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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.
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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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Linearity is a system property characterized by a direct input-output relationship, combining homogeneity and additivity.
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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.
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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.
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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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CONVERGENCE PROPERTIES OF ADAPTIVE SYSTEMS AND THE DEFINITION OF EXPONENTIAL STABILITY.

Benjamin M Jenkins1, Anuradha M Annaswamy1, Eugene Lavretsky2

  • 1Department of Mechanical Engineering, MIT, Cambridge, MA 02139 (bjenkins@mit.edu, aanna@mit.edu).

SIAM Journal on Control and Optimization
|November 28, 2019
PubMed
Summary

Persistent excitation of the regressor vector ensures exponential stability in adaptive control systems. However, excitation of the reference model only guarantees weak persistent excitation, leading to uniform asymptotic stability, not exponential stability.

Keywords:
34D2337C7593C4093D20adaptive controlasymptotic stabilityexponential stabilitypersistence of excitationweak persistence of excitation

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Area of Science:

  • Control Theory
  • Adaptive Systems
  • Nonlinear Dynamics

Background:

  • Adaptive control systems rely on excitation conditions for convergence.
  • Persistent excitation of the regressor vector ensures global exponential stability.
  • Misconceptions exist regarding excitation conditions on reference inputs/models.

Purpose of the Study:

  • Revisit the definition of persistent excitation in adaptive control.
  • Clarify the implications of reference model excitation.
  • Analyze stability properties under different excitation scenarios.

Main Methods:

  • Theoretical analysis of adaptive system dynamics.
  • Examination of persistent excitation definitions.
  • State-space analysis of convergence properties.

Main Results:

  • Persistent excitation of the reference model implies only weak persistent excitation.
  • Weak persistent excitation guarantees uniform asymptotic stability, not exponential stability.
  • An infinite region of bounded state rates exists in adaptive systems.

Conclusions:

  • Distinguish between strong and weak persistent excitation for stability.
  • Clarify stability guarantees for open-loop and closed-loop adaptive systems.
  • Highlight the existence of regions with finite convergence rates.