Page 111 - Classification Parameter Estimation & State Estimation An Engg Approach Using MATLAB

100 STATE ESTIMATION

one is much smaller, but follows the same pattern). It can be seen that
after about 3 (s) the Kalman gain (and the error covariance matrix)
remain constant. The filter has reached its steady state.

4.2.2 Suboptimal solutions for nonlinear systems

This section extends the discussion on state estimation to the more
general case of nonlinear systems and nonlinear measurement functions:

xði þ 1Þ¼ fxðiÞ; uðiÞ; iÞ þ wðiÞ
ð
ð4:31Þ
zðiÞ¼ hxðiÞ; iÞ þ vðiÞ
ð
The vector f( , , ) is a nonlinear, time variant function of the state x(i)
and the control vector u(i). The control vector is a deterministic signal.
Since u(i) is fully known, it only causes an explicit time dependency in
f( , , ). Without loss of generality, the notation can be shortened to
f(x(i),i), because such an explicit time dependency is already implied in
that shorter notation. If no confusion can occur, the abbreviation f(x(i))
will be used instead of f(x(i), i) even if the system does depend on time.
As before, w(i) is the process noise. It is modelled as zero mean, Gaussian
white noise with covariance matrix C w (i). The vector h( , ) is a non-
linear measurement function. Here too, if no confusion is possible, the
abbreviation h(x(i)) will be used covering both the time variant and the
time invariant case. v(i) represents the measurement noise, modelled as
zero mean, Gaussian white noise with covariance matrix C v (i).
Any Gaussian random vector that undergoes a linear operation retains its
Gaussian distribution. A linear operator only affects the expectation and the
covariance matrix of that vector. This property is the basis of the Kalman
filter. It is applicable to linear-Gaussian systems, and it permits a solution
that is entirely expressed in terms of expectations and covariance matrices.
However, the property does not hold for nonlinear operations. In nonlinear
systems, the state vectors and the measurement vectors are not Gaussian
distributed, even though the process noise and the measurement noise might
be. Consequently, the expectation and the covariance matrix do not fully
specify the probability density of the state vector. The question is then how
to determine this non-Gaussian density, and how to represent it in an
economical way. Unfortunately, no general answer exists to this question.
This section seeks the answer by assuming that the nonlinearities of the
system are smooth enough to allow linear or quadratic approximations.