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Nonlinear ARX (NARX) based identication and fault detec-tion in a 2 DOF system with cubic stiffness

J.S. Sakellariou and S.D. FassoisStochastic Mechanical Systems (SMS) GroupDepartment of Mechanical & Aeronautical Engineeringe-mail:

sakj,fassois ¡ @mech.upatras.grweb page: http://www.mech.upatras.gr/ ¢ sms

AbstractThis paper addresses the problem of system identication and fault detection in a two DOF nonlinear systemcharacterized by cubic stiffness. System identication is based upon Nonlinear ARX (NARX) models, while

a novel Functional Model Based Method is employed, for the rst time within the context of a nonlinearsystem, for tackling the combined problem of fault detection, identication (localization), and fault magnitudeestimation. The Functional Model Based Method utilizes Functional NARX (FNARX) models, which arecapable of accurately representing the system in a faulty state for the latter’s continuum of fault magnitudes, aswell as statistical decision theory tools. The results of the study indicate the effectiveness of both NARX basedidentication and the Functional Model Based Method in detecting, identifying, and estimating the magnitudeof faults based upon only two measured signals.

1 Introduction

This paper is concerned with the problems of systemidentication and fault detection, identication (lo-calization), and magnitude estimation in a two DOF(Degree-of-Freedom) system characterized by localpolynomial nonlinearity (cubic stiffness £ ¤ ; gure 1).

The approach postulated is based upon discrete-time NARX models [1, 2, 3], that is NonlinearAutoRegressive models with eXogenous excitation,which constitute nonlinear extensions of the conven-tional linear ARX models [4, 5]. Unlike alternativenonlinear representations, such as those based upon

Volterra or Wiener series [1, 6, 7], describing func-tions [1, 8], or neural networks [1], which are oftenused in identication, NARX models offer a numberof advantages, including accuracy and compactnessof representation (the latter leading to improved sta-tistical parsimony), physical signicance, and directcorrespondence between the NARX and the physicalsystem parameters. NARX models also feature linearregression based estimation and the availability of anumber of tools for model structure selection [9].

Once the feasibility and effectiveness of NARX

based system identication is demonstrated, the com-bined problem of fault detection, identicaton (lo-calization), and magnitude estimation is tackled viaa Functional Model Based Method . This method,

¥§ ¦ ¨ ©

!" #

$ % & ' ( ) 0 1

2 3

4 57 69 8

@ A B CE D F G H I G H

P Q

G R H S T UV W

Figure 1: Two DOF system with cubic stiffness.

which has been recently introduced by the authors

[10, 11], is used for the rst time within the context of a nonlinear system, cultivating upon the NARX basedsystem representation. The Functional Model BasedMethod achieves fault detection, identication, andmagnitude estimation in a unied way, based uponthe novel class of stochastic functional models andstatistical decision theory tools. The stochastic func-tional models, presently Functional NARX (FNARX)models , play a very central role, as they are capableof accurately representing the system in a faulty statefor the latter’s continuum of fault magnitudes.

The rest of this paper is organized as follows: Thesystem and the considered faults are described in sec-tion 2, while NARX based identication is presentedin section 3. The Functional Model Based Method

International Conference on Noise and Vibration EngineeringSeptember 16-18, 2002 - Leuven, Belgium



X Yb ad c ef Xh gE ap i i (kg)q

Y

ad c e

q

g

as r e

q t

as r e e (kN/m)u Yb as vw i ex u

t

as y e e (Ns/m)

Table 1: Physical system parameters.

Fault Descriptionmode

q

g stiffness changes(

q

ad E y b y )q

Y stiffness changes(reduction value

q

as y e )

Table 2: The considered fault modes.

for fault detection, identication, and magnitude es-timation is presented in section 4, and correspondingresults are summarized in section 5. The conclusionsof this study are summarized in section 6.

2 The system and the faults

The two degree-of-freedom system considered ischaracterized by cubic stiffness in spring

q

t

(gure

1). The rest of the system elements are linear. Thesystem’s physical parameters are indicated in table 1.

The system dynamics are described by the differ-ential equations:

X Y 9 Y s

q

Y

q

g 9 Y

q

gw g u Y b 9 YE a

Xh gw gb

q

gw 9 Y

q

gw g

q

t

t

g

u

t

g a e (1)

with designating a force externally applied on massX Y , and the -th mass displacement, veloc-ity, and acceleration, respectively.

System identication and fault detection are bothbased upon measurement of the force excitation(subsequently designated as ) and the vibration dis-placement response g (subsequently designated as

j ).System simulation is based upon discretization of

equations (1) via forward differencing [12, pp.13-22]with time step k9 l

as mn m m m o` c ew n b u (sampling fre-quency l

ad c i e e h ).The faults. Two types of faults (fault modes) are

considered (table 2 and gure 1): The rst mode

corresponds to stiffness changes inq

g

. Each indi-vidual fault is represented as , with the super-script

q

g indicating the fault mode and the subscriptq

the exact fault magnitude (changes in the range of

q

a E y b y are considered; negative/positivevalues indicate stiffening/loosening, respectively).

The second mode corresponds to stiffness changesin

q

Y . Each individual fault is similarly represented as

. A single fault mangitude of q

ad y e

(stiffnessreduction) is in this case considered.

3 NARX based identication

The manipulation of the discretized system equationsof motion leads to the following relationship betweenthe force excitation z { and the obtained displace-ment response j9 z { 1 :

j9 z { a} | ~

Y9

z {w } |

9 Y9

| ~

z {x a

a j9 z {§ a E b z {n (2)

where:

z { a Y z {

| ~ |

z { {

| ~ |

Y (3)

s a

Y

|b ~

... Y

|

| ~ |

Y

(4)

and z designates normalized discrete time ( z a

c n yn

) with absolute time beingz c

k l

.The regressors z {E a c X d X are

monomials of degrees a c n y , as indicated in ta-ble 3. With the addition of a zero mean, uncorrelated,and uncrosscorrelated with the excitation z { , noiseterm w z { in equation 2, the excitation-response rela-tionship assumes the NARX( , ) (Nonlinear Au-toRegressive with eXogenous excitation) form [1, 2]:

j9 z {§ a z {n w z { (5)

In this form represents the model parameter vec-tor, with a c X designating the -th AR(AutoRegressive) parameter and E a c X

the -th X (eXogenous) parameter, and a ,a , the AR and X orders, respectively. The

excitation-response delay (see the X term in table 3)is a , while the model includes linear, quadratic,and cubic terms, with the maximum nonlinearity de-gree being max as y (see table 3).

System identication. Identication of thehealthy system is based upon the NARX( , )

model of the form of table 3. Estimation is accom-plished via minimization of the quadratic criterion1Lower case/capital bold face symbols designate vec-

tor/matrix quantities, respectively.



Degree 9 d 9 s 9 s

Par. Monomial Par. Monomial Par. MonomialAR term n ª «9 ¬ ª w -® - ª «9 ¬¯ ª ° «9 ¬` ± ª² w ³® ³ ª «9 ¬ ª ° «w ´ ¬ ± ª

” ” ´ ´

ª «9 ¬¯ ª w µ® µ ª «

´

¬` ± ª ¶· ¶w ª «9 ¬ ª ° «

´

¬ ± ª

” ” ¸· ¸w ª «9 ¬¯ ª w ¹® ¹ ª «

¸

¬` ± ª

” ” º· ºw ª «9 ¬ ± ª

X term » ¼ - ª ½ ¬ ± ª

AR order: ¾9 ¿ ± X order: ¾ »E ± Delay: À ±

AR terms: Áh ¿ s Â X terms: Áh »E d

Table 3: The NARX( ¾9 Ã ¾9 » ) model structure.

Ä Å Æ9 Ç È

Åh É

ÅÊ Ë

Ì

´ ª , with Í designating thelength of the excitation-response signals used. Owing

to the linear dependence of the error term Ì

ª

uponthe parameter vector Î , this leads to the linear regres-sion estimator:

Ï

Î Ð

Í

Å

Ñ

Ê Ë

9 Ò

ª °

ÒE Ó

ª Ôh Õ °n Ð

Í

Å

Ñ

Ê Ë

9 Ò

ª ° «9 ª Ô

(6)

Ï

Ö

´×

Í

Å

Ñ

Ê Ë

Ï

Ì

´

ª (7)

Identication is presently based uponÍØ

Ùn Ã Ù Ù Ùb Ú Á Û

Ì

( Ùb Ú

Ì Ü

) long excitation and responsesignals. A typical vibration response signal is de-picted in gure 2, whereas the corresponding iden-tication result (model-based one-step-ahead predic-tions and prediction errors) is, for a segment of thatsignal, presented in gure 3. As it may be readilyobserved, the model-based predictions practically co-incide with the system response and the prediction er-rors are very small.

4 The fault detection and identi-cation method

The Functional Model Based Method consists of twophases (also see [10, 11]): The rst (a-priori) phaseincludes the baseline modeling (via identication) of the healthy system’s dynamics, as well as the model-ing of each fault mode, for its continuum of fault mag-nitudes, via the novel class of stochastic functionalmodels .

The second (inspection) phase is performed pe-riodically during the system’s service cycle, and in-cludes the functions of fault detection, identication(localization), and fault magnitude estimation.

0 2 4 6 8 10 12 14 16 18 20−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

Time (sec)

D i s p l a c e m e t

( m )

Figure 2: System vibration displacement response.

−0.3

−0.2

−0.1

0

0.1

0.2

D i s p l a c e m e n t

( m )

2 2.5 3 3.5 4

−1

0

1

x 10−7

Time (sec)

R e s i d u a l s

( m )

(a)

(b)

Figure 3: (a) The actual ( —) and predicted ( - - -) dis-placement responses; (b) the corresponding predic-tion errors.

4.1 Baseline and fault mode modeling(a-priori phase)

Baseline modeling. A single experiment is per-formed, based upon which an interval estimate of



a NARX( Ý9 Þ ß Ý9 à ) dynamical model, of the form of equation (5) representing the healthy system’s dy-namics, is obtained.

Fault mode modeling. The notion of fault mode

refers to the union of faults of all possible magnitudes(severities) originating from a single physical cause.

For the modeling of a fault mode, a series of á

experiments are performed (either physically or viasimulation). Each experiment is characterized by aspecic fault magnitude â , with the complete seriescovering the range of possible fault magnitudes, say

ã

â ä å æ ß â ä ç è é , via a discretization ê ân ë ß âw ì ß í í í ß âw îð ï

(in the sequel it is tacitly assumed, without loss of generality, that the healthy system corresponds to

â ñ ò ). This procedure yields a series of excitation-

response signal pairs (each of length ó ):ô õ

ã ö

é ßw ÷

õ

ã ö

é ø

ö

ñd ù ß í í í ß ó` úû ø â¿ ñs ân ë ß âw ì ß í í í ß âw îð ú

(8)Based upon these, a proper mathematical descrip-

tion of the fault mode may be constructed in the formof a stochastic Functional Model (presently Func-tional NARX – FNARX – model). A FNARX model,being a generalization of a NARX model (5), is of theform:ü

ø ý úb þÿ ÷

õ

ã ö

é ñ¡

¢

õ

ã ö

é¤ £ ¥ ø ân ú§ ¦© ¨

õ

ã ö

é â

(9)

¥ ø ân ú = Þ

ë

ø ân ú í í í Þ

ä ç

ø ân ú

... à

ë

ø ân ú9 í í í à ä ø ân ú

¢

(10)

Þ å ø ân ú ñ " ! # $

ë

Þ å

# %& #

ø ân ú à å ø â ú' ñ " ! # $

ë

à å

# %& #

ø ân ú

(11)In these expressions â designates the fault magnitude,

õ

ã ö

é is dened analogously to

ã ö

é [equation (3)],ô õ

ã ö

é , ÷

õ

ã ö

é designate the corresponding measured ex-

citation and resulting response signals, respectively,and ¨

õ

ã ö

é the corresponding stochastic model resid-ual (one-step-ahead prediction error). For an accu-rate model, the residual sequence is zero-mean, un-correlated, with variance (

ì

)

ø ân ú , and uncrosscorre-lated with the corresponding excitation. Residual se-quences corresponding to different fault magnitudesare assumed uncrosscorrelated.

As equation (11) indicates, the AR and X parame-ters Þ å ø ân ú , à å ø â ú are modeled as explicit functions of the fault magnitude â , belonging to a 0 -dimensional

functional space spanned by the (mutually indepen-dent) functions %

ë ø ân ú ß í í í ß

%

!

ø ân ú ( functional basis ).The constants Þ å

# , à å

# designate the AR and X, respec-tively, coefcients of projection.

The FNARX model of equations (9)-(11), desig-nated as

ü

ø ý ú , is thus parametrized in terms of theparameter vector (to be estimated from the measuredsignals):

ý ñ

ã

Þ å

#

... à å

#

... (

ì

)

ø ân ú é

¢

ñ ý

¢ ... (

ì

)

ø ân ú

¢2 1¤ 3

ß 4 ß â

The model of equation (9) may be then re-writtenas:

÷

õ

ã ö

é ñ6 5

¢

õ

ã ö

é8 7 9

¢

ø ân ú @A £ ý ¦B ¨

õ

ã ö

é ñ¡ C

¢

õ

ã ö

é £ ý ¦ ¨

õ

ã ö

é

(12)with:

9 ø ân úD ñ

ã

%

ëø ân ú í í í

%

!

ø ân úw é

¢E

!G F

ë H

(13)

ýI ñ Þ

ë P ë

í í í Þ

ä ç P

!

... à

ë P ë

í í í à ä P

!

¢

E Q

ä ç R ä' S T

!G F

ë H

(14)and 7 designating Kronecker product [13, pp. 27-28].

For model parameter estimation, the FNARXequation (12) gives, following substitution of the data[equation (8)] corresponding to a single fault magni-tude â :

UVW

÷

õ

ã

ù é

...÷

õ

ã

óh é

X Y

`

ñ

UVW

C

¢

õ

ã

ù é

...C

¢

õ

ã

óð é

X Y

`

£ ýa ¦

UVW

¨

õ

ã

ù é

...¨

õ

ã

óð é

X Y

` (15)

ñc be d

õ

ñg f

õ

£ ýa ¦i h

õ (16)

Stacking together these expressions for thedata corresponding to the discrete fault magni-tudes ê ân ë ß â ì ß í í í ß âw îð ï considered in the experimentsyields:

dñ

g f" £ýa ¦i h (17)

with:

dpñ

UV

V

VW

d

ë

d

ì

...d

î

X Y

Y

Y

`

q r s tE u

î

F

ë H

fvñ

UV

V

VW

f¿ ë

f ì

...f

î

X Y

Y

Y

`

q r s tE u

î

F

Q

äb ç R ä S T

!

H

hBñ

UV

V

VW

h ë

hn ì

...h

î

X Y

Y

Y

`

q r s tE u

î

F

ë H

Parameter estimation (determination of the param-eter vector ý ) may be then based upon the OrdinaryLeast Squares (OLS) criterion:

w

ñ Trace xCovã

hn é ñ

î

y

õ

$

ë

u

y

$

ë

¨

ì

õ

ã ö

é (18)



in which Cov designates sample covariance of theindicated vector. This leads to the estimators:

6 A D ¤ c §

6

B

§

j

§

a

§

k j (19)

l

mc n

oG 8

l

n for (20)

Thel

estimator is asymptotically (z | {

)Gaussian distributed with mean coinciding with thetrue parameter vector and covariance matrix } ~ ,based upon which interval estimates of the true pa-rameter vector may be constructed [14].

4.2 Fault detection, localization, and es-timation (inspection phase)

Let k§ (8

) represent the excita-tion and response signals, respectively, obtained fromthe system in its current (unknown) state.

Fault detection. Fault detection may be basedupon the re-parametrized FNARX model of any faultmode. Toward this end consider the re-parametrized(in terms of , m

n

o , which are the parameters to be esti-mated) FNARX model corresponding to the stiffness

n

fault mode [notice that the basis functions and co-efcients of projection are those of the chosen faultmode model; compare with equations (9) and (12)]:

m§ n

o '

k§

¡

8

8 §

6 D

G

¤ (21)

The estimation of , m

n

o based upon the current ex-citation and response signals is achieved via the non-

linear regression (Nonlinear Least Squares – NLS) es-timator (realized via golden search and parabolic in-terpolation [15]):

l

B arg n

l

m

n

o

l

n (22)

This estimator may be shown [14] to be asymptot-ically (

{

) Gaussian distributed, with meanequal to the true (underlying) value, say G , andvariance m

n [l

¡ 6 G

m

n ]. This may be in turn

estimated as:

l

mc n

l

m

n

o

' G

l

v

n

j

c

l

m

n

o

'

8

8

l

n

j

§

(23)

with dened by equation (3) andl

designatingthe chosen fault mode’s vector of coefcients of pro-

jection [of the form of equation (14)].Since the healthy system corresponds to ,

fault detection may be based upon the hypothesis test-ing problem:

: G

¡

(No fault has occurred).:

¡

(A fault has occurred).

which (based upon the previous results) leads to thefollowing test at the

v G

risk level ( prob-

ability of type I error, that is rejecting if it is cor-rect):

Fault detection test ¡ G

l

G

l

m

c

a

is accepted(no fault is detected).

Elsec

a

is rejected(a fault is detected).

Fault identication. Once fault occurrencehas been detected, fault localization is based uponthe successive estimation and validation of the re-parametrized FNARX models [of the form of equa-tion (21)] corresponding to the various fault modes.The procedure stops as soon as a particular model issuccessfully validated; the corresponding fault modeis then identied as current .

Model validation may be based upon statisticaltests examining the hypothesis of excitation and resid-ual sequence uncrosscorrelatedness, as well as resid-ual uncorrelatedness. The latter is presently examinedvia the statistical hypothesis testing problem:a

: n

G

¡

(the fault mode is identied as current).: Some G ª

¡

(1© «' i ¬

(the fault mode is not the current one).

in which ª

«

8

¬

designates the residualseries normalized autocorrelation at lag

«

. It may beshown [16, p. 149] that the test statistic:

-

¡

ª

l

n

ª

¯ ®

« (24)

in which designates the residual signal length (innumber of samples),

l

G ª the sample normalized resid-ual autocorrelation, and

¬

the maximum lag, follows



−40 −20 0 20 405.842

5.843

5.844

5.845

5.846

5.847

k (%)

α 2

( k )

−40 −20 0 20 40−3.854

−3.852

−3.85

−3.848

−3.846

−3.844

−40 −20 0 20 400.949

0.95

0.951

0.952

0.953

−40 −20 0 20 402

3

4

5x 10

−11

α 4

( k )

k (%)

b 1 ( k )

k (%)

α 3

( k )

k (%)

Figure 4: Theoretical ( - - -) and FNARX estimated

(—) model parameters as functions of the fault mag-nitude ° (fault mode ± ² ³

²

).

Test Incurred FaultCase

I No fault (healthy system)II ± ²

³

´ ( µ ¶ reduction in stiffness °G · )III ± ²

³

· ¸

( ¹ º ¶ reduction in stiffness ° · )IV ±

² »

´ ¼ ( µ ½ ¶ reduction in stiffness °G ¾ )

Table 4: The four test cases.

a chi-square ( ¿

· ) distribution with À Ág Â degrees of freedom.

This leads to the test (at the Ã risk level):

Fault identication test Ä6 Å

¿

·

¾ Æ8 Ç8 È É Æ § ¾ Êc Ë Ì

¼ is accepted(fault mode is current).

ElseÊc Ë Ì

¼ is rejected

(fault mode not current).

Fault magnitude estimation. Once the currentfault mode has been determined, the interval esti-mate of the fault magnitude is constructed based uponGaussianity and the Í° ,

Í

Î

·

²

estimates [equations (22),(23)] obtained from the ÏÑ Ð °c Ò

Î

·

Ó Ô model [equation(21)] of the identied fault mode. Thus:

Fault magnitude interval estimate ( Ã

Ê

½8 Õ ½G Ö ):×

Í°a Á© Â Õ º Ø

Í

Î

²

Ò Í° Ù¡ Â Õ º Ø

Í

Î

² Ú

5 Fault detection and identica-tion results

5.1 Baseline and fault mode modeling(a-priori phase).

Baseline modeling. The identication of the baseline(healthy) system via a NARX( Û§ Ü¤ Ò ÛA Ý ) model of theform of table 3 has been discussed in section 3.

Fault mode modeling. Fault mode modeling ispursued only for the ± ² ³

²

fault mode characterized bychanges in the °G · stiffness (notice that the ± ² »

²

faultmode is not presently modeled). A total of Þ

Ê

Â ß

experiments, one corresponding to the healthy sys-tem ( °

Ê

½ ¶ variation in °

· ) and the rest corre-

sponding to various fault magnitudes (faults± ² ³

²

with°¡ à6 á ÁD µ ¹G Ò µ ¹ ¶a â ; increment ã °

Êå ä

¶ ), are carriedout. The signals obtained are, in all cases, ¹8 Ò ½ ½ ½

æ

Ü ça èc é ê

æ long.The FNARX modeling procedure [14] leads to a

FNARX( Û§ Ü¤ Ò Û§ Ý ) ± ²

³

²

fault mode model, with func-tional basis consisting of the rst two (0th and 1stdegree, thus è

Ê

¹ ) Chebyshev Type II polynomials[17]. The theoretical and FNARX-based estimates of certain of the model parameter trajectories (as func-tions of the fault magnitude ° ) are compared in gure

4, from which excellent agreement is observed.

5.2 Fault detection and identication(inspection phase).

Four test cases, as indicated in table 4, are presentlyconsidered via Monte Carlo experiments ( Â ½ runs percase).

Monte Carlo fault detection results are pictoriallypresented in gure 5, fault identication (localization)results in gure 6, and a summary of the fault mag-

nitude estimation results (averages overÂ ½

runs percase) is presented in table 5. In all statistical tests theselected risk level is Ã

Ê

½8 Õ ½ Ö . Comments on eachtest case follow.

Test Case I (healthy system). In this case the faultmagnitude interval estimate includes the °

Ê

½ valuein each one of the Â ½ runs [gure 5(a)], thus no faultis (rightly) detected. In addition, the value of the

Ä

statistic is, for all Â ½ runs, below the critical point [g-ure 6(a)]. The excellent accuracy of the ° estimates isconrmed by the average (over the Â ½ runs) point and

standard deviation estimates presented in table 5.Test Case II (fault ±

²

³

´ – µ ¶ reduction in the °G ·

stiffness) . This is a small magnitude fault, yet fault de-tection is accurate in all Â ½ runs [the fault magnitude



−0.2

0

0.2

2.5

3

3.5

28.5

29

29.5

1 2 3 4 5 6 7 8 9 100

0.2

0.4

0.6

Monte Carlo experiment

F a u l t m a g n .

k ( % )

F a u

l t m a g n .

k ( % )

F a u l t m a g n .

k ( % )

F a u

l t m a g n .

k ( % )

(a)

(b)

(c)

(d)

Figure 5: Fault detection results: (a) Test case I (healthy system); (b) test case II (fault ë ì í

î ); (c) test case III(fault ë ì

í

ï ð ); (d) test case IV (fault ë ì ñ

î ò ) [10 Monte Carlo runs per case; the solid horizontal lines designate truefault magnitude, the circles corresponding point estimates, and the boxes interval estimates at the ó" ô õG ö õG ÷

level].

Test Case Fault True Fault Average Point Average Standard

Magnitude øG ùD ú û ü Estimate ýø Deviation Estimateý

þ

ì

Ië

ì

í

ò

0ÿ¡

G ö ÷¢¤ £¦ ¥

õ§

î

7.20£ ¥

õ© §

ï

II ë ì í

î 3 3.02 7.12 £ ¥õ© §

ï

III ë ì

í

ï ð 29 29.02 7.18 £ ¥õ

§

ï

Table 5: Fault magnitude estimation results (averages over 10 Monte Carlo runs).

interval estimates do not include the ø ô¡ õ value; g-ure 5(b)]. The ë ì

í

ì

fault mode is also correctly iden-tied, as the value of the statistic is, for all ¥ õ runs,below the critical point [gure 6(b)]. The excellent

accuracy of the ø estimates is, once again, conrmedby the average (over the ¥ õ runs) point and standarddeviation estimates presented in table 5.

Test Case III (fault ë ì í

ï ð – ¢ û reduction in the ø

ï

stiffness) . This is a larger magnitude fault, the detec-tion of which is also without problems in all ¥ õ runs[the fault magnitude interval estimates do not include

the ø ô6 õ value; gure 5(c)]. Fault mode identica-tion is also accurate, as the value of the statistic isbelow the critical point for all ¥ õ runs [gure 6(c)].



0

20

40

60

Q

− s t a t i s t i c

0

20

40

60

0

20

40

60

1 2 3 4 5 6 7 8 9 100

1000

2000

Monte Carlo experiment

Q − s t a t i s t i c

Q − s t a t i s

t i c

Q − s t a t i s t i c

(a)

(b)

(c)

(d)

Figure 6: Fault identication results: statistic (bars) and the critical point (- - -) at the © level (thefault mode " ! #

!

is identied as current if is lower than the critical point). (a) Test case I (healthy system); (b)Test case II (fault " ! #

$ ); (c) test case III (fault " ! #

% & ); (d) test case IV (fault ' ! (

$ ) ) [10 Monte Carlo runs per case;0

2 1© ].

The accuracy of the 3 estimates is similarly excellent(table 5).

Test Case IV (fault ' ! (

$ ) – 4 5 reduction in the3© 6 stiffness). This is a somewhat different case, as thefault considered does not belong to the modeled ' ! #

!fault mode (for this reason fault magnitude estimationis not addressed).

Yet, the obtained fault detection results are verygood, as the fault magnitude interval estimates do not,in all 7 runs, include the 38 9 value [gure 5(d)].Moreover, the fault mode identication results de-nitely suggest that the present fault does not belongto the !

#

!

mode, as the value of the statistic is farabove the critical point for all 7 runs [gure 6(d)].

6 Conclusions

This paper was concerned with system identicationand fault detection in a two DOF nonlinear systemcharacterized by cubic stiffness. System identica-tion was based upon Nonlinear ARX (NARX) mod-els, while a novel Functional Model Based Method,using Functional NARX (FNARX) models, was, forthe rst time, employed for fault detection, identica-tion, and fault magnitude estimation within the con-text of a nonlinear system.

The results of the study conrmed: (a) The effec-tiveness and accuracy of NARX based identication

for the system at hand; (b) the effectiveness and accu-racy of the Functional Model Based Method for tack-ling the combined problem of fault detection, identi-cation, and fault magnitude estimation.



The Functional Model Based Method was specif-ically demonstrated to accurately detect, identify (lo-calize), and estimate even small magnitude faults (afault as small as @ A stiffness reduction was consid-

ered) in the presence of stochastic uncertainty andonly two measured signals.

Acknowledgements

The authors acknowledge the nancial support of thisstudy in part by the General Secretariat for Researchand Technology – Greece and the European SocialFund (PENED99 Project #580), and in part by theEuropean Commission (Growth Project GRD1-2000-

25261 – ADFCSII).

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2002 cubic stiffness isma

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