On the calibration problem

First we consider estimating a constant
n
-dimensional parameter vector
x
using an
m
-dimensional observation vector
y = Hx
for
m < n
. Unless H is time-varying,
x
cannot be estimated. This is the case addressed. It is shown that the Kalman filtering approach yields an estimation algorithm equivalent to a direct deterministic approach which may be more practical to implement. Using Friedland's "separate bias" algorithm [1], we extend the analysis to the problem of indirect observations, i.e., for
\dot{z}= Az + Hx
with
y = Cz + Dx+\upsilon
(
\upsilon
=observation noise), and show that the results reduce to those for the first problem as observation noise
\upsilon
tends to zero. As an illustration, the application to the calibration of four parameters in a two-axis gyro is presented.