Online Estimation of the Adhesion Coefficient and Its Derivative Based on the Cascading SMC Observer

The adhesion coefficient of wheel-rail surface cannot be directly measured, so a cascaded sliding-mode observer is proposed to observe adhesion coefficient and its derivative. A kinetic model for running heavy-duty locomotive is also established. The state equation of wheel adhesion control system is derived from the equation of traction motor torque balance, and adhesion coefficient is proposed to be calculated by load torque. Then, the cascaded sliding-mode observer is designed, and its stability is justified by Lyapunov stability. Based on the equivalence control principle for sliding-mode variable structure, an algorithm to estimate adhesion coefficient and its derivative is established. The simulation and experimental results are used to verify the effectiveness of the observer with load variations or wheel-rail status changes.


Introduction
Traction power of heavy-duty locomotives has improved along with the rapid development of railway transportation industry.However, the traction power of locomotives is limited by wheel-rail adhesion.The actual value of locomotive wheel-rail adhesion depends on the utilization of locomotive adhesion, which is generally expressed by adhesion coefficient.The wheel-rail adhesion is highly nonlinear and varies rapidly with time because of changes in rails and creep speed.Moreover, the adhesion coefficient cannot be measured directly [1][2][3].Smart sensors are a crucial part of automatic detection and control owing to their functions of supervision and diagnosis.High measuring accuracy, quick response, working stability, and data analysis and state judgment functions are all required for sensors to guarantee safety and reliability of locomotives in motion [4].Therefore, a smart sensor is necessary to supervise and diagnose the adhesion coefficient to maintain optimal adhesion.
Adhesive control algorithms can be divided into two types according to the maximum adhesion point tracking: readhesion control and optimized adhesion control [5].
The readhesion control type [6] adjusts the motor output torque through the driver's handle to restore adhesion after idling, and obtaining a higher adhesion utilization is difficult.Meanwhile, the optimal adhesion control type is mainly the creep speed and the adhesion characteristic curve slope algorithms [7][8][9].The creep speed algorithm adjusts the adhesion coefficient repeatedly to maximize the adhesion coefficient, but it requires the measurement of the locomotive speed.Thus, its implementation is difficult.The adhesion characteristic curve slope algorithm is used to let the locomotive work at the peak after obtaining the adhesion characteristic curve in real time.Obtaining the derivative of the adhesion coefficient to the creep speed in this algorithm is difficult.To solve this problem, the adhesive derivative algorithm is proposed to estimate the actual adhesion coefficient and its derivative based on the load torque and its derivative obtained by designing the observer.Takaoka et al. [10,11] and Teramoto et al. [12] proposed observing the load torque with a reduced-dimension observer.The algorithm is simple and can satisfy the basic requirements of the system.However, it is susceptible to noise-signal interference because of its direct differential processing when obtaining the adhesion coefficient derivative.Wenli et al. [13] and Liao et al. [14] introduced an omnidirectional observer for the motor load torque and its derivative, thereby improving the system's immunity.The algorithm is based on the assumption that the traction motor load torque does not change.However, ensuring that the motor load torque remains constant with changes in track condition, creep speed, and so on during the actual running of the locomotive is difficult.
The variable structure sliding-mode control (SMC) is robust in the presence of model uncertainties, parameter variations, or external disturbances.The SMC can effectively reduce the dependence on any accurate system model, and its application in the field of state observers has widened [15][16][17].This study aims to develop a smart sensor based on the cascaded sliding-mode observer to handle the complex working conditions of locomotive.The sensor determines the load torque and its derivative and then estimates the adhesion coefficient and its derivative.
This paper is structured as follows: the second part describes the wheel-rail kinetic model in the locomotive running process.The third part presents the cascading SMC observer designed to observe the adhesion coefficient and its derivative.The fourth part introduces the simulation analysis.The fifth part provides the conclusions.

Description of the Wheel-Rail System
The wheels roll forward under the action of driving torque  when the train is running.The wheels relative to the tracks produce creep as a result of elastic deformation on the wheelrail contact surface.They also form an adhesive force on the contact surface, which is the only power for driving the wheels forward, when rolling. Figure 1 shows the simplified uniaxial wheel-rail model.
The creep speed is defined as follows [18]: Obtaining the locomotive speed V  more accurately is hard.The adhesion coefficient is defined as follows [19]:  The pulling motor motion equation is as follows [20]: The kinetic equation of the wheel set is ( The train motion equation is presented as follows: where V  is the wheel speed,   is the adhesion traction,   is the wheel-rail driving force,  is the locomotive mass,  is the axle load,   is the equivalent moment of the motor inertia,  is the moment of inertia of the wheel set,   is the electromagnetic torque of the motor,   is the motor load torque,  is the driving torque,  is the wheel set radius,   is the gearbox ratio, and   (V  ) is the drag.Figure 2 shows an adhesion characteristic curve, which indicates that the creep speed increases with the traction when the train is running in the creep region.The adhesion coefficient also increases until a peak is reached.The train traction enters a nonstable section when the train traction required exceeds the maximum adhesion force of the wheelrail.The adhesion coefficient within the section sharply decreases with higher creep speed.Moreover, the adhesion force of the wheel also reduces, hence the train idling.Locomotive is prone to wheel set idling at two transitions: (1) locomotive suddenly travels from dry high-adhesion railwheel surface to low-adhesion rail-wheel surface and (2) driver pulls up the control handle of locomotive, causing the traction beyond maximum available traction.
The peak point of the adhesion in the locomotive traction process is /V  = 0, with the stable section /V  > 0 on the left side and the idling section /V  < 0 on the right side (Figure 2).
Based on the following: the load torque   can be estimated as follows to determine the peak point of the adhesion coefficient.Using ( 9), By selecting the state variables,  1 = ,  2 = , and  3 =   , where  is the displacement angle of the motor and  is the motor speed, Ṫ is assumed as a bounded variable.Among them, only  1 is measurable.The following state equation can be obtained as follows using (3): In literature [9,10], the following state equation is often obtained based on the hypothesis that the rate of change in the motor load torque is 0: However, in actual locomotive run, change rate in motor load torque being zero is difficult to ensure.

Design of Load Torque Observer.
As for object Ṫ ̸ = 0 described in formula (11), only  1 is measurable, and load torque is observed using a cascaded sliding-mode observer as follows: The SMC item is as follows: where x1 , x2 are the observed values of  1 and  2 ,  1 ,  2 are constants to be determined, and sgn(⋅) is a sign function.
One defines the observer bias: where  1 can be obtained from  1 and x1 .And because  2 is unknown, therefore,  2 can not be directly obtained.
Step 1. Selecting the Lyapunov function  1 = (1/2) 1 2 and considering (18), the derivative of time is When  1 is large enough, and  1 is an arbitrary positive constant.Then, The system meets the conditions for the SMC.Accordingly, the system will reach the SMC plane within a limited time.After reaching the SMC plane and based on the principle of equivalence of the SMC [21], Using (18), Step 2. In selecting the Lyapunov function  2 = (1/2) 2 2 +  1 , the derivative of time is obtained as follows: Using ( 19) and ( 22), When  2 is large enough, and  2 is an arbitrary positive constant; then, Likewise, the following is derived using (19): where  2 is derived from (24).This completes the proof.

Design of Load Torque Derivative Observer.
As for object Ṫ ̸ = 0 described in formula (11), only  1 is measurable, and load torque derivative is observed using a cascaded slidingmode observer as follows: The SMC item is as follows: where ẑ1 , ẑ2 , and ẑ3 are the observed values of  1 ,  2 , and  3 .Meanwhile,  1 ,  2 , and  3 are constants to be determined, and sgn(⋅) is a sign function.
The state estimation error is defined as where  1 can be obtained from  1 and ẑ1 .And because  2 and  3 are unknown, therefore,  2 and  3 can not be directly obtained.
Step 1. Selecting the Lyapunov function  3 = (1/2) 1 2 and considering (36), the derivative of time is When  1 is large enough, and  1 is an arbitrary positive constant.Then, The system meets the conditions for the SMC.Accordingly, the system will reach the SMC plane within a limited time.After reaching the SMC plane and based on the principle of equivalence of the SMC [21], Using (36), Step 2. In selecting the Lyapunov function  4 = (1/2) 2 2 + 3 , the derivative of time is obtained as follows:  Using (37) and (41), and  2 is an arbitrary positive constant; then, Likewise, the following is derived using (37): where  2 is derived from (43).
The SMC-based observer algorithm of the adhesion coefficient and its derivative is summarized as follows: (1) Load torque observers and its derivative are established by formulas ( 14) and (30), respectively.(2) By (24),  2 can be estimated.
(5) By (48),  3 can be estimated.The sign function sgn(  ) is replaced with the relay method, sgn(  ) =   /(|  | +   ), to solve the chattering problem of the SMC observer. 1 is a very small positive constant.
Hence, the adhesion coefficient  and its derivative u can be obtained, with the corresponding SMC observer structured as Figure 3.    Case 2. Simulation environment for locomotive is always in a dry state, and initial value of   is zero, with an increase by 2000 N⋅m/s.Figures 8 and 9 show wheel-rail adhesion coefficient and adhesion coefficient derivative, respectively.In the figures, red solid, blue-dotted, and black-dotted lines represent actual value, observed value from load torque observer, and observed value from derivative observer, respectively.As shown in Figures 8 and 9  implement the proposed approach, experiments have been carried out on an OP5600 RT-LAB platform using HILS.The RT-LAB platform is shown in Figure 12, and the configuration is shown in Figure 13.The observer is a TMS320F2812 digital signal processor, which implements high-performance control and computation.The motor and wheel-rail system are simulated by RT-LAB.The sampling period is chosen as 50 s, which is the same as the SMO.The semiphysical simulation results are shown in Figures 14-17.

Simulation and Experiment
Case

Conclusions
Obtaining the wheel-rail adhesion state is extremely hard.This study designs a method to estimate the adhesion   coefficient and its derivative online based on the cascading SMC observer.Mathematical model of wheel-rail is obtained by analyzing wheel-rail adhesion mechanism, which is proposed to estimate adhesion coefficient by load torque.The derivative of the torque load in the existing algorithms is required to be 0; that is, Ṫ = 0.However, in the practical   operation of locomotives, this is difficult to guarantee because of the changing track conditions and creep speed.In the present study, this assumption is relaxed; that is, it should only be assumed as a bounded quantity.The derivative of the load torque is estimated using the principle of equivalence in SMC.The chattering problem of the SMC observer is then solved.The algorithm proposed in this study has a simple structure with high robustness and is easy to implement.By means of simulation and experiment, the observed values are consistent with the actual values, which confirms the algorithm validity

Figure 2 :
Figure 2: Characteristic curve of the adhesion.

Figure 3 :
Figure 3: System diagram of the derivative observer of the adhesion coefficient.

Figure 13 :
Figure 13: The structure of RT-LAB hardware-in-the-loop system.
[22,23]f complex dynamic systems.HILS differs from pure real-time simulation and RCP by the use of the real controller in the loop; this controller is connected to the rest of the system simulated by RT-LAB.Therefore, in HILS, the plant is simulated, but the controller is real[22,23].To 1.Experiment environment for locomotive starts with dry track surface; at 2.5 s, locomotive travels from dry to wet rail surface.Figures14 and 15are the experimental waveforms under the change of the rail surface.In Figure14, the oscilloscope shows the adhesion coefficient; in Figure15, the oscilloscope shows the adhesion coefficient derivative, CH3 channel shows the actual value, and CH1 channel shows the observed value.The rail surface changes from dry to wet state in 2.5 s, the locomotive running into the skid region.Figures16 and 17are experimental waveforms under the change of traction force.In Figure16, the oscilloscope shows