By Prof. Dr. Dimiter Driankov, Dr. Hans Hellendoorn, Dr. Michael Reinfrank (auth.)
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Extra resources for An Introduction to Fuzzy Control
Suppose, for example , that you are asked how strongly you agree that a given number x E [0,20] is a large number. Je number and if x < d then you disagree. Thus, if you place a mark on an agree-disagree scale, it might be distributed uniformly over the right half of the scale whenever x 2: d and uniformly over the left half if x < d. Such a person is called a threshold person. he agree end. hen he will be squeezed if an even larger number is presented. However such a person will use the full continuum of the scale according to how large or how small he thinks the numbers are.
In our example, E will occur if the result of throwing a dice is anyone of I, 2, 3. Then, from the frequentist point of view, the probability of E, P(E) E [0,1], is the proportion of occurrences of E in a long series of experiments. Furthermore, a conditional probability, denoted as P(E IE'), defines a probability of an event E given that we are told that the event E' has occurred. In our example, P(E I 3) = I, since we know that 3 is an element of the subset U that defines the event E, and if 3 has occurred then, according to our definition of an event, this means that E has occurred as well.
U max time Fig. 7. Type of control action desired in different regions around the set-point. (Redrawn from Sripada et al. 7 illustrates the type of control action desired when the process output is in different regions around the set-point, Although the above control objectives are typical of most industrial applications, a conventional PI(D )-controller cannot handle all three of them unless extended by some additional heuristic logic. This was the case described in the previous section. On the other hand, an experienced process operator can easily meet all three control objectives.