# Central Non-Linear Model-Based Predictive Vehicle Dynamics Control
## 4 Results
### Double Lane Change
#### 4.2.3 Pitch Behavior  
Similar to the manipulation of the self-steering behavior, the reduction of pitching movements is a subordinate control objective within the non-linear model-based predictive control algorithm. Due to the vehicle's equipment with active stabilizers and semi-active dampers, only a limited influence on the pitch behavior is possible. In order to evaluate the central predictive control in a qualitative way with respect to the control objective of reducing the pitching movements, the pitch angle is plotted over time. The pitch angle curves are shown in Figure 7.  
The focus is on the acceleration phase of the validation maneuver, since pitching movements are present here due to the gear changes and the present longitudinal accelerations. The pitching movements during the double lane change in contrast are negligibly small. The representation remains consistent. Analogous to the manipulation of the self-steering behavior, a constant reference variable \(\theta_{\text{Ref}}\) is used. This reference variable represents the stationary pitch angle of the vehicle, which corresponds to the pitch angle that is present when the vehicle is at standstill. The reference variable cannot be adjusted fully by the present vehicle setup, but it is considered in order to implement the reduction of pitching movements. In comparison to the vehicle with passive chassis elements \(\theta_{\text{Pas}}\), pitching movements of the vehicle with the central predictive control \(\theta_{\text{MPC}}\) are, therefore, reduced. The implementation of the skyhook damping within the conventional control approach also reduces pitching movements compared to the passive vehicle. By exploiting synergies, however, the implementation of the central predictive control results in a greater reduction of pitching movements compared to the conventional control approach.  
Figure 6: Control Quality Double Lane Change–Self-Steering Behavior–Frequency Spectra.  
The corresponding frequency spectra are shown in Figure 8. Here, the positive influence of the central predictive control is also evident. The non-linear model-based predictive control results in greater damping within the frequency spectrum. The amplitudes of the conventional control approach present at very low frequencies are sometimes even greater than those of the passive vehicle. In terms of ride comfort, the central predictive control, thus, provides a significant improvement in comparison to the application of the conventional control approach.  
The quantitative evaluation confirms this statement. The non-linear model-based predictive control yields a root mean squared error of 0.0106 rad, which corresponds to 0.6073\({}^{\circ}\). The conventional control approach results in a root mean squared error of 0.0119 rad corresponding to 0.6818\({}^{\circ}\). By using the central predictive control, pitching movements are, thus, reduced compared to the conventional control approach as well as compared to the passive vehicle. As a result, the ride comfort is, thus, increased.  
Figure 8: Control Quality Double Lane Change–Pitch Behavior-Frequency Spectra.  
Figure 7: Control Quality Double Lane Change–Pitch Behavior–Pitch Angle Curves.