IEEE Transactions on signal processing vol:42 issue:11 pages:3104-3113
Structured rank-deficient matrices arise in many applications in signal processing, system identification, and control theory. We discuss the structured total least squares (STLS) problem, which is the problem of approximating affinely structured matrices (i.e., matrices affine in the parameters) by similarly structured rank-deficient ones, while minimizing an L(2)-error criterion. It is shown that the optimality conditions lead to a nonlinear generalized singular value decomposition, which can be solved via an algorithm that is inspired by inverse iteration. Next we concentrate on the so-called L(2)-optimal noisy realization problem, which is equivalent with approximating a given data sequence by the impulse response of a finite dimensional, time invariant linear system of a given order. This can be solved as a structured total least squares problem. It is shown with some simple counter examples that ''classical'' algorithms such as Steiglitz-McBride, iterative quadratic maximum likelihood and Cadzow's iteration do not converge to the optimal L(2) solution, despite misleading claims in the literature.