Stability Analysis and Stabilization of Continuous-Time Linear Systems with Distributed Delays

Concetti Chiave

This monograph explores new methods for the stability and stabilization of linear systems with nontrivial distributed delays through the construction of Lyapunov-Krasovski˘ı functionals.

Sintesi

The content provides a comprehensive overview of the stability analysis and stabilization of linear systems with distributed delays (DDs). It begins by introducing the general framework of coupled differential-functional equations (CDFEs) to model systems with delays, and presents the Lyapunov-Krasovski˘ı stability criterion for CDFEs.

The main focus is on reviewing existing frequency-domain and time-domain approaches for the stability analysis of linear time-delay systems (LTDSs), with a particular emphasis on linear systems with DDs. Frequency-domain methods involve computing the spectral abscissa by finding the zeros of the characteristic equation, while time-domain approaches rely on constructing Lyapunov-Krasovski˘ı functionals (LKFs).

The content then outlines the research motivations and provides an overview of the subsequent chapters, which cover:

1. Dissipative stabilization of linear delay systems with distributed delays
2. Dissipative stabilization of uncertain linear distributed delay systems
3. Novel integral inequalities for stability analysis
4. Stability and dissipativity analysis of linear coupled differential-difference systems with distributed delays
5. Dissipative delay range analysis of coupled differential-difference delay systems
6. Dissipative stabilization of linear systems with uncertain bounded time-varying distributed delays

The proposed methods utilize the LKF approach and novel integral inequalities to address the stability and stabilization of linear systems with nontrivial distributed delays, considering dissipativity constraints.

Statistiche

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Approfondimenti chiave tratti da

Stability Analysis and Stabilization of Continuous-Time Linear Systems with Distributed Delays

by Qian Feng,Al... alle arxiv.org 04-09-2024

https://arxiv.org/pdf/2204.08353.pdf

Domande più approfondite

How can the proposed methods be extended to handle more general classes of nonlinear systems with distributed delays

The proposed methods for stability analysis and stabilization of linear systems with distributed delays can be extended to handle more general classes of nonlinear systems by incorporating nonlinear dynamics into the system models. This extension would involve developing new mathematical frameworks that can accommodate the nonlinearities present in the system dynamics. One approach could be to use nonlinear Lyapunov functions or functionals to analyze the stability of nonlinear systems with distributed delays. By formulating appropriate stability conditions based on these nonlinear Lyapunov functions, it would be possible to assess the stability of a broader range of nonlinear systems with distributed delays.

What are the potential applications of the developed techniques in emerging areas such as networked control systems, multi-agent systems, or renewable energy systems

The developed techniques for stability analysis and stabilization of linear systems with distributed delays have significant potential applications in emerging areas such as networked control systems, multi-agent systems, and renewable energy systems. In networked control systems, where communication delays and packet losses are common, the methods can be used to design controllers that ensure stability and performance despite these challenges. For multi-agent systems, the techniques can be applied to coordinate the actions of multiple agents with communication delays, leading to improved system behavior and coordination. In renewable energy systems, where distributed delays may arise due to energy storage or transmission constraints, the methods can be used to optimize energy management and ensure stability in the system.

Are there any connections between the stability analysis of linear systems with distributed delays and the study of partial differential equations or functional analysis

There are connections between the stability analysis of linear systems with distributed delays and the study of partial differential equations (PDEs) or functional analysis. In the context of distributed delays, the dynamics of the system can be modeled using PDEs, where the distributed delays correspond to spatial delays in the system. By analyzing the stability of systems with distributed delays, insights can be gained into the behavior of systems governed by PDEs with delays. Additionally, functional analysis techniques can be used to study the properties of the operators associated with the system dynamics, providing a mathematical framework for analyzing stability and control properties. The study of linear systems with distributed delays can thus serve as a bridge between control theory and the broader field of PDEs and functional analysis.