Intelligent rail lubrication system based on fuzzy group analysis

Abstract

The security of a train becomes a more critical issue as the train’s speed and the complexity of the railway conditions increases. It is especially true when the train runs on a curved radius rail when the lateral force between the train and the rail is less stable. The rail’s side grinding is a significant problem that affects the train’s safety, especially when the train passes through small radial sections in mountainous areas. The intelligent rail lubrication system is critical to enhancing rails’ safety and efficiency and reducing grease pollution along rail lines. This system is modeled with a force analysis of train curve motion and numerical simulation of wear power. The lubrication system is constructed with hardware and software. Based on fuzzy group analysis, this system and the adaptive Proportional Integration Differential (PID) controller is presented to improve the lubricative effects. The system test results show that the quality of lubrication control using this system is efficacious; the control convergence is more reliable than the conventional PID controller.

Keywords

Security curve rail adaptive PID lubrication system

1 Introduction

The upkeep of train wheels and rails is a fundamental issue for the safe operation of trains and rail transit. As the train velocity increases and railway conditions become complicated, this problem becomes more critical. For example, rail mileage exceeded 130,000 km by 2017 [1, 2]. As a result, to ensure safety and reliability, the demand for routine maintenance has become increasingly high for railways. Railway track components deteriorate more quickly along tight curves with massive loads in poor maintenance conditions [3]. The force between the rail and wheel reduces the life span, and accidents can occur if the rail lubricant system fails.

Statistical data shows that China’s industrial sector suffered 950 billion ¥ of losses due to friction and wear in 2009, according to a report issued by the China Academy of Engineering (CAE). The CAE report investigated the current situation and development strategy of tribological science and engineering applications. Such cases were observed in developed countries, in which these losses could account for approximately 2% 7% of the gross national product (GNP). These are staggering numbers. The statistical data also show that 3% to 5% of the GNP can be saved annually when advanced lubrication technology is applied to machines and equipment [4, 5].

Side wear is a prominent characteristic of the force between a train wheel and rail, especially along the small radius section of a mountainous area. Rail friction mainly includes rail crushing, rail side grinding, rail corrugating, and rail stripping. The side grinder is one of the more complex problems affecting train safety. Rail replacement frequency keeps increasing, often with a life span of 1-2 years. According to the Railway Department, wear damage sees an annual increase of up to $0.7 billion to replace and repair the rails. Notably, the train wheel and rail’s lateral force are more unstable when they run on a small radius rail [6 –9].

The system lubricates the protruding part of a wheel and the inside of the rail. Wheel-rail friction and lubrication are related to locomotive traction, energy consumption, traffic safety, wheel-rail material consumption, and maintenance cost [10]. With further improvement in train speed, the problem becomes more critical. Reasonable and practical wheel and rail lubrication can reduce wheel and rail wear, considerable significance for energy savings, locomotive traction efficiency, and train operation [11]. Wheel and rail lubrication technology can be divided into vehicle-mounted wheel rim lubrication, vehicle-mounted rail lubrication, ground rail lubrication, and vehicle-mounted wheel rim robust lubrication technologies [12, 13].

The lubricant is divided into the lubricating oil, grease, oil lubricant, and solid lubricant in wheel and rail maintenance. Due to vehicle vibration, it is difficult to control the accuracy of the vehicle-mounted rail lubrication system’s oiling position. Therefore, most similar lubricators are subject to speed limits. Specialized oiling vehicles are used in some places. For example, the United States has conducted a wide range of “ground + vehicle” wheel and rail lubrication tests. The combination of the two can save fuel consumption by 25% 35%. Lubrication on a curve with a radius of 350 m can reduce the train’s running resistance by 50%. Many aspects of the train influence rail friction during movement. Security and robustness of the equipment guarantee train safety [14, 15].

In prolonging wheel service life and reducing railways’ maintenance expenditures, it is essential to study and design the lubricating system’s concept based on the results of radius and velocity and the perspective of lubrication analysis [16, 17]. According to the model analysis of force conditions and the real-time lubrication theory foundation, this device establishes a mutual relationship between the vehicle body and the wheel-rail movement. With the adaptive PID, it can be generated within a few cycles of low-cost and low-consumption mechanical excitation. This device controls the lubrication quality and improves the process. This concept holds potential in direct current (DC) power generation and maintenance domains.

In 1965, professor Lotfi Aliasker Zadeh, an American expert on cybernetics, extended the exact group concept to fuzzy sets in information and control [18]. He asserted that the computer could not deal with the imperfection of the fuzzy concept from mathematics [19]. The fuzzy set is the inevitable reflection of the concept of objective existence. The ambiguity is in the communication of human thinking and language, which is an expression of uncertainty. In 1974, the first successful application of fuzzy systems was made by Mamdani, a professor at Imperial College London, which led to a boom in fuzzy control [20]. Compared with the traditional control theory, fuzzy control has two incomparable advantages [21]. Firstly, fuzzy control can realize human control strategy and experience effectively and quickly in many applications. This advantage has been widely emphasized since the birth of fuzzy control. Secondly, fuzzy control can achieve better control without the mathematical model of the controlled object. It is because the controlled object’s dynamic characteristics are implicit in the input and output fuzzy sets and fuzzy rules of the fuzzy controller. Furthermore, the fuzzy set was initially developed to overcome particular difficulties, especially the vague information. It is not only a combination of fuzzy mathematics and control theory but also an essential part of intelligent control [22].

Fuzzy mathematics has been applied to fuzzy control, fuzzy recognition, fuzzy decision, fuzzy cluster analysis, etc.[23 –26]. For example, fuzzy mathematics can make air conditioner temperature control more reason-able, and washing machines can save electricity and water from improving efficiency. The biggest advantage is interpretability. That is, the entire system is built on rules, and each rule can be intuitively understood. However, with the current popularity of data-driven modeling, fuzzy system rules have become more complex and difficult. There are two kinds of rules commonly used in fuzzy systems: Zadeh rules, in which the rule post is a fuzzy set, and Takagi-Sugeno-Kang (TSK for short) rules, where the fuzzy postposition is the input function. Zadeh fuzzy system consists of four parts: it is fuzzification, rule base, inference machine, and defuzzification [27]. TSK fuzzy systems do not need to be de-fuzzy because the reasoning machine’s output is directly clear values. TSK fuzzy systems have been more popular [28]. However, it is not easy to build a fuzzy system with good performance. A fuzzy system usually includes many technologies, such as optimizing, interpretability, high dimensional data, and generalization performance. How to build one reliable rail lubrication system based on the fuzzy group is a big challenge for us.

Based on fuzzy group analysis, one system is presented to improve the paper’s rail lubricative effect. This paper is organized into six sections. The authors discuss previous studies on the intelligent rail lubrication system in section 1. The Primary concept is explained in Section 2, the simulation of speed, and the curve radius is presented in Section 3, and the Fuzzy PID is presented in Section 4. The intelligent lubrication system’s design show in Section 5 and the conclusions in Section 6.

2 Primary concept

2.1 Fuzzy group

It has been shown that fuzzy control is vital to system control. This method overcomes different types of problematic uncertainty, primarily when equations and functions cannot describe the kinematics and mechanisms. Train safety is so essential that all the mechanical and physical equipment must be dealt with appropriately. Thus, the mutual correlation of the equipment must be explicit. However, in many cases, the facilities’ relationship is too complicated to pinpoint precisely [15, 29]. The safety and reliability impacts are directly linked to the vehicle.

The fuzzy group is better suited for this question. Assuming domain U describes a mapping relationship, that is $ν_{A} : Ψ \to [0, 1] \begin{matrix} φ \mapsto ν_{A} (φ) \end{matrix}$ (1)

It defines the fuzzy subgroup, which is described as M. In fuzzy control, the processes contain four steps: input and output fuzzification, fuzzy rules, fuzzy reasoning, and defuzzification resolution. In the fuzzy controller, the number of fuzzy inputs should be reasonably selected according to the actual situation. The input of the one-dimensional fuzzy controller should be based on the error between the actual set value and the sensor’s feedback.

Two-dimensional fuzzy controller input is usually based on the actual setting error between the sensor feedback value and the joint changing error rate. It not only satisfies the dynamic features of the controlled object but also has a straightforward model and does not contain complexly multivariable such as three-dimensional fuzzy controllers. It performs better than one-dimensional fuzzy controllers and is widely used in practical systems.

Fuzzy reasoning usually takes the form as a condition statement. That is, “if E is M and E_C is N, then Ψ is Q ”. In addition to the set M is the fuzzy input subset that presents error E, the set N that represents the fuzzy input subset of the error change rate E_C, and Ψ that represents the fuzzy subset of the output; if the input parameters are M and N, we can obtain Q, then the relationship is Π₁₁ between the input and output, as shown in Equation (2). $Π_{11} = {(M \times N)}^{T} \times Q$ (2)

According to the above formula, the fuzzy relations of each fuzzy rule can be obtained by this method, and then the overall fuzzy relations can be obtained to determine each fuzzy relation, which is shown by Equation (3): $Π = Π_{1} \cup Π_{2} \dots \cup Π_{k}$ (3)

By summarizing the above formula, the fuzzy relation of ΔK_p ΔK_i, and ΔK_d can be obtained, as shown in Equation (4): ${\begin{matrix} Π_{p} = \underset{i, j}{\cup} (E_{i} \times E_{C} \times Δ K_{Pij}) \\ Π_{i} = \underset{i, j}{\cup} (E_{i} \times E_{C} \times Δ K_{Iij}) \\ Π_{d} = \underset{i, j}{\cup} (E_{i} \times E_{C} \times Δ K_{Dij}) \end{matrix}$ (4) where the E_i is the i-th fuzzy state group e and E_Cj is the j - th and i - th fuzzy state of ɛc; ΔK_Pij, ΔK_Iij and ΔK_Dij are the i - th error and the j - th error rate with the fuzzy rule.

The fuzzy inputs are M₁ and N₁ after fuzzification processing of the precise quantity, and the fuzzy output Q₁ at this time can be obtained after fuzzy reasoning, as shown in Equation (5). $Q_{1} = {(M_{1} \times N_{1})}^{T} \circ Π$ (5) where Q₁ is the inference results under fuzzy rule conditions.

The result is a fuzzy quantity after fuzzy reasoning, but it can be an accurate value for controlling an object. An accurate value reflecting the result of fuzzy reasoning is called defuzzification [30]. Defuzzification is an output transformation between fuzzy inference results and precision, similar to the input transformation from fuzzy to the fuzzy controller. There are many classic methods, ranging from fuzzy to accurate. The following summarizes the advantages and disadvantages of each method and selects an appropriate method for defuzzification.

2.2 The membership degree and maximum degree of membership method

Membership degree function is a quantitative description of the fuzzy concept, and the correct determination of membership degree function is the basis of using the fuzzy set theory to solve practical problems. The determination of membership degree function is objective, but it is also subjective. It is generally determined according to the distribution rules of experience or statistical data, and can also be given by experts or authorities. The commonly used membership degree function methods include fuzzy statistical, binary comparison sorting method, expert comparison sorting, and neural network method. The common membership degree functions and membership degree result of fuzzy control are as following Figs. 1-2.

Fig. 1

Common membership degree functions.

Fig. 2

Degree of membership.

The maximum membership degree method is regarded as the accurate output result of defuzzification according to the highest membership degree’s fuzzy output value in the domain element. The minimum value can be taken as the maximum membership degree in the defuzzification [31], as shown in Equation (6): $Ψ^{*} = max Ψ_{C} (ν_{i})$ (6)

The disadvantage of the membership method is that it involves less information because it only considers the essential elements of membership degree and does not consider the secondary membership degree elements. Therefore, this kind of consideration is not comprehensive enough. Other methods of defuzzification are the weighted median method and the center-of-gravity method. The center-of-gravity method is also known as the area center method.

2.3 The average value and center-of-gravity method

When there are multiple elements with the maximum membership degree method (MMD) in the discussion field, the average value is taken as the output result of defuzzification precision, as shown in Equation (7): $Ψ^{*} = \frac{1}{k} \sum_{i = 1}^{k} max Ψ_{C} (ν_{i})$ (7)

This method takes the center of gravity of the area enclosed by the curve of the membership degree function and the x-coordinate and considers the accurate output result of defuzzification [32]. Contrasted with the MMD method and the weighted median method (WM), the output follows the input change when the input signal changes slightly, resulting in a smoother output inference control, as shown in Equation (8): $Ψ^{*} = \frac{\sum_{i = 1}^{k} Ψ_{C} (ν_{i}) ν_{i}}{\sum_{i = 1}^{k} Ψ_{C} (ν_{i})}$ (8)

2.4 Goodman-smith diagram

The Goodman-Smith diagram is an overall consideration of fatigue stress amplitude, the flat fatigue strength of uniform stress, and the mechanical property limits of materials. It is used to design the structural components of railway vehicles. The modified Goodman-Smith model is shown in Fig. 3 [33].

Fig. 3

Principle of the Goodman-Smith diagram.

The fatigue curve is constructed with the Goodman-Smith model. The random probability p of fatigue with the confidence coefficient Q and probabilistic fatigue strength η_-1N,p-Q is obtained, and the fatigue is obtained with Equation (9). ${\begin{matrix} η_{- 1 N, p - Q} = η_{- 1 N, av} - [Z_{p} + t_{1 - Q} (n - 1) η_{- 1 N, nms}] \\ η_{- 1 N, nms} = \sqrt{\frac{1}{n - 1} \sum_{i = 1}^{k} {(η_{- 1 N, i} - η_{- 1 N, av})}^{2}} \\ η_{- 1 N, av} = (\frac{1}{n}) \sum_{i = 1}^{k} η_{- 1 N, i} \end{matrix}$ (9) where k denotes the number of η_-1N, the percentage point Z_p is the normal distribution with the probability condition p, and k - 1 is freedom degree. η_-1N,av is the mean square error of fatigue life N in the joint probability distribution p - Q, and η_-1N,nms is the mean value of fatigue life N in the joint probability distribution p - Q.

The joint fatigue urban mass transit strength distribution with probability distribution p - Q can be obtained as shown in Equation (10): $η_{- 1 N, p - Q} = {\begin{matrix} \frac{lg N - M_{p - Q}}{N_{p - Q}}, N < N_{T} \\ \frac{lg N - G_{p - Q}}{E_{p - Q}}, N ⩾ N_{T} \end{matrix}$ (10) where M_p-Q, N_p-Q are the intercept and slope of the fatigue curve, and E_p-Q = K_T × N_p-Q, which separates into $G_{p - Q} = M_{p - Q} + (1 - K_{T}) N_{p - Q} lg \frac{lg N_{T} - M_{p - Q}}{N_{p - Q}} K_{T}$ where K_T is the index coefficient.

As the operation continues to a probability of p = 0.99, Q = 0.95, and the lubricating oil can firmly absorb the metal and form the boundary mask. The thickness ratio of the sheet is shown with Equation (11): $γ = \frac{χ_{min}}{\sqrt{Π_{q 1}^{2} + Π_{q 2}^{2}}}$ (11) where χ_min is the thickness of the oil mask and $Π_{q 1}$ and $Π_{q 2}$ are the root mean square deviations.

3 Simulation of speed and the curve radius

The radius of the rail curve and the speed of the train are two critical factors that cause rail wear. The parameters are shown in Tables 1 and 2.

Table 1
Wear power (Nm/s) of the railway under different speeds

Time (s) 5 10 15 20 25 30 35 40

Speed

44 6 70 96 96 94 95 94 53

48 18 84 102 102 101 102 98 7

52 26 106 109 109 108 108 40 0

56 38 113 116 116 115 88 0 0

60 49 120 120 120 120 54 0 0

64 61 123 126 126 128 10 0 0

68 70 130 130 130 90 0 0 0

72 80 135 136 136 56 0 0 0

Time (s)	5	10	15	20	25	30	35	40
44	6	70	96	96	94	95	94	53
48	18	84	102	102	101	102	98	7
52	26	106	109	109	108	108	40	0
56	38	113	116	116	115	88	0	0
60	49	120	120	120	120	54	0	0
64	61	123	126	126	128	10	0	0
68	70	130	130	130	90	0	0	0
72	80	135	136	136	56	0	0	0

Table 2

Wear power (Nm/s) of the railway at different radii

Time (s)	4	6	8	10	12	14	16	18	22	24	26
Radius
300	50	110	137	135	134	136	135	134	130	85	14
400	25	80	106	105	104	105	106	104	106	67	10
500	20	58	88	85	84	83	84	84	86	46	8
600	10	42	72	71	70	72	71	70	69	32	6
700	8	33	52	51	50	51	52	50	53	30	5
800	5	26	44	43	42	41	42	40	43	16	4
900	4	20	36	35	34	33	34	35	37	13	3
1000	3	12	29	28	27	28	27	27	29	10	2

These tables show that the rail curve radius and velocity of the train have significant influence. The wearing power increases rapidly from 2 seconds to 8 seconds. Then, these maintain a steady-state from 9 seconds to 24 seconds. Finally, they decline very quickly from 25 seconds to 28 seconds. The rail’s wearing power has a positive correlation with the velocity and a negative correlation with the radius. It can be shown that different wear powers are correlated with different velocities and radii.

4 Fuzzy PID control

4.1 PID CONTROL

The PID control method is widely used in various control systems, with a control loop feedback mechanism and continuously modulated control. In a PID controller, an error value e (t) is continuously calculated, which is the difference between the desired setpoint (SP) and a measured process variable (PV). A correction can be applied based on the proportional, integral, and derivative terms (denoted P, I, and D, respectively). Its mathematical form is expressed as Equation (12). $ν (t) = κ_{p} ɛ (t) + κ_{i} \int_{0}^{t} ɛ (t) dt + κ_{d} \frac{d ɛ (t)}{dt}$ (12) where ɛ (t) = γ (t) - η (t) shows the error between the SP and PV, K_p denotes the coefficient for the proportional term, K_i denotes the coefficient for the integral term, and K_d denotes the coefficient for the derivative term. The controller is shown in Fig. 4. where γ (t) is a setpoint or desired value and the measured process value (PV) is η (t).

Fig. 4

A PID controller diagram.

The transfer function is obtained by Laplace transform with the following equation: $H (s) = \frac{U (s)}{E (s)} = κ_{p} + \frac{κ_{i}}{s} + κ_{d}$ (13)

We can use different control algorithms as a suitable controlled object to meet specific practical needs [14]. For example, the proportional control factor can be obtained as:

$vt = κ_{p} ɛ (t)$ (14) or proportional and integral control factors obtained as $v (t) = κ_{p} [ɛ (t) + \frac{1}{T_{i}} \int_{0}^{t} ɛ (t) dt]$ (15)

4.2 Fuzzy control

Fuzzy control methodology has been applied in many fields, such as temperature control. “Fuzzy” indicates that the logic is expressed with “true” or “false” instead of as “partially true.” The advantages of fuzzy control are apparent. For example, the principle is simple, convenient to use, and has strong adaptability. The main idea of fuzzy control is to solve the standard fuzzy cognition in reality by precise mathematical methods. This relaxes the strict restriction that an object or phenomenon belongs to a set with absolute certainty or not. Especially in process control, the fuzzy controller model can deal with the process state by mapping sensor data or other inputs with appropriate membership functions. The advantage of fuzzy logic is that this solution is cast in human operators, and the experience can be explored with controller design, which makes it easier to mechanize tasks. The fuzzy controller is shown in Fig. 5.

Fig. 5

Fuzzy controller diagram.

Fuzzy controllers are simple and consist of an input stage, a processing stage, an output stage, and three steps: fuzzification, fuzzy rule design and reasoning, and defuzzification.

The fuzzy adaptive PID algorithm is a new idea based on the combination of PID and fuzzy control theory. It not only shows the high reliability and precision characteristics of PID but also the flexible and secure characteristics of fuzzy control that can adapt to complex environments. Its structure is shown in Fig. 6.

Fig. 6

Fuzzy and adaptive controller diagram.

A fuzzy set is defined with a membership function. The two parameters are the error value e and the error rate ec. These can be obtained according to the amount of lubrication oil (original empirical value) calculated from the flow sensor feedback. The input variables e ec’ fuzzy subsets are defined as sets of {NB, NM, NS, ZO, PS, PM, PB}, and the discourses are set as E ={ - 6, - 5, - 4, - 3, - 2, - 1, 0, 1, 2, 3, 4, 5, 6 } and E_C ={ - 3, - 2, - 1, 0, 1, 2, 3 }. To generate a conclusion, a logical inference or evaluation by the reasoning can be described with this method. It can generally follow the inference rules with the form: if case1 = M, and case2 = N, $\begin{matrix} then & effect = Q \end{matrix}$ , where M N, and Q are logical variables.

To simplify the calculations, the flow control system’s oil pump model can be directly equivalent to the model of the motor transfer function. This equation is: $G (s) = \frac{1 / K_{e}}{T_{m} T_{i} s^{2} + T_{m} s + 1}$ (16) where T_m and T_i are defined as: ${\begin{matrix} T_{m} = \frac{JR}{K_{e} K_{m}} \\ T_{i} = \frac{L}{R} \end{matrix},$ (17)

With Matlab Simulink, the results of the joint simulation are shown in Fig. 7.

Fig. 7

Chart of fuzzy adaptive control and general PID.

If we choose the step signal as the input signal, the simulating result is shown in Fig. 8.

Fig. 8

Simulation result (the red line is the fuzzy adaptive PID, and the blue line is the PID).

The simulation results show that the adaptive fuzzy PID method has better robustness, small overdose, faster response time, and increased stability than general PID as a control strategy. Therefore, the adaptive fuzzy PID method is more suitable to design the intelligent lubrication control of the rail curve than other methods. Based on these algorithms and the relationship between the speed and radius, the system can be realized with a microchip unit (MCU).

5 Design of the intelligent lubrication system

5.1 System design

An intelligent rail lubrication system was established based on fuzzy simulation. The intelligent lubrication system consists of an oil supply pump station, monitor and display unit, oil distributor, testing unit, and other accessories. The MCU not only realizes the detection of oil pressure, oil level, and other factors but also controls the operation of the system. The total architecture is shown in Fig. 9.

Fig. 9

System architecture diagram.

5.2 Software design

The system software includes sensor detection, system control, and monitor. The system flowchart is shown in Fig. 10.

Fig. 10

Chart of the technological process.

The process flow is as follows. First, the position and radius are set in the system. As the position sensor detects the position, the lubrication oil system runs the interrupting subprogram when the train reaches the starting position. The subprogram realizes the flow detection, fuzzy PID controlling, and parameter updating.

In different temperature and domains, the quantity of oil pump is as Table 3.

Table 3

The quantity of oil pump

Temperature	1#(g)	2#(g)	3#(g)	4#(g)
–20°	0.90	0.89	0.93	0.91
–20°	0.92	0.92	0.94	0.85
–30°	0.93	0.88	0.90	0.86
–30°	0.94	0.92	0.92	0.79
–40°	0.86	0.86	0.87	0.84
–40°	0.88	0.89	0.92	0.87

6 Conclusions

Maintenance upkeep and inspection are critical issues for a train’s wheels and rails, and these issues are more complicated at faster speeds and heavier loads. It is essential to analyze the force with theoretical and practical analysis when the train runs along curved rails, especially those with a small radius. The wearing power of the rail was simulated with the radius of the curve changing with velocity in the paper. The results showed that the force relations with the radius and speed are positive. We use the fuzzy PID controller to complete Compared with the adaptive controller, some advantages are found. Finally, with analysis and simulation, the system was designed with an MCU and sensors. As developed, the testing system is easily used as a solution to the issue of lubrication quality, but it is only the beginning of this research. The next, fuzzy adaptive finite-time fault-tolerant control for strict-feedback nonlinear systems, event-triggered robust fuzzy adaptive finite-time control of nonlinear systems with prescribed performance will be expanded.

Footnotes

Acknowledgments

This work was support by Key project of Education Department of Henan Province (21B510014), National Natural Science Foundation of China (61975015) and the Open Research Fund of Key Laboratory of Hydraulic and Waterway Engineering of the Ministry of Education (Grant No. SLK2017A02).

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Time (s)	5	10	15	20	25	30	35	40
Speed
44	6	70	96	96	94	95	94	53
48	18	84	102	102	101	102	98	7
52	26	106	109	109	108	108	40	0
56	38	113	116	116	115	88	0	0
60	49	120	120	120	120	54	0	0
64	61	123	126	126	128	10	0	0
68	70	130	130	130	90	0	0	0
72	80	135	136	136	56	0	0	0