Optimal gains of an α–β filter derived from a tracking method using a weighted measurement covariance matrix

We describe the α–β filter, which is a tracking method in one-dimensional space that estimates the true values of the position and velocity from observations of the target position. We disclose a tracking method, known as the observed noise multiplication method, that multiplies a constant l not limited to being greater than 1 by the observed noise covariance matrix in the calculation of the smoothed error covariance matrix in a Kalman filter. The α–β filter is obtained from this observed noise multiplication method. The tracking performance of this α–β filter was evaluated by the tracking accuracy of a target with constant-velocity linear motion based on the condition that the steady-state tracking error for a constant-acceleration target is constant (equivalent to a constant gain β). The gain α of the observed noise multiplication method is a function of the constant l. We see from the evaluation result that a constant l exists where the tracking accuracy of the α–β filter derived from the observed noise multiplication method is optimized. However, it is unclear whether this constant l generally optimizes the tracking accuracy of the general α–β filter or not. In this paper, we improve the α–β filter derived from the observed noise multiplication method so that the optimum gain α is uniquely determined. We see from this result that the variance of the tracking error is about one-fourth that of an α–β filter derived from the conventional observed noise multiplication method. © 2006 Wiley Periodicals, Inc. Electron Comm Jpn Pt 1, 89(10): 34–43, 2006; Published online in Wiley InterScience (www.interscience. wiley.com). DOI 10.1002/ecja.20273