WO2022191733A1 - Method for monitoring and controlling a stochastic system - Google Patents

Abstract

The invention relates to methods in dynamic systems for organizing monitoring and control of a stochastic system with an unbiased asymmetrical target distribution density of the output parameter. On the basis of a registration of the state of an object and the structure of the values of the sample parameter, coordinates of the entropy-parametric feature space of shape features for an asymmetrical unbiased distribution density of the output parameter are determined. To ensure the uniformity of optimal control at different points of the entropy-parametric space, a space of an optimal system state is created. Unification of data near to different optimal system states when choosing a model shape is achieved by setting a boundary of the control zone, determining normalized position coordinates for the distribution of the output parameter in the feature space of the optimal system state, determining a criterion for the system state, and determining position coordinates of possible models in the feature space of the optimal state. Minimizing the inconsistency of the entropy-parametric potential of the asymmetric distribution of the output parameter of the stochastic system provides control over the scale of the distribution model of the system.

Description

METHOD FOR MONITORING AND CONTROL OF STOCHASTIC SYSTEM

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 for various dynamic systems with chaotic properties.

There is a known method of control by a stochastic process, which consists in selecting a n-dimensional vector of the output parameter Y, determining and minimizing the value of the standard deviation (RMS) 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, which ensures the minimization of the variance or root mean square error of the output parameter. Methods for optimal filtering of vector random signals are implemented in solving the problems of constructing the optimal Wiener filter and the problems of constructing Kalman filter systems [1, 2].

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

There is a known method for controlling the conditional distribution density of the output signal for generalized nonlinear and non-Gaussian stochastic systems [4], in which the optimal control is achieved by embedding a recursive formula into the dynamic system and minimizing the difference between the conditional output and the target probability density. In this case, the control feature is to use a partial approximation of the trajectory as the target distribution density. The main disadvantage of control is to replace 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 registering the state of an object, forming a sample y, values of the output parameter y(t), determining the mathematical expectation M, the standard deviation s and the fourth central moment C4 of the distribution of the output parameter, the entropy potential of the output parameter D _/ *; 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 controller parameter, adjustment of the distribution law and formation of the control action.

As follows from the claims, in the known method of control and management of a dynamic system, the entropy coefficient of the symmetrical distribution and counter-kurtosis is determined; determination of the entropy-parametric criterion of the area of optimal control; determination of the value of the entropy-parametric potential of symmetric distributions; minimizing the value of the entropy-parametric potential of the dynamic system and adjusting the real parameter of the distribution law of the output parameter [5].

As disadvantages of this method of control and management of a dynamic system, the following should be noted:

- restriction of system behavior models only by a set of symmetrical forms of distributions of the output parameter;

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

- lack of estimation of the optimal zone boundary;

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

Brief description of the drawings

The figure 1 shows an example of controlling 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 for monitoring and controlling a dynamic system. The figure 3 shows a topographic diagram illustrating the state of the dynamic system of the output parameter when implementing the known method of monitoring and control.

Figure 4 shows drawings of models for setting entropy and parametric uncertainties for symmetric and asymmetric distributions: a) an uncertainty model for a symmetrical distribution; b) uncertainty model for asymmetric distribution;

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

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

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

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

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

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

The 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.

The 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 detailed description material contains explanations of 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 the class of nonlinear stochastic systems is the problem of tracking the distribution density track with respect to the target distribution density function of the output parameter [6]. For such systems, the goal of control is to minimize the difference between a partial approximation of the distribution density of the controlled parameter at the output of the system and an approximation given as the target distribution density. Among the control features, it is noted that it is necessary to choose 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].

The figure 1 shows an example of controlling the conditional distribution density of the output parameter for a generalized non-linear and non-Gaussian stochastic system. When controlling the non-Gausian 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 over a limited range 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, it should be noted the sensitivity of the distribution track to the current values of the output parameter, 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 fourth-order symmetric power series. With this approach, the appearance of instability of the distribution density track of the output parameter is due to the finite probability of the appearance of values on the tails of the real distribution, which are far from the center of the distributions of 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 a recursive algorithm to provide a 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 accepted model of the stochastic system, since the accepted approximation of the objective function is valid for a large variety of models of non-Gaussian 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 the analysis of distributions is the tendency of distributions to cluster by approximately one line to a greater extent than is permissible for the actual detection of the variability of sample observations belonging to different types of distributions [8]. For this reason, in known distribution analysis systems, after approximate identification of distributions, the distribution is selected by establishing the fairness 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 An of the output parameter, which is equal to the product of the value of the standard deviation s by the entropy coefficient A _// [1 1]:

D _// = s^ , ( 1 ) 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 D _n is defined as half of the range of uniform distribution in the interval from {An) to An, having the same entropy as that of a particular parameter.

The disadvantage of the control and management method based on the analysis and target change of the entropy potential is the use of the entropy coefficient K _n to control the degree of transformation of the distribution law. In this case, the system, when minimizing the entropy potential, 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 monitoring and managing the state of system uncertainty, based on the analysis and purposeful change in the value of the complex entropy potential 1d of the parameter of the system, equal to the ratio of the product of the entropy coefficient K _n and the standard deviation to the base value Xb, relative to which the state of uncertainty of the system is considered [13]:

_ K _H s AH ~ (2)

X b ,

The mathematical expectation of the parameter, its range, the nominal value or the base value of the entropy potential are used as the base value. Control is achieved by determining the complex entropy potential after each stage of the system evolution and controlling the system by the magnitude of the increment or differential of the complex entropy potential by changing the base value settings that specify the range of parameters, limit or nominal values, or the base value 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 _A of the parameter, is to determine the entropy potential proportional to the entropy coefficient Kn, the value of which is determined by the distribution law of the parameter. The point is that for a fixed value of the entropy coefficient, a set of both symmetric and asymmetric distributions with different shapes and properties is possible [14, 15]. For this reason, the use of the base value X _in when controlling the complex entropy potential L _A H does not guarantee the preservation of the distribution form of the output quantity.

A known method of control and management of a dynamic system based on minimizing the entropy-parametric potential of the symmetrical distribution of the output parameter in the space of features of the entropy coefficient and counter-kurtosis. The flowchart of the steps for the known method for monitoring and controlling a dynamic system is given in Figure 2, where the following designations are given: step 205 - step registration of the state of the object; step 210 - generating a sample of output parameter values; step 215 - determining the mathematical expectation, the standard deviation and the fourth central moment; step 220 - formation of a database of reference parameters of the distribution of the output parameter; step 225 - determining 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; step 235 - transformation of the distribution law; step 240 - determining the entropy-parametric potential of the symmetrical distribution; step 245 - minimizing the value of the entropy-parametric potential of the symmetrical distribution; step 250 - adjustments distribution law; step 255 - the formation of the control action; step 260 - the state is optimal.

According to the process diagram, at step 205, the state of the object is registered, including preliminary amplification, change and digitization of the output parameter of the system y(t), by registering the value y, at known discrete time intervals At. At the next stage 210, a sample of values of the output parameter y(t) is formed to construct a histogram and to estimate the probability of observing p _j the recorded values" on the interval [", "+Dy].

When monitoring and controlling a stochastic system, a continuous process of forming a sample of values of the output parameter is necessary, which preserves the entropy and parametric properties at any time. The preservation of the properties of the sample at any moment of the system operation is achieved by conducting observations during the time cycle T _p characteristic of the system, which has chaotic properties, for example, the engine operation cycle, the heart contraction cycle, and others. The number of values n of the generated sample is equal to the ratio of the cycle period of the system to the time interval of one measurement D / _and .

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 generated sample of values, the mathematical expectation is determined

_{* *} m , standard deviation (RMS) s and the fourth central moment m ₄ ^* according to the formulas:

To carry out the process of statistical measurement and control, it is necessary to assess the discrepancy between the sample distribution parameters and the reference parameters of the output parameter. For this reason, at step 220 of the diagram in figure 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 database of distribution models, the parameters of which are taken as reference parameters for the optimal state of the system.

где к и Кн - контрэксцесс и коэффициент энтропии для симметричного распределения выходного параметры, определяемые по выражениям:

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

where k and kn are the counter-kurtosis and entropy coefficient for the symmetrical distribution of the output parameters, determined by the expressions:

Here An _{s is} the entropy potential of the symmetric distribution:

где /с_{р n-> о}, - коэффициент нормального стандартного распределения. At step 230 of the process diagram in figure 2, the state of the system is checked for belonging to the area of optimal control. If the criterion g exceeds its maximum value y _max , then the dynamic system has left the region of optimal control, and for further control it is necessary to take strict measures to ensure the return of the system to the region of the optimal state. To do this, it is necessary to determine the possible transformation of the distribution law of the output parameter and form a control action for the controls of the object. The transformation of the distribution law of the controlled parameter is carried out at step 235 of figure 2 of the process diagram by changing the setting 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 A _HPs . 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

where /c _{p n-> o} , is the coefficient of the normal standard distribution.

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

D _HP s^ min (1 1)

If the entropy-parametric potential of the symmetric distribution A _H P _S does not reach its minimum value, then the analysis of the required adjustment of the distribution law is performed at step 250, based on the results of which, at step 255, a control action is formed on the controls of the object. When the minimum entropy-parametric potential is reached, the monitoring and control system takes the state of the object as the optimal state and saves 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 “Optimal state”

A feature of the implementation of the known method of control and management over a dynamic system is illustrated by the topographic diagram in figure 3, which shows:

310 - point of position 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 (dashed curve), limiting part of the space of the most probable positions for the optimal states of the dynamic system;

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

330 - curve of the position of the system with two-modal 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 in the distribution of the output parameter of the Chapeau 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 - point of the system position at the normal distribution of the output parameter (belongs to the curve 325);

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

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

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

390, 395 - points limiting the section of the curve for the positions of the system with arc-cosine distributions;

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

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

When controlling the asymmetric distribution, the state of the output parameter is displayed on the plane of symmetrical distributions of the output parameter, which causes the appearance of the ambiguity of the object's behavior. The point is that the entropy coefficient of a symmetric distribution, which is 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 the uncertainties of different intervals and becomes strongly dependent on the distribution scale parameters. This way of implementing management and control creates conditions under which the coefficient of entro- PII calculated by the symmetric distribution formula corresponds to many forms of the asymmetric distribution, which makes the choice of the form of the asymmetric distribution density of the output parameter multivalued.

Thus, the known method of distribution control is limited and comes down to control of the shape of the approximating model of the symmetrical distribution. When controlling a system with asymmetric distributions of the output parameter by replacing the real asymmetric distribution with its approximate symmetric model, the system goes into non-working states. The point is that with the same model of symmetric distribution, the ambiguity of the choice of asymmetric models is preserved.

The Importance of Controlling Skewed Unbiased Distributions

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

one . 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, rupture or fracture force, material fatigue [24].

5. Analysis of resources [24].

6. Models of distribution of survival probabilities [25, 26, 27].

7. Monitoring of income distribution of firms and individuals in statistical models of the economy, models of the stock market [28].

8. Statistical analysis in the processing of magnetic resonance images in computed tomography [29, 30].

9. Radionuclide studies and diagnostic analysis of nuclear medicine [31, 32].

10. Modeling the kinetics of pharmacological drugs in functional studies [32, 33, 34].

eleven . Modeling of 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 problem is to minimize the deviation of parameters for an asymmetric unbiased distribution density of the output parameter

The author of the alleged invention believes that control of the shape and scale of distribution models in the entropy-parametric space of signs of an 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 the targeted adjustment of the distribution law and the formation of control influences for correction of connections of stochastic systems.

It is obvious that it is necessary to display models of the 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 inoperative 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.

где m₂ ^*, m₃* и D_H ^* _h - второй начальный момент, третий центральный момент и энтропийный потенциал несимметричной несмещённой плотности распределения выходного параметра, определяемые по формулам:

где т - число интервалов группирования; Ду - интервал группирования выходного па- раметра; p_j - статистическая частота попадания отсчётов в у-й интервал группирования; формирование пространства оптимального состояния стохастической системы включа- ющее в себя следующие действия: To do this, in the method of monitoring and control by a stochastic system with an unbiased asymmetric target density distribution of the output parameter, which consists in registering the state of the stochastic system, forming a sample y, values of the output parameter y(() of the stochastic system, determining the mathematical expectation M ^* , the standard deviation o* and the central fourth moment T _{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; parameters of the controller, the adjustment of the distribution law and the formation of the control action, characterized in that they additionally determine the coordinates of the space of the attributes of the form - asymmetry, kurtosis and coefficient entropy coefficient of asymmetric distribution - for asymmetric unbiased distribution density of the output parameter of a stochastic system according to the matrix vector formula:

where m ₂ ^* , m ₃ * and D _H ^* _h - 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; Du - grouping interval of the output parameter; p _j - statistical frequency of hitting samples in the y-th grouping interval; formation of the space of the optimal state of the stochastic system, which includes the following actions:

где у_{Т1ах -} радиус зоны контроля; x_{1g -}координаты границы контроля; определение нормированных координат положения для состояния распределения вы- ходного параметра в пространстве признаков оптимального состояния стохастической системы по формуле матричного вектора:

где Sko, Exo, Кц„о асимметрия, эксцесс и коэффициент энтропии несимметричной плотности распределения для оптимального состояния системы; а , Ь, с - коэффициенты интервалов признаков для зоны контроля; определение критерия для состояния стохастической системы в пространстве признаков оптимального состояния по формуле:

- определение нормированных координат положений возможных моделей стохастиче- ской системы в пространстве признаков оптимального состояния по формуле матрич- ного вектора:

где Sk_s, Ex_s, Ки_т асимметрия, эксцесс и коэффициент энтропии несимметричного плотности распределения для оптимального состояния системы; s - номер модели стохастической системы: - 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: , max ' (16)

where y _{T1ax is} the radius of the control zone; x _{1g -} control border coordinates; determination of the normalized position coordinates for the distribution state of the output parameter in the feature space of the optimal state of the stochastic system using the matrix vector formula:

where Sko, Exo, Кц„о are the skewness, kurtosis and entropy coefficient of the asymmetric distribution density for the optimal state of the system; a, b, c - coefficients of intervals of signs for the control zone; determination of the criterion for the state of the stochastic system in the space of features 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 features of the optimal state according to the formula of the matrix vector:

where Sk _s , Ex _s , Ki _t are the skewness, kurtosis and entropy coefficient of the asymmetric distribution density for the optimal state of the system; s - model number of the stochastic system:

_ ( {1, 2, 3, ... } For possible states,

( 0 for the optimal state; determination of the distances between the positions of possible models and real states in the normalized centered space of signs of the optimal state of the stochastic system by the formula

выбор формы модели стохастической системы из условия минимума дистанции между положениями модели и реального состояний AL _S : (20)

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

AL _S ->min; (21 ) 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 А _Н ^* _Р of the asymmetric distribution density of the output parameter of the stochastic system

где тг_\ и Кн„_\ - второй начальный момент и коэффициент энтропии несимметричной мо- дели распределения выходного параметра; - formation of the entropy-parametric potential of the asymmetric model of the distribution density of the output parameter of the stochastic system

where tg _\ and Kn „ _\ - the second initial moment and the entropy coefficient of the asymmetric distribution model of the output parameter;

минимизацию рассогласований энтропийно-параметрических потенциалов модели отно- сительно оптимального состояния посредством целевой корректировки закона распреде- ления и формирования управляющего воздействия для коррекции связей стохастической системы и картирование состояния стохастической системы в энтропийно- параметрическом пространстве признаков выходного параметра. - determination of the scale parameter from the condition of the minimum difference of entropy-parametric potentials for an asymmetric model and the real state of the distribution density of the output parameter of a stochastic system |D^ _R ^HP min; (24) saving 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

minimization of discrepancies between the entropy-parametric potentials of the model relative to the optimal state by target adjustment 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 features of the output parameter.

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 signs of the form of the asymmetric distribution density 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;

- choosing the form of the distribution model of the output parameter from the condition of minimum distance between the position of possible models and real states in the normalized centered space of features in the space of the optimal state of the system;

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

- minimizing the mismatch between the model parameters and the parameters of the optimal state of the distribution of the output parameter by target correction of the links of the stochastic system.

In the intended application, the author uses a coordinate space built simultaneously on probabilistic and informational features of distributions to display and select the shape of models of nonsymmetric 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 sharpness of distribution. For the formation of informational coordinates, the author of the alleged invention proposed the entropy coefficient of asymmetric distributions, which characterizes the property of disorder in the asymmetric distribution of the output parameter. [fifteen]. Figure 4 illustrates the peculiarity of constructing the entropy coefficient of an asymmetric distribution, where drawings of models for specifying entropy and parametric uncertainties with a symmetric and asymmetric distribution are given.

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

H (X) \u003d M (- In p (x)) . (26)

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

The Shanonn uncertainty model for symmetric distributions was proposed in the works of Novitsky P.V. [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 the measurement under the condition of obtaining a reading x _p with a uniform distribution of uncertainty over the 2D interval is equal to the logarithmic measure of the 2D entropy uncertainty interval:

SHCH / x ) \u003d - f - In - <& \u003d 1h (2D) . (28) p J ? 2Dd 2 ?Dl

Then the entropy measurement error D, equal to half of the uncertainty interval 2D 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 systems of control and management of an object, minimization of the difference between the mismatch of the output parameter and the measure is ensured by constructing a negative feedback. As a measure of the uncertainty of the state of an object, Lazarev V.L. proposed using half of the entropy interval of uncertainty to build control over the object and consider it as the entropy potential of the object [9]. A drawing of the uncertainty model for a symmetrical 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 symmetrical distribution model

The entropy coefficient of the symmetric distribution is entered as the ratio of the entropy interval 420 of the mismatch uncertainty of the output parameter to its parametric interval 415, equal to twice the values of the entropy potential of the symmetric distribution 2A _Hs and the standard deviation 2s. The formula for determining the entropy coefficient of a symmetrical distribution is:

The entropy coefficient of the symmetric distribution KH _s , 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 sign 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, are based on the conditional engropy of the distribution, provided that an estimate of the mean value is 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 distribution probability density is zero for negative values of the output parameter. Since the limit expression under the Shannon integral (20) is equal to zero, lim(z In z) = Hm(ln z ^z ) = In 0° = In 1 = 0 z-> o z-*0 output parameter, an expression of the form

When using the uniform distribution density as a model of an unbiased asymmetric distribution density of the output parameter, the interval is set from 0 to its maximum value Дн„, taken as the entropy potential of the asymmetric unbiased distribution. Model drawing undefined data for asymmetric distribution is shown in figure 4, b, where the following notation is used:

425 - asymmetric uniform distribution model;

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 the entopy potential An _p of an asymmetric distribution model;

440 - 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 define different essences of distributions.

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

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

The differential entropy for determining the uncertainty of the position of the center of the asymmetric distribution is calculated as the conditional entropy of the 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 about the center of distributions is not equal to the entropy of asymmetric distribution (31 ), then the intervals of entopy uncertainty for a symmetric model 420 and for asymmetric model 435 also have different values. Interval 420 is plotted relative to the distribution center and offset from the origin.

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

The entropy coefficient Кн _p for the asymmetric distribution of the parameter is equal to the ratio of the entropy uncertainty interval 435 to the parametric uncertainty interval 430 of the output parameter, which are equal to the entropy potential of the asymmetric distribution D _/ *, and the square root of the second initial moment _\ Jm ₂ , respectively.

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

The entropy coefficient of the nonsymmetric distribution (33), equal to the ratio of the entropy potential of the nonsymmetric 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 sign of the form of the nonsymmetric distribution [15].

The entropy of a non-symmetric unbiased distribution has the property of proportionality with respect to the entropy of a symmetric distribution obtained by reflecting the non-symmetric distribution about the origin. If /„( C,a,t,l ) is the density of asymmetric distribution, then the density of the corresponding symmetric distribution has the form:

Since the range of the output parameter doubles for the symmetric distribution, linear proportionality is valid for the entropies of the symmetric and nonsymmetric distributions of the form:

H ( , a, b, l) - H _p (y, a, b, l) + In 2 (35)

Since the expression (28) for the conditional entropy and measurement is valid for the symmetric distribution, provided that the reading х _р is equal to zero, then writing the logarithm of the product as the sum of logarithms, we obtain an expression of the form for the entropy of the symmetric distribution: H _s (y, a, b, l) = In 2 + In A _WI . (36)

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

H„(g,a,b,l) = \hA _Hh . (37)

The formula for calculating the entropy potential of the asymmetric distribution А _Ип , with a known entropy #„( ,a,b,l) will take the form: n _p = exp(R„(^,а,рД)). (38)

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

то коэффициент энтропии несимметричного распределения можно получить как коэффи- циент энтропии симметрированного распределения, энтропия которого определена свой- ствами несимметричного распределения. Since the root-mean-square deviation as for the distribution of the parameter symmetric with respect to the origin is equal to the square root of the second initial moment m2, calculated for the initial asymmetric unbiased distribution of the output parameter y

then the entropy coefficient of a nonsymmetric distribution can be obtained as the entropy coefficient of a symmetric distribution, the entropy of which is determined by the properties of the nonsymmetric distribution.

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

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

Possible positions of the object in the coordinate entropy-parametric space of signs 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 distributions of the output parameter: - figure 5, a - the projection of the trajectories onto the plane specified by the signs of asymmetry and the entropy coefficient of the asymmetric distribution;

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

- figure 5, c - projection of the trajectories onto the plane specified by the signs of asymmetry and kurtosis of the asymmetric distribution.

The following designations are used on the diagram projections:

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

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

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

520 - curves of the system position when using for the output parameter the target distribution 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 system position when used for the output parameter of the target distribution from the forms of the logarithmic normal distribution;

535 - position point of the system when using the form of asymmetric exponential distribution of the output parameter, coinciding with the Pearson distribution c with three degrees of freedom;

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

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

555, 560, 565 - curves of the system position 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 possible values of which correspond to the values of 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 optimal system state 505 under asymmetric distribution is on the curve 520 of possible system positions when using the target distribution for the output parameter from the forms of the family of gamma distributions. The real 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 environment that make up the system.

The point of position of the actual state of the system 510 is near the optimal state. Since the real state does not coincide with the possible models of its state, the position point 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 reflected in the change in the shape and scale of the distribution model of the output parameter.

For this reason, when monitoring and controlling, 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 boundary of the control area 515 with known acceptable forms of models of asymmetric distributions of the output parameter is allocated, which are stored in the database of reference parameters of the distribution of the output parameter. When the system goes beyond the control area, the distribution law of the controlled parameter is transformed due to a change in the setting 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 proposed invention is convinced that displaying the position of an object in the coordinate entropy-parametric space of signs of asymmetric unbiased distributions of the output parameter allows realizing new opportunities for monitoring and controlling a stochastic system.

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

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

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

- step 610 of forming the space of the optimal state of the system; - step 615 of determining the distance between the positions of the possible models and the actual states;

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

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

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

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

- stage 640 optimization mismatch parameters of the model;

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

Determining the Coordinates of the Shape Feature Space for an Asymmetric Unbiased Distribution Density of the Output Parameter of a Stochastic System

The first action, illustrated by step 605 of the flowchart in FIG. 6, is to determine the shape feature space coordinates for the nonsymmetric distribution density. The space coordinates are given by dimensionless features of the distribution form and are calculated by the matrix vector formula (12). Asymmetry and kurtosis characterizing the skewness and peakedness of the distribution were used as parametric features of the distribution shape. The third coordinate is given by an independent information sign of the distribution form - the entropy coefficient of the asymmetric distribution. Displaying the distribution of the output parameter of the system in a three-dimensional space of attributes of the distribution forms provides control over 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 by step 610 of the flowchart in FIG. 6 is to obtain the positions of the models and the position of the state of the nonsymmetric distribution of the system output in the normalized coordinate space of the optimal state 505 with the nonsymmetric distribution of the output. The scheme 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 action of the proposed method for monitoring and controlling a stochastic system with an unbiased asymmetric target distribution density of the output parameter contains

- step 715 of 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 distribution state of the output parameter in the feature space of the optimal state of the stochastic system;

- step 725 to determine the criterion for the state of the stochastic system in the feature space of the optimal state†

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

The essence of the scheme of the process of forming the space of the optimal state of the system is to determine the coordinates of possible models near the optimal state within the control boundary.

±с) по признакам асиммет- рии, эксцесса и коэффициента энтропии несимметричных распределений. Step 715 of the process diagram in Figure 7 is to set the boundaries of the control zone in the feature space of the optimal state 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 for changing the signs for the control zone are set using the product of the value of the sign of the optimal state (Sko, Ex o, Kcn o) and the coefficients of the intervals of the signs a, b, c. Then the position of possible values in the entropy-parametric space of states of the output parameter is limited by the values of features Sk ₀

±c) on the basis of asymmetry, kurtosis and entropy coefficient of asymmetric distributions.

The boundaries of the feature intervals of possible models can be determined in two ways. The first method consists in modeling the object and obtaining the bounds of parameters based on the model using the Monte Carlo method by superimposing the output parameter and models of destabilizing factors on the models. In this case, the signs of the optimal state Sk o, Ex o, K _Np o are determined by the properties of the system model. Another approach is related to the accumulation of statistical data on the stochastic system and the estimation of the intervals of signs of the control zone, provided that the system (or similar systems) is in a healthy state for a controlled mode of operation. In this case, as the optimal entropy-parametric features of the system Sk o, Ex o and Kn _p about estimates of the average value of features obtained from previous observations are accepted. The mean square estimates of the spread are used as feature intervals in statistics. Since the methods of analysis and statistical accumulation use different methods for estimating the intervals of features, it is convenient to use the limits of the intervals of features 5Ά: ₀ (ΐ ±o) to set the border of the control zone

±с) с указанием вектора признаков оптимального состояния [5Vco, Ex о,£x ₀ (l

±с) indicating the vector of signs of the optimal state [5Vco, Ex o,

Kn _o ] ^T and vectors of dispersion coefficients [o,6,s] ^t The control zone boundary is established from the condition that at least 95% of operating states are located inside the boundary under the influence of various destabilizing factors. Going beyond the control boundary is considered as a transition to an inoperable state that requires correction of the parameters and probability matrices of the relationships of the stochastic system. When setting the control boundary in the form of a sphere, the parameter y _max of the boundary radius is in the range from 1 to 3. If the spread coefficients are determined for the values of the mean square spread, then to find 95% of the operable states, the parameter of the boundary radius y _max has the value , equal to 2.8. With known boundaries of admissible states of the system in the space of entropy-parametric features, it is convenient to use as a working space the space centered relative to a given optimal state and normalized to the spread of features of asymmetry, kurtosis, and the entropy coefficient of asymmetric distributions of the output parameter.

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 - the position point of the optimal state with an asymmetric distribution of the output parameter; 510 - position point of the real state of the system with asymmetric distribution of the output parameter; 515 - control zone border; 520, 525, and 530 are curves of the system position when target distributions from the Weibull-Gnedenko family forms, the gamma distribution family, and the lognormal distribution forms are used for the output parameter, respectively; 535 - position point of the system model when using the form of asymmetric exponential distribution of the output parameter, which coincides with the Pearson parameter c ² with three degrees of freedom; 910, 920, and 930 illustrate the position points of possible system state models using the forms of the Weibull-Gnedenko family, the gamma distribution family, and the lognormal distribution, respectively; 915, 925 and 935 - position points of the system state models when using the system position curves 520, 525 and 530 as distributions of the output parameter pa forms of the Weibull-Gnedenko family, the families of the gamma distribution and the logarithmic normal distribution, respectively. Diagram 9 additionally gives the position of the system state model 940 when using the curve of forms of the Pareto family 555. space, illustrating the possible position of the model in the inoperable state of the system.

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

To display the state of the system in the feature space of the optimal state, at step 720 of the process in Figure 7, the normalized coordinates of the state position are determined by the formula of the matrix vector (17). The state of the output parameter system on the projection of the optimal state of the system with an asymmetric distribution of the output parameter is shown on the projections of the control zone (figure 9, figure 10 and figure 11) in the form of a position point 510. For illustration, a state point with signs of asymmetry, kurtosis and entropy coefficient is used asymmetric distribution of the 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 given by the form parameter a 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 entropy coefficient 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 adopted as the form of the target distribution of the output parameter of the stochastic system. The intervals of signs of asymmetry, kurtosis and entropy coefficient calculated for the control zone are 0.58, 3.5 and 0.225. The coefficients of intervals of features a, b and c for the control zone are 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 figure 7, the criterion y ^* is determined for the state of the stochastic system in the feature space of the optimal state by the formula (eighteen). 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 feature space of the optimal state according to the 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 model parameters. In order to unify the data under different conditions, the control intervals near the optimal state should be taken into account.

In the control zone of the optimal state, the number of distinguishable forms of models is limited by the difference in the shape parameter of possible models. When controlling the shape to the third significant figure 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, and 150 models from the Weibull-Gnedenko family with shape parameters from 0.8 to 0, 95. Determining the coordinates of possible models according to the 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 a normalized centered space of signs of the optimal state of a stochastic system

The third distinctive action of the process diagram in Figure 6, illustrated by step 615, is to determine the distance between the position of the possible models and the actual states for nonsymmetric stochastic system output parameter distributions in the normalized coordinate space of the optimal state. The positions of the possible models from individual families correspond to the model 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 a non-symmetrical distribution of the output parameter to the curve of the positions of the system when using the family of possible models.

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

The fourth distinctive action, carried out by step 620 of the process diagram in figure 6, is to select the shape of the model from the condition of minimum distance between the models - pour and the real state. Models are selected from a family of distribution shapes and model shape parameter estimation from the model database.

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

Existing algorithms make it possible to estimate the shape of the distribution of the output parameter only approximately. The point is that the parametric characteristics of the form of asymmetry and kurtosis do not allow one to obtain the form when using several subfamilies of generalized distributions. The close location 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 model form.

In the proposed invention, the quality of the choice of form is carried out from the condition of the minimum (21 ) between the position of the model and the state of the system. The quality of the choice of form is ensured due to the distinguishability of asymmetric distributions in 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.

A further feature of the contemplated invention is to determine the scaling of the distribution of the output parameter of the stochastic system in step 625 of the process diagram of Figure 6. The process diagram for determining the scaling parameter of the distribution of the output parameter is given in Figure 8 as steps 805, 810, and 815. From Figure 8 it follows that the fourth distinctive action 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;

- formation at step 810 of the entropy-parametric potential of an asymmetric model of the distribution density of the output parameter; - determination at step 815 of the scale parameter l from the minimum condition (24) of the difference between the entropy-parametric potentials for the asymmetric model and the actual state of the distribution density of the output parameter.

To explain the actions taken when determining the scale of the distribution model of the output parameter of a stochastic system, figure 12 shows the projection of the space of entropy and parametric uncertainties of asymmetric distributions of the output parameter, where the following notation is given

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

- 1205 - the origin of the space of entropy and parametric uncertainties;

- 1210 - the distance of the entropy-parametric potential A ^* _w, the output parameter;

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

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

- 1225 - equipotential of the entropy-parametric potential А ^' _нр output parameter;

_ 1230 - the distance of the entropy-parametric potential A ° _NR of the optimal state;

- 1235 - equipotential of the entropy-parametric potential A _H1, the optimal state of the system;

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

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

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 Anp and the coordinate of the interval of parametric uncertainty equal to the square root of the second initial moment frn^ of the distribution of the output parameter.

In the space of entropy and parametric uncertainties of asymmetric distributions, the selected distribution model corresponds to the scale line of the 1220 model with the selected distribution form, on which different distances from the origins 1205 to the points of the line 1220 correspond to different scales l of the distribution of the output parameter. The slope of the line 1220 is completely determined by the ratio of the entropy coefficient of the non-symmetric distribution, determined from expression (33).

The point of position 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 non-symmetric 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 skewed distribution model differs from the entropy coefficient of the optimal state skewed distribution, then the scale lines of the selected model 1220 and the optimal state model 1245 exist as separate entities.

If the entropy coefficient of the skewed distribution of the output parameter Kn _p differs from the entropy coefficient of the selected skewed distribution model, then the position 510 of the actual state of the system with the skewed distribution of the output parameter is outside the scale line 1220 of the selected model. Fig. 12 of the projection of the space of entropy and parametric uncertainties of non-symmetric distributions schematically shows the scaling scheme for the distribution model with the selected shape, explaining the actions of the scheme in Fig. 8 of the process of determining the scale of the distribution model of the output parameter.

At step 805 of process diagram 8, the entropy-parametric potential of the unbiased distribution density of the output parameter is determined by 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 relation (33) in the space of entropy parametric uncertainties of asymmetric distributions of the output parameter is given in Figure 12, where the positions of the objects are shown schematically to illustrate the processes of determining and minimizing the distribution scale parameter.

The distance from point 1205 of the origin 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. Formula (22) was used to determine the entropy-parametric potential of the asymmetric distribution of the output parameter through the coordinates of the uncertainty space. In Figure 12, the entropy-parametric potential of the output parameter A ^* _H1 corresponds to the distance 1210 from the origin 1205 to the 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 positions 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 match of the scale parameter for the model 1215 is located on the scale line 1220 of 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 entropy and parametric space is not minimum certainty. At small mismatch angles, the position point 1215 of the model with the selected shape is found as the intersection of the model scale line 1220 and the equipotential entropy-parametric potential of the output parameter. To determine the scale parameter l of the model with the selected shape, the relationship between the scale parameter l of the model and the entropy-parametric potential of the output parameter is necessary.

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

The entopy-parametric potential for the distribution model 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 entropy and parametric uncertainty. The formula for determining the entopy-parametric potential of the model D ^l _HI, through the coordinates of the uncertainty space has the form:

If the form of the model is chosen, then the entropy coefficient Kn _n% for the model of asymmetric distribution is unambiguously known. The square of the entropy potential of the non-symmetric distribution is equal to the product of the entropy coefficient and the second initial moment /P2x of the non-symmetric distribution of the output parameter distribution model. For the entropy-parametric potential of the asymmetric distribution model of the output parameter, formula (23) is valid, according to which for the density of the output parameter, many models are possible that differ in different parameters of the distribution scale. For most asymmetric distributions, the entropy-parametric potential is proportional to the scale parameter of the model l. Then define- dividing the distance from the origin of coordinates 1205 to the position point of the model 1215 with a known shape of the model, we obtain the value of the scale l of the distribution of the output parameter.

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

Figure 12 illustrates the shape parameter selection process. Obviously, the best match between the model and the actual state of the system distribution can be achieved when the position points of the model 1215 coincide with the selected shape and the actual state 510 of the system. Since, due to the action of destabilizing external factors, the position of the point 510 of the actual state of the system is outside the line 1220 of the models of the selected shape, the position points of the model 1215 correspond 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 model shape line and the real state shape line does not exceed 1 ... 5°. In this case, the scale of the model is selected from the 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 save recoverable information about the state of the object with a minimum amount of transmitted data. For this reason, step 630 of the process flow in Figure 6 ensures that the model parameters are stored in a database. The preservation of features and distribution makes it possible to restore information about the state of the system.

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

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

For the optimal state, the second initial moment t _2° and the entropy potential A° _Hp of the asymmetric distribution of the output parameter, which characterize the intervals of entropy and parametric uncertainty, have their own individual values. In the space of entropy and parametric uncertainty on the figure

12 optimal state of the system with an asymmetric distribution of the output pa- the parameter corresponds to the position point 505. The distance 1230 from the origin 1205 to the 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 А° _n/, . For models with the selected distribution form, there corresponds a set of points on the scale line of the 1220 model, which differ in the distribution scale. The model is closest to the optimal state if the distance between the position point 505 of the optimal state of the system and the point 1240 of the position of the optimal state model on the scale line 1220 of the model with the selected shape is minimal. The position point 1240 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.

If the entropyino-parametric potentials of the optimal A° _n/, and real D ^* _n/, states of the stochastic system are different, then the models of the real and optimal states, given by points 1215 and 1240 , located on the figure 12 on the scale line of the model 1220, have different scale with the same shape. For this reason, at step 635 of the process diagram in Figure 6, the mismatch of the model of the real and optimal state of the system is determined. To determine the mismatch, the difference between the distances of 1210 and 1230 entropy-parametric potentials of the output parameter and the optimal state is calculated using 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:

(41)

Squaring the terms of expression (41) and regrouping the terms, we obtain 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 twice the sum of the products of the entropy and parametric uncertainties:

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

Minimization of expression (43) makes it possible to minimize the discrepancy between the real state model and the optimal state.

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: K _Np) = K _Np0 + AK _Np . 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 bound proportional 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, if the entropy-parametric potentials of the model and the optimal state of the mismatch are equal, the square of the intervals of entropy and parametric uncertainties is determined by the mismatch of the second initial moments and entropy coefficients of asymmetric distributions. Thus, expression (45) is a more stringent condition for determining and minimizing the mismatch than formula (25).

Minimizing the Mismatch of the Entropy-Parametric Potentials of the Model with Respect to the Optimal State

The next action, illustrated by step 640 of the flowchart 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

BA _1P, - > min. The minimization of the mismatch (1 1) of the entropy-parametric potentials is achieved by target adjustment 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 distribution scale parameter.

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 feature space of the distribution of the output parameter and in the form of projections of the control zone onto the normalized coordinates of the feature space of the optimal state. At step 260, a message is generated regarding the optimal state of the object.

A possible implementation of the material means, 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 the device for monitoring and controlling a stochastic system with an unbiased asymmetric target distribution density

To explain 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 notation is used:

1310 - subsystem of uncontrolled processes of the stochastic system;

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

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

1320 - controlled process of the 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 signs 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 scale parameter of the model;

1365 - device for mapping the state of the stochastic system;

1370 - database;

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

1380 - distribution transformation block;

1385 - a device for reproducing a measure of the EPG of the target distribution;

1390 - block target adjustment form distribution;

1395 - control action generation unit for adjusting the links of stochastic systems;

In the stochastic system, figure 13 highlights the subsystem of uncontrolled processes 1310 of the stochastic system, the controlled process 1320, the connections 13 5 of the uncontrolled processes of the stochastic system with the controlled process of the stochastic system, and the connections 1317 of influencing factors with the controlled process of the stochastic system. The impact on the controlled process 1320 is possible by changing the links 1315 of the uncontrolled processes of the stochastic system and by changing the links 1317 of influencing factors of artificial and natural origin.

Recording device 1330 records the output parameter of the controlled process of the stochastic system. The output parameter distribution generating device 1335 performs the process of generating a sample [Y] of the values of the output parameter of the stochastic system obtained from the recording device 1330. The form feature determination unit 1340 is designed to calculate the features of the shape of the output distribution of the stochastic system: asymmetry, kurtosis, and entropy coefficient of asymmetric unbiased distributions. The values of the shape features are transmitted to the optimal state space generator 1350, at the output of which a matrix vector [x ^* ] of normalized position coordinates for the output parameter distribution state and a matrix [x] of normalized position coordinates of possible models for the distribution states of the output parameter in the space of normalized shape features are formed optimal state of the stochastic system. The row number of the matrix [x] of normalized coordinates corresponds to the number s of a possible model of the stochastic system. The numbers of the columns of the matrix [x] determine the normalized coordinates of the positions of the possible models of standing of the stochastic system. The matrix vector [x ^* ] and the matrix [x] of the normalized coordinates of the positions of possible models are transmitted to the device for selecting the shape of the model 1355, at the output of which the number s of the shape of a 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 form are known for the models.

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 by the formula (22). The entropy-parametric potential D ^* _HR of the asymmetric unbiased distribution of the output parameter of the stochastic system characterizes the position of the mean value. Device 1385 is designed to reproduce the measure of the entropy-parametric potential of the target distribution. The discrepancy between the state of the system d DH _R is estimated from the difference between the entropy-parametric potential of the output parameter D ^* _HR and the entropy-parametric potential of the target distribution _HR . The discrepancy between the entropy-parametric potential of the real and optimal states of the system, corresponding to the difference in estimates of the average values of the output and target distributions, is used by the block 1395 for generating the control action c to adjust the links 1315 of the stochastic system. When forming the control action c, the position of the model s of the real state of the system is also taken into account. According to the model number l*, block 1390 provides a target distribution shape adjustment, which is taken into account by block 1395 when generating a control for adjusting the links of the stochastic systems. Thus, when minimizing the discrepancy between the entropy-parametric potential of the model relative to the optimal state, the deviation of the model form from the optimal state is taken into account by target correction of the stochastic system connections.

Based on the values of the entropy-parametric potential D _H ^* _R of the output parameter and the known number s of the form of the real state model, using the device 1360, the scale parameter of the model l 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 the stochastic system in the entropy-parameter feature space of the output parameter. Mapping is carried out according to the known shape parameters (a, b) of model 5 and the scale parameter l. To ensure the monitoring process, an entry is made in the database 1370 number .9, shape parameters (a, b) and scale l of the distribution model of the output parameter for the real state of the stochastic system.

The device 1375 for checking whether the state of the stochastic system belongs to the control zone of the optimal state allows you to control the exit of the system from the control zone by changing the signs of the shape of the output distribution. Distribution transformation block 1380 is used to generate a message h about the need for distribution transformation to return the system to the control zone. Distribution transformation can be achieved by changing the properties of the system by changing the setting parameters of the controller located in the subsystem of uncontrolled processes 1310 of the stochastic system, by changing the composition of influencing factors 1325 or by changing the choice of the optimal state, relative to which monitoring and control over the stochastic system is established.

It follows from consideration of the diagram in Figure 13 that to ensure the stability of the system, block 1395 generates control actions to adjust the links 1315 of the uncontrolled processes of the stochastic system and the links 1317 of the influencing factors with the controlled processes of the output parameter of the stochastic system.

The formation of the control action block 1395 to adjust the links 1315 and 1317 is subject to non-zero model number s and/or non-zero mismatch of the entropy-parametric potential. Block 1390 of the target adjustment of the form of the distribution law is designed to form 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 given by the number s. The Applicant draws attention to the fact that determining the number s of the real state model makes it possible to purposefully change the shape of the distribution to its target form by differential regulation of links 1315 and 1317 according to the S code. mismatch of the entropy-parametric potential to ensure the scale of the stochastic system model to the scale of its target distribution.

The optimal state of a stochastic system is a state tracked by the system and given 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 of time, the characteristics of which are determined by a sample of values of the output parameter. The distribution parameters of the output parameter values 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 of the estimate of the entropy-parametric potential of the controlled distribution of the output parameter for the model of the real and optimal states of the stochastic system.

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

The alleged invention can also be used to monitor and control the evolution of the system or its cyclic 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 with respect to different optimal states, for which the boundaries of the control zones are a priori set.

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

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Claims

A method for monitoring and controlling a stochastic system with an unbiased asymmetric target distribution density of the output parameter Claim A method for monitoring and controlling a stochastic system with an unbiased asymmetric target distribution density of the output parameter, which consists in registering the state of the stochastic system, forming a sample y, values of the output parameter y(t) of the stochastic system, determining the mathematical expectation M of the standard deviation o * and the central fourth moment C ₄ 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 setting parameters of the controller, adjustment of the distribution law and formation of the control action, characterized in that they additionally determine the coordinates of the space of the shape attributes - asymmetry, kurtosis and the entropy coefficient of the asymmetric distribution - for an asymmetric unbiased distribution density output parameter of the stochastic system according to the matrix vector formula:

where m , m and D^ _h are 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; Ay - grouping interval of the output parameters; p _j - statistical frequency of hitting samples in the /-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 y _max is the radius of the control zone; xi _g - control boundary coordinates; determination of the normalized position coordinates for the distribution state of the output parameter in the feature space of the optimal state of the stochastic system using the matrix vector formula:

where Sko, Ex o, Kn _o - asymmetry, kurtosis and entropy coefficient of the asymmetric distribution density for the optimal state of the system; a, b, c - coefficients of intervals of signs for the control zone; determination of the criterion for the state of the stochastic system in the space of features 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 features of the optimal state according to the formula of the matrix vector:

where Sk _s , Ex _. , Ku _ns - asymmetry, kurtosis and entropy coefficient of asymmetric distribution density for the optimal state of the system; s - model number of the stochastic system: _ ( {1, 2, 3, ... } for possible states,

I 0 for the optimal state; determination of distances between the positions of possible models and real states in a normalized centered space of signs of the optimal state of a stochastic system by the formula

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 AL _S -» min ; 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 D ^* _HR of the 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 ttk and K _Np - the second initial moment and the entropy coefficient of the asymmetric distribution model of the output parameter; - determination of the scale parameter from the condition of the minimum difference of entropy-parametric potentials for an asymmetric model and the real state of the distribution density of the output parameter of a stochastic system

min; saving 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

minimization of discrepancies between the entropy-parametric potentials of the model relative to the optimal state by target adjustment 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 features of the output parameter.