# Robust Fuzzy Adaptive Sliding Mode Stabilization for Fractional-Order Chaos
## 2 Preliminaries
### General Fractional-Order Chaotic System Description  
In this work, we consider the class of three-dimensional fractional-order chaotic systems given by the following equations [16]:  
\[\left\{\begin{array}{l}D^{(q_{1})}x=y\cdot f(x,y,z)+z\cdot\phi(x,y,z)-\alpha \cdot x\\ D^{(q_{2})}y=g(x,y,z)-\beta\cdot y\\ D^{(q_{3})}z=y\cdot h(x,y,z)-x\cdot\phi(x,y,z)-\gamma\cdot z\end{array}\right. \tag{11}\]  
where \(q_{i}(i=1,2,3)\) are fractional orders satisfying \(0<q_{i}<1\); and \(x\), \(y\) and \(z\) are state variables. Each of the four functions \(f(.)\), \(g(.)\), \(h(.)\) and \(\phi(.)\) is considered as a continuation of nonlinear vector functions belonging to \(\mathbb{R}^{3}\rightarrow\mathbb{R}\) space. \(\alpha,\beta,\gamma\) are known constants, for any negative or positive values.  
**Remark 1**.: _The fractional-order system in Equation (11) is called a commensurate fractional-order system if \(q_{1}=q_{2}=q_{3}=q\), otherwise we call the system in Equation (11) an incommensurate fractional-order system._  
**Remark 2**.: _We note that many fractional-order chaotic systems belong to the class of systems characterized by Equation (11) depending on the values of the functions \(f(.)\), \(g(.)\), \(h(.)\) and \(\phi(.)\). Table 1 details this class of nonlinear fractional-order systems [16; 30]._  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Name** & **Model** & \(f(x,y,z)\) & \(g(x,y,z)\) & \(h(x,y,z)\) & \(\phi(x,y,z)\) \\ \hline \multirow{3}{*}{[PERSON]’s system} & \(\left\{\begin{array}{l}D^{(q_{1})}x(t)=a(x(t)-y(t))\\ D^{(q_{2})}y(t)=d-x(t)z(t)+cy(t)\\ D^{(q_{3})}z(t)=x(t)y(t)-bz(t)\end{array}\right.\) & \(a\) & \(d-xz+cy\) & \(x\) & \(0\) \\ \multirow{3}{*}{[PERSON] system} & \(\left\{\begin{array}{l}D^{(q_{1})}x(t)=a(y(t)-x(t))\\ D^{(q_{2})}y(t)=x(t)(b-z(t))-y(t)\\ D^{(q_{3})}z(t)=x(t)y(t)-cz(t)\end{array}\right.\) & \(a\) & \(x(b-z)\) & \(x\) & \(0\) \\ \multirow{3}{*}{Lu’s system} & \(\left\{\begin{array}{l}D^{(q_{1})}x(t)=a(y(t)-x(t))\\ D^{(q_{2})}y(t)=-x(t)z(t)+cy(t)\\ D^{(q_{3})}z(t)=x(t)y(t)-bz(t)\end{array}\right.\) & \(a\) & \(-xz+cy\) & \(x\) & \(0\) \\ \multirow{3}{*}{Liu system} & \(\left\{\begin{array}{l}D^{(q_{1})}x(t)=-ax(t)-ey^{2}(t))\\ D^{(q_{2})}y(t)=-kx(t)z(t)+by(t)\\ D^{(q_{3})}z(t)=mx(t)y(t)-cz(t)\end{array}\right.\) & \(-cy\) & \(-kxz+by\) & \(x\) & \(0\) \\ \multirow{3}{*}{Financial system} & \(\left\{\begin{array}{l}D^{(q_{1})}x(t)=z(t)+x(t)(y(t)-a)\\ D^{(q_{2})}y(t)=1-by(t)-x^{2}(t)\\ D^{(q_{3})}z(t)=-x(t)-cz(t)\end{array}\right.\) & \(x\) & \(1-x^{2}\) & \(0\) & \(1\) \\ \hline \hline \end{tabular}
\end{table}
Table 1: Class of nonlinear fractional-order systems characterized by Equation (11).  
The fractional order version of Chen system is a particular case of class of three-dimensional fractional-order chaotic systems (Equation (11)), with  
\[\left\{\begin{array}{l}f(x,y,z)=a\\ g(x,y,z)=dx-xz\\ h(x,y,z)=x\\ \phi(x,y,z)=0\end{array}\right. \tag{12}\]  
It becomes then [31],  
\[\left\{\begin{array}{l}D^{(q_{1})}x(t)=a(y(t)-x(t))\\ D^{(q_{2})}y(t)=(c-a)x(t)-x(t)z(t)+cy(t)+u(t)\\ D^{(q_{3})}z(t)=x(t)y(t)-bz(t)\end{array}\right. \tag{13}\]  
where \(x(t)\), \(y(t)\) and \(z(t)\) are state variables and \(0<q_{1},q_{2},q_{3}<1\) are fractional orders.  
For the fractional orders' values \(q_{1}=0.9\) and \(q_{2}=q_{3}=0.95\), the system in Equation (13) without the controller exhibits a chaotic behavior, as shown in Figure 1, when \((a,b,c,d)=(35,3,28,-7)\). The simulation results are carried out using the MATLAB software.  
Figure 1: _Cont._