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Related Concept Videos

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.
The stability of an LTI system is determined by the roots of its characteristic equation, known as poles. A system is stable if it produces a bounded...
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Transmission-Line Differential Equations01:26

Transmission-Line Differential Equations

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Transmission lines are essential components of electrical power systems. They are characterized by the distributed nature of resistance (R), inductance (L), and capacitance (C) per unit length. To analyze these lines, differential equations are employed to model the variations in voltage and current along the line.
Line Section Model
A circuit representing a line section of length Δx helps in understanding the transmission line parameters. The voltage V(x) and current i(x) are measured...
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BIBO stability of continuous and discrete -time systems01:24

BIBO stability of continuous and discrete -time systems

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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.
To determine the BIBO stability, the convolution integral is utilized when a bounded continuous-time input is applied to a Linear Time-Invariant (LTI) system....
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Second Order systems II01:18

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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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Linear Approximation in Frequency Domain01:26

Linear Approximation in Frequency Domain

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Linear systems are characterized by two main properties: superposition and homogeneity. Superposition allows the response to multiple inputs to be the sum of the responses to each individual input. Homogeneity ensures that scaling an input by a scalar results in the response being scaled by the same scalar.
In contrast, nonlinear systems do not inherently possess these properties. However, for small deviations around an operating point, a nonlinear system can often be approximated as linear....
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Linear Approximation in Time Domain01:21

Linear Approximation in Time Domain

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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.
For a simple pendulum with a mass evenly distributed along its length and the center of mass located at half the pendulum's length,...
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Stability in distribution for uncertain delay differential equations based on new Lipschitz condition.

Yin Gao1, Lifen Jia2

  • 1School of Mathematics, Renmin University of China, Beijing, 100872 China.

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|April 11, 2022
PubMed
Summary

This study introduces a new Lipschitz condition for uncertain delay differential equations, enabling proofs of stability in distribution. A class of these equations is shown to be stable without prior limitations.

Keywords:
Liu processStability in distributionUncertain delay differential equationsUncertain process

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Area of Science:

  • Stochastic Analysis
  • Differential Equations
  • Control Theory

Background:

  • Existing research on stability in distribution for uncertain delay differential equations relies on a strong Lipschitz condition involving only the current state.
  • This condition is often difficult to satisfy in practice as these equations typically depend on both current and past states.

Purpose of the Study:

  • To introduce a novel Lipschitz condition that accounts for both current and past states in uncertain delay differential equations.
  • To establish a sufficient theorem for proving stability in distribution using this new condition.
  • To demonstrate the stability in distribution for a specific class of these equations without restrictive conditions.

Main Methods:

  • Development of a new Lipschitz condition that is a generalization of the existing strong Lipschitz condition.
  • Formulation and proof of a sufficient theorem for stability in distribution based on the new Lipschitz condition.
  • Application of the theorem to a specific class of uncertain delay differential equations.
  • Verification through two numerical examples.

Main Results:

  • A new Lipschitz condition is proposed, which is satisfied if the strong Lipschitz condition is met, but not necessarily vice-versa.
  • A sufficient theorem for stability in distribution of uncertain delay differential equations is derived using the new condition.
  • A class of uncertain delay differential equations is proven to be stable in distribution unconditionally.
  • Numerical examples confirm the effectiveness of the derived theorem.

Conclusions:

  • The newly proposed Lipschitz condition offers a more practical approach to analyzing the stability in distribution of uncertain delay differential equations.
  • The established theorem provides a valuable tool for determining stability in distribution, extending previous results.
  • The unconditional stability demonstrated for a specific class highlights the robustness of these systems under the new condition.