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Plane geometry and convexity of polynomial stability regions

Authors:

Didier Henrion,

Michael SebekAuthors Info & Claims

ISSAC '08: Proceedings of the twenty-first international symposium on Symbolic and algebraic computation

Pages 111 - 116

https://doi.org/10.1145/1390768.1390786

Published: 20 July 2008 Publication History

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Abstract

The set of controllers stabilizing a linear system is generally non-convex in the parameter space. In the case of two-parameter controller design (e.g. PI control or static output feedback with one input and two outputs), we observe however that quite often for benchmark problem instances, the set of stabilizing controllers seems to be convex. In this note we use elementary techniques from real algebraic geometry (resultants and Bezoutian matrices) to explain this phenomenon. As a byproduct, we derive a convex linear matrix inequality (LMI) formulation of two-parameter fixed-order controller design problem, when possible.

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Cited By

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Le HEl Din M(2021)Solving parametric systems of polynomial equations over the reals through Hermite matricesJournal of Symbolic Computation10.1016/j.jsc.2021.12.002Online publication date: Dec-2021
https://doi.org/10.1016/j.jsc.2021.12.002
Röbenack KVoßwinkel R(2020)Eigenvalue Placement by Quantifier Elimination - the Static Output Feedback ProblemActa Cybernetica10.14232/actacyb.24.3.2020.824:3(409-427)Online publication date: 16-Mar-2020
https://doi.org/10.14232/actacyb.24.3.2020.8
Ebihara YPeaucelle DArzelier DEbihara YPeaucelle DArzelier D(2014)Static Output-Feedback SynthesisS-Variable Approach to LMI-Based Robust Control10.1007/978-1-4471-6606-1_6(165-198)Online publication date: 17-Oct-2014
https://doi.org/10.1007/978-1-4471-6606-1_6
Show More Cited By

Index Terms

Plane geometry and convexity of polynomial stability regions
1. Mathematics of computing
  1. Mathematical analysis
2. Theory of computation
  1. Design and analysis of algorithms
    1. Mathematical optimization
      1. Continuous optimization
        Convex optimization

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Published In

ISSAC '08: Proceedings of the twenty-first international symposium on Symbolic and algebraic computation

July 2008

348 pages

ISBN:9781595939043

DOI:10.1145/1390768

General Chair:
J. Rafael Sendra
Universidad de Alcalá, Spain
,
Program Chair:
Laureano Gonzalez-Vega
Universidad de Cantabria, Spain

Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from [email protected]

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Published: 20 July 2008

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ISSAC '08: International Symposium on Symbolic and Algebraic Computation

July 20 - 23, 2008

Linz/Hagenberg, Austria

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Le HEl Din M(2021)Solving parametric systems of polynomial equations over the reals through Hermite matricesJournal of Symbolic Computation10.1016/j.jsc.2021.12.002Online publication date: Dec-2021
https://doi.org/10.1016/j.jsc.2021.12.002
Röbenack KVoßwinkel R(2020)Eigenvalue Placement by Quantifier Elimination - the Static Output Feedback ProblemActa Cybernetica10.14232/actacyb.24.3.2020.824:3(409-427)Online publication date: 16-Mar-2020
https://doi.org/10.14232/actacyb.24.3.2020.8
Ebihara YPeaucelle DArzelier DEbihara YPeaucelle DArzelier D(2014)Static Output-Feedback SynthesisS-Variable Approach to LMI-Based Robust Control10.1007/978-1-4471-6606-1_6(165-198)Online publication date: 17-Oct-2014
https://doi.org/10.1007/978-1-4471-6606-1_6
Delibaşı AHenrion D(2010)Hermite matrix in Lagrange basis for scaling static output feedback polynomial matrix inequalitiesInternational Journal of Control10.1080/00207179.2010.53139783:12(2494-2505)Online publication date: 13-Dec-2010
https://doi.org/10.1080/00207179.2010.531397

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Convexity of the zeros of some orthogonal polynomials and related functions

Convexity of the extreme zeros of Gegenbauer and Laguerre polynomials

Tangencies between families of disjoint regions in the plane

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