Lyapunov-Krasovskii Based Approach To Time-Delay Systems: Computational Efficiency And Feedback Control

Author

• Huan Phan Phan, Southern Illinois University CarbondaleFollow

Date of Award

5-1-2026

Degree Name

Doctor of Philosophy

Department

Engineering Science

First Advisor

Gu, Keqin

Abstract

Time-delay systems arise naturally in many engineering applications due to sensing,actuation, communication, and computation delays. The presence of delays has a significant influence on system performance both positively and negatively. Although Lyapunov--Krasovskii functional (LKF) methods provide a systematic framework for stability analysis and controller synthesis, their implementation often suffers from high computational complexity, particularly for large-scale systems and systems involving multiple feedback channels. This dissertation addresses both computational efficiency and feedback control for linear time-delay systems.The first part of this dissertation develops a decomposition-based approach to reduce the computationalburden of stability analysis. By exploiting structured invariant subspaces associated with the system dynamics, the system is decomposed into lower-dimensional subsystems. This decomposition significantly reduces the size and complexity of the resulting linear matrix inequalities (LMIs), thus reducing computational time and memory requirement. The proposed framework is particularly effective for large-scale systems with multiple feedback interconnections.The second part develops a feedback controller design method for linear time-delay systems. The controller synthesis problem is formulated using operator-based transformations, and the feedback gains are recovered constructively by approximating the inverse of the Lyapunov operator through discretization and numerical approximation methods. Under the proposed feedback structure, the resulting closed-loop dynamics naturally include distributed delay terms due to the distributed delay gains introduced through the feedback control. A generalized Lyapunov-Krasovskii functional based method is developed to conduct stability analysis for distributed delay systems in order to verify the stability of the resulting closed-loop system that may include errors due to discretization and inversion.Numerical examples demonstrate the effectiveness of the proposed methods in both efficient stability analysis and feedback controller synthesis. The results show substantial reduction in computational time for large-scale stability verification and confirm that the proposed controller design and distributed-delay stability checks perform effectively in practice.