System Identification with Multi-Step Least-Squares Methods

Sammanfattning: The purpose of system identification is to build mathematical models for dynam-ical systems from experimental data. With the current increase in complexity of engineering systems, an important challenge is to develop accurate and computa-tionally simple algorithms, which can be applied in a large variety of settings.With the correct model structure, maximum likelihood (ML) and the predictionerror method (PEM) can be used to obtain (under adequate assumptions) asymp-totically efficient estimates. A disadvantage is that these methods typically requireminimizing a non-convex cost function. Alternative methods are then needed toprovide initialization points for the optimization.In this thesis, we consider multi-step least-squares methods for identificationof dynamical systems. These methods have a long history for estimation of timeseries. Typically, a non-parametric model is estimated in an intermediate step, andits residuals are used as estimates of the innovations of the parametric model ofinterest. With innovations assumed known, it is possible to estimate the parametricmodel with afinite number of least-squares steps. When applied with an appropriateweighting orfiltering, these methods can provide asymptotically efficient estimates.The thesis is divided in two parts. In thefirst part, we propose two methods:model order reduction Steiglitz-McBride (MORSM) and weighted null-spacefitting(WNSF). MORSM uses the non-parametric model estimate to create a simulateddata set, which is then used with the Steiglitz-McBride algorithm. WNSF is a moregeneral approach, which motivates the parametric model estimate by relating thecoefficients of the non-parametric and parametric models.In settings where different multi-step least-squares methods can be applied, weshow that their algorithms are essentially the same, whether the estimates are basedon estimated innovations, simulated data, or direct relations between the modelcoefficients. However, their range of applicability may differ, with WNSF allowing usto establish a framework for multi-step least-squares methods that is quiteflexible inparametrization. This is specially relevant in the multivariate case, for which WNSFis applicable to a large variety of model structures, including both matrix-fractionand element-wise descriptions of the transfer matrices.We conduct a rigorous statistical analysis of the asymptotic properties of WNSF,where the main challenge is to keep track of the errors introduced by truncationof the non-parametric model, whose order must tend to infinity as function of thesample size for consistency and asymptotic efficiency to be attained. Moreover, weperform simulation studies that show promising results compared with state-of-the-art methods.In the second part, we consider extensions of the developed methods for appli-cability in other settings. These include unstable systems, recursive identification,dynamic networks, and cascaded systems.