and magnitude estimation in railway vehicles is tech- nologically ..... This may be in turn estimated as: & ÑÂ¥ Ñ
& Ñ. '3. 1. 3. Рдж Ñ· bc збйв c. ·ed Ñ. Â¥ Ñgf. Â¥#. ·.
On board fault detection and identification in railway vehicle suspensions via a Functional Model Based Method J.S. Sakellariou, K.A. Petsounis, S.D. Fassois Stochastic Mechanical Systems (SMS) Group Department of Mechanical & Aeronautical Engineering University of Patras, GR 265 00 Patras, Greece e-mail: sakj,kpetsoun,fassois @mech.upatras.gr internet: http://www.mech.upatras.gr/ sms
Abstract The combined problem of fault detection, identification, and magnitude estimation in railway vehicle suspensions is tackled via a novel Functional Model Based Method. This method is based upon a new class of stochastic functional models, which are capable of accurately representing the system in a faulty state for its continuum of fault magnitudes, as well as statistical decision theory tools. The method’s effectiveness in properly detecting, identifying, and estimating the magnitude of suspension faults, by using only two measured vibration signals, is demonstrated via Monte Carlo experiments with a six degree-of-freedom railway vehicle model.
1 Introduction The combined problem of fault detection, identification (localization, that is fault type determination), and magnitude estimation in railway vehicles is technologically important, as prompt fault detection and assessment may lead to improved safety, better dynamic performance, as well as improved comfort and maintenance. Fault detection in railway vehicles has been the subject of considerable research in recent years. The methods commonly employed are based on ultrasonics [1, 2], electromagnetic ultrasounds [3], acoustic wayside detectors [4, 5], and “excessive” heat detectors [6]. Vibration based methods have, thus far, received limited attention [7]. Yet, they form an interesting and promising alternative, as they are economical, work at a high (“system”) level, may be automated, and may offer the important advantage of continual, on board, operation. Two such methods, based upon non-parametric statistics and operating on a single vibration signal, were recently introduced by the present authors [7]. Both have been experimentally confirmed to be successful in detecting suspension faults by detecting statistically significant changes in the measured vibration signal’s sample second order moment. Yet, they are limited to fault detection (no fault identi-
fication/localization or fault magnitude estimation), may be sensitive to track and vehicle operating conditions, and may have difficulty in detecting “small” (incipent) faults. The goal of the present study is the introduction of a vibration based method, suitable for on board implementation, that can: (i) Achieve fault detection, as well as identification/localization and fault magnitude estimation, (ii) exhibit sufficient sensitivity to “small” (incipient) faults, (iii) exhibit robustness (insensitivity) to track and vehicle operating conditions, (iv) account for the presence of random measurement noise and uncertainty, and (v) maintain implementation simplicity. These objectives are met by a Functional Model Based Method utilizing the novel class of stochastic functional models. These models are characterized by the unique ability to accurately represent the system in a faulty state for the state’s continuum of fault magnitudes [8]. The problems of fault detection, identification, and fault magnitude estimation are then properly formulated within this framework, and tackled via statistical decision theory tools. In order to achieve the required sensitivity to “small” (incipient) faults and robustness to track and vehicle operating conditions, the method is designed to operate on the mathematical (transfer function) relationship between two measured signals.
International Conference on Noise and Vibration Engineering September 16-18, 2002 - Leuven, Belgium
T UV
Q R S
WYX Z [ \ ] ^`_ a b c d e f g h i j
Ã Ä Å
n op ¡£¢ ¤ ¥
© ª « ¬® ¯ ° ± ² ³ ´ klm
Æ Ç È É Ê Ë
% & ' ()*+ , - . / 01
! " # $
x yz w
µ ¶ · ¸`¹ º » ¼ ½ ¾ ¿ qrs
Ì ÍÎ Ú Û Ü Ý`Þ ß à á â ã ä ¦§¨
GH J IKL MNOP
À ÁÂ
>? @ABC
F ED
2 9 3 : 4 5 ; 6< 7= 8 t uv Ï Ð ÑÓÒ Ô Õ×Ö Ø Ù å æ ç è®é ê ë ì í î ï ðñò { | } ~
ó®ô õ ö ÷ ø ù ú®û ü ý þ ÿ
Figure 1: Simplified model of one-half of a railway vehicle.
2 The vehicle model, the faults, and the FDI unit
the track vertical velocity input being approximated by Gaussian white noise with spectrum equal to:
! " #%$
2.1 The vehicle model A simplified six degree-of-freedom model of one (left or right) half of a railway vehicle is depicted in figure 1. The vehicle is assumed to run on an horizontal track with constant speed . The model assumes symmetrical loading of the two rails (small roll angle) and no wheel lift. The car is modeled as a rigid body with two degrees of freedom (vertical displacement and pitch angle ), and is connected to the bogies located at its front and rear ends (leading and trailing bogies, respectively) via the secondary suspension. The secondary suspension’s elements (physically realized via air chambers and hydraulic dampers) are indicated as , (leading part), and , (trailing part). Each bogie is modeled as a rigid body with two degrees of freedom (vertical displacement or , and pitch angle or ), and is connected to two wheelsets (modeled as massless point followers [9]) via the primary suspension. The primary suspension is modeled as linear spring-dashpot elements, which are indicated as , , with designating the corresponding wheelset (figure 1). The track is assumed to be fixed and rigid, with
"
&
$ ' (*)
(1)
with representing the (horizontal) vehicle velocity (*) and a roughness factor indicative of track quality [10]. The numerical values of the system parameters, corresponding to a typical passenger vehicle of the Hellenic Railways southern network (MAN OSE/KAT4, see [7]), are provided in Appendix A. The system’s linearized equations of motion are summarized in Appendix B. System simulation is based
upon time-discretization with time step equal to +
, - , , ,6587 /. 0 1 (frequency of 2 3 4 ).
2.2 The faults Two types of faults (fault modes) are considered (table 1 and figure 1): The first mode corresponds to stiffness reduction in the secondary suspension’s trailing
airspring . Each such fault is represented as 9 : , with the subscript ; indicating the exact fault magnitude (stiffness reduction). Reductions in the range ,%- - - < , = ; are considered. The second mode corresponds to reduction in the / > primary suspension stiffness (fourth wheelset).
Fault Mode ?/@
A
Description Secondary suspension: reduction@ in C the trailing airspring stiffness B (reduction values DEGF%H H H I F J ) Primary suspension: reduction in the KM fourth wheelset stiffness B (reduction value DEGI J )
40
Magnitude (dB)
?LK A
20 0 80
−20
The fault detection and identification unit is designed to operate upon two vibration signals: The track vertical velocity profile N (which, due to the negligible wheel mass, coincides with the vertical velocity of any wheelset), and the vehicle body vertical acceleration at point A, say P O Q (right above the trailing airspring, figure 1). This design essentially aims at monitoring the measured signals’ linearized mathematical (tranfer function) relationship in order to achieve sufficient sensitivity to “small” (incipient) faults, as well as robustness to track and vehicle operating conditions. The transfer function between P O Q and N may (with the track velocity excitation being accounted for as acting on all four wheelsets with the proper delays included) be shown to be of the form:
RTS U V E
WO Q S U V U Z[ R]\ S U V_^Y` R K SU Vc XS U VYE H a/b
(2)
U R\ S U V with designating the Laplace variable, and , R K SU V
the transfer functions relating the car body’s vertical and pitch angle displacements, respectively, to the track velocity input. Figure 2 presents the transfer function’s theoretical Frequency Response Function (FRF) magnitude ? @ a function of both fault magnitude D (fault mode as A , stiffness reduction in the secondary suspension’s trailing airspring) and frequency. It is evident that the FRF magnitude is not particularly sensitive to small magnitude faults, but it does change (particularly in the lower frequency range) as the fault magnitude increases.
(%
) 15 10 Frequency (Hz)
20 5
itu
20
de
−
k
40
ag n
−60 25
indicating fault magnitude (only one fault, of magnitude DEGI %, is presently considered).
2.3 The fault detection & identification (FDI) unit
60
−40
0
0
tm
?*K Each such fault is represented as A , with D similarly
Fa ul
Table 1: The considered fault modes.
Figure 2: Theoretical model based FRF magnitude ? @ versus frequency and fault magnitude D (fault mode A ).
3 The fault detection and identification method The Functional Model Based Method consists of two phases: The first (a-priori) phase includes the baseline modeling (via identification) of the healthy system’s (nominal) dynamics using a customary discrete-time model, as well as the modeling of each fault mode, for its continuum of fault magnitudes, via the novel class of stochastic functional models. The second (inspection) phase is performed on board, during the system’s service cycle, and includes the functions of fault detection, identification, and fault magnitude estimation.
3.1 Baseline and fault mode modeling (a-priori phase) Baseline modeling. A single experiment is performed, based upon which an interval estimate of a discrete-time dynamical model representing the healthy system dynamics is obtained via standard identification procedures [11, 12]. In the present study a single-excitation single-response discretetime AutoRegressive with eXogenous excitation (ARX) model [11] of the following form is used1 :
o o n o n o dfe S g%V%h P [ i c jlk m [ i ^/t c k u [ i ^/t c j]x [ i c bP E e v bw p_qsr p
(3) 1
Lower case/capital bold face symbols designate vector/matrix quantities, respectively.
with y designating normalized discrete time (y{z | } ~ } } |
s , with absolute time being y , where
s stands for the sampling period), y , s y the measured excitation (track velocity) and response (acceleration at A) signals, respectively, _ , _ the AutoRegressive (AR) and eXogenous (X) orders, respectively, and y the stochastic model residual (onestep-ahead prediction error) which should, for an accurate model, form a zero-mean uncorrelated, and uncorrelated with y , sequence with variance . The model is parametrized in terms of the parameter vec} } tor {z , which is to be estimated from the measured signals. Fault mode modeling. The notion of fault mode refers to the union of faults of all possible magnitudes (severities) originating from a (properly defined) single physical cause or mechanism. For the modeling of a fault mode, a series of experiments are performed (either physically or via simulation). Each experiment is characterized by a specific fault magnitude , with the complete series covering the range of possible fault magnitudes, say } } } } % , via a discretization ¢¡ (it is tacitly assumed, without loss of generality, that the healthy structure corresponds to 8z£ ). This procedure yields a series of excitation-response signal pairs (each of length ¤ ):
} ¥ y
¥ y ¦ yz
| } } ¤
} } } z3
(4) with the subscript designating the corresponding fault magnitude. Based upon these, a proper mathematical description of the fault mode may be obtained in the form of a stochastic Functional Model (presently Functional ARX – FARX – model). FARX models are introduced for the first time [13], and may be thought of as generalizations of the conventional ARX form of eq. (3). They are defined as:
§
¨ z
%©
« ® ¬¯
¥ y ª
_
« ¬_
_
¥ y_T sz (5)
¥ y8 ªY ¥ y °²±²³
%´ « ¸ · } ´ « * ¸ · z¶µ · ¹ z µ · · ¬s ·_ ¬
with _ , s designating the AR and X orders, respectively, the fault magnitude (presently percentage stiffness reduction), ¥ y , ¥ y the corresponding measured excitation and resulting vibration response, respectively, and ¥ y the corresponding stochastic model residual (one-step-ahead prediction error). For an accurate model the residual sequence should be zero-mean, uncorrelated, with variance s , and uncorrelated with the corresponding excitation. Residual sequences corresponding to different fault magnitudes are considered uncrosscorrelated. As eq. (6) indicates, the AR and X parameters , are modeled as explicit functions of the fault magnitude belonging to a º -dimensional functional space spanned by the (mutually independent) ¸ } }¸ (functional basis). The functions constants · , · designate the AR and X, respecµ tively, coefficients of projection. The FARX model of eqs. (5), (6), designated as § ¨ , is parametrized in terms of the parameter vector (to be estimated from the measured signals): . . ´ }½ } ¨ zl · .. · .. ¼» The FARX model [eq. (5)] may be re-written as:
¥ y z¿¾ À ¥ y ÁYÂ
¨%ª ¥ y zÅÄ ¥ y % ¨ ª ¥ y (7)
with: . ´ | À ¥ y zÇÆ * ¥ y_ ¥ yTs .. 8
É Ê .. . ¥ y ¥ y8_ È *´ ¸ ! ¸ Â zl ¨ z
´
Æ Ï
(8)
Ë ® Ë_ Ì Í Î
É ! Í Î (9) µ É Ê ¯ Ï ®Ï È Ë ® Ë_ Ì Í Î µ (10) µ
µ
. Ï ..
µ
and Á designating Kronecker product [14, pp. 27-28]. For model parameter estimation, the FARX equation (7) gives, following substitution of the data [eq. (4)] corresponding to a single fault magnitude :
ÐÑÒ
| ÓÔ ¥ .. Õ . ¥ ¤² zsÖ¶× ¥
(6)
Ã!
ÐÑÒ
| ÓÔ Ä ¥ .. z Õ ¨]ª . Ä ¥ ¤² zÙØ¥ ¨]ªÛÚ ¥
ÐÑÒ
| ÓÔ ¥ .. Õ . ¥ ¤¢
(11)
Stacking together these expressions, for the data corresponding to the discrete fault magnitudes
Ü Ý Þ ß Ý à ß á á á ß Ý â¢ã
Ýß ,+ M
considered in the experiments,
ä8åÅæç è]éYê
yields:
à
å -
(12) with:
ä
æ
ðñ ä Þ ññ à ññ ñ
ä6ë ìíîî
æ æYë ì]îîí îî æ îîï ² â õ ÷ø ù óô
îî ä ... îïî â ò õ Þö ó ô â¢
à
Þ
ê
ð ññ ññ ññ
.. .â
ê
ê%ë ìíîî
à
Þ ð ññ ññ ññ
îî ê ... îîï â ò ó ô â¢õ Þ ö
ò
ú ûù ü û_Þ ý ö
è estimation (determination of the paramParameter eter vector ) may be then based upon the Ordinary Least Squares (OLS) criterion: å þYÿ
Trace Cov
â
êså
Þ ì
ç
ô
à Þ
ì
(13)
in which Cov designates sample covariance of the indicated vector. This leads to the estimators (estimators/estimates are indicated by a hat):
Þ è¢
å æ _æ ç æ _ä8å
å
â ì
ô
Þ
ì
å à Ý è
Þ
!
Þ
ç
â
ô ì
Þ
Þ ì
à
for
Ý
ô
å
ì
Þ
!#"%$
3.2 Fault detection, identification, and magnitude estimation (inspection phase) å ß ß * ß á á á ß ! Let ) ( ) represent the excitation and response signals, respectively, obtained from the system in its current (unknown) state. Fault detection. Fault detection may be based upon the re-parametrized FARX model of any fault à end consider the re-parametrized mode. Toward Ý this (in terms of , , which are to be estimated) model corresponding to any particular fault mode (notice that the basis functions and coefficients of projection are those of the chosen fault mode model):
Þ.
Ý
Ý
ç
ç
/10
)
/10
å
é
à å !
ô ì
Þ
à
(17) This estimator may be shown [13] to be asymptoti!#"8$ cally ( ) GaussianÝ distributed, Ý: with 9 mean equal à toà the true (underlying) value, say , and variance
;8= Ý:9 ß [Ý< ]. This may be in turn estimated as:
à ô ! ! Þ > ì à ô ! ! Þ> ì
à å
(14)
The estimator is asymptotically ( ) Gaussian distributed with mean coinciding with the true parameter vector and covariance matrix &(' , based upon which interval estimates of the true parameter vector may be constructed [13].
ù ü ì -
ô à arg 465 7 Þ ì
å Ý ÿ
(15)
-
à (16) Ý The estimation of , based upon the current excitation and response signals is achieved via the Nonlinear Least Squares (NLS) estimator (realized via golden search and parabolic interpolation [15]):
Ý Þ ß á á á ß Ý â
ù ú
ì23
å
é
Þ à
? ÝA@ C ì B à Þ ?
:E Ý ç è @ ?GF ÝIHH ìB D H ? H
(18)
è
with D being defined analogously to eq. (8), and designating the chosen fault mode’s vector of coefficients of projection [of the form of eq. (10)]. Ý å8J Since the healthy structure corresponds to , fault detection may be based upon the statistical hypothesis testing problem:
K
2: K Þ :
åLJ Ý 9 åLJ Ý 9M :
(No fault has occurred). (A fault has occurred).
J J O results, leads J J O to the which, based upon the å previous á á following test at the N risk level ( probK ability of type I error, that is rejecting 2 if it is correct):
åT Fault detection test (N
P Ý PQ# á R S
) 2 is accepted (no fault is detected). K 2 is rejected (a fault is detected).
K
åT Else
åLJ J O á
Fault identification. Once fault occurrence has been detected, fault localization is based upon the successive estimation and validation of the reparametrized FARX models [of the form of eq. (16)]
in which Wbi eCY%j k l:k m m m k g1h designates the residual series normalized autocorrelation at lag e . It may be shown [16, p. 149] that the test statistic:
n
YLop^ i orq
