Generalized System Identification with Stable Spline Kernels

Regularized least-squares approaches have been successfully applied to linear system identification. Recent approaches use quadratic penalty terms on the unknown impulse response defined by stable spline kernels, which control model space complexity by leveraging regularity and bounded-input bounded-output stability. This paper extends linear system identification to a wide class of nonsmooth stable spline estimators, where regularization functionals and data misfits can be selected from a rich set of piecewise linear quadratic penalties. This class encompasses the 1-norm, huber, and vapnik, in addition to the least-squares penalty, and the approach allows linear inequality constraints on the unknown impulse response. We develop a customized interior point solver for the entire class of proposed formulations. By representing penalties through their conjugates, we allow a simple interface that enables the user to specify any piecewise linear quadratic penalty for misfit and regularizer, together with inequality constraints on the response. The solver is locally quadratically convergent, with O(n2(m+ n)) arithmetic operations per iteration, for n impulse response coefficients and m output measurements. In the system identification context, where n << m, IPsolve is competitive with available alternatives, illustrated by a comparison with TFOCS and libSVM. The modeling framework is illustrated with a range of numerical experiments, featuring robust formulations for contaminated data, relaxation systems, and nonnegativity and unimodality constraints on the impulse response. Incorporating constraints yields significant improvements in system identification. The solver used to obtain the results is distributed via an open source code repository.