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 ż = Az + Hx with y = Cz + Dx + ν (ν = observation noise), and show that the results reduce to those for the first problem as observation noise ν tends to zero. As an illustration, the application to the calibration of four parameters in a two-axis gyro is presented.