IMPROVING THE EFFICIENCY OF THE PROCEDURE OF LYAPUNOV SPLINE-FUNCTIONS CONSTRUCTION FOR NONLINEAR NONSTATIONARY SYSTEMS

Abstract

The paper proposes a numerical algorithm for constructing Lyapunov functions for investigating the absolute stability of nonlinear nonstationary systems. In the case of asymptotic stability of the system, the implementation of the algorithm will lead to the construction of the Lyapunov function level set in the form of a smooth closed surface of dimension equal to the dimension of the original system. To construct a smooth level set of the Lyapunov function, a new type of surface has been developed. Thus, the task of constructing the level set was reduced to a series of simple optimization problems, which guarantees the convergence of the algorithm. Unlike the algorithm for constructing piecewise linear Lyapunov functions, this algorithm analyses systems located near the stability boundary in an acceptable time. The relationship of this algorithm and methods based on frequency criteria and quadratic Lyapunov functions is shown. A significant improvement in the accuracy of estimates of the stability boundary was demonstrated in comparison with the classical methods. To achieve a balance between the accuracy and speed of the algorithm, recommendations on the choice of initial conditions are given.

Keywords

• differential inclusions
• nonlinear nonstationary systems
• absolute stability
• lyapunov functions
• stability areas
• bezier splines
• bernstein polynomials