Dynamic Sliding Mode Backstepping Control for Vertical Magnetic Bearing System

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.


Introduction
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 [1,2]. The MBS have many advantages over mechanical and hydrostatic bearings. These include zero frictional wear and efficient operation at extremely high speeds.
In addition, they are considered an ideal choice for clean environments as they do not require lubrication. MBS are used in many applications, such as energy storage flywheels, high-speed turbines, compressors, pumps and jet engines. Compared with other bearings, magnetic bearings have many advantages [3], as follows: (1) It has almost no rotation resistance and the rotor speed can be much higher than other bearings; (2) They do not require complex lubrication or pneumatic system and can save space; (3) They have a long service life and low maintenance cost; (4) They can avoid the friction-induced noise; (5) They can be applied to special working environment such as under extremely low temperature or high vacuum; (6) They can provide the required rigidity through active control and effectively restraining the vibration problem caused by high-speed operation. Because of the advantages mentioned above, the magnetic floating bearings have been widely used in many fields.
In recent years, several researches have been devoted to the development of magnetic floating bearings [4]. Although many studies have determined the magnetic circuit structure and mechanical structure design of magnetic floating bearings, few researchers have analyzed and studied the nonlinear dynamic characteristics of the magnetic bearing system (MBS). Besides, two estimator based nonlinear controllers [5,6] have been proposed, which are used to estimate the total uncertainty of parameter variations and external disturbances in real time. The design of backstepping control [6,7] cannot respond to the input signals effectively, so it should be combined with the adaptive estimator [8] to estimate the unknown parameters of the system. In the self-tuning controller, the parameters are adapted and the estimated parameters are provided to the controller to solve the nonlinear problem of the MBS. The adaptive backstepping controller [9] can perform better steady-state response by using an estimation system with adaptive rules.
This research paper promotes the application of dynamic sliding mode backstepping control (DSMBC) for the MBS [10]. The slip line from the sliding mode control is used to estimate the total uncertainty of the system. The slope of the sliding line tends to make the coming speed faster, so that the response is compensated in time. Moreover, the estimated values are more effective to reach the accurate values. In this case, the system produces better results than the backstepping control. The Lyapunov function is used in the backstepping control to guarantee the convergence of the position tracking error. Two height position tracking is considered and tested. The simulation and experimental results demonstrate that the developed controller provides better tracking performance in the respect of model uncertainty.
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.

Magnetic Bearing System Architecture
The structure of MBS is illustrated in Fig. 1. The system consists of a permanent magnet part and an electromagnet part. The magnetic bearing can float in the air when the electromagnet is electrified. The output current can reduce the power consumption to better alleviate the power required when the magnetic bearing position moves near the center. After comparing the command signal with the height signal feedback, the error signal is sent to the power amplifier through the controller operation. Furthermore, the voltage of the controller is transformed into electric current by a power amplifier and sent to an electromagnet to produce magnetic force. After the magnetic bearing can be suspended in the air, the height information is measured using an infrared sensor. The servo motor is used as the load platform, through which the load is added to complete the experiment. According to the Newton's laws of motion, the dynamic system model is given by where m is the bearing mass with a parameter of 1.6 kg, f d is external interference, x is the position of the magnetic bearing, and F z is the sum of the magnetic forces produced by the top and bottom of the electromagnet. In Fig. 2, the nonlinear electromagnetic force [11,12] can be expressed as where x 0 and i 0 are the initial position of magnetic bearing and the initial electric current respectively, i c is the control current, and k is the electromagnetic force coefficient. Fig. 2 shows the current of magnetic bearing and the relationship between the positions. The nonlinear electromagnetic force is modeled and linearized by using the curve fitting method [12], and is defined as where C z is the position parameter, and C i is the current parameter. Eqs. (3) and (4) are substituted into Eq. (1), and obtain The dynamic system can be written as

Conventional Adaptive Backstepping Control
The MBS can be considered as a general second-order nonlinear system.
where f ðxÞ and bðxÞ are unknown continuous nonlinear functions, u ∈ R is the control input, y ∈ R is the system output, and is the state vector of the system, which is assumed to be available for measurement. In order for the dynamic system to be controllable, the function bðxÞ must be nonzero for vector x in certain controllability region. Without losing generality, we assume that 0 < bðxÞ < 1. Considering the effect of parameter uncertainty, the dynamic equation is written as x d is the desired height. The differential height error defined is as follows The stability function is defined as where c 1 is a positive constant. The variable z 2 is defined as where z 2 is the stability function.
The difference between the actual value of the total uncertainty and the estimated value of the total uncertainty is given as 926 IASC, 2022, vol.32, no.2 Ẽ ðxÞ ¼ EðxÞ ÀÊðxÞ The continuous Lyapunov function is defined as The time derivative of function V 1 can be expressed as 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 uðtÞ is defined as where b is a positive constant.
Substituting Eqs. (16) and (17) into Eq. (15), which can be represented as 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.

Proposed Dynamic Sliding Mode Backstepping Control
The DSMBC method is derived and developed in the MBS system. The tracking error is defined as z 1 ¼ x d À x 1 . The x d is the desired height. The error differentiation variable is given as follows The stability function can be given by where C 1 is a positive constant. The sliding surface is selected as where C 2 is a positive constant. The difference between the actual value of the total uncertainty and the estimated value of the total uncertainty is given bỹ where E is the actual value of the total set uncertainty. The error parameter of C 2 is defined as where C 2 is a constant. The variable z 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 n 1 and n 2 are positive constants. The differential operator is used and it becomes The ideal control law is defined as where b 1 is a positive constant, and is the limited coefficient.
If g d > 0, it can be expressed as with C 1 z 2 1 ¼ K > 0, g d > g Integrating the above equation with respect to time, resulting in Because V b3 ð0Þ is bounded, and V b3 ðtÞ is nonincreasing and bounded. We obtain that S is bounded, which implies S 2 L 1 [13,14]. By using Barbalat's lemma [5,13,14], S will converge to zero as t ! 1.
And the lemma implies that lim t!1 eðtÞ j j ¼ 0. Thus, the designed system is stable and the error converges to zero asymptotically. The stability of the DSMBC scheme is guaranteed.

Experimental Results
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. Fig. 3 illustrates the experimental setup of the MBS equipment.   4 describes a block diagram of an adaptive backstepping control system. The MBS system with uncertainties is considered and analyzed. An error estimator is designed to estimate the system with unknown parameters. The estimator can modify the parameters according to the signal fed back to the system. It can also provide the estimated parameters to the controller on-line and calculate the estimated values of the parameters. The adaptive estimation scheme and backstepping control are used to solve the nonlinear problem of MBS. The adaptive backstepping control can keep the robustness of the system when the parameters of the system change or external disturbance occurs.
The system block diagram of the proposed DSMBC of the magnetic bearing is shown in Fig. 5. After the height command is obtained from the system, the controller can send the control command, and know the bearing system's height and the calculation error subtracted by command. Then, the output of the bearing system enters the control card to calculate the error adjustment sliding mode control to estimate the uncertainty. After adjustment, it enters the control card operation until the command is chased. The controller tends to be based on step-return control and cooperates with the sliding. The model can estimate the uncertainty, where we adjust the parameter C 2 to achieve faster compensation when the bearing encounters external disturbance.

Simulation Results
The simulation results using adaptive backstepping control method and the proposed DSMBC method were obtained and compared to verify the control performance. Adaptive backstepping controller C z ¼ 21, m ¼ 1:6, g ¼ 9:8,  Fig. 6, the output response of command height is 2.5 mm, and an additional external load of 0.3 A is added at t ! 5 s. In addition, it is clear from Fig. 6a that the overshoot response of adaptive backstepping control method is bigger and faster. The DSMBC method has a slower transient response, but a height response can be rendered smoother and more accurate. Compared with the tracking response depicted in Figs. 6b-6d, the conventional sliding mode backstepping control has larger tracking error and estimation error. Therefore, the DSMBC approach can have better error convergence than backstepping control scheme.
As shown in Fig. 7, the adaptive backstepping control has a faster transient response, as illustrated in Fig. 7a. However, the DSMBC method achieves the robustness performance under difference inputs and load conditions. The amount of control increases immediately after the height command changes, see Fig. 7c. Meanwhile, in Fig. 7d, the DSMBC also outperforms the adaptive backstepping control for estimation comparison. It can be seen that the DSMBC method performs better in real-time compensation and robustness tracking regardless of the height command and load disturbance.

Experimental Results
To verify the practicality of the proposed DSMBC system, the experimental tests were performed and demonstrated. Fig. 8 depicts that the height command is set as 2.5 mm during 0 t < 5 s and an additional external load of 0.3 A is added during t ! 5 s. The two adaptive controllers are employed, and the adaptive laws are also developed to estimate the uncertainty of the model. The parameters of DSMBC are adjusted adaptively, so that the adaptation gain can improve the tracking accuracy effectively. Figs. 8a-8d show that the DSMBC method has a faster response time and less overshoot than the traditional adaptive method. The DSMBC method can quickly track the height command in the transient state and has a smooth response in the steady state. Fig. 9a illustrates the height position response, and the error responses are shown in Fig. 9b. Fig. 9c presents the control input signals, and Fig. 9d depicts the estimated uncertainty responses. During the development of the proposed control method, the height error can be effectively reduced to within 1.1 s. The proposed DSMBC controller guarantees the asymptotic stability and exhibits better tracking capability.   where a is the number of the sampled points, and e i ½ is the position error.
Tabs. 2 and 3 clearly demonstrate that the DSMBC method is superior to the dynamic sliding backstepping schemes under all operational conditions because its energy control input is considered in our design. Thus, the experimental results can conclusively establish the regulation ability, dynamic tracking capability and robustness of the proposed adaptive method in a wide speed range.

Conclusion
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.