Variable Universe Fuzzy Control of High-Speed Elevator Horizontal Vibration Based on Firefly Algorithm and Backpropagation Fuzzy Neural Network

To effectively suppress the horizontal vibration of a high-speed elevator car caused by uncertainties such as the irregularity of guide rails and the piston wind in the hoistway, this paper proposes a new variable universe fuzzy control method for semi-active guide shoes with magnetorheological (MR) dampers based on the firefly algorithm and backpropagation (FA-BP) algorithm fuzzy neural network (FNN). First, the dynamic model of a car system is constructed, and the FNN system based non-parametric model of the MR damper is established with the original MR damper data collected from experiments. Then, a variable universe fuzzy main controller is presented to control the input current of the MR damper. As for its variable universe contraction-expansion factors, another FNN auxiliary controller is designed to accurately adjust the universe. Furthermore, the FA-BP algorithm is utilized to train the FNN auxiliary controller. Finally, the elevator experiment is carried out to analyze the characteristics of horizontal vibration, and simulation tests are conducted using the measured excitation signal and two kinds of simulated excitation signals, respectively. The results show that the proposed controller has a lower horizontal vibration acceleration, tilt angle acceleration and other index values than the passive controller or the FNN controller, indicating that the proposed control method can effectively suppress the horizontal vibration of a high-speed elevator. This study can provide the technical foundation for intelligent vibration control of high-speed elevators.

SECTION I. Introduction

With the increasing number and height of high-rise buildings, elevators with higher speeds are inevitably being developed. However, the horizontal vibration of the lift car caused by uncertainties such as irregular excitation of guide rail, piston wind in the hoistway and the swinging of hoisting rope will also intensify, which will greatly affect the riding comfort of passengers and even cause severe reactions such as dizziness and vomiting. In addition, the instruments on the car will also be affected by vibration, which could cause life shortening or even failure, eventually leading to more serious safety accidents. Therefore, horizontal vibration suppression is one of the key problems to be solved in the field of high-speed elevators.

Guide shoes are the primary units for elevator horizontal vibration control. In which, the semi-active guide shoe refers to the guide shoe whose stiffness coefficient $K$ of the elastic element is not variable and damping coefficient $C$ of the damping element is variable. Because of their simple structure, high reliability and low energy consumption, semi-active guide shoes have been widely used in high-speed elevators. Among the adjustable dampers, the magnetorheological (MR) damper is a promising damping element with high reliability and can output continuously variable damping force. Its response time is less than 1 ms and the required control current is not more than 10 A, which currently makes it an ideal choice for a damping element in semi-active guide shoes [1], [2]. However, due to the non-Newtonian fluid characteristics of MR fluid under the action of the magnetic field, MR dampers have obvious strong nonlinear characteristics, such as hysteresis and yield, which greatly increase the difficulty of establishing an accurate parametric model and make it hard to calculate the ideal control current during operation.

To this end, Stanway [3] constructed a Bingham plastic model of an electrorheological damper. Although the model is simple and easy to analyze, it cannot describe the hysteretic characteristics of the damper. Bouc and Wen [4], [5] introduced hysteretic variables into the model, which can systematically describe the nonlinear hysteretic characteristics of MR dampers. To further analyze the dynamic characteristics of MR dampers, Spencer [6] proposed a phenomenological model based on the Bouc-Wen model, which has greatly advanced the parametric modeling of MR dampers. Although the model can accurately describe the dynamic response of the MR damper, there are too many parameters needed to be measured and identified, which leads to the increased time delays and makes it difficult to use for actual engineering. After those, the parametric modeling still cannot be used to establish a relatively accurate mathematical model with fewer parameters [7]. Therefore, it is necessary to utilize new modeling methods to ensure the accuracy of the model and to be as simple and efficient as possible, thereby providing an accurate and practical model basis for the next step in researching vibration control methods.

In our previous study [8], the main factors causing the horizontal vibration of high-speed elevator cars were rail irregularity and unsteady air flow represented by piston wind. The vibration under the influence of such complex coupling factors determines the complexity and accuracy requirements for the control method. Among the existing vibration control methods, fuzzy control takes the both requirements above into account and does not depend on the control model, which has been widely studied by scholars. Kanjuro [9] proposed a fuzzy-based multimodal vibration controller, and the experimental results show that the controller has high vibration suppression performance. Tairidis [10] proposed a fuzzy logic vibration controller of intelligent plates and used the genetic algorithm to improve the controller performance. However, the design of a fuzzy controller is highly dependent on the expert experience. When there are deviations or even errors in the existing experience, the performance of the control system will be greatly degraded [11]. In addition, the accuracy of traditional fuzzy control is limited by fuzzy partition. If the number of fuzzy partitions is added to improve the control accuracy, the steady-state error will be increased and the reliability of the controller will be reduced.

Therefore, to overcome the shortcomings of the above conventional fuzzy control, Li [12] proposed variable universe fuzzy control, pointing out that the discourse universe of fuzzy variables can be compact with decreasing system error, and vice versa. To a certain extent, it solves the problem of over-dependence on expert experience, and could precisely densify the control of some target domains. The ability of local targeted densifying control makes the variable universe fuzzy method widely studied by scholars and successfully applied to robot control [13], crane positioning control [14] and other fields. However, the range of the discourse universe depends on a parameter called the contraction-expansion factor, which is not a variable that can be obtained by expert experience. Moreover, in the current related research, the control methods for the contraction-expansion factor are not unified: mathematical functions, integral adjustment and multi-population genetic algorithm are used in [15]–​[17], respectively, to describe and optimize the scaling of the factor. Although they optimize the control effect to varying degrees, the description of the factor has a crucial impact on the control effect of the variable universe fuzzy control, and the control methods mentioned above are still not accurate.

To this end, Pang [18] used fuzzy neural network (FNN) to control the contraction-expansion factor, and improved the control effect through the learning ability of FNN. However, the commonly used backpropagation (BP) algorithm requires a large number of iterations to train the network to approximate the ideal output. This seriously reduces the engineering practicability of the controller due to the time and resources required by the training. In addition, as a result of the poor global optimization ability of the BP algorithm, once it falls into the local optimum, it will need more iterations to jump out of the pseudo-optimum point [19]. Therefore, the FNN controller training needs improved algorithm to achieve a higher convergence rate as far as possible on the guarantee of the global optimization ability.

Therefore, based on the firefly algorithm and backpropagation (FA-BP) algorithm FNN, this paper proposes a new variable universe fuzzy controller for MR damper semi-active guide shoes, referred to as FAFNN-VFC. The originality of this paper is as follows:

• The non-parametric model of an MR damper is established based on the FNN system, and its accuracy is verified by experimental comparison. This model does not depend on the damper object and is more practical in engineering.

• A variable universe fuzzy control method based on FA-BP algorithm FNN is proposed. This method is suitable for complex excitation and control models, and the accuracy and training speed of the controller are highly improved by FAFNN, which provides an effective and practical control method for the intelligent control of complex vibration, such as the horizontal vibration of high-speed elevators.

• A real elevator experiment is carried out to analyze the horizontal vibration characteristics of the elevator car, and the actual response data set is used for the controller training. Furthermore, the effectiveness of the controller is proved by simulation tests under the actual excitation, random excitation and impulse excitation of the guide rail.

SECTION II. Semi-Active Control Model of Lift Car and Non-Parametric Model of MR Damper

A. The Lift Car Semi-Active Control Dynamic Model Establishment

The dynamic model of horizontal vibration semi-active control for a high-speed elevator car system is shown in Figure 1. It is mainly composed of the lift car and 4 semi-active guide shoes, which are axisymmetrically distributed. The guide shoe is simplified as a mass-spring-damper system with mass $m_{i} (i=1\sim 4)$ and equivalent spring stiffness $k_{s}$ . The controllable damping force $f_{i} (i=1\sim 4)$ is provided by the MR damper used in this paper, and then the guide wheel with equivalent stiffness $k_{t}$ closely fits against the guide rail. The horizontal vibration displacement of the car and four guide shoes are $x_{c}$ and $x_{i} (i=1\sim 4)$ , respectively. Meanwhile, considering other uncertain excitation such as aerodynamic force, the coupling force on the car is denoted as $f_{c}$ . $m_{c}$ represents the mass of the lift car; $I_{c}$ represents the moment of inertia; $\theta$ denotes the tilt angle displacement at the gravity center of car body; $l_{1}$ and $l_{2}$ represent the longitudinal distances from the gravity center to the upper and lower guide shoes, respectively.

FIGURE 1.

Semi-active control model for horizontal vibration of car system.

For the connection points between each guide shoe and the car, their horizontal displacements have the following relationship:

$$\begin{align*} \begin{cases} x_{c1} =x_{c} -l_{1} \tan \theta \approx x_{c} -l_{1} \theta \\ x_{c2} =x_{c} +l_{2} \tan \theta \approx x_{c} +l_{2} \theta \\ x_{c3} =x_{c} +l_{1} \tan \theta \approx x_{c} +l_{1} \theta \\ x_{c4} =x_{c} -l_{2} \tan \theta \approx x_{c} -l_{2} \theta \\ \end{cases}\tag{1}\end{align*}$$ Then, the dynamic equations of the car system are as follows: $$\begin{align*} \begin{cases} m_{1} \ddot {x}_{1} =k_{s} \left ({{x_{1} -x_{c} +l_{1} \theta } }\right)+k_{t} \left ({{x_{r1} -x_{1}} }\right)+f_{1} \\ m_{2} \ddot {x}_{2} =k_{s} \left ({{x_{2} -x_{c} -l_{2} \theta } }\right)+k_{t} \left ({{x_{r2} -x_{2}} }\right)+f_{2} \\ m_{3} \ddot {x}_{3} =k_{s} \left ({{x_{3} -x_{c} -l_{1} \theta } }\right)+k_{t} \left ({{x_{r3} -x_{3}} }\right)+f_{3} \\ m_{4} \ddot {x}_{4} =k_{s} \left ({{x_{4} -x_{c} +l_{2} \theta } }\right)+k_{t} \left ({{x_{r4} -x_{4}} }\right)+f_{4} \\ m_{c} \ddot {x}_{c} =k_{s} \left ({{\sum \limits _{i=1}^{4} {x_{i}} -4x_{c}} }\right)+\sum \limits _{i=1}^{4} {f_{i} +f_{c}} \\ I_{c} \ddot {\theta }=l_{1} \left [{ {k_{s} \left ({{x_{1} +x_{3} -2x_{c}} }\right)+f_{1} +f_{3}} }\right] \\ \qquad \,\,-l_{2} \left [{ {k_{s} \left ({{x_{2} +x_{4} -2x_{c}} }\right)+f_{2} +f_{4}} }\right] \\ \end{cases}\tag{2}\end{align*}$$ Define $X_{s} =\left [{ {x_{1},x_{2},x_{3},x_{4}} }\right]^{T}$ , $\ddot {X}_{c} =\left [{ {\ddot {x}_{c}, \ddot {\theta }} }\right]^{T}$ , $X_{cs} =\left [{ {x_{c} -l_{1} \theta,x_{c} +l_{2} \theta,x_{c} +l_{1} \theta,x_{c} -l_{2} \theta } }\right]^{T}$ , $X_{r} =\left [{ {x_{r1},x_{r2},x_{r3},x_{r4}} }\right]^{T}$ , $F_{MRD} =\left [{ {f_{1},f_{2},f_{3},f_{4}} }\right]^{T}$ , $X_{ud} =\left [{ {x_{1} +x_{3} -2x_{c},x_{2} +x_{4} -2x_{c}} }\right]^{T}$ , $F_{ud} =\left [{ {f_{1} +f_{3}, f_{2} +f_{4}} }\right]^{T}$ and $F_{c} =\left [{ {f_{c},0} }\right]^{T}$ , then EQUATION (2) can be expressed in matrix form as $$\begin{align*} \begin{cases} M_{s} \ddot {X}_{s} =k_{s} \left ({{X_{s} -X_{cs}} }\right)+k_{t} \left ({{X_{r} -X_{s}} }\right)+F_{MRD} \\ M_{c} \ddot {X}_{c} =k_{s} QX_{ud} +QF_{ud} +F_{c} \\ \end{cases}\tag{3}\end{align*}$$ where each matrix is represented as follows: $$\begin{align*} M_{s} =\left [{\begin{matrix} {m_{1}} & 0 & 0 & 0 \\ 0 & {m_{2}} & 0 & 0 \\ 0 & 0 & {m_{3}} & 0 \\ 0 & 0 & 0 & {m_{4}} \\ \end{matrix} }\right], M_{c} =\left [{\begin{matrix} {m_{c}} & 0 \\ 0 & {I_{c}} \\ \end{matrix} }\right], Q=\left [{\begin{matrix} 1 & 1 \\ {l_{1}} & {-l_{2}} \\ \end{matrix} }\right].\end{align*}$$

B. Non-Parametric Modeling of MR Damper

To suppress the vibration of the car in the above model, the damping force needs to be adjusted in real time according to the excitation and the displacement of the car. The internal basic structure of the MR damper used in this paper is shown in Figure 2. To continuously control the damping force, the current in the coil should be adjusted continuously to change the magnetic field, and to further modify the damping coefficient of the MR damper.

FIGURE 2.

Internal structure of the MR damper.

However, the use of MR fluid also brings strong nonlinear characteristics such as hysteresis and yield. If the conventional parametric modeling method is used, the identification and measurement of the newly introduced parameters will increase the time delay, which will reduce the practicability and control effectiveness of the model. Compared with the MR damper widely adopted in automobile suspension, the MR damper for elevator guide shoes lacks typical products, which usually needs independent design and customization. This further increases the cost of parameter optimization and trial-and-error in parametric modeling.

Therefore, this paper uses non-parametric modeling [20] and introduces FNN to adaptively adjust the parameters, so that the model does not depend on a specific damper. It only needs to train FNN through measurement data to quickly establish the response model of the target MR damper. As shown in Figure 3, the model consists of two parts: one is based on the FNN I to switch the output level $k$ of the damping force according to the control current input into the coil; the other is based on the FNN II to estimate the damping force $u$ under standard current according to the displacement and velocity of the MR damper piston. Finally, the output damping force of the MR damper is obtained by calculating $f=k\times u$ .

FIGURE 3.

Adaptive FNN non-parametric modeling of MR damper.

To verify the accuracy of the established model, experimental data were collected for comparative analysis in this paper. Utilizing the sine wave with amplitude of 5 mm and frequency of 2 Hz, 5 Hz and 10 Hz as the excitation sources, respectively, the magnitude of MR damping force with control currents of 0, 1 A, 2 A, 3 A and 4 A was measured repeatedly on the vibration mechanics testing platform. For simplicity, this paper only shows the response comparison between the MR damping force and displacement under three kinds of control currents using typical 5 Hz frequency excitation, as shown in Figure 4.

FIGURE 4.

Comparison of model and actual response under 5 Hz excitation.

It can be seen that the relationship curve of the established model is highly consistent with the experimental response and properly reflect the nonlinear characteristics of the MR damper. This indicates that the FNN non-parametric model can accurately simulate the actual output response of the MR damper.

SECTION III. Variable Universe Fuzzy Controller Based on Firefly Algorithm Fuzzy Neural Network

The horizontal vibration of high-speed elevators is a complex coupling vibration influenced by uncertain excitation such as guide rail irregularity and piston wind in the hoistway. To study its vibration characteristics and guide the design of controllers, this paper specifically measures and analyzes the excitation signal in actual operation, as shown in Section IV-A below. It can be seen that the coupling vibration signal is characterized as high-frequency low-amplitude excitation and occasional short-time strong-amplitude excitation. To further improve the control accuracy, the control of the local universe with low-amplitude needs to be densified.

The variable universe fuzzy control method can densify the control of the local universe without increasing the number of fuzzy partitions and fuzzy rules [21]. Therefore, to realize the precise control for horizontal coupling vibration of high-speed elevators, a variable universe fuzzy controller based on FNN combined with the firefly algorithm and backpropagation algorithm is proposed in this paper. The structure of the proposed FAFNN-VFC is shown in Figure 5.

FIGURE 5.

The structure of FAFNN-VFC.

The main controller is a Takagi-Sugeno (T-S) variable universe fuzzy controller, whose inputs are the car horizontal acceleration $\ddot {x}_{c}$ and velocity $\Delta v$ between the lift car and the guide shoe. The fuzzy universe is adjusted by the variable universe contraction-expansion factors $\alpha _{1}$ and $\alpha _{2}$ to precisely control the input current of the MR damper in the guide shoe.

Then, to further increase the engineering practicability of the controller, the control accuracy and the convergence training speed of the controller need to be improved. On the one hand, the contraction-expansion factor in the main controller is the key to improve the control accuracy. Considering the lack of standard control methods for this factor, this paper designs another auxiliary controller based on the Mamdani FNN. On the other hand, due to the large number of undetermined parameters introduced by FNN, this paper combines the FA and BP algorithm to globally optimize and quickly converge, and to realize the efficient learning and training of the FNN auxiliary controller.

A. Variable Universe Fuzzy Main Controller

In essence, the fuzzy controller is a kind of interpolator [22]. When the fuzzy controller needs to approximate the ideal output more accurately, it is necessary to densify the fuzzy partition and fuzzy rules in the target universe, which will enlarge the steady-state error of the controller. By dynamically adjusting the size of the universe, densifying control can be achieved skillfully in another way. This is the principle of variable universe fuzzy control, which is highly suitable for complex excitation and control models [23], [24]. It can be expressed as follows: Set the input of the controller as $x_{i} \left ({{i=1,2,\cdots,n} }\right)$ and the corresponding n-dimensional fuzzy universe as $X_{i} =\left [{ {-E_{i},E_{i}} }\right]$ , the output as $j$ and the corresponding universe as $J=\left [{ {-U,U} }\right]$ . By changing variable universe contraction-expansion factors, the universe will be dynamically adjusted to:

$$\begin{align*} X_{i} \left ({{x_{i}} }\right)=&\left [{ {-\alpha _{i} \left ({{x_{i}} }\right)E_{i},\alpha _{i} \left ({{x_{i}} }\right)E_{i}} }\right]\tag{4}\\ J\left ({j }\right)=&\left [{ {-\vartheta \left ({j }\right)U,\vartheta \left ({j }\right)U} }\right]\tag{5}\end{align*}$$ where $\alpha _{i} \left ({{x_{i}} }\right)$ and $\vartheta \left ({j }\right)$ are contraction-expansion factors, and their dynamic influence on the universe is shown in Figure 6.

FIGURE 6.

Dynamic adjustment of the universe.

T-S fuzzy control is used in this variable universe fuzzy main controller. The T-S fuzzy controller can be divided into the premise network and the consequent network, and both of their inputs are the horizontal acceleration $\ddot {x}_{c}$ and the relative velocity $\Delta v$ between the car and the guide shoe. $\bar {\beta }_{j}$ denote the normalized output value of the premise network under each fuzzy rule, while the final output can be expressed as follows:

$$\begin{align*} y_{_{j}}^{i}=&p_{_{j0}}^{i} +p_{_{j1}}^{i} x_{1} +\cdots +p_{_{jn}}^{i}x_{n} \\&\quad \left ({{i=1,2,\cdots,r;~j=1,2,\cdots,m} }\right)\tag{6}\\ y_{i}=&\sum \limits _{j=1}^{m} {\bar {\beta }_{i} \cdot y_{ij}}\tag{7}\end{align*}$$ where $y_{j}^{i}$ represents the output of the consequent network, $p_{jn}^{i}$ represents the adaptive parameters, $x_{n}$ represents the input of networks, $x_{1} =\ddot {x}_{c}$ , $x_{2} =\Delta v$ , $r$ and $m$ represent the number of network outputs and fuzzy rules, respectively, and $y_{i}$ represent the final output of the whole network.

In this paper, the main function of the T-S variable universe fuzzy main controller is to accurately output the control current $I$ of the MR damper according to the running conditions to provide the required damping force. Considering its requirements for accuracy and stability, the initial universe is defined as $\left \{{{-1,1} }\right \}$ and divided into seven fuzzy subsets, i.e., {NL (negative large), NM (negative middle), NS (negative small), ZO (zero), PS (positive small), PM (positive middle), PL (positive large)}, and all of them use the Gaussian function as membership functions.

Of note, there are three kinds of adjustable parameters in the main controller: the Gaussian function center $c$ , the width $\sigma$ and the weight $p_{jn}^{i}$ of the consequent network. The initial value of these parameters is set based on existing expert experience to determine the approximate range of the ideal parameter value and is then gradually refined to converge to the ideal value through the following optimization and training, so that the fuzzy controller can approximate the ideal output. The evaluation function of the controller is defined as:

$$\begin{equation*} F=\textrm {RMS}(\ddot {x}_{c})+\textrm {RMS}(\ddot {\theta })+\textrm {RMS}\left({\sum \limits _{i=1}^{4} {x_{ci} -x_{i}} }\right)\tag{8}\end{equation*}$$ where RMS represent the root mean square.

Considering the complex and highly nonlinear characteristics of the lift car horizontal vibration excitation, to ensure that the final control effects meet the requirements under actual working conditions, the horizontal vibration data of the car were collected through elevator experiments in this paper. To avoid repetition, these will be specifically described in Section IV-A. Afterwards, take the collected data in training group as training samples and EQUATION (8) as evaluation function to train and optimize the main controller by importing them into MATLAB.

B. FNN Auxiliary Controller

For the contraction-expansion factors $\alpha _{1}$ and $\alpha _{2}$ of the main controller, using the FNN control method [25], [26], the auxiliary controller is designed. As shown in Figure 7, this FNN adopts a Mamdani forward network with two inputs and two outputs; $\ddot {x}_{c}$ and $\Delta v$ are the car horizontal acceleration input and velocity input between the car and the guide shoe, respectively. Each input has seven fuzzy partitions expressed as {NL, NM, NS, ZO, PS, PM, PL}.

FIGURE 7.

The structure of FNN.

The first and the second layers are the input layer and the fuzzy layer, where the Gaussian function is selected as the fuzzy membership function:

$$\begin{align*} f_{ij}^{(2)} =\exp \left ({{{-\left ({{x_{i} -c_{ij}} }\right)^{2}} \mathord {\left /{ {\vphantom {{-\left ({{x_{i} -c_{ij}} }\right)^{2}} {\sigma _{ij}^{2}}}} }\right. } {\sigma _{ij}^{2}}} }\right)\quad \left ({{i=1,2;~j=1,2,\ldots m_{i}} }\right)\!\!\!\!\! \\\tag{9}\end{align*}$$ where $x_{i}$ represent the inputs of the first layer, $x_{1} =\ddot {x}_{c}$ , $x_{2} =\Delta v$ ; $c_{ij}$ and $\sigma _{ij}$ represent the center and width of the Gaussian membership function, respectively; and $m_{i}$ represents the number of fuzzy partitions of the $i$ th input.

The third layer of the network is the inference layer, which chooses the smaller value:

$$\begin{equation*} f_{j}^{(3)} =\min \left \{{{x_{1i_{1}}^{(2)},x_{2i_{2}}^{(2)}} }\right \} \left ({{j=1,2,\cdots,m;m=\prod \limits _{i=1}^{n} {m_{i}}} }\right)\tag{10}\end{equation*}$$ The fourth and fifth layers are the normalization layer and the output layer, where the centroid method is used to realize the defuzzification: $$\begin{align*} f_{j}^{(4) }=&{f_{j}^{(3) }} \bigg /{\sum \limits _{i=1}^{m} {f_{j}^{(3) }}}\tag{11}\\ f_{i}^{(5) }=&\sum \limits _{j=1}^{m} {\omega _{ij} \cdot f_{j}^{(4)}}\tag{12}\end{align*}$$ where $\omega _{ij}$ represents the weight of the $i$ th output in the $j$ th normalization layer nodes; $f_{1}^{(5) }$ and $f_{2}^{(5) }$ denote the network outputs $\alpha _{1}$ and $\alpha _{2}$ , respectively. Therefore, it can be seen that the FNN has three kinds of learning parameters: center $c_{ij}$ , width $\sigma _{ij}$ and weight $\omega _{ij}$ , which will be trained and learned by the following algorithm.

C. FA-BP Algorithm

For FNN training, the BP algorithm is generally used to make the network outputs approximate the expectations. Assuming that $y_{i}$ and $t_{i}$ are the actual and expected outputs of the network, respectively, the supervised error function is defined as:

$$\begin{equation*} E=\frac {1}{2}\sum \limits _{i=1}^{r} {\left ({{t_{i} -y_{i}} }\right)^{2}}\tag{13}\end{equation*}$$ where the BP algorithm can be described as: $$\begin{align*} \begin{cases} \displaystyle c_{ij} (k+1)=c_{ij} (k)-\beta \frac {\partial E}{\partial c_{ij}} \\ \displaystyle \sigma _{ij} (k+1)=\sigma _{ij} (k)-\beta \frac {\partial E}{\partial \sigma _{ij}} \\ \displaystyle \omega _{ij} (k+1)=\omega _{ij} (k)-\beta \frac {\partial E}{\partial \omega _{ij}} \\ \displaystyle \end{cases}\tag{14}\end{align*}$$ where $\beta >0$ represents the learning rate. However, the training speed and output accuracy of the BP algorithm are highly dependent on the initial value of the parameters in the network, and improper initialization of the parameters will make the network fall into a local optimum [27].

In recent years, swarm intelligence algorithms such as particle swarm optimization [28], artificial bee colony algorithm [29] and firefly algorithm [30] have been widely studied by scholars to replace the BP algorithm. Among them, FA is a heuristic algorithm proposed by Yang in 2009 [31]. Compared with other swarm intelligence algorithms, FA has a higher convergence speed under the premise of the excellent global search ability, and has outstanding performance in global optimization with multiple local extremum points [32].

The basic idea of the FA can be described as: $n$ fireflies $X_{i} \left ({{i=1,2,\ldots,n} }\right)$ are randomly distributed in the D-dimensional search space, and the position $X_{i} =\left ({{X_{i1},X_{i2},\ldots,X_{iD}} }\right)^{T}$ of the $i$ th firefly represents the feasible solution. The brightness $I$ of a firefly at the current position is directly proportional to the fitness function value $f\left ({{X_{i}} }\right)$ corresponding to the feasible solution. All fireflies will attract each other regardless of gender. The relative attractiveness $\beta \left ({r }\right)$ is described as:

$$\begin{align*} r_{ij}=&\left \|{ {X_{i} -X_{j}} }\right \|=\sqrt {\sum \limits _{m=1}^{D} {\left ({{X_{jm} -X_{im}} }\right)^{2}}}\tag{15}\\ \beta \left ({r }\right)=&\beta _{0} e^{-\gamma r^{2}}\tag{16}\end{align*}$$ where $r_{ij}$ represents the distance between the fireflies $i$ and $j$ , $\beta _{0}$ represents the attractiveness at the initial position $r=0$ , and $\gamma$ represents the light absorption coefficient. In each iteration, each firefly moves toward the fireflies that are brighter than itself. The equation of firefly $i$ moving towards firefly $j$ can be expressed as: $$\begin{equation*} X_{i}^{k+1} =X_{i}^{k} +\beta _{0} e^{-\gamma r_{ij}^{2}}\left ({{X_{j}^{k} -X_{i}^{k}} }\right)+\alpha \left ({{\textrm {rand}-\frac {1}{2}} }\right)\tag{17}\end{equation*}$$ where $\alpha$ represents the compensation randomization parameter and rand denotes a random number generator uniformly distributed in $\left [{ {0,1} }\right]$ . The brightest firefly moves randomly around itself.

However, it should be pointed out that although FA has a high global optimization efficiency, like most swarm intelligence algorithms, in the late iterations, its local convergence speed slows and the optimization accuracy is insufficient. As a result, when it is used as the training algorithm of FNN alone, the final output performance is poor. Therefore, the FA-BP algorithm is used in this paper to train FNN, and its algorithm flow chart is shown in Figure 8.

FIGURE 8.

FA-BP algorithm flow chart.

In the early iterations, the FA is used for quick global optimization. When iterating to the maximum number or the output reaches the expected accuracy, the optimal firefly’s position is substituted into the FNN as the optimized solution for the initial parameter. Then, the BP algorithm is used to learn and train, so that it can quickly approximate the ideal output.

By selecting the same parameter setting and initial value, the FA-BP algorithm, the BP algorithm and the FA are used to train the FNN auxiliary controller, respectively. The error convergence comparison of the mean value from experimental results in multiple times is shown in Figure 9:

FIGURE 9.

Comparison of the training error convergence.

It can be seen that the FA-BP algorithm and the FA have good convergence speeds at the initial stage of iterations. However, after 20 iterations, the convergence speed of the FA decreases, while the FA-BP algorithm and the BP algorithm still maintain a certain decline speed. At the end of the iterations, the FA fails to make the network converge to a higher accuracy, while the FA-BP algorithm achieves that with fewer iterations than the BP algorithm. From the above analysis, the FA-BP algorithm not only ensures the convergence accuracy but also improves the training speed, which could further increase the engineering practicability of the controller.

SECTION IV. Experiment and Simulation

A. Elevator Experiment

To analyze the complex coupling horizontal vibration characteristics of a high-speed elevator and to obtain the vibration excitation data under real running conditions for training and simulation tests, the elevator experiment was carried out in this paper. As shown in Figure 10., the experiment date was October 5, 2020, the experiment site was the 120 m elevator test tower in Dezhou, Shandong Province. The experimental instrument was the DT-4A elevator acceleration tester. The measurement range of the horizontal vibration acceleration was ±1.2 m/s², and the measurement accuracy was ±0.005 m/s². The experiment object was the FJ14-90000 7 m/s high-speed elevator. During the experiment, the DT-4A tester was placed at the equivalent centroid of the elevator car, and it ran repeatedly to collect a large amount of original acceleration data.

FIGURE 10.

Experimental data measurement of 7m/s high-speed elevator.

To improve the engineering practicability of the proposed controller and to further test the effectiveness of the controller under actual excitation, the experimental data were used as shown in Figure 11.

FIGURE 11.

Data flow of the elevator experiment.

The original data after normalization are substituted into the lift car model established in Section II-A above, and then the external force of the car and ideal control current under actual running conditions are calculated as the processed vibration response data set. First, the vibration characteristics of the data set are analyzed to guide the design of the controller in Section III, so that the controller can precisely suppress the vibration. Then, the data set is divided into two groups: one group is used as the training data set to learn and train the variable universe fuzzy main controller in Section III-A above; the other group, as the actual excitation signal in Section IV-B below, is applied to the car system in the simulation test to verify the effectiveness of the controller proposed in this paper.

The detailed parameters of the 7 m/s elevator car system are shown in Table 1.

TABLE 1 Numerical Values of the Elevator Car

Based on the above parameters, the horizontal excitation of the car under actual running conditions was calculated and analyzed in the frequency domain. For simplicity, only the vibration excitation data of a 7 m/s constant speed section are shown in Figure 12.

FIGURE 12.

Excitation (a) time domain curve (b) frequency domain curve at 7 m/s constant speed section.

It can be seen from the time domain curve that the horizontal vibration excitation of the 7 m/s high-speed elevator is concentrated in the range of ±100 N for most of the time, and the excitation value reaches more than 120 N a few times. From the frequency domain curve, it is observed that the vibration excitation is mainly concentrated in the range of 2–12 Hz, and there are peaks at the frequencies of 5 Hz and 10 Hz. This shows that the excitation signals of the high-speed elevator are mostly low-amplitude and high-frequency signals with occasionally short-time strong-amplitude signals under the coupling influence of many uncertain factors.

Thus, to achieve the precise and effective control of the horizontal vibration of high-speed elevators, not only should the strong-amplitude excitation in the whole universe be fully suppressed but also the low-amplitude high-frequency excitation which occupies most of the running time should also be noted, and this local universe signal should be further densified for control. The design of the controller should, therefore, not only ensure the global control response but also have the ability to control the local universe with higher precision.

Based on the above analysis, this paper designs the following three parts of simulation tests to fully verify the control effect of the proposed FAFNN-VFC. In Section IV-B, the testing data set obtained from the elevator experiment is used as the excitation signal to measure the control ability of the controller under actual working conditions. In Section IV-C, normally distributed random signal is used as the guide rail excitation to simulate the controller in the low-amplitude and high-frequency signal universe. Finally, in Section IV-D, the guide rail pulse excitation signal is used to determine the control effectiveness of the controller for occasional strong-amplitude excitation.

B. Actual Excitation Simulation

To verify the control effect of the proposed controller under actual running conditions, the testing data set obtained from the above elevator experiment were used as excitation signals for testing.

For the comparison, the passive controller and FNN controller were selected as the comparison objects. The passive controller was the same as the passive guide shoe system used in the elevator experiment above, and its parameters are shown in Table 1. The FNN controller used T-S FNN, and its parameter settings were the same as the proposed variable universe fuzzy controller. To objectively evaluate the control effect, the same training data set as the FAFNN-VFC was used for this FNN controller, and the number of training iterations was set to be the same. The comparison controllers in Section IV-C and Section IV-D simulation tests were set in the same way, and will not be repeated below.

The excitation signal was imported into MATLAB to simulate the response of each controller with the lift car model. The results are shown in Figure 13.

FIGURE 13.

The comparisons of (a) time domain and (b) frequency domain horizontal vibration acceleration under actual excitation.

It can be seen from the time domain curve that compared with the FNN semi-active control and the passive control, the amplitude of horizontal vibration acceleration of the proposed FAFNN-VFC is greatly reduced, especially for the occasional strong-amplitude excitation. It can also be seen from the frequency domain curve that the proposed controller can effectively control the acceleration at most frequencies, and it significantly improves the high-frequency vibration in the range of 4–10 Hz.

To further verify the control effect of the proposed FAFNN-VFC, Table 2. shows the numerical comparison of the horizontal acceleration $\ddot {x}_{c}$ and the damping force $f$ of different controllers under the actual excitation. Compared with the passive controller, the root mean square (RMS) values of horizontal acceleration of the FNN controller and the proposed FAFNN-VFC are reduced by 44.4% and 74.3%, respectively, and the RMS values of the required damping force are reduced by 44.1% and 72.9%, respectively.

TABLE 2 Comparison of the Numerical Responses Under Actual Excitation

From the above data, it can be seen that the proposed FAFNN-VFC can effectively suppress the low-amplitude high-frequency vibration and the occasional strong-amplitude vibration, and the damping force of each guide shoe is also greatly reduced. It has been fully shown that the variable universe fuzzy controller based on the FA-BP FNN can densify the control in the local area of low-amplitude vibration while ensuring the global control effect. The effectiveness of the controller’s response to the actual excitation from high-speed elevator car horizontal vibration has been verified.

C. Random Excitation Simulation

The excitation signal for guide rail surface irregularity caused by manufacturing and installation is mostly a low-amplitude and high-frequency signal. To verify the control effect of the proposed controller on this kind of excitation, a standard normally distributed random signal with mean value of 0 and standard deviation of 16 mm² was used as the random excitation of guide rail irregularity in the simulation test. The results are shown in Figure 14. and Figure 15.

FIGURE 14.

The comparisons of (a) time domain and (b) frequency domain horizontal vibration acceleration under random excitation of the guide rail.

FIGURE 15.

The comparisons of (a) time domain and (b) frequency domain tilt angle acceleration under random excitation of the guide rail.

It can be seen from Figure 14. that, compared with the FNN semi-active control and the passive control, the amplitude of horizontal vibration acceleration of the proposed FAFNN-VFC is greatly reduced under the random excitation of the guide rail. The vibration of approximately 4 Hz frequency is significantly reduced, and the high-frequency vibration above 6 Hz is also effectively suppressed. Similarly, it can be seen from Figure 15. that the amplitude of tilt angle acceleration has decreased to a certain extent, and the peak value of tilt angle acceleration in the frequencies of 3–6 Hz has also been reduced.

To further verify the control effect of the proposed FAFNN-VFC, Table 3. shows the numerical comparison of horizontal acceleration $\ddot {x}_{c}$ , tilt angle acceleration $\ddot {\theta }$ and damping force $f$ under random excitation of the guide rail. Compared with the passive guide shoe, the RMS values of horizontal acceleration of the FNN controller and the proposed controller are reduced by 35.4% and 62.9%, respectively. The RMS values of tilt angle acceleration are reduced by 27.6% and 46.9%, respectively, and the RMS values of damping force are reduced by 34.9% and 61.7%, respectively.

TABLE 3 Comparison of the Numerical Responses Under Random Excitation of the Guide Rail

From the above data, it can be seen that the proposed FAFNN-VFC can effectively control the low-amplitude high-frequency signal excitation such as the random excitation of the guide rail, and the tilt angle acceleration and required control force are also improved to varying degrees. The results show that the variable universe fuzzy control based on FA-BP FNN can precisely control the low-amplitude high-frequency vibration signal represented by guide rail irregularity excitation.

D. Impulse Excitation Simulation

Considering the convex or concave welding seam and the joint step caused by uneven docking between two guide rails due to manufacturing and installation errors, the guide shoes will be subjected to a large impact load in a very short time when passing through the joint. This kind of excitation is mostly shown as a strong-amplitude signal in a very short time, which can be expressed in the form of impulse signal [33]. Therefore, a simulation test for an impulse excitation signal is designed in this section. The impulse signal is shown in Figure 16. and the simulation results are shown in Figure 17. and Figure 18.

FIGURE 16.

Impulse excitation signal of the guide rail.

FIGURE 17.

The comparisons of (a) time domain and (b) frequency domain horizontal vibration acceleration under the impulse excitation of the guide rail.

FIGURE 18.

The comparisons of (a) time domain and (b) frequency domain tilt angle acceleration under the impulse excitation of the guide rail.

It can be seen from Figure 17. that, compared with the FNN semi-active control and the passive control, the amplitude of horizontal vibration acceleration for the proposed FAFNN-VFC is greatly reduced under the impulse excitation of the guide rail, especially the convergence rate to zero after the impulse is significantly improved. In the frequency domain, the vibration peak at 2 Hz is significantly controlled, and the high-frequency vibration above 4 Hz is also effectively suppressed. Similarly, it can be seen from Figure 18. that the convergence rate of the tilt angle acceleration after the impulse is significantly improved, and the peak value of the tilt angle acceleration in the frequencies of 4–6 Hz is also greatly reduced.

To further verify the control effect of the proposed FAFNN-VFC, Table 4. shows the numerical comparison of horizontal acceleration $\ddot {x}_{c}$ , tilt angle acceleration $\ddot {\theta }$ and damping force $f$ under the impulse excitation of the guide rail. Compared with the passive controller, the RMS values of horizontal acceleration of the FNN controller and the proposed controller are reduced by 40.2% and 74.7%, respectively. The RMS values of tilt angle acceleration are reduced by 40.4% and 71.1%, respectively, and the RMS values of the required damping force are reduced by 36.9% and 73.8%, respectively.

TABLE 4 Comparison of Numerical Responses Under the Impulse Excitation of the Guide Rail

From the above data, it can be seen that the proposed FAFNN-VFC has a significant control effect on the occasional strong-amplitude signal, such as the impulse excitation of the guide rail, and the tilt angle acceleration and required control force are also greatly reduced. The results show that the variable universe fuzzy control based on FA-BP FNN has a good control effect on the occasional strong-amplitude signal represented by the impact load at the rail joint.

SECTION V. Conclusion

To suppress the horizontal vibration of a high-speed elevator lift car caused by uncertainties such as rail irregularity and piston wind in the hoistway, a new variable universe fuzzy controller based on FA-BP FNN is proposed for car system using MR dampers. The main conclusions are as follows:

1. The dynamic model of the car and the non-parametric model of the MR damper based on the FNN system are established. The accuracy of the MR damper model is verified by comparison with experimental data, which shows that the model can precisely simulate the horizontal vibration response of the car.

2. An experiment with a 7 m/s high-speed elevator is carried out, and the vibration characteristics of the experimental data are analyzed to explore the requirements for controller design. The results show that the car horizontal vibration signal mainly presents as a low-amplitude high-frequency signal and occasional strong-amplitude signal.

3. Simulation tests are carried out by three kinds of signals: the actual excitation, the random excitation and the impulse excitation. The results show that the proposed FAFNN-VFC can reduce the max and RMS values of the car’s horizontal vibration acceleration by more than 62%, and the tilt angle acceleration and required damping force are reduced by more than 46% and 61%, respectively, which are far better than the passive control and the FNN control. The effectiveness of the proposed controller is fully proved, and the control method is suitable for the horizontal vibration control of high-speed elevators.

The proposed control method provides important guidance for researchers who study vibration control for high-speed elevators and other systems with similar vibration characteristics. In future research, we will further consider the high-dimensional coupled vibration control of high-speed elevators and the development of an actual controller.