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On Nuclear Norm Minimization in System Identification
KTH, School of Electrical Engineering (EES), Automatic Control. (Systemidentifiering)ORCID iD: 0000-0002-4977-1055
2016 (English)Licentiate thesis, monograph (Other academic)
Abstract [en]

In system identification we model dynamical systems from measured data. This data-driven approach to modelling is useful since many real-world systems are difficult to model with physical principles. Hence, a need for system identification arises in many applications involving simulation, prediction, and model-based control.

Some of the classical approaches to system identification can lead to numerically intractable or ill-posed optimization problems. As an alternative, it has recently been shown beneficial to use so called regularization techniques, which make the ill-posed problems ‘regular’. One type of regularization is to introduce a certain rank constraint. However, this in general still leads to a numerically intractable problem, since the rank function is non-convex. One possibility is then use a convex approximation of rank, which we will do here.

The nuclear norm, i.e., the sum of the singular values, is a popular, convex surrogate of the rank function. This results in a heuristic that has been widely used in e.g. signal processing, machine learning, control, and system identification, since its introduction in 2001. The nuclear norm heuristic introduces a regularization parameter which governs the trade-off between model fit and model complexity. The parameter is difficult to tune, and the

current thesis revolves around this issue.

In this thesis, we first propose a choice of the regularization parameter based on the statistical properties of fictitious validation data. This can be used to avoid computationally costly techniques such as cross-validation, where the problem is solved multiple times to find a suitable parameter value. The proposed choice can also be used as initialization to search methods for minimizing some criterion, e.g. a validation cost, over the parameter domain.

Secondly, we study how the estimated system changes as a function of the parameter over its entire domain, which can be interpreted as a sensitivity analysis. For this we suggest an algorithm to compute a so called approximate regularization path with error guarantees, where the regularization path is the optimal solution as a function of the parameter. We are then able to guarantee the model fit, or, alternatively, the nuclear norm of the approximation, to deviate from the optimum by less than a pre-specified tolerance. Furthermore, we bound the l2-norm of the Hankel singular value approximation error, which means that in a certain subset of the parameter domain, we can guarantee the optimal Hankel singular values returned by the nuclear norm heuristic to not change more (in l2-norm) than a bounded, known quantity.

Our contributions are demonstrated and evaluated by numerical examples using simulated and benchmark data.

Place, publisher, year, edition, pages
Stockholm: KTH Royal Institute of Technology, 2016. , x, 94 p.
TRITA-EE, ISSN 1653-5146 ; 2016:013
National Category
Control Engineering Signal Processing
Research subject
Electrical Engineering
URN: urn:nbn:se:kth:diva-181805ISBN: 978-91-7595-859-0OAI: diva2:900592
2016-02-25, Q2, Osquldas väg 10, KTH, Stockholm, 10:15 (English)

QC 20160205

Available from: 2016-02-05 Created: 2016-02-04 Last updated: 2016-02-05Bibliographically approved

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