Particle Filter — Remaining Useful Life Estimation

A Sequential Monte Carlo (SMC) implementation for fatigue crack growth prognosis using Paris' Law, with a generic particle filter library that can be adapted to any state-space problem.

Overview

This project estimates the Remaining Useful Life (RUL) of a structural component subject to fatigue crack growth. A particle filter assimilates crack size measurements and propagates an ensemble of particles forward in time to predict when the crack will exceed a failure threshold.

Architecture

ParticleFilter.py
├── StateSpaceModel      — Abstract base class defining the model interface
├── ParticleFilter       — Generic Sequential Monte Carlo algorithm
├── CrackGrowthModel     — Paris' Law implementation of StateSpaceModel
├── RemainingUsefulLife  — Post-hoc RUL estimator
└── main()               — Driver: assimilation + prediction + plotting

StateSpaceModel (ABC)

Defines the interface any domain model must implement:

Method	Description
sample_initial(N)	Returns (N, d) array of initial particles
transition(particles)	Propagates particles one step forward, returns (N, d)
likelihood(measurement, particles)	Returns (N,) likelihoods
estimate(particles, weights)	Returns (mean, var) — has a sensible weighted default

ParticleFilter

Generic SMC filter — no domain knowledge baked in:

Method	Description
predict()	Calls model.transition() on all particles
update(measurement)	Weights particles by likelihood, normalises
resample()	Systematic resampling
resample_if_needed()	Resamples when neff < N/2
neff()	Effective sample size: 1 / Σwᵢ²
estimate()	Delegates to model.estimate()
prognosis(threshold)	Fraction of particles (crack column) exceeding threshold

CrackGrowthModel

Implements Paris' Law crack growth:

da/dN = C · (ΔS √(π·a))^m

State vector per particle: [crack_size, m, c]

Parameter	Default	Description
sigma	0.001	Measurement noise (log-normal std)
stress_range	78	Cyclic stress range ΔS
dN	50	Cycle increment per step
threshold	0.015	Failure crack size

Installation

pip install numpy scipy matplotlib

Usage

python ParticleFilter.py

This runs with 100,000 particles and two crack size measurements, producing three plots:

1. Crack mean ± 95% CI over cycles
2. Probability of failure over cycles
3. RUL histogram across all particles

Custom measurements

from ParticleFilter import ParticleFilter, CrackGrowthModel

model = CrackGrowthModel()
pf = ParticleFilter(num_particles=10000, model=model)

for measurement in my_measurements:
    pf.predict()
    pf.update(measurement)
    pf.resample_if_needed()
    mean, var = pf.estimate()
    print(f"Crack estimate: {mean:.4f} ± {var**0.5:.4f}")

Custom state-space model

Subclass StateSpaceModel to apply the filter to any domain:

from ParticleFilter import StateSpaceModel, ParticleFilter
import numpy as np

class MyModel(StateSpaceModel):
    def sample_initial(self, N):
        return np.random.randn(N, 2)          # (N, d)

    def transition(self, particles):
        particles[:, 0] += 0.1 * particles[:, 1]
        return particles

    def likelihood(self, measurement, particles):
        return np.exp(-0.5 * (particles[:, 0] - measurement)**2)

pf = ParticleFilter(5000, MyModel())

Reference

• Gordon, N.J., Salmond, D.J., Smith, A.F.M. (1993). Novel approach to nonlinear/non-Gaussian Bayesian state estimation. IEE Proceedings F.
• Paris, P., Erdogan, F. (1963). A critical analysis of crack propagation laws. Journal of Basic Engineering.

Author

Spandan Mishra