The role of parametrizations in identification of linear systems

In identification the problem is to attach to every string of data of the form y₁,...,y_T; Y_t∈R^s , a system from an a priori specified model class. Usually the model class is described by a space of free parameters. In the fully automated case, the system (or its free parameters) is attached to the data by a function ψ. If the data are assumed to be generated by an underlying stochastic process (y_t|t ∈ Z) (called the data generating process, DGP) and if ψ is measurable, then ψ is an estimator and the identification problem is an estimation problem. The special features of system identification arise from the rather complicated relation between external behavior, internal system parameters and free parameters for a given model class. For simplicity here we consider linear, finite dimensional, time-invariant, causal and stable systems only, where in addition the only inputs are unobserved white noise. We discuss state space and ARMA forms