## 4 Simulation Study and Discussion
In order to further exhibit the developed work, we performed some comparison tests to justify its implementation under the same conditions:  
(i) The influence of stochastic noise on heading control:  
This work discusses the ship heading tracking with stochastic noise, which is more in line with the actual navigation environment. In order to illustrate the influence of stochastic noise more clearly and intuitively, the corresponding test results without/with different levels of stochastic noises are presented. In Figure 7, the heading tracking control results of the ship control system for three cases are described under the developed control algorithm: the control system (3) with small stochastic noise, the control system (3) with larger stochastic noise, and the control system (3) without stochastic noise, respectively. By comparison, it can be seen that the stochastic noise has a certain effect on the heading tracking control in the actual navigation environment. Therefore, it is necessary to consider the stochastic noise in the ship control when establishing the ship heading system mathematical model and then carry out the heading tracking control;  
(ii) The comparison of the TD method with other methods:  
To better illustrate the impact of the TD method on the control performance, we performed the following experimental tests, respectively. Firstly, the method without the TD feature was applied to control the ship heading system (3), and the corresponding control result of the heading tracking with controllers is shown in Figure 8, which indicates that the controllers developed without the TD feature can still provide effective heading tracking control performance. However, it should be noted that there is the undesired problem of the \"explosion of terms\" occurring in the control design. The structure of such a heading control scheme is thus complicated, and the term of \(\sum_{j=0}^{1}\frac{\partial\mathbf{a}_{k1}}{y_{d}^{(j)}}y_{d}^{(j+1)}\) is involved, while the knowledge of \(y_{d}(t)\) and its first two-order time derivatives \(\dot{y}_{d}(t),\dot{y}_{d}(t)\) is needed; therefore, more computational work is required, which is not conducive to practical engineering applications. Secondly, the commonly used method of dealing with the \"explosion of terms\", the DSC method, was also employed to carry out the heading tracking control for System (3), and the corresponding tracking control result is described in Figure 4:3 In this way, the problem of the \"explosion of terms\" was avoided successfully; however, in the commonly used DSC method, the boundary \(s_{2d}=\alpha_{2d}-\alpha_{1}\) results from the filter are not considered and compensated, which will have a certain influence on the control performance. Our TD-based control method considers the boundary \(s_{2d}\) and introduces the compensating signals (28) and (38) to compensate this. From the above results, we can see that, under the same conditions, both of these methods can develop the heading tracking control of System (3) and achieve the control  
Figure 4:4 Control effect of System (3) without stochastic noise under TD3. goal successfully, but the TD method can give a better control performance and is more conducive to practical engineering applications;  
(iii) The effect of the TD feature on control performance:  
To illustrate the effect of the TD feature on the control performance, three TDs were chosen in the following experimental tests, defining: TD1: \(R_{2}=10,a_{21}=1.5,a_{22}=5\), \(l_{21}=l_{22}=1\); TD2: \(R_{2}=100,a_{21}=5,a_{22}=10,l_{21}=l_{22}=0.5\); TD3: \(R_{2}=1\), \(a_{21}=a_{22}=0.5,l_{21}=l_{22}=5\), respectively; the corresponding comparison results are exhibited in Figures 10-15. Firstly, Figure 10 clearly shows the heading tracking performance for System (3) with different TDs. This is followed by the corresponding controller performances obtained in Figures 5, 11 and 12.