Journal
AUTOMATICA
Volume 118, Issue -, Pages -Publisher
PERGAMON-ELSEVIER SCIENCE LTD
DOI: 10.1016/j.automatica.2020.109038
Keywords
Linear system identification; BIBO stability; Stable reproducing kernel hilbert spaces; Kernel-based regularization; Regularized least squares
Funding
- [PRIN 2015 2015PJ28EP]
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Reproducing kernel Hilbert spaces (RKHSs) are key spaces for machine learning that are becoming popular also for linear system identification. In particular, the so-called stable RKHSs can be used to model absolutely summable impulse responses. In combination e.g. with regularized least squares they can then be used to reconstruct dynamic systems from input-output data. In this paper we provide new structural properties of stable RKHSs. The relation between stable kernels and other fundamental classes, like those containing absolutely summable or finite-trace kernels, is elucidated. These insights are then brought into the feature space context. First, it is proved that any stable kernel admits feature maps induced by a basis of orthogonal eigenvectors in l(2). The exact connection with classical system identification approaches that exploit such kind of functions to model impulse responses is also provided. Then, the necessary and sufficient stability condition for RKHSs designed by formulating kernel eigenvectors and eigenvalues is obtained. Overall, our new results provide novel mathematical foundations of stable RKHSs with impact on stability tests, impulse responses modeling and computational efficiency of regularized schemes for linear system identification. (C) 2020 Elsevier Ltd. All rights reserved.
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