In our context, however, x and y are
random variables, so such a precise
functional relationship will not hold.
Figure 5 shows an example in which x
and y are scalar-valued random vari-
ables. The gray ellipse in this figure,
called a confidence ellipse, is a pro-
jection of the joint distribution of x
and y onto the (x, y) plane that shows
where some large proportion of the
(x, y) values are likely to be. Suppose
x takes the value x1. Even within the
confidence ellipse, there are many
points (x1, y), so we cannot associate
a single value of y with x1. One possi-
bility is to compute the mean of the
y values associated with x1 (that is,
There are several equivalent expres-
sions for the Kalman gain for the fusion
of two estimates. The following one,
which is easily derived from Equation
23, is the vector analog of Equation 17:
The covariance matrix of the optimal estimator y(x1, x2) is the following.
Summary. The results in this section can be summarized in terms of the
Kalman gain K as shown in Figure 4.
Best Linear Unbiased Estimator
In some applications, the state of the
system is represented by a vector but
only part of the state can be measured
directly, so it is necessary to estimate
the hidden portion of the state corresponding to a measured value of the
visible state. This section describes an
estimator called the best linear unbiased
16, 19, 26 for doing this.
Consider the general problem of
determining a value for vector y given
a value for a vector x. If there is a functional relationship between x and y (say
y=F(x) and F is given), it is easy to compute y given a value for x (say x1).
Figure 6. State estimation using Kalman filtering.
(a) Discrete-time dynamical system.
(b) Dynamical system with uncertainty.
(c) Implementation of the dataflow diagram (b).
(d) Implementation of the dataflow diagram (b) for systems with partial observability.