Розрахунок областей зовнішньої та внутрішньої стійкості для параметричних систем

Л. А. Панталієнко


UDC 681.5.07




L.A.PANTALIYENKO, Candidate of Physical and Mathematical Sciences


Abstract. The general problems of the practical stability of systems of ordinary differential equations, depending on parameters, with continuous perturbations are considered. In the framework of the formulated problems, a linear system was studied in the presence of known or perturbation-bound perturbations. The algorithms of numerical construction of the regions of external and internal stability in the structurally determined form for linear and nonlinear dynamic constraints on phase coordinates are developed. Separately, the case of compatible restrictions on the state vectors and system parameters are considered.


Keywords: рractical stability, perturbation, norm, parametric system, dynamic constraints


Topicality. Further development of constructive methods for analysis of stability  and associated tasks of design, management, adaptation , primarily due to the needs of the application nature, in particular, arising in accelerating techniques . At the same time from the practical point of view, it is important to study the stability of motion not only to change the initial conditions, but also in relation to external perturbations, which to some extent affect the behavior of the real object .

Analysis of recent research and publications. One of the directions of the development of the theory of stability is the numerical solution of stability problems on a finite period of time . Unlike other formulas , the analysis of the stability of parametric systems  allows us to expand the range of investigated problems, since as parameters we can consider the initial conditions dynamic system. In addition, such tasks are an important part of the task of designing low-sensitive control systems and guaranteed sensitivity  directly related to the calculations for

                                                                                     © L.A. Pantaliyenko, 2017

linear resonant accelerators.

         The aim of research ─ development of numerical methods for solving practical stability problems for parametric systems of differential equations in the presence of continuous perturbations.

Materials and methods of research. The methods of the theory of stability, differential equations and optimization are used in this paper.

Results of the research and their discussion. Assume that the dynamics of an object is described by a parametric system of differential equations

,                  (1)

where constantly acting disturbances  are selected from a certain area .

Definition 1. The system (1) is called internally - stability if  for arbitrary perturbations, initial conditions and parameters satisfying the relationship

, .

Definitions 2. The system (1) is externally - stability if there exists at least one point in time  for which  for any , .

By analogy, the corresponding definition of stability for the system (1) is introduced in the case of compatible restrictions on the state and parameter vectors.

In the framework of the above formulas of problems we consider the linear system of the form

                                      , .                                (2)

in the presence of known or restricted perturbations.

For numerical estimates of the set of initial conditions, consider linear and nonlinear phase constraints:

                    , ,                                 (3)

                   .                                             (4)

If dynamic constraints on the state vector and system parameters compatible type :

, ,                           (5)


,                                               (6)

numerically assess the set of initial conditions .

Criterion 1. In order for the system (2) to be internally - or - stability persistent in the presence of known perturbations, it is necessary and sufficient that one of the inequalities



, , , ;                             (7)




                   , , , .                          (8)

Here ,  − boundary of a closed convex set , , , , =,  – normed fundamental matrix of solutions of the homogeneous system (2) at .

Similar estimation criteria were obtained for the case of compatible constraints of type (5), (6) and external stability.

Research results and their discussion. For linear parametric systems of differential equations in the presence of perturbations, a numerical calculation of the areas of internal and external stability was performed. Considered cases of known and bounded perturbations for linear and nonlinear dynamic constraints.

Conclusions and perspectives of further research. The formulation of problems of practical stability of parametric systems of ordinary differential equations in the presence of constantly acting perturbations is formulated. For linear inhomogeneous systems with perturbations, optimal estimates of the regions of initial conditions in the given structures are obtained. Such an approach can be introduced to analyze the stability of other classes of parametric systems (systems with variable structure and discrete parametric systems).

Повний текст:



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