# Dynamic Reduction-Based Virtual Models for Digital Twins--A Comparative Study
## 4 Results
The structure has twelve rigid wheels with a 710 mm diameter, three of which are attached to each of its four corners. Additionally, the structure is driven by six electrical machines, three on each side, which means that three wheels on each side are driven by electrical machines while the other three are undriven. The rated power of each electrical machine is 165 kW, the rated speed is 1413 rpm, and the rated frequency is 60 Hz. An encoder is attached to one of the driving wheels on either side. The driveline system also includes gears with a gear ratio of 26.28. The powertrain control module is composed of a proportional-integral controller (PI controller) with gain parameters (\(k_{P}\) and \(k_{I}\)) that are a linear, monotonically decreasing function of the wheel's instantaneous angular velocity in the acceleration regime and a constant in the steady-state regime, and a linear, monotoni  
Figure 4: Simulation Model for the Structure in MATLAB. Figure 3: Modal Vector Orthogonality Checks of the Reduced-Order Models. cally increasing function of the wheel's instantaneous angular velocity in the deceleration regime. As coded in the control module, the piecewise reference trapezoidal trajectory is as follows:  
\[\Omega(t_{i},\Omega_{r},v,s)= \begin{cases}\Theta_{d}t_{i}^{ss}&:t_{i}=t_{i}^{ss}, s_{i}<s_{d},v_{i}<v_{m},\\ &\Omega(t_{i}^{ss})<\Omega_{r}\\ \Omega_{r}&:t_{i}=t_{i}^{ss},s_{i}<s_{d},v_{i}<v_{m},\\ &\Omega(t_{i}^{ss})=\Omega_{r}\\ \Omega_{t_{i-1}^{ss}}&:t_{i}=t_{i}^{ss},s_{i}<s_{d},v_{i}=v_{m}\\ \Theta_{d}t_{i}^{dd}&:t_{i}=t_{i}^{d},s_{i}\geq s_{d}\end{cases}  \tag{45}\]  
The simulation scenario is described as follows: the structure is allowed to travel a distance of 30 m along the Y-axis; subsequently, the braking regime is initiated, where the electrical machines reverse the direction of rotation; and ultimately, the simulation terminates as soon as the wheels come to a standstill. The structure is permitted to attain a maximum velocity of 2 m/s at any instant in time. The control modules, as illustrated in Figure 4, monitor both displacement and velocity at the output ports of the reduced-order model's state-space model representation through a feedback loop. In practice, the control module monitors the encoder signals for displacement and velocity. Initially, a fixed-interface cms-based reduced-order model with modes up to 30 Hz is used to explain the dynamics of the structure, followed by a comparison of the simulation results of the reduced-order models obtained from both dynamic reduction methods. Figure 5 illustrates the simulation outcomes for the wheel's angular velocity and displacement for both the structure's narrow and wide sides. As evident from Figure 5, the braking regime commences as soon as the structure traverses 30 m, and eventually the wheels come to a complete halt at 25.5 s.  
In Figure 6, the simulation results for the motor angular velocities and translational velocities at the narrow and wide sides are depicted. Additionally, Figure 6 depicts the acceleration phase, which lasts until the structure reaches its maximum velocity of 2 m/s, at which point it maintains a constant velocity until the braking process begins. Figure 7 depicts the torque at the gear output side for both narrow and wide sides, the angular velocities of the electrical machines on the narrow and wide sides, and the reference trajectory of the angular velocity that the structure is expected to track. In addition, Figure 7 shows the torque necessary to keep the angular velocities consistent with the reference ramp. Figure 5: Wheel Angular Velocity and Displacement for the Structure. The minimum and maximum coordinates spanned by the structure along the \(X\), \(Y\), and \(Z\)-axes are \((-19.4,63.5)\), \((-10.14,10.14)\), and \((0,35.88)\), respectively, whereas the center of mass \((X,Y,Z)\) is at \((28.317,-0.417,21.269)\), where the units are in the SI system. The minimum and maximum span of the coordinates along the \(X\)-axis are on the narrow and wide sides, respectively.