Mathematic model parameter estimation procedure

After the formulation stage, we have all the equations of the model, but they are not useful yet, because parameters in the equations do not have a particular value. Consequently, the model cannot be used to reproduce the behavior of a physical entity. The parameter estimation procedure consists of obtaining a set of parameters that allows simulation with the model. In many cases, parameters can be found in literature, but in other cases it is required to fit the model to the experimental behavior by using mathematical procedures. The easier and more used types of procedures are those based on the use of optimization algorithms to make minimum the differences between the experimental observations and the model outputs. The more frequently used criterion to optimize the values of the parameters is the least square regression coefficient. In this procedure, a set of values is proposed for all model parameters (one for every parameter) and the model is run. After that, the error criterion is calculated as the sum of the squares of the residues (differences between the values of every experimental and modeled value). Then, an optimization procedure is used to change the values of the model parameters in order to get the minimum value of this criterion. 

A more detailed description of the mathematical model presented herein, the model solution, parameter estimation procedure and the confidence limits equations are described in detail in references 5,8 and 11. 

Parameter estimation is a procedure for taking the unit measurements and reducing them to a set of parameters for a physical (or, in some cases, relational) mathematical model of the unit. Statistical interpretation tempered with engineering judgment is required to arrive at realistic parameter estimates. Parameter estimation can be an integral part of fault detection and model discrimination. 

Parameter Estimation Relational and physical models require adjustable parameters to match the predicted output (e.g., distillate composition, tower profiles, and reactor conversions) to the operating specifications (e.g., distillation material and energy balance) and the unit input, feed compositions, conditions, and flows. The physical-model adjustable parameters bear a loose tie to theory with the limitations discussed in previous sections. The relational models have no tie to theory or the internal equipment processes. The purpose of this interpretation procedure is to develop estimates for these parameters. It is these parameters hnked with the model that provide a mathematical representation of the unit that can be used in fault detection, control, and design. 

A survey of the mathematical models for typical chemical reactors and reactions shows that several hydrodynamic and transfer coefficients (model parameters) must be known to simulate reactor behaviour. These model parameters are listed in Table 5.4-6 (see also Table 5.4-1 in Section 5.4.1). Regions of interfacial surface area for various gas-liquid reactors are shown in Fig. 5.4-15. Many correlations for transfer coefficients have been published in the literature (see the list of books and review papers at the beginning of this section). The coefficients can be evaluated from those correlations within an average accuracy of about 25%. This is usually sufficient for modelling of chemical reactors. Mathematical models of reactors arc often more sensitive to kinetic parameters. Experimental methods and procedures for parameters estimation are discussed in the subsequent section. 

In order to solve the mathematical model for the emulsion hquid membrane, the model parameters, i. e., external mass transfer coefficient (Km), effective diffu-sivity (D ff), and rate constant of the forward reaction (kj) can be estimated by well known procedures reported in the Hterature [72 - 74]. The external phase mass transfer coefficient can be calculated by the correlation of Calderback and Moo-Young [72] with reasonable accuracy. The value of the solute diffusivity (Da) required in the correlation can be calculated by the well-known Wilke-Chang correlation [73]. The value of the diffusivity of the complex involved in the procedure can also be estimated by Wilke-Chang correlation [73] and the internal phase mass transfer co-efficient (surfactant resistance) by the method developed by Gu et al. [75]. 

Those based on strictly empirical descriptions Mathematical models based on physical and cnemical laws (e.g., mass and energy balances, thermodynamics, chemical reaction kinetics) are frequently employed in optimization applications. These models are conceptually attractive because a general model for any system size can be developed before the system is constructed. On the other hand, an empirical model can be devised that simply correlates input/output data without any physiochemical analysis of the process. For these models, optimization is often used to fit a model to process data, using a procedure called parameter estimation. The well-known least squares curve-fitting procedure is based on optimization theory, assuming that the model parameters are contained linearly in the model. One example is the yield matrix, where the percentage yield of each product in a unit operation is estimated for each feed component... 

Approaches based on parameter estimation assume that the faults lead to detectable changes of physical system parameters. Therefore, FD can be pursued by comparing the estimates of the system parameters with the nominal values obtained in healthy conditions. The operative procedure, originally established in [23], requires an accurate model of the process (including a reliable nominal estimate of the model parameters) and the determination of the relationship between model parameters and physical parameters. Then, an online estimation of the process parameters is performed on the basis of available measures. This approach, of course, might reveal ineffective when the parameter estimation technique requires solution to a nonlinear optimization problem. In such cases, reduced-order or simplified mathematical models may be used, at the expense of accuracy and robustness. Moreover, fault isolation could be difficult to achieve, since model parameters cannot always be converted back into corresponding physical parameters, and thus the influence of each physical parameters on the residuals could not be easily determined. 

A key factor in modeling is parameter estimation. One usually needs to fit the established model to experimental data in order to estimate the parameters of the model both for simulation and control. However, a task so common in a classical system is quite difficult in a chaotic one. The sensitivity of the system s behavior to the initial conditions and the control parameters makes it very hard to assess the parameters using tools such as least squares fitting. However, efforts have been made to deal with this problem [38]. For nonlinear data analysis, a combination of statistical and mathematical tests on the data to discern inner relationships among the data points (determinism vs. randomness), periodicity, quasiperiodicity, and chaos are used. These tests are in fact nonparametric indices. They do not reveal functional relationships, but rather directly calculate process features from time-series records. For example, the calculation of the dimensionality of a time series, which results from the phase space reconstruction procedure, as well as the Lyapunov exponent are such nonparametric indices. Some others are also commonly used ... 

Application of this procedure is illustrated by an example of analysis of unsteady-state processes in a single catalyst pellet [8, 9]. A separate consideration of this element allows estimation of the domains of parameters where certain stages of heat and mass transfer can be neglected and the mathematical model thus simplified. These criteria derived after assuming the steady-state reaction rate r are given in Tabic 2. 

Numerous reports are available [19,229-248] on the development and analysis of the different procedures of estimating the reactivity ratio from the experimental data obtained over a wide range of conversions. These procedures employ different modifications of the integrated form of the copolymerization equation. For example, intersection [19,229,231,235], (KT) [236,240], (YBR) [235], and other [242] linear least-squares procedures have been developed for the treatment of initial polymer composition data. Naturally, the application of the non-linear procedures allows one to obtain more accurate estimates of the reactivity ratios. However, majority of the calculation procedures suffers from the fact that the measurement errors of the independent variable (the monomer feed composition) are not considered. This simplification can lead in certain cases to significant errors in the estimated kinetic parameters [239]. Special methods [238, 239, 241, 247] were developed to avoid these difficulties. One of them called error-in-variables method (EVM) [239, 241, 247] seems to be the best. EVM implies a statistical approach to the general problem of estimating parameters in mathematical models when the errors in all measured variables are taken into account. Though this method requires more information than do ordinary non-linear least-squares procedures, it provides more reliable estimates of rt and r2 as well as their confidence limits. 

There are many reviews on mathematical models for hthium ion batteries. Botte et al. presented an extensive review on mathematical modeling of rechargeable lithium batteries. A review of mathematical models of lithium and nickel battery systems is discussed in hterature." " Experimental developments in the field can be found in a recent review article that describes new solutions, new measurement procedures and new materials for Li-ion batteries. " Apart from the enormous body of work on modehng of Li-ion batteries, efforts have also been made in making these continuum models more computationally efficient to simulate." Computationally efficient models can not only be used to predict battery behavior but can also be used in situations where real-time parameter estimation is needed, for example, situations where super accurate determination of State of Health (SOH) of a battery is critical, adding a new dimension to the capabilities of continuum models. 

A procedure is presented for estimation of uncertainty in measurement of the pK(a> of a weak acid by potentiometric titration. The procedure is based on the ISO GUM. The core of the procedure is a mathematical model that involves 40 input parameters. A novel approach is used for taking into account the purity of the acid, the impurities are not treated as inert compounds only, and their possible acidic dissociation is also taken into account. Application to an example of practical pK(a> determination is presented. Altogether, 67 different sources of uncertainty are identified and quantified within the example. The relative importance of different uncertainty sources is discussed. The most important source of uncertainty (with the experimental set-up of the example) is the uncertainty of the pH measurement followed by the accuracy of the burette and the uncertainty of weighing. The procedure gives uncertainty separately for each point of the titration curve. The uncertainty depends on the amount of the titrant added, being lowest in the central part of the titration curve. The possibilities of reducing the uncertainty and interpreting the drift of the pKJa) values obtained from the same curve are discussed. 

Calibration refers to the procedures used for correlating test method ontpnt or response to an amount of analyte (concentration or other quantity). The characteristics of a calibration fnnction and justification for a selected calibration model should be demonstrated dnring SLV and ILS stndies. The performance of a calibration technique and the choice of calibration model (e.g., first-order linear, cnrvifinear, or nonlinear mathematical function) are critical for minimizing method bias and optimizing precision. The parameters of the model are nsnally estimated from the responses of known, pnre materials. Calibration errors can result from failure to identify the best calibration model inaccnrate estimates of the parameters of the model errors in the composition of calibration materials or inadeqnately smdied, systematic effects from matrix components. This section focnses on the critical issne of the traceability and supply of materials used for calibration of marine biotoxin methods. 

There is an increasing interest in technologies that maximize the production of various essential enzymes and therapeutic proteins based on E. coli cultivation. The costs of developing mathematical models for bioprocesses improvements are often too high and the benefits are too low. The main reason for this is related to the intrinsic complexity and non-linearity of biological systems. The important part of model building is the choice of a certain optimization procedure for parameter estimation. The estimation of model parameters with high parameter accuracy is essential for successful model development. 

An analytical method can be represented by a point or by a region in a multidimensional "space of procedures". The coordinates correspond to the parameters of the method, like accuracy, time, cost, etc.. Kaiser C4213 applied the information theory.to the estimation of the "informing power" of analytical procedures. Pattern recognition methods have been proposed by Wold et. al. C36, 341, 3433 for an objective evaluation of analytical methods. A data matrix is obtained by the application of me-thods to a number of real samples. Mathematical models were constructed for the clusters describing the methods under consideration. 

Equation 4 was discretised by a 5-point central difference formula and thereafter first-order differential equations 1, 2, 4 and 6 were solved by a backward difference method. Apparent reaction rate was solved by summing the average rates of each discretisation piece of equation 4. The reactor model was integrated in a FLOWBAT flowsheet simulator [12], which included a databank of thermodynamic properties as well as VLE calculation procedures and mathematical solvers. The parameter estimation was performed by minimising the sum of squares for errors in the mole fractions of naphthalene, tetralin and the sum of decalins. Octalins were excluded from the estimation because their content was low (<0.15 mol-%). Optimisation was done by the method of Levenberg-Marquard. 

NN models are nonlinear mathematical structures built by summing up iteratively nonlinear transformations of linear combinations of certain input variables. NN models can assume many different configurations. In the simplest case, usually called as the feed-forward NN structure, three different layers are employed (Fig. 6.8) the input layer, the hidden layer, and the output layer. The input layer is fed by values of a number of input variables, generally the spectral data measured at certain wavelengths. The output layer provides the desired process response. The backpropagation procedure is normally used to estimate the NN model parameters [77], The nonlinear transformation generally used at each particular node of the NN model is a sigmoidal activation function, defined as... 

The book includes model formulation, i.e. how to describe a physical/chemical reality in mathematical language, and how to choose the type and degree of sophistication of a model. It is emphasized that this is an iterative procedure where models are gradually refined or rejected in confrontation with experiments. Model reduction and approximate methods, such as dimensional analysis, time constant analysis, and asymptotic methods, are treated. An overview of solution methods for typical classes of models is given. Parameter estimation and model validation and assessment, as final steps, in model building are discussed. The question What model should be used for a given situation is answered. 

An efficient slurry health monitoring tool should be able to provide both chemical as well as abrasive particle information on a continuous basis. There have been some efforts in this direction using an NIR absorption spectrum based analyzer [19]. This unit can provide oxidizer concentration and abrasive particle information in CMP slurry and operates on the principles of chemometrics, which is a two-phase process. In the first calibration phase, samples with known property values are measured by the system. A mathematical procedure then determines the correlation between the measured spectra and the true property values. The output of this phase is a model that optimally calculates the parameter values from the measured spectra of the calibration samples. In the second measurement phase, unknown samples are measured by the system, employing a model to produce estimates of the property values. 

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