A Note on Non-Stationary Stochastic Embedding for Modelling Error Quanti cation in the Estimation of Resonant Systems

In Technical Report EE98024, the authors present a closed-form expression for the quan-tiication of model error in transfer function estimation by characterising undermodelling by a random walk process in the frequency domain. The present note shows that, for systems with resonant modes, less conservative model error quantiications may be obtained by alternatively characterising undermodelling by an integrated random walk process in the frequency domain. In Goodwin et al. (1998), the authors have described an approach to transfer function model error quantiication based on non-stationary stochastic embedding in the frequency domain. The end result of their proposal is a closed-form stochastic characterisation of the model errors due to noise and un-dermodelling. This characterisation has proved to eeectively capture typical cases of undermodelling found in practice (Goodwin, 1999). The modelling error characterisation in the aforementioned paper, however, may be overly conservative at low frequencies for systems with lightly damped modes at high frequencies. Speciically, a system with a lightly damped mode displays a large peak in its frequency response, thus if un-modelled this peak will introduce large undermodelling at the frequencies where it occur. Large undermodelling at high frequencies may propagate as excessively large undermodelling at lower frequencies , because the rate at which the sizee of the undermodelling can vary with frequency is restricted by the variance growth of the non-stationary embedded process; in Goodwin et al. (1998): a random walk.