1 Introduction

Slip/slide phenomenon usually causes insufficient braking and traction performance as well as early damaging of rails and wheels and is one of the critical issues that must be solved in order to supply more reliable and acceptable way of transportation.The required traction and braking efforts are performed by adhesion forces between wheels and rail.Adhesion is a highly complex event which depends on many factors,including surface properties,material types,temperature,velocity,etc.However,experimental results show that for each specific combination of these factors,there exists an optimum slip ratio which maximizes the adhesion force between wheel and rail as a function of each velocity value.

The main objective of the re-adhesion control strategy is to maintain the slip velocity at the optimal value.It is compulsory to derive or estimate the optimum slip ratio for specific vehicle velocity and environmental conditions.The slip ratio can be defined as a relative velocity proportion in the contact area.Hence,the evaluation performance of the slip ratio is directly coupled with the integrity of the vehicle speed information.Many methods have been suggested to estimate the railway vehicle velocity by using magnetic sensors such as Hall effect gear tooth active speed sensors,which are very accurate in velocity measurement.However,these methods are very expensive and sensitive to wheel–rail contact conditions(rain,sand,snow,dew,etc.).The other way of acquiring velocity information is to utilize global positioning system(GPS)signals.There are many well-known techniques in the literature that combine GPS information with other sensors such as the wheel speed sensor,accelerometer and inertial measurement unit(IMU)[1–4].This kind of information can be also obtained by integrating indirect methods and different estimation approaches.One of these methods is detecting the excitations resulted from track irregularities with acceleration and angular rate measurements at different wheelsets,and another is through measurement of the time shift between the dynamic responses of two wheelsets with respect to the same track irregularity[5].In addition,robust velocity estimation involving establishment of a dynamic Friction model has also been investigated[6].However,the slip/slide control systems based on vehicle velocity estimation do not meet the safety concerns of industrial applications.These systems demand an additional estimation process to track the optimal slip ratio.

Among the alternative methods,there exists an indirect detection and estimation process of the slip/slide conditions by measuring the voltage,current and speed of the alternative current(AC)traction motor through an extended Kalman filter(EKF)[7].Another method is to control the slip/slide states by evaluating the wheel speed information of different axles together.If one of the axles in slip/slide state turns faster or slower than the others,the slipping or sliding axle can be detected and controlled unless all of the axles slip or slide at the same time.A trailer axle solves this kind of problem as well[8].When a wheelset is accelerated higher than the pre-defined threshold,the reference torque of this wheelset will be decreased distinctly till the adhesion situation is recovered to the micro-slip region.This axle is assumed as a reference axle,and the adhesion forces on the other axles are expected to be maximized considering its wheel speed information.

A new method used to detect the slip velocity is based on the multi-rate EKF state identification which combines the multi-rate and EKF method to estimate the traction motor load torque.This method provides a faster detection of slip and improves the reliability and traction performance[9].A re-adhesion control system based on the disturbance observer for examining the first resonant frequency of the bogie system to utilize adhesion more eff iciently was also proposed in[10,11].However,these methods are not reliable enough for industrial real-time applications because the time derivative of the adhesion force cannot be estimated correctly when the adhesion force changes quickly in case of sudden slip[12].Estimation accuracy of adhesion force among the wheel and rail is quite beneficial for the solution of slip/slide problem.The aim of the controller is to maximize the adhesion coefficient while sustaining the overall stability of the railway vehicle.The slip velocity is adjusted by keeping it in the stable region adjacent to the peak point of the adhesion coefficient.To achieve the sub-optimal slip velocity,which corresponds to the approximity of the maximum value of the adhesion coefficient,is also suitable for the controller performance.The relationship between the tractive force and the wheel slip is a nonlinear function of the wheel–rail contact condition.For this reason,an estimator that estimates the wheel–rail adhesion condition and a traction controller which adjusts the wheel slip velocity to the optimal value should be added to the control scheme.Several control methods have been proposed based on sliding mode controller,fuzzy logic,adaptive schemes in the literature[13–15].

In this paper,electric multiple units consisting of selfpropelledcarriageswereusedduetothesuperiorityfromthe viewpointof tractive effort.The railway vehicle is driven by 120 kWinductionmotorswitha750 VDCbusvoltagewhich arecontrolledusingthefield-orientedcontrol(FOC)method withanindirectvectorcontrolscheme.Arobustadaptivereadhesioncontrolscheme toregulatethecontroltorqueofthe traction motor was designed for not only suppressing the excessive wheel slip,which make the system unstable,but also optimizing the wheel slip to operate at the optimal adhesion point even if some system parameter uncertainties or disturbances exist.The adhesion coefficient and slip velocity were predicted through the second-order sliding mode observer.Optimal slip velocities at different wheel–railcontactconditionswerefoundwithahigherprecisionvia the guidance of the optimal slip velocity seeking algorithm.By creating a Lyapunov candidate function,it was proved that the ultimate boundedness of the proposed strategy with time-varying disturbance and finite-time convergence of the proposed method were satisfied.The traction capability was also maintained after recovering adhesion.A steep grade railwaywaschoseninthesimulationstotesttheperformance ofthecontrollerinlowadhesionconditions.Onsteepgrades,the friction between wheel and rail is not able to transfer sufficient adhesion to overcome gravity.

这几年来，眼睁睁看着房价攀升，易非不是没动过买房的心，只是，她觉得自己的钱似乎应该有更大的用途，她希望有朝一日能给向南开家影楼，或者成为一笔小小的原始资本，她甚至比向南更渴望功成名就，她从来没有忘记过要替爸爸活着，要为爸爸扬眉吐气。可这一次，敏之姐的话，真的在她心里留下涟漪了。

The organization of the paper is as follows.Section 2 introduces the dynamic model of the longitudinal traction system that is used in the control system scheme.The proposed robust adaptive re-adhesion control strategy is described in Sect.3.Section 4 gives the details of the optimal slip velocity estimator based on the second-order sliding mode observer,and Sect.5 presents the simulation results of the robust adaptive approach.Finally,conclusions are provided in the last section.

2 Traction system dynamics model

The slip mechanism should be modeled to construct a slip control strategy.Therefore,a traction system dynamics model with the traction motor reference torque as input and the forward speed of the railway vehicle as output is built,which concentrates on two fundamental parts,i.e.,the mechanical transmission and the outer condition.The input and output of the mechanical transmission part are the traction motor torque and angular velocities of wheels,respectively.The outer condition part produces the actual railway vehicle forward speed,which depends on the angular velocities of the wheels,actual adhesion level and various losses due to air resistance,rolling resistance,curve resistance,gradient resistance,etc.[16].The mechanical transmission consists of a traction motor,a gearbox and two wheels which are coupled with each other through a wheelset axle and a brake system.The mechanical transmission model is demonstrated in Fig.1.

The output torque of the traction motor(Tm,k)is derived as

where V is the forward speed of the railway vehicle;the slip velocity in the longitudinal direction de fined as in Eq.(10):

where Jm,kis the moment of inertia of the traction motor,ωm,kis the motor angular velocity,Tm,kis the traction motor input torque,Tt,kis the transmission output torque,the subscript k={1,2,…,l}is the serial number of the wheelset to which the traction motor are mounted,and l is the total number of traction shafts in the train.

（3）胜利油田山东地区的油藏资源所在地地质形态多变，有“地质大观园”的称号。当前除主要工作地区之外，胜利油田把工作重点转移到了寻找隐蔽油气藏方面，整个山东省隐蔽油气藏的储量十分可观，然而由于其所在地地质条件复杂，使寻找和开发这些油气贮藏点都变得十分困难，再加上相关技术的不足，影响了油气产量的增加。

By substituting Eq.(20)into Eq.(22),one obtains

where n is the total number of wheelsets in each bogie and m is the number of the bogies.Usually,l=nm-2.In our model,each bogie has two wheelset(n=2),and the train is assumed to have three bogies(m=3).The leading and trailing bogies are equipped with traction motors,while the wheelsets in the middle bogie have no traction motor.Thus,there exist four traction motors in total,which are equivalent to four traction shafts.The output torque of the traction motor is transmitted to the gearbox by a shaft,as stated in Eq.(3).The torque is then transmitted to the left and the right wheels via the wheelset axle.The distances from the gearbox to the left and right wheels are different;therefore,the spring constants and damping coefficients of the shaft on two sides are also different.Here,considering that the part of wheelset axle from the gearbox to right wheel is very short when compared with that to the left wheel,we neglect the spring constant and damping coefficient of the shaft on the motor side and the loss due to viscous friction that emerges in the gearbox.The gearbox scales the input torque and speed through a gear ratio(nk).

where the superscripts R and L refer to the right and left wheels,respectively;Jg,kis the moment of inertia of the gearbox;TRw，kandTLw，kare the traction torques of the right and left wheels,respectively;and t is time.

综上所述，妊娠糖尿病高危孕妇极容易产生巨大儿、胎膜早破、胎儿窘迫和早产等不良妊娠结局。为了防范不良妊娠结局产生，医务人员应加强对产妇各个妊娠阶段的综合管理，为其提供科学的饮食、运动干预指导，促进高危孕妇合理膳食，形成母婴安全的良好前提。

1.3 参与物质能量代谢 肠道微生物群通过糖酵解途径、磷酸戊糖途径、糖类厌氧分解途径进行糖类的代谢，生成短链脂肪酸，为宿主提供能量，为肠道菌群生长繁殖提供营养物质[2]。肠道菌群在代谢过程中产生大量的小分子多肽和蛋白质，这些物质在宿主细胞和宿主的共生微生物之间起到分子信号传递的重要作用[3-5]。

The drive shaft torque transmitted to the left wheel is given by

where Kkis the spring constant;Bkis the damping coef ficient of the wheelset axle;θLw，kand ωLw，kare the angular displacement and angular velocity of the wheelset axle to the left wheel,respectively.The wheels transmit the traction torque to the rail,and the amount of torque transmitted depends on the adhesion level between the wheel and rail,as represented in Eqs.(5)and(6).

whereis the peripheral velocity of the right or left wheel through the contact patch.Note that the superscript‘‘R,L’’of some variables throughout this paper means that the quantity is for the right or left wheel,and the related equation holds not only for the right wheel but also for the left wheel.

where JwR，kand JwL，kare the moment of inertia of the right and left wheels,respectively;ωRw，kis the angular velocity of the right wheel;Jx,kand Jbr，kare the moments of inertia of the wheelset axle and brake system,respectively;TaR，kand TaL，kare the adhesion torques of the right and left wheels,respectively:

where FaR，kand FaL，kare the adhesion forces of the right and left wheels,respectively;rwR，kand rwL，kare the effective radii of the right and left wheels,respectively.

To determine the adhesion force,the dynamic property of outer conditions must be taken into consideration as well.The maximum value of the adhesion mainly depends on the adhesion coefficient and the adhesion weight of the railway vehicle,which vary with time and vehicle’s position on the track.The adhesion coefficient is a function of the slip velocity,conditions of the wheel–rail contact,railway vehicle velocity and temperature in the contact area[17].The forces and velocities that act on a single wheel are depicted in Fig.2.

While the motor torque and the adhesion torque are adopted as input in this traction system model,the railway vehicle velocity and angular speed of the wheel are taken as output.The slip ratio,which is defined as the ratio of the relative velocity between wheel and rail in the contact area to the forward speed of the railway vehicle,is calculated for each driven wheel–rail interface as follows[18]:

7．统计学处理：采用SPSS 22.0统计软件进行数据处理。呈正态分布的计量资料以表示，组间比较采用单因素方差分析，组间两两比较采用SNK-q检验。P<0.05为差异有统计学意义。

全过程全方位的德育工作者。思政课教师要全过程全方位地参与高职学生的德育工作，真正成为学生的良师益友，而不能把自己仅仅定位于课堂教育教学完成者角色。

Slip generation is a highly nonlinear process,which varies continuously over time and can be modeled as the adhesion coefficient–slip characteristic curve depicted in Fig.3.This function can be separated into three phases.In the first phase,the adhesion coefficient increases linearly with the slip velocity;in the second part,however,the adhesion coefficient presents a nonlinear relationship with the slip velocity.In both of the two phases,the function characteristic between adhesion coefficientand slip velocity is basically stable,and hence,the range of adhesion coefficient or slip velocity covering the two phases is regarded as a stable region[17].At the endpoint of the second phase,the adhesion coefficient reaches its maximum value(μa，max),which corresponds to the optimal slip velocity(ν*s).In other words,the adhesion saturates at theIf the traction torque remains steady when the current adhesion force strongly decreases,the characteristic function switches into the third phase,which is called as a slip region or an unstable region[19].The main objective of the re-adhesion or slip control is to drive the slip velocity to the stable region of the characteristic function and to maintain the optimal slip velocity,which comes across to the maximum adhesion coefficient for best tractive effort.

In Fig.4,adhesion slip characteristics of the wheel–rail contact surface in different conditions such as ‘‘dry,’’‘‘wet,’’‘‘low’’and ‘‘very low’’[20]are shown,which are used for evaluation and simulation process of this paper.

The tractive effort of the railway vehicle is evaluated as a sum of partial tangential forces of all driven shafts whose wheels are subjected to different adhesion coefficients and adhesion weights.To compute the slip velocity,it is necessary to know the angular velocity of thethe railway vehicle longitudinal velocity(V).The angularis obtained directly from the mechanical transmission model.

两权分离度与企业技术创新实证研究——兼论产权性质与业绩偏离的调节效应..................................................................................................................................刘 玉 盛宇华（60）

In this model,the vehicle velocity is actually evaluated from the adhesion coef ficientsystem.Given the adhesion coef ficientwheel,the adhesion forceswheels individually as in Eqs.(11)and(12):

不同于慧理财，微量网作为策略提供者与用户的中介桥梁，主要产品除了提供股票策略外，还包括期货策略、组合策略。平台主要采用网页和移动端APP相结合的方式，为用户提供专业的量化投资服务和量化策略。用户需要注册微量网账户，购买策略，进行交易，并与股票或者期货等账户进行绑定，才可真正实现最终交易。

whereare the adhesion forces of the left and right wheels, respectively; g is the gravitationalare adhesion coefficients of the left and right wheels,respectively;adhesion masses of the left and right wheels,respectively;and

where mstcis the total static mass of the railway vehicle,mvbis the railway vehicle body mass,mb，iis the mass of the bogie,mm，kis the mass of the motor,mg，kis the mass of the gearbox,mx，ijis the wheelset axle mass,mbr，ijis the mass of the brake system,mw，ijis the wheel mass,the subscript j={1,2,…,n}refers to the wheelset axle number in one bogie,and the subscript i={1,2,…,m}refers to the bogie number.

The adhesion torque for each traction wheelset is described as

The adhesion torque is fed back into the mechanical transmission as described in Eqs.(7)and(8).There is also anotherway to calculate the adhesion torque by approaching from the traction motor side,as represented in Eqs.(14)and(15).

where Jeqv,kis the equivalent moment of inertia of the wheelset projected into the motor side:

To evaluate the adhesion coefficient and analyze the wheel slip ratio by comparing with the wheel velocities,the velocity of the railway vehicle should be estimated.The railway vehiclevelocity isevaluated based on the Newton’s second law of motion as in Eq.(16)[21,22]:

在五代时期出现了“把”“将”同句出现，和连用形成复合介词“把将”的这种语言现象。说明“把”的处置义表现的更为明显了，把字完全虚化，完成了由动词到介词的转变。

where mdynis the dynamic(effective)mass,which refers to the particular drive shaft to be accelerated:

η is the gear efficiency,for each wheel and is calculated as in Eq.(17):

and is the nominal wheel radius and defined as in Eq.(18):

FLis the sum of various outer losses,as represented in Eq.(19)[23]:

where Fair，Froll，Fcrvand Fgrvdenote,respectively,the losses due to air resistance,roll,cornering and the angle of the lateral slope on which the railway vehicle is running.

The loss due to roll depends on the dynamic mass of the vehicle and the current vehicle velocity:

where Cr1and Cr2are vehicle specific parameters related to the wheel characteristic.

The cornering loss is originated from the increased friction between the wheel and rail while the railway vehicle is moving along a curve.It is calculated through an empirical formula,called as Ro¨ckl formula[23,24]:

where Ryis the radius of the track curvature;k1and k2are parameters which change with the radius of track curvature as

The air and roll resistance forces can be combined into the propulsion force using Davis formula,evaluated as follows[24]:

为此，国际社会应当在网络反恐的国际立法上加强沟通和交流，积极推动联合国层面出台网络反恐国际合作的决议或宣言，为其他组织和区域性网络反恐合作提供指导。区域性组织的国家之间因为地缘、利益等相近因素，可以最大限度的在反恐问题上达成共识，也可以考虑先行在其框架内建立网络反恐合作的法律机制。如，可以将上合组织网络反恐合作演习的实践以法律形式固定下来，形成成员国之间开展网络反恐合作的固有机制，切实增强网络反恐合作的实效。

where Ap，Bp，Cpand Dpare resistance coefficients;Avis front section area of the vehicle;and Wx，kis the wheelset axle load,defined as

(4)烘干转筒内有专门的烟气通道管，沸腾炉烟气首先通过烘干转筒中间的中空管到达烘干转筒的尾部，然后又从烘干转筒尾部沿烘干转筒内圆四周布置的管道返回转筒头部，经排烟口排出。物料不与烟气接触，只与灼热的烟气管接触，不会造成二次污染；

where κ0, κ1and κ2are nonnegative constants.

where ζ is the lateral slope angle of the rail.

In order to build a realistic model,the values of parametersoughtto be determined accurately.The derivative of slip velocitystated as in Eq.(20):

where the moment of inertia Jdynis equal to mdyn¯r2.According to Eq.(20),one can deduce that the longitudinal slip velocity depends on the traction motor torque,adhesion torque and resistance torque.The slip velocity must be limited by designing a re-adhesion controller to prevent the unstable operation region due to a sudden increase in the slip velocity and must be driven to the safe and optimal operation region.The re-adhesion control can be achieved provided that the asymptotic tracking in the sense of νs，k →denotes the optimal slip velocity in the stable region of the kth wheelset.Therefore,in this paper,a robust adaptive nonlinear control with a secondorder sliding mode observer which satisfies the optimal tracking performance goal under some parameter uncertainty and unmodeled dynamic conditions is proposed.

3 Proposed control strategy

The primary objective of the re-adhesion control methodology is to adjust the slip velocity νs，kvia controlling the traction motor torque under the assumption that all signals are restricted.

In the first step of the controller design,the slip velocity tracking error(es)is defined as

The traction motor torque is controlled by considering a task driving the slip velocity tracking error convergence to the small neighborhood of zero value as time goes to infinity[25].The time derivative of the tracking error is taken as in Eq.(22):

The following inequality is always valid for all railway vehicle types:

If the exact value of the Ψ(·)is known,the slip velocity tracking can be achieved perfectly.

On the other hand,the adhesion coefficient of the railway vehicle changes with the wheel–rail interaction condition.The adhesion torque is full of uncertainty and behaves nonlinearly.The resistance force is also timevarying,and its certain value cannot be acquired.Therefore,a robust adaptive control strategy is proposed to handle the effect of Ψ(·)indirectly.This problem can be solved by bounding Ψ(·)determination of adhesion torque Ta，kand the resistance force( F L)should abide by the following inequality:

A scalar function φ(·)is defined as in Eq.(25):

Although Ψ(·)is composed of nonlinear,uncertain and time-varying terms,it can be bounded by multiplication of Eq.(25)and an unknown positive constant κ,as follows:

Sincedesign,˙L(·)≤0 is guaranteed.Thus,it can be deduced that the re-adhesion control strategy is stable.

The loss due to the lateral slope angle of the rail is described as follows:

Equation(26)is independent of any system parameters or operating condition.For this reason,the control strategy provides a highly robust and adaptive control scheme even if the adhesion conditions or system parameters of railway vehicle shift.The robust adaptive tracking control is generated by

in which k0＞0 is a free parameter,and^κ is the estimate of κ,obtained using the following adaptive algorithm:

where σ0is a positive constant for adaptation rate;then asymptotically stable trajectory tracking is fulfilled in that es→0 as t→∞[26].

From Eqs.(23)and(26),one can derive

There always exists a positive constant λ which satisfiesInspecting the Lyapunov function candidate stated as below:

we know that L(·)is a positive definite function and its derivative with respect to time is

where p and q are unknown and positive constants,and κ can be selected as in Eq.(27).

经过统计，当乐视网股价跌幅出现极端情况时，GARCH-VaR模型对股价实际损失的预测会失败。在90%的置信水平下，股价跌幅达到8.07%左右时，模型预测失败。在95%的置信水平下，股价跌幅达到8.84%左右时，模型预测失败。在99%时置信水平下，股价跌幅达到9.67%左右时，模型预测失败。模型预测失败说明风险不可控，所以当股价的跌幅较大时，为了控制投资风险，应及时止损，卖出股票。

4 Optimal slip velocity seeking method

To realize the proposed control strategy,foreknowledge about the optimal slip velocity must be acquired,but in the practical applications the optimal slip velocity cannot be accessible.There is no opportunity to get all the adhesion relationships under various railway conditions.Both of the adhesion model and the optimal slip velocity are unknown in real-time applications.For this reason,an optimal slip velocity seeking method must be developed to estimate the exact value of the optimal slip velocity[27].

According to the characteristic curve presented in Fig.3,if one knows that

The proposed estimator of optimal

where γ＞ 0 is an adaptable parameter,tkis the switching point in which the values of the estimated slip velocity^νs are switched to the current value of νsaccording to Eq.(35):

in which∈is a small threshold value to evade numerical drawbacks.Based on the estimate^νsof ν*s,Eq.(36)can be obtained:

For the estimated valueinformation ofwhich can be retrieved by utilizing the second-order sliding mode observer.The sliding mode observer method was explained in detail in the literature[29–32].The observer can be constructed independently from the controller information.

亡羊补牢，为时未晚。如今，党中央号召全面复兴中华优秀传统文化，可谓“扶正固本”之举。当此之际，不仅是“中国大妈”，全体中国人，都应该像习近平总书记在十九大报告中所说的：“从家庭做起，从娃娃抓起，深入挖掘中华优秀传统文化蕴含的思想观念、人文精神、道德规范，结合时代要求，继承创新，让中华文化展现出永久魅力和时代风采。”这是对我们每个中国人提出的要求。在中国人的传统文化中，“母亲”总是温暖慈爱、舍己为人的代名词，但不知从何时起，“中国大妈”代替了这种美好形象而受到歧视。那些以居高临下的傲慢和以偏概全的逻辑来随意抹黑“中国大妈”的人，想想你们自己的母亲，于心何忍！

5 Simulation results

The mechanical transmission,re-adhesion controller and the optimal slip velocity estimator were implemented in MATLAB and Simulink,and the simulations were carried out by using the Runge–Kutta fourth-order method with a fixed step time of 0.0001 s.It is assumed that a wheel slip was experienced after the rail condition changes suddenly in traction mode.The parameters used in simulation are listed in ‘‘Appendix 1.’’

The adhesion coefficient is related to the slip velocity as follows[33,34]:

where a,b,c and d are friction model parameters which are designed with respect to the wheel–rail contact condition,and their values are listed in ‘‘Appendix 2.’’Initial states of the railway vehicle velocity and wheel angular velocity are given as

It is assumed that the estimated value of slip velocity switches every 10 s.The trajectories of the railway vehicle forward speed and wheel speed are given in Fig.5,and trajectories of the adhesion coefficient are presented in Fig.6.

The optimal slip velocities change with the adhesion conditions.The adhesion conditions switch every 10 s,and the controller works until the optimal slip velocities for each condition are achieved.It can be seen that the value of the wheel speed changes suddenly at t=10 s and t=20 s,which correspond to the switching points of the estimated value of slip velocity.The adhesion coefficient decreases abruptly and wheel slip emerges because of the rail conditions change from ‘‘dry to wet,’’‘‘dry to low’’and‘‘wet to low’’at t=10 s and from ‘‘wet to low,’’‘‘wet to very low’’and ‘‘low to very low’’at t=20 s,respectively.The wheel velocity become much faster than the vehicle velocity,and the slip velocity goes from the safety limit to the dangerous area gradually.Afterward,the wheel slippage is detected and the re-adhesion control strategy regulates the control traction motor torque to attenuate the wheel slip and drives the railway vehicle at the optimal slip velocity.

Figure 7 shows the traction motor torque control trajectories.There exists a saturation point in the motor traction torque at 20 s,but the effect time of the peak is very short and can be negligible.Consequently,the adhesion coefficient and speed values are not affected by the sudden increase in the motor torque.

The trajectories of the actualvelocities,as shown in Fig.8,converge to some value nearslip velocities.It can be seen from the results that the actual slip velocities are able to track the optimal slip velocities with a low error band.Optimal slip velocity values are 1.1841 m/s for the dry condition,1.1852 m/s for the wet condition,1.1864 m/s for the low condition and 1.3272 m/s for the very low condition.The root mean square(RMS)values of the deviations of the actual slip velocities from the optimal slip velocities,which refer to the slip velocity tracking errors,and the RMS values of the deviations of the estimated slip velocities from the actual slip velocities,which refer to the slip velocity estimation errors,are also shown under the given wheel–rail contact conditions in Table 1.Both of the estimator and controller exhibit the worst performance when the wheel–rail contact condition switches from ‘‘dry to very low.’’The estimator achieves its best performance in the adhesion condition switching from ‘‘dry to low,’’whereas the controller performs the best in the switching from ‘‘dry to wet.’’

The whole adhesion performance is illustrated in Fig.9.The actual slip velocity can precisely track the optimal slip velocity through the adhesion characteristic map.The full detail of adhesion coefficient varying in the re-adhesion control is captured.The adhesion coefficient converges to a neighborhood of the maximum adhesion value for the‘‘dry’’condition in the initial acceleration phase;after that it suddenly declines to the safe region in the ‘‘wet,low or very low’’condition adhesion characteristic and converges to the maximum adhesion value of this characteristic function.By using a re-adhesion control,the adhesion coefficient remains in the safe operation region and high traction ability is guaranteed.The stability of the re-adhesion control is preserved through the full simulation case from the view point of the Lyapunov,and this is also verified in Fig.10.

6 Conclusion

A robust adaptive re-adhesion control scheme with secondorder sliding mode observer-based Lyapunov theorem was presented for the railway vehicles under the existence of nonlinear and uncertain parameters.The main superiority of this kind of control strategy is that it does not necessitate exact information of system parameters and is versatile to maintain high traction performance under subsequently switching adhesion conditions as well.The adaptive control scheme and optimal slip estimator,which are built with the second-order slidingmode observer,canovercomeallofthe drawbacks in the studies in the existing literature.Four different wheel–rail conditions are modeled,and the railway vehicleadhesionisexaminedduringsimulationbyswitching these conditions sequentially.Through the optimal slip velocity seeking method,the railway vehicle adhesion can rapidly converge to the optimal slip velocity even if the wheel–rail contact condition degenerates abruptly.The proposed method is validated both theoretically and numerically in the simulation by high performance and robustness criteria.Simulation results show that this kind of robust adaptive approach provides a rapid response to suddenly changing adhesion conditions and driver requests while maximizing the adhesion utilization even if poor adhesion conditions exist.It was proven that this control strategy could solve possible stability problems of the wheel acceleration-based strategies.

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License(http://creativecommons.org/licenses/by/4.0/),which permits unrestricted use,distribution,and reproduction in any medium,provided you give appropriate credit to the original author(s)and the source,provide a link to the Creative Commons license,and indicate if changes were made.

References

1.Hahn JO,Rajamani R,Alexander L(2002)GPS-based real-time identification of tire-road friction coefficient.IEEE Trans Control Syst Technol 10(3):331–343

2.Malvezzi M,Vettori G,Allotta B,Pugi L,Ridolf iA,Rindi A(2013)A localization algorithm for railway vehicles based on sensor fusion between tachometers and inertial measurement units.Proc Inst Mech Eng F J Rail Rapid Transit 228(4):431–448

3.Mirabadi A,Mort N,Schmid F(1996)Application of sensor fusion to railway systems.In:IEEE/SICE/RSJ international conference on multisensor fusion and integration for intelligent systems(Cat.No.96TH8242)

4.Yuan J,Zheng Y,Xie X,Sun G(2011)Driving with knowledge from the physical world.In:The 17th ACM SIGKDD international conference on knowledge discovery and data mining,KDD’11,New York,NY,USA,ACM

5.Mei TX,Li H(2008)A novel approach for the measurement of absolute train speed.Veh Syst Dyn 46(Supp.1):705–715

6.Geistler A,Bohringer F(2004)Robust velocity measurement for railway applications by fusing eddy current sensor signals.In:IEEE intelligent vehicles symposium

7.Zhao Y,Liang B(2013)Re-adhesion control for a railway single wheelset test rig based on the behaviour of the traction motor.Veh Syst Dyn 51(8):1173–1185

8.Yasuoka I,Henmi T,Nakazawa Y,Aoyama I(1997)Improvement of re-adhesion for commuter trains with vector control traction inverter.In:Proceedings of power conversion conference—PCC

9.Wang S,Xiao J,Huang J,Sheng H(2016)Locomotive wheel slip detection based on multi-rate state identification of motor load torque.J Franklin Inst 353(2):521–540.https://doi.org/10.1016/j.jfranklin.2015.11.012

10.Shimizu Y,Ohishi K,Sano T,Yasukawa S(2010)Anti-slip/slid re-adhesion control based on disturbance observer considering bogie vibration.Electr Eng Jpn 172(2):37–46

11.Kadowaki S,Ohishi K,Miyashita I,Yasukawa S(2006)Antislip/skid re-adhesion control of electric motor coach based on disturbance observer and sensor-less vector control.EPE J 16(2):7–15

12.Sado H,Sakai S,Hori Y(1999)Road condition estimation for traction control in electric vehicle.ISIE 99.In:Proceedings of the IEEE international symposium on industrial electronics(Cat.No.99TH8465).https://doi.org/10.1109/isie.1999.798747

13.Borelli F(2003)Constrained optimal control of linear and hybrid systems,LNCIS,vol 290.Springer,Berlin,pp 185–196

14.Hu JS,Yin D,Hu FR(2011)A robust traction control for electric vehicles without chassis velocity.In:Soylu S(ed)Electric vehicles—modelling and simulations,InTech.https://doi.org/10.5772/16942

15.Watanabe T(2000)Anti-slip readhesion control with presumed adhesion force,Method of presuming adhesion force and running test results of high speed Shinkansen train.Q Rep Railw Tech Res Inst 41(1):32–36

16.Zobory I,Bekef iE(1995)On real-time simulation of the longitudinal dynamics of trains on a specified railway line.Period Polytech Ser Transp Eng 23(1):3–18

17.Pichlik P,Zdenek J(2014)Overview of slip control methods used in locomotives.Trans Electr Eng 3(2):38

18.Ward CP,Goodall RM,Dixon R,Charles GA(2012)Adhesion estimation at the wheel–rail interface using advanced modelbased filtering.Veh Syst Dyn 50(12):1797–1816

19.Ohyama T(1990)Tribological studies on adhesion phenomena between wheel and rail at high speeds.Mech Fatigue Wheel/Rail Contact 263–275.https://doi.org/10.1016/b978-0-444-88774-0.50021-31

20.Kumar S,Krishnamoorthy PK,Rao DL(1986)Wheel–rail wear and adhesion with and without sand for a North American locomotive.J Eng Ind 108(2):141

21.Cole C(2006)Longitudinal train dynamics.In:Iwnicki S(ed)Handbook of railway vehicle dynamics.CRC,Boca Raton

22.Genin J,Ginsberg JH,Ting EC(1974)Longitudinal track–train dynamics:a new approach.J Dyn Syst Meas Contr 96(4):466

23.Steimel A(2008)Electric traction—motive power and energy supply:basics and practical experience.Oldenbourg Industrieverl,Mu¨nchen

24.Matthews V(2011)Railway construction,8th edn.Springer.https://doi.org/10.1007/978-3-8351-9107-5

25.Lu K,Song Y,Cai W(2014)Robust adaptive re-adhesion control for high speed trains.In:17th International IEEE conference on intelligent transportation systems(ITSC)

26.Landau YD,Lozano R,MSaad M,Karimi A(2011)Adaptive control.Algorithms,analysis and applications,2nd edn.Springer,London,590 pp.https://doi.org/10.1007/978-0-85729-664-1

27.Kawamura A,Takeuchi K,Furuya T,Cao M,Takaoka Y,Yoshimoto K(2004)Measurement of tractive force and the new maximum tractive force control by the newly developed tractive force measurement equipment.Electr Eng Jpn 149(2):49–59

28.Chen Y,Dong H,Lu J,Sun X,Guo L(2016)A super-twistinglike algorithm and its application to train operation control with optimal utilization of adhesion force.In:IEEE transactions on intelligent transportation systems,pp 1–10

29.Solvar S,Le V,Ghanes M,Barbot J,Santomenna G(2010)Sensorless second order sliding mode observer for induction motor.In:2010 IEEE international conference on control applications

30.Mabrouk WB,Belhadj J,Pietrzak-David M(2011)Sliding mode observer for mono-inverter bi-motors railway traction system.In:Eighth international multi-conference on systems,signals and devices

31.Davila J,Fridman L,Levant A(2005)Second-order sliding-mode observer for mechanical systems.IEEE Trans Autom Control 50(11):1785–1789.https://doi.org/10.1109/tac.2005.858636

32.Gokasan M,Bogosyan O,Sabanovic A,Arabyan A(1998)A sliding mode observer and controller for a single link flexible arm.In:Proceedings of the 37th IEEE conference on decision and control(Cat.No.98CH36171)

33.Takaoka Y,Kawamura A(2000)Disturbance observer based adhesion control for Shinkansen.In:Proceedings of the 6th international workshop on advanced motion control,Nagoya,Japan,pp 169–174

34.Kawamura A,Furuya T,Takeuchi K,Takaoka Y,Yoshimoto K,Cao M(2002)Maximum adhesion control for Shinkansen using the tractive force tester.In:IEEE 28th annual conference of the industrial electronics society.IECON 02

35.Yamazaki H,Karino Y,Nagai M,Kamada T(2005)Wheel slip prevention control by sliding mode control for railway vehicles(experiments using real size test).In:Proceedings,IEEE/ASME international conference on advanced intelligent mechatronics