Kanatnikov

pp. 23-36

pp. 1–15

pp. 1-17

pp. 307-319

One of the methods of qualitative analysis of a dynamical system is to estimate the position of its compact invariant sets closely associated with bounded trajectories of the system. As a solution to such a problem, one can use a localizing set, i.e. a set in the phase space which contains all in-variant compact sets of a system. In this paper the discrete-time dynamical Lozi system of second order was considered. This system was proposed as a piecewise linear analogue of the known discrete-time Chenon system which has a chaotic attractor for some parameter values. For positive invariant and negative invariant sets of the Lozi system a family of localizing sets was constructed and their intersections were determined. The results of this investigation were presented in figures.

One method of the qualitative analysis of a dynamical system is to estimate the position of its compact invariant sets closely associated with bounded trajectories of the system. As a solution to such a problem, one can use a localizing set, i.e. a set in the phase space containing all invariant compact sets of the system. In this article two continuous two-dimensional dynamical systems describing behavior of some biological systems are explored. For each of these systems a family of localizing sets is constructed, and then the intersection of the family is calculated. For the first system the solution was obtained analytically and for the second one the numerical procedure of constructing localizing sets was proposed. The investigation results are shown in figures.

A six-dimensional model, where the aircraft is treated as a material point, is used for planning trajectories of aircraft. In this case, the state variables are the coordinates of the aircraft in the trajectory coordinate system, and controls are longitudinal and transversal overloads and the roll angle of the transversal overload vector. In the context of this model, the author considers a terminal control problem in which it’s required to find such controls at which the aircraft is transferred from a specified initial point of the phase space to a specified destination point. Methods of solving this terminal control problem are well known if the flight time is given. Selection of flight time, if it is not known, is not an easy task, as this choice influences the shape of the flight trajectory. One of the methods to solve the terminal control problem with unknown time is the energy method based on replacement of the independent variable (time) by the normalized mechanical energy of the system. The energy method leads to flight trajectories with a monotonic variation of energy. It provides good solutions for landing and take-off of aircraft, but may not be applicable for complex maneuvers. The author considers methods of planning trajectories with non-monotonic variation of energy, which, nevertheless, are based on the energy method. These methods are based on special short-term maneuvers at the beginning and end of the trajectory (so-called transient maneuvers), and on selection of certain intermediate points through which the trajectory must pass.

The three-dimensional polynomial dynamical system dx/dt = y+z, dy/dt = -x+αy, dz/dt = x²-z with complex behavior is considered. In the particular case α = 0,5 this system reduces to one of systems with the chaotic behavior, which was found by J.C. Sprott. For the specified system the problem of localization of invariant compact sets, i.e. a problem of construction of such set in phase space of the system which contains all invariant compact sets of this system is solving. In article  the family of localizing sets for invariant compact sets is received using the functional method of localization proposed by A.P.~Krishchenko. The intersection of this family is found by numerical optimization methods.

The model of the wheeled robot with automobile configuration of wheels in a problem of path stabilisation is considered. Conditions at which transition from the Cartesian coordinates to the trajectory coordinates at the path stabilization problem is correct are investigated. Performance of these conditions for the elementary types of a trajectory of the robot is analyzed.

Trajectory planning problem of the unmanned aerial vehicle (UAV) was considered in this article. UAV should flied by the preset traveling points during the preset moments of time. State and control variables were also restricted.The main problem is to find the permissible trajectory, which satisfy given restrictions. The approach based on design of a trajectory from a certain set of sample maneuvers was proposed. These maneuvers were formed with use of a combination of analytical methods of trajectories calculation, methods of mathematical simulation and various heuristic algorithms.The sequence of traveling points breaks required trajectory into segments. Essential simplification of the problem can be obtained by the demand that calculation of the trajectory segment does not affect on calculation of the subsequent segments and depends only on values of state and control variables of UAV obtained on the previous segment. In this case planning of a trajectory was carried out consistently, from one segment to another.In this paper the maneuver planning problem of an echelon change was solved. This maneuver in a combination with rectilinear uniform movement allowed to plan those segments of a UAV trajectory where movement can take place in the vertical plane, i.e. with constant value of a traveling angle.The nonlinear mathematical model of UAV movement as material point in the trajectory coordinates was described. The proposed method of the solution of a terminal problem was based on usage of polynoms on time. Two heuristic algorithms of maneuver planning of echelon change were described. Examples were included. Results of simulation and the scheme of testing of proposed planning method were presented.