# Online Accurate Estimation of the Wheel-Rail Adhesion Coefficient and Optimal Adhesion Antiskid Control of Heavy-Haul Electric Locomotives Based on Asymmetric Barrier Lyapunov Function

Authors: Kaihui Zhao; Peng Li; Changfan Zhang; Jing He; Yanfei Li; Tonghuan Yin
Journal: Journal of Sensors
 (2018)

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## 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.


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## Body

## 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 Section2, 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 Figure1. 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 vd. During the traction operation process, wheel speed vd is always greater than vehicle speed vt. Generally, creep speed vs 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 Fμ from the creep phenomenon between the wheel and the rail.Figure 1
Simplified wheel-rail model of the HHEL.


### 2.1. Dynamic Model of the Locomotive Wheel

With the damping coefficient ignored, the traction motor equation is as follows [16]:
(1)Jmdωmdt=Tm−TL,TLRg=Fμr,where Tm is the electromagnetic torque of the traction motor N⋅m, TL is the load torque N⋅m, ωm is the rotor angular velocity rad/s, r is the radius of the wheel m, Jm is the moment of inertia of the traction motor kg⋅m2, Fμ is the adhesion force between the wheel and the rail N, and Rg is the speed ratio of the gear box, which is written as follows:
(2)Rg=ωmωd,where ωd is the angular velocity of the wheel (rad/s).Substituting (2) into (1), the dynamic model of the locomotive wheel is
(3)JmRg2r2dvddt=RgrTm−Fμ.


### 2.2. Dynamic Model of the Full Locomotive

The mechanical dynamics of the HHEL with 8 axles is defined as follows [17]:
(4)Mdvtdt=∑i=18Fμi−Fd,Fd=l+pvt+qvt2Mg,where M is the total quality of both the locomotive and load (kg); Fμi is the adhesion force for the traction motor of the i axle N; Fd is the sum of retardation forces N, which is closely related to vehicle speed vt; and l, p, and q are retardation coefficients, all of which are positive constants.


### 2.3. Adhesion Characteristic Curve of the Wheel and Rail

The adhesion coefficientμ 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 W. Then, the adhesion force Fμ is produced by the interaction between the wheel and the rail.Figure 2
Schematic of the wheel rail adhesion.The adhesion forceFμ and the adhesion coefficient μ have the following relation [18]:
(5)Fμ=μ⋅Wg,where Wg is the axle load of the locomotive.According to (5), a large adhesion coefficient μ equates to a good adhesive force Fμ. Two methods to improve the adhesion coefficient μ 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 μmax, thereby enhancing the traction power of the locomotive.The adhesion coefficientμ is not only related to the surface condition of the wheel and rail but also constrained by creep velocity vs. The following empirical equations are derived based on numerous experimental data and statistical samples [19]:
(6)vs=vd−vs,μ=ce−avs−de−bvs,where a, b, c, and d are the surface condition coefficients of the wheel and rail, which are positive constants. vs is the creep speed m/s.Figure3 shows the adhesion characteristic curves under three different wheel-rail surface conditions. Although different optimal adhesion coefficients μmax exist in different rail surface conditions, they all exhibit the same trends. That is, the adhesion coefficient increases to μmax with the increase of vs in the stable region and then decreases sharply to zero with the increase of creep vs after crossing the optimal adhesion point vsopt. If antiskid measures cannot be immediately obtained, idling or sliding fault becomes inevitable.Figure 3
Adhesion characteristic curve.According to (3), (4), and (6), the dynamic model of the HHEL is as follows:
(7)ω̇mv̇s=1JmrJmRg2Tm+−1JmTL−r2JmRg2TL−vt2.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 vsopt. This method ensures that the locomotive operates in the optimal adhesion point μmax and that the maximal adhesion force is obtained. However, the creep speed vs fluctuates around the optimal adhesion point vsopt 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.


## 2.1. Dynamic Model of the Locomotive Wheel

With the damping coefficient ignored, the traction motor equation is as follows [16]:
(1)Jmdωmdt=Tm−TL,TLRg=Fμr,where Tm is the electromagnetic torque of the traction motor N⋅m, TL is the load torque N⋅m, ωm is the rotor angular velocity rad/s, r is the radius of the wheel m, Jm is the moment of inertia of the traction motor kg⋅m2, Fμ is the adhesion force between the wheel and the rail N, and Rg is the speed ratio of the gear box, which is written as follows:
(2)Rg=ωmωd,where ωd is the angular velocity of the wheel (rad/s).Substituting (2) into (1), the dynamic model of the locomotive wheel is
(3)JmRg2r2dvddt=RgrTm−Fμ.


## 2.2. Dynamic Model of the Full Locomotive

The mechanical dynamics of the HHEL with 8 axles is defined as follows [17]:
(4)Mdvtdt=∑i=18Fμi−Fd,Fd=l+pvt+qvt2Mg,where M is the total quality of both the locomotive and load (kg); Fμi is the adhesion force for the traction motor of the i axle N; Fd is the sum of retardation forces N, which is closely related to vehicle speed vt; and l, p, and q are retardation coefficients, all of which are positive constants.


## 2.3. Adhesion Characteristic Curve of the Wheel and Rail

The adhesion coefficientμ 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 W. Then, the adhesion force Fμ is produced by the interaction between the wheel and the rail.Figure 2
Schematic of the wheel rail adhesion.The adhesion forceFμ and the adhesion coefficient μ have the following relation [18]:
(5)Fμ=μ⋅Wg,where Wg is the axle load of the locomotive.According to (5), a large adhesion coefficient μ equates to a good adhesive force Fμ. Two methods to improve the adhesion coefficient μ 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 μmax, thereby enhancing the traction power of the locomotive.The adhesion coefficientμ is not only related to the surface condition of the wheel and rail but also constrained by creep velocity vs. The following empirical equations are derived based on numerous experimental data and statistical samples [19]:
(6)vs=vd−vs,μ=ce−avs−de−bvs,where a, b, c, and d are the surface condition coefficients of the wheel and rail, which are positive constants. vs is the creep speed m/s.Figure3 shows the adhesion characteristic curves under three different wheel-rail surface conditions. Although different optimal adhesion coefficients μmax exist in different rail surface conditions, they all exhibit the same trends. That is, the adhesion coefficient increases to μmax with the increase of vs in the stable region and then decreases sharply to zero with the increase of creep vs after crossing the optimal adhesion point vsopt. If antiskid measures cannot be immediately obtained, idling or sliding fault becomes inevitable.Figure 3
Adhesion characteristic curve.According to (3), (4), and (6), the dynamic model of the HHEL is as follows:
(7)ω̇mv̇s=1JmrJmRg2Tm+−1JmTL−r2JmRg2TL−vt2.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 vsopt. This method ensures that the locomotive operates in the optimal adhesion point μmax and that the maximal adhesion force is obtained. However, the creep speed vs fluctuates around the optimal adhesion point vsopt 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 speedωm is selected as state variable x1. Equation (7) is then transformed as follows:
(8)x1=ωm,ẋ1=1JmTm−1JmTL.The sliding mode observer is designed as follows:(9)x^1=ω^m,x^̇1=1JmTm+η1sgnx1−x^1,where x^1 is the observed value of x1 and η1 is a designed gain.The observer error is defined as follows:(10)e1=x1−x^1.Substituting (9) and (8) into (10) yields
(11)ė1=−1JmT^L−η1sgne1.


### 3.2. Online Observation of the Wheel-Rail Adhesion Coefficient

The observation errore1 is selected as the sliding surface.
(12)s1=e1.Theorem 1.
Considering the system (8) and its observer (9), if the sliding surface is set as (12) and if η1 is sufficiently large enough for any initial x0∈Rn, e1 causes global asymptotic convergence to zero.Proof.
The Lyapunov function is selected as follows:(13)V1=12s12.
Substituting (11) to (13) yields
(14)V̇1=e1ė1=−e1JmTL−η1e1≤e11JmTL−η1e1=1JmTL−η1e1,where η1 satisfies η1≥1/JmTL+η2 and η2>0. η1 and η2 are constants to be determined.
The following equation is then obtained:(15)V̇1≤−η2e1.
According to the Lyapunov stability criterion and sliding mode reaching condition, the errore1 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.
(16)e1=ė1=0.Substituting (16) into (11), then will yield
(17)T^L=−Jmη1sgne1.The adhesion coefficient can be calculated by substituting (17) into (1) and (5), as shown as follows:
(18)μ^=RgrWgT^L.


## 3.1. Design of the Sliding Mode Observer

Motor rotor speedωm is selected as state variable x1. Equation (7) is then transformed as follows:
(8)x1=ωm,ẋ1=1JmTm−1JmTL.The sliding mode observer is designed as follows:(9)x^1=ω^m,x^̇1=1JmTm+η1sgnx1−x^1,where x^1 is the observed value of x1 and η1 is a designed gain.The observer error is defined as follows:(10)e1=x1−x^1.Substituting (9) and (8) into (10) yields
(11)ė1=−1JmT^L−η1sgne1.


## 3.2. Online Observation of the Wheel-Rail Adhesion Coefficient

The observation errore1 is selected as the sliding surface.
(12)s1=e1.Theorem 1.
Considering the system (8) and its observer (9), if the sliding surface is set as (12) and if η1 is sufficiently large enough for any initial x0∈Rn, e1 causes global asymptotic convergence to zero.Proof.
The Lyapunov function is selected as follows:(13)V1=12s12.
Substituting (11) to (13) yields
(14)V̇1=e1ė1=−e1JmTL−η1e1≤e11JmTL−η1e1=1JmTL−η1e1,where η1 satisfies η1≥1/JmTL+η2 and η2>0. η1 and η2 are constants to be determined.
The following equation is then obtained:(15)V̇1≤−η2e1.
According to the Lyapunov stability criterion and sliding mode reaching condition, the errore1 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.
(16)e1=ė1=0.Substituting (16) into (11), then will yield
(17)T^L=−Jmη1sgne1.The adhesion coefficient can be calculated by substituting (17) into (1) and (5), as shown as follows:
(18)μ^=RgrWgT^L.


## 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]:
(19)ẋ=fx+gxu,e=x−x∗,where x is the state variable, x∗ is a given target, u is the control law, fx and gx are smooth functions, and e is the error between the actual value x and the given value x∗.The following BLF is selected [14]:
(20)Ve=12logka2ka2−e2.Lemma 1 [15].
Ve is continuous and positive definite in an open region D=e∈R:−ka<e<ka. Ve comprises continuous first-order partial derivatives V̇e and is negative definite and continuous. If e→ka or e→−ka, Ve→∞. A positive constant p exists. When e0<D,∀t≥0, then et∈D,Ve<p.
The schematic of a BLF is shown in Figure4(a). The boundaries of constraint D 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 velocityvsopt is considered as a tracking target x∗, two problems should be considered.
(i)
Stable region: the entire stable region must be included in the boundary ofD to satisfy the initial convergence domain e0≤D. Therefore, ka≥vsopt.
(ii)
Skid region: the tracking targetx∗ is the peak point vsopt of the adhesion characteristic curve, and the creep state should be as close as possible to the peak point. ka is designed as a small constant.Figure 4
BLF schematic.
(a)
BLF
(b)
ABLFThe 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 functionqe, D is divided into two parts, that is, Da=−ka<e<0 and Db=0<e<kb. In the stable and skid regions, ka,kb 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 variablex2=vs.
(21)x2=vs,ẋ2=v̇s.
Substituting (7) into (21) yields
(22)ẋ2=fx2u+gx2,where fx2=r/JmRg, gx2=−r2/JmRg2Fμ−v̇t, and u=Tm.
(2)
Define the tracking errore2.
(23)e2=x2−vs∗.
Substituting (23) into (22) yields the following derivative of error e2.
(24)ė2=fx2u+gx2−v̇s∗.
(3)
Design the ABLF as follows:(25)Ve2=12qe2logka2ka2−e22+121−qe2logkb2kb2−e22,where ka>vsopt and kb>0.
(26)qe2=1,ase2≤0,0,ase2>0.Remark 2.
The creep speed tracking error is expressed ase2, if and only if e2=0, in which case Ve2=0. If e2∈D and e2≠0, Ve2>0 and Ve2 are continuous in e2=0. Therefore, Ve2 is continuous and positive in the open region D.For the system described by (21) and (22), the control law u is designed as follows:
(27)u=1fx2−gx2+v̇s∗−qe2k1eka2−e22−1−qe2k2ekb2−e22,where k1>0 and k2>0.Theorem 2.
For the nonlinear systems described by (23) and (24), if initial error e20∈−ka,kb, the error is always constrained in the region D∈−ka,kb. Then, variable e2 gradually converges to zero.Proof.
The stability proof of the controller is discussed in two cases.Case 1.
Whene2∈D and e2≤0 qe2=1, Ve2=1/2logka2/ka2−e22; its derivative can be obtained as follows:
(28)V̇e2=e2ė2ka2−e22=e2ka2−e22fx2+gx2u−v̇s∗.
The control lawu is changed to u1.
(29)u1=1gx2−fx2+v̇s∗−k1e2ka2−e22.
Substituting (31) to (30) yields
(30)V̇e2=−k1e22.
V̇e2≤0 for Da=e2∈R:−ka<e2≤0 if and only if e2=0 and V̇e2=0.Case 2.
Whene2∈D and e2>0 qe2=0, Ve2=1/2logkb2/kb2−e22; its derivative can be obtained as follows:
(31)V̇e2=e2ė2kb2−e22=e2kb2−e22fx2+gx2u−v̇s∗.
The control lawu is changed to u2.
(32)u2=1gx2fx2+v̇s∗−k2e2kb2−e22.
Substituting (32) to (31) yields
(33)V̇e2=−k2e22.
V̇e2≤0 for Db=0<e2<kb, and
(34)lime2→0+V̇e2=V̇0=0.
According to (30) and (33), V̇e2 is transformed into
(35)V̇e2=−qe2k1e22−1−qe2k2e22.
In summary,Ve2 is a continuous and positive definite scalar function in the open region D=e2∈R:−ka<e<kb. The derivative of Ve2 is negative definite and continuous. ∀e2→−ka or e2→kb, Ve2→∞. According to Lemma 1, initial error e20∈−ka,kb, which is always constrained in the interval e2t∈−ka,kb.
According to the stability proof of the controller in this section, the following is obtained.(1)
e2t∈−ka,kb,e2t is bounded.
(2)
V0,Ve2 is bounded, and
(36)Ve2=V0+∫0tV̇e2dt=V0−∫0tqe2k1e22+1−qe2k2e22dt,where ∫0tqe2k1e22+1−qe2k2e22dt is bounded. Obviously, ∫0te22tdt is bounded.
(3)
According to (24),
(37)ė2t=−qe2k1e2γa2−e22=−1−qe2k2e2γa2−e22,where e2t is bounded and qe2, k1, k2, γa, and γb are constants. Evidently, ė2t is bounded.
In summary, variablee2t is bounded; its square integral and derivative are bounded. According to Barbalat’s lemma [20], the variable e2t converges gradually.
This equation completes the proof.Remark 3.
According to (23),
(38)−ka+vs∗t<vst=e2t+vs∗t<kb+vs∗t,where vs∗t∈0,vsopt and ka>vsopt; η3=−ka+vs∗t<0 then yields
(39)η3<vst<kb+vsopt.
As shown in the left portion of (37), η3<0. Considering that the locomotive is running under traction conditions vst≥0, then,
(40)0≤vst<kb+vsopt.The asymmetric constraint target of the creep state can be guaranteed by the design ofka,kb. 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.


## 4.1. ABLF Theory

The following nonlinear system is considered [14]:
(19)ẋ=fx+gxu,e=x−x∗,where x is the state variable, x∗ is a given target, u is the control law, fx and gx are smooth functions, and e is the error between the actual value x and the given value x∗.The following BLF is selected [14]:
(20)Ve=12logka2ka2−e2.Lemma 1 [15].
Ve is continuous and positive definite in an open region D=e∈R:−ka<e<ka. Ve comprises continuous first-order partial derivatives V̇e and is negative definite and continuous. If e→ka or e→−ka, Ve→∞. A positive constant p exists. When e0<D,∀t≥0, then et∈D,Ve<p.
The schematic of a BLF is shown in Figure4(a). The boundaries of constraint D 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 velocityvsopt is considered as a tracking target x∗, two problems should be considered.
(i)
Stable region: the entire stable region must be included in the boundary ofD to satisfy the initial convergence domain e0≤D. Therefore, ka≥vsopt.
(ii)
Skid region: the tracking targetx∗ is the peak point vsopt of the adhesion characteristic curve, and the creep state should be as close as possible to the peak point. ka is designed as a small constant.Figure 4
BLF schematic.
(a)
BLF
(b)
ABLFThe 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 functionqe, D is divided into two parts, that is, Da=−ka<e<0 and Db=0<e<kb. In the stable and skid regions, ka,kb 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 variablex2=vs.
(21)x2=vs,ẋ2=v̇s.
Substituting (7) into (21) yields
(22)ẋ2=fx2u+gx2,where fx2=r/JmRg, gx2=−r2/JmRg2Fμ−v̇t, and u=Tm.
(2)
Define the tracking errore2.
(23)e2=x2−vs∗.
Substituting (23) into (22) yields the following derivative of error e2.
(24)ė2=fx2u+gx2−v̇s∗.
(3)
Design the ABLF as follows:(25)Ve2=12qe2logka2ka2−e22+121−qe2logkb2kb2−e22,where ka>vsopt and kb>0.
(26)qe2=1,ase2≤0,0,ase2>0.Remark 2.
The creep speed tracking error is expressed ase2, if and only if e2=0, in which case Ve2=0. If e2∈D and e2≠0, Ve2>0 and Ve2 are continuous in e2=0. Therefore, Ve2 is continuous and positive in the open region D.For the system described by (21) and (22), the control law u is designed as follows:
(27)u=1fx2−gx2+v̇s∗−qe2k1eka2−e22−1−qe2k2ekb2−e22,where k1>0 and k2>0.Theorem 2.
For the nonlinear systems described by (23) and (24), if initial error e20∈−ka,kb, the error is always constrained in the region D∈−ka,kb. Then, variable e2 gradually converges to zero.Proof.
The stability proof of the controller is discussed in two cases.Case 1.
Whene2∈D and e2≤0 qe2=1, Ve2=1/2logka2/ka2−e22; its derivative can be obtained as follows:
(28)V̇e2=e2ė2ka2−e22=e2ka2−e22fx2+gx2u−v̇s∗.
The control lawu is changed to u1.
(29)u1=1gx2−fx2+v̇s∗−k1e2ka2−e22.
Substituting (31) to (30) yields
(30)V̇e2=−k1e22.
V̇e2≤0 for Da=e2∈R:−ka<e2≤0 if and only if e2=0 and V̇e2=0.Case 2.
Whene2∈D and e2>0 qe2=0, Ve2=1/2logkb2/kb2−e22; its derivative can be obtained as follows:
(31)V̇e2=e2ė2kb2−e22=e2kb2−e22fx2+gx2u−v̇s∗.
The control lawu is changed to u2.
(32)u2=1gx2fx2+v̇s∗−k2e2kb2−e22.
Substituting (32) to (31) yields
(33)V̇e2=−k2e22.
V̇e2≤0 for Db=0<e2<kb, and
(34)lime2→0+V̇e2=V̇0=0.
According to (30) and (33), V̇e2 is transformed into
(35)V̇e2=−qe2k1e22−1−qe2k2e22.
In summary,Ve2 is a continuous and positive definite scalar function in the open region D=e2∈R:−ka<e<kb. The derivative of Ve2 is negative definite and continuous. ∀e2→−ka or e2→kb, Ve2→∞. According to Lemma 1, initial error e20∈−ka,kb, which is always constrained in the interval e2t∈−ka,kb.
According to the stability proof of the controller in this section, the following is obtained.(1)
e2t∈−ka,kb,e2t is bounded.
(2)
V0,Ve2 is bounded, and
(36)Ve2=V0+∫0tV̇e2dt=V0−∫0tqe2k1e22+1−qe2k2e22dt,where ∫0tqe2k1e22+1−qe2k2e22dt is bounded. Obviously, ∫0te22tdt is bounded.
(3)
According to (24),
(37)ė2t=−qe2k1e2γa2−e22=−1−qe2k2e2γa2−e22,where e2t is bounded and qe2, k1, k2, γa, and γb are constants. Evidently, ė2t is bounded.
In summary, variablee2t is bounded; its square integral and derivative are bounded. According to Barbalat’s lemma [20], the variable e2t converges gradually.
This equation completes the proof.Remark 3.
According to (23),
(38)−ka+vs∗t<vst=e2t+vs∗t<kb+vs∗t,where vs∗t∈0,vsopt and ka>vsopt; η3=−ka+vs∗t<0 then yields
(39)η3<vst<kb+vsopt.
As shown in the left portion of (37), η3<0. Considering that the locomotive is running under traction conditions vst≥0, then,
(40)0≤vst<kb+vsopt.The asymmetric constraint target of the creep state can be guaranteed by the design ofka,kb. 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

Figure5 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 μ^ with the motor torque signal Tm and motor speed ωm as input signals. The ABLF control module calculates the control output torque Tm∗ with the error between the creep speed vs and the given optimal creep velocity vsopt, the adhesion coefficient μ^, and the differential velocity signal v̇t of the vehicle body.Figure 5
Model of the ABLF-based antiskid controller for HHEL.Table1 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.Table 1
Parameters of the HHEL.
Parameter
Description
Value
M
Total quality of the locomotive and load
6000 t
W
Single axle load of the locomotive
30 t
Jm
Inertia moment of motor
11 kg·m2
r
Wheel radius
0.625 m
Rg
Gear speed transfer ratio
6.294Table 2
Parameters of three types of rail surfaces.
Rail surface
Optimal creep speed
Adhesion coefficient
Dry
0.2 m/s
0.29
Wet
0.085 m/s
0.16
Snow
0.153 m/s
0.06The 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 ratevsopt is 0.2 m/s. After 0.5 s, the locomotive enters the wet rail and vsopt changes to 0.085 m/s. After 1 s, the locomotive enters the snow rail and vsopt becomes 0.153 m/s. The retardation coefficients of the vehicle are l=0.5, p=0.041, and q=0.0022. The ABLF controller gains are k1=1000 and k2=1000. The barrier boundaries are kb=0.01 and ka=vsopt+kb. The switching gains of the sliding mode controller are c=0.1, ξ=10, and λ=10.


### 5.2. Simulation Results

The simulation results are shown in Figures6–10. 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.Figure 6
Actual value and observed value by SMO and FDSO of the adhesion coefficient.Figure 7
Given value and actual value of the creep speed.
(a)
ABLF control
(b)
SMC controlFigure 8
Adhesion characteristic curve.
(a)
ABLF control
(b)
SMC controlFigure 9
Motor output torque.
(a)
ABLF control
(b)
SMC controlFigure 10
RT-Lab experimental platform.Figure7 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.Figure9 shows the output torque waveform of the locomotive traction motor. At 0.5 s and 1 s, the output torque Tm 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 11
Loop system configuration diagram of RT-Lab.Figure12 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.Figure 12
Actual value and observed value of the adhesion coefficient by SMO and FDSO.
(a)
SMO
(b)
FDSOFigure 13
Given value and actual value of the creep speed.
(a)
ABLF control
(b)
SMC controlFigure 14
Motor output torque.
(a)
ABLF control
(b)
SMC control


## 5.1. Model of the ABLF-Based Antiskid Controller

Figure5 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 μ^ with the motor torque signal Tm and motor speed ωm as input signals. The ABLF control module calculates the control output torque Tm∗ with the error between the creep speed vs and the given optimal creep velocity vsopt, the adhesion coefficient μ^, and the differential velocity signal v̇t of the vehicle body.Figure 5
Model of the ABLF-based antiskid controller for HHEL.Table1 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.Table 1
Parameters of the HHEL.
Parameter
Description
Value
M
Total quality of the locomotive and load
6000 t
W
Single axle load of the locomotive
30 t
Jm
Inertia moment of motor
11 kg·m2
r
Wheel radius
0.625 m
Rg
Gear speed transfer ratio
6.294Table 2
Parameters of three types of rail surfaces.
Rail surface
Optimal creep speed
Adhesion coefficient
Dry
0.2 m/s
0.29
Wet
0.085 m/s
0.16
Snow
0.153 m/s
0.06The 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 ratevsopt is 0.2 m/s. After 0.5 s, the locomotive enters the wet rail and vsopt changes to 0.085 m/s. After 1 s, the locomotive enters the snow rail and vsopt becomes 0.153 m/s. The retardation coefficients of the vehicle are l=0.5, p=0.041, and q=0.0022. The ABLF controller gains are k1=1000 and k2=1000. The barrier boundaries are kb=0.01 and ka=vsopt+kb. The switching gains of the sliding mode controller are c=0.1, ξ=10, and λ=10.


## 5.2. Simulation Results

The simulation results are shown in Figures6–10. 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.Figure 6
Actual value and observed value by SMO and FDSO of the adhesion coefficient.Figure 7
Given value and actual value of the creep speed.
(a)
ABLF control
(b)
SMC controlFigure 8
Adhesion characteristic curve.
(a)
ABLF control
(b)
SMC controlFigure 9
Motor output torque.
(a)
ABLF control
(b)
SMC controlFigure 10
RT-Lab experimental platform.Figure7 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.Figure9 shows the output torque waveform of the locomotive traction motor. At 0.5 s and 1 s, the output torque Tm 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 11
Loop system configuration diagram of RT-Lab.Figure12 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.Figure 12
Actual value and observed value of the adhesion coefficient by SMO and FDSO.
(a)
SMO
(b)
FDSOFigure 13
Given value and actual value of the creep speed.
(a)
ABLF control
(b)
SMC controlFigure 14
Motor output torque.
(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.



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*Source: 2740679-2018-04-08.xml*