Open AccessThe paper considers a problem of the time-specified control terminal for the second order system with restrictions on the state variables.
Most developed methods for solving problems of the terminal [1, 2, 3, 4, 5] do not allow us to take into account the restrictions on the system condition. To solve such problems are widely used methods based on the concept of inverse dynamics problems [6, 7, 8, 9, 10], with one step of which being to specify a kinematic object trajectory. Some methods use an iterative [11] process of finding a desired program trajectory.
This work is based on the results presented in [12]. It is shown that the solution of the original problem is equivalent to finding the terminal phase of the trajectory that satisfies the restrictions imposed on the state variables, as well as the certain additional conditions. It is assumed that the restrictions imposed on the state variables can be represented as functions for which, in a certain class of functions, special approximations are built. A desired phase trajectory is built as a linear combination of obtained functions-approximations. Thus constructed phase trajectory is a solution to the original terminal problem. The presented formulas are true for both the upper and lower half-plane of the phase space. The paper proposes an optimization approach to a choice of the trajectory as well as the options to extend the set in which the phase trajectories are sought. It gives the numerical simulation results, a presented in [12] algorithm, and also the results of numerical solution to the optimization problem.
This approach can be used to solve the terminal problems of vector-controlled mechanical systems with restrictions on the state variables.