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Stable Polytopes for Discrete Systems by Using Box Coefficients

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Abstract

Given a discrete-time system, if all roots of corresponding characteristic polynomial belong to the open unit disc, then the corresponding system is called (Schur) stable. The stability domain of nth-order monic polynomials is defined as the set of all stable n-dimensional vectors in the coefficient space. The main hindrance of the stability and stabilization problems is the nonconvexity of the stability domain. Therefore, inner convex approximations of this domain are very important. In this paper, by using known geometrical and topological properties of the set of stable polynomials (the edge theorem, nonconvexity, openness, polytopic property and the structure of the boundary), we define a new multilinear map from the n-dimensional open cube \((-1,1)^n=(-1,1)\times \cdots \times (-1,1)\) onto the stability region and box coefficients. The main advantage of our map is that the stability of polytopes edges is equivalent to the stability of the convex combinations of third- and fourth-order factor polynomials. Inner approximations of the stability region by polytopes and applications to stabilizability problems are given.

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Acknowledgements

The author thanks the reviewers for important suggestions to improve this article.

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Authors and Affiliations

  1. Department of Mathematics Education, Faculty of Education, Burdur Mehmet Akif Ersoy University, Istiklal Campus, Burdur, 15030, Turkey

    Şerife Yılmaz

  2. Dyson School of Design Engineering, Imperial College London, London, SW7 1AL, UK

    Şerife Yılmaz

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  1. Şerife Yılmaz
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Yılmaz, Ş. Stable Polytopes for Discrete Systems by Using Box Coefficients. Circuits Syst Signal Process 41, 789–804 (2022). https://doi.org/10.1007/s00034-021-01830-6

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  • DOI: https://doi.org/10.1007/s00034-021-01830-6

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