Research on Sparsity Measures for Rotating Machinery Health Monitoring

Suppose a sparse measure $S$ as a function with the mapping as follows

where $\mathbb{X}^{n}$ represents a complex vector of length N. Metric $S$ maps complex vectors into a real number. According to previous research and inspiration from economics[23, 18], it is generally believed that ideal SMs should own six attributes.

D1. Robin Hood

Dalton’s 1st Law states that Robin Hood decreases sparsity. Robin Hood operation subtracts a positive constant $\alpha$ from any relatively large element $x_{i}$ in the vector, and adds $\alpha$ to a relatively small element $x_{j}$. After operation, new $x_{i}$ is still larger than new $x_{j}$. The mapping value of the sparse metric $S$ on the new vector becomes smaller. Robin Hood can be formally defined as follows:

for all $\alpha$, $x_{i}$, $x_{j}$ such that $x_{i}>x_{j}$ and $0<\alpha<\frac{x_{i}-x_{j}}{2}$.

D2. Scaling

Dalton’s modified 2nd Law states that Sparsity is scale invariant. After all elements in the vector are multiplied by a constant factor $\alpha$, the sparse value of the vector remains invariant. Scaling can be formally defined as follows:

D3. Rising Tide

Dalton’s 3rd Law states that adding a constant to each element in vector can reduce sparsity. After adding a positive constant $\alpha$ to all elements in the vector, the sparse value of the vector becomes smaller. We exclude the case where all elements are equal as this is equivalent to D2. Rising Tide can be formally defined as follows:

D4. Cloning

Dalton’s 4th Law states that sparsity is invariant under cloning operation. The sparse value is unchanged after the vector is joined with itself. Cloning can be formally defined as follows:

P1. Bill Gates

An element $x_{i}$ in the vector becomes infinitely large, and the sparse value of the vector also tends to infinitely large. Bill Gates can be formally defined as follows:

$\forall i$, $\exists\beta=\beta_{i}>0$, such that $\forall\alpha>0$

P2. Babies

After adding an element of zero into the vector, and the sparse value of the new vector will increase. Babies can be formally defined as follows:

In previous work, Hurely[18] proved the sparsity of kurtosis, Hoyer, Pq-mean and Gini index and other classical sparse measures. Since smoothness index and negative entropy were added to the sparsity measures family, their sparse properties have not been fully studied. The first work of this paper is to analyze the sparse quantification capability of smoothness index and negative entropy from the above six attributes. The result of mathematical derivation shows that smoothness index satisfies nonnegativity, D2 and D4; and negative entropy satisfies nonnegativity, D1, D2, D3, D4, P1, as shown in Table I. The detailed proofs are given in Appendix. Based on our research with Dr. Hurely, the following conclusion can be drawn: Pq-mean, Hoyer, Gini index and negative entropy have better sparse quantification capability among existing popular sparse measures.

Suppose a function has the form of Eq. (8), then it can be called a power mean function, where $a_{i}(i=1,2,\dots,n)$ are incomplete equal positive constants. This function is defined on the entire real number axis, and there is always $F(x)>0$. Obviously, no matter how x varies, the power mean function does not tend to infinity. Therefore, power mean function is bounded. If $min{\left\{a_{i}\right\}}=a$, $max{\left\{a_{i}\right\}}=A$, then $a\leq F(x)\leq A$.

Based on the above, we define a function with the form $\Gamma(\mathbf{x},y)=[\frac{1}{N}\sum_{n=1}^{N}\left(x_{n}\right)^{y}]^{\frac{1}{y}}$ as a multivariate power mean function (MPMF), where $\mathbf{x}=\left[x_{1},x_{2},\cdots x_{n}\right]$ represents a row vector. The classical RMS metric can be expressed as $\Gamma(\mathbf{x},2)$ in the form of MPMF. In addition, the multivariate power mean inequality needs to be introduced: If $y_{1}>y_{2}$, then $\Gamma\left(\mathbf{x},y_{1}\right)\geq\Gamma\left(\mathbf{x},y_{2}\right)$, and the equality holds if and only if $x_{1}=x_{2}=\cdots x_{n}$. The proof only needs to use Jensen inequality in mathematical analysis (i.e., up and down convexity), and take the auxiliary function $f(x)=x^{a}$.

Assume that a raw vibration signal is denoted as $x[n],n=1,2,3,\dots N$. Then, a bandpass filtered signal $x_{l,h}[n]$ can be obtained by a using bandpass filter with a passband $[l,h]$. Further, a complex signal $\overline{x_{l,h}}[n]=x_{l,h}[n]+j\cdot\text{{Hilbert}}\left\{x_{l,h}[n]\right\}$ and square envelope $SE_{l,h}[n]=\left|\overline{x_{l,h}}[n]\right|^{2}$ can be obtained by Hilbert transform. To simplify symbols, $SE_{l,h}[n]$ is replaced by $\overline{x_{l,h}}[n]$ for the rest of the paper.

Six classic sparsity measures are respectively reformulated as the ratios of different multivariate power mean functions in this section, including spectral kurtosis, spectral $Lp/Lq$ norm, pq-mean, spectral smoothness index, spectral negative entropy and Gini index. The following subsections give the details of the reformulation.

Inspired by reformulation of the six classical sparsity measures in Subection B, we propose a general paradigm based on ratios of multivariate power mean functions for designing more advanced health indexes for machine health monitoring and degradation state assessment.

Complex health indexes designed by combining ratios of the sum of multiple MPMFs:

Complex health indexes designed by combining ratios of the product of multiple MPMFs:

where $p_{m}$ and $q_{n}$ represent different weights of the power average function respectively, and satisfy $\sum_{m=1}^{M}p_{m}=1,\sum_{n=1}^{N}q_{n}=1$, $\lambda$ is a tradeoff parameter, and $c$ is an offset coefficient providing a baseline for indexes. From previous studies, it is clear that $\Gamma(\alpha\mathbf{x},y)=\alpha\Gamma(\mathbf{x},y)$ and $\Gamma(\mathbf{x}\|\mathbf{x}\|\cdots\|\mathbf{x},y)=\Gamma(\mathbf{x},y)$ , so $PHI$ and $MHI$ satisfy at least two sparse attributes: D2 and D4.

According to Eq. (18) and Eq. (19), we can give a general paradigm as follows:

The specific expressions for the four newly designed HIs as examples are as follows:

where $\tau$ represents slip coefficient which is related to shifted percentage [25]. In our case, $\tau$ is set to 0.065. At the same time, an extra property of the newly designed HIs need to be emphasized. According to multivariate power mean inequality, $HI_{1}$-$HI_{4}$ have strict upper and lower bounds $[0,1)$. Therefore, it is helpful to set a reasonable and effective failure threshold for machine health monitoring under different operating conditions.

The accelerated bearing degradation experimental dataset provided by the NSF I/UCR Center for Intelligent Maintenance Systems was firstly used to verify the effectiveness and superiority of the designed health indexes. The experimental platform is shown in Fig. 1 with four bearings installed on one shaft. The sampling frequency is set at 20KHz, and 20,480 points are collected each time with an interval of 1 second. Set No.2 collected a total of 984 files in Channel 1. The bearing finally produced an outer ring failure, and the outer ring failure frequency could be calculated as 236.4Hz according to the theoretical formula of failure frequency.

The bearing degradation curves are plotted by classic spectral kurtosis, spectral smoothness index, spectral negative entropy and Gini index as shown in Fig. 2, where the real values of spectral smoothness index are multiplied by -1 to achieve an increasing trend. From the figures, it can be seen that these classical sparsity measures can reflect the bearing degradation trend to a certain extent and help in incipient fault detection, but they cannot characterize the bearing degradation monotonically and the degradation curve is highly fluctuant. Our designed index degradation curves are shown in Fig. 3. As can be seen from the figures, $HI_{1}$-$HI_{4}$ have good monotonic degradation trends with less fluctuation. Enlarged pictures around FFOT at file No.460 to No.580 quantified by designed indexes are respectively plotted tt the top left of each figure. It can be easily concluded that $HI_{1}$-$HI_{4}$ can detect the first failure occurrence time (FFOT) more precisely. In the following subsections, we will further evaluate the performance of the designed health metrics in three aspects.

To further validate the effectiveness and generalization of our proposed framework, this section is validated on the XJTU-SY run-to-failure bearing dataset. The bearing testbed is shown in Fig. 8. The sampling frequency is set at 25.6 kHz, and 32768 points (1.28 seconds) are collected each time with an interval of 1 minute. Details are described in [27].

The curves of bearing degradation quantified by classic spectral kurtosis, spectral smoothness index, spectral negative entropy and Gini index are plotted in Fig. 9. Meanwhile, the curves of bearing degradation quantified by $HI_{1}$-$HI_{4}$ are plotted in Fig 10. From the above figures, it can be concluded that the degradation trends quantified by four classic sparsity measures are poor and can only roughly identify the moment of incipient failure of bearings. The degradation curves quantified by $HI_{1}$-$HI_{4}$ can show good degradation trends and own the ability to identify incipient and severe bearing faults. In terms of fluctuation, except for degradation curves quantified by $HI_{3}$, the vast majority of HIs show less fluctuant. The validity of the proposed indexes and generalization of the framework based on ratios of multivariate power mean functions are further verified by the XJTU-SY run-to-failure bearing dataset. Based on the general paradigm, more reasonable and advanced health indexes can be easily designed, and the specific index design formulas can be determined depending on the specific conditions of industrial sites.