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:
( 28)
The covariance matrix of the optimal estimator y(x1, x2) is the following.
( 29)
( 30)
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
(BLUE)
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
estimator (BLUE)
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.
Predictor
Predictor
Measurement Fusion
Measurement Fusion
