Regularized linear system identification using atomic, nuclear and kernel-based norms: the role of the stability constraint
This work addresses the problem of improving accuracy in linear system identification for control and signal processing applications, representing an incremental advance in regularization methods.
The paper compared regularization techniques for linear system identification, finding that stable spline kernels outperform atomic and nuclear norms by embedding stability and smoothness information, with new integral versions of these kernels outperforming classical methods even with oracle model selection.
Inspired by ideas taken from the machine learning literature, new regularization techniques have been recently introduced in linear system identification. In particular, all the adopted estimators solve a regularized least squares problem, differing in the nature of the penalty term assigned to the impulse response. Popular choices include atomic and nuclear norms (applied to Hankel matrices) as well as norms induced by the so called stable spline kernels. In this paper, a comparative study of estimators based on these different types of regularizers is reported. Our findings reveal that stable spline kernels outperform approaches based on atomic and nuclear norms since they suitably embed information on impulse response stability and smoothness. This point is illustrated using the Bayesian interpretation of regularization. We also design a new class of regularizers defined by "integral" versions of stable spline/TC kernels. Under quite realistic experimental conditions, the new estimators outperform classical prediction error methods also when the latter are equipped with an oracle for model order selection.