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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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Adaptive Critic Learning for Constrained Optimal Event-Triggered Control With Discounted Cost.

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    This study presents an optimal event-triggered control (ETC) strategy for nonlinear systems with asymmetric constraints and nonzero equilibrium points. The method ensures system stability and efficient control updates, validated by simulations.

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    Area of Science:

    • Control Theory
    • Nonlinear Systems Analysis
    • Adaptive Learning

    Background:

    • Event-triggered control (ETC) aims to reduce system resource usage by triggering control updates only when necessary.
    • Existing ETC methods often assume systems have zero equilibrium points, limiting their applicability.
    • Asymmetric control constraints pose challenges for traditional control design.

    Purpose of the Study:

    • To develop an optimal event-triggered control (ETC) strategy for nonlinear continuous-time systems with asymmetric control constraints and nonzero equilibrium points.
    • To introduce a novel approach for optimal ETC that avoids coordinate transformations.
    • To ensure system stability and performance under event-triggered conditions.

    Main Methods:

    • Introduction of a discounted cost function to handle nonzero equilibrium points.
    • Formulation of an event-triggered Hamilton-Jacobi-Bellman equation (ET-HJBE).
    • Development of an event-triggering condition to guarantee a minimum intersample time.
    • Implementation of an adaptive critic learning framework with a critic network.
    • Utilization of a modified gradient descent method for tuning critic network weights using historical and instantaneous data.
    • Application of Lyapunov stability analysis to prove uniform ultimate boundedness.

    Main Results:

    • A novel optimal ETC strategy for nonlinear systems with asymmetric constraints and nonzero equilibrium points.
    • An event-triggering condition that ensures a positive lower bound on the minimal intersample time, enhancing efficiency.
    • Proof of uniform ultimate boundedness for all signals in the closed-loop system, guaranteeing stability.
    • Validation of the proposed ETC strategy through simulations on pendulum and oscillator systems.

    Conclusions:

    • The developed optimal ETC strategy effectively addresses challenges posed by asymmetric constraints and nonzero equilibrium points in nonlinear systems.
    • The adaptive critic learning approach provides a viable method for solving the ET-HJBE.
    • The proposed method ensures system stability and efficient control, demonstrated by simulation results.