Maximizing the determinant of the information matrix with the effective independence method

A METHOD has been presented by Kammer that addresses the problem of optimally placing sensors on a large space structure for the purpose of on-orbit modal testing. The method ranks potential sensor sites according to their contribution to the independence of the target modes and iteratively deletes the sites that have the lowest ranking. It is implied that deleting sensor sites in this way tends to maximize the determinant of the Fisher information matrix (FIM). Numerical examples are given to demonstrate that the method maintains a larger determinant for the FIM when compared with the kinetic energy method of determining optimal sensor locations. However, no relationship between the effective independence ranking and the determinant of the FIM was proven. The purpose of this comment is to provide a proof that deleting the potential sensor location with the smallest effective independence distribution (ED) value will produce the smallest relative change in the determinant of the information matrix, and so this method does provide a local maximization of the determinant of the FIM. The formal statement is included in the following theorem where the notation closely follows that of the original paper. Theorem: Let A and B, respectively, be the Fisher information matrices before and after the /th sensor is deleted; then det B -(\ -EDi) det^4 , where EDi is the effective independence distribution of the /th sensor site. The theorem is a direct consequence of the following lemma. Lemma: If B=A R f R f , where Rt is a column vector, then