## 2 Mathematical Preliminaries  
### System Model  
The mathematical model of the ship system is described as:  
\[Th+h+\tau h^{3}=K\delta \tag{1}\]  
which is the Norrbin nonlinear mathematical model [3; 12], where \(T\) and \(K\) denote the time constant and rudder gain, \(h=\dot{\psi}\) is the yaw rate with \(\psi\) the heading angle, \(\tau\) denotes the Norrbin coefficient, and \(\delta\) represents the rudder angle. Considering that wind, waves, and ocean currents have a certain randomness in ship sailing, the stochastic noise was not taken into account in the above model (1), which lacks practical application. Furthermore, the dynamic equation of the nonlinear ship heading control system could be switched control at different levels of speed. In order to study the practical problems better, the aforementioned disturbance is denoted by a stochastic process \(g_{\sigma(t)}^{T}(\psi,h)\), which is unknown. Then, System (1) is rewritten into the switched stochastic form:  
\[T_{\sigma(t)}h+h+\tau_{\sigma(t)}h^{3}=K_{\sigma(t)}\delta_{\sigma(t)}+g_{ \sigma(t)}^{T}(\psi,h)\omega \tag{2}\]  
where \(\omega\) denotes the standard Brownian motion and \(\sigma(t):[0,\infty)\rightarrow\mathfrak{L}=\{1,\cdots,l\}\) describes the switching law. Let \(x_{1}=\psi\), \(x_{2}=h\), and \(u_{\sigma(t)}=\frac{K_{\sigma(t)}}{T_{\sigma(t)}}\delta_{\sigma(t)}\), then we have the controlled switched stochastic Norrbin nonlinear model:  
\[dx_{1}= x_{2}dt,\] \[dx_{2}= (u_{\sigma(t)}+f_{\sigma(t)}(x_{2}))dt+\frac{1}{T_{\sigma(t)}}g_{ \sigma(t)}^{T}(x_{1},x_{2})d\omega,\] \[y= x_{1} \tag{3}\]  
where \(f_{\sigma(t)}=-\frac{1}{T_{\sigma(t)}}x_{2}-\frac{T_{\sigma(t)}}{T_{\sigma(t) }}x_{2}^{3}\) is an unknown smoothing function. **Remark 1**: _Note that the [PERSON] model is often utilized to analyze the performance and related control design of unmanned surface vehicles [3; 12]. By contrast, the stochastic noise and switching theory have not been taken into account, and this will have certain limitations on the research of actual ship control problems. In actual navigation, uncertain stochastic influences and stochastic motion will occur in the ship control system due to the influence of wind and waves. To exhibit the influence of stochastic noise on heading tracking performance, the comparison test results are shown in the following. Considering the complexity and variability of the internal mechanisms on the switched stochastic nonlinear systems, the research of this work is more challenging and has more practical significance._  
Here, \(u_{k}=\Gamma_{k}(v_{k})\) is the nonsymmetric and switched dead-zone nonlinearity in the controlled system [35]:  
\[u_{k}=\Gamma_{k}(v_{k})=\ \left\{ \begin{array}{ll}m_{r,k}(v_{k}-b_{r,k}),&v_{k} \geq b_{r,k}\\ 0,&-b_{l,k}<v_{k}<b_{r,k}\\ m_{l,k}(v_{k}+b_{l,k}),&v_{k}\leq-b_{l,k}\end{array} \right. \tag{4}\]  
in which \(m_{r,k},m_{l,k}\) and \(b_{r,k},b_{l,k}\) are the nonsymmetric parameters of the dead-zone and are positive and unknown. Similar to the idea in [35], the following expression is formulated to carry out the tracking control smoothly:  
\[u_{k}=\Gamma_{k}^{\prime}(v_{k})+\varpi_{k} \tag{5}\]in which \(\varpi_{k}=\Gamma_{k}(v_{k})-\Gamma_{k}^{\prime}(v_{k})\) is the error and \(\Gamma_{k}^{\prime}(v_{k})\) is an unknown smoothing function, and there exists a constant \(\bar{\varpi}_{k}\) satisfying \(|\varpi_{k}|\leq\bar{\varpi}_{k}\). In order to facilitate the problem, an unknown function \(\sigma^{\prime}(v_{k})=\Gamma_{k}^{\prime}(v_{k})-v_{k}\) is defined, then one obtains:  
\[u_{k} =\Gamma_{k}^{\prime}(v_{k})-v_{k}+v_{k}+\varpi_{k}\] \[=v_{k}+\sigma^{\prime}(v_{k})+\varpi_{k}. \tag{6}\]  
For the actual controller, we let:  
\[v_{k}=v_{ck}-v_{\phi k}; \tag{7}\]  
therefore, it yields:  
\[u_{k}=v_{ck}+\sigma_{k}^{\prime}(v_{k})-v_{\phi k}+\varpi_{k} \tag{8}\]  
It should be noted that \(v_{\phi k}\) and \(v_{ck}\) denote the introduced dead-zone input's compensator and the corresponding control law. **Remark 2**.