Eric Walter

Set inversion via interval analysis for nonlinear bounded-error estimation

Abstract In the context of bounded-error estimation, one is interested in characterizing the set of all the values of the parameters to be estimated that are consistent with the data in the sense that the errors between the data and model outputs fall within prior bounds. While the problem can be considered as solved when the model output is linear in the parameters, the situation is far less advanced in the general nonlinear case. In this paper, the problem of nonlinear bounded-error estimation is viewed as one of set inversion. An original algorithm is proposed, based upon interval analysis, that makes it possible to characterize the feasible set for the parameters by enclosing it between internal and external unions of boxes. The convergence of the algorithm is proved and the algorithm is applied to two test cases. The results obtained are compared with those provided by signomial analysis.

Estimation of parameter bounds from bounded-error data: a survey

Set-membership techniques for estimating parameters from uncertain data are reviewed. Contrary to the prevailing usage, the error in the data is not considered as a random variable with known or parameterized probability density function. Instead, the error is assumed to lie between some known upper and lower bounds. One is then looking for a suitable characterization of the set of all parameter vectors consistent with the model structure, data, and bounds on the errors.

Qualitative and quantitative experiment design for phenomenological models—a survey

Abstract Designing an experiment for parameter estimation involves two steps. The first one is qualitative, and consists of selecting a suitable configuration of the input/output ports so as to make, if possible, all the parameters of interest identifiable. The second step is quantitative, and based on the optimization of a suitable criterion (with respect to the input shapes, sampling schedule,…) so as to get the maximum information from the data to be collected. When the model is nonlinear in the parameters, the typical situation for phenomenological models, both steps present specific difficulties which are discussed in this paper. The practical importance of qualitative experiment design is illustrated by a very simple biological model. Various policies presented in the literature for quantitative experiment design are reviewed. Special emphasis is given to methods allowing uncertainty on the prior information to be taken into account.

An informational approach to the global optimization of expensive-to-evaluate functions

In many global optimization problems motivated by engineering applications, the number of function evaluations is severely limited by time or cost. To ensure that each evaluation contributes to the localization of good candidates for the role of global minimizer, a sequential choice of evaluation points is usually carried out. In particular, when Kriging is used to interpolate past evaluations, the uncertainty associated with the lack of information on the function can be expressed and used to compute a number of criteria accounting for the interest of an additional evaluation at any given point. This paper introduces minimizers entropy as a new Kriging-based criterion for the sequential choice of points at which the function should be evaluated. Based on stepwise uncertainty reduction, it accounts for the informational gain on the minimizer expected from a new evaluation. The criterion is approximated using conditional simulations of the Gaussian process model behind Kriging, and then inserted into an algorithm similar in spirit to the Efficient Global Optimization (EGO) algorithm. An empirical comparison is carried out between our criterion and expected improvement, one of the reference criteria in the literature. Experimental results indicate major evaluation savings over EGO. Finally, the method, which we call IAGO (for Informational Approach to Global Optimization), is extended to robust optimization problems, where both the factors to be tuned and the function evaluations are corrupted by noise.

Ellipsoidal parameter or state estimation under model uncertainty

Ellipsoidal outer-bounding of the set of all feasible state vectors under model uncertainty is a natural extension of state estimation for deterministic models with unknown-but-bounded state perturbations and measurement noise. The technique described in this paper applies to linear discrete-time dynamic systems; it can also be applied to weakly non-linear systems if non-linearity is replaced by uncertainty. Many difficulties arise because of the non-convexity of feasible sets. Combined quadratic constraints on model uncertainty and additive disturbances are considered in order to simplify the analysis. Analytical optimal or suboptimal solutions of the basic problems involved in parameter or state estimation are presented, which are counterparts in this context of uncertain models to classical approximations of the sum and intersection of ellipsoids. The results obtained for combined quadratic constraints are extended to other types of model uncertainty.

Exact recursive polyhedral description of the feasible parameter set for bounded-error models

A method is described which exactly characterizes the set of all the values of the parameter vector of a linear model that are consistent with bounded errors on the measurements. It provides a parameterized expression of this set, which can be used for robust control design or for optimizing any criterion over the set. This approach is based on a new variant of the double description method for determining the edges of a polyhedral cone. It can be used in real time and provides a suitable context for implementation on a computer. Whenever a new measurement modifies the set, the characterization is updated. The technique is illustrated with a simple example. >

Robust experiment design via stochastic approximation

Abstract A new methodology is proposed for robust experiment design. It allows uncertainly in the nominal parameters of the model under study to be taken into account by assuming that these parameters belong to some population with known statistics. The mathematical expectation of the determinant of the Fisher information matrix over this population is here taken as a measure of optimality, but the expectation of other nonrobust criteria could have been considered as well. Stochastic approximation techniques are advocated as the simplest tools for optimizing these robust criteria. The efficiency of the proposed algorithms is demonstrated on simple examples—for which an analytical solution exists—as well as on more complex ones. A comparison is made with Landaws suboptimal approach, which supports an interesting conjecture about the robustness of replicate samples.

Model-based fault diagnosis for aerospace systems: a survey

This survey of model-based fault diagnosis focuses on those methods that are applicable to aerospace systems. To highlight the characteristics of aerospace models, generic non-linear dynamical modelling from flight mechanics is recalled and a unifying representation of sensor and actuator faults is presented. An extensive bibliographical review supports a description of the key points of fault detection methods that rely on analytical redundancy. The approaches that best suit the constraints of the field are emphasized and recommendations for future developments in in-flight fault diagnosis are provided.

On the identifiability and distinguishability of nonlinear parametric models

Testing parametric models for identifiability and distinguishability is important when the parameters to be estimated have a physical meaning or when the model is to be used to reconstruct physically meaningful state variables that cannot be measured directly. Examples are used to explain why and indicate briefly how, with special emphasis on nonlinear models.

Robust Autonomous Robot Localization Using Interval Analysis

This paper deals with the determination of the position and orientation of a mobile robot from distance measurements provided by a belt of onboard ultrasonic sensors. The environment is assumed to be two-dimensional, and a map of its landmarks is available to the robot. In this context, classical localization methods have three main limitations. First, each data point provided by a sensor must be associated with a given landmark. This data-association step turns out to be extremely complex and time-consuming, and its results can usually not be guaranteed. The second limitation is that these methods are based on linearization, which makes them inherently local. The third limitation is their lack of robustness to outliers due, e.g., to sensor malfunctions or outdated maps. By contrast, the method proposed here, based on interval analysis, bypasses the data-association step, handles the problem as nonlinear and in a global way and is (extraordinarily) robust to outliers.