To the question of the stabilization of motion metods of practical stability
DOI:
https://doi.org/10.31548/energiya2018.06.176Abstract
Topicality. When solving a number of problems associated with the design of complex control systems [1-3], it often becomes necessary to adjust the motion in the vicinity of the calculated trajectories set at the initial moment of time and belong to some dynamic sets of phase space [2, 3]. In this case, the correction requirements may relate to the individual coordinate of the calculated trajectories and their linear combination. These tasks belong to the class of tasks of stabilization of motion and reduce to the construction of regulatory influences that provide the original system a certain type of stability [1, 2, 4].
This approach is used, for example, in modeling charged particles in order to control them in dynamics in the presence of appropriate sustainability requirements. To construct constructive algorithms, the stability analysis is carried out on a finite time
interval, and the initial conditions are given in the structural form [3, 5-7]. The latter allows us to obtain numerical estimates for the stabilization of motion to practical stability in the presence of dynamic constraints on phase coordinates.
Analysis of recent research and publications. With the analysis of practical stability and sensitivity of dynamic systems, problems of optimal control of a beam of trajectories are closely related [2, 3, 6]. Thus, numerical algorithms for calculating the optimal areas of practical stability are used to estimate the area of particle capture in the acceleration process. In this case, the field of initial conditions can be in the given structures (sphere, ellipsoid), and the maximum in volume (optimal by inclusion). Then the question is raised about optimization of estimates of such sets due to the proper choice of system parameters [7].
The tasks of constructing regulatory influences that provide the desired stability requirements are an important part of a set of tasks that arise in the design of technical systems, in particular accelerating-focusing [1, 2]. Unlike the classical statements [1], the analysis of the stability of parametric systems [2-4, 6, 7] allows us to study the system's performance in real modes, taking into account the requirements for sensitivity, and to solve such problems numerically from the standpoint of stability [3].
The purpose of the research ─ development of constructive algorithms for solving stabilization problems for practical stability for linear parametric systems with disturbances.
Materials and methods of research. The paper uses methods of stability theory, differential equations and control theory.
Research results and their discussion. For linear parametric systems of differential equations with perturbations, the sequence of stabilization problems to practical stability in the given domains of dynamic constraints is investigated. Considered cases of known and limited perturbation norm. For structurally-defined areas of initial conditions, numerical estimates of the stabilization of motion to practical stability have been obtained.
Conclusions and perspectives of further research.
Formation of stabilization problems for practical stability of linear parametric systems with perturbations is formulated. On the basis of practical stability algorithms, numerical criteria for stabilizing the system motion to a certain type of stability were obtained. Such an approach can be extended to the solution of problems of stabilization of motion to asymptotic stability in regions of a more general structure.
Keywords: mathematical model, stabilization, regulator, parameters, practical stability, parametric system, dynamic constraints
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