Lyapunov Theory for Discrete Time Systems

TL;DR

This paper extends Lyapunov theory to discrete time systems, providing new proofs for averaging and separation of time scales that require only Lipschitz conditions, and guarantees semi-global exponential stability.

Contribution

It introduces a Lyapunov-based approach for discrete systems that relaxes differentiability requirements and broadens stability guarantees compared to existing methods.

Findings

Lyapunov functions with Lipschitz conditions suffice for stability analysis.

The approach guarantees semi-global exponential stability under mild interconnection conditions.

New proofs for averaging and separation of time scales in discrete systems are developed.

Abstract

In this work, we present the equivalent of many theorems available for continuous time systems. In particular, the theory is applied to Averaging Theory and Separation of time scales. In particular the proofs developed for Averaging Theory and Separation of time scales departs from those typically used in continuous time systems that are based on twice differentiable change of variables and the multiple use of the Implicit Function Theorem and Mean Value Theorem. More specifically, by constructing a suitable Lyapunov function only Lipschitz conditions are necessary. Finally, it is shown that under mild condition on the so-called "interconnection conditions" the proposed tools can guarantee semi-global exponential stability rather than the more stringent local exponential stability typically found in the literature