The safety of the cable car system is very important for the lives of the people. But, it is easily affected by the environment such as the wind which causes the cable car system to have strong vibration disturbance, thus degrading the safety of the cable car system. In this paper, a new nonlinear active disturbance rejection control (ADRC) is proposed to restrain the vibration of the cable car. First, a new two-mass-spring system model is utilized to establish the cable car system model. The new translation vibration nonlinear model is derived by a linear-invariant two-mass-spring system. Then, a special nonlinear

Cable car is a means of transportation, which is erected on the slopes of the scenic area to transport tourists. There are many types of cable cars. This paper mainly studies the ropeway cable car. It is susceptible to strong interference, such as changes in the number of passengers, airflow, and changes in the characteristics of the cable car installation system, all of which affect the safe operation of the ropeway cable car. The ropeway cable car system is mainly composed of the driving device, load-bearing rope, traction rope, carriage, walking trolley, and anchoring drum. It is a typical complex electromechanical system with the characteristics of time-varying parameters, strong coupling, and nonlinearity. The driving device and the walking trolley are the core control components of the cable car control system, as well as the main factors affecting the cable car vibration. During normal operation, this part of the control system is a flexible connection, variable mass, variable speed, and variable stiffness motion control system.

As far as the control mechanism is concerned, the vibration problem belongs to the category of motion control and is caused by the resonance mode of the system; therefore, the two have great similarities. Thus, the vibration problem and the resonance mode can be attributed to the same type of problem. The resonance mode is the inherent internal dynamic characteristics of the system. The resonance mode changes with the change of the system parameters. When the natural frequency of the resonance is close to or overlaps the system bandwidth, it may cause mechanical resonance, causing oscillation and instability of the system. It makes the personal safety of cable car tourists greatly threatened. Based on this, how to suppress mechanical resonance problem in the ropeway cable car system has become the research direction of more and more experts and scholars. To deal with the resonance problem, on the one hand, the mechanical structure of the ropeway cable car system can be improved and on the other hand, the control compensation of the control system can also be started. Therefore, studying the control methods and controller design of complex control systems with time-varying parameters, strong coupling, multivariate factors, and other nonlinear factors can effectively improve the control effect of the ropeway cable car system, which is of great practical significance.

At present, the control method used in the ropeway cable car system is still based on traditional PID and improved PID. However, the control effect of this traditional single control method is increasingly unable to meet the actual control requirements of the complex control system, and it is difficult to solve the problems of system oscillation in the cable car system. Siemens, Yaskawa, and Panasonic have all adopted notch filters for simple compensation or correction of the servo system. This passive suppression method can effectively suppress mechanical resonance problems in the system, but it is necessary to know the resonant frequency of the ropeway cable car system when using the notch filter to compensate the system, which makes it more complicated to suppress the mechanical resonance of the system. Active disturbance rejection control treats all uncertain items and disturbance items of the system as total disturbances. Regardless of the resonant frequency of the cable car system, they can be treated as total disturbances, which have certain advantages.

In order to solve the control compensation of the vibration problem in the electromechanical system, many domestic and foreign scholars have carried out a lot of research work. Wie and Bernstein [

This paper analyzes the vibration model aiming at the vibration problem of the ropeway cable car system and designs a new nonlinear ADRC to suppress various uncertain nonlinear factors in the system [

The other organization content of this paper is arranged as follows: the second part completes the analysis of the cable car vibration model and derives the general expression of the cable car vibration model; the third part analyzes and designs the algorithm of nonlinear active disturbance rejection controller (NLADRC) aiming at the control goal of the cable car vibration; the fourth part applies a certain disturbance according to the cable car model which has been established and the control algorithm which has been designed and completes the verification through simulation technology; the fifth part gives the research conclusions of this paper and follow-up research objectives.

The cable car driving device pulls the walking trolley on the load-bearing rope through the traction cable so that the carriage reciprocates around the anchor drum to transport tourists. According to the motion form of the carriage carrier, it can be regarded as a translational motion control. The common vibration and disturbance suppression problems in the translational motion control can be reduced to a benchmark problem expressed by a two-mass-spring system. Assuming that the mass of the traction cable and its accessories is

The schematic diagram of the two-mass-spring system.

Among them, the mass of the traction device is

According to Newton’s second law and Hooke’s law, ignoring friction, the force condition on the mass block

The force condition of the mass block

Ignoring noise and interference, we set

The state space representation represented by the vector matrix is shown as follows:

Among them,

The above-deduced formula is the state space model of a linear system under ideal conditions, which is similar to a fourth-order linear time-invariant system. Referring to equation (

Note that among them,

Through the analysis of the two-mass-spring system, a system expression such as equation (

According to the control system shown in equation (

There are many “integrator” paths from the control input

The schematic diagram of the integrator path.

Here, there are ten integrator paths from the control input

The first path is

The second path is

The third path is

The fourth path is

After analyzing the integrator path, the least number of integrators in the path is 3, namely, the first path, and the relative order of the system is determined to be 3.

According to the open-loop system shown in equation (

Then,

Among them,

In equation (

According to the previous analysis and combined with the characteristics of the system, the ADRC controller structure is designed as shown in Figure

The schematic diagram of the NLADRC.

Tracking differentiator (TD) can arrange the transition process for the input signal

The differential signal of each order of the input signal

Using the nonlinear state observer, if the state of

After equation (

Among them,

Here,

In equation (

According to equation (

Then, the system shown in equation (

For system (

After equation (

Among them,

Parameter sequences.

According to equations (

From the system shown in equation (

Comparing with the original system, it is relatively easy to deal with the linear integrator series system. It can adopt the linear combination method or nonlinear combination method. During the operation of the system, it is sufficient to appropriately apply control forces to make

The control quantity

Substituting equation (

Equation (

According to the above analysis, using the system errors

If the input signal

Through the previous analysis, the NLADRC algorithm in this paper mainly includes three parts: TD, LSEF, and ESO. The complete algorithm is as follows:

In this paper, for the two-mass-spring system represented using equation (

This paper uses the discretization form of equation (

TD parameter table.

0.01 | |

When the input signal

TD trend chart of the square wave.

When the input signal

TD trend chart of

When the input signal changes to

TD trend chart of

TD trend chart of

The simulation results show that the designed TD tracking algorithm can quickly track the zero-order, first-order, second-order, and third-order differential signals of the input signal, with good accuracy and robustness.

This paper uses the discretization form of the ESO algorithm in equation (

ESO parameter table.

0.01 | |

In Figure

In Figure

In Figure

The simulation results show that the designed ESO estimation algorithm can estimate the state signal very accurately and the speed is also satisfactory.

This paper uses the discretization form of complete control algorithm shown in equation (

LSEF parameter table.

0.01 | |

10 | |

100 | |

300 | |

180 |

When the input signal

Control system response of 1(

When the input signal

Control system response of 3(

When the input signal

Control system response of

When the input signal

Control system response of

The periodic square wave signal has a severe impulse in the initial stage of the signal. The error signal

The input signal

Control system response with disturbance signal

The input signal

Control system response with disturbance signal

The simulation results of the complete control algorithm show that after the control system has set the control parameters, the controlled output

Aiming at the ropeway cable car vibration problem, this paper derives the nonlinear state space representation form in the general sense based on the two-mass-spring system model of translational vibration. Combined with the established representation model, this paper analyzes the relative order of the system, which provides a basis for the order of the NLADRC algorithm. On this basis, this paper establishes the NLADRC structure and demonstrates the corresponding algorithm from the three aspects of TD, ESO, and LSEF in combination with theoretical analysis. The complete algorithm of the NLADRC controller for suppressing the vibration of the cable car is obtained. Finally, this paper verifies the feasibility and accuracy of the algorithm through simulation. The follow-up work will further analyze the movement characteristics of the cable car on the basis of suppressing the vibration of the cable car and is committed to applying the research results of this paper to the actual system.

No data were used to support this study.

The authors declare that there are no conflicts of interest.

This work was supported by the Shandong Provincial Key Research and Development Project (2019GGX101005), Shandong Provincial Natural Science Foundation (ZR2017MF048), and National Natural Science Foundation of China (61803216).