Abstract
This paper proposes a new scheme of online accurate estimation of wheel-rail adhesion coefficient and optimal adhesion antiskid control of heavy-haul electric locomotives (HHEL) based on sliding mode and asymmetric barrier Lyapunov function (ABLF) theory. To achieve optimal adhesion control of the HHEL, it is necessary to precisely estimate the wheel-rail adhesion coefficient. However, the adhesion coefficient is difficult to be measured with a conventional physical sensor. The first novelty of this paper is to design a smart adhesion coefficient sensor based on sliding mode observer (SMO). The perception of the adhesion coefficient is transformed into the observation of load torque of the traction motors, and the wheel-rail adhesion coefficient is further calculated by using the load torque observed value. The HHEL achieves maximum traction from operating in the optimal adhesion point. However, wheel skidding is most likely to occur at this point. According to the changing trend of the adhesive coefficient characteristic curve, the operating state of a locomotive can be divided into two regions: the stable and skid regions. The second novelty of this paper is the adaptation of ABLF to guarantee that the HHEL operated at a stable region and the optimal adhesion antiskid control of HHEL is achieved. Finally, the simulation and experimental results verify the feasibility and effectiveness of the proposed method.
1. Introduction
Heavy-haul electric locomotives (HHEL) that are widely used in railway freight are characterized by high tractive forces. However, the effective use of traction power of the HHEL is limited by the wheel-rail adhesion conditions, which is affected by temperature, humidity, and surface condition. To achieve the optimal adhesion control required for the HHEL, it is necessary to precisely estimate the wheel-rail adhesion coefficient. But the adhesion coefficient is difficult to be measured with a conventional physical sensor [1]. Advanced sensor is one of the essential components of the HHEL control system. The high-precision sensor can accurately monitor, quickly feedback, and be insensitive to external disturbances, which effectively improves the performance of the control system [2]. Therefore, an advanced adhesion coefficient sensor is necessary to enable the HHEL to achieve optimal adhesion control and exert maximize traction.
Recent studies have focused on obtaining the adhesion coefficient of the locomotive. The method of full-state observer is presented to observe the adhesion coefficient for subway trains in [3]. However, the feedback gain matrix has a large effect on the accuracy of the observer. The Kalman filter method is used to estimate the locomotive adhesion coefficient in [4], which requires a larger data capacity.
In practice, the observation of the adhesion coefficient can be converted into the estimate of the traction motor load torque; then, the adhesion coefficient can be calculated by the observed value of the load torque. Sliding mode control method has good robustness to parameter variations and external disturbance [5]. The method of observing the permanent magnet synchronous motor (PMSM) load torque is proposed by the sliding mode observer in [6], which obtains better observation accuracy. A cascade sliding mode observer is designed to observe the derivative of the adhesion coefficient, and a better control effect is achieved in [7].
In order to effectively utilize locomotive traction power, a considerable number of researchers have focused on obtaining optimal adhesion control [8, 9]. However, according to the changing trend of the adhesive coefficient characteristic curve, the creep velocity of locomotives must be constrained at the stable region to avoid wheel skidding. Traditional readhesion control is obtained through detecting the wheel slip rapidly and adjusting torque to make the locomotive return to the stable region. The disadvantage of this traditional approach is that it cannot completely eliminate the skidding phenomenon [10].
In recent years, the barrier Lyapunov function (BLF) state-constrained control method has been paid attention by scholars. The BLF technology is also applied to the stable control of high-speed trains, and the creep state is limited in the stable area, which ensures the safety of train operation and the comfort of passengers [11, 12]. Because the barrier boundaries of the BLF control technique are symmetric, the obstacle region inevitably includes the skid region. Therefore, the stability of control cannot be guaranteed [13]. The ABLF theory is presented in [14], in which barrier boundaries can be designed according to the requirements of different constraints; the initial conditions of the ABLF can be more relaxed than those of the BLF. The work in [15] proposes a brake control system for aircraft landing using the ABLF to avoid aircraft locking during braking.
Inspired by the above surveys, the adhesion coefficients of HHEL are estimated accurately in real time by using the sliding mode control method [6, 7], and the optimal antiskid wheel control method for the HHEL is proposed based on the ABLF algorithm [14, 15] to operate locomotives in the stable region while near the optimal adhesion point. The proposed method ensures optimal adhesion control and antiskid control and effectively utilizes the HHEL traction power.
The rest of this paper is organized as follows. In Section 2, the model of the wheel-rail system of the HHEL is described. In Section 3, a sliding mode observer is designed to observe the adhesion coefficient. In Section 4, an ABLF controller is designed, and its stability is analyzed. In Section 5, simulations and experiments are conducted to verify the effectiveness of the proposed method. In Section 6, conclusions are presented.
2. Description of the Wheel-Rail System of the HHEL
The simplified wheel-rail model of a locomotive is shown in Figure 1. The model is composed of three parts, traction motors, gear box, and wheel and rail [16]. The traction equipment’s main task is to transmit the traction torque of the traction motors to the wheelsets of the locomotive through the gearboxes and drive the wheel such that it rotates at a speed of . During the traction operation process, wheel speed is always greater than vehicle speed . Generally, creep speed can be defined as the speed difference between the wheel speed and the vehicle speed. The locomotive is driven to move forward by the adhesion force from the creep phenomenon between the wheel and the rail.
2.1. Dynamic Model of the Locomotive Wheel
With the damping coefficient ignored, the traction motor equation is as follows [16]: where is the electromagnetic torque of the traction motor , is the load torque , is the rotor angular velocity , 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 , and is the speed ratio of the gear box, which is written as follows: where is the angular velocity of the wheel (rad/s).
Substituting (2) into (1), the dynamic model of the locomotive wheel is
2.2. Dynamic Model of the Full Locomotive
The mechanical dynamics of the HHEL with 8 axles is defined as follows [17]: where is the total quality of both the locomotive and load (kg); is the adhesion force for the traction motor of the axle ; is the sum of retardation forces , which is closely related to vehicle speed ; and , , and are retardation coefficients, all of which are positive constants.
2.3. Adhesion Characteristic Curve of the Wheel and Rail
The adhesion coefficient is used to characterize the complex mechanical relationship between the wheel and the rail. The schematic of the mechanism of wheel-rail adhesion is shown in Figure 2. The microdeformation region occurs in the wheel-rail contact region when the wheels rotate as the result of the action of the axle load . Then, 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 [18]: where is the axle load of the locomotive.
According to (5), a large adhesion coefficient equates to a good adhesive force . Two methods to improve the adhesion coefficient are proposed in this study. The first solution is to increase the locomotive axle load. However, the wear between the wheel and the rail also increases due to the pressure of the axle load. The second solution is to fully utilize the current rail surface conditions to obtain the optimal adhesion coefficient , thereby enhancing the traction power of the locomotive.
The adhesion coefficient is not only related to the surface condition of the wheel and rail but also constrained by creep velocity . The following empirical equations are derived based on numerous experimental data and statistical samples [19]: where , , , and are the surface condition coefficients of the wheel and rail, which are positive constants. is the creep speed .
Figure 3 shows the adhesion characteristic curves under three different wheel-rail surface conditions. Although different optimal adhesion coefficients exist in different rail surface conditions, they all exhibit the same trends. That is, 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 adhesion point . If antiskid measures cannot be immediately obtained, idling or sliding fault becomes inevitable.
According to (3), (4), and (6), the dynamic model of the HHEL is as follows:
Equation (7) shows that the locomotive adhesion control system exhibits strong nonlinear characteristics. The traditional optimal adhesion control method is used to track the given optimal creep velocity . This method ensures that the locomotive operates in the optimal adhesion point and that the maximal adhesion force is obtained. However, the creep speed fluctuates around the optimal adhesion point during the process of tracking. Once the locomotive moves into the skid region, it becomes prone to skidding or idling. Therefore, a control strategy must be established based on the optimal adhesion control to ensure that the creep state is constrained in the stable region.
3. Online Observation of the Wheel-Rail Adhesion Coefficient
In this section, the design step of the adhesion coefficient sliding mode observer is given. First, the traction motor load torque observer is designed to observe the load torque. Then, the wheel-rail adhesion coefficient is further calculated by using the load torque observed value.
3.1. Design of the Sliding Mode Observer
Motor rotor speed is selected as state variable . Equation (7) is then transformed as follows:
The sliding mode observer is designed as follows: where is the observed value of and is a designed gain.
The observer error is defined as follows:
Substituting (9) and (8) into (10) yields
3.2. Online Observation of the Wheel-Rail Adhesion Coefficient
The observation error is selected as the sliding surface.
Theorem 1. Considering the system (8) and its observer (9), if the sliding surface is set as (12) and if is sufficiently large enough for any initial , causes global asymptotic convergence to zero.
Proof. The Lyapunov function is selected as follows:
Substituting (11) to (13) yields
where satisfies and . and are constants to be determined.
The following equation is then obtained:
According to the Lyapunov stability criterion and sliding mode reaching condition, the error converges gradually to zero.
This equation completes the proof.
Once the system reaches the sliding surface, (16) can be obtained according to the sliding mode equivalent principle.
Substituting (16) into (11), then will yield
The adhesion coefficient can be calculated by substituting (17) into (1) and (5), as shown as follows:
4. Optimal Adhesion Antiskid Control Based on Asymmetric Barrier Lyapunov Function
In this section, firstly, the principle of ABLF is introduced and then, the optimal adhesive antiskid controller is designed and the proof of stability is made.
4.1. ABLF Theory
The following nonlinear system is considered [14]: where is the state variable, is a given target, is the control law, and are smooth functions, and is the error between the actual value and the given value .
The following BLF is selected [14]:
Lemma 1 [15]. is continuous and positive definite in an open region . comprises continuous first-order partial derivatives and is negative definite and continuous. If or , . A positive constant exists. When , then .
The schematic of a BLF is shown in Figure 4(a). The boundaries of constraint are strictly symmetric in relation to the origin.
Remark 1. When the BLF is applied to the antiskid wheel control of the locomotive and the optimal creep velocity is considered as a tracking target , two problems should be considered. (i)Stable region: the entire stable region must be included in the boundary of to satisfy the initial convergence domain . Therefore, .(ii)Skid region: the tracking target is the peak point of the adhesion characteristic curve, and the creep state should be as close as possible to the peak point. is designed as a small constant.
(a) BLF
(b) ABLF
The two problems contradict each other. To solve the different requirements of the two regions, the ABLF method is proposed for the antiskid constraints of the HHEL. With the help of a switch function , is divided into two parts, that is, and . In the stable and skid regions, are designed to construct the ABLF and achieve the antiskid constraint. The corresponding ABLF schematic is shown in Figure 4(b).
4.2. Design of the ABLF-Based Antiskid Wheel Controller
The ABLF-based antiskid controller for HHEL is designed as follows. (1)Choose the state variable . Substituting (7) into (21) yields where , , and .(2)Define the tracking error . Substituting (23) into (22) yields the following derivative of error . (3)Design the ABLF as follows: where and .
Remark 2. The creep speed tracking error is expressed as , if and only if , in which case . If and , and are continuous in . Therefore, is continuous and positive in the open region .
For the system described by (21) and (22), the control law is designed as follows: where and .
Theorem 2. For the nonlinear systems described by (23) and (24), if initial error , the error is always constrained in the region . Then, variable gradually converges to zero.
Proof. The stability proof of the controller is discussed in two cases.
Case 1. When and , ; its derivative can be obtained as follows: The control law is changed to . Substituting (31) to (30) yields for if and only if and .
Case 2. When and , ; its derivative can be obtained as follows:
The control law is changed to .
Substituting (32) to (31) yields
for , and
According to (30) and (33), is transformed into
In summary, is a continuous and positive definite scalar function in the open region . The derivative of is negative definite and continuous. or , . According to Lemma 1, initial error , which is always constrained in the interval .
According to the stability proof of the controller in this section, the following is obtained.
(1) is bounded.(2) is bounded, and
where is bounded. Obviously, is bounded.(3)According to (24),
where is bounded and , , , , and are constants. Evidently, is bounded.In summary, variable is bounded; its square integral and derivative are bounded. According to Barbalat’s lemma [20], the variable converges gradually.
This equation completes the proof.
Remark 3. According to (23), where and ; then yields As shown in the left portion of (37), . Considering that the locomotive is running under traction conditions , then,
The asymmetric constraint target of the creep state can be guaranteed by the design of . Compared with that of traditional BLF methods, the initial convergence region of the ABLF contains the entire locomotive stable region. The initial convergence condition of the BLF is relaxed.
5. Simulation and Experiment
5.1. Model of the ABLF-Based Antiskid Controller
Figure 5 shows the diagram of the model of the ABLF-based antiskid controller for the HHEL used in the simulation and experiment. The model mainly includes three parts, namely, the locomotive dynamics model, the adhesion coefficient sliding mode observer, and the ABLF controller. The dynamic model of the locomotive is composed of a traction motor control model, a HHEL dynamics model, a locomotive body resistance equation, and an adhesion empirical equation model. The sliding mode observer for the adhesion coefficient estimates the locomotive adhesion coefficient with the motor torque signal and motor speed as input signals. The ABLF control module calculates the control output torque with the error between the creep speed and the given optimal creep velocity , the adhesion coefficient , and the differential velocity signal of the vehicle body.
Table 1 presents the parameters of the HHEL. The parameters of the optimal creep rate and optimal adhesion coefficient of the three types of rail surfaces are shown in Table 2.
The proposed method is compared with the traditional sliding mode control method to verify the robustness of the ABLF-based control strategy. The rail surface condition gradually deteriorates. At 0–0.5 s, the locomotive starts from the dry rail surface and the optimal creep rate is 0.2 m/s. After 0.5 s, the locomotive enters the wet rail and changes to 0.085 m/s. After 1 s, the locomotive enters the snow rail and becomes 0.153 m/s. The retardation coefficients of the vehicle are , , and . The ABLF controller gains are and . The barrier boundaries are and . The switching gains of the sliding mode controller are , , and .
5.2. Simulation Results
The simulation results are shown in Figures 6–10. Figure 6 shows the actual value, the estimation by sliding mode observer (SMO), and the estimation by full-dimension state observer (FDSO) [21] of the adhesion coefficient. Firstly, the SMO tracks the actual adhesion value at 0.05 s, and the FDSO tracks the actual adhesion value at 0.07 s. Secondly, when the rail surface switches to the wet surface, the SMO tracks the actual value of the adhesion coefficient at 0.51 s, and the FDSO achieves it at 0.54 s. Finally, the SMO performs better observation effectiveness than the FDSO again, in which the rail surface switches to the snow surface. The sliding mode observer achieves excellent observation effectiveness at the whole stage.
(a) ABLF control
(b) SMC control
(a) ABLF control
(b) SMC control
(a) ABLF control
(b) SMC control
Figure 7 shows the tracking results of the creep speed of the locomotive. Figure 8 shows the adhesion characteristic curve of the creep speed. The locomotive is driven from point A in the dry rail. The creep speed is stable at 0.2 m/s and the adhesion coefficient is stable at 0.29. At 0.5 s, the rail surface is changed into the wet rail surface, the adhesion point moved toward the adhesion point B, the creep speed is stable at 0.08 m/s, and adhesion coefficient is stable at 0.16. At 1 s, the rail surface is changed into the wet rail surface once again, the creep speed and adhesion coefficient still have good stability. Compared with the traditional sliding mode control technique, the ABLF controller not only converges quickly and minimizes the steady-state error but also avoids the chattering phenomenon of the sliding mode control after reaching the optimal adhesion point.
Figure 9 shows the output torque waveform of the locomotive traction motor. At 0.5 s and 1 s, the output torque of the traction motor changes with the change of the wheel-rail surface condition. The difference between the two methods is that the control torque of the ABFL controller changes only when the rail surface condition switches and that the control torque shows no fluctuation in the steady state. As a result of the chattering of the SMC method, the chattering of the control torque always occurs in the entire process.
5.3. Experimental Results
RT-Lab is a powerful modular real-time simulation platform, which can be used as rapid control prototyping and hardware-in-the-loop simulation (HILS) [22]. Figure 10 shows the RT-Lab experimental platform used in this work. Figure 11 presents the RT-Lab HILS configuration diagram of the ABLF control system of the HHEL. The C program code of the ABLF controller and sliding mode observer is downloaded to the DSP controller TMS320F2812 using a PC. Then, the HHEL model is compiled and downloaded to OP5600, and the HILS of the ABLF control system of the HHEL can be realized. The DSP can use the vehicle speed, wheel speed, motor torque, and optimal value of creep speed as the input signal of the controller. Sampling period is set to 10 μs.
Figure 12 shows the experimental waveforms of the actual and observed values by the SMO and FDSO. The experimental results show that the SMO can accurately track the actual adhesion coefficient, and the performance of the SMO is better than that of the FDSO. Figure 13 shows the experimental values of the given and actual values of creep velocity. The result shows that the ABLF controller can asymptotically track the given value of the creep speed. Figure 14 shows the experimental waveforms of the motor control torque using the two control strategies. The ABLF control strategy clearly shows good control precision in the steady state, thus avoiding the chattering of the traditional sliding mode control method.
(a) SMO
(b) FDSO
(a) ABLF control
(b) SMC control
(a) ABLF control
(b) SMC control
6. Conclusions
This paper proposes a new method of online accurate estimation of wheel-rail adhesion coefficient and optimal adhesion antiskid control of heavy-haul electric locomotives (HHEL) based on sliding mode and asymmetric barrier Lyapunov function (ABLF) theory. The following conclusions can be drawn from the results obtained with the proposed method. (1)A dynamic model of a HHEL was derived, and the adhesion mechanism of the HHEL is analyzed.(2)The difficulty of measuring adhesion coefficients with a conventional physical sensor is solved, by designing a sliding mode observer which can online accurately observe the adhesion coefficient in real time.(3)An optimal adhesion antiskid controller is designed based on the ABLF theory. The proposed controller achieved asymmetric control according to the different creep states of the HHEL, which ensures that the creep state is constrained in the stable region. In contrast to existing antiskid control methods, the proposed method reduces the torque ripple of motors. The adhesion control performance of the HHEL is improved.
Conflicts of Interest
The authors declare no conflict of interest.
Authors’ Contributions
Kaihui Zhao proposed the ABLF antiskid control scheme and derived the dynamic model of the locomotive. Peng Li and Kaihui Zhao designed the overall algorithms and the simulations. Peng Li, Yanfei Li, and Tonghuan Yin performed the experiments and simulations; Kaihui Zhao and Peng Li wrote the manuscript. Changfan Zhang and Jing He supervised the work and revised the paper.
Acknowledgments
This work is supported by the Natural Science Foundation of China under Grant nos. 61773159, 61473117, and 61733004, the Scientific Research Fund of the Hunan Province Education Department under Grant nos. 16A058 and 17B073, the Natural Science Foundation of China of Hunan Province under Grant nos. 2018JJ2093, 2017JJ4031, and 2016JJ5012, the Hunan Province Education Department under Grant no. 13CY018, the Teaching Reform Fund of Hunan Province Education Department under Grant no. Hunan Education Notice [2016] no. 400, the Teaching Reform of Degree and Postgraduate Education Fund of Hunan University of Technology under Grant no. JG1604, and the Key Laboratory for Electric Drive Control and Intelligent Equipment of Hunan Province Grant no. 2016TP1018.