# Robust Speed Control of a Multi-Mass System: Analytical Tuning and Sensitivity Analysis
## 2 System Modeling
### Modeling of VEL Test Rig  
The system plant consists mainly of the vehicle under the test powertrain, which is, in this case, a front-wheel-drive electric vehicle. A constant velocity (CV) shaft connects each loading machine with the powertrain's corresponding axle and wheel hub. The installed measurement devices are an angular speed sensor at each loading machine end and a moment sensor at each wheel-hub end. The resistance moments generated by the electric loading machines are delivered to the wheel hub through the power transmission of the VEL, as illustrated in Figure 2. Since the right and left sides of the front-drive system with the VEL test bench are symmetric, a reduced system of half of the total system may be considered for building the model equations. Then, a suitable control algorithm is developed based on that. Finally, the complete system (i.e., full plant and the developed controllers for both sides) is used for testing. Similar approaches are considered in [11; 22; 23]. As shown in Figure 3, the plant can be modeled as three masses and two spring-damper systems. The three masses are the loading machine plus the CV shaft inertia (\(J_{T}\)), the wheel-hub inertia (\(J_{W}\)), one axle inertia plus half of the powertrain equivalent inertia (\(J_{P1 ZW}\)). The two spring-damper systems are CV shaft-equivalent torsional spring and damper coefficients and the axle-equivalent torsional spring and damper coefficients. Due to the relative flexibility of the shaft between the loading machine and the wheel hub, the shafts will be respected as flexible coupling components with equivalent stiffness and internal damping. The system has two inputs: first, the resistance moment from the loading machine (\(M_{M}\)), which will be the manipulated variable for the speed controller; and second, half of the final moment at the differential output gear (\(M_{Sx}\)) (i.e., x = left, right). One output state variable is assigned for the system, which is the measured angular speed of the loading machine. The gearboxes of modern commercial vehicles are produced with high efficiency, so backlash losses become negligible [23; 24]. The friction influence between gears is indeed significant. Including friction in the system's equations adds nonlinear dynamic constraints to the model, making the system more stable [25]. The friction acting in the differential gearbox combines Coulomb and viscous friction [11; 23; 24]. The Coulomb friction moment has a constant value (\(M_{F0}\)) and a moment-dependent part. The total output moment from the differential equals the vehicle's motor moment (\(M_{e}\)) multiplied by the total gear transmission ratio (i.e., gearbox transmission ratio (\(i_{g}\)) multiplied by the differential transmission ratio (\(i_{d}\))). This last quantity is specified by the factor (\(k_{M}\)). The factor (\(k_{\Delta\omega}\)) and the angular speed difference between the differential outputs (\(\Delta\omega_{D}\)) determine the viscous friction moment portion. Therefore, the total gear friction moment (\(M_{F}\)), which can be considered a disturbance, is described in Equation (1) [24]  
\[M_{F}=\big{(}M_{F0}i_{d}+k_{M}\big{|}M_{e}i_{g}i_{d}\big{|}\big{)}sgn(\Delta \omega_{D})+k_{\Delta\omega}\Delta\omega_{D} \tag{1}\]  
\(M_{F}\) will be distributed equally between the left and right sides of the total system model [23; 24]. The parameterization of the mechanical part of VEL associated with the front wheels is listed in Table 2. The controlled system plant and the controller in the state