Page 144 - Classification Parameter Estimation & State Estimation An Engg Approach Using MATLAB

MIXED STATES AND THE PARTICLE FILTER 133

instance, if v(i) is zero mean, Gaussian with covariance matrix C v , the
weights are calculated as:

1 T 1

ðkÞ
ðkÞ
w ðkÞ ¼ constant exp ðzðiÞ hðx ÞÞ C ðzðiÞ hðx ÞÞ
2 v
The actual value of constant is irrelevant because of the normalization.
The drawing of new samples in the prediction step involves the state
equation. If x(i þ 1) ¼ f(x(i),u(i)) þ w(i), then the drawing is governed
by the density of w(i).
The advantages of the particle filtering are obvious. Nonlinearities
of both the state equation and the measurement function are handled
smoothly without the necessity to calculate Jacobian matrices. How-
ever, the method works well only if enough particles are used. Espe-
cially for large dimensions of the state vector the required number
becomes large. If the number is not sufficient, then the particles are
not able to represent the density p(x(i)jZ(i 1)). Particularly, if for
some values of x(i) the likelihood p(z(i)jx(i)) is very large, while on
these locations p(x(i)jZ(i 1)) is small, the particle filtering may not
converge. It occurs frequently then that all weights become zero
except one which becomes unit. The particle filter is said to be
degenerated.

Example 4.10 Particle filtering applied to volume density estimation
The problem of estimating the volume density of a substance mixed
with a liquid is introduced in Example 4.1. The model, expressed in
equation (4.3), is nonlinear and non-Gaussian. The state vector
consists of two continuous variables (volume and density), and one
discrete variable (the state of the on/off controller). The measure-
ment system, expressed in equation (4.40), is nonlinear with additive
Gaussian noise. Example 4.6 has shown that the EKF is able to estimate
the density, but only by using an approximate model of the process
in which the discrete state is removed. The price to pay for such a
rough approximation is that the estimation error of the density and
(particularly) the volume has a large magnitude.
The particle filter does not need such a rough approximation
because it can handle the discrete state variable. In addition, the
particle filter can cope with discontinuities. These discontinuities
appear here because of the discrete on/off control, but also because
the input flow of the substance occurs in chunks at some random
points in time.