The paper proposes conceptual model of a digital platform for cyber-physical management of modern enterprises in the upcoming era of Industry 5.0. Unlike Industry 4.0, which focuses on automation of physical processes, Industry 5.0 is oriented on digitization of knowledge and automation of reasoning processes for creating artificial intelligence that is able to manage enterprises. This still emerging era will be characterized by the vision of any business, including industrial production or logistics, as a complex adaptive system built on fundamental principles of self-organization and evolution, as well as interaction of artificial intelligence systems and humans. The paper shows that implementation of such production and logistics management systems will require development of new models and decision-making methods based on knowledge and semantic information processing, integration of computational and communication components, accumulation of big data and its processing for predictive analytics, blockchain technologies for fixing mutual obligations of systems components in the for m of smar t contracts, as well as human-machine and software inter faces. Existing approaches to creation of digital platforms within the digital economy of Industry 4.0 and their limitations are analyzed. The concept of digital ecosystem is developed as an open, distributed, self-organized "system of systems" of smart services capable of coming up with solutions and automatically resolving conflicts through negotiations and concessions. The concept of the digital platform within Industry 5.0 is described, which will be able to support functioning of the digital ecosystem of "smart services" of cyberphysical management of both individual objects and enterprises of humans and robots, and in the future, industries of such enterprises — implemented using self-organizing autonomous agents at all levels.

The frequency and algebraic directions of researches of robust stability are considered. Frequency or Tsypkin-Polyaka’s direction is considered briefly in a survey order. The algebraic or Kharitonov’s direction is considered more more widely, namely basic provisions and results of the Algebraic method of robust stability of interval dynamic systems developed within development algebraic or Kharitonov’s direction of robust stability are presented. Fundamental works of V. L. Kharitonov since issue have caused a huge flow of the publications connected with extreme relevance of the solution of problems of a robustness of systems. So far from a circle of problems of a robustness many issues of robust stability are resolved. Discrete analogs and versions of theorems of Kharitonov are received. Frequency conditions of robust stability are considered and solved in Ya. Z. Tsypkin, B. T. Polyak, Yu. I. Neymark works. However in a problem of robust stability not all issues are so far resolved, especially big contradictions have arisen in a continuous case. Also the tasks considered here for interval matrixes and polyhedrons of matrixes haven’t been solved. In work the theorem like the third theorem of Kharitonov which cancels counterexamples to former known results in this direction is formulated and proved and also on its basis the costal theorem for polyhedrons of matrixes is proved. The new costal theorem also cancels counterexamples for this case. To the main theorem of the considered algebraic method the specifying remark is formulated that in the absence of a full set of four angular polynoms of Kharitonov of a condition of this theorem are necessary, but can be insufficient for stability of system. Determination of angular separate coefficients of a characteristic polynom of system is generally carried out by means of use of methods of nonlinear programming. For a discrete case the discrete analog of the theorem of Kharitonov which is received on the basis of Schur’s theorem is presented. At the same time, the concepts of points and intervals of a variabless used for the theorem of an analog of a continuous case are entered. The algorithm of definition of a robustness of discrete systems is formulated. The main results, the Algebraic method of robust stability developed by the author are illustrated with cancellation of the counterexamples widely known from scientific literature.

This work is proposed to continue the discussion of the problems, theoretical foundations and practical features of the construction and synthesis of robust control systems with high gain, allowing us to control multidimensional nonlinear dynamic objects of high dimensional with functional uncertainties. If problems could not be solved at the level where they appeared, it is necessary to rise the level of understanding of the laws of nature, or in the words of master Lui-Shi Chun Qiu (China, 3rd century BC): "The boy of five chi growth leads the bull by the bridle and the bull obeys him in everything. This is because the person in this case follows naturalness" (the laws of nature). The judo philosophy ("soft way") is based on the principles of using the power and energy of the opponent to achieve victory. The purpose of this work is to demonstrate the theoretical aspects and practical features of the methods of synthesis of optimal control systems by the criterion of maximum reproduction accuracy using the example of robust systems, which allow to control dynamic objects with functional uncertainties, including unstable objects, no minimal-phase objects, neutral objects and objects with differentiation properties. The simplicity (at the level of the engineer) and universality, mathematical rigor and physical validity of this approach is based on the judo philosophy: suppressing the dynamics of a functionally uncertain object and external disturbances by the infinitely large gain with the finite control signal and at the same time maintaining sustainability. Theoretically exhaustive solution of the problem of robust control is given by the idea of constructing systems that are stable with an unlimited increase of the gain coefficient. The sustainability properties are valid for optimal systems that were synthesized using quadratic quality functionals that do not explicitly depend on the control signal, and using a restriction on the control signal. It is significant that in contrast to continuous systems with un-measurable disturbances and not well known control object (in which the conditions of invariance imply the use of infinitely large gain), in relay (discontinuous) systems the equivalent effect is achieved with the help of finite control signal. A nice bonus is the highest accuracy which leads to mathematically zero error of regulation, thus all error coefficients (of position, speed, acceleration, acceleration derivative, etc.) is also equal to zero in the presence of external and internal interferences. In fact, the optimal accuracy control system is equivalent to a system with astatism of the n-th order: the regulator contains n serial connected integrators.

This article is devoted to parametric synthesis of a control system of a two-wheeled balancing robot. From the mathematical point of view, the robot is an inverted pendulum type object with a pivot point placed on the wheel axis. This device is unstable while deenergized. Devices of this type are excellent labor atory stands for testing and debugging control algorithms of unstable nonlinear systems. The inverted pendulum math model is well studied theoretically but, when designing a particular device many additional tasks arise such as taking into account the error in measuring the tilt angle and the influence of actuator nonlinearities. In this paper, one of these tasks is solved, namely, the problem of reducing the amplitude of the robot’s oscillation around the equilibrium position. In practice, this oscillation almost always occurs in such systems and leads to various negative effects, such as increased energy consumption, increased wear of an actuator and heating of its windings, etc. Therefore, reducing the amplitude of the oscillation is an important task. To solve this task, the authors of the article propose to use the method of numerical optimization of the regulator, which is well recommended for solving many problems. The article analyzes the behavior of the device near the equilibrium position and identifies the causes of the self-oscillation. Further the method of its simulation is proposed. On the basis of numerical experiments, the main reason for the increase in the amplitude of the oscillation is revealed. The reason is an overlay of the reverse peak of the device transient process on the peak caused by a torque throw. The throw is generated by a combination of actuator backlash and static friction effects which cause the robot self-oscillation. The authors propose a technique of adjusting the regulator, aimed at reducing the magnitude of the reverse peak of the transition process and, as a consequence, reducing the amplitude of the oscillation. The effectiveness of the technique is confirmed experimentally by the results of numerical simulation of the robot’s behavior and the results of testing the coefficients obtained in a real device. The use of the technique allowed reducing the oscillation amplitude in a real device by almost three times.

The task of the study is to establish the nature of mechanical resonance, namely, it is a resonance of forces or speeds. Two definitions are introduced. Definition 1. Resonance of forces is a resonance arising at a frequency ω = (k/m)0,5 in a mechanical system including an inert body and an elastic element, at which the reactive forces developed by them are maximal and opposite. Definition 2. The velocity resonance is a resonance arising at a frequency ω = (k/m)0,5 in a mechanical system, including an inert body and an elastic element, at which the speeds developed by them are maximum and opposite. The equation of forced mechanical oscillations corresponds to a parallel connection scheme, in which the inert body and changes in the dimensions of the elastic element and damper have a uniform speed, and their reactive forces are added. The sum of the reactive forces of the consumers of mechanical power is equal to the force developed by the source of mechanical power, which, like a voltage source in electrical engineering, can be called a source of power. Theorem 1 holds. If the condition ω = (k/m)0,5 is satisfied in a mechanical system consisting of parallel-connected inert bodies, an elastic element and a damper, a resonance of forces occurs. The inert body, the elastic element and the damper can be connected not only in parallel but also in series. With a series connection, a single force is applied to the elements of the system, and the velocities of the inert body and the changes in the dimensions of the elastic element and damper are added. The sum of the speeds of consumers of mechanical power is equal to the speed developed by the source of mechanical power, which, like a current source in electrical engineering, can be called a source of speed. Theorem 2 is valid. Under the condition ω = (k/m)0,5 in a mechanical system consisting of a series-connected inert body, an elastic element and a damper, a velocity resonance occurs. The mechanical resonance described in the courses of theoretical mechanics is the resonance of forces. It corresponds to a parallel connection of an inert body, an elastic element and a damper. When these elements are connected in series, a velocity resonance occurs.

The article deals with the problem of program control design for a dynamic object defined by a nonlinear system of differential equations. Known methods of optimal control require the two-point boundary value problem solution, which in general is coupled with fundamental difficulties. Therefore, this paper proposes a technique that uses the direct method, in which the functional is minimized directly using a population-based algorithm. The use of direct methods is based on the assumption that control signals may be defined by a finite set of parameters. Then a scalar functional is formed, the numerical value of which measures the quality of the obtained solutions. In this case, the search for optimal control is reduced to the problem of single-criterion multi-parameter optimization. The practical importance of this approach is that it eliminates the need to solve a two-point boundary value problem. However, this results in another difficulty, since the approximation of control, in general, requires a large number of parameters. It is known that in this case, the effectiveness of conventional gradient numerical optimization methods decreases markedly. Therefore, it is proposed to take the next step and apply genetic or population-based optimization algorithms that have confirmed their performance in solving this class of problems. For this purpose the paper uses one of the modifications of the particle swarm algorithm. The technique is applied to a test problem describing the spatial movement of a maneuverable aircraft. The direct method is compared with two classical solutions based on the condition that the partial control derivatives of the Hamilton function are equal to zero and with the condition of Hamilton function maximum over controls (Pontryagin’s maximum principle). The presented results show the high degree of similarity between obtained controls for all considered methods of selecting the target functional. At the same time, the accuracy of classical algorithms turns out to be slightly worse, and they show a higher sensitivity to the quality of the initial approximation. Thus, the obtained results confirm the approximate equivalence of the direct method and the classical methods of program control design, at least for the class of problems under consideration. The practical significance of this research is that the use of the direct method is much simpler than solving a two-point boundary value problem necessary for classical algorithms.

In article the structure and the control algorithm are considered by diverse redundancy of the computing system of the perspective integrated modular avionics. Computing resources of the integrated modular avionics system are generally represented by heterogeneous computing systems used for information processing as part of the onboard integrated computing environment. The basis of heterogeneous computing systems are processor nodes, redundancy of computing systems is that the number of processor nodes is greater than one. The task is to synthesize such a computer system in which the automatic control of redundant computational resources would be carried out by using the own capabilities of the processor nodes and without the use of additional hardware resources. It is considered that the redundant computer system performs meaningful calculations of the problem solved by several processor nodes in parallel. All meaningful calculations for any signs initially divided into relatively short stages, providing an opportunity to assess the effectiveness of the completion of each of them. The computational system redundancy management is based on the periodic calculation and comparison of the success indicators of the stage. Pairwise arbitration of processor nodes is carried out according to a hierarchical scheme by comparing the values of the success indicators of the stages of the same name. Subsequent reconfiguration of the computer system allocates passive and leading processor nodes in pairs at all levels of the hierarchical scheme. The failure of the passive processor node does not affect the execution of the main cycle. The failure of the host processor node does not cause interruptions in the output of the results of calculations, but destroys the structure of reserves, which is restored after arbitration in the next cycle. Failure of the lead CPU top node leads to the failure output in the current cycle, the computational process is restored along with the new hierarchy of the computing system in the next cycle. The proposed solution is aimed at parrying both hardware failures and software malfunction. The methodical example based on the computer system of the modern onboard equipment complex of the transports category aircraft is resulted.