# Indirect System Identification for Unknown Input Problems : With Applications to Ships

Sammanfattning: System identification is used in engineering sciences to build mathematical models from data. A common issue in system identification problems is that the true inputs to the system are not fully known. In this thesis, existing approaches to unknown input problems are classified and some of their properties are analyzed. A new indirect framework is proposed to treat system identification problems with unknown inputs. The effects of the unknown inputs are assumed to be measured through possibly unknown dynamics. Furthermore, the measurements may also be dependent on other known or measured inputs and can in these cases be called indirect input measurements. Typically, these indirect input measurements can arise when a subsystem of a larger system is of interest and only a limited set of sensors is available. Two examples are when it is desired to estimate parts of a mechanical system or parts of a dynamic network without full knowledge of the signals in the system. The input measurements can be used to eliminate the unknown inputs from a mathematical model of the system through algebraic manipulations. The resulting indirect model structure only depends on known and measured signals and can be used to estimate the desired dynamics or properties. The effects of using the input measurements are analyzed in terms of identifiability, consistency and variance properties. It is shown that cancelation of shared dynamics can occur and that the resulting estimation problem is similar to errors-in-variables and closed-loop estimation problems because of the noisy inputs used in the model. In fact, the indirect framework unifies a number of already existing system identification problems that are contained as special cases.For completeness, an instrumental variable method is proposed as one possibility for estimating the indirect model. It is shown that multiple datasets can be used to overcome certain identifiability issues and two approaches, the multi-stage and the joint identification approach, are suggested to utilize multiple datasets for estimation of models. Furthermore, the benefits of using the indirect model in filtering and for control synthesis are briefly discussed. To show the applicability, the framework is applied to the roll dynamics of a ship for tracking of the loading conditions. The roll dynamics is very sensitive to changes in these conditions and a worst-case scenario is that the ship will capsize. It is assumed that only motion measurements from an inertial measurement unit (IMU) together with measurements of the rudder angle are available. The true inputs are thus not available, but the measurements from the IMU can be used to form an indirect model from a well-established ship model. It is shown that only a subset of the unknown parameters can be estimated simultaneously. Data was collected in experiments with a scale ship model in a basin and the joint identification approach was selected for this application due to the properties of the model. The approach was applied to the collected data and gave promising results.

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