Selection and identification of physical parameters from passive observation. Application to a winding process

Abstract

This paper deals with parameter selection and estimation of large and complex simulation models. This estimation problem is addressed in the case of passive observation, i.e. when no controlled experiment is possible. Given the lack of information in the data, an appropriate methodology is proposed to select and estimate some physical parameters of the model. Its implementation is based on a new software: Diffedge${}^{\copyright }$ which makes it possible to symbolically determine model output sensitivity functions of block diagrams. An application to a winding prototype is developed to illustrate the effectiveness of such an approach in practice.

Introduction

There are three kinds of mathematical models of dynamic processes: (i) white-box models based on first principles of physics, and sufficiently detailed to contain the representations of individual components (Maciejowski, 1997), (ii) black-box models based on generic model structures, e.g. linear, for the design of feedback controllers (Ljung, 1987) and (iii) grey-box models, a sort of compromise between the two boxes previously mentioned, i.e. a black-box model in which a part of the underlying physics is available and taken into account in the identification procedure (Bohlin, 1991). This article deals with the parameter estimation of white-box models. The term ‘calibration’ is also widely used to express the determination of a parameter set, usually from observed data, and thus provide the ‘best’ representation of the system being modeled. It is generally a misused term. Indeed, calibration means the adjustment in comparison to a standard, i.e. a noise free reference. In practice, data are noisy and the calibration process corresponds in fact to parameter estimation. Herein, the estimation problem is addressed in the case of passive observation, i.e. when no input design can be applied to the process because of economic or safety reasons (Thomassin, Bastogne, Richard, & Libaux, 2003). For an engineer with extensive experience with a specific model, manual calibration of a white-box model could probably be sufficient to some applications. However, manual adjustment of such complex models is usually time-consuming, and its results are not often reproducible. For these reasons a great effort has been devoted to the development of automatic methods in parameter estimation. Approaches like those proposed by Isaksson, Lindkvist, Zhang, Nordin, and Tallfors (2003) for the estimation of physical parameters are not appropriate to passive data since they require to carry out three dedicated experiments. Calibration tools of grey-box models like MoCaVa (Bohlin, 2003) cannot be applied either. Its principle consists in recursively fitting and testing a series of model structures. Submodels which do not contribute to significantly reduce the overall loss are eliminated from consideration, while those which do contribute are candidates for further refinements. In the case of a white-box model, its resolution, i.e. the number of physical parameters, is fixed. Its internal structure is imposed and no model reduction is possible. In such a case, the estimation problem is twofold:

• checking the uniqueness of the solution given the passive data, i.e. assessing the practical identifiability (Dochain & Vanrolleghem, 2001);

• given the number of identifiable parameters, selecting those which can be estimated using the passive data.

Using a general approach, Vanrolleghem et al. have proposed in Vanrolleghem, Van Daele, and Dochain (1995), Weijers and Vanrolleghem (1997), and Dochain and Vanrolleghem (2001) to both assess the practical identifiability and select the most identifiable parameters. This article presents three types of contribution:

• the relationship between the practical identifiability (Dochain et al., 1995, Dochain and Vanrolleghem, 2001) and the output distinguishability (Grewal & Glover, 1976) is firstly emphasized;

• a new selection mode of the most identifiable parameters is proposed;

• and a new implementation solution is applied to this approach, based on a new software: Diffedge${}^{\copyright }$ which makes it possible to symbolically determine model output sensitivity functions of block diagrams.

This paper is composed of three major sections. The identification problem is stated in Section 2. The identification procedure is presented in Section 3 and an application study to a winding process is developed in Section 4, in order to illustrate the effectiveness of the proposed approach.