Extrapolation Methods in Control and Adaptive System

Extrapolation Methods in Control and Adaptive System

DOI: 10.4018/978-1-4666-2967-7.ch008

Abstract

In the initial stages of the choice of approaches and methods, the heuristic of the investigator is very important, because in most of the cases there is a lack of measurements or even clear scales under which to implement these measurements and computations. This stage is often outside of the strict logic and mathematics and is close to the art, in the widest sense of the word. For complex systems and practical problems, the explicit description of such an algorithm is a difficult problem; often it is not solvable, but the existence of beforehand solutions and realizations allow one to make meaningful prognosis estimates. Such an approach is the method of “Multidimensional Linear Extrapolation” (MLE) described in this chapter. The main idea is that close situations give close solutions. The method is very efficient and gives many good solutions for difficult modeling problems.
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1. Theoretical Basis Of Extrapolation In Control Design

The indeterminacy of the initial data is an objective condition in which decisions are made for the creation of new samples of technical or other products in the diverse human activity. This indeterminacy is especially extensive in the initial stages and the preliminary planning, as it can be in the structure and/or the parameters of the product (Gig, 1981). From the mentioned here, it is clear that in such tasks it is necessary to use an approach of consecutive stages, phases in which programs are applied with the aim of achieving one or several intermediate objectives. The aspiration for quantity measurements, estimates, and prognosis at all phases of the investigations and the creation of the new sample products is natural. However, this task is solved with very scarce initial information, especially in the initial development phase. Precisely this objective condition has to be accounted for when choosing the mathematical methods for design and prognosis and construction of new and non-standard products or new control solutions (Pfanzagl, 1971; Raiffa, 1968).

In the initial stages of the choice of approaches and methods, the heuristic of the investigator is very important, because in most of the cases there is a lack of measurements or even clear scales under which to implement these measurements and computations.

This stage is often outside of the strict logic and mathematics and is close to the art, in the widest sense of the word, to choose the right decision among great number of circumstances and often without associative examples of similar activity (Gig, 1981; Rastrigin, 1981). Often intuitively insignificant circumstances have to be ignored and the main solution has to be chosen as a process of recognition. We call this the art to orient the scientist in this initial undetermined situation.

In these initial stages mathematical methods are used which in their concept allow for broad semantic interpretations, flexibility and iterativity in the mathematical constructs in the models as well as in the control solutions. The models which are built at this initial stage are based on preliminary experience with other, often quite different products or solutions, or are formed by training.

For this reason, the methods for design and prognosis here have to facilitate solutions in multidimensional spaces, to allow for easy correction and easy tweaking of the objectives to adequately handle the accumulating basic information in the process of the investigation and to admit iterative development of the modeling process. Such an approach is the method of “Multidimensional Linear Extrapolation” (MLE), designed by the Russian scientist Rastrigin (Rastrigin, 1981, 1986).

The solution of the problem of the identification in the broad sense of the word or the prognosis is determined by the nature of the process or the nature of the basic information for the process. The information in most general terms can be of quantitative or qualitative type. In the beginning, it is obligatory to have a stage of classification, determination of the basic scales and the way of measurements taking into account the type of basic information (Pfanzagl, 1971; Gig, 1981).

The choice of the methodology depends on whether the process is deterministic or random in nature. When the process is deterministic, the problem of prognosis and construction can be described using interpolation or extrapolation. The interpolation means determination of the value of the function or model in the points of interest using the values of the function or the solutions of the model in other points. We are looking for solution inside the interval or the segment of the observation.

In the problem of extrapolation, we are looking for the function values outside the interval of the observation. Moreover, for complex processes and new products the data is often scarce.

It is possible that the function modeling the product is known in advance or in the harder case these functions are unknown and only their approximation is sought based on scarce initial information. If we add the presence of a noise or internal initial indetermination in the data and the multidimensionality of the process then the choice of the mathematical methods and models becomes severely limited.

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