Download EM algorithm for state space model estimation: application to gear

Prognosis of gear health using stochastic
dynamical models with online parameter
estimation
10th International PhD Workshop on Systems and Control
a Young Generation Viewpoint
Hluboka nad Vltavou, Czech Republic, September 22-26, 2009
Matej Gašperin, Pavle Boškoski, Đani Juričić
“Jožef Stefan” Institute, Ljubljana, Slovenia
Motivation
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An estimated 95% of installed drives
belong to older generation
- no embedded diagnostics functionality
- poorly or not monitored
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These machines will still be in operation for some time!
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Goal: to design a low cost, intelligent condition
monitoring module
Prognosis of gear health using stochastic dynamical models with online parameter estimation
Outline
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Introduction
Experimental setup
Feature extraction
State-space model of the time series
Parameter estimation using EM algorithm
Experimental results
Conclusion
Prognosis of gear health using stochastic dynamical models with online parameter estimation
Problem setup
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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)
Prognosis of gear health using stochastic dynamical models with online parameter estimation
Experimental setup
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Experimental test bed with motor-generator pair and
single stage gearbox
Prognosis of gear health using stochastic dynamical models with online parameter estimation
Experimental Setup
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Experiment description
65 hours
390 samples
constant torque (82.5Nm)
constant speed (990rpm)
accelerated damage mechanism (decreased surface area)
Prognosis of gear health using stochastic dynamical models with online parameter estimation
Experimental Setup
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Vibration sensors
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Signal acquisition
Prognosis of gear health using stochastic dynamical models with online parameter estimation
Feature Extraction
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For each sensor, a time series of feature value
evolution is obtained
Prognosis of gear health using stochastic dynamical models with online parameter estimation
State space model of a time series
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Consider a time series as an output of a SDSSM:
x t 1  f (x t ,u,w t ,Θ)
y t  g (x t ,u,e t ,Θ)
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Assuming that functions f and g are linear and that the
system has no measurable inputs leads to the following
form:
x t 1  Ax t  w t
y t  Cx t  et
Prognosis of gear health using stochastic dynamical models with online parameter estimation
DSSM estimation via EM algorithm
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Expectation – Maximization is applied as an iterative
method to calculate a maximum likelihood estimate of
unknown parameters (Θ).
Θ  {A, C, Q, R, x0 , Σ0 }
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And the iterative scheme is given as:
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Θ k 1  arg max EX|Y, Θk log pY, X | Θ 
Θ
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Algorithm alternates between maximizing the likelihood
function with respect to hidden system states (M-step)
and with respect to unknown parameters (E-step)
Prognosis of gear health using stochastic dynamical models with online parameter estimation
E – step of EM algorithm
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In dynamic systems with hidden states, E-step directly
corresponds to solving the smoothing problem.
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Given an estimate of parameter values
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Θ k  {A k ,estimate
Ck , Q k , R of
k , xsystem
0 k , Σ 0 k } states, can be obtained by
an optimum
employing the Rauch-Tung-Striebel (RTS) smoother
EX|Y, Θ k ( x t )
EX|Y, Θ k ( x t x t ' )
Prognosis of gear health using stochastic dynamical models with online parameter estimation
M – step of EM algorithm
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Maximization of the objective function l(Θ, Θ’), rather
than data log-likelihood function
L(Θ)  l (Θ, Θ' )  V (Θ, Θ' )
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l(Θ, Θ’) is bounded from above by L(Θ)
increasing the value of l(Θ, Θ’) will increase the value of
L(Θ)
Prognosis of gear health using stochastic dynamical models with online parameter estimation
M – step of EM algorithm
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Complete data log-likelihood l(Θ, Θ’):
pΘ ( yt ,..., y N , xt 1 ,..., x N )  pΘ ( yt ,..., y N | xt 1 ,..., xN ) pΘ ( xt 1 ,..., xN )
N
N
k t
k t
 pΘ ( xt 1 ) pΘ ( xk | xk 1 ) pΘ ( yk | xk )
 2l (Θ, Θ' )  ln | Σ 0 |  (x 0  μ 0 )' Σ 01 ( x 0  μ 0 )
N
 n log | Q |   ( x t  Ax t 1 )' Q 1 (x t  Ax t 1 )
t 1
N
 n log | R |   ( y t  Cx t )' R 1 (y t  Cx t )
t 1
Prognosis of gear health using stochastic dynamical models with online parameter estimation
M – step of EM algorithm
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Taking the expected value of l(Θ, Θ’) with respect to the
current parameter estimate and complete observed data:
 
l (Θ, Θ' ) = ln | Σ 0 | tr Σ 01 P0n  (x 0  μ 0 )( x 0  μ 0 )
n
n

n
n
n
 1  n

 n ln | Q | tr Q E (x t x t  )  E (x t x t 1 ' ) A'( A Ex t 1x t ' )  A E (x t 1x t 1 ' ) A'
t =1
t =1
t =1

  t =1
n
n
n
 1  n

 n ln | R | tr R E (y t y t  )  y t E (x t ' )C'CE (x t )y t   CE (x t x t ' )C'
t =1
t =1
t =1

  t =1
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Derivatives of the function with respect to all parameters
can be calculated analytically
Prognosis of gear health using stochastic dynamical models with online parameter estimation
Estimation setup
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Model structure
x1 (t  1)  a11x1 (t )  a21x2 (t )  w1 (t )
x2 (t  1)  a22 x2 (t )  w2 (t )
y (t )  c1 x1 (t )  e(t )
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Model parameters are estimated using a time window of
100 samples
Convergence is achieved when log-likelihood increase is
less than 10-4
Prognosis of gear health using stochastic dynamical models with online parameter estimation
Online tracking of model parameters
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Time series
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Eigenvalues of system
matrix A
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Diagonal elements
of noise covariance
matrix Q
Prognosis of gear health using stochastic dynamical models with online parameter estimation
Prediction of first passage time
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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
Prognosis of gear health using stochastic dynamical models with online parameter estimation
Prediction of first passage time
Prognosis of gear health using stochastic dynamical models with online parameter estimation
Prediction of first passage time
Prognosis of gear health using stochastic dynamical models with online parameter estimation
Conclusion
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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