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Exact minimum number of bits to stabilize a linear system

Kostina, Victoria and Peres, Yuval and Ranade, Gireeja and Sellke, Mark (2021) Exact minimum number of bits to stabilize a linear system. IEEE Transactions on Automatic Control . ISSN 0018-9286. doi:10.1109/tac.2021.3126679. (In Press) https://resolver.caltech.edu/CaltechAUTHORS:20211217-98215000

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Abstract

We consider an unstable scalar linear stochastic system, X_(n+1) = aX_n + Z_n − U_n , where a ≥ 1 is the system gain, Z_n's are independent random variables with bounded α-th moments, and U_n's are the control actions that are chosen by a controller who receives a single element of a finite set {1,…,M} as its only information about system state Xi. We show new proofs that M > a is necessary and sufficient for β-moment stability, for any β < α. Our achievable scheme is a uniform quantizer of the zoom-in/zoom-out type that codes over multiple time instants for data rate efficiency; the controller uses its memory of the past to correctly interpret the received bits. We analyze the performance of our scheme using probabilistic arguments. We show a simple proof of a matching converse using information-theoretic techniques. Our results generalize to vector systems, to systems with dependent Gaussian noise, and to the scenario in which a small fraction of transmitted messages is lost.


Item Type:Article
Related URLs:
URLURL TypeDescription
https://doi.org/10.1109/TAC.2021.3126679DOIArticle
https://arxiv.org/abs/1807.07686arXivDiscussion Paper
https://resolver.caltech.edu/CaltechAUTHORS:20190204-124407625Related ItemConference Paper
ORCID:
AuthorORCID
Kostina, Victoria0000-0002-2406-7440
Peres, Yuval0000-0001-5456-6323
Ranade, Gireeja0000-0002-6747-4492
Sellke, Mark0000-0001-9166-8185
Additional Information:© 2021 IEEE. This work was supported in part by the National Science Foundation (NSF) under Grant CCF-1751356, and by the Simons Institute for the Theory of Computing. Research of Y. Peres was partially supported by NSF grant DMS-1900008. G. Ranade acknowledges the Siebel Energy Institute Seed Funding.
Funders:
Funding AgencyGrant Number
NSFCCF-1751356
Simons FoundationUNSPECIFIED
NSFDMS-1900008
Siebel Energy InstituteUNSPECIFIED
Subject Keywords:Linear stochastic control, source coding, data rate theorem
DOI:10.1109/tac.2021.3126679
Record Number:CaltechAUTHORS:20211217-98215000
Persistent URL:https://resolver.caltech.edu/CaltechAUTHORS:20211217-98215000
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:112526
Collection:CaltechAUTHORS
Deposited By: George Porter
Deposited On:18 Dec 2021 00:13
Last Modified:18 Dec 2021 00:13

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