Ill-conditioned inverse problems arising in closed loop system identification

This thesis addresses ill-conditioning that arises in several aspects of closed loop system identification. System identification is a means of obtaining a mathematical model of a process from experimental data. In particular, we view system identification as an inverse problem (i.e. inversion of experimental data to obtain a model). However, inherent to all inverse problems is the complication of ill-conditioning which can affect the accuracy of the obtained solution. In closed loop, the input of a process is dependant, in some form, on its current output which adds a further ambiguity to the inverse problem. In this context, we examine several features of the identification problem which have the potential to lead to ill-conditioning. Aspects considered include both non-parametric and parametric closed loop identification, iterative controller tuning and numerical issues arising with wide-band data. Specific new results contained in this thesis include: Use of an exclusion zone to obtain bias and variance estimates for non-parametric estimation based on finite closed loop data; confirmation of the above results by use of an alternative regularisation of Tikhonov type; analysis of the sensitivity of parametric closed loop estimates to the fidelity of the noise model; analysis of the effect of ill-conditioning on iterative control law tuning and adaptive control; methods for overcoming spurious unstable pole-zero cancellations arising in parametric closed loop identification; and a novel set of basis functions which improve numerical ill-conditioning in identification applied to wide-band systems. These results show how by appropriate use of regularisation techniques one can ameliorate the impact of ill-conditioning in closed loop system identification.