# Output Feedback Tracking Control for Vessel with Collision-Avoidance and Performance Constraints
## 4 Collision-Avoidance Switch Control Strategy
### Prescribed Performance Controller Design  
Step 1: A prescribed error function is defined as  
\[z_{1}=\frac{1}{2}\text{In}\frac{k_{b}^{2}}{k_{b}^{2}-\eta_{e}^{T}\eta_{e}}, \tag{34}\]  
where \(k_{b}\) is a constraint function that is employed to achieve the required prescribed performance, and it is given as follows:  
\[k_{b}(t)= \delta_{l}[(\rho_{0}-\rho_{\infty})e^{-kt}+\rho_{\infty}] \tag{35}\]  
where \(\delta_{l},\rho_{0},\rho_{\infty}\), and \(k\) are positive constants and \(\rho_{0}\) is satisfied such that \(\rho_{0}\geq\eta_{e}(0)^{T}\eta_{e}(0)\). \(\rho_{\infty}\) defines the steady-state value of the performance function. \(k\) is the bound of the error-convergence rate. \(\eta_{e}=\eta-\eta_{d}\). The time derivative of \(z_{1}\) is  
\[\dot{z}_{1}= \frac{\eta_{e}^{T}\eta_{e}}{k_{b}^{2}-\eta_{e}^{T}\eta_{e}}+\frac{ -k_{b}\dot{k}_{b}}{k_{b}^{2}-\eta_{e}^{T}\eta_{e}}+\frac{\dot{k}_{b}}{k_{b}}. \tag{36}\]  
To stabilize the error \(z_{1}\), the desired virtual control \(\alpha\) is given as:  
\[\alpha=R^{T}[\dot{\eta}_{d}-K_{1}\eta_{e}-\frac{k_{b}^{2}-\eta_{e}^{T}\eta_{e} }{1+\epsilon*\text{sign}(\eta_{e})}\Delta], \tag{37}\]  
where \(K_{1}\in\mathbb{R}^{3\times 3}\) is a positive definite matrix, \(\epsilon\
eq 1\) is a positive constant, and  
\[\Delta=\frac{k_{ib}^{T}k_{ib}}{k_{ib}^{T}k_{ib}-z_{1}^{T}z_{1}}-\frac{k_{b}}{ k_{b}}. \tag{38}\]  
Step 2: In this step, a kinetic controller is developed. The velocity error vector is defined as  
\[z_{2}=\hat{v}-\bar{\alpha}-\beta, \tag{39}\]  
where \(\beta\) is the parameter of the ADS. \(\bar{\alpha}\) is employed to avoid the computation of \(\dot{\alpha}\), a first-order filter \(\hat{l}\hat{\alpha}+\bar{\alpha}=\alpha\) is introduced instead of the time derivative of \(\alpha\), and \(l\) is a positive time constant. The update law of \(\beta\) is chosen as  
\[\dot{\beta}=-K_{r}\beta-K_{m}\text{sign}(\beta)-\frac{\frac{1}{2}\Delta_{ \tau}^{T}\Delta_{\tau}}{\|\beta\|^{2}}\beta-M^{-1}\Delta_{\tau}, \tag{40}\]  
where \(K_{r}\) and \(K_{m}\in\mathbb{R}^{3\times 3}\) are positive definite matrices. The time derivative of \(z_{2}\) is given as  
\[\dot{z}_{2}=-K_{o2}R^{\text{T}}(\psi)(\hat{\eta}-\eta)+\tilde{\zeta}+M^{-1} \tau-\hat{\alpha}-\dot{\beta}. \tag{41}\]  
Then, the control input is given as  
\[\tau_{ppc}=M(-K_{2}z_{2}-\frac{R(\psi)^{T}\eta_{e}}{k_{b}^{2}-\eta_{e}^{T}\eta _{e}}-\xi+\hat{\alpha}-K_{r}\beta-K_{m}\text{sign}(\beta)-\frac{\Delta_{\tau}^ {T}\Delta_{\tau}}{2\|\beta\|^{2}}\beta) \tag{42}\]  
**Lemma 2**. : _If the system contains the vessel dynamics (1) and (2), the control law (42), and the observer (6)-(8) with input saturation and the unknown term \(\zeta\), and if Assumptions 1 and 2 are satisfied, the proposed CASC guarantees that all signals in the closed-loop system are bounded._Proof. : Choose a Lyapunov function candidate as follows:  
\[V_{1}=z_{1}. \tag{43}\]  
The derivative of \(V_{1}\) is given as  
\[k_{b}(t)= \delta_{l}[(\rho_{0}-\rho_{\infty})e^{-kt}+\rho_{\infty}] \tag{35}\]0  
where \(\dot{\eta}_{e}=R(\psi)\
u-\dot{\eta}_{d}\).