Response Models

When operating conditions were changed, transient phenomena were sometimes observed that first move in one direction and in the reverse direction on going to the final steady state. To study these transients and to design an improved control strategy for the unit, a dynamic response model was needed. With the inclusion of the fast coke in the model, it became possible to extend the steady-state model to obtain useful dynamic response results by the addition of time-dependent accumulation terms (Weekman et al., 1967). 

The flow dififerential for the element dx of the kiln is given by FcMc dQ, FcA/cC dy 

To study the response of the kiln to transient conditions and to different control schemes, the set of partial differential equations (39), (40), (44), and (48) were solved using a hybrid analog-digital computer, the EAI Hydac 2000. A description of the computer and of the methods used in the solution are given in the paper by Weekman et al. (1967). The kiln conditions used in the simulation to be discussed are given in Table V. The dynamic model was first used to study the effect of fast coke on kiln stability. 

The amount of fast coke was determined from a correlation based on observation of operating kilns. As the reactor outlet temperature changes, the amount of oil on the catalyst changes. For the usual purge conditions existing in the seal leg connecting the reactor to the kiln, 

As the reactor outlet temperature rises, the amount of fast coke falls. Thus, as the kiln receives more heat from the catalyst, it receives less fast coke. This leads to a self-regulatory effect that played an important role in kiln stability. This effect is seen by a comparison of Figs. 21 and 22. 

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A key featui-e of MPC is that a dynamic model of the pi ocess is used to pi-edict futui e values of the contmlled outputs. Thei-e is considei--able flexibihty concei-ning the choice of the dynamic model. Fof example, a physical model based on fifst principles (e.g., mass and energy balances) or an empirical model coiild be selected. Also, the empirical model could be a linear model (e.g., transfer function, step response model, or state space model) or a nonhnear model (e.g., neural net model). However, most industrial applications of MPC have relied on linear empirical models, which may include simple nonlinear transformations of process variables. 

The original formulations of MPC (i.e., DMC and IDCOM) were based on empirical hnear models expressed in either step-response or impulse-response form. For simphcity, we will consider only a singleinput, single-output (SISO) model. However, the SISO model can be easily generalized to the MIMO models that are used in industrial applications. The step response model relating a single controlled variable y and a single manipiilated variable u can be expressed as... 

The step-response model is also referred to as a finite impulse response (FIR) model or a discrete convolution model. 

The step-response model in Eq. (8-63) is equivalent to the following impulse response model ... 

Memory requirements for one-dimensional eontinuum dynamies ealeulations are minimal by the standards of eurrent hardware. Thus, sufTieiently fine zoning ean be used in sueh ealeulations to eapture details of material response and provide a rigorous test of fidelity for the numerieal models employed. The ability to use fine zoning also ensures that any diserepaneies between ealeulation and experiment ean be attributed, with eonsiderable eonfidenee, to Inadequaeies in the material response model. In faet, most desktop workstations have suffieient eomputing horsepower and memory to meet the eom-putating needs in one-dimensional material response studies. 

T Ichiye. Solvent free energy curves for electron transfer A non-lmear solvent response model. J Chem Phys 104 7561-7571, 1996. 

D. J. McNaughton, G. G. Worley, and P. M. Bodner, "Evaluating Emergency Response Models for the Chemical Industry," Chent. Eng. Prog., pp. 46-51, January 1987. 

Uncertainty on tlie other hand, represents lack of knowledge about factors such as adverse effects or contaminant levels which may be reduced with additional study. Generally, risk assessments carry several categories of uncertainly, and each merits consideration. Measurement micertainty refers to tlie usual eiTor tliat accompanies scientific measurements—standard statistical teclmiques can often be used to express measurement micertainty. A substantial aniomit of uncertainty is often inlierent in enviromiiental sampling, and assessments should address tliese micertainties. There are likewise uncertainties associated with tlie use of scientific models, e.g., dose-response models, and models of environmental fate and transport. Evaluation of model uncertainty would consider tlie scientific basis for the model and available empirical validation. 

FIGURE 7.14 Effect of an allosteric modulator that increases the efficacy of the agonist but has no effect on affinity in two different systems, (a) For full agonists, increases in efficacy produce parallel shifts to the left of the concentration-response curves. Responses modeled with Equation 7.3 with a= 1, , = 5, t = 20, and Ka = 3j.lM. Curves shown for [B]/Kb = 0, 0.3, 1, 3, 10, and 30. (b) In systems with lower receptor density and/or poorer receptor coupling where the agonists does not produce the full system maximal response, an allosteric modulator increases the maximal response and shifts the curves to the left. Responses modeled with Equation 7.3 for the same agonist and same allosteric modulator but in a different tissue (parameters as for A except t= 1). 

Andersen ME, Kirshnan K. 1994. Relating in vitro to in vivo exposures with physiologically based tissue dosimetry and tissue response models. In Salem H,ed. Animal test alternatives Refinement, reduction, replacement. New York, NY Marcel Dekker, Inc., 9-25. 

Benchmark Dose Model—A statistical dose-response model applied to either experimental toxicological or epidemiological data to calculate a BMD. 

Physiologically Based Pharmacodynamic (PBPD) Model—A type of physiologically-based dose-response model which quantitatively describes the relationship between target tissue dose and toxic end points. These models advance the importance of physiologically based models in that they clearly describe the biological effect (response) produced by the system following exposure to an exogenous substance. 

Analysis of most (perhaps 65%) pharmacokinetic data from clinical trials starts and stops with noncompartmental analysis (NCA). NCA usually includes calculating the area under the curve (AUC) of concentration versus time, or under the first-moment curve (AUMC, from a graph of concentration multiplied by time versus time). Calculation of AUC and AUMC facilitates simple calculations for some standard pharmacokinetic parameters and collapses measurements made at several sampling times into a single number representing exposure. The approach makes few assumptions, has few parameters, and allows fairly rigorous statistical description of exposure and how it is affected by dose. An exposure response model may be created. With respect to descriptive dimensions these dose-exposure and exposure-response models... 

Model equations can be augmented with expressions accounting for covariates such as subject age, sex, weight, disease state, therapy history, and lifestyle (smoker or nonsmoker, IV drug user or not, therapy compliance, and others). If sufficient data exist, the parameters of these augmented models (or a distribution of the parameters consistent with the data) may be determined. Multiple simulations for prospective experiments or trials, with different parameter values generated from the distributions, can then be used to predict a range of outcomes and the related likelihood of each outcome. Such dose-exposure, exposure-response, or dose-response models can be classified as steady state, stochastic, of low to moderate complexity, predictive, and quantitative. A case study is described in Section 22.6. 

Figure 22.3 The drug dose-response model was augmented by nsing data for the comparator drug. Because the mechanism of the drugs was the same, this comprised additional data for the model. This enhanced the predictive power of the model, in a better estimate for central tendency (solid line compared with dotted line) bnt also in smaller confidence intervals. This is especially prononnced at the higher doses— precisely where data on the drug were sparse. See color plate.

Geyer, MA, Swerdlow, NR, Mansbach, RS and Braff, DL (1990) Startle response models of sensorimotor gating and habituation deficits in schizophrenia. Brain Res. Bull. 25 485-498. 

C Low dose effects usually not measurable directly In human or animal observations Need to extrapolate observed high dose effects to low or zero dose range by theoretical dose-response models ... 

The above procedure will be followed in the following three examples with two, three and four parameter linear single response models... 

Obviously, it is very important that the next experiment has maximum discriminating power. Let us illustrate this point with a very simple example where simple common sense arguments can lead us to a satisfactory design. Assume that we have the following two rival single-response models, each with two parameters and one independent variable ... 

Box and Hill (1967) proposed a criterion that incorporates the uncertainties associated with model predictions. For two rival single-response models the proposed divergence expression takes the form,... 

The fitting of electronegativities and hardnesses is done independently of each other with the help of a reformulation of the fluctuating charge model in terms of a linear response model [117, 120, 210], In the presence of an external potential the electrostatic energy defined in Eq. (9-35) is ... 

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