## 3 Main Result
### Command-Filter Based Controller Design with Prescribed Performance and Disturbance Observer  
In this subsection, a disturbance observer is proposed in order to provide a better estimation of the disturbance in the design of the command-filter based controller with prescribed performance. The structure of command-filtered backstepping integral sliding mode control with prescribed performance for ship roll stabilization strategy developed in this study is shown in Figure 2.  
From (8), it can be obtained that  
\[F=-f(x)+bu+\dot{x}_{2}. \tag{32}\]  
Therefore, the disturbance observer for (32) is designed as follows  
\[\bar{F}=\lambda+l_{f}(x_{1},x_{2}) \tag{33}\]  
\[\dot{\lambda}=-l_{o}\lambda+l_{o}\left(-f(x)+bu-l_{f}(x_{1},x_{2})\right) \tag{34}\]where \(l_{f}(x_{1},x_{2})=l_{o}x_{2}\) and \(l_{o}\) is a positive scalar. In addition, the estimation error \(\tilde{F}\) is defined as \(\tilde{F}=F-\hat{F}\). With the implement of the disturbance observer, the impact caused by disturbance can be offset.  
Due the estimation time delay, the estimation error still exists before the disturbance observer can give an accurate estimation of the disturbance. It is reasonable to assume that the estimation error \(\tilde{F}\) satisfies that \(|\hat{F}|\leq\kappa\). The proposed controller is modified as  
\[u=b^{-1}\left(-f(x)+\dot{x}_{2}^{c}-az_{2}^{m/n}-k_{3}s-\tau sign(s)-\hat{F}- \kappa sign(s)\right) \tag{35}\]  
\[\dot{k}=\epsilon s\cdot sign(s) \tag{36}\]  
where \(\kappa\) and \(\epsilon\) are positive scalars.  
Theorem 2: _For the rolling angle dynamic system (7) and (8), suppose that Assumptions 1-2 hold, the transformation error \(\epsilon(t)\) and \(z_{2}\) are asymptotically stable by the proposed virtual controller (26) and the sliding mode controller (35) with the disturbance observer (33) while \(z_{1}\) converges to the neighbor domain of \(0\) with the prescribed performance._  
Proof of Theorem 2.: Considering the following Lyapunov function  
\[V_{2}=\frac{1}{2r}\epsilon^{2}+\frac{1}{2}s^{2}+\frac{1}{2}\hat{F}^{2}+\frac{ 1}{2\epsilon}\hat{\kappa}^{2}, \tag{37}\]  
denoting \(\tilde{\kappa}=\kappa-\hat{\kappa}\), it can be obtained  
\[\begin{split}\dot{V}_{2}&\leq-\left(k_{2}-\frac{1}{ 2}\right)\cdot\epsilon^{2}+\frac{1}{2}s^{2}+s\left(f(x)+bu+F-\dot{x}_{2}^{c}+ az_{2}^{m/n}\right)-\hat{F}\hat{F}-\hat{\kappa}\hat{\kappa}\\ &\leq-\left(k_{2}-\frac{1}{2}\right)\cdot\epsilon^{2}-\left(k_{ 3}-\frac{1}{2}\right)s^{2}+s\left(F-\hat{F}-\hat{\kappa}sign(s)\right)-\hat{ \kappa}s\cdot sign(s)\\ &\quad-\hat{F}\left(-l_{o}\lambda+l_{o}\left(-f(x)+bu-l_{f}(x_{1},x_{2})\right)+l_{f}(x_{1},x_{2}).\right)\end{split} \tag{38}\]  
Figure 2: The structure of the proposed method in this paper.  
Due to (32)-(34),  
\[\dot{V}_{2} \leq-\left(k_{2}-\frac{1}{2}\right)\cdot\epsilon^{2}-\left(k_{3}- \frac{1}{2}\right)s^{2}+s\left(\tilde{F}-\kappa sign(s)\right)-\tilde{\kappa}s \cdot sign(s) \tag{39}\] \[\quad-\tilde{F}\left(-l_{o}\hat{F}+l_{o}\left(-f(x)+bu\right) \right)+\hat{l}_{f}(x_{1},x_{2})\right)\] \[\leq-\left(k_{2}-\frac{1}{2}\right)\cdot\epsilon^{2}-\left(k_{3}- \frac{1}{2}\right)s^{2}+s\left(\tilde{F}-\kappa\cdot sign(s)\right)+s\kappa \cdot sign(s)\] \[\quad-s\hat{\kappa}\cdot sign(s)-\tilde{\kappa}s\cdot sign(s)- \tilde{F}\left(-l_{o}\hat{F}+l_{o}F\right)\] \[\leq-\left(k_{2}-\frac{1}{2}\right)\cdot\epsilon^{2}-\left(k_{3} -\frac{1}{2}\right)s^{2}-l_{o}\hat{F}^{2}.\]  
**Remark 4**.: _It implies that the error system still can be stable while the upper bound of the lumped disturbance is no longer needed here. It implies that the proposed prescribed performance control strategy possesses the real industrial potential in the ship stabilizer design with the implement of the nonlinear disturbance observer._  
**Remark 5**.: _In order to eliminate the chattering phenomenon in the simulation verification phase, the discontinuous terms \(sign(s)\) in (27) and (35) are replaced by the saturation functions \(sat(s/\delta)\), where \(\delta\) is known as the boundary layer thickness of saturation function._