# Robust Fuzzy Adaptive Sliding Mode Stabilization for Fractional-Order Chaos
## 4 Sliding Mode Control Design for the General Fractional-Order Chaotic System  
### Sliding Mode Control Law  
The control input \(u(t)\) is added to the second state equation to control the fractional-order chaotic system in Equation (11). The resulting class of fractional-order model can be expressed as [25]:  
\[\left\{\begin{array}{l}D^{(q_{1})}x=y\cdot f(x,y,z)+z\cdot\phi(x,y,z)-\alpha \cdot x\\ D^{(q_{2})}y=g(x,y,z)-\beta\cdot y+u(t)\\ D^{(q_{3})}z=y\cdot h(x,y,z)-x\cdot\phi(x,y,z)-\gamma\cdot z\end{array}\right. \tag{16}\]  
To obtain the sliding mode control law, let us define the sliding surface as [35]:  
\[S(t)=D^{(q_{2}-1)}y(t)+\int_{0}^{t}\psi(\tau)d\tau \tag{17}\]  
Figure 1: Chaotic State trajectories of the fractional-order Chen system: (**a**) states \(z\)-\(y\); (**b**) states \(z\)-\(x\); and (**c**) states \(y\)-\(x\).  
where \(\psi(t)\) is a function described by  
\[\psi(t)=y\cdot f(x,y,z)+c_{1}\cdot h(x,y,z)-(1-c_{2})g(x,y,z)+\beta\cdot y \tag{18}\]  
For the sliding mode technique, the sliding surface and its derivative must satisfy  
\[S(t)=D^{(q_{2}-1)}y(t)+\int_{0}^{t}\psi(\tau)d\tau=0 \tag{19}\]  
and  
\[\dot{S}(t)=D^{(q_{2})}y(t)+\psi(t)=0\;\Rightarrow\;D^{(q_{2})}y(t)=-\psi(t) \tag{20}\]  
According to the sliding mode theory and using Equations (16) and (20), the equivalent control law is derived as  
\[u_{eq}(t)=-y\cdot f(x,y,z)-c_{2}\cdot g(x,y,z)-c_{1}\cdot h(x,y,z) \tag{21}\]  
To design the reaching mode control scheme, which drives states onto the sliding surface, a reaching law can be chosen as  
\[u_{r}(t)=-K_{r}\,sgn\left(S(t)\right) \tag{22}\]  
where  
\[sgn\left(S(t)\right)=\left\{\begin{array}{ccc}-1&\;\;\;if\;\;S(t)<0\\ 0&\;\;\;if\;\;S(t)=0\\ +1&\;\;\;if\;\;S(t)>0\end{array}\right. \tag{23}\]  
and the \(K_{r}\) is the reach gain of the controller, \(K_{r}>0\).  
Finally, the total control law can be defined as  
\[u_{s}(t)=u_{eq}(t)+u_{r}(t) \tag{24}\]  
We can then establish the following result,  
**Theorem 1**.: _The nominal fractional chaotic system in Equation (16) is asymptotically stabilized under the proposed sliding control law in Equation (24)._  
Proof.: The Lyapunov candidate function is selected as:  
\[V=\frac{1}{2}S^{2} \tag{25}\]  
The time derivative of the Lyapunov function is given by  
\[\dot{V}=S\dot{S} \tag{26}\]  
where \(\dot{V}=\frac{d}{dt}\left\{S(t)\right\}\), we obtain [35],  
\[\dot{V} = S\left(D^{(q_{2})}y(t)+\psi(t)\right) \tag{27}\] \[= S\left(g(x,y,z)-\beta\cdot y+u_{s}(t)+y\cdot f(x,y,z)+c_{1}\cdot h (x,y,z)-(1-c_{2})g(x,y,z)+\beta\cdot y\right)\] \[= S\left(g(x,y,z)+u_{eq}(t)+u_{r}(t)+y\cdot f(x,y,z)+c_{1}\cdot h (x,y,z)-(1-c_{2})g(x,y,z)\right)\] \[= S\left(-K_{r}\,sgn\left(S(t)\right)\right)\] \[= -K_{r}\left|S(t)\right|<0\]  
Thus, the closed loop system is globally asymptotically stable via the sliding mode control.  
**Remark 3**: _The same form as Equation (17) is used for the sliding surface than in [35], however the proposed function \(\psi(t)\) defined in Equation (18) is augmented by the presence of the \(g(x,y,z)\) function, leading to the different (and novel) control law in Equation (21). The coefficients \(c_{1}\) and \(c_{2}\) verifying \(c_{1}\in]0,1]\) and \(c_{2}\in[1,2]\), have been introduced in the sliding mode control law in Equation (21) as tuning parameters, allowing the control of the states convergence speed._  
**Remark 4**: _From Table 1 representing different fractional order systems included in the considered class of fractional order chaotic systems, it is obvious that the control law in Equation (21) can be easily and successfully applied to all these systems, confirming former studies on the similarities of [PERSON], [PERSON] and Chen systems (e.g., [35; 36])._