Carine Jauberthie

Identifiability of a nonlinear delayed-differential aerospace model

This technical note shows the identifiability of a nonlinear delayed-differential model describing aircraft dynamics. This result is obtained by an original combination: An extension of the results of Grewal and Glover to delay systems and an application of linear delayed-differential model identifiability of Orlov et al. . It requires initial conditions corresponding to an equilibrium state, which is implemented during the experimentation.

Fault detection and identification relying on set-membership identifiability☆

Identifiability is the property that a mathematical model must satisfy to guarantee an unambiguous mapping between its parameters and the output trajectories. It is of prime importance when parameters must be estimated from experimental data representing input–output behavior and clearly when parameter estimation is used for fault detection and identification. Definitions of identifiability and methods for checking this property for linear and nonlinear systems are now well established and, interestingly, some scarce works (Braems, Jaulin, Kieffer, & Walter, 2001; Jauberthie, Verdiere, & Trave-Massuyes, 2011) have provided identifiability definitions and numerical tests in a bounded-error context. This paper resumes and better formalizes the two complementary definitions of set-membership identifiability and μ-set-membership identifiability of Jauberthie et al. (2011) and presents a method applicable to nonlinear systems for checking them. This method is based on differential algebra and makes use of relations linking the observations, the inputs and the unknown parameters of the system. Using these results, a method for fault detection and identification is proposed. The relations mentioned above are used to estimate the uncertain parameters of the model. By building the parameter estimation scheme on the analysis of identifiability, the solution set is guaranteed to reduce to one connected set, avoiding this way the pessimism of classical set-membership estimation methods. Fault detection and identification are performed at once by checking the estimated values against the parameter nominal ranges. The method is illustrated with an example describing the capacity of a macrophage mannose receptor to endocytose a specific soluble macromolecule.

Set-membership identifiability: definitions and analysis

Abstract Definitions and methods for checking the identifiability of linear and nonlinear systems are now well established. However this property has not really been investigated for uncertain systems, in particular for set-membership models in a bounded-error context. In this paper, we propose two complementary definitions, set-membership identifiability and μ-set-membership identifiability, the first one is conceptual whereas the second can be put in correspondence with existing set-membership parameter estimation methods. The links between these definitions are exhibited and two methods to check the properties are proposed, illustrated by examples.

Optimal input design for a nonlinear dynamical uncertain aerospace system

An optimal input design technique for aircraft uncertain parameter estimation is presented in this paper. The original idea is the combining of a dynamic programming method and interval analysis for the optimal input synthesis. This approach does not imply the estimation of a nominal value for parameter and allows to include realistic practical constraints on the input and output variables. The precise description of the approach is followed by an application in aerospace sciences.

Fault detection using interval Kalman filtering enhanced by constraint propagation

In this paper, we consider an extension of conventional Kalman filtering to discrete time linear models with bounded uncertainties on parameters and gaussian measurement noise. To solve the interval matrix inversion problem involved in the equations of the Kalman filter and the over-bounding problem due to interval calculus, we propose an original approach combining the set inversion algorithm SIVIA and constraint propagation. The improved interval Kalman filter is applied in a fault detection schema illustrated by a simple case study.

Aircraft Parameter Estimation: Successive Steps

Abstract System identification based on physical laws often involves parameter estimation. Before performing estimation problem, it is necessary to investigate its identifiability. A signifiant increase in accuracy of the parameter estimation may be obtained by a suitable choice of experimental conditions. This paper considers a model describing aircraft dynamics. Our contribution consists in solving the problem of optimal design. Then two successive steps are used to solve this problem. The first one is based on dynamic programming and the second one solves an optimization problem by a gradient algorithm using the results of the previous method.

Set-membership identifiability of nonlinear models and related parameter estimation properties

Abstract Identifiability guarantees that the mathematical model of a dynamic system is well defined in the sense that it maps unambiguously its parameters to the output trajectories. This paper casts identifiability in a set-membership (SM) framework and relates recently introduced properties, namely, SM-identifiability, μ-SM-identifiability, and ε-SM-identifiability, to the properties of parameter estimation problems. Soundness and ε-consistency are proposed to characterize these problems and the solution returned by the algorithm used to solve them. This paper also contributes by carefully motivating and comparing SM-identifiability, μ-SM-identifiability and ε-SM-identifiability with related properties found in the literature, and by providing a method based on differential algebra to check these properties.

Improvements in computational aspects of interval Kalman filtering enhanced by constraint propagation

This paper deals with computational aspects of interval kalman filtering of discrete time linear models with bounded uncertainties on parameters and gaussian measurement noise. In this work, we consider an extension of conventional Kalman filtering to interval linear models [1]. As the expressions for deriving the Kalman filter involve matrix inversion which is known to be a difficult problem. One must hence find a way to implement or avoid this tricky algebraic operation within an interval framework. To solve the interval matrix inversion problem and other problems due to interval calculus, we propose an original approach combining the set inversion algorithm SIVIA and constraint satisfaction propagation. Several contractors are proposed to limit overestimation effects propagating within the interval Kalman filter recursive structure. Thus the description of our approach is followed by an application and we compare the proposed approach with interval kalman filtering developped in [1].

Identifiability and estimation of aircraft parameters and delays by using optimal input design

This paper considers test of identifiability, optimal input design and estimation problem given by aerospace domain describing aircraft nonlinear dynamics with time delays. The original idea is to use an approximation well in line with the given system and an algebraic approach to analyze identifiability. Then the approximate model is considered to obtain the optimal input and to estimate the parameters and delays of the original model. Numerical results are given and compared.

Evidential box particle filter using belief function theory

A box particle filtering algorithm for nonlinear state estimation based on belief function theory and interval analysis is presented. The system under consideration is subject to bounded process noises and Gaussian multivariate measurement errors. The mean and the covariance matrix of Gaussian random variables are considered bounded due to modeling errors. The belief function theory is a means to represent this type of uncertainty using a mass function whose focal sets are intervals. The proposed algorithm applies interval analysis and constraint satisfaction techniques. Two nonlinear examples show the efficiency of the proposed approach compared to the original box particle filter.