New stable spline estimators for robust, sparse and inequality constrained linear system identification

New regularized least squares approaches have recently been successfully applied to linear system identification. In these approaches, quadratic penalty terms are used on the unknown impulse response, defined by the so called stable spline kernels, which include information on regularity and BIBO stability. In this paper, we extend these regularized techniques by proposing new nonsmooth stable spline estimators. We study linear system identification in a very broad context, where regularization functionals and data misfits can be selected from a rich set of piecewise linear quadratic functions. Furthermore, polyhedral inequality constraints on the unknown impulse response can also be included in this framework. For any formulation inside this class, we show that interior point methods can solve the system identification problem with complexity O(n^3 +mn^2) in each iteration, where n and m are the number of impulse response coefficients and output measurements, respectively. The utility of the framework is illustrated using several numerical experiments, including scenarios with output measurements corrupted by outliers, relaxation systems, nonnegativity and unimodality constraints on the impulse response.