Fault diagnosis of complex electromechanical systems based on nonlinear spectral feature and support

Fault diagnosis of complex electromechanical systems based on nonlinear spectral feature and support vector machine

J.-F. Cao J.-L. Zhang F. Gao

State Key Laboratory for Manufacturing for Systems Engineering, Xi’an Jiao t ong University, Xi’an, 710049, People’s Republic of China

E-mail: cjf@https://www.sodocs.net/doc/0d6431193.html,

Abstract：In this paper, a novel approach of frequency domain fault diagnosis of complex electromechanical system is proposed. In order to avoid the large calculation amount of identifying general frequency respond function (GFRF), the mapping relationship between input frequency spectrum and output frequency spectrum of nonlinear system is firstly obtained, and it is approximated by least squares support vector machine(LSSVM).Then, the LSSVM reference model is constructed to produce the reference output value. The error function between real output frequency spectra and reference output frequency spectra is used to detect and diagnose fault. The proposed approach avoids direct solving generalized frequency response function so that only need to obtain reference model by learning. F inally, a fault diagnosis simulation experiment of the mine hoist completed, experimental results verify the effectiveness of the proposed approach.

1Introduction

The fault diagnosis of complex electromechanical system is a very active field in recent years. This problem is more and more important because a lot of objects are complex electromechanical systems, such as aerospace vehicle, large scale computer numerical control (CNC) equipment, wind power system and so on. Most of complex electromechanical systems have nonlinear characteristics so that the problems of fault diagnosis of these systems have not been solved radically.

Generalized frequency response function (GFRF) is an expansion of frequency response function of linear system to nonlinear system based on V olterra series, which has clear physical significance and good robustness. Many research results have been obtained in the estimation and analysis of the GFRF in the past decades [1-6]. The GFRF has already been used to fault diagnosis and obtained some progress. V olterra time domain kernels were obtained by genetic algorithm and used to acquire general frequency response function by multidimensional Fourier transform, and then, the fault diagnosis of rotor-bearing system was researched [7]. However, the equations of describing the relations between input-output data and GFRF models are very complex, so the calculation amount of solving GFRF is very large and the practicability of using GFRF to detect and diagnose fault is confined. Support vector machine (SVM) is a machine learning method based on statistical learning theory, which well solved the problems of small sample, nonlinearity, local minimum and so on. Some researchers have already used SVM in fault diagnosis in recent years [8-14]. The independent component analysis (ICA) and SVM were combined to be used in non-Gaussian multivariate processing monitoring [15]. Information entropy and SVM were used to study the problem of reciprocating compressor fault diagnosis [16]. The problem of car assembly line fault diagnosis was researched by a hybrid diagnosis model composed of particle swarm optimization (PSO) and Gaussian support vector classifier machine [17]. SVM with genetic algorithm were applied to fault diagnosis of the power transformer [18].The training of SVM need solve quadratic programming problem so that the training time is quite long. Least square support vector machine (LSSVM) transforms quadratic programming problem of SVM into solving linear equation set and

reduces the complexity of calculation by constructing lost function [19-21].

In order to avoid the problem of calculation amount large of solving GFRF directly, the mapping relationship between input frequency spectrum and output frequency spectrum of nonlinear system is obtained. The LSSVM is used to approximate the mapping relationship and constructed as the reference model The fault diagnosis of nonlinear system can be completed by the error function between reference output frequency spectrum and real output frequency spectrum and avoids direct solving GFRF. A fault diagnosis experiment of the mine hoist used to demonstrate the effectiveness and engineering significance of the proposed approach.

2 Spectral representation of nonlinear dynamic system

Nonlinear system can be described by the V olterra functional series model in general

1

()()n n y t y t ∞

==∑ (1.a)

11

1

()(,)

()n n n n n

i i

i y t h u t d ττττ∞

∞

∞

∞

-∞-∞

-∞

===?-∑?

?

?∏ (1.b)

where ()u t R ∈,()y t R ∈ are the input and output of the system respectively, and n h is the n th-order V olterra kernel of the system. The n th-order GFRF of

the system (1) is defined by taking the multiple Fourier transformation of the n th-order kernel n h

11

21

(,,)(,,)i i n n n

n

n j f i

i H f f h e d πτττ

τ∞

∞

-∞

-∞-==

???∏ (2)

Consider the multi-tone input composed of K different sinusoids, called K -tone input

120

()c o s (2)2

i K

i i i

i K

j f t

i i K

i u t A f t A A e π

π==-≠=+∠=

∑∑

(3)

where 1)1(f i f i ?-=,K i ,,3,2 =, i f represents the frequency, and i A , i A ∠ and *i A are the amplitude, phase and complex conjugate of

i A respectively; * denotes the complex conjugate of a

complex number; i A -=*i A and i i f f -=-. The frequency spectrum of )(t u is

()[()]K

i

i

i K i U f A f f πδ=-≠=

-∑ (4)

Now we calculate the output frequency spectrum of the system (1). The harmonic component with frequency f of n th order output of the system (1) can be expressed as

2,1(,)(,,)j f n n f K y t f F A A e π= (5)

,,,1,0,,,!(,,)2!()

n f i m

K i n f K n i K i n f i n n f A n F A A m H M =-≠??

= ? ????∏ (6)

,,,,,11,,11,,({},,{},

{},,{})

n f n f K K n f n f n f K K M m f m f m f m f ----= (7)

where }{,,j j f n f m denotes j f n m ,, consecutive arguments with the same frequency j f . For example,

)

,,,(311 f f f H n will be represented by

)},{1},{0},{2(321 f f f H n . In (7), the n th-order

module vector of the input, which contributes to the output frequency f , namely

,,,,1,,1,,(,,,,,)n f K n f n f n f K M m m m m --= (8)

is used, where 0>i m is the non-negative integer, and

,,,0

K

n f i i K i m n =-≠=∑

(9)

Furthermore,

,,,,,,,0

1

()i K

K

n f i i n f i n f i i i K i i m f m m f f =-=-≠==-=∑

∑ (10)

The total harmonic component with frequency f of the system output, which is the sum of

,2,1),,(=n f t y n , can be written as

21(,)(,,)j ft f K y t f F A A e π= (11)

1,11(,,)(,,)f K n f K n F A A F A A ∞

==∑ (12)

Therefore the output of the system (1) is

1211

()(;)()

(,,)()

i i L

i i L

j f t f K i y t y t f t F A A e t πεε===+=+∑∑ (13)

where i f ,L i ,,2,1 = are frequencies generated by the multi-tone input. Note that L is infinite in general, but the bandwidth of any real system is finite, so we can assume that L is a finite integer, and ε is the truncation error. The frequency spectrum of the output is

1

()2()()i L

f i i Y f Y f f f πδ?==-+∑ (14)

12(,,,)i i f f K Y F A A A = (15)

where ?denotes the error generated by ε.

Now, a conclusion can be drawn from the above discussion.

Proposition: For a nonlinear system (1), suppose multi-tone signal (3) is its input, then the output

spectrum can be obtained by (14) and (15), and the value of which in every point i f ,1,,i L = is a nonlinear function of sampled frequency spectrum of input signal, namely 1,,K A A .

3 The fault diagnosis algorithm of nonlinear system based on SVM model

The mapping relation between input frequency spectrum and output frequency spectrum is represented by (14) and (15). This mapping relation is used as reference model when picks up fault diagnosis. Fault diagnosis is conducted through the deviation of reference output frequency spectrum and real output frequency spectrum of system. One method obtains the mapping relation directly through identifying V olterra frequency kernel, but the calculation amount is too large to fulfill the timeliness of online fault diagnosis. Another method uses machine learning method to construct system reference model so that has small calculation amount. SVM is a small sample machine learning method. For complex electromechanical system, the obtaining of more fault samples means large economic loss, so the SVM regression method is used to construct reference model in this paper. Suykens proposed least square support vector machine by changing constraint condition and risk function of SVM [19]. The training of LSSVM just need solve a linear equation set, so that it is easy to achieve and has high training efficiency.

For the regression problem of LSSVM, considers training sample set is 1{(,),,}n M i i i i i S y R y R ==∈∈x x , where i x is the th i input vector, i y is the target value correspond i x , and M is sample capacity. The constrained optimization problem of LSSVM regression is transformed to the problem of solving linear equation set based on constrained optimization theory and KKT condition. The linear equation set is

1

00T

b γ-??????

?=??????+?????

??11

y I Ωα (16) where

12[,,,]M ααα= α, i R α∈is Lagrange

multiplier,I is M dimensions unit matrix , 1 is M dimensions unit column vector, Ω is M dimensions symmetry square matrix,

,(,)i j i j K Ω=x x , (,)K ?? is kernel function, b is

regression parameter, and 0γ> is penalty factor.α and b can be obtained through solving (16). The regression function is

1

?((,)M

i i i y

K b α=?+∑x)=x x (17) In accordance with above analysis, the mapping

relation between input frequency spectrum and output frequency spectrum can be represented through LSSVM

()f f f

Y ξ=Φ+X (18)

where 12[,,,]T k A A A = X is input frequency spectrum vector, f Y is output frequency spectrum correspond frequency

f , 1

()()M

f j f j f j K ,b α=Φ=+∑X X X is

regression function of LSSVM, and

f ξ is modeling

error.

LSSVM uses single kernel function in general. In order to improve its fit precision, the hybrid-kernel function (,)mix i j K X X composed of polynomial kernel function (,)poly i j K X X and radial basis kernel function (,)RBF i j K X X is adopted in this paper. The hybrid-kernel function is

(,)(,)(1)

(,),(01)

mix i j poly i j RBF i K K K λλλ=+-?≤≤X X X X X X (19)

where

(,)(1)q

poly i j i j K =?+X X X X ,

2

2(,)exp[/]RBF i j i j

K σ=--X X X X ,λ is

adjust coefficient, q is the degree of polynomial, and σ is the width of radial basis function.

On the basis of above thinking of LSSVM regression algorithm, 1

()cos m

j

j

j u t A t ω==

?∑ used to as

the excite signal of nonlinear system. Assume sampling frequency is s f , sampling length is s N , the degree of polynomial is 3q =, and the width of radial basis kernel function is 1σ=, then the steps of identify f Φ are

Step 1: Excite the system under normal state and sample input data and output data synchronously, and then obtain input data series 1,,{()}|s i N u i = and output data series 1,,{()}|s i N y i = . Use FFT to obtain input frequency domain series 1,2,,{()}s k N U k = and output frequency domain series 1,2,,{()}s N Y k of sample data. We choose input frequency spectrum that correspond ,(1,2,,)j f j m = to construct input

vector 1[,]T

m A A = X and use f Y as target value.

Put up N time experiments through changing the amplitude of excite signal and achieve N group data. Step 2: Choose λ、γ that the parameters of LSSVM. Let 1N group frequency spectrum data as training data that are obtained from step1. Train LSSVM through (16) and (17) to obtain regress function f Φ.

Step 3: Use 2N group data that are not applied

training to test the generalized ability. If the precision fulfill the demand, training is end; else, turn to step2. In this paper, n LSSVM are constructed to fit the frequency spectrum value of n frequencies of the nonlinear system output frequency spectrum. Fig.1 shows the LSSVM reference model of nonlinear system, 1[,]T m A A = X is input vector, 1,,M X X are support vectors,

1,,(1,,)i iM i n αα= denote

the Lagrange multiplier correspond support vector,

?i

f Y

is reference output frequency spectrum.

Fig.1 LSSVM reference model structure of nonlinear

system

Assume 12[,,,]n f f f Y Y Y = Y is real output frequency spectrum vector, 12????

[,,,]n f f f Y Y Y = Y is

reference output frequency spectrum vector, and ?e

is the deviation vector between Y and ?Y . Fig.2 shows

fault diagnosis structure

Fig.2 Fault diagnosis structure

After obtaining the support vector machine reference model of object, we use excitation

signal 1

()cos m

j

j j u t A

t ω==

?∑ to excite the reference

model. Suppose sampling frequency is s f , sampling length is s N then the steps of online fault detection and diagnosis are

Step 1: Sample input signal and output signal synchronously that obtain input frequency spectrum

1[,]T

m A A = X and output frequency

spectrum 12[,,,]n f f f Y Y Y = Y , and then use (17)

to

calculate

reference

output

frequency

spectrum 12????

[,,,]n f f f Y Y Y = Y .

Step 2: Choose real output frequency spectrum and reference output frequency spectrum correspond frequency i f that fulfill ,1,2,,i f Y i L α>=

to construct real output frequency spectrum feature vector 12'[,,,]L f f f Y Y Y = Y and reference

output

frequency

spectrum

feature

vector 12'????

[,,,]L f f f Y Y Y = Y . Calculate error function

1

221

?{[]}i i

L

f f i Y Y δ==-∑ (20)

Step 3: Choose

0δ as threshold, if 0δδ≤,system

under normal state, else system occur fault.

4 Fault diagnosis simulation experiment

Mine hoist is one of the principal equipments of mine production and transportation that used to transport mineral, miner, equipment and so on. One of the important factors that effect the safety of producing and miner's survival is whether the mine hoist is under normal state. Brake is the key component of the mine hoist and it is also the last safety barrier of the mine hoist. Because of the reasons of component natural characteristics, long time operation, braking disc pollution and so on, the brake of mine hoist takes easily some faults. The brake is a nonlinear unit ，the

nonlinear system fault diagnosis is a difficult problem of fault diagnosis field, so the fault diagnosis of the brake is very difficult. Fig. 3 shows the operating principle of the disc brake of the mine hoist.

Fig.3 The operating principle of the brake of mine

hoist

When there is no pressure in the oil jar, the disk

spring is in free state. When the brake disc is in contact with the brake wheel, the brake is under braking state. When the pressure in oil jar rise to some extent, under the force of hydraulic pressure, the disk spring drives the brake wheel separating from the braking disc by the connecting rod, the brake is under non-braking state. In both states of the brake system, the disk spring endures the pressure. Assume )(t u is the pressure impressed to the spring, )(t x is the shift of the connecting rod, k 1 is the linear elastic coefficient the spring and 23,k k describe the nonlinear plastic characteristic between the pressure and the shift, then the relationship between the pressure )(t u and the shift )(t x can be written as

23123()()()()u t k x t k x t k x t =+++ . The spring

may be damaged because of fatigue under long-term use. The apparent feature of the fatigue of the spring is the enhancement of the plasticity and the weakness of the elasticity, namely the diminishment of 1k and the increment of 23,k k .

Let B , m denote the viscous damping coefficient of the hydraulic oil and the total mass of the spring, connecting rod and brake disc respectively, only consider the impact of nonlinear plastic coefficient

2k , then the dynamical model of the brake system

can be written as

212()()()()()0mx

t Bx t k x t k x t u t +++-= (21) The parameters of a normal brake system of mine hoist are 1m =、14.5B =、1200k =、220k =, input signal is 12()cos(1.2)cos(2.4)u t a t a t ππ=+,

sample frequency is 20Hz, sample length is s N =100. Let 1a obey uniform distribution of [50, 60] and 2a obey uniform distribution of [100,120]. The input frequency spectrum vector

12[,]T A A =X and

output frequency spectrum vector 1[,,]n f f Y Y = Y are obtained through simulation experiment. Experiments are repeated that obtain 51 groups input and output frequency data. 50 group data as training samples are used to learn LSSVM. We use mix kernel function as kernel function that the same as Eq. (19), where, the degree of polynomial is 3q =, the width of radial basis function is 1σ=. The generalized ability of LSSVM is tested through the data that are not trained. Experiment result shows that the fitting precision is good.

Let input is ()60cos(1.2)120cos(2.4)u t t t ππ=+, sample frequency is s f =20Hz, sample length is

s N =100. We use reference model of LSSVM to fit

output frequency spectrum of normal state and fault state. When mine hoist is fault, the parameters are

1m =，14.5B =，1180k =，230k =. The output

frequency spectrum of mine hoist are shown in Fig.4~Fig7. Where, Fig.4 is real output frequency spectrum and reference output of normal state, Fig.5 is the deviation between real output frequency spectrum and reference output of normal state, Fig.6 is real output frequency spectrum and reference output of fault state, and Fig.7 is the deviation between real output frequency spectrum and reference output of fault state. The frequency spectrum deviation of fault state is greater than the normal state from Fig.5 and

Fig.7.

Fig.4 Real output and reference output of normal

state

Fig.5 The deviation between real output and reference

output of normal state

Fig.6 Real output and reference output of fault

state

Fig.7 The deviation between real output and reference

output of fault state

Let 1.5α=, choose frequency spectrum that correspond

0.2Hz,(0,1,,15)

f l l =?= to

construct real output frequency spectrum feature vector '00.23[,,,]Y Y Y = Y and reference output frequency

spectrum feature vector

'00.23

????[,,,]Y Y Y = Y . Calculate error function δ through (20) to obtain 0.29normal δ=of normal state and

3.97fault δ= of fault state. The difference of

δbetween normal state and fault state is very

observably. So we can choose threshold 0 1.5δ= to diagnosis the brake system of mine hoist.

5 Conclusions

In this paper, nonlinear spectral analysis and least square support vector machine are combined to study the fault diagnosis of complex electromechanical system. Firstly, the mapping relationship between input frequency spectrum and output frequency spectrum of nonlinear system is obtained. Then, a reference model of nonlinear frequency spectrum characteristics is constructed by LSSVM. Finally, the deviation function between reference output frequency spectrum and real output used to detect and diagnose fault of nonlinear system. The proposed fault diagnosis approach avoids the problem of calculation amount large of solving GFRF directly. Fault

diagnosis experiment results of the mine hoist verify the effectiveness of the proposed approach.

6 Acknowledgements

This work was supported by the Technology Project of Shaanxi Province, China (Grant No. 2010K08-13) and Technology Project of Xi?an, China (Grant No. CXY1013(1)).

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高中英语公开课说课稿

Unit3 Traveljournal中reading的教学设计思路 JOURNEY DOWN THE MEKONG .No.7 Middle School of wuwei wang zhongwen 一、教材的分析与处理 1、课型：阅读理解 2、教材分析： 1.地位： 本文是该单元“旅行日志”的重点文章，属精读材料。本单元的中心话题是“旅游”，可以说这是一个世界性的时尚话题，随着经济发展、社会进步、人们生活水平的提高，旅游作为现代人的一种生活方式，越来越被更多的人们所接受与喜爱。本单元讲述了一段沿湄公河而下的自行车旅行，主人公“王坤”以旅行日记的形式详细的记录了这一过程。学生通过追随这一段旅程，探讨与“旅游”相关的各种话题，如：如何为旅游做准备，怎么选择适当的旅游方式，怎样确定旅游路线，计划或日程等等。通过本单元的学习，不仅可以实使学生学到与旅游有关的语音知识和语言技能，还会使学生对旅游产生浓厚的兴趣，通过旅游了解世 2.内容：“阅读”(Reading)部分的题目使”沿湄公河而下的旅程“。文章讲到主人公王坤和姐姐王薇想骑山地车旅行，于是选定了沿着云南西部的澜沧江，也就是湄公河作为旅游路线。他们选择海拔5000多米，空气非常稀薄的高山作为旅行的起点。着注定是一次非常刺激的旅行经历。通过查

阅资料，他们知道了河流的特点和流向，沿途所要经过的地形等等，增长了地理方面的知识，开阔了视野。通过阅读，教师要让学生学到一些有关地理的单词和短语，训练他们的阅读技巧，还要让他们做好懂得如何做好旅行前的准备工作，例如选择自己感兴趣的旅行地点，确定旅游路线，通过查地图等了解沿途的相关信息。这课时主要侧重于阅读能力的培养，教给学生多种灵活多变的阅读方法。引导学生开展任务型阅读，以任务为依托，激发学生参与主体，从篇章中准确、高效地获取知识与信息。 体裁：旅行日记 3、学生分析：学生已经学了warming-up与部分单词，并且预习了这篇课文，上课会比较轻松，理解也比较容易。 4、教学目的要求： ①语言知识：理解与旅游有关的知识并掌握文中的词汇短语。 ②语言技能：培养阅读策略（跳读、查读、略读与归纳能力）；了解旅游常识；学会如何写旅行日志。 ③情感态度：感受主人公认真谨慎的态度，养成做事充分准备坚持到底的好习惯。 ④学习策略：培养查找旅游信息的能力，出行之前讨论、制定计划、查阅资料等能力。 ⑤文化意识：增强对祖国大好河山的热爱，以及在旅游中接受异国文化的能力。 5、教学重点：让学生理解本课旅游日记的内容。 6、教学难点：培养学生的阅读能力，尤其是理解归纳的能力。