Unmanned aerial vehicle (UAV), today, plays prominent role in many parts of human activity. One of serious problems exploitation UAV is accuracy of positioning. Satellite clocks, satellite orbits, ionosphere and troposphere, multipath, etc - all of this are sources of signal's uctuation. But largest uctuation of signal is created by ionosphera. One of distribution laws (Nakagami, normal distribution law, Rayleigh and Rice - most common situations) characterize Ionosphere's uctuations. For solve this problem supposed to using of adapting control system, based on Flight Control Center. At working this system there exist probability of making type I and type II errors during salvation of identi cation problems. This work proposes an approach to implementing the threshold optimization for classifying the states of satellite communication systems, when disorting of signals described by Rayleigh distribution law.
When receiving signals in satellite communication channels there frequently appears a uctuation of the signal amplitude in time. The uctuations usually take place as a random process with a qausiperiod from fractions of a second to dozens of minutes, and their main descriptive feature is fading depth. The fading depth is de ned by the deviation of the instantaneous amplitude of the signal from the set level (as a rule, median one) and can reach dozens of decibels. The level of the uctuating signal can be estimated with the statistical method. In general, the nonstationary fading process is usually divided into two stationary processes: with uctuations of the medium eld values and fast uctuations around medium values. The rst type of uctuations refers to slow fading, whereas the second type refers to the fast one [Taa16].
The most common distribution of the signal amplitude at fast interference fading is close to Rayleigh distribution.
The energy loss in a satellite communication channel, when there is Rayleigh fading, turns out to be quite substantial [Pin19]. Therefore, it is necessary to provide the probability of the wrong data reception, in order to keep the quality of communication in case of a Rayleigh fading not less than 10-5.
One of the methods to keep the set probability of the wrong data receipt at the necessary level is to use an automated quality control and management system, which will assess the state of the transitionspheric communication channels in the mode close to real time. Based on the transitionspheric communication channels state assessment, it is possible to identify the required control action which will allow to keep the necessary probability of the wrong reception of a GPS/GLONASS signal, and thereby to provide the required accuracy of remotely piloted aircraft system positioning. Thus, it is necessary to solve the task of the transionspheric communication channels state identi cation at Rayleigh fading in order to develop the appropriate control actions leading to the required remotely piloted aircraft system positioning accuracy.
During the transionspheric communication channels state identi cation, errors of the rst and second kind may appear, which may lead to an inappropriate control action or to an untimely reaction to the change in the transionspheric communication channels state, resulting in the deterioration of the remotely piloted aircraft system positioning accuracy. 2
Let us analyze the factors that in uence the remotely piloted aircraft system positioning accuracy. The main error types and the inaccuracies in the remotely piloted aircraft system positioning caused by them are given in Table 1 [Ste12].
It is obvious that ionosphere has the most in uence on the remotely piloted aircraft system positioning accuracy. Satellite communication channels are largely in uenced by ionosphere due to the electron density and its changes, which leads to high level Rayleigh fading in satellite radio-signals. The ratio signal/noise (S/N) experiences fading depending on the changes in electron density.
If the fading level in a communication channel is rather high, i.e. the number of square components is big, and none of them exceeds in its amplitude all the rest, then the amplitude signal distribution is de ned by Rayleigh distribution, which is described by [Lam10]: pR(r) = r 2 exp( where r is the received signal amplitude; 2 is the average signal energy.
The most common error metric in satellite communication channels with character oriented messaging without antinoise coding is error probability per message bit. Modern requirements of the International Communication Union set limits for this parameter in the required in practice value not less, than 10 5 [Ion16]. The error probability is de ned [Ion16]: where hb20 is a normalized ratio signal/noise at receiver input; k - is the ratio of e ective voltage of the regular and di use signal components at the receiver input, where k12 = 1 + k2.
When the regular component is absent at the receiver input (scattering in the communication channel), for a rather little error probability and high hb0 values, the probability of the wrong receipt equals [Ion16]: With typical for practice values of k = 57dB [Ion16], the losses in the Rayleigh channel are 7-14 dB higher than in channel with Rician fading, which is a considerable value. Antinoise coding, implemented in channels with permanent parameters, at slow fading provides an advantage of several dB [Ion16], which does not provide a compensation of the antinoise loss. An automatic control and management system is to carry out the controlled object condition identi cation, to work out the control action (based on the management goals and taking into account the current condition of the environment), to realize the control action in the automated mode close to real time. This will provide a timely reaction to the changes in the condition of transionospheric communication channels characterized by high uncertainty. When identifying the controlled object condition there appear errors of the rst and the second kind. In statistical hypothesis testing a type I error is the rejection of a true null hypothesis (also known as a "false positive" nding or conclusion), while a type II error is the non-rejection of a false null hypothesis (also known as a "false negative" nding or conclusion). It is necessary to determine the optimal value of the classi cation threshold as the error data directly in uence the object positioning accuracy. The article suggests a solution for the optimization of the thresholds of the rst and second type error classi cation for an automated system of control and management of transionospheric communication channels, the condition of which is described by the Rayleigh distribution law. 3
In general, the task aimed to identify objects could be reduced to the veri cation of numerous hypotheses H1; H2; : : : ; Hi; : : : ; Hn, where Hi is a hypothesis implying the objects belonging to Class Ai . Lets assume that the a priori distributions of these hypotheses probabilities are set. It is known whats the likelihood the object n P (Hi) can belong to class Ai (or how often the object of this class is appeared). Moreover, P P (Hi) = 1, as i=1 the object is to be pertained to a certain class. The conditional density of distribution is pi(x) = p(xi=Hi).
Two hypotheses H1 = N and H2 = N are used in the identi cation system under design at corresponding to them a priori probabilities of situation, emerging in the network as a normal p1 = p(H1) = p(N ) one and an abnormal p2 = p(H2) = p(N ) . And also p1 + p2 = 1.
It is required to nd a decision rule ensuring the top accuracy in the identi cation system. Using the NeymanPearson criterion we x the probability of "false alarm" Pf:a: at stable level C and claim the minimum of pass error Ppmasins of TN operating trouble. Then
Ppmasins = p2[1 n Y i=1 The activity of the monitoring error evaluation system can be described by a probability graph.
The transionospheric communication channels are characterized with the certain states of N and N , under which we will mean normal and abnormal operation in the process of its functioning which (in the monitoring system) are respectively displayed into the normal A and abnormal state.
A priori probabilities of the normal and abnormal states in transionospheric communication channels are detected correspondingly by a priori probabilities of the p1 and p2 states. For all the stages, with abnormal situation omission, were obtained resultant errors:
Ppass = p2 1 + p2 1 2 + + p2 1 2 : : : n = p2[1
(xoi)] where: i = (xoi), i = (xoi), p1 = 1 p2 a priori probability of a normal situation occurrence; p2 a priori probability of its absence.
The problem of thresholding xoi for each stage is currently central. Lets make tradeo s of thresholds. Following the Neyman-Pearson criterion we will set the probability of false alarm at certain given C level. Then, for the entire network, we get (2) (3) (4) (5) (6) (7) (8) (9) ("false alarm" and "abnormal situation" errors), are de ned as follows:
Having minimized the probability of abnormal situation omission we get As a result, the decision rule ensuring the highest level of accuracy for the identi cation system is the NeymanPearson criterion which is detected for the TN by expressions (6) and (7). The minimization problem of the function (7), where variables xoi are linked by functional dependence (6) is the constrained optimization problem. Lets form a functional of optimization where is an undetermined Lagrange multiplier.
Calculating the derivatives ddxFoi = 0 we get a system of n equations: p2 d (xo1) dxo1 (xo2) : : : (xon) (xo2) : : : (xon) = 0; : : : p2 d (xon) dxon (xo1) : : : (xo(n 1)) + p1 (xo1) : : : (xo(n 1)) = 0; which, along with equation (6) allow to nd the undetermined Lagrange multiplier and n of xoi variables.
Since the system (9) contains n of xoi variables and undetermined Lagrange multiplier , which must be de ned, while the equation system (9) has n equations and n + 1 of variables, then for nding its unambiguous solution one more equation should be added, in the capacity of which we take the tolerance probability equation for "false alarm" (6).
Equations (9) and (6) including (2) and (3) after di erentiation on the lower and higher threshold take the form: xo2 xon R f (x2=N )dx2 : : : R f (xn=N )dxn 1 1 1 1 R f (x2=N )dx2 : : : R f (xn=N )dxn xo2 xon xo1 R f (x1=N )dx1 : : : 1 1 R f (x1=N )dx1 : : : xo1 xo(n 1)
R 1 1
R xo(n 1) f (xn 1=N )dxn 1 f (xn 1=N )dxn 1 = = p1f (xo1=N ) p2f (xo1=N )
; : : : p1f (xon=N ) p2f (xon=N )
; f (xi=N )dxi = C: p1
Yn Z1 i=1xoi
Under certain distribution laws and f (xi=N ) this problem has a single value solution. In equations (
Ppmasins
At each of the following stages, the information about correct solution of i = 1 i (information about normal transionospheric communication channels functioning) is analyzed. When the Type I or Type II errors are detected, their speci cation is carried out by means of the following stage procedure. Let us de ne the procedure of error detection by the feature x as the rst stage of two-stage procedure, while for the second stage we will detect the error by the feature y. Generally, the identi cation problem solution is determined by Type I and Type II errors. Now lets write down the functions for distribution density of the feature x at transionospheric communication channels problem-free functioning f (x=I = f1(x)) and at trouble functioning f (x=I = f1(x)). Then the errors of Type I and Type II of the detector (stage 1) are : o = f1(x)dx o =
f2(x)dx p = f1(y)dy p =
f2(y)dy
Zxo 1 yo Z 1 Type II errors for the recognizer (Stage 2) are de ned:
Transionospheric communication channels has the following conditions: "1" - the system is out of order, the failure was not detected; "2" the system is operational, it was found as workable; "3" - failure was detected and recognized (abnormal situation presence); "4" the system is workable, false detection and recognition (false alarm); "5" system is out of order, failure was detected but not recognized (abnormal situation omission); "6" - the system is in order, a false detection and correct recognition.
In view of formulae (
Pf:a: = p1 f1(x)dx
f1(y)dy = C = const Ppmasins = minp2(1 f2(x)dx
f2(y)dy) dxo dyo
Since in the expression (14) the thresholds xo and yo are linked with one functional dependence xo = (yo), having di erentiated (15) by yo and set it to zero, we get: f2(xo) f2(y)dy + f2(yo) f2(x)dx = 0
At free-hand laws of x and y features distribution, particularly, at normal law, there is no possibility to obtain an exact solution of the classi cation thresholds optimization problem. However, in certain cases, at Rayleigh distribution laws, in particular, the solution could be obtained in its nal form.
Lets set densities of features x and y distribution probability in the form of Rayleigh distribution laws. The Type I and Type II errors are in the form of: o = f1(x)dx; o = f2(x)dx; p = f1(y)dy; p = f2(y)dy; x2 f1(x) = xe 2 ; f2(x) = (x a)e
y2 ; f1(y) = ye 2 ; f2(y) = (y (y b)2
2 ;
yo
Z
b
b)e
Equations (14) and (
C (xo a) + (yo
b) = 0 xo = s 2lnp1=C 1 + ( b )2 a ; yo = s 2lnp1=C 1 + ( a )2 b
Having di erentiated (17) by o by substituting the result in (18), we obtain optimal classi cation thresholds: 5
Lets analyze increasing UAV positioning errors, using program u-center. Analized data was got in real experiment from UAV.
Fig. 1 demonstrate example of UAV positioning errors distribution, when navigation area is uctuating.
Fig. 1.left demonstrating example of UAV positioning errors distribution for clear sky, i.e. normal state of navigation area. UAV positioning errors less than 0,9 m. That data was get with PDOP (Position dilution of precision) = 1, signal/noise ratio in measuring channels - not less than 45 dBm / Hz, the time delay of the signal is 1 : : : 2 ns.
Fig 1.right demonstrating example of UAV positioning errors distribution caused by ionosphere uctuation. UAV positioning errors more than 7 m. That data was get with PDOP = 1 : : : 2, signal/noise ratio in measuring channels 33 : : : 28 dBm / Hz, the time delay of the signal is 2 : : : 5 ns.
Thereby, when its uctuation of navigation area, we must establish the fact of uctuating and, after that, identify state of transionospheric communication channels for develop necessary control procedure. This will allow to hold signal/noise of GPS/GLONASS in required values.
Dependences in Fig. 2a, built according to the formula (
The problem of optimizing classi cation thresholds of the rst and second kind of errors of trans-ionospheric communication channels control and monitoring system, described by the Rayleigh distribution law of random variables is solved in this paper. The authors have obtained the common solution of problem the optimal classi cation thresholds for the anomalous situation detector and recognizer. In future work, it is advisable to conduct an investigation on the dependencies of the communication channel states for the distribution law of random variables described by Nakagami distribution functions. 6.0.1
Acknowledgements This research is supported by the scienti c project Development of a multi-rotor robotic unmanned aerial vehicle using a strapdown inertial navigation system in the Federal Target Program 2014-2020 (unique identi er RFMEFI57818X0222) with the nancial support of the Ministry of Science and Higher Education of Russian Federation. [Taa16] Taabu S.D., D'Ujanga F.M., Ssenyonga, T. Prediction of ionospheric scintillation using neural network over East African region during ascending phase of sunspot cycle 24 (2016) Advances in Space Research, 57 (7), pp. 1570-1584