# Nonlinear Control Design of a Half-Car Model Using Feedback Linearization and an LQR Controller
## 3 Modelling  
In quarter car model, only the linear motion of the body is taken without considering the rotational motion of wheel and body so, there are 4 states in this model.In half car model, rotational motion is considered along with the linear motion of the body and has 8 states while in full car model all the motions are considered so, it has 16 states in total. Better analysis of suspension system can be carried out by use of half car models as they are a fair approximation of full car model as compared to quarter car models. Half-car model presented in [8], is shown in Figure 2. It is presented as a Linear 4-Degree of Freedom(DOF) system. A single sprung mass (chassis) is connected to two unsprung masses (both wheels) at each corner. Vertical and pitch motion is applicable for sprung mass while only vertical motion w.r.t chassis is applicable for both the masses of wheels. The suspensions modelled between chassis and both the wheels are conventional springs and dampers while tires are designed as conventional springs without dampness.  
Figure 1: Flowchart of Research.  
By applying the [PERSON]'s law of motion and state of equilibrium, the following state equations are derived.  
\[\dot{x_{1}} =x_{2} \tag{1}\] \[\dot{x_{2}} =\frac{1}{m_{s}}[-(B_{f}+B_{r})x_{2}+(aB_{f}-bB_{r})x_{4}cosx_{3}-K_ {f}x_{5}\] \[+B_{f}x_{6}-K_{r}x_{7}+B_{r}x_{8}+(f_{f}+f_{r})+K_{tf}z_{rf}+K_{tr} z_{rr}]\] (2) \[\dot{x_{3}} =x_{4}\] (3) \[\dot{x_{4}} =\frac{1}{J_{y}}[(aB_{f}-bB_{r})x_{2}cosx_{3}-(a^{2}B_{f}+b^{2}B_{ r})x_{4}cos^{2}x_{3}\] \[+aK_{r}x_{5}cosx_{3}-aB_{f}x_{6}cosx_{3}-bK_{r}x_{7}cosx_{3}+bB_{ r}x_{8}cosx_{3}\] \[+(af_{f}+bf_{r})cosx_{3}+K_{tf}z_{rf}+K_{tr}z_{rr}]\] (4) \[\dot{x_{5}} =x_{2}-ax_{4}cosx_{3}-x_{6}\] (5) \[\dot{x_{6}} =\frac{1}{m_{uf}}[-K_{tf}x_{1}+B_{f}x_{2}+aK_{tf}sinx_{3}-aB_{f}x _{4}cosx_{3}\] \[+(K_{f}+K_{tf})x_{5}-B_{f}x_{6}+K_{tf}z_{rf}-f_{f}]\] (6) \[\dot{x_{7}} =x_{2}+bx_{4}cosx_{3}-x_{8}\] (7) \[\dot{x_{8}} =\frac{1}{m_{uf}}[-K_{tr}x_{1}+B_{r}x_{2}-bK_{tr}sinx_{3}-bB_{r}x _{4}cosx_{3}\] \[+(K_{r}+K_{tr})x_{7}-B_{r}x_{8}+K_{tr}z_{rr}-f_{r}] \tag{8}\]  
the state variables of the system along with their description is shown in Table 1.  
It can be observed from the state equation that the system presented above has coupled state equations which is hard to treat. Since the linear system holds the beauty of easy analysis, the complex form can be converted into a simple linearized form. Then, the linear controller can be implemented to  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**States** & **State Variables** & **Variable Names** \\ \hline \(x_{1}\) & \(z\) & Vertical Displacement \\ \(x_{2}\) & \(z\) & Vertical Velocity \\ \(x_{3}\) & \(\theta\) & Angular Displacement \\ \(x_{4}\) & \(\dot{\theta}\) & Angular Velocity \\ \(x_{5}\) & \(z_{sf}-z_{uf}\) & Front Suspension Travel \\ \(x_{6}\) & \(z_{uf}\) & Velocity of Front Unsprung Mass \\ \(x_{7}\) & \(z_{sr}-z_{ur}\) & Rear Wheel Suspension Travel \\ \(x_{8}\) & \(z_{ur}\) & Velocity of Rear Unsprung Mass \\ \hline \hline \end{tabular}
\end{table}
Table 1: State Variables and Description.  
Figure 2: Half car model of ASS [8].  
control it. the general model for this specified nonlinear system is shown in Figure 3. This system block is a nonlinear system with state equations \(x(t)\). It has force input \(u(t)\) that contain both \(f_{f}\) and \(f_{r}\). Its output is \(y(t)\). The road disturbances \(z_{rf}\) and \(z_{rr}\) effect the system states depending upon the peak value which will lead to the serious problems of ride-discomfort and poor handling performance of road. Therefore, this whole block needs to be linearized and controlled so that the effect off road disturbances on all the states should be minimum.