A Chebyshev system approach to optimal input design

This paper considers the problem of generating test signals for estimating
P
parameters in linear single-input single-output stochastic systems using a frequency domain approach. The input signals are power constrained and are optimal in the sense of maximizing system information where the criterion employed is a scalar function of the inverse Fisher information matrix. Minimal properties of test signals are discussed and a necessary and sufficient condition for local identifiability is proved based on the concept of an input design index. Conditions are sought under which optimality is achieved by test signals whoso design index is the minimum necessary to ensure identifiability. It is shown that the theory of Chebyshev systems (
T
-systems) furnishes a fruitful approach in which the set of information matrices can be represented as a convex set (moment space) induced in R
^p
by a
T
-system. These considerations lead to the concept of canonical and principal spectral representations of test signals with the required minimal properties and to sufficient conditions for the existence of optimal designs of the desired type, placing only weak restrictions on the choice of optimality criterion. A number of examples are presented that illustrate the usefulness of a geometrical approach.