Stabilization of linear time-delayed systems by delayed feedback

TL;DR

This paper presents a stability analysis method for linear time-delayed systems using system decomposition based on simultaneous block triangularization, and demonstrates stabilization via delayed feedback control.

Contribution

It introduces a novel decomposition approach for stability analysis of time delay systems and applies delayed feedback for stabilization, addressing cases with repeated roots on the imaginary axis.

Findings

Decomposition based on common invariant subspaces enables stability analysis.

Delayed feedback can stabilize systems with challenging characteristic equations.

The method extends stability analysis capabilities for complex time delay systems.

Abstract

In this paper, stability analysis of time delay systems is considered based on decomposition of the systems to subsystems. The decomposition process needs matrices of these systems to be simultaneously block triangularize. We show that a finite set of matrices are simultaneously block triangularize if and only if they have a common invariant subspace. The importance of the decomposition is highlighted by some systems that their characteristic equation has repeated roots on the imaginary axis. For such systems, the cluster treatment method and the direct method cannot be employed to analyze the stability of them, unless, we decompose the systems to subsystems. On the other hand, it has been shown that stabilization of time delay systems can be done by delayed feedback. Moreover, this kind of feedback is applied to design a controller.