The identification of an undermodeled transfer function from input-output data is stated as a constrained optimization problem. The constraints determine the identification procedure, the residual error and whether on average the magnitude of the frequency response is L2-0verhiased, L2-underbiased or L2-unbiased, as measured by a certain weighted Le-bias integral. The L2-unhiased solutions are linear combinations of L2-0verhiased and L2-underhiased solutions, which are precisely the classical least squares estimates. They can be obtained from the solution of certain eigenvalue problems.
1. Introduction: idenafication as constrained minimization
IN THIS PARER, we put some identification approaches for undermodeling (i.e. the model set does not contain the "true" system) of SISO (single-input/single-output) systems into a general framework of constrained minimization. Using an L2-error criterion, it will be shown that, depending on the constraints, the corresponding model can be L2-0verbiased, L2-underbiased or L2-unbiased, as quantified by a certain L2-bias integral, which can be elegantly derived from the Lagrangean of the optimization problem. The main purpose of this work is to investigate the interaction between the specific constraints and some properties of the resulting identified model.