## 3 Quadrotor Control Design Aided with Bioinspired Neural Dynamics
### Stability Analysis  
This paper presents a novel approach to achieve progressive trajectory tracking control by integrating a bioinspired neural network control strategy with the aerodynamics and external disturbances of a quadrotor unmanned aerial vehicle. The asymptotic stability of both control systems is verified through rigorous Lyapunov stability analysis. Additionally, by constructing a global Lyapunov candidate function, the study establishes the global asymptotic stability of the proposed control system. To demonstrate the stability of the bioinspired backstepping controller, the Lyapunov function is defined as follows:  
\[V_{P}=\frac{1}{2}e_{P}^{2}+\frac{1}{2}e_{V}^{2}+\frac{k_{P}}{2B_{P}}E_{P}^{2} +\frac{k_{V}}{2B_{V}}E_{V}^{2} \tag{22}\]Then, the derivative of \(V_{P}\) is denoted as:  
\[\dot{V}_{P}=e_{P}\dot{e}_{P}+e_{V}\dot{e}_{V}+\frac{k_{P}}{B_{P}}E_{P}\dot{E}_{P}+ \frac{k_{V}}{B_{V}}E_{V}\dot{E}_{V} \tag{23}\]  
Splitting the time derivative expression (23) of the candidate function into two components, we have \(e_{P}\dot{e}_{P}+e_{V}\dot{e}_{V}\) and \(\frac{k_{P}}{B_{P}}E_{P}\dot{E}_{P}+\frac{k_{V}}{B_{V}}E_{V}\dot{E}_{V}\). Based on the tracking errors and Equation (10), \(e_{P}\dot{e}_{P}+e_{V}\dot{e}_{V}\) can be written as:  
\[e_{P}\dot{e}_{P}+e_{V}\dot{e}_{V}=e_{P}\bigg{(}\dot{p}_{d}^{E}-\dot{P}^{E} \bigg{)}+e_{V}\bigg{(}\dot{V}_{d}^{E}-\dot{V}^{E}\bigg{)}=-k_{P}E_{P}e_{P}-k_{ V}E_{V}e_{V} \tag{24}\]  
For the second part, \(\frac{k_{P}}{B_{P}}E_{P}\dot{E}_{P}+\frac{k_{V}}{B_{V}}E_{V}\dot{E}_{V}\), of the Lyapunov function, considering the specific characteristics of the quadrotor unmanned aerial vehicle model, let us assume \(B_{P}=-D_{P}\) and \(B_{V}=-D_{V}\). Based on the biological neural dynamics model Equation (8), the second part can be written as:  
\[ \begin{array}{c}\frac{k_{P}}{B_{P}}E_{P}\dot{E}_{P}+\frac{k_{V}}{B_{V}}E_{V} \dot{E}_{V}=\frac{k_{P}}{B_{P}}E_{P}(-A_{P}E_{P}+(B_{P}-E_{P})f(e_{P})-(E_{P}- D_{P})g(e_{P}))\\ +\frac{k_{V}}{B_{V}}E_{V}(-A_{V}E_{V}+(B_{V}-E_{V})f(e_{V})-(E_{V}-D_{V})g(e_{ V}))\\ =\frac{k_{P}}{B_{P}}E_{P}^{2}(-A_{P}-f(e_{P})-g(e_{P}))+k_{P}E_{P}(f(e_{P})-g(e_{P}))\\ +\frac{k_{V}}{B_{V}}E_{V}^{2}(-A_{V}-f(e_{V})-g(e_{V}))+k_{V}E_{V}(f(e_{V})-g(e _{V}))\end{array}  \tag{25}\]  
Combining Equations (24) and (25), we can infer that:  
\[ \begin{array}{c}\dot{V}_{P}=\frac{k_{P}}{B_{P}}E_{P}(-A_{P}E_{P}+(B_{P}-E_{ P})f(e_{P})-(E_{P}-D_{P})g(e_{P}))\\ +\frac{k_{V}}{B_{V}}E_{V}(-A_{V}E_{V}+(B_{V}-E_{V})f(e_{V})-(E_{V}-D_{V})g(e_{ V}))\\ =\frac{k_{P}}{B_{P}}E_{P}^{2}(-A_{P}-f(e_{P})-g(e_{P}))+k_{P}E_{P}(f(e_{P})-g(e _{P})-e_{P})\\ +\frac{k_{V}}{B_{V}}E_{V}^{2}(-A_{V}-f(e_{V})-g(e_{V}))+k_{V}E_{V}(f(e_{V})-g( e_{V})-e_{V})\end{array}  \tag{26}\]  
According to Equation (8), we can conclude that:  
\[ \begin{cases}f(e_{i})=e_{i}&g(e_{i})=0&,\;\;e_{i}\geq 0\\ f(e_{i})=0&g(e_{i})=-e_{i}&,\;\;e_{i}<0\end{cases}  \tag{27}\]  
By using Formula (27), we obtain:  
\[-A_{i}-f(e_{i})-g(e_{i})<0\]  
\[f(e_{i})-g(e_{i})=e_{i} \tag{28}\]  
From Equations (22), (26), and (28), it can be inferred that as \(t\rightarrow\infty\), \(V_{P}\to 0\) and \(\dot{V}_{P}\) is negative definite. \(V_{P}\) can only be zero when \(e_{i}=0^{3\times 1}\). Therefore, the bioinspired backstepping position control system designed in this paper is asymptotically stable. To prove the stability of the bioinspired sliding mode controller, the designed Lyapunov function is defined as follows:  
\[V_{\Theta}=-\frac{1}{2}S^{T}S+\frac{1}{2\,B_{S}}E_{S}^{T}E_{S} \tag{29}\]  
Differentiating the Lyapunov equation, we obtain:  
\[\dot{V}_{\Theta}=-S^{T}\dot{S}+\frac{1}{B_{S}}\dot{E}_{S}^{T}E_{S} \tag{30}\]Combining the biological neural dynamics model (8) with Equation (30), we have:  
\[