#### Abstract

An optimal utilization of adhesion force based on extremum seeking with sliding mode (SMES) and asymmetric barrier Lyapunov function (ABLF) is proposed for heavy-haul electric locomotives (HHELs), which can eliminate the wheel skidding at optimal adhesion point and achieves maximum traction for HHELs. First, the state equation of wheel-rail adhesion control system is described. The optimal utilization of adhesion force and anti-slip control are analyzed considering the condition changes at the wheel-rail surface. Then, the nonsingular terminal sliding mode observer (NTSMO) is designed to achieve the accurate adhesion coefficient of the wheel-rail. Finally, the SMES method for HHEL is developed to obtain the optimal slip speed and the maximum adhesion coefficient of the uncertain wheel-rail surface. Meanwhile, the ABLF controller is designed to achieve anti-slip control for HHELs in the optimal adhesion state. Comparing with the conventional differential acceleration control (DAC) method, the simulations and experiments verify that the proposed method can achieve optimal adhesion anti-slip control with quick dynamic response, and the HHEL achieves maximum traction.

#### 1. Introduction

Heavy-haul electric locomotives (HHELs) are widely used in railway freight due to its high tractive forces. Taking HXD1F as an example, the HXD1F locomotives are twin unit Bo-Bo+Bo-Bo vehicles, that is two locomotives with 4 axles each. The model has a single axle weight of 30 tons. Single locomotive traction power reaches 9600kW [1]. However, the traction power of the HHELs is limited by the adhesion force between the rail and the wheel. Given that wheel–rail adhesion is affected by temperature, humidity, and surface condition, the adhesion force is nonlinear, uncertain, and time varying [2, 3].

To utilize locomotive traction power effectively, it is necessary to study a control strategy for the HHELs to achieve the maximum adhesion force. This paper proposes a new scheme for the HHELs to optimal utilize of adhesion Force, and the locomotive running at the optimal adhesion point.

The main purpose of optimal adhesion control algorithm is to find the maximum adhesion point to utilize the traction force as much as possible. In recent years, many control methods have been proposed to seek the optimal adhesion point of the wheel-rail. A cellular automata model is suggested and realized multi-objective optimization of automatic train operation system in [4]. An artificial neural network is proposed to determine the optimal coasting speed of train operation for the mass rapid transit system in [5]. Peng et al. seek the optimal adhesion point of wheel-rail surface by calculating the differential value of adhesion force based on differential acceleration control (DAC) method [6, 7]. However, the DAC’s disadvantage is unable to completely eliminate the skid phenomenon, and the effect of wheel-rail adhesion disturbances will amplify by differential parts. A train trajectory optimization approach is investigated for mass rapid transit systems in [8]. Most of the above works focus on the optimal adhesion utilization for the locomotive. However, the nonlinearity and parameter disturbances of the wheel-rail is not taken into consideration intensively.

Extreme seeking control (ESC) is a class of control methods, which can autonomously find an optimal system behavior [9]. Extremum seeking with sliding mode (SMES) algorithm was proposed by Davila et al. based on sliding mode control and extreme seeking control theory [10–12]. The algorithm can stabilize a class of systems with optimal parameter uncertain and fast time-varying parameter disturbance, which has applied to many real engineering problems. A new control algorithm of antilock braking system was proposed base on the SMES algorithm to seek the peak point of the tire braking force-slip rate curve [13, 14]. To enhance the dynamic and static performance of the ESC scheme, a novel fast ESC scheme without steady-state oscillation is proposed in [15]. Extreme seeking algorithm is applied to search a maximum or minimum point of the desired behavior or performance for the systems with parametric uncertainties [16]. Furthermore, the anti-slip problem of the wheel–rail at the optimal adhesion state must be considered carefully. The readhesion control strategies were proposed by monitoring the significant fault signal in [17, 18]. The disadvantage of this readhesion control approach is hard to predict the skid phenomenon completely. Recently, the state-constrained control method based on asymmetric barrier Lyapunov function (ABLF) has been paid attention to improve the stability of nonlinear dynamic systems [19]. An aircraft landing control system is designed based on ABLF to achieve unilateral anti-slip constraints and shorten the braking distance of the aircraft [20]. The adaptation of ABLF guarantees that the HHEL operated at a stable region and the optimal adhesion anti-slip control of HHEL is achieved [8]. Inspired by the above work, this paper proposes a new scheme of optimal utilization of adhesion force for the HHEL to achieve the optimal adhesion force based on SMES and ABLF, which can overcome the effects of time-varying parameters of wheel-rail. By constructing an anti-slip controller based on ABLF and SMES, the optimal utilization of adhesion force for the HHEL can be guaranteed effectively.

The remainder of this paper is organized as follows: the wheel-rail mathematical model of the HHEL is described in Section 2. Considering the wheel-rail parameter uncertain, the optimal adhesion point seeking method is proposed based on SMES algorithm in Section 3, and the ultimate convergence is proved. Considering the anti-slip problem at the optimal adhesion point, the anti-slip controller is designed based on ABLF algorithm in Section 4. Compared with the conventional differential acceleration control (DAC) method, simulations and experiments are conducted to verify the effectiveness of the proposed method in Section 5, and Section 6 concludes this paper.

#### 2. Wheel-Rail Mathematical Model for the HHEL

The locomotive model is made up of two 4-axis HDX1F locomotives. The locomotive has a single axle weight of 30 ton. The wheel-rail model is composed of three parts, traction motors, gear box, and wheel-rail [8]. The simplified wheel-rail model is shown in Figure 1. The simplified locomotive’s traction equipment transmits the traction torque to the wheelsets through the gearboxes and drives the wheel such that it rotates at a speed of . During the traction operation process, wheel speed is always greater than vehicle speed . The adhesion force between the wheel and the rail drives locomotive to move forward [21].

With the damping coefficient ignored, the equation of traction motor is as follows [22]:where is the electromagnetic torque of the traction motor ; is the load torque ; is the rotor angular velocity ; is the angular velocity of the wheel ; is the radius of the wheel ; is the moment of inertia of the traction motor ; is the adhesion force between the wheel and the rail ; is the speed ratio of the gear box, .

The mechanical dynamics of the HHEL with eight axles is defined as follows [8, 21]:where is the total quality of both the locomotive and load . is the axle’s adhesion force of the traction motor . is the sum of resistance forces in the course of locomotive operation , and , , and are resistance coefficients, which are positive constants.

The schematic of the wheel-rail adhesion is shown in Figure 2. The adhesion coefficient is used to characterize the complex mechanical relationship between the wheel and the rail. The micro-deformation region occurs in the wheel-rail contact region when the wheels rotate with the axle load [23]. The adhesion force is produced by the interaction between the wheel and the rail.

The adhesion force and the adhesion coefficient have the following relation:

The adhesion coefficient is not only related to the wheel-rail surface but also constrained by creep velocity . The following empirical equations are derived based on numerous experimental [2]:where , , , and are the contact’s constants, which are positive constants.

State variable is chosen as follows:The dynamic model of the HHEL is as follows according to (1):where , , and . is the measurable parameters of HHELs and , is an unknown load disturbance. But is a bounded quantity, and .

The adhesion characteristic curves under three different wheel-rail surface conditions are shown in Figure 3 [24, 25]. The conclusion can be drawn as follows:(i)The maximum adhesion coefficients exist in different wheel-rail surface, which exhibit the same trends.(ii)The adhesion coefficient increases to with the increase of in the stable region and then decreases sharply to zero with the increase of creep after crossing the optimal creep velocity .(iii)In order to improve the utilization of adhesion force and utilize the locomotive traction motor power effectively, it is necessary to accurately obtain optimal adhesion point of the wheel-rail surface, namely, the maximum adhesion coefficient and the optimal creep velocity .

#### 3. Optimal Adhesion Point Seeking Based SMES

Figure 4 shows the schematic of the optimal adhesion SMES method for uncertain wheel-rail surface.

During the traction operation of the HHEL, the nonsingular terminal sliding mode adhesion coefficient observer (NTSMO) is designed to obtain the real-time observation value of the wheel-rail adhesion coefficient , using the torque and the rotor speed of the traction motor. According to the adhesion coefficient observation value , the unit of extremum seeking with sliding mode iterates the wheel-rail maximum adhesion coefficient and obtains the optimal slip speed .

The unit of anti-slip controller uses the slip speed signal to give the torque command to control the locomotive, which can make optimal utilization of adhesion force, and the traction motor outputs the maximum traction force.

##### 3.1. Nonsingular Terminal Sliding Mode Adhesion Coefficient Observer

Obtaining the real-time adhesion coefficient of the wheel-rail surface is a prerequisite for the extremum seeking with sliding mode method [26]. However, the adhesion coefficient is difficult to measure with general tools. Common processing methods include full-scale state observers, Kalman Filters [27–29].

This paper proposes a nonsingular terminal sliding mode observer (NTSMO) to observe the load torque of the traction motor, and obtain the real-time adhesion coefficient of the wheel-rail [22]. According to (6), the state variables is selected as

The observer is structured as follows: where the variable represents the observed value of and is the control input vector of observer.

The state estimation error is defined as follows:

Differentiating (9) with respect to time, one obtains

In order to improve the observation accuracy, and reduce the chattering phenomenon of the traditional sliding mode observer, the nonsingular terminal sliding mode observer (NTSMO) is used to observe adhesion coefficient. This paper proposes the following nonsingular terminal sliding mode surface combining the traditional sliding mode surface with the nonsingular terminal sliding mode surface [30]:where , , , and are the parameters to be designed, which are positive constants, and , are odd number, .

Theorem 1. *For the system (7) and the observer (8), if the nonsingular terminal sliding mode surface (11) is selected and the control law (12) is designed, the observation error will convergence to zero in finite time, for any initial value is satisfied .*

*Proof. *The Lyapunov function is selected to beDifferentiating (13) with respect to time, one obtainswhere and .

Substituting (10) into (15), yieldsWhere satisfies , the following can be obtained:According to the Lyapunov stability criterion and the sliding mode reachability condition, the error gradually converges to zero in a finite time.

This completes the proof.

*Remark 2. *Once the system reaches the sliding surface, it is available according to the sliding mode equivalence principle:Substituting (18) into (10), (19) then will yieldAccording to (1) and (19), the observed adhesion coefficient can be obtained as follows:

##### 3.2. Seeking Maximum Adhesion Point by SMES

The extremum seeking with sliding mode (SMES) [16] can search the maximum adhesion coefficient using the adhesion coefficient observation value. Then, the optimal creep speed is obtained.

The specific design steps are as follows.

*Step 1. *The adhesion coefficient is selected as the input variable and the sliding surface is designed as follows:where is monotonically increasing at the rate of .

The time derivative of (21) is

*Step 2. *Sliding mode adaptive law is designed as follows:where and are positive constants, and . The symbol function is shown in Figure 5.

*Step 3. *Substituting (20) into (12) yields

*Remark 3. *The schematic of extremum seeking with sliding mode principle (SMES) is shown in Figure 6, and the search process is divided into two phases. (i)The first phase is to make move from or to by SMES. When it arrives at or points, gradually becomes smaller. The extreme seeking conditions are no longer satisfied. Then, the searching process enters the second phase.(ii)When it arrives at and , converges to the maximum adhesion point by the vibration integral effect at the second phase.

Theorem 4. *Assume that the system (21) meets the following: will gradually converge to the corresponding sliding surface or and , where is a constant determined by the initial .*

*Proof. *Assume that the initial value is satisfied as follows:The adaptive law of (23) has the following form:The slid region and stable region are analyzed as follows:*Case **1*. When the creep speed is in the skid region, that is, can make the following equivalent transformation:Defining variables yieldsThe convergence of is analyzed as follows:Hence, one can easily conclude that will gradually converge to 0 with , and one can conclude*Case **2*. When the creep speed is in the stable region, that is,According to the same analysis as (29)-(31), one can easily conclude that will gradually converge to 0 with , and one can concludeThis completes the proof.

*Remark 5. *When , according to formula (25), the following are obtained:The following conclusions are available:(i)If and , is established and then goes to the optimal adhesion point from the right.(ii)If and , is established and then goes to the optimal adhesion point from the left.

In Figure 6, and represent the critical point of . Once , according to (24), the following features exist:

In this case, there is a monotonically decreasing function. There are two deceleration rates:

The derivative of creep speed is shown in Figure 7. Due to , monotonically decreasing. In the process of decrementing, suppose that the total time spent by in Area I is and the total time spent in area II is .

In segment and , the deceleration rate of is faster in area II than area I. This means that the entire search time , therefore,Therefore, increases in segment , and it reaches the peak point .

Similarly, in the segment , decrements, and it also reaches the peak point.

*Remark 6. *Assuming that the condition of (25) is satisfied and the initial value is at , will converge to the corresponding sliding surface of or . Once the convergence is completed, . Whether is located in the left half or right half of the extreme point , the creep speed will eventually converge in the neighborhood of the optimal adhesion point .

Once the extreme seeking is completed, the tracked optimal creep speed is obtained.

#### 4. Design of the Anti-Slip Controller Based on ABLF

The anti-slip control must prevent creep speed from entering the skid region of the adhesion characteristic curves in Figure 3 [2].

According to (6), the state variables are chosen as

The state estimation error is defined as follows:

Differentiating (42) with respect to time, one obtains

Assume that the locomotive is allowed to have an anti-slip range of and is a normal number to be designed. The error is expressed as . The anti-slip control for HHEL is actually the problem of constraint control of the creep speed. In Figure 8, the adhesion curve can be divided into two parts: the stable region and the skid region .

In view of the different anti-slip constraints of the locomotive’s stable region and skid region, the following asymmetric barrier Lyapunov function (ABLF) is designed:

The switching function has the following definition:

Assume that the creep speed is in the allowable anti-slip region . If and only if , if , and if , and are continuous. is continuous and positive in the entire anti-slip region.

The following control laws are designed:where are the positive constants to be designed.

Theorem 7. *For the nonlinear system described in (41), if the initial error , the error will gradually converge to 0 and will always be constrained in the interval during the process of convergence.*

*Proof. *The skid region and stable region are analyzed by flowing two cases.*Case **1*. When the creep speed is in the stable region, that is,The time derivative of (47) isThe control law becomes :Substituting (49) into (48) yieldsIt can be seen in the interval , , and . If and only if , .*Case **2*. When the creep speed is in the skid region, that is,The time derivative of (51) isThe control law becomes The same analysis process is as (50). Therefore, in the interval , and . If and only if .

This completes the proof.

*Remark 8. *In the interval , is continuous positive definite function and is negative and continuous. When the error is close to the anti-slip boundary, or , . According to Barbalat Lemma [31], the error will be constrained in anti-slip interval .

According to (42), one obtains

Due to and , and then yields

By designing , it is ensured that the HHEL has good anti-slip performance during tracking the given optimal creep speed by SMES.

The proposed optimal adhesion control method for HHEL based on SMES and ABLF is as follows:(i)The nonsingular terminal sliding mode observer (NTSMO) is designed to observe the adhesion coefficient by (20).(ii)The observed adhesion coefficient is input to the SMES, the optimal creep speed signal is obtained by (40).(iii)The estimated creep speed is input to the ABLF anti-slip controller, and the given torque command is calculated according to (46). The optimal utilization of adhesion force for the HHEL can be guaranteed.

#### 5. Simulation and Experiment

Figure 9. shows the diagram of the model of optimal utilization of adhesion force for HHEL based on SMES and ABLF. The simulation model is divided into four parts, namely, wheel-rail model of the HHEL, the adhesion coefficient nonsingular terminal sliding mode observer (NTSMO), Extremum seeking with sliding mode (SMES) and ABLF anti-slip controller. Table 1 presents the simulation parameters of HHEL, and Table 2 shows the parameters of adhesion characteristics of three different rail surfaces of dry, wet, and snow [2, 8].

In order to verify the robustness of the proposed method in this paper, the simulated wheel-rail surface changes are simulated. The simulated wheel-rail surface gradually deteriorates, the 0-5s wheel-rail surface is dry, and the 5-10s wheel-rail surface is damp. 10-15s rail surface is rain and snow. The running resistance coefficient is , , and . The parameters of SMES are designed as , , and . The gain of ABLF anti-slip controller is designed as follows: , , and .

##### 5.1. Simulations

The simulation waveforms are shown in Figures 10–15. Figure 10 shows the observed values and actual values of the adhesion coefficient observed by the nonsingular terminal sliding mode observer. The observed value tracks the actual value of the adhesion coefficient at 0.025s. At the time of 5s and 10s, the observer maintains a good observation accuracy and achieves the desired observation target.

**(a) NTSMO**

**(b) FDSO**

**(a) SMES method**

**(b) DAC method**

**(a) SMES method**

**(b) DAC method**

**(a) SMES method**

**(b) DAC method**

**(a) SMES method**

**(b) DAC method**

**(a) Sliding mode controller**

**(b) ABLF controller**

In order to verify the performance of the proposed SMES and ABLF algorithm, the comparing simulation is conducted with the differential acceleration control (DAC) method. Figure 11 shows the wheel-rail speed and locomotive speed of the sliding mode extreme seeking algorithm and the DAC method. Figure 12 shows the creep speed of the sliding mode extreme seeking algorithm and the DAC method. From Figures 11 and 12, it can be seen that the extreme seeking unit accurately searches the best slip speed of the dry track at 0.2 (m/s) in the neighborhood after a search time of 1(s). When the rail surface switches to wet at 5(s), extremum seeking with sliding mode unit according to the adhesion coefficient observed value changes and the slip speed search value stabled in 0.08(m/s) after 0.8(s). When the rail surface is switched to snow rail surface, the sliding mode extreme unit still maintains good search results. In Figure 12(b), differential modules in the DAC method cause a sharp fluctuation in the slip speed and the search speed is slow at the start. Figure 13 shows the waveforms of the adhesion coefficients of the two search algorithms. The sliding mode extreme seeking algorithm has a faster response time.

Figure 14 shows a three-dimensional map of the adhesion characteristic. The locomotive starts from point A on the dry rail surface, the adhesion state moves toward the best adhesion point B, and the adhesion coefficient is finally in . Then, the rail surface switched into the wet rail surface. The adhesion state moves toward the optimal adhesion point C, and the adhesion coefficient is . Finally, the rail surface switched into the snow rail surface. The adhesion point moves toward the adhesion point D. Eventually, the adhesion coefficient is in. In Figure 14(b), the traditional DAC method has a sharper fluctuation due to the creep speed, which results in poor stability of the adhesion characteristic curve.

Figure 15(a) is the output torque waveform of the conventional sliding mode control anti-slip controller. The conventional sliding mode anti-slip control algorithm has larger torque fluctuations in output traction torque. Figure 15(b) is the output torque waveform of the ABLF controller designed in this paper. The output torque waveform of the ABLF anti-slip controller is stable and the amplitude is small, which effectively reduces the loss during locomotive operation and is beneficial to prolong the life cycle of the wheel-rail.

##### 5.2. Experiments

The real locomotive operating environment is difficult to rely on simulation soft. This work uses the RT-lab experimental platform to conduct semiphysical experiments. Figure 16 shows the RT-Lab experimental platform used in this work. The C program code of the SMES optimal adhesion utilization and ABLF anti-slip controller is downloaded to the DSP controller TMS320F2812 using a PC. Then, the HHEL model is compiled into OP5600, and the HILS of the optimal adhesion utilization system of the HHEL can be realized. Figure 17 presents the RT-Lab HILS configuration diagram of the SMES optimal adhesion control system of the HHEL. The experimental results are shown in Figures 18–21.

**(a) NTSMO**

**(b) FDSO**

**(a) SMES method**

**(b) DAC method**

**(a) SMES method**

**(b) DAC method**

**(a) Sliding mode controller**

**(b) ABLF controller**

In Figure 18, the experimental results of the conventional FDSO and NTSMO under the condition of wheel-rail parameter uncertain are shown. By contrast, the proposed NTSMO method exhibits a faster response speed then the conventional FDSO. Figure 19 shows the creep speed seeking response comparisons of the conventional DAC method and SMES method under rapid changes of wheel-rail. Adhesion coefficient is based on DAC method and SMES methods as shown in Figure 20. The experimental results indicate that the SMES method can improve the robustness against parameter uncertain. Figure 21 shows the torque response comparisons of the sliding mode controller and ABLF controller. The ABLF control strategy clearly shows good control precision in the steady state and avoids the chattering of the sliding mode control method.

#### 6. Conclusions

This paper proposes a novel optimal utilization of adhesion force for HHEL considering wheel-rail parameter uncertain based on SMES and ABLF. The wheel-rail adhesion coefficient can be achieved precisely by the designed nonsingular terminal sliding mode observer (NTSMO). The optimal adhesion point of the uncertain wheel-rail surface can be searched online based on the designed extremum seeking with sliding mode (SMES) method. The designed ABLF controller can achieve anti-slip control for HHELs in the optimal adhesion state. The proposed method can achieve optimal adhesion anti-slip control, and the HHEL achieves maximum traction. It ensures the safety while realizing the optimal utilization adhesion Force for HHEL.

#### Data Availability

The data of the locomotive and adhesion conditions used to support the findings of this study are included within the article. The model data of matlab and RT-lab used to support the findings of this study are available from the corresponding author upon request.

#### Conflicts of Interest

The authors declare that they have no conflicts of interest.

#### Acknowledgments

This work was supported in part by the Natural Science Foundation of China under Grant nos. 61773159 and 61503131, in part by the Hunan Provincial Natural Science Foundation of China under Grant nos. 2018JJ2093, 2018JJ4066, 2017JJ4031, and 2016JJ5012, in part by the Scientific Research Fund of the Hunan Provincial Education Department under Grant no. 17B073, and in part by the Key Laboratory for Electric Drive Control and Intelligent Equipment of Hunan Province Grant no. 2016TP1018.