Process uncertainty

Model-free adaptive (MFA) control does not require process models. It is most widely used on nonlinear applications because they are difficult to control, as there could be many variations in the nonlinear behavior of the process. Therefore, it is difficult to develop a single controller to deal with the various nonlinear processes. Traditionally, a nonlinear process has to be linearized first before an automatic controller can be effectively applied. This is typically achieved by adding a reverse nonlinear function to compensate for the nonlinear behavior so that the overall process input-output relationship becomes somewhat linear. It is usually a tedious job to match the nonlinear curve, and process uncertainties can easily ruin the effort. 

Fig. 5.7. Stability analysis of a process control loop for the reactive magnetron sputtering of high-index metal oxides. The control of discharge power to stabilize the oxygen partial pressure set point is modeled within the framework of the Berg model. A cycle time of 100 ms and process uncertainties for discharge current and oxygen partial pressure measurements are assumed, (from [71])...

The agreement in the case of the published data is not close. It is felt that this is contributed to by the process of extrapolation which is, at best, a hazardous occupation. In many cases such as this the measurements are made at elevated temperatures, and vapor pressures at room temperature are calculated from the Clausius-Clapeyron equation. There are two uncertainties in this extrapolation process uncertainty in the slope of the line of best fit and inaccuracy in the equation itself. 

Besides the deterministic context, the predicted amount of material is subjected now to a variability expressed by the second equation. This expresses the random character of the fractional flow rate, and it is known as process uncertainty. Extensive discussion of these aspects will be given in Chapter 9. 

Biological media are inhomogeneous, and the simplest way to capture structural and functional heterogeneity is to operate at a molecular level. First, one has to model the time spent by each particle in the process and second, to statistically compile the molecular behaviors. As will be shown in Section 9.3.4, this compilation generates a process uncertainty that did not exist in the deterministic model, and this uncertainty is the expression of process heterogeneity. 

So, we find that the mean behavior of the stochastic model is described by the deterministic model we have already developed. The fundamental difference between the stochastic and the deterministic model arises from the chance mechanism in the stochastic model that generates so-called process uncertainty, or stochastic error. 

The stochastic error is expressed in (9.23) by the variance Var [Aj (t)] and co-variance Cov [Nj (t) Nk (t)] that did not exist in the deterministic model. This error could also be named spatial stochastic error, since it describes the process uncertainty among compartments for the same t and it depends on the number of drug particles initially administered in the system. For the sake of simplicity, assume riQi = uq for each compartment i. From the previous relations, the coefficient of variation CVj (t) associated with a time curve Nj (t) in compartment 3 is... 

To illustrate the process uncertainty, we present the case of the two-compartment model, Figure 9.1. Equations (9.5) associated with the transfer-intensity matrix H were used to simulate the random distribution of particles, which expresses the process uncertainty. 

These profiles were normalized with respect to the initial condition in each compartment. The wider confidence intervals correspond to the initial conditions no, and the narrower confidence intervals to 10rro. Even without measurement error, fluctuations in the predicted amounts expressing the process uncertainty were observed the lower the number of molecules initially present in the compartments, the higher the observed fluctuations. 

Consequently, the observed process uncertainty may actually be an important part of the system and the expression of a structural heterogeneity. When the fluctuations in the system are small, it is possible to use the traditional deterministic approach. But when fluctuations are not negligibly small, the obtained differential equations will give results that are at best misleading, and possibly very wrong if the fluctuations can give rise to important effects. With these concerns in mind, it seems only natural to investigate an approach that incorporates the small volumes and small number of particle populations and may actually play an important part. 

Any remaining process uncertainties have been analyzed and defined. 

Moving horizon estimation is a practical strategy for designing state estimators by means of online optimization, which allows one to include constraints and nonlinearities in the state estimation [8]. In order to improve the estimation procedure, imperfect models can be augmented with other physical information, such as constraints on states variables, process disturbances or model parameters. Many process imcertainties are bounded, as well as state variables, which are also almost always positive. Unlike the process uncertainties, constraints on state variables are implicitly enforced by the model of the process, but it is not rare to face approximate models where this implicit enforcement may fail. Then, the inclusion of constraints is needed also on the state variables so as to reconcile the approximate model with the process measurements. 

Akbari AA, Karimi B (2015) A new robust optimization approach for integrated multi-echelon, multi-product, multi-period supply chain network design under process uncertainty. Int J Adv Manuf Technol 79 229-244... 

Exactly how reliable or accurate are the results we obtain from analyzing binding data This is perhaps the most important question we can ask about the results obtained—without any information about the reliability of our data, analysis is useless. The estimation of uncertainty can and should be approached from at least two directions as there are the two components that usually have the largest contribution to the estimation of uncertainties in binding studies, namely the experimental repeatability uncertainty and the data analysis (fitting process) uncertainty. If both are known, the overall uncertainly of the experiment can be estimated by comparing and possibly combining them, albeit this would have to be done in a subjective manner. 

The intermediate storage has an important role in improving operating efficiency of batch processing systems. It increases the variability of the system and reduces the process uncertainties. Over a long time horizon, it can also buffer the effects of equipment and batch failures when it is sized adequately. 

The rest of the knowledge, to the extent possible, must be generated from actual experiments (at both lab and pilot plant scale) and from mathematical modeling. Ideally the modeling should be done in intimate conjunction with the experimental testing. Models are useful tools to help design experiments since a reasonable model captures not only process understanding but also process uncertainty. 

Random uncertainties cannot be avoided they are part of the measuring process. Uncertainties are measures of random errors. These are errors incurred as a result of making measurements on imperfect apparatus which can only have a certain degree of accuracy. They are predictable, and tbe degree of error can be calculated. They can he reduced hy repeating and averaging the measurement. 

The resulting process uncertainties permanently impede your productivity. 

In summary, if the conditions of this theorem are satisfied, robust stability is guaranteed for the assumed process uncertainty description in Eqs. J-32 to J-34. The theorem... 

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In batch processing, uncertainty can emerge in operational or market-related parameters. Processing rates, duration of activities, material purchasing and product selling prices are common uncertain variables. Such uncertainty affects the throughput of a plant, the profitability and other key performance indicators (KPIs). In Monte Carlo simulation, numerous scenarios of a model are simulated by repeatedly picking values from a user-defined probability distribution for the uncertain variables. Those values are used in the model to calculate and analyze the outputs in a... 

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