We prove that if a component of the response signal of a controllable linear time-invariant system is persistently exciting of sufficiently high order, then the windows of the signal span the full system behavior. This is then applied to obtain conditions under which the state trajectory of a state representation spans the whole state space. The related question of when the matrix formed from a state sequence has linearly independent rows from the matrix formed from an input sequence and a finite number of its shifts is of central importance in subspace system identification.

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A software package is presented that computes locally optimal solutions to low-rank approximation problems with the following features:• mosaic Hankel structure constraint on the approximating matrix,• weighted 2-norm approximation criterion,• fixed elements in the approximating matrix,• missing elements in the data matrix, and• linear constraints on an approximating matrix's left kernel basis.It implements a variable projection type algorithm and allows the user to choose standard local optimization methods for the solution of the parameter optimization problem. For an m× n data matrix, with n > m, the computational complexity of the cost function and derivative evaluation is O(m 2 n). The package is suitable for applications with n ≫ m. In statistical estimation and data modeling-the main application areas of the package-n ≫ m corresponds to modeling of large amount of data by a low-complexity model. Performance results on benchmark system identification problems from the database DAISY and approximate common divisor problems are presented.

We consider the problem of approximating an affinely structured matrix, for example a Hankel matrix, by a low-rank matrix with the same structure. This problem occurs in system identification, signal processing and computer algebra, among others. We impose the low-rank by modeling the approximation as a product of two factors with reduced dimension. The structure of the low-rank model is enforced by introducing a penalty term in the objective function. The proposed local optimization algorithm is able to solve the weighted structured low-rank approximation problem, as well as to deal with the cases of missing or fixed elements. In contrast to approaches based on kernel representations (in linear algebraic sense), the proposed algorithm is designed to address the case of small targeted rank. We compare it to existing approaches on numerical examples of system identification, approximate greatest common divisor problem, and symmetric tensor decomposition and demonstrate its consistently good performance.

An adjusted least squares estimator is derived that yields a consistent estimate of the parameters of an implicit quadratic measurement error model. In addition, a consistent estimator for the measurement error noise variance is proposed. Important assumptions are: (1) all errors are uncorrelated identically distributed and (2) the error distribution is normal. The estimators for the quadratic measurement error model are used to estimate consistently conic sections and ellipsoids. Simulation examples, comparing the adjusted least squares estimator with the ordinary least squares method and the orthogonal regression method, are shown for the ellipsoid ÿtting problem.

Abstract.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.

The structured total least squares estimator, defined via a constrained optimization problem, is a generalization of the total least squares estimator when the data matrix and the applied correction satisfy given structural constraints. In the paper, an affine structure with additional assumptions is considered. In particular, Toeplitz and Hankel structured, noise free and unstructured blocks are allowed simultaneously in the augmented data matrix. An equivalent optimization problem is derived that has as decision variables only the estimated parameters. The cost function of the equivalent problem is used to prove consistency of the structured total least squares estimator. The results for the general affine structured multivariate model are illustrated by examples of special models. Modification of the results for block-Hankel/Toeplitz structures is also given. As a by-product of the analysis of the cost function, an iterative algorithm for the computation of the structured total least squares estimator is proposed.