# An Optimized Fractional-Order PID Horizontal Vibration Control Approach for a High-Speed Elevator
## 2 Dynamic Model of Horizontal Vibration of High-Speed Elevator  
The elevator car is composed of a car body and a car frame. The car body is connected to the car frame through support rubber, and the four guide shoes guide the elevator to move. The guide shoes and the support rubber work together to reduce vibration. The elevator car is a relatively complex multi-degree of freedom vibration system. Referring to the experimentally validated model [14], we optimized the complex model [25] to obtain an elevator model that meets our requirements. The car system is simplified to the following lateral vibration dynamics model, as shown in Figure 1.  
In the model, \(m_{c}\) and \(I_{c}\) are the car body's mass and rotational inertia, \(m_{f}\) and \(I_{f}\) are the car frame's mass and rotational inertia. \(k_{1}\) and \(c_{1}\) are the stiffness and damping coefficient of the active guide shoes, while \(k_{2}\) and \(c_{2}\) are the stiffness and damping coefficients of the  
Figure 1: Dynamic Model of Elevator Lateral Vibration. supporting rubber. The active control force is \(F_{i}(i=1\cdots 4)\); The guide shoe displacement excitation is \(d_{i}(i=1\cdots 4)\);  
\[X_{a}= \begin{bmatrix}x_{1}\\ x_{2}\end{bmatrix} X_{b}= \begin{bmatrix}x_{3}\\ x_{4}\end{bmatrix} X_{c}= \begin{bmatrix}x_{c}\\ \theta_{c}\end{bmatrix} X_{f}= \begin{bmatrix}x_{f}\\ \theta_{f}\end{bmatrix}  \tag{1}\]  
\(x_{i}(i=1\cdots 4)\) is the horizontal displacement of corresponding points. \(x_{c}\), \(x_{f}\) are the displacement of the car body's barycenter and frame's barycenter, respectively. \(\theta_{c}\), \(\theta_{f}\) are the rotation angle of the car body's barycenter and frame's barycenter, respectively. \[ \begin{array}{l}x_{1}=x_{c}+l_{1}\theta_{c}\\ x_{2}=x_{c}-l_{2}\theta_{c}\\ x_{3}=x_{f}+l_{3}\theta_{f}\\ x_{4}=x_{f}-l_{4}\theta_{f}\end{array}  \tag{2}\]  
\(x_{2}=l_{1}\),\(l_{2}\) is the distance between the rubber and the car body's barycenter. \(l_{3}\),\(l_{4}\) are distance between active guide shoes and the car frame's barycenter. To facilitate writing, the following transformations are made:  
\[X_{a}=H_{c}X_{c}\,X_{b}=H_{f}X_{f} \tag{3}\]  
Including:  
\[H_{c}= \begin{bmatrix}1&l_{1}\\ 1&-l_{2}\end{bmatrix} ,\;H_{f}= \begin{bmatrix}1&l_{3}\\ 1&-l_{4}\end{bmatrix}  \tag{4}\]  
Define transformation matrix:  
\[H_{p}= \begin{bmatrix}1&l_{5}\\ 1&-l_{6}\end{bmatrix}  \tag{5}\]  
\(l_{5}\) and \(l_{6}\) represents the distance from the rubber to the car frame's barycenter. The dynamic equations for the car body and car frame in the horizontal direction are as follows:  
\[M_{c}\ddot{X}_{c}=2H_{c}^{T}\Big{(}-K_{2}X_{a}-C_{2}\dot{X}_{a}+K_{2}H_{p}X_{ f}+C_{2}H_{p}\dot{X}_{f}\Big{)} \tag{6}\]  
\[ \begin{array}{l}M_{f}\ddot{X}_{f}=2H_{f}^{T}\Big{(}-K_{1}X_{b}-C_{1}\dot{X}_ {b}+K_{1}D_{l}+C_{1}\;D_{l}^{\text{-}}+K_{1}D_{r}+C_{1}\dot{D}_{r}+F_{l}-F_{ r}\Big{)}\\ \qquad\qquad\qquad-2H_{p}^{T}\Big{(}-K_{2}X_{a}-C_{2}\dot{X}_{a}+K_{2}H_{p}X_{ f}+C_{2}H_{p}\dot{X}_{f}\Big{)}\end{array}  \tag{7}\]  
In Equations (6) and (7), \(M_{c}=diag(m_{c},l_{c})\), \(M_{f}=diag\Big{(}m_{f},I_{f}\Big{)}\), \(K_{1}=diag(k_{1},k_{1})\), \(K_{2}=diag(k_{2},k_{2})\), \(C_{1}=diag(c_{1},c_{1})\), \(C_{2}=diag(c_{2},c_{2})\), \(F_{l}= \begin{bmatrix}F_{1}\\ F_{3}\end{bmatrix} \), \(F_{r}= \begin{bmatrix}F_{2}\\ F_{4}\end{bmatrix} \), \(D_{l}= \begin{bmatrix}d_{1}\\ d_{3}\end{bmatrix} \), \(D_{r}= \begin{bmatrix}d_{2}\\ d_{4}\end{bmatrix} \). By selecting appropriate state variables, the above motion equation can be transformed into a system state equation. In this paper, the state equation is obtained by fusing the differential terms of rail displacement excitation into the state equation.