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    Area of Science:

    • Control Theory
    • Nonlinear Systems Analysis
    • Applied Mathematics

    Background:

    • Traditional backstepping control is limited by the requirement of a non-zero input function at each step.
    • Block strict-feedback nonlinear systems present challenges for direct application of standard backstepping techniques.
    • Polynomial nonlinear systems can utilize Sum of Squares (SOS) for Lyapunov function construction.

    Purpose of the Study:

    • To develop a generalized backstepping control method applicable to block strict-feedback nonlinear systems.
    • To overcome the limitation of non-zero input functions in the backstepping procedure.
    • To integrate Approximate Dynamic Programming (ADP) with backstepping for enhanced control design.

    Main Methods:

    • Polynomial Lyapunov functions are constructed using the Sum of Squares (SOS) technique.
    • Sontag's feedback formula is employed to derive virtual controls, equivalent to optimal control solutions.
    • Approximate Dynamic Programming (ADP) is utilized to estimate Lyapunov (value) functions, complementing SOS.
    • Control Lyapunov Functions (CLF) are constructed for the full system using backstepping and strict-feedback properties.
    • A portion of the CLF is computed via SOS and semidefinite programming, reducing computational complexity.

    Main Results:

    • A novel backstepping control strategy is proposed for a class of block strict-feedback nonlinear systems.
    • The method successfully relaxes the requirement of non-zero input functions in backstepping steps.
    • The integration of ADP broadens the applicability of backstepping to a wider range of systems.
    • The combined approach of backstepping with surface dynamics and SOS reduces the computational load of ADP.

    Conclusions:

    • The proposed method effectively broadens the application scope of backstepping control techniques.
    • Block strict-feedback systems can be stabilized by relaxing the non-zero input function constraint.
    • The hybrid approach of ADP and SOS-based control offers a computationally efficient path to robust stabilization.