Degradation state prediction of rolling bearings using ARX ......ORIGINAL ARTICLE Degradation state prediction of rolling bearings using ARX-Laguerre model and genetic algorithms Taoufik

ORIGINAL ARTICLE

Degradation state prediction of rolling bearings using ARX-Laguerremodel and genetic algorithms

Taoufik Najeh1& Jan Lundberg1

Received: 23 April 2020 /Accepted: 24 November 2020# The Author(s) 2020

AbstractThis study is motivated by the need for a new advanced vibration-based bearing monitoring approach. The ARX-Laguerre model(autoregressive with exogenous) and genetic algorithms (GAs) use collected vibration data to estimate a bearing’s remaininguseful life (RUL). The concept is based on the actual running conditions of the bearing combined with a new linear ARX-Laguerre representation. The proposed model exploits the vibration and force measurements to reconstruct the Laguerre filteroutputs; the dimensionality reduction of the model is subject to an optimal choice of Laguerre poles which is performed usingGAs. The paper explains the test rig, data collection, approach, and results. So far and compared to classic methods, the proposedmodel is effective in tracking the evolution of the bearing’s health state and accurately estimates the bearing’s RUL. As long asthe collected data are relevant to the real health state of the bearing, it is possible to estimate the bearing’s lifetime under differentoperating conditions.

Keywords Vibration analysis . Condition monitoring . RUL . Rolling-element bearings . Through-life engineering . GAs .

ARX-Laguerremodel

1 Introduction

In many industries, prognostics and health management(PHM) are key tools for condition-based maintenance. Thebearing industry, like other industries, uses various ap-proaches to extract health indicators that help in decision-making for maintenance. One common method to determinethe remaining useful life is to use the dynamic load capacityand applied load, a method endorsed by standardISO281:1977 and the modified ISO281:2007. However, inreal operating conditions, the bearing can suffer from unex-pected circumstances, and the actual operating life could becompletely different [1]. Another method is to monitor thecondition of the bearing to determine its health state and re-maining useful life (RUL). This paper proposes a new modelto track the evolution of bearing health using vibration data tomore accurately estimate the bearing’s RUL. The concept is

based on the actual running conditions of the bearing com-bined with a new linear ARX-Laguerre representation.

By the end of the eighteenth century, bearing manufac-turers and users started to focus on bearing selection and thelife of bearings needed for a well-designed machine. In 1896,Stribeck [2] was the first to measure the mechanical fatigue ofbearings. A decade later, Goodman [3] adopted the fatigueapproach to determine load limits on cylindrical roller bear-ings. For quite some time, Palmgren’s work was the mostimportant in the field of bearing life calculations. In 1947,Palmgren and Lundberg used Palmgren’s previous work andtheWeibull distribution to develop a modified version of bear-ing life [4]. Since 1952, all standards have been based onLundberg and Palmgren’s approach [5]. It takes into accountmany parameters like the contact area and the length of theraceway, the number of stress repetitions, the probability ofsurvival, the internal stress created by the external load, andthe stressed volume.

By 1970, however, both manufacturers and users began toadmit that bearing life was much better than ISO-standardpredictions. In 1971, ASME proposed “Life AdjustmentFactors” to minimize the errors in remaining life prediction[6]. But bearing design continued to improve, and bearingnormal service life was about six times longer than the

* Taoufik [email protected]

1 Division of Operation and Maintenance, Luleå University ofTechnology, 97187 Luleå, Sweden

https://doi.org/10.1007/s00170-020-06416-1

/ Published online: 10 December 2020

The International Journal of Advanced Manufacturing Technology (2021) 112:1077–1088

predicted lifetime. In 1985, based on Lundberg andPalmgren’s theory, Harris et al. [7] added an additional pa-rameter to describe the fatigue strength of bearing material.The new contribution allowed more accurate predictions forthe ball and roller bearings. This method still works even inless than optimal operating conditions like the reduction oflubrication or contamination inside the housing. In its lastversion, international standard ISO 281: 2007 fully supportsthis theory for bearing-life calculation using the basic follow-ing equation:

L10 ¼ CP

� �p

ð1Þ

where C is the basic dynamic load capacity, P is the appliedload, and p is the life equation exponent. The dynamic loadcapacity is developed based on empirical testing under a con-stant load that allows the bearing to reach a specific number ofrevolutions without fatigue. The L10 life refers to the expectedlife of 90% of an ensemble of similar bearings which can becustomized with a life adjustment factor for reliability a1. Thebasic equation has since been adjusted and non-linear modifi-cations include the addition of a stress life modification factor,aslf , to combine the effect of the contamination level andlubrication quality of the bearing.

In addition to the C/P technique, there are various online oroffline data-driven methods for prognostics and health man-agement using the available observed failure time data andvarious health indicators [8]. The condition monitoring datacan be used directly or indirectly. For example, data on wearor cracks can give a direct estimation of the RUL, while otherdata, such as vibration data, indirectly describe the health stateof the bearing. In the latter case, a failure event is needed asadditional information to predict the RUL. Several researchershave proposed methods for RUL. Hasan et al. suggested atechnique based on wavelet packet decomposition for bearingprognostics [9]. Guo and colleagues proposed using a recur-rent neural network-based health indicator for RUL estimation[10]. Hinchi and Tkiout developed an end-to-end deep frame-work based on convolutional and long short-term memory(LSTM) recurrent units for RUL prediction [11].

More approaches to rolling element bearing RUL esti-mation are reported in the literature and differ according tothe nature of the data and the way they are collected[12–15]. This paper proposes a new perspective in the fieldof estimating a bearing’s lifetime. It uses artificial intelli-gence to solve the problem of high nonlinearity and toachieve more accurate health predictions. This state-of-the-art research is based on the ARX-Laguerre model anduses genetic algorithms (GAs). Its goal is to provide anaccurate estimation of the RUL of bearings using a newsignal processing algorithm.

2 Experimental details

Figure 1 is a photograph of the test rig used for the experiment.It has a customized bearing house for the SKF 61900 deepgroove ball bearing which is directly coupled to the shaft of75 kW, three-phase, 1650 rpm motor. This high-powered elec-trical motor was justified by the idea of extending the test rig tosupport additional functions related to gearbox fault detection.In the rig’s hydraulic system, a cylinder pulls the bearing houseto apply a radial load to the tested bearing. All parts are boltedto a large steel bed secured to concrete flooring. The choice ofthe tested bearing (SKF 61900) was justified by the low loadneeded during the test; the housing is directly mounted on theshaft, so the radial load applied on the tested bearing must beless than the maximum load supported by the original bearingsinstalled inside the electric motor.

In this experimental set-up, the rotating shaft is directly at-tached to the bearing. The bearing housing is thus stationaryand is connected to a rigid support (the steel bed) using a hy-draulic cylinder which both give the radial load and the support.This bearing arrangement is thus in principle, from the loadpoint of view, identical with the bearing housing arrangementused for an instant in machinery equipment. The advantage ofthe experimental set-up, even though different from the con-ventional test rigs, is that it is purely considering the mainparameters that are affecting the lifetime of the bearing normal-ly, angular speed, radial load in a clearly defined direction, andthe lubricant. The measurements will therefore not be disturbedby the support of the bearing, the radial load will be accuratelymeasured, and the measured parameters will be the same asused in the standardized method for bearing lifetime calcula-tion. An example of machines with this type of bearing arrange-ment can be seen in excavators and other heavy equipmentvehicles. Another advantage, with this test rig, just one bearingis considered; with the conventional test rigs, we can haveseveral bearings running simultaneously, and it will be hardto draw a consistent conclusion.

Measurement equipment and software schematics for thetest rig are presented in Fig. 2. The PXI National Instrumentsplatform was used for data acquisition, speed control of theelectric motor, and load control of the hydraulic pump. TheNIPXIe 6361 was used to measure the speed, and because thismodule has two analog outputs (2.86 MS/s), 24 digital I/Os; itwas used as a speed and load controller respectively for theelectric motor and the hydraulic pump. The NIPXI 4472Bwasused for the vibration signal acquisition and temperature mea-surement. This module had eight channels, each of which hasa sample rate of 102.4 kHz.

A triaxial accelerometer (Bruel−Kjær) with frequencyrange (0.25–3000 Hz) was mounted directly on the bearinghouse using adhesive. To get an accurate measurement of thebearing’s strain, a place was reserved as close as possible tothe housing for the strain sensor to directly measure the strain

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on the tested bearing. DTE 25 hydraulic oil (ISOV G46) wasused for lubrication, and the housing was filled to half level.The vibration data, combined with actual running conditions,are used to define the health state of the bearing and predictwhen it will reach a faulty mode. Failure is defined as a spe-cific level of the root mean square (RMS) value which willindicate the presence of cracks on the surface of raceways orrolling elements. The final goal is an accurate assessment ofthe degradation path from a healthy to a faulty mode to esti-mate the remaining useful life of a rolling element bearing.

Before beginning the experiment, to acquire a general un-derstanding of the bearing’s behavior, several random testswere done for different loads and various threshold values todecide on the selected thresholds. First, a constant load of1 kN and normal to load-bearing surfaces was used; this loadis the maximum load supported by the SKF 61900 deepgroove ball bearing. However, we never managed to createdefects, even after 20 h of running time, and the RMS neverexceeded the 1.8 m/s2 level. Figure 3a confirms the absence ofdefects; it indicates how much running time is required to

Fig. 2 Measurement equipmentand software schematic for thetest rig

Fig. 1 Test rig

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create a defect under those conditions. Next, we tried a 3-kNload. In this case, the test ended up with a failure in less than10 h, but a plastic deformation was noted from the first test(see Fig. 3b). It is impossible to get any reasonable results onthe lifetime estimation under this plastic deformation.

The problem was how to accelerate the test withoutoverloading the bearing. Arizona dust was used to solve theproblem; this had an acceptable test duration without chang-ing the failure mechanism, and it was possible to keep the loadequal or under 1 kN where degradation of the bearing healthcould be recorded from the start-up to failure and confirmedwith microscopic images, as shown in Fig. 3c and d. Table 1shows the values of the predefined levels where the thresholdRMS levels were varied from 1 to 5 m/s2.

There are several kinds of Arizona dust referring to thestandard ISO 12103-1. The ARIZ-KSL is the hardest type,moreover is not labeled as hazardous to health. For this typeof Arizona dust, there are four possible grades of fineness:ultra-fine (A1), fine (A2), medium (A3), and coarse (A4).This study only considered A2 (0.97 to 176 μm) and A3(0.97 to 284 μm) fine grade. The A1 or A2 grade refers to

the size distribution of particles. As an example, Table 2 rep-resents the distribution of A2 fine dust content in terms ofparticle size.

Based on several tests performed on the test rig, we end upwith the conclusion that the RMS is starting with a value usu-ally less than 1 m/s2 and almost constant, but suddenly, itmonotonically increases. An increase of 10% in the currentRMS value will be assumed as a triggering event to considerthe bearing running in the second phase. A value of 7 m/s2 ismore or less common for most of the tests when total failureaccrues; the time consumed to reach this threshold was signif-icantly different from one run to another. This can be ex-plained by the smoothing process caused by the continuousrotation of balls against the inner race and the outer race.

Anyhow, it is important to consider when a critical vibra-tion level has reached to avoid critical future failure. In this

)b()a(

)d()c(

Fig. 3 Microscopic images ofdegradation

Table 1 Duration and threshold values for the termination of each test

Run Load (kN) Arizona dust Duration (h) RMS (m/s2) Figure

1 1 No 20 2 3a

2 3 No 11 5 3b

3 1 Yes 14 4 3c

4 0.9 Yes 17 5 3d

Table 2 Size distributionof A2 fine dust Particles(μm) ≤ %

0.97 4.5–5.5

1.38 8.0–9.5

2.75 21.3–23.3

5.50 39.5–42.5

11.00 57.0–59.5

22.00 73.5–76.0

44.00 89.5–91.5

88.00 97.9–98.9

124.50 99.0–100.0

176.00 100.0

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study, the value 7 m/s2 will be considered the failure thresholdof the RUL predicted. Each time the measurements were ac-quired using a sampling frequency of 25.6 kHz, the measuredsample length was 3 s, and the samples were taken once perminute. The purpose of these tests was to create a database thatcould be used in the bearing life estimation or any other futurestudy connected to prognostic problems. Note that to take intoaccount the inherent randomness of end-of-life of bearings, itis crucial to collect data during all operational scenarios andcover all normal and abnormal behavior states. This can bedone using a factorial experimental design.

3 Remaining useful life prediction of bearings

The main goal is to predict the bearing’s health state andestimate upcoming degradation as early as possible. The pro-posed strategy is inspired by the RMS measurement versustime (Fig. 4). As Fig. 4 shows, there are always two distin-guishable stages in the curve. The first one lasts from thebeginning until the defect reaches an almost constant RMSlevel. The second stage is identified by a continuous and as-cendant variation of the RMS measurement. The RMS varia-tions typically show this behavior in all tests; the only differ-ences are the end time of the first phase and the slope in thesecond phase, both of which change from one bearing to an-other. In this paper, we focus on the second phase. The bearinglifetime calculation refers to the period after the initial defectoccurs. We first determine a critical level at which to startusing data; this time is considered the beginning of the failure

mode. We then use the ARX-Laguerre model to predict theRMS value at a specific point in time and estimate when thebearing will reach the end of its useful life. In terms of param-eter complexity reduction and quality approximation, theARX-Laguerre with the RMS feature shows good perfor-mance. Besides, the results confirm the efficiency of the pro-posed model for the RUL estimation with respect to the run-ning conditions. Given the good results, we propose a possibleextension of this work by combining other features.Specifically, expanding the ARX coefficients on twoLaguerre bases instead of directly using the ARXmodel givesan accurate approximation of a complex linear system.

The idea of expanding the ARX model in this way was firstproposed by Bouzrara et al. [16]. However, the capability ofthis model is highly dependent on the poles’ calculation of bothLaguerre. When we combine the ARX with two Laguerre ba-ses, we end up with a simple representation and a good estima-tion of the complex system. The new model is called the ARX-Laguerre model. The efficiency of the model is closely linkedto the choice of the poles of both Laguerre bases.

Before the lifespan prediction model can be adapted, westart by choosing a reasonable way to experiment with this testrig. It seems that a full factorial experiment design is feasiblein our case, i.e., 3 × 3 × 2 = 18 runs. The speed and load fac-tors have three levels, and the contamination size and amounthave two. The term level is used when referring to the value ofone of the independent parameters (e.g., speed = 1 means1650 rpm) and the term run when referring to a combinationof levels (e.g., load = − 1, speed = 1, contamination = 0). Theselection of the level takes into account the load capacity of

0 50 100 150 200 250 300

Number of iterations

0

1

2

3

4

5

6

RM

S (

m/s

2)

Phase 1 Phase 2

Fig. 4 RMS measurement

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the bearing and the limiting operating condition of the test rig.The factors of the test are given in Table 3. The table showstwo three-level factors and one two-level factors.

3.1 The mathematical formulation of the ARX-Laguerre model

To model a discrete-time process where u(k) and y(k) are theinput and output, respectively, the ARX model can beexpressed as follows [16]:

y kð Þ ¼ ∑naj¼1ha jð Þy k− jð Þ þ ∑nb

j¼1hb jð Þu k− jð Þ ð2Þ

where {na, nb} and {ha(j), hb(j)} are respectively the ARXmodel orders and parameters. These parameters can be devel-oped on two independent Laguerre bases formed by a set ofLaguerre functions as shown in Eqs. (3) and (4):

ha jð Þ ¼ ∑∞n¼0gn;al

an j; ξað Þ ð3Þ

hb jð Þ ¼ ∑∞n¼0gn;bl

bn j; ξbð Þ ð4Þ

where lan j; ξað Þ and lbn j; ξbð Þ are the orthonormal functions ofLaguerre bases associated with the output and the input, re-spectively, {gn, a, gn, b} are the Fourier coefficients and {ξa ,ξb} are the pole defining the orthonormal bases, respectively.

The Z-transform of the ARX-Laguerre model truncated toa finite order Na and Nb can be represented as:

Y zð Þ ¼ ∑Na−1n¼0 gn;aX n;a z; ξað Þ þ ∑Nb−1

n¼0 gn;bX n;b z; ξbð Þþ E zð Þ

ð5Þ

With :X n;a z; ξað Þ ¼ Lan zð ÞY zð ÞX n;b z; ξbð Þ ¼ Lbn zð ÞU zð Þ

�ð6Þ

Lan zð Þ ¼ffiffiffiffiffiffiffiffiffiffiffiffi1−ξa

2qz−ξa

1−ξazz−ξa

� �n

Lbn zð Þ ¼ffiffiffiffiffiffiffiffiffiffiffiffi1−ξb

2qz−ξb

1−ξbzz−ξb

� �n

8>>>>><>>>>>:

n ¼ 0; 1;…;Ni−1ð Þ ð7Þ

where Y zð Þ;U zð Þ; Lan zð Þ; Lbn zð Þ;Xn;a z; ξað Þ; X n;b z; ξbð Þ;and E zð Þ are the system output, and input, the Z-transformof the orthonormal functions defining both independentLaguerre bases, and the truncation error and the filtered outputand the filtered input respectively by Laguerre functions.

The orthonormal functions ( Lin zð Þ; i ¼ a; bÞ definingboth independent Laguerre bases satisfy the followingrecurrence:

La0 zð Þ ¼ffiffiffiffiffiffiffiffiffiffiffi1−ξi

2qz−ξi

Lin zð Þ ¼ 1−ξizz−ξi

Lin−1 z; ξið Þn ¼ 0; 1;…;Ni−1ð Þ

8>>><>>>:

ð8Þ

Based on Eq. (8), the filtered input and output Xn, a(z, ξa),Xn, b(z, ξb) can be represented as:

X 0;a z; ξað Þ ¼ffiffiffiffiffiffiffiffiffiffiffiffi1−ξa

2qz−ξa

Y zð Þ

X n;a z; ξað Þ ¼ffiffiffiffiffiffiffiffiffiffiffiffi1−ξa

2qz−ξa

X n−1;a z; ξað Þ

8>>>>><>>>>>:

ð9Þ

X 0;b z; ξbð Þ ¼ffiffiffiffiffiffiffiffiffiffiffiffi1−ξb

2qz−ξb

Y zð Þ

X n;b z; ξbð Þ ¼ffiffiffiffiffiffiffiffiffiffiffiffi1−ξb

2qz−ξb

X n−1;b z; ξbð Þ

8>>>>><>>>>>:

ð10Þ

The ARX-Laguerre filter network based on Eqs. (5), (9),and (10) is illustrated in Fig. 5.

The discrete time-recursive representation of the ARX-Laguerre model can be created from Fig. 5 as:

XNa k þ 1ð Þ ¼ AaXNa kð Þ þ bay k þ 1ð ÞXNb k þ 1ð Þ ¼ AbXNb kð Þ þ bby k þ 1ð Þy kð Þ ¼ CTX kð Þ þ e kð Þ

8<: ð11Þ

with:

C ¼ g0;a;…; gN−1;a ; g0;b;…; gN−1;b� � ð12Þ

and X(k) is defined as follows:

X kð Þ ¼ XNa kð Þt X Na kð Þt� �T ð13Þ

Table 3 Levels of variables and coding identification

Level Load (kN) Speed (rpm) Particle size Coding

High 1.03 1650 A3 11 1

Center 0.92 1450 - 00 -

Low 0.76 1250 A1 − 1-1 0

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where XNa and XNb are defined in Eqs. (14) and (15),respectively:

XNa ¼ x0;a kð ÞT XNa−1;a kð ÞTh iT

ð14Þ

XNb ¼ x0;b kð ÞT XNb−1;b kð ÞTh iT

ð15Þ

Ai for (i = a, b) are two square matrices with dimensions Ni

given by:

ai11 0 ⋯ 0ai12 ai22 ⋯ 0⋮ ⋮ ⋱ ⋮

aiNið Þ1 aiNið Þ2 ⋯ aiNið Þ Nið Þ

2664

3775 ð16Þ

with:

aism ¼ξi if s ¼ m

ξið Þ s−m−1ð Þ�1− ξið Þ2 if s ¼ m

0 if s ¼ m

8<: m ¼ 1;⋯;Nið Þ ð17Þ

and bi for (i = a,b) are defined as follows:

bi ¼ bi1 bi2…biNi

h iTð18Þ

with : bim ¼ −ξað Þm−1ffiffiffiffiffiffiffiffiffiffiffi1−ξi

2q

; m ¼ 1;⋯;Nið Þ ð19Þ

3.2 Laguerre poles optimization based on geneticalgorithms

As mentioned, when the ARX coefficients are expandedon two Laguerre bases, the capability of the resultingARX-Laguerre model is very sensitive to the choice ofthe poles defining Laguerre bases ξa and ξb. Bouzraraet al. [17] proposed an iterative algorithm to optimizeLaguerre poles based on an analytical solution of coeffi-cients defining the ARX-Laguerre model. To find the op-timum values of Laguerre poles, we choose an approachbased on genetic algorithms and proposed by Tawfik et al.which is used [18, 19].

The first step is the formulation of the objective function toevaluate any possible values of Laguerre poles by minimizingthe normalizedmean square error (NMSE) and the normalizedmean absolute error (NMAE) as a cumulative error betweenthe real and the predicted output:

NMSE ξa; ξbð Þ ¼ ∑Dk¼1 yreal kð Þ−ymodel kð Þ½ �2

∑Dk¼1 yreal kð Þ½ �2 ð20Þ

Fig. 5 ARX-Laguerre filternetwork

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NMAE ξa; ξbð Þ ¼ ∑Dk¼1 yreal kð Þ−ymodel kð Þj j

∑Dk¼1 yreal kð Þ½ � ð21Þ

where D is the observation number, y(k) is the real output ofthe system, and ymodel is the predicted output.

After the fitness function is determined, a population ofa set of possible values of ξa and ξb is randomly gener-ated. It will perform genetic operations, such as evalua-tion, mutation, crossover, and selection [20, 21]. The op-timum Laguerre poles are selected by minimizing the fit-ness function [20].

After a predefined number of generations, the algo-rithm ends up with optimal poles defining Laguerre

bases. The following algorithm summarizes the optimi-zation approach:

Selecting a measurement window of input/output (yreal(k),U(k)) and the truncating orders (Na,Nb).

A maximum number of generation Gmax are predefined

0 20 40 60 80 100 120 140

Number of iterations

-8

-7.5

-7

-6.5

-6

-5.5

-5

Str

ain

10-4

Identification phase Validation phase

Fig. 6 Input signal: strain

04102100108060402

Number of iterations

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

RM

S (

m/s

2)

Identification phase Validation phase

Fig. 7 Output signal: RMS

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4 Results and discussion

Once all combinations of the full factorial experiment designare accomplished, the ARX-Laguerre model is applied to theexperimental data to predict degraded states of the bearingreflected by the RMS increase. To build a reliable model withhigh accuracy, relevant variables covering the behavior of thebearing’s health should be collected. Two variables are

considered in our ARX-Laguerre model, the force as the input(Fig. 6) and the RMS as the output (Fig. 7). Eighteen sets ofdata with different running are generated under specific con-ditions (load, speed...). For each test, a model is built using40% of the recorded data; the remaining data are used for thevalidation of the model. We select the truncating orders Na =Nb = 6, and algorithm 1 is used for the optimization of thepoles based on GAs.

The bearing degradation can be described by the ARXmodel; 12 parameters of degradation are estimated using theleast-squares method. To validate the proposed model, we usethe output of the realRMS value and the ARX-Laguerremodeloutput as a predicted value for 18 independent runs of bear-ings (Table 4). The performance of the model is measured interms of normalized mean square error (NMSE) and normal-ized mean absolute error (NMAE), the cumulative error be-tween the measured output ym(k), and the proposed modeloutput ymodel(k). These performance metrics are used as accu-racy measures for the optimal identification of Laguerre polesξa and ξbwitch can guarantee an important reduction of ARX-Laguerre complexity. Several performance metrics (errormeasures) can be used for the evaluation in the forecastingmodel validation. In the forecasting literature using theARX-Laguerre model, the NMSE metric has been the mostcommonly, suggested metric [16–18]. The results are present-ed in Figs. 8, 9, 10, and 11 (for runs 1, 4, 8, and 12, respec-tively). From this plot, we see that for all tests, the ARX-Laguerre model can predict the RMS level with enough accu-racy to anticipate any failure just on time; thus, total failure ofthe bearing can be avoided.

The goal of the adopted approach is to predict the un-known output presented as a fault and then to detect thetime corresponding to the bearing malfunction. To

Table 4 ARX-Laguerre poles (ξa, ξb) with the NMSE and the NMAEof each test

#Data set ξa ξb NMSE NMAE

1 −0.74 0.95 0.0028 0.0013

2 0.23 0.86 0.0013 0.0006

3 − 0.86 0.21 0.0070 0.0045

4 − 0.4694 0.88 0.0242 0.0090

5 0.45 0.37 0.0185 0.0087

6 0.84 − 0.27 0.0702 0.0433

7 0.33 − 0.94 0.0046 0.0032

8 0.19 0.35 0.0330 0.0179

9 0.46 0.94 0.0924 0.0067

10 − 0.83 0.50 0.0087 0.0056

11 0.42 0.68 0.0764 0.0447

12 0.61 0.83 0.0059 0.0031

13 0.53 0.76 0.0496 0.0244

14 0.80 − 0.92 0.0092 0.0059

15 − 0.25 0.63 0.0448 0.0377

16 0.67 − 0.98 0.2731 0.1355

17 0.79 0.38 0.0624 0.0216

18 0.86 − 0.58 0.0837 0.0368

08070605040302010

Number of iterations

2.5

3

3.5

4

4.5

5

RM

S (

m/s

2)

ARX-Laguerre model outputReal output

Fig. 8 Validation of prediction.Dataset 1

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08070605040302010

Number of iterations

2.2

2.3

2.4

2.5

2.6

2.7

2.8

2.9

RM

S (

m/s

2)

ARX-Laguerre model outputReal output

Fig. 9 Validation of prediction.Dataset 2

0 10 20 30 40 50 60 70 80 90 100

Number of iterations

1.5

2

2.5

3

3.5

4

4.5

5

RM

S (

m/s

2 )

ARX-Laguerre model outputReal output

Fig. 10 Validation of prediction.Dataset 3

08070605040302010

Number of iterations

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

RM

S (

m/s

2)

ARX-Laguerre outputReal output

Fig. 11 Validation of prediction.Dataset 4

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evaluate the efficiency of the proposed method, we studythe RMS variation compared with other existing methodsin the literature, such as ARMA, support vector machine(SVM), Kalman filtering, and long short-term memorynetwork.

FromTable 5, we can conclude the prediction effectivenessbased in terms of prediction error for different approaches. Forexample, Pengfei et al. [22] and based on the built ARMAmodel, the prediction accuracy is about 96.51%. Using theSVMmethod and a standard accelerated aging platform namedPRONOSTIA, Abdenour et al. [23] end up with a predictionaccuracy between 99.4 and 98.75%. Rodney et al. [24] haveused the same test rig (PRONOSTIA) as [23] but they used adata-drivenmethodology based on extended Kalman filtering;the average prediction accuracy was about 80%. In the case ofa long short-termmemory network [25], a prediction accuracyof 45.27 % is obtained.

5 Conclusion

The paper describes how run-to-failure data can be used tobuild an ARX-Laguerre model as a decision model to monitorthe health of bearings. This type of reduced complexity modelis subject to the optimal selection of Laguerre poles obtainedfrom input-output measurements and genetic algorithms. Itfinds that the GA method for ARX-Laguerre pole optimiza-tion is more efficient in terms of mean square error, but theapproach requires a high computing time compared to othermethods (Newton-Raphson’s and Bouzrara et al.’s methods).However, the performance of the model is reasonably good,with a small deviation between the predicted and experimentalvalues. No extra hardware is needed except for a classicalaccelerometer and a load cell. Finally, the proposed methodhas high accuracy, and the estimated lifetime can be estimatedat any point in time using the present running conditions.

Authors’ contributions Experimentation: Taoufik Najeh, Jan Lundberg;numerical modeling: Taoufik Najeh; writing (original draft preparation):Taoufik Najeh; writing (review and editing): Taoufik Najeh, JanLundberg.

Funding Open access funding provided by Lulea University ofTechnology.

Availability of data and material The authors confirm that materialsupporting the findings of this work is available within the article. Thecollected data of this work are not available within the article.

Compliance with ethical standards

Competing interests The authors declare that they have no competinginterests.

Ethical approval The article follows the guidelines of the Committee onPublication Ethics (COPE) and involves no studies on human or animalsubjects.

Consent to participate Not applicable. The article involves no studieson humans.

Consent to publish Not applicable. The article involves no studies onhumans.

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