On the approximate realization of responses using transient curves

Abstract

The problem of realizing the transient response is often solved separately from the problem of realizing a given trajectory. This means that if the initial condition does not coincide with the initial position of the trajectory being realized, then, at first the transient response which permits us to hit onto the trajectory is produced by one method and then the problem of realizing this trajectory is solved by another method (see, for example, [1]). Here we shall consider a method, using the results of Barbashin [2–4] on the approximate realization of a trajectory, which allows us to solve problems by a single method. The main point of the proposed method consists of the following. It is considered that by some means there is given a family of transient curves determining a direction field in the phase coordinate space. The given system of differential equations also determines some direction field which depends on control functions. The control functions are then found from the conditions of minimization at each instant of time of the square of the deviation between corresponding vectors from the two above-mentioned direction fields.