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On the Common Linear Copositive Lyapunov Functions for Compartmental Switched Systems

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Positive Systems (POSTA 2018)

Part of the book series: Lecture Notes in Control and Information Sciences ((LNCIS,volume 480))

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Abstract

For a positive switched system, the existence of a common linear copositive Lyapunov function (CLCLF) for the family of the subsystem matrices represents an important sufficient condition for its asymptotic stability. The main necessary and sufficient condition for the existence of a CLCLF (Fornasini and Valcher, IEEE Trans Autom Control 55:1933–1937, 2010, [1], Knorn et al, Automatica 45:1943–1947, 2009, [2]) consists in the explicit evaluation of the Hurwitz property of a family of \(p^n\) matrices, where p is the number of subsystems and n the size of each subsystem. In this paper we show that, when restricting our attention to compartmental switched systems, the Hurwitz property may be checked on a smaller subset of smaller matrices. Based on this result, we provide an algorithm that allows to determine whether a CLCLF exists, by simply checking the column sums of matrix sets of increasingly lower dimension and cardinality.

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Notes

  1. 1.

    In the following, we will treat compartments of the system (3.1), or of the associated matrix A, and vertices of \({\mathcal D}(A)\) as equivalent entities.

  2. 2.

    Note that the explicit evaluation of the matrices \(B_i\) is not really necessary, as the equivalent information could be easily derived by referring to the matrices \(B_\pi \).

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Author information

Authors and Affiliations

  1. Dip. di Ingegneria dell’Informazione, Univ. di Padova, via Gradenigo 6/B, 35131, Padova, Italy

    Maria Elena Valcher

Authors
  1. Maria Elena Valcher
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Correspondence to Maria Elena Valcher .

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Editors and Affiliations

  1. Department of Mechanical Engineering, University of Hong Kong, Hong Kong, Hong Kong

    James Lam

  2. Hangzhou Dianzi University, Hangzhou, China

    Yun Chen

  3. Southwest Minzu University, Chengdu, China

    Xingwen Liu

  4. Dalian University of Technology, Dalian, China

    Xudong Zhao

  5. Hangzhou Dianzi University, Hangzhou, China

    Junfeng Zhang

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Valcher, M.E. (2019). On the Common Linear Copositive Lyapunov Functions for Compartmental Switched Systems. In: Lam, J., Chen, Y., Liu, X., Zhao, X., Zhang, J. (eds) Positive Systems . POSTA 2018. Lecture Notes in Control and Information Sciences, vol 480. Springer, Cham. https://doi.org/10.1007/978-3-030-04327-8_3

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