Gas–Solid Coupling Dynamic Modeling and Transverse Vibration Suppression for Ultra-High-Speed Elevator

Abstract

When in operation, ultra-high-speed elevators encounter transverse vibrations due to uneven guide rails and airflow disturbances, which can greatly undermine passenger comfort. To alleviate these adverse effects and boost passenger comfort, a gas–solid coupling dynamic model for ultra-high-speed elevator cars is constructed, and a vibration suppression approach is proposed. To start with, the flow field model of the elevator car-shaft under different motion states is simulated, and the calculation formula of air excitation is derived. Next, by incorporating the flow field excitation into the four degrees of freedom dynamic model of the separation between the car and the frame, a transverse vibration model of the elevator car based on gas–solid coupling is established. Finally, an LQR controller is used to suppress elevator transverse vibration, and a multi-objective optimization algorithm is applied to optimize the parameters of the weight matrix to obtain the optimal solution of the LQR controller. A set of controllers with moderate control cost and system performance meeting the requirements was selected, and the effectiveness of the controller was verified. Compared with other methods, the proposed LQR-based method has greater advantages in suppressing the transverse vibration of ultra-high-speed elevators. This work provides an effective solution for enhancing the ride comfort of ultra-high-speed elevators and holds potential for application in the vibration control of high-speed transportation systems.

1. Introduction

With growing building height, ordinary elevators are unable to meet the needs of people in super high-rise buildings anymore. Faster and safer elevators have become urgently needed for such buildings. The emergence of ultra-high-speed elevators has resolved the aforementioned needs. Nevertheless, the higher the elevator’s speed during operation, the more intense the vibration, and passengers standing inside the car can sense a decline in comfort [1]. Meanwhile, it also diminishes the elevator’s lifespan. In fact, the most notable factor causing this situation is the transverse vibration of the elevator car, which is chiefly caused by two factors: the guide system and airflow interference.

To alleviate adverse effects and upgrade passenger comfort, a multitude of methods has been advanced. Presently, vibration reduction for ultra-high-speed elevator cars chiefly depends on two means: passive and active vibration attenuation. Given that passive vibration attenuation is usually inflexible and insufficiently effective, active suppression has turned into the primary method for vibration control [2]. Zhang et al. [3] devised an active car shock absorber incorporating a linear motor, formulated a five degrees of freedom space vibration model, and implemented a back propagation neural network PID controller in conjunction with a linear prediction model. Tusset et al. [4] probed into the transverse response of vertical transportation with nonlinearities under guide rail deformation excitation and adopted an LQR control strategy using MR dampers to uplift passenger comfort. In Zhang et al.’s [5] investigation, an electro-hydraulic active guide shoe was conceived to tackle car transverse vibration triggered by guide unevenness excitation. The Takagi–Sugeno fuzzy reasoning method was utilized to approximate the nonlinear car system model, and an adaptive gain H_∞ output feedback control strategy was proposed, hinging on the parallel distributed compensation rule, to curtail transverse vibration. Su et al. [6] posited a transient response feedback control strategy with specified performance, centering on problems like subpar transient convergence and the real-time performance of current active control algorithms for car system transverse vibration. Ge et al. [7] tendered a fuzzy sliding mode-based active disturbance rejection (ADR) control method, adept at governing the horizontal vibration of high-speed elevator car systems attributable to factors such as irregular guide rails and the nonlinear damping of springs in guide shoes.

In the study of high-speed elevator transverse vibration mentioned in the above references, active control methods are mainly used to reduce transverse vibration. Yet, ultra-high-speed elevators are influenced not just by rail excitation but also by airflow disturbances. Reference [8] points out that the greater the elevator speed, the more significant the airflow impact. Currently, an increasing number of scholars have investigated the airflow disturbance of high-speed elevators. Zhang et al. [9] looked into the airflow characteristics in various hoistways. Qiao et al. [10] performed the theoretical modeling of super-high-speed elevators based on Bernoulli’s principle for unsteady flow and analyzed the parameter impact of the car-induced airflow. Zhang et al. [11] came up with a multi-region dynamic layering method to research the hoistway airflow and the rational ventilation hole opening. Cui et al. [12] set up and numerically simulated a full-scale 3D model of the shaft with different blocking ratios, aiming to probe into the aerodynamic features of the elevator car’s high-speed movement and the airflow law in the shaft. Qiu et al. [13] presented an elevator car air pressure compensation method grounded in internal–external flow field (IE-FF) coupling analysis, which is beneficial for adaptively tracking the ideal car interior air pressure curve and controlling air pressure fluctuation.

Currently, some studies have taken the gas–solid coupling method into account. Qiu et al. [14] created the high-speed elevator car transverse vibration gas–solid coupling model, integrating guide excitations and air excitations, and set up the numerical car-hoistway interface region model. Shi et al. [8] developed a gas–solid coupling model for the whole high-speed elevator operating process using the finite volume method, Lagrange’s theorem, and so on. The actual testing of a 7 m/s high-speed elevator confirmed the feasibility of the model and modeling method.

Recent advances in elevator vibration control predominantly adopt two isolated approaches: Mechanical-focused methods [3,4,5,6,7] actively suppress rail-induced vibrations but neglect aerodynamic coupling effects, which should not be ignored, especially at high speeds [8,12]. Aerodynamic-focused methods [9,10,11,12,13] optimize airflow fields yet lack integration with real-time vibration control systems. This disconnect creates a critical gap: unmodeled fluid–structure interactions may destabilize controllers at ultra-high speeds. Our work bridges this by unifying high-fidelity CFD with LQR control, providing a new framework for control under gas–solid coupling.

This paper uses a more intuitive and straightforward approach, namely, the numerical fitting of aerodynamic load, to avoid the complexity and inefficiency of calculations. Considering gas–solid coupling, the formula linking aerodynamic load to working conditions is derived via simulation. This aerodynamic load is then incorporated into the dynamic equation to finish gas–solid coupling modeling. Moreover, this paper innovatively applies a multi-objective genetic algorithm to optimize LQR control parameters, effectively suppressing ultra-high-speed elevator lateral vibration.

The key contributions of this study are outlined as follows:

• A gas–solid coupling dynamic model for ultra-high-speed elevators is established, and a control method based on the LQR has been proposed.
• Simulation analysis of the flow field characteristics inside high-speed elevator shafts is executed, leading to a calculation formula for air excitation corresponding to the elevator car motion state, with an R-square as high as 0.999.
• The key weight matrix Q and R parameters in the LQR controller are optimized utilizing a multi-objective genetic algorithm, considering both control performance and cost of the elevator.
• When the speed is 6 m/s, 8 m/s, and 10 m/s, compared with PID control, the proposed LQR-based method can significantly reduce the transverse acceleration by 7.55%, 8.47%, and 10.27%. The higher the speed, the more effective the proposed LQR-based method is.

The rest of this paper is structured as follows: Section 2 introduces external excitations, including rail excitation and air excitation. Section 3 is the dynamic modeling and state equation representation of ultra-high-speed elevators. Then, in Section 4, the method for suppressing transverse vibrations is mainly presented. Section 5 compares the obtained results with various situations to validate the effectiveness of the proposed method. Finally, the conclusion is presented in Section 6.

2. Modeling of External Excitation

Manufacturing and installation errors unavoidably lead to rail unevenness, which is a primary cause of vibration in ultra-high-speed elevators. Moreover, during the ascending or descending of ultra-high-speed elevator cars within the hoistway, they are subjected to varying aerodynamic pressures from the airflow inside the shaft, which intensifies the transverse vibrations. Therefore, this section focuses on analyzing two main elements influencing the transverse vibration of ultra-high-speed elevators: guide excitation and air excitation. The guide excitation is illustrated by white noise signals filtered by a low-pass filter, whereas the air excitation is acquired through Fluent simulation and formula fitting.

2.1. Air Excitation Modeling

2.1.1. Turbulence Model

During operation, the ultra-high-speed elevator is restricted from performing lifting movements in the shaft, and the air around the ultra-high-speed elevator in the shaft is driven by the elevator’s operation. The air part inside the shaft, excluding the elevator, is analyzed as a whole flow field, which exhibits a low Mach number (Ma < 0.3) and high Reynolds number (Re > 2300) state [15]. This flow field can be analyzed as incompressible turbulence. Therefore, the flow field inside the shaft can be described utilizing the Navier–Stokes equations, which include two parts: momentum conservation and mass conservation. The equations and coefficients of the turbulence model are provided in Appendix A.

2.1.2. Numerical Simulation of the Airflow Within the Shaft

Figure 1 presents the model of the elevator shaft and car. In an ideal scenario, the elevator car should be positioned on the vertical centerline of the shaft. If the car and car frame have uniform shape and mass distribution, the flow field’s pressure and velocity distributions around them would be symmetrical. This symmetry would cause the air-generated excitations on the car to cancel out in all directions, resulting in a zero net force on the car. However, achieving this ideal state is challenging. Installation errors and the car’s operational vibrations often cause it to deviate from the centered, symmetrical position. Consequently, the car experiences additional air excitations.

SIMPLE is a classic semi-implicit method. This paper utilizes the second-order upwind scheme and SIMPLE algorithm for steady-state flow field simulation in Ansys Fluent.

In order to analyze the factors affecting air excitation under asymmetric car positions, the three-dimensional model needs to be established based on Figure 1 and simulated and solved utilizing Ansys Fluent 2022 R2. For the convenience of calculation, the subsequent simplifications are implemented:

(1)

The elevator car and the elevator frame are considered to be symmetrically installed with a rectangular shape, and their centers of mass are located on the centerline of the hoistway.

(2)

The guide shoes and shock absorbers are simplified to a spring–damper system.

(3)

The influence of the hoisting ropes on the car’s transverse vibrations is neglected.

The boundary conditions for solving the flow field are set as velocity inlet and boundary outlet, and no-slip conditions are set for all outer surfaces of the elevator car and the inner wall of the shaft.

Boundary conditions for airflow inlet:

$$u_y=v_s,u_x=0,{\displaystyle \frac{\mathsf{\partial }p}{\mathsf{\partial }x}}=0$$

(1)

Boundary conditions for airflow outlet:

$${\displaystyle \frac{\mathsf{\partial }{v}_y}{\mathsf{\partial }y}}={\displaystyle \frac{\mathsf{\partial }{u}_x}{\mathsf{\partial }x}}=0$$

(2)

As the elevator operates within the hoistway, the airflow exerts aerodynamic drag on the car’s outer surface. As the car moves within the shaft, the variation in the flow field region complicates the solution of the flow field [16]. Based on the principle of relative motion, in the simulation, air enters from one side of the shaft entrance at the same speed as the car, flows through the car, and exits from the shaft outlet [17]. The top and bottom of the car are positioned 5 m away from the upper and lower openings of the hoistway, respectively. The boundary conditions for the flow field solution are set as velocity inlet and boundary outlet, with all the elevator car’s outer surfaces and the shaft’s inner walls set to no-slip conditions.

In order to speed up the calculation, after dividing the entire flow field into grids according to the default method, additional grid refinement processing was performed on the car wall and shaft boundary [18,19] to ensure more accurate calculation results at the boundary.

Utilize the Fluent module for solving, adjust the convergence criterion to $1\times {10}^{-5}$, and the maximum number of iteration steps is 1000.

The equivalent transverse concentrated force, and concentrated moment $M_z$ have the most significant impact on the lateral vibration. By integrating the air pressure and viscous stress acting on the surface of the car body, the equivalent transverse concentrated force and concentrated moment $M_z,$ acting on the center of the car, can be obtained as follows [20]:

$$F_y=\underset{\mathrm{S}\mathrm{c}\mathrm{a}\mathrm{r}}{\int }pd{s}_{car}+\underset{\mathrm{S}\mathrm{c}\mathrm{a}\mathrm{r}}{\int }\tau d\left|S_{car}\right|$$

(3)

$$M_z=\int \left(R_a-R_o\right)d{F}_x$$

(4)

In the equation, $p$ is the airflow pressure, τ is the viscous stress, Scar is the area of the outer surface of the car, $R_a$ represents the position of the action surface unit, and $R_o$ represents the position of the car center of mass.

As noted in reference [21], the elevator car’s aerodynamic load is closely related to its operating speed and position deviation. Therefore, the simulation (uniform speed) will focus on the transverse force and overturning moment experienced by the elevator car’s center of mass under varying conditions of transverse displacement, deflection angle displacement, and rated speed. Ultimately, this study will explore the relationships between transverse force, overturning moment, transverse displacement, deflection angle displacement, and rated speed [22].

In this study, the transverse displacements ($y$) of the car body are taken as 2 mm, 4 mm, 6 mm, 8 mm, and 10 mm, combined with the running speeds ($v$) of 4 m/s, 6 m/s, 8 m/s, 10 m/s, and 12 m/s, respectively. The deflection angulars ($\theta$) are taken as 0.5°, 1°, 1.5°, 2°, and 2.5°, combined with the running speeds (v) of 4 m/s, 6 m/s, 8 m/s, 10 m/s, and 12 m/s, respectively. The results obtained are shown in

Table 1. Equivalent aerodynamic load of the car under different speeds and transverse displacement.

$\mathbf{Velocity}/\mathbf{m}·{\mathbf{s}}^{-1}$	$\mathbf{Aerodynamic} \mathbf{Load}/\mathbf{N}$	$\mathbf{Transverse} \mathbf{Displacement}/{10}^{-3}\mathbf{m}$
		2	4	6	8	10
4	$F_y/\mathrm{N}$	1.443328	2.820529	4.433297	5.847365	7.3607995
	$M_z/\mathrm{N}·\mathrm{m}$	0.618003	1.0108336	1.9515373	2.0938242	3.1336382
6	$F_y/\mathrm{N}$	3.295254	6.7790241	10.148966	13.594114	16.945068
	$M_z/\mathrm{N}·\mathrm{m}$	1.350664	2.5176488	4.2651951	5.1110839	6.820766
8	$F_y/\mathrm{N}$	5.714808	11.594293	17.849078	27.131752	30.916064
	$M_z/\mathrm{N}·\mathrm{m}$	2.568629	4.3573907	7.6928151	10.613392	13.341611
10	$F_y/\mathrm{N}$	9.473283	18.867489	27.128074	39.512653	48.962687
	$M_z/\mathrm{N}·\mathrm{m}$	3.877174	7.3896428	11.632615	15.696517	20.812077
12	$F_y/\mathrm{N}$	13.207638	26.184425	39.162721	52.128905	65.367814
	$M_z/\mathrm{N}·\mathrm{m}$	5.472908	11.094941	16.469281	22.089928	27.486092

and

Table 2. Equivalent aerodynamic load of the car at different speeds and deflection angles.

$\mathbf{Velocity}/\mathbf{m}·{\mathbf{s}}^{-1}$	Aerodynamic Load	Deviation Angle/°
		0.5	1	1.5	2	2.5
4	$F_y/\mathrm{N}$	17.251313	34.505746	52.871793	71.051558	91.897572
	$M_z/\mathrm{N}·\mathrm{m}$	7.868474	14.97594	22.961873	30.512319	37.802049
6	$F_y/\mathrm{N}$	38.561698	77.617425	118.40733	159.92379	205.83671
	$M_z/\mathrm{N}·\mathrm{m}$	17.462975	34.716924	51.357129	68.417756	84.439622
8	$F_y/\mathrm{N}$	68.351175	137.96199	209.92972	284.15439	364.88847
	$M_z/\mathrm{N}·\mathrm{m}$	30.941648	59.233599	90.979506	121.32131	149.41584
10	$F_y/\mathrm{N}$	106.55557	215.40127	327.46022	443.79444	569.06771
	$M_z/\mathrm{N}·\mathrm{m}$	48.171635	92.279483	141.81271	189.18662	232.75207
12	$F_y/\mathrm{N}$	153.12809	309.96903	471.02111	638.66358	818.20037
	$M_z/\mathrm{N}·\mathrm{m}$	69.106236	132.5533	203.83158	271.97316	334.42207

.

Utilizing the data in Table 1 and Table 2, plot the variation in aerodynamic load with different transverse displacements and running speeds of the car, as shown in Figure 2 and Figure 3.

Figure 2 demonstrates that, as the car’s transverse displacement and speed increase, both the transverse force and overturning moment on the car’s center of mass exhibit a consistent upward trend. The transverse force and overturning moment exhibit a linear relationship with the transverse displacement. Similarly, both transverse force and overturning moment are approximately proportional to the deflection angle.

By fitting the curve in Figure 4, it can be demonstrated that there is a high R-squared in the quadratic fitting. The ratios of transverse force and overturning moment to transverse displacement at different rated speeds are designated as the influence coefficients $C_{fx}$ and $C_{mx}$. Consider taking $C_{fx}$ and $C_{mx}$ as quadratic polynomials of $v$ and solving the coefficients of the polynomials utilizing the equation shown in Equation (11). Equation (12) is the result obtained by fitting.

$$\begin{array}{c}{C}_{fx}={a_{1}v}^2+b_{1}v+c_1\\ C_{mx}={a_{2}v}^2+b_{2}v+c_2\end{array}$$

(5)

$$\begin{array}{c}\begin{array}{c}{F}_y=\left(35.93v^2+160.94v-521.67\right)y\\ M_y=\left(16.84v^2+40.88v-153.83\right)y\end{array}\\ \begin{array}{c}{F}_{\theta }=\left(124.65v^2+11.48v-17.85\right)\theta \\ M_{\theta }=\left(52.90v^2+12.55v-17.94\right)\theta \end{array}\end{array}$$

(6)

The influence of air disturbance in the elevator system can be simplified as transverse force and overturning moment acting on the elevator car, and the equivalent aerodynamic load $F_g$ and aerodynamic torque $M_g$ will be applied to the ultra-high-speed elevator car, thereby establishing a transverse vibration model for the ultra-high-speed elevator car.

2.2. Guide Excitation Modeling

In elevator installation, achieving symmetrical alignment on both sides is crucial. However, guide rail manufacturing and installation errors can impact their flatness, leading to displacement excitations from bending, tilting, and step variations [14]. These excitations can be theoretically combined to approximate real-world scenarios, though parameter selection is challenging. For easier calculations, following reference [23], real guide rail excitations mainly occur in the low-frequency range below 10 Hz, determined by the guide rail’s properties and installation section length [24]. Thus, a low-pass filter processed white noise signal is chosen as the guide rail excitation, retaining only the low-frequency components [25], as shown in Equation (13).

$$w_d={\displaystyle \frac{2\pi \times 10}{s+2\pi \times 10}}$$

(7)

3. Gas–Solid Coupling Dynamic Modeling of Ultra-High-Speed Elevator

This section establishes a dynamic model of the car’s transverse vibration caused by external excitations, including rail excitation and fluctuating aerodynamic pressure, and develops the corresponding state-space equations.

3.1. Dynamic Modeling of Elevator Transverse Vibration

The transverse vibration of elevators primarily involves the car system and the guidance system. The car system comprises the car and its frame, linked by supporting rubber, while the guidance system consists of guide rails and shoes. Guide rails stabilize the car’s position, counteracting tilting and swaying from traction rope torsion and asymmetric loads. Active guide shoe controllers, along with spring–damping systems, dampen car vibrations, enhancing operational safety and comfort [26,27].

Guide shoes are pivotal in suppressing elevator transverse vibrations. Active guide shoes reduce vibrations by applying forces opposite to them, a common method in elevator vibration control. In dynamic modeling, guide shoes can be simplified as a spring–damping system, as shown in Figure 5. The ARG (active roller guide) in Figure 5 denotes the active control mechanism that applies active control force to the car.

To simplify subsequent calculations, similar variables are merged into a matrix:

$$Z_c=\left[\begin{array}{c}{z}_{cu}\\ z_{cd}\end{array}\right],X_c=\left[\begin{array}{c}{x}_c\\ {\theta }_c\end{array}\right],Z_f=\left[\begin{array}{c}{z}_{fu}\\ z_{fd}\end{array}\right],X_f=\left[\begin{array}{c}{x}_{fc}\\ {\theta }_{fc}\end{array}\right]$$

(8)

$z_{cu}$ and $z_{cd}$, respectively, represent the transverse displacement of the upper and lower parts of the elevator car;

$x_c$ and ${\theta }_c$, respectively, represent the transverse displacement and deflection angle of the center of mass position of the elevator car;

$z_{fu}$ and $z_{fd}$, respectively, represent the transverse displacement of the upper and lower parts of the car frame;

$x_{fc}$ and ${\theta }_{fc}$, respectively, represent the transverse displacement and deflection angle of the center of mass position of the car frame.

$Z_c$ and $Z_f$, as well as $X_c$ and $X_f$, can be represented by the equations shown in Equation (9).

$$Z_c=H_c{X}_c, Z_f=H_f{X}_f$$

(9)

where

$$H_c=\left[\begin{array}{cc}1& L_{cu}\\ 1& {-L}_{cd}\end{array}\right],H_f=\left[\begin{array}{cc}1& L_{fu}\\ 1& {-L}_{fd}\end{array}\right]$$

(10)

define

$$H_{fc}=\left[\begin{array}{cc}1& L_{fcu}\\ 1& {-L}_{fcd}\end{array}\right],H_t=\left[\begin{array}{cc}0& 0\\ 0& 1\end{array}\right]$$

(11)

Considering airflow disturbance, the transverse dynamic equation of gas–solid coupling in the ultra-high-speed elevator car is as follows:

$$\begin{array}{cc}\hfill M_f{\ddot{X}}_f=& {H_f}^T\left(F_{gl}+K_{gl}{Z}_{dl}+K_{gr}{Z}_{dr}+C_{gl}{\dot{Z}}_{dl}+C_{gr}{\dot{Z}}_{dr}\right)\hfill \\ \hfill & {{-H}_f}^T\left(\left(K_{gl}+K_{gr}\right)Z_f+\left(C_{gl}+C_{gr}\right){\dot{Z}}_f+F_{gr}\right)\hfill \\ \hfill & {{+H}_{fc}}^T\left(\left(K_{rl}+K_{rr}\right)Z_c+\left(C_{rl}+C_{rr}\right){\dot{Z}}_c\right)\hfill \\ \hfill & {{-H}_{fc}}^T\left(\left(K_{rl}+K_{rr}\right)H_{fc}{H_f}^{-1}{Z}_f+\left(C_{rl}+C_{rr}\right)H_{fc}{H_f}^{-1}{\dot{Z}}_f\right)\hfill \\ \hfill & +2l^2{H}_t\left(k_v\left({H_c}^{-1}{Z}_c-{H_f}^{-1}{Z}_f\right)+c_v\left({H_c}^{-1}{\dot{Z}}_c-{H_f}^{-1}{\dot{Z}}_f\right)\right)\hfill \end{array}$$

(12)

$$\begin{array}{cc}\hfill M_c{\ddot{X}}_c=& {H_c}^T\left(Z_{dF}-\left(K_{rl}+K_{rr}\right)Z_c-\left(C_{rl}+C_{rr}\right){\dot{Z}}_c\right)\hfill \\ \hfill & {+H_c}^T\left(\left(K_{rl}+K_{rr}\right)H_{fc}{H_f}^{-1}{Z}_f+\left(C_{rl}+C_{rr}\right)H_{fc}{H_f}^{-1}{\dot{Z}}_f\right)\hfill \\ \hfill & -2l^2{H}_t\left(k_v\left({H_c}^{-1}{Z}_c-{H_f}^{-1}{Z}_f\right)+c_v\left({H_c}^{-1}{\dot{Z}}_c-{H_f}^{-1}{\dot{Z}}_f\right)\right)\hfill \end{array}$$

(13)

The coefficient matrices are derived from the dynamic model parameters (see Appendix A for full matrix definitions).

3.2. Establishment of System State Equation

Utilizing x as the state variable equation of the system,

$$x={\left[\begin{array}{cccc}{Z}_f& Z_c& {\dot{Z}}_f-H_f{M}_f{H_f}^T\left(C_{gl}{Z}_{dl}+C_{gr}{Z}_{dr}\right)& {\dot{Z}}_c\end{array}\right]}^T$$

(14)

The calculation formulas for guide rail excitation, air excitation, and active control force matrix $d_g$, $d_F$, and $u_g$ are as follows:

$$\begin{array}{c}{d}_g=\left[\begin{array}{c}{Z}_{dl}\\ Z_{dr}\end{array}\right]\\ d_F=Z_{dF}\\ u_g=\left[\begin{array}{c}{F}_{gl}\\ F_{gr}\end{array}\right]\end{array}$$

(15)

Then, the transverse vibration dynamic Equation (13) of the elevator car system can be written in the form of the system state Equation (16):

$$\dot{x}=A\left(p\right)x+B_{1g}\left(p\right)d_g+B_{1F}\left(p\right)d_F+B_{2g}\left(p\right)u_g$$

(16)

The complete coefficient matrix is detailed in the Appendix A.

The output z can be represented by Equation (17).

$$\left\{\begin{array}{c}\mathit{z}={\left[\begin{array}{ccc}{\ddot{\mathit{Z}}}_c& {\mathit{Z}}_{sd}& {\mathit{Z}}_{cd}\end{array}\right]}^T\\ {\mathit{Z}}_{sd}={\mathit{Z}}_f-{\displaystyle \frac{{\mathit{Z}}_{dl}+{\mathit{Z}}_{dr}}{2}}\\ {\mathit{Z}}_{cd}={\mathit{Z}}_c-{\mathit{Z}}_f\end{array}\right.$$

(17)

where ${\ddot{\mathit{Z}}}_c$ represents the transverse vibration acceleration experienced by the top and bottom sections of the elevator car, ${\mathit{Z}}_{sd}$ represents the displacement of the frame’s top and bottom from the central axis, and ${\mathit{Z}}_{cd}$ represents the relative lateral displacement between the elevator car and the elevator frame.

This will result in the following output Equation (18):

$$\mathit{z}={\mathit{C}}_z\left(p\right)\mathit{x}+{\mathit{D}}_{\mathrm{z}\mathrm{g}1}\left(p\right){\mathit{d}}_g+{\mathit{D}}_{\mathrm{z}\mathrm{F}1}\left(p\right){\mathit{d}}_F+{\mathit{D}}_{\mathrm{z}g2}\left(p\right){\mathit{u}}_g$$

(18)

The complete coefficient matrix is also detailed in Appendix A.

Next, it is essential to include the excitation from the changing flow field during motion into the calculations, employing the concept of time discretization [28], and to separately solve for the flow field excitation and the elevator model at each time step [29], as is shown in Figure 6.

4. Proposed LQR-Based Vibration Suppression Method

Since LQR can account for multiple control objectives at once and attain a multi-objective balance by optimizing the cost function, thereby adapting to the dynamics of complex systems, this paper uses the LQR-based method to suppress transverse vibration. Meanwhile, for better results, a multi-objective optimization algorithm was used to optimize the weight matrix parameters, obtaining the optimal LQR controller solution. The control process diagram of the gas–solid coupling model is shown in Figure 7.

4.1. Linear Quadratic Regulator

LQR (Linear Quadratic Regulator) belongs to the use of linear quadratic optimal control. Before applying LQR design, it is necessary to define a time-domain continuous performance functional. The following Equation (19) is commonly used for infinite time-domain performance functions:

$$J={\displaystyle \frac{1}{2}}{\int }_0^{\mathrm{\infty }}\left[{x\left(t\right)}^{T}Qx\left(t\right)+{u\left(t\right)}^{T}Ru\left(t\right)\right]dt$$

(19)

Optimal control aims to identify a controller that minimizes the performance function. The state cost weight matrix Q and control cost weight matrix R, typically diagonal, are to be designed. And these two terms each have their own weight coefficients. Under the same error, the larger the weight, the greater the proportion of this term in $J$ [30,31].

The optimal control law $u=-Kx$ is obtained by solving the algebraic Riccati equation:

$$\begin{array}{c}{A}^{T}P+PA+Q-PB{R}^{-1}{B}^{T}P=0\\ K=R^{-1}{B}^{T}P\end{array}$$

(20)

The derivation process is detailed in Appendix A.

As per LQR theory, the closed-loop system is asymptotically stable if (A, B) is controllable and Q and R are positive definite [32,33].

4.2. Multi-Objective Genetic Algorithm

The multi-objective genetic algorithm (MOGA) involves several key definitions [34]:

Definition 1: Pareto Solution

Consider a multi-objective optimization problem with objective function $F\left(X\right)=\left\{f_1\left(x\right),f_2\left(x\right),f_3\left(x\right),\dots f_k\left(x\right)\right\}$ where $k\ge 2$. For two solution vectors x₀ and x₁ in the solution space, x₀ dominates x₁ if there exists an $i\in \left[1,k\right]$ such that $f_i\left(x_0\right)>f_i\left(x_1\right)$, and for all $i\in \left[1,k\right]$, $f_i\left(x_0\right)>f_i\left(x_1\right).$ A solution is a Pareto optimal solution if no other solution in the space is better. The set of all such solutions is the Pareto optimal set, expressed as follows:

$$PS=\left\{u\in \Omega |\nexists v\in \Omega ,\mathrm{v}\succ \mathrm{u}\right\}$$

(21)

Here, u is a solution vector, and Ω is the solution space.

Definition 2: Non-inferior solution sorting.

This sorting is crucial in MOGA as it guides the optimization search direction. The process is as follows:

For solution space S, pairwise comparison yields the non-inferior set $S₁$.

Update $S=S-S₁$, and repeat to find $S₂$.

Continue until S is empty.

Solutions in S₁ are first-level non-inferior; those in S₂ are second-level, etc. S₁ contains the optimal solutions.

In each generation, selection, crossover, and mutation operations were applied to the population to generate a new one. Then, fitness calculation and non-dominated solution ranking were performed, repeating until completion.

This approach has been successfully applied in complex dynamic systems such as milling stability prediction [35], where Pareto-optimal solutions balance conflicting objectives like machining efficiency and vibration suppression.

4.3. Design and Parameter Optimization of LQR Controller

Based on the gas–solid coupling dynamic model of the ultra-high-speed elevator established in Section 3.2, the cost function is established as follows:

$$J={\int }_0^{\infty }{x}^{T}Qx+u^{T}Ru dt$$

(22)

Q and R are both weight matrices, where ${\mathrm{Q}=diag(q}_1,q_2,q_3,q_4,q_5,q_6,q_7,q_8$), $R=diag(r_1,r_2,r_3,r_4)$. The size of the diagonal elements in matrices $Q$ and $R$ determines the weight of each corresponding element in $x\left(t\right)$ and $u\left(t\right)$ during the optimization process, respectively.

The main evaluation criteria for vibration suppression in ultra-high-speed elevators are the car’s transverse acceleration and displacement and the displacement relative to the frame. To meet these criteria, the weight matrix Q’s corresponding elements can be set to higher values for better control effects. Some scholars have successfully optimized LQR controllers using optimization algorithms [36,37].

Adjusting Q and R matrices can tailor the system’s control performance. However, manual tuning becomes impractical when there are too many parameters, making the design process complex. Therefore, leveraging the multi-objective genetic algorithm from Section 4.2, we can design a fitting fitness function to optimize the two weighting matrices, achieving an LQR controller with desired performance.

Firstly, design the fitness function as follows:

$$\left\{\begin{array}{c}{J}_p={‖\begin{array}{c}{r}_1{\ddot{\mathit{Z}}}_{cd}\\ r_2{\mathit{Z}}_{sd}\\ r_3{\mathit{Z}}_{cd}\end{array}‖}_2\\ J_c={‖r_4\mathit{u}‖}_2\end{array}\right.$$

(23)

where $r_1, r_2, r_3$ and $r_4$ are regularization coefficients, and the performance cost $J_p$ is calculated from the H2 norm of the ${\ddot{\mathit{Z}}}_{cd}$, ${\mathit{Z}}_{sd}$ and ${\mathit{Z}}_{cd}$. The control cost $J_c$ is calculated from the H2 norm of the control force.

A simulation control system for ultra-high-speed elevators is built using the Simulink toolbox. In MATLAB, the LQR controller is solved using the lqr() function and the system state equation:

$$K=lqr\left(A,B,Q,R\right)$$

(24)

The multi-objective genetic algorithm was used to optimize the weight matrix Q and R parameters. Since the optimization’s main goal is to suppress the car’s transverse acceleration, the weight matrix parameters’ upper and lower limits were set as follows:

$l_b$ = [${10}^{-6}$, ${10}^{-6}$, ${10}^{-6}$, ${10}^{-6}$, ${10}^{-6}$, ${10}^{-6},{10}^{-6},{10}^{-6},{10}^{-8},{10}^{-8},{10}^{-8},{10}^{-8}$],

$u_b$ = [${10}^2$, ${10}^2$, ${10}^6$, ${10}^6$, ${10}^2$, ${10}^2,{10}^6,{10}^6,{10}^{-5},{10}^{-5},{10}^{-5},{10}^{-5}$].

The maximum allowable control cost $J_{c max}$ was set to 600. The population size was 400, and the maximum number of generations was 500. The regularization coefficients are $r_1={10}^3,r_2=10,r_3=2\times {10}^{-2},r_4=1\times {10}^{-3}$. Optimize and iterate the parameters in the weight matrix to obtain the Pareto optimal solution curve of the parameters in the weight matrix, as shown in Figure 8.

In Figure 8, the horizontal axis denotes the control system’s performance indicators, and the vertical axis indicates the control cost. The curve is a Pareto curve made up of optimized controller parameters. Curves closer to the vertical axis mean smaller system output errors, while those nearer to the horizontal axis mean lower control costs. The elevator structural parameters are shown in

Table 3. Elevator structure parameter values.

Parameters	Unit	Value
$m_c$	$\mathrm{k}\mathrm{g}$	1100
$I_c$	$\mathrm{k}\mathrm{g}·{\mathrm{m}}^2$	1600
$m_{fc}$	$\mathrm{k}\mathrm{g}$	2400
$I_{fc}$	$\mathrm{k}\mathrm{g}·{\mathrm{m}}^2$	8600
$k_v$	$\mathrm{N}/\mathrm{m}$	900,000
$c_v$	$\mathrm{N}·\mathrm{s}/\mathrm{m}$	3,600,000
$k_{ri}(i=1,2)$	$\mathrm{N}/\mathrm{m}$	200,000
$k_{ri}(i=3,4)$	$\mathrm{N}/\mathrm{m}$	170,000
$c_{ri}(i=1,2)$	$\mathrm{N}·\mathrm{s}/\mathrm{m}$	2300
$c_{ri}(i=3,4)$	$\mathrm{N}·\mathrm{s}/\mathrm{m}$	1800
$k_{gi}(i=1\dots 4)$	$\mathrm{N}/\mathrm{m}$	120,000
$c_{gi}(i=1\dots 4)$	$\mathrm{N}·\mathrm{s}/\mathrm{m}$	2000
$L_{cu}$	$\mathrm{m}$	1.6
$L_{cd}$	$\mathrm{m}$	1.4
$L_{fu}$	$\mathrm{m}$	3.3
$L_{fd}$	$\mathrm{m}$	4.5
$L_{fcu}$	$\mathrm{m}$	1.25
$L_{fcd}$	$\mathrm{m}$	1.75
$l$	$\mathrm{m}$	0.5

.

5. Results and Discussion

5.1. Comparison Results of Control Methods

In order to compare the performance of the LQR controller before and after optimization, empirical weight matrices are used as a benchmark. Additionally, traditional PID control and no control are selected for comparison to confirm the method’s efficiency.

From the optimization results using the MOGA, a weight matrix with a control cost of approximately 45 is chosen as the optimized LQR parameter. The corresponding Q and R matrices are given by the following:

$Q$ = diag [57.327, 40.264, 335,248.304, 370,400.807, 54.426, 67.531, 566,579.656, 822593.37]

$R$ = diag [4.782 × 10⁻⁶, 3.076 × 10⁻⁷, 3.773 × 10⁻⁶, 3.4050 × 10⁻⁷]

Based on experience and control objectives, this paper manually sets another set of weight matrices to solve the LQR controller:

$Q_1$ = diag [0, 1, 100, 100, 1, 1, 100, 100]

$R_1$ = diag [1 × 10⁻⁶, 1 × 10⁻⁶, 1 × 10⁻⁶, 1 × 10⁻⁶]

Multivariable PID is used to compare with LQR-based methods. Since the input state variable is an 8 × 1 vector and the control output is a 4 × 1 vector, the proportion P, integral I, and derivative D in multivariate PID are a 4 × 8 matrix. Here, the multivariable PID controller employs gain matrices. After multiple iterations of simulation, the relationship between the gain matrix $M$ and $P$, $I$, and $D$ is set as follows:

$$K_P=2M, K_I=3M, K_D=0.01M$$

(25)

where the base matrix $M=\left[\begin{array}{c}\begin{array}{c}\begin{array}{ccc}\begin{array}{ccc}{10}^5& {10}^5& -{10}^5\end{array}& \begin{array}{cc}-{10}^5& {10}^4\end{array}& \begin{array}{ccc}{10}^3& {10}^3& {10}^3\end{array}\end{array}\\ \begin{array}{ccc}\begin{array}{ccc}{10}^6& {10}^6& {-10}^6\end{array}& \begin{array}{cc}-{10}^6& {10}^4\end{array}& \begin{array}{ccc}{10}^4& {10}^5& {10}^5\end{array}\end{array}\end{array}\\ \begin{array}{c}\begin{array}{ccc}\begin{array}{ccc}{10}^5& -{10}^5& {10}^5\end{array}& \begin{array}{cc}{10}^5& -{10}^4\end{array}& \begin{array}{ccc}-{10}^4& -{10}^4& -{10}^4\end{array}\end{array}\\ \begin{array}{ccc}\begin{array}{ccc}{10}^6& -{10}^5& {10}^6\end{array}& \begin{array}{cc}{10}^6& -{10}^4\end{array}& \begin{array}{ccc}-{10}^4& -{10}^5& -{10}^5\end{array}\end{array}\end{array}\end{array}\right]$.

The relevant coefficients are explained as follows:

Force-dominant rows (1–2): High-magnitude terms (10⁵–10⁶) in columns 1–4 resist excitation. Displacement-focused rows (3–4): Negative coupling terms compensate for car–frame interactions. Velocity compensation: Lower-gain terms (10³–10⁴) in columns 5–8 match vibration energy at different positions.

After selecting the elevator speed, Equation (6) is used to obtain air excitation, which is then added to the control model. According to the weight matrices Q, R, Q¹, and R¹, the feedback gain matrices K and K¹ are calculated using the lqr() function in MATLAB 2023b. These matrices are then substituted into the same control model to obtain the numerical simulation results of the LQR controller before and after optimization.

The results at speed $v=10 \mathrm{m}/\mathrm{s}$ are as follows: Figure 9, Figure 10 and Figure 11 show the simulation results of the output $\mathit{z}$ (the transverse vibration acceleration of the bottom of the car, the displacement of the bottom of the frame off the centerline, the relative transverse displacement between the bottom car and the bottom frame) at elevator speed $v=10 \mathrm{m}/\mathrm{s}$.

Figure 9 compares the lateral acceleration of the elevator under four control methods. It shows that the optimized LQR controller effectively reduces the lateral acceleration. Figure 10 compares the displacement of the car frame off the centerline. The three active control methods have almost no effect on controlling this displacement because this part’s proportion in the Q weight setting is not large, and the car frame offset is almost not suppressed. Figure 11 shows the comparison of the displacement of the car relative to the frame. The peak value obtained by the optimized LQR method is the smallest among the four control cases.

The root mean square values of system performance indicators, control costs, lateral acceleration, etc., under four control methods are shown in

Table 4. Comparison of system output performance before and after optimization (v = 10 m/s).

	Optimized LQR Control	Unoptimized LQR Control	PID Control	Without Control
Performance index (${\mathit{J}}_{\mathit{p}}$)	5.192 × 103	1.621 × 104	5.787 × 103	2.141 × 104
Control cost (${\mathit{J}}_{\mathit{c}}$)	41.943	4.726	24.832	0
Displacement of the car frame off the centerline (${\mathit{Z}}_{sd}$)	8.653 × 10−4	8.705 × 10−4	8.931 × 10−4	8.894 × 10−4
Displacement of the car relative to the frame (${\mathit{Z}}_{cd}$)	2.006 × 10−4	5.538 × 10−4	3.278 × 10−4	5.702 × 10−4
Transverse acceleration at the bottom of the car (${\ddot{\mathit{Z}}}_c$)	0.037	0.114	0.041	0.130
Control force (${\mathit{F}}_{\mathbf{g}1}$)	90.558	17.502	54.633	0
Control force (${\mathit{F}}_{\mathit{g}2}$)	114.787	17.502	91.056	0
Control force (${\mathit{F}}_{\mathit{g}3}$)	215.523	16.261	168.792	0
Control force (${\mathit{F}}_{\mathit{g}4}$)	194.744	16.261	168.792	0

,

Table 5. Comparison of system output performance before and after optimization (v = 8 m/s).

	Optimized LQR Control	Unoptimized LQR Control	PID Control	Without Control
Performance index (${\mathit{J}}_{\mathit{p}}$)	4.583 × 103	1.838 × 104	5.007 × 103	2.141 × 104
Control cost (${\mathit{J}}_{\mathit{c}}$)	40.302	5.342	24.887	0
Displacement of the car frame off the centerline (${\mathit{Z}}_{sd}$)	8.706 × 10−4	8.724 × 10−4	9.082 × 10−4	9.361 × 10−4
Displacement of the car relative to the frame (${\mathit{Z}}_{cd}$)	2.131 × 10−4	5.470 × 10−4	3.349 × 10−4	5.961 × 10−4
Transverse acceleration at the bottom of the car (${\ddot{\mathit{Z}}}_c$)	0.032	0.130	0.035	0.151

and

Table 6. Comparison of system output performance before and after optimization (v = 6 m/s).

	Optimized LQR Control	Unoptimized LQR Control	PID Control	Without Control
Performance index (${\mathit{J}}_{\mathit{p}}$)	4.161 × 103	1.638 × 104	4.495 × 103	2.176 × 104
Control cost (${\mathit{J}}_{\mathit{c}}$)	38.871	5.641	24.172	0
Displacement of the car frame off the centerline (${\mathit{Z}}_{sd}$)	8.453 × 10−4	8.670 × 10−4	8.721 × 10−4	9.957 × 10−4
Displacement of the car relative to the frame (${\mathit{Z}}_{cd}$)	2.336 × 10−4	5.636 × 10−4	3.535 × 10−4	6.155 × 10−4
Transverse acceleration at the bottom of the car (${\ddot{\mathit{Z}}}_c$)	0.029	0.116	0.032	0.154

(at elevator speeds of 10 m/s, 8 m/s, and 6 m/s, respectively).

5.2. Multidimensional Performance Evaluation

In order to intuitively compare the pros and cons of each method, radar charts are built using performance index, control cost, and three transverse vibration-related outputs as evaluation indicators. The radar charts for three different speeds are presented in Figure 12, Figure 13 and Figure 14. Since smaller values of the performance index, transverse acceleration, frame displacement off the centerline, and car–frame displacement indicate better methods, their reciprocals are used and combined with cost control as radar chart indicators. As shown in Figure 12, Figure 13 and Figure 14, the optimized LQR control method has the largest range, with all indicators surpassing those of other methods. PID control is the second-best, underperforming compared to the optimized LQR in some aspects. The unoptimized LQR control method ranks third in terms of performance. All three methods outperform the no-control approach.

Taking the 10 m/s scenario as an example, comparison between the optimized and unoptimized LQR controller reveals the following: The optimized LQR controller achieved a 25.18% reduction in transverse acceleration, a 0.60% decrease in car frame displacement off the centerline, and a 63.78% decrease in relative car–frame displacement. Additionally, the root mean square values of the active control forces were all under 250 N, with the maximum control force not exceeding 850 N, as shown in Figure 15. These data indicate that the optimized LQR controller can more effectively reduce the elevator’s transverse vibration acceleration and the displacement of the car relative to the frame compared to its unoptimized counterpart. The optimized LQR controller has proven to be highly effective in controlling the elevator’s transverse acceleration, thus affirming the efficiency of the optimization approach in designing LQR controllers. By establishing appropriate objective functions and weight matrices, the optimal LQR controller can be derived.

When compared to PID control, the LQR controller demonstrated greater effectiveness in reducing the following: (1) transverse acceleration by 10.27%, (2) car–frame displacement by 3.12%, and (3) relative car–frame displacement by 38.8%. These results indicate that the proposed LQR-based method is superior to traditional PID control. In the context of ultra-high-speed elevators, which are highly nonlinear, strongly coupled, and characterized by variable parameters, PID controllers may struggle to adapt to rapidly changing control requirements due to their lack of predictive capabilities. Consequently, they may not provide sufficient control accuracy and stability for systems that demand rapid response and high precision. In contrast, the LQR is better equipped to handle multivariable control problems in such complex systems, offering faster response times and higher control accuracy.

5.3. Speed Sensitivity Research

To analyze the impact of air excitation on transverse vibration across different speeds, we have drawn transverse vibration diagrams for various speeds, presented in Figure 16, Figure 17, Figure 18 and Figure 19. To clearly illustrate how transverse vibration varies with speed, the left figures display normalized values at three different velocities using the maximum–minimum normalization method, while the right figures show the true values. Given the significant differences in numerical ranges, logarithmic scales are utilized. In these figures, PI denotes performance index, CC denotes control cost, FDOC denotes frame displacement off centerline, CFD denotes car–frame displacement, and TA denotes transverse acceleration. Since the without control method has a control cost of 0, there is no CC in Figure 19.

From Figure 16, Figure 17, Figure 18 and Figure 19, it is evident that, generally, higher speeds result in greater control costs and performance indices. When comparing LQR and PID control methods at different speeds, at v = 10 m/s, 8 m/s, and 6 m/s, the LQR outperforms PID in reducing transverse vibration acceleration by 10.27%, 8.47%, and 7.55%, respectively. Similarly, the LQR surpasses PID in suppressing car–frame displacement by 38.8%, 36.3%, and 33.9% at these speeds.

Overall, as elevator speed increases, airflow disturbance becomes more intense, exerting a greater impact on the elevator. Correspondingly, LQR control demonstrates better effectiveness in suppressing transverse vibration. However, from Figure 17, Figure 18 and Figure 19, it can be observed that, when the control method is suboptimal, transverse acceleration may slightly decrease with increasing speed. This can be explained as follows:

• Although speed increase typically heightens air resistance, in some cases, it can stabilize the airflow between the elevator car and hoistway, thereby reducing aerodynamic-induced vibration.
• At high speeds, the interaction between airflow and the guide shoe–guide rail contact may become closer, potentially mitigating vibrations caused by guide rail unevenness.

5.4. Limitations and Robustness Analysis

Though the MOGA-LQR performs well at different speeds, its engineering application boundaries still need to be explored. This section quantitatively analyzes the engineering applicability boundary of the proposed method, focusing on three dimensions: optimization robustness, computational cost, and nonlinear interference impacts.

We conducted tests on four sets of MOGA optimization algorithms for different population sizes and the number of generations and found that the final results showed almost consistent trends, as shown in Figure 20 (with an initial parameter x of 0). Meanwhile, there is not much difference in the LQR coefficients of the same points on different graphs. The results confirm MOGA’s strong convergence characteristics for our optimization problem.

The computational cost of the multi-objective genetic algorithm (MOGA) optimization is primarily determined by three key factors: population size (N), number of generations (G), and fitness evaluation time per individual ($t_f$). The time consumption of genetic operations (crossover/selection/mutation, etc.) is much smaller than simulation and can be ignored. The total optimization time can be expressed as follows:

$$T_{total}\approx N\times G\times t_f+T_A$$

(26)

where N is the population size, G is the number of generations (maximum 500 generations), $t_f$ is the time per fitness evaluation, and $T_A$ is the algorithm operation time.

The fitness evaluation time constitutes the dominant computational cost, as each evaluation requires running the full Simulink model simulation. Our analysis reveals scales with both elevator speed and model complexity:

• Speed dependency: We tested 10 times each at 6 m/s, 8 m/s, and 10 m/s, and the time results for each run are as follows: At 6 m/s: $t_f=\left[0.871\right]\pm \left[0.017\right]$ s. At 8 m/s: $t_f=\left[0.874\right]\pm \left[0.027\right]$ s;. At 10 m/s: $t_f=\left[0.883\right]\pm \left[0.016\right]$ s. It can be observed that higher operating speeds do not significantly increase aerodynamic complexity and do not prolong the simulation time for each evaluation.
• Model complexity impact: Obviously, more complex models will lead to an increase in simulation time. We tested 10 times using the LQR method, PID method, and uncontrolled conditions ($v=10 \mathrm{m}/\mathrm{s}$), and the time results for each run are as follows: LQR: $t_f=\left[0.883\right]\pm \left[0.016\right]$ s; PID: $t_f=\left[0.891\right]\pm \left[0.018\right]$ s; without control: $t_f=\left[0.788\right]\pm \left[0.011\right]$ s. In 10 tests, the PID method consumed slightly longer time than the LQR method, which is related to the differentiation and integration in the calculation process, but this part has little impact on the overall time. In short, the more complex the control method (the more complex the model), the longer the time required for a single Simulink calculation, and the longer the time consumed by the entire MOGA.

Current modeling uses Gaussian white noise filtered through a low-pass filter to represent rail excitation. However, long-term wear alters three critical statistical properties of rail irregularities:

• Mean shift increases baseline vibration.
• Variance expansion (${\sigma }^2$ grows after numerous cycles) amplifies stochastic disturbances.
• Peak deformation intensification causes intermittent high-amplitude impulses ($>3\sigma$ events will increase).

These changes significantly alter the guiding incentives, and one approach is to monitor the data of the guide rail in real time to obtain relevant attributes. However, real-time monitoring remains challenging because laser displacement sensors cannot be permanently installed in hoistways due to contamination risks, and strain gauges measure and achieve wireless transmission in the presence of vibration under the shaft.

Future implementations could embed relevant sensors in guide shoes to track wear progression and dynamically adjust the excitation model’s parameters ($\mu$, $\sigma$, kurtosis).

The quadratic approximation in Equation (6) effectively captures steady-state effects but neglects irregular flow phenomena in turbulence, such as turbulence intermittency and transient vortex shedding. The relevant content deserves further research, such as establishing more reasonable models or developing adaptive turbulence correction methods to achieve better results.

There are currently nonlinear problems, and deep neural networks are good at handling these problems. The fusion of physics-based models and data-driven neural networks can enhance the riding experience of elevators. In addition, it can also be used to predict the lifespan of elevators, assess the risks of key components, and more.

The Nonlinear Energy Sink (NES) is a promising vibration control device [38]. Combining the NES with the proposed LQR-based active control method could create a hybrid control strategy that leverages the strengths of both approaches. While the LQR controller actively adjusts the guide shoes to counteract disturbances, the NES passively absorbs and dissipates residual vibrational energy. This combination could enhance the overall effectiveness of the vibration suppression system, and this is also one of the directions that we will try in the future.

In the future, elevators will become more intelligent. Typically equipped with accelerometers and gyroscopes to measure vibration acceleration and angular velocity, elevators can enhance the accuracy and reliability of vibration monitoring and suppression through data fusion algorithms [39] that integrate data from these sensors.

6. Conclusions

This paper incorporates active control force into the elevator system via active guide shoes, constructing a four-degree-of-freedom ultra-high-speed elevator transverse vibration dynamic model. The airflow during operation is analyzed as incompressible viscous turbulence within the shaft. Fluent is adopted to analyze the flow field, iteratively solving until convergence to identify the air excitation. Quantitative analysis determines the relationship between air excitation and operating speed, car transverse displacement, and car deflection angle. Then, shaft airflow disturbances are transformed into transverse deflection forces and torques on the car, establishing a gas–solid coupling dynamic model for ultra-high-speed elevators. Using this model, the state equation is derived to solve the LQR controller with MATLAB tools. Simultaneously, a multi-objective genetic algorithm optimizes the key weight matrices Q and R of the LQR controller, using system performance and control cost as objective functions. The Pareto solution curves for parameters in these matrices are obtained, and suitable parameters are selected to solve the optimal LQR controller. Numerical simulations via Simulink show that, compared to PID control, the LQR-based method reduces transverse acceleration by 7.55%, 8.47%, and 10.27% at 6 m/s, 8 m/s, and 10 m/s, respectively, verifying the effectiveness of the controller, and the effectiveness of the proposed method increases with the increase in speed. This work provides an effective solution for enhancing the ride comfort of ultra-high-speed elevators and holds potential for application in the vibration control of high-speed transportation systems. While this study establishes a numerical framework for gas–solid coupling vibration control, further investigations are warranted: (1) experimental validation of the proposed model on full-scale ultra-high-speed elevator systems, particularly under real-world disturbances (e.g., shaft turbulence), to verify simulation accuracy; (2) integration of passive control mechanisms, such as Nonlinear Energy Sinks (NESs), to develop hybrid passive–active suppression systems.