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Recurrent Interval Type-2 Fuzzy Wavelet Neural Network with Stable Learning Algorithm: Application to Model-Based Predictive Control

Abstract

Fuzzy neural networks, with suitable learning strategy, have been demonstrated as an effective tool for online data modeling. However, it is a challenging task to construct a model to ensure its quality and stability for non-stationary dynamic systems with some uncertainties. To solve this problem, this paper presents a novel identification model based on recurrent interval type-2 fuzzy wavelet neural network (RIT2FWNN) with new learning algorithm. The model benefits from both advantages of recurrent and wavelet neural networks such as use of temporal data and fast convergence properties. The proposed antecedent and consequent parameters update rules are derived using sliding-mode-control-theory. To evaluate the proposed fuzzy model, it is utilized to design a nonlinear model-based predictive controller and is applied for the synchronization of fractional-order time-delay chaotic systems. Using Lyapunov stability analysis, it is shown that all update rules of the parameters are uniformly ultimately bounded. The adaptation laws obtained in this method are very simple and have closed forms. Some stability conditions are derived to prove learning dynamics and asymptotic stability of the network by using an appropriate Lyapunov function. The efficacy and performance of the proposed method is verified by simulation examples.

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Author information

Correspondence to Mohammad Teshnehlab.

Appendix: Proof of Theorem 1

Appendix: Proof of Theorem 1

The time derivative of (16) is calculated as follows:

$$\begin{aligned}&\dot{\tilde{\underline{w}}}_{r}=\frac{\dot{\underline{w}}_{r}\sum _{r=1}^{N}\underline{w}_{r} -\sum _{r=1}^{N}\dot{\underline{w}}_{r}\underline{w}_{r}}{\left( \sum _{r=1}^{N}\underline{w}_{r}\right) ^2} \ =\frac{\dot{\underline{w}}_{r}-\tilde{\underline{w}}_{r}\sum _{r=1}^{N}\dot{\underline{w}}_{r}}{\sum _{r=1}^{N}\underline{w}_{r}} \nonumber \\&\dot{\tilde{\overline{w}}}_{r}=\frac{\dot{\overline{w}}_{r}\sum _{r=1}^{N}\overline{w}_{r} -\sum _{r=1}^{N}\dot{\overline{w}}_{r}\overline{w}_{r}}{\left( \sum _{r=1}^{N}\overline{w}_{r}\right) ^2} \ =\frac{\dot{\overline{w}}_{r}-\tilde{\overline{w}}_{r}\sum _{r=1}^{N}\dot{\overline{w}}_{r}}{\sum _{r=1}^{N}\overline{w}_{r}} \end{aligned}$$
(51)

considering the following equations for \(\dot{\underline{w}}_{r}\) and \(\dot{\overline{w}}_{r}\):

$$\begin{aligned} \dot{\underline{w}}_{r}=\underline{J}_r \underline{w}_r ; \ \ \dot{\overline{w}}_{r} =\overline{J}_r \overline{w}_r \end{aligned}$$
(52)

thus the Eq. (51) changes as follows:

$$\begin{aligned} \dot{\tilde{\overline{w}}}_{r}&= -\tilde{\overline{w}}_{r} \overline{J}_r + \tilde{\overline{w}}_{r} \sum _{r=1}^{N}\tilde{\overline{w}}_{r}\overline{J}_r;\ \dot{\tilde{\underline{w}}}_{r}\nonumber \\&=-\tilde{\underline{w}}_{r} \underline{J}_r + \tilde{\underline{w}}_{r} \sum _{r=1}^{N}\tilde{\underline{w}}_{r}\underline{J}_r \end{aligned}$$
(53)

Furthermore, the time derivative of upper and lower of type-2 MFs are calculated as (54):

$$\begin{aligned} \dot{\underline{\mu }}_{ik}(x_i)&=-\left( \frac{\left( x_i+\xi _{ik}-c_{ik}\right) (\dot{x}_i+\dot{\xi }_{ik}-\dot{c}_{ik})}{ \underline{\sigma }_{ik}^2} \right. \nonumber \\&\quad -\left. \frac{\dot{\underline{\sigma }}_{ik}\left( x_i+\xi _{ik}-c_{ik}\right) ^2}{\underline{\sigma }_{ik}^3}\right) \nonumber \\ \dot{\overline{\mu }}_{ik}(x_i)&=-\left( \frac{\left( x_i+\xi _{ik}-c_{ik}\right) (\dot{x}_i+\dot{\xi }_{ik}-\dot{c}_{ik}}{ \overline{\sigma }_{ik}^2}\right. \nonumber \\&\quad -\left. \frac{\dot{\overline{\sigma }}_{ik}\left( x_i+\xi _{ik}-c_{ik}\right) ^2}{\overline{\sigma }_{ik}^3}\right) \end{aligned}$$
(54)

and the time derivative of the recurrent parameter is as follows:

$$\begin{aligned} \dot{\xi }_{ik}= \frac{\dot{\underline{\theta }}_{ik} (\underline{\mu }_{ik}(t-1)-\xi _{ik}) + \dot{\overline{\theta }}_{ik} (\overline{\mu }_{ik}(t-1)-\xi _{ik})}{(\underline{\theta }_{ik}+\overline{\theta }_{ik})} \end{aligned}$$
(55)

considering \(\dot{\underline{\mu }}_{ik}(x_i)=-\underline{A}_{ik} \dot{\underline{A}}_{ik}\) where \(\underline{A}_{ik}\) and \(\dot{\underline{A}}_{ik}\) are as follows:

$$\begin{aligned} \underline{A}_{ik}&=\frac{x_i}{\underline{\sigma }_{ik}} - \frac{c_{ik}}{\underline{\sigma }_{ik}}+ \frac{\underline{\theta }_{ik} \underline{\mu }_{ik}(t-1) + \overline{\theta }_{ik} \overline{\mu }_{ik}(t-1)}{\underline{\sigma }_{ik}(\underline{\theta }_{ik}+\overline{\theta }_{ik})} \nonumber \\ \dot{\underline{A}}_{ik}&=\frac{\dot{x}_i}{\underline{\sigma }_{ik}} - \frac{\dot{c}_{ik}}{\underline{\sigma }_{ik}} - \dot{\underline{\sigma }}_{ik} \left( \frac{x_i-c_{ik}+\xi _{ik} }{\underline{\sigma }_{ik}^2}\right) \nonumber \\&\quad +\frac{\dot{\underline{\theta }}_{ik}(\underline{\mu }_{ik}(t-1) -\xi _{ik})}{\underline{\sigma }_{ik}(\underline{\theta }_{ik} +\overline{\theta }_{ik})} +\frac{\dot{\overline{\theta }}_{ik}\left( \overline{\mu }_{ik}(t-1) -\xi _{ik}\right) }{\underline{\sigma }_{ik}(\underline{\theta }_{ik} +\overline{\theta }_{ik})} \end{aligned}$$
(56)

By using Maclaurin series expansion:

$$\begin{aligned}\frac{1}{\underline{\sigma }_{ik}}&=\frac{1}{\frac{\underline{\sigma }_{ik}+\overline{\sigma }_{ik}}{2}+ \frac{\underline{\sigma }_{ik}-\overline{\sigma }_{ik}}{2}}\nonumber \\& =\frac{2}{\overline{\sigma }_{ik}+\underline{\sigma }_{ik}} \left( 1 \underbrace{-\frac{\underline{\sigma }_{ik}-\overline{\sigma }_{ik}}{\overline{\sigma }_{ik}+\underline{\sigma }_{ik}} +\left( \frac{\overline{\sigma }_{ik}-\underline{\sigma }_{ik}}{\underline{\sigma }_{ik}+\overline{\sigma }_{ik}}\right) ^2 +H.O.T }_{\underline{D}_{ik}}\right) \nonumber \\ \frac{1}{\overline{\sigma }_{ik}} & =\frac{1}{\frac{\overline{\sigma }_{ik}+\underline{\sigma }_{ik}}{2}+ \frac{\overline{\sigma }_{ik}-\underline{\sigma }_{ik}}{2}}\nonumber \\ & =\frac{2}{\underline{\sigma }_{ik}+\overline{\sigma }_{ik}} \left( 1 \underbrace{-\frac{\overline{\sigma }_{ik}-\underline{\sigma }_{ik}}{\underline{\sigma }_{ik}+\overline{\sigma }_{ik}} +\left( \frac{\underline{\sigma }_{ik}-\overline{\sigma }_{ik}}{\overline{\sigma }_{ik}+\underline{\sigma }_{ik}}\right) ^2 +H.O.T }_{\overline{D}_{ik}} \right) \end{aligned}$$
(57)

\(\underline{D}_{ik}\) and \(\overline{D}_{ik}\) are limited as follows:

$$\begin{aligned} \left( 1+\underline{D}_{ik}\right) ^2+1<B_D, \ \left( 1+\overline{D}_{ik}\right) ^2+1<B_D \end{aligned}$$
(58)

thus the \(\overline{J}_r\) and \(\underline{J}_r\) in (53) are as follows:

$$\begin{aligned} \overline{J}_r= -\sum _{i=1}^I \overline{A}_{ik} \dot{\overline{A}}_{ik} \ ; \ \ \ \ \underline{J}_r=-\sum _{i=1}^I \underline{A}_{ik} \dot{\underline{A}}_{ik} \end{aligned}$$
(59)

by substituting (19)–(23) in the above equation:

$$\begin{aligned} \overline{J}_r = \underline{J}_r = I \alpha \text{sgn}(e) \end{aligned}$$
(60)

In order to check the stability condition, the following Lyapunov function is considered:

$$\begin{aligned} V=\frac{1}{2}e^2 +\frac{1}{2\gamma }\left( \alpha -\alpha ^*\right) ^2 \end{aligned}$$
(61)

The stability condition is met when the time derivative of V is negative definite (\(\dot{V}<0\)). So,

$$\begin{aligned} \dot{V}=\dot{e}e+\frac{1}{\gamma }\dot{\alpha }\left( \alpha -\alpha ^*\right) =e\left( \dot{y_N}-\dot{y}\right) +\frac{1}{\gamma }\dot{\alpha }\left( \alpha - \alpha ^*\right) \end{aligned}$$
(62)

by differentiating (17)

$$\begin{aligned} \dot{y}_N &=\dot{q} \sum _{r=1}^{N} f_{r}\underline{\tilde{w}}_{r}+q\sum _{r=1}^{N} \left( \dot{f}_{r}\underline{\tilde{w}}_{r}+f_{r}\dot{\tilde{\underline{w}}}_{r}\right) \nonumber \\ & \quad - \dot{q} \sum _{r=1}^{N} f_{r}\tilde{\overline{w}}_{r} +\left( 1-q\right) \sum _{r=1}^{N} \left( \dot{f}_{r}\tilde{\overline{w}}_{r}+f_{r}\dot{\tilde{\overline{w}}}_{r}\right) \end{aligned}$$
(63)

by substituting (53):

$$\begin{aligned} \dot{y}_N & =\dot{q} \sum _{r=1}^{N} f_{r}\underline{\tilde{w}}_{r} \nonumber \\ & \quad +q\sum _{r=1}^{N} \left( \dot{f}_{r}\underline{\tilde{w}}_{r}+f_{r}(-\tilde{\underline{w}}_{r} \underline{J}_r + \tilde{\underline{w}}_{r} \sum _{r=1}^{N}\tilde{\underline{w}}_{r}\underline{J}_r)\right) \nonumber \\ & \quad -\dot{q} \sum _{r=1}^{N} f_{r}\tilde{\overline{w}}_{r} \nonumber \\ & \quad +\left( 1-q\right) \sum _{r=1}^{N} \left( \dot{f}_{r}\tilde{\overline{w}}_{r}+f_{r}\left( -\tilde{\overline{w}}_{r} \overline{J}_r + \tilde{\overline{w}}_{r} \sum _{r=1}^{N}\tilde{\overline{w}}_{r}\overline{J}_r\right) \right) \end{aligned}$$
(64)

by considering (60):

$$\begin{aligned}\dot{y}_N & =\dot{q} \sum _{r=1}^{N} f_{r}\underline{\tilde{w}}_{r}\nonumber \\&\quad +q\sum _{r=1}^{N} \left( \dot{f}_{r}\underline{\tilde{w}}_{r}-I \alpha \text{sgn}(e) \ f_{r}\left( \tilde{\underline{w}}_{r}-\tilde{\underline{w}}_{r}\sum _{r=1}^{N}\tilde{\underline{w}}_{r}\right) \right) \nonumber \\&\quad -\dot{q} \sum _{r=1}^{N} f_{r}\tilde{\overline{w}}_{r} \nonumber \\&\quad +\left( 1-q\right) \sum _{r=1}^{N} \left( \dot{f}_{r}\tilde{\overline{w}}_{r}-I \alpha \text{sgn}(e) \ f_{r} \left( \tilde{\overline{w}}_{r}-\tilde{\overline{w}}_{r}\sum _{r=1}^{N}\tilde{\overline{w}}_{r}\right) \right) \end{aligned}$$
(65)

since \(\sum _{r=1}^{N} \tilde{\overline{w}}_{r}=1\) and \( \sum _{r=1}^{N} \tilde{\underline{w}}_{r}=1\) Eq. (65) can be summarized and then by substituting \(\dot{q}\) according to (27),

$$\begin{aligned} \dot{y}_N &= -\frac{1}{F(\tilde{\underline{W}}-\tilde{\overline{W}})^T} \alpha \text{sgn}(e) \sum _{r=1}^{N} f_{r} \left( \tilde{\underline{w}}_{r}-\tilde{\overline{w}}_{r}\right) \nonumber \\ & \quad +\sum _{r=1}^{N} \dot{f}_{r} \left( q\tilde{\underline{w}}_{r}+ (1-q)\tilde{\overline{w}}_{r}\right) \nonumber \\ & = - \alpha \text{sgn}(e) + \sum _{r=1}^{N} \dot{f}_{r}(q\underline{\tilde{w}}_{r} + (1-q)\tilde{\overline{w}}_{r}) \end{aligned}$$
(66)

and the time derivative of \({f}_{r}\) in (10) is calculated as:

$$\begin{aligned} \dot{f}_{r}&= \sum _{i=1}^{I} \dot{\rho }_{ri} \varPsi _{ri}(z)\nonumber \\&\quad +\sum _{i=1}^{I} \rho _{ri} \left( -\frac{1}{2} \dot{a}_{ri} a_{ri}^{-1} |a_{ri}|^{-\frac{1}{2}} (1-z_{ri}^2) e^{-\frac{z_{ri}^2}{2}} \right) \nonumber \\&\quad + \rho _{ri} \left( |a_{ri}|^{-\frac{1}{2}}\left[ -2\dot{z}_{ri}z_{ri} e^{-\frac{z_{ri}^2}{2}} - \dot{z}_{ri}z_{ri} e^{-\frac{z_{ri}^2}{2}} (1-z_{ri}^2) \right] \right) \end{aligned}$$
(67)
$$\begin{aligned} \dot{f}_{r}&= \sum _{i=1}^{I} \dot{\rho }_{ri} \varPsi _{ri}(z) + \sum _{i=1}^{I} \rho _{ri} (-\frac{1}{2} \dot{a}_{ri} a_{ri}^{-1}) \varPsi _{ri}(z) \nonumber \\&\quad - \sum _{i=1}^{I} \rho _{ri} \dot{z}_{ri}z_{ri} \left[ 2 \frac{\varPsi _{ri}(z)}{1-z_{ri}^2} + \varPsi _{ri}(z) \right] \nonumber \\&= \sum _{i=1}^{I} \dot{\rho }_{ri} \varPsi _{ri}(z) + \sum _{i=1}^{I} \rho _{ri} \varPsi _{ri}(z) \nonumber \\&\quad \left( -\frac{1}{2} \dot{a}_{ri} a_{ri}^{-1} - \dot{z}_{ri}z_{ri}\left( \frac{2}{1-z_{ri}^2} +1\right) \right) \nonumber \\&= \sum _{i=1}^{I} \dot{\rho }_{ri} \varPsi _{ri}(z) + \sum _{i=1}^{I} \rho _{ri} \varPsi _{ri}(z) \nonumber \\&\quad \left( -\frac{1}{2} \dot{a}_{ri} a_{ri}^{-1} - \dot{z}_{ri}z_{ri} \left( \frac{3-z_{ri}^2}{1-z_{ri}^2}\right) \right) \nonumber \\&= \sum _{i=1}^{I} \dot{\rho }_{ri} \varPsi _{ri}(z) + \sum _{i=1}^{I} \rho _{ri}\varPsi _{ri}(z) \left( -\frac{1}{2} \dot{a}_{ri} a_{ri}^{-1} \right. \nonumber \\&\quad -\left. \left( \frac{\dot{x}_{i} - \dot{b}_{ri}}{a_{ri}} - \dot{a}_{ri} \frac{x_i-b_{ri}}{a_{ri}^2}\right) z_{ri} \left( \frac{3-z_{ri}^2}{1-z_{ri}^2}\right) \right) \end{aligned}$$
(68)
$$\begin{aligned} \dot{f}_{r} &= \sum _{i=1}^{I} \dot{\rho }_{ri} \varPsi _{ri}(z) + \sum _{i=1}^{I} \rho _{ri} \varPsi _{ri}(z) \nonumber \\&\quad \left( \dot{a}_{ri} \left( -\frac{1}{2} a_{ri}^{-1} +\frac{x_i-b_{ri}}{a_{ri}^2} \frac{3-z_{ri}^2}{1-z_{ri}^2}z_{ri} \right) \right) \nonumber \\&\quad -\sum _{i=1}^{I} \rho _{ri} \varPsi _{ri}(z) \left( \frac{\dot{x}_i}{a_{ri}} \frac{3-z_{ri}^2}{1-z_{ri}^2} z_{ri} + \frac{\dot{b}_{ri}}{a_{ri}} z_{ri} \left( \frac{3-z_{ri}^2}{1-z_{ri}^2}\right) \right) \end{aligned}$$
(69)

so \(\dot{y}_N\) in (66) changes as follows:

$$\begin{aligned} \dot{y}_N&= - \alpha \text{sgn}(e) + \sum _{r=1}^{N} \sum _{i=1}^{I} \dot{\rho }_{ri} \varPsi _{ri}(z) \nonumber \\&\quad + \sum _{i=1}^{I} \rho _{ri}\varPsi _{ri}(z) \left( \dot{a}_{ri} \left( -\frac{1}{2} a_{ri}^{-1} +\frac{x_i-b_{ri}}{a_{ri}^2} \frac{3-z_{ri}^2}{1-z_{ri}^2}z_{ri} \right) \right. \nonumber \\&\quad -\left. \frac{\dot{x}_i}{a_{ri}} \frac{3-z_{ri}^2}{1-z_{ri}^2} z_{ri} + \frac{\dot{b}_{ri}}{a_{ri}} z_{ri} \left( \frac{3-z_{ri}^2}{1-z_{ri}^2}\right) \right) \nonumber \\&\quad \left( q\underline{\tilde{w}}_{r} + (1-q)\tilde{\overline{w}}_{r}\right) \end{aligned}$$
(70)

substituting \(\dot{y}_N\) (70) and then \(\dot{a}_{ri}\), \(\dot{b}_{ri}\) and \(\dot{\rho }_{ri}\) according to (24) to (26) into the time derivative of Lyapunov function (62):

$$\begin{aligned} \dot{V}&= -|e|(3IN+1)\alpha +e \left[ \sum _{r=1}^{N}\left[ \sum _{i=1}^{I} \rho _{ri} \left( -\varPsi _{ri}(z) \frac{\dot{x}_i}{a_{ri}} \right. \right. \right. \nonumber \\&\quad \left. \left. \left. \frac{3-z_{ri}^2}{1-z_{ri}^2} z_{ri}\right) \left( q\underline{\tilde{w}}_{r} + (1-q)\tilde{\overline{w}}_{r}\right) \right] -\dot{y} \right] \nonumber \\&\quad + \frac{1}{\gamma }\dot{\alpha }\left( \alpha -\alpha ^{*} \right) \nonumber \\ \dot{V}&< -|e|(3IN+1)\alpha +|e| \left[ \sum _{r=1}^{N}\left[ \sum _{i=1}^{I} B_{\rho } \right. \right. \nonumber \\&\quad \left. \left( -B_{\psi } \frac{B_{\dot{x}}}{a_{ri}} \left| \frac{3-z_{ri}^2}{1-z_{ri}^2} \right| |z_{ri}|\right) (q\underline{\tilde{w}}_{r} + (1-q)\tilde{\overline{w}}_{r}) \right] \nonumber \\&\quad \left. -B_{\dot{y}} \right] + \frac{1}{\gamma }\dot{\alpha }\left( \alpha -\alpha ^{*} \right) \nonumber \\&< -|e|(3IN+1)\alpha +|e| \left( N I B_{\rho } B_{\psi } B_{\dot{x}} B_{a} - B_{\dot{y}} \right) \nonumber \\&\quad +\frac{1}{\gamma }\dot{\alpha }\left( \alpha -\alpha ^{*} \right) \end{aligned}$$
(71)

where, \(\alpha ^{*} \ge \frac{( N I B_{\rho } B_{\psi } B_{\dot{x}} B_{a} -B_{\dot{y}})}{(3NI+1)}\) which is regarded as an unknown parameter and is determined during the tuning of learning rate. Thus,

$$\begin{aligned} \dot{V}&\le -|e|(3IN+1) \alpha +|e| \left( N I B_{\rho } B_{\psi } B_{\dot{x}} B_{a} - B_{\dot{y}} \right) \nonumber \\&\quad + |e|(3IN+1) \alpha ^* -|e|(3IN+1) \alpha ^* + \frac{1}{\gamma }\dot{\alpha }\left( \alpha -\alpha ^{*} \right) \nonumber \\&\le |e| \left( N I B_{\rho } B_{\psi } B_{\dot{x}} B_{a} -B_{\dot{y}} \right) - |e|(3IN+1)\alpha ^{*}\nonumber \\&\quad -(\alpha -\alpha ^{*}) \left( (3IN+1)|e| - \frac{1}{\gamma }\dot{\alpha } \right) \end{aligned}$$
(72)

the adaptive learning rate \((\alpha )\) changes as follows:

$$\begin{aligned} \dot{\alpha } =(3IN+1) \gamma |e|-\nu \gamma \alpha \end{aligned}$$
(73)

in which \(\nu \) has small real value. The time derivative of Lyapunov function (72) is changed as follow:

$$\begin{aligned} \dot{V}&\le |e| \left[ N I B_{\rho } B_{\psi } B_{\dot{x}} B_{a} -B_{\dot{y}} \right] \nonumber \\&\quad - |e|(3IN+1) \alpha ^{*}-(\alpha -\alpha ^{*}) \nu \alpha \nonumber \\&\le |e| \left[ N I B_{\rho } B_{\psi } B_{\dot{x}} B_{a} -B_{\dot{y}} \right] \nonumber \\&\quad - |e|(3IN+1) \alpha ^{*} - \nu \left( \alpha - \frac{\alpha ^*}{2}\right) ^2 + \frac{\nu \alpha ^{*2}}{4} \end{aligned}$$
(74)

since \(\alpha ^{*} \ge \frac{( N I B_{\rho } B_{\psi } B_{\dot{x}} B_{a} -B_{\dot{y}})}{(3NI+1)}\)

$$\begin{aligned} |e| \left( N I B_{\rho } B_{\psi } B_{\dot{x}} B_{a} -B_{\dot{y}} \right) -|e|(3IN+1) \left( \frac{\alpha ^*}{2}\right) \le 0 \end{aligned}$$
(75)

consequently,

$$\begin{aligned} \dot{V} \le - \left( \frac{\alpha ^*}{2}\right) (3IN+1) |e| + \frac{\nu \alpha ^{*2}}{4} \end{aligned}$$
(76)

In this case in order to have a negative definite time derivative of the Lyapunov function, it is required that the following condition holds.

$$\begin{aligned} |e| \le \frac{\alpha ^* \nu }{2(3IN+1)} \end{aligned}$$
(77)

Hence, it is concluded that error converges to a small neighborhood of zero, in which and stays there. Furthermore, the radius of this neighborhood can be made as small as desired using appropriate values for \(\alpha ^*\) and \(\nu \). This concludes the proof.

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Tafti, B.E.F., Teshnehlab, M. & Khanesar, M.A. Recurrent Interval Type-2 Fuzzy Wavelet Neural Network with Stable Learning Algorithm: Application to Model-Based Predictive Control. Int. J. Fuzzy Syst. (2020). https://doi.org/10.1007/s40815-019-00766-z

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Keywords

  • Identification
  • Sliding mode
  • Model predictive controller
  • Fractional-order chaotic systems
  • Synchronization