Kalman Filtering 57
with it. Let us call that R. We are now ready to formally list the steps in the
Kalman-filtering process as follows:
- Evaluate and using the state equation.
- Find the observation YtandRby observing the system.
- Evaluate Kt, also known as the Kalman gain, which will be used to ob-
tain the linear minimum error variance estimate. - Evaluate given by.
- Finally, evaluate Pt, the error variance/covariance of.
These steps are repeated again for the next time step. The formulas for
the evaluations at each step are relegated to the appendix at the end of this
chapter. Upon examining the equations in the appendix, one can say that
they do seem a little cryptic, and the reasoning and rationale behind them is
not evident. In the subsequent sections we will illustrate the ideas behind the
equations.
The Scalar Kalman Filter
In this section, we will discuss the estimation of the value of a constant. Let
us first examine how to do it in normal course. The typical method would
be to take multiple measurements of the value and use the average of the
measured values as an estimate of the constant. The reasoning behind the
approach is that the measurements could have errors associated with them;
that is, some measured values could be greater than the true value of the
constant, and others could be lower. By taking an average of the values, we
expect the errors to cancel each other out. More precisely, the standard de-
viation of the error in the average goes down by a square root of nfactor,
wherenis the number of measurements.
The consequence of this method is in fact a well-known statistical con-
cept. By increasing the number of observations of a constant variable and
taking averages, we can make the error in our estimate of the constant as
small as desired.
However, a caveat to that approach is that we will need to wait until the
last of the nmeasurements have been completed before coming up with an
estimate of the constant (which may not be a bad idea at all). The Kalman
filter, however, makes an estimate of the value of the constant based on the
current available information and updates the estimate as and when more
observations are made. Of course, the result after nobservations in both
cases will be the same.
ˆ
Xtt|ˆˆ
Xˆtt| XKYHXtt||−− 11 +−t()t ttˆ
Ptt|− 1ˆ
Xtt|− 1