Model sets and parametrizations for identification of multivariable equation error models

Abstract Equation error (or linear regression) models are known to inherently require the a priori choice for specific signal variables to be considered as regressand and/or regressor. This implies that a model set should be—a priori—restricted in some way in order to define an acceptable identification problem. In the case of approximate identification (i.e. the system to be modelled is not contained in the model set), this restriction acts as a design variable, with the identified models being dependent on its specific choice. In this paper the necessity of this restriction is quantified by the property of discriminability, i.e. the ability of an identification criterion to distinguish between all the different models in a model set. Employing a deterministic, signal-oriented framework, several sets of sufficient conditions are derived for model sets to be discriminable by a least squares identification criterion. To this end use is made of polynomial model representations in two shift operators. Although it is of a different nature, the problem discussed is shown to be closely related to the problem of constructing identifiable parametrizations for sets of rational transfer functions. It is shown that the pseudo-canonical or overlapping parametrization of all transfer functions with fixed McMillan degree constitutes a nonoverlapping set of equation error models that is discriminable by a least squares identification criterion.