Introduction

So far we have seen how polynomial Kalman filters perform when the measurement is a polynomial signal plus noise. In the real world most measurements cannot be described exactly by simple polynomials, and it is therefore of considerable practical interest to see how well the polynomial Kalman filter performs under these circumstances and if anything can be done to improve performance. Two examples will be picked in which the real world cannot be described by a polynomial signal corrupted by noise. Several possible Kalman-filter designs will be presented to illustrate key concepts.

Polynomial Kalman Filter and Sinusoidal Measurement

Suppose that the actual measurement is a pure sine wave of unity amplitude corrupted by noise or

where ? is the frequency of the sinusoidal signal and the noise is zero-mean Gaussian with a standard deviation of unity. We would like to build a Kalman filter that will be able to track the sinusoid and estimate its states based on the noisy sinusoidal measurement.

Because the true signal is the sinusoid

its derivative is given by

For the sinusoidal measurement signal we can first attempt to use the first-order polynomial Kalman filter. The polynomial Kalman filter is a reasonable choice because at this point we have not yet studied any other kind of Kalman filter.

Recall that the general formula for the Kalman filter, assuming no known deterministic inputs, is given by

For the first-order polynomial Kalman filter we have already shown that the fundamental and measurement matrices are given...