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Related Concept Videos

Stability01:28

Stability

73
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...
73
BIBO stability of continuous and discrete -time systems01:24

BIBO stability of continuous and discrete -time systems

321
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....
321
Transient and Steady-state Response01:24

Transient and Steady-state Response

134
In control systems, test signals are essential for evaluating performance under various conditions. The ramp function is effective for systems undergoing gradual changes, while the step function is suitable for assessing systems facing sudden disturbances. For systems subjected to shock inputs, the impulse function is the most appropriate test signal.
These test signals are integral in designing control systems to exhibit two key performance aspects: transient response and steady-state...
134
Linear Approximation in Time Domain01:21

Linear Approximation in Time Domain

59
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,...
59
Pole and System Stability01:24

Pole and System Stability

235
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.
Simple poles are unique roots of the denominator polynomial. Each simple pole corresponds to a distinct solution to the system's characteristic equation, typically resulting in exponential decay terms in the system's...
235
Linear time-invariant Systems01:23

Linear time-invariant Systems

202
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.
The input-output behavior of an LTI system can be fully defined by its response to an impulsive excitation at its input. Once this impulse response is known, the system's reaction to any other input can be...
202

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Semi-Global and Global Fixed-Time Stability for Nonlinear Impulsive Systems.

Fangmin Ren, Xiaoping Wang, Yangmin Li

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    This study introduces novel methods for analyzing nonlinear impulsive systems (NISs), achieving semi-global and global fixed-time stability. The research provides new criteria and controllers for complex systems, ensuring stability within a finite time.

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    Area of Science:

    • Control Theory
    • Nonlinear Dynamics
    • Systems Engineering

    Background:

    • Nonlinear impulsive systems (NISs) present unique challenges in stability analysis due to evolving integration methods caused by impulses.
    • Existing methods struggle with distinct trajectories inside and outside the semi-global attraction set (SGAS) for global fixed-time stability (GFTS).

    Purpose of the Study:

    • To investigate the semi-global fixed-time stability (SGFTS) and global fixed-time stability (GFTS) of nonlinear impulsive systems (NISs).
    • To develop theoretical frameworks and criteria for achieving fixed-time stability in NISs under various impulse conditions.

    Main Methods:

    • Dynamically partitioning the semi-global attraction set (SGAS) to address evolving integration methods for SGFTS.
    • Constructing transition dynamics of impulse points and iteratively computing them to establish SGFTS conditions.
    • Introducing the maximum-minimum impulse interval concept to ensure systems enter the SGAS for GFTS.
    • Developing criteria for GFTS under varying impulse degrees and estimating convergence time.

    Main Results:

    • Established conditions for SGFTS under both stabilizing and destabilizing impulses in NISs.
    • Derived a sufficient condition for GFTS, ensuring systems can enter the SGAS from a distance.
    • Provided a criterion for GFTS with varying impulse degrees and convergence time estimation.
    • Demonstrated the effectiveness of a designed fixed-time impulse controller for global stabilization of complex systems.

    Conclusions:

    • The study provides robust theoretical frameworks for analyzing and achieving fixed-time stability in nonlinear impulsive systems.
    • The developed methods and criteria offer significant advancements in understanding and controlling complex dynamic systems with impulses.
    • The findings have direct implications for designing advanced control systems capable of finite-time stabilization.