S.Sh.Kazhikenova

Cand.Tech.Sci., Karaganda State Technical University

THE ENTROPY-INFORMATION ANALYSIS

OF TECHNOLOGICAL SYSTEMS

Methods of calculation the information suggested by Shannon allow to reveal a ratio of quantity of the predicted information and quantities of the unexpected information which cannot be predicted beforehand, and thus to enable to define a qualitative and quantitative estimation of the certain technological circuit. As a probability of detection of the main element of technological system it is possible to accept its maintenance in a product, expressed in shares of unit. For example, let’s examine the maintenance of a considered chemical element, in our case - copper, in products of technological repartition. Also for probability of detection it is possible to take the maintenance of suitable fraction (remnants, briquettes) in a corresponding product. The same concerns the process of extraction of an element in this or that product, as in this case a parameter of extraction is equal to a probability of transition of the given element from one condition of system into another. These both parameters - the maintenance and extraction - can be equally used for an estimation of quality of a product or technological repartitions. For accounting of a various degree of unexpectedness (probability) of events C.Shannon has suggested to use probabilities' function of entropy borrowed from statistical physics, resulted as follows:

, (1)

where –is a probability of detection of any elements of system, .

The mathematical description of development of any system is set by the formula:

,

where - weight, - number of elements of technological system.

The positive second derivative testifies the accelerated development of the system. The essence of this acceleration is that at transition to a higher structural level of technological process the law or a principle of progressive increase of variety comes into effect. In mathematical understanding the principle of increase of variety means the following: with transition to higher structural levels the number of the elements forming the given structural level, having various attributes, increases under the law:

, (2)

where - number of levels, - length of a code of elements at each level of system.

More strictly this principle will be expressed as follows .

Before the publication of K. Shannon's theory R.Hartly has suggested to define quantity of the information under the formula:

. (3)

The theorem 1 Let - number of elements of - level. - capacity of the information of a zero level of technological system. Then the capacity of the information of -level counting upon one element is expressed by the formula:

.

The theorem 2 Information capacity of hierarchical system and n-level are defined by equality:

, , (3)

where - greatest possible entropy of a system.

The theorem 3 Information capacity of technological system is defined by its stochastic part.

The theorem 4 The limiting degrees of determination and of ineradicable stochasticity of technological system are defined under the formula:

, ,

where - a system determined component, - a system stochastic component, - the system maximal information.

For calculation system and level of stochastic components from a condition formulas (3) are applicable:

, . (4)

On the basis of equality (4) we shall receive formulas for definition of the system determined components:

, (5)

(6)

The maximal information of n -level with account of formula (3) is defined as:

. (7)

Total meaning of the maximal information we shall define under the formula:

. (8)

On the basis of properties of additive in entropy and information and the law of preservation we have:

(9)

For definition of a limiting degree of determination of technological repartition we shall calculate a limit:

. (10)

Degree of ineradicable stochasticity of technological repartition we shall express, using the theorem of addition of probabilities of two opposite events:

. (11)

At substitution of equality (3) in (4)-(8) we shall receive formulas for definition of all kinds of the information of hierarchical system:

,

, ,

, .

Influence of length of a code that is elements of system (target component and the basic impurity) can be revealed in the further researches. As a whole the improvement of quality of a product in process of its technological processing correlates with dynamics of growth of the determined component in abstract hierarchical system that proves the expediency of the further entropy-information analysis of similar systems. Thus, the theorems proved in the given section show indissoluble connection of the determined and stochastic components from which the first is dominating and providing stability, and the second defines the most thin changes and optimum information capacity of technological systems. In this connection we conclude, that the entropy-information approach to research of technological systems is objectively necessary.