L2-optimal linear system identification structured total least squares for SISO systems

Treats the extension of total linear least squares to single-input single-output dynamic systems. It is shown that the L₂-optimal model for a given dataset has some remarkable properties: The given data sequence can be decomposed into two sequences, the L₂-approximations and the residuals, which are orthogonal to each other (in diagonal inner products derived from given weights); It is shown how certain block Hankel matrices W˜_i constructed from the weighted residuals are always rank deficient and hence, the residuals themselves can be considered as being generated by a linear system. One can find completions Wˆ_i^com of the block Hankel matrices Wˆ_i with the L₂ -approximations, such that Wˆ_i^com W˜_i^T=0 (which is a more general type of orthogonality than the first one mentioned). The results of this paper are just a special application of a general theory to approximate in a least squares sense, a given structured matrix, by one which has the same structure and is rank deficient. The main result of this theory says that, when the matrix structure is affine, the solution is generated in terms of a Riemannian SVD, which is a `nonlinear´ generalized singular value decomposition