# Hierarchical Synchronization Control Strategy of Active Rear Axle Independent Steering System
## 3 Hierarchical Synchronization Control Strategy Design
### Steering Angle Tracking Controller  
LADRC is a control method for estimating and compensating uncertainties. The LADRC method takes the internal disturbance caused by the model parameter perturbation of the system together with the external disturbance of the system as the total disturbance of the system. It is observed by LESO and compensated to the controller. It is a kind of control method which is not completely dependent on the system model. It is suitable for angle tracking control of an ARIS system with uncertainty and nonlinearity. It can be seen from Equation (1) that the plant is a nonlinear coupled dynamic model. Firstly, the terms in the model are written in the form of equivalent total disturbance except for the second-order term and the control input term in the model. Then, the plant is decoupled into the second-order single input single output plant. Furthermore, the total disturbance is considered as an extended new state, which is estimated by LESO in real time. Finally, the observation disturbance is compensated to the linear feedback control law and the final control output is obtained. Because the left and right wheels are decoupled by the LADRC control method, the steering angle tracking controller can be designed separately. In the framework of LADRC, Equation (1) can be rewritten as  
\[\tilde{\delta}_{ri}=b_{0i}u_{i}+f_{i} \tag{13}\]  
Figure 4: The virtual rack and pinion steering gear model of the rear axle. where \(b_{0i}=b_{r}/J_{i}\),\(f_{i}=(M_{fi}+M_{zi}+M_{wi}-c_{i}\dot{b}_{ri}-k_{i}\delta_{ri})/J_{i}u_{i}=F_{i} \cdot\cos\delta_{ri}\), and \(i=1\),\(2\). The total disturbance term \(f_{i}\) in the model can be defined as an extended state, and the system state vector can be defined as  
\[x=\left[ \begin{array}{cccccccc}x_{1}&x_{2}&x_{3}&x_{4}&x_{5}&x_{6}\end{array}  \right]^{T}=\left[ \begin{array}{cccccccc}\delta_{r1}&\dot{\delta}_{r1}&f_{1}& \delta_{r2}&\dot{\delta}_{r2}&f_{2}\end{array} \right]^{T} \tag{14}\]  
Equation (13) can be written in the form of state space equation, given by  
\[\left\{ \begin{array}{l}\dot{x}=\mathbf{A}x+\mathbf{B}u+\dot{\mathbf{F}}\\ y=\mathbf{C}x\end{array} \right. \tag{15}\]  
where \(y\) is the system output, \(u=\left[ \begin{array}{c}u_{1}\\ u_{2}\end{array} \right]\) is the control input, \(f=\left[ \begin{array}{c}f_{1}\\ f_{2}\end{array} \right]\), \(\mathbf{A}=\left[ \begin{array}{cc}A_{1}&0\\ 0&A_{2}\end{array} \right]\), \(\mathbf{B}=\left[ \begin{array}{cc}B_{1}&0\\ 0&B_{2}\end{array} \right]\)\(\mathbf{C}=\left[ \begin{array}{cc}C_{1}&0\\ 0&C_{2}\end{array} \right]\), \(\mathbf{E}=\left[ \begin{array}{cc}E_{1}&0\\ 0&E_{2}\end{array} \right]\) are the coefficient matrix with \(A_{i}=\left[ \begin{array}{cc}0&1&0\\ 0&0&1\\ 0&0&0\end{array} \right]\), \(B_{i}=\left[ \begin{array}{cc}0\\ 0\\ 0\end{array} \right]\)\(C_{i}=\left[ \begin{array}{cc}1\\ 0\\ 0\end{array} \right]^{T}\), \(E_{i}=\left[ \begin{array}{c}0\\ 0\\ 1\end{array} \right]\), \(i=1,2\). Define \(z=\left[ \begin{array}{cccccccc}z_{1}&z_{2}&z_{3}&z_{4}&z_{5}&z_{6}\end{array}  \right]^{T}\) as the observed state of system. According to the theory of state observer, LESO can be designed as follows  
\[ \begin{array}{ll}\dot{z}&=\mathbf{A}z+\mathbf{B}u+\mathbf{L}(y-\mathbf{C}z)\\ &=(\mathbf{A}-\mathbf{L}\mathbf{C})z+\mathbf{B}u+\mathbf{L}y\end{array}  \tag{16}\]  
where \(\mathbf{L}_{6\times 2}=[L_{1}\ L_{2}]^{T}\) is the gain matrix of the observer, \(L_{1}=L_{2}=[\beta_{1}\ \beta_{2}\ \beta_{3}]^{T}\). Assuming that the total disturbance \(f\) is bounded, and an appropriate observer gain \(\mathbf{L}\) can be selected to make \(\mathbf{A}\)-\(\mathbf{L}\mathbf{C}\) asymptotically stable, then state \(\mathbf{x}\) can be estimated. To simplify parameter tuning process, \(L_{1}\) and \(L_{2}\) are set to \([3\omega_{0}\ 3\omega_{0}{}^{2}\ \omega_{0}{}^{3}]^{T}\) by the method of pole configuration. \(\omega_{o}\) is called the observer bandwidth. By introducing the estimated disturbance state \(z_{3}\), the controlled system is compensated as a linear integral series system.