#### 期刊菜单

Rail Corrugation Detection Method Based on Sparse Representation Research
DOI: 10.12677/MOS.2024.131086, PDF, HTML, XML, 下载: 88  浏览: 147

Abstract: Corrugation is one of the common problems in the operation of rail transit. In order to detect rail corrugation, different from the traditional corrugation detection method, the vibration signal of the vehicle axle box is processed in a MATLAB environment to obtain the rail corrugation waveform. The dynamic modeling is carried out, and the simplified model of train lumped parameters is es-tablished to derive the train vibration signal under the condition of corrugation. The signal sparse representation method is used to extract and diagnose the minor fault features. The L1 norm as the regularization method is the most commonly used method at present. However, based on this method, it is easy to underestimate the amplitude of the reconstructed signal, which may cause large errors. Therefore, the sparse representation method based on the GMC penalty function is used to solve this problem. The objective function is established, and the convexity-preserving con-ditions of the objective function are studied. The forward-backward splitting algorithm (FBS) is used to solve the sparse representation objective function, and the performance of the two methods in reconstructing the signal is compared. The results show that the GMC penalty function has better performance in signal reconstruction than the L1 penalty function. Then, the sparse representation method based on the GMC penalty function is simulated and measured to verify the effectiveness of the proposed method, and the shortcomings are analyzed.

1. 引言

2. 动力学建模与稀疏表示理论

2.1. 轨道波磨简化模型

(a) 单一谐波 (b) 多谐波

Figure 1. Simplified model of harmonic excitation

${Z}_{o}\left(t\right)=\frac{1}{2}a\left(1-\mathrm{cos}\omega t\right)\text{}0\le t\le L/\upsilon$ (1)

${Z}_{o}\left(t\right)=\frac{1}{2}a\left(1-\mathrm{cos}\omega t\right)\text{}0\le t\le nL/\upsilon$ (2)

2.2. 列车集总参数简化模型

Figure 2. Train lumping model

$\left\{\begin{array}{l}{m}_{1}{\stackrel{¨}{z}}_{1}+{k}_{1}\left({z}_{1}-{z}_{2}\right)=0\\ {m}_{1}{\stackrel{¨}{z}}_{2}+{c}_{2}{\stackrel{¨}{z}}_{2}+{k}_{2}{z}_{2}-{k}_{1}\left({z}_{1}-{z}_{2}\right)=0\end{array}$ (3)

$\left\{\begin{array}{l}{m}_{1}{\stackrel{¨}{z}}_{1}+{k}_{1}\left({z}_{1}-{z}_{2}-\eta \right)=0\\ {m}_{1}{\stackrel{¨}{z}}_{2}+{c}_{2}{\stackrel{¨}{z}}_{2}+{k}_{2}{z}_{2}-{k}_{1}\left({z}_{1}-{z}_{2}-\eta \right)=0\end{array}$ (4)

$\left\{\begin{array}{l}\frac{{\stackrel{¨}{z}}_{1}}{{\omega }_{1}^{2}}+{z}_{1}-{z}_{2}=\eta \\ \frac{{\stackrel{¨}{z}}_{2}}{{\omega }_{2}^{2}}+\frac{{c}_{2}}{{k}_{1}}{\stackrel{˙}{z}}_{2}+\left(\frac{k{}_{2}}{{k}_{1}}+1\right)-{z}_{1}=-\eta \end{array}$ (5)

$\eta \left(s\right)=\frac{\frac{1}{2}\alpha {\omega }^{2}}{s\left({s}^{2}+{\omega }^{2}\right)}$ (6)

$\left\{\begin{array}{l}{z}_{1}\left(s\right)={H}_{{z}_{1}}\left(s\right)\eta \left(s\right)\\ {z}_{2}\left(s\right)={H}_{{z}_{2}}\left(s\right)\eta \left(s\right)\end{array}$ (7)

$\left\{\begin{array}{l}\left(\frac{{s}^{2}}{{\omega }_{1}^{2}}+1\right){H}_{{z}_{1}}\left(s\right)-{H}_{{z}_{2}}\left(s\right)=1\\ {H}_{{z}_{1}}\left(s\right)-\left(\frac{{s}^{2}}{{\omega }_{2}^{2}}+\frac{{c}_{2}s}{{k}_{1}}+\frac{{k}_{2}}{{k}_{1}}+1\right){H}_{{z}_{2}}\left(s\right)=1\end{array}$ (8)

${H}_{{z}_{1}}\left(s\right)=\frac{\frac{{s}^{2}}{{\omega }_{2}^{2}}+\frac{{c}_{2}s}{{k}_{1}}+\frac{{k}_{2}}{{k}_{1}}}{\frac{{s}^{4}}{{\omega }_{1}^{2}{\omega }_{2}^{2}}+\frac{{c}_{2}{s}^{3}}{{k}_{1}{\omega }_{1}^{2}}+\left[\frac{1}{{\omega }_{2}^{2}}+\left(\frac{{k}_{1}}{{k}_{2}}+1\right)\frac{1}{{\omega }_{1}^{2}}\right]{s}^{2}+\frac{{c}_{2}s}{{k}_{1}}+\left(\frac{{k}_{2}}{{k}_{1}}+1\right)}$ (9)

$\begin{array}{c}{\stackrel{¨}{z}}_{1}={s}^{2}{z}_{1}\left(s\right)\\ ={s}^{2}{H}_{{z}_{1}}\left(s\right)\eta \left(s\right)\\ =\frac{\frac{{s}^{3}}{{\omega }_{2}^{2}}+\frac{{c}_{2}{s}^{2}}{{k}_{1}}+\frac{{k}_{2}s}{{k}_{1}}}{\frac{{s}^{4}}{{\omega }_{1}^{2}{\omega }_{2}^{2}}+\frac{{c}_{2}{s}^{3}}{{k}_{1}{\omega }_{1}^{2}}+\left[\frac{1}{{\omega }_{2}^{2}}+\left(\frac{{k}_{1}}{{k}_{2}}+1\right)\frac{1}{{\omega }_{1}^{2}}\right]{s}^{2}+\frac{{c}_{2}s}{{k}_{1}}+\left(\frac{{k}_{2}}{{k}_{1}}+1\right)}\cdot \frac{\frac{1}{2}\alpha {\omega }^{2}}{{s}^{2}+{\omega }^{2}}\end{array}$ (10)

${\stackrel{¨}{z}}_{1}\left(s\right)=\underset{i=1}{\overset{3}{\sum }}\frac{{A}_{i}}{s+{p}_{i}}+\frac{Bs+C}{{s}^{2}+{\omega }^{2}}$ (11)

${\stackrel{¨}{z}}_{1}=\underset{i=1}{\overset{3}{\sum }}{A}_{i}{\text{e}}^{-{p}_{i}t}+D\mathrm{cos}\left(\omega t+\psi \right)$ (12)

2.3. 稀疏表示模型

$x=Ac=\underset{i=1}{\overset{\underset{o}{m}}{\partial }}{a}_{i}{c}_{i}$ (13)

Figure 3. Sparse representation model

$\stackrel{^}{c}=\mathrm{arg}\underset{c}{\mathrm{min}}\left\{\frac{1}{2}{‖y-Ac‖}_{2}^{2}+\lambda P\left(c\right)\right\}$ (14)

3. 基于GMC罚函数的稀疏表示方法

GMC惩罚函数是一类具有非凸性质的函数，能够保证整个目标函数是凸的，其自身的特性能够解决传统BPD算法求解过程中对幅值计算不准确的问题，又能够更大程度地加强信号特征的稀疏性。

3.1. GMC罚函数

GMC罚函数是一种特殊的稀疏性诱导非凸罚函数，在特定情况下，保证构建的稀疏表示目标函数为凸函数，并通过凸优化算法，优化目标函数，从而获得最优解。

GMC罚函数 ${\varphi }_{B}:{R}^{N}\to R$ 定义为

${\phi }_{B}\left(x\right)={‖x‖}_{1}-{S}_{B}\left(x\right)$ (15)

${S}_{B}\left(x\right)=\underset{v\in {R}^{N}}{\mathrm{min}}\left\{{‖v‖}_{1}+\frac{1}{2}{‖B\left(x-v\right)‖}_{2}^{2}\right\}$ (16)

3.2. 建立目标函数与保凸性证明

$y=x+e$ (17)

$\begin{array}{c}J\left(c,v\right)=\frac{1}{2}{‖y-Ac‖}_{2}^{2}+\lambda {\phi }_{B}\left(c\right)\\ =\frac{1}{2}{‖y-Ac‖}_{2}^{2}+\lambda {‖c‖}_{1}-\underset{v}{\mathrm{min}}\left\{\lambda {‖v‖}_{1}+\frac{\lambda }{2}{‖B\left(c-v\right)‖}_{2}^{2}\right\}\end{array}$ (18)

$\begin{array}{c}J\left(c,v\right)=\frac{1}{2}{‖y-Ac‖}_{2}^{2}+\lambda {‖c‖}_{1}-\underset{v}{\mathrm{min}}\left\{\lambda {‖v‖}_{1}+\frac{\lambda }{2}{‖B\left(c-v\right)‖}_{2}^{2}\right\}\\ =\underset{v}{\mathrm{max}}\left\{\frac{1}{2}{‖y-Ac‖}_{2}^{2}+\lambda {‖c‖}_{1}-\lambda {‖v‖}_{1}-\frac{\lambda }{2}{‖B\left(c-v\right)‖}_{2}^{2}\right\}\\ =\underset{v}{\mathrm{max}}\left\{\frac{1}{2}{c}^{\text{T}}\left({A}^{\text{T}}A-\lambda P\right)c+\lambda {‖c‖}_{1}+\frac{1}{2}{y}^{\text{T}}y-\lambda {‖v‖}_{1}+\left(\lambda {v}^{\text{T}}P-{y}^{\text{T}}A\right)c-\frac{\lambda }{2}{v}^{\text{T}}Pv\right\}\\ =\frac{1}{2}{c}^{T}\left({A}^{\text{T}}A-\lambda P\right)c+\lambda {‖c‖}_{1}+\underset{v}{\mathrm{max}}\left\{g\left(c,v\right)\right\}\end{array}$ (19)

${A}^{\text{T}}A-\lambda P\succ =0$ (20)

$B=\sqrt{\gamma /\lambda }A,0\le \gamma \le 1$ (21)

3.3. 构造离散傅里叶变换字典

${\left[{A}_{1}c\right]}_{n}=\sqrt{K}{\left[DF{T}_{K}^{-1}\left\{c\right\}\right]}_{n}，c\in \stackrel{⌢}{K},n\in \stackrel{⌢}{N}$ (22)

${\left[{A}_{1}^{H}x\right]}_{k}=\frac{1}{\sqrt{K}}{\left[DF{T}_{K}^{-1}\left\{x\right\}\right]}_{k}，k\in \stackrel{⌢}{K},x\in \stackrel{⌢}{N}$ (23)

${‖a‖}_{2}=\sqrt{\frac{N}{K}}$ (24)

3.4. 求解算法

$\left({C}^{\text{opt}},{v}^{opt}\right)=\mathrm{arg}\underset{c}{\mathrm{min}}\underset{v}{\mathrm{max}}J\left(c,v\right)$ (25)

$J\left(c,v\right)=\frac{1}{2}{‖y-Ac‖}_{2}^{2}+\lambda {‖c‖}_{1}-\lambda {‖v‖}_{1}-\frac{\gamma }{2}{‖A\left(c-v\right)‖}_{2}^{2}$ (26)

$J\left(c,v\right)={f}_{1}\left(c,v\right)+{f}_{2}\left(c,v\right)$ (27)

${f}_{1}\left(c,v\right)=\frac{1}{2}{‖y-Ac‖}_{2}^{2}-\lambda {‖v‖}_{1}-\frac{\gamma }{2}{‖A\left(c-v\right)‖}_{2}^{2}$ (28)

${f}_{2}\left(c,v\right)=\lambda {‖c‖}_{1}$ (29)

${\omega }^{\left(k\right)}={c}^{\left(k\right)}-\mu {A}^{\text{T}}\left(A\left({c}^{\left(k\right)}+\gamma \left({\upsilon }^{\left(k\right)}-{c}^{\left(k\right)}\right)\right)-y\right)$ (30)

${c}^{\left(k+1\right)}=soft\left({\omega }^{\left(k\right)},\mu \lambda \omega \right)$ (31)

$\mu <2/\text{max}\left\{1,\gamma /\left(1-\gamma \right){‖{A}^{\text{T}}A‖}_{2}\right\}$ (32)

${u}^{\left(k\right)}={v}^{\left(k\right)}-\mu \gamma {A}^{\text{T}}A\left({v}^{\left(k\right)}-{c}^{\left(k\right)}\right)$ (33)

${v}^{\left(k+1\right)}=soft\left({u}^{\left(k\right)},\mu \lambda w\right)$ (34)

$soft\left(x,T\right)=x\cdot \mathrm{max}\left\{1-T/|x|\right\}$ (35)

Table 1. GMC regularized sparse decomposition algorithm

3.5. GMC罚函数与L1罚函数信号重构的效果对比

${\stackrel{¨}{z}}_{1}=\underset{i=1}{\overset{3}{\sum }}{A}_{i}{\text{e}}^{-{p}_{i}t}+D\mathrm{cos}\left(\omega t+\psi \right)$ (36)

Figure 4. Artificial signal

Figure 5. L1 sparse representation of fault signal

Figure 6. GMC sparsely represents fault signal

RMSE利用观测值同真实值之间的差异表明测量数据背离实际数据的程度，RMSE值越大说明对于原始信号的重构能力越差，值越小，重构性能越好。其计算表达式如式(37)所示

$\text{RMSE}=\sqrt{\frac{1}{N}{‖x-\stackrel{^}{x}‖}_{2}^{2}}$ (37)

$\text{SNR}=\frac{{P}_{\text{signal}}}{{P}_{\text{noise}}}$ (38)

${\text{SNR}}_{dB}=10{\mathrm{log}}_{10}\left(\text{SNR}\right)\left(dB\right)$ (39)

$\text{RMSE}\left(L1\right)>\text{RMSE}\left(\text{GMC}\right)$ (40)

4. 工程试验分析

4.1. 数据分析

$\stackrel{^}{c}=\mathrm{arg}\underset{c}{\mathrm{min}}\left\{\frac{1}{2}{‖y-Ac‖}_{2}^{2}+\lambda P\left(c\right)\right\}$ (41)

Figure 7. Normal segment signal processing

Figure 8. Corrugation section signal processing

Figure 9. Flow chart

4.2. 信号实测

Figure 10. Typical corrugation

Table 2. Fault diagnosis results of corrugation

5. 总结

1) 基于信号稀疏表示基本原理，根据信号分量的不同特征，构造不同的变换字典，求解目标函数，完成对故障成分和稳态成分的提取和重构，开展仿真，对比了L1范数正则化和GMC非凸罚函数的稀疏表示方法对仿真信号的效果，得出GMC罚函数在信号重构方面的性能更好，优于L1罚函数。

2) 基于GMC罚函数的稀疏表示法进行工程实验，分析重构信号的时域图和频谱图，得出故障特征，判断波磨位置，验证了所提方法的有效性并针对实验结果的不足之处加以分析。

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