Linear reduced model

For the single response linear regression model (w=l), Equations (3.17a) and (3.17b) reduce to... 

If linear (dose) models without thresholds are to be used for carcinogen (or other) risk assessment, estimation of exposure at specified levels becomes irrelevant to risk assessment or, at least, its use is nonintuitive. For example, a carcinogen risk analysis may be based on a linear, nonthreshold health effects model. The total health risk would thus be proportional to the long-term exposure summed for all affected people for the identified period, and exposure of many people at low concentrations would be equivalent to exposure of a few to high concentrations. The atmospheric dispersion that reduces concentrations would also lead to exposure of more people therefore, increments... 

This paper extends previous studies on the control of a polystyrene reactor by including (1) a dynamic lag on the manipulated flow rate to improve dynamic decoupling, and (2) pole placement via state variable feedback to improve overall response time. Included from the previous work are optimal allocation of resources and steady state decoupling. Simulations on the non-linear reactor model show that response times can be reduced by a factor of 6 and that for step changes in desired values the dynamic decoupling is very satisfactory. 

For linear plant models Crowe et al. (1983) used a projection matrix to obtain a reduced system of equations that allows the classification of measured variables. They identified the unmeasured variables by column reduction of the submatrix corresponding to these variables. 

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. 

Once the kinetic parameters have been estimated, (3.60) becomes linear in the unknown parameters A Hr . Therefore, the errors between the total heat of reaction, computed via the detailed model, and the total heat, computed via each reduced model, can be minimized by resorting to the least squares solution of a linear regression problem, discussed in Sect. 3.4. The molar heats of reaction, included in the vector of parameters... 

To compare the original system with the reduced system, the pyruvate flow in the original, the linearized, and the reduced model was simulated. Starting at steady state, with the constant input u(t) = 50 mM, 10% and 50% step increases in input glucose concentration were applied, resulting in increases in the flow of pyruvate by only 0.3% and 1.5% respectively (Fig. 15.2). Notwithstanding the small increase in flux, the balanced truncation introduced a 10% overestimation of the increase in pyruvate flux. 

Principal component analysis (PCA) and multivariate curve resolution-alternating least squares (MCR-ALS) were applied to the augmented columnwise data matrix D1"1", as shown in Figure 11.16. In both cases, a linear mixture model was assumed to explain the observed data variance using a reduced number of contamination sources. The bilinear data matrix decomposition used in both cases can be written by Equation 11.19 ... 

For obvious reasons, we shall refer to Eq. (43) as the global equation while Eq. (44) that couples the local and global scales as the local equation. Although we can leave the reduced model in this two-mode form, in this special case, we can combine the two equations to obtain a single equation either for Cm or (C). In this specific example, the model is linear in the concentration and hence both Cm and (C) satisfy the same equation (as they are linearly related). Since the cupmixing concentration (which is often measured in experiments) is more relevant in applications, we write the reduced model in terms of Cm ... 

As our first example, we consider the case of a first-order homogeneous reaction A -> B in a laminar flow tubular reactor for which the global equation is linear in c (i.e. r( c)) — (c)) and is therefore completely closed. To obtain the range of convergence of the two-mode model, we need to consider only the local equation. In this specific case, the reduced model equations to all orders of p are then given by... 

The last comer of our ideal triangle symbolizes the reduced model theory (RMl). The last few years have heard some debates on how to build up a modeling approach to molecular dynamics in the liquid state. These earlier attempts are based mainly on the generalized Langevin equation of Mori, which is linear in nature. In this book we shall illustrate a nonlinear version of this approach, the application of which implies wide use of both the AEP and CFP. 

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