# Robust Speed Control of a Multi-Mass System: Analytical Tuning and Sensitivity Analysis
## 2 System Modeling
### Modeling of VEL Test Rig
\begin{table}
\begin{tabular}{c c} \hline \hline
**Feature** & **Value** \\ \hline Maximum allowable vehicle mass & 12,000 [kg] \\ Maximum allowable wheel load & 3000 [kg] \\ Wheelbase range & 1.8–4.9 [m] \\ Vehicle’s track width range & 1.2–3.9 [m] \\ Maximum loading machine speed & 2000 [RFM] \\ Nominal loading machine torque & 2500 [N.m] \\ Nominal loading machine power & 209 [kW] \\ Steering angle range at the front wheels & \(\pm\)20 [degree] \\ Maximum steering load torque at the front wheels & 1000 [N.m] \\ Maximum air speed of the portable airstream simulator & 135 [km/h] \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Technical specification data of VEL test rig. space can be described in differential equations and in the form of matrices. Therefore, Equation (2) describes the three-mass system shown in Figure 3  
\[ \begin{array}{c}J_{T}\frac{d\omega_{M}}{dt}=M_{M}-D_{S}(\omega_{M}-\omega_{W})- K_{S}(\theta_{M}-\theta_{W})\\ J_{W}\frac{d\omega_{W}}{dt}=D_{S}(\omega_{M}-\omega_{W})+K_{S}(\theta_{M}-\theta_{ W})-D_{Ax}(\omega_{W}-\omega_{Sx})-K_{Ax}(\theta_{W}-\theta_{Sx})\\ J_{Pt2W}\frac{d\omega_{D}}{dt}=D_{Ax}(\omega_{W}-\omega_{D})+K_{Ax}(\theta_{ W}-\theta_{D})-M_{Sx}-\frac{M_{F}}{2}\\ \text{where}\;J_{Pt2W}=J_{A}+\frac{I_{Pt2W}^{2}+\left(1+I_{t}+I_{t}\right)^{2}_{ t}}{2}\;.\end{array}  \tag{2}\] \begin{table}
\begin{tabular}{c c c} \hline \hline
**Element** & **Value** & **Unit** & **References** \\ \hline Powertrain motor inertia (\(J_{E}\)) & 0.03 & Kg-m\({}^{2}\) & [26] \\ Wheel hub inertia (\(J_{W}\)) & 0.124 & kg-m\({}^{2}\) & [27] \\ Axle inertia (\(J_{Ax}\)) & \(3.7\times 10^{-4}\) & kg-m\({}^{2}\) & [27] \\ Differential inertia (\(J_{D}\)) & \(1\times 10^{-2}\) & kg-m\({}^{2}\) & [26; 27; 28] \\ Gearbox inertia (\(J_{G}\)) & 0.0524 & kg-m\({}^{2}\) & [27] \\ Gearbox transmission ratio (\(i_{g}\)) & 4 & - & Technical data \\ Differential transmission ratio (\(i_{d}\)) & 2.5 & - & Technical data \\ Loading machine inertia (\(J_{M}\)) & 0.6 & kg-m\({}^{2}\) & Technical data \\ CV shaft inertia (\(J_{S}\)) & 0.0243 & kg-m\({}^{2}\) & Technical data \\ CV shaft stiffness (\(K_{S}\)) & 1715 & N.m/rad & Technical data \\ CV shaft internal damping (\(D_{S}\)) & 5.99 & N.m/rad & Technical data \\ Axle stiffness (\(K_{Ax}\)) & 7700 & N.m/rad & [29] \\ Axle internal damping (\(D_{Ax}\)) & 3.57 & N.m.s/rad & [30] \\ Coulomb friction constant (\(M_{F0}\)) & 0.5 & N.m & [23] \\ Friction moment factor (\(k_{M}\)) & 0.06 & - & [23] \\ Viscous friction factor (\(k_{Ax}\)) & 0.06 & - & [23] \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Parameters of the VEL mechanical parts associated with front wheels. Figure 3: Three mass models. Figure 2: Complete simulation model of VEL mechanical power transmission system and a front-wheel-drive. In order to rewrite the model in a state-space (SS) presentation, the state variables need to be specified as follows:  
* The angular speed of the electrical loading machine (\(\omega_{M}\));
* The difference between the loading machine angular position (\(\theta_{M}\)) and the wheel-hub angular position (\(\theta_{W}\));
* The angular speed of the wheel-hub (\(\omega_{W}\));
* The difference between wheel-hub angular position (\(\theta_{W}\)) and the differential gear output angular position (\(\theta_{Sx}\));
* The angular speed of the differential gear output (\(\omega_{Sx}\)). Accordingly, the state vector becomes  
\[\mathbf{x}^{T}=\left[\omega_{\text{M}}\quad(\theta_{\text{M}}-\theta_{\text{W}}) \quad\omega_{\text{W}}\quad(\theta_{\text{W}}-\theta_{\text{Sx}})\quad\omega _{\text{Sx}}\right]\]  
where \(\omega_{M}\) is the reference variable, \(\omega_{W}\) and \(\omega_{Sx}\) are estimated outputs, and (\(\theta_{M}-\theta_{W}\)) and (\(\theta_{W}-\theta_{D}\)) are estimated disturbances. The model is given in matrix form as in Equation (3). \[\dot{\mathbf{x}}=\underbrace{\left[