# Intelligent Prognostics of Degradation Trajectories for Rotating Machinery Based on Asymmetric Penalty Sparse Decomposition Model

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

- (1)
- The historical data with long-terms, including normal and abnormal time series data, should be collected using sensors and other portable testing devices. Typically, good degradation data must capture the physical transitions that the rotating machinery such as rolling bearing undergo during different stages of its running life; fortunately, most of the historical data are easy to collect since the rotating machinery is frequently updated in real time.
- (2)
- The health indicators (e.g., peak-to-peak value and Kurtosis) should be extracted to assess and continuously track the health condition of rotating machines or system. The health indicators are used to design a suitable prediction model that captures the evolution of the degradation trend of rotating machines.
- (3)
- The reliable prediction models and possible failure models should be established and the remaining useful life (RUL) or future trajectory of rotating machines could be effectively predicted.

- (1)
- Generally, the penalty functions that are established in a low-rank matrix approximation model are symmetric function, e.g., absolute value function; the common drawback is that this penalty function is non-differentiable at zero point, which can lead to some numerical issues such as local optimum and early termination of algorithm.
- (2)
- In conventional SLMD methods, the convex regularizer (e.g., L1-norm) which usually underestimates the sparse signal when the absolute value function is used as a sparsity regularizer. Additionally, both the convex and nonconvex methods shrink all the coefficients equally and remove too much energy of useful information, resulting in the separation of the LFC and HFC becoming more challenging.

- (1)
- To address the issue of models merging and improve the prediction accuracy, the health indicators time series (HITS) is decomposed into different sub-components, i.e., low frequency component (LFC) and high frequency component (HFC), using the APSD algorithm.
- (2)
- To address the drawback that penalty function is non-differentiable at zero point, a new asymmetric penalty function is proposed.
- (3)
- To solve the proposed non-convex regularization problem based on asymmetric penalty function, the majorization-minimization (MM) algorithm is introduced.
- (4)
- The decomposed LFC and HFC components can be predicted using the wavelet neural network (WNN) and ARMA combined with recursive least squares algorithm (ARMA-RLS) methods, respectively.
- (5)
- The prediction accuracy is greatly improved compared with some state-of-the-art models, and the presented approach has powerful application potentials.

## 2. Asymmetric Penalty Sparse Decomposition (APSD) Algorithm

#### 2.1. Sparse Representation and Filter Banks

**D**is a matrix defined as $\mathit{D}=\left[\begin{array}{cccccc}-1,& 1& & & & \\ & & \dots & & & \\ & & & \dots & & \\ & & & & -1,& 1\end{array}\right]$, which determines the sparsity degree of the approximating value of x. Commonly, if x is a sparse component, i.e., most of the values in x tend to zero, the correspondingly optimization problem in Equation (8) can be estimated by L1-norm fused lasso optimization (LFLO) model [24,25,26,27,28,29,30,31], i.e.,

_{0}and λ

_{1}are regularization parameters. The solution of LFLO model can be obtained by the soft-threshold method [34] and total variation de-noising (TVD) algorithm [35,36,37], we have,

#### 2.2. Asymmetric Penalty Regularization Model

**D**

_{1}defined as ${\mathit{D}}_{1}=\left[\begin{array}{cccccc}-1,& 1& & & & \\ & & \dots & & & \\ & & & ...& & \\ & & & & -1,& 1\end{array}\right]$, and matrix

**D**

_{2}defined as ${\mathit{D}}_{2}=\left[\begin{array}{cccccc}-1,& 2,& -1& & & \\ & -1,& 2,& -1,& & \\ & & \dots & & & \\ & & & -1,& 2,& -1\end{array}\right]$. The innovations of the novel compound regularizer model are as follows:

- (I)
- The M-term compound regularizers to estimate the fault transient impulses;
- (II)
- The compound regularizer model consists of symmetric and asymmetric penalty functions, wherein the symmetric penalty function is a differentiable function compared with the nondifferentiable function $\Vert {x}_{i}\Vert $ at point i = 0.
- (III)
- The MM algorithm is introduced for the solution of proposed compound regularization method, i.e., the OCF.

**For the first issue**, traditional LFLO regularization approach uses the absolute value function ${\varphi}_{A}(x)=\Vert x\Vert $ as the penalty function; however, the common drawback of function ${\varphi}_{A}(x)=\Vert x\Vert $ is that this function is non-differentiable at zero point, which can cause some numerical problems. To address this problem, a non-linear approximation function of ${\varphi}_{B}(x)$ or ${\varphi}_{c}(x)$ is proposed, i.e.,

**For the second issue**, inspired by the absolute value function ${\varphi}_{A}(x)=\Vert x\Vert $, and also in contrast to the symmetric and differentiable penalty function ${\varphi}_{B}(x)$ and ${\varphi}_{c}(x)$, here, a segmented function is proposed as follows,

**The third issue**, will be solved and derived in Section 2.3 using the majorization-minimization algorithm.

#### 2.3. The Solution of Proposed Model Based on Majorization-Minimization Algorithm

- (a)
- Majorizer of symmetric and differentiable function $\varphi \left({\left[{\mathit{D}}_{i}x\right]}_{n}\right)$ based on MM algorithm.
- (b)
- Majorizer of asymmetric and differentiable function ${\theta}_{\epsilon}\left({x}_{n};r\right)$ based on MM algorithm.
- (c)
- Majorizer of objective cost F(x) based on MM algorithm.

**For problem (a)**, we first seek a majorizer $g\left(x,v\right)$ for $\varphi \left(x\right)$, i.e.,

**For problem (b)**, assume that ${g}_{0}\left(x,v\right)$ is the majorizer of the asymmetric and differentiable function ${\theta}_{\epsilon}\left({x}_{n};r\right)$, since the $f(x)=\frac{1+r}{4\epsilon}{x}^{2}+\frac{1-r}{2}x+\frac{(1+r)\epsilon}{4},\Vert x\Vert \le \epsilon $, we have,

**For problem (c)**, based on Equations (28) and (33), the majorizer of F(x) based on MM algorithm is given by,

**BA**

^{−1}in Equation (35), we have,

- (1)
**Inputs**: signal y, r ≥ 1, matrix**A**, matrix**B**, ${\lambda}_{i},i=0,1,\dots ,M$, k = 0;- (2)
- $\mathit{E}={\mathit{B}}^{T}\mathit{B}{\mathit{A}}^{-1}y-{\lambda}_{0}{\mathit{A}}^{T}b$
- (3)
- Initialize x = y;
- (4)
- Repeat the following iterations:$${\left[\mathsf{\Gamma}\left(v\right)\right]}_{n}=\left(1+r\right)/4\left|{v}_{n}\right|,\left|{v}_{n}\right|\ge \epsilon ;$$$${\left[\mathsf{\Gamma}\left(v\right)\right]}_{n}=\left(1+r\right)/4\epsilon ,\left|{v}_{n}\right|\le \epsilon ;$$$${\left[\mathsf{\Lambda}\left({\mathit{D}}_{i}v\right)\right]}_{n}=\frac{{\varphi}^{\prime}\left({\left[{\mathit{D}}_{i}v\right]}_{n}\right)}{{\left[{\mathit{D}}_{i}v\right]}_{n}},i=0,1,2,\dots ,M;$$$${\mathit{M}}^{(k)}=2{\lambda}_{0}\mathsf{\Gamma}\left({x}^{(k)}\right)+{\displaystyle \sum _{i=1}^{M}{\lambda}_{i}{\mathit{D}}_{i}^{T}\left[\mathsf{\Lambda}\left({\mathit{D}}_{i}{x}^{(k)}\right)\right]{\mathit{D}}_{i}};$$$${\mathit{Q}}^{(k)}={\mathit{B}}^{T}\mathit{B}+{\mathit{A}}^{T}{\mathit{M}}^{(k)}\mathit{A};$$$${x}^{(k+1)}=\mathit{A}{\left[{\mathit{Q}}^{(k)}\right]}^{-1}\mathit{E};$$
- (5)
- If the stopping criterion is satisfied, then output signal x, otherwise, k = k + 1, and go to step (4).
- (6)
**Output**: signal x.

#### 2.4. Parameters Selection

_{i}are set as [26]:

_{0}and β

_{1}are the constants so as to maximize the signal-to-noise-rate (SNR), β

_{0}and γ are typically set up to be constants, i.e., β

_{0}= [0.5, 1], γ = [7.5, 8], and σ is the standard deviation (SD) of the external noise. In practical engineering applications, the SD of the external noise in Equation (40) could be computed using the healthy data and fault data under the same operating environment. Moreover, when the healthy data is not available or unknown, the SD of the external noise can still be estimated by the following formula,

## 3. Wavelet Neural Network and ARMA-RLS Algorithm

#### 3.1. Wavelet Neural Network Algorithm

_{1}, X

_{2}, …, X

_{k}, and the predicted outputs are denoted by Y

_{1}, Y

_{2}, …, Y

_{m}, w

_{ij}is the link-weight between the input layer and hidden layer. When the input data is x

_{i}(i = 1, 2, …, k), the output of hidden layer can be calculated as,

_{j}is the scaling factor of function h(j), b

_{j}is the translation factor of function h(j), The output of the WNN can be expressed as,

_{ik}denotes the link-weight between the hidden layer and output layer, h(i) is the output of the i-th hidden layer, l represents the number of hidden layer and m represents the number of output layer. According to the fundamental principle of BPNN and gradient descent learning algorithm, the corresponding adjustment process of network weights and wavelet basis function is as follows,

_{n}(k) is desired outputs and y(k) predicted outputs generated by the WNN method.

_{k}and translation parameters b

_{k}of the wavelet basis function as well as the network learning rate η, the link-weights are w

_{ij}and w

_{jk}, respectively, the error threshold ε, and maximum iterations T.

#### 3.2. ARMA Combined With Recursive Least Squares (RLS) Algorithm

#### 3.2.1. ARMA Review of ARMA Model

_{t}is called an autoregressive moving-average process (ARMA) with order p and q, namely ARMA (p, q) [48,49], if the process is stationary and satisfies a linear stochastic difference equation of the form,

_{t}is white gaussian noise (WGN), i.e., ${e}_{t}\sim WN(0,{\sigma}^{2})$, parameters ${\varphi}_{1},{\varphi}_{2},\dots ,{\varphi}_{p}$ and ${\theta}_{1},{\theta}_{2},\dots ,{\theta}_{q}$ are coefficients of AR(p) and MA(q) models, and the polynomials are as follows,

#### 3.2.2. Recursive Least Squares (RLS) Algorithm

## 4. Experimental Validations

- (1)
- bearing 1: operating conditions: speed 1800 rpm and load 4000 N; whole test life: 28,030 s;
- (1)
- bearing 2: operating conditions: speed 1800 rpm and load 4000 N; whole test life: 18,020 s;
- (1)
- bearing 3: operating conditions: speed 1650 rpm and load 4200 N; whole test life: 7970 s;
- (1)
- bearing 4: operating conditions: speed 1800 rpm and load 4000 N; whole test life: 14,280 s.

## 5. Conclusions

## Author Contributions

## Acknowledgments

## Conflicts of Interest

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**Figure 3.**The decomposition results of noisy chromatogram data using proposed APSD algorithm [38]. (

**a**) The raw data with additive noise; (

**b**) The estimated LFC signal; (

**c**) The estimated HFC signal.

**Figure 7.**Raw signal of a tested bearing. (

**a**) The historical signals of the whole lifetime; (

**b**) The time-domain waveform of the bearing in the normal stage; (

**c**) The time-domain waveform of the bearing in the failure stage.

**Figure 8.**The health indicator curve of peak-to-peak value of bearing 1. (

**a**) The peak-to-peak value of the whole lifetime of bearing 1; (

**b**) The peak-to-peak value of the point 2001 from to point 2803.

**Figure 9.**The health indicator curves of different bearings. (

**a**) The equivalent vibration intensity curve of bearing 2; (

**b**) The Kurtosis curve of bearing 3; (

**c**) The equivalent vibration intensity curve of bearing 4.

**Figure 10.**Decomposition results of low frequency component (LFC) and high frequency component (HFC) based on asymmetric penalty sparse decomposition. (

**a**) The low frequency component (LFC); (

**b**) The high frequency component (HFC).

**Figure 11.**Prediction results of LFC and HFC based on WNN and ARMA-RLS method, respectively. (

**a**) Predicted result of LFC based on WNN method; (

**b**) Predicted results of HFC based on ARMA-RLS method.

**Figure 12.**Comparison of the forecasting for peak-to-peak series using benchmarking methods and proposed approach.

**Figure 14.**Decomposition results of LFC and HFC based on asymmetric penalty sparse decomposition. (

**a**) The low frequency component of bearing 2; (

**b**) The High frequency component of bearing 2; (

**c**) The low frequency component of bearing 3; (

**d**) The High frequency component of bearing 3; (

**e**) The low frequency component of bearing 4; (

**f**) The High frequency component of bearing 4.

**Figure 15.**The predicted results of remaining three bearings. (

**a**) The predicted results of bearing 2; (

**b**) the predicted results of bearing 3; (

**c**) the predicted results of bearing 4.

**Figure 16.**Quantitative performance evaluation based on absolute error boxplot of remaining three bearings.

Functions | $\mathit{\varphi}\left(\mathit{x}\right)$ | ${\mathit{\varphi}}^{\prime}\left(\mathit{x}\right)$ |
---|---|---|

${\varphi}_{A}\left(x\right)$ | $\Vert x\Vert $ | $\mathrm{Signal}\left(x\right)$ |

${\varphi}_{B}\left(x\right)$ | $\sqrt{{\left|x\right|}^{2}+\epsilon}$ | $x/\sqrt{{\left|x\right|}^{2}+\epsilon}$ |

${\varphi}_{c}\left(x\right)$ | $\left|x\right|-\epsilon \mathrm{log}\left(\left|x\right|+\epsilon \right)$ | $x/\left(\left|x\right|+\epsilon \right)$ |

Parameters β_{0} | Parameters γ | Regularization Parameter λ_{0} | Regularization Parameter λ_{1} | Regularization Parameter λ_{2} | M-Term | Iteration Times |
---|---|---|---|---|---|---|

β_{0} = 0.8 | γ = 7.5 | λ_{0} = 1.7400 | λ_{1} = 3.2625 | λ_{2} = 1.7400 | 2 | 50 |

Index | ARMA | FARIMA | WNN | L-Lyap | Proposed |
---|---|---|---|---|---|

Mean Absolute Error-MAE | 0.5880 | 17.2700 | 8.2364 | 3.2453 | 0.9770 |

Average relative error-ARE | 0.1872 | 0.4666 | 0.1583 | 0.2691 | 0.2599 |

Root-Mean-Square Error-RMSE | 5.9094 | 173.5702 | 82.7793 | 32.6164 | 9.8190 |

Normalized Mean Square Error-NMSE | 0.1657 | 2.4144 | 0.7173 | 0.3008 | 0.2244 |

Maximum of Absolute Error-MaxAE | 21.9865 | 75.0981 | 36.4112 | 28.3548 | 32.9561 |

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## Share and Cite

**MDPI and ACS Style**

Li, Q.; Liang, S.Y.
Intelligent Prognostics of Degradation Trajectories for Rotating Machinery Based on Asymmetric Penalty Sparse Decomposition Model. *Symmetry* **2018**, *10*, 214.
https://doi.org/10.3390/sym10060214

**AMA Style**

Li Q, Liang SY.
Intelligent Prognostics of Degradation Trajectories for Rotating Machinery Based on Asymmetric Penalty Sparse Decomposition Model. *Symmetry*. 2018; 10(6):214.
https://doi.org/10.3390/sym10060214

**Chicago/Turabian Style**

Li, Qing, and Steven Y. Liang.
2018. "Intelligent Prognostics of Degradation Trajectories for Rotating Machinery Based on Asymmetric Penalty Sparse Decomposition Model" *Symmetry* 10, no. 6: 214.
https://doi.org/10.3390/sym10060214