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Article:

# A statistical approach to the formulization of Lyapunov stability conditions in guided dynamical systems modelling

Keywords:

Nguyen Quang Thuong - Dr.Sc. (Eng.), Professor, State University of Management (Moscow)
E-mail: tikhonovrus@gmail.com
Nguyen Manh Cuong - Post-graduate Student, Moscow Institute of Physics and Technology (State University)
Email: nguyencuongkshk@gmail.com

This work presents a statistical approach to the formulization of Lyapunov stability conditions in guided dynamical systems modelling, in what the stability conditions are constructed by a statistical sample for rescovering Lyapunov functions, the elements of wich are defined according to built optimal conditions.
A mathematical formulization of stability conditions for a guided dynamical system (GDS) is presented. The GDS' ma-thematical model is defined by the physical existance conditions for GDS in the form of project making decision equations, time-not associated; by the functional existance conditions for GDS in the form of differential equations of a time-depending relation.
The project parameters components and nonlinear differential equation system of a unmanned aerial vehicle's (UAV's) motion for Lyapunov stability investigating are shown.
The authors have offered a Lyapunov functions constructing by a statistical sample for an estimation of UAV's model stability.
The Lyapunos function for UAV's model is taken in the form of a generalized exponential polynom, in what the regu-lations for Lyapunov functions constructing from a two-variate function are formulated.
Along to formulated regulations, the statistical samples for a Lyapunov function and its derivative, necessary for computing the appropriate regulization criterion are obtained.
A Lyapunov functions constructing by the conditions of sign-positive and derivative from a Lyapunov function in time is realised according to a regulization criterion.
In whole, according to a statistical approach to the formulization of Lyapunov stability conditions the stability estimation of a guided dynamical system (in an example of UAV's model) consists of the following tasks sequentially:
– a statistical synthethis of a Lyapunov function according to trajectory sampling points;
– a statistical synthethis of a Lyapunov function's derivative in time according to trajectory sampling points;
– a formulization of a statistical sample with the critical values of a Lyapunov function's and its derivative's in time for every trajectory sampling point;
– a Lyapunov functions constructing according to a sign-positive condition and constructing a function "a derivative of the Lyapunov function in time" according to a sign-negative.

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June 24, 2020
May 29, 2020

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