Vol 55, No 6 (2016)

Methods for solving the Lyapunov matrix differential and algebraic equations in the time and frequency domains are considered. The solutions of these equations are finite and infinite Gramians of various forms. A feature of the proposed new approach to the calculation of Gramians is the expansion of the Gramians in a sum of matrix bilinear or quadratic forms that are formed using Faddeev’s matrices, where each form is a solution of the linear differential or algebraic equation corresponding to an eigenvalue of the matrix or to a combination of such eigenvalues. An example illustrating the calculation of finite and infinite Gramians is discussed.

Based on band matrices, a new formula for finding polynomial coefficients of the numerator of the transfer function for a linear single-input single-output system is derived. Its properties are studied for analyzing and synthesizing linear systems with many outputs, as a result of which a method is developed for the synthesis of the specified polynomial of zero dynamics (specified output) for a linear system with many outputs.

We consider the following problem: using the discrete time measurements of the state variables of a continuous stochastic object and obtain the most precise estimate of these variables. To accelerate the estimating process, we synthesize the optimal structure of a discrete final-dimensional nonlinear filter with a piecewise-constant prediction, remembering the last few measurements in its state vector. The dimension of this vector (i.e., the filter memory volume) can be selected arbitrarily to balance the desired measurement precision and available processing speed for the measurements. We obtain the mean-square optimal structure functions of the filter expressed via the corresponding probability distributions and a chain of Fokker—Planck—Kolmogorov equations to find those distributions. We describe the procedure to obtain the structural functions of the filter numerically by means of the Monte-Carlo method. Also, we provide simple numeric-analytic approximations to the proposed filters: they are compared with approximations to the known filters. An example of the construction of such approximations is considered.

The optimal control of deterministic discrete time-invariant automaton-type systems is considered. Changes in the system’s state are governed by a recurrence equation. The switching times and their order are not specified in advance. They are found by optimizing a functional that takes into account the cost of each switching. This problem is a generalization of the classical optimal control problem for discrete time-invariant systems. It is proved that, in the time-invariant case, switchings of the optimal trajectory (may be multiple instantaneous switchings) are possible only at the initial and (or) terminal points in time. This fact is used in the derivation of equations for finding the value (Hamilton–Jacobi–Bellman) function and its generators. The necessary and sufficient optimality conditions are proved. It is shown that the generators of the value function in linear–quadratic problems are quadratic, and the value function itself is piecewise quadratic. Algorithms for the synthesis of the optimal closed-loop control are developed. The application of the optimality conditions is demonstrated by examples.

This paper introduces Allen’s extended interval logic whose sentences are Boolean combinations of sentences of Allen’s interval logic with metric constraints on time points. For this extended logic, a deduction method based on analytic tableaux is defined. This method is used for answering queries on ontologies specified in Allen’s extended interval logic. An example illustrating the applicability of this extended logic to the problem of workflow specification is presented.

Upgrade options for a high-performance specialized system designed to solve resourceintensive problems, such as exhaustive searches, are considered. The concept of the set of executable computational jobs of various types is introduced in order to describe the system’s functional capabilities. User requirements for the upgraded system are described by the vector whose components determine the number of assigned jobs that can be executed together in one operating cycle. The method of multiparameter sliding scheduling is used to analyze the system development options. The solution of a sequence of optimization problems makes it possible to obtain system configuration options with the minimal cost, minimal energy consumption, and maximal number of executable jobs. The selection procedure of the compromise project that best takes into account various conflicting requirements is proposed.

This study continues the series of papers devoted to the problems of autonomous operation of spacecraft in a geostationary orbit. The solution of the problem considered here assumes the formation of a set of algorithms for control processes in a closed autonomous spacecraft control and navigation system in a geostationary orbit. The paper is aimed at the formalization and solution of the new technical task of autonomous control during the spacecraft’s ascent to the given orbital position and remaining in this position. An important requirement is to provide the safe separation of several spacecraft in one orbital position. The control problem is solved using the combined optimization method developed by us; in this method, the control vector is divided into the synthesized and the programmed components taking into account the principle of the separation of the navigation and control problem in the stochastic approach. The motion’s models proposed in the previous paper are used to develop the control algorithms for a spacecraft’s ascent to the working position in a geostationary orbit and remaining in this position. The results of the algorithms simulating the ascent and maintaining for the exactly known state vector taking into account the random spread of the initial conditions and thrust are presented.

The traditional problem is discussed of an optimal spacecraft slew in terms of minimum energy costs. The spacecraft is considered as a rigid body with one symmetry axis under arbitrary boundary conditions for the angular position and angular velocity of the spacecraft in the quaternion formulation. Using substitutions of variables, the original problem is simplified (in terms of dynamic Euler equations) to the optimal slew problem for a rigid body with a spherical mass distribution. The simplified problem contains one additional scalar differential equation. A new analytical solution is presented for this problem in the class of conical motions, leading to constraints on the initial and final values of the angular velocity vector. In addition, the optimal slew problem is modified in the class of conical motions to derive an analytical solution under arbitrary boundary conditions for the angular position and angular velocity of the spacecraft. A numerical example is given for the conical motion of the spacecraft, as well as examples showing the closeness of the solutions of the traditional and modified optimal slew problems for an axisymmetric spacecraft.

A model of a mobile capsule robot that consists of the housing and internal body is considered. The internal body can move relative to the housing along a straight line. The internal body is attached to the housing by a spring. The system motion is excited by a force that acts between the housing and the internal body. The force changes in a pulse-width periodic mode. The robot’s motion along a straight line on a rough horizontal plane is investigated. It is assumed that the dry Coulomb friction acts between the housing and the plane. The dependence of the average steady state robot velocity on excitation parameters is analyzed. It is established that it is possible to control the magnitude and direction of the robot motion by changing the period and the duty cycle of the pulse-width excitation signal. The effect of the variation in the direction of the robot motion due to changing the excitation period is observed. This effect is associated with the phenomenon of resonance.

In this paper, formulations of the synthesis and analysis problems in multiterminal communication networks with variable edge capacities are considered. In contrast with traditional formulations, the notions of flow density, flow speed, and flow intensity are introduced and the time characteristics of a transfer of given volumes through a network are analyzed. Related problems are found in various fields of human activities, e.g., with studies of traffic flows in evacuation models and, e.g., in models of the requalification of people on the labor market. The algorithm for solving the transportation problem in a multiterminal network with variable capacities is presented; under certain conditions, this algorithm allows one to avoid dealing with complex nonlinear large-scale problems of mathematical programming.