Transfer function identification by minimizing the adaptive vs. optimal filter state estimates mismatch

The article is concerned with a further development of the Active Principle of parametric system identification in the class of linear, time-invariant, completely observable models. As the identification target model, the optimal Kalman filter (OKF) is designated that is present, no more than conceptually, in the system’s discretely observed response to a training excitation of the white noise type. By modifying the physically given structure into the standard observable model in both the observed response and the Adaptive Kalman Filter (AKF), a so-called Generalized Residual (GR) is constructed equaling the mismatch between the adaptive and the optimal filter state estimates plus an AKF-independent noise component. By virtue of this modification, the GR mean square becomes a new model proximity criterion for these filters. Minimizing this criterion via conventional practical optimization methods produces exactly the same result (AKF = OKF) as would be obtained by minimizing the theoretical criterion being, unfortunately, inaccessible to any AKF numerical optimization methods. The article presents a detailed step-by-step procedure explaining the above solution in terms of a parameterized transfer function. For the sake of clarity and for stimulating real world applications of the approach, the article employs the transfer function model of a twisted-pair line in a typical xDSL system. The implementation challenges of theoretical provisions of the method are discussed. The issue of extending the proposed approach to the problems of identifying linear models for nonlinear systems is outlined in the directions for further research.