Fault Diagnosis and Prognosis of Bearing Based on Hidden Markov Model with Multi-Features

Weiguo Zhao; Tiancong Shi; Liying Wang

Open Access

Fault Diagnosis and Prognosis of Bearing Based on Hidden Markov Model with Multi-Features

Weiguo Zhao

Tiancong Shi

and

Liying Wang

| Mar 30, 2020

Applied Mathematics and Nonlinear Sciences

Volume 5 (2020): Issue 1 (January 2020)

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Page range: 71 - 84

Received: Dec 05, 2019

Accepted: Jan 14, 2020

DOI: https://doi.org/10.2478/amns.2020.1.00008

Keywords
hidden Markov model, fault diagnosis, prognosis, multi-features, wavelet packet

© 2020 Weiguo Zhao et al., published by Sciendo

This work is licensed under the Creative Commons Attribution 4.0 International License.

Training curves of various fault states of HMMs.

Fault Identification results of training samples.

Identification results of various fault states of bearing for every 30 observation samples.

Identification results of various fault states of bearing for every 100 observation samples.

Some obtained observation values as the input of HMM.

States of faults	No. of observation values

	1	2	3	4	5	6	7	8	9	10
Nor	0.71	0.03	0.1	0.1	0	0.01	0	0	0.05	0
R1	0.07	0.56	.07	0.1	0	0.17	0	0	0.03	0
R2	0.23	0.08	0.35	0.2	0.01	0.06	0.04	0	0.01	0.02
R3	0.31	0.07	0.06	0.41	0.01	0.03	0.04	0	0.07	0
I1	0	0	0.02	0.01	0.84	0.07	0.01	0	0	0.05
I2	0.04	0.3	0.07	0.06	0.13	0.37	0.02	0.01	0	0
I3	0	0	0.01	0.04	0.11	0.04	0.65	0	0	0.15
O1	0	0	0	0	0	0.02	0	0.97	0	0.01
O2	0.37	0.1	0.02	0.32	0	0	0.01	0	0.18	0
O3	0	0.01	0.03	0.01	0.15	0.03	0.07	0.03	0	0.67

Statistics of test results of various fault states for different sample lengths.

Length of observation samples	States of faults

	Nor	R1	R2	R3	I1	I2	I3	O1	O2	O3
10	64.3%	82.2%	41.1%	53.9%	93.8%	81.3%	97.5%	98.3%	58.9%	85.1%
20	86.1%	88.7%	55.0%	72.3%	93.9%	97.8%	100%	100%	69.7%	92.2%
30	96.8%	95.5%	60.2%	77.8%	99.5%	99.5%	100%	100%	71.9%	98.6%
40	100%	100%	64.0%	77.3%	100%	100%	100%	100%	67.3%	100%
50	100%	100%	63.7%	85.6%	100%	100%	100%	100%	70.6%	100%
60	100%	100%	69.1%	86.9%	100%	100%	100%	100%	74.3%	100%
70	100%	100%	76.8%	90.6%	100%	100%	100%	100%	91.7%	100%
80	100%	100%	83.6%	91.2%	100%	100%	100%	100%	100%	100%
90	100%	100%	83.2%	94.4%	100%	100%	100%	100%	100%	100%
100	100%	100%	90.7%	100%	100%	100%	100%	100%	100%	100%
110	100%	100%	95.7%	100%	100%	100%	100%	100%	100%	100%
120	100%	100%	100%	100%	100%	100%	100%	100%	100%	100%
130	100%	100%	100%	100%	100%	100%	100%	100%	100%	100%
140	100%	100%	100%	100%	100%	100%	100%	100%	100%	100%
150	100%	100%	100%	100%	100%	100%	100%	100%	100%	100%

The statistics of the overall training results using HMM.

States of faults	Nor	R1	R2	R3	I1	I2	I3	O1	O2	O3
Accuracy	100%	99%	95%	90%	100%	99%	100%	100%	96%	99%

Time-domain statistical characteristics.

${ft}_{1} = \frac{1}{N} \sum_{i = 1}^{N} x_{i}$ f{t_1} = {1 \over N}\sum\limits_{i = 1}^N {{x_i}}	${ft}_{2} = \sqrt{\frac{1}{N} \sum_{i = 1}^{N} x_{i}^{2}}$ f{t_2} = \sqrt {{1 \over N}\sum\limits_{i = 1}^N {x_i^2} }	${ft}_{3} = {[\frac{1}{N} \sum_{i = 1}^{N} \sqrt{\| x_{i} \|}]}^{2}$ f{t_3} = {\left[ {{1 \over N}\sum\limits_{i = 1}^N {\sqrt {\left\| {{x_i}} \right\|} } } \right]^2}	${ft}_{4} = \frac{1}{N} \sum_{i = 1}^{N} \| x_{i} \|$ f{t_4} = {1 \over N}\sum\limits_{i = 1}^N {\left\| {{x_i}} \right\|}
${ft}_{5} = \frac{1}{N} \sum_{i = 1}^{N} x_{i}^{3}$ f{t_5} = {1 \over N}\sum\limits_{i = 1}^N {x_i^3}	${ft}_{6} = \frac{1}{N} \sum_{i = 1}^{N} x_{i}^{4}$ f{t_6} = {1 \over N}\sum\limits_{i = 1}^N {x_i^4}	${ft}_{7} = \frac{1}{N - 1} \sum_{i = 1}^{N} {(x_{i} - \bar{X})}^{2}$ f{t_7} = {1 \over {N - 1}}\sum\limits_{i = 1}^N {{{\left( {{x_i} - \bar X} \right)}^2}}	f t₈ = max{\|x_i\|}
f t₉ = min{x_i}	f t₁₀ = max(x_i) − min(x_i)	${ft}_{11} = \frac{{ft}_{2}}{{ft}_{4}}$ f{t_{11}} = {{f{t_2}} \over {f{t_4}}}	${ft}_{12} = \frac{{ft}_{8}}{{ft}_{2}}$ f{t_{12}} = {{f{t_8}} \over {f{t_2}}}
${ft}_{13} = \frac{{ft}_{8}}{{ft}_{4}}$ f{t_{13}} = {{f{t_8}} \over {f{t_4}}}	${ft}_{14} = \frac{{ft}_{8}}{{ft}_{3}}$ f{t_{14}} = {{f{t_8}} \over {f{t_3}}}	${ft}_{15} = \frac{{ft}_{5}}{{ft}_{2}^{3}}$ f{t_{15}} = {{f{t_5}} \over {ft_2^3}}	${ft}_{16} = \frac{{ft}_{6}}{{ft}_{2}^{4}}$ f{t_{16}} = {{f{t_6}} \over {ft_2^4}}

Frequency-domain statistical characteristics.

${ff}_{1} = \frac{\sum_{k = 1}^{K} s (k)}{k}$ f{f_1} = {{\sum\limits_{k = 1}^K {s(k)} } \over k}	${ff}_{2} = \frac{\sum_{k = 1}^{K} {(s (k) - {ff}_{1})}^{2}}{k - 1}$ f{f_2} = {{\sum\limits_{k = 1}^K {{{\left( {s(k) - f{f_1}} \right)}^2}} } \over {k - 1}}	${ff}_{3} = \frac{\sum_{k = 1}^{K} {(s (k) - {ff}_{1})}^{3}}{k {(\sqrt{p_{2}})}^{3}}$ f{f_3} = {{\sum\limits_{k = 1}^K {{{\left( {s(k) - f{f_1}} \right)}^3}} } \over {k{{\left( {\sqrt {{p_2}} } \right)}^3}}}	${ff}_{4} = \frac{\sum_{k = 1}^{K} {(s (k) - {ff}_{1})}^{4}}{k \cdot {ff}_{2}^{2}}$ f{f_4} = {{\sum\limits_{k = 1}^K {{{\left( {s(k) - f{f_1}} \right)}^4}} } \over {k \cdot ff_2^2}}
${ff}_{5} = \frac{\sum_{k = 1}^{K} f_{k} s (k)}{\sum_{k = 1}^{K} s (k)}$ f{f_5} = {{\sum\limits_{k = 1}^K {{f_k}s(k)} } \over {\sum\limits_{k = 1}^K {s(k)} }}	${ff}_{6} = \sqrt{\frac{\sum_{k = 1}^{K} {(f_{k} - {ff}_{5})}^{2} s (k)}{k}}$ f{f_6} = \sqrt {{{\sum\limits_{k = 1}^K {{{\left( {{f_k} - f{f_5}} \right)}^2}s(k)} } \over k}}	${ff}_{7} = \sqrt{\frac{\sum_{k = 1}^{K} f_{k}^{2} s (k)}{\sum_{k = 1}^{K} s (k)}}$ f{f_7} = \sqrt {{{\sum\limits_{k = 1}^K {f_k^2s(k)} } \over {\sum\limits_{k = 1}^K {s(k)} }}}	${ff}_{8} = \sqrt{\frac{\sum_{k = 1}^{K} f_{k}^{4} s (k)}{\sum_{k = 1}^{K} f_{k}^{2} s (k)}}$ f{f_8} = \sqrt {{{\sum\limits_{k = 1}^K {f_k^4s(k)} } \over {\sum\limits_{k = 1}^K {f_k^2s(k)} }}}
${ff}_{9} = \frac{\sum_{k = 1}^{K} f_{k}^{2} s (k)}{\sqrt{\sum_{k = 1}^{K} s (k) \sum_{k = 1}^{K} f_{k}^{4} s (k)}}$ f{f_9} = {{\sum\limits_{k = 1}^K {f_k^2s(k)} } \over {\sqrt {\sum\limits_{k = 1}^K {s(k)} \sum\limits_{k = 1}^K {f_k^4s(k)} } }}	${ff}_{10} = \frac{{ff}_{6}}{{ff}_{5}}$ f{f_{10}} = {{f{f_6}} \over {f{f_5}}}	${ff}_{11} = \frac{\sum_{k = 1}^{K} {(f_{k} - {ff}_{5})}^{3} s (k)}{{kp}_{6}^{3}}$ f{f_{11}} = {{\sum\limits_{k = 1}^K {{{\left( {{f_k} - f{f_5}} \right)}^3}s(k)} } \over {kp_6^3}}	${ff}_{12} = \frac{\sum_{k = 1}^{K} {(f_{k} - {ff}_{5})}^{4} s (k)}{{kp}_{6}^{4}}$ f{f_{12}} = {{\sum\limits_{k = 1}^K {{{\left( {{f_k} - f{f_5}} \right)}^4}s(k)} } \over {kp_6^4}}
${ff}_{13} = \frac{\sum_{k = 1}^{K} {(\| f_{k} - p_{5} \|)}^{\frac{1}{2}} s (k)}{k \sqrt{p_{6}}}$ f{f_{13}} = {{\sum\limits_{k = 1}^K {{{\left( {\left\| {{f_k} - {p_5}} \right\|} \right)}^{{1 \over 2}}}s(k)} } \over {k\sqrt {{p_6}} }}	${ff}_{14} = \frac{\sum_{k = 1}^{K} {(f_{k} - {ff}_{5})}^{2} s (k)}{\sum_{k = 1}^{K} s (k)}$ f{f_{14}} = {{\sum\limits_{k = 1}^K {{{\left( {{f_k} - f{f_5}} \right)}^2}s(k)} } \over {\sum\limits_{k = 1}^K {s(k)} }}

eISSN:: 2444-8656
Language:: English

Publication timeframe:: Volume Open
Journal Subjects:: Life Sciences, other, Mathematics, Applied Mathematics, General Mathematics, Physics

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Fault Diagnosis and Prognosis of Bearing Based on Hidden Markov Model with Multi-Features

Published Online: Mar 30, 2020

Page range: 71 - 84

Received: Dec 05, 2019

Accepted: Jan 14, 2020

DOI: https://doi.org/10.2478/amns.2020.1.00008

Keywords
hidden Markov model, fault diagnosis, prognosis, multi-features, wavelet packet

© 2020 Weiguo Zhao et al., published by Sciendo

This work is licensed under the Creative Commons Attribution 4.0 International License.

Fig. 1

Fig. 2

Fig. 3

Fig. 4

Fig. 5

Fig. 6

Fig. 7

Some obtained observation values as the input of HMM.

Statistics of test results of various fault states for different sample lengths.

The statistics of the overall training results using HMM.

Time-domain statistical characteristics.

Frequency-domain statistical characteristics.

Fault Diagnosis and Prognosis of Bearing Based on Hidden Markov Model with Multi-Features

Published Online: Mar 30, 2020

Page range: 71 - 84

Received: Dec 05, 2019

Accepted: Jan 14, 2020

DOI: https://doi.org/10.2478/amns.2020.1.00008

Keywordshidden Markov model, fault diagnosis, prognosis, multi-features, wavelet packet

© 2020 Weiguo Zhao et al., published by Sciendo

This work is licensed under the Creative Commons Attribution 4.0 International License.

Fig. 1

Fig. 2

Fig. 3

Fig. 4

Fig. 5

Fig. 6

Fig. 7

Some obtained observation values as the input of HMM.

Statistics of test results of various fault states for different sample lengths.

The statistics of the overall training results using HMM.

Time-domain statistical characteristics.

Frequency-domain statistical characteristics.

Keywords
hidden Markov model, fault diagnosis, prognosis, multi-features, wavelet packet