# Adaptive Semi-Active Suspension and Cruise Control through LPV Technique
## 3 Modeling and Control Synthesis of the Suspension System
\]  
Table 1 shows the nominal parameters of the quarter-car model for the rear and front suspension. In this study, the unmodelled dynamics \(\Delta\) are also considered with \(|\Delta(\omega_{1})|=0.25\) at low frequencies and \(|\Delta(\omega_{2})|=1\) at high frequencies, and is assumed to be stable with the norm condition \(\|\Delta\|_{\infty}<1\). It is necessary to define the performance specifications in order to achieve a desired trade-off between ride comfort and road holding, while also taking control force into consideration. The acceleration of sprung mass of the vehicle must be minimized for the sake of increasing passenger comfort with the following optimization criterion:  
\[z_{1}=\hat{q_{1}}\to 0.\]  
Stability of the vehicle is guaranteed with the minimization of the suspension deflection:  
\[z_{2}=(q_{1}-q_{2})\to 0.\]  
In order to decrease variations of side force to guarantee stability, the dynamic tire load must be minimized with the optimization criterion:  
\[z_{3}=(q_{2}-w)\to 0.\] \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Parameters** & **Front** & **Rear** & **Unit** \\
**(Symbols)** & **Suspension** & **Suspension** & \\ \hline sprung mass (\(m_{s}\)) & 214 & 336 & kg \\ unsprung mass (\(m_{u}\)) & 40 & 40 & kg \\ suspension stiffness (\(k_{s}\)) & 30 & 60 & kN/m \\ tire stiffness (\(k_{i}\)) & 220 & 220 & kN/m \\ damping (\(b_{s}\)) & 50 & 50 & N/m/s \\ time constant (\(\tau\)) & 0.01 & 0.01 & s \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Parameters of the suspension. To avoid actuator saturation, the control force must also be considered with the optimization criterion:  
\[z_{4}=F\to 0.\]  
These performances are inserted in a performance vector  
\[z= \begin{bmatrix}z_{1}&z_{2}&z_{3}&z_{4}\end{bmatrix} ^{T}.\]  
The measured signal is the relative displacement between the masses, that is, \(y=x_{1}-x_{2}\). The control input \(u\) is the vertical force generated by the MR damper, with the dynamics listed in (12). The presented high-level controller is founded on a weighting strategy formulated through a closed-loop architecture shown in Figure 4.  
Here, \(G\) expresses the quarter-car control-oriented model defined in (14), \(K\) denotes the designed Linear Parameter Varying (LPV) controller characterized with the scheduling variable \(\rho\) responsible for control reconfiguration, \(u\) is the control input, \(y\) is the measured output, \(n\) is the measurement noise, \(z\) represents the performance outputs and \(w\) is the road disturbance. The uncertainties of the quarter-car model are considered with a weighting function of \(W_{r}\) and \(\Delta\). The weighting functions need to be considered as penalty functions, hence weights should be large where small signals are desired and vice versa. The goal of weighting function \(W_{n}\) is to represent sensor noises with \(n\) measurement noise. The weighting function \(W_{w}\) denotes the road disturbances with \(w\) road disturbance. Weighting functions for performances \(W_{p}\) is responsible to keep performances which are the sprung mass acceleration (\(W_{p_{a}}\)), suspension deflection (\(W_{p_{d}}\)), tire deformation (\(W_{p_{d}}\)) and control input (\(W_{p_{a}}\)) small over the required frequency range. \(W_{p_{a}}\) represents passanger comfort, \(W_{p_{d}}\) denotes directional stability and \(W_{p_{s}}\) presents dynamic tire load. Since the predefined performance designations may collide with each other, performance weighting functions \(W_{p}\) need to be designed in a way that a relevant trade-off can be guaranteed between them. Additionally, to guarantee control reconfiguration in case one of the predefined performances becomes more important due to estimated future road conditions, a scheduling variable \(\rho\in[0.01,0.99]\) is introduced to shape weighting function  
Figure 4: Closed-loop interconnection structure. \(W_{p_{a}}\), \(W_{p_{a}}\) and \(W_{p_{b}}\).