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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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First-order systems, such as RC circuits, are foundational in understanding dynamic systems due to their straightforward input-output relationship. Analyzing their responses to different input functions under zero initial conditions reveals significant insights into system behavior.
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Proportional-Integral (PI) controllers are essential in many control systems to improve stability and performance. They are commonly used in everyday devices like thermostats to enhance system damping and reduce steady-state error. When the zero in the controller's transfer function is optimally placed, the system benefits significantly in terms of stability and accuracy.
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Proportional Integral (PI) controllers are a fundamental component in modern control systems, widely used to enhance performance and mitigate steady-state errors. They are particularly effective in applications such as automatic brightness adjustment on smartphones, where they excel at mitigating steady-state errors for step-function inputs. Unlike PD controllers, which require time-varying errors to function optimally, PI controllers leverage their integral component to address residual...
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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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    This study introduces prescribed-instant stability, a novel control method ensuring system states stabilize at a chosen time, regardless of initial conditions. This approach offers precise convergence time independent of starting states.

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    Area of Science:

    • Control Systems Engineering
    • Nonlinear Dynamics
    • Robotics

    Background:

    • Conventional control methods often struggle with precise convergence time, especially under varying initial conditions.
    • Existing prescribed-time stability approaches provide bounds or predefined times, not arbitrary instant stabilization.
    • System parameter uncertainties and initial states pose challenges for predictable state convergence.

    Purpose of the Study:

    • To develop a novel controller design for achieving prescribed-instant stability.
    • To ensure system states converge to equilibrium at an arbitrarily selected time, independent of initial states and parameters.
    • To demonstrate the efficacy of the proposed method on n-order integrator systems and a magnetic suspension system.

    Main Methods:

    • A new controller design methodology is proposed for prescribed-instant stability.
    • The controller ensures bounded control action that converges to zero at the stabilization instant.
    • Stability analysis is performed using the method of reduction to absurdity.

    Main Results:

    • The proposed method achieves stabilization at an arbitrarily selected time instant.
    • Convergence time is independent of the initial system states.
    • Simulations show distinct advantages over conventional prescribed-time control methods.
    • Experimental validation on a magnetic suspension system with disturbances confirms the method's robustness.

    Conclusions:

    • The developed controller design enables precise, arbitrary-time stabilization of system states.
    • Prescribed-instant stability offers a significant advancement over existing time-based control strategies.
    • The method is effective for n-order integrator systems and practical applications like magnetic levitation.