# Fractional-Order Sliding Mode with Active Disturbance Rejection Control for UAVs
## 3 Flight Controller Design
### External Loop Controller (Fractional Order Convergence Law Smooth Mode Controller Design)
The hyperbolic tangent function is continuous and smooth, and the use of the hyperbolic correct function instead of the discontinuous sign function can effectively reduce the jitter in the sliding mode control. The hyperbolic tangent function is as follows:  
\[s_{\theta}=c_{\theta}\times(\theta_{d}-\theta)+(\dot{\theta}_{d}-\dot{\theta}) =c_{\theta}\times c_{\theta}+\dot{e}_{\theta} \tag{5}\]1  
where \(\epsilon>0\) and the magnitude of the \(\epsilon\) value determines how quickly or slowly the hyperbolic tangent function inflection point changes. There are various forms of fractional-order calculus operators, of which the Caputo fractional-order calculus [24] is defined as  
\[s_{\theta}=c_{\theta}\times(\theta_{d}-\theta)+(\dot{\theta}_{d}-\dot{\theta}) =c_{\theta}\times c_{\theta}+\dot{e}_{\theta} \tag{5}\]2  
where \(n-1\leq\alpha<n\), \(n\) is a positive integer, and \(\Gamma(\cdot)\) is the gamma function. Caputo fractional-order differentiation is the operation of nth order derivatives and then nth order integrals on a function. Using Equations (14) and (15), the new hyperbolic tangent exponential reaching law for fractional order calculus is obtained as  
\[s_{\theta}=c_{\theta}\times(\theta_{d}-\theta)+(\dot{\theta}_{d}-\dot{\theta}) =c_{\theta}\times c_{\theta}+\dot{e}_{\theta} \tag{5}\]3where \(\varepsilon_{j}>0,k_{j}>0\) are adjustable parameters, \({}_{0}D_{t}^{-\alpha}\) is the fractional order differential integral operator, and \(\alpha\in R\), \(\alpha>0\) is the fractional order. The final fractional order reaching law sliding mode control law is obtained by replacing the sliding mode surface derivatives in the control law Equations (10)-(12) using the reaching law as  
\[\left\{ \begin{aligned} & u_{2}=I_{x}/l[C_{\phi}(\dot{\psi}_{d}- \dot{\psi}+\varepsilon_{\phi}({}_{0}D_{t}^{-\alpha}\tanh(s_{\phi}))+k_{\phi}s_ {\phi}+\ddot{\psi}_{d})]-\dot{\phi}\dot{\psi}(I_{y}-I_{z})/l\\ &-I_{y}\dot{\phi}\Omega_{r}/l+K_{4}\dot{\psi}\\ & u_{3}=I_{y}/l[C_{\theta}(\dot{\theta}_{d}-\dot{\phi}+ \varepsilon_{\theta}(0D_{t}^{-\alpha}\tanh(s_{\theta}))+k_{\theta}s_{\phi}+ \ddot{\theta}_{d})]-\dot{\phi}\dot{\psi}(I_{z}-I_{x})/l\\ &+I_{y}\dot{\phi}\Omega_{r}/l+K_{5}\dot{\psi}\\ & u_{4}=I_{z}[C_{\phi}(\dot{\psi}_{d}-\dot{\psi}+-\varepsilon_{ \phi}(0D_{t}^{-\alpha}\tanh(s_{\psi}))-k_{\phi}s_{\psi}+\ddot{\psi}_{d})]- \dot{\phi}\dot{\phi}(I_{x}-I_{y})\\ &+K_{6}\dot{\psi}\\ \end{aligned} \right. \tag{17}\]  
Stability analysis  
**Theorem 1**. : _For the attitude angle model in the attitude angle control subsystem Equation (1), the controller Equation (17) can stabilize the controlled system in a finite amount of time for a given sliding mode surface, as described in Equations (4)-(6)._  
Proofs must be formatted as follows:  
**Proof of Theorem 1. ** For the sliding mode controller, the Lyapunov function is defined as  
\[V_{j}=\frac{1}{2}s_{j}^{2}(j=\phi,\theta,\psi) \tag{18}\]  
Carried over into Equations (3)-(12) and (20), we obtain  
\[\dot{V}_{j}=s_{j}\cdot s_{j}=-\varepsilon_{j}\cdot s_{j}\cdot({}_{0}D_{t}^{- \alpha}\tanh(s_{j}))-k_{j}s_{j}^{2} \tag{19}\]  
The hyperbolic tangent function \(f(t)=\tanh(s_{j})\) is an odd function, continuous and smooth, and its value increases monotonically from \(-1\) to \(1\) as \(s_{j}\) increases. When \(s_{j}\) decreases from a large positive value, the function starts to decrease from a value close to \(1\). \(s_{j}\) decreases from a large positive value; the value of the function starts to decrease from a value close to \(1\). The fractional-order calculus operator \({}_{0}D_{t}^{-\alpha}\) takes into account the averaging effect of the hyperbolic tangent function over the integration interval \([0,t]\). If \(s_{j}\) is always positive, even when it is decreasing, \(f(t)\) is positive, only approaching \(1\) at a reduced rate, so the value of \({}_{0}D_{t}^{-\alpha}\tanh(s_{j})\) remains positive, and the result of \({}_{0}D_{t}^{-\alpha}\tanh(s_{j})\) becomes negative when \(s_{j}\) decreases to a negative value, too.