Model, mathematical input-output

Parameter estimation is one of the steps involved in the formulation and validation of a mathematical model that describes a process of interest. Parameter estimation refers to the process of obtaining values of the parameters from the matching of the model-based calculated values to the set of measurements (data). This is the classic parameter estimation or model fitting problem and it should be distinguished from the identification problem. The latter involves the development of a model from input/output data only. This case arises when there is no a priori information about the form of the model i.e. it is a black box. 

The problem of parametric identification for mathematical models using input-output or output-only dynamic measurements has received much attention over the years. One important special case is modal identification, in which the parameters for identification are the small-amplitude modal frequencies, damping ratios, mode shapes and modal participation factors of the lower modes of the dynamical system. In other words, the model class in modal identification is the class of linear modal models. Many time-domain and frequency-domain methodologies have been formulated for input excitation and output response measurements [24,48,75,81,187],... 

It must be noted after a model is identified, a new set of experimental data is required to validate it in order to have confidence in its accuracy for the purpose or application it is established for. Establishing a valid model is one of the scientific and reliable tools researchers or practitioners use to undertake system analysis and design and optimize the system configuration and performance in order to establish and validate various concepts and ideas. To this end, significant research has been dedicated to the modeling and identification of EAP transducers. The procedure for model identification, which aims to obtain a mathematical model from input-output data, is depicted in Fig. 9. 

Statistical analysis can range from relatively simple regression analysis to complex input/output and mathematical models. The advent of the computer and its accessibiUty in most companies has broadened the tools a researcher has to manipulate data. However, the results are only as good as the inputs. Most veteran market researchers accept the statistical tools available to them but use the results to implement their judgment rather than uncritically accepting the machine output. 

Numeric-to-numeric transformations are used as empirical mathematical models where the adaptive characteristics of neural networks learn to map between numeric sets of input-output data. In these modehng apphcations, neural networks are used as an alternative to traditional data regression schemes based on regression of plant data. Backpropagation networks have been widely used for this purpose. 

Those based on strictly empirical descriptions Mathematical models based on physical and chemical laws (e.g., mass and energy balances, thermodynamics, chemical reaction kinefics) are frequently employed in optimization apphcations. These models are conceptually attractive because a gener 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... 

Develop via mathematical expressions a valid process or equipment model that relates the input-output variables of the process and associated coefficients. Include both equality and inequality constraints. Use well-known physical principles (mass balances, energy balances), empirical relations, implicit concepts, and external restrictions. Identify the independent and dependent variables (number of degrees of freedom). 

A simple mathematical model is used for quantitative description of the process and consists of a set of equations relating inputs, outputs, and key parameters of the system. The model for an alcoholic fermentation fed-batch process developed by Mayer (10) and adapted with the Ghose and Tyagi (11) linear inhibition term by the product was used as the starting point for the development of a model-based substrate sensor with product (ethanol) and biomass on-line measurements. 

Control based on neural network. Similar to fuzzy logic modeling, neural network analysis uses a series of previous data to execute simulations of the process, with a high degree of success, without however using formal mathematical models (Chen and Rollins, 2000). To this goal, it is necessary to define inputs, outputs, and how many layers of neurons will be used, which depends on the number of variables and the available data. 

For many mathematical operations, including addition, subtraction, multiplication, division, logarithms, exponentials and power relations, there are exact analytical expressions for explicitly propagating input variance and covariance to model predictions of output variance (Bevington, 1969). In analytical variance propagation methods, the mean, variance and covariance matrix of the input distributions are used to determine the mean and variance of the outcome. The following is an example of the exact analytical variance propagation approach. If w is the product of x times y times z, then the equation for the mean or expected value of w, E(w), is ... 

Fig. 2 A PBPK model. The body is described as a set of compartments corresponding to organs, group of organs, or tissues. The arrows indicate inputs, outputs, and transport between those compartments by blood, lymph, or diffusion. The corresponding mathematical equations can be used to compute concentrations or quantities of one or several substances in any compartment as a function of time, dose, body, and substance characteristics...

There are different kinds of mathematical models, and they can be classified in two ways by their complexity and by the number of estimable parameters they use. The most simple models are cartoons with very few parameters. These—such as the black box that was the receptor at the turn of the century—usually are simple input-output functions with no mechanistic description (i.e., the drug interacts with the receptor and a response ensues). Another type, termed the Parsimonious model, is also simple but has a greater number of estimable parameters. These do not completely characterize the experimental situation but do offer insights into mechanism. Models can be more complex as well. For example, complex models with a large number of estimable parameters can be used to simulate behavior under a variety of conditions (simulation models). Similarly, complex models for which the number of independently verifiable parameters is low (termed heuristic models) can still be used to describe complex behaviors not apparent by simple inspection of the system. 

Some domain ontologies have been developed, which so far cover only a small portion of the chemical engineering domain They enable the representation of material properties, experiments and process recipes, as well as the structural description of mathematical models. In parallel, an ontology for the modeling of work processes and decisions has been built. Yet the representation of work processes is confined to deterministic guidelines, the input/output information of activities cannot be modeled, and the acting persons are not... 

Equation (5.1) is the mathematical model of the stirred tank heater with T the state variable, while T, and Ta are the input variables. Let us see how we can develop the corresponding input-output model. 

The mathematical models we learned to develop in Chapter 4 using state variables are not of the direct input-output type. Nevertheless, they constitute the basis for the development of an input-output model. This is particularly easy and straightforward when the state variables coincide completely with the output variables of a process. In such a case we can integrate the state model to produce the input-output model of the process. 

The theoretical models proposed in Chapters 2-4 for the description of equilibrium and dynamics of individual and mixed solutions are by part rather complicated. The application of these models to experimental data, with the final aim to reveal the molecular mechanism of the adsorption process, to determine the adsorption characteristics of the individual surfactant or non-additive contributions in case of mixtures, requires the development of a problem-oriented software. In Chapter 7 four programs are presented, which deal with the equilibrium adsorption from individual solutions, mixtures of non-ionic surfactants, mixtures of ionic surfactants and adsorption kinetics. Here the mathematics used in solving the problems is presented for particular models, along with the principles of the optimisation of model parameters, and input/output data conventions. For each program, examples are given based on experimental data for systems considered in the previous chapters. This Chapter ean be regarded as an introduction into the problem software which is supplied with the book an a CD. 

The graphical method is based on the notion that the mathematical model of a discrete-time finite-order (stationary) dynamic system is, in general, a multivariate function /( ) of the appropriate lagged values of the input-output variables... 

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