Electromagnets are commonly used as support for machine components and parts in magnetic bearing systems (MBSs). Compared with conventional mechanical bearings, the magnetic bearings have less noise, friction, and vibration, but the magnetic force has a highly nonlinear relationship with the control current and the air gap. This research presents a dynamic sliding mode backstepping control (DSMBC) designed to track the height position of modeless vertical MBS. Because MBS is nonlinear with model uncertainty, the design of estimator should be able to solve the lumped uncertainty. The proposed DSMBC controller can not only stabilize the nonlinear system under mismatched uncertainties, but also provide smooth control effort. The Lyapunov stability criterion and adaptive laws are derived to guarantee the convergence. The adaptive scheme that may be used to adjust the parameter vector is obtained, so the asymptotic stability of the developed system can be guaranteed. The backstepping algorithm is used to design the control system, and the stability and robustness of the MBS system are evaluated. Two position trajectories are considered to evaluate the proposed method. The experimental results show that the DSMBC method can improve the root mean square error (RMSE) by 29.94% compared with the traditional adaptive backstepping controller method under different position tracking conditions.

Dating back to 1842, it was first proposed to use permanent magnets or fixed current electromagnets alone. No matter how they were configured, it was impossible to suspend a magnetically guided object stably in midair, so it was necessary to find ways to better stabilize the operation. The active magnetic bearing was first produced in 1937 [

In recent years, several researches have been devoted to the development of magnetic floating bearings [

This research paper promotes the application of dynamic sliding mode backstepping control (DSMBC) for the MBS [

The rest of this paper is structured as follows. Section II introduces the construction of the MBS and system model. In Section III, the dynamic sliding backstepping control is designed and analyzed. Section IV provides the experimental results. Finally, in Section V, conclusions are drawn.

The structure of MBS is illustrated in

where

where

where

The dynamic system can be written as

where

The MBS can be considered as a general second-order nonlinear system.

where

where

The stability function is defined as

where c_{1} is a positive constant. The variable

where

The difference between the actual value of the total uncertainty and the estimated value of the total uncertainty is given as

The continuous Lyapunov function is defined as

The time derivative of function

The second Lyapunov function is selected as

where r is a positive constant. After differentiation operation, it can be expressed as

The ideal control input

where

Substituting

By using the control law, the state can always approach the sliding surface and hit it. The asymptotic stability of the system can be guaranteed.

The DSMBC method is derived and developed in the MBS system. The tracking error is defined as

The stability function can be given by

where

where

where E is the actual value of the total set uncertainty. The error parameter of

where _{2} is defined as

The candidate Lyapunov function is given by

The time derivative of the Lyapunov function is

The second Lyapunov function can be written as

where

The ideal control law is defined as

where

Substituting

If

with

Integrating the above equation with respect to time, resulting in

Because

The experimental results show the control capability of the developed algorithm for MBS. In this study, two types of control methods are compared. They are (a) the adaptive backstepping control method, (b) the proposed DSMBC method.

The system block diagram of the proposed DSMBC of the magnetic bearing is shown in

Control Methods | MBS Parameters | Controller Parameters |
---|---|---|

Adaptive backstepping controller | ||

Dynamic sliding mode backstepping controller |

The simulation results using adaptive backstepping control method and the proposed DSMBC method were obtained and compared to verify the control performance.

In

As shown in

To verify the practicality of the proposed DSMBC system, the experimental tests were performed and demonstrated.

where

Height position tracking(mm) | ||
---|---|---|

Control Methods | 2.5 mm | 2.5 mm to 2.3 mm |

Adaptive backstepping controller | 0.2153 | 0.2336 |

Dynamic sliding mode backstepping controller | 0.1776 | 0.1778 |

Height position tracking(mm) | ||
---|---|---|

Control Methods | 2.5 mm | 2.5 mm to 2.3 mm |

Adaptive backstepping controller | 0.3171 | 0.3204 |

Dynamic sliding mode backstepping controller | 0.2118 | 0.2287 |

The DSMBC method was successfully developed and used for vertical MBS in height tracking applications. The dynamic model of the MBS system was built by referring to the nonlinear characteristics, and the estimate functions of these nonlinear factors were proposed and applied to the equivalent control law of sliding mode control. The control system was designed, using the backstepping algorithm, and the stability of the MBS system was analyzed. Based on the Lyapunov theorem, the adaptive control law can be obtained and utilized for height position control application. The proposed DSMBC method achieves better tracking capability in real-time compensation and robustness tracking, and is not affected by height command and load disturbance. Two height trajectories are simulated and experimented to illustrate the robustness of the proposed system. Compared with the conventional adaptive backstepping method, the proposed DSMBC control system demonstrates more accurate performance, showing 20.70% improvement of RMSE in simulations and 29.94% improvement of RMSE in experiments. In the future, our goal is to establish a microcontroller or DSP platform, so that the proposed DSMBC method can be better implemented and widely used in industrial applications.

The authors thank TopEdit (www.topeditsci.com) for its linguistic assistance during the preparation of this manuscript.