J. Cent. South Univ. (2019) 26: 1910-1920

DOI: https://doi.org/10.1007/s11771-019-4123-6

Weak-fault diagnosis using state-transition-algorithm-based adaptive stochastic-resonance method

YIN Jin-tian(尹进田)^{1, 2}, XIE Yong-fang(谢永芳)¹, CHEN Zhi-wen(陈志文)¹,PENG Tao(彭涛)¹, YANG Chun-hua(阳春华)¹

1. School of Automation, Central South University, Changsha 410083, China;

2. Hunan Provincial Key Laboratory of Grids Operation and Control on Multi-Power Sources Area, Shaoyang University, Shaoyang 422000, China

Central South University Press and Springer-Verlag GmbH Germany, part of Springer Nature 2019

Abstract: In the early fault period of high-speed train systems, the interested characteristic signals are relatively weak and easily submerged in heavy noise. In order to solve this problem, a state-transition-algorithm (STA)-based adaptive stochastic resonance (SR) method is proposed, which provides an alternative solution to the problem that the traditional SR has fixed parameters or optimizes only a single parameter and ignores the interaction between parameters. To be specific, the frequency-shifted and re-scaling are firstly used to pre-process an actual large signal to meet the requirement of the adiabatic approximate small parameter. And then, the signal-to-noise ratio is used as the optimization target, and the STA-based adaptive SR is used to synchronously optimize the system parameters. Finally, the optimal extraction and frequency recovery of a weak characteristic signal from a broken rotor bar fault are realized. The proposed method is compared with the existing methods by the early broken rotor bar experiments of traction motor. Experiment results show that the proposed method is better than the other methods in extracting weak signals, and the validity of this method is verified.

Key words: stochastic resonance (SR); state-transition-algorithm (STA); fault diagnosis; broken rotor bar

Cite this article as: YIN Jin-tian, XIE Yong-fang, CHEN Zhi-wen, PENG Tao, YANG Chun-hua. Weak-fault diagnosis using state-transition-algorithm-based adaptive stochastic-resonance method [J]. Journal of Central South University, 2019, 26(7): 1910-1920. DOI: https://doi.org/10.1007/s11771-019-4123-6.

1 Introduction

With the rapid development and construction of railways across the world, high-speed and heavy-duty vehicles have become pioneers on oriented development. The requirements for the safety and comfort of train operations have become more important over time. For high-speed trains in particular, the systems are complex, the working environment is harsh, and their operating speeds are high. As the consequences of faults are possibly serious, these systems require effective monitoring and diagnosis of early faults to prevent further increases of the fault. Generally, early fault features are weak and may be crowded out by the heave noise, so it is hard to extract them accurately. The existing weak-fault detection methods mainly include adaptive filtering [1], signal model parameter identification [2], local mean decomposition [3], singular value decomposition [4, 5], the lifting multiwavelet method [6], and canonical correlation analysis [7, 8]. However, these methods are mainly trying to suppress noise in order to improve the signal-to-noise ratio (SNR). Although such processing methods exhibit certain good characteristics, they reduce the noise, while also weaken the characteristic signals, especially when the noise frequency and the characteristics of the signal frequency are coincident or close to each other, or when the characteristic signals are unavoidably damaged, which significantly affect the efficacy of the weak-fault detection methods [9, 10].

Stochastic resonance (SR) is a well-developed method proposed by BENZI et al, and uses noise to enhance weak characteristic signals [11-13]. As opposed to conventional denoising methods, it can transfer noise energy to a faint feature signal, reduce the noise, and strengthen weak signal that would otherwise be crowded by the noise. In recent years, it has been widely used for weak feature extraction and fault diagnosis. SR refers to a non-linear bistable system, and only under small periodic signals or noise is insufficient to make the outputs of system fluctuate between two steady states. However, under the combined action of noise and small periodic signals, a peak appears at the resonant frequency of the signal in the output spectrum. Furthermore, when the noise intensity is at a suitable value, the peak value of the output spectrum reaches a maximum, which is the quintessence of the aforementioned SR phenomenon. Owing to the limitations of adiabatic approximation theory and eigenvalue theory [14, 15], the traditional SR is only suitable for small-parameter signals [16], that is, when the amplitude and frequency of the input signal and the intensity of the noise both are very small (<<1).

When a traction motor in high-speed train system fails, the frequency of characteristic signals usually range from several tens to several thousands of hertz or even higher. In addition, the characteristic signal amplitude and noise intensity far exceed the applicable range of the traditional SR. Therefore, to detect such large-parameter signals, measures need to be taken to transform them into small-parameter signals. In Ref. [17], large- parameter signals are converted into small- parameter signals by using frequency- shifted and re-scaling double conversion. Furthermore, the detection of large-parameter signals is thus realized. The other solution that may be used to solve the aforementioned problems is to use secondary sampling SR [18], or to use a modulated SR method [19] to realize the application of SR in engineering signal detection. Existing SR methods are basically fixed-SR parameters or are only optimized for a single parameter [20, 21], ignoring the interaction between various parameters, which are actually local optimization; thus, the advantages of SR in weak- feature extraction are not fully considered.

Therefore, this paper presents a new adaptive SR method based on the state-transition-algorithm (STA) to diagnose weak-faults. The objectives are to (1) present an STA-based adaptive SR method;(2) convert large-parameter signals into small- parameter signals by frequency-shifted and re-scaling double conversion, and adopt STA to optimize parameters a and b of the frequency- shifted and re-scaling SR system simultaneously with the optimization goal of maximum SNR; and (3) apply the proposed method to the diagnosis of traction motor broken rotor bar weak faults in traction drive control system of high-speed trains.

The rest of the paper is organized as follows: Section 2 discusses the basics of SR and STA. Section 3 proposes the STA-based adaptive SR fault- diagnosis method; Section 4 presents the results of the simulation case and hardware-in-the- loop fault- injection benchmark case, and Section 5 draws the conclusion.

2 Basics of SR and STA

2.1 SR of bistable system

SR refers to the phenomenon whereby the SNR of an output signal is enhanced by cooperation between the weakly periodic signal and noise under certain non-linear conditions. SR models generally include three basic elements, a weak input signal, noise, and a non-linear system. There are many existing SR models, of which the most widely used one is the bistable system, which can be expressed by the Langevin equation as [22, 23]:

              (1)

where a and b are parameters of the bistable system of interest. Letbe the characteristic signal of the weak fault, A be the signal amplitude, f₀ be the frequency of the fault characteristic signal,be white noise, D be the noise intensity, and g(t) be white noise with a zero mean and a unit variance. Γ(t) has the statistical properties ofand Figure 1 shows the SR model of a bistable system, in which x(t) is the system output, and the non-linear bistable system is represented by a potential function as follows:

                        (2)

Figure 1 SR model of a bistable system

Figure 2 shows the curve of the bistatic potential function, having an unsteady state x=0, two steady states and a barrier height When there is no periodic signal and noise, the system is in either of two steady states, which are determined by the initial state of the system. When the system inputs the weak signal S(t), the signal energy is too small to cross the barrier, so the system can only move within a potential well. Then, if noise Γ(t) is added to the system, the noise energy will be partially transferred to the input weak signal, so that the system can overcome the potential barrier ΔU and jump between two steady states with the signal frequency f₀. Because some of the noise energy are converted into signal energy, the difference between the two stable positions ΔU is much larger than the amplitude of the input signal, which effectively amplifies the amplitude of the system output signal and improves the SNR of the output signal, that is, random resonance occurs. SNR is the most commonly used measure to quantitatively describe classical SR. The SNR is expressed as:

                   (3)

Figure 2 Bistable potential function curve

where ω is the signal frequency; S(ω) is the signal power spectral density; S_N(ω) is the noise intensity near the signal frequency.

2.2 State-transition-algorithm

STA is an effective random-search algorithm that is employed to optimize several parameters simultaneously [24]. This algorithm takes the solution of the optimization problem as a state, and makes the process of searching in the search space an evolutionary algorithm of a state-transition process. The four operators, i.e., rotation, translation, expansion and axesion, are used to solve continuous optimization problems, and this algorithm has strong global searching ability, as well as exhibits high precision and fast convergence [25].

STA can be written in the form:

                       (4)

where and x_k+1 respectively represent the current state and the transferred state; n is dimension of the system; A_k and B_k are the state-transition matrices; u_k is a function of current state x_k and the historical states; and Obj denotes the target function.

To solve the above optimal problem (4), four transformation factors are introduced.

1) Rotation operator,

                       (5)

where α is a positive constant called the rotation factor; is a random matrix that is uniformly distributed on [-1, 1]; this operator enables the algorithm to search within a hypersphere with center x_k and radius α satisfying

2) Translation operator,

                   (6)

where β is a positive constant called the translation factor, and is a random number that is uniformly distributed on [0, 1]; the operator makes the algorithm search along the positive direction of the gradient from a to b with the maximum search step β.

3) Expansion operator,

                          (7)

where γ is a positive constant called the expansion factor, is a random diagnostic matrix, and the elements in the matrix follow a Gaussian distribution. The extension operator can conduct a global search over the whole search space.

4) Axesion operator,

                         (8)

where δ is a positive constant, called the axesion factor; is a random diagnostic matrix, and there is only one random element in the matrix that is non-zero.

The flowchart of the STA operation is shown in Figure 3.

In Figure 3, best represents the optimal individuals of the population; SE is the individual number of items in the search population; α, β, γ and δ are the aforementioned four factors, and the transfer operation is embedded into the extension, rotation, and translation operations.

Figure 3 Flowchart of STA

3 STA-based adaptive SR fault diagnosis method

According to the adiabatic approximation theory, the assumption holds when the amplitude, frequency, and noise intensity of the input signal must be limited to the range of the small parameters, namely, they should be much less than 1. Therefore, before the signal is inputted into the bistable SR, the input signal needs to be modulated, normalized, frequency-shifted, and re-scaled to meet the requirements of the small parameters. Here, the frequency-shifted and re-scaling methods are used to pre-process the signal. The frequency of the measured signal is compressed linearly, then according to the Langevin equation analysis of bistable system SR spectra, the spectral characteristics of the weak signals are obtained. Finally, based on the compression scale ratio, the measured data is restored. To overcome the problem in which the traditional frequency-shifted and re-scaled SR only optimizes a single parameter, and the interaction between the system parameters is ignored, the proposed STA-based adaptive SR method can achieve the synchronized optimization of system parameters a and b. The STA takes a maximum SNR as the optimization target.

The flowchart of the STA-based adaptive SR method is shown in Figure 4, and is divided into the following steps. 1) Data are collected and filtered using a high-pass filter. 2) The acquired data are processed by the methods of frequency-shifted and re-scaling to meet the small parameters of SR. 3) The variables of STA are initialized and the objective function is established based on SNR. 4) The parameters a and b are optimized synchronously using STA. 5) Weak-fault characteristics are detected using SR. 6) The frequency and amplitude of weak-fault characteristics are recovered. 7) The fault is diagnosed according to the obtained fault characteristics.

Among them, the specific steps of frequency- shifted and re-scaling are given as follows:

1) Pass the input signal through a high-pass filter to eliminate low-frequency interference.

Figure 4 Flowchart for STA-based adaptive SR

2) Multiply the filtered output signal with the high-frequency carrier signal, which is essentially the frequency shift of the spectrum of the signal along the frequency axis, to achieve a shift of the signal spectrum to a lower frequency.

3) Set a frequency compression ratio to obtain a small-parameter signal after frequency compression. The achieved signal frequency is

                         (9)

where f_out denotes the output frequency after the frequency-shifted scale; f_in is the input signal frequency; f_c represents the high-frequency carrier frequency; and R represents the frequency compression ratio.

4 Case study

4.1 A simulation case study

To verify the validity of the proposed method, we set the input signal to be Gaussian white noise with a mean value 0 and intensity 0.02 is added to the signal; the number of sampling points is 8000; and the sampling frequency is 5 Hz. Figure 5 shows the original time-domain waveform and spectrum of the input signal, respectively. It can be seen that the frequency of the input periodic signal cannot be found in the spectrum shown in Figure 5(b).

The original signal is fed into the traditional SR system. The parameter setting is that a and b are random inputs, or we fix one of the parameters and adjust the other parameter within a certain range. The SNR is used as a standard for evaluating the merits of the SR. Figure 6 shows the fixed parameter case for a=0.1 and for b on [0, 5]. The result shows that when b=0.9440, SNR=33.6653 dB is the maximum value.

Figure 5 Input signal shown in time and frequency- domains:

Figure 7 shows the case with a fixed parameter b=1 and a on [0, 5]. The result shows that when a=0.0670, SNR=29.3099 dB is the maximum value.

When using the proposed method both parameters a and b of the SR system are optimized in parallel using the STA. Both a and b are within the range [0, 5] to find the optimal parameter combination in this SR system. The optimization results, a=0.2298, b=4.1436, give an output SNR=39.7594 dB, compared with the traditional method of fixing a and b, respectively. This proposed method has therefore increased the SNR by 18.1% and 35.7%, respectively. The time- domain signal and spectrum after processing are shown in Figure 8. By comparing Figure 8 with Figures 6 and 7, it is found that the proposed method outputs a time-domain signal that is more periodic than the conventional method; the effect of SR is more pronounced; and the characteristic frequency in the output spectrum is more prominent than when using the conventional method. Therefore, the proposed STA-based adaptive SR method is more suitable for detecting early weak-faults than the traditional method.

Figure 6 SR analysis results with fixed a=0.1 and variable b:

Figure 7 SR analysis results with fixed b=1 and variable a:

Figure 8 Results of STA-based adaptive SR analysis:

4.2 Hardware-in-the-loop fault-injection benchmark case study

Common faults of traction motor include bearing fault, broken rotor bar fault, stator winding inter-turn short circuit fault, air gap eccentricity fault, etc. Among them, broken rotor bar fault accounts for 8% of all faulty cases; other cases correspond to 44% for bearing fault, 26% for stator faults, and 22% for other faults [26]. The early detection of broken rotor bar defects cannot be detected in a timely manner. This may cause serious faults, and it is therefore important to detect these faults early in the process in order to avoid the occurrence of serious damage to the system [27].

The experimental device used here included a dSPACE-based CRH2 EMU traction drive control system controller in the loop real-time fault semi-physical platform (Figure 9). It includes a real-time simulator, a fault-injection unit (FIU), a physical traction drive control unit (TCU), and a real-time data acquisition and monitoring unit. The controller is a physical object. The dSPACE hardware includes a DS1007 CPU board, a DS5203 field-programmable gate array (FPGA) board, a DS4004 digital I/O board, and a DS2103 multi-channel high-precision D/A board. It is used to build a traction asynchronous motor and a main circuit, and establishes communication between them. The fault injection benchmark consists of four fault injection modules: traction converter, traction motor, sensor and traction drive controller TCU, which are constructed by means of signal conditioning, the references [28, 29] gives a detailed introduction, and the fault injection simulation platform can be downloaded on the website http://gfist.csu.edu.cn/ indexE.html. The FIU realizes the early weak fault injection of the broken rotor bar of the traction motor in the traction drive control system. According to the characteristics of the broken rotor bar fault of the traction motor, a specific fault signal is generated, and the signal is adjusted with the normal signal to generate the fault injection signal. Then the normal signal is replaced by the signal and injected into the fault injection point. In this way, it is realized the fault injection of the broken rotor bar fault of traction motor [30, 31].

The main parameters of the CRH2 traction drive control system are shown in Table 1. The motor speed after the platform operates stably is 3860 r/min, and the train speed is 200 km/h; the load torque of the traction motor is related to the train running speed and the train's own parameters, which is 210.7 N·m when running stably. The frequency of the broken rotor bar fault feature is calculated as[32, 33], where f_D is the characteristic frequency of the broken rotor bar, s is the motor slip rate, s=0.018, and f is the voltage frequency loaded on the three-phase winding of the motor stator, f=131 Hz. From the calculations, f_D1=135.7 Hz and f_D2=126.3 Hz. Because these two frequencies always appear in pairs, the detection of one of the fault frequencies, such as f_D1=135.7 Hz, can determine the occurrence of a broken rotor bar fault.

Figure 9 Semi-physical platform of the CRH2 EMU traction drive system

Table 1 Electrical parameters of system simulation platform

Considering the influence of background noise, Gaussian white noise is added to the measured motor stator current signal. Since the theoretical fault characteristic frequency is 135.7 Hz, the passing frequency of the high-pass filter is set to 135 Hz, and the carrier frequency is consistent with the pass frequency of the high-pass filter, which is 135 Hz, the sampling frequency is 100 kHz, and the frequency compression ratio is 1000. Figure 10 shows the waveform and frequency spectrum of the stator A-phase current primary signal in the early stage of a rotor break. It can be seen that the background noise in the time-domain signal is strong, and overpowers the characteristic features of the broken rotor bar fault. In the frequency spectrum, the 135-200 Hz region is locally amplified, and no obvious peaks can be found at the fault feature frequency. Considering that the actual traction drive system emits much stronger background noise, the background noise in the time-domain signal and frequency spectrum will completely mask the fault characteristics of the broken rotor bar, making it impossible to determine whether the rotor bar is faulty, resulting in a missed diagnosis.

Figure 10 Actual signal of broken rotor bar fault with added noise:

The signal is analyzed by the traditional frequency shifted and re-scaling SR. Figure 11(a) shows the results for the fixed parameter a=1, and when value of b is automatically optimized within [0, 10], which maximizes the system output SNR at b=0.03, giving SNR=40.4326 dB. Figure 11(b) shows the results obtained for fixed parameter b=1, and when the value of a is automatically optimized within [0, 10], which maximizes the system output SNR at a=4.52, giving SNR=45.1606 dB.

The STA-based adaptive SR is used to analyze the signal by setting the search strength to 20 and the problem dimension to 2; the range of the two parameters to be optimized is [0, 30]; the number of iterations is 30; and the output waveform after SR is shown in Figure 12, where a=1.5754, b=21.2303, and SNR=51.7158 dB.

Figure 11 Actual signal of broken rotor bar fault with added noise:

Figure 12 Results of STA adaptive SR analysis:

By comparing Figures 11(a), (b) and 12(b), it can be seen that the STA has a higher spectral peak and the SNR values increase by 27.9% and 14.5%, respectively.

4.3 Comparison with existing methods

To further clarify the diagnosis performance of the proposed STA adaptive SR method, several existing methods, namely GA adaptive SR, PSO adaptive SR, DE adaptive SR, traditional SVD and traditional wavelet transform are considered in the following comparison study. The comparison results are shown in Table 2. It can be seen from the analysis of the same input signal that the STA adaptive SR output has a higher SNR, indicating that the proposed method is more suitable for the extraction of weak characteristic signals similar to the broken rotor bar of traction motor.

Table 2 Comparisons between proposed method and exiting methods

The experimental results show that the results of the traditional SR method are better than the original spectrum for the early detection of a broken rotor bar fault signal in a traction asynchronous motor. The adaptive SR method proposed in this paper makes full use of the STA global searching characteristics, and can make use of the interaction between system parameters and their optimization. Thus, the proposed method gives better results in this case, and is more advantageous than the existing methods, including GA adaptive SR, PSO adaptive SR, DE adaptive SR, traditional SVD and traditional wavelet transform.

5 Conclusions

In this paper, a STA-based adaptive SR method is proposed to solve weak fault diagnosis in high-speed train systems. We focused on the characteristics whereby early faults of actual systems are weak and their signals are overpowered by heavy noise. Firstly, the large-parameter signals are pre-processed by frequency-shifted and re-scaling method. Secondly, a global search using the proposed STA is performed, with the maximization of the SNR as the optimization goal, along with adaptive selection and synchronization optimization of the SR parameters. Finally, the optimal extraction of the weak characteristic signal of the broken rotor bar fault is realized. This proposed method is verified on a numerical case and the hardware-in-the-loop fault injection benchmark of a high-speed train traction drive control system with a broken rotor bar fault. The achieved results show that the proposed method has obvious advantages compared with traditional SR and other exiting methods, and its effectiveness in early fault diagnosis is verified.

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(Edited by HE Yun-bin)

中文导读

基于状态转移自适应随机共振的微弱故障诊断

摘要：针对高速列车系统早期故障发生时特征信号微弱且淹没在强背景噪声之中的问题，提出了一种状态转移自适应随机共振方法。该方法解决了传统随机共振固定参数或只对单一参数进行优化、忽略参数之间交互作用的不足。首先，利用移频变尺度对实际大信号进行预处理，使信号满足绝热近似小参数的要求；然后，以信噪比作为优化目标，采用状态转移随机共振对系统参数进行同步优化；最终，实现转子断条故障微弱特征信号的最优提取和频率恢复。通过对牵引电机早期转子断条实验进行比较，结果表明，所提方法的微弱信号提取效果明显优于其他算法，验证了该方法的有效性。

关键词：随机共振；状态转移；故障诊断；转子断条

Foundation item: Projects(61490702, 61773407, 61803390, 61751312) supported by the National Natural Science Foundation of China; Project(61725306) supported by the National Science Foundation for Distinguished Young Scholars of China; Project(61621062) supported by the Foundation for Innovative Research Groups of the National Natural Science Foundation of China; Project(2017TP1002 ) supported by Hunan Provincial Key Laboratory, China; Project(6141A0202210) supported by the Program of the Joint Pre-research Foundation of the Chinese Ministry of Education; Project(61400030501) supported by the General Program of the Equipment Pre-research Field Foundation of China; Project(2016TP1023) supported by the Science and Technology Project in Hunan Province Hunan Science and Technology Agency of China; Project(2018FJ34) supported by the Science and Technology Project in Shaoyang Science and Technology Agency of China

Received date: 2018-10-08; Accepted date: 2019-03-07

Corresponding author: CHEN Zhi-wen, PhD, Lecturer; Tel: +86-18182113567; E-mail: zhiwen.chen@csu.edu.cn; ORCID: 0000-0002- 4759-0904