Title: OKID as a general approach to linear and bilinear system identification
Description: This work advances the understanding of the complex world of system identification, i.e. the set of techniques to find mathematical models of dynamical systems from measured input-output data, and exploits well-established approaches for linear systems to address nonlinear system identification problems. We focus on observer/Kalman filter identification (OKID), a method for simultaneous identification of a linear state-space model and the associated Kalman filter from noisy input-output measurements. OKID, developed at NASA, resulted in a very successful algorithm known as OKID/ERA (OKID followed by eigensystem realization algorithm). We show how ERA is not the only method to complete the OKID process, developing novel algorithms based on the preliminary estimation of the Kalman filter output residuals. The new algorithms do not only show potential for better performance, they also cast light on OKID, explicitly establishing the Kalman filter as central to linear system identification in the presence of noise, paralleling its role in signal estimation and filtering.

The Kalman filter embedded in the OKID core equation is capable of converting the original problem, affected by random noise, into a purely deterministic problem. The new interpretation leads to the extension of OKID to output-only system identification, providing a new tool for applications in structural health monitoring, and raises OKID to the level of a unified approach for input-output and output-only linear system identification. Any algorithm for linear system identification formulated in the absence of noise can now optimally handle noisy data via a preliminary step consisting in solving the OKID core equation. The OKID framework developed for linear system identification is then extended to bilinear systems, which are of interest because several natural phenomena are inherently bilinear and also because high-order bilinear models are universal approximators for a wide class of nonlinear systems. The formulation of an optimal bilinear observer for bilinear state-space models, similar to the Kalman filter in the linear case, leads to the development of an extension of OKID to bilinear system identification.

This is the first application of OKID to nonlinear problems, not only because bilinear systems are themselves nonlinear, but also because one can think of bilinear OKID as a technique to find bilinear approximations of nonlinear systems. Furthermore, the same strategy adopted in this work could be used to extend OKID directly to other classes of nonlinear models.