Variance-error quantification for identified poles and zeros

This paper deals with quantification of noise induced errors in identified discrete-time models of causal linear time-invariant systems, where the model error is described by the asymptotic (in data length) variance of the estimated poles and zeros. The main conclusion is that there is a fundamental difference in the accuracy of the estimates depending on whether the zeros and poles lie inside or outside the unit circle. As the model order goes to infinity, the asymptotic variance approaches a finite limit for estimates of zeros and poles having magnitude larger than one, but for zeros and poles strictly inside the unit circle the asymptotic variance grows exponentially with the model order. We analyze how the variance of poles and zeros is affected by model order, model structure and input excitation. We treat general black-box model structures including ARMAX and Box–Jenkins models.