Modeling and Simulation of a Double Inverted Pendulum System

This document presents a detailed methodology for modeling and simulating a double inverted pendulum system. The implementation utilizes Lagrangian mechanics to derive the system's nonlinear dynamics equations, which are then linearized around the upright equilibrium position for controller design. The simulation includes MATLAB code for implementing both classical PID control and modern LQR optimal control strategies, with comprehensive stability analysis through phase portrait visualization and root locus plots. Key technical components include: - Dynamic equations derivation using Euler-Lagrange formulation - State-space representation conversion for linear control design - Real-time animation implementation using MATLAB's plotting functions - Performance comparison between different control approaches - Stability margin analysis through Bode plots and Nyquist criteria The author acknowledges the complexity of multivariable control systems and welcomes constructive feedback for improvement. The provided code examples demonstrate practical implementation techniques for nonlinear system linearization, observer design, and real-time simulation visualization. This work serves as an educational resource for understanding underactuated mechanical systems and advanced control theory applications.