## 4 PP-NFTSMC-ESO Controller Design  
For the trajectory tracking control of a deep-sea electric oil-filled joint actuator with unmeasured system states, dynamic uncertainties, and unknown disturbances, the PP-NFTSMC-ESO controller was proposed in this paper. The block diagram of this controller is shown in Figure 2. The ESO was applied to estimate the unmeasured velocity and the lumped uncertainty, while the PPF was employed to constrain the transient and steady-state performances of the trajectory tracking error. The NFTSMC controller was then synthesized with the signals from the reference trajectory, ESO, and PPF. Define the position error \(e\) and velocity error \(z_{2}\) as  
\[e=y-y_{\text{d}}=x_{1}-x_{1\text{d}} \tag{19}\]  
\[z_{2}=\hat{x}_{2}-\alpha \tag{20}\]  
where \(y_{\text{d}}\) and \(x_{1\text{d}}\) are the desired joint position, and \(\alpha\) is the virtual control function of \(x_{2}\). Step 1: According to the results of the prescribed performance function (11), the position error \(e\) can be transformed as follows  
\[z_{1}=\frac{1}{2}\ln\frac{e/\rho+l}{h-e/\rho} \tag{21}\]  
The time derivative of \(z_{1}\) is  
\[\dot{z}_{1}=r\bigg{(}\dot{e}-\frac{\dot{\rho}e}{\rho}\bigg{)}=r\bigg{(}x_{2}- \dot{x}_{1\text{d}}-\frac{\dot{\rho}e}{\rho}\bigg{)} \tag{22}\]  
Figure 2: The block diagram of the proposed PP-NFTSMC-ESO controller. Therefore, the virtual control function of \(x_{2}\) can be designed as  
\[\alpha=-k_{0}z_{1}+\dot{x}_{1\rm d}+\frac{\dot{\rho}e}{\rho} \tag{23}\]  
where \(k_{0}\) is a positive design parameter. From Equation (20), we can have \(x_{2}=z_{2}+\alpha+\widetilde{x}_{2}\), and substituting it into Equation (22) results in  
\[\dot{z}_{1}=r\bigg{(}z_{2}+\alpha+\widetilde{x}_{2}-\dot{x}_{1\rm d}-\frac{ \dot{\rho}e}{\rho}\bigg{)} \tag{24}\]  
Define the Lyapunov function as  
\[V_{1}=\frac{1}{2r}z_{1}^{2} \tag{25}\]  
Substituting Equations (23) and (24) into (25), the time derivative of \(V_{1}\) is  
\[\dot{V}_{1}=\frac{1}{r}z_{1}\dot{z}_{1}=-k_{0}z_{1}^{2}+z_{1}z_{2}+z_{1} \widetilde{x}_{2} \tag{26}\]  
Step 2: Define NFTSMC sliding function as [30]  
\[e=y-y_{\text{d}}=x_{1}-x_{1\text{d}} \tag{19}\]0  
where \(\alpha_{1},\alpha_{2}\) are positive constants subject to \(0<\alpha_{1}<1\) and \(\alpha_{1}<\alpha_{2}\), \(|z_{i}|^{[\alpha_{i}]}=|z_{i}|^{[\alpha_{i}]}\)sgn\((z_{i})\), \(i=1,\ 2;k_{1}\) and \(k_{2}\) are positive design parameters. The time derivative of \(\sigma\) is  
\[e=y-y_{\text{d}}=x_{1}-x_{1\text{d}} \tag{19}\]1  
Substituting the second equation of (13) and (14) into (29) produces  
\[e=y-y_{\text{d}}=x_{1}-x_{1\text{d}} \tag{19}\]2  
Define the Lyapunov function as  
\[e=y-y_{\text{d}}=x_{1}-x_{1\text{d}} \tag{19}\]3  
Substituting Equations (26) and (29) into (30), the time derivative of \(V_{2}\) is  
\[ \begin{array}{ll}\dot{V}_{2}=&\dot{V}_{1}+\sigma\dot{\sigma}=-k_{0}z_{1}^{2 }+z_{1}z_{2}+z_{1}\widetilde{x}_{2}\\ &+\sigma\Big{[}\theta_{1n}u+\theta_{2n}x_{2}+x_{3}-\dot{\alpha}-\theta_{2n} \widetilde{x}_{2}-\widetilde{x}_{3}+l_{2}\omega_{0}^{2}\widetilde{x}_{1}+k_{1 }|z_{1}|^{[\alpha_{1}]}+k_{2}|z_{2}|^{[\alpha_{2}]}\Big{]}\end{array}  \tag{31}\]  
The control \(u\) of PP-NFTSMC-ESO controller is designed as  
\[u=-\frac{1}{\theta_{1n}}\big{(}u_{\alpha q}+u_{sw}+u_{b}\big{)} \tag{32}\]  
\[u_{\alpha q}=\theta_{2n}\hat{x}_{2}+\hat{x}_{3}-\dot{\alpha}+k_{1}|z_{1}|^{[ \alpha_{1}]}+k_{2}|z_{2}|^{[\alpha_{2}]} \tag{33}\]  
\[u_{sw}=\eta\text{sign}(\sigma)+k_{3}\sigma \tag{34}\]  
\[u_{b}=2\frac{|z_{1}z_{2}|}{\sigma}\text{tanh}^{2}\Big{(}\frac{\sigma}{\tau} \Big{)} \tag{35}\]  
where \(\eta\), \(\tau\), and \(k_{3}\) are positive design parameters. As seen, the PP-NFTSMC-ESO controller consists of three parts, where \(u_{eq}\) is the equivalent control law to hold the error trajectory on the sliding surface, \(u_{sw}\) is the robust term used to compensate for the estimation errors, and \(u_{b}\) is the stable feedback term to stable the tracking error.