Interval methods for nonlinear identification and robust control

International audienceInterval methods can provide guaranteed solutions to difficult nonlinear problems, such as the global optimization of non-convex criteria or the characterization of sets defined by nonlinear inequalities. They can even deal with problems involving quantifiers, as encountered for example in robust control design. For high dimensional problems; their efficiency can be considerably improved by resorting to constraint propagation. In this paper, key ideas of interval analysis and constraint propagation are presented and applied to two problems. The first one is the guaranteed characterization of the set of all parameter vectors that are consistent with experimental dat up to bounds on the acceptable errors. The second one is the design of a PI controller robustly stabilizing a set of models that may have been obtained as the solution to the first problem

Interval analysis for non-linear parameter and state estimation: contributions and limitations

Parameter or state estimation should take into account the fact that the model is an approximation of reality and that the data are corrupted by noise. In this paper, each uncertain quantity is assumed to belong to a known set. The problem, known as parameter or state bounding, is the to characterize the set of all parameter or state vectors that are consistent with the model structure, data and error bounds. A description of how interval analysis can be used to find guaranteed estimates in a nonlinear context is provided. The main notions of interval analysis are first recalled very briefly. the simpler problem of parameter estimation is then considered. State estimation, which contains parameter tracking as a special case, is treated next. A simple illustrative example is finally presented

Nonlinear state estimation using forward-backward propagation of intervals in an algorithm

International audienceThe paper deals with the estimation of the state vector of a discrete-time model from interval output data. When the model outputs are affine in the initial state vector, a number of methods are available to enclose all estimates that are consistent with data by simple sets such as ellipsoids, orthotopes or parallelotopes, thereby providing guaranteed set estimates. In the nonlinear case, the situation is much less developed and there are very few methods that produce such guaranteed estimates. In this paper, the state estimation of a discrete-time model is performed by combining a set-inversion algorithm with a forward-backward propagation of intervals through the model. The resulting methodology is illustrated on an example

Guaranteed set computation with subpavings

International audienceThis paper is about the approximate representation of compact sets using subpavings, i.e., unions of non-overlapping boxes, and about computation on these sets, with particular attention to implementation issues. Some basic operations such as evaluating the intersection or union of two subpavings, or testing whether a box belongs to a subpaving are first presented. The binary tree structure used to describe subpavings then allows a simple implementation of these tasks by recursive algorithms. Algorithms are presented to evaluate the inverse and direct images of a set described by a subpaving. In both cases, a subpaving is obtained that is guaranteed to contain the actual inverse or direct image of the initial subpaving. The effectiveness of these algorithms in characterizing possibly nonconvex on even nonconnected stes is finally illustrated by simple examples

Méthodes ensemblistes garanties pour l'estimation de grandeurs physiques

Building a parametric model from experimental data raises numerous questions that one would wish to solve in a guaranteed way, as they condition the success of the operation:- is the model structure appropriate?- is the selected sensor able to fulfill the experimenters' requirements?- have all the values of the parameters consistent with the data and the hypotheses been found?- may numerical errors have falsified the results?In order to attempt to answer these questions, the direction followed in this thesis is to formulate each of them in a set-theoretical framework. Branch and bound algorithms using interval analysis then provide guaranteed inner and outer approximations of the resulting sets. We show that the size of the problems that can be treated ùay be significantly increased by using contractors developed in the context of constraint satisfaction programming.The estimation of unidentifiable kinetic parameters of an electrochemical reaction illustrates the performances of this approach. THe characterization of a new experimental set-up to measure thermal properties of materials taking into account intrinsic uncertainty is made possible by a new algorithm to project sets in a guaranteed way.The set-theoretical formulation of the analysis of structural properties of models leads to new definitions of identifiability and distinguishability, which seem more appropriate than the traditional ones. Algorithms to test such properties are also presented. We finally propose a new way to evaluate the characteristics of a sensor in terms of accuracy.La construction d'un modèle paramétrique à partir de données expérimentales soulève de nombreuses questions auxquelles l'expérimentateur souhaiterait répondre de façon garantie, car elles conditionnent le succès de l'opération :- la structure de modèle retenue est-elle pertinente?- le capteur sélectionné permet-il de satisfaire les contraintes du cahier des charges de l'expérience?- toutes les valeurs des paramètres compatibles avec les données et les hypothèses ont-elles été trouvées?- les erreurs d'arrondi du calculateur faussent-elles les résultats?Pour tenter de répondre à ces questions, cette thèse prend le partir de les aborder dans une perspective ensembliste. L'obtention d'approximations intérieure et extérieur des ensembles résultants, à l'aide de l'analyse par intervalles, assure alors cette garantie. Nous montrons que la taille des problèmes qu'il est possible de traiter peut être significativement augmentée par l'emploi de contracteurs issus de la programmation par contraintes.L'estimation de paramètres cinétiques non-identifiables en électrochimie illustre les performances de cette approche. La caractérisation d'un nouveau dispositif de mesure en thermique,en tenant compte des incertitudes intrinsèques à ce dispositif, est rendue possible par un nouvel algorithme de projection garantie d'ensembles.La formulation ensembliste du problème de la caractérisation de structures de modèles permet également de proposer de nouvelles définitions de l'identifiabilité et de la discernabilité, qui semblent plus pertinentes que les notions traditionnelles. Des algorithmes qui permettent le test effectif de ces propriétés sont également proposés. Enfin, une évaluation des caractéristiques d'un capteur en terme de précision est présentée

Reliable Discrimination of Models Based on EIS Data: I. Calibration on Equivalent Electrical Circuits

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