The problem of synthesis of an optimal controlled system with a quadratic quality criterion having an infinite number of switching points at a finite time inter val is discussed. In the theor y of optimal control, this phenomenon is called the "Fuller phenomenon". For more than 60 years, the Fuller problem has been very attractive, relevant, and still unsolved, especially for non-linear multidimensional dynamical systems of high order, and even more so, with obtaining a solution in an explicit analytical form for practical implementation in a control system.

The purpose of this work is to demonstrate the theoretical aspects and practical features of the method of synthesis of optimal control systems by the fast acting criterion by the example of solving problems related to the Fuller phenomenon.

When solving these problems, we use in the classical variations calculus and the Pontryagin maximum principle of the method of introducing a new additional phase variable into consideration, which is defined to the integral quality criterion and expands the original phase vector of the object. As a result, if the best optimal control in terms of fast acting for the control object is known then this technique makes it ver y easy to get a worse optimal control in terms of accuracy by including the Fuller accuracy criterion in the dynamics of the control object. It should be note that an important acquisition here is to increase the accuracy to the optimal value and reduce the established control error to zero, with all error coefficients (in position, speed, acceleration, jerk, etc.) equal to zeroin the presence of external and internal interference.

Statements and solutions of the classical and modified Fuller problems are presented. As illustrative examples, we consider the traditional problems of the synthesis of optimal control in terms of speed, solved in well-known methods.

Incorrect one- and two-dimensional inverse problems of reconstructing images of objects with angular resolutionexceeding the Rayleigh criterion are considered. The technique is based on the solution of inverse problems of source reconstruction signals described Fredholm integral equations. Algebraic methods and algorithms for processing dataobtained by measuring systems in order to achieve angular superresolution are presented. Angular superresolution allows you to detail images of objects, solve problems of their recognition and identification on this basis. The efficiency of using algorithms based on developed algebraic methods and their modifications in parameterization the inverse problems under study and further reconstructing approximate images of objects of various types is shown. It is shown that the noise immunity of the obtained solutions exceeds many known approaches. The results of numerical experiments demonstrate the possibility of obtaining images with a resolution exceeding the Rayleigh criterion by 2-6 times at small values of the signal-to-noise ratio. The ways of further increasing the degree of superresolution based on the intelligent analysis of measurement data are described. On the basis of the preliminary information on a source of signals algorithms allow to increase consistently the effective angular resolution before achievement greatest possible for a solved problem. Algorithms of secondary processing of the information necessary for it are described. It is found that the proposed symmetrization algorithm improves the quality of solutions to the inverse problems under consideration and their stability. The examples demonstrate the successful application of modified algebraic methods and algorithms for obtaining images of the objects under study in the presence of a priori information about the solution. The results of numerical studies show that the presented methods of digital processing of received signals allow us to restore the angular coordinates of individual objects under study and their elements with super-resolution with good accuracy. The adequacy and stability of the solutions were verified by conducting numerical experiments on a mathematical model. It was shown that the stability of solutions, especially at a significant level of random components, is higher than that of many other methods. The limiting possibilities of increasing the effective angular resolution and the accuracy of image reconstruction of signal sources, depending on the level of random components in the data utilized, are found. The effective angular resolution achieved in this case is 2—10 times higher than the Rayleigh criterion. The minimum required signal-to-noise ratio for obtaining adequate solutions with super-resolution is 13—16 dB for the described methods, which is significantly less than for the known methods. The relative simplicity of the presented methods allows you to use inexpensive computing devices and work in real time.

The paper covers creating the algorithms for calculating the probability of various types of defects in tunnels, the development of which can lead to accidents. Tunnels are an important and complex part of the transport and communication system, through which heavy traffic is carried out. Determining the probability of defects in the latent period of their initiation in individual sections of tunnels is an important problem. The formation of defects is accompanied by the appearance of noise that distorts the useful signals coming from sensors and measuring instruments installed to control the stability of the tunnel and the reliability of its structures. Traditionally measuring instruments register noisy signals, and the technical condition of the tunnels is assessed on the basis of the values of their characteristics. It is shown in the paper that the more reliable indicators of fixing the onset of dangerous changes in the latent period of initiation are the characteristics of the noise, which cannot be extracted from the noisy signal. It is noted that the probability with which the noise takes on admissible and critical values is an indicator of changes in the technical condition of tunnels. Algorithms have been developed for calculating the probabilities of the noise values getting in the given intervals. These probabilities are stored as reference sets for the initiation of tunnel defects. After the training has been carried out, the values of the probabilities with which the noise takes on the given values at different time instants are matched to the type of defect and one of the possible technical states: serviceable, operational, partially operational, inoperable; pre-emergency; emergency, etc. It is also shown that the differences in the probabilities with which the noise takes on the same values at different times are indicators of the dynamics of changes in the malfunction in the tunnels. A database ofinformative attributes of the intensity of the development of failures is also created in the paper. For this database, the indicators of the dynamics of the development of a defect are determined, such as insignificant, slow, significant, intensive.

The proportional guidance method-based missile homing systems (MHS) have been widely used the real-world environments. In these systems, in order to destroy the targets at different altitudes, a normal acceleration stabilization system (NASS) is often utilized. Therefore, the MHS are complex and the synthesis of these systems are a complex task. However, it is necessary to synthesize NASS during the synthesis of the MHS. To simplify the synthesis process, a linear model of the NASS is used. In addition, we make use of the available commands in Control System Toolbox in MATLAB. Because the Toolbox has the commands to describe the transfer function, determine the stability gain margin, and the values of the transient respond of the linear automatic systems. Thus, this article presents two methods for synthesizing the missile homing systems, including (i) a method for synthesizing the MHS while ensuring the permissible stability gain margin of the NASS, and (ii) a method for synthesizing the MHS while ensuring the permissible stability margin of the NASS by overshoot. These techniques are very easy to implement using MATLAB commands. The synthesis of the proposed MHS is carried out by the parametric optimization method. To validate the performance of the proposed techniques, we compare them withthe MHS synthesized by ensuring the stability margin of the NASS bythe oscillation index. The results show that, two our proposed methods and the existing method provide the same results in terms of high-precision. Nevertheless, the proposed methods are simple and faster than the conventional method. The article also investigates the effect of gravity, longitudinal acceleration of the rocket, andblinding of the homing head on the accuracy of the synthesized MHS. The results illustrate that they have a little effect on its accuracy.

The article describes the methods and test results of a solid-wave gyroscope (SVG) — an angular rate sensor (ARS), developed at the Department of Control Devices, Tula State University and manufactured by the serial plant of JSC "Michurinsky Plant" Progress "according to the technology it worked out. The metal resonator SVG-ARS is made of an elinvar alloy and has a cylindrical structure of different thickness, the lower part of which, with a smaller wall thickness, acts as a suspension for the upper cylinder, the resonator itself, which has a conical shape, providing better vibration localization at its end edge. Technological manufacturing defects, different frequencies and variability, are eliminated by balancing " by mass" based on the removal of excess metal at certain points on the end edge of the resonator. The electronic module provides the second mode of primary and secondary oscillations of the resonator edge arising during rotation and creates a signal to compensate for the Coriolis and quadrature components of the output signal at the nodes. The maximum amplitudes of the excitation and compensation signals do not exceed 10 V. Therefore, at large values of mechanical influences, the compensation circuit may not work out the increased signal and the SVG-ARS loses its operability. The total processing time of the compensation signal does not exceed 1 μs. The maximum power consumption of the electronic module is not more than 4 W. When testing for mechanical and temperature effects, the norms were used that are typical for similar devices (angular rate sensors) used on board aircraft. The tests were carried out on the bench equipment of a specialized enterprise. The stability of the zero signal and the scale factor was determined under the simultaneous action of the measured speed and temperature on the SVG-ARS. The values of the random walk and the instability of the zero signal were obtained from the Allan deviation plots. Their values provide a basis for the conclusion about the possibility of using the developed SVG for several hours on board dynamic aircraft in orientation, stabilization and navigation systems. It was found that SVG-ARS possesses impact strength and restores its measuring ability after impact. Tests for vibration resistance revealed resonance frequencies and frequency rangesin which the tested VTG-DUS sample can be used without significant modification. The results of vibration tests can be used to refine the design and control electronics for the operating conditions of a particular aircraft.

In the presented work, a controlled change by dimensions of a spacecraft descending in the atmosphere of Mars is considered. The aim of the work is to obtain a method for calculating the mass and mass-geometric characteristics of a spacecraft when changing its dimensions, which provides angular velocity passive control during the descent of this spacecraft in a low-density atmosphere. In the process of solving this problem, the geometric and mass-geometric characteristics of the descent spacecraft (volume, cross-sectional area, moments of inertia) were calculated. It is assumed that the outer shape of the spacecraft posterior to the incoming flow is a one-sheet rotational hyperboloid, which changes its dimensions during the spacecraft descent in the low-density atmosphere of Mars. As a result of solving the nonlinear programming problem, the minimum and maximum values of the main axial moments of inertia are obtained, which able to spin the spacecraft relative to the longitudinal axis of symmetry. The initial data for solving the nonlinear programming problem are the minimum volume and the maximum cross-sectional area of the hyperboloid, calculated according to the specified intervals of the variable controlling the dimensions of this surface. The method for calculating the mass and mass-geometric characteristics of a spacecraft when changing its dimensions ispresented, which makes it possible to control the magnitude of the angular velocity of a symmetric spacecraft in the low-density atmosphere of Mars without the use of onboard jet engines. In particular, it is shown in the work that as the height of the hyperboloid increases, the moment of inertia about the spacecraft longitudinal axis of symmetry decreases, accompanied by an increase in the moments of inertia about the transverse axes of symmetry. It can be shown that in this case there is an increase in the angular velocity of the spacecraft about the longitudinal axis, which makes it possible to achieve a stable orientation of the spacecraft upon entering the atmosphere. However, a more detailed study of the dynamics ofthe spacecraft relative motionwith a changeable shape in the atmosphere is beyond the scope of this work, but it can be presented in further publications.