# Adaptive Semi-Active Suspension and Cruise Control through LPV Technique
## 3 Modeling and Control Synthesis of the Suspension System  
A number of studies have already been developed related to the active and semi-active suspension system based on the quarter-car model with different methods. The methods such as \(\mathcal{H}_{\infty}\) and linear-quadratic suspension control can provide guarantee good performance in both vehicle stability and passenger comfort but due to their fixed weighting of performance property, a dynamic control configuration is not possible. Hence, the Linear Parameter Varying (LPV) framework is a good solution for the dynamic control configuration to have road adaptation capabilities. Here, the semi-active suspension control is founded on the Linear Parameter Varying (LPV) framework, as already introduced in the previous studies [20; 21]. A model of the used two-degree-of-freedom quarter-car model that is more useful for researches is shown in Figure 3.  
The dynamic model of the quarter-car semi-active suspension system and the actuator is written as follows:  
\[m_{s}\dot{q_{1}}+b_{s}(\dot{q_{1}}-\dot{q_{2}})+k_{s}(q_{1}-q_{2})+F_{mr}=0 \tag{11a}\] \[m_{u}\dot{q_{2}}+b_{s}(q_{2}-q_{1})+k_{s}(q_{2}-w)+k_{s}(q_{2}-q_ {1})-F_{mr}=0, \tag{11b}\]  
where the actuator dynamics is approximated as  
\[\dot{F}_{mr}=-\frac{1}{\tau}F_{mr}+\frac{1}{\tau}u. \tag{12}\]  
Here, \(m_{s}\) and \(m_{u}\) represent the quarter vehicle's sprung mass and the unsprung mass, \(k_{t}\) and \(k_{s}\) state the stiffness of tire and spring, \(b_{s}\) is the damping rate of the shock absorber, while \(u=F\) is the control input of the system and \(F_{mr}\) is the control force generated by the actuator. Note, that \(q_{1}\) and \(q_{2}\) are the vertical displacement of the sprung mass and the unsprung mass, while \(w\) expresses road disturbance. The state vector \(x\) is selected as \(x= \begin{bmatrix}x_{1}&x_{2}&x_{3}&x_{4}&x_{5}\end{bmatrix} ^{T}\), in which the components are shown in Equation (13). \[ \begin{split} x_{1}&=q_{1}\\ x_{2}&=q_{2}\\ x_{3}&=q_{1}\\ x_{4}&=q_{2}\\ x_{5}&=F_{mr},\end{split}  \tag{13}\]  
where \(q_{1}\) and \(q_{2}\) are vertical displacement of sprung and unsprung masses, \(q_{1}\) and \(q_{2}\) are vertical velocity of sprung and unsprung masses. Thereafter, the system is given with Equation (11) is transformed into state-space representation form as follows:  
\[\dot{x} =Ax+B_{1}w+B_{2}u \tag{14a}\] \[z =C_{1}x+D_{11}w+D_{12}u\] (14b) \[y =C_{2}x+D_{21}w+D_{22}u. \tag{14c}\]  
Figure 3: Quarter-car model. The main idea in order to find matrices is arranging the Equation (11) according to state vector components, performance vector components and the measured signal. Then, results are arranged according to Equation (14). and coefficient of the \(x,w,u\) are clarified. These the coefficients(\(A,B_{1},B_{2},C_{1},C_{2},D_{11},D_{12},D_{21},D_{22}\)) give us the matrices for the control design. These matrices are found below:  
\[\text{A}= \begin{bmatrix}0&0&1&0&0\\ 0&0&0&1&0\\ -\frac{k_{x}}{m_{a}}&\frac{k_{x}}{m_{a}}&-\frac{b_{x}}{m_{a}}&\frac{b_{x}}{m_{a }}&-\frac{1}{m_{a}}\\ \frac{k_{x}}{m_{a}}&-\frac{k_{x}}{m_{a}}-\frac{k_{x}}{m_{a}}&\frac{k_{x}}{m_{a }}&-\frac{b_{x}}{m_{a}}&\frac{1}{m_{a}}\\ 0&0&0&0&-\frac{1}{\tau}\end{bmatrix} \]  
\[\text{B1}= \begin{bmatrix}0&0&0&\frac{k_{x}}{m_{a}}&0\end{bmatrix} ^{T},\quad \text{B2}= \begin{bmatrix}0&0&0&0&\frac{1}{\tau}\end{bmatrix} ^{T},\quad\text{B }= \begin{bmatrix}B1&B2\end{bmatrix} \]  
\[\text{C1}= \begin{bmatrix}-\frac{k_{x}}{m_{s}}&\frac{k_{x}}{m_{s}}&-\frac{b_{x}}{m_ {a}}&\frac{b_{x}}{m_{a}}&-\frac{1}{m_{s}}\\ 1&-1&0&0&0\\ 0&1&0&0&0\\ 0&0&0&0&1\end{bmatrix} ,\quad\text{C2}= \begin{bmatrix}1&-1&0&0&0\end{bmatrix} ,\quad\text{C}= \begin{bmatrix}C1\\ C2\end{bmatrix} \]  
\[D_{11}= \begin{bmatrix}0&0&-1&0\end{bmatrix} ^{T},D_{12}= \begin{bmatrix}0&0&0&0\end{bmatrix} ^{T},D_{21}= \begin{bmatrix}0\end{bmatrix} ,D_{22}= \begin{bmatrix}1\times 10^{-10}\end{bmatrix} ,\text{D}= \begin{bmatrix}D_{11}&D_{12}\\ D_{21}&D_{22}\end{bmatrix}