Reducible Models

Most of the modeling procedures commonly used require that the model first be reduced to a form which is linear in the unknown parameters. This procedure represents very good tactics the technique will be exploited frequently in this review, particularly in Section V. If the scope of the models to be used or the range of experimental variables to be explored is not limited when applying this philosophy, the procedure also represents good modeling strategy. 

For these reasons, a proper balance must be achieved between the linearization tactics and the overall modeling strategy. Both linear and nonlinear methods will be illustrated in the review, along with the problems encountered when relying too heavily on either single approach. First, however, some of the more common linear procedures will be discussed. 

For a single reacting component, Eq. (1) reduces to a form that can easily be compared to rate data to determine (a) if the model is adequate and, if so, (b) the best estimates of the rate constant and the reaction order. For this case, Eq. (1) becomes 

/is a measure of reaction time. To analyze data from certain reactor types using this model, one can use one of the following methods. 

the rate equation, such as Eq. (3), is integrated to yield 

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J Skolmck, A Kolinski, AR Ortiz. Application of reduced models to protein stiaicture prediction. In J Leszczynski, ed. Computational Molecular Biology. Theor Comput Chem Ser New York Elsevier Science, 1999, pp 397-440. 

The complex contains 72 atoms with 244 valence electrons distributed in 226 valence atomic orbitals. In order to reduce the computational effort, and to assess the contribution of the ligand 7r-orbitals to the overall spectrum, we examined a "reduced" model, see Figure 2, in which the benzene rings of the ligands are replaced by -HC=CH- groups. This model compound consists of... 

Based on the shapes of the responses to step changes in controller output, and reasoning from the physical configuration of the extruder barrel, a reduced order dynamic model of the process was postulated. One can think of the Topaz program as order 80 (the number of nodes in the finite element subdivision), and the reduced model of order 4 (the number of dynamic variables). The figure below illustrates the model. 

Quite often we are face with the task of reducing the order of a transfer function without losing essential dynamic behavior of the system. Many methods have been proposed for model reduction, however quite often with unsatisfactory results. A reliable method has been suggested by Luus (1980) where the deviations between the reduced model and the original one in the Nyquist plot are minimized. 

After the parameters have been estimated, generate the Nyquist plots for the reduced models and the original one. Comment on the result at high frequencies. Is N=100 a wise choice ... 

Redo this problem. However, this time assume that the reduced model is a fourth order one. Namely, it is of the form... 

In the fully reduced model, four electrons are transferred to dioxygen through sequential one-electron oxidations of heme as s iron ion, the Cub ion, the heme a iron ion, and one of the bimetallic center s Cua ions. The sequence of electron transferal differs in the mixed valence model, and a tyrosine radical (tyr) is generated. The proposed formation of a tyrosine radical during catalytic turnover arises from the known post-translational modification in most CcO s in which a covalent bond is formed between the his240 ligand of Cub... 

Let us now consider the more general case in which the reduced model contains more parameters than the single parameter Pq, We assume that we have an expanded model... 

It is possible to selectively choose a subset of 4 of the original 8 factor combinations and use these to fit the reduced model with 100% efficiency. The resulting design is called a fractional factorial design . A full 2 factorial design has two half-replicates as shown in Figures 14.4 and 14.5, or in cube plot form as ... 

Thomas, G.L., Sessions, R.B., Parker, M.J. Density guided importance sampling application to a reduced model of protein folding. Bioinformatics 2005, 21, 2839 13. 

Fig. 27. Coomans diagram for SIMCA usual (enlarged) and reduced models. The fitting of Grignolino samples to the two models of the Barolo class is shown...

Figure 27 shows an application of a Coomans diagram to a comparison between the usual (enlarged) and reduced SIMCA models, showing that the reduced model is less easy to penetrate than the usual model. 

Fig. 30. Probability density function for SIMCA (reduced model)...

The basic strategy of modal reduction approaches is to retain only certain modes of the high-order model in the low-order model. Wilson et al. (1974) summarized these techniques and showed that many of the published modal approaches are equivalent since they produce identical reduced models. Bonvin (1980) also provides a comparison of the various modal techniques with respect to their steady-state and dynamic accuracies as well as to the dependence of the reduced models on the retained state variables. 

Of course, this technique does not actually eliminate the states gj, y1 , and y2j. Instead it retains their steady-state effects and relates their dynamic behavior to the gas and thermal well temperatures. The reduced model is then... 

Figure 28 shows comparisons of the transient gas and solid axial temperature profiles for a step-input change with the full model and the reduced models. The figure shows negligible differences between the profiles at times as short as 10 sec. Concentration results (not shown) show even smaller discrepancies between the profiles. Additional simulations are not shown since all showed minimal differences between the solutions using the different linear models. Thus for the methanation system, Marshall s model reduction provides an accurate 2Nth-order reduced state-space representation of the original 5/Vth-order linear model. 

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