Технічні науки/12. Автоматизовані системи управління на виробництві

K.f.-m.n. Shiyan A.A.

Vinnitsa national technical university, Ukraine

Operators in the information space for the problem of control in industrial and organizational systems

Introduction. Control of industrial and organizational systems requires the consideration of the operator – the person who makes decisions. Therefore, the construction of an automated control system must include a model of a person activity. Previously the psychological models for a person activity were used. However, the problem on identification of the psychological model for areal person is no solving.

Thus, the problem construction of the model for description of a person management activity is urgent and has a great future for practical applications.

Literature review and problem statement. It is shown in (Novikov, 2007) that the key point for successful application of modeling human needs to establish correspondence between two spaces: A space that characterizes the strategy of human and space A₀, which describes the results of its operations. In (Shiyan, 2007) the Information Space is constructed and the definitions for Information Space and Control are presented. Information Space structures a database. It contains the eight components of information.

The article aims is to develop the mathematical apparatus for the operators in Information Space, which one can to interpret as a types of a person activity.

Information space and modeling of the person activity.

Definition 1. Transformation (change) of characteristics for components of information for the Information Space (IS) of a problem can be called a person activity.

Let’s have a look at the object which can perform a person activity in terms mentioned above. It is possible to give to it such a definition.

Definition 2. An object which accepts (identifies) components of information (from the corresponding information space of an economic problem) about stages and \ or processes in reviewed industrial and organizational system and which can transform (change, convert) stages and\or processes in it is called an information operation (IO).

Note. It should be stressed (emphasized) that IO used, in general case, two different information spaced. The first information space it “creates” before taking a decision (before a choice of strategy for a person activity), and this information space plays in it programming role. Another information space IO creates after performing by it a person activity. As a result, IO is an automorphizm for an information space for a problem (invariant with the precision to the choice of economic object and to a variety of choices for strategies in each person activity).

In a given definition is clearly stressed an ability of a subject (that performs a person activity) to a change of stages and\or processed in industrial and organizational system. Finally, AIO can be observed as an object which has such structure.

<block of perceiving (software)½block of activity>

Constructed in such a way IO by its first block (a block of software) receives certain components into components of information (generally speaking – different ones) as a result of its activity by another block (a block of activity). Thus, IO is an object which fulfills a set of methods (algorithms, regimes, models, means, technologies etc.) for fulfillment of a person activity.

Abstract informational operator in the information space. Presented above IO in a mathematics sense can be studied as an operator in IS.

For this purpose it can be used a basis in IS. Arbitrary information about an economic system can be presented in such way.

(1)

Here i_k – basis vector of space component information, which determine the names of component information; I_k – characteristic (data) which can be referred (attributed) to this component of information (for example it can be a database that attributes to this component).

Equation (1) has such a meaning. I_k presents by itself a database which refers ratio to k component of IS. It can be noted that I_k is not a number, and as a result of this an operation “step by step addition” has to be determined as an integration of two homogeneous (i.e. the ones which describe one and the same component of information) databases into one (similar elements are taken into account only once). “Step by step subtraction” is determined analogically. An operation of multiplication of a number which is necessary for creation a linear space corresponds to the change of scope for units of measurement while describing a database. In this sense an entry (1) presents a certain generalization of a linear space.

Using (1) a person activity of the agent can be described in a way of operator G that changes IS I_before for a researched problem (which had been before performed a person activity), in IS I_after for the same problem (but which appears already after a fulfilled a person activity). It can be written in the following way.

(2)

Note. Determined in (2) operator G has such a characteristic: if an information space is divided into two sub-spaces I_b1 and I_b2, which do not cross, than G(I_b1+I_b2)=G(I_b1)+G(I_b2) (operation of an addition is commutative). This feature is a result of a case that an answer (solution) to an aggregate of tasks where each of them is received by means of decomposition of a main (difficult) task in mutually correlated parts (when each of them can be resolved independently) is equivalent to an opening of a hard task. But it means that sub-spaces I_b1 and I_b2 of information space to not cross (i.e. to not have common points).

This feature can be studied as a reflection of the case that a choice of strategy by a player is done independently for each of the parts of space activity A in a situation when each of sub-sets doesn’t have common parts.

As a result of this if IS I_before is divided into a direct sum of sub-spaces.

, (3)

then an operator G acts in the following way:

. (4)

In (4) is a different way of a record (1).

From (4) it becomes clear that the operator G can be represented as a tensor operator which has n “low” and m “upper” indices. As the same time because of availability in the information space a basis, a quantity of “upper” and “low” components with a tensor G_(n)^(m) is limited to 8: n,m£8.

It can be agreed for easiness that “low” components correspond to components of information for IS I_before, and “upper” to I_after.

Then (2) can be shown in such a way:

(5)

Using feature (3) and (4) it is possible to make a conclusion that activity of any tensor operator G_(n)^(m) can be represented in a way of sum of operators which have a form gⁱ_k (operators in this sum are commutative).

This statement can be formulated in a way of the theorems.

Theorem 1. For creating a model for performing any person activity it is necessary to have only such IO which are being programmed by only one component information (from IS I_before) and activity of which can be expressed also in a change of only one component information (from IS I_after).

It means that for such IO resulting change of IS because of the result of their activity (transformations I_before to I_after ) lies in change of IS I_after of only one component (in comparison with IS I_before).

Theorem 2. Operator gⁱ_k has a feature to be commutative g_i^k+g_n^m=g_n^m+g_i^k and associative g₁+(g₂+g₃)=( g₁+g₂)+ g₃.

According to theorem 1 and 2 each operators that corresponds to AIO can be expressed as a sum of certain “binary” operators which connect between each other only two components information: one is from a space I_before, and another one – from space I_after.

(6)

Theorem 3. General quantity of operators gⁱ_k counts 64 different variants.

Binary operator gⁱ_k may have only one out of eight component from IS I_before and only one out of eight component from IS I_after. A number of different possible variants can be 8´8=64.

Definition 3. IO is called double component (abbreviated – 2IO) if it corresponds to the following conditions:

1. Each 2IO has at the input only one component information and at the output – also only one.

2. For each 2IO one component information describes statics and another one – dynamic quality.

3. For each 2IO one component information is general and another one – detailed.

Correct definition 2IO as an object that fulfills these or other means (types, algorithms, methods, ways, etc.) of a person activity can be only like that.

Theorem 4. Not resisting ruling can be done only by an aggregate of 16 types of 2IO.

Theorem 5. For fulfilling an arbitrary economic behavior in an arbitrary information space it is necessary and enough to have 16 types of 2IO.

References.

1. Novikov D.A. (2007). The theory of managing of organizational systems. Мoskow, Fizmatlit. (in Rusian).

2. Shiyan A.A. (2007). Economical Cybernetics. Lviv, Publ. House “Magnolia-2006”.