The Synchronization of Hyperchaotic Systems Using a Novel Interval Type-2 Fuzzy Neural Network Controller

This paper proposed a novel interval type-2 fuzzy neural network controller (NT2FC) to synchronize 5-D hyperchaotic systems with noise disturbance and system uncertainties. In the proposed controller, the type 2 fuzzy set is designed with the 3-dimensional Gaussian membership functions (3DGMFs) to increase the system’s ability to respond to uncertainty. The parameters of the NT2FC controller are updated online via adaptation laws, which are built based on the gradient descent approach. The system stability is ensured through the Lyapunov stability analysis. In addition, the modified Jaya algorithm (MJA) is applied to optimize the learning rates in adaptation laws. Finally, the efficiency of the proposed NT2FC is examined by the numerical simulation of the hyperchaotic system’s synchronization.


I. INTRODUCTION
In recent years, the chaotic system's synchronization has been studied and applied in many different fields because of its unique dynamic characteristics including unpredictable behaviours, nonlinear and sensitive response to initial conditions [1]. Chaotic system's synchronization is a procedure that generates the control force to make one or more slave chaotic systems follow the behavior of the master chaotic system. Some notable synchronization methods of the chaotic system have been proposed in [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], and [12]. In 2013, Lin et al. proposed a wavelet cerebellar model articulation controller for synchronization of the unified chaotic system [2]. In 2018, Mufti et al. presented a synchronization and antisynchronization using sliding mode control [3]. In 2021, Dai et al. introduced a robust synchronization of chaotic systems using a neural network approach [10]. Hyperchaotic systems are high-dimensional chaotic systems, which have at least two positives and one negative Lyapunov exponents to ensure the boundedness of the systems [13], [14]. Therefore it has The associate editor coordinating the review of this manuscript and approving it for publication was Paolo Crippa . more complicated topological structures and unpredictable dynamical behaviour than the typical chaotic system. The synchronization of 5-D hyperchaotic systems has been developed in [13], [15], [16], [17], and [18], which considers five dimensions space. Nowadays, along with the development of synchronous control techniques, the hyperchaotic system is widely applied in various fields, especially in signal processing and information security [19].
Recently, the combination of neural network (NN) and fuzzy logic system (FLS) has produced the fuzzy neural system (FNN). Therefore, FNN has the reasoning approach of FLS and the learning capability of NN. The traditional fuzzy logic systems are also known as the type-1 fuzzy logic system (T1FLS), which use the type-1 membership functions in the consequent and the antecedent space. Some notable studies using T1FLS have been introduced in [20], [21], [22], and [23]. In 2012, Faieghi et al. proposed a fuzzy sliding mode controller for controlling the fractional-order Liu system [20]. In 2015, Delavari and Shokrian presented a fuzzy modeling method for synchronization of the hyperchaotic complex system [21]. Recently, some studies have shown that the type-2 fuzzy logic system (T2FLS) can better deal with the uncertainties than the T1FLS [24], [25], [26], [27]. However, the computation cost in the T2FLS is often enormous [28]. Therefore, the interval type-2 fuzzy logic system (IT2FLS) is introduced in [29] to reduce the computational complexity. Since then, the IT2FLS has been applied in many different fields [30], [31], [32], [33], [34], [35], [36], [37]. In 2018, Jiang et al. presented an IT2FLS for for stock index forecasting [31]. In 2017, Mohadeszadeh and Delavari proposed a type-2 fuzzy sliding mode controller for synchronization of hyper-chaotic systems [37]. This paper proposed a novel IT2FLS with 3-D type-2 Gaussian membership functions to achieve better synchronization performance of the chaotic system's synchronization.
Choosing the learning rates is very important in the adaptation law for updating the network's parameters. It can make the learning process oscillate or even not converge [38]. Most studies in the literature often use trial and error to obtain the suitable learning rates, but this method is often time-consuming and difficult to achieve optimal coefficients [39]. Therefore, we applied the MJA to obtain the optimal learning rates for the designed adaptation law in this study. To enhance and extend the searchability of the traditional Jaya algorithm (JA), our MJA additionally considers second-best and third-best solutions as well as second-worst and thirdworst solutions. Moreover, random parameters are added to the updating formula, which makes the MJA algorithm more flexible than the traditional JA. Rao first introduced the original JA in 2016, which is known as the parameterless algorithm [40]. JA's optimization mechanism is asymptotically the best solution and away from the worst solution in the swarm. Unlike most other optimization algorithms, JA does not require using any particular parameters other than two general parameters, such as the number of iterations and the population size. Some notable applications of JA for optimization problems in various fields can be found in [41], [42], [43], [44], [45], and [46]. In 2019, Yu et al. presented a Jaya algorithm to solve economic load dispatch problems [43]. In 2021, Deboucha et al. introduced a modified deterministic Jaya -based MPPT to enhance the PV system's effectiveness [46].
Motivated by above discussion, this work provided a novel controller, which combines the advantages of the IT2FNN, the 3DGMFs, the self-evolving algorithm, and the modified model Jaya algorithm. The significant contributions of the proposed synchronization system are (1) The successful  design of IT2FNN with parameters can be updated online using the designed adaptation laws; (2) The design of 3-D type-2 Gaussian membership functions for better cope with the system uncertainties; (3) The design of MJA for optimization the suitable learning rates.

III. DESIGN OF NT2FC CONTROLLER A. STRUCTURE OF NT2FC
The fuzzy inference, which represents the relationship between the consequent and the antecedent, is given as follows: where X = [x 1 , x 2 , . . . , x n i ] T n i andẊ = [ẋ 1 ,ẋ 2 , . . . ,ẋ n i ] T n i are the input vector and its derivative, respectively; n i and n m are the number of inputs and outputs, respectively; n k and n j are the number of rules and the number of membership functions in each input, respectively;μ ij = µ ij µ ij andw km = w km w km respectively are the input membership functions and the output connecting weights.
The structure of the proposed NT2FC includes five layers, as shown in Fig. 1. The final output of the NT2FC is computed as the following steps: Step 1: This step prepares the vector inputs X = x 1 , x 2 , . . . , x n i T andẊ = ẋ 1 ,ẋ 2 , . . . ,ẋ n i T . Then these inputs will be directly transferred to the membership functions in the next step.
Step 2: The inputs x i andẋ i in the previous step will be fed into the 3-D Gaussian membership functions in (8) and (9). A detailed illustration can be seen in Figure 2. The equation for computing the membership grades are given as: T.-L. Le, V.-B. Ngo: Synchronization of Hyperchaotic Systems Using a Novel Interval Type-2 Fuzzy Neural Network Controller whereμ ij andv ij respectively are the upper membership grades and upper variance of the GMFs; µ ij and v ij respectively are the lower membership grades and lower variance of the GMFs; c ij is the mean of the GMFs.
Step 3: In this step, the product operation is applied to obtain the upper and lower fuzzy firing strengths as follows:f wheref k and f k respectively are the upper and lower fuzzy firing strengths.
Step 4: In this step, the centre-of-sums type-reduction is used to obtain the pre- The output connecting weights w kmw km using in this layer can be expressed as follows: Step 5: In this step, the defuzzification operator using the average of the left and right pre-output layer values   is performed as: where o l m and o r m respectively are the left and right pre-output nodes;û m NT 2FC is the control signal of m th output using the proposed NT2FC controller.
The control vectors using the proposed NT2FC can be denoted as follows:

B. THE PARAMETER LEARNING FOR NT2FC
The control vectorû NT 2FC in (16) is used to optimize the ideal synchronizer control vector u * (t) in (6). The Lyapunov function is defined as E = 1 2 (e m (t)) 2 , where m can be replaced by x, y, z, v, and w. Then, using the gradient descent approach and the chain rule, the adaptation laws ofû NT 2FC are derived as follows: whereη w ,η c ,η v are the positive learning-rates.

C. LEARNING RATE OPTIMIZATION ALGORITHM
In order to optimize the learning rates for the adaptation law of the proposed network, the modified Jaya algorithm is designed as follow: Step 1: Initialize n η set learning ratesη w ,η c ,η v .
Step 2: For i = 1 to n η ; Run the NT2FC synchronization system with i-th learning rate setη i w ,η i c ,η i v .
Step 3: Calculate the fitness function based on the synchronization errors. The root mean square error (RMSE) equation is defined as follows: where n q is the number of samples.
Step 4: Choose the best and worst learning rate sets based on the lowest and highest fitness function, respectively. Obtain η best w , η best c , η best v and η worst w , η worst c , η worst v . VOLUME 10, 2022 Step 5: For i = 1 to n η ; Update the learning rate set η w ,η c ,η v by: where can be replaced by w, c, or v; rd ∈ [1, 3], rd 1 ∈ [0, 1] and rd 2 ∈ [0, 1] are the random numbers; η best(rd) and η worst(rd) respectively are the learning rates with rd lowest fitness function and highest fitness function.
Step 6: Run the NT2FC synchronization system with the updated learning rate sets. If new solutions in (35) are better than the old solutions, then accept the new i-th learning rate setη w ,η c ,η v . Otherwise, keep the previous i-th learning rate set.
Step 7: Return to step 2 and repeat until the synchronization process is finished.  Therefore, the NT2FC synchronizer can obtain suitable learning rates for the adaptation law by applying the proposed modified Jaya algorithm.

IV. ILLUSTRATIVE EXAMPLES
In this section, two examples of hyperchaotic synchronization are given to verify the validity of the proposed synchronization method. The synchronization scheme using the NT2FC synchronizer is demonstrated in Fig. 3. The parameters for the NT2FC synchronizer are set as n i = 5, n j = 3, n η = 20; the sampling time is 0.001s; the initial parameters of hyperchaotic systems are set as [x 1 , y 1 , z 1 , v 1 , w 1 ] = [5,5,5,5,5 ] T and [x 2 , y 2 , z 2 , v 2 , w 2 ] = [0, 0, 0, 0, 0 ] T , respectively. The external disturbances and system uncertainties are chosen as

Case 1:
In this case, as same with [48], the parameters of the 5-D hyperchaotic systems are set by a = 10, b = 8/3, c = 28, h 1 = −2, h 2 = −0.09, and h 3 = 8. The state trajectories of the hyperchaotic synchronization systems in 3-D using the NT2FC are displayed in Fig. 4. The state trajectories of master-slave hyperchaotic systems in time domain using the NT2FC are shown in Fig. 5. The synchronization control signals and the synchronization error dynamics are respectively given in Fig. 6 and Fig. 7. The learning rates adjustment using the MJA are shown in Fig. 8.

Case 2:
In this example, as same with [48], the parameters of the 5-D hyperchaotic systems are set by a = 10, b = 8/3, c = 28, h 1 = −1.155, h 2 = −0.12, and h 3 = 11.3. The state trajectories of the hyperchaotic synchronization systems in 3-D are displayed in Fig. 9. The state trajectories of masterslave hyperchaotic systems using the proposed NT2FC are shown in Fig. 10. The synchronization control signals and the synchronization error dynamics are respectively given in Fig. 11 and Fig. 12. The learning rates adjustment using the MJA are shown in Fig. 13.

Case 3:
In this example, the parameters of the 5-D hyperchaotic systems are chosen as same as case 2. In order to show the efficiency of the proposed method, the initial parameters of master and slave hyperchaotic systems are set different with the two cases above as: [x 1 , y 1 , z 1 , v 1    The state trajectories of the hyperchaotic synchronization systems in 3-D are displayed in Fig. 14. The state trajectories of master-slave hyperchaotic systems using the proposed NT2FC are shown in Fig. 15. The synchronization control signals and the synchronization error dynamics are respectively given in Fig. 16 and Fig. 17. The learning rates adjustment using the MJA are shown in Fig. 18.
Remark 1: For three cases, the proposed NT2FC synchronizer has shown the effectiveness of synchronizing the 5-D hyperchaotic systems even when cosidering the external disturbances and system uncertainties. The RMSE among our proposed synchronizer and the cerebellar model articulation controller (CMAC), the interval type-2 fuzzy neural network (IT2FNN) are given in Table 1. The proposed NT2FC synchronizer has paid a little longer computation time than the other methods due to the MJA's timely processing. However, it does not disturb the synchronization performance, and the proposed NT2FC synchronizer can obtain a better result in all cases.
Remark 2: The synchronization in case 3 had shown the robustness of the proposed method even when the initial parameters of hyperchaotic systems are set differently from case 1 and case 2.
Remark 3: The design of NT2FC's structure such as the number of member functions and the number of fuzzy rules has a great influence on the performance of the system. In this study, we use the trial and error method to achieve a suitable network structure. The application of self-structured algorithms to achieve the optimal structure will be the next research direction.

Remark 4:
In this study, both online and offline MJA modes are used. First, the system will be run in offline mode 1000 times (each time 20 seconds) to calibrate the fitness function. After running the system with MJA offline mode, the best optimal learning rates will be selected to initialize the value for the MJA online mode (each time 0.1 seconds), the online MJA can learn and adapt quickly for real-time application. The fast adaptation and learning are due to the real-time feedback system.
Remark 5: In this study, the Lyapunov approach is used to find the learning rate ranges such that the system is stable. Besides, the MJA algorithm is also used to achieve optimal learning rates. Furthermore, the MJA algorithm can achieve optimal performance faster when the initial values are selected from that learning rate range.

V. CONCLUSION
In this paper, an NT2FC synchronizer is designed to synchronize a class of 5-D hyperchaotic systems. In the design of NT2FC, the 3-D type-2 Gaussian membership functions are applied to increase the ability to deal with uncertainties and disturbances. The MJA is applied to optimize the learning rates of the designed adaptation laws. Finally, the superiority of proposed method is examined on two simulation results of synchronization of the hyperchaotic systems. In summary, the proposed NT2FC is an efficient controller for the synchronization problems. Applying new algorithms to improve the system performance and using them to control other nonlinear systems will be our future work.