Total least squares for affinely structured matrices and the noisy realization problem

Structured rank-deficient matrices arise in many applications in signal processing, system identification, and control theory. The author discusses 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 L₂-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 the author concentrates on the so-called L₂-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 the Steiglitz-McBride (1965), iterative quadratic maximum likelihood and Cadzow´s (1988) iteration do not converge to the optimal L₂ solution, despite misleading claims in the literature