# Robust Fuzzy Adaptive Sliding Mode Stabilization for Fractional-Order Chaos
## 5 Fuzzy Adaptive Sliding Mode Control of Uncertain Perturbed Fractional Order Chaotic System  
Let us consider the fractional-order chaotic system in Equation (16) with uncertainties in the presence of an external disturbance. It will be illustrated that the system can be stabilized under the sliding mode and fuzzy adaptive control. \[\left\{ \begin{array}{l}D^{(q_{1})}x=y\cdot f(x,y,z)+z\cdot\phi(x,y,z)-\alpha \cdot x\\ D^{(q_{2})}y=g(x,y,z)-\beta\cdot y+\Delta g(x,y,z)+d(t)+u(t)\\ D^{(q_{3})}z=y\cdot h(x,y,z)-x\cdot\phi(x,y,z)-\gamma\cdot z\end{array} \right. \tag{30}\]  
where \(\Delta g(x,y,z)\) denotes an uncertainty and \(d(t)\) denotes an external disturbance against the performance of the system, which are bounded by some positive constants. We introduce the augmented adaptive control action given by:  
\[u(t)=u_{s}(t)+u_{f}(t) \tag{31}\]  
where \(u_{s}(t)\) is the control law of the ideal case in Equation (24), and \(u_{f}(t)\) is an additive adaptive fuzzy control law defined as,  
\[u_{f}(t)=\theta^{T}(t)\ \xi(x) \tag{32}\]  
Figure 4: The time response of the controller \(u(t)\). Figure 5: The sliding surface. Fuzzy logic systems are known as universal approximations, i.e., they are able to approximate a smooth function on a compact space [34]. Thus, we can obtain \(\theta^{\star}\) such that:  
\[W(t)=-\theta^{\star T}(t)\;\xi(x) \tag{37}\]  
and then the following result is obtained,  
\[\dot{V}=S\left[-K_{r}\;sgn\left(S(t)\right)\right]=-K_{r}\left|S\right|<0 \tag{38}\]  
_Application to the Uncertain and Disturbed Fractional-Order Chen System_  
To illustrate the performance of the proposed robust adaptive control approach, we consider the fractional-order chaotic Chen system in Equation (28), subject to model uncertainties and additional disturbances. The fractional uncertain perturbed version of Chen system is given by:  
\[\left\{ \begin{array}{l}D^{(q_{1})}x(t)=a(y(t)-x(t))\\ D^{(q_{2})}y(t)=(c-a)x(t)-x(t)z(t)+cy(t)+\Delta g(x,y,z)+d(t)+u(t)\\ D^{(q_{3})}z(t)=x(t)y(t)-bz(t)\end{array} \right. \tag{39}\]  
We use the same parameter values of the fractional-order Chen system given in Section 4.2. The uncertainty term \(\Delta g(x,y,z)\) applied to the system is given by,  
\[\Delta g(x,y,z)=0.75\sin(10x(t))\cos(3y(t))\cos(\pi z(t)) \tag{40}\]  
and the external disturbances \(d(t)\) are defined as:  
\[d(t)=1.25\cos(2 ty(t))+0.5\sin(3t) \tag{41}\]  
We define five fuzzy membership functions for each error and derivative of error \(e_{i}\), \(i=1,2\), given as  
\[\mu_{F_{1}^{i}}(e_{i}) = \exp\left[-\left(\frac{e_{i}+1.25}{0.3}\right)^{2}\right]\] \[\mu_{F_{2}^{i}}(e_{i}) = \exp\left[-\left(\frac{e_{i}+0.625}{0.3}\right)^{2}\right]\] \[\mu_{F_{3}^{i}}(e_{i}) = \exp\left[-\left(\frac{e_{i}}{0.3}\right)^{2}\right] \tag{42}\] \[\mu_{F_{4}^{i}}(e_{i}) = \exp\left[-\left(\frac{e_{i}-0.625}{0.3}\right)^{2}\right]\] \[\mu_{F_{5}^{i}}(e_{i}) = \exp\left[-\left(\frac{e_{i}-1.25}{0.3}\right)^{2}\right]\]  
For the given application, the adaptive gain is set to \(\gamma=20\). The simulation results are given in Figures 6-10. To show the robustness of the proposed AFSMC, the control effort is activated at \(t=5\) s. Figure 6 shows the state trajectories \(x,y\) and \(z\) of stabilized uncertain fractional-order Chen system. Figure 7 presents a phase portrait of the uncertain chaotic system under the control action. We can see that a fast convergence to \(0\) is achieved as the control effort is activated. The global control-effort trajectory is shown in Figure 9, and trajectory of the sliding surface \(S(t)\) is shown in Figure 10.  
We remark that the performance of the designed controller is satisfactory and the robustness of the designed fuzzy adaptive controller is also verified and approved. Even the disturbance magnitude \(\|d(t)\|\leq 1.2\) in this numerical study is widely increased when compared with similar control designs in the literature, we cite for instance the work presented in [35] with \(\|d(t)\|\leq 0.5\) and [22] with \(\|d(t)\|\leq 0.1\).