An information theoretic approach to optimization of linear observations for the Kalman-Bucy filter

We are concerned with a problem of the optimal selection of the gain matrix of a linear observation mechanism for the Kalman-Bucy filter. By introducing an information theoretic constraint, we obtain a gain matrix which maximizes the reduction speed of an weighted estimation error. In this paper, we are especially concerned with the case where the weighting matrix is not positive definite but has positive eigenvalues as many as the dimension of the observation. By this condition, we can treat an observation with any dimension. This result is more general than the one obtained by the author using a formulation in the optimal transmission framework.