Grey box models

Hence, there can be many states of grey box models. We propose the grey box modeling approach shown in Figure 4 to handle the problem of hfe extension for the subsea situation. 

The grey box model approach, i.e., a combination of the black box and white box model is proposed for remaining useful hfe prediction. Use of expert judgement is suggested to compensate for lack of operating ejqrerience and operational data. 

Xiong, Q. and Jutan, A., 2002, Grey-box modelling and control of chemical processes. Chemical Engineering Science, Volume 57, 6,1027-1039. 

In the present paper a more systematic procedure is proposed. The key idea of this procedure is to exploit the close connection between ODE models and SDE models to develop a methodology for determining the proper structure of the functional relations directly from the experimental data. The new procedure more specifically allows important trends and dependencies to be visually determined without making any prior assumptions and in turn allows appropriate parametric expressions to be inferred. The proposed procedure is a tailored application of the grey-box modelling approach to process model development proposed by Kristensen et al. (2002b), within which specific model deficiencies can be pinpointed and their structural origin uncovered to improve the model. 

Hastie, U. Tibshirani, RJ. and Friedman, J. (2001). The Elements of Statistical Learning - Data Mining, Inference and Prediction. Springer-Verlag, New York, USA. Kristensen, NR. Madsen, H. and J0rgensen, SB. (2002a). Parameter Estimation in Stochastic Grey-Box Models. Submitted for publication. 

Grey-box models, which combine the first-principle models with experimental data. In these types of models, the general form of the model is obtained using a first-principle approach, and then experimental data are used to obtain the values of the different constants. The advantage of this approach is that it combines the advantages of the other approaches. It is a very common approach in chemical engineering. 

Given the potential problems associated with both approaches, a third, middle way, has also been considered. This approach is called grey-box modelling, where the initial form of the equation determined based on the first-principle model is used for data-driven modelling. This approach has the advantage that the form of the equation has some physical meaning and could provide a reasonable description of the process. 

Grey-box modelling combines the advantages of first-principle and data-driven models. 

A reasonable trade-off between the theoretical and the empirical approach is represented by hybrid modeling, leading to a so-called grey-box model... 

The last group of methods for the exploitation of spectra of real samples involves not only factor analysis procedures and related methods but also the semi-deterministic approach. Both types of methods can be considered as statistical ones, but with a slight difference. Factors analysis and related methods are also described as black boxes, because they require no information. They function in a manner that is quite opposite to that of MLR methods, for which the response of all compounds of the sample must be known (which is obviously impossible for a real sample). Also, the semi-deterministic approach is based on a grey box type of model, where only a part of the information is needed, the other part remaining stochastic. 

Figure 2.2. Model pore size distributions that represent bimodal porous structures in CCLs, calculated with Eq. (2.2) [25]. The three distributions posses the same total porosity but varying volume portions of primary and secondary pores. The equilibrium capillary radius, is shown for the specified operating conditions. Values of the corresponding liquid water saturations (areas under the distribution functions within the grey box) are given.

Protein sequence based on intron/exon modeling performed by Todd Richmond (hHp //cellwall. stanford.edu/php/structure.php). Black boxes = putative transmembrane domains White boxes = conserved U domains Grey boxes = hydrophobic regions manually. 

This paper investigate the feasibility using of grey-box neural type models (GNM) for design and operation of model based Real Time Optimization (RTO) systems operating in a dynamical fashion. The GNM is based on fundamental conservation laws associated with neural networks (NN) used to model uncertain parameters. The proposed approach is applied to the simulated Williams-Otto reactor, considering three GNM process approximations. Obtained results demonstrate the feasibility of the use of the GNM models in the RTO technology in a dynamic fashion. 

An alternative way of solving this difficulty is the use of dynamical models, based on combinations of first principles and neural networks (NN), called grey-box neural models (GNM). A GNM normally consists of a phenomenological part (heat and/or mass balances differential equations) and an empirical part (a neural network in this work). Due to the inherent flexibility of NN, models based on this structure are well suited to represent complex functions such as those encountered in chemical reaction processes. This work proposes incorporate in the RTO system a dynamical GNM of the... 

A modular neural network model is developed in order to give on line estimations of bed water content and temperature in a SSC bioreactor. This grey box, predictive model gives accurate results for previous experiments carried out in the case of Gibberella fujikuroi SSC, and could be used for bioreactors on line monitoring. 

Also, we can use Hammerstein-Wiener model as a grey box structure to take physical knowledge about process characteristics. For instance, the input nonlinearity might represent typical physical transformations in actuators and the output nonlinearity might describe common sensor characteristics [13]. 

The grey boxes in Figure 1 show the changes that had to be made to the design model for the software to be able to deal with these responsibilities. They include three new classes, five new methods, and four new associations. Their description follows ... 

Figure 22 Schematic representation of proposed models for the fibril formation in the cases of pH 3.3 and 7.5. (A) hCT monomers in solution (B) a homogeneous association to form the a-helical bundle (micelle) (C) a homogeneous nucleation process to form the P-sheet and heterogeneous association process (D) a heterogeneous fibrillation process to grow a large fibril, a-helix, antiparallel p-sheet, and parallel p-sheet forms are shown by a box, drawn by dark grey and grey, respectively. From Ref. 163 with permission.

Figure 1. The proposed procedure for developing phenomena models. The boxes in grey illustrate tasks and the boxes in white illustrate inputs and outputs.

Fig. 1. Scheme of the procedure used to generate atomistic silica pore models used in this study. Front views and/or cross sections of the three pore models are also shown. Oxygen, silicon and hydrogen atoms are depicted in white, grey and black, respectively. For model C, we have represented two simulation boxes aligned in the axial direction z. 

At the extremes there are two types of models models that only contain physical-chemical relationships (the so-called white box or mechanistic models) and models that are entirely based on experiments (the so-called black box or experimental models). In the first category, only the system parameters are measured or known from the literatnre. In that case it is assumed that the stmcture of the model representation is entirely correct. In the second case, also the relationship(s) are experimentally determined. Between the two extremes there is a grey area. In many cases, some parts of the model, especially the balances, are based on physical relationships, whereas other parts are determined experimentally. This specifically holds for parameters in relationships, which often have a limited range of validity. However, the experimental parts can also refer to entire relationships, such as equations for the rate of reactions in biochemical processes. 

---