Nonlinear dynamic modeling and fuzzy sliding-mode controlling of electromagnetic levitation system of low-speed maglev train

3.1. Nonlinear sliding mode controller

Since the conventional linear control algorithm cannot reject the nonlinear dynamic disturbance and multi-parameters perturbations/uncertainties satisfactorily, the number of passengers in low-speed maglev train is limited strictly. In order to solve this problem, a continuous robust nonlinear control algorithm is developed in this section. As is known to all, the sliding mode is a robust control method and can effectively deal with parameter perturbations and disturbance. But for the strong nonlinear systems such as ELS, it is difficult to design the control algorithm without any linear approximation and analyze the stability on the sliding mode surface. We will design and modify the sliding mode controller based on nonlinear mathematic model without any approximation subsequently.

The state variables of system are chosen as follows:

The airgap error and error rate are defined in the following manner:

where, $r$ is the target levitation airgap.

The sliding mode surface of the system is designed as:

where, $C=[c_1,c_2]$, $R=[r,\dot{r}{]}^T$, $X=[x_1,x_2{]}^T$.

To ensure the stability of the system, the Hurwitz’s stability need to be satisfied, which is $c_1/c_2>0$.

It must meet the requirement that $\underset{s\to 0}{\mathrm{l}\mathrm{i}\mathrm{m}}sds/dt\le 0$ to guarantee the existence and reachability of the sliding mode. The moving point’s stability condition on the sliding surface is $ds/dt=0$.

The following equation should be satisfied with the system’s entrance into sliding motion:

where, $u_s$denotes the output of the controller. $f_d$ is disturbance, and $\left|f_d\right|\le D$. $D$ is the maximum upper bound of disturbance.

The controller is designed by the exponential reaching law which is shown as follow:

where, $\xi$ and $k$ denote constant reaching coefficient and exponential reaching coefficient, respectively.

Based on (13), (14) and (17), the sliding mode controller (SMC) of the system can be designed as follow:

Stability analysis:

Lyapunov function candidate is selected as:

This function is positive semi-definite, and the derivative of the both side can be obtained as:

Based on the Lyapunov theorem, it is easy to prove that the system is global asymptotically stable.

The sliding mode controller (SMC) is designed in the form of (18). Set $c_1=$ 8, $c_2=$ 1, $k=$ 7, $\xi =$ 13. The target airgap is 9 mm. The initial position is 17 mm. In order to demonstrate the increased performance over traditional linear controller, we compare the proposed controller with PID controller. The parameters of PID controller are sufficiently tuned to obtain the best performance, which yields the following values: $K_P=$ 5700, $K_I=$ 1700, $K_D=$ 2400.

The simulation results of airgap with PID controller and FSMC controller are illustrated in Fig. 2. The simulation results of control input are shown in Figs. 3. The airgap response with PID controller has oscillation. The SMC has better dynamic performance than PID, and has no overshoot. The response of levitation airgap of SMC is faster than PID controller and stability performance of SMC are better. But the control current of SMC as shown in Fig. 3 shows that high frequency oscillations exist in the controller inputs, which is called chattering phenomenon.

Fig. 2Simulation results of airgap (mm)

Fig. 3Simulation results of control current (SMC)

The dynamic disturbance shown in Fig. 4 is applied to the system. The simulation results of airgap under disturbance are shown in Figs. 5. The control current of SMC under the dynamic disturbance is shown in Fig. 6.

Fig. 4Periodic dynamic disturbance force

Fig. 5Airgap under disturbance (mm)

Fig. 6Control current under disturbance (SMC)

We can learn from the Fig. 5 that the system has no overshoot and the regulation is completed within 0.5 s. Comparing PID control performance, the SMC has stronger robustness against disturbance. However, Figs. 6 illustrates that the chattering phenomenon exist in the controller inputs. The chattering will bring mechanical wear to the actuator and increase the consumption of energy, which restrict the application of sliding mode control in the engineering practice [27].

3.2. Fuzzy-sliding mode controller

To reduce the chattering phenomenon, a fuzzy inference method is utilized to improve the sliding mode controller.

The sliding mode control strategy is expressed as follows:

where, the switching control law is $u_{sw}=\frac{m}{c_2{P}_i}\left(c_{2}D\mathrm{s}\mathrm{g}\mathrm{n}\right(S)/m-\xi \mathrm{s}\mathrm{g}\mathrm{n}\left(S\right))$.

The switching control law $u_{sw}$ contains the symbol term$\mathrm{s}\mathrm{g}\mathrm{n}(\cdot )$, which leads to the sliding mode variable structure control cannot keep the control object on the sliding surface. The control object crosses back and forth on both ends of the sliding surface to generate chattering. In order to suppress the chattering and the influence of disturbance, fuzzy control is introduced to the control law [23-26]. The output of the fuzzy control is a new switched control law $u_f$, which instead of the gain term in the switching control of the traditional sliding mode control. The control signal is smoothed by the fuzzy control, thereby reducing the chattering. To sum up, the control diagram of ELS based on variable structure control strategy with fuzzy sliding mode can be illustrated in Fig. 7.

Fig. 7Control block diagram of FSMC

$u_f$ is the output of a fuzzy controller. After adding fuzzy control, the inputs of fuzzy control are $S$ and $\dot{S}$. $S$ represents the distance between the phase locus and the sliding surface of the system, while $\dot{S}$ is expressed as the velocity of the sliding surface. The unmodeled high-frequency of the system and other disturbance will be indirectly reflected in the $\dot{S}$ through ${\dot{x}}_1$, so it is reasonable to use $S$ and $\dot{S}$ as the input of fuzzy controller to infer the value of $u_{sw}$.

In addition, the fuzzy processing method can filter out the high-frequency noise caused by internal disturbance and inaccurate measurement. The basic universe of discourse of control inputs $S\text{,}$$\dot{S}$ and output $u_{sw}$ in the two-dimensional fuzzy controller are: $\left[\begin{array}{ll}-\left|S_{\mathrm{m}\mathrm{a}\mathrm{x}}\right|& \left|S_{\mathrm{m}\mathrm{a}\mathrm{x}}\right|\end{array}\right]\text{,}$$\left[\begin{array}{ll}-\left|D{S}_{\mathrm{m}\mathrm{a}\mathrm{x}}\right|& \left|D{S}_{\mathrm{m}\mathrm{a}\mathrm{x}}\right|\end{array}\right]$, $\left[\begin{array}{ll}-\left|{u_{sw}}_{\mathrm{m}\mathrm{a}\mathrm{x}}\right|& \left|{u_{sw}}_{\mathrm{m}\mathrm{a}\mathrm{x}}\right|\end{array}\right]$, respectively. The linguistic variable of $S$, $\dot{S}$ and $u_{sw}$ are $S$, $DS$ and $UF$. The fuzzy universe of discourse is divided into 7 subsets: PB, PM, PS, ZO, NS, NM, NB. The grades of universe of discourse are expressed as follows: $S=$ {–6, –4, –2, 0, 2, 4, 6}, $DS=${–3, –2, –1, 0, 1, 2, 3}, $UF=$ {–6, –4, –2, 0, 2, 4, 6}.

The membership functions of each fuzzy linguistic value subsets are determined by triangle distribution as shown in Figs. 8-10.

Fig. 8Membership function of input S

Fig. 9Membership function of input S˙

Fig. 10Membership function of output usw

The fuzzy rule for the switching control law is shown as follows:

$R^i$: ($S\left(t\right)$ is $A^i$) and ($\dot{S}\left(t\right)$ is $B^i$) then ($u_{sw}\left(t\right)$ is $C^i$), where $i=1,\dots ,n$, $n$ denotes the rule number. $A^i$, $B^i$ and $C^i$ represent fuzzy sets of $S\left(t\right)$, $\dot{S}\left(t\right)$ and $u_{sw}\left(t\right)$, respectively. Since each input has seven fuzzy sets, the total rule number is $N=$49 fuzzy rules. Table 2 describes the fuzzy control rules for the ELS.

In order to meet the conditions of $S\dot{S}<0$, the fuzzy control schemes for $u_{sw}\left(t\right)$ is described in Table 2.

Stability analysis:

Lyapunov function candidate is selected as:

This function is positive semi-definite, and the derivative of the both side can be obtained as:

It can be seen from the Table 2, when $s=PB$ and $\dot{s}=PB$, $s\dot{s}$ is also the same as PB. At the moment, it is necessary to control the input of positive and big variable to reduce $s\dot{s}$ rapidly and close to $s\dot{s}<0$. When $s$and $\dot{s}$were positive and negative respectively, it satisfies this condition of $s\dot{s}<0$. Thus, at this time the control variable is zero (ZO). When $s=NB$ and $\dot{s}=NB$, $s\dot{s}$ is also the same as PB. At this point, $s\dot{s}>0$ is positive and big. It is necessary to control the input of negative and big variable to reduce $s\dot{s}$ rapidly and close to $s\dot{s}<0$.

The fuzzy control system is stable because of all the control rules are designed based on $s\dot{s}<0$. We can obtain the conclusion as follows:

Thus, $V\left(x\right)\ge 0$, $\dot{V}\left(x\right)\le 0$. Based on the Lyapunov theorem, it is easy to prove that the system is global asymptotically stable.

$c_1$, $c_2$, $k$ and $\xi$ remain the same. The target airgap is 9 mm. The initial position is 17 mm. The system simulation results of position tracking performance are shown in Figs. 11-12. The dynamic disturbance of Fig. 4 is added to the system and the simulation results are provided in Figs. 13-14. We can learn that the dynamic performance of FSMC is about the same as SMC. The robustness of FSMC is slightly weaker than SMC against disturbance. But, the oscillations of controller inputs reduce significantly, which can significantly reduce the energy consumption of the system.

The maglev trains also have some parameter perturbations/uncertainties. The robustness from disturbances or parameter perturbations is crucial aspect of the controller. Different from the usual practice of only considering perturbation of the mass parameter, we consider the multi-parameters perturbations simultaneously, such as inductance $L$, mass $m$ and resistance $R$. The inductance of ELS varies by ± 20 %, the mass varies by ± 40 %, and the resistance of ELS varies by ± 20 %. The response of the airgap with multi-parameters perturbation is illustrated in Fig. 15. We can learn that the linear controller is sensitive to multi-parameters perturbations.

Fig. 11Airgap with FSMC (mm)

Fig. 12Control current with FSMC (A)

Fig. 13Airgap under disturbance (FSMC)

Fig. 14Control current under disturbance (FSMC)

Fig. 15The response of the airgap with multi-parameters perturbation (PID) (L varies by ± 20 %, m varies by ± 40 %, R varies by ± 20 %)

To study the robustness of proposed control algorithm from the multi-parameters perturbations, the inductance also varies by ±20 %, the mass varies by ±40 %, and the resistance varies by ±20 %. The response of the airgap of the FSMC with multi-parameters perturbation is shown in Fig. 16. We can see that the FSMC can eliminate the mismatches between mathematical model and dynamics of the ELS with multi-parameters perturbations.

In conclusion, the simulation results indicate that the proposed controller using fuzzy inference method with sliding mode surface has strong robustness against the disturbance and multi-parameters perturbations, excellent dynamic and steady-state performance without chattering and high energy consumption.

Fig. 16The response of the airgap with multi-parameters perturbation (FSMC) (L varies by ±20 %, m varies by ±40 %, R varies by ±20 %)

4.1. Experiment 1: Levitating from stationary stat

The presented FSMC controller and PID controller are applied in the test train respectively. The maglev train is stationary on the rail during 0 s-14 s with the initial airgap of 17 mm. At 14 s, the train begins to be levitated to the target airgap of 9 mm. The dynamic response of the levitating airgap during the experiment is provided in Fig. 18.

The solid line in Fig. 18 represents the FSMC control performance and the dashed line denotes PID control performance. The first 14 s shows the train is on the rail and the initial airgap between rail and electromagnet is 17 mm. At 14 s, the train begins to track the target position. We can see from the Fig. 18 that the levitation airgap response of PID is fluctuant and the dynamic performance of FSMC is better, which can rapidly convergence to the target position. Moreover, The FSMC controller has no overshot. The control current response of PID and FSMC controller is illustrated in Fig. 19. We can see that though the FSMC is based on sliding mode control with chattering phenomenon in control inputs, the fuzzy control law has reduced the chattering effectively. The fluctuation of FSMC control current in Fig. 19 can meet the engineering application requirements.

Fig. 18Results of Experiment 1: Airgap (mm)

Fig. 19Results of Experiment 1: Control current (A)

4.2. Experiment 2: with periodic shock disturbance

When the trains arrive at station, the passengers get on or off the train will lead to inevitable shock disturbances to the levitating train. In this experiment, we take the jumping load of 10 people in the train as periodic shock disturbance. The airgap response results of the levitating train with conventional PID and proposed FSMC are provided in the Fig. 20 and Fig. 21, respectively. The experiment results show that the airgap with PID controller has large variational amplitude under the jumping disturbance, but the airgap with FSMC controller remains in the target position with small fluctuation. The acceleration of the train with FSMC is shown in Fig. 22. The response of the control current with FSMC is provided in Fig. 23. It is shown in the figure that the chattering phenomenon in control inputs is suppressed significantly, thus the FSMC can save energy consumption and reduce wear of the actuator.

Fig. 20Results of Experiment 2: Airgap with PID (mm)

Fig. 21Results of Experiment 2: Airgap with FSMC (mm)

Fig. 22Results of Experiment 2: Acceleration with FSMC (m/s2)

Fig. 23Results of Experiment 2: Control current with FSMC (A)

The fuzzy sliding mode controller presented in this paper is more suitable for practice application in the maglev train. The experimental results of the test maglev train demonstrate that the FSMC control algorithm makes the train stable levitating on a target airgap with stronger robustness, better dynamic performance and higher anti-disturbance capacity.