# Target Tracking Based on a Nonsingular Fast Terminal Sliding Mode Guidance Law by Fixed-Wing UAV
## 3 Target Tracking Based on Sliding Model Control
### Sliding Manifold Designing  
For the problem of SMC designing, a switching surface providing the desired convergence characteristic should be defined at first. Then, a controller should be designed to steer the objective states to the chosen manifold, which is not influenced by disturbances or uncertainties. Definition 3: The conventional terminal sliding mode (TSM) can be described by the following first-order nonlinear differential equation [30]:  
\[s=\dot{x}+\alpha_{1}sig^{\beta_{1}}(x)\qquad\alpha_{1},\beta_{1}>0, \tag{11}\]  
where \(x\in R\), \(sig^{\beta}(x)=sign(x)\cdot|x|^{\beta}\) and \(sign(x)\) is the signum function. \(s\) is the sliding manifold. As shown in Figure 2, let the initial state be \(x(0)=5\), and the properties of system (11) can be summed as: (i) when \(\beta_{1}=1\), the system will reduce to \(\dot{x}=-\alpha_{1}x\), and the system is exponentially stable; (ii) when \(0<\beta_{1}<1\), the convergence rate increased gradually with the increase in distance from zero. This result is interesting as it can make the system converge in finite-time; (iii) when \(\beta_{1}>1\), the system will converge to origin with a high speed at a distance from zero, but its convergence rate will slow down when close to the origin. The convergence rate is inversely proportional to the distance from zero. In addition for fixed parameters \(\beta_{1}\), convergence rate has the same monotony as \(\alpha_{1}\). Figure 2: The Parameter analysis of terminal sliding mode (TSM). **Remark 3**. : _From the finite-time differential equation, the dynamics (11) are globally finite-time stable with any given initial condition \(x(0)=x_{0}\), and the convergence time can be obtained as_  
\[T_{TSM}=\int_{0}^{|x_{0}|}\frac{1}{\alpha_{1}x^{\beta_{1}}}dx=\frac{|x_{0}|^{1- \beta_{1}}}{\alpha_{1}(1-\beta_{1})}.\quad 0<\beta_{1}<1.\]  
As mentioned above, the convergence rate has an obvious difference with different \(\alpha_{1}\) and \(\beta_{1}\). It is best to combine the two ways with \(0<\beta_{1}<1\) and \(\beta_{1}>1\). Therefore, the linear combination which is developed as follows not only improves the convergence rate, but is also easy to achieve. **Definition 4**. : _The fast terminal sliding mode (FTSM) is described by the following differential equation:_  
\[s=\dot{x}+\alpha_{2}sig^{\beta_{2}}(x)+\alpha_{3}sig^{\beta_{3}}(x)=0,\qquad \alpha_{2},\alpha_{3}>0,\beta_{2}\geq 1,0<\beta_{3}<1. \tag{12}\]  
From Definition 4, it is evident that FTSM is the combination of different \(\alpha_{2}\), \(\alpha_{3}\), \(\beta_{2}\) and \(\alpha_{4}\) cases. When the system state is far from the equilibrium point, the subitem in Equation (12) \(\alpha_{2}sig^{\beta_{2}}(x)\) dominates \(\alpha_{3}sig^{\beta_{3}}(x)\), which provides a high convergence rate. However, the main term \(\alpha_{3}sig^{\beta_{3}}(x)\) guarantees the system converge in finite-time when the system state is close to the equilibrium. Therefore, the whole FTSM will converge very quickly. Figure 3 shows that the FTSM has a better convergence rate than the TSM (when \(\alpha_{2}=0,\beta_{2}=0\)). The result is verified by the following sliding modes setting the initial state \(x(0)=6\):  
\[s_{1}=\dot{x}+sig^{0.5}x=0,\quad s_{2}=\dot{x}+sig^{3}x+sig^{0.5}x=0,\quad s_{3 }=\dot{x}+x+sig^{0.5}x=0.\]  
**Remark 4**. : _By solving the differential Equation (12), \(x=0\) will be reached in a finite time for any initial condition \(x(0)=x_{0}\)[31]:_  
\[T_{FTSM} =\int_{0}^{|x_{0}|}\frac{1}{\alpha_{2}x^{\beta_{2}}+\alpha_{3}x^{ \beta_{3}}}dx\] \[=\frac{|x_{0}|^{1-\beta_{2}}}{1-\beta_{2}}\cdot\alpha_{2}^{(1- \beta_{2})/\beta_{2}}\cdot F\left(1,\frac{\beta_{2}-1}{\beta_{2}-\beta_{3}}; \frac{2\beta_{2}-\beta_{3}-1}{\beta_{2}-\beta_{3}};-\alpha_{3}\alpha_{2}^{-1} |x_{0}k|^{\beta_{3}-\beta_{2}}\right).\]  
The definition of the Gaussian hypergeometric function \(F(\cdot)\) can be given as follows:  
\[F(a,b;\gamma;z)=\frac{1}{B(b,\gamma-b)}\int_{0}^{1}t^{b-1}(1-t)^{\gamma-b-1}(1 -tz)^{-a}dt,\]  
Figure 3: The comparison of different fast terminal sliding modes (FTSMs).