Model reduction examples

It is important to understand and be able to identify dominant poles if they exist. This is a skill that is used later in what we call model reduction. This is a point that we first observed in Example 2.6. Consider the two terms such that 0 < a < a2 (Fig. 2.4),... 

The KOCAT Society (no patent reference known by authors), in South Korea, solved the DeNO problem of big burners, by directly injecting oxygenates on the catalyst at the outlet of the burner. This process involves the third function of our model. This example shows that only one model (the present one) for DeNO reaction can be used for either mobile or stationary sources. Pathways are the same what is changing is the nature of the reductant, which has to be activated, through its partial oxidation, at the temperature when N—O bonds (dinitrosyl species) are broken. 

Practical identifiability is not the only problem that can be adressed by principal component analysis of the sensitivity matrix. In (refs. 29-30) several examples of model reduction based on this technique are discussed. 

Example 3. In Example 1, the core model was obtained using model reduction, and the mapping is thus already given... 

Systems approach borrowed from the optimization and control communities can be used to achieve various other tasks of interest in multiscale simulation. For example, Hurst and Wen (2005) have recently considered shear viscosity as a scalar input/output map from shear stress to shear strain rate, and estimated the viscosity from the frequency response of the system by performing short, non-equilibrium MD. Multiscale model reduction, along with optimal control and design strategies, offers substantial promise for engineering systems. Intensive work on this topic is therefore expected in the near future. 

Numerous examples of radical formation during model reductions with 1,4-dihydropyridines are known The overall reaction would then be (e , H ). The reverse reaction, between an alcohol and NAD, would produce the equivalent of an alkoxide, as is also required in the hydride-transfer mechanism (Eq. 35). 

NO reacts with both ferric and ferrous centers in hemoproteins to form the respective iron(II) and iron(III) nitrosyl adducts, whose structural features are similar to those observed for iron (II) and iron(III) porphyrin nitrosyls. These analogies are also reflected in similar chemical reactivity observed for nitrosylated ferri- and ferroproteins and their respective porphyrin models. For example, NO-adducts of Fe(III) undergo reductive nitrosylation in the presence of an excess of NO, and a similar process is commonly observed for synthetic Fe(III) porphyrins. The first step of this reaction involves nucleophilic attack of OH on the nitrosyl ligand coordinated to the iron center, as presented in reaction (13) (33,60) ... 

We present a novel approach to Model Predictive Control problems, which combines a model reduction scheme coupled with parametric programming. Balanced Truncation is used to first reduce the size of the original Model Predictive Control formulation, while multi-parametric programming is employed to derive the parametric control laws offline. The theoretical developments are presented with an example problem. 

A novel gradient-based optimisation framework for large-scale steady-state input/output simulators is presented. The method uses only low-dimensional Jacobian and reduced Hessian matrices calculated through on-line model-reduction techniques. The typically low-dimensional dominant system subspaces are adaptively computed using efficient subspace iterations. The corresponding low-dimensional Jacobians are constructed through a few numerical perturbations. Reduced Hessian matrices are computed numerically from a 2-step projection, firstly onto the dominant system subspace and secondly onto the subspace of the (few) degrees of freedom. The tubular reactor which is known to exhibit a rich parametric behaviour is used as an illustrative example. 

The foregoing equations have resulted from the stated assumptions of the models employed. These models are examples of many models that have been proposed for electrochemical reactions. The equations are accepted here because of their simplicity of form and the fact that they do predict relationships between exchange current density, equilibrium half-cell potential, and concentration, which are frequently observed experimentally. The theories and resulting equations are obviously more complicated for surface reactions, such as the reduction of dissolved oxygen, 02 + 4H+ + 4e — 2H20. Theories for this reaction have proposed as many as eight individual steps. 

By applying the dimension reduction procedure indicated above we are able to limit the number of states to a manageable size. At the system level we then obtain the steady state probabilities and hence system regularity by standard Markov methods. The model has been applied in relation to oil production where reservoir characteristics have to be included in the model. For example for a gas injection system a degraded performance (e.g., 66.67% injection capacity) may have no immediate effect on oil production. This is due to the fact that produced gas is sent to flare for a shorter period of time without affecting the reservoir performance. Due to environmental restriction imlimited flaring is not acceptable, hence after a period of some hours production has to be reduced... 

In the mathematical sense, simplification can be defined as model reduction, that is, the rigorous or approximate representation of complex models by simpler ones. For example, in a certain domain of parameters or times, a model of partial differential equations ( diffusion-reaction model) is approximated by a model of differential equations, or a model of differential equations is approximated by a model of algebraic equations, and so on. 

This illustrative example is divided into three studies. The first study explores the proposed model reduction procedure by showing the outputs of many of the intermediate steps leading to the proper model for a single high forward speed input. Because the proper model is scenario dependent, a second study explores the effect of changing the input to a smoother input (lower vehicle forward speed than that used in the first study) on the proper model produced. Finally, a third study explores the effect of the user-defined model reduction threshold, (/ ), on the reduced model accuracy, by generating a series of reduced models under the same input conditions. 

This section provides an example to illustrate the mechanics of the ECI-based model reduction algorithm and emphasize its advantages, namely its applicability to nonlinear systems, ability to achieve graph-level reduction, and ability to reduce the order and structure of the model, while taking into account the scenario of interest and preserving the realization of the model. 

Table 8.1 An arson risk assessment model (an example - government crime reduction toolkits)...

In this chapter several model reduction techniques will be discussed. The first method is based on firequency response matching, other methods make use oficonversion ofi the model structure to a state space model and subsequently truncating the states that have a minimum impact on the input-output relationship. The main indicator used fior this purpose is the so-called Hankel singular value. In addition, the model structure is converted to a balanced realization, afiter which the reduction techniques can be applied. Several examples are given on how to apply the dififierent methods. 

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