Page 109 - Classification Parameter Estimation & State Estimation An Engg Approach Using MATLAB

98 STATE ESTIMATION

z
The interpretation is as follows. ^ z(i)is the predicted measurement.It is an
unbiased estimate of z(i) using all information from the past. The
so-called innovation matrix S(i) represents the uncertainty of the predicted
measurement. The uncertainty is due to two factors: the uncertainty of
x(i) as expressed by C(iji 1), and the uncertainty due to the measure-
ment noise v(i) as expressed by C v (i). The matrix K(i)is the Kalman gain
T
matrix. This matrix has large, when S(i) is small and C(iji 1)H (i)is
large, that is, when the measurements are relatively accurate. When this is
the case, the values in the error covariance matrix C(iji) will be much
smaller than C(iji 1).
The prediction, i.e. the determination of p(x(i þ 1)jZ(i)) given
p((x(i)jZ(i)), boils down to finding out how the expectation x(iji) and
the covariance matrix C(iji) propagate to the next state. Using (4.11) and
(4.13) we have:

xði þ 1jiÞ¼ FðiÞxðijiÞþ LðiÞuðiÞ
ð4:28Þ
T
Cði þ 1jiÞ¼ FðiÞCðijiÞF ðiÞþ C w ðiÞ
At this point, we increment the counter, and x(i þ 1ji) and C(i þ 1ji)
become x(iji 1) and C(iji 1). These recursive equations are generally
referred to as the discrete Kalman filter (DKF).
In the Gaussian case, it does not matter much which optimality
criterion we select. MMSE estimation, MMAE estimation and MAP
estimation yield the same result, i.e. the conditional mean. Hence, the
final estimate is found as ^ x(i) ¼ x(iji). It is an absolute unbiased estimate,
x
and its covariance matrix is C(iji). Therefore, this matrix is often called
the error covariance matrix.
In the time invariant case, and assuming that the Kalman filter is
stable, the error covariance matrix converges to a constant matrix. In
that case, the innovation matrix and the Kalman gain matrix become
constant as well. The filter is said to be in the steady state. The steady
state condition simply implies that C(i þ 1ji) ¼ C(iji 1). If the notation
for this matrix is shortened to P, then (4.28) and (4.27) lead to the
following equation:

T
T
P ¼ FPF þ C w FPH T HPH þ C v 1 HPF T ð4:29Þ
The equation is known as the discrete algebraic Ricatti equation.
Usually, it is solved numerically. Its solution implicitly defines the steady
state solution for S, K and C.