Author: Simone Garatti
Advisors: Prof. Sergio Bittanti, Prof. Marco C. Campi
Abstract: Model quality assessment is a very important (and also challenging) problem in system identification. In fact, it has been widely recognized that an identified model is of little use in practical applications if an estimate of its reliability is not given along with the model itself. More precisely, if P is the data-generating system and M is the identified model, it is fundamental to characterize the system-model mismatch, i.e. the distance between P and M. One of the best-known tools for model quality assessment is the asymptotic theory of system identification. The asymptotic theory is set in a probabilistic framework and returns ellipsoidal confidence regions for P- namely, regions in the parameter space to which the data-generating system parameter belongs with a pre-assigned probability - which have been proved to be asymptotically correct, as the number of data points increases. The main advantage of using the asymptotic theory is that the confidence regions can be easily computed from the available data and, moreover, these confidence regions are often reliable and give a tight description of uncertainty. On the other hand, the asymptotic theory is rigourously correct only when the number of data points tends to infinity while in practice one is always supplied with a finite data record. As a consequence, the asymptotic theory is used as a heuristic tool for the model quality evaluation and, though it is common experience that the asymptotic theory returns sensible results in many cases, it is also true that in the case of wide uncertainty (i.e. M far from P) the computed ellipsoid could lead to misleading conclusions. This limitation of the asymptotic theory is severe because large uncertainty is common in many applications (e.g. when the identification has to be performed in closed-loop with restricted bandwidth leading to poorly exciting input signals). Moreover, at a more general level one can argue that model quality assessment becomes more important when the system-model mismatch is significant. One goal of this work is to critically discuss the applicability of the asymptotic theory in practice. The focus is on Prediction Error Methods (PEM) as well as Instrumental Variable (IV) techniques. In both cases, the theoretical conditions for a safe use of the asymptotic theory in practice are critically analyzed.Thanks to this analysis it is possible to show that in many different identification settings the asymptotic theory exhibits a certain ``robustness'' so that the ellipsoidal confidence regions prescribed by the asymptotic theory can be safely used to assess the quality of the identified model in practice. Yet, there are situations where the asymptotic theory should be used with care since it may lead to erroneous results.
As an application of this analysis, in the second part of the thesis the design of iterative control schemes is considered. Iterative control (which has been widely studied in recent years) is an efficient methodology for the design of highly-performing controllers. The main idea is to perform a sequence of closed-loop identification and control design steps so that the designer learns how to increase the control-loop performance through experience as time progresses. In this thesis an iterative control scheme which explicitly accounts for the presence of uncertainty in the plant model is studied. The uncertainty description is updated at each iteration based on the results of the first part of this thesis. In this way, the controller designed at each iteration is tuned to the present uncertainty level so that the control performance increases quickly, while preserving robust stability.