TY - JOUR
T1 - ESTIMATION ENTROPY, LYAPUNOV EXPONENTS, AND QUANTIZER DESIGN FOR SWITCHED LINEAR SYSTEMS
AU - Vicinansa, Guilherme S.
AU - Liberzon, Daniel
N1 - \ast Received by the editors April 13, 2021; accepted for publication (in revised form) September 5, 2022; published electronically February 15, 2023. Some partial preliminary versions of the results in this paper were presented at the 58th Conference on Decision and Control and at the 24th ACM International Conference on Hybrid Systems: Computation and Control. https://doi.org/10.1137/21M1411871 Funding: This work was supported by NSF grants CMMI-1662708 and CMMI-2106043 and by AFOSR grant FA9550-17-1-0236. \dagger Coordinated Science Laboratory, University of Illinois Urbana-Champaign, Urbana, IL 61801 USA ([email protected], [email protected]).
PY - 2023/2
Y1 - 2023/2
N2 - In this paper, we study connections between the estimation entropy of a switched linear system and its Lyapunov exponents. We prove lower and upper bounds for the estimation entropy in terms of the Lyapunov exponents and show that, under the so-called regularity assumption, those bounds coincide. To do that, we use a geometric object called Oseledets' filtration of the system. Further, we show how to use the exponents and the Oseledets' filtration to design a quantization scheme for state estimation of switched linear systems. Then, we prove that we can make this algorithm work at an average data-rate arbitrarily close to the upper bound we provided for the estimation entropy of the given system. Furthermore, we can choose the average data-rate to be arbitrarily close to the estimation entropy whenever the switched linear system is regular. We show that, under the regularity assumption, the quantization scheme is completely causal in the sense that it depends only on information that is available up to the current time instant. We show that regularity is a natural property of many practical systems, such as Markov jump linear systems, and give sufficient conditions for it.
AB - In this paper, we study connections between the estimation entropy of a switched linear system and its Lyapunov exponents. We prove lower and upper bounds for the estimation entropy in terms of the Lyapunov exponents and show that, under the so-called regularity assumption, those bounds coincide. To do that, we use a geometric object called Oseledets' filtration of the system. Further, we show how to use the exponents and the Oseledets' filtration to design a quantization scheme for state estimation of switched linear systems. Then, we prove that we can make this algorithm work at an average data-rate arbitrarily close to the upper bound we provided for the estimation entropy of the given system. Furthermore, we can choose the average data-rate to be arbitrarily close to the estimation entropy whenever the switched linear system is regular. We show that, under the regularity assumption, the quantization scheme is completely causal in the sense that it depends only on information that is available up to the current time instant. We show that regularity is a natural property of many practical systems, such as Markov jump linear systems, and give sufficient conditions for it.
KW - estimation entropy
KW - minimum data-rate
KW - switched systems
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U2 - 10.1137/21M1411871
DO - 10.1137/21M1411871
M3 - Article
AN - SCOPUS:85152130465
SN - 0363-0129
VL - 61
SP - 198
EP - 224
JO - SIAM Journal on Control and Optimization
JF - SIAM Journal on Control and Optimization
IS - 1
ER -