MODIFIED ALGORITHM FOR DETERMINATION OF FULL STABILITY AREAS IN NONSTATIONARY NONLINEAR SYSTEMS

Abstract

The paper proposes a numerical algorithm for constructing Lyapunov spline 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 piece-wise smooth (smooth, if additional conditions are met) closed surface of dimension equal to the dimension of the original system. It is shown that the modified algorithm significantly improves the stability boundary estimates obtained with frequency methods. Unlike the algorithm for constructing piecewise linear Lyapunov functions, the running time of the proposed algorithm for constructing the Lyapunov spline functions does not tend to infinity as the system approaches the stability boundary. This circumstance makes it possible to use a modified algorithm to determine the stability of systems that are close to the stability boundary. An estimate of the accuracy of determining the stability area using an example of a third-order system is shown. Specific recommendations on the algorithm initial conditions choice are given.

Keywords

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