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Behavioral theory for stochastic systems? A data-driven journey from Willems to Wiener and back again
Citation Link: https://doi.org/10.15480/882.9184
Publikationstyp
Journal Article
Date Issued
2023-01
Sprache
English
Author(s)
TORE-DOI
Journal
Volume
55
Start Page
92
End Page
117
Citation
Annual Reviews in Control 55: 92-117 (2023-01)
Publisher DOI
Scopus ID
Publisher
Elsevier
The fundamental lemma by Jan C. Willems and co-workers is deeply rooted in behavioral systems theory and it has become one of the supporting pillars of the recent progress on data-driven control and system analysis. This tutorial-style paper combines recent insights into stochastic and descriptor-system formulations of the lemma to further extend and broaden the formal basis for behavioral theory of stochastic linear systems. We show that series expansions – in particular Polynomial Chaos Expansions (PCE) of L2-random variables, which date back to Norbert Wiener's seminal work – enable equivalent behavioral characterizations of linear stochastic systems. Specifically, we prove that under mild assumptions the behavior of the dynamics of the L2-random variables is equivalent to the behavior of the dynamics of the series expansion coefficients and that it entails the behavior composed of sampled realization trajectories. We also illustrate the short-comings of the behavior associated to the time-evolution of the statistical moments. The paper culminates in the formulation of the stochastic fundamental lemma for linear time-invariant systems, which in turn enables numerically tractable formulations of data-driven stochastic optimal control combining Hankel matrices in realization data (i.e. in measurements) with PCE concepts.
Subjects
Behavioral systems theory
Data-based prediction
Data-driven control
Descriptor systems
Linear stochastic systems
Polynomial chaos expansions
Uncertainty propagation
Uncertainty quantification
DDC Class
621: Applied Physics
510: Mathematics
Publication version
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1-s2.0-S1367578823000093-main.pdf
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1.45 MB
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