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Dynamical analysis of an inverted pendulum with positive position feedback controller approximate uniform solution.

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This study controls the inverted pendulum (IP) using nonlinear control theory and advanced perturbation methods. Results show increased generalized force, stiffness, and magnetic fields enhance IP stability, offering insights for nonlinear system control.

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Area of Science:

  • Nonlinear Control Theory
  • Analytical Mechanics
  • Vibrational Analysis

Background:

  • The inverted pendulum (IP) is a fundamental system in classical mechanics and control theory.
  • Controlling the IP's stability, especially under external forces and resonances, is crucial for various applications.
  • Existing methods may not fully address complex dynamics and multi-excitation scenarios.

Purpose of the Study:

  • To develop and validate nonlinear control strategies for the inverted pendulum system.
  • To analyze the system's dynamic behavior and stability under various conditions, including multi-excitation forces.
  • To explore the effectiveness of perturbation methods and feedback control in stabilizing the IP.

Main Methods:

  • Derivation of the equation of motion using classical analytical mechanics.
  • Application of Taylor expansion for analysis of restoring forces.
  • Utilizing the modified Homotopy perturbation method (mHPM) for approximate periodic solutions.
  • Employing the fourth-order Runge-Kutta (RK-4) method for numerical validation.
  • Developing positive position feedback (PPF) control.
  • Applying the second-order multiple time scale perturbation technique (MSPT) for a two-degree-of-freedom model.
  • Using the Routh-Hurwitz criterion for stability analysis.

Main Results:

  • A satisfactory periodic solution for the IP was obtained using mHPM and validated numerically.
  • PPF control was effective in dampening vibrations under multi-excitation forces.
  • MSPT analysis revealed insights into the system's behavior at primary and 1:1 internal resonance.
  • Frequency response curves and numerical simulations confirmed the controlled performance.
  • Increased generalized force, torsional stiffness, and magnetic fields were found to suppress instability and enhance IP stability.

Conclusions:

  • The proposed nonlinear control methods, including mHPM and PPF, are effective for stabilizing the inverted pendulum.
  • Perturbation techniques provide accurate approximate solutions validated by numerical simulations.
  • System parameters like stiffness, external force, and magnetic fields significantly influence stability.
  • The findings have implications for the design and control of numerous other nonlinear systems.