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Linear time-invariant Systems01:23

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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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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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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.
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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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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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Stabilization of systems with interval time-varying delay based on delay decomposing approach.

Wei Qian1, Manman Yuan1, Lei Wang2

  • 1School of Electrical Engineering and Automation, Henan Polytechnic University, 454000 Henan, China.

ISA Transactions
|June 8, 2017
PubMed
Summary

This study presents a new method for stabilizing systems with time-varying delays using a novel Lyapunov-krasovskii functional (LKF). The approach reduces conservatism and improves system stabilization time.

Keywords:
Delay-decomposingInterval time-varying delayLyapunov-Krasovskii functional (LKF)Stabilization

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Area of Science:

  • Control Systems Engineering
  • Nonlinear Dynamics
  • Systems Theory

Background:

  • Systems with interval time-varying delays present significant control challenges.
  • Existing methods often suffer from conservatism and a large number of decision variables.

Purpose of the Study:

  • To develop a less conservative stability criterion for systems with interval time-varying delays.
  • To design a state feedback controller for such systems.
  • To reduce the system stabilization time.

Main Methods:

  • Decomposition of the delay interval into subintervals.
  • Definition of a novel Lyapunov-krasovskii functional (LKF) incorporating triple integral terms.
  • Application of extended integral inequality and convex combination techniques.
  • Design of a state feedback controller using linearization and linear matrix inequalities (LMIs).

Main Results:

  • A new stability criterion with reduced conservatism and fewer decision variables was derived.
  • The existence conditions for the state feedback controller were established in LMI form.
  • The proposed method demonstrated effectiveness in numerical examples, including reduced stabilization time.

Conclusions:

  • The proposed method offers an effective approach for stabilizing systems with interval time-varying delays.
  • The developed stability criterion and controller design are less conservative than existing techniques.
  • The method provides a practical way to reduce system stabilization time through parameter selection.