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Delay and delay-derivative dependent stability analysis for linear time delay systems with a cyclical delay.

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Summary

This study presents a new method for analyzing linear systems with cyclical delays, improving stability criteria using novel Lyapunov-Krasovskii functionals and a new negative-definiteness condition. The findings offer less conservative stability analysis for systems with complex delay patterns.

Keywords:
Delay-derivative-dependent inequalityGeneralized bivariate matrix polynomialLinear matrix inequalityLyapunov-Krasovskii functionalsNegative-definiteness condition

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Area of Science:

  • Control Theory
  • Systems Engineering
  • Applied Mathematics

Background:

  • Stability analysis is crucial for linear systems, especially those with time delays.
  • Cyclical delays, with alternating increasing and decreasing intervals, present unique challenges in stability analysis.
  • Existing methods may be conservative for systems with complex delay profiles.

Purpose of the Study:

  • To develop a less conservative stability criterion for linear systems with cyclical delays.
  • To introduce novel Lyapunov-Krasovskii functionals (LKFs) tailored for cyclical delay characteristics.
  • To integrate advanced mathematical conditions for improved stability analysis.

Main Methods:

  • Construction of two-loop Lyapunov-Krasovskii functionals using delay product terms.
  • Development of a novel delay-derivative-dependent inequality.
  • Application of a new negative-definiteness condition (NDC) for generalized bivariate matrix polynomials.
  • Formulation of the stability criterion as a linear matrix inequality (LMI).

Main Results:

  • A new stability criterion in LMI form was derived.
  • The proposed method effectively utilizes distinct delay monotonicity intervals.
  • Numerical examples confirmed the reduced conservatism compared to existing methods.

Conclusions:

  • The developed stability criterion offers significant improvements for linear systems with cyclical delays.
  • The novel approach enhances the accuracy and applicability of stability analysis in control systems.
  • This work provides a valuable tool for engineers and researchers dealing with time-delayed systems.