## 2 Mathematical Model of the Suspension System  
The mathematical model that will be utilized to control the system was obtained using the Lagrange-Euler approach. The quarter-car model is shown in Figure 1. Our quarter-car model has two degrees of freedom. In this model, the car is designed symmetrically and is divided into four parts. The model includes only vertical vibration motion without considering the pitch and roll motion of the chassis and wheel. The model of the suspension system is given in the following equations.  
Figure 1: Quarter-car model.  
Here \(m_{b}\) is the mass of a quarter of the car body, \(m_{w}\) is the mass of the wheel assembly, \(k_{s}\) is the spring coefficient of the suspension system, \(b_{s}\) is the damper coefficient of the suspension system, \(k_{t}\) is the spring coefficient of the tire, and \(b_{t}\) is the damper coefficient of the tire. In addition, while the state variables show the displacement (\(x_{w}\)) movements of the body (\(x_{b}\)) and the wheel assembly, the input variables \(x_{r}\) show the road roughness (distortion) and \(F\) the control force exerted by the active element applied between the body and the wheel.  
\[m_{b}\ddot{x}_{b}+b_{s}\big{(}\dot{x}_{b}-\dot{x}_{w}\big{)}+k_{s}(x_{b}-x_{w}) -F_{s}=0 \tag{1}\]  
\[m_{w}\ddot{x}_{w}+b_{t}\big{(}\dot{x}_{w}-\dot{x}_{r}\big{)}+k_{t}(x_{w}-x_{r}) -b_{s}\big{(}\dot{x}_{b}-\dot{x}_{w}\big{)}-k_{s}(x_{b}-x_{w})-F_{s}=0 \tag{2}\]  
\[\ddot{x}_{b}=-\frac{1}{m_{b}}(b_{s}\big{(}\dot{x}_{b}-\dot{x}_{w}\big{)}+k_{s} (x_{b}-x_{w})-F_{s}) \tag{3}\]  
\[\ddot{x}_{w}=-\frac{1}{m_{w}}(b_{t}\big{(}\dot{x}_{w}-\dot{x}_{r}\big{)}+k_{t} (x_{w}-x_{r})-b_{s}\big{(}\dot{x}_{b}-\dot{x}_{w}\big{)}-k_{s}(x_{b}-x_{w})-F_ {s}) \tag{4}\]  
\[x_{1}=x_{b};x_{2}=x_{w};x_{3}=\dot{x}_{b};x_{4}=\dot{x}_{w}; \tag{5}\]  
\[\dot{x}_{1}=x_{2}\]  
\[\dot{x}_{2}=-\frac{1}{m_{b}}(b_{s}(x_{3}-x_{4})+k_{s}(x_{1}-x_{2})-F_{s}) \tag{6}\]  
\[\dot{x}_{3}=x_{4}\]  
\[\dot{x}_{4}=-\frac{1}{m_{w}}(b_{t}(x_{4}-\dot{x}_{r})+k_{t}(x_{2}-x_{r})-b_{s} (x_{3}-x_{4})-k_{s}(x_{1}-x_{2})-F_{s})\]  
The dynamic model of the linearly moving system is constructed in the state space form \(\dot{x}=Ax+Bu\) as follows.  
\[\begin{bmatrix}\dot{x}_{1}\\ \dot{x}_{2}\\ \dot{x}_{3}\\ \dot{x}_{4}\end{bmatrix}=\begin{bmatrix}0&1&0&0\\ -\frac{k_{s}}{m_{b}}&-\frac{b_{s}}{m_{b}}&\frac{k_{s}}{m_{b}}&\frac{b_{s}}{m_{ b}}\\ 0&0&0&1\\ \frac{k_{s}}{m_{w}}&\frac{b_{s}}{m_{w}}&\frac{-k_{s}-k_{t}}{m_{w}}&-\frac{b_{s}}{m _{w}}\\ \end{bmatrix}\begin{bmatrix}x_{1}\\ x_{2}\\ x_{3}\\ x_{4}\end{bmatrix}+\begin{bmatrix}0&0\\ 0&\frac{1}{m_{b}}\\ 0&0\\ \frac{k_{t}}{m_{w}}&-\frac{1}{m_{w}}\\ \end{bmatrix} \tag{7}\]  
\[y=\begin{bmatrix}1&0&0&0\\ 1&0&-1&0\\ -\frac{k_{s}}{m_{b}}&-\frac{b_{s}}{m_{b}}&\frac{k_{s}}{m_{b}}&\frac{b_{s}}{m_ {b}}\\ \end{bmatrix}\begin{bmatrix}x_{1}\\ x_{2}\\ x_{3}\\ x_{4}\end{bmatrix}+\begin{bmatrix}0&0\\ 0&0\\ 0&\frac{1}{m_{b}}\\ \end{bmatrix} \tag{8}\]  
The dynamics of the hydraulic actuator used in active suspension control are shown with the transfer function given in Equation (9).  
\[H_{act}(s)=\frac{1}{1/60s+1} \tag{9}\]  
The active suspension road disturbance is modeled as a 5 cm high road bump using Equation (10) for the input signal.  
\[r(t)=0.025\times[1-\cos(8\pi t)] \tag{10}\]  
Road disturbances with different characteristics, quality of sensing sensors required for feedback, limitations on available control force, and body acceleration are some factors that affect passenger comfort, represented by \(\mathtt{a}_{b}\), and suspension deflection, represented by \(s_{d}\). This article presents the control algorithms applied to the quarter-car active suspension model in a closed-loop system with the measurement of body acceleration \(\ddot{x}_{b}\) as feedback.