Smooth trivial vector bundle structure of the space of Hurwitz polynomials

Aguirre‐Hernández, Baltazar; Frías-Armenta, Martín Eduardo; Verduzco, Fernando

doi:10.1016/j.automatica.2009.09.011

Cited by 22 publications

(17 citation statements)

References 6 publications

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“…The importance of the stability test was appreciated when Kharitonov used it to obtain his celebrated theorem. Moreover, it is worth mentioning that other important properties of the set of Hurwitz polynomials were shown with the stability test (see [6,13]). Now we use the stability test for studying the following subset of the Hadamardized Hurwitz polynomial set:…”

Section: The Stability Test the Set Of Hadamardized Hurwitz Polynomimentioning

confidence: 99%

“…Since then, a lot information has been generated and part of it can be found in [1][2][3][4]. On the other hand, in the study of Hurwitz polynomials, topological approaches have recently been reported in [5][6][7][8][9]. Now, in this paper we present some topological and geometric results about the set of Hurwitz polynomials that admit a Hadamard factorization; such set will be called the set of Hadamardized polynomials.…”

Section: Introductionmentioning

confidence: 99%

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Properties of the Set of Hadamardized Hurwitz Polynomials

Aguirre‐Hernández

Díaz-González

Loredo-Villalobos

et al. 2015

Mathematical Problems in Engineering

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We say that a Hurwitz polynomialptis a Hadamardized polynomial if there are two Hurwitz polynomialsftandgtsuch thatf∗g=p, wheref∗gis the Hadamard product offandg. In this paper, we prove that the set of all Hadamardized Hurwitz polynomials is an open, unbounded, nonconvex, and arc-connected set. Furthermore, we give a result so that a fourth-degree Hurwitz interval polynomial is a Hadamardized polynomial family and we discuss an approach of differential topology in the study of the set of Hadamardized Hurwitz polynomials.

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Section: The Stability Test the Set Of Hadamardized Hurwitz Polynomimentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Properties of the Set of Hadamardized Hurwitz Polynomials

Aguirre‐Hernández

Díaz-González

Loredo-Villalobos

et al. 2015

Mathematical Problems in Engineering

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“…There exists approaches that focus on the problem of stability check of different types of polynomial spaces. [10][11][12][13][14][15][16][17][18] One of the important theorems in this topic is the boundary crossing theorem that briefly states that, if there exist stable and unstable polynomials in a path-connected polynomial spaces, then there exists a polynomial with pure imaginary roots. López-Renteria et al generalize this theorem in order to discuss about the number of the pure imaginary roots of these critical polynomials.…”

Section: Introductionmentioning

confidence: 99%

Output feedback controller for polytopic systems exploiting the direct searching of the design space

Abolpour

Dehghani

Talebi

2019

Intl J Robust & Nonlinear

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This paper deals with the problem of designing a robust static output feedback controller for polytopic systems. The current research that tackled this problem is mainly based on LMI method, which is conservative by nature. In this paper, a novel approach is proposed, which considers the design space of the controller parameters and iteratively partitions the space to small simplexes. Then, by assessing the stability in each simplex, the solution space for design parameters is directly determined. It has been theoretically proved that, if there exists a feasible solution in the design space, the algorithm can find it. To validate the result of the proposed approach, comparative simulation examples are given to illustrate the performance of the design methodology as compared to those of previous approaches. KEYWORDS exposed edges, output feedback controller, parametric uncertain systems, polytopic systems, stability of polynomial convex spaces 5164

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“…For example, in [19] it was shown that the set of Hurwitz polynomials of degree n with positive coefficients -H + n -is contractible. With ideas of differential topology, in [4] it was proved that H + n is a smooth trivial vector bundle over H + n−k . With this last approach the set of Schur polynomials was studied (see [5]).…”

Section: Introductionmentioning

confidence: 99%

Hadamard Factorization of Stable Polynomials

Loredo-Villalobos¹,

Aguirre‐Hernández²

2011

AIP Conference Proceedings

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The stable (Hurwitz) polynomials are important in the study of differential equations systems and control theory (see [7] and [19]). A property of these polynomials is related to Hadamard product. Consider two polynomials p, q ∈ R[x]:p(x) = a n x n + a n−1 x n−1 + · · · + a 1 x + a 0. Some results (see [16]) shows that if p, q ∈ R[x] are stable polynomials then (p * q) is stable, also, i.e. the Hadamard product is closed; however, the reciprocal is not always true, that is, not all stable polynomial has a factorization into two stable polynomials the same degree n, if n ≥ 4 (see [15]).In this work we will give some conditions to Hadamard factorization existence for stable polynomials.These results open new possibilities for robust stability analysis of families with non-linear dependencies of parameters, allowing in some cases, separate parameters via Hadamard stable factorization to obtain two stable polynomials, with fewer parameters and consequently obtaining a simpler problem to solve.

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Smooth trivial vector bundle structure of the space of Hurwitz polynomials

Cited by 22 publications

References 6 publications

Properties of the Set of Hadamardized Hurwitz Polynomials

Properties of the Set of Hadamardized Hurwitz Polynomials

Output feedback controller for polytopic systems exploiting the direct searching of the design space

Hadamard Factorization of Stable Polynomials

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