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17102
09/07/2017

Lyapunov stability analysis of a linear system coupled to a heat equation

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

MAC

Manifestation avec acte : IFAC World Congress 2017 du 09 juillet au 14 juillet 2017, Toulouse (France), Juillet 2017, 6p. , N° 17102

Lien : https://hal.laas.fr/hal-01496115

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Abstract

This paper addresses the stability analysis of a system of ordinary differential equations coupled with a classic heat equation using a Lyapunov approach. Relying on recent developments in the area of time-delay systems, a new method to study the stability of such a class of coupled finite/infinite dimensional systems is presented here. It consists in a Lyapunov analysis of the infinite dimensional state of the system using an energy functional enriched by the mean value of the heat variable. The main technical step relies on the use an efficient Bessel-like integral inequality on Hilbert space leading to tractable conditions expressed in terms of linear matrix inequalities. The results are then illustrated on academic examples and demonstrate the potential of this new approach.

140055
17073
09/07/2017

Refined exponential stability analysis of a coupled system

M.SAFI, L.BAUDOUIN, A.SEURET

ISI, MAC

Manifestation avec acte : IFAC World Congress 2017 du 09 juillet au 14 juillet 2017, Toulouse (France), Juillet 2017, 6p. , N° 17073

Lien : https://hal.laas.fr/hal-01496136

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Abstract

The objective of this contribution is to improve recent stability results for a system coupling ordinary differential equations to a vectorial transport partial differential equation by proposing a new structure of Lyapunov functional. Following the same process of most of the investigations in literature, that are based on an a priori selection of Lyapunov functionals and use the usual integral inequalities (Jensen, Wirtinger, Bessel...), we will present an efficient method to estimate the exponential decay rate of this coupled system leading to a tractable test expressed in terms of linear matrix inequalities. These LMI conditions stem from the new design of a candidate Lyapunov functional, but also the inherent properties of the Legendre polynomials, that are used to build a projection of the infinite dimensional part of the state of the system. Based on these polynomials and using the appropriate Bessel-Legendre inequality, we will prove an exponential stability result and in the end, we will show the efficiency of our approach on academic example.

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14357
27/06/2017

Linear conic optimization for nonlinear optimal control

D.HENRION, E.PAUWELS

MAC

Ouvrage (contribution) : Advances and Trends in Optimization with Engineering Applications, N°ISBN 978-1-61197-467-6, Juin 2017, Chapter 10, pp.121-133 , N° 14357

Lien : http://hal.archives-ouvertes.fr/hal-01019410

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Abstract

Infinite-dimensional linear conic formulations are described for nonlinear optimal control problems. The primal linear problem consists of finding occupation measures supported on optimal relaxed controlled trajectories, whereas the dual linear problem consists of finding the largest lower bound on the value function of the optimal control problem. Various approximation results relating the original optimal control problem and its linear conic formulations are developed. As illustrated by a couple of simple examples, these results are relevant in the context of finite-dimensional semidefinite programming relaxations used to approximate numerically the solutions of the infinite-dimensional linear conic problems.

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17125
22/05/2017

Real-time control systems: feedback, scheduling and robustness

D.SIMON, A.SEURET, O.SENAME

LIRMM, MAC, GIPSA-Lab

Rapport LAAS N°17125, DOI 10.1080/00207721.2017.1316879, Mai 2017, 11p.

Lien : https://hal-lirmm.ccsd.cnrs.fr/lirmm-01515226

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Abstract

The efficient control of real-time distributed systems, where continuous components are governed through digital devices and communication networks, needs a careful examination of the constraints arising from the different involved domains inside co-design approaches. Thanks to the robustness of feedback control, both new control methodologies and slackened real-time scheduling schemes are proposed beyond the frontiers between these traditionally separated fields. A methodology to design robust aperiodic controllers is provided, where the sampling interval is considered as a control variable of the system. Promising experimental results are provided to show the feasibility and robustness of the approach.

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17114
15/05/2017

Payoff-oriented quantization and application to power control

C.ZHANG, N.KHALFET, S.LASAULCE, V.VARMA, S.TARBOURIECH

L2S, CRAN, Vandoeuvre, MAC

Manifestation avec acte : International Workshop on Resource Allocation, Cooperation and Competition in Wireless Networks ( RAWNET ) 2017 du 15 mai au 15 mai 2017, Paris (France), Mai 2017, 6p. , N° 17114

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Abstract

In many resource allocation problems, optimal allocation strategies must be determined when only a quantized version of the relevant parameters are available, for instance, power allocation in wireless communications. The contribution of this work is threefold. First, the quantization problem is revisited and a framework which encompasses the classical problem of quantization is proposed. Instead of minimizing the distortion, the goal is to minimize the gap between the maximum of a general payoff function (which would be reached by knowing all parameters of the function) and what is effectively reached when only the quantized version of the parameters is available. Then, to determine such a quantizer, the well-known Lloyd-Max algorithm is generalized. At last, we show how this framework can be applied to the problem of power control in wireless communications; the obtained numerical results clearly show the potential of such a framework.

139733
16159
03/05/2017

Positivity certificates in optimal control

E.PAUWELS, D.HENRION, J.B.LASSERRE

MAC

Ouvrage (contribution) : Geometric and Numerical Foundations of Movements, Springer, N°ISBN 978-3-319-51546-5, Mai 2017 , N° 16159

Lien : https://hal.archives-ouvertes.fr/hal-01311874

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Abstract

We propose a tutorial on relaxations and weak formulations of optimal control with their semidefinite approximations. We present this approach solely through the prism of positivity certificates which we consider to be the most accessible for a broad audience, in particular in the engineering and robotics communities. This simple concept allows to express very concisely powerful approximation certificates in control. The relevance of this technique is illustrated on three applications: region of attraction approximation, direct optimal control and inverse optimal control, for which it constitutes a common denominator. In a first step, we highlight the core mechanisms underpinning the application of positivity in control and how they appear in the different control applications. This relies on simple mathematical concepts and gives a unified treatment of the applications considered. This presentation is based on the combination and simplification of published materials. In a second step, we describe briefly relations with broader literature, in particular, occupation measures and Hamilton-Jacobi-Bellman equation which are important elements of the global picture. We describe the Sum-Of-Squares (SOS) semidefinite hierarchy in the semialgebraic case and briefly mention its convergence properties. Numerical experiments on a classical example in robotics, namely the nonholonomic vehicle, illustrate the concepts presented in the text for the three applications considered.

139713
14329
01/05/2017

Rigorous uniform approximation of D-finite functions using Chebyshev expansions

A.BENOIT, M.M.JOLDES, M.MEZZAROBBA

EXT, MAC, LIP6-CNRS

Revue Scientifique : Mathematics of Computation , Vol.86, N°305, pp.1303-1341, Mai 2017 , N° 14329

Lien : http://hal.archives-ouvertes.fr/hal-01022420

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Abstract

A wide range of numerical methods exists for computing polynomial approximations of solutions of ordinary differential equations based on Chebyshev series expansions or Chebyshev interpolation polynomials. We consider the application of such methods in the context of rigorous computing (where we need guarantees on the accuracy of the result), and from the complexity point of view. It is well-known that the order-n truncation of the Chebyshev expansion of a function over a given interval is a near-best uniform polynomial approximation of the function on that interval. In the case of solutions of linear differential equations with polynomial coefficients, the coefficients of the expansions obey linear recurrence relations with polynomial coefficients. Unfortunately, these recurrences do not lend themselves to a direct recursive computation of the coefficients, owing among other things to a lack of initial conditions. We show how they can nevertheless be used, as part of a validated process, to compute good uniform approximations of D-finite functions together with rigorous error bounds, and we study the complexity of the resulting algorithms. Our approach is based on a new view of a classical numerical method going back to Clenshaw, combined with a functional enclosure method.

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17111
27/04/2017

Nonlinear control of multi-rotor aerial vehicles based on the zero-moment direction

G.MICHIELETTO, A.CENEDESE, L.ZACCARIAN, A.FRANCHI

RIS, University of Padova, MAC

Rapport LAAS N°17111, Avril 2017, 6p.

Lien : https://hal.laas.fr/hal-01513229

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A quaternion-based nonlinear control strategy is here presented to steer and keep a generic multi-rotor platform in a given reference position. Exploiting a state feedback structure, the proposed solution ensures the stabilization of the aerial vehicle so that its linear and angular velocity are zero and its attitude is constant. The main feature of the designed controller is the identification of a zero-moment direction in the feasible force space, i.e., a direction along which the control force intensity can be assigned independently of the control moment. The asymptotic convergence of the error dynamics is confirmed by simulation results on a hexarotor with tilted propellers.

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17110
27/04/2017

A minimum-fuel fixed-time low-thrust rendezvous solved with the switching systems theory

C.GAZZINO, D.ARZELIER, L.CERRI, D.LOSA, C.LOUEMBET, C.PITTET-MECHIN

MAC, ROC, CNES, Thalès Alenia Space

Rapport LAAS N°17110, Avril 2017, 6p.

Lien : https://hal.laas.fr/hal-01511019

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In this paper, a fuel optimal rendezvous problem is tackled in the Hill-Clohessy-Wiltshire framework with several operational constraints as bounds on the thrust, non linear non convex and disjunctive operational constraints (on-off profile of the thrusters, minimum elapsed time between two consecutive firings...). An indirect method and a decomposition technique have already been combined in order to solve this kind of optimal control problem with such constraints. Due to a great number of parameters to tune, satisfactory results are hard to obtain and are sensitive to the initial condition. Assuming that no singular arc exists, it can be shown that the optimal control exhibits a bang-bang structure whose optimal switching times are to be found. Noticing that a system with a bang-bang control profile can be considered as two subsystems switching from one with control on to with control off, and vice-versa, a technique coming from the switching systems theory is used in order to optimise the switching times.

139665
17103
25/04/2017

Observer-based event-triggered control for linear systems subject to cone-bounded nonlinearities

S.TARBOURIECH, A.SEURET, L.MOREIRA, J.M.GOMES DA SILVA Jr

MAC, UFRGS

Rapport LAAS N°17103, Avril 2017, 6p.

Lien : https://hal.laas.fr/hal-01497747

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The paper presents an observer-based event-triggered control strategy for linear systems subject to input cone-bounded nonlinearities by using only available measurable variables. Sufficient conditions based on linear matrix inequalities are proposed to ensure the asymptotic stability of the closed loop and the avoidance of Zeno behavior in an emulation context. Based on these conditions, a convex optimization problem to compute the parameters of the event-trigger rule aiming at reducing the number of control updates is proposed. The approach is illustrated on a numerical example that considers the control of a linear system with a logarithmic input quantization constraint.

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