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    This study presents a new sliding-mode control (SMC) for nonlinear systems with changing conditions. The proposed method ensures system stability and precise trajectory control despite external disturbances.

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    Area of Science:

    • Control Systems Engineering
    • Nonlinear Dynamics
    • Stochastic Systems

    Background:

    • Markov jump nonlinear systems (MJNSs) present challenges due to their piecewise structure and time-varying dynamics.
    • External disturbances and asynchronous transitions complicate control design for these systems.
    • Existing control methods often struggle with time-varying transition probabilities and external disturbances in MJNSs.

    Purpose of the Study:

    • To develop a robust sliding-mode control (SMC) strategy for discrete-time, piecewise nonhomogeneous MJNSs.
    • To address the challenges posed by external disturbances and time-varying transition probabilities.
    • To ensure stochastic stability and precise trajectory tracking within a specified region.

    Main Methods:

    • Construction of a discrete-time asynchronous integral sliding surface.
    • Utilization of the mode-dependent Lyapunov function technique for stability analysis.
    • Derivation of an SMC law and an adaptive law for control and adaptation.

    Main Results:

    • Achieved matched-nonlinearity-free sliding-mode dynamics (SMDs).
    • Established a sufficient condition for stochastic stability of SMD with extended dissipation.
    • Successfully designed controller gains, an SMC law, and an adaptive law.
    • Demonstrated the effectiveness through a simulation example.

    Conclusions:

    • The proposed SMC design effectively stabilizes discrete-time piecewise nonhomogeneous MJNSs under external disturbances.
    • The method ensures system trajectories converge to a predetermined region with high precision.
    • The approach is feasible and validated through simulation, offering a robust control solution.