III-posed and well-posed problems in systems identification

Impulse response identification almost always leads to an ill-posed mathematical problem. This fact is the basis for the well-known numerical difficulties of identification by means of the impulse response. The theory of regularizable ill-posed problems furnishes a unifying point of view for several specific methods of impulse response identification. In this paper we introduce a class of input/output representations, which we call λ-representations, for linear, time-invariant systems. For many cases of practical interest the identification of one of these representations is mathematically well-posed. Its determination is thus relatively insensitive to certain experimental uncertainties, and rational error-in-identification bounds may be found, so that λ-identification is often an attractive alternative to impulse response identification in the nonparametric modeling of physical systems which must be identified from input/ output records. We investigate the effects of input and output uncertainties (noise) on λ-identification, and discuss the problem of finding minimal realizations from these representations. We illustrate the work with an example of electromagnetic pulse (EMP) threat prediction using experimental data. Hard error bounds are provided on the predicted threat. For this problem, the appropriate λ-representation turns out to be the ramp response.