Using the point mapping method, we obtained analytical expressions for the first return functions for determining simple and complex attractors in the stabilization mode by a general-purpose relay controller with the linear formation of the control signal. We investigated self-oscillations with account for the operating members’ aftereffect, the dead zone of the speed sensor, and the time-independent perturbation action. The study shows that the dead zone of the speed sensor introduces significant changes in the behavior of the system, giving it new properties. The analysis of dynamic processes on a three-sheet phase surface revealed a wide variety of limit cycles and their dependence on the system’s parameters. Complex limit cycles are represented by combining simple cycles of two types, which allowed for a simplifying approach to their search based on the theory of multidimensional transformations of Yu.I. Neymark. A more complete result was obtained in comparison with the well-known literary sources.
The article considers solving the problem of analytical construction of Poincare maps for finding simple and complex limit cycles in a relay dynamic system involving constant perturbation and delay. Application of Neymark’s theory of multivariate point transformations allows reducing the problem under consideration to the search for a multifold fixed point thus overcoming the difficulty of finding complex periodic motions. The choice of switching lines as arcs without contact taking into account delays significantly simplified the task of analytical construction of point mapping. The results of analytical constructions are confirmed by numerical simulation of movements. The results obtained can find practical application in developing reactive systems for controlling the orientation and stabilization of the spacecraft. Compared with the previously known solution, a more complete result is obtained, which is of particular importance in the study of systems with high efficiency of executive bodies.
The study introduces a method of ground conditions physical modeling of the spacecraft motion around a fixed axis. On a natural scale of parameters and variables, the dynamic modes under consideration can be implemented only with an extremely small amount of kinetic energy dissipation. The feasible minimum friction for a test bed of simple design significantly exceeds the required values. In current modes of economical limit cycles, the characteristics of the simulated process are distorted so much that the physical modeling test bed is unsuitable for practical use. The solution to this problem is usually sought by complicating the design of the test bed through the use of air or magnetic suspension. The paper proposes an innovative method of “invariant scaling”, based on the principle of dynamic similarity of self-oscillating processes. Its application makes it possible to drastically reduce the effect of friction on the characteristics of physically modeled modes during ground developmental testing of control algorithms. Computer modeling with the use of this method has confirmed its high efficiency. It has been shown analytically and numerically that the modeling accuracy can be radically improved. An example of reducing the modeling error by 200 times is given.
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