﻿ 基于分数阶Takagi-Sugeno模糊模型的分数阶Chen混沌系统的控制

# 基于分数阶Takagi-Sugeno模糊模型的分数阶Chen混沌系统的控制Stability of Fractional Order Chen Chaos Based on Takagi-Sugeno Fuzzy Model

Abstract: The control problem of fractional order Chen chaos system based on fractional order Tak-agi-Sugeno fuzzy model is studied. The Takagi-Sugeno fuzzy model for fractional order Chen chaos systems is given, and a parallel distributed compensate fuzzy controller is designed to asymptotically stabilize the model. Then effectiveness of the approach is tested on fractional order Chen chaos system with α=0.95 .

1. 引言

2. 分数阶Takagi-Sugeno模糊系统

2.1. Caputo型分数阶导数

$\begin{array}{l}{}_{a}^{C}{D}_{t}^{\alpha }f\left(t\right)=\frac{1}{\Gamma \left(r-\alpha \right)}{\int }_{a}^{t}\frac{{f}^{\left(r\right)}\left(\tau \right)}{{\left(t-\tau \right)}^{\alpha +1-r}}\text{d}\tau ,\text{\hspace{0.17em}}\left(r-1\le \alpha (1)

2.2. 分数阶Takagi-Sugeno模糊系统

Takagi-Sugeno模糊模型自产生之日起就是处理非线性系统问题的有效方法之一。混沌系统是一类典型的非线性系统，本文利用分数阶Takagi-Sugeno模糊模型来研究非线性分数阶Chen混沌系统的控制问题。首先我们介绍分数阶Takagi-Sugeno模糊系统。

Rule i: If ${z}_{1}\left(t\right)$ is ${M}_{i1}$ and … ${z}_{p}\left(t\right)$ is ${M}_{ip}$ ;

Then $\left\{\begin{array}{l}{D}^{\alpha }x\left(t\right)={A}_{i}x\left(t\right)+{B}_{i}u\left(t\right)\hfill \\ x\left(0\right)={x}_{0}\hfill \end{array}$

$\left\{\begin{array}{l}{D}^{\alpha }x\left(t\right)=\underset{i=1}{\overset{r}{\sum }}{h}_{i}\left(z\left(t\right)\right)\left({A}_{i}x\left(t\right)+{B}_{i}u\left(t\right)\right)\hfill \\ x\left(0\right)={x}_{0}\hfill \end{array}$ (2)

$\left\{\begin{array}{l}\underset{i=1}{\overset{r}{\sum }}{w}_{i}\left(z\left(t\right)\right)>0\hfill \\ {w}_{i}\left(z\left(t\right)\right)\ge 0\hfill \end{array}$

$\left\{\begin{array}{l}\underset{i=1}{\overset{r}{\sum }}{h}_{i}\left(z\left(t\right)\right)=1\hfill \\ {h}_{i}\left(z\left(t\right)\right)\ge 0\hfill \end{array}$

${A}_{i}Q-{B}_{i}{M}_{i}+{\left({A}_{i}Q-{B}_{i}{M}_{i}\right)}^{\text{T}}<0,\text{\hspace{0.17em}}\text{\hspace{0.17em}}i=1,2,\cdots ,r$, (3)

${A}_{i}Q-{B}_{i}{M}_{j}+{A}_{j}Q-{B}_{j}{M}_{i}+{\left({A}_{i}Q-{B}_{i}{M}_{j}+{A}_{j}Q-{B}_{j}{M}_{i}\right)}^{\text{T}}<0,\text{\hspace{0.17em}}\text{\hspace{0.17em}}1\le i, (4)

3. 分数阶Chen混沌系统

3.1. 分数阶Chen混沌系统

$\left\{\begin{array}{l}{D}^{\alpha }{x}_{1}\left(t\right)=a\left({x}_{2}\left(t\right)-{x}_{1}\left(t\right)\right),\hfill \\ {D}^{\alpha }{x}_{2}\left(t\right)=\left(c-a\right){x}_{1}\left(t\right)-{x}_{1}\left(t\right){x}_{3}\left(t\right)+c{x}_{2}\left(t\right),\hfill \\ {D}^{\alpha }{x}_{3}\left(t\right)=-b{x}_{3}\left(t\right)+{x}_{1}\left(t\right){x}_{2}\left(t\right).\hfill \end{array}$ (5)

Figure 1. Chaotic behaviors of the fractional order Chen system

3.2. 分数阶Chen混沌系统的Takagi-Sugeno模糊模型

Rule 1: If ${x}_{1}\left(t\right)$ is ${F}_{1}\left({x}_{1}\left(t\right)\right)$, then ${D}^{\alpha }x\left(t\right)={A}_{1}x\left(t\right)$ ;

Rule 2: If ${x}_{1}\left(t\right)$ is ${F}_{2}\left({x}_{1}\left(t\right)\right)$, then ${D}^{\alpha }x\left(t\right)={A}_{2}x\left(t\right)$.

${A}_{1}=\left[\begin{array}{ccc}-a& a& 0\\ c-a& c& -d\\ 0& d& -b\end{array}\right]$${A}_{2}=\left[\begin{array}{ccc}-a& a& 0\\ c-a& c& d\\ 0& -d& -b\end{array}\right]$

${F}_{1}\left({x}_{1}\left(t\right)\right)=\left(1/2\right)\left(1-{x}_{1}\left(t\right)/d\right)$

${F}_{2}\left({x}_{1}\left(t\right)\right)=\left(1/2\right)\left(1+{x}_{1}\left(t\right)/d\right)$

${D}^{\alpha }x\left(t\right)=\underset{i=1}{\overset{2}{\sum }}{F}_{i}\left(z\left(t\right)\right){A}_{i}x\left(t\right)$ (6)

$\left\{\begin{array}{l}\underset{i=1}{\overset{2}{\sum }}{F}_{i}\left({x}_{1}\left(t\right)\right)=1\hfill \\ {F}_{i}\left({x}_{1}\left(t\right)\right)\ge 0\hfill \end{array}$

3.3. 基于分数阶Takagi-Sugeno模糊系统的分数阶Chen混沌系统的控制

$\left\{\begin{array}{l}{D}^{\alpha }{x}_{1}\left(t\right)=a\left({x}_{2}\left(t\right)-{x}_{1}\left(t\right)\right),\hfill \\ {D}^{\alpha }{x}_{2}\left(t\right)=\left(c-a\right){x}_{1}\left(t\right)-{x}_{1}\left(t\right){x}_{3}\left(t\right)+c{x}_{2}\left(t\right)+u\left(t\right),\hfill \\ {D}^{\alpha }{x}_{3}\left(t\right)=-b{x}_{3}\left(t\right)+{x}_{1}\left(t\right){x}_{2}\left(t\right).\hfill \end{array}$ (7)

Rule 1: If ${x}_{1}\left(t\right)$ is ${F}_{1}\left({x}_{1}\left(t\right)\right)$, then ${D}^{\alpha }x\left(t\right)={A}_{1}x\left(t\right)+{B}_{1}u\left(t\right)$ ;

Rule 2: If ${x}_{1}\left(t\right)$ is ${F}_{2}\left({x}_{1}\left(t\right)\right)$, then ${D}^{\alpha }x\left(t\right)={A}_{2}x\left(t\right)+{B}_{2}u\left(t\right)$.

Rule 1: If ${x}_{1}\left(t\right)$ is ${F}_{1}\left({x}_{1}\left(t\right)\right)$, then $u\left(t\right)=-{K}_{1}x\left(t\right)$ ;

Rule 2: If ${x}_{1}\left(t\right)$ is ${F}_{2}\left({x}_{1}\left(t\right)\right)$, then $u\left(t\right)=-{K}_{2}x\left(t\right)$.

$u\left(t\right)=-\underset{i=1}{\overset{2}{\sum }}{F}_{i}\left({x}_{1}\left(t\right)\right){K}_{i}x\left(t\right),\text{\hspace{0.17em}}\text{\hspace{0.17em}}i=1,2$ (8)

${D}^{\alpha }x\left(t\right)=\underset{i=1}{\overset{r}{\sum }}\underset{j=1}{\overset{r}{\sum }}{h}_{i}\left(z\left(t\right)\right)\left({A}_{i}-{B}_{i}{K}_{j}\right)x\left(t\right)$ (9)

${A}_{1}Q-{B}_{1}{M}_{1}+{\left({A}_{1}Q-{B}_{1}{M}_{1}\right)}^{\text{T}}<0$

${A}_{2}Q-{B}_{2}{M}_{2}+{\left({A}_{2}Q-{B}_{2}{M}_{2}\right)}^{\text{T}}<0$

${A}_{1}Q-{B}_{1}{M}_{2}+{A}_{2}Q-{B}_{2}{M}_{1}+{\left({A}_{1}Q-{B}_{1}{M}_{2}+{A}_{2}Q-{B}_{2}{M}_{1}\right)}^{\text{T}}<0$ (10)

$Q=\left[\begin{array}{ccc}0.\text{1782}& 0.0\text{213}& -0.00\text{49}\\ 0.0\text{213}& 0.\text{4162}& 0.000\text{4}\\ -0.00\text{49}& 0.000\text{4}& 0.\text{5926}\end{array}\right]$

${M}_{1}=\left[\begin{array}{ccc}\text{12}.\text{8975}& \text{16}.\text{4537}& -\text{2}.\text{9931}\end{array}\right]$

${M}_{2}=\left[\begin{array}{ccc}\text{13}.0\text{224}& \text{16}.\text{2451}& \text{5}.\text{2164}\end{array}\right]$

${K}_{1}=\left[\begin{array}{ccc}\text{67}.\text{9598}& \text{36}.0\text{6}0\text{1}& -\text{4}.\text{511}0\end{array}\right]$

${K}_{2}=\left[\begin{array}{ccc}\text{69}.\text{1121}& \text{35}.\text{4868}& \text{9}.\text{352}0\end{array}\right]$

$u\left(t\right)=-\underset{i=1}{\overset{2}{\sum }}{F}_{i}\left({x}_{1}\left(t\right)\right){K}_{i}x\left(t\right),\text{\hspace{0.17em}}\text{\hspace{0.17em}}i=1,2$

Figure 2. Control results of the fractional order Chen system

Figure 3. Control curve of the fractional order Chen system

4. 结论

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