Feedback Stabilization of a Class of Perturbed Nonlinear Autonomous Difference Equations

TL;DR

This paper develops feedback control strategies for stabilizing a broad class of unstable nonlinear difference equations, including perturbed systems with different orders, using fixed point principles and incremental controllers.

Contribution

It introduces a novel stabilization approach combining nominal and incremental controllers for perturbed nonlinear difference equations, extending existing methods.

Findings

Effective stabilization of perturbed nonlinear difference equations.

Controller design based on Banach fixed point principle.

Applicable to unstable oscillatory solutions.

Abstract

This paper investigates the local asymptotic stabilization of a very general class of instable autonomous nonlinear difference equations which are subject to perturbed dynamics which can have a different order that that of the nominal difference equation. In the general case, the controller consists of two combined parts, namely, the feedback nominal controller which stabilizes the nominal (i.e. perturbation - free) difference equation plus an incremental controller which completes the stabilization in the presence of dynamics in the uncontrolled difference equation. A stabilization variant consists of using a single controller to stabilize the nominal difference equation and also the perturbed one under a smallness-type characterization of the perturbed dynamics. The study is based on Banach fixed point principle and it is also valid with slight modification for the stabilization of unstable oscillatory solutions.