# 6.9.6. A genuinely optimum fire bucket - часть 7

3) The rate of change of the account (the differential component, showing whether the client's business is on the rise or decline);

It's possible to take into account other factors, but three is a nice number!

The PID-control algorithm has imperceptibly become something of a fetish. Fuzzy management cames as a fresh breeze in the theory of automatic control, whose basic rules are now open to review. To be sure, there are other opinions. Some scientists believe that the use of FST in automatic control, and cybernetics in general, is just replacing one uncertainty with another ("trying to stitch soapsuds", the Russian expression goes) and that's all there is to it. Sceptics explain away the observed improvement in control quality to our devoting more attention to regulator technology (as if the ritual of attention could improve performance). Besides some researchers believe that FST, since it dates back 30 years (it was devised by the Iranian-American Dr Lotfi Zadeh at UC/Berkeley in the 1960s) is old and best forgotten.

Actually, the skip from a task about a tripartite duel to one about an optimum bucket wasn't entirely casual. Traditionally, precise sets are illustrated by circles with sharply delineated borders. Fuzzy sets are drawn as circles formed of separate dots, with a high density at the centre and thinning to zero (as if evaporating) towards the edges. Such 'fuzzy set' images can be seen on a firing range wall where targets are hung. The bullet traces form a probability distribution, whose mathematics is well-known. It appears that the theory for working on fuzzy sets, as probabilistic distributions, has already existed for a long time...

We keep talking about fuzzy sets. But are they – mathematically – actually sets? To be consistent, it's necessary to ascertain that the fuzzy set has elements (fuzzy subsets, fuzzy sub-subsets, etc). We'll return to a classical example: a heap of grain. An element of this fuzzy set will be, say, a million grains. But a million grains is not a precise element: it's a fuzzy subset. If you count out grains, whether manually or automatically), it's no wonder that you might mistake, say, 999 997 grains for a million. In FST terminology, you could say that the element 999 997 has a 'membership value' 0.999997 for the fuzzy set {a million grains}. Besides, even "a grain" is not a precise element, and is another fuzzy subset: it might be a high-grade grain, but there are also underdeveloped grains, grain fragments, and bits of husk. Depending which way you decide, you might count one grain as two, or vice versa.