Logic fuzzy

Fuzzy models may be slightly more difficult to program and develop than qualitative models, but they share all of the advantages. In addition, they provide smoother control. This may be a particular advantage in processes that require more rapid control decisions than autoclave curing. The cost of developing these controllers is relatively low and the data required to develop them is usually available from the development of the original process plan. Further, neural networks can be used to automate the development of control-process relationships. 

Wu and Joseph [35] incorporated fuzzy logic into a knowledge-based control system for control of composite curing. They used the fuzzy logic to interpret the sensors and adjust the amount of control reaction on a simulated process. Even though the limitations of the simulator used did not allow full evaluation of the advantages of this system, it did show that the controller could react to material and process variations and improve the process plan. 

Whereas fuzzy logic offers an improvement over qualitative reasoning in some aspects, the relationships it uses still must be derived from data. There fortunately, exists a means of automating the development of these relationships, even in the absence of a detailed analytical model. 

The term fuzzy logic is often interpreted in two ways. In a broad interpretation, fuzzy logic is viewed as a system of concepts, principles, and methods for dealing with modes of reasoning that are approximate rather than exact. In a narrow interpretation, it is viewed as a generalization of the various many-valued logics. This narrow interpretation of fuzzy logic is not of interest in this chapter. 

Fuzzy logic deals with propositions expressed in natural language. The linguistic expressions involved may contain fuzzy linguistic terms of any of the following types  

All of these linguistic terms except fuzzy modifiers are represented in each context by appropriate fuzzy sets. Fuzzy predicates are represented by fuzzy sets defined on universal sets of elements to which the predicates apply. Fuzzy truth values and fuzzy probabilities are represented by fuzzy sets defined on the unit interval [0,1]. Fuzzy quantifiers are either absolute or relative they are represented by appropriate fuzzy numbers defined either on the set of natural numbers or on the interval [0,1]. Fuzzy modifiers are operations by which fuzzy sets representing the various other linguistic terms are appropriately modified to capture the meaning of the modified linguistic terms. 

In a crude way, it is useful to distinguish the following four types of fuzzy propositions, each of which may, in addition, be quantified by an appropriate fuzzy quantifier. 

Unconditional and unqualified propositions are expressed by the canonical form 

The properties shown in Table 1 use the standard definition suitable for use in strict FL, and the Lukasiewicz definition suitable for use in fuzzy clustering. 

The basic assumption upon which two-valued logic is based, that every proposition is either true or false, has been questioned since Aristotle. For example, propositions about future events are neither actually true, nor actually false, but potentially either. Consequently, their truth-value is undetermined, at least prior to the event. Propositions whose truth-value is problematic are not limited to future events. For example, propositions that imply physical or chemical measurements depend on the limitations of mea-surement. While several types of multivalued logic have been proposed, here we will discuss the infinite-valued logic, whose truth-values are represented by all the real numbers in the interval [0,1]. This is also called standard Lukasiewicz logic L. The primitives of this logic are defined as 

It can be verified that the relations above reduce to their usual counterparts when applied to binary logic. The standard Lukasiewicz logic Li is isomorphic to fuzzy set theory based on the standard fuzzy operations in the same way the two-valued logic is isomorphic to the crisp set theory. The membership degree A(x) for x e X may be interpreted as the truth value of the proposition x is a member of the set A . The reciprocal is also valid. 

Its truth-value is determined in a way similar to the cases above. 

Naturally, this advice is of no use at all, but fortunately there is help at hand in the form of fuzzy logic, which provides a way to reason from vague information. 

The vagueness in the second rule above and, consequently, the unhelpful imagined response of the ES, reflect the way that we use woolly language in everyday conversation  

Overweight people should pay higher health insurance premiums. 

These statements offer information of a sort, but they are imprecise. We know what the first two mean, but they are vague, while, in the third statement, the information provided is not factual, but is merely an opinion thus all we know is that the speaker has a particular view about large people. None of the information that these statements provide could be incorporated into an ES using the kind of rules outlined in the last chapter because the system that we met there was not designed to deal with imprecision. 

If we need to include this rather uncertain information in an ES, our first thought might be that vague statements could be firmed up by adding some quantitative information  

In many experimental cases, a certain degree of interference occurs among the measures, which gives rise to possible collections of results however, the situation is even more complex if the input data are subjected to uncertainty or imprecision (Kaufmann and Gupta, 1991). Fuzzy logic is the only mathematical application that can properly solve problems with imprecision in input data. 

Fuzzy logic is based on the generalization of theory of sets characteristic function that Zadeh defined as membership function , //U), (Zadeh, 1965) 

But in order to build a practical system that must be precise and consistent in its behaviour, one has to impose restrictions with regard to the use of ambiguous modifiers (e.g. very not large) and the order of them in the composition (e.g. very X, fairly X, more or less X, above X, much above X) (Dubois and Prade, 1980 Kandel, 1986). 

However, the identification of the fuzziness associated with single parameter characterizing foodstuffs is not enough. The authentication process is not usually restricted to a single parameter but in fact there are often several of them. If we want to operate with their membership function (e.g. low linoleic and high 24-methylene cycloarthanol), we need to define operations on the fuzzy set. Thus, the classical rule R = IF (input) A, THEN (output) B can be extended 

Schiitze, P. and Will, J. (1993) Expert system for interpretation of x-ray diffraction spectra. Anal. Chim. Acta, 271, 287—291. 

Accuracy checks are generally performed by comparing measured data with data from certified reference materials. When measured data are not accurate because of relative or systematic errors, or a lack of precision (noise), the comparison between measured data and reference values cannot lead to any useful conclusion in an expert system. To process larger sets of potential source data for knowledge bases, a method must be used that takes inaccuracies as well as natural fuzziness of experimental data into account — ideally automatically and without the help of an expert. 

Problems of uncertainty and inaccuracy can be addressed by using statistical and stochastic methods that have been described before [29,30]. The fuzzy logic approach provides a mathematical framework for representation and calculation of inaccurate data in AI methods [31,32]. Fuzzy logic is a superset of conventional (Boolean) logic that has been extended to handle the values between exactly true and exactly false. 

The general principle in fuzzy logic is that a reference value Xq is associated with a fuzzy interval dx, and experimental data within an interval of Xq dx are identified as reference data. Since natural, or experimental, data are always inaccurate, and the representation of knowledge is quite like that in fuzzy logic, expert systems have to use fuzzy logic or some techniques similar to fuzzy logic [33]. In a computer system based on the fuzzy logic approach, fuzzy intervals for reference values are defined a priori. 

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Fuzzy sets and fuzzy logic. Fuzzy sets differ from the normal crisp sets in the fact that their elements have partial membership (represented by a value between 0 an 1) in the set. Fuzzy logic differs from the binary logic by the fact that the truth values are represented by fuzzy sets. 

All the three techniques mentioned above may make use of fuzzy sets and fuzzy logic (for fuzzy classification, fuzzy rules or fuzzy matching) but this does not effect the discussion of the applicability to NDT problems in the next section. 

To know about fuzzy sets and fuzzy logic... 

Conventional computers initially were not conceived to handle vague data. Human reasoning, however, uses vague information and uncertainty to come to a decision. In the mid-1960 this discrepancy led to the conception of fuzzy theory [14]. In fuzzy logic the strict scheme of Boolean logic, which has only two statements true and false), is extended to handle information about partial truth, i.e., truth values between "absolutely true" and absolutely false". It thus gives a mathematical representation of uncertainty and vagueness and provides a tool to treat them. 

Fuzzy logic and fuzzy set theory are applied to various problems in chemistry. The applications range from component identification and spectral Hbrary search to fuzzy pattern recognition or calibrations of analytical methods. 

An overview over different applications of fuzzy set theory and fuzzy logic is given in [15] (see also Chapter IX, Section 1.5 in the Handbook). 

Here, the application of fuzzy logic for multicomponent spectral analysis is described. 

The principle of applying fuzzy logic to matching of spectra is that, given a sample spectrum and a collection of reference spectra, in a first step the reference spectra are unified and fuzzed, i.e., around each characteristic line at a certain wavenumber k, a certain fuzzy interval [/ o - Ak, + Afe] is laid. The resulting fuzzy set is then intersected with the crisp sample spectrum. A membership function analogous to the one in Figure 9-25 is applied. If a line of the sample spec-... 

Fuzzy logic extends the Boolean logic so as to handle information about truth values which are between absolutely true and "absolutely false . 

A series of monographs and correlation tables exist for the interpretation of vibrational spectra [52-55]. However, the relationship of frequency characteristics and structural features is rather complicated and the number of known correlations between IR spectra and structures is very large. In many cases, it is almost impossible to analyze a molecular structure without the aid of computational techniques. Existing approaches are mainly based on the interpretation of vibrational spectra by mathematical models, rule sets, and decision trees or fuzzy logic approaches. 

One variation of rule-based systems are fuzzy logic systems. These programs use statistical decision-making processes in which they can account for the fact that a specific piece of data has a certain chance of indicating a particular result. All these probabilities are combined in order predict a final answer. 

While the single-loop PID controller is satisfactoiy in many process apphcations, it does not perform well for processes with slow dynamics, time delays, frequent disturbances, or multivariable interactions. We discuss several advanced control methods hereafter that can be implemented via computer control, namely feedforward control, cascade control, time-delay compensation, selective and override control, adaptive control, fuzzy logic control, and statistical process control. 

Fuzzy Logic Control The apphcation of fuzzy logic to process control requires the concepts of Fuzzy rules and fuzzy inference. A fuzzy rule, also known as a fuzzy IF-THEN statement, has the form ... 

In addition to single-loop process controllers, products that have benefited from the implementation of fuzzy logic are ... 

Sometimes fuzzy logic controllers are combined with pattern recognition software such as artificial neural networks (Kosko, Neural Networks and Fuzzy Systems, Prentice Hall, Englewood Cliffs, New Jersey, 1992). 

Fuzzy logic control systems 10.2.1 Fuzzy set theory... 

An important aspect of fuzzy logic is the ability to relate sets with different universes of discourse. Consider the relationship... 

MATLAB Fuzzy Inference System (FIS) editor can be found in Appendix 1. Figure 10.16 shows the control surface for the 11 set rulebase fuzzy logic controller. 

Self-Organizing Fuzzy Logic Control (SOFLC) is an optimization strategy to create and modify the control rulebase for a FLC as a result of observed system performance. The SOFLC is particularly useful when the plant is subject to time-varying parameter changes and unknown disturbances. 

Fig. 10.16 Control surface for 11 set rulebase fuzzy logic controller.

Fig. 10.17 Self-Organizing Fuzzy Logic Control system.

The angular positional control system shown by the block diagram in Figure 10.36 is to have the velocity feedback loop removed and controller K replaced by a fuzzy logic controller (FLC) as demonstrated by Barrett (1992). The inputs to the FLC... 

Fig. A1.8 Simulink implementation of inverted pendulum fuzzy logic control problem.

Polkinghorne, M.N. (1994) A Self-Organising Fuzzy Logic Autopilot for Small Vessels, PhD Thesis, School of Manufacturing, Materials and Mechanical Engineering, University of Plymouth, UK. 

Yan, J., Ryan, M. and Power, J. (1994) Using fuzzy logic - Towards intelligent systems, Prentice-Hall International (UK), Hemel Hempstead, UK. 

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