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Stochastic Practical Fixed-Time Input-to-State Stability and Its Applications

Ruiming Xie, Shengyuan Xu

    IEEE Transactions on Cybernetics
    |July 27, 2026
    PubMed

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    Stability01:28

    Stability

    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...
    Transient and Steady-state Response01:24

    Transient and Steady-state Response

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

    BIBO stability of continuous and discrete -time systems

    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.
    State Space Representation01:27

    State Space Representation

    The frequency-domain technique, commonly used in analyzing and designing feedback control systems, is effective for linear, time-invariant systems. However, it falls short when dealing with nonlinear, time-varying, and multiple-input multiple-output systems. The time-domain or state-space approach addresses these limitations by utilizing state variables to construct simultaneous, first-order differential equations, known as state equations, for an nth-order system.
    Consider an RLC circuit, a...
    Time-Domain Interpretation of PD Control01:07

    Time-Domain Interpretation of PD Control

    Proportional-Derivative (PD) control is a widely used control method in various engineering systems to enhance stability and performance. In a system with only proportional control, common issues include high maximum overshoot and oscillation, observed in both the error signal and its rate of change. This behavior can be divided into three distinct phases: initial overshoot, subsequent undershoot, and gradual stabilization.
    Consider the example of control of motor torque. Initially, a positive...
    Pole and System Stability01:24

    Pole and System Stability

    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 response.