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In an underdamped second-order system, where the damping ratio ζ is between 0 and 1, a unit-step input results in a transfer function that, when transformed using the inverse Laplace method, reveals the output response. The output exhibits a damped sinusoidal oscillation, and the difference between the input and output is termed the error signal. This error signal also demonstrates damped oscillatory behavior. Eventually, as the system reaches a steady state, the error diminishes to zero.
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Nonlinear systems often require sophisticated approaches for accurate modeling and analysis, with state-space representation being particularly effective. This method is especially useful for systems where variables and parameters vary with time or operating conditions, such as in a simple pendulum or a translational mechanical system with nonlinear springs.
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Observation and sliding mode observer for nonlinear fractional-order system with unknown input.

Nadia Djeghali1, Said Djennoune1, Maamar Bettayeb2

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Summary

This study introduces observability and left invertibility for nonlinear fractional-order systems, linking them to integer-order systems. It also proposes a sliding mode observer for fault detection and estimation in these systems, proving finite-time convergence.

Keywords:
Fault detection and estimationLeft invertibilityLyapunov stabilityNonlinear fractional-order systemsObservabilitySliding mode observers

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Area of Science:

  • Control Theory
  • Nonlinear Systems
  • Fractional Calculus

Background:

  • Fractional-order systems are increasingly used in modeling complex phenomena.
  • Observability and invertibility are crucial for system analysis and control.
  • Fault detection and estimation are vital for system reliability.

Purpose of the Study:

  • Introduce observability and left invertibility for nonlinear fractional-order systems.
  • Develop a sliding mode observer for fault detection and estimation.
  • Analyze the properties and performance of the proposed observer.

Main Methods:

  • Transformation to an equivalent nonlinear integer-order system.
  • Step-by-step design of a first-order sliding mode observer.
  • Lyapunov stability theory for finite-time convergence analysis.

Main Results:

  • Established observability and left invertibility for nonlinear fractional-order systems.
  • Demonstrated the deduction of these properties from equivalent integer-order systems.
  • Designed a sliding mode observer with proven finite-time convergence for fault detection.

Conclusions:

  • The proposed methods provide a framework for analyzing and controlling nonlinear fractional-order systems.
  • The sliding mode observer is effective for fault detection and estimation.
  • Numerical examples validate the theoretical findings and practical applicability.