# Central Non-Linear Model-Based Predictive Vehicle Dynamics Control
## 4 Results
### Double Lane Change
#### 4.2.2 Self-Steering Behavior  
In comparison to controlling the roll angle, influencing the self-steering behavior is a subordinate control objective. Here, a constant understeering behavior \(SSG_{\text{Ref}}\) is pursued, which corresponds approximately to the one of passive vehicle. Because the vehicle is equipped with active stabilizers and semi-active dampers, this control objective can only be pursued to a limited extent. With regard to the representability for the evaluation, the pseudo quantity \(\widetilde{a}_{i}\) is introduced:  
\[\widetilde{a}_{i}=\alpha_{t,i}-\alpha_{r,i}=a_{y}SSG_{i},i\epsilon\{\text{Ref},\text{ Pas},\text{ PID},\text{Mpc}\} \tag{37}\]  
The pseudo quantity \(\widetilde{\alpha}_{i}\) represents the difference between the slip angle of the front axle \(\alpha_{t,i}\) and rear axle \(\alpha_{r,i}\) and, thus, corresponds to the product of the self-steering gradient \(SSG_{i}\) and the lateral acceleration \(a_{y}\). The evaluation is conducted in the following based on the pseudo quantity \(\widetilde{\alpha}_{i}\). The qualitative evaluation is performed using the pseudo quantity courses for the validation maneuver. This is shown in Figure 5.  
With regard to the absence of lateral dynamics in the acceleration phase, only the section of the double lane change is considered. The representation of the pseudo quantity suggests a dynamic reference variable \(\widetilde{\alpha}_{\text{Ref}}\). However, this results from the dynamics of the lateral acceleration \(a_{y}\). With regard to safety, a constant self-steering gradient \(SSG_{\text{Ref}}\) is used. The representation remains consistent with the evaluation of the roll behavior. The vehicle with the central predictive control shows an almost identical course of the pseudo quantity to the passive vehicle. Due to the weighting of this control target and the limited possibility to manipulate, there is a deviation from the reference variable. Since the conventional control approach does not explicitly consider the influence on the self-steering behavior, only a limited evaluation can be performed. For the validation maneuver of the double lane change, the conventional control exhibits a self-steering behavior corresponding to the passive self-steering behavior, similar to the central predictive control.  
Figure 5: Control Quality Double Lane Change–Self-Steering Behavior–Pseudo Quantity Curves.  
In principle, the control objective is satisfied by the central predictive control, since the self-steering behavior corresponds to the behavior of the passive vehicle despite utilizing an active control system. This is confirmed by examining the frequency spectra, which are illustrated in Figure 6.  
The frequency spectrum for the vehicle with the central predictive control is almost identical to that of the passive vehicle. The conventional control approach features a similar frequency spectrum.  
The quantitative evaluation for the whole validation maneuver results in a root mean squared error for the central predictive control of \(8.1009\times 10^{-4}\) rad, which is equivalent to \(0.0464^{\circ}\). The conventional control approach yields a root mean squared error of \(8.1334\times 10^{-4}\) rad, which corresponds to \(0.0466^{\circ}\).