Bounds on the simulation power of equation error estimates

Several results concerning the properties of least-squares models for linear, time-invariant, discrete-time SISO systems are derived. The model set used is the set of all autoregressive moving-average (ARMA) models of order ( mu , nu ) or less. The authors consider properties of least squares (equation error) models of systems which are not in the model set. In particular, the difference between the actual and model impulse response is characterized. It is shown that although asymptotically the first mu plus 1 impulse response elements are matched exactly (where mu is the numerator order of the model transfer function) the rest of the fit can be quite poor. Several theorems are presented which describe the properties of this fit.