Generalization of nonlinear control for nonlinear discrete systems

TL;DR

This paper introduces a generalized nonlinear control method for stabilizing unstable periodic orbits in discrete nonlinear systems, improving convergence speed and reducing memory requirements, with applications demonstrated through numerical simulations.

Contribution

It extends Morgul's scalar feedback method to vector systems, combining nonlinear and semilinear feedback for enhanced stabilization and efficiency.

Findings

Reduced prehistory length in control algorithms

Faster convergence to periodic solutions

Effective stabilization demonstrated through simulations

Abstract

The problem of stabilization of unstable periodic orbits of discrete nonlinear systems is considered in the article. A new generalization of the delayed feedback, which solves the stabilization problem, is proposed. The feedback is represented as a convex combination of nonlinear feedback and semilinear feedback introduced by O. Morgul. In this article, the O. Morgul method was transferred from the scalar case to the vector one. It is shown that the additional introduction of the semilinear feedback into the equation makes it possible to significantly reduce the length of the prehistory used in the control and to increase the rate of convergence of the perturbed solutions to periodic ones. As an application of the proposed stabilization scheme, a possible computational algorithm for finding solutions of systems of algebraic equations is given. The numerical simulation results are presented.