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BIBO stability of continuous and discrete -time systems01:24

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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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Survival curves are graphical representations that depict the survival experience of a population over time, offering an intuitive way to track the proportion of individuals who remain event-free at each time point. These curves are widely used in fields such as medicine, public health, and reliability engineering to visualize and compare survival probabilities across different groups or conditions.
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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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A separable differential equation is a type of first-order differential equation where the derivative dy/dx can be expressed as a product of two functions: one that depends only on x and another that depends only on y. This allows for the rearrangement of the equation so that all terms involving y are on one side, and all terms involving x are on the other. This process, known as the separation of variables, simplifies the process of solving the equation by enabling the integration of both...
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Classification of Systems-I

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Linearity is a system property characterized by a direct input-output relationship, combining homogeneity and additivity.
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Survivability of Deterministic Dynamical Systems.

Frank Hellmann1, Paul Schultz1,2, Carsten Grabow1

  • 1Potsdam Institute for Climate Impact Research, P.O. Box 60 12 03, 14412 Potsdam, Germany.

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|July 14, 2016
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Summary
This summary is machine-generated.

We introduce survivability, a new measure of system stability, to quantify the likelihood that a system remains in desirable states. This approach offers a more comprehensive understanding of complex system resilience than traditional methods.

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Area of Science:

  • Complex Systems Research
  • Dynamical Systems Theory
  • Stability Analysis

Background:

  • The concept of a "safe operating space" or "viable region" is crucial for understanding the behavior of dynamical systems.
  • Existing stability measures often fail to capture the transient dynamics of systems within these desirable regions.

Purpose of the Study:

  • To define and introduce a novel stability measure called "survivability."
  • To assess the probability that a deterministic system, starting from a random initial condition, remains within a predefined region of desirable states during its transient phase.

Main Methods:

  • Defining survivability as a probabilistic measure of state-space containment.
  • Applying survivability analysis to models from climate science, neuronal networks, and power grids.
  • Developing and utilizing a semi-analytic lower bound for survivability in linear systems.

Main Results:

  • Demonstrated the utility of survivability across diverse complex systems.
  • Showcased the computational efficiency of the semi-analytic lower bound for power grid survivability analysis.
  • Highlighted that survivability captures a distinct aspect of stability not addressed by asymptotic measures.

Conclusions:

  • Survivability provides a novel and valuable metric for assessing the resilience of dynamical systems.
  • The developed methods enable efficient and accurate survivability analysis, particularly for large-scale systems like power grids.
  • Survivability analysis offers critical insights into system robustness beyond traditional stability definitions.