Substantial and formal deductions in logics with vector semantics

The problem of logic deduction for one class of logics with a vector semantics is discussed. In the given logics the trustiness is represented by a vector with components 〈Truth; Lie〉, which do not depend on each other. The problem of organization of “substantial” and “formal” deductions is considered. In the first case the meaning of truth (semantics) of judgements is taken into account, whereas in the second case only the structure of judgements (syntax) is considered.     

Partial orders and Jordan normal form

Consideration is given to the relation between the structure of the acyclic binary relation and the adjacency matrix of its corresponding graph. In this case, the existing methods for studying the binary relations and their corresponding graphs in terms of the spectrum, that is, the set of eigenvalues, of the adjacency matrix are inapplicable because for the acyclic relations this matrix is nilpotent and its spectrum is identically zero. Therefore, a more refined characteristic of the matrix is required. The present paper considers the Jordan normal form (JNF) as such.     

On one extremal problem of adaptive machine learning for detection of anomalies

An adaptive algorithm to solve a wide range of problems of unsupervised learning by constructing a sequence of interrelated extremal principles was proposed. The least squares method with a priori defined weights used as a starting point enabled determination of the “center” of learning sample. Next, a natural passage from the least squares method to more flexible extremal principle enabling adaptive determination of both the “center” and weights of the learning sample events was performed. Finally, a universal extremal principle enabling determination of the scaling coefficient of the membership function in addition to the “center” and weights was constructed.     

Control of single-sector economy over a finite time interval with allowance for employer consumption

The problem of optimal control of a single-sector economics over a finite time interval was studied for the class of the linear uniform production functions in terms of the criterion for maximum employer consumption under exponential growth of the manpower resources. Solution was obtained in the form of the “turnpike theorem.” Its properties were considered. The results obtained were specified for the case of the Cobb-Douglas production function.     

Models of closed multimachine computer systems with transient-fault-tolerance and fault-tolerance on the basis of replication under byzantine faults

Consideration was given to the key definitions, notions, and models that may be useful for transient-fault-tolerant and fault-tolerant computing in the unmanned multimachine computer systems having many interconnected autonomous computers without shared memory and centralized control organ and operating with high degree of computational parallelism, that is, executing on different computers simultaneously various tasks which interchange their information. These computations should establish reliable results under byzantine faults and controllable degradation of the system at detection of faults.     

Two-level orientation control system of flexible spacecraft with active stabilization of structural elastic oscillations

Active stabilization of the elastic oscillations of the flexible objects belonging to the class of flexible spacecraft (FSC) was shown to be required for stable orientation control in addition to the main goal of control. An algorithm of Kalman estimation of the FSC coordinates and parameters was constructed, and a principle was proposed for two-level orientation control based on realtime estimates of the controlled coordinates. At that, the first level of control orients the FSC by the estimated “rigid” component of the total motion, and the second level stabilizes its structural elastic oscillations using the estimated elastic modes as controls. A number of algorithms was proposed to actively damp the elastic oscillations, and the methods of mathematical modeling were used to prove that in the case of two-level control the combined FSC motion is a stable manifold closed in a bounded domain where the requirements on the desired dynamics of FSC orientation processes are satisfied.     

Analysis of probability distribution of a random variable of an execution time of the program in the presence of random failures

The mathematical model of an estimation of an execution time of the program in the presence of random failures is studied. For decreasing of influence of failures on a program execution time the known method of checkpoints is considered. The main objective consists in reception of the formula for a distribution function of a random variable of an execution time of the program and an estimation of a quantile of the received distribution. Properties of a mean value of a random variable of an execution time of the program are studied.     

Method for segregation of suspected logical malfunctions in combinatorial discrete devices

Consideration was given to the direct problem of technical diagnosis which lies in determining the technical state of a combinatorial discrete device from the results of testing. The graph and analytical models of behavior of the combinatorial discrete device allowing for the technical state of its elements were presented. A method of segregation of the suspected logical malfunctions under which observable behavior of the combinatorial discrete device is possible was proposed.     

Problems of boundary observability of elastic vibrations described by the system of telegrapher equations

Problems are solved for the observability of elastic vibrations described by various boundary problems for the system of telegrapher equations. The initial conditions that generate observable vibrations are found in the explicit form. The necessary time period of an observation is defined. The solutions of observation problems are given in the case of the classical and generalized solutions of the class L ₂ of appropriate boundary problems.     

Optimal multilevel organizations for the power cost functions

The problem of maximizing the profit of an economic organization with simultaneous selection of its composition and structure was formulated. Consideration was given to a class of the power cost functions whose parameters reflect the organization environmental characteristics. The optimal organization was analytically established for this case, and the regularities of its variation vs. the environmental conditions were investigated.     

Universal software packages to model the distributed-parameter systems

Consideration was given to the universal software packages for modeling objects and distributed-parameter systems obeying the partial differential equations. The packages may serve as important tools for industrial automation because the majority of industrial objects have distributed parameters. The review presented the results of comparing a number of commercial packages against a set of user-defined criteria. Its final part was devoted to some freeware packages. Conclusions were drawn, and recommendations for using the considered packages were given.     

Design of discrete control system of flexible spacecraft maintaining robust stability of elastic oscillations

For the flexible spacecraft with a nonlinear orientation control system using flywheel engines, an approach to making its elastic oscillations robust stable was considered. It relies on a purposeful variation of the boundaries of the stability domains in the parameter space of the spacecraft and controller with the aim of maximizing the number of robust stable elastic modes of the flexible spacecraft. To make its control stable in the large, consideration was given to the possibility of expanding the basic algorithm by adding the sum of the components using the estimated coordinates of the remaining series of low-frequency unstable modes. Presented was an example of mathematical modeling of the proposed spacecraft orientation system which corroborated operability of the approach making the multifrequency spacecraft stable to the elastic oscillations.     

Design of modified PD algorithm to control angular motion of large space structure

Consideration was given to the design of an algorithm to control orientation of the large space structure (LSS) in the presence of infralow (less than 0.01Hz) frequencies of elastic oscillations exerting considerable influence on the quality of transients due to closeness of the LSS eigenfrequencies to the frequency of control of the object’s “rigid” motion. Procedure for designing a modified PD-algorithm for orientation of the LSS with infralow frequencies of elastic structural oscillations was described. Methods were proposed for calculation of the coefficients of additional components of the modified algorithm and determination of the LSS eigenfrequency boundary to the right of which the orientation control system is robust to the deviations of the object frequencies or inaccuracy of their definition. Some examples of mathematical modeling of the processes of the relay gyroforce control of LSS orientation using the proposed algorithm were given.     

Robust control of composite beams

The structure uncertainty effect is considered on actively controllable composite beam that is a part of the structure of a civil building in a seismodangerous zone. The solution of the problem for active control of vibrations of the composite beam is based on the introduction of a simplified model of sensors and an actuator, which are implemented in the form of piezoelectric plates pasted on two opposite planes of the beam. A suggestion is made of the practical approach that affords the robust control of the composite beam and is founded on the modern H ₂- and H _∞-methods of the theory of control. The effectiveness of the suggested approach for the suppression of vibrations of the composite beam under consideration is illustrated by the examples of numerical modeling.     

Dynamic features of flexible spacecraft control in process of its transformation into a large space structure

Consideration was given to dynamics of angular motion control of the flexible spacecraft reconstructed into a large space structure. In formal terms, this transformation lies in gradual reduction of the constructive rigidity to small values giving rise to low-frequency ( $ \tilde f Consideration was given to dynamics of angular motion control of the flexible spacecraft reconstructed into a large space structure. In formal terms, this transformation lies in gradual reduction of the constructive rigidity to small values giving rise to low-frequency ( ≤ 0.05 Hz) oscillations which represent one of the attributes of the class of large space structures. The existing quantitative definition of the large space structure was specified. It was demonstrated that as the frequencies of structure’s elastic oscillations approach those of the control of object “rigid” motion, a new kind of interrelations between the motions of both types, the so-called “capture” of the controller frequency by that of the elastic oscillations, arises which impairs control efficiency to the point of losing system stability. Analytical (for the linear control systems) and computer-aided (for the discrete systems) methods for determination of the boundaries separating the two qualitatively different forms of existence of the transformed elastic object were proposed. Some results of computer simulation of the orientation control of variable objects such as flexible spacecraft and large space structure were presented. Original Russian Text ? I.N. Krutova, V.M. Sukhanov, 2008, published in Avtomatika i Telemekhanika, 2008, No. 5, pp. 41–56. This work was supported by the Russian Foundation for Basic Research, project no. 05-08-18175.     

Non-smooth guiding potentials in problems on forced oscillations

New classes of non-smooth guiding potentials are determined. The introduced concepts are used for solving the problem on periodic oscillations of controlled objects described by the system of differential equations and by the system of differential inclusions.     

On bifurcation of periodic solutions for functional differential equations of the neutral type with small delay

The class is singled out of systems described by ordinary differential equations unsolved relative to a derivative, in which a small delay leads to bifurcation of periodic solutions from the equilibrium state. The direct application of the classical results of M.A. Krasnosel’skii to these systems is made difficult in view of the complex character of the dependence on a bifurcation parameter, which is a small delay. The problem on bifurcation of periodic solutions for the stated systems is solved by methods of the theory of rotation of condensing vector fields.     

Minimum of power consumption in the problem of oscillation damping

Consideration was given to minimization of the L ¹-norm of control of the linear oscillatory system. For consumption of the energy resources required to damp oscillations, bilateral estimates were obtained.     

On the method of dynamic programming for linear-quadratic problems of optimal control in hybrid systems

For a sufficiently wide class of the linear hybrid systems, an algorithm of optimal feedback control was proposed. Consideration was given to the hybrid control systems with autonomous switching, as well as the corresponding problems of the hybrid linear-quadratic optimal control based on the recently suggested principle of maximum. Interrelations between the hybrid principle of maximum and the method of dynamic programming for the systems of this class were discussed. The classical formalism was extended, the corresponding Riccati equations were obtained, and discontinuity of the “hybrid” Riccati matrix was proved. The computational aspects of the established theoretical results were considered.     

On the properties of the identically singular Lagrange problem

We consider the identically singular Lagrange problem of the calculus of variations. It is investigated how the conjugate points and the Jacobi equation index are related to the solvability conditions of the appropriate matrix Riccati equation and the reducibility of the functional to a perfect square. The criterion of the trajectory variation smallness at a small variation of the functional in the neighborhood of the extremal is obtained.