Identification aspects of SDP based polynomial optimization relaxations

The goal of parametric system identification is to provide estimates for parameters of a certain model structure based on given measurement data. This problem can always be presented as an optimization problem with an appropriate choice of cost and constraint functions. Apart from the simplest cases the resulting optimization problems are nonconvex with multiple local minima. Due to the existence of these, usually there are no guaranties that the model resulting from a given identification method is a global minimizer. This paper applies semidefinite programming (SDP) relaxation techniques to the optimization problem arising in time domain identification. From the solution of the defined sequence of SDPs a sequence of system models can be extracted that converges to the globally optimal system model. We give a short overview of the SDP relaxation technique for polynomial optimization problems (POP), then this technique is applied to the identification problem. The properties of the resulting POP are examined in detail. The solutions of the SDP sequence usually converge to the optimizer only in the limit. Model structures where finite convergence occurs and issues regarding detecting finite convergence are also considered.