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In many practical and theoretical contexts, the exact value of a definite integral may be inaccessible. This limitation typically arises when the antiderivative of a function is either unknown or cannot be expressed in a closed mathematical form. Alternatively, it can occur when a function is defined not by a formula but by a finite set of empirical data points, such as those collected during experiments. In these cases, approximate integration techniques provide a valuable solution.One of the...
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    This study introduces a novel online optimal regulation method for control-affine systems, combining local state following (StaF) and regional model-based reinforcement learning (R-MBRL) for improved value function approximation and stability.

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    Area of Science:

    • Control Theory
    • Reinforcement Learning
    • Dynamical Systems

    Background:

    • Optimal regulation problems for control-affine deterministic systems are crucial in advanced control applications.
    • Traditional methods like regional model-based reinforcement learning (R-MBRL) approximate value functions over large sets, posing scalability challenges.
    • Local state following (StaF) offers an alternative by focusing on local state neighborhoods.

    Purpose of the Study:

    • To develop an online optimal regulation method for control-affine deterministic systems.
    • To enhance value function approximation by integrating StaF and R-MBRL techniques.
    • To establish semiglobal uniformly ultimately bounded (SGUUB) convergence to the origin.

    Main Methods:

    • An infinite-horizon optimal regulation problem is addressed using a novel approach.
    • The value function is approximated via a state-dependent convex combination of StaF and R-MBRL approximations.
    • Lyapunov-based analysis is employed to prove SGUUB convergence.

    Main Results:

    • The proposed method effectively approximates the value function by transitioning between StaF and R-MBRL approximations near the origin.
    • Semiglobal uniformly ultimately bounded (SGUUB) convergence of system states to the origin is theoretically established.
    • Simulations on systems with 2, 3, 6, and 10 states demonstrate scalability and performance.

    Conclusions:

    • The integrated StaF and R-MBRL approach provides an effective online solution for optimal regulation.
    • The method ensures system stability and convergence, outperforming traditional R-MBRL in certain aspects.
    • The demonstrated scalability suggests broad applicability in complex dynamical systems.