A review of artificial intelligence machine condition with a widespread consideration of "grey system theory"

KRZYSZTOF JUREK BOGDAN ĩÓŁTOWSKI

University of Technology and Life Sciences

Summary

The publication describes Grey System Theory (GST) and takes into account Grey differential model (GM) and Grey Generating Space (GS). Grey System Theory shows to which extent vibration signals (deriving from tested objects) influence the evaluation and analysis of machine condition. The above-mentioned theory applies under the existence of fixed, non-negative and monotonic data correlated with insuf-ficient and uncertain data sources. In relation to these circumstances, the Forecast-ing (rollForecast-ing window) method seems to be an appropriate solution, which remains the main subject of this paper.

Keywords: Grey systems, vibration diagnostics, forecasting, modelling of condition 1. Introduction

The Grey System Theory (GST) was established in 1982 by a Chinese scientist known as J.-L. Deng. At the beginning, despite an opportunity for a widespread application of GST, the theory did not attract the attention of scientists and researchers from western countries. In the early nineties of the twentieth century the circumstances changed; that is why we can observe wide popularity of the GST, which allows easy forecasting of the condition of entities, machines respec-tively. The theory is commonly used across other related areas such as social and natural sciences, demography, hydrology and economy (such as anticipation of market condition bases on particular data sources). Furthermore, the idea of GST offers practical solutions available not only for scien-tists, but also for engineers and entrepreneurs in the form of appropriate decision-making process.

One of the most adequate tools which guides the management of algorithm sequence is MATLAB program with an important Toolbox (Statistic) function. The knowledge and the ability of practical usage of MATLAB program is required to understand the algorithm sequence de-scribed in the following section.

The algorithm described below is not so complex; that is why the combination of its deep re-vision and appropriate MATLAB’ skills force the correct implementation of the algorithm. 2. The grey system algorithm towards condition forecasting

Most of the technical algorithms, including Grey System algorithm, can be expressed through mathematical shape. First of all, it seems necessary to admit that Grey Model (GM) describes the system behaviours in relation to a particular symptom defined as (x(0)_{(t)), in which t means the} following obtained symptom, for instance (t <1,2,3,…,’),

maybe presented under differential quotation at the bases on k, simultaneously forcing e (the same foundation), which can also be shown as GM (k, e). The principle states that k concerns the factor

which forces changes under differential quotation, and e takes the below-presented mathematical forms: (1) 1 (1) 1 0 1 n i e k i n j j i j

d x

a

b y

dt

− − − = =

=

¦

¦

in which: x(1)_{(k) =} ( )0 1

( )

t i

x

i

=

¦

is a variable factor of the base object yj – independent behaviour allows correct interpretation of the reviewing object ai , bi – polynomial rate estimating form time line x(0)_{(t), t = 1,2,3…,’}

In most circumstances the starting point is obtained as GM (1,1) equation, which means a differential quotation with only one forcing factor. The solution of this problem can be guided by the following mathematical process:

Stage 1

Determination of observation vector:

x(0) _{= [x}(0) _{(1), x}(0) _(2),x(0) _(3),…,x(0) _(n)], requirements: n • 4,

Stage 2

Formation of Accumulating Generating Operation (AGO): x(1)_{(t) =} ( )0 1

( ),

t i

x

i

=

¦

t = 1,2,3,…n

which clearly defines vector monotonicity and growth:

x(1) _{= [x}(1) _{(1), x}(1) _(2),x(1) _(3),…,x(1) _(n)] requirements: x(1) _{(1) = x}(0) _(1).

Stage 3

Basing on the above-defined AGO vector, it is appropriate to describe the Grey differential model (GM) in relation to the starting position GM (1,1):

(1) (1)

( )

( )

dx

t

ax

t

u

dt

+

=

in which: a – growth index,

u – controllable variable factor,

t – uncontrollable factor (e.g. time, asset depreciation). Stage 4

The solution of the above-presented differential quotation with a constantly growing variable t, as following:

ڡ(1)_{(k+1) = [x}(0)_{(1) – u/a] exp(-ak) + u/a,}

in which ڡ(1) _{describes a potential forecast of Accumulating Generating Operation} Stage 5

The replacement of differential complete accretion (Stage 3) relates to t=1 and the composition of the precedent and progressive equations.

- precedent equation: x(1)_{(k+1) – x}(1)_{(k) + ax}(1) _{(k) = u} - progressive equation: x(1)(k+1) – x(1)(k) + ax(1) (k+1) = u

The combination of equations gives us the model of: x(1)_{(k+1) – x}(1)_{(k) = -a/2[x}(1)_{(k) + x}(1)_{(k+1)] +} u

k = 1,2,3,…,n. Stage 6

The conversion of the above-mentioned model relates to the following k values (basing on previ-ously obtained observation vector - x(0)_{) to estimate the unknown extra differential quotation ratios} [a, u]. This estimation is indicated via a selection of ‘smaller squares’ to final attainment of matrix solution:

[a,u]T _{= (B} T_B)-1 _BT_Y

B = ( ) ( ) ( ) 1 (1) 1 (1) 1 (1)

[x (1)

(2)]

1

[x (2)

(3)]

1

...

...

[x (

1)

( )] 1

x

x

n

x

n

−

+

−

+

−

− +

Stage 7

The inverse transformation of AGO presents a potential forecast (basing on AGO’ vector) obtain-ing:

ڡ(0) _{(k+1) = ڡ}(1) _{(k+1) – ڡ}(1) _(k),

furthermore, the conversion of previously-presented progressive and precedent equation (Stage 5) allows to determine the final forecast in relation to starting GM (1,1) model:

ڡ(0) _{(k+1) = [x}(0) _{(1) – u/a](e}-ak _{– e}-a(k-1) ₎ k = 2,3,4,…,n

The final stage of this algorithm shows a general principle of the Grey System Theory (GST). With accordance to Stage 2, it is easy to suggest that the method can fully describe the monotonic and expanding process related to depreciation of particular asset or machine elements. That is why the above-described method can be used to properly describe and forecast machine condition basing mainly on vibration symptoms.

What is more important, sufficient forecast can be obtained after a few observations; at the same time, an increasing number of observations is positively correlated with the growth of the possibility of gathering wrong research data – decrease the efficiency of the observation.

The Forecasting (small-size rolling window) method seems to be an appropriate solution for short term prediction. which simultaneously covers and analyzes huge amounts of data. The determination of appropriate and final forecast (basing on previously-described numerical method)

can be possible by cooperation with MATLAB program; that is why adequate researcher skills are an extremely important factor which definitely determines the forecast success.

3. One – dimensional forecasting according to grey system’ method

As it was mentioned before, the already reviewed algorithm can be easily implemented across a practical basis. This stage of the report describes the influence of the Grey System method on the process of diagnosis and analysis of machines. An excellent example of this phenomenon is contained in the review of airplane turbine bearing.

As a results of asset depreciation, we can observe the rise of oscillations. The number of ob-servations regarding oscillation features – velocity of vibration (mm/s) across equal time interval (each 20 hours) – are presented below:

0 2 4 6 8 10 12 14 2 4 6 8 10 12 14 16 18 20 22 number of obserwations va lu e ( m m /s ) data

Observation results show the monotonic and expanding line; that is why it seems appropriate to adopts the Grey System Theory (GST). Furthermore, we implement this algorithm at the MAT-LAB program and upload already gathered data, which helps us to obtain the following forecast:

0 5 10 15 20 25 30 0 20 40 60 80 100 120 number of observation c o m p on en t v a lu e [ m m /s ] data forecast

A part of the evolution of the airplane turbine bearing, basing on 14 observations and the Forecasting (rolling window) method (15 continual inspections – 300 working hours)

According to above-mentioned graph, the obtained forecast contains several similarities to a quadratic function graph. This phenomenon is fully adequate to practical reality, which means that particular unit depreciation is irregular – a higher depreciation ratio drives the velocity of depreciation process. The above-mentioned forecast has visional similarities that allow to state that the forecast is fully suitable. After the measurement of forecast, failure ratio is a value slightly higher than 5%. Basing on this result, we can state that the forecast fulfills the credibility require-ment (the forecast can be accepted with around 5% of failure).

Moreover, it seems also necessary to admit that this method is mainly dedicated for a short-term forecast. What is more important, the implementation of measurement should base on contin-ual observations across eqcontin-ual periods of time.

4. Multidimensional condition forecast

Currently multidimensional condition forecast is at the stage of early development, so that the implementation of related analysis methods is very difficult. Generally, multidimensional condi-tion forecast bases on other necessary tools, connected with artificial intelligence called neural network, data fusion respectively. Moreover, the Singular Value Decomposition (SVD) is also closely related to the above-mentioned forecast. However, a thorough description of these tools and methods are specified in other reports because of their discrepancies with the Grey System algorithm basis.

0 5 10 15 20 25 -2 -1 0 1 2 3 4 5 6 7

GSago.m; X0, W indow forececast-xj0 -*- & error-edc% /10; for X0=sdi

C o m p o n e n t v a lu e Ordering No of observ. rolling window; w=5 average error; % =10.26 sdi for sil24d01

The fundamental knowledge of the multidimensional condition forecast is required to become fully capable to interpret processes related with vibration diagnostics’ studies.

0 5 1 0 1 5 2 0 2 5 3 0 3 5 4 0 4 5 - 2 0 0 2 0 4 0 6 0 P r i m . s y m p t . v e c t o r X 0 = 1 + S D i O p t , G S fo r e c a s t V p - * - & e r r o r e % / 1 0 ; fo r k r a k 1 C o m pon en t v a lu e O r d e r i n g N o o f o b s e r v . a v e r a g e e r r o r ; % = 4 6 . 4 0 6 9 T o t a l f o r e c a s t , n o r o l l i n g w i n d o w 0 5 1 0 1 5 2 0 2 5 3 0 3 5 4 0 4 5 - 1 0 0 1 0 2 0 3 0P r i m . s y m p . v e c t o r X 0 , W i n d o w f o r e c e c a s t - x j 0 - * - & e r r o r - e d c % / 1 0 ; fo r X 0 = 1 + S D i O p t C o m p o nen t v a lu e O r d e r i n g N o o f o b s e r v . R o l l i n g w i n d o w ; w = 5 a v e r a g e e r r o r ; % = 8 . 1 8 2 4 6 8 1 0 1 2 1 4 1 6 1 8 2 0 7 8 9 1 0 1 1 1 2 F o r e c a s t i n g w i n d o w s p a n - w A v era ge e rror [ % ] G S a g o . m ; A v e r a g e e r r o r - a e W i n % v s f o r e c e c a s t w i n d o w ( w ) ; fo r X 0 = 1 + S D i O p t

5. Conclusion

This report clearly states that the Grey System Theory is especially adequate to forecast one-dimensional unit conditions. The wide opportunity of application of the theory can be utilized for statistical purposes (estimation of revenue or trends), as well as in assets vibration diagnostics.

The good and suitable performance of Forecasting (rolling window) method allows to de-crease the possibility of forecasting failure; that is why appropriate interpretation of Grey System algorithm and increase in knowledge drives the regularity and conformity of the research.

Finally, the report reminds that proper utilization of above-mentioned method relates to de-preciation of machine units, with the monotonic and expanding continual values.

Bibliography

1. Cempel C., Tabaszewski M., Zastosowanie teorii szarych systemów do modelowania i prognozowania w diagnostyce maszyn, Instytut Mechaniki Stosowanej, Politechnika Po-znaĔska, Diagnostyka’2(42)/2007.

2. Deng J., Introduction to grey system theory, Journal of Grey System, 1(1), 1989 pp. 1–24. 3. Ruey-Chyn T., The Grey System Theory, Department of Finance, Hu Suan Chuang

Univer-sity, 2006.

4. Deng J-L., Control Problems of Grey Systems, Systems and Control Letters, Vol. 1, No 5, North Holland, Amsterdam, 1982.

5. Wen K.L., Chang T. C., The research and development of completed GM(1,1) model toolbox using Matlab, International Journal of Computational Cognition, 2005, Vol. 3, No 3, pp. 42–48.

PRZEGLĄD PROBLEMATYKI SZTUCZNEJ INTELIGENCJI W DIAGNOZOWANIU MASZYN

UWZGLĉDNIAJĄCEJ TEORIĉ SZARYCH SYSTEMÓW Streszczenie

W opracowaniu opisano podstawy teorii szarych systemów mającą zastosowa-nia podczas opracowywazastosowa-nia informacji z badaĔ, danych pomiarowych niepełnych I niepewnych. Teoria szarych systemów zastosowana dla sygnałów drganiowych stanu maszyn umoĪliwia ocenĊ stanu wraz z jego prognozowaniem. Pierwsze próby wyko-rzystania tej teorii w obszarze eksploatacji i diagnostyki maszyn przedstawiono w tym opracowaniu.

Słowa kluczowe: niepewnoĞci danych, szare systemy, prognozowanie, modelowanie stanu

*This paper is a part of WND-POIG.01.03.01-00-212/09 project.

Krzysztof Jurek Bogdan ĩółtowski

University of Technology and Life Sciences WIM POIG