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Paul Sundvall www.s3.kth.se/~pauls Presentation in course “Optimal filtering” Signals, Sensors and Systems, KTH November 11th 2004. An introduction to Particle filtering. Outline. Introduction Comparison with the Kalman filter Description of the algorithm Implementation Example.

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slide1
Paul Sundvall

www.s3.kth.se/~pauls

Presentation in course “Optimal filtering”

Signals, Sensors and Systems, KTH

November 11th 2004

An introduction to Particle filtering
outline
Outline
  • Introduction
  • Comparison with the Kalman filter
  • Description of the algorithm
  • Implementation
  • Example

Paul Sundvall

introduction
Introduction

Particle filtering

  • is a method for state estimation
  • is a Monte Carlo method
  • handles nonlinear models with non-Gaussian noise

Paul Sundvall

comparison to the discrete kalman filter

Kalman filter

Particle filter

State equation

Noise type

Output

Solution

Exact, optimal

Approximate

Computational speed

Fast

Slow

Comparison to the discrete Kalman filter

Any distribution, uni- or multimodal

Gaussian, unimodal

Paul Sundvall

significant property
Significant property

The particle filter gives an approximate solution to an exact model, rather than the optimal solution to an approximate model.

Paul Sundvall

algorithm
Algorithm
  • The propability density function is approximated using point weights
  • Each point is called a particle
  • Each particle has a positive weight

Basic algorithm:

  • Initialize
  • Time update (move particles)
  • Measurement update (change weights)
  • Resample (if needed)
  • Goto 2 when new measurement arrives
  • Each point is called a particle
  • Each particle has a positive weight
  • Initializew
time update
Time Update

One-step prediction of each particle

Note that a realization of the

process noise is used for every

particle.

measurement update
Measurement update
  • The weights are adjusted using the measurement
  • All weights are normalized
  • Particles that can explain the measurement gain weight
  • Particles far off the true state lose weight.
  • The density of the cloud changes
resampling
Resampling
  • It can be shown that the algorithm degenerates
  • Allt particles but one become very light

Solved by resampling so that all weights become equal

implementation
Implementation
  • Calculation demand is proportional to the number of particles
  • The approximation error decreases as the number of particles grow
  • N can easily be changed during runtime
  • One needs to know what to do with p(x)

is not a good choice for multimodal distributions!

example
Example
  • A boat travels on a one-dimensional sea
  • Noisy depth measurements are given
  • Given a perfect sea-chart d(x), estimate the position!
  • Matlab code for the example is available on www.s3.kth.se/~pauls
final comments
Final comments
  • Better the more multimodal, non-linear and non-gaussian the system is
  • The most basic variants are simple to implement
  • It is easy to add model knowledge (saturation, limit checking, nonnegativeness...)
  • Variants of the particle filters exist
    • To reduce the number of particles needed, by combining Kalman filters and particle filters
    • To ensure that states with low propability but high risk are tracked despite few particles (fault detection)
    • To use discrete states
    • To use both discrete and continuous states (hybrid)
references
References
  • A good introduction is given in”A tutorial on Particle Filters for Online Nonlinear/Non-Gaussian Bayesian Tracking”M. Sanjeev Arulampalam et. al.
  • Very good application examples can be found in”Particle Filters for Positioning, Navigation and Tracking”Fredrik Gustafsson et. al.
  • A description of how particle filters can be used for fault detection is found in”Particle Filters for Rover Fault Diagnosis”Vandi Verma et. al.