Determination of set-membership identifiability sets

International audienceThis paper concerns the concept of set-membership identifiability introduced in \cite{jauberthie}. Given a model, a set-membership identifiable set is a connected set in the parameter domain of the model such that its corresponding trajectories are distinct to trajectories arising from its complementary. For obtaining the so-called set-membership identifiable sets, we propose an algorithm based on interval analysis tools. The proposed algorithm is decomposed into three parts namely {\it mincing}, {\it evaluating} and {\it regularization} (\cite{jaulin2}). The latter step has been modified in order to obtain guaranteed set-membership identifiable sets. Our algorithm will be tested on two examples

Estimation des systèmes dynamiques incertains et des propriétés connexes. Application à la surveillance de la santé

In this work, we are mainly interested in the preventive surveillance of nonlinear systems with bounded uncertainties, that is to say systems for which uncertainties are defined only by their belonging to intervals. For this purpose, we have placed ourselves in a so-called "ensembliste" context in which we have been able to extend two important properties studied in a stochastic context that are identifiability and diagnosticability. Beyond the conceptual definitions required by this work, we proposed tools related to the differential algebra to verify these two properties. The impact of the set identifiability on the results of a parameter estimation was also analyzed, the estimation of parameters being one of the approaches chosen for the detection and the isolation of defects in this work. We also sought to improve the estimation of parameters by developing criteria for the planning of experiments, consisting in optimizing the initial conditions, inputs and / or sampling period. These criteria were applied to two case studies (pharmacokinetics and aeronautics) with very good results. In addition, we are interested in the modeling of systems with "mixed" uncertainties, combining bounded and stochastic uncertainties, notably by proposing an improvement of the Kalman filter at intervals. The work achieved formalizes problems not previously addressed and lays the groundwork for future research.Dans ce travail, nous nous intéressons principalement à la surveillance préventive des systèmes non linéaires à incertitudes bornées, c’est-à-dire des systèmes pour lesquels les incertitudes ne sont définies que par leur appartenance à des intervalles. Pour cela, nous nous sommes placés dans un contexte dit « ensembliste » dans lequel nous avons pu étendre deux propriétés importantes largement étudiées en contexte stochastique qui sont l’identifiabilité et la diagnosticabilité. Au delà des définitions conceptuelles requises par ce travail, nous avons proposé des outils liés à l’algèbre différentielle permettant de vérifier ces deux propriétés. L’impact de l’identifiabilité ensembliste sur les résultats d’une estimation de paramètres a également été analysé, l’estimation de paramètres étant l’une des approches retenues pour la détection et l’isolation de défauts dans ce travail. Nous avons également cherché à améliorer l’estimation de paramètres en développant des critères pour la planification d’expériences, consistant ici en l’optimisation des conditions initiales, entrées et/ou période d’échantillonnage. Ces critères ont été appliqués à deux cas d’étude (pharmacocinétique et aéronautique) avec de très bons résultats. Par ailleurs, nous nous sommes intéressés à la modélisation des systèmes à incertitudes « mixtes », combinant des incertitudes bornées et stochastiques, en proposant notamment une amélioration du filtre de Kalman par intervalles. Le travail réalisé formalise des problèmes non abordés auparavant et pose des jalons pour les recherches futures

Estimation des systèmes dynamiques incertains et des propriétés connexes. Application à la surveillance de la santé

In this work, we are mainly interested in the preventive surveillance of nonlinear systems with bounded uncertainties, that is to say systems for which uncertainties are defined only by their belonging to intervals. For this purpose, we have placed ourselves in a so-called "ensembliste" context in which we have been able to extend two important properties studied in a stochastic context that are identifiability and diagnosticability. Beyond the conceptual definitions required by this work, we proposed tools related to the differential algebra to verify these two properties. The impact of the set identifiability on the results of a parameter estimation was also analyzed, the estimation of parameters being one of the approaches chosen for the detection and the isolation of defects in this work. We also sought to improve the estimation of parameters by developing criteria for the planning of experiments, consisting in optimizing the initial conditions, inputs and / or sampling period. These criteria were applied to two case studies (pharmacokinetics and aeronautics) with very good results. In addition, we are interested in the modeling of systems with "mixed" uncertainties, combining bounded and stochastic uncertainties, notably by proposing an improvement of the Kalman filter at intervals. The work achieved formalizes problems not previously addressed and lays the groundwork for future research.Dans ce travail, nous nous intéressons principalement à la surveillance préventive des systèmes non linéaires à incertitudes bornées, c’est-à-dire des systèmes pour lesquels les incertitudes ne sont définies que par leur appartenance à des intervalles. Pour cela, nous nous sommes placés dans un contexte dit « ensembliste » dans lequel nous avons pu étendre deux propriétés importantes largement étudiées en contexte stochastique qui sont l’identifiabilité et la diagnosticabilité. Au delà des définitions conceptuelles requises par ce travail, nous avons proposé des outils liés à l’algèbre différentielle permettant de vérifier ces deux propriétés. L’impact de l’identifiabilité ensembliste sur les résultats d’une estimation de paramètres a également été analysé, l’estimation de paramètres étant l’une des approches retenues pour la détection et l’isolation de défauts dans ce travail. Nous avons également cherché à améliorer l’estimation de paramètres en développant des critères pour la planification d’expériences, consistant ici en l’optimisation des conditions initiales, entrées et/ou période d’échantillonnage. Ces critères ont été appliqués à deux cas d’étude (pharmacocinétique et aéronautique) avec de très bons résultats. Par ailleurs, nous nous sommes intéressés à la modélisation des systèmes à incertitudes « mixtes », combinant des incertitudes bornées et stochastiques, en proposant notamment une amélioration du filtre de Kalman par intervalles. Le travail réalisé formalise des problèmes non abordés auparavant et pose des jalons pour les recherches futures

Indicateurs d'endommagement et durée de vie des réseaux d’assainissement, modélisation

International audienceBuilding models of physical systems turns out to be an awkward task by the presence of noise and disturbance. It is not always possible to get information about disturbances and noises acting on the system. This may turn the usual stochastic framework inappropriate. In such cases, assuming bounded uncertainties may be a solution. An interesting way to go is then to use guaranteed estimation methods, which learn the state and/or parameters of the models from data. The set-membership models are well suited for fault detection and for making relevant diagnoses to improve the process evolution. A damage analysis of sewerage structures in Rennes (France) is undertaken. These systems (unit network rainwater and wastewater or separate system) consist of cement-based materials. The damage is quantified through various indicators of material properties (material extracted in the city). One hundred samples characterized by different streets and laying dates between 1892 and our days are analyzed. Thus, in this work, starting from a set-membership model of the behaviour of different sections of pipe wastewater, we use recent techniques of set-membership estimation to model and diagnose the system. A particular attention is paid to estimating the corrosion level for considering the full or partial rehabilitation

Parameter estimation procedure based on input-output integro-differential polynomials. Application to the Hindmarsh-Rose model

International audienceThis paper deals with a parameter estimation method based on input-output integro-differential polynomials. From the Rosenfeld-Groebner algorithm, some differential relations depending only on the inputs, the outputs and the parameters of the model are obtained. A pretreatment consisting in some integrations of these relations permits to obtain new ones. The latter contain essentially integrals depending only on the model inputs/outputs and their higher order derivatives are lower than the initial relations. Therefore, they are less sensitive to the noise on the measurements compared to the initial ones. Integrating permits also to attenuate the effect of the noise improving by the same the parameter estimates. However, even if the numerical estimation algorithm provides a very good value of the parameters, the latter can lead to an incorrect behavior of the model output. Indeed, in biological or physical applications, a little change of some parameter values can lead to a radical change of the model behavior as for the Hopf bifurcation. A Hopf bifurcation refers to a radical change of the model output dynamic due to a parameter crossing a reference value. Therefore, an algorithm is proposed in this paper to test the reliability of any parameter estimation procedure with respect to the dynamic of the system. More precisely, from a given noise on the output(s), it consists in calculating the probability that the result of a parameter estimation algorithm will permit to reproduce the correct behavior of the model output. Finally, this algorithm is applied on the estimation procedure based on the input-output integro-differential polynomials and on the Hindmarsh-Rose model, a slow-fast model able to reproduce the main behaviors of a neuron and presenting a Hopf bifurcation

Optimal input design for a nonlinear dynamical uncertain aerospace system

International audienceAn 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 and identification via bounded-error parameter estimation using distribution theory

International audienceIn this paper, an improvement of the bounded-error fault detection and identification method based on input-output polynomials of ([2]) is proposed. It is based on integro-differential polynomials used to estimate the fault values. The standard input-output polynomials are obtained from differential algebra elimination theory and can be used both for diagnosability analysis and fault estimation. Unfortunately, they may involve derivatives of high order whose estimation is a hard problem when system outputs are uncertain. Distribution theory allows us to transform them into integro-differential polynomials that involve lower order derivatives of the model outputs. In this paper, this method, extended to the set-membership (SM) framework, is used with the focus of achieving fault detection and identification. The original method and the new method are applied to a coupled water-tanks model and compared. It is shown that the new method significantly improves the fault detection and identification results

Bounded-Error Parameter Estimation Using Integro-Differential Equations for Hindmarsh–Rose Model

International audienceA numerical parameter estimation method, based on input-output integro-differential polynomials in a bounded-error framework is investigated in this paper. More precisely, the measurement noise and parameters belong to connected sets (in the proposed work, intervals). First, this method, based on the Rosenfeld–Groebner elimination algorithm, is presented. The latter provides differential equations containing derivatives, sometimes of high order. In order to improve the numerical results, a pretreatment of the differential relations is done and consists in integration. The new relations contain, essentially, integrals depending only on the outputs. In comparison with the initial relations, they are less sensitive to measurement noise. Finally, the impact of the size of the measurement noise domain on the estimated intervals is studied

A sufficient condition to test identifiability of a nonlinear delayed-differential model with constant delays and multi-inputs

International audienceIn this paper, an original result in terms of a sufficient condition to test identifiability of nonlinear delayed-differential models with constant delays and multi-inputs is given. The identifiability is studied for the linearized system and a criterion for linear systems with constant delays is provided, from which the identifiability of the original nonlinear system can be proved. This result is obtained by combining a classical identifiability result for nonlinear ordinary differential systems due to M.S. Grewal and K. Glover (1976) with the identifiability of linear delayed-differential models developed by Y. Orlov et al. (2002). This paper is a generalization of the results provided by L. Denis-Vidal, C. Jauberthie, G. Joly-Blanchard (2006), which deal with the specific case of nonlinear delayed-differential models with two delays and a single input

Fault detection using Upper Bound Interval Kalman Filter for unmaned aerial vehicle

International audienceThis paper deals with dynamics estimation and fault detection of unmaned aerial vehicles (uav) using the Upper Bound Interval Kalman Filter (UBIKF). An upper bound for all positive semi-definite matrices included in an interval matrix is calculated. This upper bound is used to generate envelopes for the variables to be estimated which are the dynamics of the UAV. It allows to provide a guaranteed estimation envelope for the considered dynamics. Then, the fault detection scheme is used based on a χ2 test. The faults concern sensors and actuators. Simulations on a discrete uncertain UAV model highlight the efficiency of the proposed filter for both UAV dynamics estimation and fault detection