## 3 Controller Design and Stability Analysis
### Control Algorithm  
In this section, a flight-path angle controller ADSC is designed to track the ideal signals, and the control framework is given by Figure 4.  
Step 1: Define the first error surface regarding the flight-path angle \(\gamma\) as follows [29]:  
\[\zeta_{1}=x_{1}-x_{1d} \tag{4}\]  
Then, differentiating (4) and combining (2), one has  
\[\dot{\zeta}_{1}=h_{1}(\frac{f_{1}(x_{1})}{h_{1}}+x_{2}+\frac{\Delta_{1}(x,t)- \dot{x}_{1d}}{h_{1}}) \tag{5}\]  
RBF-1 can be used to approximate unknown functions contained in (5), invoking (3)  
\[\frac{f_{1}(x_{1})}{h_{1}}=w_{1}^{\star T}\xi_{1}(x_{1})+\sigma_{1}^{\ast} \tag{6}\]  
Figure 4: Scheme of control system. Define  
\[w_{1}^{T}=[{w_{1}^{*}}^{T}\quad 1/h_{1}]^{T} \tag{7}\]  
\[\mu_{1}=\left[\xi_{1}(x_{1})\quad c_{1}\zeta_{1}+\frac{\dot{\tau}_{2}^{2}\zeta_{ 1}}{2\varepsilon}-\dot{x}_{1d}\right]^{T} \tag{8}\]  
where \(\frac{\dot{\rho}_{1}^{2}\zeta_{1}}{2\varepsilon}\) is a nonlinear damping items designed to overcome the influence of the uncertain interference term \(\Delta_{1}(x,t)\) in the system. The virtual control law is designed as  
\[\overline{x_{2}}=-{(\overset{\wedge}{w_{1}})}^{T}\mu_{1} \tag{9}\]  
Make the controller capable the ability to adjust its control parameters online, let the adaptive law of RBF-1\({}^{\prime}\)s weight estimation be  
\[\overset{\wedge}{w_{1}}=\Gamma_{1}\mu_{1}\zeta_{1}-\Gamma_{1}\eta_{1}\overset{ \wedge}{w_{1}} \tag{10}\]  
Deliver the designed virtual control \(\overline{x_{2}}\) into a first-order filter with a time constant of \(\tau_{2}\), obtain new state variable \(x_{2d}\), satisfying  
\[\tau_{2}\dot{x}_{2d}+x_{2d}=\overline{x_{2}},x_{2d}(0)=\overline{x_{2}}(0) \tag{11}\]  
Step 2: Define the second error surface regarding the AOA \(\alpha\) as follows:  
\[\zeta_{1}=x_{1}-x_{1d} \tag{4}\]0  
Then, differentiating (12) and combining (2), one has  
\[\zeta_{1}=x_{1}-x_{1d} \tag{4}\]1  
RBF-2 can be used to approximate unknown functions contained in (13), invoking (3)  
\[\zeta_{1}=x_{1}-x_{1d} \tag{4}\]2  
Define  
\[\zeta_{1}=x_{1}-x_{1d} \tag{4}\]3  
\[\zeta_{1}=x_{1}-x_{1d} \tag{4}\]4  
The virtual control law is designed as  
\[\zeta_{1}=x_{1}-x_{1d} \tag{4}\]5  
where \(\frac{\rho_{2}^{2}\zeta_{2}}{2\varepsilon}\) is a nonlinear damping item designed to overcome the influence of the uncertain interference term \(\Delta_{2}(x,t)\) in the system. Let the adaptive law of RBF-2\({}^{\prime}\)s weight estimation be  
\[\zeta_{1}=x_{1}-x_{1d} \tag{4}\]6  
Deliver the designed virtual control \(\overline{x_{3}}\) into another first-order filter with a time constant of \(\tau_{3}\) and obtain a new state variable \(x_{3d}\), satisfying  
\[\zeta_{1}=x_{1}-x_{1d} \tag{4}\]7Step 3: Define the last error surface regarding the pitch rate \(q\) as follows:  
\[\zeta_{1}=x_{1}-x_{1d} \tag{4}\]8  
Then, differentiating (20) and combining (2), one has  
\[\zeta_{1}=x_{1}-x_{1d} \tag{4}\]9  
RBF-3 can be used to approximate unknown functions contained in (21), invoking (3)  
\[\dot{\zeta}_{1}=h_{1}(\frac{f_{1}(x_{1})}{h_{1}}+x_{2}+\frac{\Delta_{1}(x,t)- \dot{x}_{1d}}{h_{1}}) \tag{5}\]0  
Define  
\[\dot{\zeta}_{1}=h_{1}(\frac{f_{1}(x_{1})}{h_{1}}+x_{2}+\frac{\Delta_{1}(x,t)- \dot{x}_{1d}}{h_{1}}) \tag{5}\]1  
\[\dot{\zeta}_{1}=h_{1}(\frac{f_{1}(x_{1})}{h_{1}}+x_{2}+\frac{\Delta_{1}(x,t)- \dot{x}_{1d}}{h_{1}}) \tag{5}\]2  
where \(\frac{\rho_{3}^{2}\xi_{3}}{2\varepsilon}\) is a nonlinear damping items designed to overcome the influence of the uncertain interference term \(\Delta_{3}(x,t)\) in this system. Here, the actual control law is developed as follows:  
\[\dot{\zeta}_{1}=h_{1}(\frac{f_{1}(x_{1})}{h_{1}}+x_{2}+\frac{\Delta_{1}(x,t)- \dot{x}_{1d}}{h_{1}}) \tag{5}\]3  
Let the adaptive law of RBF-3's weight estimation be  
\[\dot{\zeta}_{1}=h_{1}(\frac{f_{1}(x_{1})}{h_{1}}+x_{2}+\frac{\Delta_{1}(x,t)- \dot{x}_{1d}}{h_{1}}) \tag{5}\]4  
The adaptive laws for weight estimation (10), (18), (26) and control law (25) can constrain tracking errors and eliminate the influence of interference.