Backstepping Sliding Mode Control for Magnetic Levitation Position Control: Design and Analysis

The Maglev system has many applications because it eliminates energy losses caused by surface friction. The Maglev system is a technique that allows an object (the ball) to float in the air without support. Therefore, it must generate a flux by controlling the amount of a coil current. The generated magnetic flux will be utilized to make the body levitate in the air at a predetermined distance from the coil position so that the body movement within that generated magnetic flux can be negligible [1]. The magnetic field’s power is in the opposite direction to the gravity attraction of the ball, so the body will be lifted toward the coil. Using vibration isolation, this system is capable of solving the friction problem. Consequently, friction loss can be negligible in this system, which undoubtedly impacts both the desired response and the performance of the system [2].

The non-contact property with zero friction of the Maglev system makes it very popular, and it is considered as the future technology. The Maglev system has many applications, such as high-speed transportation systems, launching rockets, as well as other applications, etc. [3, 4]. Many controllers have been proposed to control the Maglev model, as discussed below.

Jose and Mija [5] proposed a fractional order sliding mode controller (FOSMC) for controlling the position of the Maglev system. The Particle Swarm Optimization (PSO) algorithm is employed to determine the fractional order switching surface and the order of the fractional derivative. Adil et al. [6] used supertwisting SMC and integral backstepping SMC for controlling the position of the Maglev model. The supertwisting SMC provides a superior dynamic response with negligible chattering and is robust against external disturbances in controlling the Maglev model compared to the integral backstepping SMC and other nonlinear controllers. Burakov [7] utilized a fuzzy PID controller for the Maglev system control system. The controller is structured with three channels, each with its own control function that characterizes the variation in the gain factor based on the input variable's value. The process of setting the controller entails performing genetic optimization algorithm in off-line mode. Wei et al. [8] used a deep neural network feedforward compensation controller based on an enhanced Adagrad optimization algorithm for the position control of the Maglev system. The control structure of the controller comprises a deep neural network identifier, a deep neural network feedforward compensator, and a PID controller. The performance proposed controller demonstrates good dynamic and static performance as well as some robustness. Humaidi et al. [9] proposed the design and analysis of the Active Disturbance Rejection Control (ADRC) approach for the control and disturbance rejection of the Maglev system. Two controllers are considered using the ADRC structure. These controllers are known as the Linear ADRC (LADRC) and Nonlinear ADRC (NADRC). A comparison of the robustness against parameter variation and the capability to reject applied disturbance of LADRC and NADRC has been conducted. The MATLAB simulation results showed that LADRC exhibits better robustness characteristics compared to NADRC. Moreover, when a specific disturbing force is applied to the ball mass, the LADRC shows superior disturbance rejection capabilities compared to the NADRC.

MohammadRidha and Kadhim [10] used an Adaptive Variable Structure Controller based on barrier function (AVSCbf) for position control of the Maglev system. The performance of the AVSCbf has been compared with that of the Adaptive Variable Structure Controller without the barrier (AVSC) and the classical Variable Structure Controller (VSC). The simulation results show that the AVSCbf outperforms the AVSC and VSC in reducing steady-state error and improving disturbance rejection. Yadav et al. [11] utilized an optimized proportional-integral-derivative (PID) controller for position control of the Maglev system. The variables of the PID controller are optimized using Grey Wolf Optimizer (GWO). The proposed controller's effectiveness has been validated through comparison with a classical PID controller tuned using Ziegler-Nichols tuning criteria.

The following points summarize the contributions of this study:

• To enhance and develop the performance of the SMC for controlling the position of the steel ball of the Maglev model by replacing the equivalent control law of the SMC with the backstepping control law.

• To employ a BSSMC in order to better cope with the effect of the external disturbances that affect the position control of the Maglev system.

• To perform a comparative study of the controlled system performance utilizing a BSSMC and other controllers. The assessment of each controller used to control the position of the ball of the Maglev system is conducted based on minimizing the settling time and the external disturbance effect.

This work is addressed as follows: the mathematical model for the Maglev system is presented in section two. The controllers design for the SMC and BSSMC are explained in section three. In section four, the computer simulation results are discussed to verify the proposed controller’s effectiveness. Finally, the conclusion is addressed in section five.

The first design step involves defining the error $z_1(t)$ between the actual position of the ball $x_1(t)$ and the desired position $x_d(t)$ as follows [15-18]:

$z_1(t)=x_1(t)-x_d(t)$           (9)

The first- and second-time derivative of Eq. (9) can be obtained as follows:

$\dot{z}_1(t)=\dot{x}_1(t)-\dot{x}_d(t)$             (10)

$\dot{z}_1(t)=x_2(t)-\dot{x}_d(t)$             (11)

$\ddot{z}_1(t)=\dot{x}_2(t)-\ddot{x}_d(t)$             (12)

where, $x_1(t)$ is the actual position of the ball, $x_d(t)$ represents the desired position of the ball, and $x_2(t)$ is the actual velocity of the ball.

Eq. (12) can be expressed as follows:

$\ddot{z}_1(t)=g-\frac{k_o k_1^2}{m} \frac{u^2(t)}{x_1^2(t)}+d(t)-\ddot{x}_d(t)$              (13)

To design a SMC, the sliding surface can be expressed using the error equations provided in Eqs. (9) and (11) as follows:

$s(t)=\lambda z_1(t)+\dot{z}_1(t)$          (14)

where, λ is a positive constant.

The first-time derivative of the equation for a sliding surface is given by:

$\begin{gathered}\dot{s}(t)=\lambda\left(x_2(t)-\dot{x}_d(t)\right)+g-\frac{k_o k_1^2}{m} \frac{u^2(t)}{x_1^2(t)}+d(t)- \ddot{x}_d(t)\end{gathered}$          (15)

By setting (i=u), the above equation is written as follows:

$\begin{gathered}\dot{s}(t)=\lambda\left(x_2(t)-\dot{x}_d(t)\right)+g-\frac{k_0 k_1^2}{m} \frac{u^2(t)}{x_1^2(t)}+d(t)- \ddot{x}_d(t)\end{gathered}$         (16)

Based on SMC theory, the control law u consists of the following:

$u_1=u_{e q}+u_{s w}$             (17)

where, $u_{e q}$ represents the equivalent control law, and $u_{s w}$ represents the switching control law.

These two control laws can be described as follows:

$u_{e q}=\sqrt{\left(\frac{m x_1^2}{k_o k_1^2}\right)\left(\lambda x_2(t)-\lambda \dot{x}_d(t)+g-\ddot{x}_d(t)\right)}$          (18)

$u_{s w}=\operatorname{Ksat}(s)$          (19)

where, K represents a scaler design constant and sat(s) represents the saturation function (boundary layer function) used instead of the signum function in the sliding mode control effort to avoid the chattering effect.

The saturation function is defined in Eq. (20) as follows:

$\operatorname{sat}(s)=\left\{\begin{array}{lr}1 & s \geq \Delta \\ \frac{s}{\Delta} & -\Delta<s<\Delta \\ -1 & s \leq-\Delta\end{array}\right.$              (20)

where, ∆ represents the boundary layer thickness.

When the sliding surface and its derivative are set to zero, the sliding mode control effort can be expressed as follows:

$u_1=\sqrt{\left(\frac{m x_1^2}{k_0 k_2^2}\right)\left(\lambda x_2(t)-\lambda \dot{x}_d(t)+g-x_d(t)+\operatorname{Ksat}(s)\right)}$               (21)

The controlled system block diagram using MSMC for controlling the ball position of the Maglev model is shown in Figure 2.

2.png

Figure 2. The block diagram of the controlled system using MSMC for the position control of the steel ball of the Maglev model

To design a BSSMC, the backstepping control law is used instead of the equivalent control law in the SMC. To design the backstepping control law, the error $Z_2(t)$ between the actual velocity of the ball and the virtual controller is defined as follows [19-22]:

$z_2(t)=x_2(t)-\alpha(t)$            (22)

where, $\alpha(t)$ is the virtual controller.

Eq. (23) is derived by substituting Eq. (22) into Eq. (11) as follows:

$\dot{z}_1(t)=z_2(t)+\alpha(t)-\dot{x}_d(t)$          (23)

Using the virtual controller as in Eq. (24), the Eq. (23) is written as follows:

$\alpha(t)=-c_1 z_1(t)+\dot{x}_d(t)$           (24)

$\dot{z}_1(t)=-c_1 z_1(t)+z_2(t)$          (25)

where, $c_1$ a positive design variable.

Taking the time derivative of Eqs. (22) and (24) gives:

$\dot{z}_2(t)=g-\frac{k_o k_1^2}{m} \frac{u^2(t)}{x_1^2(t)}-\dot{\alpha}(t)$          (26)

$\dot{\alpha}(t)=-c_1\left(x_2(t)-\dot{x}_d(t)\right)+\ddot{x}_d(t)$            (27)

The time derivative of the error $z_2(t)$ can be expressed as follows:

$\dot{z}_2(t)=-c_2 z_2(t)-z_1(t)$           (28)

where, $c_2$ is a positive design variable.

The backstepping control law is obtained by substituting Eqs. (9), (22), (24), (27), and (28) in Eq. (26) with some rearrangements as in Eq. (29). Finally, the backstepping sliding mode control law is expressed in Eq. (30). The block diagram of the controlled system using BSSMC for controlling the position of the ball of the Maglev model is shown in Figure 3.

$u_b=\sqrt{\begin{array}{c}\left(\frac{m x_1^2}{k_o k_1^2}\right)\left(\left(c_1 c_2+1\right) x_1(t)+\left(c_1+c_2\right) x_2(t)\right.  \left.-\left(1+c_1 c_2\right) x_d(t)-\left(c_1+c_2\right) \dot{x}_d(t)-\ddot{x}_d(t)+g\right)\end{array}}$                 (29)

$u_2=\sqrt{\left(\frac{m x_1^2}{k_o k_1^2}\right)\left(\left(c_1 c_2+1\right) x_1(t)+\left(c_1+c_2\right) x_2(t)-\left(1+c_1 c_2\right) x_d(t)-\left(c_1+c_2\right) \dot{x}_d(t)-\ddot{x}_d(t)+g+\operatorname{Ksat}(s)\right)}$               (30)

3.png

Figure 3. The block diagram of the controlled system using BSSMC for the position control of the steel ball of the Maglev model

This work proposes a backstepping sliding mode control strategy for controlling and stabilizing the ball of the Maglev system. The effectiveness of the BSSMC and the MSMC have been examined using computer simulation based on the MATLAB program. A comparative study has been conducted to assess the performance of the BSSMC and the MSMC in conjunction with other controllers. The simulation results show that the performance of the controlled system using the BSSMC is better than that using the MSMC and other controllers in minimizing the settling time for a desired ball position of 0.01 m when the system is exposed to external force disturbance. Moreover, the proposed controllers’ effectiveness is validated for sinusoidal input reference. In this case, the controlled system dynamic performance using the BSSMC is also better than that using the MSMC and other controllers in minimizing the tracking time and the steady-state error in the presence of an external disturbance.

For future work, this study can be extended by implementing the backstepping sliding mode control strategy in a real-time environment via LabVIEW programming software or embedded hardware design such as the FPGA [23-25]. Furthermore, another extension of this study is to propose other control techniques for controlling and stabilizing the Maglev system's steel ball to demonstrate their effectiveness and performance in comparison to the proposed BSSMC [26-30].