# Central Non-Linear Model-Based Predictive Vehicle Dynamics Control
## 4 Results
### Double Lane Change
#### 4.2.1 Roll Behavior  
For the qualitative evaluation of the control quality of the non-linear model-based predictive control algorithm with respect to the main control objective, the roll angle is plotted over time for the validation maneuver. This is illustrated in Figure 3. The roll angle \(\varphi_{\text{Mpc}}\) resulting from the execution of the non-linear model-based predictive control algorithm is represented by a dotted black line. The dynamic reference variable \(\varphi_{\text{Ref}}\) is represented by a red solid line. Furthermore, a green dashed line illustrates the roll angle curve \(\varphi_{\text{Pas}}\) of the vehicle with passive chassis elements and an orange dotted line illustrates the roll angle curve \(\varphi_{\text{PID}}\) resulting from the conventional control approach.  
The reference variable \(\varphi_{\text{Ref}}\) reduces the roll motion by about 75% compared to the passive vehicle \(\varphi_{\text{Pas}}\). The roll angle resulting by the central predictive control follows the reference with an excellent accuracy. In contrast, the conventional control approach based on the proportional integral derivate controller and the skyhook damping results in increased control deviations. Thus, with the central predictive control, not only the safety but also the comfort is increased compared to the conventional control approach as well as to the passive vehicle behavior. The rolling movements that the passive vehicle exhibits during acceleration are not present for the vehicle utilizing the central predictive control and the vehicle utilizing the conventional control approach, respectively.  
In addition, the impacts of the active roll stabilizations on the comfort are evaluated by examining the resulting frequency spectra. These are illustrated in Figure 4. Due to the use of the central predictive control, a stronger damping within the frequency spectrum is present compared to the passive vehicle. The frequency spectrum of the conventional approach also features a stronger damping than the passive vehicle. Compared to the central predictive control algorithm, however, a weaker damping is present.  
Furthermore, the root mean squared errors are determined for the quantitative evaluation:  
\[RMSE(x_{\text{Ref}},x_{i})=\sqrt{\frac{\sum_{k=1}^{n}\bigl{(}x_{\text{Ref}}(k) -x_{\text{Mpc}}(k)\bigr{)}^{2}}{n}},i\in\{\text{PID},\text{Mpc}\}. \tag{36}\]  
Figure 4: Control Quality Double Lane Change-Roll Behavior-Frequency Spectra.  
Figure 3: Control Quality Double Lane Change–Roll Behavior–Roll Angle Curves.  
In relation to the active roll stabilization, the non-linear model-based predictive control exhibits a root mean squared error of \(2.3906\times 10^{-4}\) rad. This corresponds to \(0.0137^{\circ}\). In contrast, the conventional control approach results in a root mean squared error of \(0.0010\) rad, which is equivalent to \(0.0573^{\circ}\). The quantitative evaluation, thus, confirms the qualitative analysis. The control quality of the central predictive control is excellent. In general, the central predictive control, thus, outperforms the conventional control approach.