Jtam.dvi


JOURNAL OF THEORETICAL

AND APPLIED MECHANICS

46, 4, pp. 777-797, Warsaw 2008

FORECASTING THE GLOBAL AND PARTIAL SYSTEM
CONDITION BY MEANS OF MULTIDIMENSIONAL

CONDITION MONITORING METHODS

Czesław Cempel

Poznan University of Technology, Wydział lu Instutut po angielsku, Poznań, Poland

e-mail: czeslaw.cempel@put.poznan.pl

Machines have many faults which evolve during their operation. If one
observes some number of symptoms during the machine operation, it is
possible to capture fault oriented information. One of the methods to
extract fault information from such a symptom observationmatrix is to
apply the Singular Value Decomposition (SVD), obtaining in this way
the generalized fault symptoms. The problem of this paper is to find if
the total damage symptom, being a sum of all generalized symptoms
is the best way to infer on machine condition or is it better to use the
first generalized symptom for the same purposes. There were some new
software created for this purpose, and two cases of machine condition
monitoring considered, but so far it is impossible to state that one of the
inference methods is better. Moreover, it seems to the author that both
inferencemethods are complimentary for each other, and should be used
together to increase the reliability of diagnostic decision.

Key words: condition monitoring, multidimensional observation, singu-
lar value decomposition, generalized fault symptoms, greymodels, fore-
casting

1. Introduction

The contemporary advancement in measurement technology allows us to me-
asure almost any component of the phenomenal field inside or outside the
working machine. The only condition for such diagnostic is some kind of
proportionality to gradual worsening of the machine condition which takes
place during it operation. If it is so, we can name the measured component
of the machine phenomenal field as the symptom of condition. In this way,



778 C. Cempel

we measure a dozen of would be symptoms, and our condition monitoring is
multidimensional from the beginning. Due to this situation, the application
of multidimensional machine condition observation is now a well established
fact, see Cempel (1999), Korbicz et al. (2004), Tumer and Huff (2002), Ja-
siński (2004) – for example. And there exist some differences in application
and processing of the multidimensional signals and/ or symptom observation
matrix. For signals and symptoms one can also apply the so called data fusion
(Hall and Llinas, 1997; Roemer et al., 2001; Korbicz et al., 2004), and similar
techniques developed lately. In the case of multi symptom observation, one
can applyPrincipal ComponentAnalysis (PCA), or SingularValueDecompo-
sition (SVD), looking for principal or singular components, which may have
some diagnostic meaning. For the case of Singular Value Distribution (SVD)
method, there exists the body of experimental evidence (Cempel, 2004; Cem-
pel and Tabaszewski, 2007a,b)that singular components and the quantities
created from them can be treated as generalized fault symptoms.

All that transformation and symptomprocessing starts from the data base
called the SymptomObservation Matrix (SOM). Let us explain now how the
SOM is structured and obtained.

During themachine life θwe can observe its condition bymeans of several
symptoms Sm(θ) measured at some moments of life θn, n = 0,1, . . . ,p > r,
θp <θb, (θb –anticipated breakdown time).This creates sequentially theSymp-
tomObservationMatrix (SOM), the only source of information on the condi-
tion evolution of amachine in its lifetime 0<θ<θb.We assume additionally
that the condition degradation is also multidimensional and is described by
semi-independent faults Ft(θ), t = 1, . . . ,u < r, which are evolving in the
machine body, as the expression of gradual degradation of the overall ma-
chine condition. This degradation proceeds from the not faulty condition1up
to its near breakdown state. Generalizing, one can say now that we have m-
dimensional symptomspace for condition observation, and r-dimensional fault
space (m>r), which we are trying to extract from the observation space by
using SVD or PCA.

Moreover, some of would be symptoms are redundant; it means not carry-
ing enough information on the evolving faults during themachine life. But of
course there is not a unique criterion of the redundancy. During the course of
our research, severalmeasures of redundancyhave been applied, the volume of
observation space (Vol1), pseudo Frobenius norm (Frob1) of SOM (Cempel
andTabaszewski, 2007a,b), and others. But they seem to be not good enough
with respect of the quality of the final diagnostic decision. This means addi-

1We assumemachine is new, or after the overhaul and repair process.



Forecasting the global and partial system... 779

tionally, when optimizing the observation space, we should take into account
the adequate assessment of the current and the future machine condition, in
the formof condition forecastwith a possibly small error. The paper considers
this problem, and it is done on the level of previous SVDworks of the author.
As the forecasting technique with minimal error, the grey system model with
rolling window (Yao andChi, 2004) was adopted for diagnostic purposes, and
has been applied here (Cempel andTabaszewski, 2007a). But having themul-
tidimensional problemof fault assessment and the observation, it is important
now what type of generalized symptomwe use for the forecasting. Do we use
the overall degradation symptomof themachine or some specified generalized
symptom proportional to one fault only.
Theresults of suchanewapproach tomultidimensional diagnosispresented

here were verified on the real data of machine vibration conditionmonitoring.
Concerning the software, some modification of the last program for the data
processing was needed as well. As a result it was found that this approach se-
ems to bepromising and enabling abetter understandingofmachine condition
and also better current and future condition assessment.

2. Extraction method of partial faults of the system

As it was said in the introduction, our information onmachine condition evo-
lution is contained in p · r Symptom Observation Matrix (SOM), where in
r columns are presented p rows of the successive readings of each symptom
made at equidistant system lifetimemoments θn, t=1,2, . . . ,p. The columns
of such SOM are next centered and normalized to three point average of the
three initial readings of every symptom. This is in order to make the SOM
dimensionless, to diminish starting disturbances of symptoms, and to present
the evolution range of every symptom from zero up to few times of the initial
symptom value S0n, measured in the vicinity of lifetime θ1 =0.
After such preprocessing, we will obtain the dimensionless Symptom Ob-

servation Matrix (SOM) in the form

SOM≡Opr = [Snm] Snm =
S̃nm

S0m
−1 (2.1)

where S̃nm letters indicate primarymeasuredandaveraged dimensional symp-
toms.
As was said in the introduction, we apply now to the dimensionless SOM

(2.1) the Singular Value Decomposition (SVD) (Golub, 1983; Will, 2005), to



780 C. Cempel

obtain singular components (vectors) and singular values (numbers) of SOM
in the form

Opr =UppΣprV
⊤

rr (⊤− matrix transposition) (2.2)

where Upp is a p-dimensional orthonormalmatrixof the left-hand side singular
vectors, Vrr is a r-dimensional orthonormal matrix of the right-hand side
singular vectors, and the diagonal matrix of singular values Σpr is defined as
below

Σpr = diag(σ1, . . . ,σl) (2.3)

with nonzero singular vectors

σ1 >σ2> ... >σu > 0

and zero singular values

σu+1 = . . .=σl =0
l=max(p,r)
u¬min(p,r)
u<r<p

In terms of machine condition monitoring, above (2.3) means that from
the r primarily measured symptoms (dimension of observation space) we can
extract only u ¬ r nonzero independent sources of diagnostic information,
describing the evolving generalized faults Ft(θ), t=1, . . . ,u, and creating in
this way the less dimensional fault space. But only a few faults developing
in amachine aremaking essential contribution to total fault information (are
enoughdeveloped).The rest of potential generalized faults, symbolizedhereby
small σu, are usuallybelow the standard10% level of noise.What is important
here, that such SVD decomposition can be made currently, after each new
observation (reading) of the symptom vector Sm; n = 1, . . . ,p, and in this
waywe can trace the faults evolution, and their advancement in any operating
mechanical system.

3. Diagnostic interpretation of SVD

From the current research and implementation of this idea (Cempel, 2003),
one can say that the most important fault oriented indices obtained from
SVD is the generalized fault symptom SDt t= 1,2, . . ., and also the sum of
all generalized fault symptoms SumSDi, as some equivalent symptomof total



Forecasting the global and partial system... 781

(cumulated) machine damage. In another way, the generalized fault symptom
SDt can be named also as discriminant, or the generalized symptom of the
fault order t, and one can obtain this as the SOM product and singular vec-
tor vt, or in general in matrix notation as below

SD=OprV=UΣ (3.1)

and in particular

SDt =Oprvt =σtut t=1, . . . ,u< r

Weknow fromSVD theory (Golub, 1983;Will, 2005) that all singular vectors
vt and ut, as the components of singular matrices, are normalized to one, so
the energy norm of this new discriminant (generalized fault symptom) gives
simply the respective singular value σt

Norm(SDt)≡‖SDt‖=σt t=1, . . . ,u (3.2)

Theabovedefineddiscriminant SDt(θ) can alsobenamedas the lifetime fault
profile, and the respective singular value σt(θ) as the function of the lifetime
seems to be its life advancement (energy norm) and the same the measure
of importance of the fault. That is the main reason why we use dimensional
or dimensionless singular values for the ordering of importance of generalized
symptoms (faults).
The similar fault inference can be postulated to the meaning, and the

evolution of summation quantities, the total damage profile SumSDi(θ) as
below

SDt(θ)∝F t(θ) t=1,2, . . .
(3.3)

SumSDi(θ)=
z∑

i=1

SDi(θ)=
z∑

i=1

σi(θ)ui(θ)∝F(θ)

Currently it seems that the condition inference based on the first summation
damage measure; SumSDi, (total damage measure) may stand as the first
approach to multidimensional condition inference, as it was lately shown in
the previous papers (see for example Cempel, 2004, 2005, 2006).
Goingback toSVD itself, it isworthwhile to showthat every perpendicular

matrix has such decomposition, and itmay be interpreted also as the product
of three matrices (Will, 2005), namely

Opr = (Hanger)×(Stretcher)×(Aligner)
⊤

(3.4)



782 C. Cempel

This is very metaphorical description of SVD transformation, but it seems
to be a useful analogy for the inference and decision making in our case.
The diagnostic interpretation of formulae (3.4) one can obtain very easily.
Namely, using its left hand side part, we are stretching our SOMover the life
(observations) dimension, obtaining thematrix of generalized symptomsas the
columns of the matrix SD (see below). And using its right hand side part of
(3.4), we are stretchingSOMover the observed symptomsdimensionobtaining
the assessment of contribution of every primary measured symptoms in the
matrix AL, assessing in this way the contribution of each primary symptom
to the generalized fault symptom SDi

SD=OprVrr =UppΣrr and AL=U
⊤

ppOpr =ΣrrV
⊤

rr (3.5)

Thismeans that SDmatrix is stretched along the life coordinate giving us the
life evolution of the weighted (σi) singular vectors. And ALmatrix is aligned
along the symptom dimension with the same way of weighting by σi, giving
the assessment of information contribution of each primary symptom.

We will calculate numerically the above matrices and use them for better
interpretation of monitoring results (SD), and optimization of dimension of
the observation space (AL).

4. SOM information measure and optimization

Having in mind the redundancy of some primary symptoms, i.e. the primary
observation space, some additional considerations should bemade concerning
the SOM information assessment. In terms of previous findings this can be
done by calculating the Frobenius norm (Frob) of thismatrix, and the volume
(Vol) created by u-dimensional generalized fault space identified by applica-
tion of SVD.One can calculate easily both information indices as the sumand
the product of singular values in the followingway (Golub, 1983; Kiełbasiński,
1992)

Frob(SOM)≡

√√√√
u∑

i=1

σ2i and Vol(SOM)≡
u∏

i=1

σi

But squaring the small singular values of σi (less than one) make themmuch
smaller, giving seemingly smaller contributions to the matrix information as-
set, and to the volume of the observation space. Due to this it was proposed



Forecasting the global and partial system... 783

by Cempel and Tabaszewski (2007a,b) to use not the exact Frobenius norm
but its modification as below

Frob1=
u∑

i=1

σi and Vol1=
u∏

i=1

σi (4.1)

This will give us possibility to look for small, just evolving faults, not omit-
ting them when we try to reduce the redundancy of the observation vector.
Consequently, one can get less redundancy of a new optimized SOM with a
less number of columns, but also keeping in observation the small just evolved
fault information (σi).

The use of Frobenius measure for a matrix has also mathematical valida-
tion. In general, one can understand this as the problem of approximation of
matrix B, by the so called k-rank approximation. Following the paper Ber-
ry et al. (1999), we can make the quantitative assessment of such a k-rank
approximation of the matrix B as the difference below

‖B−Bk‖F =
√
σ2
k+1+ . . .+σ

2
u (4.2)

where the subscript u stands for the maximal dimension of nonzero singular
value, i.e. the rank of our primary SOM.
This also means that instead of (3.5) we write a simplified measure of

approximation of SOM in the form of deviation from primary SOM rank, as
below

∆kFrob1≡ Frob1o− Frob1k = {σk+1+ . . .+σu} (4.3)

Using this quality index ofmatrix approximationmeasure, we can forman ad-
ditional objective measure of the SOM redundancy. And minimization of the
SOM rank may be carried out by excluding some primary measured symp-
toms Sm with low information contribution, which produces mainly small
(less than one) singular vales σu.
Such criteria of redundancyminimization we have used quite recently. But

following the last papers Cempel and Tabaszewski (2007a,b), one may notice
that after some symptom rejection, it gives an expected increase in the volume
of information space (Vol1). Also the rank approximation of SOMgives only
a slight drop in Frob1measure, but the result of prognosis is not good enough,
giving erroneous future values, even less than the previous one. How to avoid
such errors in forecasting? There seem to be one possibility more to make
the symptom rejection more objective and anticipating the goodness of the
condition forecast.We have to consider the contribution of primarymeasured
symptoms to the creation of first generalized symptoms SD1, and also the



784 C. Cempel

creation of the total damage generalized symptom SumSDi. The first overall
contribution measure can be calculated separately to each primary symptom
from the correlation matrix of our SOM (with appended lifetime in the first
column) as the centered and normalized sum of column elements. The second
measure can be obtained if we append additionally to the previousmatrix the
vector SumSDi as the first column.When calculating the covariance matrix
from these and in the first row,wewill have needed information. After needed
normalization to the first element of this row, thiswill give us the contribution
of every primary symptom to the total damage symptom SumSDi.

5. The global and partial fault inference

Wehave gathered above all necessary analytical and inference knowledge con-
cerning the processing of the symptom observation matrix, the extraction of
fault information and optimization of the SOM rank. So, there is a right mo-
ment to validate these findings and proposal by some experimental data taken
from real situations of vibration condition monitoring. In order to do this,
the last Matlab R© program svdopt1gs.m presented in Cempel and Tabaszew-
ski (2007a) has beenmodified to svdopt2gs.m. The inference basis for the first
program is the total damage generalized symptom SumSDi, while in themo-
dified program such on inference basis is the first generalized symptom SD1.
Just to catch the inference and the followed diagnostic decision difference, we
will take someuneasy case of a heavy fan (power of 3MW)working in unstable
and load uncontrolled regime (random supply of the air to the mine shaft),
serving as the source of fresh air for ventilation at a deep copper mine. The
main troubleswith this fanwere the unbalance andnonalignment between the
fan and the driving electric motor. Due to that, it was constantly monitored.

Figure 1 presents below six pictures as a result of fan data processing by
specially prepared program svdopt1gs.m2 made in theMatlab R© environment,
and here is adopted the inference according to the total damage symptom
SumSDi. The first top left picture shows results of 30-week measurements
of symptom life curves of vibration velocity at five points located on the fan
aggregate structure. Onemay notice here the great instability of symptom re-
adings, symptomNo. 4 in particular. This is better seen at the picturemiddle
left when the data are centered and normalized to the average value of the

2Author is indebted to Dr M. Tabaszewski for the help in making the program
and for the vibration conditionmonitoring data.



Forecasting the global and partial system... 785

three initial symptom readings. The picture bottom left presents the gene-
ralized symptoms as the result of SVD processing, indicating also the symp-
tom limit value calculated for the generalized symptom of the total damage
SumSDi.Wemaynotice here that the initial instability of primary symptoms
is even enlarged, giving a situation where the main generalized symptoms are
falling down at the end of the fan life, giving erroneous assessment of the fan
condition.

Fig. 1. Results of SVD processing of vibration data of a huge fan pumping air into
the copper mine shaft

From the diagnostic decision point of view, it is a very critical situation to
observe such instability of primaryand generalized symptoms.Thenextpictu-
re top right at this figure (dimensionless singular values) give us the indication
how much independent sources of diagnostic information can be observed by
these five primary symptomswith appended lifetime θ. One can see here that
in reality we can count on two independent sources of information, it may



786 C. Cempel

mean two independent faults, like shaft unbalance andmachine aggregate mi-
salignment. Thepicturemiddle right gives us the contribution of each primary
symptom (the first = lifetime) to creation of the first three generalized fault
symptoms taken directly from SVD as the product of SOM and Umatrices.
We may notice that primary symptom No. 5 gives small contribution to all
three generalized symptoms, and may serve as one of the candidates to re-
jection in the course of further processing. The last picture, the bottom right
one, gives us the result of current calculation of the symptom limit value Sl
by our concept of symptom reliability (Cempel et al., 2000). This program
calculates Sl with data of the total damage generalized symptom SumSDi
(the picture bottom left). One may notice that the limit value Sl is steadily
evolving up to the value Sl =1.0306. But the application of this limit value to
the total damage symptom SumSDi is impossible as this symptom is falling
down. Having this in mind, we should decide now which symptom we should
reject here in order to obtain stable course of the generalized fault symptomof
the total damage SumSDi in order tomake a reliable forecast and diagnostic
inference.

Let us now change the inference basis and calculate the symptom limit
value Sl not from the total damage generalized symptom SumSDi but from
the first generalized symptom SD1 only. So, let the same SOMof the fanwill
be treated bymodified program svdopt2gs.m, as below.

With this modified program and our primary data, only the last row of
pictures in Fig.2 has been changed, namely the course of the symptom limit
value calculation and its value (as it is seen at the bottom right picture).
But due to this change, the assessed value of Sl is lower and can serve us
for further elaboration of diagnostic decision, as one can see from the bottom
left picture. Let us now check if we can correct the situation by reducing the
SOM redundancy. As one can see from Fig.2, there is a big redundancy of
the observation space (SOM), which is reflected here by a very small volume
of the fault space Vol1=0.00383. This means that some primary symptoms
carry minimal fault information, which is reflected by a very small σI, much
smaller than one.

Trying to diminish this redundancy, let us now use the alreadymentioned
correlation calculation of information contribution, i.e. contribution of each
primary symptom to the overall information resource of our data and to the
generalized symptom of total damage SumSDi. The calculation of these was
described in the previous Section. Figure 3 gives the result of such a contribu-
tion assessment, andone cannotice that theprimary symptomnumber 4 is the
first candidate (not symptom No. 5) as it influences negatively (decreasing)



Forecasting the global and partial system... 787

Fig. 2. The vibration symptom observationmatrix of a huge fan (see Fig.1)
processed by the modified program, where Sl value is calculated on the basis of the

first generalized symptom SD1

Fig. 3. The assessment of overall contributions of primary symptoms (top picture)
and the contribution to the total damage generalized symptom



788 C. Cempel

the total damage symptom (bottom picture), and also makes the smallest
contribution to the overall information resource (top picture of Fig.3).

Having such strong indication which a symptom to reject as the first
(No. 4), we have performed this by the same program, and the results can
be seen in Fig.4 presented in the same manner as it was explained in detail
for Fig.1 and Fig.2. Now one can see from Fig.4 that the rejected symptom
No. 4 was the symptomwith the greatest amplitude and instability. One can
also notice that the course of all generalized symptoms (bottom left picture)
and the limit value (bottom right picture) evolves smoothly, giving a strong
basis for the future diagnostic decision. It is also good to notice here a drop of
the Frobeniusmeasure after symptom rejection; ∆Frob1=2.4073, andmore
than a double increase of the volume of the observation space Vol1=0.0082.

Fig. 4. Condition inference of a huge fan by the same program svdopt1gs, but with
rejection of primary symptomNo. 4, and total damage symptom SumSDi

It will be very interesting what result can be achieved in calculation of the
symptom limit value rejecting the same symptom No. 4, but using modified
the program svdopt2gs.m,which calculates the symptom limit value Sl on the



Forecasting the global and partial system... 789

basis of the first generalized symptom SD1. Figure 5 shows the result of such
calculations, presenting the detailed results in the samemanner as it was done
before (Fig.1 and Fig.2).

Fig. 5. The huge fan monitoring data with a change of inference basis to the first
generalized symptom SD1

Analyzing now the last two figures (Fig.1 andFig.2), one can find that the
only change is noted at the bottompictureswhere the symptom limit value Sl
has been calculated and presented against the generalized symptoms (bottom
left pictures). One can notice here that the calculation of the limit value using
the first generalized symptom SD1 gives us a lower value, and this can give
us amore safe assessment of the lifetimemoment formachine shut down, and
renewal. From the point of view of reliability of diagnostic decision, this seems
to be important to have two sources of the symptom limit vale assessment,
and to confront these values with the associated knowledge.



790 C. Cempel

6. Forecasting of global system damage and partial faults
advancement

The final quality of diagnostic decision onemay judge bymaking the forecast
of the future condition in terms of the total damage symptom SumSDi and
the first generalized fault symptom SD1. It was said in the introduction that
the forecast will be made by the Grey System Theory (GST) (Deng, 1989),
together with the rolling window method using the first order grey model
GM(1,1) (Yao and Chi, 2004).

In general, GST assumes that our incomplete and uncertain observation
canbe the outputof somedynamicmulti-input systemof highorder, described
by a grey differential or difference model (Wen and Chang, 2005). In the
condition monitoring, wemay assume that it is enough to take the first order
system described by the grey differential equation, and one forcing or control
input. This simplest case in GST is denoted as GM(1,1), which means the
grey model of order 1 with one input only. The output of the system is the
series of discrete observations (our symptom readings) denoted here as

x
(0) = {x(0)(1),x(0)(2), . . . ,x(0)(n)} (6.1)

where n­ 4 is the number of observation made on the system (machine).

We will not present GST theory here, just using the final formulae for the
forecasting, and the rollingwindowconcept is implemented into the forecasting
software only.

The application of GST to the above symptom readings gives the possibi-
lity to forecast the future symptom value, starting from a very small number
observation, and using the formula

x̂
(0)(k+1)=

[
x
(0)(1)−

u

a

]
(e−ak − e−a(k−1)) k=2,3, . . . ,n (6.2)

where u and a are parameters to be estimated by a special least squarematrix
procedure using the observed data (6.1), and the hat (̂·) in (6.2) means the
future value of the forecasted quantity.

This concept was adjusted to the purposes of vibration condition moni-
toring in one of the earlier papers Cempel and Tabaszewski (2007a,b). One
can notice now from the top left picture of Fig.6 that the total damage ge-
neralized symptom SumSDi after rejection of the primary symptom No. 4
is well evolved, enabling a good forecast even without the rolling widow. But
of course, as usually in the case of grey systemmodelling, the rolling window



Forecasting the global and partial system... 791

forecast gives the smallest error. This error can be even smaller if we diminish
the span of window (w) as it is clearly seen from the picture bottom right in
Fig.6.

Fig. 6. Grey rolling forecast of the total damage generalized symptom SumSDi for
the huge fan sier1

It is also worthwhile to focus on other pictures of this figure. The picture
top left clearly presents that the rejection of No. 4 symptom was a good
idea allowing us to determine the symptom limit value Sl and having this
information act properly to shut down the fan ahead of the breakdown time.
The picture top right presents the total forecast of the total damage symptom
SumSDiwith themodel GM(1,1). It seems tobeagood forecastwitha small
average error, but the picture bottom left with the rolling window forecast
have a smaller error and the actual forecast adapts smoothly to the course of
SumSDi.

Knowing this, let us take into consideration the first generalized symp-
tom SD1 and the way and property of inference obtained by this new appro-
ach. Next figure (Fig.7) presents the results obtained according to this new
way of thinking by themodified program svdopt2gs.m, where the basic quan-
tity for the Sl and forecast calculation is the first generalized symptom SD1.



792 C. Cempel

Fig. 7. Grey rolling forecast of the fan condition using the first generalized
symptom SD1

It is seen in the left top picture of Fig.7 that the course of SD1 symptom
is decreasing at the end of the fan life, but the assessed symptom limit value Sl
warns enough in advance to undertake shut down decision on time. However,
comparatively to Fig.6, this is some drawback in the clarity of diagnostic
decision. On the contrary, the forecasting error is smaller here, having some
minimum around the chosen window span w=5.

As one can see from the above, it is hard to decide in advance if we should
use for inference the total damage symptom SumSD1, or the first generali-
zed symptom SD1. We need more experimental validation of this important
issue. Let us take now quite another object, a small ball bearing of the type
6305, which was tested at the durability test-stand, where 7 symptoms were
measured, each hour of the test. This particular bearing (krak3) broke down
after 40 hours, and the results of observation were processed by these two
mentioned programs. Figures 8 and 9, as the analogy to the Figs. 6 and 7,
present the last stage of calculation of the forecast for the bearing krak3.One
can conclude from these data that, again, the forecasting error is smaller if we
use the first generalized symptom SD1 (Fig.9 bottom left), but the transpa-
rency of diagnostic decision is better if one takes the total damage symptom



Forecasting the global and partial system... 793

SumSDi (Fig.8 top left). Hence, it is impossible to decide in advance which
quantity we should use for diagnostic inference – the total damage generalized
symptom SumSDi or the first generalized symptom SD1.We should decide
on this important issue each time separately if we know the machine and the
data.

Fig. 8. The diagnostic decision and forecast premise for ball bearing krak3 on the
basis of the total damage symptom (see next figure)

7. Conclusions

Thepremise towrite thispaperwas supposition thataseparate inferencebased
on the first generalizedmachine symptommay be better than an inference on
the basis of the total damage generalized symptom of the machine condition.
As tipical in the multidimensional condition monitoring, we used the singu-
lar value decomposition to extract the fault information from the symptom
observation matrix. Having calculated the just mentioned generalized symp-
toms, the symptom reliability and the symptom limit value Sl were assessed
on that basis for the total damage symptom SumSDi and for the dominating
generalized symptom SD1. The last stage of inference was the forecast of the



794 C. Cempel

Fig. 9. The diagnostic decision and forecast premise for the same situation (as in
Fig.8), but on the basis of the first generalized symptom

future value of both symptomsmade by the grey system theory and GM(1,1)
model. However, taking into consideration two cases of system monitoring, it
was impossible to validate which one of these two approaches was better.Mo-
reover, it seems that they are complementary, and both calculation should be
made and the decision on machine condition carefully undertaken on a such
basis.

References

1. Berry M.W., Drmac Z., Jessup E.R. 1999, Matrices, vector spacer, and
information retrieval, SIAM Review, 41, 2, 335-362

2. Cempel C., 1987, Simple condition forecasting techniques in vibroacoustical
diagnostics,Mechanical Systems and Signal Processing, 75-82

3. Cempel C., 1991, Vibroacoustic Condition Monitoring, Ellis Horwood Press,
NewYork, p.212



Forecasting the global and partial system... 795

4. Cempel C., 1999, Innovative developments in systems condition monitoring,
Keynote Lecture, Proceedings of DAMAS’99, Dublin, Key Engineering Mate-
rials, 167-168, 172-189

5. Cempel C., 2003, Multidimensional condition monitoring of mechanical sys-
tems in operation,Mechanical Systems and Signal Processing, 17, 6, 1291-1303

6. CempelC., 2004, Implementingmultidimensional inference capability invibra-
tion conditionmonitoring,Proceedings of Conference: Acoustical andVibratory
Surveillance, Senlis, France

7. CempelC., NatkeH.G.,YaoJ.P.T., 2000, Symptomreliability andhazard
for systems condition monitoring, Mechanical Systems and Signal Processing,
14, 3, 495-505

8. Cempel C., Tabaszewski M., 2005, Multidimensional vibration condition
monitoring of non-stationary systems in operation,Proceedings of 12 Interna-
tional Congress on Sound and Vibration, Lisbon, CD-paper No. 496 (full text
inMechanical Systems and Signal Processing, 2007, 21, 3, 1233-1241)

9. Cempel C., Tabaszewski M., 2006, Averaging the symptoms in multidi-
mensional condition monitoring for machines in nonstationary operation, 13
International Congress on Sound and Vibration, Vienna, CD-paper No. 519

10. Cempel C., Tabaszewski M., 2007a, Application of grey system theory in
multidimensional machine condition monitoring,Diagnostyka, 42, 2, 11-18 [in
Polish]

11. CempelC., TabaszewskiM., 2007b,Multidimensional conditionmonitoring
of the machines in non-stationary operation, Mechanical Systems and Signal
Processing, 21, 1233-1247

12. Deng J.-L., 1982, Control problems of grey systems, Systems and Control
Letters, 1, 5, North Holland, Amsterdam

13. Deng J.-L., 1989, Introduction to grey system theory, The Journal of Grey
System, 1, 1, 1-24

14. Deng J.-L., 1990, The Course on Grey Systems Theory, Publishing House,
Huazhong University of Technology,Wuhan [in Chinese]

15. Golub G.H., VanLoan C.F., 1983. Matrix Computation, Oxford, North
Oxford Academic

16. HallD.L., Llinas J., 1997,An introduction tomultisensor data fusion,Pro-
ceedings of the IEEE, 85, 6-23

17. JasińskiM., 2004,Empirical models in gearbox diagnostics, PhDThesis,War-
sawUniversity of Technology,Warsaw [in Polish]

18. KiełbasińskiA., SchwietlickH., 1992,Numeryczna algebra liniowa,WNT,
Warszawa, pp.502



796 C. Cempel

19. Korbicz J., et al., (eds.), 2004, Fault Diagnosis – Models, Artificial Intel-
ligence, Applications, Springer Verlag, Berlin-Heidelberg, p.828

20. Roemer M.J., Kacprzyński G.J., Orsagh R.F., 2001, Assessment of da-
ta and knowledge fusion strategies for prognostic and health management,
http://www.impact-tex.com/data/Publications/f054 gjk.pdf

21. Tabaszewski M., 2006, Forecasting of residual life of the fan mill by means
of neural nets,Diagnostyka, 39, 3, 149-156 [in Polish]

22. Tumer I.Y., Huff E.M., 2002, Principal component analysis of tri-axial vi-
bration data from helicopter transmission, 56th Meeting of the Society of Ma-
chine Failure Prevention Technology

23. Wang T.C., Liou M.C., Hung H.H., 2005, Application of grey theory on
forecasting the exchange rate between TWD and USD, Internet, 1-8

24. Wen K.L., Chang T.C., 2005, The research and development of completed
GM(1,1)model toolbox usingMatlab, International Journal of Computational
Cognition, 3, 3, 42-48

25. Will T., 2005, Hanger matrix, two-thirds theorem, http://www.uwlax.edu/
faculty/will/svd/svd/index.html(seealso:SVDingredientsMathematica,April
2004)

26. Yao A.W.L., Chi S.C., 2004, Analysis and design of a Taguchi-Grey based
electricity demand predictor for energy management systems, Energy Conver-
sion and Management, 45, 1205-1217

27. Zhang H., Li Z., Chen Z., 2003, Application of grey modeling method to
fitting and forecasting wear trend of marine diesel engines, Tribology Interna-
tional, 36, 753-756

28. Zhang L., Wang Z., Zhao S., 2007, Short-term fault prediction of mecha-
nical rotating parts on the basis of fuzzy-grey optimizing method, Mechanical
Systems and Signal Processing, 21, 856-865

Prognozowanie globalnego i cząstkowego stanu za pomocą metod
wielowymiarowej diagnostyki maszyn

Streszczenie

Maszynymająwieleuszkodzeń, które ewoluująpodczas ichpracy(życia). Jeśli ob-
serwujemypewną liczbę symptomówstanupodczas pracymaszyny, to jesteśmywsta-
nie uchwycić informację uszkodzeniową zorientowaną, za pomocą tzw. symptomowej
macierzy obserwacji (SOM). Jedną z metod dalszej ekstrakcji tej informacji diagno-
stycznej jest zastosowanie rozkładu według wartości szczególnych (SVD) do SOM.



Forecasting the global and partial system... 797

Problem postawiony w tej pracy polega na rozstrzygnięciu kwestii, czy w diagnosty-
ce stanu używać uogólnionego symptomu całkowitego uszkodzenia maszyny, czy też
posłużyć się tylko uogólnionym symptomem dominującego uszkodzenia.W tym celu
stworzono dodatkowe oprogramowanie, dzięki któremu pokazano, że takie dychoto-
miczne postawienie kwestii nie jest niewłaściwe. Najlepiej używać obydwa symptomy
uogólnione, wtedy nasza wiedza o stanie maszyny jest pełniejsza.

Manuscript received February 8, 2008; accepted for print June 6, 2008