Direct space modeling

When one or more models are constructed, they are tested against the experimental diffraction data. Often some of these approaches are combined together but they always stem from the requirement that the generated model must make physical, chemical and crystallographic sense. Thus, their successful utilization requires a certain level of experience and knowledge of how different classes of crystals are built, e.g. what to expect in terms of coordination and bond lengths for a particular material based solely on its chemical composition. Direct space modeling approaches will be discussed, to some extent, in Chapter 6. 

Direct space modeling may be an especially powerful tool in the case of intermetallic and related structures, many of which are derived from close packing of incompressible spheres. Thus, when positions of large atoms in the unit cell are known, the smaller atoms will likely occupy voids of sufficient size. 

Raum-mass, n. measure of capacity or volume, cubic measure, -menge,/. amoimt of space, volume, -meter, m. cubic meter, -modell, n. space model, -orientierung,/. orientation in space, -quantelung, /. spatial quantization. -richtung,/. direction in space, -strahl,... 

Now, you may wonder if we can generate the state space model directly from a transfer function. The answer is, of course, yes. We can use... 

One of the drawbacks of ellipsometry is that the raw data cannot be directly converted from the reciprocal space into the direct space. Rather, in order to obtain an accurate ellipsometric thickness measurement, one needs to guess a reasonable dielectric constant profile inside the sample, calculate A and and compare them to the experimentally measured A and values (note that the dielectric profile is related to the index of refraction profile, which in turn bears information about the concentration of the present species). This procedure is repeated until satisfactory agreement between the modeled and the experimental values is found. However, this trial-and-error process is complicated by an ambiguity in determining the true dielectric constant profiles that mimic the experimentally measured values. In what follows we will analyze the data qualitatively and point out trends that can be observed from the experimental measurements. We will demonstrate that this... 

The formulation described above provides a useful framework for treating feedback control of combustion instability. However, direct application of the model to practical problems must be exercised with caution due to uncertainties associated with system parameters such as and Eni in Eq. (22.12), and time delays and spatial distribution parameters bk in Eq. (22.13). The intrinsic complexities in combustor flows prohibit precise estimates of those parameters without considerable errors, except for some simple well-defined configurations. Furthermore, the model may not accommodate all the essential processes involved because of the physical assumptions and mathematical approximations employed. These model and parameter uncertainties must be carefully treated in the development of a robust controller. To this end, the system dynamics equations, Eqs. (22.12)-(22.14), are extended to include uncertainties, and can be represented with the following state-space model ... 

For effective control of crystallizers, multivariable controllers are required. In order to design such controllers, a model in state space representation is required. Therefore the population balance has to be transformed into a set of ordinary differential equations. Two transformation methods were reported in the literature. However, the first method is limited to MSNPR crystallizers with simple size dependent growth rate kinetics whereas the other method results in very high orders of the state space model which causes problems in the control system design. Therefore system identification, which can also be applied directly on experimental data without the intermediate step of calculating the kinetic parameters, is proposed. 

In applying the resulting state space model for control system design, the order of the state space model is important. This order is directly affected by the number of ordinary differential equations (moment equations) required to describe the population balance. From the structure of the moment equations, it follows that the dynamics of m.(t) is described by the moment equations for m (t) to m. t). Because the concentration balance contains c(t)=l-k m Vt), at I east the first four moments equations are required to close off the overall model. The final number of equations is determined by the moment m (t) in the equation for the nucleation rate (usually m (t)) and the highest moment to be controlled. 

Several identification methods result in a state space model, eithejp by direct identification in the state space structure or by identjLfication in a structure that can be transformed into a state space model. In system identification, discrete-time models are used. The discrete-time state-space model is given by... 

The method of lines and system identification are not restricted in their applicability. System identification is preferred because the order of the resulting state space model is significantly lower. Another advantage of system Identification is that it can directly be applied on experimental data without complicated analysis to determine the kinetic parameters. Furthermore, no model assumptions are required with respect to the form of the kinetic expressions, attrition, agglomeration, the occurence of growth rate dispersion, etc. 

A better alternative is to use the difference structure factor AF in the summations. The electrostatic properties of the procrystal are rapidly convergent and can therefore be easily evaluated in direct space. Stewart (1991) describes a series of model calculations on the diatomic molecules N2, CO, and SiO, placed in cubic crystal lattices and assigned realistic mean-square amplitudes of vibration. He reports that for an error tolerance level of 1%, (sin 0/2)max = 1-1.1 A-1 is adequate for the deformation electrostatic potential, 1.5 A-1 for the electric field, and 2.0 A 1 for the deformation density and the deformation electric field gradient (which both have Fourier coefficients proportional to H°). 

The crystal potential for L-alanine calculated with Eq. (8.34) is shown in Fig. 8.1(a). The term Ospherical.atom(r) can be evaluated in direct space by the methods described in the following section. The term 0(0) for the independent-atom model [not exactly equal to the true 0(0)] was evaluated by a summation of the IAM potential over the unit cell. 

Macroscopic experiments, both reciprocal-space techniques and direct-space techniques, conclude that the (110) surfaces of Pt, Ir and Au reconstruct to a (1 x 2) structure.30 Several atomic models have been proposed,31 of which the simple missing row model agrees better with various experimental data. In this model every other [110] atomic row is missing from the reconstructed (1 X 1) surface. To transform a (1 x 1)... 

Scheme II. Direct space representation of the STM interface. The junction is composed of two adjacent metallic blocks. Each block is divided into characteristic subparts that are stacked along the Z direction Sample bulk, Sample surface, Tip apex and Tip bulk. For clarity, ethylene molecules are schematically drawn at the sample surface on an Ag-oxide overlayer. White, black, light and dark grey circles depict C, H, Ag and O atoms respectively. The tip is composed of a W 111 surface upon which a cluster of W is adsorbed to model the apex.

Physical state space models are more attractive for use with the LQP (especially when state variables are directly measurable), while multivariable black box models are probably better treated by frequency response methods (22) or minimum variance control (discussed later in this section). 

The success of MPC is based on a number of factors. First, the technique requires neither state space models (and Riccati equations) nor transfer matrix models (and spectral factorization techniques) but utilizes the step or impulse response as a simple and intuitive process description. This nonpara-metric process description allows time delays and complex dynamics to be represented with equal ease. No advanced knowledge of modeling and identification techniques is necessary. Instead of the observer or state estimator of classic optimal control theory, a model of the process is employed directly in the algorithm to predict the future process outputs. 

A GA method has been developed [92, 93] for ab initio phasing of low-resolution X-ray diffraction data from highly symmetric structures. The direct-space parameterization used incorporates information on structural symmetry, and has been applied to study the structures of viruses, with resolution as high as 3 A [93]. A GA has also been introduced [94] to speed up molecular replacement searches by allowing simultaneous searching of the rotational and translational parameters of a test model, while maximizing the correlation coefficient between the observed and calculated diffraction data. An alternative GA for sixdimensional molecular replacement searches has been described [95,96] and GA methods have also been used [97] to search for heavy atom sites in difference Patterson functions. 

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