aeroengine exhausted gas temperature prediction using process extreme learning machine

9
Aeroengine Exhausted Gas Temperature Prediction Using Process Extreme Learning Machine Ding Gang a , Lei Da b and Yao Wei c School of Mechatronics Engineering, Harbin Institute of Technology, Heilongjiang, P.R. China, 150001 a [email protected], b [email protected], c [email protected] Keywords: Process Extreme Learning Machine, Function Inputs, Time Series Prediction, Aeroengine Health Condition Prediction. Abstract. To solve the aeroengine health condition prediction problem, a process extreme learning machine (P-ELM) is proposed based on the process neural networks (PNN) and the extreme learning machine (ELM). The proposed P-ELM has an ability of processing time accumulation effects widely existing in practical systems. The proposed P-ELM has only one unknown parameter which can be calculated directly rather than in the iteration way, thus the training time can be significantly reduced. After being validated via the prediction of Mackey-Glass time series, the proposed P-ELM is utilized to predict the aeroengine exhausted gas temperature, and the test results is satisfied. It has shown by the contrast tests that the proposed P-ELM can outperform the ELM, but has equal performance with the PNN. However, with just one unknown parameter which can be calculated directly, the proposed P-ELM is much easier to use and it needs much less training time. Thus, the proposed P-ELM is more adaptable to the practical situation of aeroengine health condition prediction compared with the PNN. Introduction Artificial neural networks (ANN) have attracted significant attention in many fields including time series prediction in the last few years mainly due to their ability of approximating nonlinear mappings. Hornik [1] and Funahashi [2] proved that multilayer feedforward neural networks can approximate any continuous function. Huang [3] took a further step and proved that standard single layer feedforward neural networks are capable to learn N distinct samples with at most N hidden neurons and any bounded nonlinear activation function with zero error. It seems that ANN can have great potential in modeling nonlinear systems. However, traditional ANN actually cannot perfectly model the actual signal processing procedure of true biological neurons, whose states of the synapses are interrelated with relative time of the input impulse [4]. Thus, it’s difficult for traditional ANN to process time accumulation effect exists in practical systems directly. To overcome this limitation, He and Liang proposed process neuron model [5] in 2000, which has similar architecture to traditional neurons, but its inputs, outputs and corresponding connection weights of the process neuron can be time-varying functions. With similar working mechanism to true biological neurons, process neural networks (PNN) can often get higher precision in time series prediction compared with NN, and is widely used in such areas [6-8]. However, the training process of PNN is still complex and time consuming just as traditional ANN. Extreme learning machine (ELM) is an extended single hidden layer feedforward neural network proposed by Huang [9], which has universal approximation capability for any type of computational hidden nodes [10]. The essence of ELM is that the hidden layer of the network need not to be tune, and is given directly instead. So, there is no iteration during the application of ELM, which can greatly reduce the training time. And ELM not only tends to reach the smallest training error but also the smallest norm of output weights which means it can gain better generalization performance compared with traditional neural networks. Applied Mechanics and Materials Vols. 423-426 (2013) pp 2355-2362 Online available since 2013/Sep/27 at www.scientific.net © (2013) Trans Tech Publications, Switzerland doi:10.4028/www.scientific.net/AMM.423-426.2355 All rights reserved. No part of contents of this paper may be reproduced or transmitted in any form or by any means without the written permission of TTP, www.ttp.net. (ID: 129.186.1.55, Iowa State University, Ames, United States of America-08/10/13,05:32:07)

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Aeroengine Exhausted Gas Temperature Prediction

Using Process Extreme Learning Machine

Ding Ganga, Lei Dab and Yao Weic

School of Mechatronics Engineering, Harbin Institute of Technology,

Heilongjiang, P.R. China, 150001

[email protected],

[email protected],

[email protected]

Keywords: Process Extreme Learning Machine, Function Inputs, Time Series Prediction, Aeroengine Health Condition Prediction.

Abstract. To solve the aeroengine health condition prediction problem, a process extreme learning

machine (P-ELM) is proposed based on the process neural networks (PNN) and the extreme learning

machine (ELM). The proposed P-ELM has an ability of processing time accumulation effects widely

existing in practical systems. The proposed P-ELM has only one unknown parameter which can be

calculated directly rather than in the iteration way, thus the training time can be significantly reduced.

After being validated via the prediction of Mackey-Glass time series, the proposed P-ELM is utilized

to predict the aeroengine exhausted gas temperature, and the test results is satisfied. It has shown by

the contrast tests that the proposed P-ELM can outperform the ELM, but has equal performance with

the PNN. However, with just one unknown parameter which can be calculated directly, the proposed

P-ELM is much easier to use and it needs much less training time. Thus, the proposed P-ELM is more

adaptable to the practical situation of aeroengine health condition prediction compared with the PNN.

Introduction

Artificial neural networks (ANN) have attracted significant attention in many fields including time

series prediction in the last few years mainly due to their ability of approximating nonlinear mappings.

Hornik [1] and Funahashi [2] proved that multilayer feedforward neural networks can approximate

any continuous function. Huang [3] took a further step and proved that standard single layer

feedforward neural networks are capable to learn N distinct samples with at most N hidden neurons

and any bounded nonlinear activation function with zero error. It seems that ANN can have great

potential in modeling nonlinear systems.

However, traditional ANN actually cannot perfectly model the actual signal processing procedure

of true biological neurons, whose states of the synapses are interrelated with relative time of the input

impulse [4]. Thus, it’s difficult for traditional ANN to process time accumulation effect exists in

practical systems directly. To overcome this limitation, He and Liang proposed process neuron model

[5] in 2000, which has similar architecture to traditional neurons, but its inputs, outputs and

corresponding connection weights of the process neuron can be time-varying functions. With similar

working mechanism to true biological neurons, process neural networks (PNN) can often get higher

precision in time series prediction compared with NN, and is widely used in such areas [6-8].

However, the training process of PNN is still complex and time consuming just as traditional ANN.

Extreme learning machine (ELM) is an extended single hidden layer feedforward neural network

proposed by Huang [9], which has universal approximation capability for any type of computational

hidden nodes [10]. The essence of ELM is that the hidden layer of the network need not to be tune,

and is given directly instead. So, there is no iteration during the application of ELM, which can greatly

reduce the training time. And ELM not only tends to reach the smallest training error but also the

smallest norm of output weights which means it can gain better generalization performance compared

with traditional neural networks.

Applied Mechanics and Materials Vols. 423-426 (2013) pp 2355-2362Online available since 2013/Sep/27 at www.scientific.net© (2013) Trans Tech Publications, Switzerlanddoi:10.4028/www.scientific.net/AMM.423-426.2355

All rights reserved. No part of contents of this paper may be reproduced or transmitted in any form or by any means without the written permission of TTP,www.ttp.net. (ID: 129.186.1.55, Iowa State University, Ames, United States of America-08/10/13,05:32:07)

Thus, a process extreme learning machine (P-ELM) is proposed based on the advantages of the

PNN and the ELM in this paper, which has the ability of processing time accumulation effect and can

calculate parameters directly rather than in the iteration way. Then, the P-ELM is utilize to monitor

the aeroengine health condition by predicting the aeroengine exhausted gas temperature which can

show the gradually decrease of the engine performance as a result of time accumulation.

Process neuron and extreme learning machine

Process Neuron. The process neuron model is composed of three sections: inputs, an activation unit

and output. This is based on the fact that a biological neuron is composed of three basic parts: a

dendrite, a soma and an axonal tree. Traditionally, the inputs and the connection weights of the neuron

in a neural network are discrete values. However, the inputs and the connection weights of process

neuron are continuous time-varying functions. The process neuron architecture is depicted in Fig. 1.

)(1 tx

)(2 tx

)(txn

)(⋅f y

)(1 tω

)(2 tω

)(tnω( )K ⋅

Fig.1. Schematic diagram process neuron

According to the difference of the operator ( )K ⋅ , the process neuron can be divided into two

different types. The output of the first type process neuron model can be expressed as:

1

( ) ( ( ) ( ) )n

i i

i

y t f t x tω θ=

= − .∑ The output of the second type process neuron model can be expressed as:

01

( ( ) ( ) )n T

i i

i

y f t x t dtω θ=

= − .∑∫ Where ],0[)( TCtxi ∈ is the i-th input function, ],0[ TC denotes the space of

continuous functions on ],0[ T , )(tiω is the i-th weight function, θ is the threshold, and )(⋅f is the

activation function.

Extreme Learning Machine. ELM is an extended single hidden layer feedforward neural network

(SLFN), so the basic model of ELM can be described as an approximation model of N arbitrary

distinct samples ( ),i i

x t , where [ ]1 2, , ,

T n

i i i inx x x R= ⋅⋅ ⋅ ∈x , and [ ]1 2

, , ,T m

i i i imR= ⋅⋅⋅ ∈t t t t . For ELM with N

training samples, L hidden neurons and activation function , the mapping relationship between the

inputs and output is

1 1

( , , ) ( ), 1, ,L L

j i i j i i i j i

i i

o G b g b j N= =

= = ⋅ + = ⋅⋅ ⋅∑ ∑β w x β w x . (1)

Where i

w is the weight vector connecting the i-th hidden neuron and the inputs and

[ ]1, ,

T

i i inw w= ⋅⋅⋅w ,

iβ is the weight vector between the i-th hidden neuron and the output neurons and

[ ]1, ,

T

i i imβ β= ⋅⋅⋅β . The training goal of a traditional neural network is to minimize the error between the

outputs and the targets, which is1

N

j j

j

min o t=

∑ , where

jt is the corresponding target of

jo .It can be

rewritten in the matrix form as min -Hβ T where

1 1 1 1 1

1 1

( ) ( , , ) ( , , )

( ) ( , , ) ( , , )

L L

N N L L N N L

h G b G b

h G b G b×

⋅ ⋅ ⋅ = ⋅⋅ ⋅ = ⋅⋅ ⋅ ⋅ ⋅ ⋅

x w x w x

H

x w x w x

(2)

2356 Applied Materials and Technologies for Modern Manufacturing

1

T

T

L L m×

= ⋅⋅ ⋅

β

β

β

and

1

N N m

t

= ⋅⋅⋅

T (3)

From the view of ELM, to train SLFN is equivalent to find a least square solution to a linear system.

Since the hidden layer parametersi

w and i

b can be chosen randomly [11], the only unknown parameter

is β , which is the output weight matrix, so the equivalent linear system of SLFN is

- min -∧

H β T Hβ T (4)

And the smallest norm solution for the above linear system is

= †β H T (5)

Where †H is the Moore-Penrose generalized inverse of matrix H .Different methods such as

orthogonal projection method, iterative method, and singular value decomposition can be adopted to

calculate the Moore-Penrose generalized inverse matrix.

Since the only unknown parameter can be calculated directly without iteration, ELM does not need

a long training time. And ELM gains the smallest training error with the smallest norm of output

weights which means it can have better generalization performance compared with traditional neural

networks.

The process extreme learning machine

The process extreme learning machine proposed in this paper is a single hidden layer feedforward

process neural network (SLFPNN). For the sake of convenience, we just analyze P-ELM with single

output node in this section, whose topology structure is depicted in Fig. 2.

1( )x t

2 ( )x t

( )nx t

( )ki tω

� �

∑ y

Lβ�

, ,∑∫ g

, ,∑∫ g

, ,∑∫ g

Fig. 2. Process extreme learning machine model

It can be seen from Fig.2 that the hidden neurons of P-ELM are process neurons, and the inputs of

P-ELM are continuous functions which are different from the basic ELM. Given N samples ( )( ) ,i i

tx t ,

where [ ]1 2( ) ( ) , ( ) , , ( ) [0, ]

T n

i i i int x t x t x t C T= ⋅⋅ ⋅ ∈x , and

iR∈t , suppose that we utilize a P-ELM model with L

hidden neurons, then the output is

01 1

( ( ) ( ) ), 1, ,L n

T

j i ki jk i

i k

o g t x t dt b j Nω= =

= + = ⋅⋅ ⋅∑ ∑∫β (6)

Where ( )jk

x t is the k-th input of the j-th input vector, and ( )ki

w t is the corresponding weight vector.

According to (6) and section 2.2, we have to calculate the integration before we calculate the hidden

output matrix H, but it is always difficult and tedious. And in the practical use of PNN, the calculation

of integration is always avoided by expanding the input and weight functions based on normalized

Applied Mechanics and Materials Vols. 423-426 2357

orthogonal basis functions [12]. We use the same method to simplify the integral calculus in P-ELM.

Denote the normalized orthogonal basis functions as{ }1

( )p p

b t+∞

=, then the input and weight functions can

be expanded as

( )

1

( ) ( )M

p

jk jk p

p

x t a b t=

=∑ (7)

( )

1

( ) ( )M

m

ki ki m

m

t w b tω=

=∑ (8)

And we have0

1,( ) ( )

0,

T

p l

p lb t b t dt

p l

==

≠∫ , so Eq.6 can be rewritten as

( ) ( )

1 1 1

( ), 1, ,L n K

p p

j i jk ki i

i k p

o g a w b j N= = =

= + = ⋅⋅ ⋅∑ ∑∑β (9)

Let ( )(1) (2) ( ), , ,

TM

jk jk jk jka a a=a � , ( )(1) (2) ( ), , ,

TM

ki ki ki kiw w w=w � , 1 2

( , , , )j j j jn=a a a a� and

1 2( , , , )T

i i i ni=w w w w� ,where

1,2, ,i n= � , then Eq.9 could be rewritten as

( ) ( )1 1

( , , ) ( ), 1, ,L L

j i i j i i j i i

i i

o G t t b g b j N= =

= = ⋅ + = ⋅⋅ ⋅∑ ∑β w x β a w (10)

Where j i⋅a w is the inner product of

ja and

iw .Then we can get the equivalent linear system of

SLFPNN, and it can be described as min -Hβ T where

( )

( )

( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( )

1 1 1 1 1

1 1

( ) ( , , ) ( , , )

( ) ( , , ) ( , , )

L L

N N L L N N L

h t G t b t t G t b t t

h t G t b t t G t b t t×

⋅ ⋅⋅

= ⋅⋅ ⋅ = ⋅⋅⋅ ⋅ ⋅⋅

x w x w x

H

x w x w x

(11)

1

L

β

β

= ⋅⋅ ⋅

β and 1

N

t

t

= ⋅⋅ ⋅

T (12)

Thus, the equivalent linear system of SLFN is

- min -∧

H β T Hβ T (13)

And the smallest norm solution for the above linear system is

= †β H T (14)

Where †H is the Moore-Penrose generalized inverse of matrix H . Notice that for P-ELM, the

element hidden output matrix ( ) ( ) ( )( , , ) ( )i i j j i i

G t b t t g b= ⋅ +w x a w , where ( )g ⋅ is the activation function,

we choose sigmoid function as the activation function in this paper. The function inputs are finally

transferred into forms of vectors which help to diminish the tedious integral calculus.

The observations of actual systems are always discrete samples, so we have to fit them into

continuous functions to get the input functions before the application of P-ELM. And the steps for

using P-ELM are as follows:

2358 Applied Materials and Technologies for Modern Manufacturing

Step 1 Determine the structure of P-ELM, which is determining the hidden process neurons and the

choosing the hidden activation function,

Step 2 Choose a time interval [ ]0,T , fit the discrete sample into the continuous functions,

Step 3 Choose a set of orthogonal basis functions to expand the input functions and weight

functions as Eq.7 and Eq.8 to get j

a ,

Step 4 Randomly generate hidden node parameters i

w and i

b ,

Step 5 Calculate hidden output matrix H,

Step 6 Calculate the output weight∧

β , where∧

= †β H T .

Application test

Mackey-Glass Time Series Prediction Mackey-Glass time series is a typical chaos time series,

which is widely used to test the performance of predicting algorithms. In this section, the procedures

of utilizing P-ELM for time series prediction is explained by forecasting the Mackey-Glass time

series, and the effectiveness of P-ELM is also validated at the same time.

We generated a Mackey-Glass with 150 samples, which can be denoted as 1501{ }k kx = . We choose

1 5( , , , )i i ix x x+ +� to generate the input function iIF , 1, ,144i = � , and 6ix + to be the corresponding output.

Then, we can get 144 couples of samples which can be denoted as 1446 1{ , }i i iIF x + = , and we use the former

72 couples to train the P-ELM and the left 72 couples to test out model. The number of the hidden

neurons of the P-ELM employed is 10, and the input number is 6. We adopted the normalized

Legendre orthogonal basis functions to expand the input functions and the weight functions. The

prediction results are depicted in Fig (3). It can be seen from Fig.3 that the performance of P-ELM is

satisfied. The absolute value of the relative error is used to evaluate the performance of our model,

and the average value is 0.59%, while the max value is 2.09%, which proves the effectiveness of

P-ELM.

0 10 20 30 40 50 60 700.4

0.6

0.8

1

1.2

1.4

Sample points

Sam

ple

valu

e

Actual Value

Prediction results by P-ELM

Fig.3. Prediction results of Mackey-Glass time series by P-ELM

Aeroengine health condition prediction. Aeroengine is a complicated nonlinear system, which is

always working under extreme conditions such as high temperature, high pressure and high speed

leading to the result that the performance of its components and subsystems will degrade gradually

with time. Thus, condition monitoring is essential in terms of flight safety and also for reduction of the

preventive maintenance cost. Exhausted gas temperature (EGT) is a pivotal health index for

aeroengine because of the temperature of the engine turbines need to be strictly limited for safety

considerations, thus EGT monitoring is very important in aeroengines’ daily operation. The EGT

fluctuation and trend should be watched carefully, which lead to the needs of predicting technologies’

Applied Mechanics and Materials Vols. 423-426 2359

utilization in such areas. In this section, the P-ELM is utilized to forecast the EGT time series to

describe its application to the aeroengine condition monitoring.All text, figures and tables must be in

English.

The EGT time series is from an airline company in China. In order to reduce the influence from

environment, a basis value was subtracted from the origin data and a new time series DEGT was

generated for prediction. The DEGT time series is denoted as 86

1{ }j jDEGT = . We choose

1 5( , , , )i i iDEGT DEGT DEGT+ +� to generate the input function iIF , 1, ,144i = � , and 6iDEGT + to be the

corresponding output. Then, we can get 80 couples of samples which can be denoted

as 806 1{ , }i i iIF DEGT + = , and we use the former 70 couples to train the P-ELM and the left 10 couples to test

out model. And we predict the same time series by ELM and PNN of single hidden layers for

comparison.

The input number of all models is set as 6, and the normalized Legendre orthogonal basis functions

are also used to expand the input functions and the weight functions. However, just as traditional

artificial neural networks, the number of hidden neurons is difficult to choose. We tried several

numbers such as 10, 20 and 30, and we found that with the increase of hidden number, the

performance of ELM and P-ELM get worse, thus ,we choose the number of hidden neurons of all

models as 10. We adopted the normalized Legendre orthogonal basis functions to expand the input

functions and the weight functions for P-ELM and PNN. And the PNN is trained with

Levenberg-Marquardt [13] algorithm, which is widely used fast training algorithm for neural

networks. The prediction results are show in Table 1.

Table 1. DEGT Prediction Results

Absolute relative error/% Absolute error/℃

average max min average max min

P-ELM 4.4268 10.1314 0.0850 2.0650 4.5085 0.0392

ELM 4.6301 10.7738 0.4577 2.1663 4.7943 0.2110

PNN 4.5270 10.1243 0.2991 2.1147 4.9931 0.1379

As shown in Table1, P-ELM performs slightly better than ELM and PNN, and PNN also perform

slightly better than ELM. However, all of the results are not satisfied because that their max absolute

relative errors reach 10%, and it can hardly be adopted in practical engine health monitoring. In order

to improve the prediction precision, the cubic spline interpolation method is utilized to generate a new

DEGT time series to add more information. We choose 166 samples from the new time series which

denoted as 166

1{ }j jDEGT = , and the prediction of the origin DEGT time series is equivalent to the prediction

as the new DEGT time series. 160 samples are generated with the method aforementioned, and the

former 140 samples are utilized for training and the left are for testing. P-ELM, ELM and PNN are

also employed to predict the same time series. The results are depicted as Fig. 4.

It can be seen from Fig.4 that the performance of all three model are improved. Actually, the

average of absolute relative error of prediction results by P-ELM is 1.07% while the max value is

2.01%; the average of absolute relative error of prediction results by ELM is 1.33% while the max

value is 3.33%; and the average of absolute relative error of prediction results by PNN is 1.74% while

the max value is 4.20%. Again, P-ELM outperforms the other two methods, but PNN performs

slightly worse than ELM this time.

Three more tests are conducted with different ratio of training samples and testing samples

utilizing the time series 166

1{ }j jDEGT = , and the results are listed in Table 2. Taking both average relative

error and max relative error into consideration, P-ELM perform slightly better than ELM, and PNN

perform slightly better than ELM. Though it is difficult to judge whether P-ELM performs better than

PNN in our tests, it is worth mentioning that the training time can be significantly reduced while using

P-ELM compared with PNN.

2360 Applied Materials and Technologies for Modern Manufacturing

1 2 3 4 5 6 7 8 9 10

42

44

46

48

50

52

54

Sample times

DE

GT

/ °C

Actual Value

Prediction results by P-ELM

Prediction results by PNN

Prediction results by ELM

Fig. 4. Prediction results of DEGT after interpolation

Table 2. DEGT Prediction Results with different sample ratios

Sample ratio Max relative error/% Average relative error/%

Train Test P-ELM PNN ELM P-ELM PNN ELM

120 40 3.41 7.35 2.52 1.03 1.89 1.08

100 60 7.52 11.05 15.38 2.84 2.47 2.69

80 80 8.85 8.82 9.52 2.82 2.65 2.95

Conclusions

The process extreme learning machine (P-ELM) is proposed in this paper taking both advantages of

PNN and ELM. The P-ELM can process time accumulation effects widely exists in practical systems,

it has only one unknown parameter which can be calculated directly rather than be tuned in an

iteration way, and this can significantly reduce the training time. Its application to aeroengine health

monitoring is described by the prediction of engine exhausted gas temperature while compared with

PNN and ELM. P-ELM outperforms ELM most of the time in our tests. This can be ascribed to the

fact that P-ELM has the ability of processing time accumulation effect in complex system, and

P-ELM can get the least norm of output weight while gain the least error and then has better

generalization performance. Though we cannot conclude that P-ELM also performs better than PNN

yet, we can conclude that P-ELM performs not worse than PNN, moreover, P-ELM has only one

unknown parameter which can be calculated directly, thus it needs much less training time than PNN,

which can make P-ELM more adaptable to the practical situation of aeroengine health monitoring.

Acknowledgments

This research was supported by the Harbin City Key Technologies R & D Program under Grant No.

2011AA1BG059, and the International S&T Cooperation Projects of Heilongjiang Province under

Grant No. WB10A104.

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2362 Applied Materials and Technologies for Modern Manufacturing

Applied Materials and Technologies for Modern Manufacturing 10.4028/www.scientific.net/AMM.423-426 Aeroengine Exhausted Gas Temperature Prediction Using Process Extreme Learning Machine 10.4028/www.scientific.net/AMM.423-426.2355