## 4 Simulations
### Application to Electro Hydrostatic Actuator System
\tag{47}\]  
It can be interpreted from the equation set (46) and (47) that \(d_{1}\), which consists of the lumped friction \(f_{Fr}\) and the force disturbance \(\delta_{2}\), represents a mismatched disturbance, whereas \(d_{2}\), which includes internal leakages and pressure disturbances, is a matched disturbance. Thus, the EHA system considered here is subject to both mismatched and matched disturbances. It is assumed that these disturbances satisfy Assumptions 1 and 2. To estimate the mismatched disturbance \(d_{1}\), one can apply (34) to yield a finite-time disturbance observer as follows:  
\[\left\{\begin{array}{l}\dot{z}_{12}=v_{12}+x_{3}\\ v_{12}=-\lambda_{12}L_{2}^{1/4}|z_{12}-x_{2}|^{3/4}\text{sign}(z_{12}-x_{2})- \mu_{12}(z_{12}-x_{2})+z_{22}\\ \dot{z}_{22}=v_{22}\\ v_{22}=-\lambda_{22}L_{2}^{1/3}|z_{22}-v_{12}|^{2/3}  
where \(z_{12}=\hat{x}_{2}\), \(z_{22}=\hat{d}_{1}\), \(z_{32}=\hat{\hat{d}}_{1}\), \(z_{42}=\hat{\hat{d}}_{1}\), and \(L_{2}>0\), \(\lambda_{j2}\), \(\mu_{j2}>0\) (\(j=1,\ldots,4\)). A chattering-attenuated full-order TSMC for the EHA system can then be designed based on (36) as:  
\[\sigma=\dot{x}_{3}+c_{3}|x_{3}+z_{22}|^{\alpha_{3}}\text{sign}(x_{3}+z_{22})+c _{2}|x_{2}|^{\alpha_{2}}\text{sign}(x_{2})+c_{1}|x_{1}|^{\alpha_{1}}\text{sign} (x_{1})+z_{32} \tag{49}\]  
where \(c_{i}(i=1,2,3)\) are positive constants satisfying the requirement that the polynomial \(p^{3}+c_{3}p^{2}+c_{2}p+c_{1}\) is Hurwitz, and \(\alpha_{i}(i=1,2,3)\) are chosen as (37). Accordingly, the control law is developed with the form:  
\[u=g^{-1}\big{(}u_{eq}+u_{n}\big{)} \tag{50}\]  
where the equivalent control \(u_{eq}\) is designed as:  
\[u_{eq}=-f-c_{3}|x_{3}+z_{22}|^{\alpha_{3}}\text{sign}(x_{3}+z_{22})-c_{2}|x_{2 }|^{\alpha_{2}}\text{sign}(x_{2})-c_{1}|x_{1}|^{\alpha_{1}}\text{sign}(x_{1}) -z_{32} \tag{51}\]  
and the switching term \(u_{n}\) is defined as in (22) and (23). The nominal values of the system parameters were determined as in Table 4[56]. The cylinder stroke length is \(0.3\,\text{m}\); thus, the working range of the actuator is set to be \(x_{1}\in[-0.15,\ 0.15]\,\text{m}\). For safety purposes, the pressure threshold of the relief valve is set at \(150\,\text{bar}\), meaning that \(P_{1},\ P_{2}\in[0,\ 150\times 10^{5}]\,\text{Pa}\). To ensure the control task was challenging and to examine the effectiveness of control algorithms, both matched and mismatched disturbances were deliberately assumed to be time varying throughout the simulation period. Specifically, the force disturbance \(\delta_{2}\) and the pressure disturbances \(\delta_{31}\), \(\delta_{32}\) were selected to be dependent on both time and system state as follows. \(\delta_{2}=-1000(0.2+0.5x_{1}+x_{2}+\sin(t))\), \(\delta_{31}=-1000(0.1+3x_{2}+\sin(t))\), and \(\delta_{32}=500(0.2+x_{2}+2\sin(t))\). Moreover, for practicality, measurement noise and parameter uncertainty were added to the simulation. Specifically, the position and pressure measurements were subject to noise with a mean of 0 and a covariance of \(\{0.0005\ \text{m},\ 3500\ \text{Pa},\ 3500\ \text{Pa}\}\)[45]. Parameters used to design the controller have nominal values, whereas the parameters used to model the EHA system include variance (\(\sim\)5% of nominal values), as indicated in Table 4. The sampling time was set to be 1 ms.  
Again, the proposed control algorithm, DOBTSMC, was compared with TSMC and ESOSMC in this case study. The design control parameters of DOBTSMC were selected as \(c_{1}=315.6\), \(c_{2}=500\), \(c_{3}=270\); \(\alpha_{i}(i=1,2,3)\) are designed as \(\alpha_{1}=1/3\), \(\alpha_{2}=1/5\), \(\alpha_{3}=1/7\). The DOB parameters were chosen as \(L_{2}=500\), \(\lambda_{12}=30\), \(\lambda_{22}=65\), \(\lambda_{32}=7.5\), \(\lambda_{42}=5.5\), \(\mu_{12}=6\), \(\mu_{22}=13\), \(\mu_{32}=1.5\), \(\mu_{42}=1.1\). Figure 8 presents the position response of the EHA system with the three considered control algorithms, namely, TSMC, ESOSMC, and DOBTSMC. Predictably, TSMC fails