Nathalie Verdière

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.

A new method for estimating derivatives based on a distribution approach

In many applications, the estimation of derivatives has to be done from noisy measured signal. In this paper, an original method based on a distribution approach is presented. Its interest is to report the derivatives on infinitely differentiable functions. Thus, the estimation of the derivatives is done only from the signal. Besides, this method gives some explicit formulae leading to fast calculus. For all these reasons, it is an efficient method in the case of noisy signals as it will be confirmed in several examples.

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.

A distribution input–output polynomial approach for estimating parameters in nonlinear models. Application to a chikungunya model

Abstract This paper presents a numerical procedure based on a distribution approach for doing parameter estimation in nonlinear dynamical models. The originality of the paper is first, to present a complete study of the errors due to the method and due to the noise on the signal then, to apply it to a recent model describing the transmission of the chikungunya virus to the human population. The advantage of this numerical procedure is not to require any knowledge about the value of the parameters or about the statistics of measurement uncertainties. Furthermore, it attenuates a part of the noise improving consequently the results of the parameter estimation. The numerical results attest the relevance of this approach.

Mathematical Modeling of Human Behaviors During Catastrophic Events: Stability and Bifurcations

In this paper, we introduce a new approach for modeling the human collective behaviors in the speci c scenario of a sudden catastrophe, this catastrophe can be natural (i.e. earthquake, tsunami) or technological (nuclear event). The novelty of our work is to propose a mathematical model taking into account di erent concurrent behaviors in such situation and to include the processes of transition from one behavior to the other during the event. Here, we focus more on the sequence of behaviors since our aim is to better apprehend and handle the collective reactions. In the literature, several models have been proposed for modeling crowd dynamics, both at microscopic (through agent-based simulation frameworks), and at macro-scopic level (via PDE equations). However, generally, the human reactions are classi ed in a single category, very often that of panic and the sequences of di fferent human behaviors are rarely considered, while in Human Sciences these concepts have already been addressed. Thus, in this multidisciplinary research included mathematicians, computer scientists and geographers, we take into account the psychological reactions of the population in situations of disasters, and we study their propagation mode. We propose a SIR-based model, where three types of collective reactions occur in catastrophe situations: re flex, panic and controlled behaviors. Moreover, we suppose that the interactions among these classes of population can be realized through imitation and emotional contagion processes. Finally, numerical simulations have been carried out to validate and compare our model with the experimental data available in the literature.

Identifiability and identification of a pollution source in a river by using a semi-discretized model

This paper is devoted to the identification of a pollution source in a river. A simple mathematical model of such a problem is given by a one-dimensional linear advection-dispersion-reaction equation with a right hand side spatially supported in a point (the source) and a time varying intensity, both unknown. There exist some identifiability results about this distributed system. But the numerical estimation of the unknown quantities require the introduction of an approximated model, whose identifiability properties are not analyzed usually. This paper has a double purpose: - to do the identifiability analysis of the differential system considered for estimating the parameters, - to propose a new numerical global search of these parameters, based on the previous analysis. Another consequence of this approach is to give the unknown pollution intensity directly as the solution of a differential equation. Lastly, the numerical algorithm is described in detail, completed with some applications.

Hybrid methods for parameter estimation comparison and numerical applications

The genetic algorithm can be used directly on a model in order to estimate its parameters. However, the time of calculus can be not reasonable and its results can be conditioned by the sensitivity of the measured signal to some parameters. In this paper, a novel procedure combining the classical input-output-parameter approach and the genetic algorithm allows to overcome this problem. The classical input-output-parameter approach requires the estimation of derivatives from a noisy signal. Several methods have already been used but in this paper a new one based on the distribution theory is proposed. It allows to obtain very satisfactory results for a first parameter estimation. Afterwards, instead of improving the results with a local algorithm like Levenberg-Marquardt which does not converge in all the cases or which converges towards a local minimum and not the global one according to the quality of the first initial value, the genetic algorithm is used in taking into account the first previous estimate. Indeed, the advantage of this algorithm is to explore a larger domain and thus, is less likely to give a local minimum. The results are compared to the ones obtained using directly the genetic algorithm on the initial system. Our theoretical result is supported by an application in the pharmacokinetic domain.

Identifiability of ordinary or delayed nonlinear models: A distribution approach

An original method, combining algebraic and distribution theory approaches, is presented in this paper for analyzing the identifiability of some ordinary or delayed nonlinear models. Then, it is shown how this analysis can be used for parameter estimation. Our purpose is supported by examples and an application in aerospace domain.

Symbolic-Numeric Methods for Nonlinear Integro-Differential Modeling

This paper presents a proof of concept for symbolic and numeric methods dedicated to the parameter estimation problem for models formulated by means of nonlinear integro-differential equations (IDE). In particular, we address: the computation of the model input-output equation and the numerical integration of IDE systems.