Research Article  Open Access
Optimal Utilization of Adhesion Force for HeavyHaul Electric Locomotive Based on Extremum Seeking with Sliding Mode and Asymmetric Barrier Lyapunov Function
Abstract
An optimal utilization of adhesion force based on extremum seeking with sliding mode (SMES) and asymmetric barrier Lyapunov function (ABLF) is proposed for heavyhaul electric locomotives (HHELs), which can eliminate the wheel skidding at optimal adhesion point and achieves maximum traction for HHELs. First, the state equation of wheelrail adhesion control system is described. The optimal utilization of adhesion force and antislip control are analyzed considering the condition changes at the wheelrail surface. Then, the nonsingular terminal sliding mode observer (NTSMO) is designed to achieve the accurate adhesion coefficient of the wheelrail. Finally, the SMES method for HHEL is developed to obtain the optimal slip speed and the maximum adhesion coefficient of the uncertain wheelrail surface. Meanwhile, the ABLF controller is designed to achieve antislip 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 antislip control with quick dynamic response, and the HHEL achieves maximum traction.
1. Introduction
Heavyhaul 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 BoBo+BoBo 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 wheelrail. A cellular automata model is suggested and realized multiobjective 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 wheelrail 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 wheelrail 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 wheelrail 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 timevarying 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 forceslip rate curve [13, 14]. To enhance the dynamic and static performance of the ESC scheme, a novel fast ESC scheme without steadystate 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 antislip 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 stateconstrained 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 antislip 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 antislip 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 timevarying parameters of wheelrail. By constructing an antislip 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 wheelrail mathematical model of the HHEL is described in Section 2. Considering the wheelrail 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 antislip problem at the optimal adhesion point, the antislip 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. WheelRail Mathematical Model for the HHEL
The locomotive model is made up of two 4axis HDX1F locomotives. The locomotive has a single axle weight of 30 ton. The wheelrail model is composed of three parts, traction motors, gear box, and wheelrail [8]. The simplified wheelrail 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 wheelrail adhesion is shown in Figure 2. The adhesion coefficient is used to characterize the complex mechanical relationship between the wheel and the rail. The microdeformation region occurs in the wheelrail 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 wheelrail 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 wheelrail surface conditions are shown in Figure 3 [24, 25]. The conclusion can be drawn as follows:(i)The maximum adhesion coefficients exist in different wheelrail 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 wheelrail 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 wheelrail surface.
During the traction operation of the HHEL, the nonsingular terminal sliding mode adhesion coefficient observer (NTSMO) is designed to obtain the realtime observation value of the wheelrail 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 wheelrail maximum adhesion coefficient and obtains the optimal slip speed .
The unit of antislip 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 realtime adhesion coefficient of the wheelrail 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 fullscale 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 realtime adhesion coefficient of the wheelrail [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 concludeCase 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 AntiSlip Controller Based on ABLF
The antislip 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 antislip range of and is a normal number to be designed. The error is expressed as . The antislip 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 antislip 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 antislip region . If and only if , if , and if , and are continuous. is continuous and positive in the entire antislip 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 antislip boundary, or , . According to Barbalat Lemma [31], the error will be constrained in antislip interval .
According to (42), one obtains
Due to and , and then yields
By designing , it is ensured that the HHEL has good antislip 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 antislip 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, wheelrail model of the HHEL, the adhesion coefficient nonsingular terminal sliding mode observer (NTSMO), Extremum seeking with sliding mode (SMES) and ABLF antislip 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 wheelrail surface changes are simulated. The simulated wheelrail surface gradually deteriorates, the 05s wheelrail surface is dry, and the 510s wheelrail surface is damp. 1015s rail surface is rain and snow. The running resistance coefficient is , , and . The parameters of SMES are designed as , , and . The gain of ABLF antislip 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 wheelrail 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 threedimensional 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 antislip controller. The conventional sliding mode antislip 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 antislip 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 wheelrail.
5.2. Experiments
The real locomotive operating environment is difficult to rely on simulation soft. This work uses the RTlab experimental platform to conduct semiphysical experiments. Figure 16 shows the RTLab experimental platform used in this work. The C program code of the SMES optimal adhesion utilization and ABLF antislip 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 RTLab 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 wheelrail 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 wheelrail. 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 wheelrail parameter uncertain based on SMES and ABLF. The wheelrail adhesion coefficient can be achieved precisely by the designed nonsingular terminal sliding mode observer (NTSMO). The optimal adhesion point of the uncertain wheelrail surface can be searched online based on the designed extremum seeking with sliding mode (SMES) method. The designed ABLF controller can achieve antislip control for HHELs in the optimal adhesion state. The proposed method can achieve optimal adhesion antislip 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 RTlab 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.
References
 Y. Hu, “Current status and development trend of technology system for railway heavy haul transport in China,” Zhongguo Tiedao Kexue/China Railway Science, vol. 36, no. 2, pp. 1–10, 2015. View at: Google Scholar
 J. Li, Y. Hu, H. Peng, and L. Liu, “Key techniques and design methods of adhesion control in rail transportation,” Electric Drive for Locomotives, no. 6, pp. 1–5, 2014. View at: Google Scholar
 Y. Yao, H.J. Zhang, Y.M. Li, and S.H. Luo, “The dynamic study of locomotives under saturated adhesion,” Vehicle System Dynamics, vol. 49, no. 8, pp. 1321–1338, 2011. View at: Publisher Site  Google Scholar
 P. E. Orukpe, X. Zheng, I. M. Jaimoukha, A. C. Zolotas, and R. M. Goodall, “Model predictive control based on mixed H2/H∞ control approach for active vibration control of railway vehicles,” Vehicle System Dynamics, vol. 46, no. 1, pp. 151–160, 2008. View at: Publisher Site  Google Scholar
 H.J. Chuang, C.Y. Ho, C.S. Chen, C.S. Chen, C.H. Lin, and C.H. Hsieh, “Design of Optimal Coasting Speed for MRT Systems Using ANN Models,” IEEE Transactions on Industry Applications, vol. 45, no. 6, pp. 2090–2097, 2009. View at: Publisher Site  Google Scholar
 H. Peng, H. Chen, Y. Zeng et al., “Simulation of adhesion control method based on differential acceleration,” Electric Drive for Locomotives, no. 2, pp. 2627, 2010. View at: Google Scholar
 G. Xu, K. Xu, C. Zheng, and T. Zahid, “Optimal operation point detection based on force transmitting behavior for wheel slip prevention of electric vehicles,” IEEE Transactions on Intelligent Transportation Systems, vol. 17, no. 2, pp. 481–490, 2016. View at: Publisher Site  Google Scholar
 K. Zhao, P. Li, C. Zhang, J. He, Y. Li, and T. Yin, “Online Accurate Estimation of the WheelRail Adhesion Coefficient and Optimal Adhesion Antiskid Control of HeavyHaul Electric Locomotives Based on Asymmetric Barrier Lyapunov Function,” Journal of Sensors, vol. 2018, Article ID 2740679, 12 pages, 2018. View at: Publisher Site  Google Scholar
 C. Zhang and R. Ordonez, Extremumseeking control and applications  A Numerical OptimizationBased Approach, Springer Science and Business Media, London, UK, 2011. View at: Publisher Site  MathSciNet
 H. S. Tsien, Engineering Cybernetics, McGrawHill, New York, NY, USA, 1954.
 S. Drakunov and U. Ozgune, “Optimization of nonlinear system output via sliding mode approach,” in Proceedings of the in IEEE International Workshop on Variable Structure and Lyapunov Control of Uncertain Dynamical System, pp. 6162, Sheffield, UK, 1992. View at: Google Scholar
 B. Zuo, Y. Hu, and J. Shi, “Research and development of extremum seeking algorithm,” Journal of Naval Aeronautical and Astronautical University, vol. 21, no. 6, pp. 611–617, 2006. View at: Google Scholar
 B. Zhou, M. Xu, X. Yuan, and L. Fan, “Acceleration slip regulation based on extremum seeking control with sliding mode,” Journal of Mechanical Science and Technology, vol. 46, no. 2, pp. 307–311, 2015. View at: Google Scholar
 B. Zhou, Y. Li, X. Yuan, X. Hu, and Y. Cheng, “Sliding mode extremumseeking algorithm for control of abs with vehicle lateral stability improvement,” Journal of Vibration and Shock, vol. 35, no. 4, pp. 121–126, 2016. View at: Google Scholar
 L. Wang, S. Chen, and K. Ma, “On stability and application of extremum seeking control without steadystate oscillation,” Automatica, vol. 68, pp. 18–26, 2016. View at: Publisher Site  Google Scholar  MathSciNet
 E. Dincmen, “Extremum seeking control of uncertain systems,” TWMS Journal of Applied and Engineering Mathematics, vol. 7, no. 1, pp. 131–141, 2017. View at: Google Scholar  MathSciNet
 T. Hara and T. Koseki, “Study on readhesion control by monitoring excessive angular momentum in electric railway tractions,” in Proceedings of the 12th IEEE International Workshop on Advanced Motion Control (AMC '12), pp. 1–6, IEEE, Sarajevo, Bosnia and Herzegovina, March 2012. View at: Publisher Site  Google Scholar
 M. Yamashita and T. Soeda, “Antislip readhesion control method for increasing the tractive force of locomotives through the early detection of wheel slip convergence,” in Proceedings of the 17th European Conference on Power Electronics and Applications, EPEECCE Europe '15, pp. 1–10, Geneva, Switzerland, 2015. View at: Google Scholar
 C. Wenchuan, L. Danyong, and L. S. Yongduan, “A novel approach for active adhesion control of highspeed trains under antiskid constraints,” IEEE Transactions on Intelligent Transportation Systems, vol. 16, no. 6, pp. 3213–3222, 2015. View at: Publisher Site  Google Scholar
 X. Chen, Z. Dai, H. Lin, Y. Qiu, and X. Liang, “Asymmetric Barrier Lyapunov FunctionBased Wheel Slip Control for Antilock Braking System,” International Journal of Aerospace Engineering, vol. 2015, Article ID 917807, 10 pages, 2015. View at: Publisher Site  Google Scholar
 Q. Ren, Design and simulation of heavy haul locomotives and trains, [Master thesis], Southwest Jiaotong University, Chengdu, China, 2014.
 K. Zhao, T. Chen, C. Zhang, J. He, and G. Huang, “Sensorless and torque control of IPMSM applying NFTSMO,” Yi Qi Yi Biao Xue Bao/Chinese Journal of Scientific Instrument, vol. 36, no. 2, pp. 294–303, 2015. View at: Google Scholar
 M. Spiryagin, P. Wolfs, C. Cole, and V. Spiryagin, Design and Simulation of Heavy Haul Locomotives and Trains, CRC Press, Florida, FL, USA, 2016.
 O. Polach, “Creep forces in simulations of traction vehicles running on adhesion limit,” Wear, vol. 258, no. 78, pp. 992–1000, 2005. View at: Publisher Site  Google Scholar
 J. Li, H. Chen, Z. Zhang, J. Hu, and L. Xu, “Deload algorithm for highly effective spressing locomotive slip and slide,” Electric Drive for Locomotives, vol. 6, pp. 6–10, 2015. View at: Google Scholar
 C. Zhang, J. Sun, J. He, and L. Liu, “Online Estimation of the Adhesion Coefficient and Its Derivative Based on the Cascading SMC Observer,” Journal of Sensors, vol. 2017, Article ID 8419295, 11 pages, 2017. View at: Publisher Site  Google Scholar
 C. Zhang, Y. Huang, and R. Shao, “Robust sensor faults detection for induction motor using observer,” Control Theory and Technology, vol. 10, no. 4, pp. 528–532, 2012. View at: Publisher Site  Google Scholar  MathSciNet
 P. Pichlík and J. Zdênek, “Extended Kalman filter utilization for a railway traction vehicle slip control,” in Proceedings of the 2nd International Conference on Optimization of Electrical and Electronic Equipment, OPTIM 2017 and Intl Aegean Conference on Electrical Machines and Power Electronics, ACEMP '17, pp. 869–874, Brasov, Romania, May 2017. View at: Google Scholar
 W. Lin, Z. Liu, and Y. Fang, “Readhesion optimization control strategy for metro traction,” Xinan Jiaotong Daxue Xuebao/Journal of Southwest Jiaotong University, vol. 47, no. 3, pp. 465–470, 2012. View at: Google Scholar
 Y.M. Wang, Y. Feng, H.W. Xia, and L.Q. Shen, “Smooth nonsingular terminal sliding mode control of uncertain multiinput systems,” Kongzhi yu Juece/Control and Decision, vol. 30, no. 1, pp. 161–165, 2015. View at: Google Scholar
 J. A. Gallegos, M. A. DuarteMermoud, N. AguilaCamacho, and R. CastroLinares, “On fractional extensions of Barbalat Lemma,” Systems & Control Letters, vol. 84, pp. 7–12, 2015. View at: Publisher Site  Google Scholar
Copyright
Copyright © 2019 Kaihui Zhao et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.