## 4 Numerical Simulation and Discussion  
### Simulation Setup  
The nominal system parameters of the studied EHS provided in Table 1 are used to design the main control strategy and observers.  
To demonstrate the effectiveness of the proposed control strategy, the following controllers are employed for comparison as:  
1. [label=(0)]
2. Proposed control strategy with controller gains as \(k_{1}=350\), \(k_{2}=250\), \(k_{3}=150\). The bandwidths of the designed ESMO and ESO are chosen as \(\omega_{1}=450\) and \(\omega_{2}=450\), respectively. The time constant of filters is \(T_{f}=0.001\). For the trajectory planner, its parameters are selected based on the system specifications as \(\dot{x}_{1d}^{+}=40\), \(\dot{x}_{1d}^{-}=-40\), \(\dot{x}_{1d}^{+}=30\), \(\dot{x}_{1d}^{-}=-30\), \(C_{tp}=10^{5}\), and \(T=0.001\).
3. DESO-BC (Dual Extended State Observer-based Command Filtered Backstepping Controller): Without the integration of PPC, the control structure and controller parameters are chosen as same as the proposed controller. The two ESOs [52] are designed to simultaneously estimate the angular velocity and both lumped mismatched and matched uncertainties, and their bandwidths are picked as \(\omega_{1}=450\) and \(\omega_{2}=450\).
4. SESO-BC (Single Extended State Observer-based Command Filtered Backstepping Controller): The output feedback controller is constructed based on command filtered backstepping control (CF-BSC) framework, whose control architecture and controller gains are designed equivalent to the DESO-BC controller. In this control scheme, an ESO [44] is established to estimate immeasurable angular velocity, load pressure, and lumped matched uncertainty with the bandwidth of the ESO is \(\omega=450\).  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Parameter** & **Unit** & **Value** & **Parameter** & **Unit** & **Value** \\ \hline \(J\) & \(\mathbf{kg}\cdot\mathrm{m}^{2}\) & 0.2 & \(C_{t}\) & \(\mathrm{m}^{3}\cdot\mathrm{s}^{-1}\cdot\mathrm{Pa}^{-1}\) & \(1\times 10^{-12}\) \\ \(B\) & \(\mathrm{N}\cdot\mathrm{m}\cdot\mathrm{s}\cdot\mathrm{rad}^{-1}\) & 90 & \(P_{\mathrm{s}}\) & \(\mathrm{Pa}\) & \(1\times 10^{7}\) \\ \(D_{m}\) & \(\mathrm{m}^{3}\cdot\mathrm{rad}^{-1}\) & \(5.8\times 10^{-5}\) & \(A_{f}\) & \(\mathrm{N}\cdot\mathrm{m}\) & \(10\) \\ \(\beta_{e}\) & \(\mathrm{Pa}\) or \(\mathrm{N}\cdot\mathrm{m}^{-2}\) & \(7\times 10^{8}\) & \(V_{t}\) & \(\mathrm{m}^{3}\) & \(1.16\times 10^{-4}\) \\ \(k_{t}\) & \(\mathrm{m}^{3}\cdot\mathrm{s}^{-1}\cdot\mathrm{V}^{-1}\cdot\mathrm{Pa}^{-1/2}\) & \(1.1969\times 10^{-8}\) & & & \\ \hline \hline \end{tabular}
\end{table}
Table 1: Nominal system parameters of the studied EHS.  
To examine the robustness of the three control approaches, the time-varying lumped mismatched and matched disturbances \(d_{1}(t)\) and \(d_{2}(t)\) are purposely inserted into the control system as  
\[\begin{split} d_{1}(t)&=500\text{sin}(\pi t/2)\;( \text{rad}/\text{s}^{2})\\ d_{2}(t)&=5\times 10^{5}\text{sin}(\pi t/2)\;(\text{rad} /\text{s}^{3})\end{split} \tag{60}\]  
Performance indexes in the steady-state regime including the maximum, average, and standard deviation of the tracking errors [44] are used to measure the tracking qualifications of the above controllers. These terms are given by:  
1. Maximal absolute tracking error \[M_{e}=\underset{i=1,\dots,N}{\text{max}}\left\{\left|\epsilon_{1}(i)\right|\right\}\] (61) where \(\epsilon_{1}\) is the output tracking error and \(N\) denotes the number of samples.
2. Average tracking error \[\mu_{e}=\frac{1}{N}\sum_{i=1}^{N}\left|\epsilon_{1}(i)\right|\] (62)
3. Standard deviation of the tracking errors \[\sigma_{e}=\sqrt{\frac{1}{N}\sum_{i=1}^{N}\left(\left|\epsilon_{1}(i)\right|- \mu_{e}\right)^{2}}\] (63)