \right) + {d_i}(t) - {\hat{f}_i} \end{aligned}$$
(5)

We define \(u_i\) as \({u_i} = u_i^ * + {u_{{s_i}}}\), in which \(u_i^ *\) is designed as follows:

$$\begin{aligned} u_i^ * = {D^{{\alpha _i}}}{x_i} - {\lambda _i}{e_i} - {\hat{f}_i} \end{aligned}$$
(6)

where \({\lambda _{\,i}}\) is chosen such that stability condition \(\left| {{\mathrm{arg(}}{\lambda _i}{\mathrm{)}}} \right| {\mathrm{> }}{\alpha _i}\frac{\pi }{2}\) is satisfied. This stability condition is derived from the following theorem:

Theorem 1

([35]) The following linear autonomous system

$$\begin{aligned} {D^\alpha }x = Ax \end{aligned}$$
(7)

with \(0 < \alpha \le 1\), \(x \in {R^n}\) and \(A \in {R^{n \times n}}\) is , is asymptotically stable if and only if all the eigenvalues of matrix A satisfy \(\left| {{\mathrm{arg}}(eig\left( A \right) )} \right| > \alpha \frac{\pi }{2}\).

By substituting (6) into (4) we have:

$$\begin{aligned} {D^{{\alpha _i}}}{e_i} + {\lambda _i}{e_i} = {\varepsilon _i} + {u_{{s_i}}} \end{aligned}$$
(8)

\(u_{{s_i}}\) is designed such that the \({H_\infty }\) norm of closed-loop transfer function \(\left( {{T_{e\varepsilon }}} \right) \), from \(e_i\) to \(\varepsilon _i\) is minimized. The proposed synchronization method is used in a secure communication application. The proposed secure communication scheme is shown in Fig. 2. The input message is encrypted using signals of master hyperchaotic system and is sent to the receiver. In the receiver, after synchronization, the message signal hidden inside a hyperchaotic signal is recovered.

3 Proposed non-singleton type-2 fuzzy neural network

In this section, the structure of proposed SE-NT2FNN is introduced. As shown in Fig. 1, the unknown functions in the dynamics of the slave systems are estimated by the proposed SE-NT2FNN. The structure and the consequent parameters of SE-NT2FNN are online adjusted based on a proposed self-evolving algorithm and the gradient descent algorithm, respectively [see Sects. 4.1, 4.2]. As shown in Fig. 1, the training data are coming from the slave system at each sample time. The main merits of proposed SE-NT2FNN compared with other fuzzy neural networks are that the proposed fuzzy system has only one rule initially and new MFs and rules are added when necessary and also the unimportant rules are deleted based on a simple algorithm. Furthermore, the input uncertainties are handled by the proposed non-singleton fuzzification.

The proposed network structure has six layers as shown in Fig. 3.

Fig. 3
figure 3

Structure of SE-NT2FNN

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Each rule has the following form:

$$\begin{aligned}&Rule\,\,i:\,IF\,\,{y_1}\,is\,\,\tilde{A}_1^i\,And\, \cdots \,And\,\,{y_n}\,is\,\,\tilde{A}_n^i\,Then\,\hat{f}\nonumber \\&\qquad \in \left[ {w_l^i,w_r^i} \right] \,\,\,i = 1,\ldots ,M \end{aligned}$$
(9)

where M is the number of rules, \({y_l}\,,\,l = 1,\ldots ,n\) are inputs of SE-NT2FNN, n is the number of inputs of SE-NT2FNN, \(\tilde{A}_j^i\,\,\,j = 1,\ldots ,n\) is the i-th interval type-2 membership function (MF) for the j-th input and \(\left[ {w_l^i,w_r^i} \right] \) are the consequent parameters. \(\hat{f}\,\) is output of SE-NT2FNN. Each layer of SE-NT2FNN is explained as follows:

\({ Input} \, { layer}\): The inputs of SE-NT2FNN are the outputs of slave system.

\({ Fuzzification} \,{ layer}\): In this layer, the non-singleton fuzzification operation is performed. The uncertainties of inputs of SE-NT2FNN are modeled by type-2 MFs. The proposed type-2 MF for input \({y_j}\) has three points, in which its first-point, center-point and end-point are located at \({y_j} - \Delta \), \({y_j}\) and \({y_j} + \Delta \), respectively (as shown in Fig. 4). \(\Delta \) is a designable parameter which is constant for all inputs. By this fuzzifier, crisp inputs are mapped into a type-2 MF. The non-singleton fuzzifier changes the membership values of MFs. Consider i-th MF for j-th input, the non-singleton fuzzifier transforms \({y_j}\) to \(\vec {y}_j^{\,i}\). By using minimum inference \(\vec {y}_j^{\,i}\) is obtained as follows:

$$\begin{aligned} \vec {y}_j^{\,i} = \left[ \begin{array}{ll} {y_j} \quad \mathrm{if}\,\,\left( {{y_j} + \Delta } \right)< a_j^i\,\,\mathrm{or}\,\,\,\left( {{y_j} - \Delta } \right)> b_j^i\,\,\mathrm{or}\,\,\,{y_j} = m_j^i\,&{}\\ \displaystyle \frac{{\Delta \times a_j^i + \left| {a_j^i - m_j^i} \right| \times \left( {{y_j} + \Delta } \right) }}{{\Delta + \left| {a_j^i - m_j^i} \right| }}&{}\mathrm{else}\;\mathrm{if}\,\,{y_j} < m_j^i\,\,\,{\mathrm{see}}\,\mathrm{Fig.}~5\\ \displaystyle \frac{{\Delta \times b_j^i + \left| {b_j^i - m_j^i} \right| \times \left( {{y_j} - \Delta } \right) }}{{\Delta + \left| {b_j^i - m_j^i} \right| }}&{}\mathrm{else}\,\mathrm{if}\,\,{y_j} > m_j^i\,\,\,{\mathrm{see}}\,\mathrm{Fig.}~6 \end{array} \right. \end{aligned}$$
(10)
Fig. 4
figure 4

Modeling of the uncertainty of inputs by proposed type-2 MF

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Fig. 5
figure 5

Non-singleton fuzzifier by using minimum inference

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where \({y_j}\) is j-th input, \(a_j^i\), \(m_j^i\) and \(b_j^i\) are first-point, center-point and end-point of i-th MF for j-th input, respectively (Figs. 5, 6).

Membership layer: In this layer, the upper and lower memberships of MFs are computed. Consider i-th MF for j-th input \(y_j\), the upper and lower memberships of \(\vec {y}_j^{\,i}\) (transformed of \(y_j\) by non-singleton fuzzifier) are obtained as follows:

$$\begin{aligned} {\bar{\mu }} _{\vec {y}_j^i}^i= & {} \left\{ {\begin{array}{l@{\quad }l} \displaystyle \frac{{\vec {y}_j^{\,i} - a_j^i}}{{m_j^i - a_j^i}}&{}\mathrm{if}\quad a_j^i< \vec {y}_j^{\,i} \le m_j^i\\ \displaystyle \frac{{\vec {y}_j^{\,i} - b_j^i}}{{m_j^i - b_j^i}}&{}\mathrm{if}\quad m_j^i< \vec {y}_j^{\,i}< b_j^i\\ 0&{}\mathrm{otherwise} \end{array}} \right. \nonumber \\ \underline{\mu }_{\vec {y}_j^i}^i= & {} \left\{ {\begin{array}{l@{\quad }l} 0.5\displaystyle \frac{{\vec {y}_j^{\,i} - a_j^i}}{{m_j^i - a_j^i}}&{}\mathrm{if}\quad a_j^i< \vec {y}_j^{\,i} \le m_j^i\\ 0.5\displaystyle \frac{{\vec {y}_j^{\,i} - b_j^i}}{{m_j^i - b_j^i}}&{}\mathrm{if}\quad m_j^i< \vec {y}_j^{\,i} < b_j^i\\ 0&{}\mathrm{otherwise} \end{array}} \right. \end{aligned}$$
(11)
Fig. 6
figure 6

Non-singleton fuzzifier by using minimum inference

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Rule layer: Each node in this layer corresponds to a rule, which computes the upper and lower firing degrees. The upper and lower firing degrees of i-th rule are computed as follows:

$$\begin{aligned} \begin{aligned} {{\bar{z}}_i}= & {} {\bar{\mu }} _{\vec {y}_1^{{\, p_1}}}^{{p_1}}\, \times \,{\bar{\mu }} _{\vec {y}_2^{{\, p_2}}}^{{p_2}} \times \cdots \times {\bar{\mu }} _{\vec {y}_j^{{\, p_j}}}^{{p_j}} \times \cdots \times {\bar{\mu }} _{\vec {y}_n^{{\, p_n}}}^{{p_n}}\\ {\underline{z} _i}= & {} \underline{\mu } _{\vec {y}_1^{{\, p_1}}}^{{p_1}}\, \times \,\underline{\mu } _{\vec {y}_2^{{\, p_2}}}^{{p_2}} \times \cdots \times \underline{\mu } _{\vec {y}_j^{{\, p_j}}}^{{p_j}} \times \cdots \times \underline{\mu } _{\vec {y}_n^{{\, p_n}}}^{{p_n}} \end{aligned} \end{aligned}$$
(12)

where \({\bar{\mu }} _{\vec {y}_j^{{\, p_j}}}^{{p_j}}\) and \(\underline{\mu } _{\vec {y}_j^{{\, p_j}}}^{{p_j}}\) are the upper and lower memberships of \({p_j}\)-th MF for j-th input, respectively.

Type-\(reduction\,layer\): Based on the center of sets type reduction, \({\hat{f}_r}\) and \({\hat{f}_l}\) are as follows:

$$\begin{aligned} \hat{f}_r^{}= & {} \frac{{\sum \nolimits _{j = 1}^R {\underline{z} _j^{}w_r^j} + \sum \nolimits _{j = R + 1}^M {\bar{z}_j^{}w_r^j} }}{{\sum \nolimits _{j = 1}^R {\underline{z} _j^{}} + \sum \nolimits _{j = R + 1}^M {\bar{z}_j^{}} }},\nonumber \\ \hat{f}_l^{}= & {} \frac{{\sum \nolimits _{j = 1}^L {\bar{z}_j^{}} w_l^j + \sum \nolimits _{j = L + 1}^M {\underline{z} _j^{}w_l^j} }}{{\sum \nolimits _{j = 1}^L {\bar{z}_j^{}} + \sum \nolimits _{j = L + 1}^M {\underline{z} _j^{}} }} \end{aligned}$$
(13)

in which R and L are obtained from KM iterative algorithm [36]. \(w_r^j\) and \(w_l^j\) are the consequent parameters. \({\bar{z}_j}\) and \(\underline{z} {\,_j}\) are the upper and lower firing degrees of j-th rule, respectively, and M is the number of rules. To simplify Eq. (13), the following definitions are considered:

$$\begin{aligned} \begin{aligned}&{q_r} \buildrel \Delta \over = {\left[ {\begin{array}{*{20}{c}} {\overbrace{\begin{array}{*{20}{c}} 1&1&\cdots&1 \end{array}}^R}&{\overbrace{\begin{array}{*{20}{c}} 0&0&\cdots&0 \end{array}}^{M - R}} \end{array}} \right] ^\mathrm{T}},\\&{q_l} \buildrel \Delta \over = {\left[ {\begin{array}{*{20}{c}} {\overbrace{\begin{array}{*{20}{c}} 1&1&\cdots&1 \end{array}}^L}&{\overbrace{\begin{array}{*{20}{c}} 0&0&\cdots&0 \end{array}}^{M - L}} \end{array}} \right] ^\mathrm{T}} \end{aligned} \end{aligned}$$
(14)

where \(\overbrace{\begin{array}{*{20}{c}} 1&1&\cdots&1 \end{array}}^R\) represent that the number of ones is R. According to (14), Eq. (13), can be rewritten as follows:

$$\begin{aligned} {\hat{f}_r}= & {} \frac{{\sum \nolimits _{j = 1}^M {\left[ {{q_r}(j)\underline{z} _j^{} + \left( {1 - {q_r}(j)} \right) \bar{z}_j^{}} \right] \times w_r^j} }}{{\sum \nolimits _{j = 1}^M {\left[ {{q_r}(j)\underline{z} _j^{} + \left( {1 - {q_r}(j)} \right) \bar{z}_j^{}} \right] } }},\nonumber \\ {\hat{f}_l}= & {} \frac{{\sum \nolimits _{j = 1}^M {\left[ {{q_l}(j)\bar{z}_j^{} + \left( {1 - {q_l}(j)} \right) \underline{z} _j^{}} \right] w_l^j} }}{{\sum \nolimits _{j = 1}^M {\left[ {{q_l}(j)\bar{z}_j^{} + \left( {1 - {q_l}(j)} \right) \underline{z} _j^{}} \right] } }} \end{aligned}$$
(15)

where \({q_r}(j)\) and \({q_l}(j)\) represent the j-th element of vector \({q_r}\) and \({q_l}\), respectively. \(Output \, layer\): The defuzzified crisp output \(\hat{f}\) is the average of \({\hat{f}_r}\) and \({\hat{f}_l}\):

$$\begin{aligned} \hat{f} = \frac{{\hat{f}_r^{} + \hat{f}_l^{}}}{2} \end{aligned}$$
(16)

4 Self-evolving algorithm

In this section, consequent parameters [see Eq. (13)] and the structure of type-2 fuzzy neural network are tuned.

4.1 Parameter learning

The consequent parameters \(w_r^j,w_l^j,\,j = 1,\ldots ,M\) are tuned based on gradient descent algorithm. As shown in Fig. 1, the parameters \(w_r^j,w_l^j,\,j = 1,\ldots ,M\) are tuned such that the following cost function is minimized:

$$\begin{aligned} E = \frac{1}{2}{\left[ {{y_i} - {D^{ - {\alpha _i}}}\left( {{{\hat{f}}_i} + {u_i}} \right) } \right] ^2} \end{aligned}$$
(17)

where \(y_i\) is the output of i-th subsystem, \(D^{ - {\alpha _i}}\) is fractional integrator, \(\hat{f}_i\) is the output of i-th SE-NT2FNN and \(u_i\) is i-th control signal in i-th subsystem. Based on gradient descent algorithm, we have:

$$\begin{aligned} \begin{aligned} w_r^j(t + 1)&= w_r^j(t) - \eta \displaystyle \frac{{\partial E}}{{\partial w_r^j}}\\&= w_r^j(t) - \eta \displaystyle \frac{{\partial E}}{{\partial {{\hat{f}}_i}}} \displaystyle \frac{{\partial \hat{f}}}{{\partial w_r^j}}\\&= w_r^j(t) \\&\quad \, +\,\eta \left[ {{y_i} - {D^{ - {\alpha _i}}}\left( {{{\hat{f}}_i} + {u_i}} \right) } \right] {D^{ - {\alpha _i}}}\\&\quad \, \times \left( {\displaystyle \frac{{{q_r}(j)\underline{z} _j^{} + \left( {1 - {q_r}(j)} \right) \bar{z}_j^{}}}{{\sum \limits _{j = 1}^M {\left[ {{q_r}(j)\underline{z} _j^{} + \left( {1 - {q_r}(j)} \right) \bar{z}_j^{}} \right] } }}\,} \right) \end{aligned} \end{aligned}$$
(18)

Similar to (18), for training of \(w_l^j\) we have:

$$\begin{aligned} \begin{aligned}&w_l^j(t + 1) \\&\quad =w_l^j(t) + \eta \left[ {{y_i} - {D^{ - {\alpha _i}}}\left( {{{\hat{f}}_i} + {u_i}} \right) } \right] {D^{ - {\alpha _i}}}\\&\qquad \times \left( {\displaystyle \frac{{{q_l}(j)\bar{z}_j^{} + \left( {1 - {q_l}(j)} \right) \underline{z} _j^{}}}{{\sum \nolimits _{j = 1}^M {\left[ {{q_l}(j)\bar{z}_j^{} + \left( {1 - {q_l}(j)} \right) \underline{z} _j^{}} \right] } }}\,} \right) \end{aligned} \end{aligned}$$
(19)

4.2 Structure learning

In this section, a new structure learning algorithm is presented. Our proposed SE-NT2FNN has only one rule initially. New rules and new MFs are added, and unnecessary rules and MFs are deleted, when necessary. And the proposed algorithm limits the number of rules from indefinitely growing. The flowchart of proposed algorithm is shown in Fig. 7. When tracking error is greater than a predefined threshold, the structure is changed. Detailed explanation is given in the following.

Fig. 7
figure 7

Proposed structure learning algorithm

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4.2.1 Adding new MF

Consider input \({y_j}\), at time t, if the maximum of upper memberships of all MFs to input \({y_j}\) is less than 0.5, a new MF is added for input \({y_j}\) as shown in Fig. 8. The center-point of new MF \(m_j^\mathrm{new}\) is located at \({y_j}\), its first-point \(a_j^\mathrm{new}\) is chosen as center-point of neighboring MF and its end-point \(b_j^\mathrm{new}\) is chosen as \(b_j^\mathrm{new} = 2 \times m_j^\mathrm{new} - a_j^\mathrm{new}\). It must be noted that each input has only one type-2 MF initially.

Fig. 8
figure 8

Adding new MF for input \(y_j\)

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4.2.2 Replacing new MF

Consider input \({y_j}\), if the number of MFs for input \({y_j}\) is greater than a predefined threshold, after adding new MF, the furthest MF from newly added MF is deleted.

4.2.3 Generate new rules

After adding or replacing new MFs, all new rules, the firing degrees of which are greater than 0.5 are generated. Let the current SE-NT2FNN output value be w, the consequent parameters of new rules \(w_r^\mathrm{new},w_l^\mathrm{new}\) are initialized, as follows:

$$\begin{aligned} \left\{ \begin{array}{lll} w_r^\mathrm{new} = w\,\,,&{}w_l^\mathrm{new} = 0&{}\quad w > = 0\\ w_r^\mathrm{new} = 0\,\,\,,&{}w_l^\mathrm{new} = w&{}\quad w < 0 \end{array} \right. \end{aligned}$$
(20)

If by adding or replacing new MFs, there is no a new rule the firing degree of which is greater than 0.5, or there is not any rule in the rule database, the firing degree of which is greater than 0.5 , then a new rule is generated as follows:

For each input, one MF is found which its upper membership is maximum. The center of this MF is changed to the current input value. Also the end-point of left neighboring MF and the first-point of right neighboring MF are modified to the current input value [see Fig. 9].

Fig. 9
figure 9

Change the center of one MF which its upper membership is maximum

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For example, assume \({y_1}\), \({y_2}\) and \({y_3}\) are the inputs of SE-NT2FNN, \(\tilde{A}_{{y_1}}^1,\tilde{A}_{{y_1}}^2,\tilde{A}_{{y_1}}^3\) are the type-2 MFs for input \(y_1\), \(\tilde{B}_{{y_2}}^1,\tilde{B}_{{y_2}}^2,\tilde{B}_{{y_2}}^3\) are the type-2 MFs for input \(y_2\) and \(\tilde{C}_{{y_3}}^1,\tilde{C}_{{y_3}}^2,\tilde{C}_{{y_3}}^3\) are the type-2 MFs for input \(y_3\) . And assume the output of SE-NT2FNN at time t is w. If the center of MFs \(\tilde{A}_{{y_1}}^1\), \(\tilde{B}_{{y_2}}^3\) and \(\tilde{C}_{{y_3}}^2\) are changed [see Fig. 9], the new rule is generated as follows:

$$\begin{aligned} \begin{aligned}&{\mathrm{If}}\,\,{y_1}\,{\mathrm{is}}\,\,\tilde{A}_{{y_1}}^1,\,{\mathrm{and}}\,\,{y_2}\,\,{\mathrm{is}}\,\,\tilde{B}_{{y_2}}^3,\,\,{\mathrm{and}}\,\,{y_3}\,\,\tilde{C}_{{y_3}}^2,\,\,\,{\mathrm{Then}}\,\,\hat{f}\,{\mathrm{is}}\,\,\\&\tilde{G}\,\,{\mathrm{Centroid}}\,\,{\mathrm{of}}\,\,\tilde{G}\,\,{\mathrm{is}}\,\,\left[ {{w_l},{w_r}} \right] \,\,{\mathrm{where}}\,\,\\&\quad \left\{ \begin{array}{l@{\quad }l@{\quad }l} {w_r} = w,&{}{w_l} = 0&{}\mathrm{if}\quad w \ge 0\\ {w_r} = 0,&{}{w_l} = w&{}\mathrm{if}\quad w < 0 \end{array} \right. \end{aligned} \end{aligned}$$
(21)

4.2.4 Add new rules to the rule database

If by adding new rules to the rule database, the number of rules is greater than a predefined threshold (maximum number of rules), the new generated rules are replaced with the rules which have the smallest firing degrees. Otherwise, new rules are added to rule database. It must be noted that since the consequent parameters of new rules are initialized to SE-NT2FNN current output value, the control signal remains continues, and then, our assumption about bounded approximation error remains valid. So the stability analysis are valid when the structure changes.

5 Stability analysis

In this section, a compensator is designed such that \({H_\infty }\) norm of transfer function \({T_{e\varepsilon }}\) is minimized. We represent the Eq. (8) as a standard \({H_\infty }\) problem as shown in Fig. 10.

Fig. 10
figure 10

Configuration of standard \({H_\infty }\) problem

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where,

$$\begin{aligned}&P\left\{ {\begin{array}{*{20}{c}} {{D^{{\alpha _i}}}{e_i} = A{e_i} + B{u_{{s_i}}} + {B_\varepsilon }{\varepsilon _i}}\\ {y = C{e_i}\, + D{u_{{s_i}}} + {D_\varepsilon }{\varepsilon _i}\,\,\,\,\,\,\,\,\,\,}\\ {z = {C_z}{e_i}\, + {D_z}{u_{{s_i}}} + {D_{z\varepsilon }}{\varepsilon _i}\,\,\,\,\,\,} \end{array}} \right. \, \end{aligned}$$
(22)
$$\begin{aligned}&K\left\{ {\begin{array}{*{20}{c}} {{D^{{\alpha _i}}}{x_k} = {A_k}{x_k} + {B_k}y}\\ {{u_{{s_i}}} = {C_k}{x_k}\, + {D_k}y\,} \end{array}} \right. \end{aligned}$$
(23)

in which, \(\,A = - {\lambda _i},\,\,B = 1,\,{B_\varepsilon } = 1,\,C = 1,\,D = 0,\,\,{D_\varepsilon } = 0,\,{C_z} = 1,\,{D_z} = 0,\,{D_{z\varepsilon }} = 0\), \({e_i} \in {R^n}\) is the state of subsystem (8), \({u_i} \in {R^{{n_u}}}\) is control signal, \(y \in {R^{{n_y}}}\) and \(z \in {R^{{n_z}}}\) are outputs and \({x_k} \in {R^n}\) is the state of controller.

With definition \(\tilde{x}(t) = {\left[ {\begin{array}{*{20}{c}} {{e_i}}&{{x_k}} \end{array}} \right] ^\mathrm{T}}\), the state-space representation of the closed-loop \(T_{z{\varepsilon _i}}^{cl}\) can be written as follows:

$$\begin{aligned} T_{z{\varepsilon _i}}^{cl} = \left\{ {\begin{array}{*{20}{l}} {{D^{{\alpha _i}}}\tilde{x} = {A_{cl}}\tilde{x} + {B_{cl}}{\varepsilon _i}}\\ {z = {C_{cl}}\tilde{x}\, + {D_{cl}}{\varepsilon _i}\,} \end{array}} \right. \end{aligned}$$
(24)

where

$$\begin{aligned} \begin{aligned} {A_{cl}}&= \left[ {\begin{array}{*{20}{c}} {A + B{D_k}C}&{}{B{C_k}}\\ {{B_k}C}&{}{{A_k}} \end{array}} \right] \,,\,{B_{cl}} = \left[ {\begin{array}{*{20}{c}} {{B_\varepsilon } + B{D_k}{D_\varepsilon }}\\ {{B_k}{D_\varepsilon }} \end{array}} \right] \\ {C_{cl}}&= \left[ {\begin{array}{*{20}{c}} {{C_z} + {D_z}{D_k}C}&{{D_z}{C_k}} \end{array}} \right] ,\,{D_{cl}} = {D_{z\varepsilon }} + {D_z}{D_k}{D_\varepsilon } \end{aligned} \end{aligned}$$
(25)

Various methods have been presented for computation of the upper bound of \({H_\infty }\)-norm.

Lemma 1

([37, 38]): The \({H_\infty }\)-norm of system (24), is bounded by \(\gamma \), if and only if there exists symmetric positive definite \(X = {X^*} \in {R^{2n \times 2n}}\) such that:

$$\begin{aligned} \begin{array}{l} \left[ {\begin{array}{*{20}{c}} {{\bar{r}}A_{cl}^TX + Xr{A_{cl}}}&{}{X{B_{cl}}}&{}{{\bar{r}}\,C_{cl}^T}\\ {B_{cl}^TX}&{}{ - {\gamma ^2}I}&{}{D_{cl}^T}\\ {r{C_{cl}}}&{}{{D_{cl}}}&{}{ - \gamma I} \end{array}} \right] < 0\\ with\,\,\,r = {e^{\left( {1 - {\alpha _i}} \right) j\pi /2}} \end{array} \end{aligned}$$
(26)

Partition the solution of (26) as follows:

$$\begin{aligned} X = \left[ {\begin{array}{*{20}{c}} Y&{}N\\ {{N^T}}&{} * \end{array}} \right] ,\,{X^{ - 1}} = \left[ {\begin{array}{*{20}{c}} Z&{}M\\ {{M^T}}&{} * \end{array}} \right] \end{aligned}$$
(27)

in which, Z and Y are symmetric and \(n \times n\). By using (26) and linearizing change of variables, Theorem 2, [14] has been derived:

Theorem 2

([14]): The fractional-order system (22) with \(1< {\alpha _i} < 2\), by output feedback control (23), is Bounded-Input Bounded-Output (BIBO) stabilizable and \(\left\| {T_{zw}^{cl}} \right\| < \gamma \), if there exist \(Z = {Z^T} \in {R^{n \times n}}\), \(Y = {Y^T} \in {R^{n \times n}}\), \({\hat{A}} \in {R^{n \times n}}\), \({\hat{B}} \in {R^{n \times {n_y}}}\), \({\hat{C}} \in {R^{{n_u} \times n}}\) and \({\hat{D}} \in {R^{{n_u} \times {n_y}}}\) such that LMI (29) is feasible with

$$\begin{aligned}&\left[ {\begin{array}{*{20}{c}} Z&{}I\\ I&{}Y \end{array}} \right] \succ 0 \end{aligned}$$
(28)
$$\begin{aligned}&\begin{aligned}&\left[ {\begin{array}{*{20}{c}} {{\bar{r}}\left( {AZ + B{\hat{C}}} \right) + r\left( {Z{A^T} + {{{\hat{C}}}^T}{B^T}} \right) }&{}{{\bar{r}}\left( {YA + {\hat{B}}C} \right) + r\left( {{A^T}Y + {C^T}{{{\hat{B}}}^T}} \right) }\\ {r{{\left( {A + B{\hat{D}}C} \right) }^T} + {\bar{r}}{\hat{A}}}&{}{{{\left( {Y{B_w} + {\hat{B}}{D_w}} \right) }^T}}\\ {{{\left( {{B_w} + B{\hat{D}}{D_w}} \right) }^T}}&{}{r\left( {{C_z} + {D_z}{\hat{D}}C} \right) }\\ {r\left( {{C_z}Z + {D_z}{\hat{C}}} \right) }&{}{{\bar{r}}\left( {YA + {\hat{B}}C} \right) + r\left( {{A^T}Y + {C^T}{{{\hat{B}}}^T}} \right) } \end{array}} \qquad \left. {\begin{array}{*{20}{c}} {{B_w} + B{\hat{D}}{D_w}}&{}{{\bar{r}}{{\left( {{C_z}Z + {D_z}{\hat{C}}} \right) }^T}}\\ {Y{B_w} + {\hat{B}}{D_w}}&{}{{\bar{r}}{{\left( {{C_z} + {D_z}{\hat{D}}C} \right) }^T}}\\ { - {\gamma ^2}I}&{}{{{\left( {{D_{zw}} + {D_z}{\hat{D}}{D_w}} \right) }^T}}\\ {{D_{zw}} + {D_z}{\hat{D}}{D_w}}&{}{ - I} \end{array}} \right] < 0\right. \quad \; \end{aligned} \end{aligned}$$
(29)

where

$$\begin{aligned} {\hat{A}} \buildrel \Delta \over =&\, N{A_k}{M^T} + N{B_k}CZ + YB{C_k}{M^T} + Y\left( {A + B{D_k}C} \right) Z\nonumber \\ {\hat{B}} \buildrel \Delta \over =&\, N{B_k} + YB{D_k}\\ {\hat{C}} \buildrel \Delta \over =&\, {C_k}{M^T} + {D_k}CZ\nonumber \\ {\hat{D}} \buildrel \Delta \over =&\, {D_k}\nonumber \end{aligned}$$
(30)

The results of Theorem 2 is derived from [38] and can be easily extended to case of fractional-order system (22) with \(0< {\alpha _i} < 1\).

After solving (29), matrices M and N must be find such that:

$$\begin{aligned} M{N^T} = I - ZY \end{aligned}$$
(31)

By using (31) and (30), the controller matrices \({A_k},{B_k},{C_k},{D_k}\) can be derived as follows:

$$\begin{aligned} \begin{aligned} {D_k} \buildrel \Delta \over =\,&{\hat{D}}\\ {C_k} \buildrel \Delta \over =\,&\left( {{\hat{C}} - {D_k}CX} \right) {M^{ - T}}\\ {B_k} \buildrel \Delta \over =\,&{N^{ - 1}}\left( {{\hat{B}} - YB{D_k}} \right) \\ {A_k} \buildrel \Delta \over =\,&{N^{ - 1}}( {{\hat{A}} {-} N{B_k}CX - YB{C_k}{M^T} {-} Y( {A {+} B{D_k}C} )X} ){M^{ - T}} \end{aligned} \end{aligned}$$
(32)

6 Simulations

In this section, two examples are presented to evaluate the performance of proposed controller for the synchronization of uncertain fractional hyperchaotic systems and its application to secure communication.

Many numerical methods, based on the approximation of fractional derivative (integral) operators, have been presented for solving the fractional differential Eqs. [39,40,41,42]. In this paper, for simulation the Caputo derivative of fractional-order, we use Simulink block nid, which has been created by Duarter Valerio [43]. The use of Caputo derivative is more popular in real applications.

Example 1

In this example, the proposed controller is applied to synchronize the fractional-order hyperchaotic Novel system and fractional-order hyperchaotic Chen system [44]. The master and slave systems dynamics are given as follows:

$$\begin{aligned} \mathrm{Master}\,\mathrm{system}{:}\,\left\{ \begin{array}{l} {D^{{\beta _1}}}x_1^{} = 35\left( {{x_2} - {x_1}} \right) + 35{x_2}{x_1}\\ {D^{{\beta _2}}}x_2^{} = 25{x_1} - 5{x_1}{x_3} + {x_2} + {x_4}\\ {D^{{\beta _3}}}x_3^{} = {x_1}{x_2} - 4{x_3}\\ {D^{{\beta _4}}}x_4^{} = - 100{x_2} \end{array} \right. \end{aligned}$$
(33)

and

$$\begin{aligned} \mathrm{Slave}\,\mathrm{system}{:}\left\{ \begin{array}{l} {D^{{\alpha _1}}}y_1^{} = 35\left( {{y_2} - {y_1}} \right) + {y_4} + {d_1} {+} {u_1}\\ {D^{{\alpha _2}}}y_2^{} = 7{y_1} - {y_1}{y_3} + 12{y_2} {+} {d_2} {+} {u_2}\\ {D^{{\alpha _3}}}y_3^{} = {y_1}{y_2} - 3{y_3} + {d_3} + {u_3}\\ {D^{{\alpha _4}}}y_4^{} = {y_2}{y_3} + 0.3{y_4} + {d_4} + {u_4} \end{array} \right. \end{aligned}$$
(34)

where \({u_i},i = 1,2,3,4\) are controllers, \({d_i},i = 1,2,3,4\) are external disturbance which are considered to be white noise with zero mean and variance 0.1. The fractional derivative orders are \({\alpha _i} = {\beta _i} = 0.97\,,i = 1,\ldots ,4\). Initial conditions are chosen as \({x_1}(0)=1\), \({x_2}(0)=1\), \({x_3}(0)=1\), \({x_4}(0)=1\), \({y_1}(0)=0.61\), \({y_2}(0)=0.21\), \({y_3}(0)=0.61\) and \({y_4}(0)=0.21\). The parameters of self-evolving algorithm are given in Table 1. The input message is considered as \(\sin (t)\).

Table 1 Parameters of self-evolving algorithm
Full size table

To design the controller, we rewrite slave system (34) as follows:

$$\begin{aligned} \begin{aligned} {D^{{\alpha _1}}}y_1^{}&= {f_1}\left( {\underline{y} } \right) + {u_1}\\ {D^{{\alpha _2}}}y_2^{}&= {f_2}\left( {\underline{y} } \right) + {u_2}\\ {D^{{\alpha _3}}}y_3^{}&= {f_3}\left( {\underline{y} } \right) + {u_3}\\ {D^{{\alpha _4}}}y_4^{}&= {f_4}\left( {\underline{y} } \right) + {u_4} \end{aligned} \end{aligned}$$
(35)

in which \(\underline{y} = \left[ {{y_1},{y_2},{y_3},{y_4}} \right] \) and \({f_1},{f_2},{f_3},{f_4}\) are nonlinear and unknown but bounded functions. The controllers \({u_1},{u_2},{u_3},{u_4}\) are designed as follows:

$$\begin{aligned} \begin{aligned} {u_1}&= {D^{{\alpha _1}}}{x_1} - {\lambda _1}{e_1} - {{\hat{f}}_1} + {u_{{s_1}}}\\ {u_2}&= {D^{{\alpha _2}}}{x_2} - {\lambda _2}{e_2} - {{\hat{f}}_2} + {u_{{s_2}}}\\ {u_3}&= {D^{{\alpha _3}}}{x_3} - {\lambda _3}{e_3} - {{\hat{f}}_3} + {u_{{s_3}}}\\ {u_4}&= {D^{{\alpha _4}}}{x_4} - {\lambda _4}{e_4} - {{\hat{f}}_4} + {u_{{s_4}}} \end{aligned} \end{aligned}$$
(36)

where \({\hat{f}_i}\,,\,i = 1,2,3,4\) are SE-NT2FNNs which estimate \({f_i}(\underline{y} ) + {d_i}(t)\,,\,i = 1,2,3,4\) [see Fig. 1]. By substituting (36) into (34), we have:

$$\begin{aligned} \begin{aligned} {D^{{\alpha _1}}}{e_1}&= - {\lambda _1}{e_1} + {\varepsilon _1} + {u_{{s_1}}}\\ {D^{{\alpha _2}}}{e_2}&= - {\lambda _2}{e_2} + {\varepsilon _2} + {u_{{s_2}}}\\ {D^{{\alpha _3}}}{e_3}&= - {\lambda _3}{e_3} + {\varepsilon _3} + {u_{{s_3}}}\\ {D^{{\alpha _4}}}{e_4}&= - {\lambda _4}{e_4} + {\varepsilon _4} + {u_{{s_4}}} \end{aligned} \end{aligned}$$
(37)

where \({\varepsilon _i}\,,i = 1,2,3,4\) are approximation errors [see Eq. (5)], by considering \({\lambda _i} = 100,\,i = 1,2,3,4\), after solving LMI (29), the matrices of controllers \({u_{{s_i}}},\,i = 1,2,3,4\) [see Fig. 10 and Eq. (23)] are obtained as follows:

$$\begin{aligned} \left\{ \begin{array}{l} {A_{{k_i}}} = - {\mathrm{18}}{\mathrm{.4916,}}\,\,{B_{{k_i}}} = - {\mathrm{1}}{\mathrm{.3122}}\\ {C_{{k_i}}} = {\mathrm{8}}{\mathrm{.4653,}}\,\,{D_{{k_i}}} = {\mathrm{99}}{\mathrm{.2928}}\\ {\gamma ^2} = {\mathrm{0}}{\mathrm{.7602}} \end{array} \right. \end{aligned}$$
(38)

Output trajectory of \({x_i}\) and \({y_i}\), \(i = 1,2,3,4\) is shown in Fig. 11. The results of the encryption and decryption are shown in Fig. 12. It can be seen that the synchronization performance is desired, and the input message is recovered with a good accuracy. The fractional-order hyperchaotic systems of this example have been synchronized in [44], by a new nonlinear technique. The comparison values of root-mean-square error (RMSE) are given in Table 2. It can be seen that RMSE values for the synchronization error of our method are significantly less than [44]. It must be noted that dynamic of slave and master systems is assumed to be unknown in our method, and furthermore, external disturbance has been considered. But in [44], dynamic of slave and master system assumed to be known, and there is no external disturbance.

Fig. 11
figure 11

Output trajectory of master and slave systems, Example 1

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Fig. 12
figure 12

Results for the encryption and decryption, Example 1

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Example 2

In this example, the proposed controller is applied to synchronize the uncertain fractional-order hyperchaotic Lorenz system as the slave system, and fractional-order hyperchaotic Chen system as the master system. The hyperchaotic Lorenz system is as follows:

$$\begin{aligned} \begin{aligned} D^\alpha {y_1}&= 10\left( {{y_2} - {y_1}} \right) + {y_4} + d_1^s + {u_1}\\ D^\alpha {y_2}&= 28{y_1} - {y_2} - {y_1}{y_3} + d_2^s + {u_2}\\ D^\alpha {y_3}&= {y_1}{y_2} - 8/3{y_3} + d_3^s + {u_3}\\ D^\alpha {y_4}&= - {y_2}{y_3} - {y_4} + d_4^s + {u_4} \end{aligned} \end{aligned}$$
(39)

Hyperchaotic chen system is:

$$\begin{aligned} \begin{aligned} D^\beta {x_1}&= 35\left( {{x_2} - {x_1}} \right) + {x_4} + d_1^m\\ D^\beta {x_2}&= 7{x_1} + 12{x_2} - {x_1}{x_3} + d_2^m\\ D^\beta {x_3}&= {x_1}{x_2} - 8{x_3} + d_3^m\\ D^\beta {x_4}&= {x_2}{x_3} + 0.3{x_4} + d_4^m \end{aligned} \end{aligned}$$
(40)
Table 2 Comparison results of proposed method in this paper and the proposed nonlinear technique in [44], Example 1
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Fig. 13
figure 13

Output trajectory of master and slave systems, Example 2

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where [45]

$$\begin{aligned} d_1^s= & {} 0.25\cos (6t){y_1} - 0.15\sin (t),\nonumber \\ d_1^m= & {} - 0.25\sin (4t){x_1} + 0.1\sin (7t)\nonumber \\ d_2^s= & {} - 0.2\cos (2t){y_2} + 0.1\sin (3t),\nonumber \\ d_2^m= & {} 0.1\cos (t){x_2} + 0.15\cos (3t)\\ d_3^s= & {} 0.15\sin (3t){y_3} + 0.2\cos (5t),\nonumber \\ d_3^m= & {} 0.25\sin (4t){x_3} - 0.15\sin (5t)\nonumber \\ d_4^s= & {} - 0.2\cos (2t){y_4} - 0.15\cos (t),\nonumber \\ d_4^m= & {} - 0.15\sin (t){x_4} + 0.2\cos (2t)\nonumber \end{aligned}$$
(41)

Initial conditions of master and slave systems are as \({y_1}(0) = 1\), \({y_2}(0) = 2\), \({y_3}(0) = 3\), \({y_4}(0) = 4\), \({x_1}(0) = 3\), \({x_2}(0) = 1\), \({x_3}(0) = 4\) and \({x_4}(0) = -1\). The fractional derivative orders are \({\alpha _i}={\beta _i} = 0.98\,,i = 1,\ldots ,4\). Other controller parameters are the same as the Example 1. The synchronization performance is shown in Fig. 13. The results of the encryption and decryption are shown in Fig. 14. The comparison results of proposed method and fractional non-singular terminal sliding mode technique [45] are given Table 3. It can be seen that the proposed controller shows better performance.

Fig. 14
figure 14

Results for the encryption and decryption, Example 2

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Table 3 Comparison results of proposed method in this paper and fractional non-singular terminal sliding mode technique [45], Example 2
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Example 3

In this example, the proposed controller is applied for synchronization of two uncertain fractional-order Liu systems with different initial conditions. The obtained results are compared with the results of [46]. In [46], the synchronization of two fractional-order Liu systems has been used in a secure communication scheme. The Liu system is as follows:

$$\begin{aligned} \begin{aligned} D^\alpha {y_1}&= 10\left( {{y_2} - {y_1}} \right) \\ D^\alpha {y_2}&= 40{y_1} - {y_1}{y_3} \\ D^\alpha {y_3}&= - 4{y_1^2}-2.5{y_3} \end{aligned} \end{aligned}$$
(42)

where \(\alpha =0.9\), the initial conditions of master and slave systems are \(y_1(0),y_2(0),y_3(0)=(2,1,3)\) and \(y_1(0),y_2(0),y_3(0)=(15,6.5,7)\) [46]. The input message is considered as sin(t). Other controller parameters are the same as the Example 1. The synchronization performance is shown in Fig. 15. The results of the encryption and decryption are shown in Fig. 16. The comparison results of proposed method and the method of [46] are given in Table 4. As the same as the previous examples, this example also shows that the proposed controller shows better performance. It must be noted that the image encryption based on the fractional-order chaotic systems has been studied in many papers. For instance in [47, 48] the image encryption scheme are proposed based on a fractional-order chaotic logistic systems.

Fig. 15
figure 15

Output trajectory of master and slave systems, Example 3

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Fig. 16
figure 16

Results for the encryption and decryption, Example 3

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Table 4 Comparison of our results and the results of [46], Example 3
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7 Conclusion

A new robust control strategy is proposed in this paper for synchronization of fractional-order hyperchaotic systems. Furthermore, a new self-evolving non-singleton type-2 fuzzy neural network (SE-NT2FNN) presented, in which the structure of fuzzy neural network is not fixed and is modified when necessary and the proposed SE-NT2FNN has ability to identify dynamic of hyperchaotic systems. The effect of approximation error and external disturbance can be eliminated by \({H_\infty }\) LMI-based control approach. It is shown that the closed-loop system is stable. The proposed method is successfully applied to synchronize two identical and nonidentical fractional-order hyperchaotic systems. The simulation results verified that proposed controller shows good performance in the presence of unknown functions and external disturbances. Also the results for the encryption and decryption of the input message in the secure communication system, confirmed the effectiveness of the proposed synchronization scheme. As we know, in all of works in the field of synchronization of the fractional-order chaotic systems and also in this paper, the value of the fractional-order is assumed to be known. In the future works, we will extended the results to the cases that the controller is not depend on the value of fractional-order.