87

Fault classification of fluid power systems using

a dynamics feature extraction technique and
neural networks

T T Le, J Watton and D T Pham

Cardiff School of Engineering, University of Wales Cardiff

Abstract: Multilayer perceptron (MLP) type neural networks and dynamic feature extraction tech-
niques, namely linear prediction coding (LPC ) and LPC cepstrum, are used to classify leakage type
and to predict leakage flowrate magnitude in an electrohydraulic cylinder drive. Both single-leakage
and multiple-leakage type faults are considered. A novel feature is that only pressure transient
responses are employed as information. In addition, the feature extraction technique used to detect
faults can result in a large data dimensionality reduction. The performance of two MLP models,
namely serial and parallel, are studied to reflect the importance of the way data are presented to
the MLP.

Keywords: fault classification, leakage detection, dynamic feature extraction, linear prediction,
artificial neural networks, fluid power system

NOTATION 1 INTRODUCTION

c , c(t) LPC cepstra coefficients There have been a number of approaches to fault diag-
i
E total square prediction error nosis of fluid power systems ranging from complex
F−1 the inverse Fourier transform system analysis to a more pragmatic artificial intelligence
g(t), G(v) source or excitation function and its approach. The failure modes and effect analysis method
Fourier transform considers all the possible failure modes in a system and
h(t), H(v) impulse response of a linear system and uses primitive relations between quantities to deduce the
its Fourier transform effect on other component characteristics (1, 2). Rule-
N window length based expert system techniques embrace all possible fail-
P line 1 pressure ure modes within the system, often incorporating some
1
P line 2 pressure knowledge of system behaviour, and usually resulting in
2
Q ,Q external leakages in line 1 and line 2 a large rule base (3–5). Artificial neural-networks-based
e1 e2
Q internal leakage across the cylinder seal approaches have been applied to servovalve/cylinder
i
R( j ) autocorrelation function drives (6, 7). These approaches consider systems with
w(n) windowing function little or no dynamic data content, and in these cases the
x(t), X(v) output parameter of a linear system and steady state signals obtained from the pressure trans-
its Fourier transform ducers and flowmeters are used to classify faults. In this
x̂ predicted value of the output discrete study a system having significant dynamics, i.e. a closed-
t
parameter x loop position control system, is considered and the
t dynamic information contained in the system transient
z independent parameter in the z-trans-
form domain pressure responses are extracted and fed to a neural
network fault classifier.
a LPC coefficient Since neural networks can perform essentially arbi-
i
e residual error in linear prediction trary non-linear functional mappings between sets of
t
variables they can, in principle, be used to map the
raw input data directly on to the required final output
The MS was received on 6 October 1997 and was accepted for publication values. However, as Bishop (8) has reported, it is import-
on 13 March 1998. ant to preprocess any raw and/or dynamic data before
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88 T T LE, J WATTON AND D T PHAM

classifying them to improve the network performance. line 2), the pressure responses for internal leakage
Preprocessing usually involves extracting relevant and ( line i ) tail off towards the same steady state level as the
discriminating information and in so doing reducing non-leakage case. This implies that if only steady state
data dimensionality. This is called the ‘feature extrac- signals are used then the internal leakage is likely to be
tion’ process. Two feature extraction techniques, namely more difficult to detect than external leakage, as reported
linear prediction coding (LPC ) and LPC cepstrum, are by Le et al. (7). However, it would appear that this
studied in this application to leakage detection and situation can be improved if dynamic information is
classification of an electrohydraulic system. The distinc- used.
tive feature of the work is that only the system pressure
transient responses are used as information to identify
leakage types and leakage flowrate, and follows the 2 FEATURE EXTRACTION APPROACHES
desire to use a minimum number of sensors in practice.
This is also an attractive approach since the position 2.1 Linear prediction coding
step responses show negligible change as the faults are
introduced, particularly at an early stage of formation. The term ‘linear prediction’ was introduced by Wiener
In addition, two neural network models are investigated, (9) and the technique was first used for speech analysis
namely serial and parallel structures. by Atal and Schroeder (10). The most important feature
In this study, a servovalve is used in the closed-loop of linear prediction (LP) is that a small number of
position control of a cylinder, as shown in Fig. 1. Two parameters, i.e. the LP coefficients, can be used to pre-
pressure transducers were placed adjacent to the servo- cisely and efficiently represent the signal waveform and
valve output ports to measure the system transient its spectrum characteristics (11). In addition, these
response, namely P and P . The operating temperature coefficients may be obtained by a relatively simple
1 2 calculation.
varied between 45±5 °C and the pressure data were
Let the discrete representation of the signal x(t) be x .
sampled at 1.0 kHz. Flowmeters were placed in the leak- t
age paths to set the appropriate leakage levels, and leak- The first-order linear equation between the present
sample x and the preceding p samples is as follows:
ages from both lines (external leak) and across the t
cylinder seal (internal leak) were artificially introduced x +a x + · · · +a x =e (1)
t 1 t−1 p t−p t
together with combinations of all three. Data were
where e is an uncorrelated statistical variable. When
acquired via a data acquisition card and a microcom- t
considering linear system identification in modern con-
puter system which also controlled the extension and
trol theory, the process exemplified by equation (1) is
retraction strokes of the cylinder. A Windows-based
called the autoregressive method, in which e and x are
graphics environment was used to initiate extension, t t
the system input and output respectively.
acquire 512 data points for each pressure, retract the
This linear difference equation can be used to predict
cylinder and acquire further data. One test cycle there-
the present sample value x from the previous p samples.
fore resulted in four transient pressure measurements. t
Given that the predicted value of x is
An example of data for each leakage type is shown in t
Fig. 2. For three different leakage flowrates, the graphs p
x̂ =− ∑ a x (2)
show that unlike the external leakage cases ( line 1 and t i t−i
i=1

Fig. 1 Hydraulic circuit diagram of the test rig

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 FAULT CLASSIFICATION OF FLUID POWER SYSTEMS 89

Fig. 2 Examples of system pressure data and its corresponding 10 LPC coefficients and 10 LPC cepstra
coefficients for different leakage types and flowrates. The number of data points for each pressure
response is 512

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90 T T LE, J WATTON AND D T PHAM

then equation (1) can be rewritten as where X(v), G(v) and H(v) are the Fourier transforms
of x(t), g(t) and h(t) respectively. Taking logarithms of
x −x̂ =e (3)
t t t the magnitude of both sides of equation (6) gives
Equation (1) is the linear prediction model equation of
order p, with linear prediction coefficients a , and e is log |X(v)|=log |G(v)|+log |H(v)| (7)
i t
referred to as the residual error. There are a number of
The cepstrum, which is the inverse Fourier transform of
approaches for calculating the LPC coefficients and the
log |X(v)|, is
most popular, which is adopted in this work, is the least
squares autocorrelation method (11). The derivation c(t)=F−1 log |X(v)|
procedure of the LPC coefficients is shown in Appendix 1
and an example of 10 LPC coefficients is shown in Fig. 2. =F−1 log |G(v)|+F−1 log |H(v)| (8)
This prediction model is also referred to as an all-pole
where F−1 is the inverse Fourier transform. The indepen-
model since it has a system transfer function given by
dent parameter of the cepstrum, t, is called quefrency.
(11)
As the cepstrum is the inverse transform of the frequency
1 domain function, t corresponds to a time domain
H(z)= (4)
1+ ∑ p a z−i parameter.
i=1 i It may be seen from equation (8) that the character-
Some typical predictions are shown in Fig. 3 and relate istics of both the source g(t) and the system h(t) can be
to the line 2 medium leakage condition shown in Fig. 2. linearly separated. The first function on the right-hand
It may be seen that the LPC technique used, with 10 side of equation (8) forms a peak in the high-quefrency
coefficients, gives excellent predictions for each pressure region, indicating the fundamental period of g(t), and
and in a situation where large fluctuations are evident. the second function yields a concentration in the low-
The maximum prediction error is less than 0.5 bar at the quefrency region, representing characteristics of the
point where P is around zero pressure. linear system h(t): this is an extremely important feature
1
of cepstrum. Furui (11) states that ‘the special feature
of the cepstrum is that it allows for the separate represen-
2.2 LPC cepstrum tation of the spectra envelope and the fine structure’. In
this work, only the first cepstra coefficients in the low-
The cepstrum, with associated cepstra coefficients c(t),
quefrency region, which reflects the characteristics of the
is defined as the inverse Fourier transform of the logar-
system, are chosen and the performance for leakage
ithmic magnitude spectrum (11). Considering linear
identification is compared to that using LPC coefficients.
system theory, the output x(t) is given by the convol-
The derivation procedure for the LPC cepstra
ution of the source g(t) and the system impulse response
coefficients is shown in Appendix 2 and an example of
h(t) as
extracted coefficients is shown in Fig. 2. In practice a
x(t)=g(t)h(t) (5) step input to the closed-loop control system is sufficient
to provide an error signal to the servovalve having a
which is equivalent to
frequency excitation component that is adequate for
X(v)=G(v)H(v) (6) cepstrum analysis.

Fig. 3 Prediction of pressures for line 2 medium leakage condition and using 10 LPC coefficients

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 FAULT CLASSIFICATION OF FLUID POWER SYSTEMS 91

Fig. 4 Overall configuration of the leakage detection system

3 NEURAL NETWORKS FOR LEAKAGE in this investigation and is given by

DETECTION

q A B
2pn
0.54+0.46 cos for 0∏n∏N−1
3.1 Approach w(n)= N−1
0 otherwise
The overall configuration of the leakage detection system
(9)
with the application of feature extraction and neural net-
works is shown in Fig. 4. Initially the pressure transient where N is the length of the window.
responses are passed through the feature extraction pro- Each pressure transient response is captured at a fixed
cess to produce either LPC coefficients or LPC cepstra length of 512 data points and the windowing length when
coefficients. These coefficients are then fed to the appro- used in feature extraction can be the whole data length
priate neural network to identify the leakage type. This or any fraction of it. The results presented examine the
output is then used to select the corresponding flowrate effect of the window length on the leakage-type identifi-
magnitude neural network, which is subsequently fed cation performance, the lengths used being 512, 256, 128
with feature coefficients to predict the leakage flowrate. and 64. In addition, the frame rate is chosen to be half
the window length.

3.2 Windowing
In practical signal processing, only part of a time signal 3.3 Neural network model
is used. The extraction of a frame of a signal is equivalent
to multiplying the signal by a finite-duration window For every cycle of cylinder extension and retraction there
function w(n). By advancing the window along the time will be four pressure response profiles to be captured.
axis, more information from the signal can be examined Each pressure profile is then used for feature extraction,
and analysed. This is illustrated schematically in Fig. 5. producing a series of coefficient vectors (see Fig. 5).
The Hamming window, which is widely used, is adopted There are two ways of presenting these coefficient vectors
to the neural network. One approach is to present the
coefficient vectors in a ‘serial’ manner, one vector at a
time. In this case, the number of input nodes to the
neural network is equal to the number of coefficients in
one vector. In testing, the network output is summed
over the number of total vectors, T, generated in one
cycle. The identification output would then be decided
on the basis of this summation and the approach is illus-
trated schematically in Fig. 6.
Another way of presenting coefficient vectors to the
neural networks is in a ‘parallel’ manner. All vectors
generated in one cycle are applied to the network at the
same time. The identification decision is made instantly
based upon this output. In this case, the number of input
nodes of the neural network is the total number of
coefficients generated in one cycle, i.e. the number of
Fig. 5 Windowing in feature extraction. P is the number of coefficients in one vector times the number of vectors,
coefficients generated in one vector T, in one cycle. The output nodes of the neural network
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92 T T LE, J WATTON AND D T PHAM

Fig. 7 The parallel neural network model

Fig. 6 The serial neural network model

performance of two MLP models, serial and parallel.
Two feature extraction techniques were used, namely
correspond to the number of faults being considered.
LPC and LPC cepstra, for each frame of data, and 10
This approach is shown in Fig. 7.
coefficients were extracted for both schemes. The total
number of coefficients generated in one cycle and the
corresponding data dimensionality reduction ratio, for
3.4 Learning
all four pressure responses and for each window size,
There are several varieties of artificial neural networks, are shown in Table 1.
each having different properties. For this work the In this work, both single-leakage and multiple-leakage
multilayer perceptron (MLP) network is used. In all faults are considered and the leakage flowrates for each
experiments, the MLPs are trained using standard error leakage type are shown in Table 2. These values should
back-propagation with conjugate gradient and batch be compared with maximum transient flowrates of typi-
update algorithms (7, 12, 13). The cross-validation cally 20 L/min generated within the system during step
method is used to terminate the learning process to limit response testing. Together with the non-leakage possibil-
the possibility of overtraining (8). ity, there are eight different cases or classes in total. The
MLPs are trained to identify these classes, the input data
being either LPC coefficients or LPC cepstra coefficients.
4 LEAKAGE-TYPE IDENTIFICATION The network topologies for both serial and parallel
EXPERIMENTAL RESULTS MLPs are also shown in Table 1, the number of neurons
in the hidden layer being set to 50 from experience and
4.1 Window size following just a few trials to obtain satisfactory network
convergence.
The first set of experiments was designed to study the For each leakage type at each leakage flowrate, the
effect of the previously discussed window size on the pressure responses from 10 to 50 cycles were arbitrarily

Table 1 The total number of coefficients generated in one cycle for various window sizes and the corresponding network topology
for both serial and parallel MLP models

Window size 512 256 128 64

Number of coefficients per frame 10 10 10 10

Total coefficients extracted per cycle 40 120 280 600
Data dimensionality reduction ratio 515 1 1751 751 3.451
Serial MLP topology 10–50–8 10–50–8 10–50–8 10–50–8
Parallel MLP topology 40–50–8 120–50–8 280–50–8 600–50–8

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 FAULT CLASSIFICATION OF FLUID POWER SYSTEMS 93

Table 2 The leakage flowrate for each leakage type used in serial model is 93 per cent at the window length of 512.
both training and testing However, for the parallel model the identification per-
Leakage (L/min)
formance is insignificantly affected as the window length
is changed, with the best performance of 98 per cent
Type Line 1 Line 2 Line i being at the window length of 256. These results reflect
No leak
the different ways of presenting information between the
two MLP models. This becomes particularly evident for
Line 1 1
3 the serial model when the window length is reduced.
6 There is simply less information available; hence the
Line 2 1 accuracy of coefficient extraction is reduced and the net-
3 work performance deteriorates.
6
For the parallel model, all discriminating information
Line i 1 is presented to the MLP despite the change in window
3
6 length. In other words, the results have shown the impor-
Lines 1 and 2 1 1 tance of presenting the extracted information from the
3 3 whole pressure profile to the MLP to discriminate
6 6 between different types of leakage, especially when
Lines 1 and i 1 1 involving multiple leaks. From Fig. 8, it can also be
3 3
6 6
observed that LPC cepstra coefficients consistently give
better performance than LPC coefficients, as has been
Lines 2 and i 1 1
3 3 reported with other applications (14). This confirms the
6 6 important feature of using LPC cepstra coefficients
Lines 1, 2 and i 1 1 1 rather than LPC coefficients which reflects more of the
3 3 3 system characteristics due to separation of excitation and
6 6 6
system cepstra coefficients, as explained in Section 2.2.
In addition, the results have also shown that with a fea-
selected to produce the training and testing data sets ture extraction technique, relevant information can be
respectively, resulting in a total of 220 cycles used in extracted from the raw dynamic pressure responses and
training and 1100 cycles used in testing. The testing con- used to detect leakages. With a frame size of 512, a ratio
dition, namely the working temperature and the leakage of 5251 in data dimensionality could be achieved and
flowrates, were kept as close as possible to those experi- still give a 95 per cent correct leakage identification
enced in training. performance.
The percentage of correctly identified faults for the An identification confusion matrix for the parallel
two MLP models, serial and parallel, using two feature MLP approach using LPC cepstra coefficients for a
extraction methods, LPC and LPC cepstra, with differ- window length of 256, taken from Fig. 8, is shown in
ent window sizes is shown in Fig. 8. It is observed that Table 3. The diagonal of the matrix shows the number
the performance for the serial model worsens as the of correct identifications of leakage type and the non-
window length is decreased. The best performance of the diagonal elements show where the errors have occurred.

Fig. 8 Leakage-type identification of serial and parallel MLP models using LPC coefficients and LPC cepstra
coefficients. Ten coefficients are extracted from each data frame for all window sizes

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94 T T LE, J WATTON AND D T PHAM

Table 3 Identification confusion matrix for the parallel MLP approach using LPC cepstra coefficients with a window length of 256

Identified leakage type

Total
True leakage Lines 1 Lines 1 Lines 2 Lines 1, cycles
type No leak Line 1 Line 2 Line i and 2 and i and i 2 and i tested

No leak 50 0 0 0 0 0 0 0 50
Line 1 0 150 0 0 0 0 0 0 150
Line 2 0 1 144 0 4 0 1 0 150
Line i 0 0 0 148 0 2 0 0 150
Lines 1 and 2 0 0 5 0 139 0 0 6 150
Lines 1 and i 0 0 0 0 0 147 0 3 150
Lines 2 and i 0 0 0 0 0 0 150 0 150
Lines 1, 2 and i 0 0 0 0 0 1 0 149 150

It can be seen that most errors are associated with coefficients, further evidence of LPC cepstra features.
multiple-fault conditions. For example, in lines 1 and 2 LPC cepstra gave the best performance using 10
leakage testing, there are five misclassifications as leak- coefficients, of 98 per cent, while LPC coefficients gave
age line 2 and six times as leakage lines 1, 2 and i. the best performance using 12 coefficients, of 96 per cent.
However, the results suggest that these errors can be
considered as tolerable.
5 LEAKAGE FLOWRATE MAGNITUDE
PREDICTION
4.2 The number of coefficients
Previous experiments have shown the performance of The experimental procedures are discussed in
MLP models on leakage-type identification for various Section 4.1. Only the parallel MLP model using the LPC
window sizes with a fixed number of coefficients cepstra feature extraction method is considered. The task
extracted from one data frame. The experiments in this of predicting the leakage flowrate magnitude is based on
section are designed to study the identification perform- an assumption that the leakage type has been success-
ance using a different number of coefficients extracted fully identified and then only the corresponding neural
from a fixed window length of 256. The total coefficients network is applied.
generated in one cycle and the corresponding MLP In total, seven leakage flowrate neural networks are
topologies are shown in Table 4. Only the parallel required to represent the seven different leakage types.
MLP model is considered. The neural network topology in every case follows pre-
The experimental procedure is the same as discussed vious experience and the number of input nodes must
previously. The results for leakage-type identification match the number of coefficients, in this case 10. The
with different numbers of coefficients extracted from one number of neurons in the hidden layer was again set to
frame are shown in Fig. 9. It is observed that the per- 50 and the single output was trained to predict the leak-
formance improves as the number of coefficients is age level. The performances of these networks in terms
increased, but then worsens as the number of coefficients of the percentage errors in flowrate prediction using 10
continues to increase. This trend is as expected from coefficients from 256 data points are shown in Fig. 10.
basic statistical considerations regarding the difference In addition, the average performances of all networks
between the number of data points and the number of for different window sizes is shown in Fig. 11.
estimated LP coefficients: the model eventually becomes From Fig. 10, it is observed that the larger the leakage
inaccurate. In this study it would appear that 10 the better the performance. The best performance (1.8
coefficients produce the optimum model for the sampling per cent error) occurs at 6.0 L/min leakage and the worst
interval used. Note, however, that correct predictions performance (32.2 per cent error) occurs at 1.0 L/min
are still better than 96 per cent using 14 coefficients and leakage. Considering the worst performance case for all
both feature extraction techniques. leakage flowrates, the error is 1.3 per cent for a leakage
The results have also shown that LPC cepstra of 1.0 L/min, 3.3 per cent for a leakage of 3.0 L/min and
coefficients consistently perform better than LPC 6.3 per cent for a leakage of 6.0 L/min. In practical

Table 4 The total number of coefficients generated in one cycle for a frame length of 256 and the corresponding network topology
for parallel MLP models

Number of coefficients per frame 6 8 10 12 14

Total coefficients per cycle 72 96 120 144 168
Parallel MLP topology 72–50–8 96–50–8 120–50–8 144–50–8 168–50–8

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 FAULT CLASSIFICATION OF FLUID POWER SYSTEMS 95

Fig. 9 Leakage-type identification using the parallel MLP model and the effect of the number of coefficients
extracted from one data frame. The frame length is 256

Fig. 10 Leakage flowrate prediction performance for different leakage types. Ten LPC cepstra coefficients
from frames of 256 data points are used

situations these results are considered as acceptable. In

addition, it was found that the window size of 256 gives
the best average performance with an error of 11.3 per
cent, as shown in Fig. 11.
Throughout this study, the leakage levels were set at
specific values of 1.0, 3.0 and 6.0 L/min. This is not too
critical in the practical situation since it has been shown
that the correct type of fault can be predicted, provided
it is within the training range. If a more accurate predic-
tion of the leakage magnitude is required, it is a simple
matter to set thresholds midway between those used in
training. This has already been addressed in parallel
Fig. 11 Average leakage flowrate prediction performance work, albeit using quasi steady state data from the same
for different window sizes using 10 LPC cepstra test rig, which showed that such a generalized testing
coefficients extracted from each frame approach also gave acceptable results (7).
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96 T T LE, J WATTON AND D T PHAM

6 CONCLUSIONS Proceedings of 10th International Fluid Power Symposium,

Brugge, 1993, pp. 277–280.
5 Watton, J., Lucca-Negro, O. and Stewart, J. C. An on-line
This study has demonstrated that with only pressure
approach to fault diagnosis of fluid power cylinder drive
transient information, neural networks combined with system. Proc. Instn Mech. Engrs, Part I, Journal of Systems
feature extraction techniques can be used successfully to and Control Engineering, 1994, 208(I4), 249–262.
detect leakage types and to predict the leakage flowrate 6 Stewart, J. C. The application of artificial intelligence to
magnitudes for both multiple-leak and single-leak types fault detection in hydraulic cylinder drive systems. PhD
in an electrohydraulic position control system. thesis, University of Wales Cardiff, 1995.
With the appropriate technique, relevant and discrimi- 7 Le, T. T., Watton, J. and Pham, D. T. An artificial neural
nating information can be extracted and used to detect network-based approach to fault diagnosis and classifi-
leakage, resulting in a large data dimensionality cation of fluid power systems. Proc. Instn Mech. Engrs,
reduction. The results have shown that the LPC cep- Part I, Journal of Systems and Control Engineering, 1997,
strum technique consistently performs better than LPC, 211(I4), 307–317.
8 Bishop, C. M. Neural Networks for Pattern Recognition,
as has been reported for other applications. The parallel
1995 (Oxford University Press).
MLPs were found to give superior performance over the 9 Wiener, N. Extrapolation, Interpolation and Smoothing of
serial ones and were less affected by the change in Stationary Time Series, 1966 (MIT Press, Cambridge,
window length. This has also reflected the importance Massachusetts).
of presenting all extracted information to the MLP in 10 Atal, B. and Schroeder, M. Predictive coding of speech
order to achieve the best performance. signals. In Proceedings of 6th International Congress on
Prediction of fault types was extremely good and typi- Acoustics, 1968, paper C-5-4.
cally better than a 95 per cent success rate using a parallel 11 Furui, S. Digital Speech Processing, Synthesis, and
data approach with LPC cepstra. Predictions for the Recognition, 1989 (Marcel Dekker, New York).
leakage flowrate magnitude required additional neural 12 Le, T. T., Mason, J. S. and Kitamura, T. Characteristics of
networks and showed promise as leakage rates increased multi-layer perceptron models in enhancing degraded
speech. IEICE Trans. on Information and Systems, June
beyond 1.0 L/min, i.e. 5 per cent of typical system
1995, E78-D(6), 744–750.
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ACKNOWLEDGEMENT APPENDIX 1

The authors wish to acknowledge the EPSRC for sup- LPC coefficients determination
porting this study. The first-order linear equation between the present
sample x and the preceding p samples is as follows:
t
x +a x + · · · +a x =e (10)
REFERENCES t 1 t−1 p t−p t
where e is called the prediction error. The total energy
t
1 Atkinson, R. M., Montakhab, M. R., Pilley, F. D. A., in the prediction error, from equation (10), is given by
Woollons, D. J., Hogan, P. A., Burrows, C. R. and Edge,

A B
2 2 p 2
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Part 1: fundamentals. Proc. Instn Mech. Engrs, Part I, t t i t−i
t=−2 t=−2 i=1
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2 Hogan, P. A., Burrows, C. R., Edge, K. A., Atkinson, R. M., derivatives to zero and produces p linear equations:
Montakhab, R. M. and Woollons, D. J. Automated fault p 2 2
diagnosis for hydraulic systems. Part 2: applications. Proc. ∑ a ∑ x x =− ∑ x x for 1∏ j∏p
i t−i t−j t t−j
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3 Creber, D. J. and Watton, J. Development of a low cost Assuming x =0 for t<0 and tN as a consequence of
expert system shell for condition monitoring of fluid power t
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systems. In Proceedings of 9th International Fluid Power the signal x ,
Symposium, Cambridge, 1990, pp. 281–291. t
4 Stewart, J. C. and Watton, J. Fault diagnosis of a cylinder N−1
R( j )= ∑ x x (13)
drive using on-line expert systems techniques. In t t−j
t=j
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 FAULT CLASSIFICATION OF FLUID POWER SYSTEMS 97

then equation (12) becomes Taking derivatives on both sides of equation (17) with
p respect to z−1 gives
∑ a R( j−i )=R( j ) for 1∏ j∏p (14)
i ∑ p ia z−i+1
i=1 2
∑ nc z−n+1=− i=1 i (18)
This system of linear equations can be solved using n 1+ ∑ p a z−i
Durbin’s recursion procedure to yield the predictor n=1 i=1 i
coefficients {a }. or
i

A B
p 2 p
APPENDIX 2 1+ ∑ a z−i ∑ nc z−n+1=− ∑ ia z−i+1 (19)
i n i
i=1 n=1 i=1
Deriving the cepstrum from the LPC model The relationship between c and a can be derived by
n n
equating the terms of various powers of z−1 on both
Let X(z) and C(z) be the z-transform of x(t) and c(t) sides of equation (19) as follows:
respectively. Therefore
C(z)=log |X(z)| c =−a (20)
1 1

A B
C(z) may be represented as follows: n−1 m
c =−a − ∑ 1− a c
2 n n n m n−m
C(z)= ∑ c z−n (15) m=1
n
n=1 for 1<n∏p (21)
Then, from equation (4),

A B
p m
1 c =− ∑ 1− a c for p<n (22)
X(z)= (16) n n m n−m
1+ ∑ p a z−i m=1
i=1 i
or The cepstrum derived through this method is referred to
as the LPC cepstrum in order to distinguish it from the

A B
2 1 cepstrum obtained via the direct fast Fourier transform
∑ c z−n=log (17)
n 1+ ∑ p a z−i (FFT ) route.
n=1 i=1 i

I05697 © IMechE 1998 Proc Instn Mech Engrs Vol 212 Part I

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