5 research outputs found
Bayesian and regularization approaches to multivariable linear system identification: the role of rank penalties
Recent developments in linear system identification have proposed the use of
non-parameteric methods, relying on regularization strategies, to handle the
so-called bias/variance trade-off. This paper introduces an impulse response
estimator which relies on an -type regularization including a
rank-penalty derived using the log-det heuristic as a smooth approximation to
the rank function. This allows to account for different properties of the
estimated impulse response (e.g. smoothness and stability) while also
penalizing high-complexity models. This also allows to account and enforce
coupling between different input-output channels in MIMO systems. According to
the Bayesian paradigm, the parameters defining the relative weight of the two
regularization terms as well as the structure of the rank penalty are estimated
optimizing the marginal likelihood. Once these hyperameters have been
estimated, the impulse response estimate is available in closed form.
Experiments show that the proposed method is superior to the estimator relying
on the "classic" -regularization alone as well as those based in atomic
and nuclear norm.Comment: to appear in IEEE Conference on Decision and Control, 201
Regularized linear system identification using atomic, nuclear and kernel-based norms: the role of the stability constraint
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
Maximum Entropy Vector Kernels for MIMO system identification
Recent contributions have framed linear system identification as a
nonparametric regularized inverse problem. Relying on -type
regularization which accounts for the stability and smoothness of the impulse
response to be estimated, these approaches have been shown to be competitive
w.r.t classical parametric methods. In this paper, adopting Maximum Entropy
arguments, we derive a new penalty deriving from a vector-valued
kernel; to do so we exploit the structure of the Hankel matrix, thus
controlling at the same time complexity, measured by the McMillan degree,
stability and smoothness of the identified models. As a special case we recover
the nuclear norm penalty on the squared block Hankel matrix. In contrast with
previous literature on reweighted nuclear norm penalties, our kernel is
described by a small number of hyper-parameters, which are iteratively updated
through marginal likelihood maximization; constraining the structure of the
kernel acts as a (hyper)regularizer which helps controlling the effective
degrees of freedom of our estimator. To optimize the marginal likelihood we
adapt a Scaled Gradient Projection (SGP) algorithm which is proved to be
significantly computationally cheaper than other first and second order
off-the-shelf optimization methods. The paper also contains an extensive
comparison with many state-of-the-art methods on several Monte-Carlo studies,
which confirms the effectiveness of our procedure