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1510documents trouvés

15302
01/10/2017

Computing Gaussian & exponential measures of semi-algebraic sets

J.B.LASSERRE

MAC

Revue Scientifique : Advances in Applied Mathematics, Vol.91, pp.137-163, Octobre 2017 , N° 15302

Lien : https://hal.archives-ouvertes.fr/hal-01185641

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Abstract

We provide a numerical scheme to approximate as closely as desired the Gaussian or exponential measure μ(\om) of (not necessarily compact) basic semi-algebraic sets \om⊂\Rn. We obtain two monotone (non increasing and non decreasing) sequences of upper and lower bounds (ω¯¯¯d), (ω−d), d∈\N, each converging to μ(\om) as d→∞. For each d, computing ω¯¯¯d or ω−d reduces to solving a semidefinite program whose size increases with d. Some preliminary (small dimension) computational experiments are encouraging and illustrate the potential of the method. The method also works for any measure whose moments are known and which satisfies Carleman's condition.

140474
17311
21/09/2017

Hybrid adaptive control of the boost converter

S.HADJERAS, C.ALBEA SANCHEZ, G.GARCIA

MAC

Rapport LAAS N°17311, Septembre 2017, 6p.

Lien : https://hal.laas.fr/hal-01588081

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Abstract

This work proposes a control law for the boost converter with unknown constant resistive load. It focuses on Hybrid Dynamical System (HDS) theory, which considers the voltage and current signals as continuous-time variables and the switching signal as discrete-time variable. In several applications the voltage has to be constant. To ensure that the voltage value is robust with respect to any reference, an adaptive scheme is proposed. This adaptation is accomplished using a state observer and assuming that all states are accessible. Then, the full system stability can be established by using a singular perturbation analysis. The hybrid adaptive controller is tested in simulations.

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17279
08/09/2017

Sparse polynomial interpolation: compressed sensing, super resolution, or Prony?

C.JOSZ, J.B.LASSERRE, B.MOURRAIN

MAC, INRIA Sophia

Rapport LAAS N°17279, Septembre 2017, 23p.

Lien : https://hal.archives-ouvertes.fr/hal-01575325

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Abstract

We show that the sparse polynomial interpolation problem reduces to a discrete super-resolution problem on the n-dimensional torus. Therefore the semidefinite programming approach initiated by Candès & Fernandez-Granda [7] in the univariate case (and later extended to the multi-variate setting) can be applied. In particular, exact recovery is guaranteed provided that a geometric spacing condition on the " supports " holds and the number of evaluations are sufficiently many (but not many). It also turns out that the (compressed sensing) LP-formulation of ℓ 1-norm minimization is also guaranteed to provide exact recovery provided that the evaluations are made in a certain manner and even though the Restricted Isometry Property for exact recovery is not satisfied. (A naive compressed sensing LP-approach does not offer such a guarantee.) Finally we also describe the algebraic Prony method for sparse interpolation, which also recovers the exact decomposition but from less point evaluations and with no geometric spacing condition. We provide two sets of numerical experiments , one in which the super-resolution technique and Prony's method seem to cope equally well with noise, and another in which the super-resolution technique seems to cope with noise better than Prony's method, at the cost of an extra computational burden (i.e. a semidefinite optimization).

140696
17263
06/09/2017

Randomized and robust methods for uncertain systems using R-RoMulOC, with applications to DEMETER satellite benchmark

M.CHAMANBAZ, F.DABBENE, D.PEAUCELLE, C.PITTET-MECHIN

ARAKUT, CNR-IEIIT, Torino, MAC, CNES

Revue Scientifique : AerospaceLab Journal, N°13, 15p., Septembre 2017 , N° 17263

Lien : https://hal.archives-ouvertes.fr/hal-01570588

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Abstract

R-RoMulOC is a freely distributed toolbox which aims at making easily available to the users different optimization-based methods for dealing with uncertain systems. It implements both deterministic LMI-based results, that provide guaranteed performances for all values of the uncertainties, and probabilistic randomization-based approaches, that guarantee performances for all values of the uncertainties except for a subset with arbitrary small probability measure. The paper is devoted to the description of these two approaches for analysis and control design when applied to a satellite benchmark proposed by CNES, the French Space Agency. The paper also describes the modeling of the DEMETER satellite and its integration into the R-RoMulOC toolbox as a challenging test example. Design of state-feedback controllers and closed-loop performance analysis are carried out with the randomized and robust methods available in the R-RoMulOC toolbox.

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17273
06/09/2017

Control of Anesthesia Based on Singularly Perturbed Model

S.TARBOURIECH, I.QUEINNEC, G.GARCIA, M.MAZEROLLES

MAC, CHU Toulouse

Ouvrage (contribution) : Positive Systems - Theory and Applications (POSTA 2016), Springer, N°ISBN 978-3-319-54210-2, Septembre 2017, Chapter 2, pp.17-29 , N° 17273

Lien : https://hal.laas.fr/hal-01573825

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Abstract

This chapter deals with the control of anesthesia taking into account the pos-itivity together with the upper limitation constraints of the variables and the target interval tolerated for the depth of anesthesia during a surgery. Due to the presence of multiple time scale dynamics in the anesthesia model, the system is re-expressed through a singularly perturbed system allowing to decouple the fast dynamics from the slow ones. Differently from general approaches for singularly perturbed systems , the control objective is then to control and accelerate the fast system without interest in modifying the slow dynamics. Thus, a structured state feedback control is proposed through quasi-LMI (linear matrix inequalities) conditions. The characterization of domains of stability and invariance for the system is provided. Associated convex optimization issues are then discussed. Finally, the theoretical conditions are evaluated on a simulated patient case.

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17247
04/09/2017

Stability analysis of a system coupled to a heat equation

L.BAUDOUIN, A.SEURET, F.GOUAISBAUT

MAC

Rapport LAAS N°17247, Septembre 2017, 8p.

Lien : https://hal.archives-ouvertes.fr/hal-01566455

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Abstract

As a first approach to the study of systems coupling finite and infinite dimensional natures, this article addresses the stability of a system of ordinary differential equations coupled with a classic heat equation using a Lyapunov functional technique. Inspired from recent developments in the area of time delay systems, a new methodology to study the stability of such a class of distributed parameter systems is presented here. The idea is to use a polynomial approximation of the infinite dimensional state of the heat equation in order to build an enriched energy functional. A well known efficient integral inequality (Bessel inequality) will allow to obtain stability conditions expressed in terms of linear matrix inequalities. We will eventually test our approach on academic examples in order to illustrate the efficiency of our theoretical results.

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17212
30/08/2017

Lyapunov stability analysis of a string equation coupled with an ordinary differential system

M.BARREAU, A.SEURET, F.GOUAISBAUT, L.BAUDOUIN

MAC

Rapport LAAS N°17212, Août 2017, 8p.

Lien : https://hal.laas.fr/hal-01548256

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This paper considers the stability problem of a linear time invariant system in feedback with a string equation. A new Lyapunov functional candidate is proposed based on the use of augmented states which enriches and encompasses the classical Lyapunov functional proposed in the literature. It results in tractable stability conditions expressed in terms of linear matrix inequalities. This methodology follows from the application of the Bessel inequality together with Legendre polynomials. Numerical examples illustrate the potential of our approach through three scenari: a stable ODE perturbed by the PDE, an unstable open-loop ODE stabilized by the PDE and an unstable closed-loop ODE stabilized by the PDE.

140441
17216
30/08/2017

Algebraic-geometric techniques for the feedback classification and robustness of the optimal control of a pair of Bloch equations with application to Magnetic Resonance Imaging

B.BONNARD, O.COTS, J.C.FAUGERE, A.JACQUEMARD, J.ROUOT, M.SAFEY EL DIN, T.VERRON

IMB, Bourgogne, LIP6-CNRS, MAC, IRIT

Rapport LAAS N°17216, Août 2017, 38p.

Lien : https://hal.inria.fr/hal-01556806

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The aim of this article is to classify the singular trajectories associated with the optimal control problems of a pair of controlled Bloch equations. The motivation is to analyze the robustness of the optimal solutions to the contrast and the time-minimal saturation problem, in magnetic resonance imaging, with respect to the parameters and B1-inhomogeneity. For this purpose, we use various computer algebra algorithms and methods to study solutions of polynomial systems of equations and inequalities which are used for classification issues: Gröbner basis, cylindrical algebraic decomposition of semi-algebraic sets, Thom's isotopy lemma.

140458
17204
29/08/2017

Lebesque and Gaussian measure of unions of basic semi-algebraic sets

J.B.LASSERRE, Y.EMIN

MAC

Rapport LAAS N°17204, Août 2017, 21p.

Lien : https://hal.archives-ouvertes.fr/hal-01543361

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Abstract

Given a finite Borel measure µ on R n and basic semi-algebraic sets Ω_i ⊂ R n , i = 1,. .. , p, we provide a systematic numerical scheme to approximate as closely as desired µ(\cup_i Ω_i), when all moments of µ are available (and finite). More precisely , we provide a hierarchy of semidefinite programs whose associated sequence of optimal values is monotone and converges to the desired value from above. The same methodology applied to the complement R n \ (\cup_i Ω_i) provides a monotone sequence that converges to the desired value from below. When µ is the Lebesgue measure we assume that Ω := \cup_i Ω_i is compact and contained in a known box B and in this case the complement is taken to be B \ Ω. In fact, not only µ(Ω) but also every finite vector of moments of µ_Ω (the restriction of µ on Ω) can be approximated as closely as desired, and so permits to approximate the integral on Ω of any given polynomial.

140412
17195
25/08/2017

Stability of linear systems with time-varying delays using Bessel-Legendre inequalities

A.SEURET, F.GOUAISBAUT

MAC

Rapport LAAS N°17195, Août 2017, 8p.

Lien : https://hal.laas.fr/hal-01537184

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Abstract

This paper addresses the stability problem of linear systems with a time-varying delay. Hierarchical stability conditions based on linear matrix inequalities are obtained from an extensive use of the Bessel inequality applied to Legendre polynomials of arbitrary orders. While this inequality has been only used for constant discrete and distributed delays, this paper generalizes the same methodology to time-varying delays. We take advantages of the dependence of the stability criteria on both the delay and its derivative to propose a new definition of allowable delay sets. It is shown that a light and smart modification in the definition of this set leads to relevant conclusions on the numerical results.

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