# Research on the Attitude Control Strategy of TBM Digging with V-Shaped Propulsion System
## 4 Joint Simulation and Result Analysis  
### Joint Simulation Platform Construction  
To conduct a simulation analysis of the V-TBM digging attitude, the V-TBM propulsion hydraulic system model is developed in AMESim, and the control strategy model is implemented in Simulink. Data exchange between the two platforms is facilitated through the AMESim-Simulink joint simulation interface. The electro-hydraulic proportional speed control valve is adjusted to achieve closed-loop position control of the propulsion hydraulic cylinder. The AMESim-based hydraulic system model is presented in Figure 5, and the relevant simulation parameters are provided in Table 1.  
To assess the stability of the propulsion system during TBM excavation, the dynamic characteristics of the propulsion hydraulic cylinder within the hydraulic system are analyzed. In addition to nonlinear controllers, widely used PID controllers are selected for comparative simulation and verification. The proportional coefficient \(kp\) determines the system's response intensity to the current error; the integral coefficient \(ki\) adjusts the control output by accumulating past errors to eliminate steady-state error; and the differential coefficient \(kd\) predicts error variation trends to suppress system oscillations. The PID con  
\begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline
**Component** & **Motor** & **Pump** & \multicolumn{2}{c}{**Balancing Valve**} & \multicolumn{2}{c}{**Hydraulic Cylinder**} & \multicolumn{2}{c}{**Matrix**} \\ \hline physical quantity & speed/ & displacement/ & check valve & flow & shaft & & coefficient & & \\  & (rev/min) & (cc/rev) & pressure & gradient/ & bore/mm & diameter/ & course/mm & \begin{tabular}{c} coefficient \\ direction/ \\ (N\({}_{\text{s}}\)m\({}^{-1}\)) \\ \end{tabular} & \
\begin{tabular}{c} equivalent \\ mass/ \\ \end{tabular} &
\begin{tabular}{c} maximum \\ load/ \\ \end{tabular} \\ parameters & 1000 & 85 & 5 & 80 & 240 & 200 & 1000 & 1000 & 1073.9 & 10\({}^{6}\) \\ \hline \hline \end{tabular}
\end{table}
Table 1: AMESim simulation parameters.  
Figure 5: Hydraulic system model based on AMESim.  
troller parameters (\(kp=185\), \(ki=0.1\), \(kd=1\)) are obtained through self-tuning in MATLAB's (R2017a) built-in PID toolbox, further refined via simulation-based optimization. A sinusoidal signal, defined as y = 10 sin(0.3t), is used as the system input. A 200 kN disturbance is applied to the hydraulic cylinder at t = 45 s, with a total simulation duration of 90 s. The system's response curve is presented in Figure 6, while the tracking error is depicted in Figure 7.  
The simulation results indicate that the velocity of the sinusoidal signal reaches its maximum at t = 0; however, the pressure-flow relationship within the valve-controlled cylinder has not yet been established, resulting in initial tracking errors and oscillations. The tracking differentiator-based nonlinear PID predicts the error trend by differentiating the signal (Equation (12)), thereby accelerating the pressure establishment process and enabling a faster transition to a stable following state. After stabilization, the maximum steady-state tracking error under tracking differentiator-based nonlinear PID control (0.43 mm) was 27% lower than that of the conventional PID (0.59 mm), indicating a higher closed-loop gain and reduced phase lag. When a disturbance was applied at t = 45 s, the oscillation amplitude (\(\pm\)0.15 mm) and recovery time (0.11 s) under tracking differentiator-based nonlinear PID  
Figure 6: Propulsion hydraulic cylinder control response curve.  
Figure 7: Propulsion hydraulic cylinder control tracking error.  
control were significantly improved compared to those of the conventional PID (\(\pm\)0.23 mm, 0.3 s), demonstrating the strong robustness of the control loop.