## 3 Control Design
### Adaptive Sliding Mode Trajectory Tracking Control Design
#### 3.2.2 Surge Motion Control Law  
In this section, the main purpose is that the surge control law is designed to stabilize \(u_{e}\). Meanwhile, the neural network MLP, the adaptive technology and the auxiliary dynamic system are used to handle unmodeled dynamics, unknown disturbance bounds and input saturation, respectively. Define the velocity error sliding surface \(s_{1}\) as  
\[s_{1}=u_{e}+\gamma_{1}\int_{0}^{t}u_{e}\left(i\right)du \tag{12}\]  
where \(\gamma_{1}\) is a positive constant. To avoid the differential expansion, a neural shunting model is introduced to realize the differential of \(\alpha_{u}\), which is expressed as  
\[\dot{\beta}_{u}=-A_{u}\beta_{u}+\left(B_{u}-\beta_{u}\right)f\left(\alpha_{u} \right)-\left(D_{u}+\beta_{u}\right)g\left(\alpha_{u}\right) \tag{13}\]  
where \(\beta_{u}\) denotes the output of the neural shunting model. **Remark 2**. : _Neural shunting model [40] originally belonged to the biological category, which has been applied to the field of robot control [41; 42], and it can be expressed in the following form_  
\[\dot{\beta}_{j}=-A_{j}\beta_{j}+\left(B_{j}-\beta_{j}\right)f\left(\alpha_{j }\right)-\left(D_{j}+\beta_{j}\right)g\left(\alpha_{j}\right) \tag{14}\]  
_where \(A_{j}\), \(B_{j}\) and \(D_{j}\)\(\left(j=u,r\right)\) denote three design parameters, \(\alpha_{j}\) and \(\beta_{j}\) represent the input and output of the system, respectively. In addition, \(f\left(\alpha_{j}\right)\) and \(g\left(\alpha_{j}\right)\) represent the threshold functions of model, in which \(f\left(\alpha_{j}\right)=\left\{ \begin{array}{l}\alpha_{j},\ \alpha_{j}\geq 0\\ 0,\ \ \alpha_{j}<0\end{array} \right.,g\left(\alpha_{j}\right)=\left\{ \begin{array} []{l}0,\ \ \ \alpha_{j}>0\\ -\alpha_{j},\ \alpha_{j}\leq 0\end{array} \right..\)_The derivative of \(s_{1}\) is written as  
\[\dot{s}_{1}=\frac{1}{m_{11}}\left(m_{11}f_{u}+\Delta f_{u}+\tau_{u}+d_{u}\right)- \dot{\beta}_{u}+\gamma_{1}\left(u-\alpha_{u}\right) \tag{15}\]  
To reduce the chattering problem and the influence of disturbances, a sliding mode control law with hyperbolic tangent function is adopted to make the sliding surface converge to zero, which can be written as  
\[\dot{s}_{1}=-\hat{d}_{u}^{*}\tanh\left(s_{1}/\varepsilon_{1}\right)-\rho_{1}s _{1} \tag{16}\]  
where \(\rho_{1}\) and \(\varepsilon_{1}\) represent two design parameters, \(\hat{d}_{u}^{*}\) is the estimate of \(d_{u}^{*}\) bound. In addition, the corresponding estimated error is \(\hat{d}_{u}^{*}=d_{u}^{*}-\hat{d}_{u}^{*}\), and the adaptive law with \"\(\kappa\)-correction\" is proposed as  
\[\hat{d}_{u}^{*}=\Gamma_{1}\left(s_{1}\tanh\left(s_{1}/\varepsilon_{1}\right)- \kappa_{1}\left(\hat{d}_{u}^{*}-d_{u0}^{*}\right)\right) \tag{17}\]  
where \(\Gamma_{1}\) and \(\kappa_{1}\) are positive design parameters, \(d_{u0}^{*}\) is the initial value. Then the surge motion control law is designed as  
\[\tau_{cu}=-m_{11}f_{u}-\Delta\hat{f}_{u}+m_{11}\dot{\beta}_{u}-\gamma_{1}m_{11 }u_{e}-\hat{d}_{u}^{*}\tanh\left(s_{1}/\varepsilon_{1}\right)-\rho_{1}s_{1} \tag{18}\]  
where \(\Delta\hat{f}_{u}=\frac{1}{2}s_{1}\hat{\theta}_{1}\Theta^{T}\Theta\) is the estimate of \(f_{u}\), the corresponding adaptive law is  
\[\dot{\hat{\theta}}_{1}=\varsigma_{1}\left(\frac{1}{2}s_{1}^{2}\Theta^{T} \Theta-\phi_{1}\hat{\vartheta}_{1}\right) \tag{19}\]  
where \(\varsigma_{1}\) and \(\phi_{1}\) are positive constants. In addition, the estimated error is \(\hat{\vartheta}_{1}=\hat{\vartheta}_{1}-\hat{\vartheta}_{1}\). In order to deal with the problem of input saturation (2), an auxiliary dynamic system [43; 44] is constructed as  
\[\zeta_{u}=\left\{ \begin{array}{ll}-K_{\psi u}\zeta_{u}-\frac{|s_{1}\Delta \tau_{u}|+0.5\Delta\tau_{u}^{2}}{\left\|\zeta_{u}^{2}\right\|}\cdot\zeta_{u}+ \Delta\tau_{u},&\left\|\zeta_{u}\right\|\geq\sigma_{u}\\ 0,&\left\|\zeta_{u}\right\|<\sigma_{u}\end{array} \right. \tag{20}\]  
where \(\zeta_{u}\) is a state variable, \(\Delta\tau_{u}=\tau_{u}-\tau_{cu}\), \(K_{\psi u}\) is a positive constant, and \(\sigma_{u}>0\) is a small design constant.