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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.
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In automotive engineering, car suspension systems often employ Proportional Derivative (PD) controllers to enhance performance. PD controllers are utilized to adjust the damping force in response to road conditions. A controller, acting as an amplifier with a constant gain, demonstrates proportional control, with output directly mirroring input.
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Updated: Apr 23, 2026

Design and Application of a Fault Detection Method Based on Adaptive Filters and Rotational Speed Estimation for an Electro-Hydrostatic Actuator
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Prescribed-Performance Control of Variable-Order Fractional-Order Nonlinear Systems Through Adaptive Dynamic

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    This study introduces a novel adaptive dynamic programming method using control barrier functions for variable-order fractional nonlinear systems. The approach ensures system safety and performance constraints are met, overcoming challenges in fractional calculus control.

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    Area of Science:

    • Control Systems Engineering
    • Nonlinear Dynamics
    • Fractional Calculus

    Background:

    • Variable-order fractional-order nonlinear systems (VOFONSs) present unique control challenges due to time-dependent and memory characteristics.
    • Traditional control methods for integer-order systems are not directly applicable to VOFONSs.
    • Prescribed-performance control (P2C) aims to ensure system states remain within specified bounds.

    Purpose of the Study:

    • To develop a control strategy for achieving prescribed-performance control (P2C) in variable-order fractional-order nonlinear systems (VOFONSs).
    • To integrate safety constraints into an optimal control framework using control barrier functions (CBFs) and adaptive dynamic programming (ADP).
    • To address the complexities arising from the time-varying nature of VOFONSs and CBF-embedded cost functions.

    Main Methods:

    • Variable-order fractional calculus and integration-by-parts formula were used to approximate VOFONSs as time-varying integer-order systems.
    • A control barrier function (CBF)-embedded adaptive dynamic programming (ADP) method was developed to handle constrained optimal control.
    • A neural network with a time-varying activation function approximated the time-varying discounted cost function.
    • A time-varying policy iteration algorithm solved the Hamilton-Jacobi-Bellman (HJB) equation.

    Main Results:

    • The CBF-embedded ADP method successfully incorporated safety constraints into the optimal control framework without explicit state transformation.
    • A novel control policy was derived, guaranteeing that the tracking error adheres to predefined constraints.
    • The proposed method effectively addressed the challenges posed by time-varying dynamics and memory effects in VOFONSs.

    Conclusions:

    • The CBF-embedded ADP control scheme is effective for achieving prescribed-performance control in VOFONSs.
    • The developed method provides a robust approach for handling safety-critical control problems in complex nonlinear systems.
    • Simulation results validated the efficacy and practical applicability of the proposed control strategy.