# 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 lateral vibration state equation can be expressed as:  
\[\dot{X}=AX+BU+EW \tag{8}\]  
\[Y=CX+DU+E_{1}W \tag{9}\]The state variable of the system is: \(X=\left[X_{b}X_{a}\dot{X}_{b}-H_{f}M_{f}H_{f}^{T}C_{1}(D_{l}+D_{r})\dot{X_{a}}\right]\); the active control of the system is: \(U=[F_{l}F_{r}]^{T}\); the disturbance of the system is: \(W=[D_{l}D_{r}]^{T}\); the output of the system is: \(Y=\left[\ddot{X_{a}}\quad X_{d}\quad X_{e}\right]\); where:  
\[X_{d}=X_{b}-\frac{(D_{l}+D_{r})}{2}X_{e}=X_{a}-X_{b} \tag{10}\]  
In Equation (10), \(\ddot{X_{a}}\) represents the acceleration of the car body, \(X_{d}\) represents the offset from the centerline, \(X_{e}\) represents the offset between the car body and frame. In Equations (8) and (9):  
\[A= \begin{bmatrix}0&0&I_{2\times 2}&0\\ 0&0&I_{2\times 2}&I_{2\times 2}\\ -2H_{f}M_{f}^{-1}\left(H_{f}^{T}K_{1}+H_{p}^{T}K_{2}H_{p}H_{f}^{-1}\right)&2H _{f}M_{f}^{-1}H_{p}^{T}K_{2}&-2H_{f}M_{f}^{-1}\left(H_{f}^{T}C_{1}+H_{p}^{T}C_ {2}H_{p}H_{f}^{-1}\right)&2H_{f}M_{f}^{-1}H_{p}^{T}C_{2}\\ 2H_{c}M_{c}^{-1}H_{c}^{T}K_{2}H_{p}H_{f}^{-1}&-2H_{c}M_{c}^{-1}H_{c}^{T}K_{2}& 2H_{c}M_{c}^{-1}H_{c}^{T}C_{2}H_{p}H_{f}^{-1}&-2H_{c}M_{c}^{-1}H_{c}^{T}C_{2} \end{bmatrix} \]  
\[B= \begin{bmatrix}0&0\\ 0&0\\ H_{f}M_{f}^{-1}H_{f}^{T}&-H_{f}M_{f}^{-1}H_{f}^{T}\\ 0&0\end{bmatrix} \]  
\[C=\begin{bmatrix}2H_{c}M_{c}^{-1}H_{c}^{T}K_{2}H_{p}H_{f}^{-1}&-2H_{c}M_{c}^{- 1}H_{c}^{T}K_{2}&2H_{c}M_{c}^{-1}H_{c}^{T}C_{2}H_{p}H_{f}^{-1}&-2H_{c}M_{c}^{- 1}H_{c}^{T}C_{2}\\ I&0&0&0\\ -\mathrm{I}&1&0&0\\ D= \begin{bmatrix}0&0\\ 0&0\\ 0&0\end{bmatrix} \mathrm{E}_{1}= \begin{bmatrix}2H_{c}M_{c}^{-1}H_{c}^{T}C_{2}H_ {p}M_{f}^{-1}H_{c}^{T}C_{1}&2H_{c}M_{c}^{-1}H_{c}^{T}C_{2}H_{p}M_{f}^{-1}H_{f}^ {T}C_{1}\\ -\mathrm{I}&0&0\\ H_{f}M_{f}^{-1}H_{f}^{T}C_{1}&H_{f}M_{f}^{-1}H_{f}^{T}C_{1}\\ 0&0\\ H_{f}M_{f}^{-1}H_{f}^{T}C_{1}&0\\ 0&0\end{bmatrix} \]  
\[E=\begin{bmatrix}H_{f}M_{f}^{-1}\left(H_{f}^{T}\left(K_{1}-2C_{1}H_{f}M_{f}^{- 1}H_{f}^{T}C_{1}\right)-2H_{p}^{T}C_{2}H_{p}M_{f}^{-1}H_{f}^{T}C_{1}\right)&H _{f}M_{f}^{-1}\left(H_{f}^{T}\left(K_{1}-2C_{1}H_{f}M_{f}^{-1}H_{f}^{T}C_{1} \right)-2H_{The sensitivity of the human body to horizontal vibration is higher than that of vertical vibration[14]. Therefore, when suppressing vibration, it is sufficient to use the horizontal vibration as the control target.  
The RMS of acceleration at the car body's bottom and the relative displacement is selected as \(J_{1}\). The RMS of the control force output by the actuator is taken as the control cost \(J_{2}\).  
\[J_{1}=\left\|w_{1}\ddot{X}_{a}\quad w_{2}X_{d}\quad w_{3}X_{e}\right\|_{2}J_{2}= \left\|w_{4}U\right\|_{2} \tag{11}\]  
where \(w_{1}\), \(w_{2}\), \(w_{3}\), \(w_{4}\) are the regularization coefficients. We can adjust \(w_{1}\), \(w_{2}\), \(w_{3}\) to change the weight of the corresponding output in the objective function \(J_{1}\). \(w_{4}\) is used to adjust \(J_{2}\) for subsequent analysis. The multi-objective integrated design model for elevator lateral vibration control is as follows:  
\[ \begin{array}{c}min:J_{1}\text{ and }J_{2}\\ subjectto:p=max(Re(eig(LFT(G,C))))<0\\ q\in(q_{0},q_{1})\\ Designvariable:q\end{array}  \tag{12}\]  
where \(q\) represents the corresponding parameter of C (s), including Kp, Ki, Kd, the integral order \(\lambda\) and differential order \(\mu\), constraint \(p\) represents the maximum real part of the eigenvalue of \(LFT(G,C)\) to ensure system stability; \(q_{0}\) and \(q_{1}\) represent the corresponding parameters' upper and lower boundaries.