Geometric and Numerical Methods in the Contrast Imaging Problem in Nuclear Magnetic Resonance

Bonnard, Bernard; Claeys, Mathieu; Cots, Olivier; Martinon, Pierre

doi:10.1007/s10440-014-9947-3

Cited by 21 publications

(25 citation statements)

References 26 publications

(52 reference statements)

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“…Note the use of a scalar product, denoted by a dot, between the gradient vector ∂v ∂x and the controlled dynamics. In [40], it is proposed then to relax strong problem (4) by optimizing over all Borel measures satisfying (7) instead of simply occupation measures, yielding the following "weak" problem:…”

Section: Occupation Measuresmentioning

confidence: 99%

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Modal occupation measures and LMI relaxations for nonlinear switched systems control

Claeys¹,

Daafouz

Henrion

2016

Automatica

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This paper presents a linear programming approach for the optimal control of nonlinear switched systems where the control is the switching sequence. This is done by introducing modal occupation measures, which allow to relax the problem as a primal linear programming (LP) problem. Its dual linear program of Hamilton-Jacobi-Bellman inequalities is also characterized. The LPs are then solved numerically with a converging hierarchy of primal-dual moment-sum-of-squares (SOS) linear matrix inequalities (LMI). Because of the special structure of switched systems, we obtain a much more efficient method than could be achieved by applying standard moment/SOS LMI hierarchies for general optimal control problems.through an adequate switching strategy, in order to impose global stability and/or performance. Interested readers may refer to the survey papers [17,31,44,33] and the books [32,46] and the references therein.In this context, these systems are generally controlled by intrinsically discontinuous control signals, whose switching rule must be carefully designed to guarantee stability and performance properties. As far as optimality is concerned, several results are available in two main different contexts.The first category of methods exploits necessary optimality conditions, in the form of Pontryagin's maximum principle (the so-called indirect approaches), or through a large nonlinear discretization of the problem (the so-called direct approaches). The first contributions can be found in [8,22,36] where the problem has been formulated and partial solutions provided through generalized Hamilton-Jacobi-Bellman (HJB) equations or inequalities and convex optimization. In [38,47,43], the maximum principle is generalized to the case of general hybrid systems with nonlinear dynamics. The case of switched systems is discussed in [4,42,2]. For general nonlinear problems and hence for switched systems, only local optimality can be guaranteed , even when discretization can be properly controlled [51]. The subject is still largely open and we are far from a complete and numericaly tractable solution to the switched optimal control problem.The second category collects extensions of the performance indices H 2 and H ∞ originally developed for linear time invariant systems without switching, and use the flexibility of Lyapunov's approach, see for instance [20,16] and references therein. Even for linear switched systems, the proposed results are based on nonconvex optimization problems (e.g., bilinear matrix inequality conditions) difficult to solve directly. Sufficient convex Linear Matrix Inequality (LMI) design conditions may be obtained, yielding computed solutions which are suboptimal, but at the price of introducing a conservatism (pessimism) and a gap to optimality which is hard, if not impossible, to evaluate. Since the computation of this optimal strategy is a difficult task, a suboptimal solution is of interest only when it is proved to be consistent, meaning that it imposes to the switched system a performance not worse th...

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Section: Occupation Measuresmentioning

confidence: 99%

“…For instance, the numerical approach of [40] proposes to consider a finite subset of the countably many linear constraints on µ given by (7). Then, as a consequence of Tchakaloff's theorem [29,Th.…”

Section: Modal Occupation Measures and Primal Lpmentioning

confidence: 99%

Modal occupation measures and LMI relaxations for nonlinear switched systems control

Claeys¹,

Daafouz

Henrion

2016

Automatica

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“…The main difference comes from the fact that the final time is now a parameter and one may study its influence on the optimal trajectories. This was already done in [6], where a state unconstrained and affine scalar control problem of Mayer form was analyzed taking into account the influence of the final time. Moreover, the state constrained case should be clarified with the same tools as those presented in this paper.…”

Section: Resultsmentioning

confidence: 99%

“…To construct this synthesis we catch the changes along the homotopies and determine the new strategy using the theoretical results. However, if we consider a general optimal control problem depending on a parameter λ, then to get a better synthesis, for a given λ we should compare the cost associated to each component of h −1 ({0}) ∩ {λ = λ}, for each homotopic function h. This approach is crucial when the optimal control problem has for example many local solutions, see [6]. x3(t f ) = +1 19.…”

Section: Synthesis With Respect To I Max and V Maxmentioning

confidence: 99%

Geometric and numerical methods for a state constrained minimum time control problem of an electric vehicle

Cots¹

2017

ESAIM: COCV

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Abstract. In this article, the minimum time control problem of an electric vehicle is modeled as a Mayer problem in optimal control, with affine dynamics with respect to the control and with state constraints. The candidates as minimizers are selected among a set of extremals, solutions of a Hamiltonian system given by the maximum principle. An analysis, with the techniques of geometric control, is used first to reduce the set of candidates and then to construct the numerical methods. This leads to a numerical investigation based on indirect methods using the HamPath software. Multiple shooting and homotopy techniques are used to build a synthesis with respect to the bounds of the boundary sets.Mathematics Subject Classification. 49K15, 49M05, 90C90, 80M50.

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Section: Modal Occupation Measures and Primal Lpmentioning

confidence: 99%

Optimal switching control design for polynomial systems: an LMI approach

Henrion

Daafouz

Claeys

2013

52nd IEEE Conference on Decision and Control

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Geometric and Numerical Methods in the Contrast Imaging Problem in Nuclear Magnetic Resonance

Cited by 21 publications

References 26 publications

Modal occupation measures and LMI relaxations for nonlinear switched systems control

Modal occupation measures and LMI relaxations for nonlinear switched systems control

Geometric and numerical methods for a state constrained minimum time control problem of an electric vehicle

Optimal switching control design for polynomial systems: an LMI approach

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