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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.