Deep Hankel matrices with random elements

Nathan Lawrence, Philip Loewen, Shuyuan Wang, Michael Forbes, Bhushan Gopaluni
Proceedings of the 6th Annual Learning for Dynamics & Control Conference, PMLR 242:1579-1591, 2024.

Abstract

Willems’ fundamental lemma enables a trajectory-based characterization of linear systems through data-based Hankel matrices. However, in the presence of measurement noise, we ask: Is this noisy Hankel-based model expressive enough to re-identify itself? In other words, we study the output prediction accuracy from recursively applying the same persistently exciting input sequence to the model. We find an asymptotic connection to this self-consistency question in terms of the amount of data. More importantly, we also connect this question to the depth (number of rows) of the Hankel model, showing the simple act of reconfiguring a finite dataset significantly improves accuracy. We apply these insights to find a parsimonious depth for LQR problems over the trajectory space.

Cite this Paper

BibTeX
@InProceedings{pmlr-v242-lawrence24a, title = {Deep {H}ankel matrices with random elements}, author = {Lawrence, Nathan and Loewen, Philip and Wang, Shuyuan and Forbes, Michael and Gopaluni, Bhushan}, booktitle = {Proceedings of the 6th Annual Learning for Dynamics & Control Conference}, pages = {1579--1591}, year = {2024}, editor = {Abate, Alessandro and Cannon, Mark and Margellos, Kostas and Papachristodoulou, Antonis}, volume = {242}, series = {Proceedings of Machine Learning Research}, month = {15--17 Jul}, publisher = {PMLR}, pdf = {https://proceedings.mlr.press/v242/lawrence24a/lawrence24a.pdf}, url = {https://proceedings.mlr.press/v242/lawrence24a.html}, abstract = {Willems’ fundamental lemma enables a trajectory-based characterization of linear systems through data-based Hankel matrices. However, in the presence of measurement noise, we ask: Is this noisy Hankel-based model expressive enough to re-identify itself? In other words, we study the output prediction accuracy from recursively applying the same persistently exciting input sequence to the model. We find an asymptotic connection to this self-consistency question in terms of the amount of data. More importantly, we also connect this question to the depth (number of rows) of the Hankel model, showing the simple act of reconfiguring a finite dataset significantly improves accuracy. We apply these insights to find a parsimonious depth for LQR problems over the trajectory space.} }
Endnote
%0 Conference Paper %T Deep Hankel matrices with random elements %A Nathan Lawrence %A Philip Loewen %A Shuyuan Wang %A Michael Forbes %A Bhushan Gopaluni %B Proceedings of the 6th Annual Learning for Dynamics & Control Conference %C Proceedings of Machine Learning Research %D 2024 %E Alessandro Abate %E Mark Cannon %E Kostas Margellos %E Antonis Papachristodoulou %F pmlr-v242-lawrence24a %I PMLR %P 1579--1591 %U https://proceedings.mlr.press/v242/lawrence24a.html %V 242 %X Willems’ fundamental lemma enables a trajectory-based characterization of linear systems through data-based Hankel matrices. However, in the presence of measurement noise, we ask: Is this noisy Hankel-based model expressive enough to re-identify itself? In other words, we study the output prediction accuracy from recursively applying the same persistently exciting input sequence to the model. We find an asymptotic connection to this self-consistency question in terms of the amount of data. More importantly, we also connect this question to the depth (number of rows) of the Hankel model, showing the simple act of reconfiguring a finite dataset significantly improves accuracy. We apply these insights to find a parsimonious depth for LQR problems over the trajectory space.
APA
Lawrence, N., Loewen, P., Wang, S., Forbes, M. & Gopaluni, B.. (2024). Deep Hankel matrices with random elements. Proceedings of the 6th Annual Learning for Dynamics & Control Conference, in Proceedings of Machine Learning Research 242:1579-1591 Available from https://proceedings.mlr.press/v242/lawrence24a.html.

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