Modeling and Balancing Control of Inverted Pendulum

The cart-pendulum model plays a critical role in real-time control systems across diverse domains including robotics (balancing mechanisms and surgical manipulators), aerospace (missile/Unmanned Aerial Vehicle stabilization), industrial automation (crane sway mitigation and conveyor vibration suppression), energy infrastructure (wind turbine oscillation damping), and structural engineering (seismic dampers). Its inherent nonlinear dynamics necessitate rapid control interventions to maintain stability and operational efficiency. This paper presents a comprehensive development of the nonlinear dynamic model for an inverted pendulum using dual methodological approaches: Newtonian mechanics employing force-balance equations and Lagrangian formulation utilizing kinetic and potential energy principles. Linearization was performed about the upright equilibrium position, followed by derivation of a state-space representation. Controllability and observability were rigorously analyzed using MATLAB computational tools. Stability Analysis for closed loop system under Proportional-Integral-Derivative (PID) control is investigated using Routh-Hurwitz’s criterion. This analysis is applied in four common control modes P, Proportional-Integral (PI), PD, and PID. The stability analysis approved that P and PD modes are not stabilizing modes, while PI and PID are stabilizing modes for inverted pendulum. Therefore, simulations in MATLAB/Simulink evaluated closed-loop behavior under PID control modes P, PI, PD, and PID. Extensive simulation results using MATLAB/SIMULINK affirmed the analytical study and showed that the inverted pendulum can be stabilized only using PI and PID control modes where tracking and regulation to reject disturbance effects are achieved.