## 4 Simulation  
To validate the proposed command-filter based sliding mode controller and evaluate prescribed performance of the compensated tracking error, the corresponding simulation is carried out and the roll stabilization performance is also tested, as shown in Figures 3-8 in this section. The particulars of the ship are shown as follows: the ship length is 175 m, the ship speed is 12.43 Kn, the fin area is 12 m\({}^{2}\), the acting force arm of the stabilizer fin is 16 m, the lift coefficient of fin stabilizer is 2.1, the initial metacentric height is 0.3 m, the inertia \(I_{x}^{\prime}=0.0000176\), the additional inertia \(J_{x}^{\prime}=0.0000034\), and the damping coefficient \(A^{\prime}=-1.2552\times\mathrm{e}^{-5}\), \(B^{\prime}=-4.725\times\mathrm{e}^{-8}\).  
In addition, the parameters of the proposed controller are selected as \(k_{1}=4\), \(k_{2}=4\), \(k_{3}=3\), \(\rho_{0}=0.001\), \(M=0.5\), \(\rho_{\infty}=0.0005\), \(\tau=0.1\), \(a=0.5\), \(b=-7.165\), and \(\epsilon=2\). The original state of the ship rolling angle is assumed as \(\phi(0)=0^{\circ}\). In addition, the initial state of the nonlinear disturbance observer is selected as 0 as well.  
In addition, the six-level sea wave momentum that acts on the ship body is gathered from a real environment and depicted in Figure 3. In order to offset the impact caused by the sea wave momentum, the corresponding estimation of the sea wave is demonstrated in Figure 4. It can be noticed that the estimation of the external disturbance, which is also known as sea wave momentum, can fit the real time curve of the sea wave with desired performance at the time point of 10 s.  
In order to validate the effectiveness of the proposed prescribed performance control strategy, the time response of the compensated tracking error \(z_{1}\) is shown in Figure 5. It can be noticed that the amplitude of the compensated tracking error \(z_{1}\) is bounded by the prescribed performance function \(\rho(t)\). Hence, the dynamic performance and the steady-state of the compensated tracking error \(z_{1}\) is guaranteed. Consequently, the ship's rolling angle can provide a desired dynamic process and the steady-state output, which is demonstrated in Figure 6, with respect to the impact caused by the six-level sea wave momentum.  
Figure 4: The value of the disturbance observer.  
Figure 5: The time response of the compensated tracking error \(z_{1}\).  
Figure 3: The six-level sea wave momentum.  
As is shown in Figures 6-8, different control strategies are examined through numerical simulations. When the control input is absent, the roll angle cannot be stabilized with respect to the external disturbance within a reasonable amount of time and the amplitude of the rolling angle varies within a relatively large range. The traditional PID controller improves performance significantly compared to the previous situation without control, however, the roll motion response of the ship still remains unchanged.  
Figure 8: The time response of fin rate.  
Figure 6: The time curves of ship rolling angle with different control methods.  
Figure 7: The time response of fin control angle \(\alpha_{f}\).  
changes within an undesirable range. The conventional CBC exhibits better performance than PID with the smaller ship rolling angle. In contrast, the proposed controller in this paper performs better than the PID controller and CBC, and it can be seen that the ship's roll angle in the proposed controller operation is evidently constrained in the neighborhood of 0 owing to the introduction of prescribed performance. In addition, Figures 7 and 8 indicate that the control input generated by the proposed control strategy is bounded to less than 10\({}^{\circ}\) and 5\({}^{\circ}\)/s, which is known as the fin control angle \(a_{f}\) and fin rate, which demonstrates our strategy's superior performance when compared to the traditional PID controller and CBC.