## 3 Controller Design
### Stability Analysis and Finite Time Convergence  
This section analyzes the stability of the proposed controller. The proposed controller uses the adaptive robust term and the reverse saturation function in a steady state. When the adaptive robust term is larger than the reverse saturation function threshold value, stability is guaranteed according to the Equation (27), and used in the adaptive robust term design. When the adaptive robust is below the threshold value, it does not generate control force at a steady state. Therefore, stability can be guaranteed according to Equation (19) used in the previous RBF neural network controller. In this section, finite time convergence is investigated to strictly prove the stability of the system including the reverse saturation function. First, we consider when the adaptive robust term is larger than the threshold value. To investigate stability, and finite time convergence, we ponder the Lyapunov function candidate as in Equation (35). \[V_{4}=s^{2}/2+\widetilde{W}^{T}\widetilde{W}/2\gamma+\tilde{\varepsilon}^{T} \tilde{\varepsilon}/2\beta \tag{35}\]  
Its time derivative is  
\[\dot{V}_{4}=\dot{s}s+\widetilde{W}^{T}\dot{\widetilde{W}}/\gamma+\tilde{ \varepsilon}^{T}\dot{\tilde{\varepsilon}}/\beta \tag{36}\]  
By substituting the proposed control input and using \(\widetilde{W}=\hat{W}-W^{*}\), and \(\tilde{\varepsilon}=\hat{\varepsilon}-\varepsilon^{*}\), we can obtain Equation (37)  
\[\dot{V}_{4}=-\lambda\dot{e}^{\lambda-1}\cdot k\cdot s\cdot\text{sgn}(s)- \sigma(\widetilde{W}^{T}\widetilde{W}+\hat{W}^{T}\hat{W}-{W^{*}}^{T}W^{*})/2- \tilde{\xi}(\tilde{\varepsilon}^{T}\tilde{\varepsilon}+\tilde{\varepsilon}^{ T}\tilde{\varepsilon}-\varepsilon^{*T}\varepsilon^{*})/2 \tag{37}\]  
\[\dot{V}_{4}\leq-\lambda\dot{e}^{\lambda-1}\cdot k\cdot s^{2}/|s|-\sigma( \widetilde{W}^{T}\widetilde{W})/2-\xi(\tilde{\varepsilon}^{T}\tilde{ \varepsilon})/2+\sigma({W^{*}}^{T}{W^{*}})/2+\tilde{\xi}(\varepsilon^{*T} \varepsilon^{*})/2 \tag{38}\]  
By replacing \(\kappa=\lambda\dot{e}^{\lambda-1}\cdot k/|s|\), \(\kappa>0\) and \(\rho_{1}=\sigma({W^{*}}^{T}{W^{*}})/2+\tilde{\xi}(\varepsilon^{*T}\varepsilon^ {*})/2\), we can obtain Equation (39)  
\[\dot{V}_{4}\leq-\kappa s^{2}-\sigma(\widetilde{W}^{T}\widetilde{W})/2-\xi( \tilde{\varepsilon}^{T}\tilde{\varepsilon})/2+\rho_{1} \tag{39}\]  
Figure 4: Proposed controller block diagram. \(\sigma\geq\kappa/\gamma\), and \(\beta\geq\kappa/\xi\) can obtain Equations (40) and (41)  
\[\dot{V}_{4}\leq-\kappa s^{2}-\kappa\cdot\widetilde{W}^{T}\widetilde{W}/\gamma- \kappa\cdot\widetilde{z}^{T}\widetilde{\varepsilon}/\beta+\rho_{1} \tag{40}\]  
\[\dot{V}_{4}\leq-2\kappa(s^{2}/2+\widetilde{W}^{T}\widetilde{W}/2\gamma+ \widetilde{z}^{T}\widetilde{\varepsilon}/2\beta)+\rho_{1} \tag{41}\]  
By substituting Equation (35), \(\psi\geq 2\kappa\) we can obtain Equation (42)  
\[\dot{V}_{4}\leq-\psi V_{4}+\rho_{1} \tag{42}\]  
From Equation (42) we can obtain Equation (43)  
\[V_{4}(t)\leq(V_{4}(t_{0})-\rho_{1}/\psi)e^{-\psi(t-t_{0})}+\rho_{1}/\psi,\, \forall t\geq t_{0} \tag{43}\]  
\(V_{4}(t_{0})\) is the Lyapunov function candidate's initial value. \(V_{4}\) is bounded as \(\rho_{1}/\psi\), and all system variables related to \(V_{4}\). So, the closed loop system is bounded; therefore, the system is stable. Next, to define the finite time, we use Equation (42), and we can obtain Equations (44)-(46)  
\[\dot{V}_{4}=\dot{s}s+\widetilde{W}^{T}\dot{\widetilde{W}}/\gamma+\tilde{ \varepsilon}^{T}\dot{\tilde{\varepsilon}}/\beta \tag{36}\]0  
\[\dot{V}_{4}=\dot{s}s+\widetilde{W}^{T}\dot{\widetilde{W}}/\gamma+\tilde{ \varepsilon}^{T}\dot{\tilde{\varepsilon}}/\beta \tag{36}\]1  
\[\dot{V}_{4}=\dot{s}s+\widetilde{W}^{T}\dot{\widetilde{W}}/\gamma+\tilde{ \varepsilon}^{T}\dot{\tilde{\varepsilon}}/\beta \tag{36}\]2  
where \(t_{N}\) is the finite time to reach the threshold value.