Optimum Design of Function-Link Type-2 Fuzzy Asymmetric CMAC Based on Self-Organizing Algorithm and Modified Jaya Algorithm

In this work, a novel self-organizing interval type-2 fuzzy cerebellar model articulation controller design for chaotic systems synchronization is presented. A function-link network and an asymmetric-membership function are employed to improve the learning ability and flexibility of the proposed network. The gradient-descent method is used to derive the adaptive laws for online updating the network parameters. Suitable learning rates for adaptive laws can be achieved by employing the modified Jaya algorithm. Furthermore, the proposed self-organizing algorithm is used to generate new fuzzy rules or delete less-active fuzzy rules. The control system's exponential stability is analyzed by applying the Lyapunov theory. Finally, numerical simulations on the synchronization of four-dimensional chaotic systems (considering external disturbances and system uncertainties) are conducted to validate the efficiency and performance of the proposed method.

SECTION I. Introduction

During the last decades, the cerebellar-model articulation controller (CMAC) has attracted increasing attention by many researchers working in various fields [1]–​[4]. This controller, which was provided by Albus in 1975, is a special type of neural network [5]. It is inspired by the mammalian cerebellum, and it is classified as a non-fully connected perceptron-like associative-memory network with overlapping receptive fields. The CMAC’s characteristic structure is mapping input space to a group of memory locations. Then, the quantized state value can be obtained from the sum of the data stored in the memory cells associated with the hypercubes covering this input [6]. Several researchers have incorporated fuzzy inference into the CMAC to obtain an architecture known as fuzzy CMAC (FCMAC) that incorporates association layers as fuzzy-membership function layers, and each associative cell performs a fuzzy rule [7]–​[9]. However, the FCMAC with type-1 fuzzy-membership functions (T1FMFs) is incapable of dealing well with uncertainties [10]. Many studies in the literature have demonstrated that the performance of type-2 fuzzy-logic systems (T2FLSs), which use type-2 fuzzy-membership functions (T2FMFs), is better than that of type-1 fuzzy-logic systems (T1FLSs), particularly in the presence of internal and external uncertainties [11]–​[13]. The high-complexity computation of T2FLSs can be reduced by the interval type-2 fuzzy-logic systems (IT2FLSs), proposed by Liang and Mendel in 2010 [14]. Since this first work, the development of IT2FLSs has been significantly extended and widely applied to various research fields [15]–​[17].

Symmetric-membership functions in fuzzy structures capable of facilitating the design of adaptation laws have been presented in the literature [18]. Recently, asymmetric-membership functions have been presented in an effort to improve the network-learning ability [19]–​[22]. In this work, a type-2 asymmetric Gaussian-membership function (T2AGMF), which is constituted by two lower Gaussian-membership functions and two upper Gaussian-membership functions, is employed.

A function-link network (FLN) was first introduced by Pao in 1989 [23]. This network is capable of expanding the input information using function expansion. Generally, the FLN is a type of feed-forward neural network, which can be efficiently employed to approximate nonlinear functions with less computational load and rapid convergence speed [24]. The FLN outputs are generated by trigonometric functions and nonlinear combinations of the input variables [25]. In the past, FLN was applied in network design to achieve better control performance [25]–​[28].

The network structure of the CMAC and fuzzy controller significantly affects the control performance. A trial-and-error method to obtain a suitable network structure has often been used in the literature. However, this is a time-consuming method, and the network’s performance needs to be improved [29]. Recently, a self-organizing algorithm has been proposed to automatically construct a network’s structure [30]–​[32]. In this work, the self-organizing algorithm is applied to obtain a suitable network structure for the designed network.

In the design of many neural networks and neural-fuzzy systems, the determination of suitable learning rates is very important as it significantly affects the system’s performance. In this work, the modified Jaya algorithm is used to optimize the suitable learning rates for the proposed network. Recently, the Jaya algorithm has been used to solve the optimization problem in various research fields [33]–​[36]. It is a simple metaheuristic algorithm proposed by Rao in 2016, and it does not require any tuning of the algorithm-specific parameters [37]. In contrast, most of the well-known metaheuristic algorithms, such as the genetic algorithm, particle swarm optimization, gray wolf optimization, and ant lion optimization, often require tuning of their algorithm-specific parameters to improve their performance. In the Jaya algorithm, only two parameters (population size and number of iterations) are required. In this work, a modified Jaya algorithm (MJA) is proposed. Compared with the original Jaya algorithm in [37], the proposed MJA considers the second and the third of the worst and the best solutions. Therefore, the searchability of the proposed MJA can be enhanced.

Chaotic systems are nonlinear dynamic systems, which exhibit highly-complex dynamics, inner randomness, self-similarity, strange attractor, and sensitivity to initial conditions [38]. The synchronization of chaotic systems is a fundamental problem in nonlinear science that has attracted the attention of many researchers in various fields. Many approaches for synchronizing chaotic systems have been proposed in the past, such as the fuzzy controller, neural network controller, robust controller, and CMAC controller [39]–​[41]. Synchronization can be defined as the process where two or more chaotic systems are coupled or when a chaotic system drives another chaotic system [42]. According to the study in [43], the sources of uncertainties in fuzzy logic systems can be listed as the unknown nonlinear characteristics of the systems, the noise, the precision, and the environmental conditions of the measurement devices.

Taking into account the above literature review, a self-organizing interval type-2 fuzzy CMAC using a FLN and an asymmetric-membership function (SFIT2FAC) is presented in this work. The proposed synchronizer can be used to synchronize chaotic systems. Compared with existing research work [44], [45], the proposed system exhibits certain advantages. For example, the network structure can be easily designed, the learning rates can be optimized, and the learning capability and network flexibility can be improved. The major contributions of this work can be summarized as follows: (1) Design of a SFIT2FAC synchronizer with adaptive parameters, which can be tuned online; (2) Design of a FLN to update the weight for the SFIT2FAC network; (3) Design of a type-2 asymmetric Gaussian-membership function to improve the network’s learning ability; (4) Design of a self-organizing algorithm to autonomously construct the proposed network structure. (5) Design of a MJA to obtain the suitable learning rates for the SFIT2FAC network.

The organization of this study is presented as follows. Section II presents the problem formulation of chaotic systems. Section III describes the structure of the proposed SFIT2FAC synchronizer. Section IV presents the simulation results of chaotic system synchronization. Section V concludes the work.

SECTION II. Problem Formulation

Consider a four-dimensional Lorenz–Stenflo chaotic system [46], which includes the master and slave systems as follows: $$\begin{align*} \dot {x}_{1} (t)=&\alpha \left ({{x_{2} (t)-x_{1} (t)} }\right)+\beta x_{4} \\[2pt] \dot {x}_{2} (t)=&\gamma x_{1} (t)-x_{1} (t)x_{3} (t)-x_{2} (t) \\[2pt] \dot {x}_{3} (t)=&x_{1} (t)x_{2} (t)-\lambda x_{3} (t) \\[2pt] \dot {x}_{4} (t)=&-x_{1} (t)-\alpha x_{4}, \tag{1}\\[2pt] \dot {y}_{1} (t)=&\alpha \left ({{y_{2} (t)-y_{1} (t)} }\right)+\beta y_{4} +\zeta _{1} (t)+\Delta f(y_{1})+u_{1} (t) \\[2pt] \dot {y}_{2} (t)=&\gamma y_{1} (t)\!-\!y_{1} (t)y_{3} (t)\!-\!y_{2} (t)+\zeta _{2} (t)+\Delta f(y_{2})+u_{2} (t) \\[2pt] \dot {y}_{3} (t)=&y_{1} (t)y_{2} (t)-\lambda y_{3} (t)+\zeta _{3} (t)+\Delta f(y_{3})+u_{3} (t) \\[2pt] \dot {y}_{4} (t)=&-y_{1} (t)-\alpha y_{4} +\zeta _{4} (t)+\Delta f(y_{4})+u_{4} (t),\tag{2}\end{align*}$$ where $\boldsymbol {x}(t)=\left [{ {x_{1} (t),x_{2} (t),x_{3} (t),x_{4} (t)} }\right]$ is the master chaotic state vector; $\boldsymbol {y}(t)=\left [{ {y_{1} (t),y_{2} (t),y_{3} (t),y_{4} (t)} }\right]$ is the slave chaotic state vector; $\boldsymbol {u}(t)=\left [{ {u_{1} (t),u_{2} (t),u_{3} (t),u_{4} (t)} }\right]$ is the synchronization control vector; $\boldsymbol {\zeta }(t)=[\zeta _{1} (t),\zeta _{2} (t),\zeta _{3} (t), \zeta _{4} (t)]$ is the external disturbances vector; $\Delta \boldsymbol {f}(y(t))=\left [{ {\Delta f(y_{1}),\Delta f(y_{2}),\Delta f(y_{3}),\Delta f(y_{4})} }\right]$ is the uncertainties vector; $\alpha$ , $\beta,\gamma$ , and $\lambda$ denote the parameters for defining the chaotic attractor.

The synchronization error vector $\boldsymbol {e}(t)=[e_{1} (t),e_{2} (t),e_{3} (t), e_{4} (t)]$ can be obtained as follows: $$\begin{align*} e_{1} (t)=&y_{1} (t)-x_{1} (t) \\ e_{2} (t)=&y_{2} (t)-x_{2} (t) \\ e_{3} (t)=&y_{3} (t)-x_{3} (t) \\ e_{4} (t)=&y_{4} (t)-x_{4} (t).\tag{3}\end{align*}$$ Applying (1) and (2) into (3), the dynamic-error vector $\dot {\boldsymbol {e}}(t)=\left [{ {\dot {e}_{1} (t),\dot {e}_{2} (t),\dot {e}_{3} (t),\dot {e}_{4} (t)} }\right]$ is obtained as follows: $$\begin{align*} \dot {e}_{1} (t)=&\alpha \left ({{e_{2} (t)-e_{1} (t)} }\right)+\beta e_{4} (t)+\zeta _{1} (t)+\Delta f(y_{1})+u_{1} (t) \\ \dot {e}_{2} (t)=&\gamma e_{1} (t)-e_{2} (t)-y_{1} (t)y_{3} (t)+x_{1} (t)x_{3} (t) \\&+\,\zeta _{2} (t)+\Delta f(y_{2})+u_{2} (t) \\ \dot {e}_{3} (t)=&y_{1} (t)y_{2} (t)-x_{1} (t)x_{2} (t)-\lambda e_{3} (t)+\zeta _{3} (t) \\&+\,\Delta f(y_{3})+u_{3} (t) \\ \dot {e}_{4} (t)=&-e_{1} (t)-\alpha e_{4} +\zeta _{4} (t)+\Delta f(y_{4})+u_{4} (t).\tag{4}\end{align*}$$ Eq. (4) can be formulated in the state space as follows: $$\begin{equation*} \dot {\boldsymbol {e}}(t)=\boldsymbol {Fe}(t)+\boldsymbol {\zeta }(t)+\Delta \boldsymbol {f}(y(t))+\boldsymbol {u}(t),\tag{5}\end{equation*}$$ where $$\begin{align*} \boldsymbol {F}=\left [{ {\begin{array}{ccccc} {-\alpha } &\quad {\alpha } &\quad {0} &\quad {\beta } \\ {\left ({{\gamma -y_{3} (t)} }\right)} &\quad {-1} &\quad {-x_{1} (t)} &\quad {0} \\ {x_{2} (t)} & \quad {y_{1} (t)} & \quad {-\lambda } & \quad {0} \\ {-1} &\quad {0} &\quad {0} &\quad {-\alpha } \\ \end{array}} }\right]\!.\end{align*}$$ In (5), if $\boldsymbol {\zeta }(t)$ and $\Delta \boldsymbol {f}(y(t))$ are precisely known, then the ideal synchronizer control vector can be determined as follows: $$\begin{equation*} \boldsymbol {u}^{\ast }(t)=-\boldsymbol {Fe}(t)-\boldsymbol {Ge}(t)-\boldsymbol {\zeta }(t)-\Delta \boldsymbol {f}(y(t)),\tag{6}\end{equation*}$$ where $\dot {\boldsymbol {e}}(t)=-\boldsymbol {Ge}(t)$ and $\boldsymbol {G}=diag\left ({{g_{1},g_{2},g_{3},g_{4}} }\right)$ is the feedback gain vector.

If the terms $\boldsymbol {\zeta }(t)$ and $\Delta \boldsymbol {f}(y(t))$ are precisely known, $\boldsymbol {u}^{\ast }(t)$ in (6) can be obtained. However, in practical applications, $\boldsymbol {\zeta }(t)$ and $\Delta \boldsymbol {f}(y(t))$ cannot be precisely known. Therefore, the proposed SFIT2FAC synchronizer, which does not request $\boldsymbol {\zeta }(t)$ and $\Delta \boldsymbol {f}(y(t))$ to be known quantities, is designed in the following section to estimate the $\boldsymbol {u}^{\ast }(t)$ .

SECTION III. Controller Design

A. The Structure of the SFIT2FAC

The $p^{\mathrm {th}}$ fuzzy inference rule for the SFIT2FAC synchronizer can be described as follows: $$\begin{align*}&\hspace {-1.5pc}Rule~p^{th}:~IF~i_{1} ~is~\tilde {\chi }_{1jk} ~and~i_{2}~is~\tilde {\chi }_{2jk},\ldots,~and~i_{n_{i}} ~is~\tilde {\chi }_{n_{i} jk} \\&\qquad \quad Then~\tilde {\psi }_{jk}^{q} =\left [{ {\underline {\psi }_{jk}^{q} ~\bar {\psi }_{jk}^{q}} }\right], \\&\qquad \quad for~i=1,2,\ldots,n_{i};\quad j=1,2,\ldots,n_{j}; \\&\qquad \quad \qquad k=1,2,\ldots,n_{k};~q=1,2,\ldots,n_{q}.\tag{7}\end{align*}$$ where $n_{i}$ and $n_{q}$ are the input and output dimensions, respectively; $n_{j}$ is the number of layers for each input dimension; $n_{k}$ is the number of blocks for each layer; $\tilde {\chi }_{ijk}$ and $\tilde {\psi }_{jk}$ denote the input and output membership functions, respectively.

The architecture of the proposed SFIT2FAC synchronizer is presented in Fig. 1. The synchronizer consists of five spaces as follows:

1. Input space: This space is used to obtain the input vector $\boldsymbol {i}=\left [{ {i_{1},i_{2},\ldots,i_{n_{i}}} }\right]$ . Each node in this space corresponds to each input.

2. Association memory space: This space is used to compute the membership grade. Each node in this space includes a type-2 asymmetric membership function (T2AMF), which is presented in Fig. 2. This function is given as follows: $$\begin{align*} \bar {\chi }_{ijk}=&\begin{cases} \displaystyle \textrm {exp}\left \{{{\frac {-\left ({{i_{i} -\bar {\rho }_{ijk}^{l}} }\right)^{2}}{2\left ({{\bar {\upsilon }_{ijk}^{l}} }\right)^{2}}} }\right \}\!,&i_{i} \le \bar {\rho }_{ijk}^{l} \\ 1,&\bar {\rho }_{ijk}^{l} \le \,i_{i} \le \bar {\rho }_{ijk}^{r} \\ \displaystyle \textrm {exp}\left \{{{\frac {-\left ({{i_{i} -\bar {\rho }_{ijk}^{r}} }\right)^{2}}{2\left ({{\bar {\upsilon }_{ijk}^{r}} }\right)^{2}}} }\right \}\!,&\bar {\rho }_{ijk}^{r} \le i_{i}, \end{cases}\tag{8}\\ \underline {\chi }_{\,ijk}=&\begin{cases} \displaystyle \sigma \ast \textrm {exp}\left \{{{\frac {-\left ({{i_{i} -\underline {\rho }_{ijk}^{l}} }\right)^{2}}{2\left ({{\underline {\upsilon }_{\,ijk}^{l}} }\right)^{2}}} }\right \},&i_{i} \le \underline {\rho }_{\,\,ijk}^{l} \\ \sigma,&\underline {\rho }_{\,\,ijk}^{l} \le \,i_{i} \le \underline {\rho }_{\,\,ijk}^{r} \\ \displaystyle \sigma \ast \textrm {exp}\left \{{{\frac {-\left ({{i_{i} -\underline {\rho }_{\,\,ijk}^{r}} }\right)^{2}}{2\left ({{\underline {\upsilon }_{\,ijk}^{r}} }\right)^{2}}} }\right \},&\underline {\rho }_{\,\,ijk}^{r} \le i_{i}, \end{cases} \\{}\tag{9}\end{align*}$$ where $\bar {\chi }_{ijk}$ and $\underline {\chi }_{ijk}$ are the upper and lower membership functions; $\bar {\rho }_{ijk}^{l}$ , $\bar {\rho }_{ijk}^{r}$ and $\bar {\upsilon }_{ij}^{l}$ , $\bar {\upsilon }_{ij}^{r}$ are the means and variances of the two upper Gaussian membership functions, respectively; $\underline {\rho }_{ijk}^{l}$ , $\underline {\rho }_{ijk}^{r}$ and $\underline {\upsilon }_{\,ij}^{l}$ , $\underline {\upsilon }_{\,ij}^{r}$ are the means and variances of the two lower Gaussian membership functions, respectively.

   The following condition is applied to ensure a reasonable T2AMF: $$\begin{align*} \begin{cases} {\bar {\rho }_{ijk}^{l} \le \underline {\rho }_{ ijk}^{l} \le \underline {\rho }_{ ijk}^{r} \le \bar {\rho }_{ijk}^{r}} \\ {\underline {\upsilon }_{ ij}^{l} \le \bar {\upsilon }_{ij}^{l} \le \underline {\upsilon }_{ ij}^{r} \le \bar {\upsilon }_{ij}^{r} } \\ {0.5\le \sigma \le 1. } \end{cases}\tag{10}\end{align*}$$

3. Receptive-field space: This space is used to compute the multidimensional receptive-field function, which is defined as follows: $$\begin{equation*} \bar {\tau }_{jk} =\mathop \prod \limits _{i=1}^{n_{i}} \bar {\chi }_{ijk} \quad \text {and} \quad \underline {\tau }_{jk} =\mathop \prod \limits _{i=1}^{n_{i}} \underline {\chi }_{ijk},\tag{11}\end{equation*}$$ where $\bar {\tau }_{jk}$ and $\underline {\tau }_{jk}$ are the upper and lower firing strengths, which are associated with the $j^{th}$ layer and $k^{th}$ block. This space can be expressed in vector form as follows: $$\begin{align*} \boldsymbol {\underline {\tau }}=&\left [{ {\underline {\tau }_{ 11}, \ldots,\underline {\tau }_{ 1n_{k}},\underline {\tau }_{ 21}, \ldots,\underline {\tau }_{ 2n_{k}},\ldots,\underline {\tau }_{ n_{j} 1}, \ldots,\underline {\tau }_{ n_{j} n_{k}}} }\right]^{T} \\\in&\Re ^{n_{j} n_{k}} \\=&\left [{ {\boldsymbol {\underline {\tau }}_{ 1}, \ldots,\boldsymbol {\underline {\tau }}_{ r}, \ldots,\boldsymbol {\underline {\tau }}_{ n_{p}}} }\right]^{T}\in \Re ^{n_{p}},\tag{12}\\ \bar {\boldsymbol {\tau }}=&\left [{ {\bar {\tau }_{ 11}, \ldots,\bar {\tau }_{ 1n_{k}},\bar {\tau }_{ 21}, \ldots,\bar {\tau }_{ 2n_{k}},\ldots,\bar {\tau }_{ n_{j} 1}, \ldots,\bar {\tau }_{ n_{j} n_{k}}} }\right]^{T} \\\in&\Re ^{n_{j} n_{k}} \\=&\left [{ {\bar {\boldsymbol {\tau }}_{ 1}, \ldots,\bar {\boldsymbol {\tau }}_{ r}, \ldots,\bar {\boldsymbol {\tau }}_{ n_{p}}} }\right]^{T}\in \Re ^{n_{p}},\tag{13}\end{align*}$$ where $n_{p} =n_{j} n_{k}$ is the number of fuzzy rules.

4. Weight memory space: This space is used to compute the fuzzy memory weights, which are used to connect the receptive-field space with the output space.

   Initially, the FLN with a trigonometric function is applied to expand the inputs, as shown in Fig. 3. The function expansion vector is defined as follows: $$\begin{align*} \boldsymbol {\varphi }=&\left [{ i_{1},\sin \left ({{\pi i_{1}} }\right),\cos \left ({{\pi i_{1}} }\right),\ldots,i_{n_{i}},\sin \left ({{\pi i_{n_{i}}} }\right),}\right. \\&\qquad \left.{\cos \left ({{\pi i_{n_{i}}} }\right),i_{1} i_{2},\ldots,i_{1} i_{n_{i}}}\right]^{T} \\=&\left [{ {\varphi _{1},\varphi _{2},\ldots,\varphi _{n_{h}}} }\right]^{T}\in \Re ^{n_{h}},\tag{14}\end{align*}$$ where $n_{h}$ is the number of function expansion outputs.

   Then, the fuzzy memory weights for the $q^{th}$ output are given as follows: $$\begin{align*} \bar {\boldsymbol {\psi }}^{q}=&\left [{ {{\begin{array}{cccccccccccccccccccc} {\bar {\psi }_{11}^{q}} \\ \vdots \\ {\bar {\psi }_{1n_{k}}^{q}} \\ {\bar {\psi }_{21}^{q}} \\ \vdots \\ {\bar {\psi }_{2n_{k}}^{q}} \\ \vdots \\ {\bar {\psi }_{n_{j} 1}^{q}} \\ \vdots \\ {\bar {\psi }_{n_{j} n_{k}}^{q}} \\ \end{array}}} }\right]=\left [{ {{\begin{array}{cccccccccccccccccccc} {\bar {\theta }_{111}^{q}} &\quad \!\! {\bar {\theta }_{211}^{q}} &\quad \!\! \cdots &\quad \!\! {\bar {\theta }_{n_{h} 11}^{q}} \\ \vdots &\quad \!\! \vdots &\quad \!\! \ddots &\quad \!\! \vdots \\ {\bar {\theta }_{11n_{k}}^{q}} &\quad \!\! {\bar {\theta }_{21n_{k}}^{q} } &\quad \!\! \cdots &\quad \!\! {\bar {\theta }_{n_{h} 1n_{k}}^{q}} \\[0.3pc] {\bar {\theta }_{121}^{q}} &\quad \!\! {\bar {\theta }_{221}^{q}} &\quad \!\! \cdots &\quad \!\! {\bar {\theta }_{n_{h} 21}^{q}} \\ \vdots &\quad \!\! \vdots &\quad \!\! \ddots &\quad \!\! \vdots \\ {\bar {\theta }_{12n_{k}}^{q}} &\quad \!\! {\bar {\theta }_{22n_{k}}^{q} } &\quad \!\! \cdots &\quad \!\! {\bar {\theta }_{n_{h} 2n_{k}}^{q}} \\ \vdots &\quad \!\! \vdots &\quad \!\! \ddots &\quad \!\! \vdots \\ {\bar {\theta }_{1n_{j} 1}^{q}} &\quad \!\! {\bar {\theta }_{2n_{j} 1}^{q} } &\quad \!\! \cdots &\quad \!\! {\bar {\theta }_{n_{h} n_{j} 1}^{q}} \\ \vdots &\quad \!\! \vdots &\quad \!\! \ddots &\quad \!\! \vdots \\ {\bar {\theta }_{1n_{j} n_{k}}^{q}} &\quad \!\! {\bar {\theta }_{2n_{j} n_{k}}^{q}} &\quad \!\! \cdots &\quad \!\! {\bar {\theta }_{n_{h} n_{j} n_{k} }^{q}} \\ \end{array}}} }\right]\left [{ {{\begin{array}{cccccccccccccccccccc} {\varphi _{1}} \\ {\varphi _{2}} \\ \vdots \\ {\varphi _{n_{h}}} \\ \end{array}}} }\right]\!\!\! \\\equiv&\bar {\boldsymbol {\theta }}^{q}\boldsymbol {\varphi },\tag{15}\\ \boldsymbol {\underline {\psi }}^{q}=&\left [{ {{\begin{array}{cccccccccccccccccccc} {\underline {\psi }_{11}^{q}} \\ \vdots \\ {\underline {\psi }_{1n_{k}}^{q}} \\ {\underline {\psi }_{21}^{q}} \\ \vdots \\ {\underline {\psi }_{2n_{k}}^{q}} \\ \vdots \\ {\underline {\psi }_{n_{j} 1}^{q}} \\ \vdots \\ {\underline {\psi }_{n_{j} n_{k}}^{q}} \\ \end{array}}} }\right]=\left [{ {{\begin{array}{cccccccccccccccccccc} {\underline {\theta }_{111}^{q}} &\quad \!\! {\underline {\theta }_{211}^{q}} &\quad \!\! \cdots &\quad \!\! {\underline {\theta }_{n_{h} 11}^{q}} \\ \vdots &\quad \!\! \vdots &\quad \!\! \ddots &\quad \!\! \vdots \\ {\underline {\theta }_{11n_{k}}^{q}} &\quad \!\! {\underline {\theta }_{21n_{k}}^{q}} &\quad \!\! \cdots &\quad \!\! {\underline {\theta }_{n_{h} 1n_{k}}^{q}} \\[0.3pc] {\underline {\theta }_{121}^{q}} &\quad \!\! {\underline {\theta }_{221}^{q}} &\quad \!\! \cdots &\quad \!\! {\underline {\theta }_{n_{h} 21}^{q}} \\ \vdots &\quad \!\! \vdots &\quad \!\! \ddots &\quad \!\! \vdots \\ {\underline {\theta }_{12n_{k}}^{q}} &\quad \!\! {\underline {\theta }_{22n_{k}}^{q}} &\quad \!\! \cdots &\quad \!\! {\underline {\theta }_{n_{h} 2n_{k}}^{q}} \\ \vdots &\quad \!\! \vdots &\quad \!\! \ddots &\quad \!\! \vdots \\ {\underline {\theta }_{1n_{j} 1}^{q}} &\quad \!\! {\underline {\theta }_{2n_{j} 1}^{q}} &\quad \!\! \cdots &\quad \!\! {\underline {\theta }_{n_{h} n_{j} 1}^{q}} \\ \vdots &\quad \!\! \vdots &\quad \!\! \ddots &\quad \!\! \vdots \\ {\underline {\theta }_{1n_{j} n_{k}}^{q}} &\quad \!\! {\underline {\theta }_{2n_{j} n_{k}}^{q}} &\quad \!\! \cdots &\quad \!\! {\underline {\theta }_{n_{h} n_{j} n_{k}}^{q}} \\ \end{array}}} }\right]\!\left [{\! {{\begin{array}{cccccccccccccccccccc} {\varphi _{1}} \\ {\varphi _{2}} \\ \vdots \\ {\varphi _{n_{h}}} \\ \end{array}}}\! }\right]\!\! \\\equiv&\boldsymbol {\underline {\theta }}^{q}\boldsymbol {\varphi },\tag{16}\end{align*}$$ where $\bar {\boldsymbol {\theta }}^{q}\in \Re ^{n_{j} n_{k} \times n_{h}}$ and $\boldsymbol {\underline {\theta }}^{q}\in \Re ^{n_{j} n_{k} \times n_{h}}$ are the upper and lower functional weights matrix; $\bar {\psi }_{jk}^{q}$ and $\underline {\psi }_{jk}^{q}$ are the upper and lower connecting weights for the $p^{th}$ rule, and $q^{th}$ output.

5. Output space: This space is used to perform the product operation of the receptive-field space and the weight memory space. Then, the $q^{th}$ output is given as follows: $$\begin{align*}&\hspace {-1.5pc}\hat {u}_{SFIT2FAC}^{q} \\=&\frac {u_{q}^{l} +u_{q}^{r}}{2} \\=&\frac {1}{2}\left ({{\omega \sum \limits _{j=1}^{n_{j}} {\sum \limits _{k=1}^{n_{k}} {\underline {\psi }_{jk}^{q} \underline {\tau }_{jk}}} +\left ({{1-\omega } }\right)\sum \limits _{j=1}^{n_{j}} {\sum \limits _{k=1}^{n_{k}} {\bar {\psi }_{jk}^{q} \bar {\tau }_{jk}}}} }\right)\!,\tag{17}\end{align*}$$ where $\omega$ is the gain for adjusting the contribution of the upper and lower T2AMF.

FIGURE 1.

Architecture of the SFIT2FAC synchronizer.

FIGURE 2.

The proposed T2AGMF (a) the upper membership function (UMF); (b) the lower membership function (LMF); (c) the combined membership function.

FIGURE 3.

Structure of the FLN.

This space can be expressed in vector form as follows: $$\begin{align*} \hat {\boldsymbol {u}}_{SFIT2FAC}=&\left [{ {\hat {u}_{SFIT2FAC}^{1},\hat {u}_{SFIT2FAC}^{2},\ldots,\hat {u}_{SFIT2FAC}^{n_{q}}} }\right]^{T} \\\in&\Re ^{n_{q}}.\tag{18}\end{align*}$$

B. Parameter Learning Algorithm

Assume that an optimal controller $\boldsymbol {u}_{SFIT2FAC}^{\ast }$ , which is an approach to the ideal controller $\boldsymbol {u}^{\ast }(t)$ in (6), exists, such that $$\begin{align*}&\hspace {-0.5pc}\boldsymbol {u}^{\ast }(t)=\boldsymbol {u}_{SFIT2FAC}^{\ast } (\underline {\theta }^{\ast },\bar {\theta }^{\ast },\underline {\rho }^{l\ast },\bar {\rho }^{l\ast }, \underline {\rho }^{r\ast },\bar {\rho }^{r\ast },\underline {\upsilon }^{l\ast },\bar {\upsilon }^{l\ast }, \\& \qquad\qquad\qquad\qquad\qquad {{\underline {\upsilon }^{r\ast },\bar {\upsilon }^{r\ast },t)+\boldsymbol \varepsilon (t),}}\tag{19}\end{align*}$$ where $\underline {\theta }^{\ast },\bar {\theta }^{\ast }, \underline {\rho }^{l\ast },\bar {\rho }^{l\ast },\underline {\rho }^{r\ast },\bar {\rho }^{r\ast },\underline {\upsilon }^{l\ast },\bar {\upsilon }^{l\ast },\underline {\upsilon }^{r\ast }, \bar {\upsilon }^{r\ast }$ are the optimal parameters for $\underline {\theta },\bar {\theta },\underline {\rho }^{l},\bar {\rho }^{l},\underline {\rho }^{r},\bar {\rho }^{r},\underline {\upsilon }^{l},\bar {\upsilon }^{l},\underline {\upsilon }^{r},\bar {\upsilon }^{r}$ ; $\boldsymbol {\varepsilon }(t)$ is the approximation error.

Since the optimal controller $\boldsymbol {u}_{SFIT2FAC}^{\ast }$ described in (19) cannot be directly obtained, the estimation controller $\hat {\boldsymbol {u}}_{SFIT2FAC}$ described in (18) and the compensator controller can be used to estimate it. Then the estimation control signal can be expressed as follows: $$\begin{align*}&\hspace {-0.5pc}\hat {\boldsymbol {u}}(t)= \hat {\boldsymbol {u}}_{SFIT2FAC} (\hat {{\underline {\theta }}},\hat {{\bar {\theta }}},\hat {{\underline {\rho }}}^{l},\hat {{\bar {\rho }}}^{l},\hat {{\underline {\rho }}}^{r},\hat {{\bar {\rho }}}^{r}, \hat {{\underline {\upsilon }}}^{l},\hat {{\bar {\upsilon }}}^{l},\hat {{\underline {\upsilon }}}^{r},\hat {{\bar {\upsilon }}}^{r},t) \\& \qquad\qquad\qquad\qquad\qquad\qquad\qquad {{+\,\hat {\boldsymbol {u}}_{RB} \left ({t }\right),} }\tag{20}\end{align*}$$ where $\hat {{\underline {\theta }}},\hat {{\bar {\theta }}},\hat {{\underline {\rho }}}^{l},\hat {{\bar {\rho }}}^{l},\hat {{\underline {\rho }}}^{r},\hat {{\bar {\rho }}}^{r},\hat {{\underline {\upsilon }}}^{l},\hat {{\bar {\upsilon }}}^{l},\hat {{\underline {\upsilon }}}^{r},\hat {{\bar {\upsilon }}}^{r}$ are the estimation of $\underline {\theta }^{\ast },\bar {\theta }^{\ast },\underline {\rho }^{l\ast },\bar {\rho }^{l\ast },\underline {\rho }^{r\ast },\bar {\rho }^{r\ast },\underline {\upsilon }^{l\ast },\bar {\upsilon }^{l\ast },\underline {\upsilon }^{r\ast }, \bar {\upsilon }^{r\ast }$ ; $\hat {\boldsymbol {u}}_{RB}$ is the robust compensator controller, which is defined as $$\begin{equation*} \hat {\boldsymbol {u}}_{RB}(t)=\hat {\boldsymbol {\Omega }}(t)sgn(\boldsymbol {s}(t)) \quad \text {for}~ \dot {{\hat {\boldsymbol \Omega }}}~\,(t)=\eta _{\Omega } \left |{ {\boldsymbol {s}(t)} }\right |,\tag{21}\end{equation*}$$ where $\hat {\boldsymbol {\Omega }}(t)$ is the estimated value of $\boldsymbol {\Omega }(t)$ ; $\boldsymbol {\Omega }$ is the uncertainty boundary, which is assumed to be an upper limit for the approximation error ($0\le \boldsymbol {\varepsilon }(t)\le \boldsymbol {\Omega })$ ; $\boldsymbol {s}(t)$ denotes the high-order sliding surface, which is given as follows: $$\begin{equation*} \boldsymbol {s}(t)=\boldsymbol {e}^{\left ({{n-1} }\right)}+\boldsymbol {g}_{1} \boldsymbol {e}^{\left ({{n-2} }\right)}\ldots +\boldsymbol {g}_{n} \int _{0}^{t} {\boldsymbol {e}\left ({\tau }\right)} d\tau,\tag{22}\end{equation*}$$ where $\boldsymbol {G}=\left [{ {\boldsymbol {g}_{n},\ldots,\boldsymbol {g}_{2},\boldsymbol {g}_{1} } }\right]^{T}$ is the positive gain matrix, where $\boldsymbol {g}_{\boldsymbol {i}} =diag\left ({{g_{i1}, g_{i2},\ldots,g_{in}} }\right)$ is a positive gain vector; $n$ is the order of the sliding surface.

The sliding surface can be expressed in vector form as $\boldsymbol {s}(t)=\left [{ {s_{1} (t), s_{2} (t),\ldots, s_{n} (t)} }\right]^{T}$ . Consider a Lyapunov function as follows: $$\begin{equation*} V_{1} \left ({{\boldsymbol {s}\left ({t }\right)} }\right)=\frac {1}{2}\boldsymbol {s}^{T}\left ({t }\right)\boldsymbol {s}\left ({t }\right).\tag{23}\end{equation*}$$ Taking the time-derivative of (22), yields the following: $$\begin{equation*} \dot {\boldsymbol {s}}(t)=\boldsymbol {e}^{(n)}+\boldsymbol {G}^{T}\boldsymbol {E},\tag{24}\end{equation*}$$ where $\boldsymbol {E}=\left [{ {\boldsymbol {e},\dot {\boldsymbol {e}},\ldots,\boldsymbol {e}^{(n-1)}} }\right]$ is the error matrix; $\boldsymbol {e}=\left [{ {e_{1}, e_{2},\ldots,e_{n}} }\right]$ is the synchronization error vector in (3).

Taking the time-derivative of (23) and using (5), (20), and (24), yields the following: $$\begin{align*} \dot {V}_{1} (\boldsymbol {s}(t))=&\boldsymbol {s}^{T}(t)\dot {\boldsymbol {s}}(t) \\=&\boldsymbol {s}^{T}(t)\left [{ {\boldsymbol {e}^{(n)}+\boldsymbol {G}^{T}\boldsymbol {E}} }\right] \\=&\boldsymbol {s}^{T}(t)\left [{ \boldsymbol {Fe}(t)+\boldsymbol {\zeta }(t)+\Delta \boldsymbol {f}(y(t))+\hat {\boldsymbol {u}}_{SFIT2FAC} }\right. \\&\qquad \qquad \left.{+\hat {\boldsymbol {u}}_{RB} +\boldsymbol {G}^{T}\boldsymbol {E} }\right]\!.\tag{25}\end{align*}$$

Using the chain rule and the gradient descent approach, the online parameter learning algorithms for $\hat {\boldsymbol {u}}_{SFIT2FAC}$ can be obtained as follows: $$\begin{align*}&\hspace {-1.7pc}\hat {\underline {\theta }}_{hjk}^{q} \left ({{t+1} }\right) \\=&\hat {\underline {\theta }}_{hjk}^{q} \left ({t }\right)-\hat {\eta }_\theta \frac {\partial \dot {V}_{1}(t)}{\partial \hat {\underline {\theta }}_{hjk}^{q}} \\=&\hat {{\underline {\theta }}}_{hjk}^{q} \left ({t }\right)-\hat {\eta }_\theta \frac {\partial \dot {V}_{1}(t)}{\hat {u}_{SFIT2FAC}^{q}}\frac {\partial \hat {u}_{SFIT2FAC}^{q}}{\partial u_{q}^{1}}\frac {\partial u_{q}^{1}}{\partial \underline {\psi }^{q}_{jk}}\frac {\partial \underline {\psi }^{q}_{jk}} {\partial \hat {\underline {\theta }}_{hjk}^{q}} \\=&\hat {\theta }_{hjk}^{q} \left ({t }\right)+\frac {1}{2}\hat {\eta }_\theta s_{k}(t)\omega \underline {\tau }_{jk}\varphi _{h},\tag{26}\\&\hspace {-1.7pc}\hat {{\bar {\theta }}}_{hjk}^{q} \left ({{t+1} }\right) \\=&\hat {{\bar {\theta }}}_{hjk}^{q} \left ({t }\right)-\hat {\eta }_\theta \frac {\partial \dot {V}_{1}(t)}{\partial \hat {\bar {\theta }}_{hjk}^{q}} \\=&\hat {{\bar {\theta }}}_{hjk}^{q} \left ({t }\right)- \hat {\eta }_\theta \frac {\partial \dot {V}_{1}(t)}{\hat {u}_{SFIT2FAC}^{q}}\frac {\partial \hat {u}_{SFIT2FAC}^{q}} {\partial u_{q}^{r}}\frac {\partial u_{q}^{r}}{\partial \bar {\psi }^{q}_{jk}}\frac {\partial \bar {\psi }^{q}_{jk}} {\partial \hat {\bar {\theta }}_{hjk}^{q}} \\=&\hat {{\bar {\theta }}}_{hjk}^{q} \left ({t }\right)+\frac {1}{2}\hat {\eta }_\theta s_{k}(t)(1-\omega) \bar {\tau }_{jk}\varphi _{h},\tag{27}\\&\hspace {-1.7pc}\underline {\hat {\rho }}_{ ijk}^{l} \left ({{t+1} }\right) \\=&\underline {\hat {\rho }}_{ ijk}^{l}\left ({t }\right)-\hat {\eta }_\rho \frac {\partial \dot {V}_{1}(t)}{\partial \hat {\underline {\rho }}_{ijk}^{l}} \\=&\underline {\hat {\rho }}_{ ijk}^{l} \left ({t }\right)- \hat {\eta }_\rho \frac {\partial \dot {V}_{1}(t)}{\partial \hat {u}_{SFIT2FAC}^{q}} \frac {\partial \hat {u}_{SFIT2FAC}^{q}} {\partial u_{q}^{l}}\frac {\partial u_{q}^{l}}{\partial \underline {\tau }_{jk}}\frac {\partial \underline {\tau }_{jk}} {\partial {\underline {\chi }}_{ijk}}\frac {\partial {\underline {\chi }}_{ijk}}{\partial \hat {\underline {\rho }}^{l}_{ijk}} \\=&\underline {\hat {\rho }}_{ ijk}^{l} \left ({t }\right)+\frac {1}{2}\hat {\eta }_\rho s_{k}(t)\omega \underline {\psi }^{q}_{jk} \frac {\underline {\tau }_{jk}} {{\underline {\chi }}_{ijk}}\frac {\partial {\underline {\chi }}_{ijk}}{\partial \hat {\rho }^{l}_{ijk}},\tag{28}\\&\hspace {-1.7pc}\underline {\hat {\rho }}_{ ijk}^{r} \left ({{t+1} }\right) \\=&\underline {\hat {\rho }}_{ ijk}^{r}\left ({t }\right)-\hat {\eta }_\rho \frac {\partial \dot {V}_{1}(t)}{\partial \hat {\underline {\rho }}_{ijk}^{r}} \\=&\underline {\hat {\rho }}_{ ijk}^{r} \left ({t }\right)- \hat {\eta }_\rho \frac {\partial \dot {V}_{1}(t)}{\partial \hat {u}_{SFIT2FAC}^{q}} \frac {\partial \hat {u}_{SFIT2FAC}^{q}} {\partial u_{q}^{l}}\frac {\partial u_{q}^{l}}{\partial \underline {\tau }_{jk}}\frac {\partial \underline {\tau }_{jk}} {\partial {\underline {\chi }}_{ijk}}\frac {\partial {\underline {\chi }}_{ijk}}{\partial \hat {\underline {\rho }}^{r}_{ijk}} \\=&{\hat {\underline {\rho }}}_{ ijk}^{r} \left ({t }\right)+\frac {1}{2}\hat {\eta }_\rho s_{k}(t)\omega \underline {\psi }^{q}_{jk} \frac {\underline {\tau }_{jk}} {{\underline {\chi }}_{ijk}}\frac {\partial {\underline {\chi }}_{ijk}}{\partial \hat {\underline {\rho }}^{r}_{ijk}},\tag{29}\\&\hspace {-1.7pc}\hat {{\bar {\rho }}}_{ ijk}^{l} \left ({{t+1} }\right) \\=&\hat {{\bar {\rho }}}_{ ijk}^{l} \left ({t }\right)-\hat {\eta }_\rho \frac {\partial \dot {V}_{1}(t)}{\partial \hat {\bar {\rho }}_{ijk}^{l}} \\=&\hat {{\bar {\rho }}}_{ ijk}^{l} \left ({t }\right)- \hat {\eta }_\rho \frac {\partial \dot {V}_{1}(t)}{\partial \hat {u}_{SFIT2FAC}^{q}} \frac {\partial \hat {u}_{SFIT2FAC}^{q}} {\partial u_{q}^{r}}\frac {\partial u_{q}^{r}}{\partial \bar {\tau }_{jk}}\frac {\partial \bar {\tau }_{jk}} {\partial {\bar {\chi }}_{ijk}}\frac {\partial {\bar {\chi }}_{ijk}}{\partial \hat {\bar {\rho }}^{l}_{ijk}} \\=&\hat {{\bar {\rho }}}_{ ijk}^{l} \left ({t }\right)+\frac {1}{2}\hat {\eta }_\rho (1-\omega) s_{k}(t) \bar {\psi }^{q}_{jk} \frac {\bar {\tau }_{jk}} {{\bar {\chi }}_{ijk}}\frac {\partial {\bar {\chi }}_{ijk}}{\partial \hat {\bar {\rho }}^{l}_{ijk}},\tag{30}\end{align*}$$ $$\begin{align*}&\hspace {-1.7pc}\hat {{\bar {\rho }}}_{ ijk}^{r} \left ({{t+1} }\right) \\=&\hat {{\bar {\rho }}}_{ ijk}^{r} \left ({t }\right)-\hat {\eta }_\rho \frac {\partial \dot {V}_{1}(t)}{\partial \hat {\bar {\rho }}_{ijk}^{r}} \\=&\hat {{\bar {\rho }}}_{ ijk}^{r} \left ({t }\right)- \hat {\eta }_\rho \frac {\partial \dot {V}_{1}(t)}{\partial \hat {u}_{SFIT2FAC}^{q}} \frac {\partial \hat {u}_{SFIT2FAC}^{q}} {\partial u_{q}^{r}}\frac {\partial u_{q}^{r}}{\partial \bar {\tau }_{jk}}\frac {\partial \bar {\tau }_{jk}} {\partial {\bar {\chi }}_{ijk}}\frac {\partial {\bar {\chi }}_{ijk}}{\partial \hat {\bar {\rho }}^{r}_{ijk}} \\=&\hat {{\bar {\rho }}}_{ ijk}^{r} \left ({t }\right)+\frac {1}{2}\hat {\eta }_\rho (1-\omega) s_{k}(t) \bar {\psi }^{q}_{jk} \frac {\bar {\tau }_{jk}} {{\bar {\chi }}_{ijk}}\frac {\partial {\bar {\chi }}_{ijk}}{\partial \hat {\bar {\rho }}^{r}_{ijk}},\tag{31}\\&\hspace {-1.7pc}\hat {\underline {\upsilon }}_{ijk}^{l} \left ({{t+1} }\right) \\=&\hat {\underline {\upsilon }}_{ijk}^{l} \left ({t }\right)-\hat {\eta }_{\upsilon } \frac {\partial \dot {V}_{1}(t)}{\partial \hat {\underline {\upsilon }}_{ijk}^{l}} \\=&\hat {{\underline {\upsilon }}}_{ijk}^{l} \left ({t }\right)-\hat {\eta }_\upsilon \frac {\partial \dot {V}_{1}(t)}{\partial \hat {u}_{SFIT2FAC}^{q}} \frac {\partial \hat {u}_{SFIT2FAC}^{q}} {\partial u_{q}^{l}}\frac {\partial u_{q}^{l}}{\partial \underline {\tau }_{jk}}\frac {\partial \underline {\tau }_{jk}} {\partial {\underline {\chi }}_{ijk}}\frac {\partial {\underline {\chi }}_{ijk}}{\partial \hat {\underline {\upsilon }}^{l}_{ijk}} \\=&\hat {\underline {\upsilon }}_{ijk}^{l} \left ({t }\right)+\frac {1}{2}\hat {\eta }_\upsilon s_{k}(t)\omega \underline {\psi }^{q}_{jk} \frac {\underline {\tau }_{jk}} {{\underline {\chi }}_{ijk}}\frac {\partial {\underline {\chi }}_{ijk}}{\partial \hat {\upsilon }^{l}_{ijk}},\tag{32}\\&\hspace {-1.7pc}\hat {\upsilon }_{ijk}^{r} \left ({{t+1} }\right) \\=&\hat {\underline {\upsilon }}_{ijk}^{r} \left ({t }\right)-\hat {\eta }_{\upsilon } \frac {\partial \dot {V}_{1}(t)}{\partial \hat {\underline {\upsilon }}_{ijk}^{r}} \\=&\hat {{\underline {\upsilon }}}_{ijk}^{r} \left ({t }\right)-\hat {\eta }_\upsilon \frac {\partial \dot {V}_{1}(t)}{\partial \hat {u}_{SFIT2FAC}^{q}} \frac {\partial \hat {u}_{SFIT2FAC}^{q}} {\partial u_{q}^{l}}\frac {\partial u_{q}^{l}}{\partial \underline {\tau }_{jk}}\frac {\partial \underline {\tau }_{jk}} {\partial {\underline {\chi }}_{ijk}}\frac {\partial {\underline {\chi }}_{ijk}}{\partial \hat {\underline {\upsilon }}^{r}_{ijk}} \\=&\hat {{\underline {\upsilon }}}_{ijk}^{r} \left ({t }\right)+\frac {1}{2}\hat {\eta }_\upsilon s_{k}(t)\omega \underline {\psi }^{q}_{jk} \frac {\underline {\tau }_{jk}} {{\underline {\chi }}_{ijk}}\frac {\partial {\underline {\chi }}_{ijk}}{\partial \hat {\underline {\upsilon }}^{r}_{ijk}},\tag{33}\\&\hspace {-1.7pc}\hat {{\bar {\upsilon }}}_{ijk}^{l} \left ({{t+1} }\right) \\=&\hat {{\bar {\upsilon }}}_{ijk}^{l} \left ({t }\right)-\hat {\eta }_{\upsilon } \frac {\partial \dot {V}_{1}(t)}{\partial \hat {\bar {\upsilon }}_{ijk}^{l}} \\=&\hat {{\bar {\upsilon }}}_{ijk}^{l} \left ({t }\right)-\hat {\eta }_\upsilon \frac {\partial \dot {V}_{1}(t)}{\partial \hat {u}_{SFIT2FAC}^{q}} \frac {\partial \hat {u}_{SFIT2FAC}^{q}} {\partial u_{q}^{r}}\frac {\partial u_{q}^{r}}{\partial \bar {\tau }_{jk}}\frac {\partial \bar {\tau }_{jk}} {\partial {\bar {\chi }}_{ijk}}\frac {\partial {\bar {\chi }}_{ijk}}{\partial \hat {\bar {\upsilon }}^{l}_{ijk}} \\=&\hat {{\bar {\upsilon }}}_{ijk}^{l} \left ({t }\right)+\frac {1}{2}\hat {\eta }_\upsilon (1-\omega) s_{k}(t) \bar {\psi }^{q}_{jk} \frac {\bar {\tau }_{jk}} {{\bar {\chi }}_{ijk}}\frac {\partial {\bar {\chi }}_{ijk}}{\partial \hat {\bar {\upsilon }}^{l}_{ijk}},\tag{34}\\&\hspace {-1.7pc}\hat {{\bar {\upsilon }}}_{ijk}^{r} \left ({{t+1} }\right) \\=&\hat {{\bar {\upsilon }}}_{ijk}^{r} \left ({t }\right)- \hat {\eta }_{\upsilon } \frac {\partial \dot {V}_{1}(t)}{\partial \hat {\bar {\rho }}_{ijk}^{r}} \\=&\hat {{\bar {\upsilon }}}_{ijk}^{r} \left ({t }\right)- \hat {\eta }_\upsilon \frac {\partial \dot {V}_{1}(t)}{\partial \hat {u}_{SFIT2FAC}^{q}} \frac {\partial \hat {u}_{SFIT2FAC}^{q}} {\partial u_{q}^{r}}\frac {\partial u_{q}^{r}}{\partial \bar {\tau }_{jk}}\frac {\partial \bar {\tau }_{jk}} {\partial {\bar {\chi }}_{ijk}}\frac {\partial {\bar {\chi }}_{ijk}}{\partial \hat {\bar {\upsilon }}^{r}_{ijk}} \\=&\hat {{\bar {\upsilon }}}_{ijk}^{r} \left ({t }\right)+\frac {1}{2}\hat {\eta }_\upsilon (1-\omega)s_{k}(t) \bar {\psi }^{q}_{jk} \frac {\bar {\tau }_{jk}} {{\bar {\chi }}_{ijk}}\frac {\partial {\bar {\chi }}_{ijk}}{\partial \hat {\bar {\upsilon }}^{r}_{ijk}},\tag{35}\end{align*}$$ where $\hat {\eta }_\theta$ , $\hat {\eta }_\rho$ , $\hat {\eta }_\upsilon$ are the positive learning-rates.

In (28)–(35), the derivative terms $\frac {\partial \underline {\chi }_{ijk}}{\partial {\hat {\underline {\rho }}}_{ ijk}^{l}}$ , $\frac {\partial \underline {\chi }_{ijk}}{\partial {\hat {\underline {\rho }}}_{ ijk}^{r}}$ , $\frac {\partial \underline {\chi }_{ijk} }{\partial \hat {{\underline {\upsilon }}}_{ ijk}^{l}}$ , $\frac {\partial \underline {\chi }_{ijk}}{\partial \hat {{\underline {\upsilon }}}_{ ijk}^{r}}$ and $\frac {\partial \bar {\chi }_{ ijk}}{\partial \hat {{\bar {\rho }}}_{ijk}^{l}}$ , $\frac {\partial \bar {\chi }_{ ijk}}{\partial \hat {{\bar {\rho }}}_{ijk}^{r}}$ , $\frac {\partial \bar {\chi }_{ ijk}}{\partial \hat {{\bar {\upsilon }}}_{ijk}^{l}}$ , $\frac {\partial \bar {\chi }_{ ijk}}{\partial \hat {{\bar {\upsilon }}}_{ijk}^{r}}$ can be determined depending on the input regions, which are given in Tables 1, 2.

TABLE 1 The Derivative Terms for Updating the Lower Gaussian Parameters

TABLE 2 The Derivative Terms for Updating the Upper Gaussian Parameters

Convergence Analysis: Considering the change of the Lyapunov function in (23), yields the following: $$\begin{align*} \Delta V_{1} (t)=&V_{1} (t+1)-V_{1} (t) \\=&\frac {1}{2}\left [{ {s^{2}(t+1)-s^{2}(t)} }\right] \\=&\Delta s(t)\left [{ {\frac {1}{2}\Delta s(t)+s(t)} }\right]\!.\tag{36}\end{align*}$$ Applying the gradient descent method for the sliding surface and then using the Taylor expansion, yields the following: $$\begin{equation*} s(t+1)=s(t)+\Delta s(t)\cong s(t)+\left [{ {\frac {\partial s(t)}{\partial \boldsymbol {\Xi }}} }\right]\Delta \boldsymbol {\Xi },\tag{37}\end{equation*}$$ where parameter $\boldsymbol {\Xi }$ can be replaced by $\hat {{\underline {\theta }}},\hat {{\bar {\theta }}},\hat {{\underline {\rho }}}^{l},\hat {{\bar {\rho }}}^{l},\hat {{\underline {\rho }}}^{r},\hat {{\bar {\rho }}}^{r}, \hat {{\underline {\upsilon }}}^{l},\hat {{\bar {\upsilon }}}^{l},\hat {{\underline {\upsilon }}}^{r}$ or $\hat {{\bar {\upsilon }}}^{r}$ ; $\Delta s$ and $\Delta \boldsymbol {\Xi }$ represent the change in $s$ and $\boldsymbol {\Xi }$ , respectively.

From (26)–(35), yield the following: $$\begin{align*} \Delta \boldsymbol {\Xi }=&-\eta _{\boldsymbol {\Xi }} \frac {\partial \dot {V}_{1} (t)}{\partial \boldsymbol {\Xi }} \\=&\eta _{\boldsymbol {\Xi }} s(t)\frac {\hat {u}_{SFIT2FAC}^{q}}{\partial \boldsymbol {\Xi }}=\eta _{\boldsymbol {\Xi }} s(t)\boldsymbol {F}_{\boldsymbol {\Xi }} (t),\tag{38}\end{align*}$$ where $\boldsymbol {F}_{\boldsymbol {\Xi }} (t)=\frac {\hat {u}_{SFIT2FAC}^{q} }{\partial \boldsymbol {\Xi }}$ in (38) is defined as follows: $$\begin{align*} \boldsymbol {F}_{\hat {{\underline {\boldsymbol {\theta }}}}} (t)=&\frac {\partial \hat {u}_{SFIT2FAC}^{q}}{\partial \hat {{\underline {\boldsymbol {\theta }}}}} \\[-2.5pt]=&\left [{\! {\frac {\partial \hat {u}_{SFIT2FAC}^{q}}{\partial \hat {{\underline {\theta }}}_{h11}^{q}}, \ldots, \frac {\partial \hat {u}_{SFIT2FAC}^{q} }{\partial \hat {{\underline {\theta }}}_{h1n_{k}}^{q}},\frac {\partial \hat {u}_{SFIT2FAC}^{q}}{\partial \hat {{\underline {\theta }}}_{h21}^{q}}, } \ldots,}\right. \\[-2.5pt]&\left.{ { \frac {\partial \hat {u}_{SFIT2FAC}^{q}}{\partial \hat {{\underline {\theta }}}_{h2n_{k}}^{q}},\ldots,\frac {\partial \hat {u}_{SFIT2FAC}^{q}}{\partial \hat {{\underline {\theta }}}_{hn_{j} 1}^{q} }, \ldots, \frac {\partial \hat {u}_{SFIT2FAC}^{q}}{\partial \hat {{\underline {\theta }}}_{hn_{j} n_{k}}^{q}}} }\right]\!,\!\!\!\!\!\! \\[-0.5pt]\tag{39}\\[-2.5pt] \boldsymbol {F}_{\hat {{{\bar {\boldsymbol {\theta }}}}}} (t)=&\frac {\partial \hat {u}_{SFIT2FAC}^{q}}{\partial \hat {{{\bar {\boldsymbol {\theta }}}}}} \\[-2.5pt]=&\left [{\! {\frac {\partial \hat {u}_{SFIT2FAC}^{q}}{\partial \hat {\bar{\boldsymbol {\theta }}\,}_{h11}^{~\,q}}, \ldots, \frac {\partial \hat {u}_{SFIT2FAC}^{q}}{\partial \hat {{\bar{ \boldsymbol {\theta }}}\,}_{h1n_{k}}^{~\,q}},\frac {\partial \hat {u}_{SFIT2FAC}^{q}}{\partial \hat {{\bar {\boldsymbol {\theta }}}\,}_{h21}^{~\,q}},} \ldots,}\right. \\[-2.5pt]&\left.{ { \frac {\partial \hat {u}_{SFIT2FAC}^{q}}{\partial \hat {{\bar {\theta }}}_{h2n_{k}}^{q}},\ldots,\frac {\partial \hat {u}_{SFIT2FAC}^{q}}{\partial \hat {{\bar {\theta }}}_{hn_{j} 1}^{q} }, \ldots, \frac {\partial \hat {u}_{SFIT2FAC}^{q}}{\partial \hat {{\bar {\theta }}}_{hn_{j} n_{k}}^{q}}} \!}\right]\!,\!\!\!\!\!\! \\[-0.5pt]\tag{40}\\[-2.5pt] \boldsymbol {F}_{\hat {{\underline {\boldsymbol {\rho }}}}^{l}} (t)=&\frac {\partial \hat {u}_{SFIT2FAC}^{q}}{\partial \hat {{\underline {\boldsymbol {\rho }}}}^{l}} \\[-2.5pt]=&\left [{\! {\frac {\partial \hat {u}_{SFIT2FAC}^{q}}{\partial \hat {\underline {\rho }}_{i 11}^{l}}, \ldots, \frac {\partial \hat {u}_{SFIT2FAC}^{q} }{\partial \hat {\underline {\rho }}_{ i1n_{k}}^{l}},\frac {\partial \hat {u}_{SFIT2FAC}^{q}}{\partial \hat {\underline {\rho }}_{i 21}^{l}},}\ldots, }\right. \\[-2.5pt]&\left.{ { \frac {\partial \hat {u}_{SFIT2FAC}^{q}}{\partial \hat {{\underline {\rho }}}_{ i2n_{k}}^{l}},\ldots,\frac {\partial \hat {u}_{SFIT2FAC}^{q}}{\partial \hat {{\underline {\rho }}}_{i n_{j} 1}^{l} }, \ldots, \frac {\partial \hat {u}_{SFIT2FAC}^{q}}{\partial \hat {\rho }_{ in_{j} n_{k}}^{l}}} \!}\right]\!,\!\!\!\!\!\!\!\! \\[-0.5pt]\tag{41}\\[-2.5pt] \boldsymbol {F}_{\hat {{\bar {\rho }}}^{l}} (t)=&\frac {\partial \hat {u}_{SFIT2FAC}^{q}}{\partial \hat {{\bar {\rho }}}^{l}} \\[-2.5pt]=&\left [{\! {\frac {\partial \hat {u}_{SFIT2FAC}^{q}}{\partial \hat {{\bar {\rho }}}_{i 11}^{l}}, \ldots, \frac {\partial \hat {u}_{SFIT2FAC}^{q} }{\partial \hat {{\bar {\rho }}}_{ i1n_{k}}^{l}},\frac {\partial \hat {u}_{SFIT2FAC}^{q}}{\partial \hat {{\bar {\rho }}}_{i 21}^{l}},} \ldots,}\right. \\[-2.5pt]&\left.{ { \frac {\partial \hat {u}_{SFIT2FAC}^{q}}{\partial \hat {{\bar {\rho }}}_{ i2n_{k}}^{l}},\ldots,\frac {\partial \hat {u}_{SFIT2FAC}^{q}}{\partial \hat {{\bar {\rho }}}_{i n_{j} 1}^{l} }, \ldots, \frac {\partial \hat {u}_{SFIT2FAC}^{q}}{\partial \hat {{\bar {\rho }}}_{ in_{j} n_{k}}^{l}}} \!}\right]\!,\!\!\!\!\!\!\!\! \\[-0.5pt]\tag{42}\\[-2.5pt] \boldsymbol {F}_{\hat {\underline {\boldsymbol {\rho }}}^{r}} (t)=&\frac {\partial \hat {u}_{SFIT2FAC}^{q}}{\partial \hat {\underline {\boldsymbol {\rho }}}^{r}} \\[2pt]=&\left [{\! {\frac {\partial \hat {u}_{SFIT2FAC}^{q}}{\partial \hat {\underline {\rho }}_{i 11}^{r}}, \ldots, \frac {\partial \hat {u}_{SFIT2FAC}^{q} }{\partial \hat {\underline {\rho }}_{ i1n_{k}}^{r}},\frac {\partial \hat {u}_{SFIT2FAC}^{q}}{\partial \hat {\underline {\rho }}_{i 21}^{r}},\ldots,} }\right. \\[2pt]&\left.{ { \frac {\partial \hat {u}_{SFIT2FAC}^{q}}{\partial \hat {{\underline {\rho }}}_{ i2n_{k}}^{r}},\ldots,\frac {\partial \hat {u}_{SFIT2FAC}^{q}}{\partial \hat {\underline {\rho }}_{i n_{j} 1}^{r} }, \ldots, \frac {\partial \hat {u}_{SFIT2FAC}^{q}}{\partial \hat {{\underline {\rho }}}_{ in_{j} n_{k}}^{r}}} }\right]\!,\!\!\!\!\!\!\!\! \\ \tag{43}\end{align*}$$ $$\begin{align*} \boldsymbol {F}_{\hat {\bar {\boldsymbol {\rho }}}^{r}} (t)=&\frac {\partial \hat {u}_{SFIT2FAC}^{q}}{\partial \hat {\bar {\boldsymbol {\rho }}}^{r}} \\=&\left [{\! {\frac {\partial \hat {u}_{SFIT2FAC}^{q}}{\partial \hat {\bar {\rho }}_{i 11}^{r}}, \ldots, \frac {\partial \hat {u}_{SFIT2FAC}^{q}}{\partial \hat {\bar {\rho }}_{ i1n_{k}}^{r}},\frac {\partial \hat {u}_{SFIT2FAC}^{q} }{\partial \hat {\bar {\rho }}_{i 21}^{r}},} \ldots,}\right. \\&\left.{ { \frac {\partial \hat {u}_{SFIT2FAC}^{q}}{\partial \hat {{\bar {\rho }}}_{ i2n_{k}}^{r}},\ldots,\frac {\partial \hat {u}_{SFIT2FAC}^{q}}{\partial \hat {{\bar {\rho }}}_{i n_{j} 1}^{r} }, \ldots, \frac {\partial \hat {u}_{SFIT2FAC}^{q}}{\partial \hat {{\bar {\rho }}}_{ in_{j} n_{k}}^{r}}} \!}\right]\!,\!\!\!\!\!\! \\ \tag{44}\\ \boldsymbol {F}_{\hat {{\underline {\upsilon }}}^{l}} (t)=&\frac {\partial \hat {u}_{SFIT2FAC}^{q}}{\partial \hat {\underline {\boldsymbol {\upsilon }}}^{l}} \\=&\left [{\! {\frac {\partial \hat {u}_{SFIT2FAC}^{q}}{\partial \hat {{\underline {\upsilon }}}_{i 11}^{l}}, \ldots, \frac {\partial \hat {u}_{SFIT2FAC}^{q} }{\partial \hat {{\underline {\upsilon }}}_{ i1n_{k}}^{l}},\frac {\partial \hat {u}_{SFIT2FAC}^{q}}{\partial \hat {{\underline {\upsilon }}}_{i 21}^{l}},} \ldots,}\right. \\&\left.{ { \frac {\partial \hat {u}_{SFIT2FAC}^{q}}{\partial \hat {{\underline {\upsilon }}}_{ i2n_{k}}^{l}},\ldots,\frac {\partial \hat {u}_{SFIT2FAC}^{q}}{\partial \hat {{\underline {\upsilon }}}_{i n_{j} 1}^{l} }, \ldots, \frac {\partial \hat {u}_{SFIT2FAC}^{q}}{\partial \hat {{\underline {\upsilon }}}_{ in_{j} n_{k}}^{l}}} \!}\right]\!,\!\!\!\!\!\! \\ \tag{45}\\ \boldsymbol {F}_{\hat {\bar {\boldsymbol {\upsilon }}}^{l}} (t)=&\frac {\partial \hat {u}_{SFIT2FAC}^{q}}{\partial \hat {\bar {\boldsymbol {\upsilon }}}^{l}} \\=&\left [{\! {\frac {\partial \hat {u}_{SFIT2FAC}^{q}}{\partial \hat {{\bar {\upsilon }}}_{i 11}^{l}}, \ldots, \frac {\partial \hat {u}_{SFIT2FAC}^{q}}{\partial \hat {{\bar {\upsilon }}}_{ i1n_{k} }^{l}},\frac {\partial \hat {u}_{SFIT2FAC}^{q}}{\partial \hat {{\bar {\upsilon }}}_{i 21}^{l}},} \ldots,}\right. \\&\left.{ { \frac {\partial \hat {u}_{SFIT2FAC}^{q}}{\partial \hat {{\bar {\upsilon }}}_{ i2n_{k}}^{l}},\ldots,\frac {\partial \hat {u}_{SFIT2FAC}^{q}}{\partial \hat {{\bar {\upsilon }}}_{i n_{j} 1}^{l}}, \ldots, \frac {\partial \hat {u}_{SFIT2FAC}^{q}}{\partial \hat {{\bar {\upsilon }}}_{ in_{j} n_{k}}^{l}}} \!}\right]\!,\!\!\!\!\!\! \\ \tag{46}\\ \boldsymbol {F}_{\hat {\underline {\upsilon }}^{r}} (t)=&\frac {\partial \hat {u}_{SFIT2FAC}^{q}}{\partial \hat {\underline {{\boldsymbol {\upsilon }}}}^{r}} \\=&\left [{\! {\frac {\partial \hat {u}_{SFIT2FAC}^{q}}{\partial \hat {{\underline {\upsilon }}}_{i 11}^{r}}, \ldots, \frac {\partial \hat {u}_{SFIT2FAC}^{q} }{\partial \hat {{\underline {\upsilon }}}_{ i1n_{k}}^{r}},\frac {\partial \hat {u}_{SFIT2FAC}^{q}}{\partial \hat {{\underline {\upsilon }}}_{i 21}^{r}}, } \ldots,}\right. \\&\left.{ { \frac {\partial \hat {u}_{SFIT2FAC}^{q}}{\partial \hat {{\underline {\upsilon }}}_{ i2n_{k}}^{r}},\ldots,\frac {\partial \hat {u}_{SFIT2FAC}^{q}}{\partial \hat {\underline {\upsilon }}_{i n_{j} 1}^{r} }, \ldots, \frac {\partial \hat {u}_{SFIT2FAC}^{q}}{\partial \hat {{\underline {\upsilon }}}_{ in_{j} n_{k}}^{r}}} \!}\right]\!,\!\!\!\!\!\!\!\! \\ \tag{47}\\ \boldsymbol {F}_{\hat {\bar {\boldsymbol {\upsilon }}}^{r}} (t)=&\frac {\partial \hat {u}_{SFIT2FAC}^{q}}{\partial \hat {\bar {\boldsymbol {\upsilon }}}^{r}} \\=&\left [{\! {\frac {\partial \hat {u}_{SFIT2FAC}^{q}}{\partial \hat {{\bar {\upsilon }}}_{i 11}^{r}}, \ldots, \frac {\partial \hat {u}_{SFIT2FAC}^{q}}{\partial \hat {{\bar {\upsilon }}}_{ i1n_{k} }^{r}},\frac {\partial \hat {u}_{SFIT2FAC}^{q}}{\partial \hat {{\bar {\upsilon }}}_{i 21}^{r}},} \ldots,}\right. \\&\left.{ { \frac {\partial \hat {u}_{SFIT2FAC}^{q}}{\partial \hat {{\bar {\upsilon }}}_{ i2n_{k}}^{r}},\ldots,\frac {\partial \hat {u}_{SFIT2FAC}^{q}}{\partial \hat {{\bar {\upsilon }}}_{i n_{j} 1}^{r}}, \ldots, \frac {\partial \hat {u}_{SFIT2FAC}^{q}}{\partial \hat {{\bar {\upsilon }}}_{ in_{j} n_{k}}^{r}}}\! }\right]\!. \\{}\tag{48}\end{align*}$$ Applying the chain rule in (37), yields the following: $$\begin{equation*} \frac {\partial s(t)}{\partial \boldsymbol {\Xi }}=\frac {\partial s(t)}{\partial \hat {u}_{SFIT2FAC}^{q}}\frac {\partial \hat {u}_{SFIT2FAC}^{q}}{\partial \boldsymbol {\Xi }}=\boldsymbol {F}_{\boldsymbol {\Xi }} (t).\tag{49}\end{equation*}$$ Using (38) and (49), (36) becomes as follows: $$\begin{align*} \Delta V_{1} (t)=&\Delta s(t)\left [{ {\frac {1}{2}\Delta s(t)+s(t)} }\right] \\[-2pt]=&\left ({{\left [{ {\frac {\partial s(t)}{\partial \boldsymbol {\Xi }}} }\right]^{T}\Delta \boldsymbol {\Xi }} }\right)\left [{ {\frac {1}{2}\left ({{\left [{ {\frac {\partial e(t)}{\partial \boldsymbol {\Xi }}} }\right]^{T}\Delta \boldsymbol {\Xi }} }\right) +s(t)} }\right] \\[-2pt]=&\left [{ {\boldsymbol {F}_{\boldsymbol {\Xi }} (t)} }\right]^{T}\eta _{\boldsymbol {\Xi }} s(t)\boldsymbol {F}_{\boldsymbol {\Xi }} (t) \\[-2pt]&\times \left [{ {\frac {1}{2}\left [{ {\boldsymbol {F}_{\boldsymbol {\Xi }} (t)} }\right]^{T}\eta _{\boldsymbol {\Xi }} s(t)\boldsymbol {F}_{\boldsymbol {\Xi }} (t)+s(t)} }\right] \\[-2pt]=&\frac {1}{2}\eta _{\boldsymbol {\Xi }} s^{2}(t)\left \|{ {\boldsymbol {F}_{\boldsymbol {\Xi }} (t)} }\right \|^{2}\left [{ {\eta _{\boldsymbol {\Xi }} \left \|{ {\boldsymbol {F}_{\boldsymbol {\Xi }} (t)} }\right \|^{2}-2} }\right].\tag{50}\end{align*}$$ From (50), if $\eta _{\boldsymbol {\Xi }}$ is chosen to satisfy $0 < \eta _{\boldsymbol {\Xi }} < \frac {2}{\left \|{ {\boldsymbol {F}_{\boldsymbol {\Xi }} (t)} }\right \|^{2}}$ , then $\Delta V_{1} (t) < 0$ . Consequently, the stability of the proposed SFIT2FAC synchronization system can be ensured.

C. Structure Learning Algorithm

The proposed methodology for the self-evolution of the SFIT2FAC synchronization structure is described in this section. A flowchart of the self-organizing algorithm is described in Fig. 4.

FIGURE 4.

The flowchart of the proposed self-organizing algorithm.

The generation criterion for adding a new layer and a new T2AGMF is based on the current membership grade in each input as follows: $$\begin{align*} \chi _{\max }^{i} < &G_{g},\tag{51}\\[-2pt] \chi _{\max }^{i}=&max\big ({{max\big [{ {\chi _{i11},\chi _{i12},\ldots,\chi _{i1n_{k}};\chi _{i21},\chi _{i22},} }} }\ldots., \\[-2pt]&{ {{ {\chi _{i2n_{k} };\ldots;\chi _{in_{j} 1},\chi _{in_{j} 2},\ldots,\chi _{in_{j} n_{k}}} }\big]} }\big),\tag{52}\\[-2pt] \chi _{ijk}=&\frac {\chi _{ijk} +\bar {\chi }_{ijk}}{2},\tag{53}\end{align*}$$ where $G_{g}$ and $\chi _{\max }^{i}$ respectively are the generating threshold and the maximum membership grade in $i^{\mathrm {th}}$ input; $\chi _{ijk}$ is the average of the upper and lower membership function.

If (51) is satisfied, the new layer is generated, and the initial parameters for new T2AGMF are given as: $$\begin{align*}&\hspace {-1.7pc}\left [{ {\bar {\rho }_{ijk}^{l},\underline {\rho }_{ ijk}^{l},\underline {\rho }_{ ijk}^{r},\bar {\rho }_{ijk}^{r}} }\right] \\=&\left [{ {\left ({{i_{i} -2\kappa } }\right),\left ({{i_{i} -\kappa } }\right),\left ({{i_{i} +\kappa } }\right),\left ({{i_{i} +2\kappa } }\right)} }\right]\!,\tag{54}\\&\hspace {-1.7pc}\left [{ {\underline {\upsilon }_{ ijk}^{l},\bar {\upsilon }_{ijk}^{l},\underline {\upsilon }_{ ijk}^{r},\bar {\upsilon }_{ijk}^{r}} }\right] \\=&\left [{ {\left ({{\upsilon _{init} -2\vartheta } }\right),\left ({{\upsilon _{init} -\vartheta } }\right),\left ({{\upsilon _{init} +\vartheta } }\right),\left ({{\upsilon _{init} +2\vartheta } }\right)} }\right]\!, \\{}\tag{55}\end{align*}$$ where $k$ and $\vartheta$ are the adjusting parameters of T2AGMF; $\upsilon _{init}$ is the initial variance.

The pruning criterion for deleting an inappropriate layer is based on the current membership grade in each input as: $$\begin{align*} \chi _{\min }^{i} < &G_{d},\tag{56}\\ \chi _{\min }^{i}=&min\left ({{max\left [{ {\chi _{i11},\chi _{i12},\ldots,\chi _{i1n_{k}};\chi _{i21},\chi _{i22},} }\right.}...,\chi _{i2n_{k} };}\right.\!\!\!\!\! \\&\qquad \left.{ {\left.{ {...;\chi _{in_{j} 1},\chi _{in_{j} 2},\ldots,\chi _{in_{j} n_{k}}} }\right]} }\right)\!,\tag{57}\end{align*}$$ where $G_{d}$ and $\chi _{\min }^{i}$ respectively are the pruning threshold and the minimum membership grade in $i^{\mathrm {th}}$ input.

If (56) is satisfied, the inappropriate layer and its corresponding T2AGMF will be deleted.

D. Modified Jaya Algorithm

The selection of the learning rates, $\hat {\eta }_\theta$ , $\hat {\eta }_\rho$ , $\hat {\eta }_\upsilon$ for the proposed SFIT2FAC synchronizer significantly affects the system’s performance. If too large or too small learning rates are selected, the gradient-descent algorithm may be trapped into a local minimum or even not converge. In this section, an MJA is proposed for optimizing these learning rates. The optimal mechanism of the Jaya algorithm is to approach the best solution and avoid the worst solution. In this algorithm, the best and worst solutions are calculated in each iteration based on the defined fitness function. The Jaya algorithm is a simple optimization algorithm, which does not require the tuning of any algorithm-specific parameters. The parameters required by the Jaya algorithm are only the population size and the number of iterations. The performance of the original Jaya algorithm can be improved by the proposed MJA, which provides a new updating equation. This equation also considers the second and third of the worst and best solutions, as shown in (59). A flowchart of the proposed MJA is presented in Fig. 5.

FIGURE 5.

Flowchart of the proposed modified Jaya algorithm.

The fitness function can be selected as follows: $$\begin{equation*} f_{n} =\left ({{\left ({{\boldsymbol {e}_{1} (t)} }\right)^{2}+\left ({{\boldsymbol {e}_{2} (t)} }\right)^{2}+\left ({{\boldsymbol {e}_{3} (t)} }\right)^{2}+\left ({{\boldsymbol {e}_{4} (t)} }\right)^{2}} }\right)\!.\tag{58}\end{equation*}$$ The learning rates are updated as follows: $$\begin{align*}&\hspace {-0.5pc}\eta _{\Theta }^{p} (t+1)=\eta _{\Theta }^{p} (t)+r_{1} \left [{ {\eta _{\Theta }^{best(rd)} (t)-\eta _{\Theta }^{p} (t)} }\right] \\& \qquad\qquad\qquad\qquad\quad {{-\,r_{2} \left [{ {\eta _{\Theta }^{worst(rd)} (t)-\eta _{\Theta }^{p} (t)} }\right]\!,}}\tag{59}\end{align*}$$ where $\Theta$ is replaced by $\theta$ , $\rho$ , and $\upsilon$ ; $\eta _{\Theta }^{p} (t)$ and $\eta _{\Theta }^{p} (t+1)$ are the current solution and the updated solution, respectively; $r_{1}$ and $r_{2}$ are two random numbers in the range of [0, 1]; rd is a random integer from 1 to 3, and $\eta _{\Theta }^{best(1) }$ , $\eta _{\Theta }^{best(2) }$ , $\eta _{\Theta }^{best(3) }$ are the solution learning rates with the lowest, the second-lowest and the third-lowest fitness function values, respectively; $\eta _{\Theta }^{worst(1) }$ , $\eta _{\Theta }^{worst(2) }$ , $\eta _{\Theta }^{worst(3) }$ are the solution learning rates with the highest, the second-highest and the third-highest fitness function values, respectively.

Pseudocode for the MJA

Input:

Population size: $n_{p}$ ; Number of iterations: $n_{g}$ ;

Output:

Learning rates $\hat {\eta }_\theta$ , $\hat {\eta }_\rho$ , $\hat {\eta }_\upsilon$ ;

for $p$ :$=1$ to $n_{p}$ do

Initialize $\hat {\eta }_\theta ^{p}$ , $\hat {\eta }_\rho ^{p}$ , $\hat {\eta }_\upsilon ^{p}$ ;

end

repeat

for $g$ :$=1$ to $n_{g}$ do

for $p$ :$=1$ to $n_{p}$ do

Run simulation with $\hat {\eta }_\theta ^{p}$ , $\hat {\eta }_\rho ^{p}$ , $\hat {\eta }_\upsilon ^{p}$

Calculate fitness function using Eq. (58)

Obtain $\hat {\eta }_\theta ^{best(rd)}$ , $\hat {\eta }_\rho ^{best(rd)}$ , $\hat {\eta }_\upsilon ^{best(rd)}$ and $\hat {\eta }_\theta ^{worst(rd)}$ , $\hat {\eta }_\rho ^{worst(rd)}$ , $\hat {\eta }_\upsilon ^{worst(rd)}$

end

for $p$ :=1 to $n_{p}$ do

Update solutions $\hat {\eta }_\theta ^{p}$ , $\hat {\eta }_\rho ^{p}$ , $\hat {\eta }_\upsilon ^{p}$ using Eq. (59)

Run simulation with updated solutions

Calculate fitness function using Eq. (58)

if (updated solutions better than previous) then

Accept updated solutions;

else

Keep previous solutions;

end

end

end

until termination criterion satisfied;

SECTION IV. Results and Discussion

In this section, the efficiency of the SFIT2FAC is verified by conducting several simulations on the synchronization of chaotic systems. A block diagram of the proposed synchronizer’s chaotic systems is shown in Fig. 6. The initial conditions for the master and slave systems are respectively chosen as $[x_{1} (t)$ , $x_{2}(t)$ , $x_{3} (t)$ , $x_{4} (t)]=[0.028, 0.02, 0.03, 0.048]$ and $[y_{1} (t)$ , $y_{2} (t)$ , $y_{3} (t)$ , $y_{4} (t)]=[0.01, 0.037, 0.029, 0.008]$ . The system’s uncertainties and external disturbances are selected as $[\Delta f(y_{1}),\Delta f(y_{2}),\Delta f(y_{3}),\Delta f(y_{4})]=r_{rd} [0.1y_{1}, 0.1y_{2},0.1y_{3},0.1y_{4}]$ and $[\zeta _{1} (t)$ , $\zeta _{2} (t)$ , $\zeta _{3} (t)$ , $\zeta _{4} (t)]=[0.2\cos \pi t,0.5\cos \pi t,0.3\cos \pi t,0.4\cos \pi t]$ , respectively; where $r_{rd} \in [{0, 1}]$ is the random uncertainty value. The root mean square error (RMSE) can be used to evaluate the system’s performance as follows: $$\begin{equation*} RMSE=\sqrt {\frac {1}{n_{r}}\sum \limits _{r=1}^{n_{r}} {\left ({{\left ({{e_{1}^{r}} }\right)^{2}+\left ({{e_{2}^{r}} }\right)^{2}+\left ({{e_{3}^{r}} }\right)^{2}+\left ({{e_{4}^{r}} }\right)^{2}} }\right)}},\tag{60}\end{equation*}$$ where $n_{r}$ is the number of samples; $e_{x}^{r}, e_{y}^{r}$ and $e_{z}^{r}$ are the tracking errors for the $r^{th}$ sample.

FIGURE 6.

Block diagram of the proposed synchronizer’s chaotic systems.

The overall steps required to acieve the synchronization of the proposed chaotic systems are given as follows:

• Step 1:

  Optimize the learning rates using the proposed MJA.

• Step 2:

  Calculate the synchronization error and its high-order sliding surface using Eqs. (3) and (22).

• Step 3:

  Generate the control signal using Eqs. (17), (20), and (21).

• Step 4:

  Update network parameters using Eqs. (26)–(35).

• Step 5:

  Construct the network using the self-organizing algorithm.

• Step 6:

  Repeat Step 1 until the synchronization time is expired.

  • Case 1:

    In this case, the parameters for defining chaotic system’s attractor are defined as $\alpha =0.5$ , $\beta =20, \gamma =30$ , $\lambda =8/3$ . The chaotic attractor of the four-dimensional Lorenz–Stenflo chaotic system is presented in Fig. 7 in a three-dimensional view. The state trajectories of the chaotic systems’ synchronization using the proposed SFIT2FAC synchronizer are shown in Fig. 8. The time history of the control signal and the tracking errors are shown in Figs. 9 and 10, respectively. From Fig. 10, it can be observed that the proposed SFIT2FAC synchronizer is capable of synchronizing the four-dimensional Lorenz–Stenflo chaotic system well, even when the system uncertainties and external disturbances are considered. The change in the number of layers using the proposed self-organizing algorithm is shown in Fig. 11, whereas the change in the learning rates using the proposed MJA is shown in Fig. 12.

  • Case 2:

    In this case, the parameters for defining the chaotic attractor of the chaotic system are taken as $\alpha =10$ , $\beta =30,\gamma =40$ , $\lambda =8/3$ . The chaotic attractor of the four-dimensional Lorenz–Stenflo chaotic system is presented in Fig. 13 in a three-dimensional view. The state trajectories of the chaotic systems’ synchronization using the proposed SFIT2FAC synchronizer are shown in Fig. 14. The time history of the control signal and the tracking errors are shown in Figs. 15 and 16, respectively. From Fig. 16, it can be observed that the proposed SFIT2FAC synchronizer is capable of synchronizing the four-dimensional Lorenz–Stenflo chaotic system well, even when the system uncertainties and external disturbances are considered. The change in the number of layers using the proposed self-organizing algorithm is shown in Fig. 17, whereas the change in the learning rates using the proposed MJA is shown in Fig. 18.

FIGURE 7.

The chaotic attractor of the four-dimensional Lorenz–Stenflo chaotic system in a three-dimensional view. (a) $x_{1} y_{1} -x_{2} y_{2} - x_{3} y_{3}$ , (b) $x_{1} y_{1} - x_{2} y_{2} -x_{4} y_{4}$ , (c) $x_{1} y_{1} - x_{3} y_{3} -x_{4} y_{4}$ , (d) $x_{2} y_{2} - x_{3} y_{3} -x_{4} y_{4}$ .

FIGURE 8.

The state trajectories of chaotic systems synchronization using the proposed SFIT2FAC synchronizer. (a) $x_{1} y_{1}$ , (b) $x_{2} y_{2}$ , (c) $x_{3} y_{3}$ , and (d) $x_{4} y_{4}$ .

FIGURE 9.

The control signals. (a) $u_{1}$ , (b) $u_{2}$ , (c) $u_{3}$ , and (d) $u_{4}$ .

FIGURE 10.

The tracking errors. (a) $e_{1}$ , (b) $e_{2}$ , (c) $e_{3}$ , and (d) $e_{4}$ .

FIGURE 11.

The change in the number of layers using the proposed self-organizing algorithm.

FIGURE 12.

The change in learning rates using the modified Jaya algorithm.

FIGURE 13.

The chaotic attractor of the four-dimensional Lorenz–Stenflo chaotic system in a three-dimensional view. (a) $x_{1} y_{1} - x_{2} y_{2} - x_{3} y_{3}$ , (b) $x_{1} y_{1} - x_{2} y_{2} - x_{4} y_{4}$ , (c) $x_{1} y_{1} - x_{3} y_{3} -x_{4} y_{4}$ , (d) $x_{2} y_{2} - x_{3} y_{3} -x_{4} y_{4}$ .

FIGURE 14.

The state trajectories of chaotic systems synchronization using the proposed SFIT2FAC synchronizer. (a) $x_{1} y_{1}$ , (b) $x_{2} y_{2}$ , (c) $x_{3} y_{3}$ , and (d) $x_{4} y_{4}$ .

FIGURE 15.

The control signals. (a) $u_{1}$ , (b) $u_{2}$ , (c) $u_{3}$ , and (d) $u_{4}$ .

FIGURE 16.

The tracking errors. (a) $e_{1}$ , (b) $e_{2}$ , (c) $e_{3}$ , and (d) $e_{4}$ .

FIGURE 17.

The change in the number of layers using the proposed self-organizing algorithm.

FIGURE 18.

The change in learning rates using the modified Jaya algorithm.

In both cases, the proposed SFIT2FAC synchronizer exhibits synchronization efficiency in the chaotic system. Figures. 8 and 14 confirm that the proposed synchronizer is capable of synchronizing well the chaotic system, even when the system’s uncertainties and external disturbances are considered. Figs. 9 and 15 show that the control signals promptly respond to adapt to the changes that occur during the synchronization process. Figs. 10 and 16 show that tracking errors quickly converge to small values. Figs. 11 and 17 indicate that using the proposed self-organizing algorithm, the number of layers promptly changes to adapt to changes in the input signals. Figs. 12 and 18 show that using the proposed MJA, the learning rates quickly change to achieve the optimal values. Finally, comparison results in the RMSE among the CMAC, the interval type-2 fuzzy neural network (IT2FNN), and the proposed synchronizer with and without the MJA algorithm are shown in Table 3.

TABLE 3 Comparison Results in RMSE

Remark 1:

By applying the MJA algorithm, the proposed network is capable of quickly obtaining suitable learning rates. Therefore, it is capable of achieving the smallest RMSE values. However, its computation time is somewhat longer than that of other methods.

Remark 2:

By applying the self-organizing algorithm, the proposed network is capable of quickly obtaining a suitable network structure. The new fuzzy rules can be generated if the entire current rules cannot cover well the input changes. The unused fuzzy rules can be deleted if their contribution is less than a predefined threshold.

Remark 3:

The selection of the generating and pruning thresholds for the self-organizing algorithm significantly affects the system’s performance. A high-generating threshold hardly leads the algorithm to generate new rules. In contrast, a low-generating threshold leads to the generation of huge rules and requires a huge computation time. A high-pruning threshold leads the algorithm to rarely delete the unused rules. In contrast, a low-pruning threshold leads to the deletion of huge rules and may not cover well the input changes. In this work, trial-and-error was used to obtain suitable thresholds. Further studies should investigate methods to optimize the threshold values.

SECTION V. Conclusion

In this paper, an SFIT2FAC synchronizer was proposed for the synchronization of a four-dimensional Lorenz–Stenflo chaotic system. The adaptive laws were derived to online update the network parameters based on the gradient-descent approach. It was found that the network’s learning ability can be improved by the proposed asymmetric membership function. A FLN was applied to adjust the lower and upper weights of the type-2 asymmetric Gaussian membership function. Then, a self-organizing algorithm was employed to autonomously construct the SFIT2FAC network. Finally, numerical simulation results on the synchronization of four-dimensional chaotic systems were presented to verify the validity of the proposed method.