Tracking Control of Magnetic Levitation System Using Model-Free RBF Neural Network Design

This paper focuses on the tracking control problem of Magnetic Levitation System (MLS). MLS is highly unstable nonlinear system, in the process of operation, MLS needs to have strong robustness and anti-interference ability. Due to disturbance and parameter uncertainties in MLS, it is complex to obtain the exact dynamics of MLS, and it is difficult to design a suitable controller. The dynamic equation of the system is established by Lagrange equation, and then we propose an adaptive Sliding Mode Control (SMC) based on Radial Basis Function Neural Network (RBFNN). Because of the parameter uncertainties and disturbance in MLS, RBFNN is used to approximate the unknown dynamics in MLS. The stability of the closed-loop system is strictly proved by using the Lyapunov stability theory, which can achieve the uniform ultimate boundedness (UUB) of the signals of the closed-loop MLS. MATLAB environment is used to verify the performance of the proposed controller. Considering disturbance, parameter change, or unmodeled dynamics in MLS, proposed controller is compared with other nonlinear controllers, simulation results verify the effectiveness of the proposed approach.


I. INTRODUCTION
MLS is a highly unstable nonlinear system with single input and single output (SISO) [1]- [5]. The basic principle of magnetic levitation system is to balance the gravity of electromagnet through the electromagnetic force generated by current. The mathematical model is represented by the third-order nonlinear differential equation. The main purpose of MLS control is to make the system have strong tracking and anti-interference ability. MLS has the advantages of low energy consumption, no friction and so on. It has been widely used in engineering and education, such as magnetic levitation bearing [6], magnetic levitation train [7]- [9], magnetic levitation wind turbines [10], etc. The controller design is very important for engineering application, and the control of MLS is an important and attractive research direction in the control field.
In recent years, the problem of tracking control has been widely studied [11], [12], for instance, a fuzzy tracking controller is applied to completely non-affine uncertain switched pure-feedback nonlinear systems [13]. Reference [14] con-The associate editor coordinating the review of this manuscript and approving it for publication was Manuel Rosa-Zurera. siders the problem of fuzzy output-feedback tracking control for switched stochastic nonlinear systems in pure-feedback form. Reference [15] considers the quantized static output feedback dissipative tracking control problem for a class of discrete-time nonlinear networked systems based on Takagi-Sugeno fuzzy model approach. In order to improve the tracking ability of MLS, a variety of control methods have been applied to the control of MLS. Parameter estimation and generalized proportional integral control (GPIC) are proposed [16], and the exponential asymptotic stability of the system is guaranteed. Under the premise of nonlinear model, a hybrid control based on disturbance observer is designed to cancel the mismatched disturbance in the system and improves the anti-interference ability of the system [17]. A controller based on feedback linearization model is developed, and the control effect is better than the traditional cascade control of voltage and current [18]. The exponential tracking problem of MLS is studied under the condition of parameter uncertainty and external disturbance [19]. A Fractional Order PID controller is proposed, and the parameters of the controller are optimized by ant colony algorithm [20]. An Integral Backstepping controller is used to realize the suspension control, and the steady-state error of the system is reduced [21]. According to the parameter estimation observer, the problem of sensorless state observation of MLS is solved [22]. The chattering in traditional sliding mode control is eliminated by using supertwisting algorithm, and the asymptotic stability of tracking error is guaranteed in the presence of time-varying uncertain parameters and external disturbances [23]. Considering the nonlinear characteristics of the nonlinear MLS and the actual working environment, if the uncertainty or disturbance is large, the performance of MLS will deteriorate rapidly, so it is necessary to design a more robust controller. RBFNN Sliding Mode Control (RBFNNSMC) is proposed to deal with the unknown dynamics of MLS. Due to the universal approximation ability of neural networks, it is usually used to approximate unknown functions in control systems [24]. A RBFNN is directly used to design controller to deal with disturbance and parameters change in coupled motor drives [25]. Fuzzy wavelet neural network is proposed to improve tracking control accuracy of manipulator [26]. Nonlinear dynamical model of friction is approximated by neural networks [27]. In order to deal with the main issues including the communication time delay, various nonlinearities and uncertainties, an RBF adaptive controller is addressed for nonlinear bilateral teleoperation manipulators [28]. Reference [29] proposes a force sensorless control scheme based on neural network for interaction between robot manipulators and human arms in physical collision.
The main contributions of this paper include the following. An affine nonlinear model of MLS is established by Lagrange equation. In order to reduce the complexity of the model, the magnitude of magnetic flux is selected as the independent variable. The conventional methods need to use the precise mathematical model of MLS, but the precise model is not easy to establish. The conventional methods are sensitive to the system parameter change and disturbance. The proposed method does not need to know the precise model of the system, RBFNN are used to approximate the unknown dynamics in MLS. The proposed controller guarantees that the signals of the closed-loop MLS are UUB. Parameter change, disturbance, or unmodeled dynamics will be used to analyze the robustness and anti-interference ability of the proposed controller later.
The rest of this paper is arranged as follows: Section II establishes the affine nonlinear model of MLS by using Lagrange equation, in order to reduce the complexity of the model, the magnitude of magnetic flux is selected as variable. the input-output feedback linearization controller is designed in Section III. RBFNNSMC is proposed and system's stability is proved in Section IV. Section V analyzes the simulation results of closed-loop MLS. Section VI is the conclusion.

II. MATHEMATICAL MODELING OF MLS BASED ON LAGRANGE EQUATION
Generally, Newton's law and Kirchhoff's law are used to establish the nonlinear model of MLS. Considering the advantages of Lagrange equation modeling, that is, only  considering the energy of the whole system, and selecting reasonable generalized coordinates for MLS, modeling the system from the angle of energy can reduce the difficulty of the problem, and quickly get the nonlinear differential equation of the system. A simple single degree of freedom MLS block diagram is shown in Fig. 1. table 1 shows the parameters of Fig. 1.
Assumption 1: The magnetic resistance of the magnetic circuit formed by electromagnet and guide rail is mainly concentrated in the air gap between them.
Assumption 2: Ignore flux leakage. It can be seen from Fig. 1 that the input is voltage, and the current in the coil is used to generate electromagnetic force. The electromagnetic force keeps the controlled plant suspended at a given position. By changing the voltage, different electromagnetic forces are generated to make the system suspend in different positions. The main purpose of control is to make the system have strong tracking ability and robustness.
It can be obtained from Assumption 1 From Assumption 2, we get magnitude of magnetic flux λ where the coil winding inductance Electromagnetic energy of the system is From (4), we get f 0 Since the Lagrange equation is used in the modeling process, the following will briefly introduce the Lagrange equation. For an energy dissipative system with degree of freedom n, the Lagrange equation with dissipation function is [30] d dt where L is the Lagrange function, r andṙ are the generalized coordinates and their derivatives respectively, G is the dissipation function of the system, and Q j is a generalized force that is non-potential force except the dissipative force. Take the MLS as the research plant, According to (6), r 1 = z and r 2 = q are the generalized coordinate andż andq = i are corresponding derivative.
The kinetic energy of MLS can be written as The potential energy of MLS can be written as The energy dissipation function in MLS is For generalized coordinates r 1 and r 2 , we have Substituting (9)-(11) into (6), we get (12) Equation (12) can be written as Combining (2)- (3) and (12)- (13), we have Defining state variables Remark 1: According to (12)-(15), selecting λ instead of i as variable reduces the complexity of the model [31] simplifies the complicated mathematical derivation in the process of controller design. In (14), λ 2 and λz are nonlinear function, it shows that MLS is nonlinear system.

III. FEEDBACK LINEARIZATION CONTROL OF MLS
The SISO nonlinear system in the form of (16) is considered.
where y is the system output, u is the control variable, and χ is continuous function. The feedback linearization control law can be easily obtained by (16), and the expected closed-loop system is shown in (17) where k 0 , k 1 , . . . , k n−1 is the expected coefficient, x r is reference input. According to (16)-(17), we have χ y (n−1) , y (n−2) , . . . ,ẏ, y, u From (18), we get u = ψ y (n−1) , y (n−2) , . . . ,ẏ, y − y r For the MLS in this paper, we have Taking time derivative of (19) Taking time derivative of (20) Taking time derivative of (21) The control u appears on the right side of (22). Design the ideal closed-loop differential equation as shown in (23) ... Defining error e The basic objective of control is to ensure that the MLS can track reference input within limited time t 0 , i.e., e → 0, as t → t 0 .
From (22)- (24), we have where The feedback linearization control structure diagram of MLS is shown in Fig. 2.
Feedback linearization is generally combined with the classical pole placement method to design the controller. According to the pole assignment method, the settling time is defined as T s , and β = 6 T s , controller satisfies [k 0 , k 1 , k 2 ] = [β 3 , 3β 2 , 3β], Different T s corresponding controllers are shown in table 2.
In Fig. 3, z r is the reference input. It can be seen from table 2 that when T s = 0.2, the coefficient of the controller is the largest, the initial overshoot of the system response is the smallest; when T s = 0.4, the coefficient of the controller is the smallest, the corresponding initial overshoot of the system response is the largest. Assuming that the system has a change in model coefficient, the specific change is shown in (26). The curve z 1 is most affected by the change of parameters, and the maximum change is 0.0016m. Curve z 3 is the least affected by parameter change, and its maximum variation is 0.0005m. Although the decrease of T s can reduce the initial overshoot and the stronger the anti-interference ability, but we can't set the settling time of the system to zero, which is not in line with the actual situation. In the process of feedback linearization control design, the accurate model of the system is needed, which can't deal with the problems such as unknown dynamics and parameters change. In the next section, RBFNN combined with SMC is introduced to solve the control problems under the condition of unknown dynamics or parameters change.
where b 0 is the nominal value of the coefficient b.

IV. DESIGN AND STABILITY ANALYSIS OF RBFNN SLIDING MODE TRACKING CONTROLLER
In this Section, firstly, RBFNN is introduced, and then RBFNNSMC is designed for MLS with unknown dynamics.

A. RADIAL BASIS FUNCTION NEURAL NETWORK
RBFNN can approximate any continuous function with arbitrary precision [24], Fig. 4 describes the single implicit layer structure of RBFNN, SMC is simple and robust, and it is widely used in control system, especially in deterministic system which can establish accurate mathematical model [32]. In this paper, the RBFNNSMC is designed considering the advantages of neural network and SMC.
Because of the disturbance and parameters change in the system, f (x) and l(x) cannot be accurately obtained. Considering that f (x) and l(x) are unknown in MLS (22), RBFNN can be used to approximate f (x) and l(x) under the condition of satisfying assumption 3 and 4, as shown in (28). Although SMC can deal with the disturbance and uncertainty in the feedback system, it can not provide a satisfactory scheme when system's dynamics is unknown. When the system dynamics is unknown, RBFNNSMC method is proposed. The stability proof based on Lyapunov theory is provided. The weight coefficient of neural network is obtained by using the adaptive law of online learning to ensure the tracking ability of MLS.
where σ j is the radial basis function, as shown in (27) σ Assumption 3: x is bounded, f (x) and l(x) are smooth functions, and |f (x)| ≤ f max , |l(x)| ≤ l max , ∀x ∈ U 1 .

B. CONTROLLER DESIGN AND STABILITY PROOF OF CLOSED-LOOP SYSTEM
Suppose that f and l are known in (22) where c i > 0, i = 1, 2, And the polynomial λ 2 + c 2 λ + c 1 is Hurwitz, so e → 0 as s → 0.
Considering that f and l are unknown in system (22), sliding mode controller u and neural network parameter vectorN andŴ is designed, makes e → 0 as s → 0.
Theorem 1: Considering that f and l are unknown in system (22), and satisfy Assumption 3-4, the controller is (33), the adaptive law is (34), then the closed-loop system is UUB.
wheref andl are estimates of f and l, respectively, satisfying W =Ŵ − W * ,Ñ =N − N * . Remark 2: In order to avoid the influence of RBFNN on the system stability, a robust term τ sgn(s) is added. ξ and δ are positive constant, which determines the convergence rate ofŴ andN, γ and ς are small positive constants, which are mainly used to reduce the error of RBFNN approximation, and can also avoid the influence of excessive weight drift on system performance. The scheme of RBFNNSMC is shown in Fig. 5.

Equation (43) can be rearranged aṡ
Make some mathematical transformations on (44), we havė where exp(αt) is exponential function. From (45), we have From (36) and (46), we have From (46)-(48), we can know that s,W ,Ñ is bounded and converges to s , W , Ñ respectively, and the closed-loop MLS is UUB. where s , W , Ñ is given as Remark 3: Considering that system (22) is mainly composed of (33)-(35), the initial values of these parameters are bounded, and the proposed controller makes the system meet the Lyapunov stability, so t > 0, All signals in MLS are still bounded.

V. SIMULATION
In order to verify the effectiveness of the proposed controller (namely RBFNNSMC), the simulation model of the proposed controller is built in Simulink [33], [34]. In order to avoid the chattering of the closed-loop MLS near the sliding surface, (50) is used to approximate sgn(s) where υ = 0.05.
In order to analyze the performance of the proposed controller, we compare the proposed controller with SMC and Backstepping control, and compare their response performance index to verify the superiority of the proposed controller. The proposed controller is not compared with linear controller, because MLS is a highly nonlinear unstable system, linear control needs to linearize the nonlinear dynamics of MLS, the dynamics of MLS is modified, and linear control cannot fully reflect the information of MLS. The proposed controller is a nonlinear controller, and SMC and Backstepping control are nonlinear control, so it is compared with SMC and Backstepping control. The tracking performance of RBFNNSMC, SMC and Backstepping control are shown in Fig. 6, The control input of RBFNNSMC, SMC and Backstepping control are shown in Fig. 7, and performance analysis are shown in table 4.
As can be seen from Fig. 6 and table 4, all three controllers can track the reference input in finite time. To be more exact, the proposed controller has the smallest initial   overshoot which is 0.0015 mm, SMC's initial overshoot is 0.0145 mm, Backstepping controller has the biggest initial overshoot which is 0.069 mm. The proposed controller can reach steady-state in 0.67 s, and the steady-state time of both SMC and Backstepping control is greater than 1 s. As shown in Fig. 7, the control input of the proposed controller has fast convergence compared with SMC and Backstepping control.
To be more precise, from in table 4, in 0 s-.003 s, the control input of the proposed controller is from 23.05 to 8.03, the control input of SMC is from 10.37 to 8.51, and the control input of Backstepping control is from 8.88 to 7.34. On the basis of analysis, we can see that the proposed controller has VOLUME 8, 2020  batter tracking capability compared with SMC and Backstepping control. During the operation of MLS, MLS may be influenced by disturbance, parameters change or unmodeled dynamic, and the corresponding control structure diagram is shown in Fig. 8. The anti-interference ability and robustness of the proposed controller will be analyzed later.
For the purpose of analyzing the behavior of the proposed controller under the condition of external constant disturbance, external constant disturbance is added to MLS in 5 s-5.2 s, the response of MLS in case of external constant disturbance is shown in Fig. 9.
According to Fig. 9, it can be observed that the anti-interference ability of Backstepping control is weakest in case of external constant disturbance. To be more specific, after adding disturbance, the overshoot of the proposed controller is 0.3668 mm, the overshoot of Backstepping control is 1.999 mm, and the overshoot of SMC is 0.5739 mm. Compared with SMC and Backstepping control, the overshoot of the proposed controller is the smallest. This result shows that the proposed controller has stronger anti-interference ability and faster convergence compared with Backstepping control and SMC.
In order to assess the robustness of the proposed controller, considering the mass of MLS changes slowly over time, the response of MLS in the presence of mass change is shown in Fig. 10 and Fig. 11, the performance analysis of the three controllers is given in table 5. From Fig. 10,    To be more specific, the response of Backstepping control fluctuates between 2.79 mm and 3.19 mm, if the parameter changes increases further, MLS maybe become unstable. the response of SMC fluctuates between 2.95 mm and 3.05 mm, the fluctuation of proposed controller's response is almost zero. The proposed controller uses neural networks to approximate unknown dynamics, so the influence of parameter changes on the system is extreme small. On the ground of analysis, we can see that proposed controller has better robustness compared with SMC and Backstepping control under the condition of mass change.   In addition to the problems of disturbance and parameter change during the operation of MLS, the unmodeled dynamics of MLS should be taken into consideration. It is assumed that there is the high frequency unmodeled dynamic in MLS, an unmodeled dynamic (0.3 sin(10πt)) is added to MLS, the response of MLS in case of unmodeled dynamic is shown in Fig. 12. From Fig. 12, it can be seen clearly that all three controllers can make MLS run stably in the presence of unmodeled dynamic. More specifically, the response of SMC and Backstepping control has larger chattering than the proposed controller, if unmodeled dynamic increases further, SMC and Backstepping control maybe make MLS unstable. In term of analysis, we can see that the proposed controller has better effect on handling with unmodeled dynamic compared with SMC and Backstepping control.
The simulation of the proposed controller tracking sinusoidal signal will be given following. The reference input is given as x r = 0.003 + 0.001 sin(4t).
When selecting the reference input, the physical meaning should be considered, instead of blindly selecting the complex nonlinear reference input. As shown in Fig. 13, the proposed controller tracks the reference input nicely. According to Fig.14, there is no obvious chattering in the control input. The result shows that the proposed controller can nicely track the sinusoid input.

VI. CONCLUSION
In this paper, RBFNNSMC has been addressed for tracking problem of MLS with unknown dynamics. The unknown dynamics of MLS is approximated by RBFNN. The proposed controller guarantees that the closed-loop MLS is UUB. Simulation results have shown that the proposed controller has faster convergence and stronger robustness compared with SMC and Backstepping control. Due to the problem of parameters selection in the process of controller design, the intelligent optimization algorithm will be considered in the future research to optimize parameters.