# Curved Path Following Control for a Small Fixed-Wing UAV with Parameters Adaptation
## 3 Controller Design
### Kinematic Controller Design with Time-Variable Control Parameters
Thus, \([0,\ 0]^{T}\) is the only equilibrium point for the nonlinear system \(h(t,\mathbf{x})\). By choosing the candidate Lyapunov function \(U_{1}(t,\mathbf{x})\) as  
\[U_{1}(t,\mathbf{x})=k_{1}(t)\int_{0}^{e_{d}}f_{sat}(y)dy+(1-cos(e_{\chi})) \tag{20}\]  
The derivative of \(U_{1}(t,\mathbf{x})\) along the trajectories of (19) is given by  
\[ \begin{array}{ll}\dot{U}_{1}(t,\mathbf{x})&=\frac{\partial U_{1}}{\partial t}+ \frac{\partial U_{1}}{\partial\mathbf{x}}h(t,\mathbf{x})\\ &=\frac{1}{\tau}(k_{1,\text{max}}-k_{1,\text{max}})e^{-(\frac{t}{\tau})}\int_ {0}^{e_{d}}f_{sat}(y)dy\\ &+k_{1}(t)\dot{e}_{d}f_{sat}(e_{d})+\dot{e}_{\chi}\sin\left(e_{\chi}\right)\\ &=\frac{1}{\tau}(k_{1,\text{min}}-k_{1,\text{max}})e^{-(\frac{t}{\tau})}\int_ {0}^{e_{d}}f_{sat}(y)dy\\ &+k_{1}(t)V\|\
abla f\|f_{sat}(e_{d})\sin(e_{\chi})+\\ &\left(-k_{1}(t)\|\
abla f\|Vf_{sat}(e_{d})-k_{2}(t)V^{2}\|\
abla f\|\sin(e_{ \chi})\right)\sin(e_{\chi})\\ &=-\frac{1}{\tau}(k_{1,\text{max}}-k_{1,\text{min}})e^{-(\frac{t}{\tau})}\int _{0}^{e_{d}}f_{sat}(y)dy-k_{2}(t)V^{2}\|\
abla f\|\sin^{2}(e_{\chi})\\ &\leq-\frac{1}{\tau}(k_{1,\text{max}}-k_{1,\text{min}})e^{-(\frac{t}{\tau})} \int_{0}^{e_{d}}f_{sat}(y)dy-k_{2,\text{min}}V_{p2}^{2}\lambda\sin^{2}(e_{ \chi})\\ &<-k_{2,\text{min}}V_{p2}^{2}\lambda\sin^{2}(e_{\chi})=-W_{1}(\mathbf{x})\end{array}  \tag{21}\]  
Furthermore, with the results shown as Equations (16) and (17), it can be derived that  
\[ \begin{array}{ll}U_{1}(t,\mathbf{x})&\geq k_{1,min}\int_{0}^{e_{d}}f_{sat}(y)dy +(1-cos(e_{\chi}))\\ &=k_{1,min}\int_{0}^{e_{d}}f_{sat}(y)dy+2\text{sin}^{2}\Big{(}\frac{e_{\chi}}{ 2}\Big{)}=W_{2}(\mathbf{x})\end{array}  \tag{22}\]  
\[ \begin{array}{ll}U_{1}(t,\mathbf{x})&\leq k_{1,max}\int_{0}^{e_{d}}f_{sat}(y)dy +(1-cos(e_{\chi}))\\ &\leq\frac{k_{1,max}}{2}e_{d}^{2}+2 sin^{2}\Big{(}\frac{e_{\chi}}{2}\Big{)} \leq\frac{k_{1,max}}{2}e_{d}^{2}+\frac{e_{\chi}^{2}}{2}=W_{3}(\mathbf{x})\end{array}  \tag{23}\]  
It can be found that \(W_{1}(\mathbf{x})\) is a continuous positive semidefinite function on \(D_{1}\), \(W_{2}(\mathbf{x})\) and \(W_{3}(\mathbf{x})\) are both continuous positive definite on \(D_{1}\). It can be found from (22) and (23) that \(U_{1}(t,\mathbf{x})\) is lower bounded. Since \(V\), \(\|\
abla f\|\), and their derivatives are all bounded, by (21) \(\dot{U}_{1}(t,\mathbf{x})\) is negative semidefinite and uniformly continuous in time. By [PERSON]'s lemma (p. 323, [31]), \(\dot{U}_{1}(t,\mathbf{x})\to 0\) as \(t\to\infty\). Moreover, it has \(sin(e_{\chi})\to 0\), and \(\dot{e}_{d}\to 0\) as in (19). Since \(e_{d}\) is bounded, it follows that \(e_{d}\) tends to a finite limit \(\overline{e}_{d}\) as \(t\to\infty\). Since \(e_{\chi}\to 0\) as \(t\to\infty\), and \(\dot{e}_{\chi}\) in (19) is uniformly continuous, it derives that \(\dot{e}_{\chi}\to 0\) as \(t\to\infty\). Hence, it follows that \(\lim\limits_{t\to\infty}f_{sat}(e_{d})\to 0\), and \(\overline{e}_{d}=0\). Therefore, the kinematic control law (14) can asymptotically drive \(e_{d}\) and \(e_{\chi}\) toward zero. \(\square\)