Design of optimal probing signals for vector parameter estimation

In the design of optimal estimators it is common to consider some function of the error covariance matrix, usually the trace, as a criterion of optimality. In the design of optimal inputs, or probing signals, for parameter estimation it is more natural to consider functions of the Fisher information matrix instead. The input which maximizes the Fisher information measure for efficient estimation of a scalar parameter also provides the minimum error variance. The information is thus a logical choice for the optimality criterion in scalar problems. No such obvious choice is apparent for vector parameter estimation, however. In this paper we examine and compare a number of performance measures and select a useful criterion. The design of an optimal probing signal using this criterion is shown to be equivalent to an optimal control problem in which certain equality constraints must be satisfied. This problem may be solved by conventional techniques of deterministic or stochastic optimal control.