Estimation Entropy, Lyapunov Exponents, and Quantizer Design for Switched Linear Systems

Abstract

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.