RU2758638C1 - Method for monitoring and controlling a stochastic system with an unbiased asymmetric target density of the output parameter distribution - Google Patents

Abstract

FIELD: monitoring systems.SUBSTANCE: invention relates to methods for organizing monitoring and control over a stochastic system with an unbiased asymmetric target density of the output parameter distribution and can be used in monitoring and control systems of various dynamic systems with chaotic properties. Based on the registration of the state of the object and the formation of the values of the sample parameter, the coordinates of the entropy-parametric space of shape features for the asymmetric unbiased density of the output parameter distribution are determined. To ensure the uniformity of optimal control at various points of the entropy-parametric space, the formation of the space of the optimal state of the system is carried out. The unification of data near various optimal states of the system when choosing the model form is achieved by setting the boundary of the control zone, determining the normalized position coordinates for the distribution of the output parameter in the space of signs of the optimal state of the system, determining the criterion for the state of the system and determining the coordinates of the position of possible models in the space of signs of the optimal state. Minimization of the mismatch of the entropy-parametric potential of the asymmetric distribution of the output parameter of the stochastic system relative to the optimal state is provided by means of a targeted correction of the distribution law and the formation of a control action to correct the connections of the stochastic system, which provides control of the scale of the distribution model of the system. Continuous recording of the parameters of the output parameter distribution preserves the possibility of restoring the state of the system during analysis. Visualization of processes is achieved by mapping the state of a stochastic system in the entropy-parametric space of features of the output parameter.EFFECT: increase in the stability of the functioning of a stochastic system by monitoring and controlling the shape and scale parameters of an unbiased asymmetric distribution density of the output parameter.1 cl, 13 dwg

Description

The invention relates to methods for organizing monitoring and control over a stochastic system with an unbiased asymmetric target distribution density of the output parameter and can be used in monitoring and control systems of various dynamic systems with chaotic properties.

There is a known method of control by a stochastic process, which consists in the selection of the n-dimensional vector of the output parameter Y, the determination and minimization of the value of the standard deviation (RMSD) of the output parameter. A typical example of the implementation of the known method is the problem of determining the optimal transfer function of the system that minimizes the variance or root-mean-square error of the output parameter. The methods of optimal filtering of vector random signals are implemented when solving problems of constructing an optimal Wiener filter and problems of constructing Kalman filter systems [1, 2].

A device for measuring the characteristics of a random process is known, in which an increase in the reliability of determining the form of a symmetric distribution law of the probability of random parameters is ensured by determining the entropy coefficient and the value of the counter kurtosis [3]. Despite the fact that this device, due to the procedures for comparing the coefficient of entropy and counter-excess, makes it possible to establish normal, triangular, arcsinusoidal, trapezoidal, antimodal and uniform distributions, for all other forms the device issues the message “distribution is not defined”. In the device, due to the inclusion of the procedure for comparing the characteristics of asymmetry, the selection of an asymmetric distribution is realized without establishing its form.

There is a known method for controlling the conditional distribution density of the output signal for generalized nonlinear and non-Gausic stochastic systems [4], in which optimal control is achieved by embedding a recursive formula into a dynamic system and minimizing the difference between the conditional output and target probability density. In this case, a feature of the control consists in using a partial approximation of the trajectory as the target distribution density. The main disadvantage of the control consists in replacing the real probability density with its linear or parabolic approximation. In this case, the distribution law of the parameter remains unknown.

The closest to the proposed invention is a method for monitoring and controlling a dynamic system [5], which consists in the fact that the state of the object is registered, the sampling y _{i of the} values of the output parameter y (t) is formed, the mathematical expectation M, the standard deviation σ and the fourth central moment μ _{4 of the} distribution of the output parameter, the entropy potential of the output parameter Δ _Hs ; formation of a database of reference parameters of the distribution law of the output parameter, checking the state of the object belonging to the area of the optimal state; transformation of the distribution law of the controlled parameter by changing the settings of the regulator parameter, adjusting the distribution law and forming the control action.

As follows from the claims, in the known method for monitoring and controlling a dynamic system, the entropy coefficient of symmetric distribution and counter-excess are determined; determination of the entropy-parametric criterion for the area of optimal control; determination of the value of the entropy-parametric potential of symmetric distributions; minimization of the magnitude of the entropy-parametric potential of the dynamic system and correction of the real parameter of the distribution law of the output parameter [5].

The disadvantages of this method of monitoring and controlling a dynamic system should be noted:

- limiting the models of the system's behavior only by a set of symmetric forms of the distribution of the output parameter;

- the lack of a way to select the form of asymmetric distribution, which affects the degree of uncertainty of the state of the system;

- lack of assessment of the boundary of the optimal zone;

- the lack of an algorithm for choosing model parameters such as shape parameters and parameters of the scale of the distribution of the output parameter.

Brief Description of Drawings

Figure 1 shows an example of control of the conditional distribution density of the output parameter for a generalized nonlinear and non-Gaussian stochastic system.

Figure 2 shows a diagram of a process that implements a known method of monitoring and control of a dynamic system.

Figure 3 shows a topographic diagram illustrating the state of the dynamic system of the output parameter when implementing a known method of monitoring and control.

Figure 4 shows drawings of models for setting entropy and parametric uncertainties for symmetric and asymmetric distributions:

a) uncertainty model for symmetric distribution;

b) uncertainty model for asymmetric distribution;

Figure 5 shows a plot of the coordinate space of features, illustrating the state of the system with an unbiased asymmetric target density of the distribution of the output parameter.

Figure 6 shows a diagram of a process that implements the proposed method of monitoring and control over a stochastic system with an unbiased asymmetric target distribution density of the output parameter.

Figure 7 shows a diagram of the process of forming the space of the optimal state of the stochastic system.

Figure 8 shows a diagram of the process for determining the scale of the model of the distribution of the output parameter for a stochastic system

Figure 9 shows the projection of the control zone onto the normalized coordinates of the asymmetry and the entropy coefficient in the normalized space of signs of the optimal state of the system.

Figure 10 shows the projection of the control zone on the normalized coordinates of kurtosis and the entropy coefficient in the normalized space of signs of the optimal state of the system.

Figure 11 shows the projection of the control zone on the normalized coordinates of asymmetry and kurtosis in the normalized space of signs of the optimal state of the system.

Figure 12 shows the projection of the space of entropy and parametric uncertainties of asymmetric distributions.

Figure 13 shows a functional diagram of a device for monitoring and controlling a stochastic system with an unbiased asymmetric target distribution density.

Detailed description of the invention

The material of the detailed description explains the embodiments of the invention with reference to the drawings, where like reference numbers represent the same or similar elements.

At the present stage of development of technology, a relatively new direction of control for a class of nonlinear stochastic systems is the problem of tracking the track of the distribution density with respect to the objective function of the distribution density of the output parameter [6]. For such systems, the purpose of the control is to minimize the difference between the partial approximation of the distribution density of the controlled parameter at the output of the system and the approximation given as the target distribution density. Among the features of the control, it is noted the need to select the current value of the controlled parameter so that the piecewise approximation of the distribution density of the output parameter can follow the target distribution density [7].

Figure 1 shows an example of control of the conditional distribution density of the output parameter for a generalized nonlinear and non-Gausic stochastic system. When controlling the non-Gaussian conditional distribution density, the real distribution density function of the output parameter is replaced by its nonlinear model in the form of a power series on a limited interval of values of the observed parameter y. In this case, the approximation of the target distribution density is specified up to the fourth order from the condition that the Gaussian distribution is taken as the desired distribution of the output parameter.

Among the disadvantages of the known control method, the sensitivity of the distribution track to the current values of the output parameter should be noted, due to the replacement of the real distribution density of the parameter with long tails by its model, which approximates the distribution over a finite interval using a symmetric fourth-order power series. With this approach, the emergence of instability in the track of the distribution density of the output parameter is due to the finite probability of the appearance of values on the tails of the real distribution, which are significantly distant from the center of the distributions of the random values of the output parameter. For this reason, to ensure stability in such systems, control schemes are used that implement recursive methods for choosing the current value of the output parameter, i.e. replacing the real value with the result of the recursive algorithm to provide the target symmetric distribution model.

As a lack of control, one should also note the lack of information about the change in the form of the adopted model of the stochastic system, since the adopted approximation of the objective function is valid for a wide variety of models of non-Gausic stochastic systems.

The use of high-order moments in order to obtain the shape of the distribution also leads to an unstable result [6]. The main disadvantage of using probabilistic features for analyzing distributions is the tendency of distributions to cluster by approximately one line to a greater extent than is acceptable for real detection of variability of sample observations belonging to different types of distributions [8]. For this reason, in the known systems for the analysis of distributions, after an approximate identification of distributions, a distribution is selected by establishing the validity of distributions using specialized criteria [9, 10].

There is a known method for monitoring and controlling a dynamic system based on the analysis and target change in the value of the entropy potential Δ _{H of the} output parameter, which is equal to the product of the value of the standard deviation σ by the entropy coefficient K _H [11]:

characterized in that the minimization of the value of the entropy potential of the output parameter of the system is achieved by transforming the distribution law of the controlled parameter by changing the parameters of the controller. In this case, the degree of transformation of the distribution law of the controlled parameter is estimated by the value of the entropy coefficient. The value of the entropy potential Δ _H is defined as half of the range of uniform distribution in the interval from (-Δ _H ) to Δ _H , which has the same entropy as a particular parameter.

The disadvantage of the method of control and management based on the analysis and target change in the entropy potential is the use of the entropy coefficient K _H to control the degree of transformation of the distribution law. In this case, the system, when the entropy potential is minimized, can pass due to a change in the entropy coefficient into a low-entropy state, in which the signal parameters remain unchanged [12].

The development of the method for controlling the entropy potential is a method for controlling and managing the state of uncertainty of the system, based on the analysis and purposeful change in the value of the complex entropy potential L _{ΔH of} the system parameter, equal to the ratio of the product of the entropy coefficient K _H and the standard deviation to the base value X _B , relative to which the state is considered system uncertainties [13]:

The mathematical expectation of the parameter, its range, nominal value or the basic value of the entropy potential is used as a base value. Control is achieved by determining the complex entropy potential after each stage of the evolution of the system and controlling the system by the magnitude of the increment or differential of the complex entropy potential by changing the settings of the base value that specify the range of parameters, limit or nominal values, or the base value of the entropy potential.

The main disadvantage of the method for monitoring and managing the state of uncertainty of the system, based on the analysis and purposeful change in the value of the complex entropy potential L _{Δ of the} parameter, is to determine the entropy potential proportional to the entropy coefficient K _H , the value of which is determined by the distribution law of the parameter. The fact is that at a fixed value of the entropy coefficient, a multitude of both symmetric and asymmetric distributions with different forms and properties is possible [14, 15]. For this reason, the use of the base value X _B when monitoring the complex entropy potential L _ΔH does not guarantee the preservation of the shape of the distribution of the output quantity.

A known method for monitoring and controlling a dynamic system, based on minimizing the entropy-parametric potential of the symmetric distribution of the output parameter in the feature space of the entropy coefficient and counterexcess. A diagram of the process of steps for a known method of monitoring and controlling a dynamic system is given in figure 2, where the following notation is given: step 205 - the step of registering the state of the object; step 210 - forming a sample of values of the output parameter; step 215 - determining the mathematical expectation, standard deviation and fourth central moment; step 220 - forming a database of reference parameters of the distribution of the output parameter; stage 225 - determination of the entropy-parametric criterion of the area of optimal control; step 230 - checking the state of the system for belonging to the area of optimal control; stage 235 - transformation of the distribution law; step 240 - determining the entropy-parametric potential of the symmetric distribution; step 245 - minimizing the value of the entropy-parametric potential of the symmetric distribution; step 250 - adjusting the distribution law; stage 255 - the formation of a control action; step 260 - the state is optimal.

According to the process diagram, at step 205, the state of the object is registered, including pre-amplification, change and digitization of the output parameter of the system y (t), by registering the value of y _i at known discrete time intervals Δt. At the next step 210, a sample of the values of the output parameter y (t) is formed to construct a histogram and to estimate the probability of observation p _{j of the} recorded values y _j in the interval [y _j , y _j + Δy].

When monitoring and controlling a stochastic system, a continuous process of forming a sample of the values of the output parameter is required, which retains the entropy and parametric properties at any time. The preservation of the properties of the sample at any moment of the functioning of the system is achieved by observing during the time cycle T _c , characteristic of the system, which has chaotic properties, for example, the cycle of the engine, the cycle of heart contraction, and others. The number of values n of the formed sample is equal to the ratio of the system cycle period to the time interval of one measurement Δt _and .

по формулам:The distribution parameters of one cycle contain information about both the internal state of the system and external influences. Therefore, at the next stage 215 for the formed sample of values, the mathematical expectation M ^* , the standard deviation (RMS) σ ^* and the fourth central moment are determined

by the formulas:

To carry out the process of statistical measurement and control, it is necessary to estimate the mismatch of the sample distribution parameters with the reference parameters of the output parameter. For this reason, at block 220 of the circuit in FIG. 2, a database of reference parameters of the distribution of the output parameter is generated. At this stage, a reference model is selected from the base of distribution models, the parameters of which are taken as reference parameters for the optimal state of the system.

Since the steady state of the system is possible in a limited area near its optimal state, then to control the steady state of the system at step 225, the entropy-parametric criterion of the area of optimal control is determined by the formula:

where κ and K _Hs are the counterexcess and the entropy coefficient for the symmetric distribution of the output parameters, determined by the expressions:

Here Δ _Hs is the entropy potential of the symmetric distribution:

At step 230 of the flowchart of FIG. 2, the system state is checked to see if it belongs to the optimal control region. If the criterion γ exceeds its maximum value γ _max , then the dynamic system has left the area of optimal control and for further control it is necessary to take strict measures to ensure that the system returns to the area of the optimal state. To do this, it is necessary to determine the possible transformation of the distribution law of the output parameter and to form a control action for the object control bodies. The transformation of the distribution law of the controlled parameter is carried out at step 235 of Fig. 2 of the process diagram by changing the tuning parameters of the controller. If the dynamic system is in the zone of optimal control and management, then the subsequent optimization of the state of the system is carried out by minimizing the entropy-parametric potential of the symmetric distribution Δ _НРs . To do this, at step 240 of the scheme in figure 2, the entropy-parametric potential of the symmetric distribution is determined by the formula

- коэффициент нормального стандартного распределения.where

- coefficient of normal standard distribution.

The process of minimizing the magnitude of the entropy-parametric potential of the symmetric distribution of the dynamic system is illustrated by stage 245 of the process diagram in Figure 2 in accordance with the condition:

If the entropy-parametric potential of the symmetric distribution Δ _HPs does not reach its minimum value, then the analysis of the required correction of the distribution law is performed at step 250, based on the results of which at step 255 a control action is generated on the object's controls. When the minimum of the entropy-parametric potential is reached, the monitoring and control system takes the state of the object as the optimal state and retains the settings of the previous control cycle. The transition of the dynamic system to the optimal state in figure 2 of the process diagram of the method for monitoring and controlling the dynamic system corresponds to step 260 "State is optimal"

The peculiarity of the implementation of the known method of monitoring and control over a dynamic system is illustrated by the topographic diagram in figure 3, which shows:

310 - position point of the optimal state of the system;

315 - curve, limiting the area of space, which corresponds to the position of the system at its optimal functioning;

320 - curve (dotted curve), limiting part of the space of the most probable positions for optimal states of the dynamic system;

325 - the curve of the position of the system with exponential distributions of the output parameter with form indices from 0 to ∞;

330 - curve of the position of the system with bimodal distributions of the output parameter, the occurrence of which is due to the appearance of various forms of hysteresis;

335 - curve of the position of the system with the distribution of the output parameter of the Chapot type with a shape index equal to 0.5;

340 - curve of the position of the system with an exponential distribution with a shape parameter equal to 0.33;

345, 350 - curves of the position of the system with compositions of discrete and exponential distributions of the output parameter;

355, 360 - curves of the position of the system with compositions of exponential and discrete two-valued distributions;

365 is the point of the position of the system with a normal distribution of the output parameter (belongs to curve 325);

370 is the point of position of the system in the two-sided exponential Laplace distribution (belongs to curve 325).

375 - point of position of the system with a uniform distribution of the output parameter (belongs to curve 325);

380, 385 - point of position of the system with a two-sided exponential distribution of the output parameter with form indices 1/2 and 1/3, respectively (belong to curve 325);

390, 395 - points limiting the portion of the curve for the positions of the system with arccosine distributions;

All curves and points of the position of the system, illustrated in the topographic diagram of the known method of monitoring and controlling a dynamic system, correspond to symmetric distributions. Any mapping of the distributions to the counter-excess plane and the entropy coefficient of symmetric distributions can be replaced by a symmetric distribution model. For this reason, the method for monitoring and controlling a dynamic system in accordance with the diagram in figure 2 is formally reduced to monitoring and controlling a system with a symmetric distribution density of the output parameter. When controlling and monitoring a system with an asymmetric distribution, an implicit replacement occurs with the help of a corresponding symmetrical analogue.

To control the shape of symmetric distributions when implementing the known method for monitoring and controlling a dynamic system, the entropy-parametric criterion of the optimal control area is determined at stage 225 of the scheme in Figure 2 in the space of signs of symmetric distributions. This approach provides effective control of the state of the object with a symmetric distribution density of the output parameter.

When monitoring an asymmetric distribution, the state of the output parameter is displayed on the plane of symmetric distributions of the output parameter, which causes the appearance of ambiguity in the behavior of the object. The fact is that the entropy coefficient of a symmetric distribution, equal to the ratio of the entropy potential calculated by the formula of a symmetric distribution to the standard deviation of an asymmetric distribution, when applied to a symmetric distribution expresses the ratio of uncertainties of various intervals and becomes strongly dependent on the parameters of the distribution scale. This way of implementing control and monitoring creates conditions under which the entropy coefficient, calculated by the formula of symmetric distribution, corresponds to many forms of asymmetric distribution, which makes the choice of the form of asymmetric distribution density of the output parameter multivalued.

Thus, the known method for controlling the distribution is limited and comes down to controlling the shape of the approximating symmetric distribution model. When monitoring a system with asymmetric distributions of the output parameter by replacing the real asymmetric distribution of its approximate symmetric model, the system transitions to inoperative states. The fact is that with the same model of symmetric distribution, the multivaluedness of the choice of asymmetric models remains.

Control value of asymmetrical unbiased distributions

In many areas of technology, control of the shape of the asymmetric unbiased distribution density of the output parameter is an effective method for analyzing and monitoring a stochastic system. As an example, the following areas of application of the present invention can be cited.

1. Analysis of the results of measuring the volumetric activity of radon [16, 17]. 2. Models for monitoring the distribution of harmful substances in the environment [18, 19, 20]. 3. Monitoring the reliability of technical systems, instruments and devices [21, 22, 23]. 4. Monitoring and control of material characteristics: creep, force of rupture or fracture, material fatigue [24]. 5. Resource analysis [24]. 6. Survival probability distribution models [25, 26, 27]. 7. Monitoring the distribution of income of firms and individuals in statistical models of the economy, stock market models [28]. eight. Statistical analysis in the processing of magnetic resonance images in computed tomography [29, 30]. nine. Radionuclide studies and diagnostic analysis of nuclear
medicine [31, 32]. ten. Modeling the kinetics of pharmacological drugs in functional studies [32, 33, 34]. eleven. Modeling biokinetic processes [35] 12. Management of descriptor systems [36].

For this reason, from the variety of stochastic systems, one should single out systems in which the control task is to minimize the deviation of parameters for an asymmetric unbiased distribution density of the output parameter

The author of the proposed invention believes that control of the shape and scale of distribution models in the entropy-parametric space of signs of asymmetric distribution of the output parameter will ensure the stability of monitoring and control of the stochastic system and exclude the possibility of its transition to non-working states due to targeted adjustment of the distribution law and the formation of control actions for correcting connections stochastic systems.

It is obvious that it is necessary to display models of states of a stochastic system in the entropy-parametric space of signs of asymmetric distributions of the output parameter to ensure an unambiguous separation of stable operable and unworkable states of the system.

The proposed invention is aimed at optimizing the monitoring and control of a stochastic system with an unbiased asymmetric target distribution density by ensuring that the shape and scale of the stochastic system model correspond to its real state in the entropy-parametric space of features of the asymmetric distribution of the output parameter.

For this, in the method of monitoring and control by a stochastic system with an unbiased asymmetric target density of the distribution of the output parameter, which consists in the fact that the state of the stochastic system is registered, the sampling y _{i of the} values of the output parameter y (t) of the stochastic system is formed, the mathematical expectation M ^* , the mean square deviation σ * and the central fourth moment μ _{4 of the} distribution of the output parameter; formation of a database of reference parameters of the distribution law of the output parameter of the stochastic system, checking the state of the stochastic system belonging to the control zone of the optimal state; transformation of the distribution law of the controlled parameter by changing the tuning parameters of the controller, correcting the distribution law and forming the control action, characterized in that they additionally determine the coordinates of the space of shape features - asymmetry, kurtosis and the entropy coefficient of an asymmetric distribution - for an asymmetric unbiased distribution density of the output parameter of a stochastic system by the matrix vector formula:

- второй начальный момент, третий центральный момент
и энтропийный потенциал несимметричной несмещённой плотности распределения выходного параметра, определяемые по формулам:where

- second starting moment, third central moment
and the entropy potential of the asymmetric unbiased distribution density of the output parameter, determined by the formulas:

where m is the number of grouping intervals; Δy is the grouping interval of the output parameter; p _j is the statistical frequency of counts falling into the j-th grouping interval;

formation of the space of the optimal state of the stochastic system, which includes the following actions:

- setting the border of the control zone in the space of signs of the optimal state with an asymmetric distribution of the output parameter of the stochastic system:

where γ _max is the radius of the control zone; ξ _iγ - coordinates of the control border;

- determination of the normalized position coordinates for the state of the distribution of the output parameter in the space of signs of the optimal state of the stochastic system according to the matrix vector formula:

where Sk ₀ , Ex ₀ , K _Hn0 - asymmetry, kurtosis and entropy coefficient of asymmetric distribution density for the optimal state of the system; a, b, c - coefficients of the intervals of signs for the control zone;

- determination of the criterion for the state of the stochastic system in the space of signs of the optimal state according to the formula:

- determination of the normalized coordinates of the positions of possible models of the stochastic system in the space of signs of the optimal state according to the formula of the matrix vector:

where Sk _s , Ex _s , K _Hns - asymmetry, kurtosis and entropy coefficient of asymmetric distribution density for the optimal state of the system; s - model number of the stochastic system:

determination of distances between the positions of possible models and real states in the normalized centered space of signs of the optimal state of a stochastic system according to the formula

choice of the form of the stochastic system model from the condition of the minimum distance between the positions of the model and real states

determination of the scale of the distribution model of the output parameter of the stochastic system, which includes the following actions:

несимметричной плотности распределения выходного параметра стохастической системы- determination of the entropy-parametric potential

asymmetric distribution density of the output parameter of the stochastic system

- formation of the entropy-parametric potential of the asymmetric model of the distribution density of the output parameter of the stochastic system

where m _2λ and K _Hnλ are the second initial moment and the entropy coefficient of the asymmetric model of the distribution of the output parameter;

- determination of the scale parameter from the condition of minimizing the difference in entropy-parametric potentials for the asymmetric model and the real state of the distribution density of the output parameter of the stochastic system

storing the parameters of the stochastic system model in the database;

determination of the mismatch of the entropy-parametric potentials of the model of the real and optimal states of the stochastic system

minimizing the mismatch of the entropy-parametric potentials of the model relative to the optimal state by means of targeted correction of the distribution law and the formation of a control action for correcting the connections of the stochastic system and mapping the state of the stochastic system in the entropy-parametric space of the output parameter features.

The introduced actions provide monitoring and control over the stochastic system with an unbiased asymmetric target distribution density of the output
parameter by

- determination of the position of the system in the entropy - parametric space of the features of the form of the asymmetric density of the distribution of the output parameter;

- formation of the space of the optimal state of the system to ensure the unity of the optimality of control at various points of the entropy-parametric space;

- selection of the form of the model of the distribution of the output parameter from the condition of the minimum distance between the position of possible models and real states in the normalized centered feature space, the space of the optimal state
systems;

- minimization of the difference between the entropy-parametric potentials of the unbiased asymmetric model and the real state of the distribution density of the output
parameter to define the parameter of the scale of the model;

- minimization of the mismatch between the parameters of the model and the parameters of the optimal state of the distribution of the output parameter by means of targeted correction of the links of the stochastic system.

In the proposed application, the author uses a coordinate space, built simultaneously on probabilistic and information signs of distributions, to display and select the shape of models of asymmetric distributions of the output parameter. In this case, the first and second signs of the distribution form are taken as probabilistic coordinates: asymmetry and kurtosis, which characterize the skewness and peakedness of the distribution. For the formation of information coordinates, the author of the present invention proposed the coefficient of entropy of asymmetric distributions, which characterizes the property of disorder with an asymmetric distribution of the output parameter. [15]. The peculiarity of constructing the entropy coefficient of an asymmetric distribution is illustrated in Figure 4, which gives drawings of models for specifying entropy and parametric uncertainties for a symmetric and asymmetric distribution.

The entropy of information was used by K. Shannon as a measure for measuring the amount of information in an ensemble of discrete messages. If the source of information produces a random sequence of independent messages x _i with probability P (x _i ), then the entropy of the source of messages X is determined by the mathematical expectation of the logarithm of the probability of messages using the expression [37]:

For a continuous distribution density of the parameter y, the expression for calculating the entropy of the message source X has the form

The Chenonne uncertainty model for symmetric distributions was proposed in the works of P.V. Novitsky. [14, 38] as a model for an independent description of the measurement process. According to the model of Novitsky P.V. the conditional entropy of measurement, provided that the reading x _{p is} obtained with a uniform distribution of uncertainty over the interval 2Δ is equal to the logarithmic measure of the entropy interval of uncertainty 2Δ:

Then the entropy measurement error Δ, equal to half of the uncertainty interval 2Δ and expressed through the conditional entropy of change, provided that the reading x _{p is} obtained, is given by an expression of the form:

In control and management systems of an object, the minimization of the difference in the mismatch between the output parameter and the measure is ensured by constructing a negative feedback. As a measure of the uncertainty of the state of the object, V.L. proposed to use half of the entropy uncertainty interval to construct control over the object and consider it as the entropy potential of the object [9].

A drawing of the uncertainty model for the symmetric distribution of the mismatch of the output parameter is given in figure 4, a, where the following notation is used.

410 - symmetrical uniform distribution model;

415 and 420 - parametric and entropy uncertainty intervals for a symmetric distribution model

The entropy coefficient of the symmetric distribution is introduced as the ratio of the entropy interval 420 of the uncertainty of the mismatch of the output parameter to its parametric interval 415, equal to the doubled values of the entropy potential of the symmetric distribution 2Δ _Hs and the standard deviation 2σ. The formula for determining the entropy coefficient of a symmetric distribution is:

The entropy coefficient of the symmetric distribution K _Hs , equal to the ratio of the entropy potential of the symmetric distribution of the output parameter to the standard deviation of the output parameter relative to the mean value, is a dimensionless normalized feature of the symmetric distribution model used to estimate the position of the mathematical expectation. The entropy coefficient of a symmetric distribution, as well as the entropy potential of a symmetric distribution, is based on the conditional entropy of the distribution, provided that an estimate of the mean value is obtained, equal to the mathematical expectation, the uncertainty of which is modeled by a uniform symmetric distribution.

A distinctive feature of distributions with asymmetric unbiased distribution density is that the probability density of the distribution is zero for negative values of the output parameter. Since the limit expression under the Shannon integral (20) is equal to zero,

then to determine the entropy of an asymmetrically distributed unbiased output parameter, an expression of the form

When using the uniform distribution density as a model of the unbiased asymmetric distribution density of the output parameter, the interval is set from 0 to its maximum value Δ _Hn , taken as the entropy potential of the asymmetric unbiased distribution. A drawing of the uncertainty model for an asymmetric distribution is shown in Figure 4, b, where the following notation is used:

425 - asymmetric uniform distribution model;

и энтопийному потенциалу Δ_Hn несимметричной модели распределения;430 and 435 - intervals of parametric and entropy uncertainty for an asymmetric distribution model, equal to the square root of the second initial moment

and entopic potential Δ _{Hn of the} asymmetric distribution model;

440 - the distance of the position of the distribution center relative to the origin, equal to the mathematical expectation M;

From the consideration of figure 4, a and figure 4, b it follows that the intervals of parametric uncertainty 415 and 430 set different essences of the distributions.

From the consideration of figure 4, a it follows that the interval 415 equal to 2σ determines the root-mean-square spread of the distribution relative to the center of the distribution, given by the position of the mathematical expectation M. From the consideration of figure 4, b it follows that the interval 430 sets the distance from the origin to the root-mean-square value. The interval 430 is equal to the square root of the second starting moment of the asymmetric distribution. For an unbiased skewed distribution, the origin coincides with the origin. For an asymmetric distribution, the interval of parametric uncertainty 415, equal to twice the root-mean-square spread 2σ of the output parameter, is plotted symmetrically about the center of the distribution, which is set by the distance 440, postponed from the origin and equal to the mathematical expectation M. As follows from the model of figure 4, b, the distance 415 and 430 have different entities for the distribution of the output parameter.

The intervals 420 and 430 for entropic uncertainty are considered similarly. The interval of entropy uncertainty 420 in figure 4, a is plotted symmetrically, relative to the position of the center of the distribution, given by its mathematical expectation. The interval of entropy uncertainty 435 in figure 4, b is plotted from the origin of the asymmetric distribution.

The differential entropy for determining the uncertainty of the position of the center of an asymmetric distribution is calculated as the conditional entropy of measurement, provided that the reading x _p is equal to the mathematical expectation M:

Since the conditional entropy (32) for determining the interval of entropy uncertainty relative to the center of distributions is not equal to the entropy of the asymmetric distribution (31), the intervals of entropy uncertainty for the symmetric model 420 and for the asymmetric model 435 also have different values. Spacing 420 is relative to the center of the distribution and is offset from the origin.

Thus, the intervals of entropy uncertainty 420 and 435 for symmetric and asymmetric distribution models have different essences in relation to the distribution of the output parameter.

, соответственно. The entropy coefficient K _Hn for the asymmetric distribution of the parameter x is equal to the ratio of the interval of entropic uncertainty 435 to the interval of parametric uncertainty 430 of the output parameter, which are equal to the entropy potential of the asymmetric distribution Δ _Hn and the square root of the second initial moment

, respectively.

The formula for determining the entropy coefficient of an asymmetric distribution is:

The entropy coefficient of the asymmetric distribution (33), equal to the ratio
the entropy potential of the asymmetric distribution of the output parameter to the square root taken from the second initial moment of the distribution of the output parameter is a dimensionless normalized feature of the asymmetric distribution shape [15].

The entropy of a nonsymmetric unbiased distribution has the property of proportionality with respect to the entropy of a symmetric distribution obtained by reflecting the nonsymmetric distribution about the origin. If f _n (X, α, τ, λ) is the density of the asymmetric distribution, then the density of the corresponding symmetric distribution has the form:

Since the range of the output parameter is doubled for a symmetrical distribution, the linear proportionality is valid for the entropies of symmetric and asymmetric distributions of the form:

Since for the symmetric distribution, expression (28) is valid for the conditional entropy and measurement, provided that the reading x _{p is} equal to zero, then, having written the logarithm of the product as the sum of logarithms, we obtain for the entropy of the symmetric distribution an expression of the form:

Comparing expression (35) and expression (36), we obtain that for the model of asymmetric distribution of the output parameter, the equality of the entropy of the distribution and the logarithm of the entropy potential of the asymmetric distribution should be ensured:

The formula for calculating the entropy potential of the asymmetric distribution Δ _Hn , with a known entropy H _n (x, α, β, λ), takes the form:

Thus, from expression (38) it follows that the entropy potential of the asymmetric unbiased distribution is equal to the exponent with an exponent equal to the entropy of the asymmetric unbiased distribution of the output parameter.

Since the standard deviation σ _S for the distribution of the parameter x symmetric relative to the origin is equal to the square root of the second initial moment m ₂ , calculated for the original asymmetric unbiased distribution of the output parameter y

then the entropy coefficient of an asymmetric distribution can be obtained as the entropy coefficient of a symmetrical distribution, the entropy of which is determined by the properties of the asymmetric distribution.

Since the intervals of entropy and parametric uncertainty are proportional to the distribution scales, the ratio of the intervals for unbiased asymmetric distributions of the output parameter does not depend on the distribution scale and is completely determined by the shape parameters.

Thus, the entropy coefficient of asymmetric distributions is an independent indicator of the shape of unbiased asymmetric distributions [15]. Combining the information feature of the form, which was used as the coefficient of entropy of asymmetric distributions, with probabilistic features of the shape of asymmetry and kurtosis made it possible to obtain an entropy-parametric space of features for displaying the coordinates of trajectories of the possible position of an object with an asymmetric unbiased distribution of the output parameter.

Possible positions of the object in the coordinate entropy-parametric space of features of asymmetric unbiased distributions of the output parameter are given in figure 5 in the form of a diagram of the projections of the trajectories of the distribution of the output parameter:

- figure 5, a - the projection of the trajectories on the plane, given by the signs of asymmetry and the entropy coefficient of an asymmetric distribution;

- figure 5, b - the projection of the trajectories on the plane, given by the signs of kurtosis and the coefficient of entropy of the asymmetric distribution.

- figure 5, c - the projection of the trajectories on the plane, given by the signs of asymmetry and kurtosis of an asymmetric distribution.

The following designations are used on the projections of the diagram:

505 - point of position of the optimal state of the system with an asymmetric distribution of the output parameter;

510 - point of position of the real state of the system with an asymmetric distribution of the output parameter;

515 - border of the control zone in the space of entropy - parametric signs of asymmetric distribution of the output parameter;

520 - curves of the position of the system when using the target distribution for the output parameter from the forms of the Weibull-Gnedenko family;

525 — Curves of the position of the system when used for the output parameter of the target distribution from the forms of the family of gamma distributions;

530 — Curves of the position of the system when used for the output parameter of the target distribution from the forms of the logarithmic normal distribution;

535 - point of position of the system when using the form of asymmetric exponential distribution of the output parameter, which coincides with the Pearson distribution χ ² with three degrees of freedom;

540 - point of position of the system when using for the output parameter the normal distribution reflected relative to the center;

545, 550 - points of system position when using symmetric forms of logistic and normal distributions for the output parameter;

555, 560, 565 - curves of the position of the system when using for the output parameter the target distribution from the forms of Pareto distributions at various shifts x _{0 of the} left border of the possible values of which correspond to the values 0.9, 1.1 and 1.3.

In the space of the entropy-parametric potential, the position point of the optimal state of the system 505 is determined by the target density of the asymmetric distribution of the output parameter and coincides with the position of the a priori known asymmetric distribution model. In figure 5, the point of the optimal state of the system 505 with a skewed distribution is on the curve 520 of possible positions of the system when using for the output parameter the target distribution from the forms of the family of gamma distributions. The actual state of the system 510 differs from its optimal position 505 due to the action of external influencing destabilizing factors and constantly changing internal properties of the system: internal connections, parameters of the elements and the environment that make up the system.

The position point of the actual state of the system 510 is near the optimum state. Since the real state does not coincide with the possible models of its state, the point of position of the real state 510 is located near the curves of the possible positions of the system. For the real state 510, it is possible to change the properties of the system, which are displayed in changing the shape and scale of the distribution model of the output parameter.

For this reason, when carrying out monitoring and control, it is important to determine the transition of the system to a new state due to changes in the properties of the model. Since the possible states of the model are close to the optimal state, then in the entropy-parametric space of features, the border of the control area 515 with the known admissible forms of models of asymmetric distributions of the output parameter is distinguished, which are stored in the database of reference parameters of the distribution of the output parameter. When the system leaves the control area, the distribution law of the controlled parameter is transformed by changing the tuning parameters of the controller. For the real state of the system, the most closely located model is selected, after which its parameters are determined.

The author of the present invention is convinced that the mapping of the position of the object in the coordinate entropy-parametric space of the features of asymmetric unbiased distributions of the output parameter makes it possible to implement new opportunities for monitoring and control over a stochastic system.

Description of the algorithm for monitoring and control over a stochastic system with an unbiased asymmetric target distribution density of the output parameter

The process diagram in figure 6 illustrates the new possibilities and features of the proposed method for monitoring and control over a stochastic system with an asymmetric unbiased target distribution density of the output parameter. To implement new possibilities in the proposed invention, the following actions are carried out, illustrated in the form of stages of the monitoring and control process in figure 6:

- step 605 of determining the coordinates of the shape feature space for the asymmetric distribution density;

- step 610 of forming the space of the optimal state of the system;

- step 615 determining the distance between the positions of the possible models and real states;

- step 620 of selecting the shape of the model from the condition of the minimum distance between the model and the real state;

- step 625 determining the scale of the model of the distribution of the output parameter of the stochastic system;

- step 630 save the parameters of the distribution model in the database;

- step 635 of determining the mismatch of the parameters of the model of the real and the optimal state of the stochastic system;

- step 640 of optimization of the mismatch of the parameters of the model;

- step 645 for mapping the position of the system in the feature space of the distribution of the output parameter.

Determination of the coordinates of the shape feature space for an asymmetric unbiased distribution density of the output parameter of a stochastic system

The first act, illustrated at block 605 of the flowchart of FIG. 6, is to determine the coordinates of the shape feature space for the asymmetric distribution density. The space coordinates are specified by dimensionless features of the distribution shape and are calculated using the matrix vector formula (12). The asymmetry and kurtosis, which characterize the skewness and peakedness of the distribution, were used as parametric features of the distribution shape. The third coordinate is given by an independent informational sign of the distribution shape - the entropy coefficient of an asymmetric distribution. Displaying the distribution of the output parameter of the system in a three-dimensional space of signs of distribution forms provides control of the position of the system in relation to possible functional states. Parameters such as the second initial moment, the third central moment and the entropy-parametric potential of the asymmetric distribution density of the output parameter of the system are determined by formulas (13), (14) and (15), respectively

Formation of the space of the optimal state of the stochastic system

The main purpose of the operation, illustrated at block 610 of the flowchart in FIG. 6, is to obtain the positions of the models and the position of the asymmetric distribution state of the system output in the normalized space of coordinates of the optimal state 505 with the asymmetric distribution of the output parameter. A diagram of the process of forming the space of the optimal state of the stochastic system is given in figure 7 in the form of steps 715, 720, 725 and 730.

From figure 7 it follows that the first distinctive effect of the proposed method of monitoring and control over a stochastic system with an unbiased asymmetric target distribution density of the output parameter contains

- step 715 setting the boundaries of the control zone in the space of signs of the optimal state with an asymmetric distribution of the output parameter of the stochastic system;

- step 720 of determining the normalized position coordinates for the state of the distribution of the output parameter in the feature space of the optimal state of the stochastic system;

- step 725 of determining a criterion for the state of the stochastic system in the space of signs of the optimal state;

- step 730 of determining the normalized coordinates of the positions of the possible models in the space of signs of the optimal state;

The essence of the scheme of the process of forming the space of the optimal state of the system consists in determining the coordinates of possible models near the optimal state within the limits of control.

,

,

по признакам асимметрии, эксцесса и коэффициента энтропии несимметричных распределений. Step 715 of the process diagram in FIG. 7 consists in setting the boundaries of the control zone in the space of the optimal state signs using expression (16). For the optimal state, there is a zone of healthy states, which are tracked by the form of the output parameter. The intervals of change of signs for the control zone are set using the product of the value of the sign of the optimal state (Sk ₀ , Ex ₀ , K _Hn0 ) by the coefficients of the intervals of signs a, b, c. Then the position of possible values in the entropy-parametric state space of the output parameter is limited by the values of the features

,

,

by the signs of asymmetry, kurtosis and entropy coefficient of asymmetric distributions.

,

,

с указанием вектора признаков оптимального состояния [Sk₀, Ex₀, K_Hn0]^T и вектора коэффициентов разброса [a,b,с]^T. Граница зоны контроля устанавливается из условия, что внутри границы находится не менее 95 % рабочих состояний при воздействии различных дестабилизирующих факторов. Выход за границу контроля рассматривается как переход в неработоспособное состояние, требующее коррекции параметров и матриц вероятностей взаимосвязей стохастической системы. При задании границы контроля в виде сферы, параметр γ_max радиуса границы в диапазоне от 1 до 3. Если коэффициенты разброса определены для значений среднего квадратического разброса, то для нахождения 95 % работоспособных состояний параметр радиуса границы γ_max имеет значение, равное 2,8. При известных границах допустимых состояний системы в пространстве энтропийно-параметрических признаков в качестве рабочего пространства удобно использовать пространство, центрированное относительно заданного оптимального состояния и нормированное к разбросу признаков асимметрии, эксцесса и коэффициента энтропии несимметричных распределений выходного параметра. The boundaries of the intervals of features of possible models can be determined in two ways. The first method consists in modeling the object and obtaining the boundaries of the parameters based on the model using the Monte Carlo method by imposing the output parameter and models of destabilizing factors on the model. In this case, the signs of the optimal state Sk ₀ , Ex ₀ , K _{Hn0 are} determined by the properties of the system model. Another approach is associated with the accumulation of statistical data on the stochastic system and the estimation of the intervals of the control zone signs, provided that the system (or similar systems) is in an operable state for a controlled mode of operation. In this case, as the optimal entropy-parametric features of the system Sk ₀ , Ex ₀ and K _Hn0 , estimates of the mean value of features obtained from earlier observations are taken. RMS scatter estimates are used as feature intervals in statistics. Since the methods of analysis and statistical accumulation use various methods for evaluating the intervals of attributes, it is convenient to use the restrictions of the intervals of attributes to set the border of the control zone.

,

,

indicating the vector of signs of the optimal state [Sk ₀ , Ex ₀ , K _Hn0 ] ^T and the vector of scatter coefficients [a, b, c] ^T. The border of the control zone is established from the condition that at least 95% of the operating states are inside the border under the influence of various destabilizing factors. Going beyond the control boundary is considered as a transition to an inoperative state that requires correction of the parameters and probability matrices of interconnections of the stochastic system. When setting the control border in the form of a sphere, the parameter γ _{max of} the border radius is in the range from 1 to 3. If the scatter coefficients are determined for the values of the root-mean-square spread, then to find 95% of the operable states, the parameter of the border radius γ _max has a value of 2.8. Given the known boundaries of the permissible states of the system in the space of entropy-parametric features, it is convenient to use the space centered relative to a given optimal state and normalized to the spread of asymmetry, kurtosis, and entropy features of asymmetric distributions of the output parameter as a working space.

Examples of projections of the control zone on the normalized space of signs of the optimal state are shown in Figures 9, 10 and 11, where the following designations of objects are used: 505 - point of position of the optimal state with an asymmetric distribution of the output parameter; 510 - point of position of the real state of the system with an asymmetric distribution of the output parameter; 515 - border of the control zone; 520, 525 and 530 - curves of the position of the system when using for the output parameter target distributions from the forms of the Weibull-Gnedenko family, the family of the gamma distribution and the forms of the logarithmic normal distribution, respectively; 535 - point of position of the system model when using the form of asymmetric exponential distribution of the output parameter, which coincides in the Pearson parameter χ ² with three degrees of freedom; 910, 920 and 930 illustrate the position points of possible models of system states using the Weibull-Gnedenko family forms, the gamma distribution family and the logarithmic normal distribution, respectively; 915, 925 and 935 are the position points of the system state models when using the curves of the system positions 520, 525 and 530 as the distributions of the output parameter of the Weibull-Gnedenko family forms, the gamma distribution family and the logarithmic normal distribution, respectively. Plot 9 additionally gives the position of the state model of the system 940 when using the curve of the forms of the Pareto family 555. Since the forms of the Pareto distribution are outside the control zone, the position curve 555 is absent in Figure 10 and Figure 11 of other projections of the entropy-parametric space, illustrating the possible position of the model when the system is inoperative.

The next step 720 of the process flow diagram in FIG. 7 is to determine the normalized position coordinates for the state of the distribution of the output parameter in the feature space of the optimal state of the stochastic system according to the matrix vector formula (17).

To display the state of the system in the space of signs of the optimal state at step 720 of the process in figure 7, the normalized coordinates of the position of the state are determined by the formula of the matrix vector (17). The state of the system of the output parameter on the projection of the optimal state of the system with an asymmetric distribution of the output parameter is shown in the projections of the control zone (Figure 9, Figure 10 and Figure 11) as a position point 510. For illustration, a state point with signs of asymmetry, kurtosis and the entropy coefficient of asymmetric distribution is used output parameter equal to 2.1, 9 and 1.53, respectively. When constructing projections, the position of the optimal state of the system in the entropy-parametric space is set by the shape parameter α of the distribution from the variety of forms of the gamma family, equal to 0.692. The projections of the space of the optimal state are constructed for the signs of asymmetry, kurtosis and the coefficient of entropy of the asymmetric distribution of the optimal state, equal to 2.405, 8.674, and 1.65, respectively. The form of the optimal state model is taken as the form of the target distribution of the output parameter of the stochastic system. The intervals of the signs of asymmetry, kurtosis and the coefficient of entropy, calculated for the control zone, are equal to 0.58, 3.5 and 0.225. The coefficients of the intervals of signs a, b and c for the control zone are equal to 0.241, 0.404 and 0.136, respectively. The coefficients and signs of the position point 510 of the real state of the system with an asymmetric distribution of the output parameter change during monitoring and control.

At step 725 of the process diagram in FIG. 7, the γ ^* criterion is determined for the state of the stochastic system in the space of features of the optimal state according to the formula (18). The criterion allows you to check the state of the object belonging to the control zone.

Step 730 consists in determining the normalized coordinates of the position of possible models in the space of signs of the optimal state according to formula (19).

The formation of the space of the optimal state is necessary to ensure the unity of the optimality of control at various points of the entropy-parametric space. The fact is that in the entropy-parametric space, the minimum distance depends on the values of the distribution signs and does not take into account the intervals of permissible changes relative to the optimal state, which reduces the quality of control when choosing the model parameters. In order to unify the data under different conditions, control intervals near the optimal state should be taken into account.

In the control zone of the optimal state, the number of distinguishable shapes of models is limited by the difference in the shape parameter of possible models. When the shape is controlled up to the third significant digit in the control zone, 300 models from the family of gamma distribution shapes with shape parameters in the range from 0.55 to 0.85 are possible, and 150 models from the Weibull-Gnedenko family with shape parameters from 0.8 to 0.95. Determining the coordinates of possible models by formula (19) in the space of the optimal state refers to the formation of space.

Determination of distances between the positions of possible models and real states in the normalized centered space of signs of the optimal state of a stochastic system

A third distinctive action of the process diagram in FIG. 6, illustrated by block 615, is to determine the distance between the position of possible models and real states for asymmetric distributions of the output parameter of the stochastic system in the normalized coordinate space of the optimal state. The positions of possible models from individual families correspond to models at the intersection of the curves of the family of distributions and the perpendicular from the point of position 510 of the real state of the system with an asymmetric distribution of the output parameter to the curve of positions of the system when using a family of possible models.

The choice of the form of the model of the stochastic system from the condition of the minimum distance between the positions of the model and the real states.

A fourth distinctive action, performed by block 620 of the process diagram in FIG. 6, is to select the shape of the model such that the distance between the model and the real state is as small as possible. Models are selected from a family of distribution shapes and model shape parameter estimates from the model database.

When constructing monitoring and control by a stochastic system with an unbiased target distribution density of the output parameter, the distribution form and its range are taken as the main control parameters. Control is achieved through the use of distribution families containing shape and span parameters.

The existing algorithms make it possible to estimate the shape of the distribution of the output parameter only approximately. The point is that the parametric features of the asymmetry and kurtosis form do not allow one to obtain a form when using several subfamilies of generalized distributions. The close arrangement of the trajectories of the curves in the space of only parametric signs of asymmetry and kurtosis does not provide an unambiguous choice of the shape of the model.

In the proposed invention, the quality of the choice of the form is carried out from the condition
minimum (21) between the position of the model and the state of the system. The quality of the choice of the form is ensured due to the distinguishability of asymmetric distributions of the entropy-parametric space of features and the transition to the space of the optimal state of the system. When you select a distribution model, the shape parameters are assigned their specific values.

Determination of the scale of the distribution model of the output parameter of the stochastic system.

The next distinctive effect of the proposed invention is to determine the scale of the distribution of the output parameter of the stochastic system at step 625 of the process diagram of Figure 6. The diagram of the process for determining the scale parameter of the distribution of the output parameter is given in figure 8 in the form of steps 805, 810 and 815. From figure 8 it follows that the fourth the distinctive effect of the proposed method for monitoring and controlling a stochastic system with an unbiased asymmetric target distribution density contains

- determining at step 805 the entropy-parametric potential of the asymmetric distribution density of the output parameter;

- forming at step 810 the entropy-parametric potential of the asymmetric model of the distribution density of the output parameter;

- determination at step 815 of the scale parameter λ from the minimum condition (24) of the difference in entropy-parametric potentials for the asymmetric model and the real state of the distribution density of the output parameter.

To clarify the actions carried out in determining the scale of the model of the distribution of the output parameter of the stochastic system, figure 12 shows a projection of the space of entropy and parametric uncertainties of asymmetric distributions of the output parameter, where the following designations are given

505 and 510 - points of the position of the optimal and real state of the system with an asymmetric distribution of the output parameter;

- 1205 - origin of coordinates of the space of entropic and parametric uncertainties;

выходного параметра;- 1210 - distance of the entropy-parametric potential

output parameter;

- 1215 - point of position of the model with the selected form of distribution and the entropy-parametric potential of the output parameter;

- 1220 - scale line of the model with the selected shape of distribution;

выходного параметра;- 1225 - equipotential of entropy-parametric potential

output parameter;

оптимального состояния;- 1230 - distance of the entropy-parametric potential

optimal condition;

оптимального состояния системы;- 1235 - equipotential of entropy-parametric potential

the optimal state of the system;

- 1240 - point of position of the model with the selected form of distribution and the entropy-parametric potential of the optimal state of the system;

- 1245 - scale line of the model with the shape of the optimal state.

распределения выходного параметра. The space of entropy and parametric uncertainties of asymmetric distributions of the output parameter in figure 12 is given by the projection of the coordinate of the entropy potential of the asymmetric distribution Δ _HP and the coordinate of the interval of parametric uncertainty equal to the square root of the second initial moment

distribution of the output parameter.

In the space of entropic and parametric uncertainties of asymmetric distributions, the selected distribution model corresponds to the scale line of the model 1220 with the selected distribution shape, at which different distances from the origin of coordinates 1205 to points of the straight line 1220 correspond to different scales λ of the distribution of the output parameter. The slope of the 1220 line is completely determined by the ratio of the coefficient of entropy of the asymmetric distribution, determined from expression (33).

The position point of the optimal state of the system 505 is on the scale line of the model 1245 with the shape of the optimal state of the distribution, which is determined by the entropy coefficient of the asymmetric distribution of the optimal state. For the shape of the optimal state model, many models with different scales are possible. If the entropy coefficient of the selected asymmetric distribution model differs from the entropy coefficient of the asymmetric distribution of the optimal state, then the scale lines of the selected model 1220 and the optimal state model 1245 exist as separate objects.

отличается от коэффициента энтропии выбранной модели несимметричного распределения, то точки положения 510 реального состояния системы при несимметричном распределении выходного параметра находится вне линии масштабов 1220 выбранной модели. На фигуре 12 проекции пространства энтропийной и параметрической неопределённостей несимметричных распределений схематично показана схема определения масштаба для модели распределения с выбранной формой, поясняющая действия схемы на фигуре 8 процесса определения масштаба модели распределения выходного параметра.If the entropy coefficient of the asymmetric distribution of the output parameter

differs from the entropy coefficient of the selected model of asymmetric distribution, then the position point 510 of the real state of the system with an asymmetric distribution of the output parameter is outside the scale line 1220 of the selected model. In figure 12, the projection of the space of entropic and parametric uncertainties of asymmetric distributions schematically shows a scale determination scheme for a distribution model with a selected shape, explaining the actions of the circuit in figure 8 of the process for determining the scale of the output parameter distribution model.

At step 805 of the process diagram 8, the entropy-parametric potential of the unbiased distribution density of the output parameter is determined by the formula (14).

According to formula (33), the entropy coefficient of the asymmetric distribution of the output parameter is proportional to the ratio of the entropy potential of the unbiased distribution to the square root of the second initial moment of the distribution. An illustration of the relation (33) in the space of entropic parametric uncertainties of asymmetric distributions of the output parameter is given in Figure 12, where the positions of objects are shown schematically to illustrate the processes of determining and minimizing the parameter of the distribution scale.

The distance from point 1205 of the origin of coordinates of the space of entropy and parametric uncertainties in figure 12 to the position of point 510 of the real state of the system is equal to the entropy-parametric potential of the stochastic system. To determine the entropy-parametric potential of the asymmetric distribution of the output parameter in terms of the coordinates of the uncertainty space, we used formula (22).

соответствует дистанция 1210 от начала координат 1205 до точки положения реального состояния системы 510. Пунктирная линия 1225 является эквипотенциалью энтропийно-параметрического потенциала выходного параметра. Эквипотенциаль соответствует точкам положения моделей, для которых модуль энтропийно-параметрического потенциала равен модулю энтропийно-параметрического потенциала выходного параметра.Figure 12 shows the entropy-parametric potential of the output parameter

corresponds to the distance 1210 from the origin of coordinates 1205 to the point of position of the real state of the system 510. The dotted line 1225 is the equipotential of the entropy-parametric potential of the output parameter. The equipotential corresponds to the points of position of the models, for which the modulus of the entropy-parametric potential is equal to the modulus of the entropy-parametric potential of the output parameter.

For the real state 510, the best correspondence of the scale parameter for the model 1215, located on the line 1220 of the scales of the models with the selected shape, in such a way that the distance between the points 510 of the position of the real state and 1215 of the position of the model with the selected shape in the space of entropic and parametric uncertainty is minimal. At small angles of mismatch, the point 1215 of the position of the model with the selected shape is found as the intersection of the line 1220 of the model scales and the equipotential of the entropy-parametric potential of the output parameter. To determine the scale parameter λ of the model with the selected shape, it is necessary to interconnect the scale parameter λ of the model and the entropy-parametric potential of the output parameter.

For this reason, at step 810 of the process diagram in FIG. 8, the formation of the entropy-parametric potential of the asymmetric model of the distribution density of the output parameter is carried out.

через координаты пространства неопределённостей имеет вид:The entopy-parametric potential for the model of the distribution of the output parameter is equal to the distance between the origin of coordinates 1205 and the position of the model 1215 in the space of entropic and parametric uncertainty. Formula for determining the entopy-parametric potential of the model

through the coordinates of the uncertainty space has the form:

If the form of the model is chosen, then the entropy coefficient K _Hnλ for the model of asymmetric distribution is uniquely known. The squared potential of the entropy of the asymmetric distribution is equal to the product of the entropy coefficient by the second initial moment m _{2λ of the} asymmetric distribution of the model of the distribution of the output parameter. For the entropy-parametric potential of the asymmetric model of the distribution of the output parameter, the formula (23) is valid, according to which for the density of the output parameter there are many models that differ in different parameters of the distribution scale. For most asymmetric distributions, the entropy-parametric potential is proportional to the model scale parameter λ. Then, having determined the distance from the origin of coordinates 1205 to the point of position of the model 1215 with the known form of the model, we obtain the value of the scale λ of the distribution of the output parameter.

Step 815 of the process diagram in FIG. 8 consists in determining the scale parameter λ from the condition (24) of the minimum difference in the entropy-parametric potential for the asymmetric model and the real state of the distribution density of the output parameter.

The shape parameter selection process is illustrated in Figure 12. Obviously, the best match between the model and the real state of the distribution of the system can be achieved when the position points of the model 1215 with the selected shape coincide with the real state 510 of the system. Since, due to the action of destabilizing external factors, the position of the point 510 of the real state of the system is outside the line 1220 of the models of the selected shape, the position point of the model 1215 corresponds to the minimum distance between the points of the position of the real state of the system 510 and the line 1220 of the selected shape of the model. With a small difference between the entropy coefficient of the model and the real state, the angle between the line of the model form and the line of the form of the real state does not exceed 1 ... 5 °. In this case, the scale of the model is chosen from condition (24) of the minimum difference between the entropy-parametric potentials of the asymmetric model and the real state of the distribution of the output parameter

Saving the parameters of the stochastic system model in the database.

The purpose of monitoring is to preserve recoverable information about the state of the object with a minimum amount of transmitted data. For this reason, at block 630 of the flowchart of FIG. 6, the model parameters are stored in the database. Preservation of features and distribution allows to ensure the recovery of information about the state of the system.

Determination of the mismatch of the entropy-parametric potentials of the model of the real and optimal states of the stochastic system.

The next step, illustrated at block 635 of the process diagram in FIG. 6, is to determine the mismatch between the real and optimal state model.
stochastic system.

и энтропийный потенциал

несимметричного распределения выходного параметра, характеризующие интервалы энтропийной и параметрической неопределённости, имеют свои индивидуальные значения. В пространстве энтропийной и параметрической неопределённости на фигуре 12 оптимальному состоянию системы при несимметричном распределении выходного параметра соответствует точка положения 505. Дистанция 1230 от начала координат 1205 до точки 505 положения оптимального состояния системы равно энтроппийно-параметрическому потенциалу оптимального состояния системы

. Для моделей с выбранной формой распределения соответствует множество точек на линии масштабов модели 1220, отличающихся масштабом распределения. Модель наиболее приближена к оптимальному состоянию, если дистанция между точкой 505 положения оптимального состояния системы и точкой 1240 положения модели оптимального состояния на линии 1220 масштабов модели с выбранной формой минимальна. Точка 1240 положения модели оптимального состояния определяется пересечением линии масштабов модели 1220 и эквипотенциали 1235 энтропийно параметрического потенциала оптимального состояния системы. For an optimal state, the second initial moment

and entropy potential

asymmetric distribution of the output parameter, characterizing the intervals of entropy and parametric uncertainty, have their own individual values. In the space of entropy and parametric uncertainty in figure 12, the position point 505 corresponds to the optimal state of the system with an asymmetric distribution of the output parameter. Distance 1230 from the origin 1205 to point 505 of the position of the optimal state of the system is equal to the entropy-parametric potential of the optimal state of the system

... For models with the selected shape of distribution, there are many points on the scale line of model 1220 that differ in the scale of the distribution. The model is closest to the optimal state if the distance between the point 505 of the position of the optimal state of the system and the point 1240 of the position of the model of the optimal state on the line 1220 of the scale of the model with the selected shape is minimal. The point 1240 of the position of the optimal state model is determined by the intersection of the scale line of the model 1220 and the equipotential 1235 of the entropy parametric potential of the optimal state of the system.

и реального

состояний стохастической системы различны, то модели реального и оптимального состояний, заданные точками 1215 и 1240 , расположенные на фигуре 12 на линии масштабов модели 1220, имеют различный масштаб при одной той же форме. По этой причине на этапе 635 схемы процессов на фигуре 6 проводится определение рассогласования модели реального и оптимального состояния системы. Для определения рассогласования рассчитывается разница дистанций 1210 и 1230 энтропийно-параметрических потенциалов выходного параметра и оптимального состояния по формуле (25).If the entropyino-parametric potentials of the optimal

and real

states of the stochastic system are different, then the models of real and optimal states, specified by points 1215 and 1240, located in figure 12 on the scale line of the model 1220, have different scales with the same shape. For this reason, at step 635 of the process flow diagram of FIG. 6, the mismatch of the model of the real and the optimal state of the system is determined. To determine the mismatch, the difference between the distances 1210 and 1230 of the entropy-parametric potentials of the output parameter and the optimal state is calculated using the formula (25).

To estimate the distance between the position points of the model 1215 with the entropy-parametric potential of the real state and the position point 505 of the optimal state of the system, the mismatch of the squares of the entropy and parametric uncertainty intervals is calculated using the formula:

Having squared the terms of expression (41) and rearranging the terms, we find that the mismatch of the squares of the entropy and parametric uncertainties is determined by the difference between the sum of the squares of the entropy-parametric potentials and the doubled sum of the products of the entropy and parametric uncertainties:

In expression (42), the first two terms are the entropy-parametric potentials for the model of real and optimal states. Replacing the entropy uncertainties using expression (40) through the intervals of parametric uncertainties, we obtain a formula for calculating the mismatch of the squares of the intervals of entropic and parametric uncertainties:

Minimizing expression (43) allows minimizing the mismatch between the real state model and the optimal state.

. Второго начального момента модели равен второму начальному моменту оптимального состояния и его рассогласования с учётом приближения для корня квадратного:The entropy coefficient of the asymmetric distribution of the model is equal to the sum of the entropy coefficient of the optimal state and the mismatch of the entropy coefficients:

... The second initial moment of the model is equal to the second initial moment of the optimal state and its mismatch, taking into account the approximation for the square root:

The square of the entropy-parametric potential is related in proportion to the second initial moment:

Then, after transforming expression (44), taking into account equality (44), we obtain a formula for the mismatch of the squares of the intervals of entropy and parametric uncertainty of the form:

It follows from the obtained expression that when the entropy-parametric potentials of the model and the optimal state of the mismatch of the square of the intervals of the entropy and parametric uncertainties are equal, the mismatch of the second initial moments and the entropy coefficients of asymmetric distributions is determined. Thus, expression (45) is a more stringent condition for determining and minimizing mismatch than formula (25).

Minimization of the mismatch of the entropy-parametric potentials of the model with respect to the optimal state

The next action, illustrated by block 640 of the process diagram in figure 6, is to minimize the mismatch between the entropy-parametric potentials of the model and the optimal state of the system, carried out in accordance with the conditions

The minimization of the mismatch (11) of the entropy-parametric potentials is achieved by means of a targeted correction of the distribution law and the formation of a control action to correct the connections of the stochastic system.

The positions of objects in the space of entropy-parametric uncertainty in figure 12 are given schematically to illustrate the processes of determining and minimizing the parameter of the distribution scale.

Mapping the state of a stochastic system in the entropy-parametric space of features of the output parameter

To visualize the state of the system, at step 645 of the process diagram in Figure 6, the position of the stochastic system is mapped into the space of signs of the distribution of the output parameter and in the form of projections of the control zone onto the normalized coordinates of the space of signs of the optimal state. At 260, a message is generated regarding the optimality of the object state.

A possible implementation of the material with the help of which the action of the claimed method is performed is illustrated by the diagram of the technical device shown in figure 13,

Description of the scheme of a device for monitoring and controlling a stochastic system with an unbiased asymmetric target distribution density

To clarify the proposed invention, figure 13 shows a functional diagram of a device for monitoring and controlling a stochastic system with an unbiased asymmetric target distribution density, where the following designations are used:

1310 - subsystem of uncontrolled processes of the stochastic system;

1315 - connections of uncontrolled processes of a stochastic system with a controlled process of the output parameter of a stochastic system.

1317 - the relationship of influencing factors with the controlled process of the output parameter of the stochastic system.

1320 is a controlled process of a stochastic system.

1325 - influencing factors of artificial and natural origin.

1330 - recording device;

1335 - device for generating the distribution of the output parameter;

1340 - block for determining the features of the distribution form of the output parameter;

1345 - block for determining the entropy-parametric potential of the output parameter;

1350 - device for forming the space of the optimal state;

1355 - device for selecting the shape of the model;

1360 - block for determining the parameter of the scale of the model;

1365 — device for mapping the state of a stochastic system;

1370 - database;

1375 - device for checking the belonging of the state of the stochastic system to the control zone of the optimal state;

1380 - distribution transformation block;

1385 — device for reproducing the EPI measure of the target distribution;

1390 - block of target adjustment of the distribution shape;

1395 - block for generating a control action for adjusting the connections of stochastic systems;

In the stochastic system in figure 13, a subsystem of uncontrolled processes 1310 of a stochastic system, a controlled process 1320, connections 1315 of uncontrolled processes of a stochastic system with a controlled process of a stochastic system and connections 1317 of influencing factors with a controlled process of a stochastic system are highlighted. The impact on the controlled process 1320 is possible by changing the connections 1315 uncontrolled processes of the stochastic system and by changing the connections 1317 influencing factors of artificial and natural origin.

The recorder 1330 records the output parameter of the controlled process of the stochastic system. The device for forming the distribution of the output parameter 1335 carries out the process of forming the sample [Y] of the values of the output parameter of the stochastic system, obtained from the registering device 1330. The block for determining the features of the form 1340 is designed to calculate the features of the form of the output distribution of the stochastic system: asymmetry, kurtosis and the entropy coefficient of asymmetric unbiased distributions. The values of the shape features are transmitted to the device for forming the space of the optimal state 1350, at the output of which the matrix vector [ξ ^* ] of the normalized position coordinates for the state of the distribution of the output parameter and the matrix [ξ] of the normalized coordinates of the positions of possible models for the states of the distribution of the output parameter in the space of the normalized features of the form the optimal state of the stochastic system. The row number of the matrix [ξ] of the normalized coordinates corresponds to the number s of a possible model of the stochastic system. The column numbers of the matrix [ξ] determine the normalized coordinates of the positions of possible models of the state of the stochastic system. The matrix vector [ξ ^* ] and the matrix [ξ] of the normalized coordinates of the positions of the possible models are transmitted to the model 1355 shape selection device, at the output of which the number s of the shape of the possible model of the state of the stochastic system is set. Since the model numbers have their own specific number s, the parameters and features of the distribution shape are known for the models.

несимметричного несмещённого распределения выходного параметра стохастической системы характеризует положение среднего значения. Устройство 1385 предназначено для воспроизведения меры энтропийно-параметрического потенциала целевого распределения. Рассогласование состояния системы δΔ_HP оценивается по разнице энтропийно-параметрического потенциала выходного параметра

и энтропийно-параметрического потенциала целевого распределения Δ_HP. Рассогласование энтропийно-параметрического потенциала реального и оптимального состояний системы, соответствующее разнице оценок средних значений выходного и целевого распределений, и используется блоком 1395 формирования управляющего воздействия χ для регулировки связей 1315 стохастической систем. При формировании управляющего воздействия χ так же учитывается положение модели s реального состояния системы. Согласно номеру модели s блок 1390 обеспечивает целевую корректировку формы распределения, которая учитывается блоком 1395 при формировании управляющего воздействия для регулировки связей стохастической систем. Таким образом, при минимизации рассогласования энтропийно-параметрического потенциала модели относительно оптимального состояния посредством целевой коррекции связей стохастической системы учитывается отклонение формы модели от оптимального состояния.Block 1345 of the diagram in figure 13 illustrates the process of determining the entropy-parametric potential of the output parameter of the stochastic system according to the formula (22). Entropy-parametric potential

asymmetric unbiased distribution of the output parameter of the stochastic system characterizes the position of the mean value. The device 1385 is designed to reproduce the measure of the entropy-parametric potential of the target distribution. The mismatch of the state of the system δΔ _HP is estimated from the difference in the entropy-parametric potential of the output parameter

and the entropy-parametric potential of the target distribution Δ _HP . The mismatch of the entropy-parametric potential of the real and optimal states of the system, corresponding to the difference in the estimates of the mean values of the output and target distributions, is used by the block 1395 for generating the control action χ to adjust the connections 1315 of the stochastic systems. When forming the control action χ, the position of the model s of the real state of the system is also taken into account. According to the model number s, block 1390 provides a targeted adjustment of the distribution shape, which is taken into account by block 1395 when generating a control action to adjust the connections of stochastic systems. Thus, when minimizing the mismatch of the entropy-parametric potential of the model relative to the optimal state by means of targeted correction of the connections of the stochastic system, the deviation of the model form from the optimal state is taken into account.

выходного параметра и известному номеру s формы модели реального состояния с помощью устройства 1360 проводится определение параметра масштаба модели λ из условия минимизации разницы энтропийно-параметрических потенциалов модели с номером s и состояния плотности распределения выходного параметра.According to the values of the entropy-parametric potential

of the output parameter and the known number s of the form of the real state model using the device 1360, the model scale parameter λ is determined from the condition of minimizing the difference in the entropy-parametric potentials of the model with number s and the state of the distribution density of the output parameter.

The device 1365 is used to map the state of a stochastic system in the entropy-parametric space of the characteristics of the output parameter. Mapping is carried out using the known shape parameters (α, β) of the model s and the scale parameter λ. To ensure the monitoring process, the number s, form parameters (α, β) and scale λ of the model of the distribution of the output parameter for the real state of the stochastic system are recorded in the database 1370.

The device 1375 for checking the belonging of the state of the stochastic system to the control zone of the optimal state makes it possible to control the exit of the system from the control zone by changing the features of the shape of the output distribution. Block 1380 transformation of the distribution is designed to generate a message η about the need to transform the distribution to return the system to the control area. The transformation of the distribution can be achieved by changing the properties of the system by changing the tuning parameters of the controller located in the subsystem of uncontrolled processes 1310 of the stochastic system, changing the composition of influencing factors 1325 or changing the choice of the optimal state, relative to which monitoring and control over the stochastic system is established.

From a consideration of the diagram in figure 13, it follows that to ensure the stability of the system, block 1395 generates control actions to adjust the connections 1315 of uncontrolled processes of the stochastic system and connections 1317 of influencing factors with the controlled processes of the output parameter of the stochastic system.

The formation of the control action by the block 1395 for adjusting the links 1315 and 1317 is carried out under the condition of a nonzero model number s and / or a nonzero mismatch of the entropy-parametric potential. Block 1390 target correction of the shape of the distribution law is designed to generate the code S of the control action, depending on the position of the model of the real state of the system in the entropy-parametric space and specified by the number s. The applicant draws attention to the fact that determining the number s of the real state model allows you to purposefully change the shape of the distribution to its target form by differentiated regulation of links 1315 and 1317 according to the code S. At the same time, proportional regulation of links taking into account the code S ensures minimization of the mismatch of the entropy-parametric potential to ensure the scale model of a stochastic system to the scale of its target distribution.

The optimal state of a stochastic system is a state tracked by the system and specified by a target distribution density with known shape and scale parameters. The entropy-parametric potential of the optimal state of the stochastic system is also given by the target distribution of the output parameter.

The real state of a stochastic system is the state of the system at the current moment in time, the characteristics of which are determined by a sample of the values of the output parameter. The parameters of the distribution of the values of the output parameter are set for the real state model. Therefore, the choice of the form and scale of the model is carried out in relation to the real state of the stochastic system.

The control is aimed at ensuring the minimum mismatch in the estimate of the entropyino-parametric potential of the controlled distribution of the output parameter for the model of the real and optimal states of the stochastic system.

The system is optimized by minimizing the mismatch of the entropy-parametric potential between the optimal and the real state by changing the parameters of the real state model and generating a control action to regulate links 1315 and 1317, shown in figure 13.

The proposed invention can also be used for monitoring and control during the evolution of the system or its cyclical change. Since for such systems there is a transition from one optimal state to another optimal state, it is obvious that at different stages of the evolution of the system, control is carried out relative to various optimal states, for which the boundary of the control zones is set a priori.

Thus, the proposed method of monitoring and control over a stochastic system with an unbiased asymmetric target distribution density eliminates the transition of the system to an inoperative state due to the ambiguity of the choice of forms of asymmetric models and ensures the stability of the system by monitoring the conformity of the shape and scale of the model to its real state in the entropy-parametric space signs of asymmetric distribution of the output parameter.

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Claims (38)

A method of monitoring and control over a stochastic system with an unbiased asymmetric target distribution density of the output parameter, which consists in the fact that the state of the stochastic system is registered, a sample y _{i of the} values of the output parameter y (t) of the stochastic system is formed, the mathematical expectation M ^* , the standard deviation σ * and the central fourth moment μ _{4 of the} distribution of the output parameter; formation of a database of reference parameters of the distribution law of the output parameter of the stochastic system, checking the state of the stochastic system belonging to the control zone of the optimal state; transformation of the distribution law of the controlled parameter by changing the tuning parameters of the controller, correction of the distribution law and the formation of a control action, characterized in that additionally carry out determination of the coordinates of the space of form features - asymmetry, kurtosis and the entropy coefficient of an asymmetric distribution - for an asymmetric unbiased distribution density of the output parameter of a stochastic system using the matrix vector formula:

where

- the second initial moment, the third central moment and the entropy potential of the asymmetric unbiased distribution density of the output parameter, determined by the formulas:

where m is the number of grouping intervals; Δy is the grouping interval of the output parameter; p _j is the statistical frequency of counts falling into the j-th grouping interval; formation of the space of the optimal state of the stochastic system, which includes the following actions: - setting the border of the control zone in the space of signs of the optimal state with an asymmetric distribution of the output parameter of the stochastic system:

where γ _max is the radius of the control zone; ξ _iγ - coordinates of the control border; - determination of the normalized position coordinates for the state of the distribution of the output parameter in the space of signs of the optimal state of the stochastic system according to the matrix vector formula:

where Sk ₀ , Ex ₀ , K _Hn0 - asymmetry, kurtosis and entropy coefficient of asymmetric distribution density for the optimal state of the system; a, b, c - coefficients of the intervals of signs for the control zone; - determination of the criterion for the state of the stochastic system in the space of signs of the optimal state according to the formula:

- determination of the normalized coordinates of the positions of possible models of the stochastic system in the space of signs of the optimal state according to the formula of the matrix vector:

where Sk _s , Ex _s , K _Hns - asymmetry, kurtosis and entropy coefficient of asymmetric distribution density for the optimal state of the system; s - model number of the stochastic system:

determination of distances between the positions of possible models and the real state in the normalized centered space of signs of the optimal state of the stochastic system according to the formula

choice of the form of the stochastic system model from the condition of the minimum distance between the positions of the model and real states

determination of the scale of the distribution model of the output parameter of the stochastic system, which includes the following actions: - determination of the entropy-parametric potential

asymmetric distribution density of the output parameter of the stochastic system

, - formation of the entropy-parametric potential of the asymmetric model of the distribution density of the output parameter of the stochastic system

where m _2λ and K _Hnλ are the second initial moment and the entropy coefficient of the asymmetric model of the distribution of the output parameter; - determination of the scale parameter from the condition of minimum difference in entropy-parametric potentials for the asymmetric model and the real state of the distribution density of the output parameter of the stochastic system

storing the parameters of the stochastic system model in the database; determination of the mismatch of the entropy-parametric potentials of the model of the real and optimal states of the stochastic system

minimizing the mismatch of the entropy-parametric potentials of the model relative to the optimal state by means of targeted correction of the distribution law and the formation of a control action for correcting the connections of the stochastic system and mapping the state of the stochastic system in the entropy-parametric space of the output parameter features.

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