Isabelle Braems

Guaranteed nonlinear estimation using constraint propagation on sets

Bounded-error estimation is the estimation of the parameter or state vector of a model from experimental data, under the assumption that some suitably defined errors should belong to some prior feasible sets. When the model outputs are linear in the vector to be estimated, a number of methods are available to contain all estimates that are consistent with the data within simple sets such as ellipsoids, orthotopes or parallelotopes, thereby providing guaranteed set estimates. In the non-linear case, the situation is much less developed and there are very few methods that produce such guaranteed estimates. In this paper, the problem of characterizing the set of all state vectors that are consistent with all data in the case of non-linear discrete-time systems is cast into the more general framework of constraint satisfaction problems. The state vector at time k should be estimated either on-line from past measurement only or off-line from a series of measurements that may include measurements posterior to k . Even in the causal case, prior information on the future value of the state and output vectors, due for instance to physical constraints, is readily taken into account. Algorithms taken from the literature of interval constraint propagation are extended by replacing intervals by more general subsets of real vector spaces. This makes it possible to propose a new algorithm that contracts the feasible domain for each uncertain variable optimally (i.e. no smaller domain could be obtained) and efficiently.

Linear and Bayesian Planet Detection Algorithms for the Terrestrial Planet Finder

Current plans call for the first Terrestrial Planet Finder mission, TPF-C, to be a monolithic space telescope with a coronagraph for achieving high contrast. The coronagraph removes the diffracted starlight allowing the nearby planet to be detected. In this paper, we present a model of the planet measurement and noise statistics. We use this model to develop two planet detection algorithms, one based on matched filtering of the point-spread function (PSF) and one using Bayesian techniques. These models are used to formulate integration time estimates for a planet detection with desired small probabilities of false alarms and missed detections.

Interval methods for nonlinear identification and robust control

Key ideas of interval analysis and constraint propagation are presented and applied to two problems frequently encountered in control. The first one is the guaranteed characterization of the set of all parameter vectors that are consistent with experimental data up to bounds on the acceptable errors. The second one is the guaranteed characterization of the set of all PI controllers robustly stabilizing a set of models that may have been obtained as the solution to the first problem.

Tight-binding variable-charge model for insulating oxides: Application to TiO2 and ZrO2 polymorphs

We have developed a new variable-charge model aimed at performing large-scale realistic simulations of oxide surfaces and interfaces. This model is based on the charge equilibration (QEq) method and explicitly takes into account the mixed iono-covalent character of the metal-oxygen bond by means of a tight-binding analytical approach. We present the first results obtained for TiO2 and ZrO2 polymorphs, which are in very good agreement with the experimental data and recent ab initio results.

Guaranteed Numerical Computation as an Alternative to Computer Algebra for Testing Models for Identifiability

Testing parametric models for identifiability is particularly important for knowledge-based models. If several values of the parameter vector lead to the same observed behavior, then one may try to modify the experimental set-up to eliminate this ambiguity (which corresponds to performing qualitative experiment design). The tediousness of the algebraic operations involved in such tests makes computer algebra particularly attractive. This paper describes some limitations of this classical approach and explores an alternative route based on new definitions of identifiability and numerical tests implemented in a guaranteed way. The new approach is illustrated in the context of compartmental modeling, widely used in biology.

Guaranteed Set Computation with Subpavings

This 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 sets is finally illustrated by simple examples.

Guaranteed numerical alternatives to structural identifiability testing

Testing models for structural identifiability is particularly important for knowledge-based models. If several values of the parameter vector lead to the same observed behavior of the model, then one may try to modify the experimental setup to eliminate this ambiguity (qualitative experiment design). The tediousness of the algebraic manipulations involved makes computer algebra particularly attractive. The purpose of the paper is to explore an alternative route based on guaranteed numerical computation. A new definition of identifiability in a domain allows testing to be cast into the framework of constraint-satisfaction problems, and makes it possible to use the tools of interval analysis and interval constraint propagation to get guaranteed answers. When the data have already been collected, the notion of structural identifiability may not be the most pertinent concept. The paper shows how interval analysis and interval constraint propagation can again be used to bypass the identifiability study and estimate even parameters that are not identifiable uniquely.

Nonlinear State Estimation Using Forward-Backward Propagation of Intervals in an Algorithm

The 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 characterization of thermal conductivity and diffusivity in presence of model uncertainty

A crucial problem that occurs when estimating physical parameters from experimental data is the computation of reliable uncertainty bounds for the estimated parameters, while accounting for uncertainty in the model and data. We introduce a new numerical method that contributes to the solution of this problem. We show how to deal with uncertain nuisance parameters located within prior intervals. The method advocated in this article makes it possible to detect the absence of solution when the model hypotheses are inconsistent with the data. An analysis of the sensitivity of estimated uncertainty bounds to the nuisance parameters is also conducted. These features are illustrated with actual data collected on a thermal device used to estimate simultaneously the conductivity and diffusivity of materials.

Prior characterization of the performance of software sensors

Abstract Sensor performance is usually evaluated a posteriori after numerous essays, in a probabilistic framework were unknwon quantities are modeled by random variables. This paper addresses the problem of evaluating a priori the limits of the performance that can be achieved with a software sensor in a given range of operation. This is done in a context of bounded-error estimation, worst-case design and MinMax optimization. An algorithm based on interval analysis is used to obtain guaranteed results, and the procedure advocated is illustrated on a simple example of saturating vapour pressure thermometers.