Derivation of Mathematical Model for Double Inverted Pendulum with Control Implementation

This article provides a comprehensive derivation of the mathematical model for double inverted pendulum systems, covering state-space representation, implementation of state feedback control, and state observer design. We analyze the impact of state observers on system dynamics through comparative simulations, demonstrating how to select optimal control strategies for practical applications. The implementation includes MATLAB code examples for Lagrange equation formulation and LQR controller design, with detailed explanations of key functions like 'lqr()' for feedback gain calculation and 'obsv()' for observability checks. Furthermore, we explore parameter adjustment techniques to enhance system stability and reliability, including pole placement methods and observer gain optimization. Experimental validation using Simulink models confirms the effectiveness of the proposed control approaches, with data visualization techniques showing real-time system responses. The content provides practical guidance for implementing these algorithms in embedded systems, including discretization methods and real-time controller deployment considerations.