Algebraic Equation Models

Equation 2.1, the mathematical model of the process, is very general and it covers many cases, namely, 

A single experiment consists of the measurement of each of the m response variables for a given set of values of the n independent variables. For each experiment, the measured output vector which can be viewed as a random variable is comprised of the deterministic part calculated by the model (Equation 2.1) and the stochastic part represented by the error term, i.e., 

If the mathematical model represents adequately the physical system, the error term in Equation 2.3 represents only measurement errors. As such, it can often be assumed to be normally distributed with zero mean (assuming there is no bias present in the measurement). In real life the vector e, incorporates not only the experimental error but also any inaccuracy of the mathematical model. 

A special case of Equation 2.3 corresponds to the celebrated linear systems. Linearity is assumed with respect to the unknown parameters rather than the independent variables. Hence, Equation 2.3 becomes 

A brief review of linear regression analysis is presented in Chapter 3. 

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In algebraic equation models we also have the special situation of conditionally linear systems which arise quite often in engineering (e.g., chemical kinetic models, biological systems, etc.). In these models some of the parameters enter in a linear fashion, namely, the model is of the form,... 

The category of algebraic equation models is quite general and it encompasses many types of engineering models. For example, any discrete dsmamic model described by a set of difference equations falls in this category for parameter estimation purposes. These models could be either deterministic or stochastic in nature or even combinations of the two. Although on-line techniques are available for the estimation of parameters in sampled data systems, off-line techniques... 

The adaptation of the original LJ optimization procedure to parameter estimation problems for algebraic equation models is given next. 

If we have very little information about the parameters, direct search methods, like the LJ optimization technique presented in Chapter 5, present an excellent way to generate very good initial estimates for the Gauss-Newton method. Actually, for algebraic equation models, direct search methods can be used to determine the optimum parameter estimates quite efficiently. However, if estimates of the uncertainty in the parameters are required, use of the Gauss-Newton method is strongly recommended, even if it is only for a couple of iterations. 

Under certain conditions we may have some prior information about the parameter values. This information is often summarized by assuming that each parameter is distributed normally with a given mean and a small or large variance depending on how trustworthy our prior estimate is. The Bayesian objective function, SB(k), that should be minimized for algebraic equation models is... 

The proposed step-size policy for differential equation systems is fairly similar to our approach for algebraic equation models. First we start with the bisection rule. We start with g=l and we keep on halving it until an acceptable value, pa, has been found, i.e., we reduce p until... 

Let us consider constrained least squares estimation of unknown parameters in algebraic equation models first. The problem can be formulated as follows ... 

Let us consider the case of an algebraic equation model (i.e., y = f(x,k)). The problem can be restated as "find the best experimental conditions (i.e., xN+)) where the next experiment should be performed so that the variance of the parameters is minimized."... 

As mentioned in Chapter 4, although this is a dynamic experiment where data are collected over time, we consider it as a simple algebraic equation model with two unknown parameters. The data were given for two different conditions (i) with 0.75 g and (ii) with 1.30 g of methanol as solvent. An initial guess of k =1.0 and k2=0.01 was used. The method converged in six and seven iterations respectively without the need for Marquardt s modification. Actually, if Mar-quardt s modification is used, the algorithm slows down somewhat. The estimated parameters are given in Table 16.1 In addition, the model-calculated values are... 

Although this is a dynamic experiment where data are collected over time, it is considered as a simple algebraic equation model with two unknown parameters. 

Programs are provided for the following examples dealing with algebraic equation models ... 

By discretizing the differential algebraic equations model using some standard discretization techniques (Liebman et al., 1992 Alburquerque and Biegler, 1996) to convert the differential constraints to algebraic constraints, the NDDR problem can be solved as the following NLP problem ... 

Process-scale models represent the behavior of reaction, separation and mass, heat, and momentum transfer at the process flowsheet level, or for a network of process flowsheets. Whether based on first-principles or empirical relations, the model equations for these systems typically consist of conservation laws (based on mass, heat, and momentum), physical and chemical equilibrium among species and phases, and additional constitutive equations that describe the rates of chemical transformation or transport of mass and energy. These process models are often represented by a collection of individual unit models (the so-called unit operations) that usually correspond to major pieces of process equipment, which, in turn, are captured by device-level models. These unit models are assembled within a process flowsheet that describes the interaction of equipment either for steady state or dynamic behavior. As a result, models can be described by algebraic or differential equations. As illustrated in Figure 3 for a PEFC-base power plant, steady-state process flowsheets are usually described by lumped parameter models described by algebraic equations. Similarly, dynamic process flowsheets are described by lumped parameter models comprising differential-algebraic equations. Models that deal with spatially distributed models are frequently considered at the device... 

In this chapter we provide listing of two computer programs with the corresponding input flies. One is for a typical algebraic equation model (Example... 

PARAMETER ESTIMATION ROUTINE FOR ALGEBRAIC EQUATION MODELS Based on Gauss-Newton method with Pseudoinverse and Marquardt s Modification. Hard BOUNDARIES on parameters can be imposed. 

Algebraic Equations Modeling Two Electrical Circuits, One with a Capacitor and a Resistor in Parallel (Left Column), the Other with a Self-Inductance in Series with a Resistor (Right Column)... 

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