An alternative approach for stability analysis of discrete time nonlinear dynamical systems

TL;DR

This paper introduces a novel method using G-functions to determine the stability of nonlinear discrete-time systems without relying on Lyapunov functions, providing new necessary and sufficient conditions.

Contribution

It proposes the concept of G-functions for stability analysis, offering an alternative to Lyapunov functions and establishing new criteria for asymptotic stability in nonlinear discrete systems.

Findings

Derived necessary and sufficient stability conditions

Established methods to estimate stability domains

Validated results with illustrative examples

Abstract

The paper endeavours to solve the problem of the necessary and sufficient conditions for testing asymptotic stability of the equilibrium state without using a positive definite or semi-definite Lyapunov function for time-invariant nonlinear discrete-time dynamical systems. The solution is based on the concept of the $\bf G$-functions introduced in this paper. As a result, new necessary and sufficient conditions for asymptotic stability of such systems and an estimation or the exact determination of the asymptotic stability domain of the state $x=0$ are established. Examples are worked out to illustrate the results.