prognosis of gear health using stochastic dynamical models with online parameter estimation

Prognosis of gear health using stochastic dynamical models with

online parameter estimation

10th International PhD Workshop on Systems and Controla Young Generation Viewpoint

Hluboka nad Vltavou, Czech Republic, September 22-26, 2009

Matej Gašperin, Pavle Boškoski, Đani Juričić“Jožef Stefan” Institute, Ljubljana, Slovenia

Motivation An estimated 95% of installed drives belong to older generation

- no embedded diagnostics functionality- poorly or not monitored

These machines will still be in operation for some time!

Goal: to design a low cost, intelligent condition monitoring module

Prognosis of gear health using stochastic dynamical models with online parameter estimation

Outline Introduction Experimental setup Feature extraction State-space model of the time series Parameter estimation using EM algorithm Experimental results Conclusion


Problem setup Gear health prognosis using feature values

from vibration sensors Model the time series using discrete-time

stochastic model Online parameter estimation Time series prediction using the estimated

model Prediction of first passage time (FPT)


Experimental setup Experimental test bed with motor-generator

pair and single stage gearbox


Experimental Setup Experiment description• 65 hours• 390 samples• constant torque (82.5Nm)• constant speed (990rpm)• accelerated damage mechanism (decreased

surface area)


Experimental Setup Vibration sensors

Signal acquisition


Feature Extraction

For each sensor, a time series of feature value evolution is obtained


State space model of a time series Consider a time series as an output of a

SDSSM:

Assuming that functions f and g are linear and that the system has no measurable inputs leads to the following form:

)(

)(

Θeuxy

Θwuxx

ttt

tt1t

,,,g

,,,f

ttt

tt1t

eCxy

wAxx


DSSM estimation via EM algorithm Expectation – Maximization is applied as an

iterative method to calculate a maximum likelihood estimate of unknown parameters (Θ).

And the iterative scheme is given as:

Algorithm alternates between maximizing the likelihood function with respect to hidden system states (M-step) and with respect to unknown parameters (E-step)

}{ 00 Σ,xR,Q,C,A,Θ

ΘXYΘ ΘY,|XΘ

|,logmaxarg1 pEkk


E – step of EM algorithm In dynamic systems with hidden states, E-step

directly corresponds to solving the smoothing problem.

Given an estimate of parameter values

an optimum estimate of system states, can be obtained by employing the Rauch-Tung-Striebel (RTS) smoother

}{ kkkkkkk 00 Σ,x,R,Q,C,AΘ

)'(

)(

ttk

t

E

Ek

xx

x

ΘY,|X

ΘY,|X


M – step of EM algorithm Maximization of the objective function l(Θ,

Θ’), rather than data log-likelihood function

l(Θ, Θ’) is bounded from above by L(Θ) increasing the value of l(Θ, Θ’) will increase

the value of L(Θ)

)',()',()( ΘΘΘΘΘ VlL


M – step of EM algorithm Complete data log-likelihood l(Θ, Θ’):

)()'(||log

)()'(||log

)()'(||ln)',(2

1

1

11

11

0010000

tt

N

ttt

tt

N

ttt

n

n

l

CxyRCxyR

AxxQAxxQ

μxΣμxΣΘΘ

N

tkkk

N

tkkkt

NtNtNtNtNt

xypxxpxp

xxpxxyypxxyyp

)|()|()(

),...,(),...,|,...,(),...,,,...,(

11

111

ΘΘΘ

ΘΘΘ


M – step of EM algorithm Taking the expected value of l(Θ, Θ’) with

respect to the current parameter estimate and complete observed data:

Derivatives of the function with respect to all parameters can be calculated analytically

')'()(')'()(||ln

')'()'(')'()(||ln

))((||ln=)',(

1=1=1=1=

1

111=

11=

11=1=

1

00100

CxxCyxCCxyyyRR

AxxAxxAAxxxxQQ

μxμxPΣΣΘΘ 00n0

tt

n

ttt

n

ttt

n

ttt

n

t

tt

n

ttt

n

ttt

n

ttt

n

t

nn

EEEEtrn

EEEEtrn

trl


Estimation setup Model structure

Model parameters are estimated using a time window of 100 samples

Convergence is achieved when log-likelihood increase is less than 10-4

)()()(

)()()1(

)()()()1(

11

22222

12211111

tetxcty

twtxatx

twtxatxatx


Online tracking of model parameters

Time series

Eigenvalues of system

matrix A

Diagonal elements of noise

covariance matrix Q


Prediction of first passage time


The goal of prediction was to determine when the extracted feature will exceed a certain value

For each time window, system parameters were estimated

Using estimated model, MC simulation was performed and time when the predicted value becomes greater than a certain threshold was obtained

For each time window, the simulation was performed 1000-times, so the result is a distribution of time

Prediction of first passage time


Conclusion Second order dynamical system is sufficient to

model the dynamical behavior of vibration feature value

Using the ML estimate of the parameters, accurate prediction of first passage time can be made 15-20 hours in advance

Comparison to linear regression:


prognosis of gear health using stochastic dynamical models with online parameter estimation

Documents

stochastic dynamical

time seriesconsider

estimated model prediction

feature values

older generation

nmconstant speed

motorgenerator pair

ani juriijoef stefan