Mean stability and $${\varvec{L}}_\mathbf{1 }$$ L 1 performance
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Mean stability and L1 performance of a class of two-time-scale Markov jump linear systems Felipe O. dos Santos1
· Marcos G. Todorov1
Received: 20 April 2020 / Accepted: 3 November 2020 © Springer-Verlag London Ltd., part of Springer Nature 2020
Abstract This paper addresses the mean stability analysis and L 1 performance of continuoustime Markov jump linear systems (MJLSs) driven by a two time-scale Markov chain, in the scenario in which the temporal scale parameter tends to zero. The jump process considered here is bivariate, with slow and fast components. Our approach relies on a convergence analysis involving the semigroup that generates the firstmoment dynamics of the MJLS when the switching frequency of the fast part of the Markov chain tends to infinity. In this setup, we introduce a new definition of stability in a limit case, and connect it with the mean stability of an averaged MJLS. In the particular case where the averaged MJLS is positive, we also derive suitable criteria for assessing mean stability and L 1 performance. These criteria are expressed in terms of the Hurwitz stability of a matrix (whose dimension is independent of the cardinality of the state space of the fast switching component), of linear programming, and of the 1-norm of a certain transfer matrix, which makes them suitable for computational purposes. We also establish comparisons between our (two-time-scale) approach and existing one-time-scale approaches from the literature, and show that our criteria are based on matrices of relatively smaller dimensions, which do not depend on the scale parameter . The effectiveness of the main results is discussed through numerical examples of epidemiological and compartmental models. Keywords Stochastic systems · Linear systems · Markov processes · Stability criteria · Positive systems · L 1 performance.
Partially supported by CAPES and the CNPq Grants 421486/2016-3 and 314537/2020-1.
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Marcos G. Todorov [email protected] Felipe O. dos Santos [email protected]
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National Laboratory for Scientific Computing - LNCC/MCTI, Av. Getúlio Vargas 333, Petrópolis, Rio de Janeiro CEP 25651-070, Brazil
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Mathematics of Control, Signals, and Systems
1 Introduction Since their appearance in the literature, dynamical systems with Markov jumps have been widely recognized as an alternative in the study of systems subject to abrupt changes. Such changes occur, for example, due to failures in system components, switching in communications signals, rapid variations in the economy, abrupt variations in genetic drift due to environmental disturbances, etc. When modeling these phenomena, this class of systems has been considered in several strategic areas, such as robotics, aeronautics, and communication systems (see, for example, [10,18]). The interest in these systems is also due, in part, to the availability of a vast repertoire of theoretical results, which combine elements of operator theory with Markov stochastic processes, in addition to structural concepts from linear systems theory [22,31,33]. MJ
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