A Method of Optimization of Linear Observations for the Kalman Filter based on a Generalized Water Filling Theorem
Yoshiki TAKEUCHI, Akihiro HIRATA · Proceedings of the ISCIE International Symposium on Stochastic Systems Theory and its Applications · 2008
We are concerned with a problem of the optimal selection of the gain matrix, over a given time interval, of a linear observation for the Kalman filter. The innovations process included in the Kalman filter has the same structure as the model of a set of parallel transmission channels with the optimal output feedback. In the linear coding problem for this set of channels, it is well-known that the optimal output feedback which minimizes the power of the encoded signal is given by the least-squares estimate of the linear term and that the channel output then becomes the innovations process. By applying a solution of the optimal transmission problem for this model, we obtain, at any time points, a set of the gains which maximize the mutual information between the observation and the signal under a constraint on the power of the innovations process. Finally, the optimal selection of the gain to minimize the estimation error is done by an optimization which is local in time.