# Target Tracking Based on a Nonsingular Fast Terminal Sliding Mode Guidance Law by Fixed-Wing UAV
## 3 Target Tracking Based on Sliding Model Control
### Stability Analysis  
**Lemma 1**. : _Suppose the continuous positive definite function \(V(t)\) satisfies the following inequality:_  
\[\dot{V}(t)+aV(t)+bV^{c}(t)\leq 0,\quad\forall t>t_{0},a>0,b>0,0<c<1, \tag{21}\]  
_where \(t_{0}\) is the initial time (see [30]). Then, the system converges to the equilibrium point in finite time \(t_{f}\), and the setting time can be given by_  
\[t_{f}\leq t_{0}+\frac{1}{a(1-c)}\ln\frac{aV^{1-c}(t_{0})+b}{b}. \tag{22}\]  
**Theorem 1**. : _For the loitering algorithm described by Definition 2, if the NFTSM manifold is provided by Equation (15), then the distance error \(r\) in Equation (5) converges to zero under the guidance law defined in Equation (19) in finite-time time._  
Proof. : From [33], considering a Lyapunov function candidate \(V_{1}=s^{2}/2\). Applying Equations (17)-(19), the derivative of \(V_{1}\) with respect to time yields  
\[\dot{V}_{1}=s\dot{s}=s(-k_{2}s-k_{3}sig^{\gamma_{1}}(s))=-k_{2}s^{2}-k_{3}|s|^ {\gamma_{1}+1}\leq 0. \tag{23}\]  
From \(\dot{V}_{1}\leq 0\), \(V_{1}(t)\) is bounded. In addition, Equation (23) can be written as  
\[\dot{V}_{1}=-2k_{2}V_{1}-k_{3}(\sqrt{2})^{\gamma_{1}+1}V_{1}^{\frac{\gamma_{1 }+1}{2}}. \tag{24}\]  
Equation (24) has a similar form to Equation (21). Consequently, the sliding manifold can be reached \(s=0\) in finite time and the convergent time is given by  
\[T_{NFSTM}\leq\frac{1}{k_{2}(1-\gamma_{1})}\ln\frac{2k_{2}V_{1}^{\frac{1-\gamma _{1}}{2}}(s(0))+k_{3}(\sqrt{2})^{\gamma_{1}+1}}{k_{3}(\sqrt{2})^{\gamma_{1}+ 1}}. \tag{25}\]  
Case 1: when \(|\ell|>\eta\) with \(\ell=\arctan{(k_{1}r)}\), letting the sliding manifold \(s=0\), the equality can be obtained from Equation (17)  
\[\dot{\psi}_{a}=-\alpha_{2}sig^{\beta_{2}}(\ell)-\alpha_{3}sig^{\beta_{3}}(\ell). \tag{26}\]  
Due to \(\dot{\psi}_{a}\in(-\pi,\pi]\), we can then get the inequality as follows:  
\[|\dot{\psi}_{a}|\leq\alpha_{2}|\ell|^{\beta_{2}}+\alpha_{3}|\ell|^{\beta_{3}} \leq\pi. \tag{27}\]According to \(\ell\in(-\pi/2,\pi/2]\), it is reasonable to set the following inequality:  
\[\alpha_{2}|\ell|^{\beta_{2}}+\alpha_{3}|\ell|^{\beta_{3}}\leq\alpha_{2}\left( \frac{\pi}{2}\right)^{\beta_{2}}+\alpha_{3}\left(\frac{\pi}{2}\right)^{\beta_{3 }}\leq\pi. \tag{28}\]  
From Equation (7) and Remark 1, we obtain  
\[\bar{\psi}_{a}^{d}\leq\arcsin\xi\quad\text{with}\quad\bar{\varsigma}=\frac{v_{ t}+v_{w}}{v_{a}}>0. \tag{29}\]  
Let \(\bar{\psi}_{a}\in(-\pi/2,\pi/2]\), the range in Equation (28) can be further reduced by using the Equation (26) as follows:  
\[|\bar{\psi}_{a}|\leq\ |\bar{\psi}_{a}^{d}|+\alpha_{2}|\ell|^{\beta_{2}}+\alpha_{3 }|\ell|^{\beta_{3}}\leq\arcsin\xi+\alpha_{2}\left(\frac{\pi}{2}\right)^{\beta_ {2}}+\alpha_{3}\left(\frac{\pi}{2}\right)^{\beta_{3}}<\frac{\pi}{2}. \tag{30}\]  
Let \(V_{2}=r^{2}/2\) be a Lyapunov candidate. Applying Equations (8) and (26), on the manifold \(s=0\), the time derivative of \(V_{2}\) can be written as  
\[ \begin{split}\dot{V}_{2}&=rt=2v_{a}\sin\left(\frac {\bar{\psi}_{a}}{2}\right)\cos\left(\frac{\bar{\psi}_{a}+\bar{\psi}_{a}^{d}}{2 }\right)r\\ &=-2v_{a}\sin\left(\frac{\alpha_{2}sig^{\beta_{2}}(\ell)+\alpha_ {3}sig^{\beta_{3}}(\ell)}{2}\right)\cos\left(\frac{\bar{\psi}_{a}+\bar{\psi}_{ a}^{d}}{2}\right)r\\ &=-2 sign(\ell)v_{a}\sin\left(\frac{\alpha_{2}|\ell|^{\beta_{2}}+ \alpha_{3}|\ell|^{\beta_{3}}}{2}\right)\cos\left(\frac{\bar{\psi}_{a}+\bar{ \psi}_{a}^{d}}{2}\right)r\\ &=-2v_{a}\sin\left(\frac{\alpha_{2}|\ell|^{\beta_{2}}+\alpha_{3} |\ell|^{\beta_{3}}}{2}\right)\cos\left(\frac{\bar{\psi}_{a}+\bar{\psi}_{a}^{d} }{2}\right)|r|\\ &\leq 0.\end{split}  \tag{31}\]  
Consequently, the distance error \(r\) will make \(\ell\) converge to the region \(|\ell|\leq\eta\) asymptotically in finite-time. By using Equation (26), we can obtain  
\[|\bar{\psi}_{a}|\leq\alpha_{2}\eta^{\beta_{2}}+\alpha_{3}\eta^{\beta_{3}}. \tag{32}\]  
Case 2: if \(|\ell|\leq\eta\), it shows that \(\ell\) will converge to the region \(|\ell|\leq\eta\) in finite-time. By using Equation (17), we obtain  
\[