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On the stability radii of continuous-time infinite Markov jump linear systems

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Abstract

In this paper we introduce the subject of stability radii for continuous-time infinite Markov jump linear systems (MJLS) with respect to unstructured perturbations. By means of the small-gain approach, a lower bound for the complex radius is derived along with a linear matrix inequality (LMI) optimization method which is new in this context. In this regard, we propose an algorithm to solve the optimization problem, based on a bisectional procedure, which is tailored in such a way that avoids the issue of scaling optimization. In addition, an easily computable upper bound for the real and complex stability radii is devised, with the aid of a spectral characterization of the problem. This seems to be a novel approach to the problem of robust stability, even when restricted to the finite case, which in turn allows us to obtain explicit formulas for the stability radii of two-mode scalar MJLS. We also introduce a connection between stability radii and a certain margin of stability with respect to perturbations on the transition rates of the Markov process. The applicability of the main results is illustrated with some numerical examples.

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Authors and Affiliations

  1. National Laboratory for Scientific Computing, LNCC/MCT, Av. Getulio Vargas, 333, Petrópolis, RJ, 25651-070, Brazil

    Marcos G. Todorov & Marcelo Dutra Fragoso

Authors
  1. Marcos G. Todorov
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  2. Marcelo Dutra Fragoso
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Correspondence to Marcelo Dutra Fragoso.

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A preliminary version of this paper was presented at the 2008 American Control Conference.

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Todorov, M.G., Fragoso, M.D. On the stability radii of continuous-time infinite Markov jump linear systems. Math. Control Signals Syst. 22, 23–38 (2010). https://doi.org/10.1007/s00498-010-0046-3

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  • DOI: https://doi.org/10.1007/s00498-010-0046-3

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