The modeli̇ng and feedback control of an inverted pendulum wi̇th posi̇ti̇on and veloci̇ty sensors an appli̇cati̇on

This study presents the modeling and feedback control of an inverted pendulum system using position and velocity sensors. The inverted pendulum is widely regarded as a fundamental benchmark problem in control engineering due to its inherent instability and nonlinear dynamics. The objective of this research is to model the system through the integration of Matlab-Simulink and Comsol Multiphysics and stabilize it using a Proportional-Integral-Derivative (PID) controller. The mathematical model of the inverted pendulum system was derived using Newton-Euler and Lagrange methods, and linearized equations were obtained. The physical model was developed in Comsol Multiphysics based on solid mechanics principles, and real-time communication with Simulink was established. The performance of the PID controller was evaluated under different initial conditions and perturbation scenarios. Simulation results demonstrated that the PID controller successfully stabilized the system's angular displacement, cart position, and velocity. The angular deviation converged to zero within 5 seconds, while the cart position reached the reference point in 4 seconds. The time-dependent behavior of the control force was analyzed, confirming that the system achieved equilibrium with minimal energy consumption in steady state. The findings indicate that the PID controller provides an effective control strategy for unstable systems such as the inverted pendulum. This study offers an integrated approach to the modeling and control of inverted pendulum systems in the field of control engineering. Future work may include incorporating friction and external disturbances into the model and implementing advanced control algorithms for further optimization.