# Central Non-Linear Model-Based Predictive Vehicle Dynamics Control
## 3 Central Predictive Control
### Optimization  
Following the prediction of the vehicle dynamics as a function of the manipulated variables, the optimization is executed with regard to the control objectives. The optimization is performed using the entire prediction horizon \(n_{\text{p}}\). The prediction horizon equals 0.15 s.  
A major advantage of the non-linear model-based control algorithm is that constraints can be taken into account within the optimization. In this contribution, the manipulated variables are constrained. Thus, the physical limits of the actuators can be considered within the optimization. This results in the restriction of the counter roll torques \(u_{1}\) and \(u_{2}\) between a minimum counter roll torque \(T_{\text{min}}\) and a maximum counter roll torque \(T_{\text{max}}\)  
\[T_{\text{min}}\ \leq u_{i}\leq T_{\text{max}},\ \ i\in\{1,2\}, \tag{20}\]  
as well as the restriction of the damping factors \(u_{3}\), \(u_{4}\), \(u_{5}\) and \(u_{6}\) in between a minimum damping factor \(d_{\text{min}}\) and a maximum damping factor \(d_{\text{max}}\)  
\[d_{\text{min}}\ \leq u_{i}\leq d_{\text{max}},\ \ i\in\{3,4,5,6\}. \tag{21}\]  
For the description of the manipulated variables, temporal polynomials defined over the prediction horizon are used to take into account the temporal course within the optimization [5]. The definition of the polynomial degree is done with respect to the desired characteristics. The manipulated variables of the counter roll torques at the front and rear axle \(u_{1}\) and \(u_{2}\), respectively, are defined as cubic polynomials:  
\[u_{1}(k)=a_{11}+a_{12}k+a_{13}k^{2}+a_{14}k^{3}, \tag{22}\]  
\[u_{2}(k)=a_{21}+a_{22}k+a_{23}k^{2}+a_{24}k^{3}. \tag{23}\]  
The manipulated variables representing the damping factors \(u_{3}\), \(u_{4}\), \(u_{5}\) and \(u_{6}\) are specified by quadratic polynomials:  
\[u_{3}(k)=a_{31}+a_{32}k+a_{33}k^{2}, \tag{24}\]  
\[u_{4}(k)=a_{41}+a_{42}k+a_{43}k^{2}, \tag{25}\]  
\[u_{5}(k)=a_{51}+a_{52}k+a_{53}k^{2}, \tag{26}\]  
\[u_{6}(k)=a_{61}+a_{62}k+a_{63}k^{2}. \tag{27}\]  
For the further description, the following notation is used:  
\[\mathbf{u}(k)=(u_{1}(k),\dots,u_{6}(k))^{\text{T}}, \tag{28}\]  
\[\mathbf{a}=(a_{11},\dots,a_{63})^{\text{T}}. \tag{29}\]  
The manipulated variables are grouped in the vector \(\mathbf{u}\) and the parameters of the polynomials are grouped in the vector \(\mathbf{a}\). Furthermore, the predicted vehicle dynamic states of the roll angle \(\varphi\), the pitch angle \(\theta\) and the self-steering gradient \(SSG\) are summarized in the vector \(\mathbf{x}\):  
\[d_{\text{min}}\ \leq u_{i}\leq d_{\text{max}},\ \ i\in\{3,4,5,6\}. \tag{21}\]0  
The reference variables of the central predictive control are given in \(x_{\text{Ref}}\). These result from the generation of the reference trajectories:  
\[d_{\text{min}}\ \leq u_{i}\leq d_{\text{max}},\ \ i\in\{3,4,5,6\}. \tag{21}\]1A dynamic roll angle specification is used for the control objective of the active roll stabilization. A non-linear roll model with passive chassis elements is used for this purpose:  
\[ \begin{array}{l}\frac{1}{l_{xx}}[h_{GR}ma_{y}\cos\varphi_{p}(k)+h_{GR}mg\sin \varphi_{p}(k)-2(s_{S,f}^{2}c_{S,f}+s_{S,r}^{2}c_{S,r})\sin\varphi_{p}(k)\\ -\left(d_{p,f}s_{D,f}^{2}+d_{p,r}s_{D,r}^{2}\right)\dot{\varphi}_{p}(k)\cos \varphi_{p}(k)\\ -2\frac{c_{S,f}s_{D,f}}{b_{S,f}}\arcsin\left(\frac{a_{S,f}}{B_{S,f}}\sin \varphi_{p}(k)\right)\\ -2\frac{c_{S,f}s_{D,f}}{b_{S,f}}\arcsin\left(\frac{a_{S,f}}{B_{S,f}}\sin \varphi_{p}(k)\right)\end{array}  \tag{32}\]  
Here, \(d_{p,i}\) represents the damping factors of the passive dampers. The passive stabilizers are characterized by the stiffnesses \(c_{S,i}\), the effective lengths \(a_{S,i}\) and the lever arms \(b_{S,i}\). The double integration of the passive roll acceleration \(\ddot{\varphi}_{p}\) by the explicit Euler method yields the corresponding roll angle \(\varphi_{p}\).