Vibration Based Fault Diagnosis for Railway Vehicle ... - CiteSeerX

Vibration Based Fault Diagnosis for Railway Vehicle Suspensions via a Functional Model Based Method: a feasibility study∗ J.S. Sakellarioua,⋆ , K.A. Petsounisb , S.D. Fassoisa a Stochastic Mechanical Systems & Automation (SMSA) Laboratory, Department of Mechanical & Aeronautical Engineering University of Patras, GR 265 04 Patras, Greece Internet: http://www.smsa.upatras.gr b Mentor Hellas LTD, 53 Marathonos Avenue, Pallini 153 51, Greece September 4, 2014

Abstract The design of a vibration based Fault Detection and Isolation (FDI) unit that may tackle the combined problem of fault detection, isolation (or identification) and magnitude estimation (collectively known as fault diagnosis), in railway vehicle suspensions is presented. The unit is initially “trained” in a baseline phase based on data obtained from a simplified physics-based model of a railway vehicle suspension. Fault diagnosis is subsequently achieved in an inspection phase through a single, properly preselected, pair of vibration signals acquired from the vehicle, and a recently introduced data-based method, referred to as the Functional Model Based Method (FMBM), without resorting on the physics-based model of the baseline phase. The method’s cornerstone is the novel class of stochastic ARX-type models which are capable of accurately representing a system in a faulty state for its continuum of fault magnitudes. Fault diagnosis feasibility in a railway vehicle suspension is demonstrated via Monte Carlo simulations using different types and magnitudes of faults in the physics-based model and generating vibration signals corresponding to the healthy and faulty suspension. Two vibration signals are used by the diagnosis unit: the track velocity profile and the vehicle body acceleration above the trailing airspring. Fault diagnosis based on the FMBM is effective in a compact and unified statistical framework accounting for experimental and modelling uncertainty through appropriate interval estimates and hypothesis testing procedures. The unit is shown to exhibit high sensitivity and accurate estimation of even very small fault magnitudes, to detect and isolate unknown faults for which it has not been trained, and to be robust to high measurement noise, car body mass variations, and varying track irregularity.

Keywords: fault detection and isolation, fault diagnosis, railway vehicle suspensions, vibration based method, data based method, statistical time series method, functional models

[No. of pages: 22; No. of figures: 9; No. of tables: 6]

c 2014 by J.S. Sakellariou, K.A. Petsounis and S.D. Fassois. All rights reserved.

author. Tel/Fax: (++ 30) 2610 969 494, 2610 969 495 / (++ 30) 2610 969 492, 2610 969 495. E-mail addresses: [email protected] (J.S. Sakellariou), [email protected] (K.A. Petsounis), [email protected] (S.D. Fassois). ∗ Copyright

⋆ Corresponding

Journal of Mechanical Science and Technology, Springer, 2014, in press.

Fault Detection and Identification in Railway Vehicle Suspensions

2

1 Introduction The combined problem of fault detection, isolation (fault type determination or identification), and magnitude estimation, collectively referred to as fault diagnosis, in railway vehicles is technologically important as prompt fault detection and assessment may lead to improved safety, better dynamic performance, reduced maintenance costs as well as improved riding comfort and quality. Fault detection in railway vehicles has been the subject of considerable research the last two decades [1]. Many methods which are based on ultrasonics [1, 2, 3, 4], electromagnetic ultrasounds [5], acoustic wayside detectors [1, 6, 7], and “excessive” heat detection systems [1, 8] have been presented for fault detection in wheels, axles and roller bearings. However limited work has been done for fault detection and isolation (FDI) in railway vehicle suspensions. Faults in the dampers and springs of the primary and secondary suspension has been studied through methods which are based on vibration acceleration signals collected from one up to twelve locations on the vehicle. These may be divided in two main categories: (i) the methods which utilize vehicle physics-based models and, (ii) the data-based methods which achieves FDI through signal processing techniques applied on vibration acceleration responses measured on the vehicle without resorting on physics-based models. The first category includes methods that combine an analytical vehicle model with Kalman filter techniques for building and monitoring a sensitive to damage “residual” sequence [9, 10, 11, 12, 13]. Such methods have been used successfully for FDI in simulations with faults corresponding to 20% (or greater) reduction in the damper and/or spring characteristics. In principle these methods have the potential to be very effective but in practice building accurate physics-based vehicle/suspension models is often difficult as the complete set of railway parameters are not available as well as the overall vehicle’s complexity does not allow for a precise model. Additionally the measurement and modelling uncertainties are accounted for only in the extent of certain types of track irregularities or low level measurement Gaussian white noise of 1% in terms of the noise to signal variance ratio. Other issues of these methods such as the selection of the state vector and initial values, the Extended Kalman Filter (EKF) algorithm convergence [10] and the increased computational complexity that requires high CPUs power for online implementation constitute further difficulties. On the other hand the second category of methods are simpler as they don’t utilize vehicle analytical models and need only some vibration, commonly acceleration, signals acquired from the vehicle. These signals are appropriately processed for the extraction of the necessary information/feature the monitoring of which may lead to the detection of a potential fault. This processing include Principal Component Analysis (PCA) and/or Canonical Variate Analysis (CVA) [12], the estimation and monitoring of the discrepancies in the signals sample second order moment [14], the monitoring of the cross correlation of two signals at different locations on the vehicle [15, 16] as well as the monitoring of the vehicle’s natural frequencies and damping ratios estimated based on the available signals [17]. These methods have been mainly used for fault detection in simulations with faults corresponding to 20% (or greater) reduction in the damper and/or spring characteristics outperforming in some cases the more advanced previous category of methods [12]. The fault isolation subproblem based on a method of this category is only presented through a feasibility study of a recently introduced conditioned monitoring scheme that detects


3

the instability of high speed railway bogies and classify the occurred faults to predetermined fault types, based on Neural Networks and changes on the vehicle’s natural frequencies and damping ratios [17]. Additionally it is noted that methods of this category have been also used for fault detection in wheels, axles and roller bearings [1, 18, 19, 20, 21]. The goal of this study is the design and feasibility assessment of an FDI unit that may tackle the combined problem of fault detection, isolation and magnitude estimation in railway vehicle suspensions via a data-based method (no need for physics based models during the unit’s operation / inspection phase). The unit achieves fault diagnosis based, for the first time, on a recently introduced data-based method, the Functional Model Based Method (FMBM) [22], that operates in a unified and compact statistical framework that accounts for experimental and modelling uncertainty through few (even a single pair) vibration signals from a simplified railway vehicle model. Some early results were presented in a preliminary study [23]; several improvements on issues such as the unit’s sensitivity to incipient faults and its robustness to experimental and operational uncertainties are presently implemented. The FMBM is based on a very special class of AutoRegressive with eXogenous input (ARX) type [24, pp. 81-93] Functional Models and it is characterized by some important advantages: (i) FDI is achieved based on data-based, partial system models, identified exclusively from measured vibration signals, (ii) there is no need for physics-based, state space, finite element or other type of analytical model in the inspection phase where the system state is unknown and fault diagnosis shall be achieved, (ii) a minimal number of vibration signals may be adequate and, (iii) all types and admissible magnitudes of faults may be identified without resorting to any classification at predetermined fault types and/or magnitudes. On the other hand the method’s performance depends on the quality of the identified partial system functional models that represent the railway vehicle under different types of faults in a continuum range of fault magnitudes necessitating caution and expertise in the identification procedure. Furthermore, from its nature as a data-based method [25, 26], it needs data from the healthy and faulty vehicle for its training in a baseline phase. This is the main practical difficulty of most data-based methods as data from the faulty system is not often available. Such vibration signals may be acquired from an in scale prototype of the actual system [22], or from the real vehicle by inserting “safe” fault types and magnitudes as it is for instance the reduction of the air pressure in an airspring, the loosening of bolts on the mounting of a damper and so on (also see [14]), or through a physics-based model such as a finite element model [27] and so on. In the present study data signals corresponding to the healthy and faulty vehicle under various types and magnitudes of faults in the damping and stiffness characteristics are obtained through a simplified physics-based model of a railway vehicle suspension – which corresponds to a passenger vehicle of the Hellenic Railways (Fig. 1) – that is exclusively developed for this purpose. The assessment of the designed FDI unit is demonstrated based on vibration signals which are obtained by inserting various types and magnitudes of faults in the physics-based model and 20 Monte Carlo simulations per case. Furthermore the unit’s robustness is examined under high measurement noise, car body mass increment and different levels of track irregularity. The rest of this article is organized as follows: The physics-based railway vehicle model is presented in Section 2. The considered faults and the FDI unit are presented in Section 3. A short overview of the Functional Model Based Method is presented in Section 4. The fault diagnosis results are presented in Section 5 and the conclusions


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Figure 1: Hellenic Railways southern network type MAN OSE/KAT.4 passenger train. of the study are summarized in Section 6.

2 The Railway Vehicle Model In this section the development of a simplified physics-based model of the railway vehicle that is exclusively used for vehicle’s simulation and data generation corresponding to the healhty and faulty vehicle under various types and magnitudes of faults – used for the training of the FMBM (also see section 4) – is presented. The six degreeof-freedom model of one (left or right) half of a vehicle is depicted in Fig. 2. The vehicle is assumed to run on an horizontal track with constant speed u. The model assumes symmetrical loading of the two rails (small roll angle) and no wheel lift. The car is modelled as a rigid body with two degrees of freedom (vertical displacement yb and pitch angle θb ), 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 Ksl , Csl (leading part), and Kst , Cst (trailing part). Each bogie is modelled as a rigid body with two degrees of freedom (vertical displacement ytl or ytt , and pitch angle θtl or θtt ), and is connected to two wheelsets (modelled as massless point followers [28]) via the primary suspension. The primary suspension consists of coil springs and shock absorbers modelled as linear spring-dashpot elements, which are indicated as Kpi , Cpi , with i = 1, 2, 3, 4 designating the corresponding wheelset (see Fig. 2). The track is assumed to be fixed and rigid, with the track vertical velocity input being approximated by Gaussian white noise with spectrum equal to: Svv (ω) = (2π)2 Ar u

(1)

with u representing the (horizontal) vehicle velocity and Ar a roughness factor indicative of track quality [29]. The model parameters, corresponding to a typical passenger vehicle of the Hellenic Railways southern network (MAN OSE/KAT.4, Fig. 1, also see [14]), are provided in Appendix A. The vertical dynamics of the vehicle model are, for small displacements, described by the linear differential


5

u Lc FDI UNIT

MB, ΙB

Car body θb(t)

A * yb(t)

Cst

Kst

Ksl

Secondary Suspension

Output Acceleration

Csl

FkKst Mtt, Ιtt

Mtl, Ιtl

Trailing bogie

Leading θtl(t) bogie

θtt(t)

Kp4

Cp4

Kp3

Cp3

4th wheelset

Kp2

3rd wheelset

2nd wheelset

Cp2

Kp1

FkCp1

Cp1

1st wheelset

yw3(t) yw4(t)

Lbw

Primary Suspension

ytl(t)

ytt(t)

FkKp4

yw1(t)

L

yw2(t) Velocity Input

Figure 2: The model of the one-half railway vehicle and the FDI unit. equation∗: ¨ (t) + C · y(t) ˙ M·y + K · y(t) = B · xw (t)

(2)

with y(t) = [ yb (t) θb (t) ytl (t) θtl (t) ytt (t) θtt (t) ]T designating the displacement vector (see Fig. 2), xw (t) the input vector, B an input shaping matrix, and M, C, K the mass, damping, and stiffness matrices, respectively. It should be noted that the elements xwi (t) (i = 1, 2, 3, 4) of the input vector xw (t) are linear combinations of delayed versions of the track vertical displacement x(t) and velocity v(t) [= x(t)], ˙ that is: ywi (t)

xwi (t) = Kpi

z }| { x(t − di ) + Cpi v(t − di )

(3)

with di designating the delay between the time instants at which the first and the i − th wheelset pass over a specific track point (see Fig. 2). These time delays are: di = li /u

(i = 1, 2, 3, 4)

(4)

where li is the distance of each wheelset from the first one. Thus:

∗ Lower

d1

=

0

d2

=

2Lbw /u

d3

=

2L/u

d4

=

2(Lbw + L)/u

case/capital bold face symbols designate vector/matrix quantities, respectively.

(5)

x


6

Table 1: The considered fault modes. Fault Mode Description Kst Fk Secondary suspension trailing airspring stiffness Kst FkKp4 Primary suspension fourth wheelset stiffness Kp4 FkCp1 Primary suspension first wheelset damping Cp1

System simulation is based on time-discretization of the dynamics and sampling with a period Ts = 0.004 s (sampling frequency fs = 250 Hz).

3 The Fault Modes and the FDI Unit 3.1 The fault modes The notion of fault mode refers to all admissible fault magnitudes of a certain fault type. Faults from three different fault modes are investigated in the present study (also see Table 1 and Fig. 2). The first fault mode corresponds to stiffness changes in the secondary suspension’s trailing airspring Kst , and physically corresponds to loss of air (softening) from the secondary air chamber (airsping) due to a malfunction or due to changes in the environmental temperature that may lead to airspring softening or stiffening (air expansion at high temperatures). Each such fault is represented as FkKst , with the subscript k indicating the exact fault magnitude (stiffness changes in the range k = −16 . . . 80 % are presently considered; negative/positive values indicate stiffening/softening, respectively). The second fault mode corresponds to stiffness changes in the primary suspension’s elasticity Kp4 (fourth wheelset). This fault may arise when the coil spring is softened due to wear and/or aging of the coils or due to the environmental temperature (softened/stiffened). Each such fault is represented in the following as FkKp4 with k taking values in the range k = −16 . . . 80 % are considered). The third fault mode corresponds to reductions in the characteristics of the primary suspension’s damping Cp1 (first wheelset). Physically the damper’s performance degrades from oil leaking due to wear on its seals as well as from loosen screws at its mounting on the vehicle. Each such fault is represented as FkCp1 . This fault mode is not modelled in the present study and a test case of this fault type is used in section 5 for testing the unit’s capability in fault diagnosis of unknown fault types.

3.2 The fault detection & identification (FDI) unit The fault detection and identification unit is designed to operate upon two vibration signals: The track vertical velocity profile v – 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 yÄ (right above the trailing airspring, Fig. 2). This design essentially aims at monitoring a proper transfer function that adequately describes the vehicle dynamics and offers high sensitivity to small (incipient) faults. The transfer function between yÄ and v may (with the track velocity input being accounted for as acting on all four wheelsets with the proper delays included) be


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Figure 3: FRF magnitude of H(s) versus frequency and fault magnitude k: (a) Fault mode FkKst ; (b) fault mode FkKp4 . shown to be of the form: H(s) =

YÄ (s) = s2 [Hv (s) − 6.4 · Hp (s)] V (s)

(6)

with s designating the Laplace variable, and Hv (s), Hp (s) the transfer functions relating the car body’s vertical and pitch angle displacements, respectively, to the track velocity input which may be obtained by the appropriate manipulation of the Laplace transform of Eq. (2). Figs. 3(a), (b) depict the variability of the Frequency Response Function (FRF) magnitude that corresponds to H(s) as a function of frequency and fault magnitude k of the fault modes FkKst , FkKp4 . It is evident that as the fault magnitude increases the main changes in the dynamics are observed in the lower frequencies and that small magnitude faults have negligible effects on the vehicle dynamics and their diagnosis will be particularly challenging.

4 The Functional Model Based Method The Functional Model Based Method consists of two phases: The baseline phase (a-priori phase) that includes the identification (modelling) of the healthy vehicle’s dynamics using a customary discrete-time data-based model as well as the corresponding modelling of each fault mode, for its continuum of fault magnitudes, via the novel class of stochastic functional models. The inspection phase is performed on board, during the system’s service cycle and includes the functions of fault detection, identification and magnitude estimation that utilize the identified functional model of the baseline phase. A concise presentation of the FMBM follows; for full details the reader is referred to [22].

4.1 Baseline phase 4.1.1 Modelling of the healthy system A single experiment (or simulation as in the present study) with the healthy system is performed and the measured vibration signals are used for the identification of a discrete-time dynamical model representing the healthy system


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dynamics via standard identification procedures [30, 24]. In the present study a single-input single-output discretetime AutoRegressive with eXogenous input (ARX) model [30, 24] of the following form is used: M0 (ϑ) : y[t] +

na X i=1

ai · y[t − i] =

nb X i=0

bi · x[t − i] + e[t]

(7)

with t designating normalized discrete time (t = 1, 2, 3, . . ., with absolute time being (t − 1) · Ts , where Ts stands for the sampling period), x[t], y[t] the measured input (track velocity) and output (acceleration at point A) signals, respectively, na, nb the AutoRegressive (AR) and eXogenous (X) orders, respectively, and e[t] the stochastic model residual (one-step-ahead prediction error) which should, for an accurate model, form a zero-mean uncorrelated, and uncorrelated with x[t], sequence with variance σe2 . The model is parameterized in terms of the parameter vector ϑ = [ai , bi , σe2 ]T , which is to be estimated from the measured signals. ARX order estimation is based on successive estimation of increasingly higher order models and examination of criteria such as the Residual Sum of Squares (RSS) or the Bayesian Information Criterion (BIC) for a minimum value [30]. Final model acceptance is based on formal verification of the model residual uncorrelatedness hypothesis [30, 24]. 4.1.2 Fault mode modelling For the modelling of a fault mode, a series of M experiments (or simulations) are performed. Each experiment is characterized by a specific fault magnitude k, with the complete series covering the range of possible fault magnitudes, say [kmin , kmax ], via a discretization {k1 , k2 , . . . , kM } (it is tacitly assumed, without loss of generality, that the healthy system corresponds to k = 0). This procedure yields a series of input-output signal pairs (each of length N ): xk [t], yk [t] (t = 1, . . . , N ) (k = k1 , k2 , . . . , kM )

(8)

with the subscript k designating the corresponding fault magnitude. The pooling of the above data sets is used in the following for the estimation of a proper mathematical description for each fault mode based on the form of a stochastic Functionally Pooled ARX (FP-ARX) model. FP-ARX models may be thought of as generalizations of the conventional ARX form of Eq. (7) with the important difference that their parameters are functions of the fault magnitude k. These models are defined as: M(θ) :

yk [t] +

na X i=1 ∆

ai (k) =

ai (k) · yk [t − i] =

p X j=1

ai,j · Gj (k),

nb X i=0

bi (k) · xk [t − i] + ek [t] ∆

bi (k) =

p X j=1

k∈R

bi,j · Gj (k)

(9)

(10)

with na, nb designating the AR and X orders, respectively, k the fault magnitude (presently percentage stiffness reduction), xk [t], yk [t] the corresponding measured input and resulting vibration output, respectively, and ek [t] the corresponding stochastic model residual (one-step-ahead prediction error). For an accurate model the residual sequence should be zero-mean, uncorrelated, with variance σe2 (k), and uncorrelated with the corresponding input. Residual sequences corresponding to different fault magnitudes are considered uncrosscorrelated.


9

As Eq. (10) indicates, the AR and X parameters ai (k), bi (k) are modelled as explicit functions of the fault magnitude k belonging to a p-dimensional functional space spanned by the (mutually independent) functions G1 (k), . . . , Gp (k) (functional basis). The constants aij , bij designate the AR and X, respectively, coefficients of projection. The FP-ARX model of Eq. (9) is re-written as:

with:

yk [t] = ϕTk [t] ⊗ g T (k) · θ + ek [t] = φTk [t] · θ + ek [t]

(11)

T . ∆ ϕk [t] = −yk [t − 1] . . . − yk [t − na] .. xk [t] . . . xk [t − nb]

(12)

[(na+nb+1)×1]

∆

T

g(k) = [ G1 (k) . . . Gp (k) ][p×1] T .. ∆ θ = a1,1 . . . ana,p . b0,1 . . . bnb,p

(13) (14)

[(na+nb+1)p×1]

and ⊗ designating Kronecker product [31, pp. 27-28]. For model parameter estimation, the FP-ARX of Eq. (11), following substitution of the data (8) corresponding to a single fault magnitude k, leads to:    T    φk [1] ek [1] yk [1]       .. .. ..  =⇒ y k = Φk · θ + ek = ·θ+  . . . T ek [N ] yk [N ] φk [N ]

(15)

Pooling together these expressions for the data corresponding to the discrete fault magnitudes {k1 , k2 , . . . , kM } considered in the experiments yields: y =Φ·θ+e with:



 ∆  y= 

y1 y2 .. . yM

    



[N M×1]

 ∆  Φ= 

Φ1 Φ2 .. . ΦM

    

(16) 

[N M×p(na+nb+1)]

 ∆  e= 

e1 e2 .. . eM

    

(17) [N M×1]

Thus, an FP-ARX model may be estimated based on the following Ordinary Least Squares estimator [22]: "M N #−1 " M N # −1 XX XX T T T ˆ θ= Φ Φ ·Φ y = φk [t]φk [t] · φk [t] yk [t] k=1 t=1

with

σ ê2 (k) =

N 1 X 2 eˆ [t] N t=1 k

(18)

k=1 t=1

for k = k1 , . . . , kM

(19)

4.2 Inspection phase Let x[t], y[t] (t = 1, 2, . . . , N ) represent the input and output signals, respectively, obtained from the system in its current (unknown) state via a fresh experiment. The fault detection and isolation (fault type determination or identification) problems are treated almost simultaneously and are based on the re-parametrized FP-ARX models of the considered fault modes. The current


10

signals are driven through the FP-ARX(na, nb) models of the baseline phase (of each fault mode), which are now re-parametrized in terms of the unknown k and the corresponding residual variance σe2 of the current data. Toward this end the re-parametrized FP-ARX model of any particular fault mode may be: M(k, σe2 ) :

y[t] +

na X i=1

ai (k) · y[t − i] =

nb X i=0

bi (k) · x[t − i] + e[t]

(20)

In a next step the estimation of k, σe2 of the unknown state of the vehicle (which shall be characterized) is achieved via the Nonlinear Least Squares (NLS) estimator (presently realized via golden search and parabolic interpolation [32]): ∆ kˆ = arg min k

N X

e2 [t]

σ ê2 =

t=1

N 1 X 2 eˆ [t] N t=1

(21)

The estimator kˆ is asymptotically (N → ∞) Gaussian distributed with mean equal to the true (underlying) k

value, say k o , and variance σk2 [kˆ ∼ N (k o , σk2 )] which may be in turn estimated as (see Appendix B): σ ˆk2

σ ˆ2 = e N

"

" 2 #−1 2 #−1 N N 1 X ϑˆ e[t] σ ê2 1 X ϑg T (k) T ˆ = ϕ [t] ⊗ ·θ N t=1 ϑk k=kˆ N N t=1 ϑk k=kˆ

(22)

ˆ includes the coefficients of projection of ϕ[t] is defined analogously to Eq. (12) based on the current data, and θ the FP-ARX model that is validated as follows. Each kˆ obtained via the left part of Eq. (21) is validated based on statistical tests examining the hypothesis of input and residual sequence uncrosscorrelatedness, as well as residual uncorrelatedness. The latter is presently used, for each re-parametrized FP-ARX model separately, via the statistical hypothesis testing problem: H0 : ρi = 0 i = 1, 2, . . . , h

(FP-ARX model is validated)

H1 : ρi 6= 0 for some i (1 ≤ i ≤ h)

(FP-ARX model is not validated)

(23)

in which ρi (i = 1, 2, . . . , L) designates the residual series normalized autocorrelation at lag i. It may be shown [33, p. 149] that the test statistic: Q = N · (N + 2) ·

L X i=1

(N − i)−1 ρˆ2i

(24)

in which N designates the residual signal length (in number of samples), ρî the sample normalized residual autocorrelation, and L the maximum lag, follows a chi-square (χ2 ) distribution with L − 1 degrees of freedom. This leads to the test (at the α risk level): Q < χ21−α,h−1

=⇒ H0

is accepted

Else

=⇒ H1

is accepted

(25)

with χ21−α,L−1 designating the distribution’s corresponding critical point. 4.2.1 Fault detection As each fault mode includes the healthy system (for k = 0), fault detection may be based on any validated reparametrized FP-ARX model through the following statistical hypothesis testing problem: H0 : k = 0 (healthy system) H1 : k 6= 0 (faulty system)

(26)


11

Under the null hypothesis the following statistic follows a t-distribution with N − 1 degrees of freedom [22]: t=

kˆ σ ˆk

∼ tN −1

(27)

which leads to the following test at the α risk level (probability of false alarm, or Type I error, that is of accepting H1 although H0 is true being equal α): t α2 ≤ t ≤ t1− α2

=⇒ H0

is accepted (healthy system)

Else

=⇒ H1

is accepted (faulty system)

(28)

with tα designating the t distribution’s (with the indicated degrees of freedom) α critical point [defined such that Prob(t ≤ tα ) = α]. Remark: If any FP-ARX model of the considered fault modes is not validated then a fault is straightforwardly detected. Conversely, if two or more FP-ARX models are simultaneously validated then there is no fault. 4.2.2 Fault identification Once a fault is detected then fault identification is based on the outcome of the model validation procedure mentioned above. The re-parametrized FP-ARX model of the fault mode that leads to white residuals describes also the type of the unknown fault. In case where none of the considered FP-ARX models are validated then a fault is detected but its identification and magnitude estimation is not possible through the FP-ARX models of the baseline phase. 4.2.3 Fault magnitude estimation Once the current fault mode has been identified, a fault magnitude interval estimate is constructed based on the ˆ σ k, ˆk2 as obtained from the M(k, σe2 ) model of Eq. (20) through Eqs. (21) and (22), respectively. Thus the fault magnitude interval estimate is constructed based on Eq. (27) at the α risk level: k

interval estimate:

h i kˆ − t α2 · σ ˆk , kˆ + t1− α2 · σ ˆk

(29)

5 Fault Detection, identification and magnitude estimation results 5.1 Baseline phase 5.1.1 Modelling of the healthy system Data from the physics-based vehicle model of section 2 corresponding to the healthy system are used for the successive fitting of ARX(k,k + 149) models for k = 8, 9, . . . , 30, based on 8.188 s (N = 2, 048 sample) long input and output signals leads to a decreasing RSS exhibiting relative small reduction after k = 12, while the BIC attains its minimum for k = 12 (Fig. 4). The simulated signals are corrupted by experimental random noise at the 40% standard deviation level,

σe σy %

= 40% [also see Eq. (7)]. Thus an ARX(12,161) model is identified and

successfully validated as the uncorrelatedness hypothesis is confirmed at the α = 0.05 level (Fig. 5). This model is characterized by unit time delay [b0 = 0 in the exogenous polynomial; also see Eq. (7)] and a high degree exogenous (X) polynomial with one hundred thirteen parameters preset to zero. It is worth noted that although the additional measurement noise the identification procedure led to an ARX model that resembles with


12

−24

−10

10

−28

−12

10

−32

−14

10

−36

−16

10

−40

−18

10

BIC

RSS/SSS (%)

−8

10

8

10

12

14

16

18

20

22

24

26

28

30

−44

AR−order

Autocorrelation

Figure 4: BIC (−✷−) and RSS (− ◦ −) normalized with respect to the Series Sum of Squares (SSS) versus AR order (healthy system). 0.1 0.05 0 −0.05 −0.1

0

20

40

60

80

100

Lag Figure 5: Normalized autocorrelation of the ARX(12,161) model residual series (healthy system). Table 2: Fault mode modelling details. Fault Mode Estimated model No. of Projection Coeffs. θ FkKst FP-ARX(12,161) 240 FkKp4 FP-ARX(12,161) 240 Estimation method: Ordinary Least Squares (OLS) Number of experiments: M = 49 (baseline phase) Range of fault magnitude k: [−16, 80]% (increment δk = 2%) Type of functional basis: Chebyshev Type II polynomials Sampling frequency: fs = 250Hz Signal length: N = 1024 samples (4.096 s)

Funct. Space Dim. p 4 4

the particular structure being dictated by the discretization of the theoretical transfer function [also see Eq. (6)] with any other ARX model structure to lead at higher values of the BIC and RSS criteria. 5.1.2 Fault mode modelling Fault mode modelling (for both fault modes FkKst and FkKp4 characterized by elasticity reduction in the trailing airspring Kst and the primary suspension Kp4 , respectively) is based on 1, 024 sample long signals for each considered fault magnitude k. A total of M = 49 experiments are performed based on the railway vehicle model of section 2, one corresponding to the healthy system (k = 0 % reduction in Kst or Kp4 ) and the rest corresponding to various fault magnitudes (see details in Table 2). The FP-ARX modelling procedure [22], for both FkKst and FkKp4 fault modes leads to two FP-ARX(12,161) models with functional basis consisting of the first p = 4 Chebyshev Type II polynomials [34] and 113 × 4 parameters pre-set to zero. These models achieve a reasonably low RSS in both cases (see Fig. 6). The Frequency Response Function (FRF) magnitude curves for the above models are, as functions of both frequency and fault magnitude k, depicted in Figure 7.


13

Table 3: The suspension states considered. Suspension State I II III IV V

Suspension condition No fault (healthy) F1Kst (1% reduction in trailing airspring) Kst F19 (19% reduction in trailing airspring) F3Kp4 (3% reduction in 4th wheelset stiffness) F7Cp1 (7% reduction in 1st wheelset damping)

3

3

10

10

−1

−1

10

−12.39

10

−12.4

10

−5

RSS

RSS

10

10

3

4

5

6

7

8

−9

−9

10

−13

10

−5

10

10

−13

(a) 1

10 2

3

4

5

6

Functional basis dimensionality

7

8

(b) 1

2

3

4

5

6

7

8

Functional basis dimensionality

Figure 6: RSS (− ◦ −) versus functional basis dimensionality: (a) Fault mode FkKst : selected model FPARX(12,161) with p = 4; (b) fault mode FkKp4 : selected model FP-ARX(12,161) with p = 4.

5.2 Inspection phase The effectiveness of the FDI unit is assessed via 20 Monte Carlo simulations for each suspension state of Table 3 under: (i) added measurement noise at the 40% standard deviation level (also see section 5.1), (ii) car body mass Mb increment (mass variability is not considered in fault mode modelling) from 1% up to 8%, which corresponds to 328.04 − 2624.32 kg and, (iii) higher level of track irregularities than those used in fault mode modelling through various values of Ar (see section 2) that correspond to input variance increase from 5% up to 100%. Furthermore all fault magnitudes (and the corresponding data series) which are considered as unknown in this phase has not been used in fault mode modelling (FP-ARX identification). Fig. 8 depicts the sensitivity of the theoretical H(s) transfer function FRF magnitude to the faults of the considered suspension states. The effects of each fault to the FRF magnitude are negligible which means that the considered faults are very small or incipient. Additionally the suspension State V includes a fault type from fault mode FkCp1 which is not modelled in the baseline phase. This case is intentionally selected for testing the capability of the FDI unit to: (i) properly detect a completely unknown fault type as fault isolation and magnitude estimation is not possible in this case, and, (ii) overrule the fault modes which is designed to monitor and indicate a different fault type. Damage detection, isolation (or identification) and magnitude estimation results for each suspension state over the 20 Monte Carlo simulations for the case of the added measurement noise are pictorially presented in Fig. 9. The first two columns include the FP-ARX model validation results through the re-parametrized FP-ARX models of the considered fault modes FkKst , FkKp4 and the Q statistic of Eq. (24), which shall be under the critical point for model validation. The third column of the same figure includes the fault detection results based on the validated FP-ARX


14

Figure 7: FP-ARX model based FRF magnitude versus frequency and fault magnitude: (a) Fault mode model FkKst ; (b) fault mode model FkKp4 . model and the t-statistic of Eq. (27) that shall be within the dashed lines for a healthy state and beyond them when a fault has occurred. The fourth column of Fig. 9 depicts the true fault magnitudes and the corresponding point estimates along with their interval estimates for each Monte Carlo experiment. Each row of Fig. 9 corresponds to a separate suspension state. A summary of the fault magnitude estimation results in terms of average point estimates over the 20 Monte Carlo simulations (per case) along with the corresponding standard deviations is presented in Table 4. For all statistical tests the risk level is α = 0.05. It is evident that in all suspension states the FDI unit detects and correctly identifies the considered faults. It is worth noting that in Suspension State I with the healthy system both re-parametrized FP-ARX models of the considered fault modes are, as expected, validated. This indicates that there is no fault in the suspension which is also confirmed from the corresponding fault detection results based on any FP-ARX model (this of the FkKst fault mode is presently used) as the fault magnitude interval estimates include k = 0 in all of the 20 simulations [see Fig. 9(c)]. Furthermore both FP-ARX models of the considered faults modes cannot be validated in Suspension State V as the Q-statistic is far above the critical points for all 20 Monte Carlo simulations, as shown in Figs. (9)(q), (r). Based on this result the FDI unit transmits an alarm for a new fault in the vehicle suspension but it can’t identify the type of the fault and estimate its magnitude via the FP-ARX models of the baseline phase. This means that the unit may achieve the detection of unknown types of faults which are not included in the baseline phase without any conflict with those for which is trained to monitor. The accurate estimates of the fault magnitudes in all considered cases with very narrow interval estimates are also evident via Figs. 9(d), (h), (l), (p) and Table 4. For the case of mass increment in the range of 328.04 − 2624.32 kg on the car body the fault detection and isolation results are similar with the previous case with all faults to be correctly detected and identified. On the other hand fault magnitude estimation is significantly affected as it is shown in Table 5 with the estimated confidence intervals to be close to the actual fault magnitudes. It is noted that for more than 8% mass increment both FPARX models are not validated which means that the method detects an unknown vehicle state than the healthy


15

40 Healthy FKst 1 Kst

30

F19

Kp4

F3

Cp1 F7

20

Magnitude (dB)

10

16.46

0

33 16.44

16.42

32

−10

16.4 31 16.38

−20

30 16.36

16.34

29

−30

16.32 28 16.3 1.5

−40 0

2

2

2.5

4

3

14.55

6

8

14.6

14.65

10

14.7

14.75

12

14.8

14.85

14

14.9

14.95

16

15

15.05

18

20

Frequency (Hz)

Figure 8: FRF magnitude for the nominal (—) and faulty states of the vehicle based on the transfer function H(s). and the specific faulty states for which it is trained to monitor. This happens because car body mass variability is not included in fault mode modelling (FP-ARX identification) and thus the method is not trained to monitor and distinguish significant mass increment effects from faults. For larger variability in mass increment corresponding simulations may be included in fault mode modelling procedure for appropriately training the method. A case where Functional Models are trained under various aircraft weights and positions of the center of gravity for the design of an Angle-of-Attack virtual sensor has been presented in [35]. In contrast to the previous case the significantly higher track irregularities than those used in fault mode modelling affects slightly the unit’s effectiveness in fault diagnosis with similar results with those shown in Fig. 6 for the case of added measurement noise. The excellent fault magnitude estimation results for this case are shown in Table 6. It is noted that even higher levels of track irregularity were examined at individual test cases with no significant changes in the fault diagnosis results.

6 Conclusions The feasibility of fault diagnosis in railway vehicle suspensions through the design of a novel FDI unit and various types and magnitudes of faults inserted in the suspension characteristics of a simple passenger physics-based railway vehicle model of the Hellenic Railways, was presented. The unit was trained based on measurement signals obtained from the physics-based vehicle model in the baseline phase, while it achieved fault diagnosis in the


Q statistic − Fault mode FKst k

Q statistic − Fault mode FKp4 k

60

t statistic

60

40

40

20

20

60

2000

0

40

1000

x 10

60

FKst 19

(f)

(e)

20

0 2000

20

0

2000

60

10

1000

40

40

1000 (j)

(i)

FKp4 3

3 2 1 0 −1 8 6 4 2 0 −2

−8

Fault magnitude − k (%) (d)

FKst

0

−2

Kst

x 10

(c)

(b)

(a)

F1

2

2

Kst

F0

16

−2

8

k

(± 2 x 10−8)

FKst

1

k

(h)

(g)

x 10

(± 2 x 10−8)

9

Kst

Fk

19 (k)

x 10

(l)

(± 3 x 10−9)

9

5

Kp4

Fk

3 0 (n)

(m)

FCp1 7

0

20

2000

2000

1000

−5 0

10

15

20

(p)

0

Monte Carlo experiment

5

10

15

20


1000 (r)

(q)

0 0

(o)

5

5

10

15

20

0 0


5

10

15

20


Figure 9: Fault detection, identification, and magnitude estimation results via the modelled fault modes: (a-d) Suspension State I, (e-h) Suspension State II, (i-l) Suspension State III, (m-p) Suspension State IV, (q-r) Suspension State V [20 Monte Carlo simulations per suspension state; Q statistic (×) and the critical point (- - -) with risk level a = 0.05 and h = 40; t statistic (+) and the critical point (- · -) with risk level a = 0.05 and N = 1023; the solid horizontal lines designate true fault magnitude and the boxes interval magnitude estimates].

Table 4: Fault magnitude estimation - Added measurement noise case (average sample mean & standard deviation estimates over the 20 Monte Carlo simulations per suspension state). Suspension

True Fault

State I II III IV

Magnitude (%) 0 1 19 3

Average Point Estimate kˆ −4.0745 × 10−10 1.0000 19.0000 3.0000

Average Standard Deviation Estimate σ ˆk 4.4741 ×10−9 4.4240 ×10−9 3.6089 ×10−9 0.6515 ×10−9

Table 5: Fault magnitude estimation - Mass increment on the car body case (average sample mean & standard deviation estimates over the 20 Monte Carlo simulations per suspension state). Suspension

True Fault

State I II III IV


Average Point Estimate kˆ 0.1729 0.0322 16.7074 3.3979

Average Standard Deviation Estimate σ ˆk 0.0003 0.0163 0.0991 0.0022

inspection phase based on the advanced Functional Model Based Method (FMBM) that operates in a unified and compact statistical framework accounting for experimental and modelling uncertainty through a pair of vibration signals without using the physics-based model of the baseline phase. Various types and magnitudes of faults are


17

Table 6: Fault magnitude estimation - Different levels of track irregularity case (average sample mean & standard deviation estimates over the 20 Monte Carlo experiments per suspension state). Suspension

True Fault

State I II III IV


Average Point Estimate kˆ −1.0012 × 10−9 0.9999 18.9999 2.9999

Average Standard Deviation Estimate 1.96 × σ ˆk 4.736 × 10−9 4.6821 × 10−9 3.7845 × 10−9 0.6930 × 10−9

used for the assessment of the method’s effectiveness and its robustness was examined under added measurement noise, mass increment in the car body and different levels of track irregularity. The main conclusions from this study are summarized as follows: • The unit achieved fault detection, isolation, and accurate magnitude estimation in all considered suspension states and for all Monte Carlo experiments under added measurement noise and different levels of track irregularity. Damage detection and isolation were flawless under car body mass increment but fault magnitude estimation was significantly affected in this case. More accurate fault magnitude estimation results covering a wider range of car body mass variability may be achieved if the unit is trained for various vehicle weights in the baseline phase. Furthermore, suspension State V for which the unit has not been trained to monitor was successfully detected and its isolation from the two known fault types was straightforward. It is also noted that the unit may accurately estimate any fault magnitude in the continuous range of magnitudes considered in the baseline phase without resorting to classification at predefined fault magnitudes. • The unit exhibited sensitivity to very small fault magnitudes, as those of Suspension States I and III, as well as robustness to operational and experimental uncertainties such as the addition of significant measurement noise at 40% in terms of signal to noise standard deviation ratio, added mass in the range of 328.04 − 2624.32 kg on the car body which was not used in the training of the unit and significantly higher levels of track irregularity that reached the 100% increase on the input variance compared with the corresponding inputs used in the training procedure. • Fault diagnosis was achieved based on a single pair of vibration signals and partial data-based functional models of the vehicle’s dynamics without the need of: (i) an array of sensors on the vehicle that increases the cost and the complexity of the experimental set-up, (ii) physics-based complete mathematical representations in the inspection phase where fault diagnosis is achieved. • The most involved stage in the design of the FDI unit is the baseline phase, where the unit is trained to monitor the investigated fault modes. At this point user expertise is necessary for the identification of suitable FP-ARX models for each fault mode as well as it is important to have qualitative vibration signals from the healthy and faulty state of the vehicle. Such signals may be collected from the actual railway vehicle where faults may be “simulated” by removing bolts from various suspension components mountings, oil from dampers, air from airsprings (also see [14]), or through in scale laboratory prototypes, or through analytical models


18

appropriately tuned with experimental data as in [17]. • Provided that the unit will be adequately trained in the baseline phase its real time operation does not require high CPU power and massive data transfer as fault diagnosis is based on the low computational complexity processing of the preselected pair of vibration signals (of the current, unknown, state of the structure) which is achieved in the inspection phase.

Acknowledgements The authors wish to thank Hellenic Railways, in particular Dr. K. Giannakos, Mr. E. Satlas and Mr. D. Papakostas, for useful discussions and fruitful collaboration. They also wish to acknowledge the financial support of this study, in part by the General Secretariat for Research and Technology – Greece and the European Social Fund (PENED99 Project #580).

References [1] Barke, D., and Chiu, K., 2005, “Structural Health Monitoring in the Railway Industry: A Review”, Structural Health Monitoring, 4(1), pp. 81-93. [2] Ward, M., 1989, “Development of a Semi Automatic / Automatic Rolling Stock Wheel Ultrasonic Testing Facility”, Proc. 4th International Heavy Haul Railway Conference, Brisbane, Australia, pp. 474-479. [3] Hirakawa, K., Toyama, K., and Kubota, M., 1998, “The Analysis and Prevention of Failure in Railway Axles”, International Journal of Fatigue, 20(2), pp. 135-144. [4] Yohso, J., 1995, “Development of Automatic Ultrasonic Testing Equipment for General and Bogie Inspection of Shinkansen Hollow Axle”, Proc. 11th International Wheelset Congress, National Conference, Australia, 2, (1995), pp. 47-52. [5] Bauer, G., Kellerer, H., and Schultes, G., 1996, “Maintenance of Rail Vehicles - Use of Technical Diagnostics”, Proc. World Congress on Railway Research 1996, Pueblo, CO, USA, pp. 417 - 424. [6] Anderson, G.B., Stone, D.H., Cline, J.E., and Smith, R.L., 1996, “New Detection Technique to Identify Defective Railroad Bearings”, Proc. of the ASME International Mechanical Engineering Congress and Exposition, New York, USA, 12, pp. 31-33. [7] Cline, J.E., Bilodeau, J.R., and Smith, R.L., 1998, “Acoustic Wayside Identification of Freight Car Roller Bearing Defects”, Proc. of the ASME/IEEE Joint Railroad Conference, Philadelphia, PA, USA, 14, pp. 79-83. [8] Wang, J.M., Anderson, G.B., and Smith, R.L., 1996, “Burn-Off Simulation Analysis of a Railroad Roller Bearing,” Technology Digest TD 96-005, Association of American Railroads. [9] Bruni, S., Goodall, R., Mei, T.X. and Tsunashima, H., 2007, “Control and monitoring for railway vehicle dynamics”, Vehicle System Dynamics: International Journal of Vehicle Mechanics and Mobility, 45(7-8), pp. 743–779. DOI: 10.1080/00423110701426690


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[10] Hayashi, Y., Tsunashima, H., and Marumo, Y., 2008, “Fault detection of railway vehicle suspension systems using multiple model approach”, Mechanical Systems for Transportation and Logistics, 1(1), pp. 88-98. [11] Mori, H., and Tsunashima, H., 2010, “Condition monitoring of railway vehicle suspension using multiple model approach”, Mechanical Systems for Transportation and Logistics, 3(1), pp. 243-258. [12] Wei, X., Jia., L., and Liu, H., 2013, “A comparative study on fault detection methods of rail vehicle suspension systems based on acceleration measurements”, Vehicle System Dynamics: International Journal of Vehicle Mechanics and Mobility, 51(5), pp. 700–720. DOI: 10.1080/00423114.2013.767464 [13] Jesussek, M., and Ellermann, K., 2013, “Fault detection and isolation for a nonlinear railway vehicle suspension with a Hybrid Extended Kalman filter”, Vehicle System Dynamics: International Journal of Vehicle Mechanics and Mobility, 51(10), pp. 1489–1501. DOI: 10.1080/00423114.2013.810764 [14] Sakellariou, J.S., Petsounis, K.A., and Fassois, S.D., 2001, “Vibration Analysis Based On Board Fault Detection in Railway Vehicle Suspensions: A Feasibility Study”, paper ANG1/P080, Proc. 1st Natl. Conf. on Recent Advances in Mech. Engr., Patras, Greece. [15] Mei, T.X. and Ding, X.J., 2008, “A model-less technique for the fault detection of rail vehicle suspensions”, Vehicle System Dynamics: International Journal of Vehicle Mechanics and Mobility, 46:51, pp. 277–287. DOI: 10.1080/00423110801939154 [16] Mei, T.X. and Ding, X.J., 2009, “Condition monitoring of rail vehicle suspensions based on changes in system dynamic interactions”, Vehicle System Dynamics: International Journal of Vehicle Mechanics and Mobility, 47:9, pp. 1167–1181. DOI: 10.1080/00423110802553087 [17] Gasparetto, L., Alfi, S. and Bruni, S., 2013, “Data-driven condition-based monitoring of high-speed railway bogies”, International Journal of Rail Transportation, 1(1-2), pp. 42–56. DOI: 10.1080/23248378.2013.790137 [18] Jia, S., and Dhanasekar, M., 2007, “Detection of Rail Wheel Flats using Wavelet Approaches”, Structural Health Monitoring, 6, pp. 121–131. DOI: 10.1177/1475921706072066 [19] Donelson III, J., and Dicus, R.L., 2002, “Bearing Defect Detection Using On-Board Acccelerometer Measurements”, Proc. ASME/IEEE Joint Railroad Conference 2002, Washington, DC, USA, pp. 95–102. [20] Sneed, W.H., and Smith, R.L., 1998, “On-board real-time railroad bearing defect detection and monitoring”, Proc. IEEE/ASME Joint Railroad Conference 1998, Philadelphia, PA, USA, pp. 149-153. [21] VibroAcoustical Systems and Technologies (VAST), Inc., Application of Bearing Diagnostics on Russian Railways, Petersburg, Russia; available on VAST website at http://www.vibrotek.com/notes/rusrail/index.htm [22] Sakellariou, J.S., Fassois, S.D., 2008, “Vibration based fault detection and identification in an aircraft skeleton structure via a stochastic functional model based method”, Mechanical Systems and Signal Processing, 22, pp. 557-573.


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[23] Sakellariou, J.S., Petsounis, K.A., and Fassois, S.D., 2002, “On Board Fault Detection and Identification in Railway Vehicle Suspensions via a Functional Model Based Method”, Proc. International Conference on Noise and Vibration Engineering (ISMA), Leuven, Belgium. [24] Ljung, L., 1999, System Identification: Theory for the User, Second Edition, Prentice Hall PTR. [25] Fassois, S.D., and Sakellariou, J.S., 2007, “Time series methods for fault detection and identification in vibrating structures”, The Royal Society Philosophical Transactions: Mathematical, Physical andEngineering Sciences, 365, pp. 411-448. [26] Fassois, S.D. and Sakellariou, J.S., 2009, “Statistical time series methods for structural health monitoring”, in: C. Boller, F. K. Chang, Y. Fujino (Eds.), Encyclopedia of Structural Health Monitoring, John Wiley Sons Ltd., pp. 443-472. [27] Hee-Young Ko, Kwang-Bok Shin, and Sung-Ho Hahn, 2011, “The suggestion of the standardized finite element model through the experimental verifications of various railway vehicle structures made of sandwich composites”, J. Mechanical Science and Technology, 25, pp. 121–131. DOI: 10.1007/s12206-011-0228-z [28] Dukkipati, R.V., and Amyot, J.R., 1998, Computer-Aided Simulation in Railway Dynamics, Marcel-Dekker, New York. [29] Foo, E., and Goodall, R.M., 2000, “Active Suspension Control of Flexible - Bodied Railway Vehicles using Electro - Hydraulic and Electro - Magnetic Actuators”, Control Engineering Practice, 8, pp. 507 - 518. [30] Fassois, S.D. , 2001, “Parametric Identification of Vibrating Structures,” in Encyclopedia of Vibration, Braun, S.G., Ewins, D.J., and Rao, S.S., eds., Academic Press, pp. 673-685. dx.doi.org/10.1006/rwvb.2001.0121 [31] Magnus, J.R., and Neudecker, H., 1988, Matrix Differential Calculus, John Wiley and Sons. [32] Forsythe, G.E., Malcolm, M.A., and Moler, C.B., 1976, Computer Methods for Mathematical Computations, Prentice-Hall. [33] Wei, W.W.S., 1990, Time Series Analysis: Univariate and Multivariate Methods, Addison-Wesley Publishing Company. [34] Abramowitz, M., and Stegun, I.A., 1970, Handbook of Mathematical Functions, New York: Dover. [35] Samara, P.A., Sakellariou, J.S., Fouskitakis, G.N., Hios, J.D. and Fassois, S.D., 2013, “Aircraft Virtual Sensor Design Via a Time-Dependent Functional Pooling NARX Methodology”, Aerospace Science and Technology, 29, pp. 114-124.

Appendix A: Railway Vehicle Parameter values The parameter values of the railway vehicle are summarized in Table A.1.


21

Table A.1: Model parameters of the passenger Hellenic Railways vehicle. Property Symbol Value Vehicle body mass Mb 32,804.00 Vehicle body inertia Ib 0.67 × 106 Leading bogie mass Mtl 2,363.50 Leading bogie inertia Itl 1,026.36 Trailing bogie mass Mtt 2,261.50 Trailing bogie inertia Itt 972.54 Leading/trailing secondary suspension elasticity Ksl /Kst 2.20 × 106 Leading/trailing secondary suspension damping Csl /Cst 4.00 × 104 Primary suspension stiffness (i-th element) Kpi (i = 1, 2, 3, 4) 5.56 × 106 Primary suspension damping (i-th element) Cpi (i = 1, 2, 3, 4) 1.80 × 104 Track roughness factor Ar 0.52 × 10−7 Distance between center of vehicle body and bogies L 6.40 Distance between center of vehicle body and secondary dampers Lc 6.85 Distance between center of bogie and wheelsets Lbw 1.05 Vehicle velocity u 25.00

Units kg kg × m2 kg kg kg kg N/m N s/m N/m N s/m m m m m m/s

Appendix B: Cramer Rao bound of kˆ The residual sequence e[t] of Eq. (20) is, for a valid fault mode model, independent and identically distributed. Assuming that follows a Gaussian distribution with zero mean and variance σe2 , [∼ N (0, σe2 )], the natural logarithm of its likelihood function L, is given by (for notational convenience λ = σe2 ):

N

L = ln L(k, λ) = −

N 1 X 2 N ln 2π − ln λ − e [t, k] 2 2 2λ t=1

(B.1)

with e[t, k] designating the residual sequence as a function of k. ˆ λ ˆ are then obtained as the maximizing elements of L. First maximize The Maximum Likelihood estimates k, with respect to λ. The first two derivatives of Eq. (B.1) with respect to λ are: N ϑ ln L(k, λ) N 1 X 2 =− + 2 e [t, k], ϑλ 2λ 2λ t=1

N ϑ2 ln L(k, λ) N 1 X 2 = − e [t, k] ϑλ2 2λ2 λ3 t=1

(B.2)

There is only one point where the first derivative is zero and the second one is negative: λ=

N 1 X 2 e [t, k] N t=1

(B.3)

Consequently, this point is a maximum. The estimate of λ is thus given by this form when the optimal value of k, that has to be determined, is used. Inserting Eq. (B.3) into (B.1) gives: L = constant −

N 1 X 2 N ln e [t, k] 2 N t=1

(B.4)

Maximizing (B.4) is equivalent to minimizing the residual sum of squares with respect to k and the minimum ˆ obtained by the Nonlinear Least Squares estimator of Eq. (21). By point is the Maximum Likelihood estimate k, ˆ is obtained. inserting kˆ into Eq. (B.3) the Maximum Likelihood estimate of λ


Thus, the random variable

22

√ N (kˆ − k) converges in distribution to the normal distribution with zero mean and

variance given by the Cramer Rao bound [24, pp. 214-215]: √ d N (kˆ − k) → N (0, σk2 ) with

−1 2 ϑ ln L(k, λ) σk2 = − E ϑk 2

(B.5)

d

with → denoting convergence in distribution. The first two derivatives of Eq. (B.1) with respect to k are given by the following equations: " # 2 N N ϑ ln L(k, λ) 1X ϑe[t, k] ϑ2 ln L(k, λ) 1X ϑe[t, k] ϑ2 e[t, k] =− e[t, k] , =− + e[t, k] ϑk λ t=1 ϑk ϑk 2 λ t=1 ϑk ϑk 2

(B.6)

with ϑ2 e[t, k] ϑ2 G1 (k) ϑ2 Gp (k) = − −y[t − 1] . . . − y[t − na] | ... | ϑk 2 ϑk 2 ϑk 2 ϑ2 G1 (k) ϑ2 Gp (k) ˆ x[t] . . . x[t − nb] θ ϑk 2 ϑk 2

(B.7)

The substitution of Eqs. (B.6), (B.7) into (B.5) gives: σk2

"

" ( 2 ) ##−1 N 1X ϑe[t, k] ϑ2 e[t, k] = E + E e[t, k] λ t=1 ϑk ϑk 2

(B.8)

As e[t, k] is a white sequence with zero mean uncorrelated with the input, the second term is equal to zero and the relation above becomes: σk2

=λ

"N X t=1

E

(

ϑe[t, k] ϑk

2 )#−1

(B.9)

ˆ is replaced by σ For a signal of N data points, the variance of kˆ may be estimated as follows (λ ê2 ): σ ˆ2 σ ˆk2 = e N

"

" 2 #−1 2 #−1 N N T 1 X ϑˆ e[t, k] σ ê2 1 X ϑg (k) = ϕT [t] ⊗ · θˆ N t=1 ϑk k=kˆ N N t=1 ϑk k=kˆ

(B.10)

with ϕ[t] including data of the current state of the structure:

T . ϕ[t] = −y[t − 1] . . . − y[t − na] .. x[t] . . . x[t − nb]

[(na+nb+1)×1]

and g(k) being defined by Eq. (13).

(B.11)