Engineering problems linear models

The high dimensional models and representations that most chemical/biological engineering problems require cannot be solved analytically even when they are linear. 

Several related problems have been previously considered in the literature. In addition to the afore mentioned statistical approaches for structural change detection in data sets and their application for linear system identification [7], the joint problem of model structure determination and parameter estimation was addressed by [8-10]. A related approach was used by [11-13] in the context of data reconciliation. Additional aspects of model selection in chemical engineering are covered in [14]. Although the present problem shares common features with the all of the previous applications, it also presents unique characteristics that require a specific formulation. 

Engineers develop mathematical models to describe processes of interest to them. For example, the process of converting a reactant A to a product B in a batch chemical reactor can be described by a first order, ordinary differential equation with a known initial condition. This type of model is often referred to as an initial value problem (IVP), because the initial conditions of the dependent variables must be known to determine how the dependent variables change with time. In this chapter, we will describe how one can obtain analytical and numerical solutions for linear IVPs and numerical solutions for nonlinear IVPs. 

The centering of the Tuckerl model used by MacGregor ensures that the main nonlinear dynamics of the data are removed. Hence, it can be considered as a linearization step, which is a common tool in engineering, and this is a very sensible preprocessing step when linear models are used subsequently. The centering of the Tuckerl model used by Wold is performed over two modes simultaneously, which might introduce extra multilinear components (see Chapter 9). Hence, multilinear models may have problems with Wold s arrangement of the data. 

Linear models are the simplest form of equations commonty used to describe a wide rai of engineering situations. In this section, we first discuss some examples of engineerii problems where linear mathematical models are found. We then oplain the basic characteristics of linear models. 

At times, the formulation of an engineering problem leads to a set of linear equations that must be solved simultaneously. In Section 18.5, we will discuss the general form for such problems and the procedure for obtainii a solution. Here, we will discuss a simple graphical method that you can use to obtain the solution for a model that has two equations with two unknowns. For example, consider the follovnng equations with x andy as unknown variables. 

In the case of educating scientists and engineers about solutions to social problems, this is even more of a challenge as there is a tendency to assume that linear progress apphes as equally to the social world as it does to the natural world. The default assumption is the linear model of scientific and technological progress, whereby each advance or solution to a problem represents the culmination of trial and error in all previous attempts, and therefore there is no need to understand the history of previous attempted solutions. 

The unknown model parameters will be obtained by minimizing a suitable objective function. The objective function is a measure of the discrepancy or the departure of the data from the model i.e., the lack of fit (Bard, 1974 Seinfeld and Lapidus, 1974). Thus, our problem can also be viewed as an optimization problem and one can in principle employ a variety of solution methods available for such problems (Edgar and Himmelblau, 1988 Gill et al. 1981 Reklaitis, 1983 Scales, 1985). Finally it should be noted that engineers use the term parameter estimation whereas statisticians use such terms as nonlinear or linear regression analysis to describe the subject presented in this book. 

In numerical studies it turned out that the MILP problem can not only be solved much faster than the MINLP problem, but for most of the model instances it provides solutions of significantly higher solution quality. Certainly, the engineered linearization of the nonlinear problem causes a loss in model precision, but on the other hand it enables a globally optimal solution. Since the MILP solutions are feasible for the MINLP problem, it is clear that the inferior quality of the MINLP solutions originates from the fact that only local minima were found. 

The Excel Solver. Microsoft Excel, beginning with version 3.0 in 1991, incorporates an NLP solver that operates on the values and formulas of a spreadsheet model. Versions 4.0 and later include an LP solver and mixed-integer programming (MIP) capability for both linear and nonlinear problems. The user specifies a set of cell addresses to be independently adjusted (the decision variables), a set of formula cells whose values are to be constrained (the constraints), and a formula cell designated as the optimization objective. The solver uses the spreadsheet interpreter to evaluate the constraint and objective functions, and approximates derivatives, using finite differences. The NLP solution engine for the Excel Solver is GRG2 (see Section 8.7). 

The above discussion shows the importance of petrochemical network planning in process system engineering studies. In this chapter we develop a deterministic strategic planning model of a network of petrochemical processes. The problem is formulated as a mixed-integer linear programming model with the objective of maximizing the added value of the overall petrochemical network. 

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