Derivation of Mathematical Model for Double Inverted Pendulum with Control Implementation

The double inverted pendulum represents a classic nonlinear multi-variable control system research object, comprising two pendulum rods and a movable cart. The system maintains vertical balance of both pendulum rods by controlling the cart's horizontal motion. Mathematical Model Derivation The system dynamics are established using Lagrange equations. By substituting kinetic and potential energy expressions into the Lagrangian formulation, we obtain nonlinear differential equations describing the system motion. For controller design purposes, linearization around the equilibrium point is typically performed. In MATLAB implementation, this involves using symbolic math toolbox to derive equations and the 'jacobian' function for linearization. State-Space Representation The linearized equations are converted into state-space form by selecting appropriate state variables including cart displacement, both pendulum angles, and their derivatives. Output variables typically include directly measurable physical quantities. The state-space model can be expressed in standard state and output equations, which can be implemented using MATLAB's 'ss' function for system analysis and simulation. State Feedback Control A state feedback controller is designed based on pole placement methods. By properly configuring closed-loop system poles, we can ensure system stability and satisfactory dynamic performance. Controller design must consider system controllability and stability requirements. In code implementation, the 'place' or 'acker' functions in MATLAB are commonly used for pole placement calculations. State Observer Implementation When some state variables cannot be directly measured, state observers are designed to estimate these states. The Luenberger observer serves as a common implementation method, where observer poles are configured to converge faster than system dynamics. Programming implementation typically involves designing observer gain matrices and implementing estimation algorithms in real-time control loops. Performance Comparison Analysis Comparative analysis between direct state feedback and observer-based control approaches. Theoretically, direct feedback achieves optimal performance when all states are measurable. However, observer integration increases system order and may affect response speed and control accuracy. Simulation studies using MATLAB/Simulink can thoroughly analyze performance differences under various conditions, including noise sensitivity and parameter variations.