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Stability01:28

Stability

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The time response of a linear time-invariant (LTI) system can be divided into transient and steady-state responses. The transient response represents the system's initial reaction to a change in input and diminishes to zero over time. In contrast, the steady-state response is the behavior that persists after the transient effects have faded.
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The transfer function is a fundamental concept representing the ratio of two polynomials. The numerator and denominator encapsulate the system's dynamics. The zeros and poles of this transfer function are critical in determining the system's behavior and stability.
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System stability is a fundamental concept in signal processing, often assessed using convolution. For a system to be considered bounded-input bounded-output (BIBO) stable, any bounded input signal must produce a bounded output signal. A bounded input signal is one where the modulus does not exceed a certain constant at any point in time.
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A system is linear if it displays the characteristics of homogeneity and additivity, together termed the superposition property. This principle is fundamental in all linear systems. Linear time-invariant (LTI) systems include systems with linear elements and constant parameters.
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Root loci often diverge as system poles shift from the real axis to the complex plane. Key points in this transition are the breakaway and break-in points, indicating where the root locus leaves and reenters the real axis. The branches of the root locus form an angle of 180/n degrees with the real axis, where n is the number of branches at a breakaway or break-in point.
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The Hartley oscillator is a positive feedback system that sustains oscillations by feeding the output back to the input in phase, thereby reinforcing the signal. Positive feedback systems can be viewed as negative feedback systems with inverted feedback signals. In these systems, the root locus encompasses all points on the s-plane where the angle of the system transfer function equals 360 degrees.
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Robust observer-based absolute stabilization for Lur'e singularly perturbed systems with state delay.

Wei Liu1, Yanyan Wang2

  • 1School of Mathematics and Statistics, Zhoukou Normal University, Zhoukou 466001, China; Center for Applied and Multidisciplinary Mathematics, Department of Mathematics, East China Normal University, Shanghai 200241, China.

ISA Transactions
|July 30, 2016
PubMed
Summary
This summary is machine-generated.

This study presents a robust observer-based control method for stabilizing complex Lur'e systems with delays and singular perturbations. The approach guarantees absolute stability using input-to-state stability (ISS) criteria.

Keywords:
Input-to-state stability (ISS)Linear matrix inequality (LMI)Lur’e singularly perturbed systemsTime-delay systems

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Area of Science:

  • Control Systems Engineering
  • Nonlinear Systems Analysis
  • Singular Perturbations

Background:

  • Lur'e systems with time delays and singular perturbations pose significant control challenges.
  • Robust observer-based control is crucial for ensuring system stability despite uncertainties.

Purpose of the Study:

  • To design a robust observer-based control law for absolute stabilization of Lur'e singularly perturbed time-delay systems.
  • To develop delay-dependent conditions for observer error system stability.

Main Methods:

  • Construction of a full-order state observer.
  • Application of linear matrix inequality (LMI) techniques for stability analysis.
  • Introduction of slack matrices to ensure input-to-state stability (ISS) of the closed-loop system.

Main Results:

  • A delay-dependent sufficient condition for the absolute stability of the observer error system was derived.
  • A sufficient condition for the input-to-state stability (ISS) of the closed-loop system was established.
  • Stability criteria were developed that are independent of the small parameter.

Conclusions:

  • The proposed observer-based control strategy guarantees absolute stabilization for the targeted systems.
  • The developed criteria provide a workable algorithm for obtaining upper bounds on absolute stability.
  • Numerical examples validate the effectiveness of the presented control methods.