Positivity and Linear Matrix Inequalities

Genin, Y.; Hachez, Yvan; Nesterov, Yurii; Stefan, R.; Dooren, Paul Van; Xu, Shengyuan

doi:10.3166/ejc.8.275-298

Cited by 27 publications

(16 citation statements)

References 39 publications

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“…Actually, the strong link between SOS formulation and S-procedure was well highlighted in de Oliveira (2005). In the same fashion, 1-D KYP lemma can be seen as the equivalence between nonnegativity of a univariate polynomial matrix (over a curve ∂D) and an LMI corresponding to the existence of a SOS-formulation of this matrix (Genin et al 2002(Genin et al , 2003. The expression in the condition (i) of Theorem 1 involves an n-variate polynomial matrix (with variables in a specified region).…”

Section: Discussion About Perspectivesmentioning

confidence: 97%

On the Kalman–Yakubovich–Popov lemma and the multidimensional models

Bachelier

Paszke

Mehdi

2008

Multidim Syst Sign Process

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This paper focuses on Kalman-Yakubovich-Popov lemma for multidimensional systems described by Roesser model that possibly includes both continuous and discrete dynamics. It is shown that, similarly to the standard 1-D case, this lemma can be studied through the lens of S-procedure. Furthermore, by virtue of this lemma, we will examine robust stability, bounded and positive realness of multidimensional systems.

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Section: Discussion About Perspectivesmentioning

confidence: 97%

On the Kalman–Yakubovich–Popov lemma and the multidimensional models

Bachelier

Paszke

Mehdi

2008

Multidim Syst Sign Process

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“…Generally speaking, it is believed that convex LMI optimization problems arising from properties of positive polynomials can be solved more efficiently than general LMI problems, see the recent results and discussions in [Nesterov, 2000, Genin, 2002 and also [Alkire, 2001]. Numerical conditionning improvement and the use of alternative polynomial bases such as Chebyshev polynomials are however still subjects of further investigation.…”

Section: Resultsmentioning

confidence: 99%

“…The advantage of formulating the sufficient condition of stability for a polynomial as an LMI arising from positivity conditions on polynomials is that these convex optimization problems have a special structure (Toeplitz structure for discrete-time polynomials, or Hankel structure for continuous-time polynomials) which may be exploited to design efficient algorithms, see [Nesterov, 2000, Genin, 2002 or [Alkire, 2001]. Numerical properties of these algorithms must be however studied in further detail, since it is known for example that positive definite Hankel matrices are exponentially ill-conditioned.…”

Section: Lemma 4 Primal Lmi (3) Is Feasible If and Only If Dual Lmi (mentioning

confidence: 99%

Positive polynomials and robust stabilization with fixed-order controllers

Henrion

Šebek

Kučera

2003

IEEE Trans. Automat. Contr.

183

177

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Recent results on positive polynomials are used to obtain a convex inner approximation of the stability domain in the space of coefficients of a polynomial. An application to the design of fixed-order controllers robustly stabilizing a linear system subject to polytopic uncertainty is then proposed, based on LMI optimization. The key ingredient in the design procedure resides in the choice of a central polynomial, or desired nominal closed-loop characteristic polynomial. Several numerical examples illustrate the relevance of the approach.

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“…Of course, direct applications of Youla's Theorem in control theory are well-known and are some the earliest of such applications. The paper of Genin et al [12], for instance, shows how SD-representability of polynomial matrix pencils can be used to design filters. Here we sketch a few other applications.…”

Section: Further Applicationsmentioning

confidence: 99%

Semidefinite Characterization of Sum-of-Squares Cones in Algebras

Papp

Alizadeh

2013

SIAM J. Optim.

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Abstract. We extend Nesterov's semidefinite programming characterization of squared functional systems to cones of sum-of-squares elements in general abstract algebras. Using algebraic techniques such as isomorphism, linear isomorphism, tensor products, sums and direct sums, we show that many concrete cones are in fact sum-of-squares cones with respect to some algebra, and thus representable by the cone of positive semidefinite matrices. We also consider nonnegativity with respect to a proper cone K, and show that in some cases K-nonnegative cones are either sum-of-squares, or are semidefinite representable. For example we show that some well-known Chebyshev systems, when extended to Euclidean Jordan algebras, induce cones that are either Sum-of-Squares cones or are semidefinite representable. Finally we will discuss some concrete examples and applications, including minimum ellipsoid enclosing given space curves, minimization of eigenvalues of polynomial matrix pencils, approximation of functions by shape-constrained functions, and approximation of combinatorial optimization problems by polynomial programming.

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Positivity and Linear Matrix Inequalities

Cited by 27 publications

References 39 publications

On the Kalman–Yakubovich–Popov lemma and the multidimensional models

On the Kalman–Yakubovich–Popov lemma and the multidimensional models

Positive polynomials and robust stabilization with fixed-order controllers

Semidefinite Characterization of Sum-of-Squares Cones in Algebras

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