Probabilistic characteristics of signals. Characteristics of random signals Parameters of random signals
The properties of random signals are evaluated using statistical(probabilistic) characteristics. They are non-random functions and (or) numbers, knowing which, one can judge the patterns that are inherent in random signals, but appear only when they are repeatedly observed.
7.4.1. Characteristics of random signals that do not change in time
The main statistical characteristics of the signal represented by a random variable (7.2) are: distribution function 
, probability distribution density 
(PDF), mathematical expectation 
, dispersion 
, standard deviation (RMS) 
and confidence interval 
. Let's consider these characteristics.
,
(7.64)
where 
- event probability symbol 
.
.
(7.65)
Dimension of PDF 
reciprocal of the dimension of quantity 
.
,
(7.66)
The result of calculations according to this formula differs from mean value random variable and coincides with it only in the case of symmetric distribution laws (uniform, normal, and others).
The quantity is called a centered random variable. The mathematical expectation of such a value is zero.
4. Dispersion random variable determines the weighted average of the squared deviation of this variable from its mathematical expectation. The dispersion is calculated by the formula
(7.67)
and has the same dimension as the square of the quantity 
standard deviation calculated by the formula
and, in contrast to the dispersion 
, has a dimension coinciding with the dimension of the measured physical quantity. Therefore, RMS turns out to be a more convenient indicator of the degree of dispersion of possible values of a random variable relative to its mathematical expectation.
In accordance with the "three sigma" rule, almost all values of a random variable that has normal distribution law, fall inside the interval 
, adjacent to the mathematical expectation of this quantity.
6. Confidence interval
the range of possible values of a random variable is called, in which this variable is located with a predetermined confidence level
. This range can be written as 
, or in the form
those. the boundaries of the confidence interval are located symmetrically with respect to the mathematical expectation of the signal , and the area of the curvilinear trapezoid with the base 
is equal to the confidence level 
(Fig. 7.7). With growth 
confidence interval 
also increases.
half confidence interval 
can be determined by solving the equation
.
(7.70)
In the practice of engineering calculations, the most widely used among the listed statistical characteristics of a random signal is the PDF 
. Knowing PDF, it is possible to determine all other statistical characteristics of the signal. Therefore, the function 
is complete statistic random signal.
Let us point out the main properties of the PDF:
2.
and
, i.e., knowing the PDF 
, we can define the distribution function of the random variable 
and, conversely, knowing the distribution function, one can determine the PDF;
,
(7.71)
this implies normalization condition PRV
.
(7.72)
because the probability of an event 
is equal to one. If all possible values of the measured random variable occupy the interval 
, then the PDF normalization condition has the form
,
(7.73)
In any case, the area of the curvilinear trapezoid formed by the PDF plot is equal to one. This condition can be used to determine the analytical form (formula) of PDF 
, if only the shape of the graph or only the form of this function is known (see Appendix 5, task 7.6) .
7.4.2. Characteristics of the random signal system
The measurement process is characterized by the presence of many random variables and events involved in the formation of the measurement result. In addition to the measured value itself, this includes non-informative parameters of the control object, parameters of the measuring instrument, environmental parameters, and even the state of the consumer of measurement information. Their cumulative effect on the measurement result is expressed in the fact that this result, obtained again under (seemingly) unchanged measurement conditions, differs from the previous result. By conducting repeated measurements and accumulating data (statistics), one can, firstly, get an idea of the degree of scatter of the measurement results and, secondly, try to find out the influence of each factor on the error of the measurement result.
If several are considered
(two or more) random variables , then they form system of random variables. Such a system, in addition to the characteristics listed above, for each random variable separately has additional characteristics, allowing to estimate the level of statistical relationships between all random variables that form the system. These characteristics are correlation moments(covariance) for each pair of random variables, 
. They are calculated according to the formula
,
(7.74)
where 
-two-dimensional PDF system of two random variables and (with mathematical expectations and, respectively), characterizing joint distribution these quantities.
In the absence of a statistical relationship between the quantities and, the corresponding correlation moment is equal to zero (i.e., 
). Such random variables are called statistically independent.
When performing mathematical operations with random variables that have known statistical characteristics, it is important to be able to determine the statistical characteristics of the results of these operations. Below such characteristics are given for the simplest mathematical operations:
If the values are statistically independent, then . those. the variance of the sum of independent random variables is equal to the sum of the variances of these variables.
Table 7.2. formulas are given for determining the characteristics of the sum two random variables. In this case , 
, and the variance 
and North Kazakhstan 
summation results significantly depend on the value of the relative correlation coefficient of the summed values 
, where 
.
Table 7.2.
Statistical characteristics of the sum of two random variables
|
Relative coefficient correlations |
Dispersion |
SKO |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Equality 
corresponds to the case when a change in quantity always entails a change in quantity and always in the same direction as, i.e. 
. If the signs of changes in these quantities are always opposite to each other, then 
. Finally, if the quantities u have finite variances and are statistically independent of each other, then 
. The converse is true only for normally distributed random variables.
If the values are statistically independent, then
,
.
,
Similarly, if 
- known function two continuous random variables , whose joint (two-dimensional) PDF 
is known, then the mathematical expectation and variance of such a random variable can be determined by the formulas
,
(7.80)
All previous formulas for calculating the results of mathematical operations with random variables can be obtained from these general formulas.
7.4.3. Typical distributions of random signals
Consider the statistical characteristics of continuous random variables having typical distribution.
7.4.3.1. Uniform distribution.
In the case of a uniform distribution, the random variable (7.2) with the same probability density falls into each point of the limited interval . PRV 
and distribution function 
such a random variable have the form (Fig. 7.8)
(7.81)
Other (private) statistical characteristics of such a random variable can be calculated using the formulas
,
,
,
.
(7.82)
7.4.3.2. Triangular distribution (Simpson distribution)
In this case, the PDF graph has the shape of a triangle with a vertex at the point 
, and the graph of the integral distribution law is a smooth conjugation of two parabolas at the point 
, where, 
,
(Fig. 7.9).
(7.83)
The mathematical expectation and variance of such a random variable can be calculated using the formulas
,
.
(7.84)
If a 
, then the Simpson distribution becomes symmetrical. In this case
,
,
,
.
(7.85)
7.4.3.3. Normal distribution (Gaussian distribution)
The normal distribution is one of the most common distributions of random variables. This is partly due to the fact that the distribution of the sum of a large number of independent random variables with different distribution laws, which is often encountered in practice, approaches a normal distribution. In this case, the PDF and distribution function have the form
,
.
(7.86)
RMS and mathematical expectation of such a value coincide with the parameters 
distributions, i.e. 
,.
Confidence interval 
is not expressed in terms of elementary functions, but can always be found from equation (7.70). The result of solving this equation for a given value of confidence probability 
can be written in the form 
, where 
- quantile, the value of which depends on the level of confidence 
.
There are tabular function values 
. Here are some of them:
,
,
,
,
........
This shows that with a fairly high probability ( 
) almost all values of a random variable with a normal distribution fall into the interval 
, having a width 
. This property is the basis of the "three sigma" rule.
On fig. 7.10 shows the plots of PDF and the integral law of normal distribution for two different values of RMS ( 
) and the same mathematical expectation.
It can be seen that the PDF plot is a single-hump "resonant" curve with a maximum at the point 
located symmetrically with respect to the mathematical expectation. This curve is the “sharper” the smaller the RMS. Accordingly, the smaller the spread of possible values of a random variable relative to its mathematical expectation. However, in all cases, the area of the curvilinear trapezoid bounded by the PDF plot is equal to unity (see (7.72)).
In probability theory, in addition to the characteristics discussed above, other characteristics of a random variable are also used: the characteristic function, kurtosis, counter-kurtosis, quantile estimates, etc. However, the characteristics considered are quite sufficient for solving most practical problems of measuring technology. Let us show an example of solving such a problem.
Example 7.4.: It is required to determine the parameter A (vertex coordinate) of the probability distribution density of a random measuring signal, the graph of which is shown in Fig. 7.11 (assuming that known only form this chart).
It is also required to determine the probability that the magnitude (module) of the signal will be greater than its standard deviation, i.e. it is required to determine the probability of an event 
.
Solution: Parameter value BUT we determine from the normalization condition for PDF (7.73), which in this case has the form
.
Here the first term corresponds to the area of the rectangle in Fig. 7.11 under the PRV schedule to the left dotted line 
, the second - the area of a right-angled triangle lying to the right this line. From the resulting equation we find 
. Given this result, the probability distribution density can be written as
Now you can calculate the mathematical expectation , variance and standard deviation of the signal. According to formulas (7.66), (7.67) and (7.68), respectively, we obtain: In fig. 7.11 dash-dotted lines show the boundaries of the interval 
.
In accordance with the normalization condition (7.71), the desired probability is equal to the sum of the areas under the PDF plot located to the left of the point 
(in this example, this area is equal to zero) and to the right of the point 
, i.e.
.
7.4.4. Characteristics of random signals that change over time
A random signal that varies in time generally contains a deterministic (systematic) and a centered random (fluctuation) component, i.e.
.
(7.87)
On fig. 7.12 shows a graph one from a number of possible realizations of such a signal. The dotted line shows its deterministic component 
, near which all other realizations of the signal are grouped and around which they oscillate.
A complete picture of the characteristics of such a signal is given by the general (complete) set of all its implementations. In practice, it is always finite. Therefore, the characteristics of a random signal found empirically should be considered estimates of its actual characteristics.
At each moment of time (that is, in each section of the signal), the values of the random function of time (7.87) are a random variable 
with the corresponding statistical characteristics discussed above. In particular, the deterministic component of a random signal at each moment of time coincides with mathematical
anticipation corresponding random variable 
, i.e.
,
(7.88)
where 
- one-dimensional PDF of the random process (7.87), which, in contrast to the PDF of the random variable (7.65) considered above, depends not only on, but also on time.
The degree of scatter of realizations of a random signal relative to its systematic component (7.88) characterizes the maximum value of the modulus of the fluctuation component of the signal 
and is estimated by the value of the standard deviation of this component, which in the general case also depends on time
.
(7.89)
where 
- variance of a random signal, calculated by the formula
.
(7.90)
For each point in time, you can determine the confidence interval 
(see (7.70)), and then construct trust region, i.e. such an area in which the realization of a random signal 
hit with a predetermined confidence probability 
(Fig. 7.13).
The three characteristics considered ( 
and 
) is sufficient to form a general idea of the properties of a random measuring signal (7.87). However, they are not enough to judge internal composition(spectrum) of such a signal.
On fig. 7.14, in particular, the graphs of the implementations of two various random signals from
the same mathematical expectation 
and North Kazakhstan 
. The difference between these signals is expressed in the different spectral (frequency) composition of their realizations, i.e. in varying degrees of statistical connection
between the values of a random signal at two points in time 
and 
, separated from each other by a value. For the signal shown in fig. 7.16, a this relationship is stronger than for the signal in Fig. 7.14, b.
In the theory of random processes, such a statistical relationship is estimated using autocorrelation function random signal (ACF), which is calculated by the formula
,
(7.91)
where 
-two-dimensional PRV signal.
Distinguish stationary and non-stationary random signals. If the signal (7.87) is stationary, then its mathematical expectation (7.88) and variance (7.90) do not depend on time, and its ACF (7.91) does not depend on two arguments 
and 
, but only from one argument - the value of the time interval 
. For such a signal
,
,
, where 
.
(7.92)
In other words, a stationary random signal is uniform in time, i.e. its statistical characteristics do not change when the time reference point changes.
If, in addition to stationarity, a random signal is also ergodic, then 
, and its autocorrelation function can be calculated by the formula
,
(7.93)
which does not require knowledge of the two-dimensional PDF 
since in this formula as can be used any implementation signal. The dispersion of such a (stationary and ergodic) signal can be calculated by the formula
,
(7.94)
A sufficient condition for the ergodicity of a random signal is that its ACF tends to zero 
with an unlimited growth of the time shift.
The ACF of a random signal is often normalized to the variance. In this case, dimensionless normalized ACF is calculated by the formula
.
(7.95)
On fig. 7.15 shows a typical graph of such an ACF.
Knowing this function, we can determine correlation interval
, i.e. time after which the values of a random signal can be read statistically independent from each other
.
(7.96)
It follows from this formula that the area under the plot of the normalized ACF coincides with the area of a rectangle of unit height, which has a doubled correlation interval at the base 
(See Figure 7.15).
Let us explain the physical meaning of the correlation interval. If information about the behavior of a centered random signal "in the past" is known, then its probabilistic forecast is possible for a time of the order of the correlation interval 
. However, the prediction of a random signal for a time exceeding the correlation interval will turn out to be unreliable, since the instantaneous values of the signal, which are so "far" separated from each other in time, are practically uncorrelated (i.e., statistically independent of each other).
Within the framework of the spectral-correlation theory of random processes, to describe the properties of a stationary random signal, it is enough to know only its ACF 
, or only energy spectrum signal 
. These two functions are related to each other by the Wiener–Khinchin formulas
,
(7.97)
,
(7.98)
those. each frequency function 
corresponds to a well-defined time shift function 
and vice versa, each ACF corresponds to a well-defined power spectral density of a stationary random signal. Therefore, knowing the energy spectrum of the fluctuation component 
random signal (7.87) 
, we can determine the ACF of this component 
and vice versa. This confirms that the frequency and correlation characteristics of a stationary random signal are closely related to each other.
Properties of the ACF of a random signal 
are similar to the properties of the ACF of a deterministic signal 
.
Autocorrelation function 
characterizes statistical connection between the values of a stationary random signal at times that are separated from each other along the time axis by the value . The smaller this relationship, the smaller the corresponding value of the ACF. energy spectrum 
characterizes the distribution along the frequency axis of the energies of the harmonic components of a random signal.
Knowing the energy spectrum 
, or ACF 
fluctuation component of the signal (7.1) 
, we can calculate its dispersion and effective spectrum width (frequency band) 
formulas
,
(7.99)
,
(7.100)
where 
- ordinate of the maximum point on the graph of the function 
.
Effective spectrum width of a random spectrum of a random signal 
similar to active spectrum width 
deterministic signal, that is, like the latter, it determines such a frequency range within which the overwhelming part of the average signal power is concentrated (see (7.55)). Therefore, by analogy with (7.55), it can be determined from the relation
.
(7.101)
where 
- a constant coefficient that determines the proportion of random signal power per frequency band 
(for example, 
=
0,95).
On fig. 7.16 is a graphical illustration of formulas (7.100) and (7.101). In the first case, the frequency band 
coincides with the base of the rectangle having height 
and area 
(Fig. 7.19, a), in the second - with the base of a curvilinear trapezoid having an area 
(Fig. 7.16, b). The frequency band of a narrow-band random process is located in the region 
, where 
- the average frequency of the spectrum (Fig. 7.16, in), and is calculated from the relation
.
The effective spectrum width of a random signal can be determined in many other ways. In any case, the values 
and 
must be related by a relation similar to the relation 
, which holds for deterministic signals (see Section 7.3.3).
a B C
Table 7.3 shows the spectral-correlation characteristics for three stationary random signals.
The first paragraph of this table shows the characteristics of the so-called white noise - a specific random signal, the values of which, located arbitrarily close to each other, are independent random variables. The white noise ACF has the form - functions, and its energy spectrum contains harmonic components of any (including arbitrarily high) frequencies. The dispersion of white noise is an infinitely large number, i.e. the instantaneous values of such a signal can be arbitrarily large, and its correlation interval is zero.
Table 7.3.
Characteristics of stationary random signals
|
Autocorrelation |
Interval correlations |
energy spectrum |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
The second paragraph of the table indicates the characteristics of low-frequency noise, and the third paragraph - narrow-band noise. If a 
, then these characteristics of these noises are close to each other.
The random signal is called narrowband if the frequency 
much less than the average frequency of the spectrum 
. A narrow-band random signal can be written as (see (7.12)), where the functions 
and 
change much more slowly than the function 
.
The properties of the spectral-correlation characteristics of a stationary random signal are similar to the properties of the amplitude spectrum and the ACF of a deterministic signal. In particular, 
and 
- even functions, 
etc. There are also differences. The difference between the correlation functions lies in the fact that the ACF of a deterministic signal 
characterizes the connection of the signal 
and its copies 
, and the ACF of a random signal 
- connection of signal values 
and 
at different points in time.
The difference between functions 
and 
is that the function 
represents an inaccurate frequency image of a random signal 
, but the average characteristic of the frequency properties of the whole ensemble of different realizations of this signal. This fact, as well as the absence in the energy spectrum 
information about the phases of the harmonic components of a random signal does not allow restoring the shape of this signal from it.
From formulas (7.97) and (7.98) it follows that the functions 
and 
are related to each other by Fourier transforms, i.e. (see (7.46))
and 
.
Therefore, the wider the spectrum of a random signal (the more 
), the narrower its ACF and the smaller the correlation interval 
.
Measuring signals, being random signals, cannot be described by a mathematical function of time with complete certainty.
Accordingly, one can only say about the probability appearance at any given moment of one or another signal values.
With such an approach the object of study is not the characteristics of a particular signal, but the probabilistic statistical characteristics of the totality of telecommunication signals of a particular type of communication.
On the statistical characteristics of a random signal s(t) relate:
mean(constant component)
where T- time of observation of a random process;
instant power random signal s(t)in the moment t by definition equals
energy random signal s(t) is equal to the integral of the power over the entire time interval of the existence or setting of the signal. In the limit:
average power random signal s(t) in the interval t 2 -t 1
The concept of average power can also be extended to the case of an unlimited interval T= t2-t 1 ⟹∞. A strictly correct determination of the average signal power should be made according to the formula:
The square root of the average power characterizes current (rms) signal value (220 V is the effective value of a harmonic oscillation with an amplitude of 380 V).
With regard to electrophysical systems, these concepts of power and energy correspond to quite specific physical quantities. Assume that the function s(t) displays the electrical voltage across a resistor whose resistance is equal to R Ohm. Then the power dissipated in the resistor, as you know, is (in volt-amperes):
w(t) = |s(t)| 2/R,
In signal theory, in the general case, signal functions s(t) have no physical dimension, and can be a formalized representation of any process or distribution of any physical quantity, while the concepts of signal energy and power are used in a wider sense than in physics . They represent the metrological characteristics of signals
If in the expression for energy
take not the square of the signal modulus, but the product of the signal and its own, but shifted by time τ, then we get the autocorrelation function
In the case of periodic signals, the ACF is calculated over one period T, averaging the scalar product and its shifted copy within the period:
energy spectrum(average power spectral density)
Function G(ω ) represents the spectral density of the average power of the process, i.e., the power contained in an infinitely small frequency band.
The power contained in the final frequency band between ω 1 and ω 2 is determined by integrating the function G(ω ) within the appropriate limits:
3.3. Dynamic range and crest factor of signals.
The instantaneous power of communication signals can take on various values within the widest range. To characterize these limits, we introduce the concepts dynamic range and signal crest factor.
Signal dynamic rangedB, is given by
where W max and Wmin - maximum and minimum instantaneous power values.
Under W max usually understand the value of the instantaneous signal power, the probability of exceeding which is small enough (for example, equal to 0.01). The value of this probability is agreed for each specific signal.
crest factor signal is called the ratio of its maximum power to the average. In logarithmic units
In some cases, the dynamic range and crest factor are determined not in logarithmic, but in absolute units (in "times").
The mathematical apparatus for analyzing stationary random signals is based on the ergodicity hypothesis. According to the ergodicity hypothesis, the statistical characteristics of a large number of arbitrarily chosen realizations of a stationary random signal coincide with the statistical characteristics of one realization of a sufficiently large length. This means that averaging over a set of realizations of a stationary random signal can be replaced by averaging over time of one rather long realization. This greatly simplifies the experimental determination of the statistical characteristics of stationary signals and simplifies the calculation of systems under random influences.
Let's determine the main statistical characteristics of a stationary random signal, given as one implementation in the interval (Fig. 11.1.1, a).
Numerical characteristics. Numerical characteristics of a random signal are the mean value (mathematical expectation) and variance.
The average value of the signal over a finite time interval is equal to
If the averaging interval - the implementation length T tends to infinity, then the time average value according to the ergodicity hypothesis will be equal to the mathematical expectation of the signal:
Rice. 11.1.1. Implementations of stationary random signals
In what follows, for brevity, we will omit the sign of the limit in front of integrals over time. In this case, either instead of the sign = we will use the sign , or by the calculated statistical characteristics we mean their estimates.
In practical calculations, when the final implementation is given as N discrete values separated from each other by equal time intervals (see Fig. 8.1), the average value is calculated using the approximate formula
A stationary random signal can be considered as the sum of a constant component equal to the mean value , and a variable component corresponding to the deviations of the random signal from the mean:
The variable component is called a centered random signal.
Obviously, the average value of the centered signal is always zero.
Since the spectrum of the signal x(t) coincides with the spectrum of the corresponding centered signal , then in many (but not all!) problems of calculating automatic systems, instead of the signal x(t), we can consider the signal .
The dispersion D x of a stationary random signal is equal to the average value of the squared deviations of the signal from the mathematical expectation , i.e.
Dispersion D x is a measure of the spread of instantaneous signal values around the mathematical expectation. The greater the ripple of the variable component of the signal relative to its constant component, the greater the dispersion of the signal. The variance has the dimension of x squared.
Dispersion can be considered in the same way as the average value of the power of the variable component of the signal.
Often, the standard deviation is used as a measure of the spread of a random signal.
For the calculation of automatic systems, the following property is important:
the variance of the sum or difference of independent random signals is equal to the sum (!) of the variances of these signals, i.e.
Mathematical expectation and variance are important numerical parameters of a random signal, but they do not fully characterize it: they cannot be used to judge the rate of change of the signal over time. So, for example, for random signals x 1 (t) and x 2 (t) (Fig. 11.1.1, b, c), the mathematical expectations and variances are the same, but despite this, the signals clearly differ from each other: signal x 1 (t) changes more slowly than the signal x 2 (t).
The intensity of change of a random signal in time can be characterized by one of two functions - correlation or spectral density function.
correlation function. The correlation function of a random signal x(t) is the mathematical expectation of the products of the instantaneous values of the centered signal, separated by the time interval, i.e.
where m is a variable shift between the instantaneous values of the signal (see Fig. 11.1.1, a). The shift varies from zero to some value. Each fixed value corresponds to a certain numerical value of the function.
The correlation function (also called autocorrelation) characterizes the degree of correlation (tightness of connection) between previous and subsequent signal values.
As the shift increases, the connection between the values and weakens, and the ordinates of the correlation function (Fig. 11.1.2, a) decrease.
This basic property of the correlation function can be explained as follows. For small shifts, the integral sign (11.1.12) includes products of factors that, as a rule, have the same signs, and therefore most of the products will be positive, and the value of the integral will be large. As the shift increases, more and more factors with opposite signs will fall under the integral sign, and the values of the integral will decrease. For very large shifts
Rice. 11.1.2. Correlation function (a) and spectral density (b) of a random signal
the factors and are practically independent, and the number of positive products is equal to the number of negative products, and the value of the integral tends to zero. It also follows from the above reasoning that the correlation function decreases the faster, the faster the random signal changes in time.
It follows from the definition of the correlation function that it is an even function of the argument , i.e.
therefore, only positive values are usually considered.
The initial value of the correlation function of the centered signal is equal to the signal dispersion, i.e.
Equality (8.14) is obtained from expression (11.1.12) by substituting .
The correlation function of a specific signal is determined by the experimentally obtained implementation of this signal. If the signal realization is obtained in the form of a continuous diagram record of length T, then the correlation function is determined using a special computing device - a correlator (Fig. 11.1.3, a), which implements formula (11.1.12). The correlator consists of a BZ delay block, a BU multiplication block, and an I integrator. To determine several ordinates, the delay block is tuned in turn to various shifts
If the implementation is a set of discrete signal values obtained at regular intervals (see Fig. 11.1.1, a), then the integral (11.1.12) is approximately replaced by the sum
which is calculated using a computer.
Fig 11.1.3 Algorithmic schemes for calculating the ordinates of the correlation function (a) and spectral density (b)
To obtain sufficiently reliable information about the properties of a random signal, the implementation length T and the discrete interval must be selected from the conditions:
where T n t h and T in h - periods, respectively, of the lowest and highest frequency components of the signal.
Spectral density. Let us now define the spectral characteristic of a stationary random signal . Since the function is not periodic, it cannot be expanded into a Fourier series (2.23). On the other hand, the function is non-integrable due to its unlimited duration, and therefore cannot be represented by the Fourier integral (2.28). However, if we consider a random signal on a finite interval T, then the function becomes integrable, and there is a direct Fourier transform for it:
The Fourier image of a non-periodic signal x(t) characterizes the distribution of the relative signal amplitudes along the frequency axis and is called the amplitude spectral density, and the function characterizes the signal energy distribution among its harmonics (see 2.2). Obviously, if the function is divided by the duration T of a random signal, then it will determine the distribution of the power of the final signal among its harmonics. If now let T tend to infinity, then the function will tend to the limit
which is called the power spectral density of the random signal. In what follows, the function will be called abbreviated as the spectral density.
Along with the mathematical definition (11.1.18) of the spectral density, a simpler - physical interpretation can be given: the spectral density of a random signal x (t) characterizes the distribution of the squares of the relative amplitudes of the signal harmonics along the axis.
According to definition (11.1.18) spectral density is an even function of frequency. When the function usually tends to zero (Fig. 11.1.2, b), and the faster the signal changes in time, the wider the graph.
Individual peaks in the spectral density graph indicate the presence of periodic components of a random signal.
Let us find the relation between spectral density and signal dispersion. We write Parseval's equality (2.36) for the final realization and divide its left and right parts by T. Then we obtain
When the left side of equality (8.19) tends to the dispersion of the signal D x [see. (11.1.10)], and the integrand on the right side - to the spectral density , i.e. instead of (8.19) we get one of the main formulas of statistical dynamics:
Since the left side of equation (11.1.20) is the total dispersion of the signal, then each elementary component under the integral sign can be considered as the dispersion or square of the amplitude of the harmonic with frequency .
Formula (11.1.20) is of great practical importance, since it allows one to calculate its dispersion from the known spectral density of a signal, which in many problems of calculating automatic systems serves as an important quantitative characteristic of quality.
The spectral density can be found from the experimental implementation of the signal using a spectral analyzer (Fig. 11.1.3, b), consisting of a bandpass filter PF with a narrow bandwidth , a squarer Kv and an integrator I. To determine several ordinates, the bandpass filter is tuned in turn to different transmission frequencies .
The relationship between the functional characteristics of a random signal. N. Viner and A. Ya. Khinchin showed for the first time that the functional characteristics of a stationary random signal are related to each other by the Fourier transform: the spectral density is an image of the correlation function, i.e.
and the correlation function, respectively, is the original of this image, i.e.
If we expand the factors using the Euler formula (11.1.21) and take into account that , and are even functions, and is an odd function, then expressions (11.1.21) and (11.1.22) can be transformed to the following form, more convenient for practical calculations:
Substituting the value into the expression (11.1.24), we obtain the formula (11.1.20) for calculating the variance.
The relations connecting the correlation function and the spectral density have all the properties inherent in the Fourier transform. In particular: the wider the graph of the function, the narrower the graph of the function, and vice versa, the faster the function decreases, the slower the function decreases (Fig. 11.1.4). Curves 1 in both figures correspond to a slowly changing random signal (see Fig. 11.1.1, b), the spectrum of which is dominated by low-frequency harmonics. Curves 2 correspond to a rapidly changing signal x 2 (t) (see Fig. 11.1.1, b), the spectrum of which is dominated by high-frequency harmonics.
If a random signal changes in time very sharply, and there is no correlation between its previous and subsequent values, then the function has the form of a delta function (see Fig. 11.1.4, a, line 3). The spectral density graph in this case is a horizontal straight line in the frequency range from 0 to (see Fig. 11.1.4, b, straight line 3). This indicates that the amplitudes of the harmonics are the same over the entire frequency range. Such a signal is called ideal white noise (by analogy with white light, in which, as is known, the intensity of all components is the same).
Fig 11.1.4 Relationship between correlation function (a) and spectral density (b)
Note that the concept of "white noise" is a mathematical abstraction. Physical signals in the form of white noise are not feasible, since according to formula (11.1.20) an infinitely wide spectrum corresponds to an infinitely large dispersion, and, consequently, an infinitely large power, which is impossible. However, real signals with a finite spectrum can often be roughly considered as white noise. This simplification is justified in cases where the signal spectrum is much wider than the bandwidth of the system affected by the signal.
For all random signals operating in real physical systems, there is a correlation between previous and subsequent values. This means that the correlation functions of real signals differ from the delta function and have a finite, non-zero decay duration. Accordingly, the spectral densities of real signals always have a finite width.
Communication characteristics of two random signals. To describe the probabilistic relationship that appears between two random signals, use the cross-correlation function and the mutual spectral density.
The mutual correlation function of stationary random signals x 1 (t) and x 2 (t) is determined by the expression
The function characterizes the degree of connection (correlation) between the instantaneous values of the signals x 1 (t) and x 2 (t), separated from each other by a value. If the signals are not statistically related (not correlated) with each other, then for all values of the function .
For the cross-correlation function, the following relation follows from definition (8.25):
The correlation function of the sum (difference) of two correlated signals is determined by the expression
The mutual spectral density of random signals x 1 (t) and x 2 (t) is defined as the Fourier image of the cross-correlation function:
It follows from definition (11.1.28) and property (11.1.26) that
Spectral density of the sum (difference) of random signals x 1 (t) and x 2 (t)
If the signals x 1 (t) and x 2 (t) are not correlated with each other, then expressions (11.1.27) and (11.1.29) are simplified:
Relations (11.1.31), as well as (11.1.11), mean that the statistical characteristics and D x of a set of several random signals uncorrelated with each other are always equal to the sum of the corresponding characteristics of these signals (regardless of the sign with which the signals are summed into this set).
Typical random impacts. Real random impacts affecting industrial control objects are very diverse in their properties. But resorting to a certain idealization in the mathematical description of influences, one can single out a limited number of typical or typical random influences. Correlation functions and spectral densities of typical actions are fairly simple functions of the arguments and . The parameters of these functions, as a rule, can be easily determined from the experimental realizations of the signals.
The simplest typical impact is white noise with limited bandwidth. The spectral density of this effect (Fig. 11.1.5, a) is described by the function
Where is the intensity of white noise. Signal dispersion according to (11.1.20)
The correlation function according to (11.1.24) in this case has the form
Taking into account (11.1.33), the function (11.1.34) can be written in the following form:
The graph of the function (11.1.35) is shown in fig. 11.1.5, b.
Rice. 11.1.5. Spectral densities and correlation functions of typical random signals
Most often in practical calculations there are signals with an exponential correlation function (Fig. 11.1.5, d)
Applying the transformation (11.1.23) to the correlation function (11.1.36), we find the spectral density (Fig. 11.1.5, c)
The larger the parameter a x, the faster the correlation function decreases and the wider the spectral density graph. The ordinates of the function decrease with increasing ax. At , the considered signal approaches ideal white noise.
In approximate calculations, the parameter а x can be determined directly from the realization of the signal - the average number of intersections of the time axis by the centered signal: .
Often a random signal contains a latent periodic component. Such a signal has an exponential-cosine correlation function (Fig. 11.1.5, e)
The parameter of this function corresponds to the average value of the “period” of the latent component, and the parameter a x characterizes the relative intensity of the remaining random components that are superimposed on the periodic component. If the exponent is , then the relative level of these components is small, and the mixed signal is close to harmonic. If the indicator is , then the level of random components is commensurate with the "amplitude" of the periodic component. At , the correlation function (8.38) practically coincides (with an accuracy of 5%) with the exponent (11.1.36).
1. Features of the study of ACS under random impacts
Under deterministic predetermined influences, the state of the ACS at any moment t is determined by the initial state of the system at some point in time t0 and the actions applied to the system. This problem is determined by solving the corresponding differential equation
anx (n)+an-1x(n-1)+…+a0x=bmg(m)+bm-1g(m-1)+…+b0g. (26.1)
If ai, bj are constant coefficients, and g is a definite function of time, then the solution of this equation for given initial conditions will be unique and definite for the entire time interval.
However, in real conditions, external influences often change randomly, i.e. not foreseen in advance. For example:
daily changes in the load of the power system;
gusts of wind acting on the aircraft;
wave impacts in hydrodynamic systems;
radar signals;
noise in radio devices, etc.
Random influences can be applied to the system from the outside (external influences) or occur inside some of its elements (internal noise).
Obviously, if in equation (26.1) g - the input action is not predetermined, i.e. is a random function, or the system parameters ai, bj change randomly, then it is impossible to obtain a solution to this equation in a deterministic (i.e., definite) form.
Of course, you can set some maximum values of these parameters and solve the problem (calculation of the system for a given accuracy with maximum values of random effects). But since the maximum values of a random variable are rarely observed, in this case, the system will be subject to obviously more stringent requirements than it is caused by reality.
True, this approach is sometimes the only possible one (high-precision production, otherwise - marriage). Therefore, in most cases, the calculation of the system under random influences is carried out not according to the maximum, but according to the most probable value of random variables, i.e. according to the value that occurs most frequently.
In this case, the most rational technical solution is obtained (lower system gain, smaller dimensions of individual devices, lower energy consumption), although for unlikely values of the driving force, system performance will deteriorate.
Calculation of ACS under random impacts using special statistical methods that operate on the statistical characteristics of random impacts, which are not random, but deterministic values.
ACS designed on the basis of statistical methods will provide the appropriate requirements not for one, deterministic impact, but for the whole set of these impacts specified using statistical characteristics (if the ACS error is random, then its exact value at any point in time at cannot be obtained statistically).
Statistical methods for calculating ACS are based on the calculations and works of Soviet scientists: Khinchina A.Ya., Kolmogorova A.N., Gnedenko V.V., Solodovnikova V.V., Pugacheva V.S., Kazakova I.E. and others, as well as foreign scientists - N. Wiener, L. Zadeh, J. Ragotsine, Kalman, Busy and others.
2. Brief information about random processes.
A random function is a function that, for each value of the independent variable, is a random variable. Random functions for which time t is the independent variable are called random processes. Since processes in ACS occur in time, in the future we will consider only random processes.
The random process x(t) is not a certain curve, it is a set of certain curves x i (t) (i=1,2,…,n) obtained as a result of individual experiments (Fig. 26.1). Each curve of this set is called a realization of a random process, and it is impossible to say which of the realizations the process will follow.
Rice. 26.1. Graphs of realizations and mathematical expectation of a random process
For a random process, as well as for a random variable, to determine the statistical properties, the concept of a distribution function (integral distribution law) F(x, t) and a probability density (differential distribution law) w(x, t) are introduced. These characteristics depend on a fixed observation time t and on some selected level x, that is, they are functions of two variables - x and t.
The functions F(x, t) and w(x, t) are the simplest statistical characteristics of a random process. They characterize a random process in isolation in separate sections, without revealing the connection between the sections of a random process.
The main characteristics of random processes that are most widely used in the study of control systems include: mathematical expectation, variance, mean value of the square of a random process, correlation function, spectral density, and others.
BUT. Expected value m x (t) is the mean value of the random process x(t) over the set and is determined
(26.2)
where w 1 (x, t) - one-dimensional probability density of a random process x(t) .
The mathematical expectation of a random process x(t) is a certain non-random function of time m x (t), around which all implementations of this random process are grouped and relative to which they fluctuate (Fig. 26.1).
The mean value of the square of a random process is the value
(26.3)
Often introduced into consideration is a centered random process, which is understood as the deviation of the random process X (t) from its average value m x (t), or
(26.4)
B. Dispersion. To take into account the degree of dispersion of the implementations of a random process relative to its average value, the concept of the dispersion of a random process is introduced, which is equal to the mathematical expectation of the square of a centered random process
(26.5)
The variance of a random process is a non-random function of time D x (t) and characterizes the spread of a random process X(t) relative to its mathematical expectation m x (t).
In practice, statistical characteristics are widely used that have the same dimension as a random variable, which include:
Root mean square value of a random process
equal to the value of the square root of the average value of the square of the random process;
RMS of a random process
(26.7)
equal to the value of the square root of the variance of the random process.
Mathematical expectation and dispersion are important characteristics of a random process, but do not give a sufficient idea of the internal connections of a random process, which have a significant impact on the nature of its implementation within a given time interval.
One of the statistical characteristics that reflect the features of the internal connections of a random process is the correlation function.
AT. correlation function random process X (t) is called a non-random function of two arguments R x (t 1, t 2), which for each pair of arbitrarily chosen values of time points t 1 and t 2 is equal to the mathematical expectation of the product of two random variables -X (t 1) and X (t 2), corresponding sections of the random process:
where w 1 (x 1 , t 1 , x 2 , t 2) is the two-dimensional probability density.
Random processes, depending on how their statistical characteristics change over time, are divided into stationary and non-stationary. Distinguish stationarity in the narrow and broad sense.
A random process X(t) is called stationary in the narrow sense if its n-dimensional distribution functions and probability density for any n do not depend on the position of the time reference t.
Stationary in a broad sense is a random process X(t), the mathematical expectation of which is constant:
M[X(t)]= m x =const, (26.9)
and the correlation function depends on only one variable - the difference of the arguments t=t 2 -t 1:
In the theory of random processes, two concepts of averages are used: average over a set and average over time.
The mean value over the set is determined on the basis of observation over the set of realizations of a random process at the same moment in time, i.e.
(26.11)
The average value over time is determined on the basis of observations of a separate implementation of a random x(t) over a sufficiently long time T, i.e.
(26.12)
It follows from the ergodic theorem that for the so-called ergodic stationary random processes, the average value over the set coincides with the average value over time, i.e.
(26.13)
In accordance with the ergodic theorem for a stationary random process with mathematical expectation m 0 x =0, the correlation function can be defined
where x(t) is any implementation of a random process.
The statistical properties of the connection of two random processes X(t) and G(t) can be characterized by the cross-correlation function R xg (t 1 ,t 2), which for each pair of arbitrarily chosen values of the arguments t 1 and t 2 is equal to
According to the ergodic theorem, instead of (26.15) we can write
(26.16)
where x(t) and g(t) are any realizations of stationary random processes X(t) and G(t).
If random processes X(t) and G(t) are not statistically related to each other and have zero mean values, then their mutual correlation function for all t is zero.
Let us present some properties of correlation functions.
1. The initial value of the correlation function is equal to the average
the value of the square of the random process:
2. The value of the correlation function for any t cannot exceed its initial value, that is
3. The correlation function is an even function of t, i.e.
(26.18)
Another statistical characteristic that reflects the internal structure of a stationary random process X(t) is the spectral density S x (w), which characterizes the distribution of the energy of a random signal over the frequency spectrum.
G. Spectral density S x (w) of the random process X(t) is defined as the Fourier transform of the correlation function R(t),
(26.19)
Consequently,
since the spectral density S x ( a) is a real and even function of the frequency w.
Relations (26.19) and (26.20) allow us to establish some dependencies between the structure of the random process X (t) and the type of characteristics R x (t) and S x (w) (Fig. 26.2).
It follows from the given graphs that with an increase in the rate of change in the implementation of X(t), the correlation function R x (t) narrows (sharpenes), and the spectral density S x (w) expands.
The invention relates to computer technology and control systems, can be used to build adaptive fuzzy controllers for solving problems of managing objects, the mathematical model of which is not a priori defined, and the purpose of functioning is expressed in fuzzy terms. The aim of the invention is to expand the functionality. The probabilistic automaton contains: the first memory block 2, the second memory block 3, the state selection block 6, the third memory block 7, the first switch 9, the output signal selection block 10, the second switch 12, the clock pulse generator 13, the first random code generation block 14, the second random code generation unit 15, the fourth memory unit 16, the first maximum code determination unit 18, the fifth memory unit 20, the second maximum code determination unit 22. 6 c.p. f-ly, 21 ill.
The invention relates to computer technology and control systems, can be used to build adaptive fuzzy controllers for solving problems of managing objects, the mathematical model of which is not a priori defined, and the purpose of functioning is expressed in fuzzy terms. A probabilistic automaton is known (A. S. USSR N 1045232, class G 06 F 15/36, 1983), containing a random code generation unit, a state selection unit, a clock generator, an AND element, a switch, a memory unit, a waiting time setting unit , OR element, random voltage generator, moreover, the group of outputs of the random code generation block is connected to the inputs of the group of information inputs of the state selection block, the group of outputs of which is connected to the group of information inputs of the switch, the group of outputs of which is connected to the group of inputs of the memory block, the group of outputs of which is connected to a group of control inputs of the state selection block and with a group of inputs of the block for setting the waiting time, the group of outputs of which is connected to the group of outputs of the automaton and to the inputs of the OR element, the output of which is connected to the inverse input of the AND element and to the first clock input of the random code generation block, the output of the clock generator pulses is connected to the first clock input of the block for setting the waiting time and with the direct input of the AND element, the output of which is connected to the clock input of the switch, to the second clock input of the random code generation unit and to the second clock input of the waiting time setting unit, the output of the random voltage generator is connected to the control input of the waiting time setting unit. Signs coinciding with the signs of the proposed technical solution is a random code generation unit, a state selection unit, a clock generator, a switch, a memory unit. The disadvantage of this device is limited functionality, since in this device it is not possible to compare the qualitative characteristics of the latter to the states of the automaton. The reasons hindering the achievement of the required technical solution lie in the particular implementation of the known device, in which it is possible to generate states and output signals only in clear terms. A probabilistic automaton is known (AS USSR N 1108455, class G 06 F 15/20, 1984), containing the first memory block, the state selection block, the random code generation block, the clock generator, the switch, the second memory block, and the inputs groups of control and setting inputs of the first memory block are connected, respectively, to the outputs of groups of control inputs and groups of installation inputs, and the group of inputs is connected to the first group of information inputs of the state selection block, the output group of which is connected to the first group of information inputs of the state selection block, the second group of information inputs which is connected to the group of outputs of the random code generation block, the group of outputs of which is connected to the group of inputs of the switch, the group of outputs of which is connected to the group of inputs of the second memory block, the group of outputs of which is connected to the outputs of the device and to the group of control inputs of the state selection block, the output of the clock pulse generator connected to the clock inputs of the block random code generation and switch. Features coinciding with the features of the proposed technical solution are a random code generation unit, a state selection unit, a first memory unit, a clock generator, a switch, a second memory unit. The disadvantage of this device is the limited functionality associated with the fact that when the fuzzy determination of the output states, the device does not allow you to set on a clear set (output signals) fuzzy sets of qualitative characteristics of these signals. The reasons hindering the achievement of the required technical solution are in particular the implementation of a probabilistic automaton, in which the generation of states and output signals belonging to well-defined sets is carried out. Of the known devices, the closest to the claimed fuzzy probabilistic automaton in terms of the totality of design and functional features is a probabilistic automaton (AS USSR N 1200297, class G 06 F 15/20, 1985), containing the first memory block, the state selection block, the block generating a random code, a switch, a second memory block, an output signal selection block, a third memory block, a clock pulse generator, wherein the inputs of the groups of control and setting inputs of the first memory block are connected to the inputs of the groups of control inputs and groups of setting inputs, respectively, and the group of outputs is connected to the first group of information inputs of the state selection block, the group of outputs of which is connected to the first group of inputs of the switch, the group of outputs of which is connected to the group of installation inputs of the second memory block, the group of outputs of which is connected to the group of control inputs of the state selection block and to the first group of control inputs of the block for selecting the output signal, output group the output of which is connected to the group of inputs of the third memory block, the group of outputs of which is connected to the group of outputs of the device, the output of the clock pulse generator is connected to the clock inputs of the switch, the block for selecting the output signal and the block for generating a random code, the group of outputs of which is connected to the second group of information inputs of the block of choice states. Features coinciding with the features of the proposed technical solution are a random code generation unit, a state selection unit, a first memory unit, a clock generator, a switch, a second memory unit, an output signal selection unit, a third memory unit. The disadvantage of the known device is limited functionality due to the fact that the known device cannot be used to solve problems of modeling and control of objects with a priori uncertainty and a fuzzy (qualitative) description of the parameters and the purpose of modeling. This is primarily due to the fact that the known device does not perform the function of establishing a correspondence between clear concepts (a set of outputs and inputs) and fuzzy concepts (qualitative characteristics of inputs and outputs) specified in the form of fuzzy variables. The reasons hindering the achievement of the required technical solution are in particular the implementation of a probabilistic automaton, in which states and output signals are generated that belong to well-defined sets, in accordance with the specified transition and output functions for modeling stochastic objects. The problem to be solved by the invention lies in the possibility of generating states and output signals in accordance with the given functions of transitions and outputs, as well as generating fuzzy variables given on sets of states and output signals in accordance with expert assessments for further use in modeling and control of complex objects in the absence of a priori information about the mathematical model. To achieve the technical result, which consists in expanding the functionality by generating fuzzy variables specified on sets of states and output signals using expert information, it is proposed to fuzzy probabilistic automaton containing a clock pulse generator, the first block for generating a random code, a block for selecting states, an output signal selection block, the first, second and third blocks and a switch, wherein the M outputs of the group of control inputs of the device are connected to the M inputs of the first groups of control inputs of the first memory block, the inputs (NxNxM) of the groups of the first setting inputs of the device are connected respectively to the inputs (NxNxM) of the groups installation inputs of the first memory block, N inputs of groups of the second control inputs of which are connected to N outputs of the group of outputs of the third memory block, the output of the first clock pulse generator is connected to the clock inputs of the first random code generation block, K outputs of the group of outputs of which connected to K inputs of the second group of information inputs of the state selection block, additionally introduce the second block for generating a random code, the fourth and fifth memory blocks, the second switch, the first and second blocks for determining the maximum code, and the inputs (NxPxM) of the groups of installation inputs of the second memory block are connected with inputs (NxPxM) of groups of the second installation inputs of the device, M inputs of the group of first control inputs are connected to M inputs of the group of control inputs of the device and to M inputs of the group of first control inputs of the first memory block, N inputs of the group of second control inputs are connected to N inputs of the group of second control inputs inputs of the first memory block, N outputs of the output group of the third memory block and N inputs of the group of control inputs of the first switch, the outputs of the P groups of information outputs are connected to the corresponding inputs of the P groups of information inputs of the output signal selection block, and the clock input is connected to the output of the clock pulse generator and to tact New inputs of the first memory block, first and second random code generation blocks, N outputs of the group of information outputs of the state selection block are connected to the corresponding N inputs of the group of first information inputs of the third memory block, K outputs of the group of outputs of the second random code generation block are connected to K inputs of the second group information inputs of the block for selecting the output signal, the output (NxL) of the groups of information inputs of the first switch is connected to the outputs (NxL) of the groups of information outputs of the fourth memory block, (NxL) of the groups of information inputs of which are connected to the inputs (NxL) of the third groups of the installation inputs of the device, the outputs L groups of information outputs of the first switch are connected to the inputs of L groups of information inputs of the first block for determining the maximum code, the outputs of the group of information outputs of which are connected to the outputs of the third group of outputs of the device, P outputs of the group of outputs of the block for selecting output signals are connected to P inputs of the groups s control inputs of the second switch, the inputs (PxF) of the groups of information inputs of which are connected to the outputs (PxF) of the groups of information outputs of the fifth memory block, the inputs (PxF) of the groups of information inputs of which are connected to the inputs (PxF) of the fourth groups of the installation inputs of the device, the outputs of P groups the information outputs of the second switch are connected to the inputs F of the groups of information inputs of the second block for determining the maximum code, the groups of information outputs of which are connected to the outputs of the fourth group of outputs of the device. The presence of a causal relationship between the technical results and the features of the claimed invention is proved by the following logical premises. And the basis of the work of the probabilistic automaton is the assumption that the formal task of the fuzzy probabilistic automaton (NVA) can be represented as where X, Y, Z - respectively, the set of input, output signals 
- a set of conditional probabilities that determine the presence of the NVA in the time step t in the state z t, provided that the signal x t is applied to the input in this cycle and the presence of the NVA in the previous (t-1) cycle in the state 
- a set of conditional probabilities that determine the presence of a signal y t at the output of the automaton, provided that there is a signal x t at the output in this cycle and the NVA stays in the previous (t-1) cycle in the state x t-1 ; linguistic variable (LP) "state choice" specified by the set (,T(),Z) , where - LP name, T () - LP term-set, Z - base set; LP "output parameter selection", specified by the set (,T(),Y), where - LP name, T () - LP term-set , Y - base set. If and are linguistic variables, and T() = ( 1 ,..., L ) and T() = ( 1 ,..., F ) is a term-set, where 
- the names of the NP, then with the help of an expert survey, you can set and - membership functions of fuzzy variables. The fuzzy probabilistic automaton generates states, output signals, as well as linguistic variables given on the sets of states and output signals. In FIG. 1 and FIG. 2 shows a diagram of the claimed object; in fig. 3 is a functional diagram of the first memory block 2; in fig. 4 is a functional diagram of the second memory block 3; in fig. 5 is a block diagram of the state selection block 6; in fig. 6 is a functional diagram of the third memory block 7; in fig. 7 is a functional diagram of the first switch 9; in fig. 8 is a functional diagram of the block for selecting the output signal 10; in fig. 9 is a functional diagram of the second switch 12; in fig. 10 is a functional diagram of the first block for generating random code 14; in fig. 11 is a functional diagram of the second block for generating random code 15; in fig. 12 is a block diagram of the fourth memory block 16; in fig. 13 is a functional diagram of the first block for determining the maximum code 18; in fig. 14 is a block diagram of the fifth memory block 20; in fig. 15 is a functional diagram of the second block for determining the maximum code 22; in fig. 16 is a functional diagram of the decoder of the first block for determining the maximum code; in fig. 17 is a block diagram of each of the comparison blocks of the first maximum code determination block, FIG. 18 is a functional diagram of the decoder of the second block for determining the maximum code; in fig. 19 is a functional diagram of each of the comparison blocks of the second block for determining the maximum code; in fig. 20 - graphs of membership functions of fuzzy variables 1 , 2 ,..., L ; in fig. 21 - graphs of membership functions of fuzzy variables 1 , 2 ,..., F . Structural diagram of a fuzzy probabilistic automaton (Fig. 1 and 2) contain: 1 1 -1 M - group of control inputs; 2 - the first block of memory; 3 - second memory block; 
- (NxNxM) groups of the first installation inputs; (NxPxM) - groups of the second installation inputs; 6 - state selection block; 7 - third memory block; 8 1 -8 N -group of outputs of the third memory block 7 and control inputs of the first switch 9; 10 - output signal selection block; 11 1 -11 P - group of the second outputs of the device and control inputs of the second switch 12; 13 - clock pulse generator; 14 - the first block of random code generation; 15 - the second block of random code generation; 16 - fourth memory block; , (NxL) groups of third groups of installation inputs of the device; 18 - the first block for determining the maximum code; 19 1 - 19 L - outputs of the third group of device outputs; 20 - the fifth block of memory; - (PxF) groups of the fourth installation inputs of the device; 22 - the second block for determining the maximum code; 23 1 -23 F - outputs of the fourth group of outputs of the device. Functional diagram of the first memory block 2 (Fig. 3) contains: - M inputs of the first group of control inputs; 
- (MxNxN) groups of installation inputs; - N inputs of the second group of control inputs; 
- registers; (25 1m i1 -25 Km iN) - (NxM) groups of elements AND; 26 - clock input; - (MxN) groups of outputs of elements AND 25 and (MxN) groups of inputs (MxN) groups of elements OR 
outputs of N groups of outputs of the memory block 2. Functional diagram of the second memory block 3 (Fig. 4) contains: - M - groups of inputs of the first group of control inputs; - N inputs of the second group of control inputs; - (MxNxP) groups of the first installation inputs; 26 - clock input; 
- registers; (31 1m ip -31 Km ip) - (NxP) groups of elements AND; (32 1m ip -32 Km ip) - (MxN) groups of outputs of elements AND 32 and groups of inputs of elements OR - outputs P groups of outputs of the memory block 3. The block diagram of the state selection block 6 (Fig. 5) contains: 
- N group of inputs of the first group of information inputs; - N comparison nodes; 36 1 - 36 K - inputs of the second group of information inputs; - N outputs of the state selection block 6; 38 1 - 38 N-1 - elements And. The block diagram of the third memory block 7 (Fig. 6) contains: 8 1 - 8 N - outputs; 37 1 - 37 N - group of inputs; 38 1 - 38 N - triggers; 39 1 - 39 N - OR elements. Functional diagram of the first switch 9 (Fig. 7) contains: - N groups of control inputs; - (LxN) groups of elements AND, D elements in each; 
- (LxN) groups of D-bit information inputs; 
- L group of elements OR, D elements in each; 
- L groups D - bit outputs of the first switch 9. Functional diagram of the output signal selection block 10 (Fig. 8) contains: - outputs; 
inputs of the first group of information inputs; - comparison nodes; 45 1 - 45 K - inputs of the second group of information inputs; 46 1 - 46 p-1 - elements P. The functional diagram of the second switch 12 (Fig. 9) contains: - P groups of inputs of the group of control inputs; 
(FxP) groups of elements AND, D elements in each; 
(FxP) groups D - bit inputs of information inputs group; 
- F groups of elements OR, D elements in each; 50 1 f -50 D f - F groups D - bit outputs of the second switch 12. Functional diagram of the first block generating random code 14 (Fig. 10) contains: 36 1 - 36 K - outputs; 51 - clock input; 52 - the first element And; 53 1 - 53 Z second elements And; 54 - code converter; 55 - Poisson pulse stream generator; 56 - cyclically closed shift register. Functional diagram of the second block generating random code 15 (Fig. 11) contains: 45 1 - 45 K - outputs; 51 - clock input; 57 - the first element And; 58 1 - 58 Z - second elements And; 59 - code converter; 60 - Poisson pulse stream generator; 61 - cyclically closed shift register. Structural diagram of the fourth memory block 16 (Fig. 12) contains: 
- (LxN) groups D - bit information inputs; 62 1i - (LxN) groups of registers; 41 1 l i -41 D l i - (LxN) groups D - bit outputs of block 16. Functional diagram of the first block for determining the maximum code 18 (Fig. 13) contains: 19 1 - 29 L - group of outputs; 
- L groups D - bit inputs; - group of registers; 65 1 - 64 D group of decoders of states; 65 1 l -65 D l - L groups of elements And, D elements in each; 66 1 - 66 D - group of analysis nodes; 67 1 - 67 L - a group of OR elements. Structural diagram of the fifth memory block 20 (Fig. 14) contains: 
(FxP) groups D - bit information inputs; 68 fp - 68 fp - F groups of registers, P in each group; 
- (FxP) groups D - bit outputs. Functional diagram of the second block for determining the maximum code 22 (Fig. 15) contains: 23 1 - 23 F - a group of outputs; 
- F groups D - bit inputs; 69 1 - 69 F - group of registers; 70 1 - 70 D - a group of state decoders; 
- F groups of elements AND, D elements in each; 72 1 - 72 D - analysis nodes; 73 1 - 73 F - a group of OR elements. The functional diagram of the decoder of the first block for determining the maximum code (Fig. 16) contains 
- the first groups of inputs; 
- groups of elements OR, L - elements in each; 76 1 - 76 D - the first elements of And; 
- second groups of inputs; 78 1 - 78 D - second elements And; 
- groups of decoder outputs 64. The functional diagram of each of the d, analysis nodes 66 of the first block for determining the maximum code 18 (Fig. 17) contains 
- D-1 groups of the first L - bit inputs; 
- D-1 groups of the second L - bit inputs; 
- D-1 first groups of elements And, L elements And in each; 
- D-1 first groups of OR elements, L OR elements in each; 
- D-1 groups of second OR elements, with L OR elements in each 
- D-1 second groups of elements AND, L elements each; 
- D-1 second groups of NOT elements, L elements in each group; 
- D-1 third groups of elements AND, L elements each; 
- D-1 third groups of elements OR, L elements in each group; 
- D-1 fourth groups of elements AND, L elements each; 
- D-1 groups L - bit outputs; 
- D-1 groups of third L - bit inputs; 
- D-1 second groups of NOT elements, L in each group; 
- D-1 third groups of NOT elements, L in each group. Functional diagrams of the decoders 70 of the second block for determining the maximum code 22 (Fig. 18) contains: 
- the first groups of inputs; 
- groups of OR elements, F elements in each; 94 1 - 94 D - the first elements of And; 
- second groups of inputs; 96 1 - 96 D - second elements And; 
- D groups of decoder outputs. The functional diagram of each of the d, analysis nodes 72 of the second block for determining the maximum code 22 (Fig. 19) contains: 
- D-1 groups of the first F - bit inputs; 
- D-1 groups of the second F - bit inputs; 
- D-1 first groups of elements And, F elements And in each; 
- D-1 first groups of OR elements, F OR elements in each; 
- D-1 groups of second OR elements, F OR elements each; 
- D-1 second groups of elements AND, F elements in each; 
- D-1 second groups of NOT elements, F elements in each group; 
- D-1 third groups of elements AND, F elements in each; 
- D-1 third groups of elements OR, F elements in each group; 
- D-1 fourth groups of elements AND, F elements in each; 
- D-1 groups F - bit outputs; 
- D-1 groups of third F - bit inputs; 
- D-1 second groups of NOT elements, F in each group; 
- D-1 third groups of NOT elements, F in each group. The elements of the fuzzy automaton are interconnected as follows. The inputs of the group of control inputs 1 1 - 1 M of the device are connected to the inputs of the first groups of control inputs of the first memory block 2 and the second memory block 3, the inputs 
(NxNxM) - groups of the first installation inputs of the device are connected respectively to the inputs of the groups of installation inputs of the first memory block 2, inputs (NxPxM) - groups of the second installation inputs of the device are connected to the inputs of the groups of installation inputs of the second memory block 3, outputs of N groups of information outputs of the first memory block 2 are connected to the corresponding inputs of N groups of the first group of information inputs of the state selection block 6, the outputs of the group of information outputs of the state selection block 6 are connected to the corresponding inputs of the group of information inputs of the third memory block 7, the outputs 8 1 - 8 N of the group of outputs of the third memory block 7 are connected to corresponding inputs 8 1 - 8 N of the group of control inputs of the first switch 9, with the inputs of the groups of the second control inputs of the first 2 and second 3 memory blocks, and with the outputs 8 1 - 8 N of the first group of outputs of the device, the outputs of P groups of information outputs of the second memory block 3 connected to the corresponding inputs P groups of information inputs of the output signal selection block 10, outputs 11 1 - 11 P of the group of control outputs of which are connected to the corresponding inputs 11 1 - 11 P of the group of control inputs of the second switch 12, with outputs 11 1 - 11 P of the second group of outputs of the device, the output of the clock generator 13 connected to the clock inputs of the first 2 and second 3 memory blocks, the first 14 and second 15 random code generation blocks, the outputs of the group K information outputs of the first random code generation block 14 are connected to the corresponding inputs of the second group of information inputs of the state selection block 6, the outputs of the second group of outputs the random code generation unit 15 is connected to the corresponding inputs of the second group of information inputs of the output signal selection block 10, the inputs (NxL) of the groups of the second information inputs of the first switch 9 are connected to the outputs (NxL) of the groups of information outputs of the fourth memory block 16, (NxL) of the groups of information inputs which are connected to the inputs (NxL) third x groups of installation inputs 
device, the outputs of L groups of information outputs of the first switch 9 are connected to the inputs of L groups of information inputs of the first block for determining the maximum code 18, the outputs of the group of information outputs of which are connected to the outputs 19 1 - 19 L of the third group of outputs of the device, the inputs (PxF) of the groups of the second information inputs the second switch 12 is connected to the outputs (PxF) of the groups of information outputs of the fifth memory block 20, the inputs (PxF) of the groups of information inputs of which are connected to the inputs (PxF) of the fourth groups of installation inputs 
device, the outputs F groups of information outputs of the second switch 12 is connected to the inputs F groups of information inputs of the second block for determining the maximum code 22, the groups of information outputs of which are connected to the outputs 23 1 - 23 F by the fourth group of outputs of the device. In the first memory block 2, each of the K inputs 
(i, j, m)-th group of installation inputs are connected to the recording inputs of the corresponding registers 24 1m ij , the outputs of the registers 
connected to the first inputs of the corresponding elements AND (25 1m i1 -25 Km i1)-(25 1m iN -25 Km iN) (im)-th group, the second inputs of the elements AND are combined and connected to the clock input 26 of the memory block 2, the third inputs elements And 25 1m 11 -25 Km NN of each of the m groups are combined and connected to the m inputs 1 m of the group of the first control inputs of the first memory block 2, the fourth inputs of the elements AND (25 1m i1 -25 Km i1) - (25 1m iN -25 Km iN) (the im-th group are combined and connected to the i-th input 8 i of the second group of control inputs of the memory block 2, the outputs of the AND element 25 are connected to the corresponding inputs (N x M) of the groups of elements OR 
, the outputs of which are connected respectively to the outputs of N groups of outputs 29 1 j -29 K j of the memory block 2. In the second memory block 3, each of the K inputs of the (i, p, m)-th group of installation inputs is connected to the write inputs of the corresponding registers 30 m i p , the outputs of the registers 30 m i 1 -30 m i P are connected to the first inputs of the corresponding elements AND (31 1m i1 -31 Km i1)-(31 1m iP -31 Km iP) of the (im)-th group, the second inputs of the elements AND are combined and connected with a clock input 26 of the memory block 2, the third inputs of the elements And 31 1m i1 -31 Km NP of each of the m groups are combined and connected to the m-and inputs 1 m of the first group of control inputs of the second memory block 3, the fourth inputs of the elements And (31 1m i1 -31 Km i1)-(31 1m iP -31 Km iP) (im)-th group are combined and connected to the i-th input 8 i of the second group of control inputs of the memory block 3, the outputs of the And 31 elements are connected to the corresponding inputs (N x M) groups of elements OR 
, the outputs of which are connected respectively to the outputs of P groups of outputs 34 1 p -34 K p of the memory block 3. In the state selection block 6, the inputs 
the first groups of information inputs are connected to the inputs of the first groups of inputs of the j-th comparison nodes 35 j , the same-name inputs of the second groups of inputs of which are combined and connected to the corresponding inputs 36 1 -36 K of the second group of information inputs of the state selection block 6, the output of the comparison node 35 1 is connected with the output 37 1 of the block 6 and with the first inverse inputs of the elements And 38 1 -38 N-1 , the outputs of the comparison nodes 35 i are connected to the direct inputs of the corresponding elements And 38 i-1 and with i-and inverse inputs of the elements And 38 i 37 i +1 block 6. In the third memory block 7, the inputs 37 1 - 37 N are connected to the single inputs of the corresponding flip-flops 38 1 - 38 N , the zero inputs of which are connected to the outputs of the corresponding elements OR 39 1 - 39 N , and the single outputs are connected to the outputs 8 1 - 8 N block 7 and the corresponding inputs of the respective elements OR 39 1 - 39 N , and the single output of the trigger 38 i connected to the output 8 i block 7 and the corresponding inputs of the elements OR 39 1 - 39 i-1 , 39 i +1 - 39 N . In the first switch 9 i-th inputs 8 i groups of control inputs are connected to the first inputs of elements AND 
groups of information inputs, outputs of elements AND 
, the outputs of which are connected to the outputs 
the first switch 9. In the output signal selection block 10, the inputs 
of the first group of information inputs are connected to the inputs of the first groups of inputs of the p-th comparison nodes 44 P , the same-name inputs of the second groups of inputs of which are combined and connected to the corresponding inputs 45 1 - 45 K of the second group of information inputs of the block for selecting the output signal 10, the output of the comparison node 44 1 connected to the output 1 1 of the block and to the first inverse inputs of the elements AND 46 1 - 46 p-1, the outputs of the comparison nodes 44p are connected to the direct inputs of the corresponding elements And 46 p-1 and with p-and inverse inputs of the elements And 46 p, the outputs of which connected to the outputs 11 p+1 block 10. In the second switch 12 p-th inputs 11 p group of control inputs connected to the first inputs of elements B 
groups, the second inputs of which are connected to the inputs 
groups of information inputs, outputs of elements AND 
connected to the corresponding inputs of the OR elements 
, the outputs of which are connected to the outputs 
the second switch 12. In the first block generating a random code 14, the clock input 52 is connected to the inverse input of the first element And 52 and to the first inputs of the second elements And 53 1 - 53 Z , the outputs of which are connected to the corresponding inputs of the code converter 54, the outputs of which are connected to the outputs 36 1 - 36 K block, the output of the Poisson pulse stream generator 55 is connected to the direct input of the first element And 52, the output of which is connected to the clock input of the cyclically closed shift register 56, the bit outputs of which are connected to the second inputs of the corresponding elements And 53 1 - 53 Z . In the second random code generation block 15, the clock input 51 is connected to the inverse input of the first element AND 57 and to the first inputs of the second elements AND 58 1 - 58 Z , the outputs of which are connected to the corresponding inputs of the code converter 59, the outputs of which are connected to the outputs 45 1 - 45 K block, the output of the Poisson pulse stream generator 60 is connected to the direct input of the first element And 57, the input of which is connected to the clock input of the cyclically closed shift register 61, the bit outputs of which are connected to the second inputs of the corresponding elements And 58 1 - 58 Z . In the fourth memory block 16 inputs 17 1 1 i -17 D l i (l, i)-groups of installation inputs 
connected to the corresponding inputs of the (li)'s registers 62 li 
, the outputs of which are connected respectively to the outputs of the (l, i)-th group of outputs of block 16. In the first block 18 for determining the maximum code, the inputs of l groups 
connected to the write inputs of the registers 63 l , direct d-th outputs of which are connected to the first group of inputs of the decoders 64 d and to the first inputs by the AND element 
, the first inverse outputs of the registers 63 l are connected to the first inputs of the second group of inputs of the decoder 64 1 , the remaining inverse outputs of the registers 63 l are connected to the inputs of the second group of inputs of the decoders 64 d and with the first groups of inputs (D-1)-th analysis nodes 66 d , the output group of the first decoder 64 1 is connected to the second group of inputs of the analysis node 66 1 b the output groups of the remaining decoders 64 d are connected to the third groups of inputs of the analysis nodes 66 d , the outputs of the dth analysis nodes 66 d are connected to the second group of inputs (d+1) -x analysis nodes 66 j+1 , L outputs of the (D-1)-th analysis node 66 D-1 
the outputs of the elements AND 65 1 l -65 K l are connected to the inputs of the elements OR 67 l , the outputs of which are connected to the outputs 19 l of the block issuing the maximum code 18. In the fifth memory block 20 inputs 
(f, p)-x groups of information inputs are connected to the corresponding inputs (fp) - registers 68 fp 
groups, the outputs of which are connected respectively to the outputs 
(f, p)-th group of outputs of block 20. In the second block 22 for determining the maximum code, the inputs of f groups of information inputs 
connected to the register write inputs, the direct d-th outputs of which are connected to the first group of inputs of the decoders 70 d and to the first inputs of the AND elements 
, the first inverse outputs of the registers 69 f are connected to the first inputs of the second group of inputs of the decoder 70 1 , the remaining inverse outputs of the registers 69 f are connected to the inputs of the second group of inputs of the decoders 70 d and with the first groups of inputs (D-1)-th analysis nodes 72 d , the output group of the first decoder 70 1 is connected to the second group of inputs of the analysis node 72 1, the output groups of the remaining decoders 70 d are connected to the third groups of inputs of the analysis nodes 72 d, the outputs of the d-th analysis nodes 72d are connected to the second group of inputs (d + 1) - x analysis nodes 72 d+1 The outputs of the (D-1)-th analysis node are connected to the second inputs of the elements AND 
, the outputs of the elements And 71 1 f -71 D f are connected to the inputs of the elements OR 73 f , the outputs of which are connected to the outputs 23 f of the second block for issuing the maximum code 22. In the decoders 64d of the first block for determining the maximum code 18, the inputs 
and with the inputs of the first elements AND , the outputs of which are connected to the second inputs of the corresponding elements OR 
, the inputs of the second group of inputs are connected to the inputs of the second elements AND , the outputs of which are connected to the third inputs of the corresponding elements OR 
, the outputs of which are connected to the outputs 
decoders 64 d , . At the analysis nodes 66 d , the first block for determining the maximum code 18 inputs 
the first group, the outputs of which are connected to the corresponding q-th inputs of the elements OR 81 l of the second group, the outputs of which are connected to the first inputs of the corresponding elements AND 
of the second group and with the inputs of the corresponding elements NOT 84 d l of the first group, the outputs of which are connected to the first inputs of the elements AND 
the third group, respectively, the outputs of which are connected by the first inputs of the OR elements 
d-th analysis node 66 d , inputs of the second group of inputs 
connected to the second inputs of the AND elements of the first group, to the second inputs of the AND elements 
first group, inputs 
connected to the second inputs of the elements AND 
the second group, the outputs of which are connected to the second inputs of the OR elements 
the third group. In the decoders 70 d of the second device for determining the maximum code 22 inputs 
of the first group of inputs are connected to the first inputs of the corresponding elements OR 
and with the inputs of the first elements AND 94 d whose outputs are connected to the second inputs of the corresponding elements OR 
inputs 
the second group of inputs are connected to the inputs of the second elements AND , the outputs of which are connected to the third inputs of the corresponding elements OR 
, the outputs of which are connected to the outputs 
decoders 70 d . In the analysis nodes 72 d of the second device for determining the maximum code 22 inputs 
of the first group of inputs are connected to the first inputs of the corresponding elements AND 
the first group, the outputs of which are connected to the first inputs of the corresponding elements OR 
the first group, the outputs of which are connected to the corresponding q-th inputs of the elements OR 100 f of the second group, the outputs of which are connected to the first inputs of the corresponding elements AND 
of the second group and with the inputs of the corresponding elements NOT 
the first group, the outputs of which are connected to the first inputs of the AND elements 
of the third group, respectively, the outputs of which are connected to the first inputs of the OR elements 
the third group, the outputs of which are connected to the first inputs of the AND elements 
the fourth group, the outputs of which are connected to the outputs 
d-th analysis node 72 d , inputs of the second group of inputs 
connected to the second inputs of the elements AND 
of the first group, with the second inputs of elements AND 
the fourth group and with the inputs of elements NOT 
the second group, the outputs of which are connected to the second inputs of the OR elements 
first group, inputs 
third group of inputs of analysis nodes 
connected to the second inputs of the elements AND 
of the third group and with the inputs of elements NOT 
the third group, the outputs of which are connected to the second inputs of the AND elements 
the second group, the outputs of which are connected to the second inputs of the OR elements 
the third group. The purpose of the fuzzy probabilistic automaton is to generate state signals and output signals, as well as to generate fuzzy variables defined on sets of states and outputs. The formal mathematical model of a fuzzy probabilistic automaton has the form: , (,T(),Z),(,T(),Y) , where X, Y, Z - sets of input, output parameters and state parameters; 
- a set of conditional probabilities that determine the stay of the probabilistic automaton in the time step t in the state z t, provided that the parameter x t is supplied to the input in this cycle and the probabilistic automaton stays in the previous time step t-1 in the state z t-1 ; 
- a set of conditional probabilities that determine the presence of the parameter y t at the output of the probabilistic automaton in the time step t, provided that the parameter x t is supplied to the input at this cycle and the fuzzy probabilistic automaton is in the previous cycle in the state z t ; (,T(),Z) - assignment of a linguistic variable , where - the name of the fuzzy variable "state selection", T () - term-set of a linguistic variable, Z - base set; (,T(),Y) - specification of a linguistic variable , where - name of the linguistic variable "output signal selection", T () - term-set of a linguistic variable, Y - base set. For example, let , where the variables: 1 - "selection of the best states", 2 - "selection of good states", 3 - "selection of bad states", are set in triplets 
- fuzzy subsets on the base set Z; 1 - "selection of the best output signal", 2 - "selection of a good output signal", 3 - "selection of a bad output signal" are set by the set 
- fuzzy sets defined on the base set Y. Membership functions are set based on a survey of experts. When preparing a fuzzy probabilistic automaton for operation, the following operations should be performed. According to the installation inputs, they are written to the registers 
(Fig. 1 and 3) of the first memory block 2 codes of the reduced matrices of transition probabilities 
. The installation inputs are recorded in the registers 30 m i p (Fig. 1 and 4) of the second memory block 3 codes of matrices of probabilities of selecting the output signal 
. According to the installation inputs 17 1 1 i -17 D l i of the fourth memory block 16 are recorded in the registers (Fig. 1 and 12) the values of the membership degrees of fuzzy variables 1 . . By installation inputs 
are written in the registers 68 fp of the fifth memory block 20 the values of the membership degrees of fuzzy variables f . . matrices 
look like: 
where
P m i j - the probability that when a signal x m arrives at time t, the automaton will go to state z j, provided that at time t-1 it was in state z i . Reduced matrices 
look like: 
,
where
Probability matrices 
are set in the following form: 
,
where
P m i p is the probability that when a signal x m arrives at time t, the automaton will generate a control action y p, provided that at time t-1 it was in state z i . Reduced matrices 
are set in the following form: 
,
where
When writing codes to registers 24, the probability of the matrix P m z will be written to the K-bit register 24 m i j of memory block 2, and the probability of the matrix P m y will be written to the K-bit register 31 m i j of memory block 3. Information about the membership function is entered according to the following rule . The power of the set is , and the range (0,1) of the values of the membership functions is quantized (in Fig. 20 quantization is shown in seven levels). For each state z i there are L values of membership functions
. For the example under consideration, L = 3. In the registers 62 l1 - 62 lN of the fourth memory block 26 will be written codes . Similar reasoning is also valid for writing quantized values of membership functions. In the registers 68 f1 - 68 fp of the fifth memory block 20 will be entered codes. The fuzzy probabilistic automaton operates according to the following algorithm. Synchronization of the fuzzy probabilistic automaton is carried out by the generator 13 clock pulses. The inputs 1 1 - 1 M supply input signals x t that control the operation of the fuzzy probabilistic automaton. The state of the automaton is stored in the third memory block. Upon receipt of the input 1 m at the time t of the control action x m, depending on the state z i was the automaton at the time t-1, i.e., depending on the signal at the output 8 i coming from the third memory block 7 to input 8 i of the memory block 2 and input 8 i of the memory block 3, the outputs of the memory block 2 are supplied with the codes i-q row of the matrix , and the outputs of the second memory block 3 are the codes of the i-th row of the matrix . It happens in the following way. Since block 2 has a potential at inputs 8 i , 2 m , as well as at input 26, the elements AND (25 1m il -25 Km i1) (25 1m iN -25 Km iN) and register codes 24 i1 - 24 iN through these elements AND and elements OR 28 will be applied to the groups of outputs (29 1 1 -29 K l)(29 1 N -29 K N), respectively. In the same way, in the second memory block 3, the codes of the probabilities of registers 30 i1 - 30 ip through open elements AND (31 1m il -31 Km i1) (31 1m iP -31 Km iP) and elements OR 33 will be fed to output groups (34 1 1 -34 K 1)(34 1 P -34 K P). . The first 14 and second 15 random code generation blocks generate codes of numbers uniformly distributed over the interval (0,1). The state selector 6 generates the current state z t in accordance with the test rule in the random event circuit. Also, in the output signal selection block 10, an output signal y t is generated in accordance with the test rule in the random event circuit. Defined for time t signals z t and y t are fed to the inputs 8 of the switch 9 and the inputs 11 of the switch 12, respectively. Depending on the received signal z i at time t, from the outputs of the first switch 9, the values of the degrees of membership of fuzzy variables corresponding to the signal z i are sent to the block for determining the maximum code. Block 18 for determining the maximum code analyzes the values received at its input code combinations, and the output 19 l receives a signal, index l which corresponds to the largest value of the degree of membership of the variable 1 . . When the output signal y P arrives at the input 11 p of the second switch 12 at the time t, the values of the membership degrees of the fuzzy variables for the element y p of the base set Y arrive at the outputs of the switch 12. a single signal is received, corresponding to the largest code combination. Consider the operation of a fuzzy probabilistic automaton in more detail. Let, for example, it is known that the set of states has three elements Z = ( z 1 , z 2 , z 3 ), the set of output signals also has three elements Y = ( y 1 , y 2 , y 3 ), and let at the moment of time t the control signal x 2 is applied to the input 12. Let the transition probability matrix look like: 
Registers 
designed to store K = 8-bit values of probabilities. Let at the moment of time (t-1) the automaton was in the state z 1 , therefore, a single signal was received from input 8 1, which made it possible to read the contents of the first row of the matrix when a synchronizing signal was received from the generator 13 of clock pulses at input 26 from registers 24 2 1 1 -24 2 3 1 through elements AND 25 12 11 -25 82 11 25 12 33 -25 82 33 , OR 28 to outputs 29 1 1 -29 8 1 -29 1 3 -29 8 3 . . That is, at the outputs 29 1 1 -29 8 1 there will be a binary code of the number 0, 1, at the outputs 29 1 2 -29 8 2 - the binary code of the number 0, 4, and at the outputs 29 1 3 -29 8 3 - the binary code code number 1. The circuit implementation of the second memory block 3 is identical to the circuit implementation of the first memory block 2. The operation of block 3 will proceed in the same way as the operation of block 2. The first block of random code generation 14 operates as follows. Random pulses from the generator 55 of the Poisson stream of pulses arrive through the open (at the time intervals corresponding to the state of the automaton in i-th states) the first element And 52 to the clock input of the cyclically closed shift register 56, in one of the bits of which one is written, and in the rest zeros . The intensity of the random pulses of the generator 55 significantly exceeds the frequency of the poll at the input 51. Then the recorded unit repeatedly "runs around" the shift register 56 between the moments of polling its states at the input 51 by the pulses of the clock generator 13. Under this condition, the unit will be at the time of the poll at any of the outputs of the shift register 56 with a probability equal to one divided by the number of outputs of the register 56. The code converter converts the code for one combination into a binary code of a number uniformly distributed over the interval (0,1). The second random code generation block 15 works in a similar way. In the state selection block 6 (Fig. 5), each i-th comparison node 35 i analyzes the code combination received from the inputs 29 1 i -29 K i of the first group of inputs and the code combination, received from the block generating a random code inputs 36 1 - 36 K of the second group of inputs. Comparison nodes operate similarly to those given in (Design of microelectronic digital devices / Edited by S.A. Mayorov. - M .: Sov. radio, 1977, pp. 127 - 134). If the value of the code combination coming through the inputs 36 1 - 36 K is less than or equal to the value coming through the i-th group of inputs 29 1 i -29 K i to the i-th comparison node, then to the inputs of the elements I38 i-1 corresponding to the comparison nodes i, and for the first element AND38 1 - to the output 37 1 of the state selection block 6. a single signal arrives, and the subsequent elements 38 g receive a zero signal that closes these elements. Thus, the state selection block 6 determines the state z i , into which the fuzzy probabilistic automaton passes at time t. Let's assume that in our case a single signal arrived at the output 37 3 , and this means that the automaton passed at the time t to the state z 3 . The third memory block (see Fig.6) delays a single signal z i received at the input 37 i from the state selector 6, one clock cycle of the generator 13, and then outputs it to the output 8 i . It happens in the following way. A single signal applied to the input 37 3 flips the trigger 38 3 in a single state. The potential from the single output of the trigger 38 3 resets the triggers 38 1 , 38 2 to the zero state through the OR elements 39 1 , 39 2 and is fed to the output 8 3 of the fuzzy probabilistic automaton and the input 8 3 of the switch 9. Similarly to the state selection block 6, the output selection block functions. signal 10. The output signal Y p determined by the block 10 is fed to the output 11 p of the fuzzy automaton and the input 11 p of the second switch 12. When a signal z i , , at the time t from the output 8 i of the third memory block 7 is read, L D-bit values of the functions accessories from the registers of the first memory block 6. The potential at the output 8 i will open the elements AND 
. The values of the contents of the registers 62 li are read, which is fed from the outputs of the switch 9 to the inputs of the first block 18 for determining the maximum code in the form of L groups of D-bit codes of the values of the membership function of fuzzy variables 1 at the point z i . When a signal Y p is received from the block for selecting the output signal 10 at time t, the F D-bit values of the membership functions are read from the registers of the second memory block 20. The potential at the output 11 p will open the elements AND 
. The contents of the registers 68 fr through the switch 12 is fed to the inputs of the second block 22 to determine the maximum code in the form of F groups of D-bit codes of the values of the membership function of fuzzy variables f at the point Y p . Block 18 determines the maximum code analyzes coming from the switch 9 L D-bit code combinations, which are respectively the degrees of membership of fuzzy variables , i.e. determines which of the fuzzy variables has the greater value of the membership function for the current state, and outputs a signal about the number of the largest code combination. L code combinations are supplied to the input tires 43 1 - 43 L (Fig.13), from which the device for determining the maximum code must select the maximum code combination, and if there are k identical in value in the inputs 43 1 - 43 L codes and maximum among L code combinations, then such a case should also be recognized. Each 1st code combination is served on the input tires 43 1 1 -43 d L in the corresponding register 63 l . Code combinations are recorded in the cells of the register 63 1 - 63 L in parallel in time, but sequentially by digits. First, pulses will be applied to input buses 43 1 1 ,43 1 2 ,43 1 3 ,...,43 1 L , then to input buses 43 2 1 ,43 2 2 ,43 2 3 ,...,43 2 L , etc. until the final supply of pulses of code combinations through the input buses 43 D 1 ,43 D 2 ,43 D 3 ,...,43 D L , . Parallel-sequential recording of code combinations in registers 63 provides sequential operation in time of state decoders 64 and analysis nodes 66. The algorithm of the block for determining the maximum code consists in sequential analysis of parallel (of the same name) bits of code combinations recorded in registers 63 1 - 63 L with sequential detection large codes in parallel (of the same name) digits, starting from the most significant digit up to the youngest. Moreover, the analysis of parallel bits of code combinations of registers 63 is carried out both by state decoders 64 and analysis nodes 66. Identification of code combinations larger in value than the smallest is performed by the first state decoder 64 1 and analysis nodes 66 1 - 66 D-1, and the latter node analysis 66 D-1 detects the maximum (one or more) code combinations of N, recorded in the registers 63. The essence of the algorithm of the block determine the maximum code is as follows. First, consider the parallel high bits a 1 1 -a 1 L registers 63. Obviously, the following events are possible here. The symbols of all digits a 1 1 -a 1 L are equal to zero, the symbols of all digits a 1 1 -a 1 L are equal to one, or there are symbols equal to zero and one. In the first two cases, the outputs 79 1 1 -79 1 L of the decoder 64 1 must have single potentials, and in the third case, the single potentials must be at those outputs 79 1 1 -79 1 L that correspond to the lower index registers 63 in the older cells of which a 1 1 -a 1 L are written single values of code bits, i.e. the logic function that determines the signal at the 1st output 79 1 l of the first decoder 64 1 can be written as follows:
. To determine the signal at the l-th output of the d-th decoder 64 d , based on the method of mathematical induction, we can write the following logical function
. Equality is a sufficient condition, but a necessary one, to determine that there can be a maximum number in register 63 l, i.e. decoders 64 d allocated registers 63 l in which the characters a l are equal to one. The first determining the state of the l-th output 88 d l of the d-th analysis node 66 d is the event: what is the state of the l-th output 88 d l -1 (d-1)-th analysis node 66 d-1 , and for the first analysis node 66 1 the state of the l-th output 88 1 l is determined by the state of the l-th output 79 1 l of the first decoder 64 1 . The second determining the state of the l-th output 88 d l of the d-th analysis node 66 d is an event determined by the inversion of the equivalent of two statements d l and some logical function d l , which is determined by the expression: 
And 
is always equal to zero if G d l -1 either , or one of the (L-1) disjunctions included in the conjunctive normal form (2) is equal to zero. Function 
determining the state of the l-th output of the d-th analysis node 66 d (one or zero at the output 88 d l), is written as:
Equations (1), (2) and (3) imply that 
is always zero if either d l or G d 1 or G d 2 etc. to G d 1 -1 are equal to zero. From the outputs of the analysis node 66 D-1 comes the code combination G D l -1 , and each output 88 D l -1 is connected to the second group of inputs of the elements And 65 1 l -65 D l .. The unit potential at the output 88 D l -1 allows open that group of elements AND 65 1 l -65 D l , which received the maximum code from register 63 l . Then the maximum code combination is fed to the input of the elements OR 67 1 l -67 D l , after which the signal about the index of the maximum code appears at one of the outputs 19 1 - 19 L of the first block for issuing the maximum code 18. This generates the value of the fuzzy variable having the largest the value of the membership degree on the given state. The second maximum code determination unit 22 operates in the same way as the first maximum code determination unit 18, so its operation will not be described in detail. So, at the outputs 19 l of the first block for determining the maximum code 18, the potential will be fixed, which determines the index l of the fuzzy variable 1 , the most preferable for the current state. At the outputs 23 f of the second block for determining the maximum code 22 there will be a potential that determines the index f of the fuzzy variable f , the most preferable for the current state. The technical and economic efficiency of the proposed device in relation to the known one (AS USSR N 1200297, class G 06 F 15/20, 1985) can be determined from the expansion of functionality, namely, the proposed device generates not only states, output signals, but also linguistic variables defined on the base sets of states and output signals. Membership functions of fuzzy variables are set by the expert survey method. The functions of transitions and outputs of the automaton are given in the form of randomized rules. If we evaluate the costs of developing and manufacturing the proposed device through the value of C 1, the costs of research - through the value of C 2, then the total costs for solving the problem will be determined
CI = C1 + C2. When using a known device for solving control problems, the costs of manufacturing special additional devices and conducting full-scale experiments are necessary. These costs will be determined by the value of CN. Note that the costs CN will significantly exceed the value CI, since field tests already require significant economic costs.
Claim
1. A fuzzy probabilistic automaton containing a clock generator, the first block for generating a random code, a state selection block, an output signal selection block, the first, second and third memory blocks and a switch, with M inputs of the control input group of the device connected to M inputs of the first control groups inputs of the first memory block, inputs (N x N x M) of the groups of the first installation inputs of the device are connected respectively to the inputs of N x N x M groups of installation inputs of the first memory block, N inputs of the groups of the second control inputs of which are connected to N outputs of the output group of the third memory block , the group of information outputs of the first memory block is connected to the inputs of the first group of information inputs of the state selection block, the output of the clock pulse generator is connected to the clock input of the first random code generation block, the K outputs of the group of outputs of which are connected to the K inputs of the second group of information inputs of the state selection block, which differs the fact that it additionally about the second block for generating a random code, the fourth and fifth memory blocks, the second switch, the first and second blocks for determining the maximum code are introduced, and the inputs of N x P x M groups of installation inputs of the second memory block are connected to the inputs of N x P x M groups of the second installation inputs device, M inputs of the group of first control inputs are connected to M inputs of the group of control inputs of the device and M inputs of the group of first control inputs of the first memory block, N inputs of the group of second control inputs are connected to N inputs of the group of second control inputs of the first memory block, N outputs of the group of outputs of the third of the memory block and N inputs of the group of control inputs of the first switch, the outputs of the P groups of information outputs of the second memory block are connected to the corresponding inputs of the P groups of information inputs of the output signal selection block, and the clock input of the second memory block is connected to the output of the clock generator and the clock inputs of the first memory block , the second generation block and random code, N outputs of the group of information outputs of the state selection block are connected to the corresponding N inputs of the group of first information inputs of the third memory block, K outputs of the group of outputs of the second block for generating random code are connected to K inputs of the group of second information inputs of the block for selecting the output signal, inputs N x L groups of information inputs of the first switch are connected to the outputs of N x L groups of information outputs of the fourth memory block, N x L groups of information inputs of which are connected to the inputs of N x L of the third groups of installation inputs of the device, the outputs of L groups of information outputs of the first switch are connected to the inputs of L groups information inputs of the first block for determining the maximum code, the outputs of the group of information outputs of which are connected to the outputs of the third group of outputs of the device, the P outputs of the group of outputs of the block for selecting output signals are connected to the P inputs of the group of control inputs of the second switch, the inputs P x F of the groups of information whose inputs are connected to the outputs P x F of the groups of information outputs of the fifth memory block, the inputs P x F of the groups of information inputs of which are connected to the inputs P x F of the fourth groups of installation inputs of the device, the outputs of P groups of information outputs of the second switch are connected to the inputs of F groups of information inputs the second block for determining the maximum code, the groups of information outputs of which are connected to the outputs of the fourth group of outputs of the device. 2. The automaton according to claim 1, characterized in that the first memory block contains registers, N x M groups of elements AND, N x M groups of elements OR, and each of the k inputs (i, j, m)-th 
groups of installation inputs are connected to the recording inputs of the corresponding (i, j, m)-x registers, the outputs of which are connected to the first inputs of the corresponding elements of the (i, j, m)-th groups of elements AND, the second inputs of the AND elements are combined and connected to the clock input of the memory block, the third inputs of the AND elements of each of the m groups are combined and connected to the m-th inputs of the group of the first control inputs of the block, the fourth inputs of the elements of the AND (im)-th group are combined and connected to the i-th input of the second group of control inputs of the block, the outputs elements AND - with the corresponding inputs of N x M groups of OR elements, the outputs of which are connected respectively to the outputs of N groups of outputs of the block. 3. The automaton according to claim 1, characterized in that the state selection block contains N comparison nodes, N - 1 elements And, and k inputs j of the first group of information inputs of the connection with the inputs of the first groups of inputs of the j-th comparison nodes, the same inputs of the second groups whose inputs are combined and connected to the corresponding k inputs of the second group of information inputs of the block, the output of the first comparison node is connected to the first output of the block and to the first inverse inputs of the AND elements, the outputs of the i-th comparison nodes are connected to the direct inputs of the corresponding (i - 1)-x elements AND and with the i-th inverse inputs of the i-th elements AND, the outputs of which are connected to the (i + 1)-th outputs of the block. 4. The automaton according to claim 1, characterized in that the third memory block contains N triggers and N OR elements, and its inputs are connected to the unit inputs of the corresponding triggers, the zero inputs of which are connected to the outputs of the corresponding OR elements, and the unit outputs are connected to the outputs of the block and the corresponding inputs of the corresponding OR elements, and the single output of the i-th trigger is connected to the i-th output of the block and to the corresponding inputs (1 - (i - 1) - (i + 1) - N) of the OR elements. 5. The automaton according to claim 1, characterized in that the output signal selection block contains P comparison nodes and P - 1 AND elements, moreover, k inputs of p first groups of information inputs are connected to the inputs of the first groups of inputs of p-th comparison nodes, the same inputs of the second groups of inputs of which are combined and connected to the corresponding k inputs of the second group of information inputs of the block, the output of the first comparison node is connected to the first output of the block and to the first inverse inputs of the AND elements, the outputs of the p-th comparison nodes are connected to the direct inputs of the corresponding (p - 1) - x elements AND and with p-th inverse inputs of p-th elements AND, the outputs of which are connected to the (p + 1)-th outputs of the block. 6. The machine according to claim 1, characterized in that the first block for generating a random code contains the first and a group of second AND elements, a code converter, and the clock input is connected to the inverse input of the first AND element and to the first inputs of the group of second AND elements, the outputs of which are connected to corresponding inputs of the code converter, the outputs of which are connected to the outputs of the block, the output of the Poisson pulse stream generator is connected to the direct input of the first AND element, the output of which is connected to the clock input of the cyclically closed shift register, the bit outputs of which are connected to the second inputs of the corresponding second elements of the AND group. 7. The automaton according to claim 1, characterized in that the first block for determining the maximum code contains L registers, D decoders, D - 1 analysis nodes, L groups of D elements AND and a group of L OR elements, and the l-th group of inputs are connected with write inputs of the l-th registers, the direct d-th outputs of which are connected to the first group of inputs of the d-th decoders and to the first inputs of the d-th elements AND the l-th group, the first inverse outputs of the l-th registers are connected to the first inputs of the second group inputs of the l-th decoders, the remaining inverse outputs of the l-th registers are connected to the inputs of the second group of inputs of the d-th decoders,
The invention relates to information-measuring technology and is intended to simultaneously obtain a pair of probabilistic characteristics, representing a two-dimensional histogram of the duration of exceeding emissions and dips of various durations of various levels of analysis
The invention relates to information-measuring and computer technology, is intended to obtain a two-dimensional histogram of the level and voltage derivative and can be used in the electric power industry to assess the variability of voltage in industrial electrical networks, as well as in other fields of technology, for example, to study and evaluate the behavior of various swinging objects: ship decks, tank platforms during movement, etc.
The invention relates to computer technology and control systems, can be used to build adaptive fuzzy controllers for solving problems of managing objects, the mathematical model of which is not a priori defined, and the purpose of functioning is expressed in fuzzy terms
