Stability of Matrix Polytopes with a Dominant Vertex and Implications for System Dynamics
Joint Authors
Pastravanu, Octavian
Matcovschi, Mihaela-Hanako
Source
Issue
Vol. 2013, Issue 2013 (31 Dec. 2013), pp.1-11, 11 p.
Publisher
Hindawi Publishing Corporation
Publication Date
2013-06-05
Country of Publication
Egypt
No. of Pages
11
Main Subjects
Abstract EN
The paper considers the class of matrix polytopes with a dominant vertex and the class of uncertain dynamical systems defined in discrete time and continuous time, respectively, by such polytopes.
We analyze the standard concept of stability in the sense of Schur—abbreviated as SS (resp., Hurwitz—abbreviated as HS), and we develop a general framework for the investigation of the diagonal stability relative to an arbitrary Hölder p-norm, 1≤p≤∞, abbreviated as SDSp (resp., HDSp).
Our framework incorporates, as the particular case with p=2, the known condition of quadratic stability satisfied by a diagonal positive-definite matrix, i.e.
SDS2 (resp., HDS2) means that the standard inequality of Stein (resp., Lyapunov) associated with all matrices of the polytope has a common diagonal solution.
For the considered class of matrix polytopes, we prove the equivalence between SS and SDSp (resp., HS and HDSp), 1≤p≤∞ (fact which is not true for matrix polytopes with arbitrary structures).
We show that the dominant vertex provides all the information needed for testing these stability properties and for computing the corresponding robustness indices.
From the dynamical point of view, if an uncertain system is defined by a polytope with a dominant vertex, then the standard asymptotic stability ensures supplementary properties for the state-space trajectories, which refer to special types of Lyapunov functions and contractive invariant sets (characterized through vector p-norms weighted by diagonal positive-definite matrices).
The applicability of the main results is illustrated by two numerical examples that cover both discrete- and continuous-time cases for the class of uncertain dynamics studied in our paper.
American Psychological Association (APA)
Pastravanu, Octavian& Matcovschi, Mihaela-Hanako. 2013. Stability of Matrix Polytopes with a Dominant Vertex and Implications for System Dynamics. Abstract and Applied Analysis،Vol. 2013, no. 2013, pp.1-11.
https://search.emarefa.net/detail/BIM-468870
Modern Language Association (MLA)
Pastravanu, Octavian& Matcovschi, Mihaela-Hanako. Stability of Matrix Polytopes with a Dominant Vertex and Implications for System Dynamics. Abstract and Applied Analysis No. 2013 (2013), pp.1-11.
https://search.emarefa.net/detail/BIM-468870
American Medical Association (AMA)
Pastravanu, Octavian& Matcovschi, Mihaela-Hanako. Stability of Matrix Polytopes with a Dominant Vertex and Implications for System Dynamics. Abstract and Applied Analysis. 2013. Vol. 2013, no. 2013, pp.1-11.
https://search.emarefa.net/detail/BIM-468870
Data Type
Journal Articles
Language
English
Notes
Includes bibliographical references
Record ID
BIM-468870