ASSESSING THE INFORMATIVENESS OF THE CONTROLLED PARAMETERS IN THE TASK OF IDENTIFYING THE STATE OF THE SYSTEM

terms of fuzzy mathematics, a symmetric criterion is proposed, which is easily calculated. Examples of the criterion calculation are given. The possibilities of increasing the level of informativeness of the criterion using analytical descriptions of membership functions of fuzzy values of the controlled parameter for different states of the system are considered.

Introduction. Assessing the state of the system based on the results of processing a set of controlled parameters is a typical task of everyday practice. The elementary mathematical model of this problem is formulated as follows.
It is assumed that the system can be in one of the many are used. Let from theoretical considerations (or based on the results of processing preliminary observations) a matrix of conditional distribution densities of random values of controlled parameters for possible states of the system be obtained: ( / ), 1, 2, , ; 1, 2, , It is clear that the task of analyzing the results of observations is the simpler, the smaller the number of controlled parameters.
The natural way to reduce this number is to estimate the informational value of the parameters and select the best ones.
Analysis of known results. One of the traditional approaches is the calculation and comparison of the "distance" between the distributions of random values of controlled parameters for various system states using the Kullback measure [1,2]. This measure is introduced as follows. To assess the information value of a specific controlled parameter x used to identify system states, for example, 1 H and 2 H the Kullback numerical criterion is calculated by the formula (1): The approach proposed by Kullback to assessing the informativeness of indicators has obvious shortcomings. The most significant of them is the asymmetry of the introduced ratio, which leads to unpredictable differences in the results of calculating the measure  when the nature of the occurrence of 1 ( x H in formula (1) changes. Secondly, the measure of informativeness of (1) is not normalized. Numerical value is equal to zero, if the distribution densities x H coincide, and can take on an arbitrarily large, not limited from above, value otherwise [3][4][5]. Thirdly, the analytical complexity of the construction of criterion (1) leads in many practical cases to the need to use numerical integration [6,7]. Finally, fourthly, the Kullback measure is designed to distinguish between distributions of random variables and cannot be used directly if the uncertainty of the initial data is described differently, for example, in terms of fuzzy set theory. This situation requires special consideration. The point is the following. The functioning of real systems, as a rule, occurs in changing conditions. At the same time, the mechanism of formation of the observed values of the controlled parameters of the environment and the system changes. As a result, the axiomatic requirements of probability theory are violated. Under these conditions, the level of adequacy of empirical distribution densities obtained by continuous approximation of histograms formed from the results of experiments may turn out to be unsatisfactory. The natural correct alternative is to use fuzzy mathematics formalisms.
In accordance with this, the purpose of the research is to develop a criterion for the distinguishability of distributions of fuzzy values, free from the shortcomings of the Kullback criterion, in the problem of choosing parameters for identifying the state of systems.
Development of a criterion for the distinguishability of distributions of fuzzy values. Let's consider the standard procedure for using some parameter of an object to diagnose its state. Let an object be in one of two states 1 H or 2 H . Let's introduce membership functions of () LR  type of fuzzy controlled parameter x for the states 1 H and 2 H : The simplest special case, when these functions are triangular, is shown in fig. 1. x is divided into subareas:  x . At the same time, for describing the uncertainty of the initial data, it is proposed to use membership functions of () LR  type that are convenient for carrying out calculations. Thus, the task is solved. It is clear that the level of informativeness of the criterion clearly depends on the length of the compatibility interval 1,2 I . Wherein, if the observed value of the controlled parameter turns out to be within this interval, then this fact, in itself, does not contain any information regarding the state of the object. However, this information can be extracted using analytical descriptions of the membership functions of controlled parameters for different states of the object 1 H and 2 H . The desired effect is achieved as follows.
For each of the membership functions of the values of the controlled parameter, we define its probabilistic counterpart. To this end, we determine the areas under the curves given by the functions 1 Let's introduce () ( ) .
The functions given by formulas (4) have all the properties of the distribution densities of random values [8,9]: they are nonnegative and  Now it can be noted that the results of the carried out research for the case when the set of possible states of the object contains only two states can easily be extended to the general case.
Let, for example, in the task of assessing the quality of an object, the following states be possible: At the same time, since then, substituting (10) into (9), we obtain: which corresponds to the Bayesian theorem [10,11]. Thus, relations (8)- (11) provide the possibility of constructive use of analytical descriptions of membership functions of fuzzy values of the controlled parameter from the compatibility interval to identify the probability x . Finally, it should be noted that some additional contribution to the assessment of the informativeness of a controlled parameter can be made by differences in the level of membership functions of the observed value of this parameter for different states of the system.
The direction of further research is the assessment of the informativeness of the controlled parameters in a situation where they are used to evaluate the effectiveness of the system in a multicriteria problem. A possible approach is proposed in [12].
Conclusions. A method for identifying the state of systems under conditions of fuzzy initial data has been developed.
A symmetrical criterion for evaluating the informativeness of the controlled parameters of the system, the values of which are not clearly specified, is proposed. The situation is considered when the parameters of membership functions of a fuzzy controlled parameter are themselves fuzzy. A method for solving the binetch problem that arises in this case is proposed.