## 2 Active Suspension Systems and Control Systems
### LQG Controller Design of Active Suspension  
The two main functions of the vehicle suspension are to ensure that the vehicle has good ride comfort and handling stability when driving. At the same time, the working space of suspension needs to be controlled within a certain range and meet the requirements of body posture during accelerating, decelerating, braking, or steering. Therefore, for the 2-DOF 1/4 vehicle suspension model, the three parameters of body vertical acceleration (BA), suspension working space (SWS), and dynamic tire deflection (DTD) are selected as performance indexes to quantitatively evaluate the suspension performance.  
The state vector of the system is:  
\[X=\left[\begin{array}{cccccc}\dot{z}_{b}&\dot{z}_{w}&z_{b}&z_{w}&z_{g}\\ \end{array}\right]^{T} \tag{4}\]  
The output vector of the system is:  
\[Y=\left[\begin{array}{cccccc}\ddot{z}_{b}&z_{b}-z_{w}&z_{w}-z_{g}\\ \end{array}\right]^{T} \tag{5}\]  
where \(\ddot{z}_{b}\) is body vertical acceleration, \(z_{b}-z_{w}\) is suspension working space, \(z_{w}-z_{g}\) is dynamic tire deflection.  
By combining Equations (1)-(3), and writing the system motion equation and the road input equation in the form of a matrix, the space state equation of the system can be obtained:  
\[\dot{X}=AX+BU+FW \tag{6}\]  
\[Y=CX+DU \tag{7}\]  
Figure 3: Time domain response curve of the composite road.  
\[\text{where, }A=\left[\begin{array}{cccc}-\frac{C_{s}}{m_{0}}&\frac{C_{s}}{m_{0}}&- \frac{K_{s}}{m_{0}}&\frac{K_{s}}{m_{0}}&0\\ \frac{C_{s}}{m_{0}}&-\frac{C_{s}}{m_{0}}&\frac{K_{s}}{m_{0}}&\frac{-K_{s}}{m_{0} }&\frac{K_{s}}{m_{0}}\\ 1&0&0&0&0\\ 0&1&0&0&0\\ 0&0&1&0&-2\pi f_{0}\end{array}\right],\,B=\left[\begin{array}{c}\frac{1}{m_ {0}}\\ -\frac{1}{m_{0}}\\ 0\\ 0\\ 0\end{array}\right],\,F=\left[\begin{array}{c}0\\ 0\\ 0\\ 2\pi n_{0}\sqrt{G_{0}u}\end{array}\right],\]  
where \(A\) is the state matrix of the system, \(B\) is the input matrix of the control vector, \(F\) is the road input matrix, \(C\) is the output matrix, \(D\) is the transfer matrix, \(U=[U_{a}(t)]\) is the control input matrix of active suspension, and \(W=[w(t)]\) is the gaussian white noise input matrix.  
The comprehensive index \(J\) of the LQR controller is defined with the three performance indexes as follows:  
\[J=\lim_{T\rightarrow\infty}\frac{1}{T}\int_{0}^{T}\big{[}q_{1}(z_{w}(t)-z_{g} (t))^{2}+q_{2}(z_{b}(t)-z_{w}(t))^{2}+q_{3}\bar{z}_{b}(t)\big{]}dt \tag{8}\]  
where, \(q_{1}\), \(q_{2}\), and \(q_{3}\) are the weighting coefficient of DTD, SWS, and BA. The selection of the weighting coefficient reflects the designer's tendency towards different performances. Due to these three weights essentially representing the relative values among them, the weighting coefficient \(q_{3}\) of BA is usually set to 1 for the convenience of calculation.  
Rewrite the above expression (8) into a matrix form:  
\[J=\lim_{T\rightarrow\infty}\frac{1}{T}\int_{0}^{T}\big{(}X^{T}QX+U^{T}RU+2X^{ T}NU\big{)}dt \tag{9}\]  
where, \(Q=\left[\begin{array}{cccc}\frac{C_{s}^{2}}{m_{0}^{2}}&-\frac{C_{s}^{2}}{m_ {0}^{2}}&\frac{C_{s}K_{s}}{m_{0}^{2}}&-\frac{C_{s}K_{s}}{m_{0}^{2}}&0\\ -\frac{C_{s}^{2}}{m_{0}^{2}}&\frac{C_{s}^{2}}{m_{0}^{2}}&-\frac{C_{s}K_{s}}{m_ {0}^{2}}&GA is a global search algorithm developed by imitating the mechanism of biological evolution, which is based on natural selection and genetic theory. This algorithm adopts the principle of survival of the fittest, elimination of inferior. In the iterative process, the population evolves towards the optimal direction through the steps of selection, crossover, and mutation and finally obtains the optimal solution that meets the requirements. Therefore, we applied GA to optimize the LQR controller of active suspension and take the solution as the reference value for the weighting space construction in our GKL algorithm.