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Complex three-order cumulant has different definition forms. Different forms conclude different information. For studying the effection of frequency in the coupled signals to fault diagnosis, the differential method to the three order cumulants of coupled signals is adopted. By using the differential of complex three order cumulants before and after respectively, then their dimensional spectrum is calculated, and the results are used to fault diagnosis. The experimental results show that, the increase frequency item in three order cumulants after differentiated impacts on the results of fault diagnosis and the degree of effection is relative to the differential times. And the correct rate of fault diagnosis can be raised by changing the differential times of three order cumulants.

The High-order statistics method which has very good inhibition for various noises is gradually becoming a new hot spot in signal processing. It is not only nonsensitive to additive noise of auto-correlation, and also is nonsensitive to non-Gauss colored noise. So it plays an important role in non-Gaussian, non-linear, non-minimum phase, non-stationary, Gaussian colored noise processing.

At present, the study of High-order cumulants has been very popular in mechanical fault diagnosis. For instance, Shao Ren-ping and others have applied bispectrum to gear damage detection [

The applications of high-order spectrum above are in the range of real number signal, W. R. Raghuveer and Chinese scholars such as Wang Shuxun have made an intensive study of high order cumulants of complex signal in the amount of coupling properties, and explicitly pointed out the different definition forms of coupling characteristics of all kinds of complex high order cumulantes^{ } [

Because 1 1 2 dimension spectrum can well response signals’ coupling, if 1 1 2 dimension spectrum differentiated is applied to the fault diagnosis, the performance of three-order cumulant before or after it is differentiated can be studied. The experiments are carried out on many types of fault, and most results improve with the increase of differential times in a certain degree.

Let { x ( n ) } be a real, discrete, zero-mean and k-order stationary random process, its three order cumulant can be defined as:

c 3 x ( τ 1 , τ 2 ) = E { x ( n ) x ( n + τ 1 ) x ( n + τ 2 ) } (1)

where τ 1 = τ 2 = τ , then its 1 1 2 dimensional spectrum is defined as the Fourier transform, i.e.:

B ( ω ) = ∑ τ = − ∞ ∞ c 3 x ( τ , τ ) e − j ω τ

Let x(n) in expression (1) be coupling signal, i.e.: x ( n ) = ∑ i = 1 3 A i cos ( ω i n + ϕ i ) , where ϕ 3 = ϕ 2 + ϕ 1 and ω 3 = ω 1 + ω 2 , so X(n) is a coupling signal, its three-order cumulant is

c 3 x ( τ 1 , τ 2 ) = E { x ( n ) x ( n + τ 1 ) x ( n + τ 2 ) } = A 1 A 2 A 3 4 ⋅ [ cos ( ω 1 τ 2 + ω 2 τ 1 ) + cos ( ω 3 τ 1 − ω 1 τ 2 ) + cos ( ω 1 τ 1 + ω 2 τ 2 ) + cos ( ω 3 τ 1 − ω 2 τ 2 ) + cos ( ω 1 τ 1 − ω 3 τ 2 ) + cos ( ω 2 τ 1 − ω 3 τ 2 ) ] (2)

where its diagonal three-order cumulant is:

c 3 x ( τ , τ ) = A 1 A 2 A 3 2 ⋅ [ cos ( ω 1 τ ) + cos ( ω 2 τ ) + cos ( ω 3 τ ) ] (3)

For x(n) Differential after:

d x ( n ) / d n = ∑ i = 1 3 A i ω i cos ( ω i n + ϕ i + π 2 )

where its three-order cumulant becomes:

c 3 x ( τ 1 , τ 2 ) = A 1 A 2 A 3 ω 1 ω 2 ω 3 4 ⋅ [ cos ( ω 1 τ 2 + ω 2 τ 1 + π 2 ) + cos ( ω 3 τ 1 − ω 1 τ 2 + π 2 ) + cos ( ω 1 τ 1 + ω 2 τ 2 + π 2 ) + cos ( ω 3 τ 1 − ω 2 τ 2 + π 2 ) + cos ( ω 1 τ 1 − ω 3 τ 2 + π 2 ) + cos ( ω 2 τ 1 − ω 3 τ 2 + π 2 ) ]

For x(n) k time’s differential after its three-order cumulant is:

c 3 x ( τ 1 , τ 2 ) = A 1 A 2 A 3 ω 1 k ω 2 k ω 3 k 4 ⋅ [ cos ( ω 1 τ 2 + ω 2 τ 1 + k π 2 ) + cos ( ω 3 τ 1 − ω 1 τ 2 + k π 2 ) + cos ( ω 1 τ 1 + ω 2 τ 2 + k π 2 ) + cos ( ω 3 τ 1 − ω 2 τ 2 + k π 2 ) + cos ( ω 1 τ 1 − ω 3 τ 2 + k π 2 ) + cos ( ω 2 τ 1 − ω 3 τ 2 + k π 2 ) ]

its diagonal three-order cumulant is:

c 3 x ( τ , τ ) = A 1 A 2 A 3 ω 1 k ω 2 k ω 3 k 4 ⋅ [ cos ( ω 1 τ + k π 2 ) + cos ( ω 2 τ + k π 2 ) + cos ( ω 3 τ + k π 2 ) ] (4)

Because of the three order cumulant of formula (3) can be considered as a function of τ, so that it can directly be differentiated by τ, the result is:

c 3 x ( τ , τ ) = A 1 A 2 A 3 2 ⋅ [ ω 1 cos ( ω 1 τ + π 2 ) + ω 2 cos ( ω 2 τ + π 2 ) + ω 3 cos ( ω 3 τ + π 2 ) ]

after k times differentiation, its diagonal three-order cumulant is:

c 3 x ( τ , τ ) = A 1 A 2 A 3 2 ⋅ [ ω 1 k cos ( ω 1 τ + k π 2 ) + ω 2 k cos ( ω 2 τ + k π 2 ) + ω 3 k cos ( ω 3 τ + k π 2 ) ] (5)

In Equation (1), assume that x(n) is a Complex signal, which is

x ( n ) = ∑ i = 1 3 A i exp ( j ( ω i n + ϕ i ) ) ,

where ϕ 3 = ϕ 2 + ϕ 1 and ω 3 = ω 1 + ω 2 , so X(n) is a coupling signal. According to literature (7), x(n) takes its conjugate or not, its third-order cumulant will have different definitions, different definitions will contain different types of coupling information, seeing

B ( ω ) = ∑ τ = − ∞ ∞ c 3 x ( τ , τ ) e − j ω τ (6).

As can be seen from the comparison of

Definition mode | Definition of three-order cumulant | Three-order cumulant | Diagonal of three-order cumulant |
---|---|---|---|

Definition 1 | E { x ( n ) x ( n + τ 1 ) x ( n + τ 2 ) } | 0 | 0 |

Definition 2 | E { x ∗ ( n ) x ( n + τ 1 ) x ( n + τ 2 ) } | A 1 A 2 A 3 [ exp j ( ω 1 τ 1 + ω 2 τ 2 ) + exp j ( ω 2 τ 1 + ω 1 τ 2 ) ] | 2 A 1 A 2 A 3 [ exp j ( ω 3 τ ) ] |

Definition 3 | E { x ∗ ( n ) x ( n + τ 1 ) x ∗ ( n + τ 2 ) } | A 1 A 2 A 3 [ exp j ( ω 3 τ 1 − ω 1 τ 2 ) + exp j ( ω 3 τ 1 − ω 2 τ 2 ) ] | A 1 A 2 A 3 [ exp j ( ω 2 τ ) + exp j ( ω 1 τ ) ] |

Definition mode | Definition of three-order cumulant | Three-order cumulant | Diagonal of three-order cumulant |
---|---|---|---|

Definition 1 | E { y ( n ) y ( n + τ 1 ) y ( n + τ 2 ) } | 0 | 0 |

Definition 2 | E { y ∗ ( n ) y ( n + τ 1 ) y ( n + τ 2 ) } | A 1 A 2 A 3 ω 1 ω 2 ω 3 [ exp j ( ω 1 τ 1 + ω 2 τ 2 + π / 2 ) + exp j ( ω 2 τ 1 + ω 1 τ 2 + π / 2 ) ] | 2 A 1 A 2 A 3 ω 1 ω 2 ω 3 [ exp j ( ω 3 τ + π / 2 ) ] |

Definition 3 | E { y ∗ ( n ) y ( n + τ 1 ) y ∗ ( n + τ 2 ) } | A 1 A 2 A 3 ω 1 ω 2 ω 3 [ exp j ( ω 3 τ 1 − ω 1 τ 2 − π / 2 ) + exp j ( ω 3 τ 1 − ω 2 τ 2 − π / 2 ) ] | A 1 A 2 A 3 ω 1 ω 2 ω 3 [ exp j ( ω 2 τ − π / 2 ) + exp j ( ω 1 τ − π / 2 ) ] |

Definition mode | Definition of three-order cumulant | Three-order cumulant | Diagonal of three-order cumulant |
---|---|---|---|

Definition 1 | E { z ( n ) z ( n + τ 1 ) z ( n + τ 2 ) } | 0 | 0 |

Definition 2 | E { z ∗ ( n ) z ( n + τ 1 ) z ( n + τ 2 ) } | A 1 A 2 A 3 ω 1 k ω 2 k ω 3 k [ exp j ( ω 1 τ 1 + ω 2 τ 2 + k π / 2 ) + exp j ( ω 2 τ 1 + ω 1 τ 2 + k π / 2 ) ] | 2 A 1 A 2 A 3 ω 1 k ω 2 k ω 3 k [ exp j ( ω 3 τ + k π / 2 ) ] |

Definition 3 | E { z ∗ ( n ) z ( n + τ 1 ) z ∗ ( n + τ 2 ) } | A 1 A 2 A 3 ω 1 k ω 2 k ω 3 k [ exp j ( ω 3 τ 1 − ω 1 τ 2 − k π / 2 ) + exp j ( ω 3 τ 1 − ω 2 τ 2 − k π / 2 ) ] | A 1 A 2 A 3 ω 1 k ω 2 k ω 3 k [ exp j ( ω 2 τ − k π / 2 ) + exp j ( ω 1 τ − k π / 2 ) ] |

same in all above three definition ways. So do the affection to 1 1 2 dimension, shown in Tables 1-3.

Through the above analysis, whatever the signal is either real or complex signal, the affect of the frequencies in coupling signals amplifies with the increases of diffenential times in fault diagnosis, and this paper will make use of this characteristic in speed control valve’s fault diagnosis.

According to document [

N ( r ) ∝ ( 1 r ) D c → lim N ( r ) = ( 1 r ) D c

After taking logarithm, capacity dimension is obtained:

D c = lim ( log N ( r ) log ( 1 r ) )

The mechanical vibration component studied in the experiment is speed control valve. Vibration signals are collected from this valve in normal state and different fault states successively. In the experiment, 5 kinds of faults are set up artifically. In every state, datum in each state collected depending on working oil pressure (1 - 5 MPa) is divided into 5 groups. 5 kinds of artifical faults are as follows:

Fault one: spring in back of throttle valve deformed;

Fault two: spring added with foreign objects;

Fault three: cylinder iron core pulled out in back of throttle valve;

Fault four: cylinder iron core replaced with a gasket;

Fault five: combined fault two and fault four.

The number of each group of datum used in the experiment is 1024 (

In order to identify fault, according to formula (2), the 21 groups of experiment datum in normal state and fault state are used. All of their 1 1 2 dimension spectrum are drawn, and three of them are shown in

In

Then the 1 1 2 dimensional spectrum of the same above-mentioned three groups of datum when differentiated 1, 4 and 8 times are shown in Figures 3-5. Because the 1 1 2 dimension spectrum affected by the frequency components, as shown above, with the increase of the differential times, spectrum peak of each data become more and more concentrated, more and more acute. This should be the result of frequency components affecting in three order cumulants with the increase of differential times, the 1 1 2 dimension spectrum peak distribution of location, density and strength have become more and more obviously distinctive between two groups of datum in normal state and fault state.

Normal state | 1.1910 | 1.1747 | 1.1796 | 1.1716 | 1.2193 | 1.1820 | 1.1178 | … |
---|---|---|---|---|---|---|---|---|

Fault 1 state | 1.1513 | 1.2631 | 1.1052 | 1.1516 | 1.1470 | 1.1299 | 1.1636 | … |

In order to effectively distinguish faults, fractal theory is used as a tool to calculate the capacity dimensions of complex signals’ 1 1 2 dimension spectrum, and all the datum which include above three different definition forms both in normal and fault state are calculated, the complex signals are obtained from original signals through Hilbert transform [

1): In definition two, the capacity dimensions of dotum’s 1 1 2 dimension spectrum (a total of 21 data, the 10 groups of normal state, 11 groups of fault one) are calculated, as shown in

2): The same experiments are done in definition one, and when the differential times is 5, the diagnostic correct rate reaches the maximum value of about 70%. The obtained results of definition three are better than that of definition one, but worse than that of definition two.

According to Literature (7) and (8), 1 1 2 dimensional spectrum can reflect well the signal frequency coupling features, and capacity dimension is used to calculate the similar ratio of complex graphics. In this paper, normal data or fault data is considered as one group respectively, capacity dimension of 1 1 2 dimensional spectrum which contains the coupling properties is used to judge the differences in self similarity between the above two groups of datum. For fault one state, the experimental results show that, according to definition two mode, the best effect is reached, but in definition one mode is not so. As can be seen in Tables 1-3, in definition one mode, both three order cumulant and its diagonal slice are 0, but in definition two, diagonal slice of cumulant is 2 A 1 A 2 A 3 ω 1 ω 2 ω 3 [ exp j ( ω 3 τ + π / 2 ) ] . And with the increase of differential times, the amplitude becomes A 1 A 2 A 3 ω 1 n ω 2 n ω 3 n . Because diagnostic results get best after differentiated eight times, so it can be deduced that the frequency factors differented in amplitude do play a role in fault diagnosis. Fathomer, it can also be inferred that the frequencies difference must exist between normal state and fault one state, so with the increase of differential times, the differences become more obvious. The same experiments are carried out on other types of fault, and most results improve with the increase of differential times in a certain degree, but for a few of the above faults, the correct rates only improve little, even reduce. Considering the above experimental results, it can be extrapolated that, because of the complexity of mechanical vibration signals, only one coupling theory cannot fully explain their characteristics. Even so, the differential method in this paper succeeds in a certain degree. Because differential times are not limited, the methods provide an infinite choice of fault diagnosis. In

The different definition forms of complex three order cumulants must contain

Method | Method in literature 10 | Method in literature 13 | Method in this paper |
---|---|---|---|

highest correct rate | 85% | 80% | 95% |

different coupling information, which is bound to react to the 1 1 2 dimension spectrum derived from it. In the paper, differential operation is first performed on complex coupling signals, then the amplitude of three order cumulants derived from these differentiated signals includes the frequency components in the

primary coupling signals, and these components are enlarged with the increase of differential times, afterwards, 1 1 2 spectrums are obtained. Fault diagnosis based on computing capacity dimension of 1 1 2 spectrum is performed, the results show that the frequency components included in three order cumulants after differentiated help to improve fault diagnostic correct rate.

This paper is supported by National Natural Science Foundation of China (61562063), Key Science and Technology Project of Jiangxi Provincial Education Department (GJJ161234).

Wu, W.B., Xiong, J.Q., Wu, Y.P. and Liu, R.H. (2018) The Application of Coupled Three Order Cumulants’ Differential Feature in Fault Diagnosis. International Journal of Modern Nonlinear Theory and Application, 7, 77-87. https://doi.org/10.4236/ijmnta.2018.72006