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Particle Filtering for Diagnosis, Prognosis and On Condition Maintenance

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Title: Particle Filtering for Diagnosis, Prognosis and On Condition Maintenance


1
Particle Filtering for Diagnosis, Prognosis and
On Condition Maintenance
Francesco Cadini Milan October, 7th 2009
2
Introduction
  • Diagnosis, Prognosis and Maintenance of
    industrial and structural components are
    recognized valuable tasks for safely operating
    and managing with profit industrial plants and
    structures
  • In practice, such activities may be difficult
    since
  • some dynamical states of the system may not be
    directly observable
  • the state observations (measurements) may be
    affected by noise and disturbances

3
Objective of our research work
  • AIM
  • To develop an efficient technique for
  • estimating the dynamics of a component or a
    structure
  • estimating the system remaining lifetime
  • devising on-line procedures for condition-based
    maintenance
  • METHODOLOGY
  • Particle Filtering ? Non-linear and non-Gaussian
    model-based algorithm which allows estimation of
    system dynamic states based on a sequence of
    noisy measurements
  • MAIN FEATURES
  • Real-time operation capabilities
  • Quantification of the associated uncertainty

4
Model-Based Dynamic Degradation State Estimation
(discrete)
  • HYPOTHESES
  • System model
  • x hidden degradation state vector
  • ? i.i.d. random process noise vector
  • f non-linear dynamics function vector
  • k time step index
  • Measurement equation
  • ? i.i.d. random measurement noise vector
  • h non-linear measurement function vector

Hidden Markov process
Available
Recursive estimate of the posterior pdf
5
The state estimate and its uncertainty
  • From a Bayesian perspective, the posterior
    distribution
  • provides a degree of belief in the hidden
    degradation state based on the noisy
    measurements
  • The posterior pdf allows to compute various
    quantities related to the stochastic properties
    of the system, e.g.
  • state mean (estimate)
  • state variance (uncertainty)
  • transition probabilities (failure!)
  • failure times

6
Detailed analytical approach to the problem
  • Given Initial state
  • Suppose the required pdf is known at k-1
  • Prediction stage
  • Update stage

Chapman-Kolmogorov equation
Likelihood
?
Bayes rule
7
Optimal Bayesian solution
  • Pdf normalization requires
  • The prediction-update recurrence forms the basis
    for the optimal Bayesian solution
  • Unfortunately, the solution is only conceptual!
    It cannot in general be determined analytically!

8
Available model-based filtering techniques
PARTICLE FILTERING Numerical solution which, in
the limit, tends to the exact posterior pdf
9
Particle filtering the logical procedure for
state estimation
Time step k
not yet collected
available
Observation Likelihood (particle weights)
Monte Carlo prediction of N state trajectories (
particles) by importance sampling
BAYES RULE
System model
measurement equation
Posterior (updated) distribution of the current
system states
10
Particle filtering the intuitive description
Particles
k1
k2
k3
k4
Actual state dynamics
11
Particle filtering the details
  • Suppose N i.i.d. samples from are available

PARTICLES
Discrete pdf representation
Problem How do we sample the required ?
12
Importance sampling theory
  • Let us draw the samples from a known distribution
  • Define the importance weights for the i-th
    particle

Importance function
?
13
Sequential Importance Sampling (SIS) algorithm
  • Let us choose the denominator of to
    factorize as
  • One can obtain samples by augmenting
    with the new state

It allows recursive (sequential) weight
computation
14
Weight update equation
  • Let us express the numerator of
  • Bayes rule
  • Markov process
  • Conditionally independence of the measurements

15
Particle filter operative procedure
  • FOR k 1Nt (number of time steps on line
    estimation)
  • FOR i 1Ns (number of particles)
  • Sample the particle
  • Append to the existing trajectory
  • Collect measurement (if available)
  • Assign the weight to the particle
  • END FOR
  • Normalize weights
  • END FOR

16
Degeneracy problem
  • Fact can only increase with time!!
  • After few iterations all particles but one will
    have negligible weights!
  • Possible strategies
  • Good choice of the importance density q
  • Resampling

17
1. Good choice of the importance density q
  • Optimal choice minimization of given
  • Convenient choice (our)

?
Does not solve the problem!
18
2. Resampling
  • Generate the new set of particles by
    resampling with replacement Ns times from
  • so that
  • Weights are re-set to

19
Sampling Importance Resampling (SIR) algorithm
  • Assumptions
  • Resampling applied at each time step
  • We need to sample from
  • Easy!

State estimation
Failure probability estimation
Failure time distribution
20
APPLICATIONS
21
Application 1 estimation of nuclear reactor
dynamics
Chernick model
? neutron flux Xe Xenon concentration I
Iodine concentration
? and ? are Gaussian noises with zero mean and
1000 particles
F. Cadini and E. Zio, Application of Particle
Filtering for Estimating the Dynamics of Nuclear
Systems, IEEE Transactions on Nuclear Science,
55, n. 2, pp. 748-757, 2009
22
Application 2 estimation of PWR steam generator
dynamics
Irving model
x x1 x2 x3 x4 sytem states z Water
level measurement
x1 mm contribution to the water level due to
the total mass of water in the SG x2 mm
reverse dynamic effect related to the shrink and
swell phenomenon x3 mm mechanical oscillation
caused by the manometer effect x4 mm/s dummy
variable for reducing the differential equation
order
? is a non-Gaussian bimodal noise vector with
zero mean and such that
100 particles
F. Cadini and E. Zio, Application of Particle
Filtering for Estimating the Dynamics of Nuclear
Systems, IEEE Transactions on Nuclear Science,
55, n. 2, pp. 748-757, 2009
23
Application 3 Crack growth evolution
Paris-Erdogan model
Discretization of the dynamics
  • x hidden degradation state (crack depth)
  • ? i.i.d. non additive Gaussian process noise
  • N load cycle
  • C, ? and n constants related to the material
    properties

d
d
d
x
0
FAILURE x ? d
24
Application 3 Crack growth evolution
Logit model non-destructive ultrasonic
inspections
  • zk degradation observations at predefined time
    steps
  • ?k i.i.d. non additive measurement noise
  • ?0, ?1 constants related to the material
    properties

Observation Likelihood
25
Application 3 Crack growth evolution
  • 5000 particles
  • 5 measurements at k1 100 k2 200 k3 300
    k4 400 k5 500
  • ? and ? are non-additive noises with zero mean

F. Cadini, E. Zio, D. Avram Monte Carlo-based
filtering for fatigue crack growth estimation,
Probabilistic Engineering Mechanics,
doi10.1016/j.probengmech.2008.10.002, 24, n. 3,
pp. 367-373, 2009
26
Application 4 Maintenance optimization
Objective Estimate of optimal replacement time
l Criterion Minimization of Method Particle
Filtering for estimating the expected cost
5000 particles 4 measurements at k1 100 k2
200 k3 300 k4 400
? and ? are non-additive noises with zero mean
F. Cadini, E. Zio Model-based Monte Carlo state
estimation for condition-based component
replacement, Reliability Engineering and System
Safety, doi10.1016/j.ress.2008.08.003, 94, n. 3,
pp. 752-758, 2009
27
References
  • F. Cadini, E. Zio, D. Avram Monte Carlo-based
    filtering for fatigue crack growth estimation,
    Probabilistic Engineering Mechanics,
    doi10.1016/j.probengmech.2008.10.002, 24, n. 3,
    pp. 367-373, 2009
  • F. Cadini and E. Zio, Application of Particle
    Filtering for Estimating the Dynamics of Nuclear
    Systems, IEEE Transactions on Nuclear Science,
    55, n. 2, pp. 748-757, 2009
  • F. Cadini, E. Zio Model-based Monte Carlo state
    estimation for condition-based component
    replacement, Reliability Engineering and System
    Safety, doi10.1016/j.ress.2008.08.003, 94, n. 3,
    pp. 752-758, 2009
  • A. Doucet, On Sequential Simulation-Based Methods
    for Bayesian Filtering, Technical Report,
    University of Cambridge, Dept. of Engineering,
    CUED-F-ENG-TR310.
  • A. Doucet, J.F.G. de Freitas and N.J. Gordon, An
    Introduction to Sequential Monte Carlo Methods,
    in Sequential Monte Carlo in Practice, A. Doucet,
    J.F.G. de Freitas and N.J. Gordon, Eds., New
    York Springer-Verlag, 2001.
  • A. Doucet, S. Godsill and C. Andreu, On
    Sequential Monte Carlo Sampling Methods for
    Bayesian Filtering, Statistics and Computing
    (2000), Vol 10, pp. 197-208.
  • M.S. Arulampalam, S. Maskell, N. Gordon and T.
    Clapp, A Tutorial on Particle Filters for Online
    Nonlinear/Non-Gaussian Bayesian Tracking, IEEE
    Trans. On Signal Processing, Vol. 50, No. 2,
    2002, pp. 174-188.
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