Lagrangian Modeling, Linearization, and Optimal State-Feedback Control of a Nonlinear Inverted-Pendulum System

This paper presents a simulation study of a cart–pole (inverted pendulum) system. Starting from first-principles physics, we derive the nonlinear equations of motion and linearize them about the upright equilibrium to obtain a state-space model for control design. Using this model, we synthesize an optimal Linear Quadratic Regulator (LQR) that balances regulation performance and control effort. All evaluations are performed in MATLAB/Simulink on the full nonlinear plant. Simulations show reliable recovery to the upright position with short settling time, limited overshoot, and negligible steady-state error. The workflow—modeling, linearization, and LQR design—is transparent and readily transferable to hardware.