Simulation of an exponentially stabilized trajectory for linear dynamic system

A completely controlled linear stationary dynamical system is considered. The
problem of program motion stabilization is solved for the system.

The program movement is calculated in such a way that under the action of some control
action, the movement trajectory (state), leaving the starting point, passes through arbitrarily
given points and arrives at a given end point. Passing the trajectory of a dynamic system
through control points allows the system to be in the right place at the right time. Besides, with
appropriately defined control points, the trajectory of the system can take on various forms.
This is important, for example, when flying an aircraft apparatus in mountainous terrain at a
sufficiently low altitude.

The initially given initial state allows us to calculate the program control and the program
trajectory. However, if the dynamic system cannot realize the given initial state at the initial
moment of time, due to some external circumstances that have arisen, then the problem of
stabilization arises. In such a case, the trajectory is calculated with a new initial value. This new
(stabilized) trajectory is calculated in such a way that over time it approaches the program
trajectory. In this article, not only the condition of exponential approach to the program
trajectory is set, but also the coincidence of trajectories at control points, including the end
point, is required. For the analytical solution of such a problem, the method of indeterminate
coefficients is used here for the first time. The difference between the values of the programmed
trajectory and the stabilized trajectory, as well as difference between program control values and
stabilized, are formed as linear combinations of exponentially decreasing scalar functions with
vector indeterminate coefficients. The resulting expressions are substituted into the equation
relating such differences and into the corresponding multi-point conditions for these differences.

As a result, a linear algebraic system for finding vector coefficients is formed. The peculiarity
of the obtained algebraic system is that at first the vector coefficients can be found only for
the stabilizing trajectory. As a result, a stabilized trajectory is built first. In the case when
the resulting algebraic system is undetermined, a set of stabilized trajectories is constructed;
then, according to some additional criteria, the most appropriate trajectory is selected from the
obtained set.

Further, the vector coefficients for stabilizing control and the corresponding stabilized
control is constructed. If all the obtained stabilized trajectories do not satisfy any additional
requirements, then it is possible to change the exponents in the exponential functions in the
above linear combinations. In this paper, a complete condition is obtained under which the
determinant of an algebraic system does not vanish.

Examples of sets of exponential functions for which this condition is satisfied are given. An
example of constructing a stabilized trajectory of a material point in a vertical plane under the
action of a reactive force for punching a tunnel in a mountainous area is given. The trajectory is
constructed for the case when the initial position of the launcher has been moved to another point.
This example shows that building a stabilized trajectory is much more economical than building
a new program trajectory with a new initial condition. Moreover, this example demonstrates the
effectiveness of the method of undefined coefficients.

Keywords: complete control, stabilization, control points, method of indeterminate coefficients.