Block-Toeplitz/Hankel Structured Total Least Squares

A structured total least squares problem is considered in which the extended data matrix is partitioned into blocks and each of the blocks is block-Toeplitz/Hankel structured, unstructured, or exact. An equivalent optimization problem is derived and its properties are established. The special structure of the equivalent problem enables us to improve the computational efficiency of the numerical solution methods. By exploiting the structure, the computational complexity of the algorithms (local optimization methods) per iteration is linear in the sample size. Application of the method for system identification and for model reduction is illustrated by simulation examples.

[1]  M. Levin.  Estimation of a system pulse transfer function in the presence of noise , 1964 .

[2]  B. Moore.  Principal component analysis in linear systems: Controllability, observability, and model reduction , 1981 .

[3]  J. Vandewalle,et al.  Analysis and properties of the generalized total least squares problem AX≈B when some or all columns in A are subject to error , 1989 .

[4]  Jerry M. Mendel,et al.  The constrained total least squares technique and its applications to harmonic superresolution , 1991, IEEE Trans. Signal Process..

[5]  Sabine Van Huffel,et al.  Total least squares problem - computational aspects and analysis , 1991, Frontiers in applied mathematics.

[6]  B. Moor.  Structured total least squares and L2 approximation problems , 1993 .

[7]  Bart De Moor,et al.  Total least squares for affinely structured matrices and the noisy realization problem , 1994, IEEE Trans. Signal Process..

[8]  B. De Moor,et al.  L/sub 2/-optimal linear system identification structured total least squares for SISO systems , 1994, Proceedings of 1994 33rd IEEE Conference on Decision and Control.

[9]  Rik Pintelon,et al.  A Gauss-Newton-like optimization algorithm for "weighted" nonlinear least-squares problems , 1996, IEEE Trans. Signal Process..

[10]  J. Ben Rosen,et al.  Total Least Norm Formulation and Solution for Structured Problems , 1996, SIAM J. Matrix Anal. Appl..

[11]  Sabine Van Huffel,et al.  Recent advances in total least squares techniques and errors-in-variables modeling , 1997 .

[12]  Sabine Van Huffel,et al.  SLICOT—A Subroutine Library in Systems and Control Theory , 1999 .

[13]  S. Huffel,et al.  Direct and neural techniques for the data least squares problem , 2000 .

[14]  Sabine Van Huffel,et al.  Fast Structured Total Least Squares Algorithm for Solving the Basic Deconvolution Problem , 2000, SIAM J. Matrix Anal. Appl..

[15]  S. Huffel,et al.  Total Least Squares and Errors-in-Variables Modeling : Analysis, Algorithms and Applications , 2002 .

[16]  M. Peruggia.  Total Least Squares and Errors-in-Variables Modeling: Analysis, Algorithms and Applications , 2003 .

[17]  Structured weighted low rank approximation , 2004, Numer. Linear Algebra Appl..

[18]  Sabine Van Huffel,et al.  Fast algorithm for solving the Hankel/Toeplitz Structured Total Least Squares problem , 2004, Numerical Algorithms.

[19]  Sabine Van Huffel,et al.  On the computation of the multivariate structured total least squares estimator , 2004, Numer. Linear Algebra Appl..

[20]  J. Willems,et al.  Application of structured total least squares for system identification and model reduction , 2005, IEEE Transactions on Automatic Control.

[21]  I. Markovsky,et al.  Consistency of the structured total least squares estimator in a multivariate errors-in-variables model , 2005 .

[22]  Gene H. Golub,et al.  An analysis of the total least squares problem , 1980, Milestones in Matrix Computation.