A Robust Levitation Control of Maglev Vehicles Subject to Time Delay and Disturbances: Design and Hardware Experimentation

applied sciences

Article A Robust Levitation Control of Maglev Vehicles Subject to Time Delay and Disturbances: Design and Hardware Experimentation

You-gang Sun 1,2 , Si Xie 3, Jun-qi Xu 2 and Guo-bin Lin 2,* 1 College of Transportation Engineering, Tongji University, Shanghai 201804, China; [email protected] 2 National Maglev Transportation Engineering R&D Center, Tongji University, Shanghai 201804, China; [email protected] 3 Logistics Engineering College, Shanghai Maritime University, Shanghai 201306, China; [email protected] * Correspondence: [email protected]; Tel.: +86-021-69580145

Received: 2 December 2019; Accepted: 5 February 2020; Published: 10 February 2020

Abstract: Maglev vehicles have become a new type of transportation system with higher speed, lower noise, and commercial appeal. Magnetic-suspension systems, which have high nonlinearity and open-loop instability, are the core components of maglev vehicles. The high-performance control of maglev vehicles has been the focus of numerous studies. Encountering challenges in the levitation control of maglev vehicles in the form of uncertain time delays and disturbances is unavoidable. To cope with these problems, this study presents the design of an adaptive robust controller based on the Riccati method and sliding-mode technology, simultaneously taking into account the inﬂuence of time delays and disturbances. The asymptotic stability of the closed-loop system with the proposed control law is proved by the Lyapunov method. Control performances of the proposed controller are shown in the simulation results. Together with the consistently stabilizing outputs, the presented control approach can handle time delays and disturbances well. Finally, experiments were also implemented to examine its practical control performance of the robust levitation-control law.

Keywords: maglev vehicles; magnetic-suspension systems; Riccati approach; adaptive robust control; time delay

1. Introduction With the substantial improvement of economies and the extensive cooperation and communication between different cities, demands for faster intercity transport are increasing, which has led to numerous outstanding achievements with regard to the high-speed railway. However, traditional high-speed trains are constrained by problems such as wheel–rail adhesion, hunting instability, running noise, and speed limit [1]. At the same time, energy consumption and mechanical friction wear increase with speed. It is well known that the maximum economic and technical speed of traditional high-speed trains is about 400 km/h. In addition, the noise and vibrations caused by the trains’ wheels also need to be considered—rail contact dynamics not only affect ride quality but also the surrounding buildings and residents. The adoption of magnetic-suspension technology can solve wheel–rail contact-induced problems such as wheel–rail adhesion, friction, vibration, and high-speed current. Maglev transportation systems have the advantages of a strong climbing ability, a small turning radius, low noise, and low maintenance cost. In recent years, they have been widely recognized and have become a new and efficient intercity mode of transportation. Studies on maglev vehicles in developed countries, especially Germany and Japan, began in the 1970s, and several test lines were also built [2,3]. In addition, China has had its share of achievements in maglev-vehicle technology [4,5]. Shanghai’s

Appl. Sci. 2020, 10, 1179; doi:10.3390/app10031179 www.mdpi.com/journal/applsci Appl. Sci. 2020, 10, 1179 2 of 17 high-speed maglev train, as the world’s first high-speed commercial maglev line, has been successfully open to traffic since December 2002, with a maximum operating speed of 431 km/h. In May 2016, the Changsha maglev line was put into trial operation and it reached the international leading level. In December 2017, Beijing’s first medium- and low-speed maglev train, S1, began operations. A number of other cities and regions are also planning to introduce maglev technology. The official opening of the commercial operation line has met the basic requirements of engineering application, but there are still many technical difficulties, especially for the analysis and design of a levitation-control system. During complex working conditions and long-term passenger service, some problems occur in the levitation system that do not occur in laboratories or the short-term test-line assessments. These emerging problems seriously affect the stability and reliability of the levitation system and even cause the partial suspension-point failure of the vehicle, which affects the comfort and riding experience of the maglev train, and even hinders the application of more commercial maglev-transportation systems. These problems are closely related to the performance of the levitation-control system. In the past few decades, studies on magnetic-suspension systems have received much interest. Yan J D [6] utilized a Back Propagation (BP) neural network to adjust online parameters with a PI (proportional -integral) controller according to electromagnetic suspension acceleration and operation speed, and simulation results showed the effective suppression of electromagnetic vibration and the improvement of vehicle riding comfort. Sun [7] proposed an adaptive neural-fuzzy controller with sliding-mode technology that could achieve excellent dynamic performance under disturbance and uncertainty. He et al. [8] designed a nonlinear controller with LQR (linear quadratic regulator) theory and a nonlinear disturbance observer. Li [9] proposed an active controller with a virtual energy harvester to suppress vehicle-guideway coupling vibration for maglev train. The simulation and experiment results proved the effectiveness of the proposed control method. Zhou et al. [10] proposed an active control method with a FIR (finite impulse response) filter for the maglev train to deal with track irregularity. However, there is linear processing during controller design and analysis. MacLeod C [11] et al. realized an LQR-based levitation optimal control and frequency domain weighting method to improve the ability to suppress disturbances in the track during the operation. Xu et al. [12] proposed an adaptive robust control algorithm to realize stable suspension by considering air-gap constraints. Wang et al. [13] designed a state feedback controller by using a state-estimation function of a Kalman filter to solve the dependence problem of track-rail stiffness, while control performance was reduced when the system was far away from the equilibrium point owing to the linear approximation model adopted by the control object. Most existing maglev-control methods are based on the assumption that the control signal is transmitted without delay. Nevertheless, the levitation system must have time delay due to the existence of signal transmission, inductance delay, and controller calculation. Time delay leads to more complex dynamic characteristics, including homogeneous bifurcation, Hopf bifurcation, and limit cycle oscillation [14,15]. The maglev train CMS04 presented by the National University of Defense Technology experienced severe rail–vehicle resonance caused by excessive time delay during testing [16,17]. Tongji University also carried out a real vehicle test that showed that suspension failure occurred when the air-gap feedback time delay of the static suspension exceeded the critical value [18]. However, they have not proposed any control algorithm that could suppress the influence of the time delay. These phenomena may seriously affect the stability and reliability of the suspension system and may hinder the further popularization and application of the maglev transportation system. Moreover, time delay widely exists in various engineering systems. Studies have shown that a time delay can lead to performance degradation or even failure. Therefore, stability analysis and control research with time delay in maglev vehicles could significantly guide other nonlinear systems. In this paper, an adaptive robust levitation control method was developed to deal with time delay and uncertainty, but bounded disturbances for a maglev train. The nonlinear dynamic models of magnetic-suspension systems were first derived. The proposed controller utilizes a discontinuous structure to improve robustness against time delay and uncertainty on the basis of the Riccati method and sliding-mode technology. In order to eliminate the chattering phenomenon in the control input, Appl. Sci. 2020, 10, 1179 3 of 17 an adaptive update law is proposed to regulate adaptive gain online. Otherwise, the stability of the overall system was assured with rigorous Lyapunov-based analysis. Both the simulation and experiment results were included to demonstrate the effectiveness of the proposed control strategy. The main contribution of this work is summarized as follows:

(1). The proposed controller can ensure better stable levitation of maglev system tackling the problems of disturbance and time delay simultaneously (2). The designed method does not require the magnitude of external disturbances, and it can attenuate chattering in control inputs eﬀectively. (3). As veriﬁed by the experimental results, the proposed method shows increased control performance in a time-delay environment.

The rest of the paper is organized as follows: in Section2, the control problem of interest is formulated. The controller design and the closed-loop stability analysis are described in Sections3 and4, respectively. Sections5 and6 detail the simulation and experimental results and analysis. Section7 summarizes the entire paper.

2. Problem Statement

2.1. System Dynamics The maglev system and single-electromagnet system can be illustrated in Figures1 and2. According to Maxwell’s equation and Biot–Savart’s theorem:

R t (i (t) x (t))dt 0 ψm m , m Fm(im(t), xm(t)) = (1) ∂xm(t) where Fm(im(t), xm(t)) denotes electromagnetic force; xm(t) represents the airgap between the electromagnet and the rail; and im(t) denotes the current. According to Kirchhoﬀ magnetic-circuit law:

Nmim(t) ψm(im(t), xm(t)) = Nm , (2) R(xm) where Nm represents the coil number of turns; µ0 is Air permeability; Am denotes the eﬀective magnetic pole area; and R(xm) can be expressed as

2xm(t) R(xm) = (3) µ0Am

Substituting Equations (2) and (3) into Equation (1), we can obtain the electromagnetic-force equation of the magnetic-levitation system as:

2 " #2 µ0AmNm im(t) Fm(im(t), xm(t)) = . (4) − 4 xm(t)

The relationship between voltage and current across the solenoid coil is:

. um(t) = im(t)Rm + Nmφm, (5) where Lm is the instantaneous inductance of the electromagnet that can be expressed as:

2 µ0NmAm Lm Lm = , φm = im(t). (6) 2xm(t) Nm Appl. Sci. 2020, 10, x FOR PEER REVIEW 3 of 17 Appl. Sci. 2020, 10, x FOR PEER REVIEW 3 of 17 In this paper, an adaptive robust levitation control method was developed to deal with time In this paper, an adaptive robust levitation control method was developed to deal with time delay and uncertainty, but bounded disturbances for a maglev train. The nonlinear dynamic models delay and uncertainty, but bounded disturbances for a maglev train. The nonlinear dynamic models of magnetic-suspension systems were first derived. The proposed controller utilizes a discontinuous of magnetic-suspension systems were first derived. The proposed controller utilizes a discontinuous structure to improve robustness against time delay and uncertainty on the basis of the Riccati structure to improve robustness against time delay and uncertainty on the basis of the Riccati method and sliding-mode technology. In order to eliminate the chattering phenomenon in the method and sliding-mode technology. In order to eliminate the chattering phenomenon in the control input, an adaptive update law is proposed to regulate adaptive gain online. Otherwise, the control input, an adaptive update law is proposed to regulate adaptive gain online. Otherwise, the stability of the overall system was assured with rigorous Lyapunov-based analysis. Both the stability of the overall system was assured with rigorous Lyapunov-based analysis. Both the simulation and experiment results were included to demonstrate the effectiveness of the proposed simulation and experiment results were included to demonstrate the effectiveness of the proposed control strategy. The main contribution of this work is summarized as follows: control strategy. The main contribution of this work is summarized as follows: (1). The proposed controller can ensure better stable levitation of maglev system tackling the (1). The proposed controller can ensure better stable levitation of maglev system tackling the problems of disturbance and time delay simultaneously problems of disturbance and time delay simultaneously (2). The designed method does not require the magnitude of external disturbances, and it can (2). The designed method does not require the magnitude of external disturbances, and it can attenuate chattering in control inputs effectively. attenuate chattering in control inputs effectively. (3). As verified by the experimental results, the proposed method shows increased control (3). As verified by the experimental results, the proposed method shows increased control performance in a time-delay environment. performance in a time-delay environment. The rest of the paper is organized as follows: in Section 2, the control problem of interest is The rest of the paper is organized as follows: in Section 2, the control problem of interest is formulated. The controller design and the closed-loop stability analysis are described in Sections 3 formulated. The controller design and the closed-loop stability analysis are described in Sections 3 and 4, respectively. Sections 5 and 6 detail the simulation and experimental results and analysis. and 4, respectively. Sections 5 and 6 detail the simulation and experimental results and analysis. Section 7 summarizes the entire paper. Section 7 summarizes the entire paper. 2. Problem Statement 2. Problem Statement Appl. Sci. 2020, 10, 1179 4 of 17 2.1. System Dynamics 2.1. System Dynamics

Figure 1. Structure of maglev system. Figure 1. StructureStructure of of maglev maglev system.

FigureFigure 2. 2. Single-electromagnetSingle-electromagnet system. system. Figure 2. Single-electromagnet system. Finishing Equation (6) and deriving its time at both ends:

. µ0N Am dim(t) µ0N Amim(t) dxm(t) φ = m m (7) m 2 2xm(t) dt − 2xm(t) dt

By Equations (5) and (7), we can obtain the electrical electromagnet equation:

2 2 µ0N Am dim(t) µ0N Amim(t) dxm(t) u (t) = i (t)R + m m (8) m m m 2 2xm(t) dt − 2xm(t) dt

Assuming fd is the disturbance force applied to the maglev-train system, the model of the magnetic-suspension system can be rewritten as follows:

2 " #2 .. µ0AmNm im(t) mxm(t) = + mg + fd. (9) − 4 xm(t)

Convert Equations (8) and (9) to state-space expressions. Here, the system state is x1(t) = xm(t), . x2(t) = xm(t), and y3(t) = im(t), and the state-space expression of the ﬂoating system is:  .  x1(t) = x2(t)   . 2 2  µ0AmNm x3(t) 1  x2(t) = + g + fd  − 4m x1(t) m  . (10)  ( ) = x2(t)x3(t) + 2x1(t) ( ( ) ( ) )  x3 t x (t) 2 um t x3 t Rm  1 µ0NmAm −  Appl. Sci. 2020, 10, 1179 5 of 17

Remark 1. In an actual maglev-train levitation system, choppers are utilized to supply power to the levitation electromagnet by the principle of the current following to adjust the current for the electromagnet. As a consequence, only the current loop was considered while designing the controller. In this way, third-order dynamic equations can be degenerated into second-order nonlinear diﬀerential equations, as shown.

 .  x1(t) = x2(t)  2  . A N2 i (t)  x (t) = µ0 m m m + g + 1 f (11)  2 4m x (t) m d  − 1 

Remark 2. As shown in Equation (4), the relative order between nonlinear electromagnetic force Fm(t) and 2 im(t) current im(t) is zero. can be taken as u to design the controller. Since air gap x1(t) can be measured at x1(t) any time, it is convenient to calculate control current im(t) according to the following formula, as long as the required F (t) is determined. m r r 2 2 x1 u x1 Fm im(t) = = , (12) κ κ 2 µ0AmNm where κ = 4m denotes the electromagnetic-force transmission coeﬃcient.

Remark 3. In Equation (9), considering problems such as passenger ﬂuctuation and parameter perturbation, the actual dynamic equation of the magnetic-suspension system can be rewritten as follows:

2 .. µ0AmNm (m + ∆m)xm(t) = u + (m + ∆m)g + f (13) − 4 d

.. Lumped uncertainty term is deﬁned as nu = ∆mg ∆mxm(t) + f , so Equation (13) can be − d rewritten as follows: 2 .. µ0AmNm mxm(t) = u + mg + nu. (14) − 4 Then, Equation (11) can be rewritten as:

" . # " #" # " # 2 x1 0 1 x1 0 µ0AmNm . = + 1 ( u + mg + nu) (15) x2 0 0 x2 m − 4 or 2 . µ0AmNm X = A X + B( u + mg + nu), (16) 1 − 4 " # " # " # 0 1 1 x1 where A1 = , B = 1 and X = . 0 0 m x2

Remark 4. In the actual system, signal transmission in the maglev control system would be delayed, which may cause serious safety accidents [12–14]. In this paper, time delay was introduced into the dynamic modeling. Time delay in states was only considered to facilitate controller design, and time delay in the control signal caused by inductance was not considered. Thus, the magnetic-levitation-system dynamics model shown in Equation (16) can be written as follows:

2 . µ0AmNm X = A X + A X(t τ) + B( u + mg + nu), (17) 1 2 − − 4

+ 2 2 where τ R represents time delay in this closed-loop system. A2 is a known matrix, and A2 R × . ∈ ∈ Appl. Sci. 2020, 10, 1179 6 of 17

2.2. Control Object The control plant can be described in Equation (17) with a time-delay environment. u is the desired controller, and the desired system states can be denoted by Xd. The control objective is to develop a controller u such that, the maglev system states X of the time-delay system (17) can asymptotically track the desired system states Xd facing disturbance and time delay simultaneously. In the next section, the designed control law is proposed to satisfy the above control objective.

3. Controller Design

2 1 To achieve the previous control objectives, the system error vector is deﬁned as E(t) R × . ∈ E(t) = X X , (18) − d h iT where Xd = x1d 0 and x1d are the target air gaps of the maglev vehicle. Taking the time derivative of Equation (18), we can obtain the following relationship:

. . . E = X X (19) − d The following condition must be met if the system is to be stable at the target air-gap position.

limE(t) = 0 (20) t →∞ To facilitate subsequent controller design, actuator saturation is not considered. Taking into account the actual working conditions of a maglev train, the following reasonable assumption is given:

Assumption 1. There exist matrix P and positive deﬁnite matrices Q, J, and K2 that satisfy the following Riccati equation: T 1 T (A BK ) P + P(A BK ) + PA J− A P = Q J (21) 1 − 2 1 − 2 2 2 − − On the basis of the desired state in Equation (18) and the system in Equation (17), the ideal response of the system is given by the following reference model:

. X = φmX + A X (t τ) + ϕmσ, (22) d d 2 d − 2 1 2 2 where σ R × is a bounded reference signal; φm, ϕm R × and their equations are as follows: ∈ ∈

φm = A + B(K K ), ϕm = BK , (23) 1 1 − 2 3 1 2 where K1, K2, and K3 R × , and they are optional. ∈ From Equation (23), matrices φm A + BK and ϕm are a linear combination of matrix B. If matrix − 1 2 B is linearly independent of matrix ϕm, which can avoid redundancy in inputs, matrices K1 and K3 are as follows: T 1 T T 1 T K = (B B)− B (φm A + BK ) = (ϕ B)− ϕ (φm A + BK ) 1 − 1 2 m m − 1 2 T 1 T T 1 T K3 = (B B)− B ϕm = (ϕmB)− ϕmϕm For the sake of improving system adaptability and robustness, an adaptive sliding-mode controller is proposed as follows:

4 h i u = K X K X K σ + ηˆsgn(BTPE) , (24) 2 2 1 d 3 µ0AmNm − − Appl. Sci. 2020, 10, 1179 7 of 17 where sgn( ) is the signum functional; ηˆ is the adjustable estimate of η and it was designed as follows: · . ηˆ = ξ ETPB , (25) where ξ is the adaptive factor and ξ R+, which was used to adjust the estimated speed of ηˆ. ∈ Choosing suitable adaptive gain can eﬀectively reduce the chattering phenomenon and remove the requirement of knowing the bound of uncertainties.

4. Closed-Loop Stability Analysis

n n Theorem 1. There exist matrices X and X R × , any positive constant a, symmetric positive deﬁnite matrix n ∈ Y, and x1, x2 R ; so, we have: ∈

T T 1 T 1 T 2x Xx ax XY− X x + x Yx 1 2 ≤ 1 1 a 2 2

Proof. For any constant a > 0 and any symmetric matrix Y > 0,

T T 1 1 T 1 T 1 T T 1 T 0 a X x Yx Y− X x Yx = ax XY− X x 2x Xx + x Yx . ≤ 1 − a 2 1 − a 2 1 1 − 1 2 a 2 2

In combination with Equations (17), (22), and (23), the ﬁrst-order derivative of the error is:

. . . 2 µ0AmNm E = X X = A X + A X(t τ) + B u + mg + nu φmX A X (t τ) ϕmσ − d 1 2 − − 4 − d − 2 d − − 2 (26) µ0AmNm = A E + A E(t τ) BK σ BK X + BK X + B u + mg + nu 1 2 − − 3 − 1 d 2 d − 4 Equation (26) can be rewritten with controller (24) as follows:

. h T i E = (A BK )E + A E(t τ) B ηˆsgn(B PE) mg nu (27) 1 − 2 2 − − − −

Theorem 2. The proposed adaptive levitation control law (24) with adaptive law (25) can guarantee that ηˆ has an upper bound and can also drive error vector E(t) to be zero within a ﬁnite time.

Proof. The Lyapunov–Krasovskii function was selected as:

Z t τ 1 T 1 − T 1 2 V = E PE + E (δ)JE(δ)dδ + eη , (28) 2 2 t 2ξ where eη = η ηˆ and ξ is a positive constant. The derivative of Equation (28) with respect to time yields − can be obtained as follows:

. . . . 1 T 1 T 1 T 1 T 1 V = E PE + E PE + E JE E (t τ)JE(t τ) eηηˆ. (29) 2 2 2 − 2 − − − ξ

Substitute error Equation (27) into Equation (29) and obtain the following: . . 1 T 1 T 1 T T V = E JE E (t τ)JE(t τ) eηηˆ ηˆ E PB + E PB(mg + nu) 2 − 2 − − −h ξ − i (30) +ETPA E(t τ) + 1 ET P(A BK ) + (A BK )TP 2 − 2 1 − 2 1 − 2 Appl. Sci. 2020, 10, x FOR PEER REVIEW 8 of 17

Then, utilize Theorem 1 and obtain the following:

Appl. Sci. 2020, 10, 1179 11− 8 of 17 EPAEtTT()−≤τττ E()() t − JEt −+ EPAJ TT1 APE. (31) 22222

ByThen, combining utilize Theorem Equations 1 and(30) obtainand (31), the the following: following can be obtained: 1 T −  ≤−+−++TT()() 1 VEPABKABKPJPAJAP1212 22 T 2 1 T 1 T 1 T E PA2E(t τ) E (t τ)JE(t τ) + E PA2 J− A PE. (31) 1 − ≤ 2 − − 2 2 −−ηη ˆˆ η EPBTT + EPBmgn() + ξ u By combining Equations (30) and (31), the following can be obtained:

11 (32) . ≤−TTT −ηη ˆˆ − η +() + 1 TEQEh EPB EPBmgnT u 1 T i V E2 P(A1 ξBK2) + (A1 BK2) P + J + PA2 J− A P ≤ 2 − . − 2 1 T T 11TTTeηηˆ  ηˆ E PB + E PB(mg + nu) ≤−EQE− −ξ ηη−ˆˆ − η. EPB + EPBmgn + (32) 1 T ξ 1 T T u 2 E QE eηηˆ ηˆ E PB + E PB(mg + nu) ≤ − 2 − ξ . − 1 T 1 T T On the basis of Theorem 1, EquationE QE (32)eηηˆ canηˆ E bePB rewritten + E PBas follows: mg + n u ≤ − 2 − ξ −  11TT On the basis of Theorem 1, EquationVEQEEPB≤− (32) can +η be rewritten − asηˆ follows: (33) 2 ξ . 1 1 . Equation (34) can be elicited byV incorporationETQE + η with ET PBEquations ηˆ (33) and (25). (33) ≤ −2 − ξ 1 VEQE ≤− T , (34) Equation (34) can be elicited by incorporation2 with Equations (33) and (25). since Q is a positive definite matrix in the. Riccati1 equation. V is negative semidefinite, and V ETQE, (34) ()ηη() ≤ ()() η ≤ −2 η ∈ + VtV0 , which proves that ˆ is bounded. That is, there exists a constant d R that ∀>tt0,ηˆ ( ) ≤η . satisfiessince Q is a positive definited . matrix in the Riccati equation. V is negative semidefinite, and V(eη(t)) + ≤ V(eη(0)), which proves1 thatT ηˆ is bounded. That is, there exists a constant ηd R that satisfies Define Ψ=−()tEQEV ≤− and integrate it with respect to time yields. Then,∈ we can obtain t > 0, ηˆ(t) η . 2 ∀ ≤ d . the following:Define Ψ (t) = 1 ETQE V and integrate it with respect to time yields. Then, we can obtain − 2 ≤ − the following: t Ψ≤()ττdV() η ()0 − V() η() t (35) Z0t Ψ(τ)dτ V(eη(0)) V(eη(t)) (35) The result can be obtained because0 V ()η≤()0 is bounded− and Vt()η () is not ever-increasing and bounded.The result can be obtained because V(eη(0)) is bounded and V(eη(t)) is not ever-increasing and bounded. t limZ tΨ≤∞()ττd (36) →∞ 0 limt Ψ(τ)dτ (36) t ≤ ∞ Ψ ()t is also bounded, so it can be→∞ proven0 that limΨ= (t ) 0 by utilizing Barbalat’s lemma t→∞ Ψ(t) is also bounded, so it can be proven that lim Ψ(t) = 0 by utilizing Barbalat’s lemma [19–21]. [19–21]. Therefore, limE = 0 . QED. t Therefore, limE = 0.t→∞ QED. →∞ In summary,t the control architecture of the proposed controller can be described in Figure 3. In summary,→∞ the control architecture of the proposed controller can be described in Figure3.

FigureFigure 3. 3. ControlControl architecture architecture of of the the proposed proposed robust levitation controller. 5. Simulation Results In order to demonstrate the superiority of the proposed adaptive robust controller while coping with an uncertain time delay for a maglev-train system, the conventional PID (proportion integration diﬀerentiation) controller, the traditional sliding mode controller (SMC) [22–28], and the proposed Appl. Sci. 2020, 10, 1179 9 of 17 controller were respectively applied to the maglev system in the simulation model. MATLAB/Simulink was utilized to provide the simulation environment. The values of the system parameters are outlined in Table1.

Table 1. Parameter values for the magnetic-suspension system.

Physical Quantity Value Levitation mass (kg) 100 Coil resistance (Ω) 0.420 Coil inductance (mH) 177.8 Number of coil turns Nm 340 Magnetic-pole width (mm) 28 Magnetic-pole length (mm) 700 Air permeability µ /(H m 1) 4π 10 7 0 · − × − Target levitation air gap (mm) 8.5

Without loss of generality, the values of nu and A2 were selected as below:

0.5t nu = e− sin(t) h i A2 = diag( 0.03randi(10) 0.001randi(20) ) h i For the proposed controller in this paper, gains were chosen as: K1 = 0, K2 = 35.4 8.12 , h i h iT K2 = 0.065 0.0971 , and ξ = 66.72. The reference signal was selected as: σ = 0.6 0 . The initial h iT h iT state of the system was X0 = 0.0016 0 . The control target trajectory was Xd = 0.0085 0 . The traditional PID controller is selected, and the control parameters are kp = 1800, ki = 150, and kd = 1150 based on the Root Locus method. The parameters of the SMC controller are chosen as: c1 = 20, c2 = 5, ksmc = 15, ξsmc = 20. The performance of the proposed controller, the SMC controller, and the PID controller were simulated under the two following cases. Case 1: Levitation without disturbance. Case 2: Levitation with disturbance. (1) Levitation without disturbance and with uncertain time delay. The Matlab/SimEvent® is utilized to build the CAN model to represent the signal transmission, which CAN reﬂect the bus load rate and the uncertain time delay. (2) Levitation with disturbance under uncertain time delay. The simulation results for the PID controller without disturbance (air-gap response, air-gap changing velocity, and control current) are illustrated in Figure4. The control current has ﬂuctuations in Figure4, which is related to the excessive gain. However, a small gain will cause the system to be unstable. The essential reason is that the maglev system is a highly nonlinear system, and the PID is a controller based on linearization theory, which is easy to fail when applied to a nonlinear maglev system. The simulation results for the SMC controller without disturbance are included in Figures5 and6 depicts the simulation results for the proposed adaptive robust levitation controller without disturbance. Figures4–6 show that the performance of the PID controller deteriorated with 29.4% overshoot, 0.248 mm steady-state error, and 2.1 s settling time and the SMC controller with 0.248 mm steady-state error and 1.05 s settling time. The performance of the proposed adaptive robust levitation controller was excellent without overshoot and steady-state error. Appl. Sci. 2020, 10, x FOR PEER REVIEW 10 of 17

Appl.controller Sci. 2020 , without10, x FOR PEERdisturbance. REVIEW Figures 4–6 show that the performance of the PID controller10 of 17 deteriorated with 29.4% overshoot, 0.248 mm steady-state error, and 2.1 s settling time and the SMC controllercontroller withoutwith 0.248 disturbance. mm steady-state Figures error4–6 show and 1.05that sthe settling performance time. The of performancethe PID controller of the deterioratedproposed adaptive with 29.4% robust overshoot, levitation 0.248 controller mm steady-s was tateexcellent error, withoutand 2.1 s overshootsettling time and and steady-state the SMC controllererror. with 0.248 mm steady-state error and 1.05 s settling time. The performance of the proposed adaptive robust levitation controller was excellent without overshoot and steady-state Appl.error. Sci. 2020, 10, 1179 10 of 17 0.02 0.02

0 0.02 0.02 0.015 -0.02 0 0.015 -0.04 -0.02 Airgap (m) Airgap 0.01 velocity (m/s) -0.06 -0.04

Airgap (m) Airgap 0.01 0.005 velocity (m/s) -0.08 0 1 2 3 4 5 6 7 8 9 10 -0.06 0 1 2 3 4 5 6 7 8 9 10 Time (S) Time (S)

0.005 -0.08 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 (a) airgapTime (S) (b) velocityTime of (S) airgap

100 0.02 (a) airgap (b) velocity of airgap 80 0 100 0.02 60 80 -0.020 x1 40 -0.04 60 -0.02 reference

Control inputs (A) 20 x1x2 40 -0.04-0.06 reference 0

Control inputs (A) 20 0 1 2 3 4 5 6 7 8 9 10 -0.06-0.08 x2 Time (S) 0 1 2 3 4 5 6 7 8 9 10 Time (S) 0 0 1 2 3 4 5 6 7 8 9 10 -0.08 0 1 2 3 4 5 6 7 8 9 10 Time (S) (c) x1 and x2 (d) control inputs Time (S) Figure 4. Simulation results for the PID controller without disturbance (dashed line: the reference (c) x1 and x2 (d) control inputs airgap). FigureFigure 4. 4.Simulation Simulation results results for for the the PID PID controller controller without with disturbanceout disturbance (dashed (dashe line:d the line: reference the reference airgap). airgap). 0.02 100

80 0.02 100 0.015 60 80

0.015 40 Airgap (m) 0.01 60

Control inputs (A) 20 40 Airgap (m) 0.01

0.005 Control inputs (A) 0 0 1 2 3 4 5 6 7 8 9 10 20 0 2 4 6 8 10 Time (S) Time (S) 0.005 0 0 1 2 3 (a4) airgap5 6 7 8 9 10 0 2 (b) control4 inputs6 8 10 Time (S) Time (S) Figure 5. Simulation results for the sliding mode controller (SMC) without disturbance (dashed line: Figure 5. Simulation(a) resultsairgap for the sliding mode controller (SMC) without (b) control disturbance inputs (dashed line: the reference airgap). the reference airgap). Figure 5. Simulation results for the sliding mode controller (SMC) without disturbance (dashed line: the reference airgap). Appl. Sci. 2020, 10, 1179 11 of 17 Appl. Sci. 2020, 10, x FOR PEER REVIEW 11 of 17

0.02 0.016 0 0.014 -0.02 0.012 -0.04 0.01 -0.04 Airgap (m) Airgap Airgap (m) Airgap velocity (m/s) velocity velocity (m/s) velocity 0.008 -0.06

0.006 -0.08 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 Time (S) Time (S)

(a) airgap (b) velocity of airgap

0.02 100

80 0 60 x1 reference 40 -0.02 0.5*x2 control inputs (A) inputs control control inputs (A) inputs control 20

-0.04 0 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 Time (S) Time (S)

FigureFigure 6. 6. Simulation results results for for the the proposed proposed robust robust levitation levitation controller controller without without disturbance disturbance (red dashed line: the reference airgap). (reddashed dashed line: line: the thereference reference airgap). airgap).

TheThe simulation simulation results results forfor thethe PIDPID controllercontroller with with external external disturbances disturbances (the (the air-gap air-gap response, response, thethe air-gap air-gap changing changing velocity, velocity, and and the the controlcontrol current)current) are are shown shown in in Figure Figure 7.7 .The The simulation simulation results results for the SMC controller with disturbances are included in Figures 8 and 9 and depict the simulation forfor the the SMC SMC controller controller with with disturbances disturbances are includ includeded in in Figures Figures 88 andand 9 and depict depict the the simulation simulation results for the proposed adaptive levitation controller with external disturbances. Figure 7 shows results for the proposed adaptive levitation controller with external disturbances. Figure7 shows that that levitation failure appeared with the PID controller. The air gap of the system with the SMC levitation failure appeared with the PID controller. The air gap of the system with the SMC controller controller in Figure 8 fluctuated above 3.5% and the current response had chattering phenomenon. in Figure8 fluctuated above 3.5% and the current response had chattering phenomenon. The proposed The proposed adaptive levitation controller could still effectively levitate the train. After adaptive levitation controller could still effectively levitate the train. After stabilization, the air gap stabilization, the air gap of the system fluctuated within 1%. The control performance indexes can be ofsummarized the system fluctuated in Table 2 as within below. 1%. The control performance indexes can be summarized in Table2 as below.

2000 0.015

1000 levitation failure 0.01 0 Fd (N) Fd Fd (N) Fd Airgap (m) Airgap Airgap (m) Airgap 0.005 -1000

0 -2000 0 2 4 6 8 10 0 2 4 6 8 10 Time (s) Time (s) (a) disturbance (b) airgap

Figure 7. Simulation results for the PID controller with external disturbance. (a) Nonlinear external disturbance. (b) Airgap response. Appl.Appl. Sci.Sci. 20202020,, 1010,, xx FORFOR PEERPEER REVIEWREVIEW 1212 ofof 1717

FigureFigure 7.7. SimulationSimulation resultsresults forfor thethe PIDPID controcontrollerller withwith externalexternal disturbance.disturbance. ((aa)) NonlinearNonlinear externalexternal Appl. Sci.disturbance.disturbance.2020, 10, 1179 ((bb)) AirgapAirgap response.response. 12 of 17

100100 0.0160.016 8080 0.0140.014 6060

0.0120.012 4040 Airgap (m) Airgap Airgap (m) Airgap Control (A)inputs Control

0.01 (A)inputs Control 0.01 2020

0.0080.008 00 00 22 44 66 88 1010 00 22 44 66 88 1010 Time (S) Time (S) Time (S) .. Time (S) ((aa)) airgapairgap ( (bb)) controlcontrol inputsinputs Figure 8. Simulation results for the PID controller with external disturbance. Figure 8. SimulationSimulation results results for for the the PID PID cont controllerroller with external disturbance.

0.010.01

0.0160.016 00 0.0140.014

0.012 0.012 -0.01-0.01 velocity (m/s) velocity (m/s) Airgap (m) Airgap Airgap (m) Airgap 0.010.01 -0.02 -0.020 2 4 6 8 10 0.0080.008 0 2 4 6 8 10 00 22 44 66 88 1010 TimeTime (s)(s) Time (s) Time (s) ((aa)) airgapairgap ( (bb)) velocityvelocity ofof airgapairgap

100100 20002000 8080 10001000 6060 00

40 (N) Fd

40 (N) Fd -1000

Control inputs (A) inputs Control -1000 Control inputs (A) inputs Control 2020

00 -2000-2000 00 22 44 66 88 1010 00 22 44 66 88 1010 Time (s) Time (s) Time (s) Time (s) ((cc)) controlcontrol inputsinputs ( (dd)) externalexternal disturbancedisturbance

FigureFigure 9. 9. SimulationSimulation resultsresults forfor thethe proposed proposed robust robust levi levitationlevitationtation controller controller with with external external disturbance. disturbance. Appl. Sci. 2020,, 10,, x 1179 FOR PEER REVIEW 1313 of of 17

Table 2. Summary of numerical values. Table 2. Summary of numerical values. Experiment (Time Delay) PID SMC Proposed Method Experiment (Time Delay)Time PID 2.10 s 1.05 s SMC 0.82 s Proposed Method TimeError 2.10 0.248 s mm 0.07 mm 1.05 s <0.01 mm 0.82 s Case 1: normal Errorovershoot 0.248 mm29.4% 0 0.07 mm 0 <0.01 mm Case 1: normal overshootChattering 29.4%None yes 0 None 0 Chattering None yes None Time 2.18 s 1.12 s 0.91 s TimeMax Error 2.18 Unstables 0.3 mm 1.12 s <0.1 mm 0.91 s Case 2: disturbanceMax Error Unstable 0.3 mm <0.1 mm Case 2: disturbance Air-gap fluctuation Unstable >3.5% <1% Air-gap ﬂuctuation Unstable >3.5% <1% Chattering Chattering / / yes yesnone none

The simulation results demonstrated the effectiveness of the presented robust levitation controllerThe simulation to deal with results uncertain demonstrated time delay the and eﬀectiveness external disturbances of the presented while robust staying levitation asymptotically controller stable.to deal with uncertain time delay and external disturbances while staying asymptotically stable. 6. Hardware Experiments 6. Hardware Experiments After suﬃcient numerical simulation, hardware experiments were implemented to examine After sufficient numerical simulation, hardware experiments were implemented to examine the the performance of the presented adaptive levitation controller on a single magnetic-suspension performance of the presented adaptive levitation controller on a single magnetic-suspension system, system, which is illustrated in Figure 10. The basic working principle of a single magnetic-suspension which is illustrated in Figure 10. The basic working principle of a single magnetic-suspension system system is described in Figure2. The test bed was composed of dSPACE ®, a levitation controller, a is described in Figure 2. The test bed was composed of dSPACE®, a levitation controller, a power-distribution cabinet, an eddy-current sensor, and a magnetic-suspension system. In particular, power-distribution cabinet, an eddy-current sensor, and a magnetic-suspension system. In the MATLAB/Simulink® Real-Time Windows Target (RTWT) was utilized to set up the proposed particular, the MATLAB/Simulink® Real-Time Windows Target (RTWT) was utilized to set up the control strategy in the loop-system hardware. proposed control strategy in the loop-system hardware.

Figure 10. The experimental platform of the magnetic suspension system with dSPACE. Figure 10. The experimental platform of the magnetic suspension system with dSPACE.

Experiment results of the PID levitation controller with an uncertain time delay are provided in Figures 11–13.11–13. We We can learn from the Figure 10 that the system becomes unstable after 14 s, at which point the instability causing the rail and electromagnetic isis hittinghitting repeatedly.repeatedly. Appl.Appl. Sci. 2020Sci. 2020, 10,, 117910, x FOR PEER REVIEW 1414 of of 17 17 Appl.Appl. Sci. Sci. 2020 2020, ,10 10, ,x x FOR FOR PEER PEER REVIEW REVIEW 1414 of of 17 17 Appl. Sci. 2020, 10, x FOR PEER REVIEW 14 of 17

15 1515 15

10 1010 10 Airgap (mm) Airgap Airgap (mm) Airgap Airgap (mm) Airgap 5

Airgap (mm) Airgap 55 5

0 000 5 10 15 000 55 1010 1515 0 5 Time10 (s) 15 TimeTime (s) (s) Time (s) Figure 11. Experiment results with PID: airgap. FigureFigure 11. 11. Experiment Experiment results results with with PID: PID: airgap. airgap. FigureFigure 11. 11. ExperimentExperiment results with with PID: PID: airgap. airgap. 80 8080 80 60 6060 60 40 4040 40 20 2020

20 (A)inputs Control Control (A)inputs Control Control (A)inputs Control Control (A)inputs Control 0 00 0 0 5 10 15 00 55 1010 1515 0 5 Time10 (s) 15 TimeTime (s) (s) Time (s) FigureFigure 12.12. ExperimentExperiment resultsresults withwith PID:PID: current.current. FigureFigure 12. 12.Experiment Experiment resultsresults withwith PID: current.

60 6060 60

） 40 2 ）） 4040 2 2 ） 40 2 m/s m/s m/s 20 （ m/s 2020 （（ 20 （ 0 00 0 -20

Acceleration -20Acceleration -20 Acceleration Acceleration

Acceleration Acceleration -20 Acceleration Acceleration -40 -40-400 5 10 15 -4000 55 1010 1515 0 5 Time10 (s) 15 TimeTime (s) (s) Time (s) FigureFigureFigure 13. 13.13.Experiment ExperimentExperiment results resultsresults with withwith PID: PID:PID: electromagnetic electromagneticelectromagnetic acceleration. acceleration.acceleration. Figure 13. Experiment results with PID: electromagnetic acceleration.

TheTheThe experiment experimentexperiment results resultsresults of the ofof proposedthethe proposedproposed robust robustrobust levitation levitationlevitation controller controllercontroller (RLC) (RLC)(RLC) with withwith an uncertain anan uncertainuncertain time The experiment results of the proposed robust levitation controller (RLC) with an uncertain delaytimetime are delaydelay provided areare providedprovided in Figures inin FiguresFigures14–16. 14–16.14–16. time delay are provided in Figures 14–16. 1818 18

161616

16））））

mm 14

mm 14 （（（（ 1212 12 Airgap Airgap Airgap 10 Airgap 10 10

888 8000 555 101010 151515 0 5 Time （10s） 15 TimeTime （（ss）） Time （s） Figure 14. Experiment results with the robust levitation controller (RLC): airgap. Appl.Appl. Sci. Sci. 2020 2020, 10, 10, x, FORx FOR PEER PEER REVIEW REVIEW 1515 of of 17 17 Appl. Sci. 2020, 10, 1179 15 of 17 FigureFigure 14. 14. Experiment Experiment results results with with the the robust robust levitation levitation controller controller (RLC): (RLC): airgap. airgap.

5050

4040

3030

2020

1010 Control inputs (A) Control inputs (A) 0 0

-10-10 0 0 5 5 1010 1515 TimeTime （（s）s）

FigureFigure 15. 15. ExperimentExperiment Experiment results results with with RLC: RLC: current. current.

3 3 ) ) 2 2 2 2

1 1

0 0 Acceleration (m/s Acceleration (m/s

-1-1 0 0 100100 200200 300300 TimeTime (s) (s)

FigureFigure 16. 16. ExperimentExperiment Experiment results results results with with RLC: RLC: electromagnetic electromagnetic acceleration. acceleration.

InIn summary, summary, summary, Figures FiguresFigures 14–16 1414–16–16 show showshow that that that the the the air air gap air gap gapof of the the of magnetic themagnetic magnetic suspension suspension suspension system system system was was stable, stable, was andstable,and the the andtime-delay time-delay the time-delay influence influence inﬂuence could could be be couldsignificantl significantl be signiﬁcantlyy yreduced reduced with reduced with the the proposed with proposed the proposedrobust robust levitation levitation robust controller.levitationcontroller. controller.

7.7. Conclusions Conclusions Conclusions InIn this this paper, paper, a a novelnovel novel adaptiveadaptive adaptive robust robust levitationlevitati levitationon control control approach approach was was developed developed to to deal deal with with uncertainuncertain time timetime delays delaysdelays and external andand external disturbances.external disturbances.disturbances. The mathematical TheThe modelmathematicalmathematical of the magnetic-suspension modelmodel ofof thethe magnetic-suspensionsystemsmagnetic-suspension was constructed. systems systems The proposed was was constructed. constructed. adaptive levitationThe The proposed proposed controller adaptive adaptive could levitation improve levitation robustness controller controller against could could improvetimeimprove delay robustness androbustness uncertainty against against by utilizingtime time delay delay the Riccatiand and uncertainty methoduncertainty and by sliding-modeby utilizing utilizing the technology.the Riccati Riccati Themethod method proposed and and sliding-modeadaptivesliding-mode update technology. technology. law could The also The proposed eliminate proposed theadaptive adaptive chattering update update phenomenon law law could could without also also eliminate eliminate knowing the the chattering boundchattering of phenomenonuncertaintyphenomenon or without disturbancewithout knowing knowing in practice. the the bound bound The of veriﬁcationof uncertaint uncertaint ofy yor the or disturbance disturbance theoretical in derivations in prac practice.tice. Thewas The verification supportedverification ofbyof the rigorous the theoretical theoretical Lyapunov-based derivations derivations analysis.was was supported supported The simulation by by ri gorousrigorous results Lyapunov-based Lyapunov-based were provided analysis. toanalysis. show The the The superioritysimulation simulation resultsofresults the proposed were were provided provided controller to to show over showthe the the conventional superiority superiority of control of the the proposed methodproposed when controller controller facing over uncertainover the the conventional timeconventional delays controlandcontrol external method method disturbances. when when facing facing Hardware uncertain uncertain experimenttime time delays delays and results and external external were includeddisturbances. disturbances. to show Hardware Hardware that the experiment experiment proposed resultsrobustresults levitationwere were included included controller to to show show achieved that that the satisfactory the proposed proposed control robust robust performance levitation levitation controller controller against uncertainachieved achieved timesatisfactory satisfactory delays. controlIncontrol future performance performance work, we will against against focus uncertai onuncertai the timen ntime time delay delays. delays. problem In In future future for maglevwork, work, we vehicle—guidewaywe will will focus focus on on the the interactiontime time delay delay problemsystems—whichproblem for for maglev maglev considers vehicle—guideway vehicle—guideway guideway elasticity interaction interaction by utilizing systems—which systems—which the robust levitationconsiders considers controlguideway guideway strategy. elasticity elasticity byby utilizing utilizing the the robust robust levitation levitation control control strategy. strategy. Author Contributions: Y.-g.S. designed the controller and drafted the paper. S.X. performed the simulations. J.-q.X. and G.-b.L. help stability analysis and carried out the experiments. All authors have read and agreed to the AuthorpublishedAuthor Contributions: Contributions: version of the Y.-G.S. manuscript.Y.-G.S. designed designed the the controller controller and and drafted drafted the the paper. paper. S.X. S.X. performed performed the the simulations. simulations. J.-Q.X.J.-Q.X. and and G.-B.L. G.-B.L. help help stability stability analysis analysis and and carried carried out out the the experiments. experiments. Funding: This work was supported by the National Natural Science Foundation of China under Grant 51905380, Funding:byFunding: China This Postdoctoral This work work was was Science supported supported Foundation by by the the under National National Grant Natural Natural 2019M651582, Science Science Foundation by Foundation National of Keyof China China R&D under Programunder Grant Grant of 51905380,51905380, by by China China Postdoctoral Postdoctoral Science Science Foundation Foundation under under Grant Grant 2019M651582, 2019M651582, by by National National Key Key R&D R&D Program Program Appl. Sci. 2020, 10, 1179 16 of 17

China under Grant 2016YFB1200600 and National Key Technology R&D Program of the 13th Five-year Plan, (2016YFB1200602). Acknowledgments: The authors would like to acknowledge H. Y Qiang and S. H. Wang from Shanghai Maritime University for technical support in this study. Conﬂicts of Interest: The authors declare no conﬂict of interest.

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