Classical System Identification

Abstract

AbstractSystem identification as a field has been around since the 1950s with roots from statistical theory. A substantial body of concepts, theory, algorithms and experience has been developed since then. Indeed, there is a very extensive literature on the subject, with many text books, like [5, 8, 12]. Some main points of this “classical” field are summarized in this chapter, just pointing to the basic structure of the problem area. The problem centres around four main pillars: (1) the observed data from the system, (2) a parametrized set of candidate models, “the Model structure”, (3) an estimation method that fits the model parameters to the observed data and (4) a validation process that helps taking decisions about the choice of model structure. The crucial choice is that of the model structure. The archetypical choice for linear models is the ARX model, a linear difference equation between the system’s input and output signals. This is a universal approximator for linear systems—for sufficiently high orders of the equations, arbitrarily good descriptions of the system are obtained. For a “good” model, proper choices of structural parameters, like the equation orders, are required. An essential part of the classical theory deals with asymptotic quality measures, bias and variance, that aim at giving the best mean square error between the model and the true system. Some of this theory is reviewed in this chapter for estimation methods of the maximum likelihood character.