Finite-time disturbance observer-based levitation control for vehicle-guideway coupling systems

Abstract

In this study, a novel composite control scheme for the vehicle-guideway coupling systems is proposed, consisting of FTDOs and a FTC, aiming to address the challenges of unknown disturbances and vibration suppression. Specifically, this method adopts a single magnet-track coupling model and introduces a finite-time disturbance observer (FTDO) that utilizes only measured electromagnet-side signals to estimate unmeasurable states and unknown disturbances. Based on the estimated information provided by the FTDO, a finite-time control (FTC) scheme is developed, which simultaneously handles the problems of disturbance compensation and finite-time tracking control. Additionally, the finite-time stability of the levitation system is analyzed and proven. Finally, simulation and experimental results are given to demonstrate the feasibility and superiority of the proposed control approach.

Keywords

Maglev vibration suppression finite-time disturbance compensation finite-time stabilization

1. Introduction

In recent years, magnetic levitation (maglev) vehicles have attracted considerable attention for their characteristics such as no wear, low noise, and strong capability of curve passing [1,2]. Unlike traditional wheel-rail trains, maglev vehicles are constructed by introducing electromagnetic levitation forces between the electromagnets and the track, enabling non-contact operation. This non-contact structure is highly suitable for long-term operation, where a levitation control mechanism to ensure safety and ride comfort is required when the maglev trains operate in complex environments [3].

Generally, the levitation system is nonlinear due to the interaction of the three-dimensional magnetic fields. The introduction of flexible beams makes the basic dynamics of the levitation system complicated. Additionally, in practical operations, the levitation system is often subjected to various unpredictable disturbances, such as internal parameter uncertainties, external disturbances, and coupling effects. These factors pose significant challenges to the design of levitation controllers, stimulating the exploration of robust control algorithms for maglev systems in this sturdy.

Various control methods have been explored to address these issues in the levitation system, including state feedback control [4], adaptive control [5–7], robust control [8], predictive control [9], sliding mode control [10,11], and model-free control [12,13]. Although these control strategies can improve levitation performance from different perspectives, they often neglect the flexibility of the track beam, which prevents the fundamental problem of vehicle-guideway coupling vibrations from being effectively resolved [14]. In order to overcome these limitations, it is proposed to incorporate track beam vibration signals into the feedback control law to actively attenuate track vibrations [15,16]. However, most of these controllers only guarantee asymptotic stability without addressing the robustness issues of the control system.

Finite-time control methods have been widely applied in various fields due to their robust performance and finite-time convergence. For instance, a continuous terminal sliding mode controller combined with high-order sliding mode observers has been applied to flexible-joint robots, demonstrating its effectiveness in handling mismatched disturbances [17]. In [18], an output feedback finite-time control method was proposed to stabilize a disturbed vehicle active suspension system. These references indicate that finite-time control methods can effectively improve system performance.

Furthermore, as the levitation performance inevitably suffers from unpredictable disturbances, disturbance observation techniques and active disturbance control methods have been introduced. A disturbance observer (DOB) was developed in [8] to estimate slow-varying disturbances exerted on the levitation system. In [19], a generalized proportional integral observer (GPIO) was designed to observe time-varying disturbances. Additionally, a finite-time disturbance observer was constructed in [20] to observe unmeasurable states and unknown disturbances of the system within a finite time. This observer framework can also be applied to other systems [21,22].

Inspired by the above literature, this paper proposes a finite-time control (FTC) scheme applicable to magnet-track coupled systems subject to external disturbances. Firstly, a magnet-track coupling dynamic model composed of electromagnets and a flexible track system is proposed. Then, a finite-time disturbance observer (FTDO) is proposed to estimate the unmeasurable states and unpredictable disturbances solely based on electromagnet-side measurement information. Finally, a robust FTC scheme is designed based on the estimated information. The proposed controller can handle unpredictable disturbances in finite time, thus enhancing the levitation stability of the electromagnets on variable-stiffness track beams. The main contributions of this work can be summarized as follows:

This paper proposes an FTC scheme to address time-varying disturbances affecting the electromagnetic magnets and the flexible track sides of the levitation system.

The proposed controller is simplified as it only utilizes electromagnet-side measurements and requires minimal mechanical parameter information, significantly reducing the design complexity.

The FTDO is introduced to observe the unmeasurable states and unpredictable disturbances, and a self-zeroing integrator is introduced to calculate the absolute velocity of the magnet.

Through experimental validation, the robustness of the controller to uncertainties is demonstrated, and superior vibration suppression performance is achieved on the flexible beam.

The article is structured as follows. In Section 2, the dynamics of a single magnet-track coupling system is described. Section 3 elaborates on the construction of the FTDO and FTC control laws, along with the proof of their stability. Section 4 presents comparative results between the proposed controller and two other controllers. Finally, a brief summary is provided in Section 5. It is important to note that the following definition will be used later: ⌈x ⌋^𝛾 = sign(x)|x|^𝛾, where 0 < 𝛾 < 1, and sign(⋅) denotes the standard sign function.

2. Model description

This section describes the dynamics model of the flexible track beam and the electromagnet.

2.1. Flexible track beam model

As shown in Fig. 1, the maglev vehicle is levitated on the track beam through electromagnetic forces provided by electromagnets mounted on levitation frames. Due to the decoupling principle described in [23], the magnet-track coupling model can be regarded as the minimum analysis unit. The stability of this unit is fundamental to ensure the stable levitation of the maglev vehicle.

Fig. 1.

Vehicle-track coupling system.

As illustrated in Fig. 1, z ₁ is the vertical displacement of the beam, and z ₂ denotes the vertical displacement of the electromagnet. d = z ₂ − z ₁ is the air gap. l _g is the length of the beam. x ₀ is the position of the electromagnet, and m ₂ is the electromagnet’s mass. F _m(t) represents the electromagnetic force.

In this model, the flexible beam is described in the form of Bernoulli-Euler beam, which can be expressed as [24]: $\begin{eqnarray}EI_{g}\frac{\partial ^{4}z_{1}}{\partial x^{4}}+{\rho}_{g}\frac{\partial ^{2}z_{1}}{\partial t^{2}}+C_{g}\frac{\partial z_{1}}{\partial t}=F_{m}(t){\sigma}(x-x_{0})\end{eqnarray}$ (1) where EI _g and 𝜌_g denote bending rigidity and line density of the track beam, respectively. C _g is the structural damping coefficient. 𝜎(⋅) is the Dirac delta function.

According to the field test results [25], the first vibration frequency of the track corresponds to the maximum amplitude. Therefore, in this study, we only consider the first mode of the flexible beam. By using the modal superposition theory, the differential equation (1) can be solved, and the displacement of the beam can be expressed as follow $\begin{eqnarray}z_{1}(x,t)={\phi}_{1}(x)q_{1}(x)\end{eqnarray}$ (2) where ${\phi}_{1}(x)=\sqrt{(2/{\rho}_{g}l_{g})}\sin ({\pi}x/l_{g})$ is the first-order modal shape function. q ₁(t) is the generalized coordinate.

Substituting (2) into (1) and then multiplying both sides by 𝜙_i(x), integrating the equation from 0 to l _g, it gives $\begin{eqnarray}\ddot{q}_{1}(t)+2{\xi}_{1}{\omega}_{1}\dot{q}_{1}(t)+{\omega}_{1}^{2}q_{1}(t)=F_{m}(t){\phi}_{1}(x_{0})\end{eqnarray}$ (3) where 𝜔₁ and 𝜉₁ represent the first mode natural frequency and damping ratio, respectively. And they can be described as $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle \displaystyle {\omega}_{1}=\left(\frac{{\pi}}{l_{g}}\right)^{2}\sqrt{\frac{EI_{g}}{{\rho}_{g}}}\end{eqnarray}$ (4) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle \displaystyle 2{\xi}_{1}{\omega}_{1}=\frac{C_{g}}{{\rho}_{g}}.\end{eqnarray}$ (5)

Multiplying both sides of Eq. (3) by 𝜙₁(x ₀), we can obtain $\begin{eqnarray}\ddot{z}_{1}(t)+2{\xi}_{1}{\omega}_{1}{\dot{z}}_{1}(t)+{\omega}_{1}^{2}z_{1}(t)=A_{1}F_{m}(t)\end{eqnarray}$ (6) where $A_{1}=∼{\phi}_{1}^{2}(x_{0})$ .

2.2. Electromagnetic model

For the levitation system, assuming no magnetic leakage and neglecting the magnetoresistance of the iron core and track, we can obtain the equation of motion, electromagnetic force equation, and electrical equation. $\begin{eqnarray}\displaystyle \left\{\begin{array}{@{}l@{}}\displaystyle m_{2}\ddot{z}_{2}=m_{2}g-F_{m}\quad \\ \displaystyle F_{m}=K\left(\frac{I}{d}\right)^{2},\quad K=-\frac{{\mu}_{0}N^{2}S}{4}\quad \\[5.0pt] \displaystyle u(t)=\mathit{IR}+\frac{d(LI)}{dt},\quad L=\frac{{\mu}_{0}N^{2}S}{2d}\quad \\[5.0pt] \displaystyle F_{m}(I_{0},d_{0})+m_{2}g=0.\quad \end{array}\right. & & \displaystyle\end{eqnarray}$ (7) The levitation force equation in (7) exhibits strong nonlinearity. To facilitate the design of controller, F _m can be linearized around the equilibrium point (I ₀, d ₀) $\begin{eqnarray}\displaystyle F_{m}(I,d)=F_{m}(I_{0},d_{0})+F_{i}(I_{0},d_{0})(I-I_{0})+F_{d}(I_{0},d_{0})(d-d_{0})+F_{h}(I,d) & & \displaystyle\end{eqnarray}$ (8) where $F_{i}=\frac{\partial F}{\partial I}|_{(I_{0},d_{0})}=2K\frac{I_{0}}{d_{0}^{2}}$ , $F_{d}=\frac{\partial F}{\partial d}|_{(I_{0},d_{0})}=-2K\frac{I_{0}^{2}}{d_{0}^{3}}$ , F _h(⋅) denotes the high-order term of F _m(I, d). Substituting (8) into (7), it gives $\begin{eqnarray}\ddot{z}_{2}=-\frac{F_{d}}{m_{2}}(z_{2}-z_{1})-\frac{F_{i}}{m_{2}}I-\frac{1}{m_{2}}F_{h}(I,d).\end{eqnarray}$ (9)

To proceed, we define new state variables as x ₁ = d, $x_{2}={\dot{d}}$ , $x_{3}=\ddot{d}$ , $x_{4}={\dot{z}}_{2}$ . Combining (6) and (9), we have $\begin{eqnarray}\left\{\begin{array}{@{}l@{}}{\dot{x}}_{1}=x_{2}\quad \\ {\dot{x}}_{2}=x_{3}\quad \\ {\dot{x}}_{3}=a_{1}x_{4}+{\varphi}_{1}(i,x,f_{d},t)\quad \\ {\dot{x}}_{4}=b_{1}u+b_{2}x_{1}+{\varphi}_{2}(i,x,f_{d},t)\quad \end{array}\right.\end{eqnarray}$ (10) where $a_{1}={\omega}_{1}^{2}$ , b ₁ = −F _i∕(m ₂ R), b ₂ = −F _d∕m ₂.

${\varphi}_{1}(i,x,f_{d},t)=-\left(\frac{1}{m_{2}}+A_{1}\right)\left(F_{i}\frac{di}{dt}+\frac{dF_{h}}{dt}\right)+2{\xi}_{1}{\omega}_{1}\ddot{z}_{1}+\frac{1}{m_{2}}\frac{df_{d}}{dt}-\left(\frac{1}{m_{2}}+A_{1}+{\omega}_{1}^{2}\right)x_{2}$ and 𝜑₂ (i, x, f _d, t) $=\frac{F_{i}L}{m_{2}R}\frac{di}{dt}-\frac{1}{m_{2}}F_{h}+\frac{1}{m_{2}}f_{d}$ represent the total disturbance which consists of external disturbances and unknown internal dynamics. The output of system (10) is y = [x ₁, x ₄]^T, where x ₄ can be obtained by integrating the acceleration signal through a self-zeroing integrator [26].

TheoremA .

For the system (10), two assumptions are proposed: (1) These disturbances, 𝜑₁ and 𝜑₂ are unknown and time-varying, but bounded by unknown constant. (2) The external disturbance f _d is bounded and Lipschitz continuous.

3. Controller design and stability analysis

In this section, we propose an observer-based FTC scheme. We begin with the design of the observer. And then the stability of the closed-loop levitation system is proven.

3.1. Observer design

Inspired by [20,27], for the system (10) under Assumption 1, two continuous FTDOs are designed to observe 𝜑₁, 𝜑₂ and their derivatives. Let 𝜁₀ = 𝜑₁, ${\zeta}_{1}=\dot{{\varphi}}_{1},\ldots ,{\zeta}_{m-1}={\varphi}_{1}^{(m-1)}$ , 𝜂₀ = 𝜑₂, ${\eta}_{1}=\dot{{\varphi}}_{2},\ldots ,{\eta}_{p-1}={\varphi}_{2}^{(p-1)}$ , the following observers can be given: $\begin{eqnarray}\displaystyle \left\{\begin{array}{@{}l@{}}\dot{\hat{x}}_{1}=\hat{x}_{2}-{\lambda}_{1,m+3}\lceil \hat{x}_{1}-x_{1}\rfloor ^{{\alpha}_{m+3}}\quad \\[4.0pt] \dot{\hat{x}}_{2}=\hat{x}_{3}-{\lambda}_{1,m+2}\lceil \hat{x}_{1}-x_{1}\rfloor ^{{\alpha}_{m+2}}\quad \\[4.0pt] \dot{\hat{x}}_{3}=a_{1}x_{4}+\hat{{\zeta}}_{0}-{\lambda}_{1,m+1}\lceil \hat{x}_{1}-x_{1}\rfloor ^{{\alpha}_{m+1}}\quad \\[4.0pt] \dot{\hat{{\zeta}}}_{i}=\hat{{\zeta}}_{i+1}-{\lambda}_{1,m-i}\lceil \hat{x}_{1}-x_{1}\rfloor ^{{\alpha}_{m-i}},\quad i=0,1,\ldots ,m-2\quad \\[4.0pt] \dot{\hat{{\zeta}}}_{m-1}=-{\lambda}_{1,1}\lceil \hat{x}_{1}-x_{1}\rfloor ^{{\alpha}_{1}}\quad \end{array}\right. & & \displaystyle\end{eqnarray}$ (11) $\begin{eqnarray}\displaystyle \left\{\begin{array}{@{}l@{}}\dot{\hat{x}}_{4}=b_{1}u+b_{2}x_{1}+\hat{{\eta}}_{0}-{\lambda}_{2,p+1}\lceil \hat{x}_{4}-x_{4}\rfloor ^{{\beta}_{p+1}}\quad \\[4.0pt] \dot{\hat{{\eta}}}_{j}=\hat{{\eta}}_{j+1}-{\lambda}_{2,p-j}\lceil \hat{x}_{4}-x_{4}\rfloor ^{{\beta}_{p-j}},\quad j=0,1,\ldots ,p-2\quad \\[4.0pt] \dot{\hat{{\eta}}}_{p-1}=-{\lambda}_{2,1}\lceil \hat{x}_{4}-x_{4}\rfloor ^{{\beta}_{1}}\quad \end{array}\right. & & \displaystyle\end{eqnarray}$ (12) where x ₁ and x ₄ are the outputs of system (10); $\hat{x}_{1},\hat{x}_{2},\hat{x}_{3},\hat{{\zeta}}_{0},\hat{{\zeta}}_{1},\ldots ,\hat{{\zeta}}_{m-1}$ are the observations of $x_{1},x_{2},x_{3},{\varphi}_{1},\dot{{\varphi}}_{1},\ldots ,{\varphi}_{1}^{(m-1)}$ , respectively; $\hat{x}_{4},\hat{{\eta}}_{0},\hat{{\eta}}_{1},\ldots ,\hat{{\eta}}_{p-1}$ are the observations of $x_{4},{\varphi}_{2},\dot{{\varphi}}_{2},\ldots ,{\varphi}_{2}^{(p-1)}$ , respectively; 𝜆 _{1, i} > 0, (i = 0,1, …, m +3) , 𝜆_{2, j} > 0, (j = 0,1, …, p +1), 𝛼_i > 0, (i = 0,1, …, m +3), 𝛽_j > 0, (j = 0,1, …, p +1) are the parameters of FTDOs (11) and (12), respectively.

By defining $e_{x_{i}}=\hat{x}_{i}-x_{i}$ , $e_{{\zeta}_{i}}=\hat{{\zeta}}_{i}-{\zeta}_{i}$ , $e_{{\eta}_{i}}=\hat{{\eta}}_{i}-{\eta}_{i}$ and subtracting (10) from the observer (11) and (12), the error dynamics can be derived: $\begin{eqnarray}\displaystyle \left\{\begin{array}{@{}l@{}}{\dot{e}}_{x_{1}}=e_{x_{2}}-{\lambda}_{1,m+3}\lceil e_{x_{1}}\rfloor ^{{\alpha}_{m+3}}\quad \\[4.0pt] {\dot{e}}_{x_{2}}=e_{x_{3}}-{\lambda}_{1,m+2}\lceil e_{x_{1}}\rfloor ^{{\alpha}_{m+2}}\quad \\[4.0pt] {\dot{e}}_{x_{3}}=e_{{\zeta}_{0}}-{\lambda}_{1,m+1}\lceil e_{x_{1}}\rfloor ^{{\alpha}_{m+1}}\quad \\[4.0pt] {\dot{e}}_{{\zeta}_{i}}=e_{{\zeta}_{i+1}}-{\lambda}_{1,m-i}\lceil e_{x_{1}}\rfloor ^{{\alpha}_{m-i}},\quad i=0,1,\ldots ,m-2\quad \\[4.0pt] {\dot{e}}_{{\zeta}_{m-1}}=-{\lambda}_{1,1}\lceil e_{x_{1}}\rfloor ^{{\alpha}_{1}}\quad \end{array}\right. & & \displaystyle\end{eqnarray}$ (13) $\begin{eqnarray}\displaystyle \left\{\begin{array}{@{}l@{}}{\dot{e}}_{x_{4}}=e_{{\eta}_{0}}-{\lambda}_{2,p+1}\lceil e_{x_{4}}\rfloor ^{{\beta}_{p+1}}\quad \\[3.0pt] {\dot{e}}_{{\eta}_{j}}=e_{{\eta}_{j+1}}-{\lambda}_{2,p-j}\lceil e_{x_{4}}\rfloor ^{{\beta}_{p-j}},\quad j=0,1,\ldots ,p-2\quad \\[3.0pt] {\dot{e}}_{{\eta}_{p-1}}=-{\lambda}_{2,1}\lceil e_{x_{4}}\rfloor ^{{\beta}_{1}}.\quad \end{array}\right. & & \displaystyle\end{eqnarray}$ (14)

It follows from [28] that the error dynamics of (11) and (12) can converge to the origin in finite time by choosing coefficients 𝜆_{1, i} and 𝜆_{2, j} reasonably, that are: (1) ∀t ≤ T _s, observed errors e _{x
_i}, e _{𝜁_i}, and e _{𝜂_i} are bounded; (2) ∀t > T _s, e _{x
_i} = e _{𝜁_i} = e _{𝜂_i} = 0.

3.2. Controller design

Under the support of FTDOs, it is possible to observe the unmeasurable states and time-varying disturbances 𝜑₁, 𝜑₂ separately. Then, an FTC scheme can be developed as follows: $\begin{eqnarray}u=(a_{1}b_{1})^{-1}\left[z_{r}^{(4)}+\sum _{i=1}^{4}c_{i}\lceil e_{x_{i}}\rfloor ^{{\gamma}_{i}}-a_{1}b_{2}x_{1}-\hat{{\zeta}}_{1}-a_{1}\hat{{\eta}}_{0}\right]^{T}\end{eqnarray}$ (15) where e ₁ = z _r − x ₁, $e_{2}={\dot{z}}_{r}-\hat{x}_{2}$ , $e_{3}=\ddot{z}_{r}-\hat{x}_{3}$ , $e_{4}=z_{r}^{\left(3\right)}-(a_{1}x_{4}+\hat{{\zeta}}_{0})$ . c _i > 0, (i = 1,2,3,4), which can be chosen by setting characteristic roots in this equation p _c(s) = s ⁴ + c ₄ s ³ + c ₃ s ² + c ₂ s + c ₁ to have negative real parts. 𝛾_i > 0, (i = 1,2,3,4) should satisfy the following condition: 𝛾_i−1 = 𝛾_i∕(2 −𝛾_i∕𝛾_i+1), (i = 2,3,4), with 𝛾₅ = 1, 𝛾₄ = 𝛾, 𝛾 ∈ (1 −𝛿,1), 𝛿 ∈ (0,1).

3.3. Stability analysis

TheoremB .
First, the finite-time stability of the observer variables is guaranteed through the above-mentioned analysis.

Next, we will prove the system states are finite-time bounded.

With the above definition of error variables e _i in mind, we have $\begin{eqnarray}\displaystyle \left\{\begin{array}{@{}l@{}}{\dot{e}}_{1}={\dot{z}}_{r}-x_{2}={\dot{z}}_{r}-\hat{x}_{2}+e_{x2}=e_{2}+e_{x2}\quad \\[3.0pt] {\dot{e}}_{2}=\ddot{z}_{r}-\dot{\hat{x}}_{2}=\ddot{z}_{r}-\hat{x}_{3}+{\lambda}_{1,m+2}\lceil e_{x_{1}}\rfloor ^{{\alpha}_{m+2}}=e_{3}+{\lambda}_{1,m+2}\lceil e_{x_{1}}\rfloor ^{{\alpha}_{m+2}}\quad \\[3.0pt] {\dot{e}}_{3}=z_{r}^{(3)}-(a_{1}x_{4}+\hat{{\zeta}}_{0})+{\lambda}_{1,m+1}\lceil e_{x_{1}}\rfloor ^{{\alpha}_{m+1}}=e_{4}+{\lambda}_{1,m+1}\lceil e_{x_{1}}\rfloor ^{{\alpha}_{m+1}}\quad \\[3.0pt] {\dot{e}}_{4}=z_{r}^{(4)}-a_{1}{\dot{x}}_{4}-\dot{\hat{{\zeta}}}_{0}=a_{1}e_{{\eta}0}-\sum _{i=1}^{4}c_{i}\lceil e_{i}\rfloor ^{{\gamma}_{i}}+{\lambda}_{1,m}\lceil e_{x_{1}}\rfloor ^{{\alpha}_{m}}.\quad \end{array}\right. & & \displaystyle\end{eqnarray}$ (16)

Choose a finite-time bounded function as [29] $\begin{eqnarray}V(e_{i})=\frac{1}{2}\mathop{\sum }_{i=1}^{4}e_{i}^{T}e_{i}.\end{eqnarray}$ (17)

From (16), $\dot{V}$ can be converted into the following form $\begin{eqnarray}\displaystyle \dot{V} & = & \displaystyle e_{1}^{T}{\dot{e}}_{1}+e_{2}^{T}{\dot{e}}_{2}+e_{3}^{T}{\dot{e}}_{3}+e_{4}^{T}{\dot{e}}_{4}\nonumber\\ \displaystyle & \leq & \displaystyle |e_{1}||e_{2}|+|e_{1}||e_{x_{2}}|+|e_{2}||e_{3}|+{\lambda}_{1,m+2}|e_{2}||e_{x_{1}}|^{{\alpha}_{m+2}}\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle +∼|e_{3}||e_{4}|+{\lambda}_{1,m+1}|e_{3}||e_{x_{1}}|^{{\alpha}_{m+1}}+a_{1}|e_{4}||e_{{\eta}_{0}}|+c_{4}|e_{4}||e_{4}|^{{\gamma}_{4}}\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle +∼c_{3}|e_{4}||e_{3}|^{{\gamma}_{3}}+c_{2}|e_{4}||e_{2}|^{{\gamma}_{2}}+c_{1}|e_{4}||e_{1}|^{{\gamma}_{1}}+{\lambda}_{1,m}|e_{4}||e_{x_{1}}|^{{\alpha}_{m}}.\end{eqnarray}$ (18)

Using the inequality |x|^r <1 + r|x| <1 + |x|, r ∈ (0,1), $\dot{V}$ can be further transformed into the following $\begin{eqnarray}\displaystyle \dot{V} & {<} & \displaystyle |e_{1}||e_{2}|+|e_{1}||e_{x_{2}}|+|e_{2}||e_{3}|+{\lambda}_{1,m+2}|e_{2}|1+|e_{x_{1}}|\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle +\displaystyle ∼|e_{3}||e_{4}|+{\lambda}_{1,m+1}|e_{3}|1+|e_{x_{1}}|+a_{1}|e_{4}||e_{{\eta}_{0}}|\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle +\displaystyle ∼|e_{4}|\sum _{i=1}^{4}c_{i}1+|e_{i}|+{\lambda}_{1,m}|e_{4}|1+|e_{x_{1}}|\nonumber\\ \displaystyle & \leq & \displaystyle \displaystyle \frac{e_{1}^{2}+e_{2}^{2}}{2}+\frac{e_{1}^{2}+e_{x_{2}}^{2}}{2}+\frac{e_{2}^{2}+e_{3}^{2}}{2}+{\lambda}_{1,m+2}\frac{1+e_{2}^{2}}{2}\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle +\displaystyle ∼{\lambda}_{1,m+2}\frac{e_{2}^{2}+e_{x_{1}}^{2}}{2}+\frac{e_{3}^{2}+e_{4}^{2}}{2}+{\lambda}_{1,m+1}\frac{e_{3}^{2}+e_{x_{1}}^{2}}{2}\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle +\displaystyle ∼{\lambda}_{1,m+1}\frac{1+e_{3}^{2}}{2}+a_{1}\frac{e_{4}^{2}+e_{{\eta}_{0}}^{2}}{2}+{\lambda}_{1,m}\frac{e_{4}^{2}+e_{x_{1}}^{2}}{2}\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle +\displaystyle ∼{\lambda}_{1,m}\frac{1+e_{4}^{2}}{2}+\sum _{i=1}^{4}c_{i}\frac{1+e_{4}^{2}}{2}+\sum _{i=1}^{4}c_{i}\frac{e_{4}^{2}+e_{i}^{2}}{2}\nonumber\\ \displaystyle & \leq & \displaystyle N_{1}V+N_{2}\end{eqnarray}$ (19) where $N_{1}=\max 2+c_{1},2+2{\lambda}_{1,m+2}+c_{2},2+2{\lambda}_{1,m+1}+c_{3},1+a_{1}+2\sum _{i=1}^{4}c_{i}+2{\lambda}_{1,m}$ , $N_{2}=\frac{1}{2}e_{x_{2}}^{2}+{\lambda}_{1,m+2}+{\lambda}_{1,m+1}+{\lambda}_{1,m+1}e_{x_{1}}^{2}+{\lambda}_{1,m+2}e_{x_{1}}^{2}+a_{1}e_{{\eta}_{0}}^{2}+\sum _{i=1}^{4}c_{i}+{\lambda}_{1,m}+{\lambda}_{1,m}e_{x_{1}}^{2}$ . According to (13) and (14), the estimation errors e _{x
_i}, e _{𝜁_i} and e _{𝜂_i} are finite-time convergent, which implies that N ₁ and N ₂ are bounded. And we can conclude that V and e _i will not diverge to infinity in finite time.

Finally, we will prove the finite-time convergence of tracking errors.

With the definition of error variable e _i and its derivative ${\dot{e}}_{i}$ in mind, we have $\begin{eqnarray}\left\{\begin{array}{@{}l@{}}{\dot{e}}_{1}=e_{2}+e_{x_{2}}\quad \\[3.0pt] \ddot{e}_{1}=e_{3}+e_{x_{3}}\quad \\[3.0pt] e_{1}^{(3)}=e_{4}+e_{_{{\zeta}_{0}}}\quad \\[3.0pt] e_{1}^{(4)}={\dot{e}}_{4}-{\lambda}_{1,m}\lceil e_{x_{1}}\rfloor ^{{\alpha}_{m}}+e_{_{{\zeta}_{1}}}.\quad \end{array}\right.\end{eqnarray}$ (20)

Substituting (20) into (16), the error dynamics can be rewritten as $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle {\dot{e}}_{4}+c_{4}\lceil e_{4}\rfloor ^{{\gamma}_{4}}+c_{3}\lceil e_{3}\rfloor ^{{\gamma}_{3}}+c_{2}\lceil e_{2}\rfloor ^{{\gamma}_{2}}+c_{1}\lceil e_{1}\rfloor ^{{\gamma}_{1}}-a_{1}e_{{\eta}_{0}}-{\lambda}_{1,m}\lceil e_{x_{1}}\rfloor ^{{\alpha}_{m}}\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle =e_{1}^{\left(4\right)}-e_{{\zeta}_{1}}+c_{4}\lceil e_{1}^{\left(3\right)}-e_{{\zeta}_{0}}\rfloor ^{{\gamma}_{4}}+c_{3}\lceil \ddot{e}_{1}-e_{x_{3}}\rfloor ^{{\gamma}_{3}}\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \quad +∼c_{2}\lceil {\dot{e}}_{1}-e_{x_{2}}\rfloor ^{{\gamma}_{2}}+c_{1}\lceil e_{1}\rfloor ^{{\gamma}_{1}}-a_{1}e_{{\eta}_{0}}\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle =0.\end{eqnarray}$ (21)

When t > T _s, e _{x
_i} = 0, e _{𝜁_i} = 0 and e _{𝜂_i} = 0, (i = 1,2,3,4) are achieved. Thus, (21) can be written as $\begin{eqnarray}e_{1}^{(4)}+c_{4}\lceil e_{1}^{(3)}\rfloor ^{{\gamma}_{4}}+c_{3}\lceil \ddot{e}_{1}\rfloor ^{{\gamma}_{3}}+c_{2}\lceil {\dot{e}}_{1}\rfloor ^{{\gamma}_{2}}+c_{1}\lceil e_{1}\rfloor ^{{\gamma}_{1}}=0.\end{eqnarray}$ (22)

According to [30], it can be concluded that the tracking error e ₁ will converge to zero in finite time. This completes the proof.

4. Simulation and experimental results

This section implements the proposed FTC scheme to stabilize the levitation system (10) and compares it with two other control methods through simulations and experiments. The physical parameters of the maglev system are shown in Table 1.

Table 1
Physical parameters of the maglev system

Parameter Description Values

m ₁ Mass of the beam 50000 kg

l _g Length of the beam 24 m

𝜔₁ First mode frequency of the beam 7.52 Hz

𝜉₁ Damping ratio of the beam 0.005

m ₂ Mass of the electromagnet 750 kg

R Resistance 1.2 Ω

𝜇₀ Vacuum permeability 4𝜋 × 10⁻⁷ H/m

S Area of the magnetic poles 0.024 m²

N Coil turn 340

d ₀ Equilibrium point 0.01 m

I ₀ Equilibrium point current 30 A

Parameter	Description	Values
m ₁	Mass of the beam	50000 kg
l _g	Length of the beam	24 m
𝜔₁	First mode frequency of the beam	7.52 Hz
𝜉₁	Damping ratio of the beam	0.005
m ₂	Mass of the electromagnet	750 kg
R	Resistance	1.2 Ω
𝜇₀	Vacuum permeability	4𝜋 × 10⁻⁷ H/m
S	Area of the magnetic poles	0.024 m²
N	Coil turn	340
d ₀	Equilibrium point	0.01 m
I ₀	Equilibrium point current	30 A

4.1. Simulation results

To demonstrate the superiority of the proposed FTC scheme, comparisons are made with the full-state feedback flexible control method [31] and the active disturbance rejection control (ADRC) strategy [32].

Control scheme 1: The proposed FTC scheme. The coefficients 𝛾_i of controller (15) are selected as 𝛾₁ = 7∕11, 𝛾₂ = 7∕10, 𝛾₃ = 7∕13, 𝛾₄ = 7∕12. The coefficients c _i of controller (15) are chosen as c ₁ = 5600, c ₂ = 5000, c ₃ = 1000, c ₄ = 1000. The gains of the observer (11) are selected as the 7th-order polynomial form: $p_{o}(s)=(s+p_{o})(s^{2}+2z_{o}{\omega}_{o}s+{\omega}_{o}^{2})^{3}$ with ${\lambda}_{1,1}=p_{0}{\omega}_{0}^{6},{\lambda}_{1,2}={\omega}_{o}^{6}+6p_{o}z_{o}{\omega}_{o}^{5},{\lambda}_{1,3}=6z_{o}{\omega}_{o}^{5}+12p_{o}z_{o}^{2}{\omega}_{o}^{4}+3p_{o}{\omega}_{o}^{4},{\lambda}_{1,4}=12z_{o}^{2}{\omega}_{o}^{4}+3{\omega}_{o}^{4}+8p_{o}z_{o}^{3}{\omega}_{o}^{3}+12p_{o}z_{o}{\omega}_{o}^{3}$ , ${\lambda}_{1,5}=8z_{o}^{3}{\omega}_{o}^{3}+12z_{o}{\omega}_{o}^{3}+12p_{o}z_{o}^{2}{\omega}_{o}^{2}+3p_{o}{\omega}_{o}^{2},{\lambda}_{1,6}=12z_{o}^{2}{\omega}_{o}^{2}+3{\omega}_{o}^{2}+6p_{o}z_{o}{\omega}_{o},{\lambda}_{1,7}=p_{o}+6z_{o}{\omega}_{o}$ , where p _o = 𝜔_o = 15, z _o = 1. 𝛼_i = (7 − i +1)𝜏 +1, (i = 1,2, …,7), 𝜏 = −2∕11. Correspondingly, the gains of the observer (12) are chosen as the 3rd-order polynomial form: $p_{c}(s)=(s+{\omega}_{c})(s^{2}+2z_{c}{\omega}_{c}s+{\omega}_{c}^{2})$ with ${\lambda}_{2,1}=p_{c}{\omega}_{c}^{2},{\lambda}_{2,2}={\omega}_{c}^{2}+2p_{c}z_{c}{\omega}_{c},{\lambda}_{2,3}=p_{c}+2z_{c}{\omega}_{c}$ , where p _c = 𝜔_c = 15, z _c = 1, 𝛽_i = (3 − i +1)𝜏 +1, (i = 1,2,3), 𝜏 = −2∕11.

Control scheme 2: The full-state feedback controller $U=-K_{1}\hat{z}_{1}-K_{2}\dot{\hat{z}}_{1}-K_{3}\hat{z}_{2}-K_{4}\dot{\hat{z}}_{2}-K_{5}\ddot{\hat{z}}_{2}$ , where K _i, (i = 1,2,3,4,5) is the feedback coefficient and can be calculated with the linear quadratic optimal method. K ₁ = 19845, K ₂ = 499, K ₃ = −72413, K ₄ = −1948, K ₅ = −16.

Control scheme 3: The ADRC algorithm. $I=k_{1}d+k_{2}{\dot{d}}+k_{3}z_{2}+k_{4}{\dot{z}}_{2}-(f_{g}+z_{g3})/b_{1}$ , where k _i, (i = 1,2,3,4) is the feedback coefficient k ₁ = 1511, k ₂ = 0.01, k ₃ = 2, k ₄ = 48.3, $f_{g}=[(\frac{1}{m_{1}}+\frac{1}{m_{2}})F_{d}-{\omega}_{1}^{2}]z_{g1}-2{\xi}_{1}{\omega}_{1}z_{g2}+{\omega}_{1}^{2}z_{2}+2{\xi}_{1}{\omega}_{1}{\dot{z}}_{2}$ . z _g1 and z _g2 are the estimation value of d and ${\dot{d}}$ , z _g3 is the estimation of the lumped disturbance. The extended state observer is: $\begin{eqnarray}\left\{\begin{array}{@{}l@{}}{\dot{z}}_{g1}=z_{g2}-{\mu}_{1}e_{g}\quad \\[3.0pt] {\dot{z}}_{g2}=z_{g3}-{\mu}_{2}\text{fal}(e_{g},a_{1},{\delta}_{1})+f_{g}+b_{1}I\quad \\[3.0pt] {\dot{z}}_{g3}=-{\mu}_{3}\text{fal}(e_{g},a_{2},{\delta}_{1})\quad \end{array}\right.\end{eqnarray}$ (23) where e _g = z _g1 − d, 𝛿₁ = 1, a ₁ = 0.1, a ₂ = 0.3, 𝜇₁ = 1047, 𝜇₂ = 2925, 𝜇₃ = 21280, fal (⋅) is a nonlinear function with linear interval: $\begin{eqnarray}\left\{\begin{array}{@{}l@{}}\displaystyle \text{fal}({\vartheta},a,{\delta}_{\text{c}})=\frac{{\vartheta}}{{\delta}_{c}^{1-a}}{\varpi}+|{\vartheta}|^{a}\text{sign}({\vartheta})(1-{\varpi})\quad \\[15.0pt] \displaystyle {\varpi}=\frac{\text{sign}({\vartheta}+{\delta}_{c})-\text{sign}({\vartheta}-{\delta}_{c})}{2}.\quad \end{array}\right.\end{eqnarray}$ (24)

The following three scenarios are implemented to compare the control performance of the three control schemes.

Fig. 2.

Three types of track irregularities.

Case 1: In this case, the robustness of the levitation system relative to short-wave (𝜆 = 2 m) and long-wave (𝜆 = 30 m) sinusoidal excitations was investigated, as shown in Fig. 2. The operating speeds of the maglev train were set to 80 km/h and 160 km/h for the short-wave and long-wave excitations, respectively. For ease of discussion, the responses of the closed-loop levitation system in the time domain and frequency domain are shown in Fig. 3 and Fig. 4, respectively. From the results of the air gap, it can be observed that the ADRC can suppress high-frequency excitations and follow low-frequency excitations, while the LQR exhibits the opposite behavior. However, under the short-wave and long-wave excitations, the air gap tracking error of the FTC-FTDO can converge to a much smaller range compared to the other two methods. Moreover, to illustrate the vibration response performance of the electromagnet, the vertical vibration output spectrum was obtained using Fast Fourier Transform (FFT), as shown in Fig. 4. It is worth noting that the FTC-FTDO control exhibits excellent vibration suppression capabilities both at low frequencies and high frequencies. Clearly, all three methods exhibit some peaks around the natural frequency of the system and disturbance frequencies, resembling local maximum.

Fig. 3.

Time domain response of levitation system under Case 1.

Fig. 4.

Frequency domain response of levitation system under Case 1.

Case 2: In this case, we set the first-order frequency of the track beam to 3.5 Hz to explore the robustness of the coupled levitation system on a low-stiffness track beam. The electromagnet is in a static levitation state. The research results are shown in Fig. 5. Firstly, we investigated the response of the low-stiffness track beam under rigid control, which means that the track flexibility was not considered when designing the control law. From Fig. 5(a), it can be observed that the phase locus of the track beam takes the form of a limit cycle, indicating the occurrence of periodic vibration in the coupled levitation system, resulting in coupled vibrations. In contrast, the other three controllers belong to flexible control methods. From Fig. 5(b), it can be seen that after incorporating vibration information into the control law, the beam displacement eventually converges to zero. Furthermore, the vibration amplitudes of the LQR and ADRC are much larger than that of the FTC-FTDO, indicating that the designed controllers exhibit satisfactory flexibility.

Fig. 5.

Dynamic response of levitation system under Case 2.

Case 3: This case considers compound disturbances, including load variations and track irregularities, as shown in Fig. 2. The operating speed of the magnetic levitation train is set to 80 km/h. The dynamic responses of the levitation system are shown in Fig. 6. From Fig. 6, it can be observed that the air gap controlled by the FTC-FTDO converges to a small range of ±0.5 × 10⁻³ m subject to complex disturbances, indicating that FTC-FTDO can compensate for external disturbances more effectively than the other two methods. Specifically, under the same parameters, even with load variations, the air gap controlled by FTC-FTDO converges quickly to the desired value of 10 mm, demonstrating the satisfactory adaptability of the designed controller. However, the air gap controlled by LQR deviates from the equilibrium point due to load variations. Meanwhile, under irregular excitations, LQR and ADRC have almost the same amplitude, while the FTC-FTDO produces a smaller amplitude, indicating a significant improvement in maintaining a safe suspension gap, thanks to the compensation effect of FTDO. From the frequency domain analysis results in Fig. 6, it can be observed that under multi-wavelength superimposed excitations, the FTC-FTDO improves vibration suppression performance throughout the frequency band. In Fig. 6(c), a comparison of the vertical beam displacement responses produced by the three methods is shown. Clearly, both LQR and ADRC have higher amplitude levels compared to FTC-FTDO. Additionally, ADRC and FTC-FTDO exhibit convergence trends, while LQR remains in equal-amplitude oscillation due to model mismatch caused by load variations.

Fig. 6.

Dynamic response of levitation system under Case 3.

Quantitative analysis of the performance improvement of the FTC scheme is conducted, and the calculation results of the maximum values and root mean square (RMS) values of the response signals, including beam acceleration, beam displacement, electromagnet acceleration, electromagnet displacement, and air gap tracking error, are shown in Table 2. Furthermore, the performance ratios of the FTC-FTDO scheme relative to the LQR scheme are illustrated in Fig. 7. Based on the results, it can be concluded that the FTC-FTDO scheme brings significant performance improvements in all dimensions.

Table 2

The quantitative indicators of the levitation system

	Track beam z ₁		Electromagnet z ₂		Air gap error d _e
Types	\|z ₁\|_max (mm)	\|z̈₁\|_rms (m/s²)	\|z ₂\|_max (mm)	\|z̈₂\|_rms (m/s²)	\|d _e\|_max (mm)	\|d _e\|_rms (m)
LQR	1.5	0.006	5.3	0.48	1.9	0.82
ADRC	1.5	0.0054	3.4	0.44	1.1	0.36
FTC-FTDO	0.34	0.0011	2.3	0.41	0.49	0.15

Fig. 7.

Ratios of the quantization performance relative to LQR.

4.2. Experimental results

The effectiveness of the FTC scheme was further validated on a magnetic levitation testbed, as shown in Fig. 8. The model parameters are provided in Table 3. The testbed was designed based on similarity theory, where the specific parameters were determined based on the principle of maintaining the same first-order natural frequency before and after similarity transformation [14]. As shown in Fig. 8, the electromagnet is connected to one end of the bracket arm, while the other end is connected to the base, thus restricting the vertical motion of the electromagnet. The track beam is connected to the bracket through springs, simulating the elastic vibrations of the track beam. By replacing springs with different stiffness levels, the natural frequency of the track beam can be adjusted. The real-time simulation system dSPACE was used to deploy the control algorithm. The control parameters related to the testbed are selected as c ₁ = 1800,c ₂ = 500, c ₃ = 55, c ₄ = 200, p _o = 𝜔_o = 5, p _c = 𝜔_c = 5, and the selection of remaining parameters is consistent with the simulation parameters.

Fig. 8.

Single-magnet levitation testbed.

Fig. 9.

Dynamic response of levitation system under Experiment 1.

Table 3

Parameters of the levitation bench

Parameter description	Value
Mass of the beam	2 kg
Damping coefficient of the beam	0 N⋅ s⋅m
Stiffness of the springs	10–30 kN/m
Mass of the electromagnet	1.8 kg
Air gap coefficient, F _d	3560 N/m
Current coefficient, F _I	19.8 N/A

The experimental comparison of LQR and the proposed method is carried out under the following two scenarios. The parameter settings for LQR are K ₁ = 1501.6, K ₂ = 167.5, K ₃ = −7256.2, K ₄ = −318.3, K ₅ = −6.6.

Experiment 1: In this case, the performance of two controllers in response to load variations is compared. After the suspension stabilizes, a 1 kg load is applied to the levitation system for 3 seconds, followed by unloading. Figure 9 shows the variations of suspension gap and the electromagnet acceleration over time. It can be observed that the air gap controlled by LQR deviates from the equilibrium position after loading, posing a risk of exceeding the safe suspension range under heavier loads. Conversely, under the control of FTC-FTDO, the maximum fluctuation peaks of suspension gap and electromagnet acceleration caused by this load are 0.42 mm and 0.31 m/s², respectively. From Fig. 9(a), it can be seen that the system response re-enters the 2% stable region after approximately 0.2 seconds. This demonstrates that FTC-FTDO can effectively resist load disturbances. Additionally, the vibration of the electromagnet using the proposed controller is slightly lower than that using the LQR controller.

Fig. 10.

Dynamic response of levitation system under track stiffness of S ₁.

Fig. 11.

Dynamic response of levitation system under track stiffness of S ₂.

Experiment 2: The condition of static suspension consists of two types of spring stiffness supports. S ₁ = 31570 N/m, S ₂ = 20611 N/m. Figure 10 and Fig. 11 show a comparison between LQR and the proposed control method with different spring stiffness, respectively. Under the stiffness of S ₁, Fig. 10 shows that the air gap and electromagnet acceleration in both LQR and FTC-FTDO fluctuate within an acceptable range, indicating that the system stability is guaranteed. It can also be observed that the amplitude of the measured signals in FTC-FTDO is much smaller than that in LQR. From Fig. 11, we can observe that reducing the spring stiffness results in the inability of the LQR-controlled system to guarantee stability, as the vibration of the track beam exacerbates the vibration of the electromagnet under low support stiffness. However, it is evident that under the stiffness of S ₂, FTC-FTDO exhibits better vibration suppression capability, thereby enhancing levitation stability. From this experiment, we can conclude that when the stiffness of the track beam is too low, it is more susceptible to external disturbances. In such cases, sensor noise and environmental vibrations are more likely to affect system stability. Therefore, under low stiffness conditions, higher demands are placed on the control precision and disturbance rejection capability of the levitation system.

5. Conclusion

This study proposes a novel composite control scheme for a levitation system, which includes FTDOs and an FTC, to address the issues of unknown disturbance compensation and fast dynamic response. The effectiveness of the proposed approach is evaluated through numerical simulations considering three different scenarios. The results demonstrate that the FTC scheme achieves finite-time convergence and significantly improves vibration suppression performance. Additionally, experimental results show that under low support stiffness, the FTC approach outperforms LQR. The use of the FTC scheme will enhance levitation safety and extend the lifespan of electromagnetic levitation systems.

Footnotes

Acknowledgements

This work was supported by the Shanghai Collaborative Innovation Center for Multi-network and Multi-Mode Rail Transit under Grant 28002360012.

References

Lee

H.W.

Kim

K.C.

and Lee

, Review of Maglev train technologies, IEEE Trans. Magn. 42 (2006), 1917–1925.

Yan

, Development and application of the Maglev transportation system, IEEE Trans. Appl. Supercond. 18 (2008), 92–99.

Zhao

C.F.

and Zhai

W.M.

, Maglev vehicle/guideway vertical random response and ride quality, Veh. Syst. Dyn. 38 (2002), 185–210.

Kim

K.J.

Han

J.B.

Han

H.S.

and Yang

S.J.

, Coupled vibration analysis of Maglev vehicle-guideway while standing still or moving at low speeds, Veh. Syst. Dyn. 53 (2015), 587–601.

Wai

R.-J.

and Lee

J.-D.

, Adaptive fuzzy-neural-network control for maglev transportation system, IEEE Trans Neural Networks 19 (2008), 54–70.

Zhou

Wang

and Li

, An adaptive vibration control method to suppress the vibration of the maglev train caused by track irregularities, J. Sound Vib. 408 (2017), 331–350.

Chen

Lin

Sun

and Zhao

, Sliding mode bifurcation control based on acceleration feedback correction adaptive compensation for maglev train suspension system with time-varying disturbance, IEEE Trans on Transportation Electrification 8(2) (2022), 2273–2287.

Yang

Zolotas

Chen

W.-H.

, Robust control of nonlinear MAGLEV suspension system with mismatched uncertainties via DOBC approach, ISA Trans. 50 (2011), 389–396.

Zhou

Zhang

and Fujita

, Model predictive control for hybrid levitation systems of maglev trains with state constraints, IEEE Trans Veh. Technol. 70 (2021), 9972–9985.

10.

Wang

Yang

and Zhou

, Continuous sliding mode control for permanent magnet synchronous motor speed regulation systems under time-varying disturbances, J. Power Electron. 16 (2016), 1324–1335.

11.

Sun

Qiang

, Adaptive sliding mode control of maglev system based on RBF neural network minimum parameter learning method, Measurement 141 (2019), 217–226.

12.

Zhao

You

Song

, Suspension regulation of medium-low-speed maglev trains via deep reinforcement learning, IEEE Trans Artif. Intell. 2 (2021), 341–351.

13.

Gao

Sun

Luo

, Deep learning controller design of embedded control system for maglev train via deep belief network algorithm, Des. Autom. Embed Syst. 24 (2020), 161–181.

14.

Cui

Shen

and Wang

, Maglev vehicle-guideway coupling vibration test rig based on the similarity theory, J. Vib. Control 22 (2016), 286–295.

15.

Wang

and Shen

, Research on control method of Maglev vehicle-guideway coupling vibration system based on particle swarm optimization algorithm, J. Vib. Control 25 (2019), 2237–2245.

16.

Wang

Zhong

and Shen

, Analysis and experimental study on the MAGLEV vehicle-guideway interaction based on the full-state feedback theory, J. Vib. Control 21 (2015), 408–416.

17.

Wang

Zhang

Sun

, Continuous terminal sliding-mode control for FJR subject to matched/mismatched disturbances, IEEE Trans. Cybern. 52 (2021), 10479–10489.

18.

Pan

and Sun

, Nonlinear output feedback finite-time control for vehicle active suspension systems, IEEE Trans. Ind. Informatics 15 (2019), 2073–2082.

19.

Wang

Rong

and Yang

, Adaptive fixed-time position precision control for Magnetic Levitation Systems, IEEE Trans. Autom. Sci. Eng. 20 (2022), 458–469.

20.

Levant

, Higher-order sliding modes, differentiation and output-feedback control, Int. J. Control 76 (2003), 924–941.

21.

Zhao

Cao

Yang

and Wang

, High-order sliding mode observer-based trajectory tracking control for a quadrotor UAV with uncertain dynamics, Nonlinear Dyn. 102 (2020), 2583–2596.

22.

Wang

Zhang

, Finite-time observer based accurate tracking control of a marine vehicle with complex unknowns, Ocean Eng. 145 (2017), 406–415.

23.

Qiang

Sun

and Liu

, Levitation chassis dynamic analysis and robust position control for maglev vehicles under nonlinear periodic disturbance, J. Vibroengineering 19 (2017), 1273–1286.

24.

Kong

Song

J.S.

Kang

B.B.

and Na

, Dynamic response and robust control of coupled maglev vehicle and guideway system, J. Sound Vib. 330 (2011), 6237–6253.

25.

Wang

Liu

, Dynamic analysis of the interactions between a low-to-medium-speed maglev train and a bridge: Field test results of two typical bridges, Proc. Inst. Mech. Eng. Part F J. Rail Rapid Transit 232 (2018), 2039–2059.

26.

RM G. Dynamics and control requirements for EMS Maglev suspensions, in: 18th International Conference on Magnetically Levitated Systems and Linear Drives, Loughborough University, Shanghai, 2004, pp. 926–934..

27.

Wang

Zhang

Zhao

, Finite-time disturbance observer-based trajectory tracking control for flexible-joint robots, Nonlinear Dyn. 106 (2021), 459–471.

28.

Perruquetti

Floquet

and Moulay

, Finite-time observers: Application to secure communication, IEEE Trans. Automat. Contr. 53 (2008), 356–360.

29.

and Tian

Y.-P.

, Finite-time stability of cascaded time-varying systems, Int. J. Control 80 (2007), 646–657.

30.

Bhat

S.P.

and Bernstein

D.S.

, Geometric homogeneity with applications to finite-time stability, Math. Control Signals Syst. 17 (2005), 101–127.

31.

Wang

Shen

and Zhong

, Study on the Maglev vehicle-guideway coupling vibration system, Proc. Inst. Mech. Eng. Part F J. Rail Rapid Transit 229 (2015), 507–517.

32.

Xiao

, Research of maglev train suspension algorithm based on active disturbance rejection control, in: 2016 IEEE International Conference on Information and Automation, IEEE, Ningbo, 2016, pp. 1296–1300.