The paper describes two modified implementations of unscented Kalman filter (UKF) and unscented particle filter (UPF) to solve nonlinear filtering problem for discrete-time dynamic space model (DSSM). DSSM is supposed to be nonlinear with additive Gaussian noise. The considered algorithm modifications are based on combination of UD-factorization of covariance matrices with sequential Kalman filter. The solution of tracking problem is illustrated for two cases. In the first case the problem of estimate of movable target coordinates from observed noised bearing is considered (a problem of passive location). In the second case the problem of an active location is described when noisy values of a distance to the accompanied object besides a bearing are available to the observer. Moreover, in the second case the motion model is extended by means of introducing a new parameter (a maneuver) such as an angle of velocity direction. To examine robustness of the considered algorithms in active target tracking problem (the second case) an arbitrary maneuver that differs from the initially given one in the motion model is considered as an observation. 

The research proposes a mathematical optimization approach of arriving aircraft traffic at the aerodrome zone. The airfield having two parallel runways, capable of operating independently of each other, is modeled. The incoming traffic of aircraft is described by a Poisson flow of random events. The arriving aircraft are distributed by the air traffic controller between two runways. There is one approach flight path for each runway. Both approach paths have a common starting point. Each approach path has a different length. The approach trajectories do not overlap. For each of the two approach procedures, the air traffic controller sets the average speed of the aircraft. The given model of airfield and airfield zone is considered as the two-channel system of mass service with refusals in service. Each of the two servicing units includes an approach trajectory, a glide path and a runway. The servicing unit can be in one of two states – free and busy. The probabilities of the states of the servicing units are described by the Kolmogorov system of differential equations. The number of refusals in service on the simulated time interval is used as criterion for assessment of mass service system quality of functioning. This quality of functioning criterion is described by an integral functional. The functions describing the distribution of aircraft flows between the runways, as well as the functions describing the average speed of the aircraft, are control parameters. The optimization problem consists in finding such values of the control parameters for which the value of the criterion functional is minimal. To solve the formulated optimization problem, the L.S. Pontryagin maximum principle is applied. The form of the Hamiltonian function and the conjugate system of differential equations is given. The structure of optimal control has been studied for two different cases of restrictions on the control of the distribution of incoming aircraft traffic. The regularities of the control parameters influence on the value of the functional criterion are revealed.

 

This article shows the relationship between filters based on modeling of the random process paths with terminating and branching and a continuous particle filter that are related to sequential Monte Carlo methods. Different variants for calculation of weight coefficients in the particle filter for stochastic continuous systems (stochastic diffusion systems) are given. Along with the representation by a continuous function, it is shown that the path of the weight function can be presented by a piecewise constant function with nonnegative real values and also by a piecewise constant function with nonnegative integer values. This representation is based on paths modeling of the general Poisson process. The relation with the differential equation for the weight function is indicated. All the given variants for weight coefficients calculation in the particle filter do not require a complex software development, they are suitable for the particle filter software using various parallel programming technologies for high-performance computing systems. The continuous particle filter considered in this paper can be used in various applied estimation tasks, for example, tracking applications, restoring the motion trajectory from observations, restoring a signal from the noise, identifying the dynamic system parameters, and many others. In the future, it is planned to expand the use of the particle filter for stochastic jump-diffusion systems. In addition, it is planned to develop algorithms for predicting the states of stochastic diffusion and jump-diffusion systems based on the calculation of weight coefficients in the particle filter considered in this article.

The paper discusses issues connected with the use of an artificial neural network (ANN) to approximate the experimental data. One of the problems in the development of the ANN is the choice of an appropriate activation function for neurons of the hidden layer and adjusting the parameters of the function in the learning process of the network. The article discusses a three-layer perceptron with one hidden layer, each neuron of which has the activation function in the form of a Gaussian curve. The choice of radial basis activation function allows the use of the direct method of determining the weight coefficients – method of least squares in the process of network training. Thus the quality of the approximation depends on the correct choice of the value parameter of the activation function, which in this case is the width of the Gaussian bell curve. In practice, this parameter is determined by conducting numerical experiments. This is a rather time-taking process. In this paper we propose to define the value of this parameter by the training set, representing the coordinates of the test curve points set with the desired properties. These properties are based on the a priori data of the approximated functions (linear, quadratic, logarithmic, exponential relationship). Because the test curve is given in explicit form, the parameter of activation function is determined from the condition of reaching the minimum of the integral from the squared difference between the values of the test functions and the output of the network. This approach guarantees obtaining the approximating curve with good properties, in particular, it is characterized by the absence of so-called "oscillations" – many inflection points in its graph.

 

The applicability of the V. Lychagin "manual" integration method is analyzed with respect to systems of two quasilinear hyperbolic differential equations of the first order with two independent variables t, x and two unknown functions u = u (t, x) and v = v (t, x). The systems under consideration are a special case of Jacobi systems, for which V. Lychagin proposed an analytical method for solving the initial-boundary value problem. Each of the equations of the system is associated with a differential 2-form on four-dimensional space. This pair of forms uniquely determines the field of linear operators, which, for hyperbolic equations, generates an almost product structure. This means that the tangent space of four-dimensional space in each point is a direct sum of two-dimensional own-subspaces of the given operator and, thus, two 2-dimensional distributions are defined. If at least one of these distributions is completely integrable, then it is possible to construct a vector field along which shifts keep the solution of the original system of equations. Thus, the solution of the initial-boundary value problem for the system under consideration can be obtained analytically by shifting the initial curve along the trajectories of the given vector field. As an example, the Buckley-Leverett system of equations describing the process of nonlinear one-dimensional two-phase filtration in a porous medium is considered. To construct the solution of the Cauchy problem, a curve of the initial data is chosen; the solution of the Buckley-Leverett system is obtained by shifting this curve along the trajectories of the vector field (this vector field is defined up to multiplication by a function). The cross-sections of the components of this graph for different instants of time are brought in the figure. The graph shows that at some point of time the solution stops being unambiguous. At this point, the solution breaks and a shock wave appears.

The application of hybrid memetic algorithm of global constrained optimization in the search optimal control of nonlinear stochastic systems problem is researched in the paper. The suggested approach is based on control parametrization, which gives an opportunity to reduce the search optimal control problem to nonlinear programming problem. The global optimization memetic algorithm is suggested to solve the nonlinear programming problem. The term "memetic algorithms" is wildly used to denote method, which is based on evolutional, cultural-evolutional or another approach, that uses notions like population and individual learning process or other local improvement procedure to determine global extremum. In the developed algorithm the cultural evolution component is implemented by means of the solution of sub-task optimization using ant colony method or simulated annealing algorithm. Herewith mems (unit of cultural information) are used to generate perfect individual. Several alternative forms of parametrization are considered: in the form of expansion by Legendre polynomials system and in the form of cosine-waves. The efficiency of the proposed algorithms was analyzed by means of created software complex. The problem of damping rotational satellite motion by means of installed engine was solved as the example. Verification of obtained results using the solution detected with local variations method was done. The described results show the sufficient efficiency level and reflect the necessity of further research in this direction.

 

This paper analyzes the optimal of constant proportion index portfolio strategies. They are also called passive strategies which are becoming more common in Russia and abroad. They are significantly cheaper to implement than active strategies. In addition, as practice shows, in the long term they are more profitable and less risky. The main problem in these strategies is the choice of the proportions in which the investor allocates his capital between risky and risk-free assets. In constant proportion index portfolio the weight of risk asset remains constant throughout investment period. For this purpose, the investor with a certain frequency restores the desired balance between risky and risk-free assets. Each period at the beginning of which such recovery occurs is called the re-balancing period. In the case of strategies with index portfolios, risky assets are the shares of the index fund, and risk-free assets are the deposits in reliable bank or government bonds. According on the daily value of units of these funds and the annual interest rate for the 11-year period, using a specially developed program optimal weight index funds in the portfolios has been found. Parameters of the analyzed portfolios are: length of the investment period (from one year to 10 years) and the frequency of weight rebalancing (month, quarter, year). The sequence of optimal weights and the corresponding optimum yield for consecutive investment periods with a specified frequency of re-balancing were determined for each fund. It was found that in almost all cases, the optimal weights of fund equals the extreme values 0 or 1. Also, the frequencies of these values in the selected sequence is about the same for all funds. This empiric fact can be conventionally called the principle of extremeness or “all or nothing” principle.

 

Air transportation industry traditionally stays on the cutting edge of technology. Feats of engineering are introduced during aircraft design and operation, airfields and airways arrangement, radio technology development. In order to raise the economic efficiency, air transportation providers increase airspace capacity. Experts of the International Civil Aviation Organization (ICAO) are planning to triple the achieved quantitative indicators by 2025 both on the basis of growing scientific and technical potential and through the improvement of air traffic management. Technically, such a rise is due to the development of the satellite navigation, but it requires updating of the flight regulation rules. This gap is filled in the ICAO regulatory documents where the technology which guarantees the required air space capacity indicators on the basis of the operational coordination of specialists (the totality of the so-called "planning horizons") has been developed. At present, there is no analytical apparatus for the optimal choice of new technologies practical realization; only universal means are known (for example, simulation modeling of aviation systems). This article discusses several related analytical models of flight control processes and air space capacity assessments on a flight priority basis are given. Each flight corresponds to the state priority of its service, which can vary depending on the development of the situation in the air (distress on board, hazardous atmospheric phenomena, equipment failures, etc.). The problem of the interacting planning horizons capacity assessment is analyzed using a model of a multichannel system with priorities. Each planning horizon is a service channel that performs the functions of air traffic flow management, restructuring the space and the management process on its territory, as well as maintaining the integrity of information in the interacting horizons. The evaluation criteria which worked well in practice are indicators of the average service waiting time and the probability of failure which is thought of such events as the direction of the aircraft to the alternate aerodrome, missed approach procedure, departure delay and other significant deviations from the balanced daily plan of airspace use.

 

For the generalized Rapoport-Leas equations, algebra of differential invariants is constructed with respect to point transformations, that is, transformations of independent and dependent variables. The finding of a general transformation of this type reduces to solving an extremely complicated functional equation. Therefore, following the approach of Sophus Lie, we restrict ourselves to the search for infinitesimal transformations which are generated by translations along the trajectories of vector fields. The problem of finding these vector fields reduces to the redefined system decision of linear differential equations with respect to their coefficients. The Rapoport-Leas equations arise in the study of nonlinear filtration processes in porous media, as well as in other areas of natural science: for example, these equations describe various physical phenomena: two-phase filtration in a porous medium, filtration of a polytropic gas, and propagation of heat at nuclear explosion. They are vital topic for research: in recent works of Bibikov, Lychagin, and others, the analysis of the symmetries of the generalized Rapoport-Leas equations has been carried out; finite-dimensional dynamics and conditions of attractors existence have been found. Since the generalized RapoportLeas equations are nonlinear partial differential equations of the second order with two independent variables; the methods of the geometric theory of differential equations are used to study them in this paper. According to this theory differential equations generate subvarieties in the space of jets. This makes it possible to use the apparatus of modern differential geometry to study differential equations. We introduce the concept of admissible transformations, that is, replacements of variables that do not derive equations outside the class of the Rapoport-Leas equations. Such transformations form a Lie group. For this Lie group there are differential invariants that separate its regular orbits, which allow us to classify the generalized Rapoport-Leas equations.

 

We consider the Landau-Lifshitz equation on a three-dimensional torus. The equation is reduced to the form of the Euler equation for the geodesic left-invariant metric on the infinite-dimensional Lie algebra of the current group. The group of currents is given by a pointwise mapping of the three-dimensional torus into a three-dimensional orthogonal group. In Lie algebra we use the non-standard commutator introduced earlier. The solutions of the Landau-Lifshitz equation can be expanded in terms of the orthonormal basis of the left-invariant metric in the currents algebra. For the expansion coefficients of the solution of the Landau-Lifshitz equation, the explicit form of the evolution equations is deduced in the framework of the constructed model. To do this, we use the expressions obtained earlier for the sums of the adjoint and coadjoint action operators in an infinite-dimensional Lie algebra of currents with nonstandard commutator. The compactness property of the indicated sum operators makes it possible to obtain the asymptotic form of the Landau-Lifshitz equation on a threedimensional torus. Evolution equations are found on the subspace of flows consisting of vector fields whose Fourier expansions contain only simple harmonics of the form cos (kØ) Such vector fields form a subalgebra of the currents algebra which is also closed under the action of coadjoint operators. In this case, an arbitrary Landau-Lifshitz equation for which the vector of initial conditions lies in this subalgebra remains in it for all t for which this solution is defined. We note that to study the Landau-Lifshitz equation the currents algebra with the standard commutator turned out to be ineffective: in particular, the Landau-Lifshitz equation is not an Euler equation on the current algebra with a standard commutator. Thus, for the Landau-Lifshitz equation on the three-dimensional torus, the explicit form of the evolution equations for the coefficients of the Fourier expansion of its solutions by means of operators representing the sum of the operators of the adjoint and co-adjoint action of the current algebra on a three-dimensional torus with nonstandard commutator is obtained. Moreover, it is the property of compactness of the indicated sum operators (while, separately, their components, the operator of the adjoint action operator as well as the coadjoint one are not even continuous) made it possible to obtain the indicated asymptotic form.

 

The work is a continuation of the research begun in previous works of the authors. At present, the theory of nonlinear waves is experiencing rapid development, and its results find numerous practical applications. One can mention the direction associated with the study of the origin and evolution of shock waves, solitary waves, kinks, periodic and quasiperiodic oscillations (for example, cnoidal waves) and many others. In this series, problems with the motion of solitons in a nonhomogeneous medium remain insufficiently studied; in this paper we consider the simplest model of such a medium: layered-inhomogeneous. The behavior of solutions of the single-wave type for the KdV-Burgers equation at various dissipative medium nonhomogeneities is considered. Various kinds of finite obstacles, as well as the transition from a dissipative medium to a free one are scrutinized. Numerical models of the solution behavior are obtained. The simulation was carried out using the Maple mathematical program through the PDETools package. The tasks considered in the paper are computationally-intensive and require a great deal of computer time. Of particular interest is the case of increasing the height of the obstacle while maintaining its width. When analyzing numerical experiments, the unexpected effect of increasing the wave height with increasing obstacle height is observed, and this may be the subject of further research. Along with this, as the height of the obstacle increases, ripples run ahead of the wave. It should be noted that in the previous work of the authors, another situation related to the appearance of a ripple was described. If, however, when the height of the obstacle remains constant, we again increase the width, then we observe an appreciable decrease in the wave amplitude, as demonstrated in the model charts. Thus, by this work of an experimental nature some new interesting properties of quasi-soliton motion are demonstrated on the basis of numerical simulation; they depend on the type and size of the dissipative obstacles.

 

Modern information and control systems cannot be imagined without synchronization subsystems. These are the basic elements that provide tracking of the frequency and phase of reference and information signals, the evaluation of information parameters, and the synthesis of reference and clock signals. Frequency synthesizers (FS) are widely used due to the high speed of frequency setting, a wide range of frequency grids and minimal phase noise in the operating frequency range. Since with the mass appearance of specialized microprocessors and with the improvement of automatic design systems, the feasibility and repeatability of products has become simpler, digital FS are increasingly being used. The most widely used are FS with a frequency divider on digital elements, which serves to convert the signal of a reference oscillator and a controlled generator. For FS using a divisor with an integer division factor in the feedback loop, there are a number of limitations, such as the lower frequency of the FS and the frequency step of the FS. To solve this problem, divisors with fractional-variable division factors in the feedback loop are used, which allow to obtain the required range and the grid frequency step of the FS. The methods of improving the quality of spectral and dynamic characteristics of digital synthesizers in a given band of frequency detuning are analyzed. The principles of the FS operation with a divisor with a fractionalvariable fission coefficient are described, and structural schemes are given. The results of imitation simulation in the Simulink system of the software package MATLAB of frequency synthesizers with a divisor with a fractional-variable fission factor implemented in various ways are presented, and a comparative analysis of the spectral characteristics of the obtained models is carried out.

 

When solving the problems of predicting the durability of aircraft structures elements in case of a bird strike, in a collision with large hail, foreign objects, it is necessary to know the dependence of the probability of the barrier penetration on the characteristics of the impactor and the impact speed. The results of the experiments have the form of a dichotomous variable having only two possible values (the barrier is punched or not), so the required dependence can be represented in the form of a logistic regression given by the integral function of the logistic distribution. To evaluate the parameters of logistic regression with a small number of experiments, it is advisable to use the Bayesian approach. One of the components of the Bayesian parameter estimation is the calculation of the likelihood function. The resulting expressions for the likelihood function and its logarithm were used to estimate the logistic regression parameters using the maximum likelihood method, which was realized by Newton method. The calculated iterative scheme of the maximum likelihood method is given. Based on the results of experimental data, estimates are obtained that are used to compare with the results obtained by the Bayesian approach. A special case is considered – the Bayesian estimation of the shift parameter of the logistic curve for a given value of the scale parameter, and also a general case – the estimation of the shift and scale parameters. A normal distribution is used as a priori distribution for the shift parameter, and a classical (two-parameter) gamma distribution is used for the scale parameter. Point estimation of logistic regression parameters was carried out by the modal value of a posteriori distribution. Using the modal rather than the mean value, made it possible to significantly reduce the amount of calculations due to the fact that in this case there is no need to calculate the integral probability in the denominator of the Bayesian formula. 

In the conditions of contemporary world market, the maintenance process to keep the required level of aircraft serviceability in airlines should be economically efficient in relation to labour, time and material expenses. The solution of the problem can be achieved by planning and managing the activity of airlines with the aim to improve the efficiency of civil aircraft maintenance process. Planning and implementation of activities for process improvement of aircraft maintenance efficiency in particular such factors as aircraft (A/C) reliability and safety, flight regularity and A/C timewise operation efficiency, A/C intensities and economy require collaboration and interaction of developer, manufacturer and operator. This work proves the need to develop the planning and performance technology of actions related to the airlines activity allowing step by step according to developer, manufacturer and operator activities to establish the cooperation and influence on aircraft serviceability, thereby increasing the maintenance process by all possible managing actions. The complex technology to improve the efficiency of aircraft maintenance process including activities on “A/C Reliability”, “Technology”, “Personnel”, “Equipment”, “Materials” has been developed with the aim to increase the A/C fleet serviceability level. The complex technology is represented as a route technology with brief description of destinations in the route chart with input/output indications at each stage.The advantages of complex technology application are shown in the example of improving one of the efficiency measures in aircraft maintenance process.

The paper considers the problem of finding the power flux density of signals reflected from two objects with different symmetric scattering matrices when they are irradiated with a completely polarized wave. The authors consider the case when the eigenvalues of the matrix are different given the form of the unitary diagonalization of the matrix for this case. The relation defining the diagonal elements of the factor is given. A comparison is made between the power flux density of the signals reflected from two objects with different scattering matrices when they are irradiated with a completely polarized wave. The power flux density of the electromagnetic wave reflected by this object is determined mathematically. Based on the definition of the scattering matrix, a transition to incident waves is performed. A parameter characterizing the degree of polarization anisotropy of the fluctuating object is given. The ratio for radar contrast is given. It is concluded that if the vector of the incident electromagnetic wave differs only in the scalar multiplier from the eigenvector of the matrix, the radar contrast will reach its maximum value. When the vector of the incident wave is proportional to the eigenvector, the value of the radar contrast reaches its minimum value. A problem is considered when the scattering matrices of two objects are simultaneously reduced to a diagonal form by means of a congruence transformation. Conditions are determined under which the Graves matrix of two scattering objects is reduced to diagonal form by means of a congruent transformation. A necessary and sufficient condition for the existence of a polarization basis is obtained in which the scattering matrices of two objects will simultaneously have a diagonal form.

 

The problem of two spherical conductors is studied quite in detail with bispherical coordinates usage and has numerous appendices in an electrostatics. The boundary-value problem about two ferromagnetic spheres enclosed on homogeneous and infinite environment in which the lack of spheres exists like homogeneous magnetic field is considered. The solution of Laplace's equation in the bispherical system of coordinates allows us to find the potential and field distribution in all spaces, including area between spheres. The boundary conditions in potential continuity and in ordinary density constituent of spheres surfaces induction flux are used. It is supposed that spheres are identical, and magnetic permeability of their material is expressed in  >> 0. The problem about falling of electromagnetic plane wave on the system of two spheres, which possesses electrically small sizes, can be considered as quasistationary. The scalar potentials received as a result of Laplace's equation solution are represented by the series containing Legendre polynomials. The concept of two spheres system effective permeability is introduced. It is equal to the advantage in magnitude of magnetic induction flux vector through a certain system’s section arising due to its magnetic properties. Necessary ratios for the effective permeability referred to the central system’s section are obtained. Particularly, the results can be used during the analysis of ferroxcube core clearance, which influences on the magnetic antenna properties.

 

The article describes the results of the numerical experiments on the mathematical model of the correlation-extreme navigation system (CESN) of the aircraft (LA) using microwave radiation of the earth's cover. The aim of the numerical experiments was the analysis of the influence of the main parameters (characteristics of the radiometer and antenna, a means of reviewing the space, parameters of the current image and the reference image, methods for image processing (algorithms for image correlation), conditions of the equipment operation (the speed and altitude of aircraft, the evolution of media) on the efficiency of CESN. The experiments were carried out with the fields of the underlying surface of three types-with an artificially synthesized map (CLAIM) containing several objects of different thermal contrast; with a homogeneous random field (OSP), with fragments of a digital map object structure (TSKOS) of real surface area of the earth. As a result of numerical experiments the author studied the influence on exactness characteristics of CESN navigation parameters (bank angles, pitch, yaw), flight altitude and speed, the noise of the radiometer, the pattern width, the width of the review sector, mis-scaling and angular misalignment of the current and reference images. Comparison of different methods of surface scanning was made based on the simulation results. During the experiments, the variation of one of the parameters with respect to the base variants of the parameters was carried out and the values and variances of errors of the CESN were estimated. All three main methods of beam scanning (longitudinal with a multi-beam radiometer, conical and transverse) were considered. The operation of the maximum search was made up of two procedures: searching for the global maximum area of the correlation matrix by enumerating all matrix entries and refining the location of the true maximum point by quadratic interpolation of the function. The implementation of errors is considered for various trajectories of the aircraft's motion, as well as for various angular variances. The implementation processing showed that an increase in the angular dispersion leads to a simultaneous increase in both the error modulus and the variance of the error modulus. The analysis of the obtained dependences shows a fairly stable tendency to reduce errors of CESN with a decrease in navigation angles. This dependence is demonstrated graphically. In order to estimate only the influence of the fluctuation noise of the radiometer on the magnitude of the CESN errors, a series of experiments was performed in which all other random fluctuations in the channel were eliminated.