Hélène Piet-Lahanier

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.

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

A method is described to exactly characterize the set of all the values of the parameter vector of a linear model that are coherent with bounded errors on the measurements. It provides a parameterized expression of this set that can be used afterwards for robust control design or for optimizing any criterion over the set. The method can be used in real time. Whenever a new measurement modifies the set, the characterization is updated.

Robust nonlinear parameter estimation in the bounded noise case

When modeling a system, it is of importance to assess the uncertainty on the estimated values of the parameters. This is usually done by taking advantage of the asymptotic properties of maximum likelihood estimators. However, the significance of the results obtained can be questionned when little information is available on the noise statistical properties and when the number of data points is limited. Membership set theory then appears as a promising tool to overcome some of these difficulties. Its purpose is to characterize the set of all the parameter vectors that are consistent with the assumed knowledge of bounds on the acceptable errors between the data and the model outputs. However, the membership set estimators presented in the literature so far are restricted to model linear in the parameters. The approach described here has been designed for handling nonlinear models as well as linear ones. It involves the maximisation of the number of data points that do not have to be considered as outliers and the characterization of the boundary of the domain of the parametric space where this number is maximum. The method is applied to two examples. It is shown to be extremely robust to outliers and to be able to handle even models that are not uniquely Identifiable.

Stability Analysis of an UAV Controller using Singular Perturbation Theory

Abstract This paper presents the stability analysis of a hierarchical controller for an Unmanned Aerial Vehicle, using singular perturbation theory. Position and attitude control laws are successively designed by considering a time-scale separation between the translational dynamics and the orientation dynamics of a six degrees of freedom Vertical Take Off and Landing UAV model. In addition, for the design of the position controller, we consider the case where the linear velocity of the vehicle is not measured. A partial state feedback control law is proposed, based on the introduction of virtual states in the translational dynamics of the system.

Estimation of non-uniquely identifiable parameters via exhaustive modeling and membership set theory

A new methodology for estimating parameters that are not structurally globally identifiable is presented. The set of all the models which have exactly the same input–output behavior is first generated. The influence of the measurement noise and structural error is then taken into account by assuming that upper and lower bounds of the acceptable output error are available. The set of all the models compatible with this hypothesis is characterized, and ranges for the possible values of the estimated parameters are provided. An example is treated involving two real–life model structures used to describe the behavior of an isotopic tracer in a reactor producing methane from carbon monoxide and hydrogen.

Worst-case global optimization of black-box functions through Kriging and relaxation

A new algorithm is proposed to deal with the worst-case optimization of black-box functions evaluated through costly computer simulations. The input variables of these computer experiments are assumed to be of two types. Control variables must be tuned while environmental variables have an undesirable effect, to which the design of the control variables should be robust. The algorithm to be proposed searches for a minimax solution, i.e., values of the control variables that minimize the maximum of the objective function with respect to the environmental variables. The problem is particularly difficult when the control and environmental variables live in continuous spaces. Combining a relaxation procedure with Kriging-based optimization makes it possible to deal with the continuity of the variables and the fact that no analytical expression of the objective function is available in most real-case problems. Numerical experiments are conducted to assess the accuracy and efficiency of the algorithm, both on analytical test functions with known results and on an engineering application.

Exact recursive characterization of feasible parameter sets in the linear case

Set-membership estimation (or parameter bounding) assumes that the error corrupting the data is described only by upper and lower bounds between which its realizations must lie. It aims at providing a description of the set of all parameter vectors that are consistent with the data, error bounds and model structure. For models linear in their parameters, an algorithm that gives an exact recursive characterization of this set is described. The problems raised by the implementation of the method on a computer are considered. Solutions are proposed and discussed. An application of the improved algorithm to a real example is finally presented.

Attitude tracking of rigid bodies on the special orthogonal group with bounded partial state feedback

A solution to the attitude tracking problem of rigid bodies with kinematic representation directly on the special orthogonal group SO(3) of rotation matrices is proposed. A dynamic partial state feedback controller is designed to address the case where no angular velocity measurements are available. In addition, the gains in the control design can be tuned in advance to ensure that the torque inputs satisfy arbitrary saturation bounds. Stability conditions are provided based on Lyapunov function analysis and Barbalats lemma. Simulation results are presented to illustrate the performance of the proposed control scheme.

Estimation of the parameter uncertainty resulting from bounded-error data

A procedure is described for characterizing the set of all parameter vectors that are consistent with data corrupted by a bounded noise. The method applies to any parametric model that can be simulated on a computer when upper and lower bounds for the noise are known a priori. The convergence properties of the associated estimator are considered, as well as its behavior in the presence of outliers. To illustrate the versatility of the technique, problems are considered where (i) the set of the true values of the parameter vector does not reduce to a singleton, (ii) the model is not uniquely identifiable, (iii) the hypotheses on the noise bounds are not satisfied, and (iv) the data contain a majority of outliers.

Further results on recursive polyhedral description of parameter uncertainty in the bounded-error context

When the (prediction) error is only known to be bounded, it is interesting to characterize the set of all values of the parameters to be estimated that are consistent with the data, error bounds, and model structure. When the error is affine in the parameters, this set is a convex polyhedron which can be fully characterized by enumerating its vertices and supporting hyperplanes. The contribution of this work is threefold. First, a new algorithm for an exact recursive description of the polyhedron is described. Second, a new method for determining the intersection of several polyhedrons (obtained, for example, from different data sets) is proposed. Third, the polyhedral-description approach is extended to output-error models. The procedure is illustrated by an example.<<ETX>>