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    Area of Science:

    • Control Systems Engineering
    • Systems Theory
    • Applied Mathematics

    Background:

    • Partial state estimation is crucial for many practical applications.
    • Designing functional observers (FOs) for systems with unknown time delays remains a significant challenge.
    • Existing methods struggle with time-varying delays and uncertainties.

    Purpose of the Study:

    • To develop a robust functional observer for linear time-invariant systems with bounded time-varying delays.
    • To address the design difficulties associated with unknown and uncertain time delays.
    • To establish delay-dependent stability conditions for the proposed observer.

    Main Methods:

    • A sliding mode functional observer structure is proposed, incorporating an auxiliary delay function.
    • Lyapunov-Krasovskii approach is utilized to derive delay-dependent stability conditions.
    • Stability conditions are formulated using linear matrix inequality (LMI) and rank conditions.
    • Analysis of a delay-free observer structure with necessary and sufficient stability conditions.

    Main Results:

    • A sliding mode functional observer design is presented that is robust to delay uncertainties.
    • Sufficient conditions for observer stability, dependent on delay bounds, are derived.
    • The observer design is validated through two numerical examples and simulation results.
    • Effectiveness of the proposed observer design algorithm is demonstrated.

    Conclusions:

    • The proposed sliding mode functional observer effectively handles time-varying delays in linear systems.
    • The developed delay-dependent stability conditions provide a rigorous framework for observer design.
    • The method offers a robust solution for partial state estimation in systems with uncertain time delays.