Simulation of an exponentially stabilized trajectory for linear dynamic system

K. A. Raetsky

Taurida Journal of Computer Science Theory and Mathematics, no. 2 (2024), pp. 57-74

Résumé

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.

Détail

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