Fundamentals of Kalman Filtering: A Practical Approach, Second Edition
Containing numerous real-life problems and examples presented in detail, this text demonstrates the many ways in which Kalman filters can be designed.
TABLE OF CONTENTS Fundamentals of Kalman Filtering: A Practical Approach, Second Edition
Chapter 10: Tracking a Sine Wave
Introduction
In this chapter we will attempt to apply extended Kalman filtering to a problem we briefly investigated with a linear Kalman filter. We will revisit the problem of tracking a sinusoidal signal measurement corrupted by noise. We have already shown that if the frequency of the sinusoid was known in advance the signal could be tracked quite well with a linear Kalman filter. However, if the frequency of the sinusoidal signal is unknown either our model of the real world or our measurement model becomes nonlinear, and we must resort to an extended Kalman filter. Several possible extended Kalman filters for this application, each of which has different states, will be explored in this chapter.
Extended Kalman Filter
In Chapter 5 we attempted to estimate the states (i.e., derivatives) of a sinusoidal signal based on noisy measurement of the sinusoidal signal. We showed that a linear polynomial Kalman filter was adequate for estimation purposes, but a much better linear Kalman filter, which made use of the fact that we knew that the true signal was sinusoidal, could be constructed. However, it was also demonstrated that if our a priori information was in error (i.e., knowledge of the frequency of the sinusoid is in error) the performance of the better linear Kalman filter deteriorated to the point where the estimates were no better, and possibly worse, than that of the linear polynomial Kalman filter (i.e., linear Kalman filter required no a priori information at all). In this section we will...
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